Logical Nihilism
Some more curiosities from recent developments in logic. This video is most interesting:
Logical laws are supposed to work in every case. Modus Tollens, non-contradiction, identity - these work in any and all cases. A logical nihilist will reject this.
There are two ways to deal with this argument.
A logical monist will take the option of rejecting the conclusion, and also the second premise. For them the laws of logic hold with complete generality.
A logical pluralist will reject the conclusion and the first premise. For them laws of logic apply to discreet languages within logic, not to the whole of language. Classical logic, for example, is that part of language in which propositions have only two values, true or false. Other paraconsistent and paracomplete logics might be applied elsewhere.
A few counter-examples of logical principles that might be thought to apply everywhere.
Identity: ? ? ?; but consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"
And elimination: ? & ? ? ?; But consider "? is true only if it is part of a conjunction".
Other and tighter examples can be found in the video.
Especially appealing is the application of Lakatos' method to logic; choosing logical pluralism over logical monism leads to more fruitful discussions.
Logical laws are supposed to work in every case. Modus Tollens, non-contradiction, identity - these work in any and all cases. A logical nihilist will reject this.
Gillian Russell:To be a law of logic, a principle must hold in complete generality
No principle holds in complete generality
____________________
There are no laws of logic.
There are two ways to deal with this argument.
A logical monist will take the option of rejecting the conclusion, and also the second premise. For them the laws of logic hold with complete generality.
A logical pluralist will reject the conclusion and the first premise. For them laws of logic apply to discreet languages within logic, not to the whole of language. Classical logic, for example, is that part of language in which propositions have only two values, true or false. Other paraconsistent and paracomplete logics might be applied elsewhere.
A few counter-examples of logical principles that might be thought to apply everywhere.
Identity: ? ? ?; but consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"
And elimination: ? & ? ? ?; But consider "? is true only if it is part of a conjunction".
Other and tighter examples can be found in the video.
Especially appealing is the application of Lakatos' method to logic; choosing logical pluralism over logical monism leads to more fruitful discussions.
Comments (722)
I'm assuming the idea is that identity, non-contradiction and excluded middle remain tautologies but are less certain even if it is generally held that these are the preconditions for sensible communication.
I found it fairly clear. But here's the article from which it derives.
https://gilliankrussell.files.wordpress.com/2018/05/logicalnihilism-philissues-v3.pdf
So this conclusion is not a law of logic. Okay.
Nothing to see here, move along.
What?
Mathematics in its entirety lacks a foundation. Yep.
Spill on aisle Real Number.
Sorry, Prof. Russell, I'm an embodied cognitivist with respect to abstract formal constructs.
I'm looking forward to the video at leisure but this leapt out for attention. Isn't it an indexical (context-dependent) utterance? The second utterance expresses a different proposition from the first - so they are not identical propositions and we would not expect identity to hold. The expression "this sentence" refers to a different sentence in the first utterance from the second.
(Just as my saying "I am Cuthbert" expresses a different proposition from someone else's saying "I am Cuthbert", which explains why the same utterance can be either true or false depending upon context.)
But there's only one sentence. Ah. Shut ma mouth. I will watch the video. I know it's all a trick, though, so I'm not going to be taken in. Not that I'm prejudiced, of course.
What about if I apply the conclusion (There are no laws of logic) to Gillian Russell's argument? The argument seems to be self-refuting - it relies on the laws of logic i.e. it assumes there are laws of logic but claims there are no laws of logic. Contradiction! Gillian Russell is contradicting herself full tilt.
Logic can't be used to negate itself for to do that is to affirm itself. Contradiction!
:grin: You poor bugger. Feel for you.
Have a look when you have time - and see the article cited, too. It's curious stuff. There's something going on.
Roger!
proof.
Really? We can assume they are all monist, which I'm hoping will gloss as absolutist? I.e. not tolerant, or relativist, inasmuch as (not) regarding logics as horses for courses? [Thought I could safely use this figure without implying anything was a race, nvm.]
I can see how someone of that persuasion (far more prevalent than I knew) might survey the totality of courses and decide that the only horse suitable is Humpty Dumpty, or worse. But a pluralist (relativist? or am I unaware of a recognised distinction?) already allows such a choice for a slowest [having fewest laws] variety of horse:
70
I wonder if that is true to some extent here, too.
So, pretty much, am I. I don't see any prima facie contradiction between the two. Embodied cognitivism does not rule out abstract formal constructs.
The horse will win, or not. There are no other options. So binary logic is applicable.
But these would be odd bedfellows, since it seems clear that Feyerabend would side with the logical nihilists against both logical monism and Russell.
My suspicion is that Feyerabend would treat binary logic as a natural interpretation, as so closely connected with the craft of logic as "to need a special effort to realise their existence and to determine their content" (Ch 6 of Against Method).
We are left with the critique found in Chapter 16, that Lakatos' method is disguised anarchy. On this account lemma incorporation and monster barring are equally valid, the choice being arbitrary - at best an aesthetic or moral preference.
Curiously Russell has another published article arguing that logic is not normative. Unfortunatly I have not been able to access it.
So Logical Nihilism has me returning to what I had taken as pretty much settled; that scientific progress does not result from a more or less algorithmic method - induction, falsification and so one - but is instead the result of certain sorts of liberal social interaction - of moral and aesthetic choice.
And this is why the article is worthy of consideration. It's ramifications are broad.
No one addressed this; I think it quite funny. So I'll spell it out.
1. This sentence is true only if it is not part of a compound sentence
2. Snow is white
therefore, since 1) stands alone, and snow is indeed white, both premises are true. So by &-introduction:
3. This sentence is true only if it is not part of a compound sentence and snow is white
...which is false, since "This sentence is true only if it is not part of a compound sentence" now occurs in a compound sentence. (A compound sentence is formed by adding two independent clauses together using a conjunction.
This is a variant on SOLO from p.9.
So who will bar the monster, and who will befriend him?
Not unlike jazz improv.
Edit: Taking that further, jazz can be cleverness that hides ugliness. I suppose on that account Russell' s work might be seen as clever but ugly. Perhaps that is what the objection is.
"Arguments can be good in all kinds of ways even when they are not logically valid".
But also, "If someone gives you a classically valid argument for the view that there are no valid arguments, that ought to be seriously worrying to someone who accepts classical logic".
...and then the discussion of modus ponens. How confident are you that there are no counter instances? But note, from the article, that logical nihilism is not the view that there are no logical truths; it is the view that there are always counter instances:
Russell invents what we might call a ladder of interpretation, starting with a single interpretation, then two, three, and presumably so on.
On a single interpretation, "T", the fallacy assuming the consequent is true. Sot he only way to avoid AC being a logical law is to widen the interpretation to include "F"...
AC looks like a logical law because the interpretation is limited.
Russell asks why we should not apply this reasoning to the law of excluded middle. Why not similarly conclude that it only looks like a logical law because we arbitrarily limit the interpretations to "T" or "F"?
Expanding the "library of interpretations" provides similar counterexamples for various other supposed logical laws.
Quoting Banno
What's Gillian Russell's citeria of a good argument? Whatever it is, it doesn't seem to be about validity and if not, how does she know the conclusion is true given some premises are?
Her issue seems to be with deductive logic (validity above) but then she isn't saying anything we already don't know - there are cogent inductive arguments that aren't deductively valid - the conclusion is probable but not necessary.
1. 99.99% of Indians are mathematcian = M
2. Y is an Indian = I
Ergo,
3. Y is a mathematician = Y
In modus ponens (the sticking point insofar as the OP is concerned) form the inductive argument looks like:
4. If (M & I) then Y
5. M & I
Ergo,
6. Y
As you can see, statement 4 even if it isn't completely true, it's truer than false. Let's just say you would bet big on it.
However, 6 doesn't follow deductively from 4 and 5 i.e. it's invalid but still, and notably, the argument is good.
Do that, then. Or read the article.
See my comments above regarding Feyerabend. The issue you raise has been of interest to me for the last forty years, and remains unresolved.
Lakatos' research programs are comparable to Wittgenstein's word games. Both are interactions between what we say, what we do and what happens next. Feyerabend was to ba a student of Wittgesntein's but moved to Popper after Witti's death. Feyerabend's criticism of Lakatos migh be applied to Wittgenstein's language games, if Wittgenstein had adopted some normative approach.
SO this potentially comes back to asking if logic is normative. I'm thinking that it isn't. That is, it sets out what we can think, but does not set out what we ought think.
It's not clear how something can be a problem of logic itself, when it can more easily be explained by certain concepts being internally incoherent.
Quoting Banno
Yes, I read them, and I agree with what you wrote:
Quoting Banno
Did you ever see this comment Lakatos supposedly made to British philosopher Donald Gillies? âWittgenstein was the biggest philosophical fraud of the twentieth century".
This would seem to support the idea that Wittgenstein would be more sympathetic to Feyerabend than Lakatos on the grounding of logic.
It is morning.
(If it was said in the morning, then it is true. But in the afternoon, it is false.)
I am Elvis Presley.
(If Elvis Presley said it, then it is true. Anyone else said it, false)
It is late summer.
(In the Northern hemisphere, it is true. In Southern hemisphere locations, it is false)
etc etc
Is it not the main reason why Logical Positivism failed? They tried to reduce the world, language and all its objects into logic, and were trying to represent and resolve all the worldly problems using Logical Analysis. But the world, its objects and language are far more rich and complicated for that to work.
...and you are doing logic.
That was from here, it seems, although no explanation is given. Lakatos perhaps disliked pokers.
Prima facie Popper, and his defender Lakatos, might fall into the logical monist camp; Wittgenstein into the logical pluralist camp. But oddly Russell is here using Lakatos to defend the pluralist camp. Things we can only guess.
Paradigms, research projects, language games, conceptual schema, and so on, treat aspects of language as discrete entities. The extent to which these might be incommensurate is an interesting topic for discussion.
I suppose that those who think logic normative might be more incline to think of conceptual schema as discreet and incommensurable. I think they are neither.
That link is really substantive material for the topic. Thanks. :up:
Could you give examples of a treatment of schemes as discrete and incommensurable vs non-discrete and commensurable?
Were Lakatos' research projects incommensurable? My recollection is that they were, but I'm not sure without looking it up. The classic example of incommensurable paradigms is from The Structure of Scientific Revolutions. Feyerabend at first takes a strong approach to incommensurability, but later disavows it. Davidson argues, I think quite convincingly, in On the very idea of a conceptual scheme, that the notion of incommensurability is unreasonable.
All sorts of other schemes can be found.
âQuoting Banno
Kuhnâs notion of incommensurability evolved over time.
âSince 1962 Kuhn's concept of incommensurability has undergone a process of transformation. His current account of incommensurability has little in common with his original account of it. Originally, incommensurability was a relation of methodological, observational and conceptual disparity between paradigms. Later Kuhn restricted the notion to the semantical sphere and assimilated it to the indeterminacy of translation. Recently he has developed an account of it as localized translation failure between subsets of terms employed by theories.â(H. Sankey)
Putnam had this to say about Kuhnâs changing notions:
âIn more recent work one finds him expressing admiration for the work of Joseph Sneed and Wolfgang Stegmuller. The notion of incommensurability still appears in his writing, but now it seems to signify nothing more than intertheoretic meaning change, as opposed to uninterpretability. According to Sneed and Stegmiiller, who build on ideas that go back to Carnap, the theoretical terms in a theory' refer to complex logical constructions out of the set of models of that theory, which in turn depend on an open set of "intended applications." I shall not go into details. But one point is worth mentioning: When two theories Con-flict, then, although the common theoretical terms generally have dif-ferent meanings and a different reference on the Sneed-Stegmiiller account (that is what "incommensurability" becomes), that does not mean that there is no "common language" in which one can say what the theoretical terms of both theories refer to.
In fact, if we have avail-able the "old terms," that is, the terms which existed in the language prior to the introduction of the specific new terms characteristic of the two theories, and enough set-theoretic vocabulary, we can express the empirical claim of both theories, and we can say what the admis-sible models of both theories are. Kuhn still maintains that we cannot interpret the term phlogiston in the language that present-day scientists use; but what this in fact means is that we must use a highly indirect mode of interpretation, which involves describing the entire phlogiston theory, its set of intended applications, and its set of admissible models in order to say what phlogiston means. A serious residual difficulty still faces Kuhn: he has long maintained that the meaning of old terms (say, observa-tion terms) is altered when new theories are constructed.
But the whole assumption of Sneed and Stegmiiller is precisely that this is not the case. Their sets of admissible models are well defined only if we can assume that the old terms have fixed meanings which are not altered by theory construction. It is precisely the aim of neopositivism to view scientific theories as constructed in levels in such a way that the terms of one level may depend for their meaning on the terms of a lower level, but not vice versa. Neopositivism denies that there is a two-way dependence between observation terms and theoretical terms, whereas Kuhn has long agreed with Quine that the dependence goes both ways. Even if I cannot make full sense of Kuhn's current position, I think that I have said enough to indicate the general nature of the development.
This might be summed up in three stages. Stage 1: There is a doctrine of radical incommensurability, that is, impossibility of interpretation. Stage 2: The doctrine is softened. We can, it turns out, say something about theories which are incommensurable with our own, and we can use some notions (justification, rationality) across paradigm changes. Stage 3: Something which is thought to be better than interpretation is embraced and propounded, namely, the structural description of theories.â(Realism with a Human Face)
Personally, I support Lyotardâs differend.
As Gallagher describes the problem with Robertâs âconversation of manikinsâ,
â The conversation of mankind fails as a model of postmodern hermeneutics not only because it is a
metadiscourse and worthy of our incredulity, but because it hides exclusionary rules beneath a rhetoric of inclusion. The overarching conversation of mankind aspires to resolve all differends.
But by requiring what is genuinely incommensurable (i.e., incommensurable with the conversation itself) to be voiced within the conversation, it denies it expression and helps to constitute it as a differend at the same time that it disguises it as a litigation. The very attempt to include something which cannot be included makes the conversation of mankind a terrorist conversation.
This is one of the issues between Lyotard and Rorty.â
I do have problems with Davidsonâs argument against incommensurability of schemes. For one thing , memory is reconstructive. There is no veridical past to re-access and compare with the present. To do so is already to be dealing with a re-interpretation. As regards the translation of concepts not only between languages but within a given language, if one wants to argue that general agreement on what is the case is always possible , then I would assent to that as long as this must be a pragmatic agreement. More importantly, I would add that in many cases , such as political polarization , agreement may be theoretically possible , but for all intents and purposes is impossible. This is because there can be no translation from one political camp to another without an enormously difficult work of transformation and expansion of political concepts in order to glimpse the opposing political viewpoint in a way that is recognizable to the other side of the conflict.
Hmm. I have difficulty seeing why this is a problem for Davidson. Where does memory fit in his argument?
Actually, I was thinking more of Putnam here. Davidson wants us to believe the idea of a conceptual
scheme presupposes a dualism of scheme and an uninterpreted reality. It doesnât. Reference doesnât have to be made to âthe way things really areâ , only to pragmatic differences in behavior. If we impute to the other an incommensurable scheme, we are anticipating a whole range of behaviors on the r part of the other that we are unable to make sense of in the way we can with someone who shares our scheme. Thus the notion of conceptual scheme validates itself via the behavior over time of the person who we claim holds this scheme.
To unmuddy things a bit , we could rename conceptual scheme ingrained habits of thought. We would also have to assume that Davidsonâs suggestion of locating a shared background of beliefs would fail miserably in dealing with anything but the most superficial level of thought. As we have learned in our current polarized world , differences in political worldview are sweeping in the areas of thought that they encompass.
Quoting Joshs
Hmm. Does Davidson assume that? Or are you saying he accuse his antagonists of so doing? Frankly I agree there is a crisis of relevance in Davidson, but I am not sufficiently familiar with Lyotardâs writings; I'll remedy that.
He argues that that the assumption of âconceptual schemeâ requires the above presumption:
â In giving up dependence on the concept of an uninterpreted reality, something outside all schemes and science, we do not relinquish the notion of objective truth -quite the contrary Given the dogma of a dualism of scheme and reality, we get conÂceptual relativity, and truth relative to a scheme. Without the dogma, this kind of relativity goes by the board.â( On the very idea of âŚ)
What is a good horse for a course (logic for a discourse)? Not, one might naively assume, one whose principles allow inference to exactly all the sentences that are said in the discourse? Or which agree with all the inferential steps or patterns that are claimed in the discourse? Although that does rather sound like G Russell's view.
I'm not clear whether the view tends to arise from the narrower example of proving a logic sound or complete for a perfectly determined 'discourse', as here:
... where the 'discourse' on the left contains no controversies. Everyone is agreed (no diagrams are denying), in this example, that if everyone loves themselves then everyone loves someone. That would be a principle that needs including in a suitable logic for the discourse. The maths, complicated enough even for such an ideal discourse, is about determining which other principles (LEM, LNC etc) are also required: either for their own sake, or in order to save others from apparent threats like 'explosion' etc.
In informal discourse, by contrast, we are generally faced with controversies, and the usual, classical logic is clearly valued for its ability to help us take sides. Which side to take, which sentences to save, it never tells. But it shows up some combinations as being either mutually compatible or not so. The compatibility is of course relative to the chosen logic, the chosen set of laws. We choose a logic which we hope will, by showing up compatibilities and incompatibilities (relative to it), have a positive influence on our choices to save and reject.
Thus Popper and Lakatos are rightly fixated on counter-examples, which are signs of incompatibility. At least one of these three will have to be rejected or revised:
Lakatos investigates all the choices, to see better what's at stake. But he is completely satisfied with ordinary logic as a test of compatibility. Nowhere does a paradox, superficial or deep, tempt him to bring a more exotic (stronger or weaker) logic on board. Paradoxes are to be resolved by better understanding the vagaries and ambiguities of, and subsequent clarifications and alterations to the reference of, (specific occurrences of) terms such as x, polyhedron and Eulerian.
I suggest it's worth noticing how people so often feel the opposite duty: e.g.
https://thephilosophyforum.com/discussion/comment/550407
https://thephilosophyforum.com/discussion/comment/566367
Which (hey, we must need a more fancy logic) is an attitude that maybe G Russell would identify as pluralist (and my protesting in those places "please not" as correspondingly monist), I'm not sure. I think I protest only because people are seeing logic as a means of revelation, instead of a (standard of) discipline. Reforming premises to meet present standards of compatibility should be tried before reforming the standards to allow all the premises.
Quoting G Russell
I would rather say that it (the principle, the discipline) must be feasible and/or appropriate for imposing in complete generality. Which of course it can't be. Witness art and poetry. Horses for courses.
Still, going with G Russell's flow, what's the analogy with ordinary counterexamples? Is it, e.g.,
?
80
I think the evidence shows that you have this backward. Often people think illogically. So thinking is definitely not contained by logic. We can, and do think in ways far outside of logic (yours truly being your living example). So if logic gives any directional influence to thinking, it must be normative.
:up: I second that if only because it frees us from being tied down to one normative system of thinking. Never realized that there could be more than one way to think purposively towards the truth. I guess the old adage - there's more than one way to skin a cat - has to be taken seriously.
The whole idea of rejecting classical logic provides a fresh perspective on madness/idiocy - they're simply different schemes of logic neither better nor worse than what has been shoved down our throats as logical orthodoxa.
I'm just wondering though how such variants of classical logic or even completely novel systems thereof look like if applied in everyday life - I recall Gillian Russell cautioning that logicians are extremely reluctant to make their systems weak; unfortunately, she doesn't clarify the term in the lecture.
Logical nihilism reminds me of the law paradox: There is one law and that law is there are no laws.
I want to put logical nihilism into practice but not just for the heck of it; I want to blur the line between sense and nonsense, between sanity and insanity, between wisdom and foolery, between affirmation and negation, :lol:
I guess what I really mean is I want to be myself - TheMadFool :lol:
Unrelated but I never understood how statements like this are paradoxical. Just add a âexcept this oneâ at the end and the paradox is resolved.
I'm glad it's not just me. I never got that either. It crops up everywhere as if it were a law of nature, and yet I've never heard anyone explain why it's a problem.
Like "Nothing is really 'true' (except this statement)".
...
All that's really happening is that a claim is being made about a grouping of entities into two sets; one containing a single unique case, and one containing 'all the others'. I can't see anything which would prima facie make that impossible, or even improbable.
But I didn't!
Socrates said "All I know is that I know nothing" and didn't see any paradoxes because it was clear to his listeners that what he meant literally was "All I know is that I know nothing except this"
I guess I don't fall in that category. Thanks though. Your point is worth noting.
The problem I see with this is not the scope so much as the "really".
Take it out and the statement is clearly wrong: "Nothing is 'true', except this statement.
So I thinked.
I'm not aware of Davidson writing anything explicitly political. Are you? So mention of political ramifications goes outside the purview of his thinking.
So this should be interesting.
Quoting Joshs
Well, i won't sell him off so quickly. His interest is in statements of what is the case, and in that regard he limits his discourse, but we can have some fun extending it. One way to proceed while keeping some of his conniving relevant would be to look at direction fo fit, as discussed in Anscombe and Searle and elsewhere. One might characterised Davidson's interest as word-to-world rather than world-to-word.
But in politics we change the world to fit the word.
So can the notion of incommensurability he is working with be used in a world-to-word language game?
I use chess games as a test case much too often. But it fits, and is at hand. Davidson might be understood as pointing out that we agree on the presence of a board and the pieces; on the squares, and perhaps even on the initial arrangement of the pieces on the board. But suppose someone does not recognise castling. The disagreement here is not as to how the world is, but how the world might be changed.
One might describe the situation as incommensurable; one player wishes to castle; the other does not recognise this as a legitimate move. This is not a disagreement as to what is the case, but as to what is to be done.
Ah, yes. I really only put that in as an example - to say that I didn't (contrary to a lot of arguments I've read) find anything wrong with the form of the proposition.
As to it's content...well I agree, my inverted commas are doing a lot of work there. As you may recall (I believe we've discussed this before?) I come from an entirely linguistic approach to truth - 'true' is just a word and it's meaning varies depending on the use it's put to in various language games. So here it's being applied to the state of the world (by which I mean all that is the case) and being used to denote uniquely high confidence, wherein there is only one thing of which we can be absolutely confident, and that is that the world is such that we cannot be absolutely confident about any of its states (except that one). Perhaps "Nothing is certain" might have been a better choice.
I agree actually. The amount of stuff we can believe to be the case without any problems arising massively outweighs the amount of stuff about which some doubt is useful. As we've encountered before, I think, my job requires I have a model which allows for that level of uncertainty. otherwise our best models of cognition don't work. Day-to-day (and I suppose philosophically too), it might well be useless and better replaced with a model of naive realism with occasional exceptions.
Gillian Russell, I'm sure, has many counter-examples for every logical law there is but all of them seem rather contrived. She reminds me of contortionists assuming odd positions - some funny, others painful - just so that fae can fit inside the box of logical nihilism.
The end result is both amazing - flexibility par excellence - and repugnant - the contortionist looks like fae's been in a horrible accident!
I don't know whether to congratulate Gillian Russell or offer her my condolences.
Quoting Banno
I havenât read Anscombe and Searle on this, but the phenomenologically informed enactivist work I follow wouldnât accept that the one direction ever proceeds independently of the other. Here perceptual processes may be instructive. When I perceive a visual pattern as something , I recognize it. Re-cognition implies two
dynamics at once. From subject to world, there is expectation derived from previous experience of what I am likely looking at. This expectation is as much intersubjectively shaped as it is subjective. The other side of the coin is the direction from world to anticipating subject. My expectations concerning what I am seeing do not univocally determine the sense for me of the phenomenon. The world contributes a novel factor that makes recognition and representation always a contextually new sense of what is being recognized.
Quoting Banno
Or one could say we interpret the world according to our subjectively and intersubjectively formed expectations. But that is not limited to âpoliticsâ unless you want to expand olp rica to include perception and cognition generally.
Quoting Banno
If we agree on the things you mention, it is likely because we abstract these particulars from our understanding of their role in the playing of the game called chess by based what matters to us about it. The game is a temporal unfolding guided by rules of procedure, an agreed upon way of going on, with an agreed upon goal. When one recognizes the pieces and board as belonging to chess , one is implicitly drawing upon this background knowledge of the unfolding activity called chess. In other words , the details get their relevant sense from their relation to the larger purpose of the game as one interprets it. If I do not recognize castling, that belief forms part of the superordinate scheme that frames my sense of the details. When we begin the game, having tacitly âagreedâ on the pieces, board , etc, my background belief about castling is already operative in my recognition of the pieces and other subordinate details. But since this belief retains only an implicit role in our activity until the point where it becomes explcit, when I say âhey, you canât do that!â, it doesnât initially affect our agreement.
This is what I mean about agreements at a superficial level masking deeper discrepancies in outlook.
Quoting Banno
I think the issue comes down to how integrated the pieces of our knowledge are in relation to overarching pragmatic purposes and goals. Davidson seems to allow for a compartmentalization and independence in components of cognitive and language schematics that the enactivists reject.
Playing along with Prof. Gillian Russell's general idea,
The logical law of Logical Nihilism: All logical laws have exceptions (counterexamples). This is a logical law because, we can, by expanding the interpretation, demonstrate that all logical laws have counterexamples (exceptions).
Therefore,
All logical laws have exceptions (counterexamples) must itself have (an) exception(s).
Ergo,
Some logical laws have no counterexamples (exceptions). In other words there are universal logical laws.
Ergo, logical nihilism is untenable.
What say you?
A copy arrived yesterday.
I feel GĂśdel lurking here. The law that all laws have exceptions can't be applied to itself.
One Logic, Or Many?
Maybe they could re-title the article to âOne Logic, Or Many, Or Just talking about something elseâ
It looks like logical nihilism is going to hinge on the Liar. It's supposed to be violating the LONC? I haven't seen other examples.
Am I understanding that correctly?
I don't see why you would think that.
Logical monism claim that there are logical laws that hold in absolutely all case. Logical pluralism claims that no law holds in absolutely all cases. Logical nihilism holds that logical laws do not hold in any case.
Quoting Wayfarer
A performative contradiction?
On that account "there are no logical laws" is not of the form ???, avoiding the problematic.
You are right, and in your terms, for a logical nihilist, the truth of "there are no logical laws" cannot be the result of a strict inference.
That leaves open other forms of ratiocination. If, as they argue, for every given logical law a counterexample can be presented, then one might induce that there are no logical laws.
Quoting Tom Storm
Some post modernists might well reject deduction. It takes all sorts to make a world.
That's just the first example given of shooting down so-called laws of logic in G Russell's article.. The PhilosophyNow article focuses on what strikes me as word games. Statements of opinion aren't true or false, so bivalence is defied?
Quoting Banno
Does the situation compare to moral nihilism? A logical nihilist recognizes that there are logical laws in play, but they hold by fiat? So there may not be a huge difference between nihilism and pluralism?
It might also indicate that logic has limits, which is not the same as to say that it isn't universally applicable within those limits. Graham Priest's diathetheism comes to mind although that too I interpret as an exploration of the limitations of logic.
You'd love that.
Rather I take the flow of the argument here to be that there are a multiplicity of logics, to be applied in many and various cases. It's more about the removal of limits to logic. Roughly, if you come across a case in which logic seems not to apply, then you are using the wrong logic.
Quoting frank
Why not? "Frank thinks statements of opinion are neither true nor false" seems to be true...
Quoting frank
The Philosophy Now article draws that analogy. I don't think it goes very far, partly for the reasons given above. Perhaps the logical monist says "this is how you ought think", the nihilist says "It doesn't matter what you think", the pluralist, "this is how we show if your thinking is consistent"
That was in the PhilosophyNow article. We could do a read through.
Quoting Banno
Russell mentioned that logical nihilists don't tend to measure up to the name. They do recognize the application of something like logical laws.
Happy to. Might be best if you take the lead, so you can highlight the points you see as salient.
Kinda like Sliding Doors, right? Multiverse stuff? That kind of thing? Am I warm?
First we look at case-based logical pluralism. This is the GTT:
Generalised Tarski Thesis (GTT):
An argument is valid-x if and only if in every case-x in which the premises are true, so is the conclusion.
Case based logical pluralism is saying that the terms in the GTT are not precise enough to rule out a plurality of meanings for "valid" and "case." Different senses of these terms will give us different logics.
I don't think this is actually the kind of logical pluralism I was thinking of though. This is just an issue with terminology. It's no threat to logical monism as far as I can see. What follows is arguments for and against it.
"One way to object to logical pluralism via cases is to agree that âcaseâ is underspecified and admits of various interpretations, while rejecting the further step that those interpretations correspond to different relations of logical consequence. One way to do this is to insist on the largest domain for the quantifier âeveryâ in the GTT. There is a tradition in logic that holds that for an argument to be logically valid, the conclusion must be true in unrestrictedly all cases in which the premises are true; if there are any cases at allâanywhere, of any kindâin which the premises are true and the conclusion not, then the argument is invalid. The One True Logic, then, is the one that describes the relation of truth-preserv is uyt[ation over all casesâwhere âallâ is construed as broadly as possible." --SEP
The fly in the ointment: the Liar. Up next.
I'm not sure why the liar is your focus. As the article suggests, a para-consistent logic might assign it "both true and false" and move on.
Fixing the typo: 'The One True Logic, then, is the one that describes the relation of truth-preservation over all casesâwhere âallâ is construed as broadly as possible'. That section then goes on to set out that construing "all" as broadly as possible may well lead to there being no valid arguments left. Logical monism would lead to logical nihilism.
The interesting issue here is, if there is One True Logic, which logic is it?
Just to give a taste of what is being discussed, here are some of the articles on "alternate" logics in SEP:
Classical Logic
Connexive Logic
Dialetheism
Free Logic
Infinitary Logic
Intuitionistic Logic
Modal Logic
Paraconsistent Logic
Relevance Logic
Second-order and Higher-order Logic
Substructural Logics
Not a complete list, but you might get the idea. Each of these is useful in some circumstance, and each is studied in its own right. Logical monism must in some way make sense of the many and varied treatments of validity, domain and truth in these logics, and either rejecting or incorporating them.
That there are multiple logics is a fact. The meta issue is how to understand the relation between them. Monism rules some in and some out - but which? Nihilism rules them all out - so no logic. Pluralism suggests we might use each as appropriate.
Now it seems to me that Pluralism is the better of these options, but the devil is in the detail, and the discussion is on-going.
What is relevant for the discussion at A challenge to Frege on assertion is that one can no longer simply presume that there is One True Logic, which is what @Leontiskos appears to do.
Would you want to flesh that out?
Quoting Banno
Sounds fair. Is there a risk with pluralism that one might simply select the logic one wants to suit ourselves? How do we determine which logic is appropriate for a given situation/problem? Sorry if this is a banal quesion.
If we were to take an investigation into the logical soundness of theism, for instance, which alternate logic would we use? Classical logic seem traditional.
But why? That's what I was asking.
Well, theism is interesting. So take the sorts of arguments that treat existence as a first-order predicate - ?!. Free logic takes this seriously, but goes on to show that one cannot deduce the existence of some individual in a valid free logic without question-begging. It seems existence is presumed, not proven. That is, taking the theistic supposition seriously does not lead to the desired conclusion. What it does do is clarify what is going on when one claims that something exists. (See Inexpressibility of Existence Conditions)
But how we might deal with a case where, say, two logics over the same domain reach opposite conclusions remains an interesting question.
And one ring to rule them all...
Actually I still don't know what logical pluralism is supposed to mean. I'll continue trudging through the article.
That's part of what the Russell article in this thread is addressing - the idea being, roughly, that if we reached the sort of impasse you describe, we might do well to develop a logic that frames the problem by adding more bits - "lemma incorporation" in the article. A logic to decide between competing logics.
The discussion would then be ongoing, keeping Logicians in paid work...
Given the benefits of the various logics I see no downsides.
Goodness. I'll leave this to the pros.
Quoting Banno
That's fascinating. As above.
Thank you.
Banno looks like the cat who has climbed and climbed and now cannot get down, and does not know where he is. What is logic? Banno thinks it is something like the arbitrary manipulation of symbols - and of course there are many ways to arbitrarily manipulate symbols. But that's not what logic is.
Historically logic is the thing by which (discursive) knowledge is produced. When I combine two or more pieces of knowledge to arrive at new knowledge I am by definition utilizing logic. If logical pluralism were true then you could know X and I could know ~X, and we would both have true knowledge, which is absurd. When, "two logics over the same domain reach opposite conclusions," we do not arrive at an "interesting question." We arrive at contradictory conclusions and conflicting arguments, one of which must be wrong.
It's called dialetheism. I thought about doing a thread on it. Probably not.
Logical pluralists seem to argue that different contexts require different logics and this seems to be determined by the kinds of reasoning or the goals of inquiry involved. So, for the most part, I'm not sure if the result is different conclusions for the same matter, more like different logics used for different situations. But I am just a curious amateur, so for me it's all about the questions.
Quoting Banno
How common would this be and how do we determine which logic to employ?
Quoting Banno
This is a slightly scary idea. Could we end up with an infinite regress?
No, that's really not it. See:
Quoting SEP | Logical Pluralism
For example, someone who believes in deductive, inductive, and abductive reasoning is not a logical pluralist. It is in no way controversial that there are different ways of reasoning.* Even SEP's phrase, "getting things right," is weasel language. The controversy and uniqueness of logical pluralism arise with the idea that there are conflicting logics that are all correct.
Each time I look into these theories they turn out to be smoke and mirrors. It looks a lot like the pseudoscience of the logical world. But even on TPF this is largely acknowledged, so there seems little reason to argue.
* Similarly, someone who utilizes different logical languages or formalisms for different arguments is also not a logical pluralist.
Quoting Tom Storm
Both interesting questions. I don't have an answer - this is a developing area of enquiry.
Russell borrows lemma incorporation from Lakatos, who was student of Popper and involved in a notary altercation with Feyerabend. In the process she is inviting comparisons between the logic of scientific discovery and meta-logic, and perhaps anticipating a response along the lines of Feyerabend's "Anything goes".
Where that leads, well...
Quoting frank
From the SEP article:
And that is where we stand. Presuming that there is one true logic is no longer viable.
Lol. I suppose that's where things stand if you just ignore the rest of the article and/or appeal to SEP as some sort of normative source, setting out what is allowed and what is not, even though it doesn't present itself that way. (Michael has that difficulty as well). In your case it is less excusable given what I have already pointed out to you. Dialetheism qua dialetheism is the flat-earthism of the logical world. Yet the inquiries of dialetheists can and have been interesting, even if they don't ultimately achieve their purported aim.
:lol:
Welcome back. The thread became denecrotised as a result of a discussion elsewhere.
This?
Quoting Banno
Quoting Banno
Yes, I can see this to some extent.
Doesn't Susan Haack argue a somewhat tamer version of this?
https://core.ac.uk/download/pdf/131210177.pdf
pp.13-15
To what extent does your disagreement on this involve, perhaps, one being a conservative and the other liberal?
"Anything goes" is a recipe for conservatism, since if anything goes then the way things are is as viable as the way they might be, and there is no sound reason for change. Think of the new-found love for free speech amongst those advocating for autocracy in recent politics. The confusion of voices shouts out rational discourse.
I supose a blanket rejection of even considering the possibility of alternate logics might be considered conservative. There might be a closer relation to views on normativety.
I think we have to differentiate "doing science" and presenting scientific evidence. Inspirational moments and the willingness to try anything isn't the same thing as establishing support for a conclusion. Popper is pretty clear on this point. He encourages people - sometimes - often times the wrong people to question scientific fact and make bold guesses. And insists they try to prove themselves wrong with tests. So, perhaps the thought process is as it's quoted by you but the rigor might be closer to algorithmic.
In relation to Haack, she seems to be saying that the scientific method is more like 'methods' - a diversity of approaches including creativity, but it is not quite 'anything goes'.
Quoting Banno
Yes, Chomsky says this is the effect of postmodernism (as you say a 'recipe') - radical skepticism about truth and objectivity has insulated the intelligentsia from popular movements and activism. But isn't the conservative approach per-say one where orthodoxy rules, where there is a right way and a wrong way to do pretty much everything? In the case of our question about logic, I'd imagine a conservative might balk against the possibility of logical pluralism. Just a thought.
It's an understandable trope, but in this case I think it is just that Banno is concerned with what I call metalogic/metamathematics and I am concerned with what I call logic. He was trained in that emphasis and so he thinks of it as logic. Would Banno actually bite the bullet and accept full-blown logical pluralism? I doubt it. I think he is just flirting with it as a contrarian who discovered an exotic idea. And I don't see enough support for that position on TPF or elsewhere to expend much effort critiquing it. Srap's logical pragmatism is an example of an approach which is much better represented.
But the substantive question relates to knowledge, which is why my first post in this thread concentrated on that topic.
(At the end of the day the principle of non-contradiction is the issue, and Aristotle showed long ago why attacks on the PNC can never succeed.)
This is a fine question, but I want to say that the better question along these same lines is this: How do we differentiate an argument which is invalid from an argument which is merely pluralistically different? There is no differentiation between the two at the level of the object language, and this inevitably pushes the formalists into a metalanguage.
Stated more simply, if different approaches to logic are just different tools, are there nevertheless tools that won't work? Are there any bad arguments at all? And can someone who says that there are no bad (or good) arguments really call themselves a logician?
Quoting Leontiskos
Well, yes, in the end that's what all this leads to. Fair point.
Quoting Tom Storm
Ok - I'd be more comfortable calling that authoritarian, a word I nearly used in the place of "conservative" in what you quoted. The normatively of telling someone "This is how you ought think..." differs from the normatively of "If you think in that way, then this will be your conclusion..." That is, the logics here are systematic, not arbitrary - what "full-blown logical pluralism" might be remains unclear until Leon addresses the issue instead of my failings. If Aristotle showed long ago why attacks on PNC cannot work it should be a small thing to show why paraconsistent logic is flawed; yet instead it is an area of growth.
Just as the apparent contradictions between classical physics and quantum physics might be about how reality manifests in different scales, perhaps logics may vary depending on the calibration of the problem they are applied to. Or something like that.
Yep, interesting stuff. In classical logic, A,~A ? B (From A and not A you can derive whatever you want). This would cause all sorts of problems. Paraconsistent logics remove this problem, usually while maintaining the Law of Noncontradiction. One can get a handle on the idea by looking at many-valued logics.
Supose we allow three truth values - "true", "false" and "buggered if I know" - abbreviated to T, F, B. Then we set up truth tables with three values instead of two. With a bit of fiddling we can make it so that A ^ ~A (A and not-A) gives the truth value "buggered if I know". I'm cheating here, but the idea heads roughly in the right direction. A contradiction does not lead to just everything being true. If you want more, see here or Chapter Ten of Open Logic.
Point is, there are formally developed logics that are coherent, if inconsistent. So fears of Woo are dissipated...
Well, for some. Perhaps those feeling less conservative?
It is a thing.
But what Russell is doing is a bit beyond all this - the next generation, if you will. She is considering:
And the approach is the antagonistic one of "You give me a law you think holds in complete generality, and I'll give you a counter-instance". The playfulness and creativity are appealing. Compare it to A nice derangement of epitaphs. The bit about undermining the law certainly will stick in some folk's craw.
She does not wish to conclude that there are no laws of logic, and so argues that a principle need not hold in complete generality. Instead, they hold in given logics.
(sorry - lots of edits.)
Well, that's logical...
Quoting Banno
A likely concession! Well, it's pretty much off limits to me, I have no knowledge of logic or philosophy, so I'll need to leave it to the cognoscenti. Thanks for the clear explanations.
I would just chime in that many people who oppose logical nihilism (and many, but not all forms of pluralism), would rather say that material logic has priority over formal logic in some important respects. Formal logic is about "ways of speaking," but logic is not about "ways of speaking" tout court.
There is the "discourse of language" which is constrained by the "discourse of the mind." As Aristotle says in the Posterior Analytics, we might very well say "square circle," or "x both is and is not, in precisely the same way, without respect to time," but we cannot think it true. But there is also the "discourse of being," the matter of logical statements. These must have form to be intelligible, but their form-"whatness/quiddity"- is not necessarily going to be found solely in the stipulated signs developed for communicating that form (e.g., an embrace of tripartite Augustinian/Scholastic/Piercean semiotics will entail a sort of realism here, where objects are relevant to the sign relation and signs not arbitrary).
Anyhow, to the extent that logical nihilism will tend to imply that things have no causes, that there is no metaphysical truth, etc. I think it's open to the criticism that:
A. This seems demonstrably false on all the evidence of sense experience, the natural sciences, etc.;
B. No one actually has the courage of their convictions on this matter and really acts as if causes and truth are "just games," and;
C. This makes the world inherently unintelligible and philosophy pointless.
Plus, to the extent that someone still tries to justify logic on "pragmatic" grounds it seems to be the case that any "pragmatic" standards bottom out in arbitrariness, there being no truth about what is truly a better standard or what truly ranks higher on any given standard. Hence appeals to the "usefulness of certain games," are unsupportable.
What are your thoughts on Russell's paradox? Is it like Witt thought, from a transcendental logic? Or what?
I haven't given Russell's paradox two much thought, at least as it respects logic as a whole. I think Wittgenstein gets something right in his early work re the necessary conditions for intelligibility and meaningful speech about the world. Whereas I take his later work to be useful in terms of rejecting a narrow view of truth and language that had become prominent in analytic philosophy in the early 20th century. Unfortunately, Wittgenstein never undertook a study of earlier philosophy, and so we don't get to see how he might have engaged with other views of truth and intelligibility, which is a shame because it could have been quite interesting.
IMO, early analytic philosophy has unfortunately become a sort of popular strawman for continental philosophy and pro-deflation analytics.
Some might say that if you have strong feelings about logical monism, you would probably have some way of dealing with paradoxes. Would you agree?
Perhaps, with the caveat that how one approaches paradoxes depends on how one views logic in the first place. If we follow the peripatetic axiom that "nothing is in the intellect that was not first in the senses," my question is "where are the paradoxes in the senses or out in the world?" I have never experienced anything both be and not be without qualification, only stipulated sign systems that declare that "if something is true it is false," and stuff of that sort.
Griffiths and Paseau's isomorphic invariance accounts of "true logics" seems like a step in the right direction, but still seems likely to founder on the equivalence of logic with formalisms.
Deflationists are often quick to point out that they are just talking about "games" and "ways of speaking," lest they step on the toes of the dominant naturalist paradigm and common sense, but they seem to invariably want to start making philosophical/scientific claims based on the study of completely abstracted formalism eventually. It's all just talk of "systems" until it isn't, e.g. "truth cannot be relational because in classical logic it only takes one argument (adicity)."
Specifically, it's provided by Statistical mathematics which reaches for an approximation to the truth. Which is probably why it's reliable, unlike syllogism which fails to account for unknown error. Which points to my earlier misadventures of pointing out that knowing A; entails the possibilty of being wrong about A and asserting it is true. The problem isn't in the system of logic but the flux of the evidence.
'What is, is' only works if you're correct about what it is initially.
A thought came to mind about Kant's (still useful) way of breaking up the world. Logic is a way of recognizing rules. This is how information is parsed out. Scientific principles regard distilling correlations to a point of being able to distill rules (of the empirical). The two logics are different- one has to do with language pattern, and one has to do with empirical patterns. However, they are both intertwined, as the rules of logic seem embedded in language, something that comes prior to the empirical correlation-distillation that takes place in the cultural practice of scientific research.
Quoting Count Timothy von Icarus
I agree with both statements in acknowledging the difference between logic as a transmission protocol and logic as it happens about the mind. Saying our rules for making statements are imperfect doesn't establish that the world can't make sense.
Quoting Count Timothy von Icarus
Yes, very good. In my opinion this all gets a little tricky because what is at stake is a ratio, not a concept. For instance, to use a formal logical system is not thereby to commit oneself to the view that logic is formal logic. Lots of people who used and even created formal systems recognized that their formal system is not identical to logic itself.
Quoting Count Timothy von Icarus
Very good. :up:
Quoting Count Timothy von Icarus
Agreed. :up:
(I am tagging @Srap Tasmaner given that we were talking about similar issues elsewhere.)
A paradox is not the type of thing that has a location.
Quoting Count Timothy von Icarus
Not having experienced it so far doesn't rule it out, though.
Right, and it is very important that we keep our eyes peeled for square circles. They are probably lurking just around the corner.
I was looking for a 'it can't happen because it's illogical.'
Care to step up to the plate?
Quoting frank
Frank, how would a square circle look? That is how would you know something was a square circle?
Perfectly round with four corners.
A new term to me - no mentions in SEP or in IEP. Not just no article, but no use of the phrase. so I googled it. A couple of blogs, none of them very clear, and with a few obvious errors. Merriam-Webster gives "logic that is valid within a certain universe of discourse or field of application because of certain peculiar properties of that universe or field âcontrasted with formal logic". I gather it means informal logic or possibly applied logic.
So I could find no justification for your claim that "many people who oppose logical nihilism (and many, but not all forms of pluralism), would rather say that material logic has priority over formal logic in some important respects.". Many people do not talk of "material logic".
Logical nihilism is not the view that things have no causes. It is the view that there are no laws of logic. But also, despite the title, it is not the conclusion being argued for in the Russell article. The article that this thread concerns, and which neither you nor Leon have so far addressed. It is also not concerned with any form of pragmatism.
Logic has moved on a bit since Aristotle.
A circle is a drawing or something imagined. it doesnt have a "back" since it is a representation of a two dimensional object. So it's not clear what you are proposing.
Well, in terms of priority, it would seem that perception is prior to speech, both in evolutionary terms and in the development of the individual. But then we would do well to remember Aristotle's dictum that "what is best known to us," are the concrete particulars (the "Many") whereas what is "best known in itself" are the generating principles/principles of unity (the "One"). Prima facie, it seems that the intelligibility of being must be prior to knowledge in the order of being/becoming, while the reverse is true in the order of becoming.
Indeed, although the paradoxes I find most interesting are paradoxes that might be said to have many instantiations, e.g. the sorties paradox, the ship of Theseus, the problem of the many. The issue here seems to lie in predication, and so it's more obvious that there has to be a metaphysical side to the investigation. Now, this is also true of Russell's paradox, since we're talking about proper predication of group membership, but I feel the issue tends to get muddled due to the degree of abstraction involved and the difficulty when it is simply assumed that, because groups can be arbitrarily stipulated, group membership is properly thought of as arbitrary. The issue at stake that the genus and species of the logician are not those of the philosopher and scientist, the latter deal with generating principles "at work" in a multitude in the world, the former with merely the possible forms of predication.
You see a similar split in the application of information theory to the sciences. What is the proper distribution to use in determining the information content of a message? This is an issue that cannot be solved by looking at the formalism is isolation.
Thanks. I should probably add that it's obvious that tackling formalism alone can be very useful. The idea that there is "nothing but formalism" is the problem. I think you can trace these problems back pretty far, to the confusion mentioned above above about the species and genus proper to the philosopher/scientist versus the logician. In the late-medieval period, these two got combined and species and genus were turned into logical constructs of a sort, which in turn fostered all sorts of arguments for a thorough-going nominalism. But if you take nominalism far enough, then of course logic is going to reduce to formalism. Frege's idea of an "empty subject," where predication has nothing to do with what is being predicated of is a step in this direction.
"Material logic," is not an esoteric term, it was part of all logic curricula for over two thousand years. The form/matter distinction is where we get the term "formal logic" from. It's a going concern for some 20th century philosophers who are less convinced about the reduction of logic to form (e.g Peirce and through him Deeley.) The term is less in vouge now, probably because of the hylomorphic distinction it implies, but obviously the relation between human discourse, the discourse of the soul, and this discourse of being is still something people talk about all the time.
Of course "logic has advanced since Aristotle," nothing I said suggested otherwise. However, I wouldn't take it as a badge of honor to be entirely ignorant of the basics of logic prior to the 20th century on account of this fact.
As for the other comments, I was just pointing out the assumptions that seemed implicit in the opening post. Logical nihilism and a deflationism vis-ĂĄ-vis truth and a denial of causes certainly seem to go together as a package deal much of the time. I've don't think I've ever seen logical nihilism not paired with deflation; who would be a counter example here?
Sure, I didn't say that perception/basic experiential sensation isn't prior to language. Rather, I am simply saying that language seems to have a logic and so do the "empirical rules" that one can distill from repeated testing/correlation-distillation. These are different but related. Prior to the scientific/empirical rules, language, and its adjacent abilities (conceptual-thinking, capacity for inference, etc.) seem to need to be in place. Both need to be explained for a proper metaphysics, and in some theories (like information theories), they aren't so separated as part of the same type of thing going on.
:up:
I want disagreeing BTW, just chiming in.
Cool. :smile: :up:
Making an argument for impossible things it seems. I maintain that a square circle ought to be perfectly round and have four corners regardless of how it appears. And if one was found then it would meet that criteria. Logically it can't exist by definition, but neither can a single point that's a wave and here we are.
Well, the little evidence I could find says otherwise. Here's an Ngram of interest.
The point is moot, since it is so off-topic.
And yes, the topic here is logic. Not Aristotle. Quoting Count Timothy von Icarus
And not deflationary theories of truth nor a denial of causation, neither of which have any relevance to the arguments offered here. And nothing about square circles, either.
Do you think the possibilities for this universe are limited by what strikes us as conceivable?
:grin:
Quoting Count Timothy von Icarus
Yes, but as I said earlier, I don't see much support for it generally or on TPF. Most people who think about this for more than 15 seconds realize that "nothing but formalism" is a complete dead end. Frank wants his square circles and Banno wants his logical pluralism. I would need to see other voices taking up such bizarre positions before I would be interested in engaging, and I don't see any. The same cannot really be said for things like nominalism or logical pragmatism, which have a wider base of support.
I think the problem there is that we are trying to understand micro quantum phenomena using macro concepts. So is a quantum particle anything like a particle of sand, or a quantum wave anything like macro wave phenomena? It seems to be not a true paradox and in part at least a terminological issue.
:wink: Yep. Unfair advantage.
Think on i. [math]\sqrt -1[/math] isn't a thing except it is. What would we get if we just assumed a perfectly round square circle with four corners? What would be the implications? Could we construct a geometry that was interesting, if somewhat divergent? When we assumed the three angles of a triangle add to more than 180 degrees we were able to develop a geometry to navigate the globe.
The properties that define circles make shapes that appear as squares in taxicab space. But the geometry jettisons our concept of roundness, unfortunately.
It's the corners that screw you up in trying to come up with such a square circle object, I think. For something to be a corner, two lines must meet at a right angle. Two lines meeting at a right angle doesn't produce a differentiable function (along the shape the lines meet in) regardless of how you rotate the shape or embed it in another one's surface, so you've got to choose between jagged edge to allow corners, and roundness.
You use the above, and the taxicab thing in my previous post (quoted below), to stipulate the following:
Quoting fdrake
I could guess the principle: every circle with corners is not round. Specifying
1) A circle is shape resulting from constant distance around a point.
2) A corner is a meeting of two lines at a right angle.
3) A round shape is smooth along its curve.
And hope to prove that there's no such shape. But I could've misspecified the underlying concepts. I imagine there's something odd about "corner" and "smooth", because "corner" relies upon "right angle", and "right angle" depends upon "angle", which depends upon the concept of an inner product, and the privileged connection between inner product and metric is something we get from usual Euclidean space. Moreover, "smooth" could also be generalised to reference a different metric.
So perhaps there is some space that has a metric related to an inner product in which there are round circles with corners, but I've not thought of such a counterexample myself.
Me going through the maths there isn't an attempt to side with over @Banno, because being able to explore the conceptual content of the allegedly logically impossible should tell you that logical impossibility isn't all it's cracked up to be. You do have to ask "which logic and system?", and "what concept am I not formalising right?" or "what concept is making the weird shit I'm imagining weird?".
The historical fact that "formal logic" is not called such because it is being set over and against "informal logic," but rather because the term refers to the study of the form of arguments as abstracted from their contents (matter) seems pretty relevant to understanding what formal logic actually is. If you think the difference is "formal versus informal" it seems easier to make the mistake of thinking that the study of form is simply all there is to logic (or that there is no debate to be had on this issue.)
This is the second time you've pulled out charts in this ridiculous way. The first was when you were telling others that "Russell had widely been seen as dispensing with causes in the sciences." Professional philosophers widely disagree with this sentiment, even partisans of Russsell. That time I shared multiple literature reviews by Neo-Russellians who themselves admit that Russell's premise that scientists don't speak of causes is false as of the 70s, false today, and likely false when Russell made the claim (although Russell bought himself some wiggle room by making an ambiguous appeal to an undefined set of "advanced sciences.") You produced a word count chart as a counter to well cited reviews in the field... Asking GPT would probably be more profitable, and I have a pretty low opinion of that as well.
You might consider that perhaps your interlocutors have some level of expertise on what they speak and that word searches are neither good arguments nor good ways of informing yourself about philosophy.
I assure you, I am not trying to trick you here. This is simply a fact, and in arguing against it I assure you that you look every bit as silly as the folks who disagree with you on the basics of classical logic and refuse to change their minds on it. Material logic is about as relevant to the history of logic as "eidos" or "form" in metaphysics. Perhaps those also fare poorly on word searches, but they are hardly esoteric.
I am not surprised that people fail to use the term, since the distinction is more apt to be phrased in terms of "form and content" today because "matter" had gained new connotations from physics. Yet clearly the subject area comes up all the time. Scientists regularly mention "the logic of thermodynamics," the "logic of natural selection," etc., and this is clearly not looking at form divorced from content.
How could this not be relevant to logical nihilism? If the form is abstracted from its contents, that's obviously going to be a much different basis for logic than if it's "form all the way down."
Quoting Banno
âAnything goesâ is also the common strawman argument against a logical pluralism that is taken disparagingly to imply a ârelativismâ or or ânihilismâ, a view that those accused of relativism never actually hold, according to Rorty.
Quoting Count Timothy von Icarus
What is true is true in relation to a normed pattern. Perception, as pattern recognition, is conceptually based. This means that expectations guide recognition of perceprual objects. It also means that in assimilating the world to our expectations we at the same time modify those expectations to accomodate to the novel aspects of what we perceive. Put differently, in a certain sense what we perceive both is and is not what we anticipated. This not the same as saying that it is both true and false, since the sense of meaning of a conceptual pattern is being qualitatively adjusted in perceiving something. Thus the thing we continually recognize continues to be true differently. With regard to formal logic, if we think of a logic as producing a rule, then in following a rule we operate the same as we do in perceiving. The criteria of rule-following no more guarantees a criterion for correctly following it than our previous experience with a perceived object tells us how to recognize it correctly now.
Is it that formal logic outlines how one statement follows from another, and material logic looks at the limits of thought and language?
For my part I don't see much need for a transcendental logic because I don't think our sensible intuition conforms to the categories in the manner which Kant seems to believe -- in some sense what Kant does is define the absurd as outside of the scope of cognition, and yet the world remains absurd for all that: We can choose the categories we want to use in describing the world, and they change far more than what is desirable in a logical system.
As evidence of this I reference the difference between Kant's categories and the most general scientific theories -- I don't see any need for a group of categories to make sense of science. I don't think the structure of the mind or the minds relationship to being is the site of knowledge, but of comfort.
Â
Basically I see the appeal of Aristotle and common sense as a mistaken appeal -- it makes sense of the world, but need not hold for all empirical cases: There are times when a person is in contradiction with themself, or an organism has a contradictory cancer, or a social organism is composed of two opposite poles (hence Hegel's use of contradiction in attempting to understand a social body or mind).
And I, for one, take up the liar's paradox as a good example of an undeniable dialetheia: A true contradiction.
Especially because the liar's sentence gives justification to P2 in the original argument: No principle holds in complete generality.
Yep
Sure, if by "pure" we mean "ignoring the content and purpose of logic." But even nihilists and deflationists don't totally ignore content and the use case of logic. If you do this, you just have the study of completely arbitrary systems, and there are infinitely many such systems and no way to vet which are worth investigating. To say that some systems are "useful" is to already make an appeal to something outside the bare formalism of the systems themselves. "Pure logic" as you describe it could never get off the ground because it would be the study of an infinite multitude of systems with absolutely no grounds for organizing said study.
One might push back on Aristotle's categories sure, but science certainly uses categories. The exact categories are less important than the derived insights about the organization of the sciences. And the organization of the sciences follows Artistotle's prescription that delineations should be based on per se predication (intrinsic) as opposed to per accidens down to this day.
This is why we have chemistry as the study of all chemicals, regardless of time, place, etc. and biology as the study of all living things as opposed to, say: "the study of life on the island of Jamaica on Tuesdays," and "the study of chemical reactions inside the bodies of cats or inside quartz crystals, occuring between the hours of 6:00am and 11:00pm," as distinct fields of inquiry. Certain sorts of predication (certain categories) are not useful for dividing the sciences or organizing investigations of phenomena (but note that all are equally empirical).
Of course, there have been challenges to this. The Nazis had "Jewish physics" versus "Aryan physics." The Soviets had "capitalist genetics" and "socialist genetics," for a time. There are occasionally appeals to feminist forms of various sciences. But I think the concept that the ethnicity, race, sex, etc. of the scientist, or the place and time of the investigation, is (generally) accidental to the thing studied and thus not a good way to organize the sciences remains an extremely strong one.
That said, if all categories are entirely arbitrary, the result of infinitely malleable social conventions, without relation to being, then what is the case against organizing a "socialist feminist biology" and a "biology for winter months," etc ?
They certainly wouldn't be useful, but that simply leads to the question "why aren't they useful?" I can't think of a simpler answer than that some predicates are accidental and thus poor ways to organize inquiry.
I agree with this. Roughly what I'm thinking is that consciousness evolves and that this involves both changes in environmental conditions and native mental flexibility. So, for instance, if the people who inhabit a two-dimensional world evolve into beings who can experience three dimensions, it will be partly because the environment makes it so they need to, but long before the general population changes, there will be those who have been expressing flexibility, even though it may have seemed pointless to those around them. These will be people who denied that their traditional logic limited them or the world.
Therefore it's ok to do pointless investigations. It's always been part of what we are, since at least 60,000 years.
We don't tend to talk about form and matter the same way today, so I would just thinking of it as the study of "content" in the "form versus content" distinction.(The term "subject matter," comes from this same distinction. The matter is the information in a subject or discipline, as opposed to the subject's formal definition, which defines which matter falls underneath it).
This could obviously include a discussion of psychology and the "laws of thought," and, depending on one's epistemic commitments, maybe it ends there. However, for most realists/naturalists it will also extend to things in the world (e.g. the real leaf we predicate "green" of).
For example, we can say "red" or "angry" of the number "4," in ways that are entirely correct vis-ĂĄ-vis form. Yet obviously such talk is nonsensical because if one considers the content of: "the number four is angry and red," it is clear that the subject is not of the sort that it can possibly possess these predicates (obviously, this implies we are speaking of the number, not some drawing of 4 in a children's book, which might indeed be angry and red).
This distinction gets trickier when we get into analogous predication, which formal logic tends to ignore because it has proven difficult to formalize. Nonetheless, we cannot totally ignore it, because we use it in natural language and the sciences constantly.
For instance, economic recessions are an empirical phenomena that are studied by the sciences. But when we predicate "double dip" of recessions we obviously don't do so in the same way that we would say a road has a "double dip." Likewise, branching processes in population genetics don't "branch out" the way tree branches and veins do, although the use here is not totally equivocal either. It seems to me that analagous predication has to involve material logic to the extent that the content defines the sort of analogy we are speaking of.
As much as I dislike GPT, it does a fine job on the basics here.
I think this is about competent language use. Russell's paradox isn't about language use. It's not nonsensical.
I asked you before: are you saying that if X is paradoxical, it can't exist? I guess I'm wondering if this is a question you don't want to address for some reason?
Anyhow, Kant's distinction is an interesting one, but it's guided by his metaphysics and epistemology. If we want to speak of why the mind is the way it is in terms of evolution, neuroscience, physics, etc., we are already leaving Kant behind.
For some, this is a bridge to far. Personally, I think the natural sciences, the study of phenomena, tell us about things other than phenomenal awareness (e.g. "the sun is made up largely of hydrogen gas," is not just about our phenomenal awareness, but expressing something true about the sun). And if this is true, then we can speak of the relationship between logic and being as opposed to just logic and experience or the necessary prerequisites of experience.
Sure, that was just an example on the relevance of content to meaningful predication. But Russell's paradox is about stipulated sign systems, "languages," no?
I guess I'm just not sure what you're asking? Of course paradoxes exist, Russell's paradox is one of them. You can observe it. But it exists in a stipulated sign system. Ditto for the Liar's Paradox. I mean, consider the common instantiated version of Russell's paradox. In this version the solution is simple, it simply is not true that "every man in the village either shaves himself or (exclusive or) is shaved by the barber." Either the barber doesn't shave or he is shaved by both (or maybe someone else shaves him). The paradox does not imply actual occasions of things that both do and do not do something in an unqualified way (although I will grant that the possibility of error and falsity itself are mysterious in a way).
What would be an example of a paradox in nature? To be sure, we call all sorts of natural phenomena "paradoxes," e.g. the Fermi paradox, the level of plankton in the Arctic given the amount of sun it gets, etc., but these seem like they could be resolved completely unparadoxically if we just knew more.
The only thing I can think of would be a case where something both is and is-not in an unqualified way, and no I don't think such a thing can exist (...and not exist :rofl: )
Tarski is stipulated sign systems. Set theory is fairly intuitive. Even the foundations, which obviously directly defy Aristotle, are fairly easy to embrace, especially after you've studied calculus. I guess you could target set theory's foundations in favor of finitism. Is that what you're thinking?
Quoting Count Timothy von Icarus
I don't know. My consciousness might have to evolve some before I can see it. My question, though, is do you think the possibilities of our universe are limited by what appears inconceivable to us?
Actually, I will correct what I said above, is this just about competent language use? Does the fact that it doesn't make sense to speak about something "moving greenly," "economic recessions being pink," or "plants being prime," only have to do with the rules of competent language use and not with what those things actually are?
To be sure, the proximate issue might be competent language use, but is language itself arbitrary or a brute fact such that it isn't the way it is due to other causes? It seems to me that it is improper to speak of recessions being pink because they aren't the sort of thing that has color.
Good question.
Quoting Moliere
What if in place of Kantâs Transcendental categories we substituted normative social practices? Doesnât that stay true to Kantâs insight concerning the inseparable role of subjectivity in the construction of meaning while avoiding a solipsistic idealism? Donât we need to think in terms of normative social practices in order to make sense of science?
Well, that's partly what material logic is concerned with. Semiotics, through Aquinas, John Poinsot, C.S. Perice, and John Deeley is one particularly developed area that has a lot of overlap with this question (Sausser-inspired and post-modern semiotics largely considers the question unanswerable/meaningless and so ignores it though).
To us? No. What is inconceivable to one man might be properly conceivable to another. I don't think toddlers can fathom many things adults can for instance. But can something exist that is inconceivable and unintelligible in an unqualified sense? I am not sure what that would mean. Something lacking not only in any possible explanation, but in any quiddity/whatness? Something that both is and is not in an unqualified sense?
Eric Perl raises the related question of: "what is meant by 'being' if 'being' is not to refer to what is apprehended by or 'given' to thought?" I think it's a good one.
Ironically, the positing of unintelligible noumena seems to have had the strange historical effect of resurrecting Protagoras' old doctrine that "man is the measure of all (meaningful/intelligible) things." I disagree with this; man is rather the proper measure of men, horse of horses, etc.
I've gathered that you're just not going to answer that question. That's cool. :up:
You seem to summon the philosophy of apokrisis. The all-encompassing "information" of the language-species AND the universe versus the context-dependent post-modernists.
Edit: I see we've engaged with this briefly before: https://thephilosophyforum.com/discussion/comment/825333
https://thephilosophyforum.com/discussion/14334/adventures-in-metaphysics-2-information-vs-stories/p1
An overlap in interest maybe. His version is idiosyncratic though.
Funny, because that's the exact word I was going to use :lol:
Nice. That's the sort of playfulness we get by adopting these considerations. I can't help you with re-defining smoothness for Taxicab space, but since every point is on a corner I don't see how the path can be differentiable, and hence smooth.
This Interactive Mathematics page shows the problem, under "An interesting question arrises". There are two values for the limit - 2 and ?2. So the space is not smooth, unless we re-define "smooth".
Quoting fdrake
That's the take-away. It's related to what I was trying to show with Banno's game - in which any rule can be undermined; but also, and yet again, to the analysis of language in A nice derangement of epitaphs.
Thanks.
Seems to me that it remains unclear what "material logic" is. But that is not true of formal logic.
A tale. One of the pre- socratics - I forget which - "proved" that air becomes colder under pressure by blowing on his figure. The breath feels cold. And we all know that a wind is cold. Hence, he disproved that gases under pressure increases in temperature. Do we take this as a refutation of thermodynamics?
Seems to me that you are truing to do something similar with formal logic. It just doesn't work. So:Quoting Moliere
Put bluntly, I do not see that you have differentiated formal and material logic in a way that can be maintained beyond "material logic is an over-simplification of formal logic".
And in any case, this does not address Russell's case.
Is this what you mean by material logic?
Quoting Leontiskos
Or we could say that logic is that by which correct inference is achieved.
Quoting Count Timothy von Icarus
Indeed! It is also a symptom of conceiving everything in terms of technicalities, technical terms, and stipulations.
Exactly. We don't use logic to tell us what's in the world. If we did, we'd still be in the stone age.
Well, your post would appear obtuse to the layman, and maybe it just is. Maybe the argument is much simpler than you are making it:
Or even simpler:
These arguments are not any less powerful for their simplicity, and most objections would be little more than quibbles. For example, someone might offer the counterargument of a shape like 'D', and claim that it is both circular and square. That quibble of course could be addressed, but need not be.
More formal:
It is very odd to question such arguments. If these are not good arguments, then there probably is no such thing as a good argument. There seems to be a point at which trying to be charitable towards a dubious thesis crosses over into sophistry, no? Logicians have a difficult time saying that some claim or argument is false or unsound, as opposed to merely invalid. In these cases one must recognize that falsity can enter into a concept; that someone can simply fail to understand what a circle or square is.
More than OK.
When we do come across a "paradox in nature", so to speak, what we do is change the way we talk about what we see. Perhaps the commonest example now is the supposed paradox of the dual nature of particles and waves. Instead of talking about particles and waves we use SchrĂśdinger's equations and things work nicely. Similar tales can be told about heliocentrism and the speed of light in a vacuum and many other adjustments to our understanding. We don't come across "paradoxes in nature", not because the world is made so as to avoid paradoxes, but becasue we change the way we describe things in order to accomodate what was previously spoken of paradoxically.
That is, we adapt our logic to match what we see.
But these are so far from counterexamples to Aristotle that they are all things he explicitly takes up.
Quoting Moliere
Every time I have seen someone try to defend a claim like this they fall apart very quickly. The "Liar's paradox" seems to me exceptionally silly as a putative case for a standing contradiction. For example, the pages of <this thread> where I was posting showed most everyone in agreement that there are deep problems with the idea that the "Liar's paradox" demonstrates some kind of standing contradiction.
Quoting Moliere
A good example of how re-thinking how we phrase the apparent paradox can provide new insight. We have "This sentence is false". It seems we must assign either "true" or "false" to the Liar â with all sorts of amusing consequences.
Here is a branch on this tree. We might decide that instead of only "true" or "false" we could assign some third value to the Liar - "neither true nor false" or "buggered if I know" or some such. And we can develop paraconsitent logic.
Here's another branch. We might recognise that the Liar is about itself, and notice that this is also true of similar paradoxes - Russell's, in particular. We can avoid these sentences by introducing ways of avoiding having sentences talk about themselves. This leads to set theory, for Russell's paradox, and to Kripke's theory of truth, for the Liar.
Again, we change the way we talk about the paradox, and the results are interesting.
And again, rejecting an apparent rule leads to innovation.
Quoting TonesInDeepFreeze
It is obtuse, but I don't think it just is.
A metric is a way of assigning distances to pairs of points. When you consider a space, it has a metric. The usual distance people think of is called the Euclidean distance, and it's the one you're thinking of and measure with a ruler on a piece of paper.
The thing is that the choice of metric is just that, a choice, and you can write down various other spaces with various other metrics. One of those other metrics is called the taxicab metric. Contrasting that to the Euclidean metric:
Imagine you start at a point, and you go 1 step north and 1 step northeast
The taxicab metric says you've travelled 2 total units - you add the steps.
The euclidean metric says you've travelled sqrt(2) total units - you measure the line.
Because a metric defines the concept of an interpoint distance, circles in taxicab geometry are different from circles in euclidean geometry. A circle in taxicab geometry, a set of points defined as equidistant from a single point, looks a lot like a square in euclidean space. 4 corners, 4 right angles, 4 equal sides.
So it is a circle, if a circle is defined by the property of being equidistant from a point. But perhaps it is not a circle, because... well, like you, you could insist that we're not talking about a circle when we're talking about sets equidistant from a point in the taxicab metric. So for you, you'd have to do something to block what we're talking about as a circle in taxicab geometry being a "real" circle.
That places a burden on you to study the concepts of circularity and square-iness, and to say why the first blocks the latter and vice versa. Which is what I did in the post. I'll go through it for nonmathematicians.
For something to count as a square, it needs to have:
S 1) Four sides of equal length.
S 2) Each side meets exactly two other sides at right angles.
Let's just take that as a given, that is what a square is. Now we need to think about a circle. What's a circle?
C 1 ) A circle is a set of points equidistant from one point.
If ( C 1 ) is the only defining property, the taxicab circle is indeed a circle, it's just a circle in taxicab space. Clearly you don't want it to be a circle, so you need to stipulate a restriction. I could also insist that it is a circle, and how are we to decide between your preference and my preference? Anyway, onwards:
Quoting Leontiskos
You specified such a restriction with "what is round is not pointy", which is something similar to what I formalised with the idea of smoothness. The "corners" form the "pointy bits" of the square because the function that defines a square is not smooth at the exact corner point.
There is an ambiguity regarding pointiness, which is similar to the above ambiguity regarding equidistance. In thinking about the corners of the square thing (the taxicab circle) in taxicab space as pointy in the above sense, that requires specifying the roundness concept in terms of the measure of size - smoothness is typically characterised with respect to a measure of size.
Something is differentiable when its derivative exists at every point.
The derivative of a curve exists at a point if and only if at that point the limit of the ratio of the function evaluated at the endpoints of an arbitrarily small interval divided by the length of that interval exists (IE it becomes just a number).
A curve is smooth if you can apply the procedure above to it arbitrarily many times.
The concepts of "interval" and "length" there are also doing a lot of work, since they're distance and size flavoured. And should we expect them to work as our prior Euclidean flavour intuitions would in taxicab geometry? What gives us the right to insist that we think of smoothness as we would in a Euclidean space and transfer it onto smoothness in a taxicab space?
Clearly you would want to insist that they do, my intuitions also run that way. But my intuitions can also side with circles not necessarily being smooth since I'm used to dealing with this stuff!
Where we can agree, though, is with lemma incorporation. In which we specify a set of properties that say exactly what counts as a circle (in your sense) and why it can't be a square.
So for you:
A circle is, by definition, a set of points Euclidean equidistant from one central point.
And thus we've revealed what sneaky hidden presumption you had through lemma incorporation. What we haven't done is decided why that must be accepted as the definition of a circle.
If you want to join in with this exercise of lemma incorporation, I invite you to stipulate a definition of pointy! And we will see where it goes.
@Leontiskos
As an aside, here are some possible counterexamples.
Take all the points Euclidean distance 1 from the point (0,0) in the Euclidean plane. Then delete the point (0,0) from the plane. Is that set still a circle? Looks like it, but they're no longer equidistant from a point in the space. Since the point they were equidistant from has been deleted.
Another one. Take the circle with radius 5 centred at the point (0,0). Then remove all points in the space which have coordinates which are both natural numbers - like (1,2), (7,8). Removing all those points removes the point (3,4), which lays upon that circle (since 3^2+4^2=5^2). That doesn't do anything to change the smoothness of the circle either, since every point on it is the same as before. So it's still smooth, no corners, all points equidistance... It's just missing a point. So, all points in that space which are Euclidean distance 5 from the origin are in the set - so is it a circle?
These would mean you have to come up with some constraint on how hole filled the space, or the circle, could be, and think about holiness itself in order to restore the fact both are clearly circles... Or maybe they're not circle at all at this point. Or neither of them are real counterexamples - it could be my specification's shite.
See what I mean?
Quoting fdrake
Importantly, doing this would not be wrong, as such. It's just one approach amongst many. The error here, if there is on, would be to presume that this was the only, or the correct, approach - that it's what we ought do.
Indeed.
Mathematics papers absolutely call taxicab-circles circles. I just wouldn't call them circles to my students learning shapes.
We had a related discussion here.
My explanation for the weirdness of the staircase paradox. The tl;dr of it is that the length you get by placing a measuring tape along a curve doesn't respect the process of infinitely refining shapes. So it's nothing to do with the shape, it's to do with the concepts of length and limit.
I honestly don't have the maths to try to think about volume and rate concepts in taxicab geometry. Other than my intuition that they're the same as the Euclidean ones... even though the length is different.
Fair enough. But is our preference for systems arbitrary? It seems very easy to have a system where "circle" can be "square." You can even make it axiomatic.
If the presupposition is that all systems are equal, our preferences for them arbitrary, then of course logical impossibility is pretty much meaningless.
But we don't pick systems arbitrarily. It's not the case that the Earth, baseballs, and basketballs are all just as triangular as they are spherical just because it is possible to define a system where this is so. To affirm that would be to default on the idea that any statement about the world having priority over any.
Quoting EnPassant
But that's cheating, of course. "Monster barring" in Russell's terms.
What about the summary here is unclear? https://thephilosophyforum.com/discussion/comment/939603
There is indeed debate over what the proper object of study is here, sure. That's also true of mathematics though. To quote Andrew D. Irvine:
[I]One of the most striking features of [modern] mathematics is the fact that we are much more certain about what mathematical knowledge we have than about what mathematical knowledge is knowledge of. Mathematical knowledge is generally accepted to be more certain than any other branch of knowledge; but unlike other scientific disciplines, the subject matter of mathematics remains controversial. In the sciences we may not be sure our theories are correct, but at least we know what it is we are studying.â[/I]
I don't think it is.
Quoting Count Timothy von Icarus
I agree. They are picked to reflect, capture or illustrate certain ideas. If you came up with a system of arithmetic that couldn't prove 1+1=2, it'd be a shitty system of arithmetic.
I agree. The everyday conceptual content of Earth (the concept), baseballs (the concept) and basketballs (the concept) are that they are round.
I disagree. I think you missed the case that priority can also be seen as purpose and context relative. Here's a series of examples regarding roundness and sphericality.
I prioritise the notion of roundness when considering the Earth on an everyday basis, and I might while calculating its surface area - fuckit it's a sphere and that'll do. But on a day to day basis, my body treats the Earth by and large as flat. And that has priority over a merely intellectual commitment to its roundness as far as my feet are concerned. If I'm trying to stand on a bosu ball, now that fucker is round.
If I were studying variations in the acceleration due to gravity on the Earth's surface, I couldn't treat the Earth as a sphere - since it's roughly oblate, it's a spheroid. And crap like Mt Everest sticks out of it, so it's pointy. If we go by @Leontiskos intuition that round things cannot be pointy in any context, well the Earth is in trouble.
More generically, the role specifying a system has might be thought of as setting out some concept for some purpose. That allows you to see whether the system specification is fit for task.
How do you decide whether it's fit for task? Well I suppose you decide on a task by task basis. Thinking of Earth strictly as a sphere, with the assumption that a sphere is like a circle where every point on its surface is equidistant from its centre... That doesn't work as soon as your legs move. So that's not fit for walking.
But it is fit for a quick and dirty calculation of volume. Or an explanation for how it attained its shape due to gravity.
Here are more abstract examples.
Those tasks are quite concrete - there are harder ones. Like how might we consider fitness for task of a concept of logic in the context of arguing with a salesperson? Their responses aren't going to follow propositional logic... So something informal is required, they're definitely trying to persuade you. Emotional appeals? Reframing? Motivational speech? We could speak of a "logic of sales" that consists of such chicanery. And it would be nuts to think of the salesperson's behaviour solely terms of syllogisms and propositions.
How might we consider the laws of addition when considered from the perspective of raindrops? Well one raindrop alongside another raindrop might be two raindrops, but it could be one larger raindrop depending upon the distance between them. So "raindrop addition" might be way more complicated than adding discrete units of things...
Here's what I think is the general principle.
The rough trick is the same in each case, you have some conceptual content you want to specify, you try to set out a collection of rules that specify the conceptual content, then you shit test the rules to see if you got anything wrong. Or you can maybe prove all and only the results that you want - or solve all your problems - then you've succeeded beyond your wildest dreams.
So it's bits of applied logic and ontology and model theory and metalogic. Fine.
But this isn't right. The Euclidean metric says you've traveled 2 total units. Yet the distance of a straight line between your starting point and your ending point is sqrt(2). Apparently taxicab geometry measures the distance between points differently.
Quoting fdrake
Not really. Only if the radius is a single unit. The larger the radius, the more circular it will be.
Quoting fdrake
You're presuming that your made up "taxicab geometry" is on a par with Euclidean geometry. But it's not. What you've done is engaged in equivocation. You want to say, 'A circle is the set of points equally "distant" from a single point.' Scare quotes are required, because we both know that your artificial definition of "distance" is not the accepted definition. Similarly, 'This figure is a "circle" in taxicab geometry.' But I was talking about circles, not "circles."
Quoting fdrake
We could say that a circle is a figure whose roundness is perfectly consistent.* There is no part of it which is more or less round than any other. In calculus that cashes out as a derivative, but folks do not need calculus to understand circles. Calculus just provides one way of conceptualizing a circle.
Quoting fdrake
Is it more "sneaky" to think that circles go hand in hand with Euclidean geometry, or to think that Euclidean geometry and taxicab geometry are on a par? Not only are they not on a par; taxicab geometry presupposes Euclidean geometry.
Quoting fdrake
But they are. You have an odd assumption that points are stipulative, as if we could delete a point or as if a point could have spatial extension. The set of points is still equidistant from a point. This idea of "deleting" points mixes up reality with imagination.
Quoting fdrake
I think you are falling into the exact sort of quibbling and sophistry that I warned against. The answer here is simple: the Earth is not perfectly spherical or perfectly round. A cross-section of the Earth is circular, but is not truly a circle.
Quoting fdrake
And the reason why is very important.
* And of course also possesses roundness
And why is this? Is it not because of what those things actually are? If not, why did this become the everyday concept?
Sure. So with the "raindrop" addition example, isn't the appropriateness of the system determined by the real properties of rain drops?
I am all on board with the idea that the tools will vary with the job, but it seems to me that to explain why some tools are better for some jobs than others requires including properties of "things in the world."
Even when we speak of "concepts," it seems to me that there is plenty of evidence to support the claim that our cognitive apparatus is shaped by natural selection, and this in turn means our thinking and our preferences, relate to "how the world is."
To put it succinctly, there are causes for our preferences and what we find useful. And I would also argue that these causes cannot all be traced exclusively to our minds/concepts, that our minds and concepts themselves have prior causes.
He didn't make it up.
Yep. If everything were arbitrarily stipulated, then all of the strange ideas in this thread would be gold. ...Or at least as valuable as everything else.
Let's change track. You tell me exactly what you mean by a circle with an intensional definition, and we'll go with that. Then do the same for roundness and pointy!
I think so, relative to tasks.
Quoting Count Timothy von Icarus
Yes.
Quoting Count Timothy von Icarus
Yeah that's a hard one. I don't know if there's a hard and fast answer for systems generically! This seems to be a root level epistemological issue - what it means for a description to be adequate.
Quoting Count Timothy von Icarus
Indeed. Though there are lots of ways what we create can model, describe or explain stuff. Maybe even mirroring different aspects of stuff. Maybe it doesn't need to do any of these things to still be important.
But then, who would ever consider approximating curves with straight lines? Ridiculous idea.
I hope I'm not the only one who recognizes that you are more interested in this conversation than me. :grin:
I am fine with taking Euclid's definition:
Quoting Circle | Wikipedia
And we can say that a square is a plane figure with four equal sides and four right angles.
Something like "roundness" I take to be a simple concept, not especially reducible to further explication. We could say that it is something like the curvature of a line.
Aye.
Quoting Circle | Wikipedia
Euclid says: not a circle. The great circle is not a plane figure.
Why do you think this? And what is "the great circle"?
Read the definition:
Quoting Circle | Wikipedia
A circle is a plane figure... so something which is not a plane figure cannot be a circle.
Quoting Leontiskos
The great circle is the circle I've highlighted on the surface of the sphere. Since the circle is confined to the surface of the sphere, and the surface of the sphere is not a plane, it is not a plane figure.
The cross-section of a sphere is a circle. A circle is always "confined" by its circumference, but it does not follow that it is not a plane figure.
@Leontiskos as student Delta: Quoting Lakatos, as quoted in Russell
Exactly.
Quoting Leontiskos
Well who said anything about cross sections? I was talking about the sphere's surface. You chided me before about extraneous points and operations, and now you've given yourself the liberty of splitting a shape in two, taking an infinitely small cut of it, how exuberant. I just gave you a sphere's surface, not a cross section so...
You'll now need to tell me in what circumstances can you take a cross section of a volume and have it work to produce a circle. Let's assume that you can take any volume and any cross section and that will produce a circle...
Therefore those squares and rectangles are circles. Which is absurd. So your principle must have caveats. What are they, you've got some explaining to do!
You depicted one. I even asked what you were depicting and you weren't very forthcoming.
Quoting fdrake
Just read what I already wrote:
Quoting Leontiskos
Again, you seem to be resorting to sophistry, and I don't see this as a coincidence in the least. In order to try to draw an absurd conclusion you are helping yourself to false premises, such as assuming that planes are bounded, or points can be deleted, or that rectangular prisms are spheres. Why are you doing this sort of thing?
You do this sort of thing because stipulating a definition and then shit-testing it is standard mathematical practice.
I showed you the great circle on the surface of a sphere because I expected you would see it as a circle - it is - but it does not satisfy Euclid's definition of one verbatim, which you were clearly inspired by. And with maths words, verbatim is all anyone has. That's how you test the boundaries of your definitions and the consequences of ideas.
In picking out the great circle as a circle, you in fact sided with the example over the definition you stipulated. Which is the right thing to do, I think. You could also have ardently insisted that indeed, the great circle was not a great circle because it was not a plane figure. But you did not.
So now that you've abandoned Euclid's verbatim definition of a circle, you've got work to do in telling us what you mean by one.
As for me, I mean a set of points equidistant from a point. And by the by that also makes the great circle a circle. Score one for the thing which includes the taxicab circle over Euclid!
I think it does. You've only asserted otherwise, you haven't shown it.
Quoting fdrake
It is a plane figure. What do you think a plane figure is? Did you delete the interior of the circle from the plane in the same way you deleted the point from the center of the circle? Deleting points or sections of planes is not possible.
To be clear, the cross-section of a sphere fulfills Euclid's definition of a circle. We could also define a circle as the cross-section of a sphere, but I was only saying that every (planar) cross-section of a sphere will in fact fulfill the definition I already set out.
Well I can tell you what I think a plane figure is. [hide=*](the definition below looks to me to be a necessary but not sufficient condition for a plane figure)[/hide]
A plane figure is closed curve which is inside a subset of [math]\mathbb{R}^2[/math]. By that definition the great circle is not a plane figure, as it's not inside a subset of [math]\mathbb{R}^2[/math] - that circle instead would be a closed curve inside a subset of [math]\mathbb{R}^3[/math], or with extra precision the surface of the sphere. [hide=*](let's not talk about the surface of a sphere being something noneuclidean here)[/hide]
What do you think a plane figure is?
He doesn't need to. The sphere is a 2-manifold, and his great circle is a set of points on that manifold. There are no planes here, nothing else, only the points on the surface of the ball.
You are imagining the sphere embedded in the usual 3d Euclidean space. Now, imagine it isn't. There is no point the points on this great circle are equidistant from.
Quoting fdrake
But don't you need to specify coplanar? If we're in 3d space, you've defined a sphere, in 4th I guess some sort of hypersphere, I don't know, blah blah blah.
We took our definition from Euclid, and the term there means a figure that lies entirely on a flat plane.
Quoting fdrake
Do you think the "great circle" (which you have yet to define) lies in three dimensional space rather than two dimensional space? That ambiguity is why I asked you to be more clear about what you were depicting in the first place.
Cutting to the case a bit, it seems that you want to talk about "circles" instead of circles and "plane figures" instead of plane figures, etc. Now if we define "distance" in an idiosyncratic way, then of course there are taxicab circles. If we define "distance" in the commonly accepted way, then there aren't. Are we disagreeing on something more profound than that?
Planes and points cannot be stipulated to exist or not exist. Your word "imagine" is on point given my earlier claim that "This idea of 'deleting' points mixes up reality with imagination." The points in question are coplanar, and therefore the figure they enfold is a plane figure.
I did no such stipulating. Look again.
Quoting Leontiskos
And you are ignoring the fact that I used it twice.
There's a pattern...
Yeah you're right. Circle, n-sphere, all the same thing in my head. Coplanarity works. A set of coplanar points equidistant from a point in their plane of coplanarity. Thanks! [hide=*](could repeat previous definition regarding smoothness and point deletion here)[/hide]
Quoting Leontiskos
Quoting Leontiskos
Fair enough. There's two things though:
Either you consider the sphere as embedded in 3-space, and the cross section plane isn't "flat" in some sense - it's at an incline. Or you consider the surface as a 2-dimensional object, in which case there's not even a plane to think about. Pick your poison. The latter is the original counter example and is much stronger, the former is easier to remedy.
Quoting Leontiskos
You're behaving like you know what these things are so well you've got them baked into your cerebellum. But clearly that's not true, as the definition you provided doesn't match something you clearly recognised as a circle! So yes, we could insist on your pretheoretical intuition, but it's no longer Euclid's... so I'm wondering what's wrong with it? How will you parry my counterexample?
Quoting Banno
It is a lot like something from Proofs and Refutations.
So? The cross-section of a hollow sphere will be a circle regardless of whether I imagine a point at the center or not.
Quoting fdrake
I had thought the example, Eulerâs formula, a bit obtuse. But perhaps Lakatos chose it so as to minimise the number of auxiliary hypotheses that his students could produce.
I still think you're just plain wrong. Namely, a 2-dimensional object lies on a plane. Pretending that there is no plane is a curious move. How do we query whether a plane is present or not? A plane is an abstract object, much like a circle. It makes no more sense to say that the cross-section of a sphere does not lie on a plane than it does to say that one can delete the point in the middle of a circle, at which point it magically becomes a non-circle.
Quoting fdrake
But it does match it, as I've already noted. Your mere assertions are getting old.
Quoting fdrake
I'm waiting for you to present one.
Quoting Banno
You artificially inserted an extraneous conversation into your own thread and then invited me here, remember?
@fdrake if you like: a circle is the two-dimensional subset of a sphere. A sphere is the set of points equidistant from a point in 3-space and a flat cross-section of a sphere is necessarily a circle, namely a set of points equidistant from a point in 2-space. As I've already said, a cross-section of a sphere conforms to the definition of a circle that I originally gave.
Does that point need also to be coplanar? Is there a counterexample I'm missing?
Quoting Leontiskos
You realize that on the sphere it's just a straight line, I hope.
?? I don't know why I'm participating in this.
Me neither. Banno's baiting into this thread is itself something I wished to avoid long before he resurrected this thread. If you had created a real thread on logical pragmatism we wouldn't be here. Blame's on you. :razz:
I'm gonna bugger off now too.
Quoting Srap Tasmaner
I was imagining putting the point away from the plane and bending the underlying surface we're trying to draw the circle on. I'm pretty sure we'd end up with some other shapes possible if we inclined the plane, never mind if we corrugated the fucker.
But I suppose that would also apply if we chose the coplanar point far away from the candidate point set... I wish I knew what circles were.
If all circles are plane figures, then the great circle is not a circle.
Hueston, we have a problem.
It would if you give yourself the liberty of hammering the cross section down onto a flat plane. Which is an exercise of the imagination, and not something set out in Euclid's axioms. Is the point. You end up having to mathematise all the stuff you do to make it work. The operative distinction is you're relying on a lot of extra-mathematical intuition and not putting in the work to make it precise. Which is mostly fine, it's just in such imprecision where lots of allegedly undesirable plurality can hide.
Do trust me that the counterexamples work verbatim though!
Quoting Srap Tasmaner
I'm glad you dropped in, at Leon's invitation, I think?
It wasn't I who engaged in necromancy - that was @frank. And you do not have to be here, if you find it too arduous.
Quoting fdrake
Cheers. I'm glad someone looked at the Russell article.
I take it that a cross-section is flat (i.e. two-dimensional) by definition. But this all goes back to the ambiguity of your figure. If the cross-section you indicated is not two-dimensional then I would of course agree that it is not a circle.
I had comments I really wanted to make about the original article but considering that a Proofs and Refutations style chat about square circles was right there it seemed a better opportunity to illustrate the intuitions behind lemma incorporation.
Quoting fdrake
For fdrake it would seem that when we see a shape he has drawn on a piece of paper, which looks like a circle, we must ask him if he "deleted the point at the center" before drawing the conclusion that it is a circle. Apparently in order to identify a circle, formally or materially, we must worry about whether the center point has been "deleted." This is taking the subjectivism and relativism a bit far.
(Like points, apparently planes can also be "deleted.")
Yes! The set {1,2,3} can have the element 3 deleted, giving the subset {1,2}. Is what I meant. The plane without the origin. This is a perfectly cromulent thing to do with sets.
Yeah I was only thinking about the point being away from the plane, no other fiddling. If I've ever considered that, it was so long ago I've forgotten.
It's just a curiosity that talking about the center of a circle is a little over-committal. It's the center, coplanar, only under a particular projection onto the plane of the circle. But under other projections, the "center" lands elsewhere, which for some reason seems really cool and even useful to me.
I suppose that means the great circle isn't a circle, since there's no coplanar points on it... Since there's no way to form a plane out of the points on a sphere's surface when you're only allowed to consider those.
But if indeed you can form a cross section, allowing yourself the exuberance of 3-space, then they are indeed coplanar and form a circle.
I suppose it's then an odd question why the same set of points can be considered a circle or not depending upon whether you consider them as part of a larger space.
Regardless though, there's no word for "coplanar" in Euclid's definition of a circle either. So we've needed to go beyond Euclid regardless. It would be odd if Euclid ever had need of the word, considering his is the geometry of the plane.
Can you show me one please?
It seems that we mean different things with the words "point" and "plane." On my view you have reified abstract realities, making them, among other things, delete-able.
Quoting fdrake
These objections are too subtle, such as supposing that I meant to confine myself to Euclid in an especially strict manner, or that the cross-section of an abstract sphere cannot be an an abstract circle.
Deletion is shorthand for considering different sets - or using the set division operation. The sets I'm referring to were [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^2/\{0\}[/math].
Are you not used to this sort of maths?
It's been too long to do much more than mildly jog the memory.
Fairy muff.
As I understand it, the "plane" in the definition of a circle is not a space, at least in the sense that your term "larger space" indicates. The cross-section of a sphere conceived as two-dimensional is planar in one sense and non-planar in another.
So is there some impediment to taking the basic definition of a circle given and saying that the cross-section of a sphere conforms to this? I don't see any real impediment. Any three-dimensional translation that occurs will not be contentious. If we interpret the abstract space presupposed by the definition of a circle to be incommensurable with the abstract space presupposed by the cross-section of a sphere, then there is clearly an impediment, but this sort of exclusion is less plausible than the alternative. How exactly do the three-dimensional points of a sphere translate to the two-dimensional points of its cross-section? I don't know, but it doesn't strike me as a great problem.
In any case we are very far from demonstrating square circles, which was the original topic.
I'll draw if I have to, but I think I can clarify it verbally.
1. Pick a point and a length.
These together determine a bunch of circles in 3-space.
2. Pick one.
If you picked one that isn't coplanar, there's a projection of the "measuring point" onto the plane the circle is in that preserves the property of being equidistant from points on the circle, in fact preserves it as you move the point toward the plane, shrinking your originally chosen length until it's the radius of the circle.
But there are other projections where that original point will land off-center, or on the circle, or outside it.
If you want to go backwards, you need an additional constraint**, because there's a whole line of possible "measuring points" through the center of a circle, perpendicular to its plane, like an axel. Your measuring point could be projected to anywhere in the plane, and any point in the plane could be projected to anywhere on that axel line.
You could also play with projecting the circle and the point onto yet another plane.
It's just curious that you can separate the point that generates the circle from its center, that those are two different properties, and there are projections that will separate them in a plane.
** The original length gives you two, I think
I guess once you have the "axel" in mind, you could say that choosing the point where that line intersects the plane of the circle as the point that "determines" the circle is natural and convenient, but just a convention. The radius and center and plane of a circle determine it, but so would an infinite number of pairs of points and distances.
*** If you think of the determining point as the vertex of a cone, there are an infinite number of cones, all sharing an axis, the circle is a section of.
I would quite like you to draw this. I don't think I am imagining it accurately.
Quoting Srap Tasmaner
I was imagining a cone, yeah. But now the variability makes sense given that there's an infinity of them. Am I right in thinking that the "correct" visualisation regarding picking the vertex is also equivalent to picking the gradient of the lines bounding the cone? Insofar as it constraints the circle in the plane's radius anyway.
I'm glad you came back to this, and I'm going to draw some pictures. I had decided last night there was nothing here and I don't know why I was going on about it, but I have an idea now!
The difference I intend between pure (as such) logic and applied (transcendental) logic is that we can do logic without addressing questions of being, whereas the latter gets into the weeds of various philosophical questions (but simultaneously presupposes a logic to get there). Logic is an epistemic endeavor dealing with validity whereas the question of the relationship of logic to being is getting more into metaphysics rather than logic.
Quoting Count Timothy von Icarus
And to highlight why this is difference -- this line of questioning you're exploring here will be an interesting question whether we are logical monists, logical pluralists, or logical nihilists. Deciding the first question doesn't necessitate a relationship between logic, the mind, being, and knowledge. We could be logical monists on the basis that there is one true logic, but we don't know what that one true logic is yet -- inferred from the conflicting accounts of logical laws -- but retain the notion that there must be One Logic to Rule them All (or, that, in fact, one logic does rule them all, if you just incorporate this already implicit Lemma....)
And simultaneously hold that there is no relationship between logic and being -- i.e. that the One True Logic is the result of the structure of knowledge requiring this or that axiom, but could still be anti-realist projections which have no relationship to being.
The purpose and scope of logic is certainly being considered by logicians, it's just that these are different questions. (also -- I, for one, am all for a socialist feminist biology for the winter months :D )
That's a lot closer to home to my way of thinking -- and why I like Feyerabend's deconstruction of Popper as a kind of object lesson for all philosophies of science which try to encapsulate the whole within some system: what I'd call totalizing.
Though at that point we would be kind of in the realm of both Hegel and Marx -- the historical a priori looks a lot like those big theories of history to me. And that's getting close to a similar totalizing project, at least on its face.
Quoting Moliere
Thatâs what pragmatist-hermeneutical and poststructural models of practice are for. For Hegel and Marx the dialectic totalizes historical becoming. In these latter models cultural becoming is contextually situated and non-totalizable. It is normativity all the way down.
Do they need to be counterexamples to Aristotle?
I don't think so. I think that I'd simply have to want to utilize some other logic -- and there are some good reasons for putting Aristotle aside in these cases. First and foremost because we're not strictly utilizing Aristotle's logic here. The Logical nihilist or pluralist or monist isn't putting together All/Some statements into the classical forms -- The Background here has incorporated parts of Aristotle (classical logic is still taught!), but isn't appealing to Aristotle's commonsensical intuition about the logic of objects.
But I don't think statements behave exactly like objects do (and I am terribly allergic to commonsense -- it's not that I don't get it, but if the appeal is to commonsense then one need not study logic in the first place. There are far more lucrative and stable careers than academia)
Basically we don't need to explicitly refute Aristotle in how we do logic. We are free insofar that we create something interesting.
Quoting Leontiskos
(1) is false. (1)
Read that as (1) being the name of the sentence so that the sentence references itself like we can do in plain English.
At face value it's clear to see that if 1 is false then it is true. And if it is true then it is false. If we combine this with the law of the excluded middle we must conclude that (1) is both true and false.
This is the notion of a dialethia. I went for a review before posting here and want to reference the SEP bit on paraconsistent logic in the liar's paradox article because just below it has an entry on dialetheism.
He has some interesting examples, but this would take us very far astray.
It's more that here seems a reasonable approach to the liar's paradox that produces interesting and novel results in logic.
Then it seems we're more or less in agreement. :up:
I would also tend to suppose that there may indeed be many ways to "skin a cat," different systems that are equally good for x purpose, but then these systems will have similarities, mappings to one another.
Right!
And far from rejecting classical logic it seems to me to give clarity to its underlying intuitions. These extensions of logic aren't so much an Undermining of All Thought, but in the critical tradition which explores terra incognita.
Super cool stuff.
Yeah, I agree. Links form here to a whole lot of other stuff.
I agree that you can study logic in total abstraction from content.
I am not sure if you can have an "epistemic endeavour," that is unrelated to being though. What is our knowledge of in this case? Non-being?
I also don't think we can have such an abstract study without the concepts provided by experience and sense awareness. For if we had no experience of the world, of encountering falsity, how would we even know what terms like "truth-preserving" meant? Likewise, how does one even have a concept of existential quantification without a concept of existence? That is, we can only abstract away the world so much.
Which is a good thing IMO. If we totally leave the world behind we'd have an infinite number of systems and no way to judge between them vis-ĂĄ-vis which are deserving of study.
Suppose we had a formal system that answered all our questions about physics, or maybe some area of it like fluid dynamics. How could it have "no relation" to being? At the very least, it would have a relation to our experiences, which are surely part of being.
I want to do leap year physics. You get a nice three year break.
Yes. Like Hamiltonians and Lagrangians. Do the same thing differently.
You mean if we leave the world behind after discovering the systems? :wink:
They are supposed to be objections to Aristotle, so yes, of course they do. You might as well have objected to Mr. Rogers by telling us that you prefer people who put on shoes. Mr. Rogers puts on shoes in every episode.
Quoting Moliere
As has been pointed out numerous times, this is just gibberish. What do you mean by (1)? What are the conditions of its truth or falsity? What does it mean to say that it is true or false? All you've done is said, "This is false," without telling us what "this" refers to. If you don't know what it refers to, then you obviously can't say that it is false. You've strung a few words together, but you haven't yet said anything that makes sense.
The 'great circle' looks elliptical to me. "Circle" is being used in the same argument in two different senses.
A great circle is the longest possible straight line on a sphere. No midpoint and diameter in that definition.
Ah. Understood. I need to read more carefully. Thanks. I appreciatcha!
:yum:
I can't wait until tomorrow, when we show that 2+2=5.
Quoting Leontiskos
I thought we agreed, formal logic is conventionalized ways of thinking :p. It can only be an approximation of our thinking, but not our thinking itself.
What's the "foundation" mean here?
Presumably, natural human reasoning, something akin to inferencing, let's say, is of an imprecise nature. It just needed to be "good enough". However, the kind of reasoning we developed- generally intertwined with linguistic capacity, and certain kinds of episodic memory, can get formalized culturally into more precise logical thinking. This is especially helped by the ability to write out the symbols.
From here, these more precise "crisp" arguments, might be said to have a foundation, perhaps Platonically (pace Frege and Plato). And thus, you might mean some kind of transcendental foundation (Platonic). Or, perhaps, like Kant, you think that it is internally a priori, and simply part of the human cognitive faculties. I challenge this, as evolutionary vagueness seems to be at play. Math is contingent on cultural preciseness, not internal preciseness. However, even math's preciseness and internal logic in its own system, doesn't necessarily have a foundation outside itself. Newton's Calculus system is not as accurate as Riemann's system, for example. And thus "foundation" can thus mean:
1) Human cognition- I challenge this usually works in vague approximations, not crisp exactitude.
2) Platonic transcendentalism- I am not sure what this would mean other than logical truths are somehow existent in some real way.
3) Naturally occurring patterns- this might be physical laws, for example. But this isn't really the logic itself. Logical systems, like mathematics, are applied to observable phenomenon, and "cashes out" in experiments and technological use.
There's nothing much to the geometry, but here's a picture to start with.
(There's other ways to look at this. You could of course go ahead and treat the "determining point" as a center and make a circle on a plane right there, then project that circle onto a parallel plane. Blah blah blah.)
Having separated the point that determines the circle from the center of the circle, it just occurred to me that you could treat it separately, do a lot of stuff with it. To start with, you don't have to project to the center of the circle in the plane, you don't have to use that orthogonal projection, but could send it (translate it) to any point A, B, or C, anywhere in the plane.
Then I thought there might be something interesting if you grouped these projections into buckets, those that send it into the circle, those that send it far away, and so on. And I thought there might be some interesting stuff there ? maybe allowing the axis to wobble a little, and see how stable your buckets were, and lots of other stuff.
But then it occurred to me what probably caught my eye about this.
If instead of thinking of the points A, B, and C as being projections of the "determining point", what if you went the other way, and thought of any point in the plane translating to the point off the plane that determines this circle.
Suddenly that cone looks like a field of vision, and the other points are other actors who are triangulating their view of ? in this case ? a tree (or whatever) with the red guy at the "determining point". (We'd probably want to move the red guy onto the plane with the A, B, and C, and create a new notional plane orthogonal to this one to represent Red's f.o.v., but whatever. At this point the whole setup is merely suggestive.)
And then it should be obvious there is a meaningful difference between being in the circle and outside it, because that determines whether you are also in Red's cone of vision.
It happens I've been reading about triangulation and joint and shared intentionality in apes and humans (Michael Tomasello), so it was probably on my mind, and that's why the whole arrangement, splitting one point into two (center/determiner), then splitting that second point into two as well (determiner/projected) ? it all suggested something to me, and this was probably it.
I wonder if there is something else interesting just to the geometry, but that's no doubt above my paygrade.
If you want something more universally foundational, I would point to the principle of non-contradiction, and ultimately its unique character of being simultaneously subjective and objective, which Kimhi alludes to. A lot of the silliness in this thread is either a direct or indirect attack on the PNC.
Few implementations of propositional logic start with modus ponens. It's most often just a theorem.
Contemporary logicians like Enderton and Gensler begin the exact same way. Other starting points are possible, but they are not all on a par if one wants to do actual logic. Of course for metamathematics the starting point is arbitrary. Banno, under the spell of metamathematics, will be at a complete loss before your question about how true reasoning and logic interrelate. As Apokrisis has pointed out numerous times, Banno begins and ends with nothing more than a bit of posturing.
This was cool. I would need to sit down with some algebra to understand it properly though. Regarding the projection - there will be a lot of degrees of freedom if you get to choose an arbitrary projection onto the plane, so I suppose picking a specific projection to the centre point in the plane and looking at its preimage under that projection is the idea you had in mind?
Yep! It turned out a property that uniquely characterised straight lines in our normal kind of space also applied to spheres, and it makes great circles. It's the taxicab circle thing again. Straight lines are only the things we expect in Euclidean ("flat") space. But that's an artificial restriction.
Edit: even flatness. The volume in the room you're in is flat.
I'll quote Gillian Russell here from the opening of her One True Logic?:
[quote=Gillian Russell]
Logic is the study of validity and validity is a property of arguments. For
my purposes here it will be sufficient to think of arguments as pairs of sets and
conclusions: the first members of the pair is the set of the argumentâs premises
and the second member is its conclusion. An argument is valid just in case
it is truth-preserving, that is, if and only if, whenever all the members of the
premise-set are true, so the conclusion is true as well.
The domain of logic, then, might be thought of as a great collection of
arguments, divided into two exclusive and exhaustive subcollections, the valid
and the invalid, the good and the bad, and the task of the logician as that of
dividing one from tâother.
[/quote]
Quoting Count Timothy von Icarus
Humean skepticism comes to mind -- it could be that our logical discourse is constrained by our mental habits rather than by being. So it goes with causation: We cannot help but to draw causal inferences by our habits of thought, but the inference we draw is unjustified (insofar that we accept Hume's notion of causation, at least - but here I'm trying to point out how an anti-realism is possible, so that's enough).
I'm more tempted to say that if we have no more questions about physics this says more about our lack of curiosity than it does about our knowledge of being.
Quoting Count Timothy von Icarus
Here's the bit where reality kicks in: You can do leap year physics. But you won't be paid for it.
What you'll be paid for is tracking patterns which people like to track, which usually involves manipulating the world in some way which we perceive as regular. It's this social bit that stops the infinite possibilities, though that's not exactly a pure rational reason or a philosophical gatekeeper.
I don't exactly object to classical logic, though -- I'm saying it has limitations, not that it's wrong in every case.
To clarify -- the wiki on syllogism has a clear rendering of what I mean by classical logic:
[quote=wikipedia]
There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M â Middle, S â subject, P â predicate.):
Major premise: All M are P.
Minor premise: All S are M.
Conclusion/Consequent: All S are P.
The premises and conclusion of a syllogism can be any of four types, which are labeled by letters[14] as follows. ...
[/quote]
etc. etc.
Notice how these can be rendered in predicate logic in that article. These things aren't at odds, exactly. It's only that they are different.
And so it goes with non-classical logics. These aren't opposed, per se -- they rely upon a different set of assumptions and look for the patterns of validity after that.
Now in a given philosophy we'll want a particular logic, or particular logics for particular ends, but the logician need not adhere to one philosophy. Why would they? What would the point be, given that here the logicians are doing their thing without Aristotle's assumptions?
Quoting Leontiskos
It's the name for a sentence.
A name denotes an individual.
The individual is an English sentence.
The sentence is "This sentence is false"
(1) is a shorthand to make it clear what "This sentence" denotes.
In a logical sense there's no reason to exclude this individual if we want our theories of logic to be entirely general -- to apply to all individuals. Denoting a sentence is surely not violating logical possibility -- it's the nefarious choice of self-reference with the "... is false" predicate which breaks the logical ambition and creates a paradox that calls for an answer.
One answer, which you've provided, is that the sentence means nothing.
It's not the only one though.
Right. Logical pluralism is saying that there is no one logic that applies to all cases. A logical pluralist would agree that the LONC is useful... where it's useful.
Just a helpful point of clarification, "classical logic," is confusingly the logic developed by Frege and co. relatively recently. There is no good catch-all term for logic before the late 19th century. People call it "Aristotlean," but then this tends to miss everything between Aristotle and 1850 or so.
There was a Stoic logic distinct from Aristotle's, but it disappeared. The big difference I recall is that Artistotle primarily saw logic as an instrument of science/philosophy whereas the Stoics thought it was a proper field of study. The dominant modern view seems to blend these two.
And why do we perceive it as regular? That's the key problem I see here. If the answer is "for no reason at all," that's a problem. If "it just is," is acceptable some places, it seems acceptable any place. Yet people almost always give up on "it just is," when they feel they have a good explanation for something, making it simply a catch-all to fill gaps and end discussions.
I'm also not sure what "being" is supposed to be if it isn't what is given to thought.
As for the quote, the same problem seems to remain. It defines logic in terms of truth, "truth-preserving," etc. I don't think these are terms are unproblematic or explainable solely in terms of formalism. And it certainly seems that logic cannot be the study of non-being either.
If I'm doing something dumb, it's okay to just say that.
Quoting fdrake
Yes exactly.
Here again is how I got here.
In school, we learn to think of circles this way:
1. You've got a plane.
2. Pick a point in the plane.
3. Find all the points in the plane equidistant from that point.
4. That set of points is your circle.
5. The point you picked in (2) is the center of your circle.
But it needn't be that way.
Your great circle example, or the conic sections we learn in Algebra II, are different.
1'. Pick a point in 3-space.
2'. Find all the points equidistant from that point.
3'. That set is a sphere, or a 2-sphere.
4'. Any coplanar subset of the points in (2') is a circle, or a 1-sphere.
If you now look at the plane of the the circle in (4'), there is a subtle difference from the plane in (4): the center is not marked. No point in the plane was used to generate the circle ? although, of course, the circle has a center you can find. But in the schoolboy's circle, you never have to go find the center ? you pick that point to start with.
(There's a direction-of-fit thing here: in one case, the center determines the circle; in the other, the circle determines the center.)
When you find the center, you might ask, is it related in any special way to the point in 3-space we picked (1')? And of course it is. There is exactly one line orthogonal to the plane that passes through that original generating point, and it passes through the plane at the center of the circle as well.
And you might then think of the center of the circle as a projection of the center of the sphere. And it is, but it's entirely optional. That projection comes after we already have the circle. It's the canonical projection alright, but you could also project that point to any point on the plane, because this projection is just a thing you're doing ? the circle doesn't need it, isn't waiting for this projection, you see?
Cool.
I mean logic prior to Frege. The square isn't found in Prior Analytics, but I would consider the likes of Frege, Peirce, and Cantor as part of the new logic which encompasses Aristotle's studies on validity, if not his entire project.
Quoting Count Timothy von Icarus
I think that's a question for metaphysics rather than logic -- and which explanation one chooses will complement this or that metaphysic. These are different questions because we can reconcile various kinds of anti/realism with various kinds of monism/pluralism/nihilism in logic. This isn't to take a side on realism or anti-realism, but to demonstrate that the question of realism isn't the same as the question between logical monism, pluralism, or nihilism.
The nihilist account seems to get along with anti-realism, but it's possible to reconcile a realist metaphysic with a nihilist logic, and an anti-realist metaphysic with a monist logic. If that's the case I conclude that they are different questions and logicians need not answer the metaphysical question in exploring monism, pluralism, and nihilism.
Even on a realist account, though, I'd say we frequently find patterns that are not real -- we find regularities because we like them so much that we find one's that are false as well as true. This is what we mean by delusions and hallucinations and such.
Which is really just to convert the question of ontology -- what is it that we know about? -- to epistemology -- how do you know the true from the false?
Quoting Count Timothy von Icarus
It's a concept in metaphysics whose meaning cannot be articulated, but only approached. If I take a page from Sartre Being is transphenomenal. If I take a page from Heidegger, then the question of the meaning of being is itself an unarticulated assumption of all philosophy prior which relies upon the notion of presence.
Is non-being somehow not-known? If I am looking for someone in a bar because we said we'd meet and I do not see them then isn't this an account of absence-in-presence?
Either way I'd say that the question of being is not a question of validity -- another demonstration from logic.
If the moon is made of green cheese then Alfred is the president
The moon is made of green cheese
Therefore Alfred is the president
The actual truth-value of these sentences isn't in question when talking about logic. It's the form between the sentences under the assumption that if the premises are true that the conclusion follows. But since the moon is not made of green cheese the question of being -- what is -- differs from the question of validity, and logic is this study of validity.
There are different ways to rationally conceive or define (and draw) a circle. Equidistance from a point is one. Aristotle prefers another, "The locus of points formed by taking lines in a given ratio (not 1 : 1) from two given points (KM1 : GM1 = KM2 : GM2 = ...) constitute a circle":
But what a circle is and how a circle is drawn are two different things. Similarly, two different ways of conceiving a circle are immaterial to the question at hand when they are formally equivalent, as is the case here. When I gave some arguments against square circles, I suggested that one could quibble with the arguments, but not oppose them in any way that goes beyond a quibble. I think that has turned out to be right. Aristotle's circle and Euclid's circle are formally equivalent. The definition of a circle is not specifying the manner in which a circle is created; it is specifying what a circle is.
So my problem again here is the use of "foundational". This is a slippery word. The way you are all using it is basically "axiomatic". I take "axiomatic" to mean "don't ask me anything further, this is as far as I'm going", or simply "duh!". It really doesn't mean much except that we need to start "somewhere" and "this seems like a good place to start". Without getting into the obvious rejoinder of the problem of circularity or "brute fact", I see the problem as more complicated.
Axioms themselves are grounded in something. One might call them "intuitions". One might call them "Platonic truths" living in some divine realm (above the divided line!). Either way, it is that I believe to be foundational. Axioms then become a digital/crisper version of the intuition/natural reasoning. From THERE, you can then work out a whole bunch of complicated formal language rules. But only after the initial FOUNDATIONAL translation from NATURAL reasoning to the "crisp" axiomatic ones of formal logic.
Eh. If you take it to mean axiomatic, then it has nothing to do with a good place to start. If you take it to mean a good place to start, then it is not axiomatic. Axioms are not good places to start except in a purely formal or economical sense. This chimera is understandable, given that my use of "foundational" was nothing like "axiomatic." Quite the opposite.
Again, the PNC is a more universal foundation or first principle than modus ponens. It is a foundation in the same sense that the first few feet of the trunk of a Redwood is a foundation. It is stable in a way that the upper branches are not, and folks never directly contravene the PNC. They only do so indirectly when they have climbed out onto limbs and lost track of where they are.
As you like.
It seems to me you think this is a question that can only ever be asked in one way and in one context, and therefore it only ever has one answer.
You can do that, and you can be right. Your response to a counterexample is "Well I didn't mean that, I meant this" and your honor is preserved. In the context you had in mind, you're still right. The counterexample isn't one.
Pick up a length of pipe. Look at it from the side and it's rectangular. Look at it straight on, it's circular. Done. "But I didn't mean that."
But you also seem to think the context you have in mind for any question that arises is the only context it can possibly arise in. I tend to have less confidence in my own omniscience, but you do you.
A circle does not have a depth dimension. If we were talking about ropes we would have a different case.
I mean, if we define circles as squares, then sure, we can have square circles. But that's not what circles are. Redefining words in an attempt to achieve substantive conclusions does not strike me as good philosophy. We can talk about whether a material "instantiation" is ever a circle or circular, and I of course concede that in a strict sense there are no material instantiations of circles (and that if the great circle is conceived in this way then it is not a circle). But that is a far cry from the conclusion that there are square circles.
At the bottom of this whole thing are important questions about philosophical motivations. When I asked @fdrake about his motivations he said that, "shit-testing is standard mathematical practice." In other words, he has to adopt the persona of an extreme skeptic to see if his ideas hold up. This is a Cartesian mentality through and through, and I submit that it is a bad one. Granted, it is more applicable to mathematics than philosophy generally, but I tend to think it conflates science and mathematics in important ways. Beyond that, when I introduced the term "square circle" into the thread, it was as a metaphor for non-mathematical contexts. In such contexts "shit-testing" really is just Cartesian method, the age-old error of mistaking philosophy for mathematics or indubitable knowledge.
I get what you are saying, but I still think you are using foundation as "axiomatic", in the definitions I described- that is to say, "This seems like a good place to start". But really you must sus out the actual "foundation" from which this axiom derives. That takes a meta-theory beyond the axiom itself (of the PNC let's say). If we sus out what your particular theory is, it seems like something akin to either an evolutionary intuition or a Platonic necessity. Either way, the foundation is deeper than the principle itself.
Edit: Notice, I am not saying the axiomatic foundation is arbitrary. There is good reason it is selected. It seems to be the case everything revolves around it in logical workings, let's say. But I am saying what is this then grounded in? That is the foundation.
This was Mandelbrot's key insight in coming up with fractional geometry. What is "smooth" at one scale is not at others, etc.
Likewise, a miter saw cutting wood is not generally considered a "chaotic" process. It's results are extremely regular on the scales we tend to care about for carpentry. Yet at a fine enough grain, it becomes extremely susceptible to strong variance due to minor changes in initial conditions.
Personally, I love C.S. Peirce on this issue. He's a big forerunner on these sorts of insights.
However, an I may have lost track of the point of the conversation, these do not seem like instances of contradictions to me, nor of particularly difficult cases for either logical realism or logical monism. TBH though, once they are properly caveated I find a lot of "logical monisms," and "logical pluralism" to be pretty much indistinguishable. If there is a material difference it goes over my head.
I guess a "strong" pluralism would declare that there are multiple equally valid/applicable logics but no morphisms between them? I just find it hard to imagine how this could be the case, since it seems that, by definition, they must have similarities in virtue of the fact that they are equally applicable to the same things.
And then a "strong monism," would presuppose a "one true formal system?" But that doesn't seem particularly plausible either.
As I was saying to Leon, the "foundation" to logic would be a meta-logical theory, not the axioms/logical systems themselves.
-
Quoting schopenhauer1
Sure, if you like. Whether the binding between reality and logic is metalogical is largely dependent on how you conceive of logic. On my view something with no relation to reality (and therefore knowledge) is not logic. Ergo: something without that binding is not logic. It is just the symbol manipulation that Banno mistakes for logic. More precisely, it is metamathematics.
When you want to call the binding metalogical that makes me think that you take logic to be something that is not necessarily bound to reality in any way at all. What I would grant is that it is a somehow different part of logic, but I do not think that these parts are as easily distinguishable as the modern mind supposes.
Glad we are on a philosophy forum and can adjust to the big picture and zoom in where necessary (and not stay in the weeds unnecessarily because- logic) then! :wink:.
Quoting Leontiskos
Nice idea. So for your understanding here you are saying that different mathematics are basically "arbitrary" forms of logic (that sometimes map to reality)? And then of course, my main question is "what is/how is it mapping to reality?"
Quoting Leontiskos
I'm unclear what you are saying here...
Yep. At least that's the hope. :grin:
Quoting schopenhauer1
Metamathematics, not mathematics. Something like "game formalism (SEP). It is something like the study of the logic of symbol manipulation.
I tried to set out my view of logic in my first post here:
Quoting Leontiskos
On this view the "binding" is part of logic, given that discursive knowledge cannot be produced without it. But there is a distinction between intellection and combination/separation, and we justifiably think of the latter as logic.
To try to get at it in just a few words, we usually think of knowledge of simples as one thing and the manipulation of that knowledge of simples as another thing. That's fine; they are distinct. I call this knowledge of simples "intellection" as opposed to "ratiocination." But even when all the simple pieces on the board are set and ready for manipulation, I would contend that we have still not left intellection behind. Why? Because an inferential move or rule involves intellection. The manner in which we move from premises to conclusions is not endlessly discursive, or not entirely related to ratiocination. We must understand that the inference is valid in order to undertake it, and this understanding is part of intellection. Logic of course tends to calcify or standardize rules of inference, thus forgetting the importance of understanding them. Basically, the closer we move to that "binding" between the formal logical system and reality, the more immersed we are in intellection, and this includes an understanding of inference.
But this isn't how logic is studied. For instance, take Curry's paradox as an example. The problem is that the common idea that "valid arguments with true premises yield true conclusions," results in absolute absurdities like "if this sentence is true then Albany, New York is in Mongolia," being used to prove "Albany, New York is in Mongolia." Harty Field and J.C. Beall have written a lot on this one. Yet if we totally abstracted all content away from "truth" (something I'd argue we aren't even mentally capable of) it seems impossible to recognize these sorts of problems. If you don't consider content at all, how do you even recognize when you're able to prove the absolutely absurd and have a problem? Sure, we could recognize triviality (i.e. when we can affirm every claim that is expressible in the language of the theory) in some abstract sense, but we wouldn't have any idea [I]why[/I] this is problematic (and paradoxes of absurdity exist in non-trivial contexts anyhow). Hemple's ravens, and probably a lot of stuff involving logic and induction would be other examples where there will be similar issues. Or accounts of implication.
That said, I get the distinction, and I think it's a useful one to some extent. Nevertheless, when logicians want to discuss truth, and validity as "truth preserving," one has to understand what is meant by "truth." One can declare one's logic "pure" and free from metaphysics, but honestly it seems that all this accomplishes is making one's presuppositions opaque and immune to scrutiny (and, relevant to this topic, does so in a way that I think is often question begging re logical nihilism).
If "truth" is just left as an empty lable, "existence" in existential quantification likewise just a symbol with rules attached to it, etc. what exactly are we preserving? An AI can spit out systems without any regard to truth. Would it be doing the purest form of logic by jettisoning all metaphysical baggage? But then why even say it has anything at all to do with truth preservation; logic is just reduced to computation.
I think it should be obvious though that this begs the question on logical nihilism, since it deflates truth and the realist is trying to make a claim about what is true universally.
We might try to get around this by divorcing "truth in logic" from "metaphysical truth," but I am not sure how effective this will be if the topic of debate is logic itself, as in the context of this thread, lol.
I prefer to think of it as putting it to the side as something that can be discussed separately -- which isn't to say our choice of a logic is metaphysically innocent or anything. When you're trying to put it all together into some kind of coherent picture usually you can see how there are some natural implications of an idea; some ideas seem to "get along" better together than others.
My task here is to point out that the nihilism isn't absurd on the basis of anti-realism/realism, that nihilism is different from pluralism, that pluralism is a worthy contender whether we are realists or anti-realists, and that logical monism isn't obviously true.
I've read you as saying that logical nihilism leads to a lack of knowing -- that we would be unable to track what is relevant with respect to knowledge if there were no logical rules. I think the case of Humean skepticism is a good one to point to for demonstrating that knowledge need not have anything to do with our habits of inference -- we build knowledge around causation, but it could very well be that we find out we were mistaken in that knowledge.
Now, just because we were mistaken that does not then mean that things weren't real. It just means that our knowledge doesn't necessarily track what's real. So if we wash our hands before treating a bleeding wound to remove the humors from our hands since it causes diseases we will know something which is false, act on it, and in the process eliminate microorganisms which cause diseases.
The whole causal mechanism is a myth, but we manage because we are the ancestors of those who were lucky enough to reproduce in this environment (and they didn't know much either, so I'd guess -- though I don't know)
In fact we could look at induction as a survival strategy which violates the basics of logic all the time since it's an invalid inference. :D
Or, at least, I put those sorts of things under the heading "informal logic" which is the study of how people actually make inferences which includes a lot more on the "content" side (since that's how you demonstrate why such and so is a fallacy). It just seems that we'd be able to accommodate informal logic, or this kind of "content based" logic regardless of our position with respect to monism, pluralism, and nihilism in logic.
I'd only used "foundational" in response to posts here. Even in propositional logic, axiomatic systems are but one of many, and in those systems there are ma y variations as to which axioms are chosen. Modus Ponens is common, but not essential. Sequent Calculus does not rely on Modus Ponens, but derives it. Natural deduction usually has modus ponens as a propositional rule. Tableaux has a rule concerning what we can write after an implication, more or less in place of modus ponens.Lambda Calculus has nothing analogous to Modus Ponens.
And most certainly, not all logics are axiomatic.
All this by way of suggesting that proposing a foundation for all logics is to invite logicians to undermine that very foundation.
That's kinda the point of logical pluralism.
Sure, but wouldn't that be if we believed that logic was completely conventional? Here we can split up something like "natural logic" (the rationalizing we can do as a certain species regarding the world), and "formal logic" (the kind of axiomatic (or non-axiomatic) based logics that we formalize with symbols and rules?
I was proposing that the foundation for formal logic can perhaps be found in a natural logic, or something like this.. a foundation outside the formalized logics themselves.
Yes so I guess to equate with your terminology, "Whence intellection"?
If you like. "Natural logic" will collapse into "formal logic" as soon as you take it seriously. The "rationalisations we make" are the very subject of formal logic.
Interestingly though, your joke post in the Lounge kind of proves a point where formal logics can lead to errors by simply abiding by the rules without interpretation (possibly the natural logic?) used to make the content work (become sound/make sensible). And thus something else is going on that isn't just the formal logic (natural logic that is)...
Also, being a bit of a devil's advocate from my past positions (contra evolutionary psychology), there is no way our species evolved "to use formal logic", rather we have rationalization capacities that happened to be able to form formal logic. It is this rationalization capacity that I am interested in- empirically understood through various methods of anthropologists, evolutionary biologists, cognitive scientists, and the like (possibly).. I'll take even armchair theories as stand-ins for now, but that is the foundation I mean.
...mostly shows how poorly folk hereabouts deal with logic.
I get it, but I think this point still stands and is important:
Quoting schopenhauer1
I'm with you in terms of, I'm not much for evolutionary psychological "just so" theories, but if it's not some sort of naturalistic/biological reason we can reason, we still have some capacity that is there by the very fact that we can develop logic, so whatever way it got there, something is happening internally/cognitively that is going on prior to the formalization process of symbolic logic.
Language.
-
This is how I want to see a disagreement between Banno and myself:
1. If we have discursive knowledge, then there is a true/correct logic.
L1. We have discursive knowledge.
L2. Therefore, there is a true/correct logic.
1. If we have discursive knowledge, then there is a true/correct logic.
B1. There is no true/correct logic.
B2. Therefore we do not have discursive knowledge.
Then the question is simply whether L1 or B1 is more plausible. The problem with Banno's approach is that, even for any merits it has, it precludes knowledge, and this is much more absurd than the alternative. Of course B1 is not exactly logical nihilism as presented in the OP, but I see no real reason to engage G. Russell's theories on their own terms. I am here because of a tangent that was redirected to this thread, not because of the OP. I would be more likely to address an argument if Banno presented it himself.
That'd be logical nihilism. What is being suggested is logical pluralism.
You might try
1. If we have discursive knowledge, then there are true/correct logics
L1. We have discursive knowledge.
L2. Therefore, there are true/correct logics.
Yep, that's what I said.
Quoting Banno
I think Count has addressed this nicely:
Quoting Count Timothy von Icarus
So we end up with this:
Pick your poison. Your thesis is that there are true/correct logics with nothing in common, such that we cannot call their similarity logic in a singular sense, and we cannot apply a rational aspect under which they are the same. But the natural language itself betrays this, for simply calling them logics indicates that they belong to a singular genus.
As I said:
Quoting Leontiskos
The idea that different formal logics can each yield sound arguments without contradicting one another is not in any way controversial, and I would not call it logical pluralism.
Which is just to give primacy to PNC, and so to beg the question.
Where you used it to adjudicate over logics:
Quoting Leontiskos
Quoting Leontiskos
You are not here to addressing the topic of this thread, by your own account. [hide="Reveal"]Quoting Leontiskos[/hide] You do not have to be here, and I am not under any obligation to address your posts.
Do you agree or disagree with that inference? There is no adjudication, just a consequence.
Quoting Banno
And yet you are the one who transplanted a different conversation into this thread. You are also the one who abandoned the OP of logical nihilism in favor of logical pluralism when I brought it up.
I'm curious, if you support that position, in virtue of [I]what[/I] would true/correct logics be true/correct and false/incorrect ones not be?
Well, on that consequence it seems possible that logical pluralism, nihilism, monism, whatever have you, could be both true and false. So everyone wins... and loses.
Yep, haha. But maybe that's the point.
Great posts of late. Your time away has served you well. :up:
"Support" is what one gives a football team. I find the ideas here very interesting, and they fit in with a bunch of other stuff. I reject logical nihilism, but there are also good reasons to reject logical monism. The article that this thread is concerned with tries to show a third path. Some have understood that.
I'll just point out what a great question this is, and how it becomes even greater after being dodged. :smile:
The only doge here is your refusal to engage with the content.
How so? "If the 'true/correct logics' contradict one another, then the PNC has been destroyed."
I have to accept the PNC to accept that claim? I think everyone can see that you are wrong here. Maybe stop dancing and start answering the simple questions being asked?
Edit: Unless you are actually presenting Aristotle's argument in Metaphysics IV, but I doubt it. If that is what you are doing you should be more forthcoming. More transparent. More philosophical.
Are we confusing true/correct with simply consistent? All of our ideas agree therefore the symbols we use to represent them must construct actual truth, feels like a reach.
You may be right, but True/correct is Leon's term. There's plenty in the detail, and looking to it would turn this thread away form the mere bitch session it is becoming.
To your point though I think the detail is in the nature or a better word for how our reality is both seemingly real in a naive sense and yet participatory. I can acknowledge a logical argument, note the premises are correct and concede the conclusion while believing it's wrong. Logic is contractual discourse.
Seems so. Various systems offer alternatives.
Quoting Cheshire I could go along with your suggestion as a way-point, but not as a conclusion. If the argument is sound and the premises true, then if the conclusion is false something is amiss and must eventually be addressed.
I have come across the paper before and Russell's other stuff. I'm not sure exactly how what you've quoted is supposed to address the question.
So replace it with "affirm." I assume you understand what I meant.
I don't see it. It doesn't say "pluralism implies a contradiction, thus not-pluralism" but rather "if pluralism then not-PNC.*" How does this give priority to PNC? One might affirm pluralism here and just deny PNC.
And then "if PNC, then not-pluralism." (But this seems irrelevant, and would seem to depend on how pluralism is defined.)
* Whether this premise is true is another question.
But I never did that, so that makes you wrong four times in a row now. Shoot. I can't have begged the question with a claim I never made.
( - Yep)
Quoting Count Timothy von Icarus
Again, true/correct is not my choice of terminology. A logic might be appropriate rather than true. Hence it depends on the interpretation given it. So, as i quoted, "? ? ? is true i? whatever.. interpretation is given to the non-logical expressions in ? and ?, if every member of ? is true, then so is ?." For extensional logics, satisfaction will suffice.
That's all I can offer, since there not being a general case is kinda the point.
Quoting Count Timothy von Icarus
I was reading this part as making the PNC conclusive. "Destroyed"? Things will remain contradictory even if there exists more than one way to arrive at a conclusion following a self-consistent process. In any other thread I would agree, but if the matter is logic itself then it cost a bit of the contextual adherence to the assumption that the PNC is a boundary that can't be crossed. Granted it's a nuanced question begging, but I'm curious about the follow through.
Quoting Leontiskos
Quoting Leontiskos
They aren't logical without total adherence seems strong, not incorrect or unintuitive.
Where do you find that claim, "They aren't logical without total adherence"?
I have asked Banno multiple times whether he agrees or disagrees with the argument, but he is being his usual coy self.
Can you answer the question? Do you agree with the argument? If you disagree then please explain which premise you oppose.
Quoting Leontiskos
To be an argument, words as premises and words as conclusions must be related with [laws of] logic.
Gillian Russell made an argument.
______________________
There is [laws of] logic.
You want to talk about logical pluralism without talking about the PNC? All that means is that you don't want to talk about logical pluralism. You are pretending.
Quoting Banno
Bitch session? It's just another rerun of, "Banno refuses to do philosophy." This is why I said I wanted a thread on Srap's logical pragmatism instead of Banno's logical nominalism. I've seen the episode too many times.
Quoting Leontiskos
I haven't encountered all the P logics, so it's inductive. Very persuasive, easy to corroborate, sound, etc.
If two proper logical systems arrive at a contradiction I think we just call it a singularity and move right along. I don't think the argument, no one would normally have to make, has been made. A counter-example of the PNC doesn't destroy it in the sense it hasn't been demonstrated.
I disagree with the first premise. They could have systematic disagree and remain consistent in their conclusions. Somehow, presumably. Or how they couldn't is the argument and this premise is the conclusion. Hence, light begging of the question.
Quoting Banno
Okay, so you think the PNC can be violated without being destroyed?
Quoting Cheshire
I'm not really following. Presumably you think the first premise presents a false dichotomy.
Again, I would suggest focusing on the argument I gave, not some argument you are afraid I will give at some point in the future.
I think we don't know that it can't. Things are certainly going to remain contradictory in many cases.
Quoting Leontiskos
Not presupposing anything other than you don't get to assume the PNC is a LNC in an argument about whether logical systems can find themselves in opposition and remain true. Does it break a lot of rules about doing philosophy? Yes and no, ironically.
Quoting Leontiskos
When you choose to enguage with the articles cited, I'll be happy to join in. In the mean time, consider:
Quoting Your logical fallacy is...
And in virtue of what is a logic appropriate?
I'm not sure how the proposed interpretation of logical consequence is supposed to answer this question.
Anyhow, I would assume the default answer (the one Russell seems to assume as well) is that logics are correct if they are truth preserving, i e., true premises will lead to true conclusions.
Now, if there are multiple correct logics, and they contradict each other, what exactly are they both preserving? (Earlier you said pluralism has nothing to do with deflation. This question is precisely why I think the two are related. If one correct logic affirms PNC and is contradicted by another correct logic, then it seems that "truth" has to be deflated and relativized.)
Russell leads with intuitionists' and the denial of LEM for a reason, and presumably it is because there are good arguments, reasons [I]in virtue of which,[/I] one might think it is true that some propositions might lack a truth value. But if truth is allowed to be defined entirely arbitrarily, it seems trivial to generate counter examples to modus ponens, disjunctive syllogism, LEM, you name it. We could have a "Protagoras logic," where every premise and conclusion always has the value true for instance; its truth tables would be very easy to develop.
This is what I mean by saying that refusing to allow any metaphysical notion of truth in logic (presumably something all about truth and its preservation) comes close to begging the question re nihilism, or at the very least it makes things very opaque. We wouldn't want to say it's a matter of democratization, but it seems easy for it to head in that direction (e.g. the removal of LEM is introduced by noting that "many philosophers" think it is plausible.)
Can't you do philosophy in your own words, and answer simple questions put to you?
This shit just happens over and over and over. The double standards are wild. I have a reminder from August 6, "Put Banno on ignore." I had some technological difficulties in the meanwhile, but it's probably time to honor that reminder and start focusing on people who are sincerely interested in philosophy. ...Interested in engaging ideas other than their own.
Quoting Count Timothy von Icarus
Let's look at the example that Russell gives:
But then
What Russell seems to be suggesting is that the difference in interpretation leads to our assigning "valid" and "invalid" to 'the very same' argument. It's not so much that one interpretation is correct, and the other not so. Instead, for ?, ? ? ? is true i? in a given interpretation, every member of ? is true, then so is ?; and for some other system, ?', ?' ? ?' is true i? in a given interpretation, every member of ?' is true, then so is ?'.
You don't need to look at the counter example to see how she answers the question, in the opening paragraph she lays it out in that paper: "Logic is the study of validity and validity is a property of arguments... We say an argument is valid just in case it is truth-preserving."
So, again, if two "valid" logics contradict one another, what are they preserving? Can something be true and not true tout court? Or does the truth and validity depend on the system being used? If the latter, how is this position not the very definition of deflationism?
I fail to even see the relevance of the counterexample for the question I asked for this question.
Of the sort that basically fails to allow for any substantial difference between pluralism and monism (a "weak" pluralism), sure. Same with the claim that the existence of multiple truth-preserving logics might be taken as evidence for pluralism. But this is obviously a far cry from a strong pluralism where:
-Gillian is in New York
-I am Gillian
-Therefore, I am in New York
Can be used to construct equally "truth-perserving" arguments that prove that the conclusion is true and false.
The validity of Russel's argument depends upon the interpretation mechanism you apply to the sentences in it, and their terms. The first formalisation of it is:
1) Gillian is in Banf,
2) Therefore, I am in Banf.
In standard predicate logic, there would be nothing saying that Gillian=I, because all you can do is assign symbols based on what's in the argument. "I" and "Gillian" are distinct referential symbols, therefore they must be parsed as different entities. In standard predicate logic, something being red does not imply that it is coloured.
If you're thinking "that's nuts", because the argument clearly is "valid" in some sense, you need to come up with a reason why. And you did just that, you mapped the argument as presented to another argument:
Quoting Count Timothy von Icarus
Which is clearly valid in the original predicate logic. However the mapping between the arguments:
1) Gillian is in Banf.
2) Therefore, I am in Banf.
to
1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf
is not an operation available to you in original predicate logic. It's an extra logical operation to map argument to argument like that, through the means of natural language comprehension. In effect you've supplemented the original predicate logic with an extra rule, in which you resolve coreference classes of each denoting term in the argument's sentences prior to evaluating whether the premises can be true and the conclusion nevertheless false.
You could then prove a meta-theorem that states that any argument of the first form is valid so long as it's valid in your new logic that resolves the coreference classes - any one where "I" and "Gillian" co-denote.
That is, you stipulate a set of equivalent denoting terms prior to evaluating it - in this case, you would stipulate that "I" would denote the same entity as "Gillian", which makes sense since Gillian was understood to be author.
In another interpretation of that same argument, the argument would be invalid, since when fdrake writes the argument, fdrake is the author, so we don't belong in the same coreference class.
There's a considerable ambiguity in natural language terms and concepts, which gives them a kind of cohesion through fuzzy boundaries, which can then be interpreted as a coherent unity, which seems to be @Leontiskos's method of argument in this thread, to my reckoning.
Maybe there's a basic imperative to gather everything into a single framework.
It's just a question of understanding the detail for me.
Quoting Srap Tasmaner
1) So I pick a point A in 3 space A={0,0,10} as {x,y,z} coords.
2) I place a plane cutting the point {0,0,0} with unit normal vector {0,0,1} (that's the xy plane). The axel is parallel to the z-axis, it points in the direction of the unit normal.
4) I then pick a circle in the x-y plane, let's just say it's centred at the origin O={0,0,0} with radius r=sqrt(10), which I think is the appropriate distance to make your construction with the cone work.
The center circle O is equivalently determined by the distance sqrt(10), the point A and the choice of the x-y plane.
That connotes a more general construction.
1) I form the sphere of radius R around A.
2) I pick a projection P and a point A. I constrain the projection P that it projects onto a plane whose normal vector is parallel to the sphere radius and that norm(PA)<=R. Intuitively, you travel along a sphere radius and blow up a plane orthogonal to the radius from a point on the radius.
3) I apply P to A, producing PA.
4) I collect the points this plane intersects the sphere's surface together, this will be a circle of radius... sqrt(d(A,PA)^2 + d(PA,intersection point of plane with sphere surface)^2)
5) I have more than enough degrees of freedom in the distance expression in 4, when I can pick A and P and R, to define any circle centred at any point.
Was that the construction?
Thanks for the attempted clarification, but this seems to entirely miss the context of the quoted part of my post, which is not about Russell's thesis.
To clarify, for Russell (and I would suppose most) a "correct logic," is one that is truth-preserving, where true premises lead to a true conclusion.
If it is the case that different "correct (truth preserving) logics" contradict one another, what exactly are they preserving?
If it is assumed that truth is relative, with many unrelated types of truth, this seems to come close to begging the question re logical monism. There will not be a single set of valid, truth-preserving arguments, but many sets that vary according to what "truth is" or "which truth" we are using.
The problem here is that questions regarding logical monism are questions about what is true tout court, analyzed in a discipline that tries to avoid discussions about what exactly truth is. But ignoring this just seems to allow people to talk past each other or engage in less than obvious question begging.
I can't think of any other context where a conversation like this would be considered good philosophy:
Jack: My thesis is that the relationship between these two sets is empty.
Jill: Interesting, how are the two sets defined?
Jack: Hey, stop trying to do metaphysics!
Or alternatively:
Jack: I don't know. We know a member when we see one... except lots of people disagree about membership.
I guess I wasn't sure what this meant. You don't think it is appropriate to judge logics based on whether or not they are truth preserving? If not, what is the measure of appropriateness? The rest of your post doesn't really help me figure out what this is supposed to be. A definition of logical consequence helps us determine the appropriateness of logic how? Just in case the relation isn't empty?
Quoting Count Timothy von Icarus
It breaks down ambiguities in a concept, attempts to clarify and resolve them, and if the resolutions contradict each other they are presented with their merits and drawbacks. That seems like standard flavour "conceptual analysis" to me. There's just no presupposition that there's one right way of doing things, even if there is a presupposition that people can come to understand the same things with sufficient thought and chatting.
And regarding truth, truth as a concept applies to both.
Gillian is in New York
Therefore, I am in New York.
will have true premises and conclusion when and only when "I" and "Gillian" refer to the same entity. It's thus not a valid argument in the standard sense, as it can be false (the author need not be understood to be Gillian).
vs
Gillian is in New York
I am Gillian.
Therefore, I am in New York.
will be valid, as you've plugged the hole in the previous argument.
Quoting Count Timothy von Icarus
Referencing the above, what they preserve is truth of conclusion given true premises. That is just what truth preservation means. Stipulate what you like, see what follows from it.
Whether you have true premises is a different issue. When you stipulate axioms, you treat them as true. Are they true? Upon what basis can they be considered as such?
Whether you have a true axiomatic system is a different issue again, and I don't really know what it means. How would you compare Peano Arithmetic and Robinson Arithmetic, for example? Which one is true? Is one "more true" than another? What about propositional logic and predicate calculus? These aren't rhetorical questions btw.
I would posit that axioms can be considered to be correct when they entail the intended theorems about the object you've conceived. That is, when they reflect the imagination. For example with @Leontiskos using Euclid's characterisation of circles as a plane figure, it would entail that a great circle on a sphere surface is not a circle... whereas it seems to be "contained in the intended concept" (scarequotes) of a circle. Which might lead you to reject the axioms, or insist upon them... Hence the method adopted in the paper and my dialogue with Leontiskos.
And there is a formalistic definition of truth, a statement is true in a theory when that statement holds in every model of that theory. Like "swans are birds" is true because there are no swans which are not birds, but "swans are white" is false because there are swans which are not white. Every collection of swans is a model of the term "swan", and all you need is one collection with a black swan in it to show the latter is false. Similarly if you wrote down the axioms of a group, something would be true of groups when it is true of every model of the theory induced by group axioms - the sets the groups are made of, and the set operations the group mappings use.
You can think of the latter as related to my dialogue with Leon in the following way - the intuition of a circle makes you want to put the great circle into every theory of circles, everything which describes what circles are, so if you think it should be in the theory, you have to reject (or repair) Euclid's definition.
Edit: or, my preferred option, acknowledge that "circle" is an imprecise concept in natural language and also that there are lots of different useful ways of fleshing it out.
Alright, forget New York because we're just talking past each other. There is no disagreement there and clearly the example is not making what I intended clear
And you don't think assuming that this definition is what is meant by "truth preserving," is question begging? Don't logical monists generally claim that their position is true tout court?
I don't see how these two together don't presuppose a deflationary theory of truth. We could debate the merits of deflation, but its presupposition seems to be very relevant.
An excellent question for a field that revolves around truth, no?
I have no qualms with setting aside metaphysical considerations of truth for formal analysis. And this is perhaps rightly the norm for cases. But it seems inappropriate in this case.
I'm not sure we have to choose between these. We're talking about truth relative to some stipulated sign system. There are multiple stories about what happened to Luke Skywalker after the original films. Are any of these more true than any other? However, it seems to be something quite different to claim that [I]all[/I]claims are true only relative to stipulated systems and that none are more true than any other.
Quoting Count Timothy von Icarus
Indeed. It doesn't seem meaningful to claim that the axiomatic systems are true or false in toto. But nevertheless, if there is a single unifying, bivalent truth concept, and two systems have incompatible theories, we should be able to say which is true and which is false. If we did not need to, we'd have to suspend that some claims are not evaluable as true or false in principle - and thus jettison bivalence by destroying the assignment mechanism of statements to truth values. And if we did need to, we'd have to claim that some systems are... false.... somehow, even when they seem to adequately represent concepts in precisely the same manner as others, just different concepts.
Quoting Count Timothy von Icarus
I don't think logical pluralists are committed to that. Everyone agrees what follows from what stipulations. So it's true to say that "not every group is abelian". You can think of stipulations as disambiguations - which is what lemma incorporation works like.
The underlying issue seems to be that everyone can agree that eg groups have certain properties, but if you stipulate the definitions differently you change the properties. But you don't change the properties of the intended object when that object is the group, you perhaps change what the intended object is tout court.
I think that is in the direction of the intended thread topic. Because the ability to stipulate lemmas that make an axiomatic system better track an intended object's properties thereby lets you make more universal judgements about more precisely demarcated structures. Everyone will agree that Euclid's definition of circle captures plane circles, but not all pre-theoretically intuited circles are plane circles.
In effect this is a way of massaging the "complete generality" predicate in the OP's argument. You can restore a sense of "complete generality" by using lemmas, by speaking about something ultra specific and formalised you can guarantee that it works in that way for that system, the latter applies without exception. Applies without exception in the sense that "fdrake is sitting drinking tea now" is true at time of writing, and thus applies at that time without exception forever. Only "now" for those refined systems is a new lemma, allowing them to better specify their intended conceptual content.
Not necessarily, as I noted before, many "weak" versions of logical pluralism start to look indistinguishable from weak forms of monism (something Russell discusses as well). And I would imagine most don't [I]want[/I] to be committed to this view. It's a different question whether is this essentially presupposed as a background assumption though.
I mean, in your response to the question of: "in virtue of what are logics to be considered correct," you presented a textbook deflationary account of truth. Now I understand that you might not advocate that view as absolute. But if we "roll with it for the purposes of analysis," it seems like it will play a key role in seemingly deciding the issue.
Do they? Isn't one of the questions at issue whether anything follows from anything else?
To quote Russell:
[I]
arguments are often said to be neither true nor false, but
rather valid or invalid. This is correct as far as it goes, but a principle containing a turnstile as its main predicate can be regarded as a sentence making claim about the relevant argument. Such a claim will be true if the argument is valid, false if it is not. Hence the nihilist can be said to believe that there are no true atomic claims attributing logical consequence.
[/I]
An interesting practical approach, to be sure.
I don't think that's an issue at stake at all. If no principle holds in complete generality, they may still hold in certain well understood and well demarcated cases and contexts. Such as modus ponens in propositional logic.
The idea that (nothing follows from anything else in virtue of a valid argument) if (there are no principles which hold in complete generality) is ultimately not a premise in the OP's argument. It could be, and I believe Gillian Russel lectures as if, there are valid arguments even if there are no principles which hold in complete generality. Because she specifies what context she's speaking in. It then remains to be seen if a sense of complete generality can be restored by supplanting restricted statements - like Euclid's definition of a circle - with disambiguating phrases - like "in plane geometry".
Quoting Count Timothy von Icarus
Does everyone who understands a system and a proof in it believe the conclusion if the proof is correct and understood? Yes. Everyone agrees that P & P=>Q allows you to derive Q in propositional calculus. That's less about there being rules which cover everything, and more about there being followable rules. That's a followable, derivable rule.
I think this is the simplest version of what I was thinking.
Given a sphere centered about A,
pick any three points in the sphere,
those three points determine a unique plane,
the intersection of that plane and the sphere is a circle.
We're just taking a section of the sphere, without any further reference to the point A, which has already done everything needed to guarantee that its coplanar subsets are circles. In particular, we did not need to project A onto the plane that sections the sphere. (We can project it onto that plane, using the obvious orthogonal projection, or anything we like.)
Am I getting something wrong here?
Nah it looks fine. I'm just confused, it's doing away with the centre by providing an equivalent construction of the centre. Which is also fine, I just want to see what you're seeing in it.
Quoting Srap Tasmaner
Are those points in the interior of the sphere or on its surface?
Sphere, not ball. The surface. The 2-manifold.
With you. Yeah.
Way back when we started, what interested me was decoupling the point with reference to which the circle is constructed from the plane within which it is constructed.
[s]Then I noticed you can decouple the point used to construct the circle from the (in-plane) center of the circle, because that's a projection, but it's a projection you don't need to do to construct the circle. Which means you can project the circle's originating point anywhere in the circle's plane.[/s]
I guess it would be better, and simpler, to say we can decouple the projection onto the plane of the originating point from the center of the circle.
And I thought there might be something interesting there, just in the geometry, but then realized the model I was creating was suggestive of stuff I've been thinking about a lot. That happens to me all the time.
That makes sense. Equivalence classes of pre-images of projections under some relation seems like a cool idea.
Nice. That cleared something that I was puzzling over. A Great Circle is defined by only two points on the surface. It can do this becasue it is a straight line. So as on a plane, a line can be defined by two points and a circle by three.
IDK, that's how I've often seen nihilism defined. Per Russell it is "the claim that there are no laws of logic, i.e., no pairs of premise sets and conclusions such that premises logically entail the conclusion."
Yes, and this makes sense if deflation vis-ĂĄ-vis truth is presupposed. You can have nihilism and truth preservation via entailment because truth is just defined in terms of the formal context.
And it might make sense in other contexts as well. Just thinking back to philosophical history, there is certainly a long history of concepts of vertical realityâsome things being "more real," or "more true." True might be predicated analogously like being and might not be fully captured by language and discursive human reason (e.g. Plato's Seventh Letter).
I'd have to think about it more but my intuition it would play havoc with other theories of truth. For example, in simple correspondence theories, X is true just in case X actually is the case. Now I'm not sure what it means for "truth preservation" if it is possible to have valid arguments that persevere truth while variously affirming and denying that "X is actually the case." I suppose people might counter that logic is now properly the study of formalism, not truth qua truth, or even natural language, to which I would disagree, the former will always sneak in the back door if left unacknowledged.
But... P & P => Q entails Q in propositional logic, who is denying this? It does not seem Russell is:
And footnote ten:
The logical consequence relation is preserved, even if the intended objects it's supposed to refer to can be taken as counter models. Like "This sentence is false" might be taken as a countermodel for the law of excluded middle, or the great circle might be taken as a countermodel for Euclid's definition of a circle.
Consider Russell's proof and refinement of LEM:
I underlined "bivalent" in the final bit, since you produced a similar repair to the argument:
1 ) Gillian is in Banf
2) Therefore, I am in Banf.
by understanding "I" as "Gillian", then adding this as a specification in the argument:
1 ) Gillian is in Banf
2 ) I am Gillian.
3 ) Therefore, I am in Banf.
Your repair could well have read "For all I-s who are Gillian", just like Russell's repair of LEM reads "for all bivalent ?".
It's also worth noting that Russell's countermodels, monsters and context specifying information (eg "for all bivalent") aren't necessarily in the object language in question. EG propositional logic just
assumes bivalent ?, so LEM applies, so you couldn't formulate a "neither" valued statement in its standard operation.
And since her countermodel of a statement which evaluates to "neither" does not have an interpretation in terms of standard propositional logic, she expands what ought to clearly be the scope of any logic of propositions to include that statement, goes "bleh, any logic worth its salt should account for this...". marks down on the page "eh, propositional logic as is works fine for bivalent ?" and then moves onto new pastures of polyvalent ?.
Russell's approach is largely telling logical nihilists not to throw the baby out with the bathwater, just because they expect logical laws to behave like The One Law To Rule Them All, a kind of context invariant divine providence.... and when they don't, why not just say they work when they work and find out where they work?
In effect the nihilist doubt machine gets going by noticing that there's arbitrary degrees of contextual variation, and throws every available piece of crap against the expectations of logical form a universalist has (like @Leontiskos and I's discussion earlier), when ultimately only the universalist need read the nihilist doubt machine as nihilist - it's just a doubt machine, you can tell it to sod off by specifying the exact mess you're in.
Something's not right here, which is just sloppiness and rustiness on my part.
In general, three non-colinear points in 3-space determine a unique plane, a unique triangle, and a unique circle. And then it takes a fourth point, not in the plane of the first three, to pick out a unique sphere.
When I was talking about sectioning a sphere ? after I realized that using a non-coplanar point to determine a circle could be thought of this way ? I reached for three points to pick out the sectioning plane out of habit, thinking that the section is guaranteed to be a circle because it is (a) planar and (b) a subset of a sphere.
Which is super super dumb. What the sphere guarantees is that the points selected are non-colinear, which hadn't even occurred to me.
All this sphere business ran roughshod over my original thinking, which was very cone oriented, as the drawings show.
Sheesh.
No one. But logical nihilism is not a position about "what is true in propositional logic." It seems like you're still presupposing deflation here, truth has to be "truth relative to this formalism."
It perhaps comes down to what is meant by "truth-preserving". A sentence in a classical extensional logic is consistent if there is at least one interpretation in which it is satisfied. If there is no such interpretation, then it is contradictory. If it is satisfied under every interpretation, then it is valid. Here, "truth-preserving" is replaced by satisfaction.
A given sentence is neither true nor false until given an interpretation. "? ? ?" is understood as "? satisfies ?". So since Tarski, truth and validity are defined in terms of satisfaction.
Logical nihilism, is the view that "there are no laws of logic, where a law of logic takes the form "? ? ?"(p.4). That is, logical nihilism is the view that there are no cases in which ? satisfies ?.
Russell lists three approaches, as follows:
She adopts the interpretations approach, but for simplicity. She gives the impression that her argument might be made using the other two approaches. She proceeds to show how P ?Q,Q ?P is truth-preserving if the interpretation includes only T; but not if it includes both T and F. That is, it is a logical law under one interpretation, but not under another. She then shows how the law of excluded middle is a logical law in the interpretation (T,F), but not in (T,F,N).
Now what this shows is that truth-preservation is a function of the interpretation. So yes, in your rough terms, truth and validity do depend on the system being used, since that system includes the interpretation.
Now I am not at all sure what you mean by 'deflation". But I am confident that all of the above could, at least for extensional cases, be put in terms of satisfaction, without mention of truth-preservation. If that for you is deflation, than so be it.
I'm not sure where that leaves our chat.
I think the univocalist extreme of splicing everything apart and analyzing it separately is representative of sophistry (or nihilism?). Namely, the methodology precludes reasoning and knowledge. If one does not admit analogical predication in one form or another then they can deny but they can never affirm. They have created a method that can only deny; a skepticism machine.
For example:
Quoting fdrake
Has it been fixed? The "sophist" would say no, and can quibble endlessly. They might ask you to specify what exactly "I am Gillian" means; what 'I' means; what a name is; what the predication of amness means (all difficult questions). They might splice (1) and (2) into different contexts, pointing out that (1) is a third-person description and (2) is a first-person description, and that it is not clear that these two discrete contexts can produce a conclusion that bridges them. "Shit-testing" seems to have no limits and no measure.
There is an interesting question about the great circle, but the method which outright denies that the great circle is a circle can outright deny anything it likes. It is the floodgate to infinite skepticism. I think we need to be a bit more careful about the skeptical tools we are using. They backfire much more easily than one is led to suppose.
Edit:
Quoting fdrake
Can you? There is an idea that floats around, according to which one can give quibble-proof arguments. I don't think this is right. I'd say the idea that there is some quibble-proof level of exactness won't cash out.
Anyway, the relevance was the difference between the maths of a sphere in [math] \mathbb {R}^3 [/math] and intrinsic spherical geometry.
Logical nihilism is not a claim about what is true in classical extensional logic. It is presumably a claim about all truth preserving arguments.
Likewise, if truth can be defined arbitrarily, if we follow Carnap in the claim that: "in logic there are no morals. Everyone is at liberty to build his own logic, i.e. his own language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments," it seems logical nihilism is trivial, but the question is effectively begged.
As for deflation: https://plato.stanford.edu/entries/truth-deflationary/
Those are quite different I believe. There's no attempt to change the verbatim meanings of argument terms in @Count Timothy von Icarus's repair, in fact there's an insistence on representing the conceptual content of what's said in spite of the means of its representation (predicate logic vs "I"). In effect, Timothy's takes the truth of the argument for granted and treats the inability of the verbatim machinery of propositional logic to reflect that truth as a failing of the logic... thus repairing the argument by explicitly spelling out the context sensitivity of "I".
Whereas your examples do not insist on taking the conceptual content of what's said for granted, indeed they're attempting to distort it. Allegorically, the logic of shit testing is that of a particularly sadistic genie - taking someone at their word but exactly at their word, using whatever pretheoretical concepts they have. The logic of your sophist is closer to doubting the presuppositions which are necessary for the original problem to be stated to begin with.
Our dispute was similar to the former - we both have the same pretheoretical intuitions about what a circle is. Agreeing on Euclid's and on the great circle's satisfaction of it. And we'd probably agree on the weird examples containing deleted points too, they would not be circles even though if you drew them they'd look exactly like circles. The issue we were having is that Euclid's definition clearly did not accurately represent our (mostly) shared pretheoretical intuition regarding what a circle was - what it looked like -, and I kept asking you to repair it.
Remember even Euclid saw fit to define a circle axiomatically. And his works exactly as planned in the plane. Just circles also live outside the plane, and thus are not bound by Euclid's plane figure definition of them verbatim.
"For all circles in the plane... (Euclid's theorems follow)" - another example which could've been in Russell's paper.
Sure. I drew attention to that. But that's were it starts. We can move on to formal intensional logics, if you like, and their algorithmic interpretations. Probably should leave that until we have a bit more agreement, though. It's important to understand that this is an area of development, and not all questions have been answered. For intensional logics, use is made of Kripke's theory of truth, but I certainly don't have the details.
I don't understand why you are talking about truth being defined arbitrarily. Tarski's definition is far from arbitrary.
And yes, I have a rough idea of what deflation is with regard to truth. I'm just not sure what part you take it to play here. For extensional languages we can define truth in terms of satisfaction. I gather you understand that as deflationary? Fine. What's the problem? Is it that you object to such an approach setting up truth in terms of interpretation? But it works.
Sure. Here is a quibble proof argument.
Where's the issue?
To be clear you would have been compelled to deny the great circle was a circle by only using Euclid's definition of it verbatim, I would not have!
This is what always seems to happen with these shiny new theories. It is motte and bailey. The controversial claims that stimulated attention dissipate upon closer examination.
I'm not really sure what you are arguing, fdrake. It doesn't sound like you hold to logical nihilism or logical pluralism in any strong or interesting sense. Am I wrong in that?
You talk a lot about the great circle:
Quoting fdrake
Let's suppose it is a countermodel. How does the logical pluralism arise? I can only see it arising if we say that a "circle" means both Euclid's definition and the great circle countermodel, and that these two models are incompatible. Is that what you hold?
-
Quoting fdrake
Given that I disagree with all of this, does it follow that you were the sophist and not the sadistic genie?
Quoting fdrake
I kept asking you to offer a reason why it needs to be repaired, because it "clearly" was fine. You are begging the question in your own favor with words like "clearly."
Why are we to believe that a three-dimensional abstraction (i.e. the great circle) does not contain a two-dimensional abstraction (i.e. a circle)? In any case, the easier disagreement here is over the question of whether one can delete a point.
Quoting fdrake
This is helpful, but I'm not convinced it is cogent. The sadistic genie is not taking them at their word by being overly pedantic, he is just being a sophist. I see the distinction you are making, but I would say that the sadistic genie is a sophist, even if not every sophist is a sadistic genie.
I saw my cousin who has Asperger's, "Your hair is long, how long has it been growing?" "Since I was born!" He is fun, and this is an example of the sadistic genie, but it is not a non-example of a sophist. Taking someone "exactly at their word" is a good way not to take them at their word.
Quoting fdrake
To take a few, you haven't defined the operations, commutativity relations, numbers, variables, etc.
Quoting fdrake
I don't follow, but you seem to think "verbatim" is a fix; a quibble-proof solution; a univocal meaning. I don't think the buck stops there or anywhere else. Literal meaning is a puzzle as much as anything else. To use the word "verbatim" and assume you have won the argument will not do.
Good posts, though. I have to run but I hope to come back to this soon.
Understand them as you usually would. + and times are spelled out in the field axioms (see classical definitions). Add that subtraction of a is equivalent to adding -a. IE x-a=x+(-a)
I think it's a confusion regarding the connection of meaning to truth, and about truth. It might not be a confusion, it could be an insistence on a unified metalanguage having a single truth concept in it which sublanguages, formal or informal, necessarily ape.
It's quite suspicious that you can talk about "for all bivalent phi" in Russell's paper but also "for all phi which are true, false or neither" in natural language, and the reader will understand some birthing of new context and propagate their understanding into that context. As if there's some big Understanding Truth Machine that gazes through the eyes as soon as you see someone write down a new system of axioms.
All the while you know there's a wealth of intended objects for the symbols to capture.
Quoting Banno
They're always going to need semantics, too. I've no idea how to specify the connection between a syntax and a semantics without using some informal metalanguage, so there will always be some unformalised remainder I think!
I suppose the question is whether you read the necessity of that unformalised remainder as a sign that all systems should be thought of univocally, or whether you can erect little fortresses of axioms and interpretations amid the sea of chaos whose waves are one voice.
(Quine's rejection of modal logic)
I suppose there's a distinction between "having the same underlying concepts of truth and meaning and law" and "having different laws", maybe all the systems we've created, despite proving different theorems, have proof and truth as analogous family-resemblance style concepts in them. Maybe they have a discoverable essence.
Not that I'm persuaded.
(I did read somewhere that modal logic could be "reduced" to first order logic...)
Only one type of logical law, all systems provide instances of? Only one type of truth, all systems provide instances of? I don't like it, or believe it, but it's possible.
A good move away from the strawmen. :up:
Quoting Leontiskos
Logic is that which reliably produces knowledge, via rational motion or inference. This is not limited to a single formal system - that is Banno's strawman. But knowledge and truth are one. There cannot simultaneously be knowledge both of X and ~X. Therefore logical pluralism is false.
And again, there is the challenge set up by that very specification, to find a logic that does not meet it. Monism again restricts development.
And yet Dialetheism. You at least need to make a case, rather than an assertion.
Er, do you ever take your own advice?
Banno has so thoroughly poisoned the well that it becomes difficult. Here is what I said to this idea:
Quoting Leontiskos
-
Quoting Moliere
So again:
Quoting Leontiskos
In order for a sentence to be true or false it must say something. That is what it means to be a sentence. "This sentence is false," does not say anything. It is not a sentence. It is no more coherent than, "This sentence is true," or, "This sentence is that."
Quoting Moliere
If you think that answer is wrong then you'll have to tell us what the sentence means.
:up: This is what I was getting at with the reference to historical philosophy, although I think, in general, most thinkers I can think of would say that truth itself is the unifying and generating principle (genus vs species).
I suppose the flip-side would be that there is no relationship between concepts of truth. I can't help but think this would make truth arbitrary, or at least have major philosophical ramifications, maybe not.
It is also another departure from natural language. We do not speak of truth as having various species with no relation to each other. Nor does the term "logics" jibe with the idea that the various logics have nothing in common.
Quoting Leontiskos
Quoting Leontiskos
What does it mean to "say something"?
I'll say more, though it's fair to ask what are the conditions you're after here -- what I have in mind is that English cannot refer to itself but must refer to objects. Is that so? Some sort of extensional theory of meaning?
Because I'd say that just from a plain language sense "This sentence is false" is clear to a point that it can't be clarified further. "This sentence" is a pronoun being used to refer to the entire phrase which the pronoun is a part of. "... is false" is the sort of predicate we apply to statements.
"...is false" is the predicate which yields the value "true" for sentences which are false in a truth-functional sense, which seems to me to be pretty clear that this is the sort of background assumptions which are part of Russell's paper. (though what I'm advancing is different from Russell's, I'm in favor of her conclusion for logical pluralism)
But neither of these things rely upon truth-conditions or states-of-affairs.
And paraconsistent logic certainly seems to me to be a worthy candidate for being significantly different from bi-valent logic since it rejects the principle of explosion, and accepts dialethia.
So be honest. When you say, "This sentence is true/false," do you think you are saying something meaningful? Would you actually use that phrase, speak it aloud, and expect to have said something meaningful?
Quoting Moliere
A sentence says something if it presents a comprehensible assertion. It says something if its claim is intelligible.
Now when you say, "X is false," I can think of X's that fit the bill. I might ask what you mean by X, and you might say, "2+2=5." That's fine. "...is false" applies to claims or assertions. If there is no claim or assertion then there is no place for "...is false." For example, "Duck is false," "2+3+4+5 is false," "This sentence is false."
Whether or not a sphere's line of circumference looks like a circle on an actual sphere presupposes a vantage point of origin. Sometimes it can and does. Other times, not. One can gradually change their own position relative to an actual sphere that has a visible line of circumference around it in such a way that the line of circumference[hide="Reveal"](great circle)[/hide] only seems to change it's shape. It doesn't. That change is one of perspective(the way the line of circumference looks to the observer).
If all circles are located on 'perfectly flat' planes, that occupy no space at all, then the line of circumference around a sphere is not a circle. All lines of circumference encircle space. So, either something that does not occupy space can encircle space or the line of circumference is not equivalent to a circle...
...despite the fact that that line of circumference can look like a circle to an observer.
Is that wrong somehow?
I don't see why one must accept this:
Quoting creativesoul
Nevertheless, if the great circle is a torusâa three-dimensional objectâthen it is not a (Euclidean) circle. If it is not a torus then it may well be a circle. Yet perhaps it is not a torus but is nevertheless a set of coplanar points, falling on an implicit plane which possesses a spatial orientation. Is it a circle then? Not strictly speaking, because two-dimensional planes do have not a spatial orientation.
But what is the point here? Recall that @fdrake's desired conclusion was that there are square circles.
I don't think this would be the way to put it. Presumably some systems are not commensurable unless we have some criteria for what will count as a correct logic.
From Griffiths and Paseau:
But I think there are multiple forms here,
e.g. "McSweeney: â[T]he One True Logic is made true by the mind-and-language-independent worldâŚ[which]âŚmakes it the case that the One True Logic is better than any other logic at capturing the structure of reality [2018, Abstract].â So, the logical pluralist denies that any one consequence relation is metaphysically privileged,"
Quoting Leontiskos
Fair enough. Part of the issue here is whether pluralism can be set out clearly. As the SEP article sets out, the issue is as relevant to monism as for pluralism. The question is how the various logics relate. It remains that monism must give an account of which logic is correct. You've made it plain that you don't accept Dialetheism, and will give no reason, so the point is moot.
, "This sentence is false" is about that sentence. It says that it is false. It's like "This sentence has six words" in some ways, and "That sentence is false" in others. There is no obvious reason to think it meaningless.
Not all paraconsistent logics accept dialetheism, but dialethiests are pretty much obligated to accept paraconsistent logic.
Interesting. Thanks for this. I'm a bit surprised by you referring to this, since I had taken it that you had a dislike for formalism.
But taking it at face value, how can we be sure that only one logic will "capture all and only consequences that obtain among meaningful sentences." If one logic has "? ? ?" and another has ?' ? ?, what is our basis for choosing which is the One, True? Not either ? or ?', without circularity. Some third logic? And again, Which? Does the monograph address this? Are we faced with an explosion of logics?
Point well-made and taken. That should have been further qualified as all spherical lines of circumference. That's what I meant. That's what I was thinking. Evidently a few synapses misfired.
Quoting Leontiskos
Just wondering if I've understood something.
Quoting Leontiskos
My interest was piqued by the claim that a line of circumference around a sphere was a circle. The shame of this all is that the term "circle" can mean whatever we decide. Then we can equivocate. Sorry for the interruption. Have at it.
My position was that there are circumstances in which it makes sense to say there are square circles, perhaps even that there are circumstances in which one can correctly assert that there are square circles, not "there are square circles" with an unrestricted quantification in "there are". Quantifying into an undifferentiated, uncircumscribed domain is a loaded move in this game. I do not imagine myself hacking into the mainframe of being to view the source code.
Absolutely crystal quote, thank you.
Well, one might accept it. I don't see any of these objections as straightforward. I don't think there is a "verbatim" meaning, to use @fdrake's word.
Does the circumference of a (Euclidean) circle encircle space? Yes, two-dimensional space. But then does the great circle's encompassing space make it a non-circle? Apparently not. Unless what we mean is that the great circle encompasses three-dimensional space, in which case this does make it a non-circle.
Quoting creativesoul
Fair enough, and I meant to ask in a broader way and include fdrake.
Quoting creativesoul
I am quite fine with that claim. Apparently I think the coplanar points of the great circle contain a circle (and a two-dimensional plane).
fdrake effectively puts words in my mouth in declaring victory, "Ah, when you say 'great circle' you mean something which does not contain a two-dimensional plane, therefore when you say 'great circle' you don't mean a Euclidean circle." But I never assented to any of these sorts of interpretations.
---
Quoting fdrake
So you are ("perhaps") willing to say that there are circumstances in which one can correctly assert that there are square circles, but you won't commit yourself to there being square circles. This is odd.
The idea behind this sort of thinking seems to be that every utterance is limited by an implicit context, and that there are no context-independent utterances. There is no unrestricted quantification. There is no metaphysics. I take it that this is not an uncontroversial theory. Here is an example of a statement with no implicit formal context, "There are no Euclidean square circles." You would presumably agree. But then to be wary of the claim that there are no square circles, you are apparently only wary of ambiguity in the terms. You might say, "Well, maybe someone would say that without thinking of Euclidean geometry." But we both know that there is no verbatim meaning of "square" and "circle," at least when subjected to this level of skepticism. This is a nominal dispute, but it won't touch on things like logical pluralism, for that question has to do with concepts and not just names. A new definition of "circle" will not move the needle one way or another with respect to the question of logical pluralism. As noted, the taxicab case involves equivocation, not substantial contradiction.
I am still wondering:
Quoting Leontiskos
Again, for Aristotle logic is the solution to the problem of the Meno. It is how discursive knowledge is achieved. It is primarily a matter of inference. Aristotle was quite clear that his formalization was not identical to logic in this fundamental sense.
If someone wants to argue for logical pluralism I would want to know exactly what they mean by that term, because it has been unhelpfully ambiguous all throughout this thread.
No, not really. You really ought to read Rombout on the way that Frege and Wittgenstein mean different things by "logic." Your whole frame is mistaken. I am not a "logical monist," and I don't think Timothy is either. If every logic is on the same level, then pluralism must be true. Logical monism and logical pluralism strike me as equally silly.
Quoting Banno
You've made it plain that you won't offer any arguments, only assertions. Moliere tried and I answered his.
Quoting Banno
"In some ways."
Unlike "...is false," "...has six words" does not require an assertion/claim.
(Moliere and yourself are doing what I would call Dialetheist apologetics. You've heard objections to the "Liar's paradox" and you are responding to those objections, regardless of the fact that my objection is quite different.)
I don't dislike formalism, I just think it is frequently called on to do things it is ill-suited for or retreated into to avoid difficulties that should rather be brought front and center. That said, I don't agree with the framing here (I haven't made it far anyhow), but it seems to me like it captures the intuition that monism is going to be about correct logics.
Their target is a natural language (or "cleaned up natural language"), or maximally, all natural and scientific languages. The analogy they draw is to physical geometry. The physicist is interested in physical geometry, not any and all geometries. They are only even potentially interested in a few of the geometries that might be dreamed up. Likewise, the applied logician is interested in logical consequence in the languages we actually use to discuss meaningful truths. Which I think is a useful analogy.
I'm sure they do in the second half, but I haven't made it that far (in part because I'm not sure about the project, but it's quite readable and got good reviews). The first half is objections to pluralism. They do foreshadow this a bit, because it is going to be a problem for pluralists too, since they generally don't want to say that [I]all[/I] logics are legitimate either. Additionally, presumably pluralists will want to convince others to be pluralists by making a valid argument for pluralism. But they're going to likely to find this impossible to do in [I]all[/I] the logics they accept as correct (at least per popular formulations of pluralism). Yet if they work with just one correct logic then inconsistency issues arise in the metalogic. That and the choice of a metalogic will be arbitrary (which Shapiro's account owns up to).
As they put it:
"In fact, it would be quite odd to suppose that there isnât a single underlying argument for pluralism, but that it must be recast in different ways from different perspectives. By far the most natural thing to say is that if there is a good argument for pluralism, then that same argument should be frameable in any true logicâand so much the worse for any logic that does not allow for its expression."
You forgot that Euclid specifies a circle as a plane figure. I realise you're not going to accept that a great circle is not a Euclid circle, or that a circle in a plane at an angle isn't a Euclid circle without a repair of his definition - but please, trust someone who's wishy washy on logic that you're just wrong that Euclid's definition encompasses all circles.
I've been using the word "verbatim" to try to mean a couple of things:
A ) At face value.
B ) Using only the resources at hand in a symbolic system.
Thus Euclid's definition of a circle, verbatim, would exclude the great circle. And I keep bringing that up because it neatly illustrates the interplay between formalism and intuition and also a pluralism vs monism point.
Quoting Leontiskos
And if you want to just talk about your intuitions without recourse to formalism, I don't know if this topic of debate is even something you should concern yourself with. You might not even be a logical monist in the OP's sense, since the kind of logic it's talking about is formal?
Quoting Leontiskos
If you actually want my perspective on things, rather than trying to illustrate points from the paper: I'm very pragmatist toward truth. I prefer correct assertion as a concept over truth (in most circumstances) because different styles of description tend to evaluate claims differently. As a practical example, when I used to work studying people's eye movements, I would look at a pattern of fixation points on an image - places people were recorded to have rested their eyes for some time, and I would think "they saw this", and it would be correctly assertible. But I would also know that some subjects would not have had the focus of their vision on some single fixation points that I'd studied, and instead would have formed a coherent image over multiple ones, in which case they would not have "seen" the area associated with the fixation point principally, they would've seen some synthesis of it and neighbouring (in space and time) areas associated with fixation points (and other eye movements). So did they see it or didn't they?
So I like correctly assertible because it connotes there being norms to truth-telling, rather than truth being something the world just rawdogs into sentences regardless of how they're made. "There are 20kg of dust total in my house's carpet"... the world has apparently decided whether that's true or false already, and I find that odd. Because it's like I'm gambling when I whip that sentence out.
I apply the same kind of thought to maths objects, though they're far easier to build fortresses around because you can formalise the buggers. I'm gambling a lot less.
Quoting Leontiskos
I would agree that every quantification is into a domain, and I don't think there are context independent utterances. I do not think it follows that there is no metaphysics. I'm rather fond of it in fact, but the perspective I take on it is more like modelling than spelling out the Truth of Being. I think of metaphysics as, roughly, a manner of producing narratives that has the same relation to nonfiction that writing fanfiction has to fiction. You say stuff to get a better understanding of how things work in the abstract. That might be by clarifying how mental states work, how social structures work, or doing weird concept engineering like Deleuze does. It could even include coming up with systems that relate lots of ideas together into coherent wholes! Which it does in practice obv.
I do also agree that there are no square circles in Euclidean geometries as the terms are usually understood.
I think this goes too far, you can do your best to interpret someone accurately and what they say can still be too restrictive or too expansive. Good shit testing requires accurate close reading. This is how you come up with genuine counterexamples.
I would have thought it clear how it relates to logical pluralism. If you model circles in Euclid's geometry, you don't see the great circle. But if you look for models of the statement "a collection of all coplanar points equidistant around a chosen point", you'll see great circles on balls (ie spheres, if you don't limit your entire geometry to the points on the sphere surface). They thus disagree on whether the great circles on balls are circles.
If you agree that both are adequate formalisations of circlehood in different circumstances, this is a clear case of logical pluralism.
Quoting Leontiskos
The taxicab example is designed as a counterexample to the circle definition "a collection of all coplanar points equidistant around a chosen point", since the points on the edge of the square in Euclidean space are equidistant in the taxicab metric on that Euclidean space. It isn't so much an equivocation as highlighting an inherent ambiguity in a definition. And mathematicians can, and do, call those taxicab squares circles when they need to.
You can side with the thing as stated, or refine it to mean "a collection of all coplanar points Euclidean equidistant around a chosen point". Which would still fall pray to the great circle on the hollow sphere considered as is own object, since the point they're equidistant about is no longer part of the space.
The point isn't to say that we don't know what a circle is - that's sophistical - the point is to show that there are mutually contradictory but fruitful understandings of what a circle is. Which is a pluralist point par excellence.
Even going by @Count Timothy von Icarus's excellent reference:
The extensional difference between all of these different formalisms are the scope of what counts as a circle. A pluralist could claim that some definitions work for some purposes but not others, a monist could not.
To put it in super blunt terms, Euclid's theory would have as a consequence that the great circle on a ball is not a circle. The equidistant coplanar criterion would prove that the great circle on a ball is a circle. Those are two different theories - consequence sets - of meaningful statements. A pluralist would get to go "wow, cool!" and choose whatever suits their purposes, a monist would not.
I agree more with the second quote I provided (albeit the "mind and language independent" part is not unproblematic), but it's worth noting that G&P allow for multiple true logics, what they argue for is one logical consequence relationship consistent with natural language, and the justification of the "one true logic" will be broadly epistemic. The "one true logic," is in a sense the "least true logic," that covers logical consequence.
The reason I thought of it though is because I think their focus on application is likely to be relevant across many forms of monism. Of course, there are a dazzling number of systems to consider, but I think the intuition is that "truth in this system" sometimes has a status akin to fiction. It doesn't have to do with how we get true inferences at all.
Quoting fdrake
Just pulling this for context. The OP is three years old. The recent discussion is not about the OP. After frank bumped the thread Banno brought in an external conversation, and pigeon-holed the discussion into one of those interminable, internecine Analytic disputes (Pluralism vs. Monism).
The external conversation revolves around this post from Srap:
Quoting Srap Tasmaner
This was Srap's attempt to frame it, but we went on to ask whether that framing was neutral or not.
I tried to continue the conversation in that thread, but Banno insisted on bringing it here. If Srap had continued the conversation in that thread I would have simply ignored Banno's transplant, given how insubstantial it was bound to become.
My position has never been logical monism's program of a single true formalization. That's just something Banno falsely pinned on me. For example:
Quoting Leontiskos
Do we need different accounts of logical consequence to have different geometries, etc.? Wouldn't pluralism be more something like: "we start with Euclid's postulates and end up with differing geometric propositions that can be deduced as true?"
No I didn't.
Quoting fdrake
See:
Quoting Leontiskos
Quoting fdrake
But it is here illustrative that I am not familiar with the concept "great circle," especially as to its specific geometrical properties, and I did query you about the picture you posted. You thought there was a verbatim sense of "great circle," but you were mistaken. You would have to explain what you mean by it in order to achieve your contradiction, because "great circle" says very little, verbatim.
Quoting fdrake
I think you're moving too fast. Formalisms have limits. What are the specific properties of lines, points, circles, great circles, two-dimensional planes, three-dimensional planes, etc.? How do they relate to each other? For example, can points be deleted or not? Is the great circle a torus, and if not is it three-dimensional at all? You're making a bunch of assumptions in all of this and drawing a fast conclusion.
But the deeper issue is that I don't see you driving anywhere. I don't particularly care whether the great circle is a Euclidean circle. If you have some property in your mind, some definition of "great circle" which excludes Euclidean circles, then your definition of a great circle excludes Euclidean circles. Who cares? Where is this getting us?
Quoting fdrake
Okay, thanks. And I agree with this. I am interested in knowledgeâincluding justificationâas opposed to just truth. Very often justified knowledge is precisely that which has been (correctly) logically inferred. I would define logic as that thing that gets you to (discursive) knowledge, or at least to justified assertion.
Quoting fdrake
And this sounds a lot like Srap's approach. I was encouraging him to write a new thread on the topic.
Plato's phrase, "carving nature at it's joints," seems appropriate here. I would say more but in this I would prefer a new or different thread (in the Kimhi thread I proposed resuscitating the QV/Sider thread if we didn't make a new one). I don't find the OP of this thread helpful as a context for these discussions touching on metaphysics.
Quoting fdrake
So:
Quoting Leontiskos
For the univocalist the two definitions are incommensurably different. For the analogical thinker there is an analogy between a great circle and a circle. I think both adhere to the definition, "A set of coplanar points equidistant around a single point," but this also involves analogical equivocity between 2D planes and 3D planes.
That also lines up just fine with my view of logic. If logical pluralism means there are incommensurably different logics which are true/correct, then I disagree. If it means there are analogically similar logics which are true/correct, then I agree. But I don't think that all true logics are isomorphic. "Incommensurably" is meant as strong incommensurability, in the sense of excluding analogical equivocity.
Quoting fdrake
Again, I think there is an equivocation on "distant." Equidistant qua circularity pertains to straight lines. The taxicab circle is premised on an extreme redefinition of "distance" - an equivocation.
Quoting fdrake
Although I don't hold to logical monism, this doesn't seem right. You are claiming that for the logical monist a token such as 'circle' can mean only one thing. I don't think that's right.
The Analytic dispute between logical pluralism and monism strikes me as a superficial dispute. The deeper question is univocal vs. analogical predication. That source abandons the more interesting question as soon as it limits itself to, a "model-theoretic definition." Pluralism looks like a poor man's analogicity, like trying to draw a perfect circle with pixels. My guess is that most versions of soft pluralism and monism are not even differentiable, unless there is some precise concept of "equally correct" logics or arguments (which I highly doubt).
Quoting fdrake
If they are different theories then they define different things, i.e. different "circles." The monist can have Euclidean circles and non-Euclidean circles. He is in no way forced to say that the token "circle" can be attached to only one concept.
Nice. Now we are getting to an interesting bit, that the difference is not about the nature of logic but about logical method.
Have a quick look at What is Logical Monism?. I suspect you would enjoy it, since it draws on the parallels with mathematics that you are using here.
Yes
Here I am using it, no? Its use-case is philosophical, rather than pragmatic, but I don't think that makes it meaningless.
Also I've changed over to the plain language version of the paradox to accommodate fears of formalism -- it's an example that arises from natural language use. What's so hard to comprehend about it?
To use 's division, this example is in (1). A child can understand the sentence.
How one answers the paradox is the interesting philosophical part, and also demonstrates the virtue of the analytic approach. The idea here is that we ought not poison the well because the implications of changing a logic are philosophically wide-reaching, at least with respect to some traditions of philosophy.
So it's not that metaphysics or knowledge are entirely ignored, but the hope is to find some implicating hint from an exposition of the conceptual map. The conceptual map doesn't represent battlelines as much as possible distinctions one can take up.
Quoting Leontiskos
Intelligible to whom?
I don't think it's so incomprehensible. I think it's very simple. "Duck is false" and "2+3+4+5 is false" don't work because "Duck" and "2+3+4+5" are not assertions at all, but nouns. Now if by "This is false" I indicated a duck perhaps I'd be using "...is false" in the place of "...is fake", but it wouldn't be the "...is false" which we use when talking about statements.
The pronoun in "This sentence is false" points to itself, which is a statement. And the statement utilizes a predicate normally reserved for statements, so there's no category error as you're implying. It's not nonsensical for this reason at least.
It may be nonsensical because it flies in the face of the principle of non-contradiction, or the principle of explosion. These are normal metrics for judging whether something is sensible or not -- the funny thing with this topic is that we can't rely upon those norms to decide the question since they are the things in question.
Do you agree that at least paraconsistent logic is significantly different enough from either Aristotelian or symbolic logic that one would count as a logical pluralist if they subscribed to the belief that both logics are valid or true in their own way or domains? That is the reason I brought up dialetheia and paraconsistent logic, after all: It seemed to be an obvious case of logical pluralism that is significant.
Cool, got it. Makes sense. One doesn't have to accept true contradictions to abandon the principle of explosion -- it could be that contradictions still always lead to falsity, but not explosion, or something like that.
I did enjoy it. It is also written in a very entertaining way. I would need to read it a few more times to follow the argument though.
I've seen that paper before. I give it credit for at least addressing the issue of metaphysical truth, but it is a prime example of implicit question begging re the deflation of truth. "Truth just is something to do with formalism, and how can you pick between formalisms? According to which one is true? Well, you have to use a formalism to discuss truth, and different formalisms say different things."
The background assumption throughout, and what the arguments routinely rely upon, is that truth is simply formalism.
But what's good for the goose is good for the gander. This is exactly the same charge leveled at pluralists by G&P. "Show pluralism is the case in your correct logics," or more strongly "show us it's the case in all of them. We think you'll find that quite impossible"
The response from Shapiro and others is, "well, the argument for pluralism is abductive." Fair enough (although G&P still point out that abduction involves deduction). But it's hard to think of a thing it is easier to make a strong abductive argument for then "things can be actually true, not just true as respects an arbitrary formalism." How do you choose between logics in this respect? The issue is epistemic, it cannot be handled by formal systems, at best they are an aid. And this is demonstrated that whenever the author wants to bring up a case of apparent conflict, they always resort to examples from formal systems, even when discussing the metaphysical view.
"The Goldbach Disjunction is a logical truth" and the like are simply ambiguous. They are claims about stipulated sign systems without reference to which system. I think the retreat into formalism covers up the obvious here. If bishops could move to the left in Pakistani chess, we could say the truth of "the bishop cannot change its color" is ambiguous and varies with context. Different systems, different logical truths. But the issue here is simply that the term "chess" is unclear.
This is not the case when we move to "all men are mortal," which isn't situated in a stipulated system. If we ask, "what does being mortal actually entail?" then "it depends," is hard to swallow as a good answer. So, the case the pluralism has to make in this respect is that there is no one intelligible pattern unifying the preservation of truth vis-a-vis this sort of (metaphysical) truth.
The study of form cannot tell us about things like "all men are mortal," but this doesn't mean that what constitutes a correct logic is unrelated to them since we care about "truth-preservation," not "truth-preservation relative to x formalism."
The Open Logic Project is a Wiki of sorts, designed to provide a free textbook on logic. It works thorough Naive set theory, propositional an predicate logic, model theory, computability, second-order logic, Lambda Calculus, many-valued logics, modal logic, intuitionistic logic and set theory.
At the end of the section on first-order logics is a short chapter named "Beyond First-order Logic". It ends with this admonition to creativity:
The commendation to the student is to be creative. This is a methodological pluralism.
Quoting Count Timothy von Icarus
Am I to read this as you saying truth is something to do with formalism, or as you saying that the flaw in the paper is that it considers truth only to be something to do with formalism?
Or do I just need more coffee?
I guess the obvious question is, if you know what truth is, apart from formal systems, then tell us. Otherwise, it seems to me that we could do far worse than Tarski's account of truth in terms of satisfaction.
And I am still not too sure what you mean by "deflation". Do you think Tarski's account is necessarily deflationary?
There are two questions with this pluralism/monism debate: What the heck is the thesis supposed to be, and Who has the burden of proof in addressing it? The answers seem to be, respectively, "Who knows?" and "The other guy!" :lol:
By rephrasing it in terms of the puzzle of the Meno and the possibility of discursive knowledge I sought to avoid such swamps, and I did that before this thread was necrobumped. The problem with this thread is that Banno and G. Russell want to say something controversial and novel and are therefore always moving between their motte and their bailey. The first question is to ask what the thesis is supposed to be, and what 'logic' means for the person proposing a thesis.
I have invented a logic in which there is no other guy and no one knows who they are.
My hypothesis is that there's a deep seated drive in most people to insist on logical monism. I think it's related to unity of consciousness: one self, one world, one logic. I think pluralists are using the term differently.
On deflationary accounts, âall that can be significantly said about truth is exhausted by an account of the role of the expression âtrueâ... in our [speech] or thought,â and we might add formal systems here. Thus, notions of truth are neither âmetaphysically substantive nor explanatory.â
This is clearly going to be a problematic background assumption to have going into an analysis of a metaphysical case for a single entailment relation applicable to being.
Ah, but this is perhaps the cardinal sin of contemporary philosophy! "X is difficult to define or account for, let's eliminate it." We've seen this done with Goodness, Beauty, Truth, meaning, and finally, in eliminitivism, our own consciousnesses. What philosophy worth doing shall be left?
Not to mention, consider this same question on other finicky definitions, such as "life." We might very well run with some sort of formal definition for expediency on some issues, but it clearly won't do to for others. A bad definition can be worse than an ambiguous one.
Now I get, the metaphysical and scientific sections are just two parts of the article. It's too much to expect a deep dive into different theories. But just consider a very influential one, Aristotle. For Aristotle, "being" is said many ways, but it is said most primarily of substances. Mathematical entities aren't substances. They don't exist simplicitier, but with qualification. So obviously the arguments in those sections that use "exists" univocally throughout are problematic, particularly since this is hardly unique to Aristotle, but common, I would guess, to most thinkers.
It doesn't seem that different from looking at contradictory stories told about superheros, saying both "exist" and declaring an exception to LNC. This is missed if one supposes that we're talking about a blanket prohibition on "a and not-a" as opposed to a prohibition on something actually being and not-being, without qualification.
As a side note, while I know the example of different mathematical objects is intuitive, but I am not sure if a lot of these even require different entailment relations.
This goes back to the discussion with Tom:
Quoting Tom Storm
Monism, and authoritarianism, offer certainty.
Which means it can't be defeated.
So what's the problem? It's not as if deflationary accounts say that there are not truths.
In Model theory truth isn't eliminated, but given a firm grounding in satisfaction.
Issues of "being" are not ignored by formal logic, either, but explicated by quantification, predication and equivalence.
If I am candid, it seems to me that your fears are ill conceived and unfounded.
The saying is "Be open minded, but not so open minded your brain rolls out."
And thus the moralistic undercurrents driving this silliness have finally become fully explicit. It's hard to put so much effort into defending an undefined thesis without this sort of moralistic self-righteousness. But of course it was there all along.
:grin:
Generalized. Return to the basic principle things ought to make sense. How that is accomplished may vary.
So I'm to blame for and @Tom Storm's questions. Fine.
From the SEP article...
That's were I came across the Clarke-Doane article and the discussion of approaching the issue as one of attitude.
But you are right, that things would be a lot simpler if we were just to go back to Aristotle.
Yep. And if there are more than one set of basic principles, then we have one form of pluralism.
And if a set of basic principles is found, then the challenge is set to see what happens if we change them, try different basic principles, or diagonalise in some way... to look at logic differently and undermine it to see what happens.
Put another way, how could we ever be sure that some set of basic principles is sufficient for all of logic?
The story at least since Russell's paradox and GĂśdel seems to indicate that this is not what happens.
Isn't it though? What did they both do but modify their systems. Russell decided you can't have self-referential sets and Godel concluded that no system really has a foundation. And both did it based on the generalized principle that things should make sense. I wasn't being dismissive, if you want a one stop shop for logic that's it. Things ought be sequitur when explained.
If that fits in catagory A or catagory B, I'm not asserting. So, if we need to translate it due the massive hurry philosophers are always in call it
For Every X is some Y
if you want to make it a party
For Every X is some Y or not some Y.
And if we can't agree on that, then what's the point of breaking it down further. All knowledge is likely probabilistic and referential and yet facts exist. Why? Some Y. Or not.
Not really pluralistic. Discovery of the undeniable rejection of monism would be one. If pluralism entails the monism of pluralism then logic has to be pluralistic and essentially monistic in that fact. The error is thinking they're two things.
First , I didn't say formal logic ignores being I said the arguments in the paper use "exist" univocally in a way that makes them facile.
Second, there seems to be a pretty strong abductive argument for "there are many cases where truth does not depend on how we choose speak."
One of the benefits of STT is that is based on notions of correspondsnce truth, and it is certainly often used it with the idea in mind that there is a "real truth." However, stripped down to mere form and taken alone as the final word on the issue it is relativistic. IIRC, Tarski claims truth is "meaningless" outside formalism. If we accept this, not as a useful tool, but as a claim about truth tout court, what exactly makes STT a better theory of truth than any other? Can it be truly better? True relative to what, itself? If we say its more useful, we might ask "is it truly more useful? Truly more useful relative to what? Why not any other theory that might justify itself?
Well, that makes sense if you read the post as "I don't think logic has existential quantifiers."
True relative to something else some one could assert. It's an approximation with an arrow toward truth.
Well, it's right. "P" is true iff P is about as direct as you can get.
That wasn't reframing. We were talking about why a monist might insist on a logic for all cases when it's not clear what that logic would be.
I am considering making a new thread on a related topic, but I am wondering what you actually mean by "shit testing"? Originally I thought you meant something like, "Throwing all the shit you can think of at a wall and seeing if anything sticks. Submitting an idea to a shitstorm of objections and seeing if it is still standing in the end." Yet now as you refine the idea we seem to be getting further and further from that idea, even to the point that I am wondering whether "shit testing" is an appropriate name.
(I suppose you might have meant, "Testing an idea to see if it is shit," except that that is much too far away from the quibbling that I complained of.)
That's my issue with the monistic approach. There's only one correct way to think about it and no one seems to know what that is exactly.
It's faith.
, I was responding honestly to questions asked.
So now the thread is about me? Nice.
Quoting Cheshire
Good summation.
I meant it as two complementary aspects - treating a definition exactly at its word to see what it entails. Sometimes this will entail something that seems very pathological. Eg here's an example of a curve which is discontinuous but you could draw without lifting your pen off a piece of paper or instantaneously changing the angle you're drawing at. Shit testing allows you to distinguish concepts, in the case of that curve, it provides an example that distinguishes continuity from the intermediate value property, by finding a curve which is not continuous but has the intermediate value property.
Since counterexamples like that let you distinguish concepts engendered by formalisations, they also let you try to distinguish what concept a collection of definitions are trying to capture from what concept they actually capture.
Philosophy has analogues, like Gettier cases exemplify shit testing of the justified true belief theory of knowledge. The concept "a rock a being cannot lift" is an attempted counterexample to an unrestricted concept of omnipotence. Lord of the Rings might serve as a counterexample to a strictly coherentist view of truth, since it may satisfy the definition of a self consistent and expansive set of propositions which nevertheless is not the one we live in. There is no Walmart in Middle Earth.
What I was calling shit testing is the process of finding good counterexamples. And a good counterexample derives from a thorough understanding of a theory. It can sharpen your understanding of a theory by demarcating its content - like the great circle counterexample serves to distinguish Euclid's theory of circles from generic circles. Counterexamples of this form have a modus tollens impact on the equivalence of a target concept from concepts in terms of a theory targeted at that concept understood at face value in its stated terms.
I don't think the sphere cross section's circumference is a "good" counterexample like that, since the thing cutting the sphere to make a cross section definitely is a plane, some Euclid fan will be able to talk about "enclosing space" like the disk the cross section whose boundary is the great circle is is, or the fact the circle lays in a plane, but just an incline one. But the circles you make on the surface of a sphere alone are a good counter example in that sense, because there's no centre point and no enclosed space.
I switched counterexamples mid explanation because it became evident you weren't familiar with the difference in geometry between sphere surfaces and planes, in virtue of reading the great circle as the boundary of a cross section of the sphere. And also weren't comfortable playing around with weird subsets of the plane. Those latter examples were attempts to make similar flavour counterexamples without the... nuclear levels of maths... that help you distinguish the surface of a sphere from flat space.
The incline plane does let you see something important though, you might need to supplement Euclid's theory with something that tells you whether the object you're on is a plane. Which is similar to something from Russell's paper... "For all bivalent...", vs "For any geometry which can be reduced to a plane somehow without distortion...". The incline plane can be reduced to a flat plane without distortion, the surface of the sphere can't - so I chose the incline plane as another counterexample since it would have had the same endpoint. But you get at it through "repairs" rather than marking the "exterior" of the concept of Euclid's circles. Understanding from within rather than without.
Someone who was familiar with the weirdness of sphere surfaces, eg @Srap Tasmaner, will have seen the highlighted great circle, said something like "goddamnit, yeah", and understood that the intention of presenting the image in the context of your reference to Euclid was to reference only the circle on its surface, since they will have had the understanding that the surface of a sphere has nothing like a working concept of a "planar figure" applicable to it at all.
Ironically enough this is similar to one of Lakatos' quips in Proofs and Refutations. I can't remember the exact wording, but he pokes fun at mathematicians for the amount of assumed knowledge supposedly self contained and fully rigorous proofs they write have. Which is also unavoidable when building on top of theories.
That isn't strictly speaking true, it's just that the generalisation of the concept of planar figure which applies to circles is so vast it doesn't resemble Euclid's one at all. You can associate planes with infinitely small regions of the sphere - the tangent plane just touching the sphere surface at a point. And your proofs about sphere properties can include vanishingly small planar figures so long as they're confined to the same vanishingly small region around a point.
Edit: or alternatively I guess you could think of shapes on a sphere's surface, but they have much different properties than those on the plane. Like triangle angles adding up to more than 180, the analogue of lines being great circles, and thus there's no parallel lines on the sphere surface.
This is simply using unclear terms. It's "P is true in L iff P is true in L." Whereas "P is true it and only if P," would simply be meaningless or ambiguous.
It's a sort of relativism. Perhaps not a pernicious sort in its original context, where the idea was to model correspondence, but the very paper we're discussing turns it into a cultural relativism of "communities."
Shapiro's eclectic pluralism says a logic is correct so long as it is useful for any "interesting" application. Trivial systems are interesting though. I assume the bar for "interesting" must be tightened up somewhat so it isn't the case that "correct logics," that is "logics that preserve-truth," are inclusive of those that show that anything expressible is true.
"P" is true IFF P is a formulation of redundancy among other things. It would be cool if @Nagase stopped by, for a number of reasons.
You're telling me I don't have to keep consulting my truth tables for statements like "P"? :rofl:
I don't think it's redundant in the context of trying to model correspondence though, since it's saying "the sentence P is true if what P claims is actually true." The claim and what makes the claim true are (often) distinct. But perhaps we should instead say something like: "S(P) iff P" However, it seems problematic for correspondence truth if logical nihilism is the case and there is no logical consequence relationship, such that P cannot entail S(P).
Of course, the history of philosophy is full of challenges to the correspondence formulation as well.
It doesn't model correspondence theory. For Tarski, it was a way of handling the truth predicate in formal languages. Maybe he would have wished he could resurrect correspondence, but he knew he hadn't.
What makes you say that?
I kind of thought of Tarski's paper, that I still struggle with reading, was basically a correspondence theory of truth?
Either way, what I'm hoping to convey is that logical theories like Russell's are attempting to accommodate any metaphysics of truth -- else it would be begging the question on truth.
I'm basing that on what Scott Soames and Susan Haack said about it. Tarski's truth predicate doesn't even mean truth in the common sense. It's more like satisfaction.
Quoting Moliere
I'm not sure, but it leads me to this question: Frege's account of the indefinability if truth is a logical brick house. Why couldn't a pluralist say, "that's not helping me, I think it would be more interesting to create a logic that eliminates Frege's concerns."
AP would have gone in an entirely different track, possibly into a ditch. How does that work?
That's how it's generally been interpreted and how it was originally presented, but yes, I agree, it need not be interpreted that way and often isn't.
Well there I wholeheartedly agree. However, the thesis that there is no truth preserving logical consequence is necessarily going to be at odds with many conceptions of truth. What is coherence truth of nothing follows from anything else?
The difficulty here is that the strongest arguments for nihilism, or at least the most popular, implicitly deflate truth.
Well, if we follow the evidence it suggest that self-reference isn't a reliable source of truth, in the sense the system breaks down per Russell and Godel. So, Popper's principle that we can know the truth about things, but not when in a technical sense has always seemed reasonable to me. It preserves truth and seems to model the evidence available.
Apparently the controversy stems from some comments from Popper. The fact that this is not the prevailing interpretation is reflected in two articles in the SEP about Tarski and his definition.
Notice that they don't use "correspondence" to describe his definition, but focus on logical consequences and satisfaction.
If you have university access you can read Susan Haack's article, which lays out explicitly how we know Tarski did not see himself as offering any definition for truth in natural languages. Just Google Haack on Tarski.
I've always wondered if Russell's paradox is coming from the foundations of set theory: the contradiction of fencing in infinity. Maybe when I land on a deserted island all by myself I'll sit and figure it out. :razz:
:up:
Yeah, as I mentioned, I recall reading somewhere where he says truth in natural language was "meaningless," but I wasn't sure if this was a later position. So this would make sense to me.
So, STT is originally/intended to be deflationary I guess, which jives with how it is often used.
I don't think he meant meaningless, but definitely indefinable: too basic to define.
Quoting Count Timothy von Icarus
In his paper he basically says that the concept of truth had disappeared from math. He felt like it could be brought back in some form, and he is ground zero for renewed interest in truth. It's just not correspondence, because that concept resists clarification sufficient for math and logic.
Deflation can be truth skepticism, which is what redundancy is. @Nagase explained once that some use the T-sentence rule without being skeptics, emphasizing that indefinable isn't the same as meaningless.
You're saying it's like a bubble universe?
It's always true or false or maybe otherwise relative to some context. Thinking you can establish truth without a point on the map seems like the radical approach. So, nihilism is atheism. A label given to people for being correct.
We seem to think about mathematics very differently. You think that a point can be deleted; that a set of coplanar points might not lie on a plane, etc. Those strike me as the more crucial disagreements. Whether something can be "reduced to" a Euclidean plane or "contains" a Euclidean plane seems less crucial and more arbitrary.
At the heart of this thread seems to be the question of whether we can actually say that someone is wrong. In mathematics the point becomes protracted. For example, you might say that I am wrong about the great circle only if I am determined to bind myself to purely Euclidean constraints. Your notion of "correctly assertible" seems to be something like a subjective consistency condition, in the sense that it only examines whether someone is subjectively consistent with their own views and intentions. For example, given that someone says something contradictory, on this theory one can only say that they are wrong and disagree if there is good reason to believe that the person accepts the PNC. If there is no good reason to believe that the person accepts the PNC, then one cannot call them wrong or disagree. The logical monist, among others, will say that someone can be wrong for contradicting themselves even if they don't subjectively claim to accept the PNC.
As I have noted many times, whether the great circle is a circle seems to be a mere matter of names, or stipulated definitions. Not so with the PNC. We can't just change a name and resolve that conflict.
A paper that I often return to in this regard is Kevin Flannery's, "Anscombe and Aristotle on Corrupt Minds," although this paper is about practical reason, not speculative reason.
Quoting fdrake
Okay, but I still don't understand why you are calling this "shit testing." Why does it have that name? It sounds like you want to give counterexamples that highlight subjective inconsistencies. Fine, but why is it called "shit testing?"
If you are just trying to give good counterexamples, then my critique of Cartesianism does not hold, but in that case I have no idea why it would be called "shit testing."
(The other possibility here is that someone's counterexample is more method than argument. For example the ancient Skeptics would argue with everyone who made a strong claim in order to try to demonstrate that strong claims cannot ultimately be made. That is apparently part of what is going on here, for the great circle has no direct bearing on square circles, but if one can generate a strong enough skepticism about circles then all claims about circles become mush, including claims about square circles.)
Well, logical nihilism is not the position that true and false are always relative, it's the position that nothing follows from anything else. It is certainly easier to argue for it if truth is relative, but it's the claim that truth cannot be inferred. You could presumably claim that there are absolute truths, just not that there is anyway to go from one truth to another.
In terms of a puzzle analogy, this seems more like claiming the pieces don't fit together, in which case it doesn't even seem like a puzzle any more.
So you use phrases like that in conversation?
Quoting Moliere
Bollocks. It is absurd to claim that such a sentence pertains to, "everyday language use and reasoning," or that a child could understand it.
Quoting Moliere
Well, 2+3+4+5 doesn't seem to be a noun, but okay.
Quoting Moliere
You haven't managed to address the argument. Let's set it out again:
Now here's what you have to do to address the argument. You have to argue against one of the premises or the inference. So pick one and have a go.
-
Note too that, "This sentence is false," is different from, "This sentence is false is false," or more clearly, " 'This sentence is false' is false. " Be clear on what you are trying to say, if you really think you are saying something intelligible at all. Be clear about what you think is false.
---
Edit:
Quoting Moliere
Or if you like, why is it false, whatever "it" is supposed to be? How do we know that it is false? Is it because you said so? But you saying so does not make a thing false, so that's a dead end. Even Wittgenstein understood that a sentence cannot prove or show its own truth or falsity.
It is as interesting to say, "2+2=4 is false." Have we thus proved Dialetheism? That 2+2=4 is both true and false? Of course not. :roll:
In both cases the only takeaway is that the speaker is confused.
I don't know what to tell you other than you learn that stuff in final year highschool or first year university maths. If you're not willing to take that you can do those things for granted I don't know if we're even talking about maths.
A set of coplanar points could have a plane drawn through them if you had the ability to form a set in that space which was a plane... and contained them. So they wouldn't even be coplanar if you couldn't draw the plane, no? Like how would coplanarity even work if you've just got three points {1,2,3}, {4,5,6} and {7,8,9} embedded in no space.
Maybe we're talking about Leontiskos-maths, a new system. How does this one work? :P
Quoting Leontiskos
Of course you can. If someone tells you that modus ponens doesn't work in propositional logic, they're wrong.
Quoting Leontiskos
More normative. It's not correct to assert that modus ponens fails in propositional logic because how propositional logic works has been established. And modus ponens works in it.
Quoting Leontiskos
I used it as a joke and then ran with it. And they aren't subjective inconsistencies, they're norms of comprehension, and intimately tied up with what it means to correctly understand those objects.
Counterexamples that I've been giving don't just refute stuff, they mark sites for theoretical innovation and clarification.
Shit-testing? I think you're just pulling shit out of your ass out of desperation at this point. You're a few inches away from Amadeus', "I'm right because I'm right, and you're wrong because I said so!" ...Which is ironic given that you meant to demonstrate that being right about math is not as easy as one supposes. Have you succeeded, then?
I've had plenty of university math. You strike me as someone who is so sunk in axiomatic stipulations that you can no longer tell left from right, and when you realize that you've left yourself no rational recourse, you resort to mockery in lieu of argument.
Quoting fdrake
Maybe "propositional logic" is as slippery as "circle."
Quoting fdrake
"Established"? A bit like, "verbatim"? All you mean is, "If you mean what I mean then you will conclude what I have concluded." You vacillate on the question of whether one should or does mean what you mean, and that's a pretty serious problem. It seems like you haven't thought about these issues as much as you thought you had.
Quoting fdrake
So are there rational norms or aren't there? What does it mean to "correctly understand a stipulated object"? One minute you're all about sublanguages and quantification requiring formal contexts, and the next minute you are strongly implying that there is some reason to reject some sublanguages and accept others. I suggest ironing that out.
Quoting fdrake
The problem is that if you hold that mathematics has no unconditional or "unquantified" relevance, then you can't give a top-level mathematical critique. You say the point at the center of a circle can be "deleted" and I say it can't, but you presuppose that there is no way of adjudicating this question. You want to be right while also holding that there is no right or wrong in such things. Hence the bluster.
Pretty sure that's just a conclusion some would assert about it. Saying there's no general rule that universally ties evidence to truth is a bit different than, no logic. And I disagree, if I'm arguing there are multiple routes to a true conclusion then I'm discussing a relativistic system. If I'm just wrong by definition then it's business as usual I suppose, but those sound like secondary assumptions.
Quoting fdrake
Quoting fdrake
is denying mathematician the right to write [math]\mathbb{R}^2/\{0\}[/math], and hence deny them the right to think about [math]\mathbb{R}^2/\{0\}[/math]. I think this an interesting case study in what we have been discussing. Monism would have it that "you can't think that".
Quoting Count Timothy von Icarus
What if there were several puzzles mixed up? Then sometimes, some pieces would not fit together, being from different puzzles. But that does nto make the puzzles unsolvable. (Nice analogy,
Quoting Leontiskos
Of course there is no way of adjudicating this question. Removing the centre point is a stipulation, of the sort that mathematicians and logicians do as a matter of course. "What happens if we consider [math]\mathbb{R}^2/\{0\}[/math]? Well, then we have a whole, cool new puzzled to play with..."
I'm saying that one can understand a language without being committed to whether it is a "correct language", and be able to say whether a given statement in it is correct or incorrect. Because the norms of the sublanguage are fixed. Like all the statements in propositional logic are bivalent, the LEM holds etc.
Where this breaks down is the intuition that propositional logic "ought" apply to all meaningful sentences. Hence the Liar and indeterminate truth values now serving as "counterexamples" in this context. They can be understood as counterexamples when one expects propositional logic to work for all meaningful sentences. This was analogised with our circle discussion.
We were talking about circles as a concept, and they have associated formalisms, we've now seen that there are different formalisms for it in different contexts, and sometimes they disagree. How can you insist that one is more correct than another? Which one is baked in the metaphysics? I don't really need you to know the final answer on it, I just want to know how you'd go about deciding it even in principle.
Quoting Leontiskos
Alright. It just surprises me that you survived all of these different things to do with maths concepts with a strong intuition remaining that there's ultimately one right way of doing things in maths and in logic, and that understanding is baked right into the true metaphysics of the world. And also seem to align this understanding with Aristotle?
Quoting Leontiskos
Neither of them is particularly slippery. The slippery thing is a pretheoretical conception of logic, or circles, which might be better exemplified in some ways by some theories and in other ways by others. There's wide agreement on what the theorems are in propositional logic, how it's used etc. I don't believe it makes sense to say something is slippery when the norms of its use are so well enshrined that it's taught to people the world over.
Neither of us disagree on what Euclidean, taxicab or great circles are at this point, I think. So they're not "slippery", their norms of use are well understood. The thing which is not understood is how they relate to the, well I suppose your, intuition of a circle. I seem to have a spectrum of intuitions about circles that apply in different contexts. Maybe you don't?
I am getting the impression that you have quite an all or nothing perspective on this - either there is a single unified objective system or there is a sea of unrestrained relativism and mere subjectivity over what theorems are provable in what circumstances. I would suggest that people can agree on what theorems are provable in what circumstances without an opinion about whether they're the "right" theorems. It seems to be knowing what theorems something should satisfy and having the right formalism to prove them are inextricably related in mathematical creativity and reasoning - eg:
Which brings us onto understanding a stipulated object.
Quoting Leontiskos
I would say that someone correctly understands a mathematical object when they can tell you roughly what theorems it should satisfy, give some examples of it, and has ideas about proof sketches for theorems about it. That means they know how it behaves and what contexts it dwells in. They know how it ought to be written down and how to write it. They know how what they imagine is captured by how they write it down, and that what's written down captures all it should capture about the object.
That's also quite contextually demarcated - eg I would say I understand differentiable bijections in terms of real analysis objects but my understanding of their role in differential geometry is much much worse, despite their major role in the latter context.
There's a bit of graph theory I work on in my spare time, regarding random fields on graphs with an associated collection of quotient graphs, and I have an idea of what I want that contraption to do, but I've yet to find a good formalism for it. Every time I've come up with one it ends up either proving something which is insane, and I reject it, or I realise that the formalism doesn't have enough in it to prove what I need to. Occasionally I've had the misfortunate of making assumptions so silly I can prove a contradiction, then have to go back to almost square one. I wouldn't say I understand the object well yet, nor what theorems it needs to satisfy, but I have a series of mental images and operations which I'm trying to be able to capture with a formalism. I would call this object "slippery", but that's because I haven't put it in a cage of the right shape yet. Because I don't have the words or the insight yet. Perhaps I never will!
Terence Tao has a blogpost on stages of mathematical comprehension in a domain of competence, if you're interested I can dig it up.
I also don't want to say that all objects are "merely" stipulated, like a differential equation has a physical interpretation, so some objects seem to have a privileged flavour of relation to how things are, even if there's no unique way of writing that down and generating predictions. I had an old thread on that, which was not engaged with due to poor writing and technical detail, called "Quantitative Skepticism and Mixtures". It's just a recipe for making largely useless models that produce the same predictions as useful ones, but have pathological properties. And the empirics aren't going to distinguish them if you choose the numbers right.
A final comment I have is that we should probably talk about the development of formalism also changing what counts as a pretheoretical intuition - cf the way of reading general relativity that undermines Kant's transcendental aesthetic, since noneuclidean geometries aren't just intelligible, they're baked into the reality of things. Also people who overdose on topology come out changed.
This seems like a useful clarification of terms. Where I have seen the term used, and how it is used in the papers we have been discussing, the idea is that there is no logical consequence relationship. It is not that there is no general consequence relationship that obtains in all cases. The idea that there are truth-preserving rules of logical consequence but that they might vary is called logical pluralism.
This is why deflationism is question begging. You can set up the argument like so:
1. Truth is defined relative to different formalisms.
2. Different formalisms each delete some supposed "laws of logic," such that there are no laws that hold across all formalisms.
3. The aforementioned formalisms each have their own definition of truth and their systems preserve their version of truth.
C: There are no laws vis-ĂĄ-vis inference from true premises to true conclusions.
A deflationary pluralist could well say this equivocates between "truth tout court" (which doesn't exist) and qualified truth relative to some system, and that the nihilist is just a deflationary pluralist with an edgy name.
The non-deflationist of any variety can say the entire argument hinges on the premise of deflation and that we are only speaking of "correct logics," which preserve truth qua truth, not a stipulated truth condition that is defined arbitrarily.
Sounds like pluralism. You need to find the structure of each discrete puzzle.
Nihilism seems more to me like we all have wood blocks and jigsaws and we can cut out whatever we please. Which, as an analogy for "how does one derive conclusions from true premises," seems like a poor one if one has any notion that truth is not some sort of post-modern "creative act."
I don't follow this, and I don't think it is only becasue you appear to have left out a few quote marks. So let's make it clearer.
"P is true in L iff P is true in L" is a simple tautology, and nothing like the sort fo thing Tarski used. The sort of thing he would have said is more like "'P' is true in L iff S" where S is a sentence in a language other than L, carefully defined so that the S is satisfied only when P is satisfied. That's what that long bit in Tarski's paper that no one reads does - it matches the names in the object language with new names in the meta language.
Quoting oneself is becoming de rigueur...
Quoting Banno
There's no "unclear terms" here - indeed, it is clear to the point of being pernickety. Hence the improt of the paper.
I believe that Tarski did not say that truth was nonsense in natural languages, but that it was indefinable. That would be a natural consequence of his theorem that a language cannot contain it's own definition of truth.
Kripke subsequently showed that a language can contain it's own definition of truth, provided one makes use of paraconsistent logic.
So with Tarski we have truth in layers of language, each one talking about the one below it. This is, speaking roughly, what is used in the iterative conception of set theory.
Speaking generally, on the one hand we have clean and clear definitions of truth within formal systems and in terms of satisfaction, and on the other hand we have a broad, ill-defined notion of truth that is supposed to be useful in adjudicating between differing logics as well as in natural languages.
Is that conclusion supposed to follow? That there are no universal laws does not deny that there are laws specific to each logic.
It is maybe worth pointing out that if someone proposes a new logic, they are obliged to set it out for us to see it, and we can judge it's consistency within itself, as well as its applicability to various situations in comparison to other logics.
Suppose that the liar's sentence is false. Then the liar's sentence is true because it says that it is false.
Suppose that the liar's sentence is true. Then the sentence is false because it says that it's false and we're saying it is true.
In either case you end up with the circuit of evaluation which yields both "...is true" and "...is false" regardless of its starting truth value.
Though I can see you're not having it.
Do you at least agree that paraconsistent logic is different enough to count as pluralism?
Quoting Leontiskos
I'll start with your first premise. "...is false" presupposes no such thing as an assertion or claim -- like I noted earlier "This duck is false" could mean "This duck is fake", right?
So it follows that the meaning of a clause depends upon the name and the predicate -- "...is false", outside of everyday, has no meaning.
I agree that "This sentence is false" differs from "This sentence is false is false" -- I think once we introduce substitution we're no longer in everyday reasoning, but it works at any level from what I can tell.
"This sentence is false" is all I need. It's a nefarious sentence. Or a purposefully chosen set that play with the notion of true and false and self-reference.
Also, even if we introduce subsitution the liar's works -- it's the extended liar's sentence. (the "strengthened" liar's sentence is what convinced me that it cannot be assigned some third value, as in many-valued logics)
Actually that's another example that I'm wondering about with respect to pluralism -- do logics with more than 2 values count as plural logics, or no?
***
Also I can just drop this point here. We're starting to getting into liar's paradox points and if it's something that doesn't really jive with you then there's no point in continuing here since the point isn't the liar's sentence but pluralism.
Thank you!
Pretty much. Even including infinite-valued logics.
The liar is clear, in the way you have argued. Rejecting it as a "nonsense" is a failing of nerve, rather than an act of rationality. There are three ways of dealing with it that I think worth considering. Tarski would say that it is a mistake to assign truth values to sentences within the same language, but permissible between languages, so the problem with the liar is that it tries to say something about the falsity of a sentence within it's own language. Kripke would say that we can assign truth values within one language, but that we shouldn't assign them to every sentence, the liar being an example of a sentence to which we cannot assign a truth value. Revision theories would have us say "this sentence is true" is true on the first iteration, false and the second, true on the third... and so on.
Here we have three examples of how accepting and facing the liar enables the development of new and interesting approaches, of creativity. Whereas simply rejecting it as a nonsense closes of such play.
Perhaps that's a nice example of the methodological difference between pluralism and monism. I don't actually think this is quite right, but at the least it shows a difference in approach.
Interesting. I like this approach of defining the difference as a matter of method.
Quoting Banno
I notice a distinct lack of dialetheism in your approach ;)
The way I understand Tarski's attempt to deal with it is the distinction between meta- and object- language. I think that's the neatest way to deal with it, but upon reading Priest I've reconsidered.
I'm not sure I understand the difference between Tarski and Kripke, though. By your sentences they look the same to me, so I'm missing something.
Revision theories sound like they can't make a decision. Not that I'd know anything about that ;D
Try this:
Quoting Banno
So you are quite right that they both use the notion that if "Adam is English" is true, then so is '"Adam is English" is true'. But whereas Tarski uses layered languages, Kripke gives a methodical way to assign truths and avoid liars in the same language.
Yes, that's the pluralist response. Like I said, I think they can accuse the nihilist of equivocating here to the extent that their argument relies on assuming deflation. But nihilism ultimately [I]has[/I] to be about a broader notion of truth preservation across all correct logics, else it is demonstrably false. LNC holds "generally" if we only look within one context for very many contexts, etc.
Hence my example, statements like "propositions must be either true or false" are ambiguous in a deflationary context. The answer is: "it depends, LEM and bivalance aren't universal." It's like saying "marijuana is legal," without specifying a jurisdiction, and then equivocating on the relevant context.
I don't know how to respond to the rest of what I wrote because you keep on responding to things that obviously are not what I'm saying, e.g. "This paper uses 'exists' univocally" for "I don't think logic has existential quantifiers."
I point out that STT allows for relativity in the context of discussing a paper that is almost entirely using examples of such relativity and you suppose that I am confused and referring to the level where it isn't relative.
Suffice to say, STT can be interpreted in a deflationary manner and was developed with that in mind. If the point in question the existence of a general logical consequence relationship applicable to truth preservation vis-ĂĄ-vis science or to metaphysical truth it is question begging to assume deflation.
Either all logics are correct logics, in which case nihilism is "true" but truth becomes essentially meaningless or there are just [I]some[/I] correct logics. Since many people are not willing to embrace the former (full deflation, truth is arbitrary) they need some criteria for deciding which logics are correct. So, we are back to ambiguous definitions anyhow, we've just obfuscated this fact.
Notice the bit that says
Now what do you make of this? I've understood you as saying Tarski is unavoidably deflationary, and that this is a bad thing.
For my part, talking off the top of my head, I agree with it, and add that deflation is pretty much the only description of truth generally, inflationary accounts only be of use in somewhat special cases.
THis by way of looking for common ground.
I think I've been pretty clear that I don't think one is more than correct than another, at least in the face of a skepticism or a univocity like your own. For instance:
Quoting Leontiskos
In common usage there are no square circles, but if we redefine either one then there could be. I've said this many times now.
-
Quoting fdrake
I don't know where you're getting these ideas. This started with an offhand comment to frank about "square circles lurking just around the corner," and then you launched into an extended argument in favor of square circles. Early on I asked about your motivations, and you said something in favor of "shit-testing" and then tried to repair that idea in favor of "counterexamples based on accurate close reading." But it is not coincidental that shit-testing is something like the opposite of close reading, and that your posts haven't engaged in much close reading at all.
I mean, what would a university math professor think if they saw someone arguing that they can delete the point in the center of a circle and make it a non-circle? I think they would call it sophistry. They might say something like, "Technically one can redefine the set of points in the domain under consideration, but doing this in an ad hoc manner to try to score points in an argument is really just sophistry, not mathematics."
Quoting fdrake
It is petitio principii to simply insist that, say, an inclined plane is not reducible to a Euclidean plane qua circles. You haven't offered anything more than arguments from your own authority for such premises. Beyond that, I see misreading, not close reading. I have said things like this many times:
Quoting Leontiskos
-
Quoting fdrake
But how do you know that when I talk about a circle I am restricting myself to a very strictly interpreted Euclidean conception, such that an inclined plane is not reducible to a Euclidean plane? You are the one who is insisting that there is a right answer to questions like these, not me.
Quoting fdrake
But it's odd to talk about an "object" here. As you go on to say, you don't even know if the "object" exists. You're just attempting to solve a problem or create a model.
Quoting fdrake
J's new thread seems on point.
The interesting question I see here is something like, "Why should we disagree?" What is a sufficient reason to disagree with someone? You seem to have fallen into the odd trap of claiming that mathematics is all arbitrary and that I have nevertheless committed some grievous sin by supposing that an inclined plane can be reduced to a Euclidean plane. If all mathematics is arbitrary, then there are no grievous sins. There is just ignorance of stipulations (such as the "great circle"). So then perhaps I am ignorant of the precise properties of a commonly-known stipulation in the math world (i.e. a "great circle"). But is that really a problem? Does someone really need to have a Masters in mathematics and understand the stipulated metaproperties of great circles in order to claim that there are no square circles lurking around the corner? I really doubt it.
Granted, I realize you think some mathematical constructs are more applicable than others, but I won't press you on that unless you somehow think that it bears on this question of the great circle.
I'm not having it because you keep begging the question. You say there is a sentence/claim but you won't say what the sentence is.
It's not much different to say, "Suppose there is a sentence that is true and false. Therefore the PNC fails."
Or else, "Suppose there is a sentence that is true if it is false and false if it is true. Therefore the PNC fails." But that's not an argument. It's, "Suppose the PNC fails; therefore the PNC fails." In order to make an argument you would actually have to identify such a sentence, and I have already pointed out the problems with the "Liar's sentence."
-
Quoting Moliere
"If false doesn't mean 'false', but instead means 'fake', then
Do you see how silly this is? You redefined falsity as something other than falsity in order to try to make a substantive point about falsity. Do you see why I feel that I am wasting my time? These are the sort of moves that so-called "Dialetheists" routinely engage in, at least on TPF.
Quoting Leontiskos
Quoting Moliere
So what do you think is false?
There's not much point continuing this if you feel like it's the same thing over and over.
Can you give me a lot more words on the phrase "an incline plane is reducible to a Euclidean plane qua circles"? I'd really like to understand the predicate:
X is reducible to Y qua Z
Quoting fdrake
So suppose we are talking about the cross-section of a sphere, which is what I originally thought you were pointing at. Is that something like a circumscribed inclined plane? It is certainly a set of coplanar points. Now you say, "The incline plane can be reduced to a flat plane without distortion." This captures what I said by, "an inclined plane is [...] reducible to a Euclidean plane." "Qua circles," meant to indicate the idea that an inclined cross-section of a sphere could be reduced to a Euclidean circle or else a flat circle." Or to use my own language, the inclined cross-section of a sphere "contains" a Euclidean circle.
Now does such a cross-section really contain a Euclidean circle? Trying to gain a great deal of precision on the answer to this question seems futile, but it seems to me that it is "correctly assertible" that it does (whatever your "correctly assertible" is exactly meant to mean :razz:).
You seem to identify different mathematical representations with the definition of a circle in a curious way. This strikes me as odd, but I don't mean to imply that a consensus of mathematicians would favor my view. So to nail it down a bit:
(We are implicitly talking about a plane figure.)
Do Euclid and Aristotle disagree on what a circle is? That sort of question is what I think lurks behind much of our disagreement, such as the deletion of points. If two people draw something differently, can they both have drawn a circle?
I think it contains a circle. It's just that the contraption you use to show that it contains a circle also means you need to go beyond Euclid's definition. An incline plane in a Euclidean space is definitely a Euclidean plane. An incline plane can't contain a circle just rawdogging Euclid's definition of a circle, since an incline plane is in a relevant sense 3D object - it varies over x and y and z coordinates - and thus subsets of it are not 'planar figure's in some sense. However, for a clarified definition of plane that lets you treat a plane that is at an incline as a standard flat 0 gradient 2D plane, the "clearly a circle" thing you draw in it would be a circle.
Quoting Leontiskos
I have had a similar experience to this. It was a discussion about rotating an object 90 degrees in space, and having to consider it as a different object in some respects because it is described by a different equation. One of the people I spoke about it with got quite frustrated, rightly, because their conception of shape was based on intrinsic properties in differential geometry. I believe their exact words were "they're only different if you've not gotten rid of the ridiculous idea of an embedding space". IE, this mathematician was so ascended that everything they imagine to be an object is defined without reference to coordinates. So for him, circles didn't even need centres. If you drop a hoop on the ground in the NW corner of a room, or the SE, they're the same circle, since they'd be the same hoop, even though they have different centres.
Which might mean that a car has a single wheel, since shapes aren't individuated if they are isomorphic, but what do I know. Perhaps the set of four identical wheels is a different, nonconnected, manifold.
Quoting Leontiskos
I can't tell if you're just being flippant here (which is fine, I enjoyed the razz), or if you actually believe that something really being the case is impossible to demonstrate in maths (or logic). Because that would go against how I've been reading you all thread.
I agree, but that's why I would not say that an incline plane in a Euclidean space is definitely a Euclidean plane. I don't see that there are incline planes in Euclidean space.
Quoting fdrake
But here too, I would say that you are confusing a "flat" plane with a Euclidean plane. A Euclidean plane is not a "0 gradient plane," it is a plane without any gradient dimension whatsoever. I have been overlooking these sorts of errors, but if you are going to be persnickety about what you see as my errors then I suppose I should return the favor, especially given that you haven't shown interest in trying to mete out the question of why/when we should disagree.
Quoting fdrake
Yep, I sympathize with him.
Quoting fdrake
People really will say that they have four of the same tires.
But the same question about Euclid's Circle vs. Aristotle's Circle is arising here. If there is no right answer to these questions then there are no real questions, and in that case I don't know why we're arguing.
Quoting fdrake
I'm being flippant, but not "just." :wink:
But no, I take it that your "correctly assertible" means something like "justifiably assertible," and on that reading I think it is correctly assertible that the cross-section contains a Euclidean circle. At the same time, I think the phrase "correctly assertible" is only a placeholder for further explication, because justification doesn't have food to eat unless there is a truth of the matter, at least on the horizon.
Then we're using Euclidean space differently. To me a Euclidean space is a space like R^3, or R^2. If you push me, I might also say that their interpoint distances must obey the Euclidean metric too. Neither of these are Euclid's definition of the plane. "A surface which lies evenly with straight lines upon itself" - R^2 isn't exactly a surface, it's an infinite expanse... But it's nice to think of it as the place all of Euclid's maths lives in. R^3 definitely is not a surface, but it is a Euclidean space.
Quoting Leontiskos
You also disagree with him strongly if you like Euclid or Aristotle's definition of a circle. I actually prefer his, since you can think of the car wheels as its own manifold, and the one he would give works for the great circle on a hollow sphere too. I think in that respect the one he would give is the best circle definition I know. Even though it individuates circles differently from Aristotle and Euclid.
Quoting Leontiskos
I'm not familiar with Aristotle's definition of a circle at all. I might not even understand it. Though, if I understand it, I think the two definitions are equivalent in the plane. So there's no disagreement between them. Which one's right? Well, is it right to pronounce tomato as tomato or tomato?
I believe that I do, and I'm happy that you continue to respond in spite of the frustration.
Gonna call it for tonight and rethink stuff, though obviously not in your favor :D
I'd appreciate you answering my question about whether or not paraconsistent logic would count as a plural logic insofar that we accept both paraconsistent logic and classical logic.
Okay, so R^3 is a Euclidean space and R^2 is the place where all of Euclid's mathematics lives. I mean, your early insistence on locating Euclidean circles in R^2 is why I am thinking of R^2 as Euclidean space. Apparently you are making the "...ean" of Euclidean do a lot of work here.
Edit: And why can't a quibbler say that R^3 and even R^2 spaces are not Euclidean? What's to stop him? When is a disagreement more than a quibble?
Quoting fdrake
None of this matters much to me. I only took Euclid's definition as a point of departure or something I would be comfortable with. But I view Euclid's definition as describing a relative property of a continuous curved line that forms an enclosed shape, which is probably why I don't think the center can be "deleted."
Quoting fdrake
But why couldn't a quibbler say that their definitions disagree on account of the formal differences between them?
Because every Aristotle Circle can be shown to be a Euclid Circle and vice versa.
Suppose the quibbler has "deleted" the center, and therefore it can only be shown to be an Aristotle Circle?
Interesting. But yes.
You stipulated that we've got to understand them in the plane in Euclid's sense, which I'll assume is R^2, and that has every point in it. So the "deletion" doesn't provide a counter model, this is similar to the "for all bivalent" thing from the paper. If we understand the definitions both to apply to the whole of R^2, if you deleted a point from R^2 we're just not dealing with R^2.
If you take the definitions and apply them on arbitrary sets, they can disagree. So, you'd begin the proof of their equivalence like "In R^2, consider...".
Fair enough. :wink:
Quoting Moliere
Yes, I didn't really understand it, and it seems like neither you nor I have a firm grasp on what it means for something to be a paraconsistent logic. Like probably everyone on TPF, I have read about paraconsistent logic as I read about animals in a far off land, but I have never worked with it or made use of it. They seem to be used mostly in the way that Aldous Huxley used his encyclopedia entries.
Are you asking me whether I think that accepting both paraconsistent and explosive logic results in the robust kind of logical pluralism? My guess is that I would answer 'no.' Paraconsistency does not entail Dialetheism. And paraconsistent logic is often used informally in everyday life (if that counts). I also haven't seen anyone in this thread who favors logical pluralism embrace Dialetheism - other than yourself, of course. They seem to be mostly Augustinians, "Lord, give me logical pluralism, but not yet!"
Perhaps you can see my complaint. Given that the sort of mathematics we are engaged in is in an important sense limited only by our imaginations, so too quibbles are limited only by our imaginations. For example:
Quoting Leontiskos
The flip side of this is that mathematical concepts seem to become purely stipulative and imaginary when viewed in such a way. In that case the ground rules for something like propositional logic lose all coherence and plausibilityâas do all conceptsâonce we have dispensed with the notion of the true or useful. It then becomes nothing more than Banno's "symbol manipulation." That's why I keep asking things like this:
Quoting Leontiskos
Oh. Because the definition of a Euclidean space, in the modern sense, includes both. They're infinite expanses of points whose interpoint distances are given by straight line distance. In the old sense, in Euclid's sense, only R^2 could be, since R^3 isn't a surface.
Quoting Leontiskos
It's interesting really. Since deleting the point from the plane impacts lots of possible circles. There will be Euclid circles in that space which are not Aristotle circles too, I believe. Though I'm not totally convinced.
Quoting Leontiskos
The discussion about capturing the intended concept is relevant here. The interplay between coming up with formal criteria to count as a circle and ensuring that the criteria created count the right things as the circle. That will tell us what a circle is - or in my terms, what's correctly assertible of circles (simpliciter).
That's the kind of quibble we've been having, right? Which of these definitions captures the intended object of a circle... And honestly none of the ones we've talked about work generically. I believe "A closed curve of constant positive curvature" is the one the differential geometry man from above would've said, but that doesn't let you tell "placements" of the circle apart - which might be a feature rather than a bug.
Sure.
Quoting fdrake
Yes, or:
Quoting Leontiskos
Quoting fdrake
But what is the "intended concept"? Presumably it is an intuitive concept, and are intuitive concepts mathematical formalisms? I wouldn't think so. So:
Quoting fdrake
Why think that the intended concept is a formalism, a mathematical equation? Similarly, why think that logic is a formalism, a logical system? Perhaps logic is as I've said: that which produces discursive knowledge. It is a natural or anthropological reality, not a prepackaged formalism.
That reads disingenuously to me. Your use of "roundness" previously read as a completely discursive+pretheoretical notion. If you would've said "I think of a circle as a closed curve of constant curvature" when prompted for a definition, and didn't give Euclid's inequivalent definition, we would've had a much different discussion. I just don't get why you'd throw out Euclid's if you actually thought of the intrinsic curvature definition... It seems much more likely to me that you're equating the definition with your previous thought now that you've seen it.
The latter of which is fair, but that isn't a point in the favour of pretheoretical reasoning, because constant roundness isn't a concept applicable to a circle in Euclid's geometry, is it? Roundness isn't quantified...
Quoting Leontiskos
Mathematical concepts tend to be expressible as mathematical formalisms, yeah. And if they can't, it's odd to even think of them as mathematical concepts. It would be like thinking of addition without the possibility of representing it as +.
Quoting Leontiskos
And therein lies a relevant distinction. Formalisms aren't prepackaged at all. In fact I believe you can think of producing formalisms as producing discursive knowledge!
Rather, if the context is different then the geometrical response is different, and I have no dog in the fight over the question of "family resemblances" as applied to geometrical abstractions. I have claimed that there are not square circles, not that "circle" can only ever be utilized within a single context.
I gave that option before giving Euclid's. You are the one who brought up Euclid in the first place, but I really don't see the two descriptions as competing.
Quoting fdrake
Pretheoretical or intuitive reasoning need not be quantified, does it? In making that comment I was making the point that pretheoretical reasoning represents the same basic idea as the calculus definition you gave. "...In calculus [consistent roundness] cashes out as a derivative, but folks do not need calculus to understand circles. Calculus just provides one way of conceptualizing a circle."
Quoting fdrake
Well then I would ask whether the intuitive concept that is the intended concept is a mathematical concept. When a child learns to place circle-shaped blocks in circle-shaped holes they are not involved in formal geometry.
Quoting fdrake
Or rather, producing a thing that can produce discursive knowledge. And knowing a true logical system is a kind of knowledge, which is probably discursive. I think that's right. But they are prepackaged in a very relevant sense, particularly for those of us who are not their inventors.
But I also don't think a logic like Frege's is merely a model, nor that it could be. To invent a logical system is to attempt to capture a (or the?) bridge to discursive knowledge, and I don't know that any success or failure is complete.
Ah. That's unfortunate. Euclid's definition makes the great circle not a circle. The closed curve one makes it a circle.
Quoting Leontiskos
It's the same basic idea, yeah. When understood in the context of a circle. You can think of curvature as a more general concept than roundness, since curvature's also "pinchiness" and "pointiness". and "flatness" etc all rolled into one. So it's sort of like roundness is to curvature as apples are to fruit.
Quoting Leontiskos
It's both innit. Getting the definitions right is one thing - yay, you have found the commonality between circles. Using the definitions to produce even more knowledge is another.
I don't think any of the examples we've discussed so far is "merely" a model, since the different frameworks place much different commitments and demands on the behaviour of people that use them.
One of the great things about producing formalisms is that they're coordinative. If you and I operated on the constant curvature definition, we'd be committed to the same beliefs about circles. The same with the Euclid one. When you add that to our ability to mathematise abstractions expressively in a common language, you end up being able to write down the mathematical rules one must follow when dealing with an abstraction - just in case you have successfully defined it in the symbols. At that point, whether it is the right abstraction for the job seems a different issue.
I certainly wouldn't tell my students that a circle is a closed curve of constant curvature, I'd show them examples of circles and just say "like this". Roll them about. Measure them. I wouldn't even show them Euclid, or try to define the shape. For a lot of things you can get an okay idea of what they are without a formalism, but that loses its charm when you need to explore things that have less straightforward intuitions associated with them.
Like the example I gave of continuous functions vs Darboux functions (functions with the intermediate value property). Mathematicians thought those were equivalent for a long time based on pretheoretical notions.
As I've said repeatedly, STT need not be deflationary. It is often taking as a means of [I]modeling[/I] correspondence truth and this leaves the door open for judging "correct logics" in terms of their ability to preserve correspondence truth not simply truth relative to some formal context.
But STT can also be rendered deflationary, and Frank has given us some sources indicating that this is more how Tarski himself considered the theory (which jives with what I've read of his work).
As for it being a "bad thing," that's an entirely different conversation. The question is: "is deflation question begging or at the very least a highly relevant and contested premise when considering logical nihilism vs pluralism vs monism, such that its implicit assumption is problematic?"
I don't see how the answer could possibly be anything but "yes." If one starts with a strong deflationary position it seems trivial to show that no laws of logic hold with generality. But monists are normally arguing for monism in a non-deflationary context, in terms of "correct logics." Monism is true for "actual truth preservation" not "truth preservation relative to an arbitrary context."
For example, G&P's target is the natural language logical consequence relation. The scientific position's target is entailment in the sciences. The metaphysical position is talking about logical consequence from the perspective of metaphysical truth.
And nihilism also seems to need to avoid deflation because nihilism is a position about logic in general or "all correct logics." If the nihilist adopts a strong deflationary position for the purposes of undercutting monism then they are guilty of equivocating when they try to tell the pluralist that there are no laws of truth preservation for logic as a whole. Deflationary nihilism is simply pluralism, the nature of truth preserving logical consequence varies by context.
But again, virtually no one wants to claim that truth should be both deflated and allowed to be defined arbitrarily. So we still have the question (even in the permissive case of Shapiro) about what constitutes a "correct logic." The orthodox position is that this question is answered in terms of the preservation of "actual truth." But we also see it defined in terms of "being interesting" (e.g. Shapiro). Either way, we are right back to an ambiguous metric for determining "correct logics," hence to common appeals to popular opinion in these papers.
I agree, speaking the same language always helps. Based on this I would fall more in the nilishist camp I suppose. The truth of the conclusion isn't a consequence of the premises. I could make most arguments backwards. Any assertion of truth comes with the 'consequence' and it is true or and it is false.
Quoting Count Timothy von Icarus
Isn't this just an attempt to dismiss the idea out of hand? I suppose if I thought nihilism was wrong I would posit that the relationship between ideas can be by free association(under nihilism), but I wouldn't confuse it with a compelling position.
Quoting Count Timothy von Icarus
This sounds more like arguing against the "no general" bit of the definition that you claimed doesn't apply. I guess I'm confused as to what question I'm begging. Do people think that ordering statements they've asserted as true eliminates the possibilty of error? Something more than a persuasive assertion?
There's no formula for making a false statement true when it isn't. So, formulation doesn't cause the truth of something. It simply presents the reasoning in an arguably unnatural way. The truth of things is constrained by the facts and the state of affairs, not the way I choose to write it down. What question is that unfair to? Thanks for the explanation though, I tried to parse it best I can.
I agree with underlined point completely. The scientific and metaphysical arguments for monism tend to be abductive arguments based on this idea. This is why deflation is problematic as a background assumption. It needs to be an explicit premise, else we end up talking past each other, since the disagreement is really about what is properly "truth-preserving" in the most perfect* sense, not about what is true of formal systems and the logical consequence relationships each uses.
As for the bolded part, I think this is something many monists, pluralists, and nihilists would agree with. Logical consequence is about truth preservation in arguments, not causation, or "that in virtue of which something is true."
Yet, we might ask, "is cause unrelated to logical consequence?" That's a common presupposition in contemporary discussions of logic. It was not a popular position for most of the history of logic though. The ideal argument is propter quid, explaining why something is true (demonstrative syllogism). Not all arguments are thought to be of this sort of course, only some.
This sort of thinking is still alive and well in relevance logic and occasional attacks on material implication.
Anyhow, I think you get at a good point, in that I can imagine that many who subscribe to "classical metaphysics" (i.e. the serious "neo-neoplatonists" today, or Thomists) might actually agree with the nihilist that laws, as in short, stipulated formulae, are incapable of capturing the logical consequence relationship because they cannot capture analogical predication of truth and being properly. But I think they would disagree in concluding that the logical consequence relationship can be either arbitrary or unintelligible as a unity. Just for an example, I don't think Eriugena's four-fold distinction of being where "to say 'angels exist' is to negate 'man exists'" (when using exists univocally) is going to fit nicely into formal context. You could add four distinct existential quantifiers related by some sort of formalism of analogy, but I don't think that's going to cut it.
* I couldn't think of a better term here than "perfect" in the sense that scholastic logic uses it. In this context, blindness is a perfect privation for a dog or a man because, by nature, these things see. Whereas we can say "non-seeing" of a rock or tree, but this is not perfect privation. The differentiation here is that truth might be said analogically of something being "true relative to some stipulated formulation of truth," but this is not true in the same way "George Washington is dead," is made true "by the world."
Right. As I said earlier, a basic challenge for the pluralist is to show which logics are acceptable/correct and which are not. I haven't seen anyone in the thread attempt such a feat, and if that can't be done then I'm not sure a serious position is being put forward. The same could be said for nihilism or monism, but no one has claimed such positions.
Well Beall & Restall at least have a tighter definition. Shapiro's "eclectic pluralism" is based on "being interesting." But triviality is interesting. Does this mean logics where everything expressible can be shown to be true are "truth-preserving?"
I think you need to assume deflation here for that to make any sense. If we aren't willing to go that far then we can still speak of how they "preserve truth," internally, in an equivocal sense, but that's it.
Seems right.
There is also a really odd thing that happens constantly on TPF (and it usually happens with SEP). Someone will champion a position like logical pluralism or dialetheism or something like that, but when it comes down to the question of what exactly they are promoting they are at a loss for words. They don't have any clear definition of, say, logical pluralism.
So we go to a secondary source like SEP or Griffiths and Paseau. But as soon as the content of the position is being taken from SEP and not from the TPFer we are no longer engaging/arguing with that TPFer. The TPFer had superficially identified with logical pluralism without being able to say what logical pluralism is, or what they mean by it, and when one flies over to SEP they have overlooked the crucial nature of this conundrum. SEP is not going to tell us what the TPFer thinks; it is only going to tell us what the author of SEP thinks. The thread becomes the discussion of a position that no one in particular holds, and that no one in particular has a stake in. In my opinion this outlines one of many misuses of SEP on TPF. Yet there is a fascination in our contemporary culture with labeling and labels!
...And to be specific, after this thread was necro-bumped Banno did the thing, "Yay for logical pluralism! Boo for Leontiskos and his logical monism!" What did Banno mean by logical pluralism? He had no real idea. Why did he think I was committed to logical monism? Again, probably no idea, although everyone took him at his word (!). It was a half-baked thought meant to stir up controversy, and that is the heart of the problem. Bringing in something like SEP is not going to make that initial move impressive or substantive.
I would be pretty happy to defend logical nihilism as set out in Russell's paper.
Thanks for the generous read and I'm still looking up some of these references. I suppose I have cake and eat it to approach to deflation. I think we we can know the truth of things. I don't think we have complete access to when that is the case and when it isn't. So its deflationary in the sense that truth claims are only assertions, but the truth itself isn't. Its a thing to be approximated. A type of perfect in this same sense.
Quoting Count Timothy von Icarus
I'm still missing the jump from a symbolic system lacks the richness of embedding found in natural language to - logical consequence is arbitrary or unintelligible as a unity. Following your discussion with Leontiskos I get how "interesting" might be on the right path. To say there's a link between a formulated argument and the compulsion to accept it doesn't seem outrageous by any stretch. I just think it's naturally limited to saying, this is why I think I'm right versus why I must be right. The 'right' part is still "truth" properly inflated. But, it's relative to a person and we come with mistakes. Not to say logic doesn't get us closer and contradictions don't indicate a likely error, but neither are flawless indications of inflated truth. So, nihilistic with respect to guarantees, but realistic in thinking ideas ought to be consistent.
I've always noted that disagreements about 1 thing, imply a disagreement about another. Is that a concession to anti-nihilism?
- Good points.
Quoting fdrake
Coordination, cooperation, intersubjective agreement, etc., really tends to be the goal and limit of contemporary thinking. I think such things are useful, but I also think that at some point we have to venture out beyond the bay and into the open sea.
I do enjoy the open sea, I just tend to think its openness is necessary. If you'll forgive me the excess of portraying metaphysical intuition through vagueness.
Quoting fdrake
Quoting Leontiskos
It seems to me that Sider's thread is the better place for this, but what you describe here doesn't really sound like metaphysics at all. The only point that sounds like metaphysics is the fanfiction metaphor, but if the fanfiction cannot be good or bad then one cannot be doing metaphysics, and are you willing to say that the fanfiction can be good or bad?
Yes. Harry Potter and the Methods of Rationality is definitely better written than My Immortal.
Well at least we have agreement that the Semantic Theory is noncommittal as to deflation or correspondence. I'm at a bit of a loss as to what happens next. You say that the correct logic is the one that "preserves correspondence truth". Does that mean that correspondence gets to decide between logics? How could that work? Is the correctness of logic to be decided empirically?
And I'm now not sure if you are claiming that there is only one logic, and hence monism, or if you are saying that there are indeed multiple logics, only one of which "preserves correspondence truth".
And it remains unclear to me why you introduced deflation into the conversation.
I'm sorry, I just have not been able to follow what you are claiming.
One of the issues in this thread is indeed the nature of logical pluralism. Deal with it as you will, but repeatedly attacking me is petty.
The IEP article ends with
When people writing on this topic discuss "correct logics," what exactly is it you think they are referring to? If all logics are correct logics then nihilism is obvious.
Yes, this is why G&P refer to the challenge as "broadly epistemic." Personally, I think the correspondence theory of truth is deficient, I only use that as an example because that is how it is most often conceived of.
If you assume deflation, I don't get how nihilism isn't a consequence. Truth just is truth as defined by some system. There are systems that both define a notion of truth and variously dispense with each of the proposed "laws of logic." Ergo, there are no laws of logic. What else more is there to say? If there is a logic that dispenses with LNC, then LNC cannot be a law of logic, etc.
To be sure, it's not a term I would use. Logics are useful, applicable, valid, consistent, incomplete and so on, but not so much "correct".
Quoting Count Timothy von Icarus
Why? If Quoting Count Timothy von Icarus
what follows is that there are logical laws that apply within each system. What does not follow is that there are no logical laws.
Quoting Count Timothy von Icarus
So there are multiple logics?
In the same way moral pluralism is nihilism? Yes.
Quoting Count Timothy von Icarus
Truth deflationists usually think of truth as having a social function. It's just something people say. That's different from using the truth predicate in a technical way as Tarski did.
How you decide that the cat is on the mat - by observation, deduction or consulting a clairvoyant, is beside the point.
Cirtangles for the win
I don't think so. The pluralist says there are multiple logical consequence relationships that preserve truth in different contexts. The nihilist would be saying there is no logical consequence, or put another we "we decide which logical consequence we want to consider correct." Or, as Russell puts it: "there is no logic."
You can see the difficulty of equivocating or refusing to elaborate on what the "truth" in "truth-preserving" means here.
Indeed, there are different flavors of deflation. "Using the truth predicate in a technical way" isn't deflation at any rate. Deflation would involve the claim that truth just is whatever technical definition one decides to use. If one justifies STT with the claim that it "mirrors correspondence," "is the closest we can get to truth," or something to that effect, one isn't being deflationary.
In virtue of what is a logic "applicable"?
How about this, why don't you explain to me why you think pluralism and nihilism are even different positions? And why do you think monism remains the dominant position?
This is an ambiguous question (which the articles shared here generally tend to point out in the introduction). If the question is "have people created systems with different logical consequence relationships?" the answer is obviously yes. But given your line of questioning this seems to be what you think the debate is about.
When Russell call nihilism "the view that there is no logic," do you think she is denying that any logics exist?
Am I being trolled here?
This isn't an answer to the question though. What do you think is being meant by "correct logic" in these articles?
To clarify, this is the opening sentences of the article you wanted to discuss:
"Logical monists and pluralists disagree about how many correct logics there are; the monists say there is just one, the pluralists that there are more. Could it turn out that both are wrong, and that there is no logic at all?"
You're acting as if this is some bizarre concept it is impossible to understand though.
Why does it matter?
Maybe @fdrake will explain what logical nihilism is?
Come on. When it has a use.
Quoting Count Timothy von Icarus
From the core article:
1) To be a law of logic, a principle must hold in complete generality.
2) No principles hold in complete generality.
3) There are no laws of logic.
Monists hold that (2) is false. Pluralists hold that (1) is false. Nihilists hold that the argument is sound. On this account pluralism is different from nihilism.
Quoting Count Timothy von Icarus
So we are back to puzzling over whether there are principles that hold in complete generality.
Russell, to be sure, is in that article giving an account of how pluralism can be maintained in the face of nihilism. She is not a nihilist, so far as I can make out.
Do some logics lack "a use?" Or do they all have one?
What does it mean to hold in generality?
On your understanding of this, why would monism remain the dominant position? It seems obviously false.
Because what it means to be "truth-preserving" and thus a "correct logic" will depend on what is being preserved.
I think it's ok for people to add on whatever significance they like to the word truth in truth-preserving. In the same way, if you lean toward ontological realism or anti-realism, you can add that onto whatever shenanigans you're doing. It doesn't change the shenanigans either way.
Perhaps. Although a logician's presenting a logic would be their making use of it.
Quoting Count Timothy von Icarus
In all logical systems, presumably. But I would be happy to consider any other options you might offer.
Quoting Count Timothy von Icarus
Appeal to popularity? So you are seeing the traction in the arguments here.
I've not seen any evidence one way or the other, although I suspect most logicians accept that there are a range of logics - that's pretty undeniable.
No, I'm just trying to figure out your understanding of the topic.
Which is why I ask, what exactly do you think the monist is claiming? That every logical system people have created has the same entailment relation? Isn't this very obviously false? I'm mystified as to why you think this is a subject of controversy given your understanding.
Well, I've been trying to work out what you are claiming, on the presumption that you are advocating monism.
So again, a monist holds that there are logical laws that are common to every system of logic.
No, not that "every logical system people have created has the same entailment relation".
And so it is up to monists to show what it is that all logical systems have in common. I don't see that it can be done.
(edited)
Its sound if complete generality is a thing. Does it follow that it must hold in partial specificity? If following things applies. Is obfuscation a system of logic?
The idea of a correct logic is endemic to logical monism. I'm not sympathetic to monism, and so I'm not the one to ask this question of.
But presumably correct logic for a monist would be only those logics that make use of the general laws of logic, whatever they might be.
Does that help?
Well, even "necessary" has differing interpretations depending on which logical system one chooses - S1 through S5 for a start. And we have logical systems that are incomplete. I'm not sure what to say.
It seems odd to define something as what it can't be. Like a 'law of aviation' can only exist if it applies to lead plane flight. There are no lead planes. There are no laws of aviation.
Bit suspect is all.
Doesn't that sound a bit tautological to you? If correct logics are just those logics that utilize the general laws then monism is true by definition.
Your understanding of each of the positions seems to make them trivial rather than controversial.
If there are general laws...
That's the issue.
Quoting Count Timothy von Icarus
How so?
Well, in virtue of what would a law be considered a "general law?" The monist says the general laws are those which hold in "correct logics," which is why they aren't forced to abandon their position on, say, LNC, due the mere existence of dialthiest systems.
Great posts. :up:
Quoting Leontiskos
Right, which is why their position is generally something like G&P's, which is that correct logics are those which capture the logical consequence relationship at work in natural language and scientific discourse, or perhaps "preserves-truth" relative to some metaphysical notion of truth, etc.
But you have acted like this is unfathomable, so I'm not really sure what you think this debate is about. Feel free to describe what you think the difference between the three views would even be in your view.
But we have plenty of criteria and that's what matters.
A pluralist will say that there is a certain type of logical consequence that is appropriate for a particular context. A nihilist will deny this.
A monist will claim there is only one logical consequence relationship, though no doubt they are aware that consistent logics have been constructed with other consequence relationships.
So why do you think there is any controversy here?
So you call a logic "correct" when I might call it "applicable". And Paraconsistent logic is for you "correct" when used for processing images and signals, while Lambda Calculus is "correct" when used for cryptography or AI.
Quoting Count Timothy von Icarus
What one? Set it out.
The impression I got was that "complete generality" doesn't commit you to quantifying over logics. A principle holding in complete generality, being understood as the entailment relation being the same for all logics, would need to contend with the fact that you can arbitrarily make systems that prove a claim and corollary systems that prove its negation when they share the same set of symbols.
So you can come up with a logic where modus ponens holds, and come up with a logic where modus ponens does not hold. Which would mean that if you wanted to find The Logic Of All and Only Common Principles (tm), you'd need to jettison modus ponens. Since it is not a common principle, since two logics disagree on whether it is a theorem.
The paper gives lots of strategies for coming up with schematic counter examples to many, many things. You can come up with scenarios where even elementary things like "A & B... lets you derive A" don't hold. So much would need to be jettisoned, thus, if The Logic Of All and Only Common Principles was taken exactly at its word, in the sense of intersecting the theorems proved by different logics.
And that's kind of a knock down argument, when you consider X is true in system Y extensionally at any rate (which is AFAIK the standard thing to do)
Phrasing it in terms of "complete generality" thus gives a whole lot of wiggle room regarding what it would mean for a principle to hold in complete generality, like you might be able to insist somehow that any logic worth its salt must have LEM, or any logic worth its salt must have modus ponens as a theorem. So that sense of "complete generality" (NOT completeness of a logic) might mean "in every style of reasoning", so it would let you think of some logics as styles of reasoning and some as not.
You also have the opportunity to think of informal logical principles as holding in "complete generality", as eg if someone believes that the No True Scotsman fallacy is fallacious in some sense, an argument definitively establishing its fallaciousness might be considered a theorem of The Logic of All and Only Common Principles, even though No True Scotsman doesn't admit an easy formalisation. Next paragraph is just extra detail supporting that it doesn't have an easy formalisation.
Just for extra detail, No True Scotsman doesn't admit of an easy formalisation in terms of predicate logic because deductively it kind of works. If x is always p( x ), and someone provides an example of x such that ~p( x ), it should be taken as a refutation. But the fallacy corresponds to interpreting the person providing the counterexample as instead providing an example of someone for whom some distinct property q( x ) holds where q( x ) != ~p( x ). Which isn't exactly a fallacy, it's a reinterpretation, and sometimes it's a good thing to do when arguing - sometimes people make bad counterexamples. But what makes it a fallacy is somehow that the suggested q( x ) only has irrelevant distinctions from ~p( x ), like a true Scotsman is just a Scotsman. You could also read it like the the asserter that x is always p( x ), upon receiving the counterexample, clarifies their position to some predicate q( x ) such that the counterexample given does not apply to it while still using the same predicate label ("Scotsman"). In that case the fallacy consists of revising the content of the claim to "just" exclude counterexample for no other reason, which deductively is without problem, but provides another irrelevant distinction. In either case, the sense of irrelevance of distinction is the thing which is so norm ladened and contextually situated that you're not going to be able to put it into a logic without (unknown to me) profound insight about logical form in natural language.
So if you wanted to have the fact that No True Scotsman is a fallacy as a "theorem" of The Logic Of All and Only Common Principles, maybe your whole logic needs to be informal to begin with.
:up: :up: :up:
Thank you, I am glad someone else also seems to understand what the topics is about and why there is even debate. I felt like I was going insane here lol.
It is interesting that you bring up the No True Scotsman because I think the monist can often be accused of something like this.
Anyhow, this is why I think avoiding any trace of metaphysics entirely seems impossible here. The CD paper uses counterexamples that involve abstract objects almost exclusively (occasionally propositions about proofs), and people's willingness to accept these as strong counterexamples seems linked to the sense in which they can be said to "exist." CD seems to suppose that if they exist in any formalization that they "exist" in a univocal sense. I imagine monists are generally going to just deny this, because monism is about logical consequence relative to some non-arbitrary context (although which one varies).
Maybe no "metaphysical" notion is needed and we just speak in terms of "plausibility" and "usefulness" but these seem to easily become even murkier notions. The two most common versions of pluralism (Beall and Restall and Shapiro) cited have very different notions of which logics should "count" for instance.
Just for extra detail - how easy it is to come up with logics that disagree on theorems is a good argument for nihilism if you agree, with a stipulated logical monist of a certain sort, that there is only one entailment relation which all of these logics ape.
I underlined "any" and "there is a case" above to highlight something about their scope of quantification. What collection is being quantified over? It must generically include arbitrary cases, premises, conclusions etc. IE, "complete generality" in a manner that allows the arbitrary representation of statements in formal languages. It's thus a metalinguistic notion with respect to any object formalism, it lays beyond and out with them.
It's, furthermore, a semantic notion:
The turnstile with two lines above means that Russell wants to find counterexamples to principles through interpreting the logic, which is a way of finding a "syntactically appropriate" mappings from its symbols to other objects - like propositions to truth values - to see in what conditions the proposed principle holds. Mucking about with interpretations like that is what makes the kind of logical nihilism she's playing with a semantic argument.
So what Russell is doing, when she's finding counterexamples, is taking "syntactically appropriate" expressions, throwing them into a formalism, then evaluating them in that formalism through an interpretation. If she can find an expression and an evaluation that fit the rules of the logic that is also a counterexample to one of its candidate principles, then it's not a principle of the logic for all expressions in it - and so is not a logical law.
So the sense of "complete generality" also allows Russell to consider variations over interpretations and the relationship of interpretations with syntactical elements of languages - it's thus a highly metalinguistic notion. Which is not surprising, as the Logic Of All And Only Universal Principles would need to have its laws apply in complete generality, and thus talk about every other logical apparatus in existence.
Which is an incredibly, incredibly strong thing to want. It's practically alchemical, one must have in mind a procedure in which the complexities and ambiguities of natural language, every inference, can be stripped, dissolved, distilled into gold. The true atoms of rationality. The story hooks in the book of divine law. In some respects it's even stronger than the petty desire to take the intersection of all logics, at least that has a precedent in each logic. And you need to claim that this holy book of divine order is spoken in one voice, the true semantical derivation symbol of the cosmos, that admits no quibbling, sophistry or perversion.
Or you could refuse the above notion and take the path Russell does, by applying metalinguistic restrictions to the space of interpretations of a theory. As in, "yes, we know the Liar blows this logic up, so let's just say for all bivalent ?", hence the method from proofs and refutations, lemma incorporation, in which a system is mapped to another system with an additional lemma in order to constrain its space of syntactically valid interpretations.
In formal terms, the latter is what distinguishes @Leontiskos's sophist from someone who finds good counterexamples, someone who finds good counterexamples ensures that they are syntactically valid - that is, obeys all and only the stipulated rules, both intended and written. If you can jam something between the intention and the written word, while playing by all the rules stipulated, you've shown that the conceptual content of the formalism does not reflect the intended object. Or alternatively the intended object is the wrongly represented in the formalism, conceived in a confusing or inopportune manner etc.
No worries. I do think your insistence that the extensional understanding of truth is deflationary in this context is imprecise. If I understand correctly, you're using "deflationary" to mean restricting the interpretations of a theory to all and only the ones which are syntactically appropriate and clearly within the logic's intended subject matter. Like propositional logic and non-self referential statements. Effectively removing everything that could be seen as contentious from the "ground" of those systems. Which would then ensure the match of their conceptual content with whatever objects they seek to model, (seemingly/allegedly) regardless of the principles used to form them. Which 'deflates' truth into unanalysable, but jury rigged, coincidence.
By contrast, correspondence would consider truth as a relation between the conceptual content of a theory and its intended object.
My intuition is that the rules which bind coming up with mathematical formalisms are the same as those which govern writing fiction. They're in general loose, murky, descriptive, but you can tell a good description from a bad one. I'd also want to liken the relationship of formalisms to their intended objects, or intended conceptual content, to the relationship some writers have with their characters. They don't always know what the character wants, how the character would react, but they'd be able to work through how they'd feel and act if they put them in a scenario. That lets them write in a manner true to the character. I think formalisms have a similar "true to the character" expressive flavour, and the concept of an interpretation lets you come up with "scenarios" and "story beats" to flesh out the understanding of the concepts and what's written about them. Interpreting your own symbols in that extensional sense is a way of finding the meaning of what you've written. And just like writing fiction, you can find the conceptual content very resistive to your expression. A theorem may escape you just like how to put a scene.
My intuition is also that there are other principles that set up relations between the practice of mathematics and logic and how stuff (including mathematics) works, which is where the metaphysics and epistemology comes in. But I would be very suspicious if someone started from a basis of metaphysics in order to inform the conceptual content of their formalisms, and then started deciding which logics are good or bad on that basis. That seems like losing your keys in a dark street and only looking for them under street lamps.
It could be thought of as a regulative principle -- here we have multiple logics, but we would like them to cohere: the monist would then be more of a project than a position, the attempt to build a logic which contains all logics. (one could presumably derive the LNC from this meta-logic, for instance -- but it's just an idea)
Quoting fdrake
I called the pluralism/monism debate an internecine debate between Analytics because they are all univocalists. Your word "ape" here is doing a lot of work, but it seems that for both pluralists and monists such an entailment relation will be purely univocally predicable. This is why the whole game is so boring. The interesting question is an adjudication between two different paradigms, and folks like Banno and probably G. Russell are eternally stuck in a single paradigm, interpreting the other paradigm in their own terms.
My definition of logic via the Meno is something like, "That which creates discursive knowledge" (or perhaps just knowledge). Now is knowledge or discursive knowledge a univocal concept? I don't think so, and therefore there can be no univocal "entailment" relation that holds for all knowledge. For the univocalist this means that each kind of knowledge and each accompanying entailment relation are hermetically separate from every other kind, and that is precisely what analogical predication denies. This is probably something like Wittgenstein's "family resemblances," although I am not overly familiar with Wittgenstein. (And note again how drastically this univocal analysis deviates from natural language use.)
In the realm of circles we are asking about the relation between the pretheoretical grasp or notion of a circle and the formalization. We could say that the formalizations "ape" that pretheoretical notion, but the scrutiny here is entirely on the manner of aping. Yet in the case of knowledge there is something more concrete and even practical at stake in the question.
Quoting fdrake
Yep.
Quoting Count Timothy von Icarus
I think this is part of it too.
Quoting fdrake
That is the big equivocation for me. Is it a relation to a non-mental reality or merely a conceptual content? Timothy's point about non-arbitrary contexts hinges on the answer.
Quoting fdrake
Sure, and I don't think this is controversial. But I don't think you've given a straight answer to the other side of the coin: are some formalisms truer than others? Is there better and worse metaphysical fan fiction? That's the nub. (And some grandchild of logical positivism is operative here, because the formalists are liable to say, "This question is not formally adjudicable, therefore there is no better or worse metaphysical fan fiction.")
(This central topic has been sidestepped in all sorts of ways. Wanting to talk about modeling or "correctly assertible" rather than truth is one of those ways. If some metaphysical fan fiction is better than others, then it is truer than others, and there is (non-deflationary) truth to be had.)
I think Leontiskos thinks logic is the Anima Mundi. Very medieval.
Yes, I would agree with this.
Well, as you say:
It is a knock down argument, but it seems to miss what monists are claiming (at least from what I've seen). Or even what the pluralists say; Beall and Restall only endorse classical logic and a few sub-classical logics.
And I agree in terms of the standard, at least that seems to be a very common way to look at it in the discipline. But I am not sure it is a useful standard in this context since it seems to allow for refuting the dominant position(s) in terms in which its advocates wouldn't recognize it.
For instance, G&P frame the position they want to argue against as: "we define logical pluralism more precisely as the claim that at least two logics provide extensionally different but equally acceptable accounts of consequence between meaningful statements."
Well, Quoting Leontiskos
made me laugh out loud.
Quoting Leontiskos
People create knowledge. I'm not following what his claims are here. Is he suggesting that we remember logic from our previous lives?
Your chat with him puts me in mind of Kripke's lecture on the surprise test paradox, such that he might reason as follows:
Or
Or
All quite sound reasoning.
Yep.
Excellent post.
Can you link me this paper please? If it hasn't been done already.
Could be. Meno is part of Neoplatonic project building which wouldn't get much more than a blank stare from AP.
It's a book, so sadly not wholly available from what I can see. Google books sometimes has a decent number of pages. There is a review by Erik Stei though and his recent book would be another example for how monists frame their own case.
https://ndpr.nd.edu/reviews/one-true-logic-a-monist-manifesto/
I have to say, I love the cheekiness of the cover.
It's:
If you know something, there is no need to inquire about it because you already know it.
If you donât know something, you wouldnât know what to inquire about or how recognize the answer when you find it.
It sometimes gets brought up in discussions of systematic search.
It is sort of related to P = NP as well. You might be able to tell if you have a correct answer easily if you are provided with one, but finding that answer can be effectively impossible, even if you have a description of what you are looking for that is a ridged designator.
Thanks. It looks a little bit like a Chuck Tingle cover.
Medhurst's Moses
https://commons.wikimedia.org/wiki/File:The_Phillip_Medhurst_Picture_Torah_457._Moses_breaks_the_tables_of_the_Law._Exodus_cap_32_v_19._after_Parmagiano.jpg
"L?G?S" looks more like the brandname for an aftershave than a worthy hypothesis.
My brush with dialethiesm, and thereby paraconsistent logic, came from my studies of the liar's paradox. So for me it's the result of reading arguments about the liar's and thinking dialetheism provided the most satisfactory answer. And actually this might be related since I read you here:
Quoting Leontiskos
First to answer the question, yes that's what I'm after: attempting to define what would count as a robust kind of logical pluralism. Here it seems you indicate that, supposing a defense of dialetheism holds, logical pluralism would count? Rather than paraconsistent logic, just the notion of true contradictions would at least ask for a different kind of logic, even if not paraconsistent, and so we'd be justified in saying there is at least two kinds of logic: the ones which reject contradictions, and the ones which utilize them in some way.
Also I don't mean to say I'm an expert by any means. Just an interested reader who thinks about these things.
I also have ulterior reasons for taking dialetheism seriously, namely Marx and Hegel. Marx's notion of contradiction I have a good feel for (but because it's more extensional it's easier to untangle Marx's notion of contradiction from the logical one by dividing wholes into parts that differ), but Hegel's continues to mystify me.
And then one day I came across Priest in reading through the Liar's sentence and as odd as it is on its face it kind of slowly grew on me. I'm not sure of extensions of [s]the[/s] dialetheism beyond the liar's, though Priest lists several (also including some Eastern philosophy too), but I think I like dialetheism as a solution [s]the[/s] to the liar's paradox because it's a queer conclusion that comes from the plainest understanding of the liar's: no fancy logic is really needed. I can understand thinking the liar's is incoherent -- once upon a time I thought that because it's hard to imagine an empirical use for it-- but since this concerns logic alone, and may provide some inroads to other interests I have, I find it worthwhile in trying to comprehend and use. (Also, I think it could be a promising theory to develop in fleshing out the absurd, which is where I began originally -- taking the absurd as a metaphysics seems to indicate that logic cannot contain reality, especially the absurd parts -- logic's whole thing is making sense!)
The other response to the liar's I held was that the liar's sentence is simply false. It's telling you exactly what it is on its face. there is no evaluation necessary.
But the strengthened liar's sentence persuaded me that there is at least an interesting formal concern.
Now I sit and wonder what it takes [s]the[/s] to contain explosion, if anything rational could be proposed in empirical (rather than conceptual) cases.
To address your concern about knowledge and logic's relation to it: I think this exercise demonstrates that we can't contain the world with logic, but rather we invent the logic to fit the world. It works because we've seen this or that enough times and so we follow the habits which reward us and call it truth*.
What's interesting about this line of thinking is that it's not denying even a metaphysical truth. But rather is showing how knowledge is produced: Guess and check. There is no method that guarantees knowledge. You just have to work things out the best you can.
So it's not entirely a dry academic consideration, to me. I see lots of interesting inroads with these ideas to other things I'm interested in, and the creative nature of it all gets along with what I think knowledge generation takes: making up new things and seeing if they work.
EDIT: An afterthought -- in a way the pluralist is actually more anti-nihilist than the monist. The monist has to hold that contradictory statements cannot be logically comprehended which is, in a way, a baby nihilism: Here is the field of inquiry where no logical rules hold. The pluralist says "Well, so far, perhaps... but what if we...."
*EDIT2: That looks dangerously close to a pragmatic theory of truth. It's off topic but I'm not a pragmatist, in spite of these sayings which would easily cohere with pragmatist theories of truth.
Almost like I read philosophy to figure things out that I still wonder about ;)
Hegel's contradiction is pretty far from most paraconsistent logics, given the unity and "development" of opposites. If you're interested though, formalization attempts have run through category theory and Lawvere is the big name here.
https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://philarchive.org/archive/CORMAA-3v1&ved=2ahUKEwjrxdPIz6CJAxURlIkEHUmyEkcQFnoECCEQAQ&usg=AOvVaw3XxnDtBEih45jE5c2zfW2d
Nlab has some stuff on this too.
I have read [I]many[/I] commentaries on the Logic at the point. Houlgate and Wallace are my favorites (Wallace isn't quite a commentary, but he does focus on the Logic), but Taylor was useful too. Despite this and now years of effort, I find the essence chapter largely impenetrable lol. But better minds then mine might have more success.
I agree. I came to the same conclusion, and was disappointed. "Further research needed" :D
I enjoy the phenomenology, but only got 1/2 through the logic and couldn't say I understand it. I could tell it was not time to climb that mountain.
Quoting Count Timothy von Icarus
Thank you! Next time I feel like trying the Kilimanjaro of philosophy I'll be referencing these ahead of time to prepare.
Quoting Joshs
Yours has been the hardest to respond to for me. Hence my tardiness.
If we substitute normative social practices for Kant's Transcendental categories, what does that look like? In a very literal sense, which I don't think you mean but this is why I'm asking for clarification, I could substitute a model of practice for quality, quantity, relation, and modality -- substitution seems to need some relation of sameness, if not strict equality, and I'm not sure how practices would work within Kant's categorical frame.
I'd reach more for the ethics, but it becomes even more confusing there lol. So I'm reaching for what's making sense to me right now to respond in kind.
How does this claim escape the charge of totalizing?
A professor I had told me about a reading of Kant more in line with an Averroist "material intellect" shared by all men. That's another solution for the slide towards solipsism I suppose. And I believe it was also somewhat normative too, the constructive mind is the "mind of Europe," in which all participate and which has been so marked by Newton and modern science
Unfortunately, I don't recall the name of the person advancing it.
Sider calls this "hostile translation." From the QV/Sider thread:
Quoting Leontiskos
@fdrake wants to talk about "good counterexamples," and he relies on notions of "verbatim" or "taking someone exactly at their word" (even in a way that they themselves reject). The problem is that if these are still hostile translations then they haven't managed to do what they are supposed to be doing: they haven't managed to produce good counterexamples.
I disagree that that is what is going on. When someone stipulates a definition, they are committed to that definition insofar as it relates to the intended concept. Rejecting a criticism of a definition on the grounds that the criticism doesn't portray your intents is a fine thing, so long as it isn't pointing out something which your stated commitments entail. Isn't this a basic idea in reasoning itself, playing out in how people codify ideas?
Indeed, you offered an alternative informal definition of logic:
Quoting Leontiskos
Which could equally mean "mind", "minds", "people", "institutions", "thought processes", "scientific experiments", "scientific theory", "perceptions", "deductive reasoning", "deductive reasoning using formalisms" and so on. Which are perhaps in the intended scope, and perhaps not.
But something like a research institution creates knowledge in a sense, and I doubt that is in the scope. And we could play the same game as we played with the formalisms out in natural language. What would make a research institution fail to be logic?
Yes. I thought it went without saying. Some things people think of are more appropriate than others in some contexts, and strictly better by some metrics. Some fiction is more valuable than others. If a thingy works better than another thingy on every relevant facet, the first thingy is better than the second thingy.
How would you judge that for a given context? Well I suppose you'd look for examples, see what pans out, provide definitions of things to see if they capture the relevant phenomena... Maybe you'd refine your criteria for what counts as a good thing in a given context based on the what you've seen and what's been created, too.
I still have the impression that you think of this is as an Objectively Correct vs Subjective-Relativist sense, and I don't want to accept the Subjective-Relativist role in the discussion since the proofs and refutations inspired epistemology of mathematics isn't relativist in the slightest, because its emphasis is on communities of people agreeing on what follows from what by following coordinating norms and demarcating those norms' contexts of application. Minimally then, it's intersubjective, and communities create knowledge about collectively understood subject matters.
If you read through Proofs and Refutations, which is an amazing book, the most clear cut resolution and associated proof of the book's central topic is offered using an entirely separate formalism than what had been considered up until that point. It was a substantial theoretical innovation and reframing that cleared away the old problems, but was nascent within them. Lakatos' approach has a dialectical flavour in that regard.
I think Meno's paradox shows that some knowledge is innate. The story we surround that with probably reflects worldview. For Plato, it meant transmission from another level of reality. We might be mysterian about it and call it hinge propositions, or we can decide it must have come from evolution.
He entered this thread with an attack not on the topic but on on me: , and maintained that personal abuse throughout. He sets out to frame the topic in strict Aristotelian terms, not talking of formal logic as it is now understood, indeed showing a neglect of that topic.
In the discussion of mathematics with @fdrake and others he repeatedly refused to consider the alternative maths on offer, insisting on framing each part of the discussion in Euclidean terms.
And now he talks of "hostile translation".
Might leave it at that. What more can one do but laugh.
It's clever, she's avoiding a semantic counter argument by using an essentially open ended term. But not in the sense of fallacy. What does this have to do with logical nihilism? Were people still under the impression there were perfect things and that needed to be addressed?
If thatâs right, then thereâs nothing in the concept of a law of logic that demands it must always apply universally. That just seems to be a characteristic these laws often exhibit, but itâs not essential to what they are. So, if it turns out that all proposed logical laws have exceptions, it doesnât mean there are no laws of logicâonly that they are more specific than we once thought.
Well to be clear, I don't think:
this is what she is doing. To do this would be to ignore what the most popular pluralists (B&R) and what most monists say about their own positions. As fdrake says, if one is allowed to appeal to "every other logical apparatus in existence," and its self-defined capacity to produce valid inferences, then it is very easy to come up with "knock down arguments" demonstrating nihilism. But Russell is willing to admit that nihilism is a slim minority opinion that is often considered "absurd," which would be strange indeed if it was a position that is easily demonstrable. Hence the argument focuses on the plausibility and [I]popularity[/I] of counterexamples, not their mere existence.
I don't really know what else to say here, SEP, IEP, the books I've referenced, and similar resources point out that this is not how the debate is defined; there is wide agreement that people have created logical systems that alternatively dispense with all of the "laws of logic" (or more accurately, would render the logical consequence relationship empty).
I feel like part of the confusion here is that this question is one of what holds for valid inference (true premises ensure a true conclusion) as a whole, across all logics, which in turn means that the common way of thinking of validity in a purely internal sense essentially begs the question here. (Russell doesn't do this BTW, although it seems this could have been made clear. Her intro on logical nihilism is clearer.)
Part of the confusion is that just how one wants to define these might vary quite a bit, although they are generally not going to be defined in terms of "every logical apparatus in existence," since I think everyone is going to agree here making the debate a bit trivial.
That's on the mark. appears to think this amounts to nihilism. It doesn't. Nihilism would have it that there are no laws of logic, that logic is at best a rhetorical device. That is not what Russell, or I think, @Clearbury, is claiming.
But Tim's view remains obscure to me. I don't see a confusion in Russell or SEP or IEP.
It seems to me that the idea that there are specific logics (specific entailment relations) for specific areas would be pluralism (at least as they define it.) Nihilism would reject this and claim that, depending on our goals and uses, we might use any logic in any setting. That is, there is, strictly speaking, no correct or singularly appropriate logic for any subject. In particular, there is no correct logic for modeling entailment in natural language.
I haven't even advocated for a position here, I have tried to clarify the monist position and how some arguments are poor responses for it. And I'd argue that if you're unable to understand the dominant position here (i.e., if it seems trivial to dispatch) then you really don't understand the debate at all.
I personally wouldn't consider myself a monist because the formalisms they advance are wholly inadequate for capturing natural language reasoning, particularly in the dimension of analogous predication, while also flattening out truth.
Articles on this topic generally refer to it as such at any rate. B&R is normally brought up as the landmark case for pluralism and it is fairly recent. Shapiro is from 2014.
Yes, which maybe should make you question if you have any clue what the debate is about.
This comment makes me question if I know what the debate is about.
What's the debate about?
It's about what "Logical Monism" is about. :wink:
In that case, clearly stipulate-able.
Thus far I've gathered that they both would like to relate to knowledge generation? I think?
This is what motived by earlier response about why the problem is interesting with respect to knowledge generation.
I'm a defender of dialetheism, thus far.
Which rules out the LNC.
Hence, the notion of pluralism -- at least so far no one has said that the logics which include the LNC are the same as the logics which exclude the LNC.
Can you explain how dialetheism rules out the LNC? My point was that within any valid logical argument of whatever stripe there must be consistency between the premises and the conclusion. If a premise contradicts another premise or the conclusion then the argument cannot be valid. That sort of thing.
Your choice of words here has me wondering if I can or not.
But I can give a straightforward answer to your question which may be aside from the point.
Quoting Janus
Quoting the SEP here:
I've been advancing the argument that the LEM holds -- because there is nothing in between truth and falseness so we cannot choose some in-between or other -- but the liar's sentence is best treated as a dialetheia.
If one accepts that then the LNC cannot hold because the LNC says "A & ÂŹA" is false. Since ÂŹA and A, as a dialetheia, are both true and false, together, the LNC is rejected.
Can you think of any examples of a sentence wherein both A and not-A are true in the same sense or context? For example I could be said to be both old or tall and not old or tall but not in the same senses or contexts.
The liar's sentence.
"This sentence is false" is the liar's sentence.
This doesn't fit your "for example", though, because it's not about a person, but a sentence.
Since the sentence can be said in any context, and it's basically about words and how we describe them, we can place them within the sense of logic.
The sense of logic can be informal or formal, and insofar as we understand one another well enough it need not be specified.
Though I'm wondering if I've just lost you at this point?
Not lost. For me the liar sentence is neither true nor false, not both true and false.
So it's para-consistent?
I would only consider dialetheism to be justified if I could think of an example of a sentence which is demonstrably true and false in the same sense and context. That said if it were somehow justified I guess that might justify logical pluralism.
Perhaps. What do you think?
That there are such logics makes it difficult to maintain that all logics must be valid and consistent.
And these logics do have some uses.
I don't know much about formal logic and perhaps there are formal ways of making invalid consistent and valid inconsistent logical posits work and even do work. I am interested only in what can be parsed in ordinary language.
Then presumably you conclude that paraconsistent logic is not logic proper? And isnât the liar in ordinary language?
All this simply to show what the interest here is.
Right it seems that is what my position entails. The liar is in ordinary language and as I said for me it is implicitly self-contradictory from which it follows that it is inconsistent and invalid and neither true nor false.
Can you think of a propositional sentence in ordinary language which is not self-contradictory that is both true and false or neither true nor false?
I doubt it and thus conclude the LNC holds in all valid logics.
It's not a long read.
Tell me what you think fo the notion of "overloading" logic with expectations.
It starts with the present King of France and why it's false that he's wise. How do we evaluate a proposition whose subject doesn't refer? Meinong attempts to help by inventing the idea of possible objects, which subsist instead of exist. The present King of France is such an object, and so the subject does have a reference. I actually like this view, but it was objectionable to Russell, who felt like this theory would cause the universe would become overcrowded, but also because this theory leads to misconceptions about what people actually think and intend to say.
Russell decided that it must be that this proposition is compound. When you start a sentence with The present King of France, you're asserting of the universe that it contains this object. And next, you're asserting of this object that it's wise. So now that we've broken the P down into q and r, we have a way to explain why P is false: because one of its parts is false. Everybody loved Russell for coming up with this way of looking at it.
@Banno
So one of the strawmen I think Peregrin is lighting ablaze is the idea that someone somewhere thought logic is the end-all to what goes on in the human psyche. No. It wasn't supposed to be that. I'd like to introduce Peregrin to analytical philosophy: the land of temperance and little tiny answers to little tiny questions.
Quoting fdrake
Whether or not it is what is going on, it is what is at stake, and thatâs the point. Your construals avoid the problem of intent, and intent is the crucial aspect (e.g. when you talk about âverbatimâ or âtaking someone âexactlyâ at their wordâ).
Quoting fdrake
I agree, but I really donât think your approach in the discussion of square circles manifested anything like an attempt at close reading or an investigation of intended concepts. It was more an exercise in interpreting utterances as they suited your purpose (of arguing for square circles). Granted, it is no wonder that a polemical and insubstantial thread continued in polemics and lack of substance. You and I were just following Banno's lead in this, and it is why Banno should not be allowed to set the pace.
Quoting fdrake
And that was quite intentional on my part. When dealing with people prone to misrepresentation it is best to give a starting point which either makes them think or ask a question. If they do neither one then they show themselves to be uninterested in philosophical discussion. It is in no way surprising that Banno managed to do neither, and after dealing with this for long enough Iâve just put him on ignore. Indeed, my earlier definitions were more specific, and the later ones became more general in proportion to my realization that the instigators were not willing to look outside their paradigm.
-
Quoting fdrake
Iâm still not seeing a straight answer. Why? Because you claim to be talking about metaphysics but then you qualify everything by words like âcontext,â âvalue,â and notions of artifice. Earlier when I asked if there is better and worse âfan-fictionâ you again cleaved to the metaphor and gave examples of literal fan fiction.
Quoting fdrake
So then do you think intersubjective agreement is metaphysics? Is that the goal? To try to garner agreement? The democratization of science?
Iâm perplexed at how impossibly difficult it is for folks on this forum to think about metaphysics and to escape modern immanentism. Truth has been so thoroughly deflated that folks around here canât even recognize the notion of truth when it shows up at the party. âCommunities of people agreeing on what follows,â is a very common substitute, but also a very bad substitute! When peer review and intersubjective consensus shifts from a helpful aid to truth, to truth itself, something very problematic and bizarre has occurred. What began as, âA number of instruments agree, therefore they are probably telling us the truth,â shifted to, âA number of instruments agree, and weâll just define that as truth qua truth.â This is substituting truth with agreement; metaphysics with intersubjectivity. This is a significant misstep. Einsteinâs physics is not superior to Newtonâs physics because more people agree with Einstein. It is superior because it has more purchase on what is actually occurring in reality; because it is truer. Agreement is an epistemic criterion, not a metaphysical criterion.
The modern world is merely anthropocentric. We have made everything about ourselves, our desires, and our values, so that this is all that even exists. To talk about something beyond that is not allowed. Science, metaphysics, and truth are barred at the gate, even to the point that we cannot say what a woman is.
That's an interesting background explanation for why the "Liar's paradox" tempts you, but what I am hearing is that you are interested in playing a game that has nothing to do with reality. You have not answered the objections, and I don't see that Marx and Hegel have much at all to do with this issue. When you talk about "truth" and "falsity" you are not talking about truth and falsity; you are equivocating. We could play an arbitrary game and call the Liar's paradox "false," but we cannot call it false, and I have explained why.
I think this is all symptomatic of the decadence of contemporary philosophy, which is more a matter of novelties and entertaining oneself than actual philosophical engagement. On this point, there was a recent article about the filmmaker Terrence Malick and his encounter with droll contemporary philosophy, "Malick the Philosopher." This form of philosophy will be made to reckon with its own vacuity.
Quoting Moliere
Something like that. I would say that the so-called "monist" accepts that people can be wrong about things, and that that is probably at the pragmatic core of this thread. Truly, there is a mystery about how error can occur. But this was never a real thread. The people behind it were never interested in giving real arguments for their position, or even attempting to distinguish "monism" from "pluralism."
Edit: I realize this was curt, but I don't see the conversation going anywhere and so I am just setting out my view. I take it that Epictetus is much more interesting, substantial, and philosophical than the "Liar's paradox."
:blush:
Reality is what's interesting here -- what I don't want to do is define reality within my logic, though. And I don't think that logic needs to restrict itself to objects since reality is not composed of objects and objects only -- it also contains sentences.
As I see it right now the objection is we don't agree on what a pluralist logic would even mean. I've asked you if you'd accept a defense of dialetheism, the belief that there are true contradictions, as a basis for making the inferences that there is more than one logic.
Unless you answer that question it becomes rather hard to answer your objections since we don't have an agreed upon notion of pluralism. I've already laid out, with the liar's sentence, why I accept dialetheism. Marx and Hegel are philosophers which, like the liar's, utilizes contradiction in their reasoning. My thinking here is to ask if you'd accept that as a basis for dialetheism.
So what do you say?
Well you can't say what it means, you can't say what a sentence is, you can't say why it would count as a sentence, you can't say how it would ever have any purchase on reality, and you don't seem to think it would ever be utterable in real life. That's a pretty problematic place to be. Again, it looks to me that you are playing a game that has nothing to do with reality.
Quoting Moliere
The objection was given <here>. You tried to answer it by redefining "false" as "fake," and I think we both agreed that that answer failed. That's where things stand, as you never made another attempt.
Quoting Moliere
Sure: if dialetheism is true, then strong logical pluralism is true.
Quoting Moliere
No, they don't. This is equivocation. Neither one has anything like the standing contradictions of dialetheism. Tensions which go on to get resolved are nothing like the stable contradictions of dialetheism.
What I said was
Quoting Moliere
It's the object that's different which changes the meaning of "...is false", which is why these examples don't work. Since the liar's sentence is a sentence the usual meaning of "...is false" works just fine.
I didn't redefine the predicates but pointed out how your counter-example didn't stick.
Quoting Leontiskos
Cool. Then it seems that an argument for dialetheism is very much on topic then, and the liar's sentence is what I'm proposing as a dialetheia
Quoting Leontiskos
I disagree. The moment of sublation in either Hegel or Marx is not a singular moment which is separable from their negations, but is rather the composition of negations and the negation of that composition. Without recognizing the unity of the opposites -- contradiction -- sublation wouldn't be recognizable as a distinct moment in the logical process.
Now that may very well be the case in fact, but conceptually speaking it seems you at least have to accept contradictions which are operable in some fashion in the logics of each philosopher -- not two opposing things that happen to yield a third thing, but rather the two opposing things very opposition is connected to this third thing in a relationship of inference, where the contradiction is part of the inference, and is not a reductio.
I set out the meaning of the liar's here:
[quote]
Quoting Moliere
It's about the number of correct logics (i.e. logics that ensure true conclusions follow from true premises). In general, it's a position about [I]applied logic[/I], which is why monists and pluralists often justify their demarcation of correct logic(s) in terms of natural language, scientific discourse, etc. Nihlism would, by contrast, say there are no correct logics (and also no incorrect ones). This is not to say that reasoning is entirely arbitrary, presumably there are [I]some[/I] standards for what constitutes appropriate reasoning. But there is no logical consequence relationship that is appropriate or correct for any particular topic. So, for instance, the intuitionist and his rival in mathematics are both wrong in that neither are "right."
You could think of this as similar to how there are very many geometries, and unfathomably many possible ones. One can identify what "follows" from their axioms according to whatever logical consequence relationship one cares to use, but this doesn't necessitate that the geometry of the physical world is infinitely variable or that it lacks any "correct" geometries. We tend to think that there would be just one geometry for physics (at least physicists normally do), or that, if there were many, there would be morphisms between them. The claims of the monist in particular are roughly analagous to the claims of the physicist re geometry. For instance, when Gisin recommends intuitionist mathematics for quantum mechanics, he does not mean to suggest that this is merely interesting or useful, but that it in some way better conforms to physics itself in ens reale, not just ens rationis.
Normally it gets framed in terms of the entailment relationship. This avoids unhelpful "counterexamples," like competing geometries that use some different axioms, but nonetheless have the same underlying entailment relationship. These are unhelpful because the question isn't about "what specifically is true/can be known to be true given different axioms" but rather "how does one move from true premises to true conclusions." This is why monists might also allow for multiple logics that are "correct," the "correct logic" being more a "weakest true logic."
So, is a fine example of the basic intuition at work in rejecting some logics for some contexts (pluralism) or holding to one logic as truth-preserving (monism) vis-ĂĄ-vis natural language, a metaphysical notion of truth, etc.
And 's "a thing can't really be otherwise or not," would be a similar sort of reasoning. Dialetheism is normally argued for in the context of paradoxes related to self-reference (as has been the case in this thread). I think critics would argue that these are no more mysterious than our ability to say things that aren't true (which perhaps IS mysterious). At any rate, the "actual" true contradictions that get thrown out, in the SEP article for example, etc. tend to be far less convincing. For example, "you are either in a room or out, but when you are moving out of a room, at one point you will be in, out, both, or neither."
I don't think Hegel is really a good example here because the Absolute is the whole process of its coming into being, in which contradiction is resolved, and contradictions contain their own resolution. It's examples of contradiction, being's collapse into nothing, etc. are very much unlike the standard examples meant to define dialetheism.
I think for Hegel a thing contains its opposition. So for redness, non-redness is part of what it is. Everything you think about is like that. You think in oppositions. But dialetheism would be a mystical state of mind?
Correct, although not everything in the Logic follows the formula of "thing" ? "negation" ? "negation of negation," some get a good deal more complex.
I think Pinkard is right that Hegel is in some sense very Aristotlean (even if I think Pinkard generally deflates Hegel for modern tastes). Hegel wants to track down the necessity in everything, the intelligibility of concepts. In his book on Hegel, Robert Wallace uses "red" as an example. We don't just have "red" implying "~red," but rather red implies the entire category of color and the things that can be colored (primarily light; nothing is red in total darkness).
You can see the strong Aristotelian bent in the last paragraph. But Hegel, living in a time where atomism is ascendant, cannot leave things as "unpacked" as Aristotle does with his vaguer concepts that cover more ground. However, maybe Big Heg should have listened to Slick A's advice in the Ethics re "don't demand that your explanations be more exact than your subject matter allows."
I agree with Wallace. I think the same idea is in Phaedo as the Cyclical Argument.
Me too. However, I also think the sense of "contradiction" here is quite far from that invoked by religiously motivated dialetheism or those motivated largely by problems of self-reference. It's quite different. But to 's point, I am not sure how much this carries over to Marx. I have read a lot of Marxists but not much Marx, so I am not really in a position to have a strong opinion on that front.
At any rate, Hegel affirms LNC in its usual contexts, but I think it's fair to call him a monist if anyone is. The role he has for logic is deeply ontological.
The situation Hegel is pointing out isn't paradoxical, if that's what you mean.
Quoting Count Timothy von Icarus
The secondary source I read said that Marx didn't use dialectics much, but I'd be interested to see a case where he did.
Quoting Count Timothy von Icarus
Sounds about right.
I'm not sure I'd go as far as to say "correct" in describing a logic. What would it possibly mean for a logic to be correct in a non-question begging way? "Correct" seems to already presume some standards of coherency, and I'd say validity is a species of coherency.
That is, we'd be presuming some logic in setting out the correct logic. Now if there were only one logic that would at least be consistent, but then we get to the part on begging the question -- which, I think, is why the puzzle is interesting: Either answer can be made self-consistent (monism or pluralism), but in what sense can the two camps speak to one another?
[/quote]
Can you fill out this analogy?
Geometry:Physics :: Logic:D
What takes the place of "D" here? I understand the relation between geometry and physics, but also by the time we're talking geometry and physics it seems a logic, an epistemology, an ontology are already in play for the purposes of producing knowledge -- Also I'm not sure that the analogy serves the monist very well because geometers do geometry outside of the bounds of physics, and so we'd presume the same would hold for the logicians?
Quoting Count Timothy von Icarus
I'm not sure the entailment relationship ends up being any more stable than the LNC or the principle of explosion. Pick your hinge and flip it!
When you say
There's a quibble I feel that may indicate some miscommunication (or not, we'll see).
The question for logic, IMO, is not "How does one move from true premises to true conclusions?" -- I'd say that's a question for epistemology more broadly -- but rather logic is the study of validity. The big difference here from even introductory logic books is that the truth of the premises aren't relevant, which I'm sure you know already -- the moon being made of green cheese and all that.
So we don't care if the premises are true or not. We only care that if they are true, due to the form of inferences, that the conclusion must be true.
Do you see a difference between the questions?
I'd say your question asks for evidence or rationation, whereas the study of validity will depend upon how we define our logic.\
Quoting Count Timothy von Icarus
Just to be clear, and I have not been so sorry, I'm not presenting Hegel as a dialetheist, but rather as a philosopher that uses contradiction in his reasoning -- since the conclusion to a contradiction is not "Meaningless" or "simply false" it strikes me as different from the older assumption of the LNC.
Also, you've mentioned it but, what makes Hegel an interesting case is his simultaneous acceptance and modification to the LNC. He accepts the LNC in its own context (i.e. outside of time), but when time gets involved he introduces a new inference -- sublation -- to manage the contradictions of becoming.
This isn't to say that he's a pluralist, either. I agree that if Hegel were anything that "monist" makes sense. It's only to say that in order for us to make sense of Hegel we have to be able to evaluate contradictions without rejecting them out of hand, and so at least the logic which makes sense of Hegel must reject the LNC.
Perhaps the core issue is whether there are logical laws that hold in every case. Given boundless human creativity, it is at least conceivable that whatever one posits as a logical law, a counterexample can be constructed. Russell gives examples of counter instances for identity, And elimination, excluded middle, and modus ponens. Whether these are thought successful or not, to rule out the construction of such counter instances is claiming that there is a one true logic that permits such a ruling. Exactly how and if such a logical monism might stand is one of the themes of this thread.
The opposite view would be that there are no rules that hold in any case. On this account logical reasoning has no compulsion, being little more than a rhetorical device. Exactly how and if such logical nihilism might stand is one of the themes of this thread.
Contradicting both these is the view that while no laws that apply in every case, there may well be laws that apply in some cases. On this account there might be a logic applicable to particular case or situations, but not in all cases or situations.
Russell proceeds by considering examples of mooted laws of logic and offering counter instances. You can get an idea of these by reading the paper or watching the video mentioned on Page One. The discussion concerns formal logic, and presumes some familiarity with that terminology and method. Those seriously considering the issues of the paper, video and of this thread should have at least some background in formal logic.
The logic talks at a meta level, so it talks about sentences, represented by greek letters such as ? and ?, phi and psi, which are part of a language ?, together with the usual connectives logical connectives. In addition she uses the Turnstile, ?. This represents the logical truth of sentences, so that "??" can be read as "Phi is true", and "???" can be read as "Phi is true in Gamma". The topic presumes an understanding of the idea of truth as satisfaction, and there is some mention of possible worlds. These are things that folk who presume to philosophical discussion ought at least have some clear grasp.
The argument presented is a defence of the use of logic in the face of the strength of logical nihilism. If you have an interest in the topic, please take some time to look at the video or read the article. Some who have commented here have done so without that due diligence, for reasons of their own, and so entirely miss what is going on.
Just to be clear, this isn't my term, but the term employed through much of the literature on this topic, including the papers discussed earlier in this thread.
However, I suppose the response would be: Are there not inference rules that allow us to move from true premises to true conclusions, such that if our premises are true our conclusion will be as well? If so, then it seems there are "correct" logics. Unless we want to say that all inference rules lead to true conclusions, making the distinction meaningless (this seems hard to defend), or that no inference rules lead us from true premises to true conclusions (this also seems hard to defend, for how would one show that such an argument makes legitimate inferences?)
I'm not sure how this would be question begging. Logic deals with valid inference, how we get from true premises to true conclusionsâtruth preservation. Presumably, it doesn't define truth itself, so the criteria for determining which inference rules (if any) preserve truth in which contexts (if any) is external.
How would you define validity?
"A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid," is the textbook answer from IEP. The textbooks I've used give the same definition.
Stanford's open introduction to logic puts it thus: "Valid: an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false."
I am aware that some scholars have tried to redefine validity in normative terms, e.g. that it is "what we should or shouldn't accept." The Clarke-Doane paper Banno shared is from this camp. However, I have never seen such a view presented that does not assume a deflationary account of truth, that "truth " as most people think of it, does not exist.
Well, that's a fine argument to have. But it gets to the point I tried to make to Banno and fdrake that one cannot retreat into formalism and ignore discussions of truth on this topic. If it would be question begging to assume that logic is about truth-preservation then it would be equally question begging to say that truth depends on / is defined by normative or formal contexts. If the latter is accepted, then of course nihilism is true (or rather true relative to some contexts and false relative to others, depending on our normative games.)
Now the arguments for deflation are abductive (what would it even mean to "prove" such a thesis?) But like I said before, it's hard to think of things it's easier to make a strong abductive argument for than: "in many cases what is true does not depend entirely on how we choose to speak or which formal system we use. It is true that if you dip your hand in boiling water you will be burned in a sense that transcends social practice or formalism." And if we take logic to be wholly normative, e.g. "you ought not stick your hand in boiling water if you don't want to be burned," it seems that we will still have the question "why ought we not do this?" The answer: "because it is true that boiling water causes burns," seems like the most plausible one, but then we are back to truth.
Yes, this is soundness versus validity. However, this distinction need not (and normally isn't) taken to imply that logic isn't about truth-preservation. The debate is about the rules of truth preservation, not about the truth of any particular premises in an argument.
Well, that's at least normally how it has been defined and it's been defined that way because the mainstream view of logic is that it is (largely) the study of validity, with validity being about truth preservationâi.e., how one goes from true premises to necessarily true conclusions. Obviously if we redefine validity this might make less sense.
But I think there is maybe a misunderstanding here because if you remove LNC you are changing the logical consequence relationship. What follows from what (the logical consequence or entailment relationship) depends on LNC, LEM, relevance conditions for implication, etc. The nihilist claims this relationship is empty, nothing follows from anything else (in any correct sense, i.e. ensuring truth preservation).
These work just fine by me which indicates we're just talking past one another.
What say you to 's proposal? Does it seem to sidestep something important, in your view?
The reason I've been delving into historical examples, and I [s]have[/s] hope I haven't gone too far afield @Banno, was to tie some of the above into the argument for pluralism: if we accept contradiction into our reasoning, and we also reject contradiction in some of our reasoning then we are pluralists.
Moving to that because of the incredulity of dialetheia, which is where originally I staked my flag in defense of pluralism.
Sorry if it was too off topic though.
What would be an example of that?
If the liar's sentence is true then the liar's sentence is false.
If the liar's sentence is false then the liar's sentence is true.
The law of excluded middle states there can be no other values for a sentence than true or false.
Therefore the liar's sentence must be true and false, or not-true and not-false.
Though this doesn't get over the hurdle of relevance, which I think has what's mostly been at stake in various responses here -- the liar's sentence isn't useful in some context outside of philosophy and so it seems like a toy which ought to be viewed as such, whereas the knowledge we actually use in the real world is girded by a firm bivalency we not only can stand atop but have not choice but to do so or be in error.
Now that's when we're doing philosophy! :D
Quoting Count Timothy von Icarus
One thing I'm guessing is that arguments for any logic, due to the generality of the topic, will by their very nature always beg the question -- otherwise the logic wouldn't be consistent with itself! And that's a terrible place for a candidate logic to be.
The point from there would simply be to demonstrate more than one logic -- one which results in a "F", where the other results in a "T" or there's perhaps another value other than "F" or "T". The trick is in being able to evaluate the logic without the logic. How can it be done? I think that's the puzzle, in a nutshell.
What do you think the term "correct logic" means in Russell's papers, G&P, Clarke-Doane's paper, etc.? I know you don't like the term, but you refused to elaborate on what you think means.
If "correctness" was simply satisfaction there wouldn't be any debate.
If you want to make use of the term, then you can set out what you take it to mean.
I haven't started excluding middles quite yet. Not suggesting a paradox either. I'm saying a tautology is the truth relative to your point of view. Which is the case where you don't know the truth about P. P is not (P or ~P), you are (P or ~P) about P. If we have to jump to cases of ambiguous Ps to support the tautology this early, we may have another pointy circle.
Unlike the brickhouse arguments in Objective Spirit?
Well no, it's how she defines the entire problem, and it's how she defines it in her introduction on nihilism. It's also how Clarke-Doane defines it, and C&P , and SEP, etc. CD cites the following paper as representative on its opening page: "Logical monism is the view that there is only one correct logic or, alternatively, the view that there is only one genuine consequence relation, only one right answer to the question." SEP opens with: "Logical pluralism is the view that there is more than one correct logic. Logics are theories of validity: they tell us which argument forms are valid. " Or in defining pluralism: "Logical pluralism takes many forms, but the most philosophically interesting and controversial versions hold that more than one logic can be correct, that is: logics L1 and l2 can disagree about which arguments are valid, and both can be getting things right."
No idea at all?
It's the term used to define the problem. I have tried explaining what people mean by it and you have acted like this is unfathomable. So I am curious exactly what you think you're reading about or discussing when you bring up this topic?
"Correct logic" is not a term defined in formal logic. That's rather the point here. You will not, for example, find a definition of "Correct Logic" in the Open Logic text. But you will find definitions of validity, satisfaction, truth and so on. These are the terms used by logicians when doing logic.
If you are so sure that there is a correct logic, all you need do is present it. What is the "consequence relation" that is to be found in all logics that renders them either correct or... what?
No, but you do see the term all over articles written by logicians on the topic of logical pluralism vs monism vs nihilism. Beale and Restall define their pluralism in these terms for instance (and as there being "multiple true logics"), Paseau and Griffith's define their monism in these terms.
No clue?
I have to say, the inability to answer strikes as akin to someone staking out a position in favor of nominalism and being unable to even define what a universal is. One need not think universals exist, or even be able to give a "philosophically adequate account" of them to defend nominalism of course, but it seems necessary to understand what is generally meant by the concept to even understand the basics of the debate. This is similar.
Where?
On the opening pages of their respective books, for the most obvious example.
Do you really need to check new sources and not the papers you yourself cited in this thread? Where those lines in the opening paragraphs just an incomprehensible muddle to you?
A rough outline of a direction in which a discussion will go does not amount to a definition.
In summary, in a discussion of logic, you are demanding I define a term that is not defined formally, for something that I doubt exists, but is central you the account of a One True Logic, that you have been unable to present.
Why would I take that seriously?
I'm not asking you to give a philosophical account, I'm asking you to show you have a basic understanding of the topic. It's an outline... "of what?"
I don't buy into trope nominalism, but I can explain what it is. You seem unable to do this for the positions under discussion.
As things stand, I doubt you have the capacity to tell who has a " basic understanding of the topic".
You are trying to play "gotcha", but you've fumbled the ball.
Ok, so you cannot define logical monism or pluralism.
From my OP
Quoting Banno
Don't be a goose.
Yes, I am aware you can copy and paste. You apparently cannot define what the term "correct logic" used in definitions of the problem in all these papers means though.
I doubt that a suitable definition of "correct logic" can be provided. Therefore I will refrain from providing one.
If Tim wishes, he may present one for our inspection.
Well no, the most cited monograph on pluralism, Beale and Restall, says there are multiple "correct/genuine" logics. The opening sentence of Russell's article for SEP on Logical Pluralism is: "Logical pluralism is the view that there is more than one correct logic."
If pluralism denied that there were any correct logics, how would it be distinguishable from nihilism exactly?
Anyhow, this is really not a "gotcha question." Or it shouldn't be.
Well, yes. "The notion of A correct logic" - singular.
Ok, can you give a definition in the plural?
That's a question I ought take up, given I'm defending pluralism and poo-poo-ing the idea of correct logics, at all.
Nihilism states there's no logical laws. Pluralism states there are more than no logical laws, and more than one logical law. Though "law", by the pluralist, is funny here. My thought is that "law" is stipulative -- my suspicion being that all arguments for a logic must beg the question the only way to evaluate a logic is to develop and utilize it in some fashion.
I'm thinking that the monist thinks there is, at the end of the day (ultimately?), only one set of logical laws that cohere together. The pluralist can accept laws insofar that they are limited in a non-lawlike(logical inference rule that fits within the logic) fashion. The nihilist states that all logical so-called laws are matters of preference -- something like a poetry of rhyme, but with ideas.
Well, validity is decided by giving a logic an interpretation. So that's pretty much correct.
I think thinking in terms of "laws" is probably unhelpful here and I have never seen a monist argument that tries to define itself in this way. If by laws we mean "true for all existing logics," then there are clearly no such laws. The monist doesn't argue that such laws "hold in generality," except insofar as they hold for "correct logic" (as they variously define it; note also that most monists embrace many logics, the question is more about consequence). So, Russell's paper is fine overall, but I think this part has just confused people because it's easy to read it in a way that seems to make the answer trivial. But based on the fact that even pluralists themselves very often claim that they are in the minority, it should give us pause if monism seems very obviously false.
This is the right intuition from my understanding.
I asked this question in some venues that are more restrictive about who can answer questions and here are the replies (pace Banno, no one found this question leading or question begging):
This is in line with how G&P and Priest define their arguments for monism and how B&R define their argument for pluralism (i.e. with reference to natural language). These are, of course, far more technical as they try to make these notions more precise, but that's the basic jist.
In terms of what constitutes "correct" logics, people do have other answers aside from using natural language as a target. Some use scientific discourse/formal theories, etc. It isn't cut and dry, which is why you frequently find appeals to popularity and more ambiguous "plausibility" arguments. Just for a good outlier example, some logics are trivial. One can prove anything expressible in them. They might have a notion of satisfaction, but there is clearly a plausibility issue when a system that allows you to prove anything is said to have correct rules for "truth-preservation."
But some people frame logic as a normative practice, as being about what we "ought" to affirm. Others, influenced by Wittgenstein, think of it in terms of assertability criteria. This response gets at that:
I think part of the confusion is that, just as idealism is much more popular on TPF than in metaphysics as a discipline, highly deflationary conceptions of logic's subject matter are also much more common. But one might agree to a deflation of truth for the purposes of doing logic without embracing any robust notion of deflation, e.g. that "on 9/11 the Pentagon was struck by an airliner not a cruise missile," is true or false in a sense transcending any formal construct or social practice. Maybe not, I only know of two surveys on this question, but they do seem to bear this out, as does the way authors actually talk about non-classical logics (i.e. they spend a lot of time making plausibility arguments, which are superfluous of logic is just about formalism).
To quote B&R:
The response here is quite good too:
https://www.reddit.com/r/askphilosophy/comments/ggklhq/what_are_the_arguments_against_logical_pluralism/
I would just add that the background assumption for looking at natural language and scientific discourse seems to be that reasoning here deals with some notion of truth qua truth (even if we think the notion ambiguous ).
"Correct" in that quote basically means appropriate. It has nothing to do with truth.
It's about the appropriateness of a logic in mirroring natural language notions of logical consequence and validity.
How is validity defined in most natural language explanations? Something like: "an argument is valid if the truth of the premises logically guarantees the truth of the conclusion."
To say something "follows from" or is "entailed by" something else in natural language is to make a statement about the relationship between the truth of the first claim and the truth of the second.
Entailment has variously been described in terms of sentences, facts, states of affairs, etc.
It seems a bit much to say that notions of "reasoning in the vernacular," re validity and entailment have "nothing to do with truth."
For instance, the metaphysical argument for monism of Sider has "nothing to do with truth?"
What is "appropriateness" then?
The preceding paragraph actually deals with just your conception of logic.
You've got the appropriate logic if it fits your purposes with regard to a specific domain.
And can one have correct purposes, or can one's purposes be defined arbitrarily? The purpose here is to capture natural language understandings of good reasoning and valid argument. Is:
"You've got the appropriate logic if it fits your purposes with regard to a specific domain."
The vernacular understanding of what is meant by "good/correct reasoning?"
I feel like the response I linked answers this pretty well:
https://www.reddit.com/r/askphilosophy/comments/ggklhq/what_are_the_arguments_against_logical_pluralism/
It's arbitrary that you want logic to capture natural language good reasoning. If I need faster than light travel, I may need an alternative to natural language. Are you saying I can't have that because of your sensibilities?
Is it arbitrary?
See:
I do not think it's plausible to say that trivial logics in which everything expressible can be proven true are only arbitrarily bad for inference for instance. Do you disagree?
The opening lines of the SEP article on logical pluralism acknowledge that the idea seems crazy at first glance, but that it becomes more plausible on further examination. I found myself getting more of a handle on it when reading the objections to it. It's all pretty technical, and that's not really something I'm super familiar with, but I did get that logical pluralism isn't taking anything away from the regular logic.
I get that you're preoccupied with issues surrounding truth, but that's not a significant aspect of this issue. Check out the SEP article if you want.
You can certainly argue for nihilism from robust deflation, but the position that it is obvious or widely accepted that validity and logical consequence "have nothing to do with truth," is belied by a look at any introductory text on logic.
You could refer to the open source ForAllX (which is very much focused on formalism), but which still defines consequence in its opening pages thus:
"For the conclusion to be a consequence of the Ppremises, the truth of the premises must guarantee the truth of the conclusion. If there is a counterexample, the truth of the premises does not guarantee the truth of the conclusion."
Or you could look at a more advanced text like the Routledge Philosophical Logic, which distinguishes between "truth simpliciter" and a "relativized notion of truth: truth in a model," and how the latter was historically developed from as a means of capturing the former.
Notions of truth outside formalism are called on all the time though. For instance, this highly cited piece by Priest (one of the major figures on dialtheism) on paradoxes of material implication.
You're just not going to read anything about it. That's cool. :up:
Read what? Obviously I can't read the sources I just quoted since they disagree with you.
I mean, on your view that "virtually all logicians embrace deflationary theories of truth," don't you think it is a little strange that:
A. They largely responded to a survey rejecting that position and;
B. That the most used introductory text book for logic in the English speaking world begins by discussing validity in terms of true conclusions or relating formalism to states of affairs on its opening pages, with nary a single mention of deflation in the whole text?
Neither of the two most cited arguments for pluralism, Beale and Restall or Shapiro argue that trivial logics should be considered correct.
Beale and Restall only endorse a few sub-classical logics and Shapiro based his "eclectic pluralism" on use cases in mathematics.
Pluralism is not the position that all logics are correct. It is the position that more than one is.
The position that any logic is correct is more in line with nihilism, although the nihilist will simply reject the idea of a correct logic.
Good posts. I would still say that until someone proffers logical pluralism, it will just be a moving target. When we talk about "logical pluralism" we are apparently talking about something that no one on TPF holds. And if someone on TPF wants to say that they hold and defend "logical pluralism," then they are the one who needs to tell us what the hell they mean by it, lol. Until that happens the wheels will continue to spin without any traction.
I don't think it's that hard to define at all. Some posters in this thread seemed to pick it up intuitively. Aside from B&R's book, they have shorter articles, and this question has been answered succinctly in many places. Their argument is roughly that the intuitive/informal notion of logical consequence is multiply-realizable (granted it is more technical in its details).
I find this version of pluralism quite plausible. The most obvious example of ambiguity in natural language is propositions about the future, which, given some (fairly popular) assumptions, are indeterminate, rather than true or false. And this is one that has a long pedigree, being discussed since antiquity (arguably being endorsed by Aristotle from the very outset on the readings of many commentators).
One could also argue that the intuitive/informal notion of logical consequence is irrelevant. I think that's a tough argument to make, but it's a possible one. What is bizarre to me is claiming that this must be the case and that anyone who disagrees has utterly failed to understand what the topic and logic as a whole.
This is a typical response, but based on some takes in this thread typical answers to this question are all way out in left field.
Active academic philosophers and logicians have been surveyed on this BTW:
https://survey2020.philpeople.org/survey/results/4858?aos=37
As Chalmers notes in the paper on this, "pluralism" was the most popular write in option so this overstates the commitment to classical logic, but not entirely.
I'm fine with another rendition other than "laws" -- that's just usually the word that comes up. I don't think they are literal laws though, not even in the "laws of nature" sense.
I ought say that I don't think monism is obviously false. I'd say monism is kind of the "default" position when we start logic, if there is a default position at all -- strictly speaking it seems to me that monism/pluralism/nihilism are more philosophies about logic than logic proper. It seems when we're doing strictly logic it wouldn't matter for the purposes of pursing the logic whether there are one or many logics (or consequence relations, as you put it). But the impression that logic gives with its generality seems to indicate there would not be another set of logical rules that lead to different consequences -- that would violate the law of non-contradiction.
I'm thinking this (very consistent!) holding onto the LNC is a part of why these developments have taken so long to be achieved.
Quoting Count Timothy von Icarus
Oh, certainly*. For my part I think the metaphysics of truth ought to be set to the side for purposes of the question -- I'd say if our metaphysics of truth can't accommodate our logic then it's our metaphysics that are in error. Hence the motivation to develop a logic sans-metaphysics, insofar that it's possible. It seems to me that acknowledging the implications of a logic without commitment is about as close as we can get there. I agree with the part of your quote here:
*EDIT: Certainly, the positions on TPF are a niche that's not representative of the academic community. And though I respect and even rely upon the academy I'm pretty sure my philosophical sympathies are not exactly academic.
I've mentioned the absurd as my metaphysical stance to kind of hint at why this is interesting to me -- I take the absurd as something of a starting point now-a-days. Reality at least seems chaotic and random enough to support a multiplicity of necessities that disagree.
So, no, my stance is not metaphysically innocent at all. In some ways Priest was appealing because he laid out a more coherent way of talking about these absurdities that seem real but are difficult to put into philosophical words.
I'm very much avoiding basing logic on either science or natural language reasoning even though I think natural language reasoning -- or informal reasoning -- is the origin of formal logic.
It seems to me logic is a bit like math (while not being reducible to math) in the way that it can be developed or "discovered".
My background epistemology of "guess and check", very much inspired by my understanding of science, definitely feeds into my motivation for a pluralistic philosophy of logic -- but I'm trying to avoid claiming either the mantle of science or the common sense of natural language reasoning in making my point. Which is probably why it falls flat.
Providing links or references to sources and quotes, and linking mentions, are basic courtesies.
https://www.reddit.com/r/askphilosophy/comments/1ggtpw0/what_is_a_correct_logic_re_monismpluralismnihilism/
https://philosophy.stackexchange.com/questions/118553/what-is-a-correct-logic-re-the-logical-monism-pluralism-nihilism-debate
But that has not been done.
So it remains that logical monism is an act of faith rather than a conclusion.
The discussion of monism, pluralism and even nihilism is ongoing, not settled.
A curiosity I came across in the Philpapers survey. The analysis examines correlations with other questions, most of which are to do with anti-realism and contradictions and such, where the correlations seem related. But then there was this:
There is a correlation between philosophers who reject abortion and accept only classical logic. What to make of that?
Well I should note that the quotations I shared are from some other places as well, I was collating them because I discovered that (almost) the same question had been answered several times before.
But as far as I can tell, they are all generally saying the same thing (feel free to search for "logical pluralism" though), which jives with my understanding of the question.
But the view that there are multiple correct logics or none wouldn't require act of faith?
Quoting Count Timothy von Icarus
Not at all.
Despite being accused of engaging in a "polemic" by , I continue to think the issue both interesting and open:
Quoting Banno
Do you think that the discussion is closed?
That is interesting.
Is there a correlation (from what you have seen) between those philosophers who privilege the classical tradition (ancient Greeks) and conservative politics?
After a bit more searching, there was also this:
Philosophers who reject god seem more willing to reject classical logic. Not unexpected, perhaps.
There is a lot of interesting stuff in there, and I really wish they had it on a platform that made it easier to slice and dice the data, because you could also look at the correlations by specialty area and I think that's almost as interesting.
The one trend I find amusing is that as one goes further back in time for historical specialties philosophers get less and less confident that "philosophy makes progress."
It would be interesting to do a similar survey of amateur philosophy spaces and compare. Idealism, epistemic nihilism, and anti-realism for instance are much more popular here it seems.
I would expect some big variance on key philosophers:
Aristotle and Kant make sense for people who go into academia. I imagine Nietzsche would be vastly more popular writ large. Judging from what bookstores carry he is by far and away the most popular.
Quoting Cheshire
No. A tautology is a formula that is satisfied by every interpretation. No contradiction is satisfied by any interpretation. Therefore, no tautology is a contradiction.
And
P v ~P
is not a contradiction.
Quoting Cheshire
What specific remarks by Godel are you referring to?
Quoting Cheshire
From Godel-Rosser we have certain systems that have self-reference and are (if consistent) incomplete. What do you mean by "break down"?
/
Quoting frank
The paradox pertains to any 2-place relation, not just the 'member of' relation.
Irrespective of set theory:
For any 2-place relation R, there is no x such that x bears R to y if and only if y does not bear R to y.
/
I don't recall the post, but in this thread (or another?) someone mentioned LEM in relation to the liar paradox. We don't need to refer to LEM for the liar's paradox. The contradiction is obtained even without LEM.
We don't need to argue this way:
L -> ~L
~L -> L
L v ~L
therefore L & ~L ... contradiction
Rather we can argue this way:
L -> ~L
so ~L
~L -> L
so L
therefore L & ~L ... contradiction
Or, intuitionistically:
L -> ~L
so ~L
~L -> L
so ~~L
therefore ~L & ~~L ... contradiction
Yes, I'd enjoy being able to interrogate the data, although the sample size is a bit small.
A similar survey could be done here using SurveyMonkey or some other.
It's way at the bottom of the demographics.
https://survey2020.philpeople.org/survey/results/demographics
While others may have done so, in this thread that's been me aping Priest.
The idea is to point out a difference between LNC and LEM, as well as to prove that the [s]dialeithic[/s] dialethic answer to the liar's is still valid in the sense of using some classical logical laws.
What are your thoughts on logical pluralism?
Of course LNC and LEM are different.
I can't find the post about the liar paradox; my own point was merely the technical one that the contradiction of the liar does not require LEM.
I'm not inclined to compose a post about it.
None. I thought that was the result of his numbering system for mathematical proofs. The Godel numbers, lead to a conclusion that you can't in fact provide support for every mathematical assertion. Without reaching some paradox. I don't remember the details.
Quoting TonesInDeepFreeze
Fair point. Trying to see if I could argue it. Boolean logic is pretty solid.
"provide support for" is vague in supposedly explaining the vague "has no foundation"
The main branches of classical mathematics are formalizable in set theory. However, if set theory is consistent then there are statements in the language of set theory such that neither the statement nor its negation is a theorem of set theory. Moreover, if set theory is consistent then set theory does not prove the consistency of set theory.
It's interesting that the proof of "if set theory is consistent then set theory does not prove that set theory is consistent" is not so much analogous to the logic of the liar or barber paradox but rather to a different paradox, viz. Curry's paradox aka 'the Santa Claus paradox'.
Heh. Well, I'd expect that from you :D -- I'm not sure that the differences between them are at the level of "of course" for the participants here.
I agree. I don't think the liar's needs anything technical at all. For thems who prefer utterances we can frame it in plain language as "I am telling a falsehood right now"
I haven't seen anyone define any of the positions in a clear and non-vacuous way, much less go on to argue in favor of one or another.
Quoting Count Timothy von Icarus
"There are multiple formal ways of realizing the informal notion of logical consequence." I suppose this gives us something, but I don't think it is very substantial. If, for example, everyone agrees that Aristotelian syllogistic and propositional logic are two ways of formalizing the informal notion of logical consequence, then where does the actual disagreement lie?
Again, what is needed is someone who believes they disagree and is willing to set out a substantial argument. The polemicists disagree without substance, and the rest of us are not sure what we are supposed to be disagreeing about.
If dialethism is true then pluralism is true.
Dialethism is true as it resolves the liar's paradox in a clear, non-vacuous way.
Therefore, pluralism is true.
But I don't really intend to continue this conversation about dialetheism, especially given my earlier demonstrations of the incoherence of the "Liar's paradox." From what I have seen, people are dialetheists for the same reason they dye their hair purple. :grin:
Perhaps partly, but I think the other big factor would be that it is not actually easy to remove LNC and not end up with triviality. You seem to have to get rid of disjunctive syllogism, reductio arguments, or disjunctive introduction, and on many early attempts to understand this, all three.
But these all make sense, e.g. disjunctive syllogism intuitively seems right, so if a contradiction lets us prove anything from it, the contradiction seems to be the problem.
So even if people further back in history wanted to remove it, they couldn't without making everything true.
First part, I'd agree, although I think it will end up being relevant if arguments for nihilism (or a pluralism bordering on nihilism) are made from the assumption of relativism and deflation re truth (which I suppose are metaphysical positions of a sort, but can be presented as "anti-metaphysical"). However, I don't think one needs any sort of in-depth metaphysical theory to say, "good reasoning has something to do with leading to truth and logic is meant to model/enhance good reasoning." Normally, the move to define "correct logics" in terms of natural language, or in the more common sense formulation of "good reasoning" seems like a way to get at this without having to make any metaphysical commitments. Normative views of logic accomplish the same thing. I just think that if we interrogate the normative views, we end up finding some notion of truth further back (maybe not, it's irrelevant to the pluralism debate anyhow).
The second part doesn't make sense to me. On this view, if we accept using truth in a model as truth for pragmatic purposes in logic we should dismiss non-relative truth in metaphysics. But I don't see how there is any connection here. The first move is a pragmatic bracketing of a thorny question, not producing an answer to that question.
Yup, which is why I imagine they have very similar sorts of debates.
Anyhow, it seems possible to both affirm and deny pluralism/monism in the terms laid out without contradiction, since there is equivocation in the "subject matter of logic." I don't think it's particularly implausible to say something like:
"If you are interested in logic primarily as an abstract formal system, there are no correct logics, but there are uninteresting ones. (A sort of nihilism). If you are interested in logic as good reasoning this answer is less obvious, but there are clearly many incorrect logics, since it is not good reasoning to affirm everything, almost everything, or almost nothing."
I think a difficulty, even in published articles, is equivocating on just this issue relative to one's opponents. It's one thing to disagree with how they define the subject matter of logic, but obviously another to use arguments based on one such definition to attempt to refute a position based on another.
Early in this thread I mentioned the older distinction between formal and material logic. This distinction is similar, although not identical, to claims that consequence might vary by domain. I think the furthest advances in material logic in Poinsot, CSP, etc. do offer at least a plausible explanation of why exactly consequence might vary when we move to consider signs (which of course introduce self-reference), particularly stipulated signs systems. This is relevant if the very point in question is if logic is about reasoning by beings or about stipulated sign relations.
Whereas if one conceives signs from the post-modern perspective that grew out of Sausser then it might seem obvious that formal relations are the only thing to consider. So, one could frame the debate in terms of the proper understanding of signs I think, and probably argue towards either position depending on how one understands signification.