Truth preserving or simply playing with symbols?
This idea is only partially fleshed out, but in other parts of the forum people are discussing truth and concerns about the relation of logic to truth. What is interesting to me is the disconnect between theories of truth and the notion of logic as truth preserving.
On an ordinary account of truth, there is some criteria which makes a truth bearer true (let’s call these things “facts” and “propositions” respectively). Let’s take a simple example: the cat is on the mat. So we look for the cat on the mat and if it is there, it is true, and if not, not. That seems simple enough.
On an ordinary account of logic, if I assume
1) 2+2=4
2) if 2+2=4 then the cat is on the mat
Then it follows
3) the cat is on the mat.
A valid argument to be sure, but so what? What makes the cat is on the mat true is neither my assumptions nor the rules, but the cat being on the mat. (That is, while the argument form may be valid, the interesting bit - whether the consequence is true - is directly evaluated by reference to the facts. In this respect, soundness is a coincidence of valid form and fact.)
If we agree so far, at what point is logic elevated from a set of propositions joined by some logical connectives to a tool by which the consequence can be made true by virtue of the ordering of those propositions and connectors, i.e. from something whose premises are constrained by the facts to something that dictates what the facts are or can be?
In some ways, it seems that when dealing with truth, certain facts are accessible while others are not accessible (or less so). In such a case, if we can establish a “necessary” relationship between the accessible fact and the non-accessible fact (call it entailment if you like) and that the accessible fact is true, does the logic make the non-accessible fact true or does the fact make the proposition about it true?
In essence, I am asking what our motivation to logic is if we accept the ordinary account of truth described. If we find that a logic does not establish the proper relationship of truth between propositions about discrete facts, should we be hesitant to adopt a logic that establishes such a relationship? And should we hesitate to change logics as befits the circumstance?
On an ordinary account of truth, there is some criteria which makes a truth bearer true (let’s call these things “facts” and “propositions” respectively). Let’s take a simple example: the cat is on the mat. So we look for the cat on the mat and if it is there, it is true, and if not, not. That seems simple enough.
On an ordinary account of logic, if I assume
1) 2+2=4
2) if 2+2=4 then the cat is on the mat
Then it follows
3) the cat is on the mat.
A valid argument to be sure, but so what? What makes the cat is on the mat true is neither my assumptions nor the rules, but the cat being on the mat. (That is, while the argument form may be valid, the interesting bit - whether the consequence is true - is directly evaluated by reference to the facts. In this respect, soundness is a coincidence of valid form and fact.)
If we agree so far, at what point is logic elevated from a set of propositions joined by some logical connectives to a tool by which the consequence can be made true by virtue of the ordering of those propositions and connectors, i.e. from something whose premises are constrained by the facts to something that dictates what the facts are or can be?
In some ways, it seems that when dealing with truth, certain facts are accessible while others are not accessible (or less so). In such a case, if we can establish a “necessary” relationship between the accessible fact and the non-accessible fact (call it entailment if you like) and that the accessible fact is true, does the logic make the non-accessible fact true or does the fact make the proposition about it true?
In essence, I am asking what our motivation to logic is if we accept the ordinary account of truth described. If we find that a logic does not establish the proper relationship of truth between propositions about discrete facts, should we be hesitant to adopt a logic that establishes such a relationship? And should we hesitate to change logics as befits the circumstance?
Comments (52)
I'm having difficulty following the issue...
Quoting Ennui Elucidator
IS it that (2) already seems to evaluate "the cat is on the mat"?
2. The cat is not in the tree.
Ergo,
3. The cat is on the mat. [1, 2 Disjunctive Syllogism]
The statement "the cat is on the mat" is logically true i.e. it's an inference.
The statement "the cat is on the mat" is definitionally true by virtue of the meanings of the words in the statement. This too requires logic i.e. it's an inference.
See anything relevant?
You're addressing the discrepancy with natural language?
Ordinarily we understand "If X then Y" as asserting that if X is true then Y is true because X is true, whereas in formal logic this isn't the case?
Quoting TheMadFool
This is exactly contrary to the way I am using “true.” Something is true when it is the state-of-affairs. When we find the cat is on the mat, we aren’t satisfying definitions, it is simply the case that the cat is on the mat. Part of what this post is about is highlighting the equivocation between true by virtue of state of affairs and true by virtue of definition (but entirely avoiding the analytic/synthetic framework). Maybe we can refer to this version of truth as “rTrue” (for realist truth).
Quoting Michael
You aren’t completely off here, but I had hoped that my reference to entailment (rather than logical implication) might help avoid focusing on the difficulty of mapping natural language to symbolic logic. I am more focused on logical consequence, i.e. what it means for a conclusion to follow from its premise, and trying to contrast it with the requirements of a realist account of truth. In order to facilitate this distinction, I will add another term, “lTrue” to mean the conclusion of a valid deductive argument (by use of whatever logical system you desire).
Perhaps we can think of this post as something like, what is the relation between rTrue and lTrue for any proposition, P. Critically, I am not interested in the semantic content of P, whether “P” can stand in relation to rTrue, or what the magic is that causes P to make “P” rTrue (that is, for us to articulate a truth rather than just observe, or potentially observe, one).
As a side note, much like the linked article, I am going to focus on deductive validity rather than inferential validity. Also, this is a maybe helpful quote:
[quote=“SEP on Logical Consequence”]
A closely related analysis for formality is that formal rules are totally abstract. They abstract away from the semantic content of thoughts or claims, to leave only semantic structure. The terms ‘mother’ and ‘cousin’ enter essentially into argument (5). On this view, expressions such as propositional connectives and quantifiers do not add new semantic content to expressions, but instead add only ways to combine and structure semantic content. Expressions like ‘mother’ and ‘cousin’, by contrast, add new semantic content.
[/quote[
.Quoting Benj96
I like this question, but I think it is a bit aside from my current focus. This seems like a problem of constructing a useful logic that can account for such things more than investigating the relation of logical consequence to the state of affairs in the first instance (i.e. logic may not have to account for all rTruths but I am asking it can account for some).
Quoting Banno
I’m not sure what you mean by “seems to evaluate” in this case. My hope was that by using a mathematical truth as the antecedent, that we could highlight that any true premises plus valid form makes “P” true of logical necessity. Logic is, perhaps, about establishing (discovering?) the rules by which the truth value of one proposition relates to the truth value of another. Classical logic, where any proposition can stand in for any other proposition with the same truth value, leads to many intuitively unsatisfactory “proofs”.
And harkening to your questions about realism, here is another snip from the logical consequences article:
[quote=“SEP on Logical Consequence”]
Perhaps there is a reason to allow the notion of logical consequence to apply even more broadly. In Gentzen’s proof theory for classical logic, a notion of consequence is defined to hold between multiple premises and multiple conclusions. The argument from a set X of premises to a set Y of conclusions is valid if the truth of every member of X guarantees (in the relevant sense) the truth of some member of
Y. There is no doubt that this is formally perspicuous, but the philosophical applicability of the multiple premise—multiple conclusion sense of logical consequence remains an open philosophical issue. In particular, those anti-Realists who take logical consequence to be defined in terms of proof (such as Michael Dummett) reject a multiple conclusion analysis of logical consequence. For an Anti-realist, who takes good inference to be characterised by the way warrant is transmitted from premise to conclusion, it seems that a multiple conclusion analysis of logical consequence is out of the question. In a multiple conclusion argument from A to B, C, any warrant we have for A does not necessarily transmit to B or C: the only conclusion we are warranted to draw is the disjunction B or C, so it seems for an analysis of consequence in terms of warrant we need to understand some logical vocabulary (in this case, disjunction) in order to understand the consequence relation. This is unacceptable if we hope to use logical consequence as a tool to define that logical vocabulary. No such problems appear to arise in a single conclusion setting. (However, see Restall (2005) for a defence of multiple conclusion consequence for Anti-realists; and see Beall (2011) for a defence of certain sub-classical multiple-conclusion logics in the service of non-classical solutions to paradox.)
[/quote]
Truthfully, this whole truth bit is complicated, but I don’t find logical proof among the things that make propositions true in articles like this. and yet it so often seems that people say that some proposition is “true” by virtue of some logical argument.
I'm not sure there's really a problem then. If premise 1 is true then premise 2 is true only if the cat is on the mat. So the realist can say that the realist account of truth is required for premise 2 to be true.
That is the backwards part. Yes, rTrue for the world. But can 1 and 2 make 3 rTrue? If we want to know if 3 is rTrue, how is a proof used as a proxy for the world?
Anytime you look to rTrue for the premises, you are focusing on the wrong spot. Pick any argument you want,
If my finger isn’t in my nose then my hand has five fingers.
My finger isn’t in my nose.
Therefore my hand has five fingers.
P then Q
P
Therefore Q
I assure you, my hand has five fingers and my finger isn’t in my nose. Why is Q true?
You claim that a proposition is true when it is a "state of affairs". What exactly do you mean by that? I ask because you seem to be implying there's no logic involved in such truths. The only variety of truths I know of that require no arguments for their veracity are so-called self-evident propositions.
Come now to self-evident propositions. Let's take your "cat on the mat" example. You claim that this (the cat is on the mat) is a state of affairs deal. I'll give you that but you have to admit that "the cat is on the mat" can be true only if that sentence is semantically loaded. That's not all, the meanings (of the words) come togther to give the meaning of the sentence and that I believe is an inference (logic).
If 1 and 2 are rTrue then 3 is rTrue. But 1 and 2 being rTrue isn't what makes 3 rTrue. Rather 1 and 3 being rTrue is what makes 2 rTrue.
It isn't. The argument is only a proof if you can prove 1 and 2 to be true. A deductive argument can have false premises after all. How do you prove 2 to be true? By proving that 2 + 2 = 4 and that a cat is on the mat. You need "the world" to do this.
But that is what my example shows you, I provided what is essentially a tautology in 1 and a claim about the world as a consequent of a conditional. The only way to show that 2 is false (and thereby show the argument is unsound) is to evaluate whether the consequent is true, which is precisely what the proof appears to be proving. So the proof adds nothing to the rTrue of the consequent and indeed is just a symbols game.
We can take 1 as proven to be true, but you haven't proved that 2 is true. How do you prove that if 2 + 2 = 4 then the cat is on the mat?
It is a premise, I don't have to prove it to assume it.
You have to prove your premises true to prove your conclusion.
That isn't how logic works. A premise is assumed, not proven. The valid conclusion is of the form "If the premises are true, the conclusion must be true".
It's not enough to present a valid argument. You have to show that the argument is sound, i.e. that its premises are also true.
My name is Michael
If my name is Michael then I am the King of England
Therefore, I am the King of England
This is a valid argument. The first premise is also true. But I haven't proven that I am the King of England because I haven't proven that the second premise is true (and I can't because it isn't).
So how do you evaluate soundness?
I'll even quote myself in the OP.
By proving that the conclusion follows from the premises and that the premises are true. In your case to prove the second premise I must confirm that 2 + 2 = 4 (which I can do by understanding maths) and confirm that a cat is on the mat (which I can do by looking at the world to see if I can see such a thing).
I've said; the premises being true are what make it sound.
Nothing. Your argument has no practical use. I have to evaluate the truth of your conclusion (by referring to the world) to evaluate the truth of your premise.
In my mathematical understanding soundness merely means coherence, that is, freedom of contradictions. Logic can determine that a set of premises /cannot/ be true or show that the truth of a given conclusion holds under given premises and derive such already implied conlusions.
This seems to have the advantage, that we can separate "discussable" statements like if the cat is really on the mat or if it is not because two paws are outside already from the purely deductive part of reasoning.
In the OPs example one can disagree with (2) or even with (1), but if agreement is reached on both points the conclusion should be implied. That or we have a very different way of thinking...
This feels much like a circle. Yes, there is no point to a logical argument about states of affairs because what makes propositions about states of affairs true are the states of affairs, not our rules of logic. Nevertheless, people want to use logic to dictate how states of affairs can be. What lTruth can you produce that relates to an rTruth? If you are happy to concede from go that logic is just the formalistic manipulation of symbols, I won't complain.
That is the bit that most people lose, I think, that all conclusions are of necessity assumed in the premises. Logic is useful revealing novel relations given a particular set of rules, but cannot reveal rTruths. Logic then is descriptive and not normative, no matter how well the logic (or math) predicts the rTruths. But is this a problem for logic or just for non-logicians?
In such occasions we don’t need to assess the truth of the conclusion to assess the truth of the premise, and so the argument tells us something about the world.
But in your case that 2 + 2 = 4 doesn’t imply that a cat is on the mat, and so we must assess the truth of the conclusion to assess the truth of the premise. Your material “implication” makes for a useless argument.
In the type of arguments we deal with, an argument is valid if the truth of the premises entails the truth of the conclusion, and an argument is sound if it is valid and its premises are true.
You seem stuck in social conventions and definitions. Forget social conventions and focus on mind-independent stuff, you know, the pre-interpreted states-of-affairs. If you are a realist and you wish to avail yourself of the power of logic to determine rTruth, can it do so?
They don’t need to. Your questions make no sense. It’s the brick hitting the window that breaks the window, not the argument “if the brick hit the window then the window is broken and the brick hit the window, therefore the window is broken.”
This made me think a little.
If for example, the grass is wet, it has to have become wet somehow. If we were in a world where this can only happen by rain, the conclusion clearly is that it must have rained.
Though this is different from "it rains" or "it rained" it is an extension knowledge. Both "rules", seen for themselves, do not have a relation to each other and do not tell you that you have to apply them together. The system is more than the sum of the parts, so to speak. If taking the already implied as "given" we forget that someone still has to do the conclusion.
You will not feel the (past) rain because of such a conclusion, of course.
If that is so clear, why do you say "the truth of the premises" and not just "the premises"?
https://en.m.wikipedia.org/wiki/Validity_(logic)
https://en.m.wikipedia.org/wiki/Soundness
Yes - read the linked document. That is, what we are talking about.
And this is the type of tension that I think the realist has - that logic is supposed to give warrant for the belief in rTruth, but the world doesn't function deductively (from generalizations to specifics), rather it just is. Probabilistic, stochastic, etc. logics which can account for the variance between premises and conclusions are deemed insufficient to give warrant (let alone certainty) for the realist, but logic can never demonstrate other than which what was assumed. So if we demand binary truth values (for unitary propositions or compound propositions), but our best methods for predicting/accounting for the world are not perfectly accurate, what is the role of such logics for the realist? It cannot do what it sets out to do (make an rTruth) and its predictive and truth preservation properties are inherently disconnected from the way that rTruths appear to relate to themselves, so what is the allure?
I am looking for an argument of what logic does for the realist besides act as a useful heuristic. In particular, I am looking for an argument as to why anyone should feel compelled to accept classical logical (or minor variations) as somehow more useful as a heuristic than any other logic.
It provides a guarantee of understandability for other beings. I guess you could get an agreement of any realist when showing the fundamental set operations - e.g. "O O is disjoint" and he will agree that no point in one O is also in the other. It is not so much the content but the basic rules of logic themselves that have a certain type of "reality".
Other than that - I do not understand your concept of rTruth completely: E.g. If you feel a poke in the back, is there "really" something that pokes you? I guess the answer is "no": it is a conclusion that everything has cause. I am not sure that all realists would reduce reality to just the given content of consciousness. In another thread I pointed out that (following e.g. Heidegger) reality seems to be purely negative - that it is mainly what _prevents_ you to assume somehing. In logics this would be a statement not(x) where x is the "state of affairs as assumed".
Realism, as being used here, is about people who think that what makes a proposition true is its relation to the states of affairs. I am not a realist. I am, however, interested in non-binary, relevant, paraconistent, etc. logics and how they solve real world problems that classical logic (however slightly modified) cannot (or perhaps they solve such problems more efficiently). I see logic more as a tool of thinking than a thing capable of governing the world (i.e. logical necessity/entailment/possibility does not preclude otherwise efficacious behavior).
Okay, nice to meet you :)
In fact, every now and then I have to deal with such "problems".
( Warning: possibly boring stuff ahead )
When evaluating a logical system of facts and rules there are (at least) 2 possible ground-laying assumptions.
"closed world": which means the system is complete in that "it is known, that _all_ true statements can be derived from it" and
"open world" which means that "some things are known, some are not".
When dealing with an open world there is a problem with "tertium non datur" (a or not a) - it might well be that not enough facts are known to prove or disprove "a". Classical logic assumes that "a" or "not a" MUST be true. This is a reality of logics itself but does paradoxically not hold if talking about e.g. mathematical systems (Goedel's incompleteness theorem).
Now the discussion must be lifted one step higher and we arrive at the question what it means (in logics) to say, for example, "a is the case". There are some alternative mathematical logics which account for the (un-)provability problem by eliminating the tertium-non-datur and the law of the double-negation by saying "x" means that "a proof can be contructed for x" and "not(x)" means "a proof can be constructed for not(x)". Doing this a failure to construct a proof for "not(x)" no longer necessarily implies "x", which makes the logic weaker (and suitable for an open world).
I get the intention of your distinction between l- and r-truth, but the world does not speak. In logics one is dealing with rigidly defined concepts. A "matter of formulation" can make the difference between true or false. I am not sure if humans act logically without a reflection in "words".
Quoting Ennui Elucidator
Then I'm not at all sure what your task is in this thread.
But then, logic is simply playing with symbols, isn't it? What more was expected?
Thanks for the mention of antirealism, but truth-bearers confuse me even more. I can't see how they help.
Quoting Ennui Elucidator
My inclination is to say simply that we can choose whatever logic suits our purpose. DO you thin this somehow incompatible with realism?
Trump and Trumpistas are realists then?
Indeed, why not make Eristische Dialektik our Bible?
I am suggesting quite the opposite - that if truthmakers are states of affairs, then logic should not be faulted for its failure to ensure rTruth. In other words, realists should feel free to use whatever logic they want including the acceptance or denial of any particular rule no matter how much of a sacred cow it is. If a logic occasionally produces an useful result, we just use a different one. If our concern is about preserving lTruth, then we apply a different criteria to our logic than if our concern is predicting the behavior of light through a slit as it spins our radiometer.
Universal logic which is responsible both for rTruth and lTruth feels much like ancient baggage. Thinking can be dynamic and free and so long as our methods achieve our ends, what more can we ask of them? If you want realism, it doesn’t require the baggage of a particular logic.
Banno may tell you that he never laughs at my jokes.
[quote=“The Art of Being Right”] Persuade the Audience, Not The Opponent
This is chiefly practicable in a dispute between scholars in the presence of the unlearned. If you have no argument ad rem, and none either ad hominem, you can make one ad auditores; that is to say, you can start some invalid objection, which, however, only an expert sees to be invalid. Now your opponent is an expert, but those who form your audience are not, and accordingly in their eyes he is defeated; particularly if the objection which you make places him in any ridiculous light. People are ready to laugh, and you have the laughers on your side. To show that your objection is an idle one, would require a long explanation on the part of your opponent, and a reference to the principles of the branch of knowledge in question, or to the elements of the matter which you are discussing; and people are not disposed to listen to it. For example, your opponent states that in the original formation of a mountain-range the granite and other elements in its composition were, by reason of their high temperature, in a fluid or molten state; that the temperature must have amounted to some 480 degrees Fahrenheit; and that when the mass took shape it was covered by the sea. You reply, by an argument ad auditores, that at that temperature - nay, indeed, long before it had been reached, namely, at 212 degrees Fahrenheit - the sea would have been boiled away, and spread through the air in the form of steam. At this the audience laughs. To refute the objection, your opponent would have to show that the boiling-point depends not only on the degree of warmth, but also on the atmospheric pressure; and that as soon as about half the sea-water had gone off in the shape of steam, this pressure would be so greatly increased that the rest of it would fail to boil even at a temperature of 480 degrees. He is debarred from giving this explanation, as it would require a treatise to demonstrate the matter to those who had no acquaintance with physics.[/quote]
So this is a type of motivation of mine - to understand that where we have warrant for both X and ~X, that we shouldn’t somehow dismiss the warrant of one or the other for fear of contradiction. Although not fully explicated here, the thought is that the way that we speak of truth is neither about coherence nor correspondence, but about achieving our ends. We can grant the realist his state of affairs for rTruth, move into why rTruth is utterly meaningless for epistemology, and then get on with the business of thinking well about solving our problems. It would be nice, however, if the realist would cease their reproach of logics that don’t meet their aesthetic based upon the faulty belief that logic is about the state of affairs.
It seems that for all practical intents and purposes, (formal) logic is a response or reaction to what is intuitively felt as wrong thinking. In other words, for many practical intents and purposes, the history of logic is the history of addressing informal logical fallacies (and then explicating principles of proper ways of thinking).
The issue is relevant when it comes to persuasion, when one person tries to persuade another person to accept the proposed view and act accordingly.
Ideally, fending against informal logical fallacies should protect one against being duped (and, if one is very nice, make one refrain from duping others).
Agreed.
Unless this belief (or, to the point: asserting this belief) is part of the realist's strategy to achieve his ends.
It's not like it is really possible to distinguish between a religious preacher who claims to have the Truth, the How Things really Are, the State of Affairs, from a realist who does the same (they just differ in what they state that Truth to be).
I want very much to like this, but there are times when informal fallacies are useful - like appeals to authority or ad hominem when it is so much more trouble to show why the person is wrong. If an informal fallacy gets you to the same end with more expedience, I question why they shouldn’t be given the same status as any other heuristic.
Indeed.
From your link:
The question is whether we should fully give in to the natural baseness of human nature, whether we should deem it absolute, the only thing that matters.
I fear the systems I was talking about are those, where there is neither a sufficient reason for X nor ~X.
_Paraconsistency_ (allowing contradiction) is seldomly seen as allowable for a single subject, but (accoring to wikipedia) a model for what can be derived from the statements of _multiple_ speakers, i.e. a social context. There is no contradiction in two people disagreeing, but the contradiction exists in the social context.
Regarding tertium-non-datur and rTruth:
Imagine the sentence "At point X on the ground of a deep ocean there are the ruins of Atlantis". Nobody has ever been at the given point X. What can we say about the truth value of that sentence apart from "might be"? Tertium-non-datur means ideally "we can get there, have a look and then _know_ if this is true". It has to be true or not when we know everything. But this is not the state-of-affairs. We do _not_ know this. That is, tertium-non-datur, realistically only applies where the truth of one of the negations is already known. "The sun is shining, therefor 'The sun is shining or not' is true".
Is "It is not raining" rtrue on a sunny day? I do not know.
I would not go so far as to say, we choose the logic that serves our ends (as in egoism etc.). But we have to choose the right logic for modeling different realities. The fitting logic is still determined by the nature of things - which leads back to the epistemological starting point and social contradictions.
Your "quite the opposite" is the same as I sugested, so it seems you are a man of your word.
I was responding to this question. Perhaps I misunderstood the tone.