God exists, Whatever thinks exists, Fiction: Free Logic
Free logic: logical systems that do not insist that individual variables denote items in the domain of discourse.
So in our usual logic, "We detect no motion of the earth relative to the ether" is false, because the ether does not exist. In a free logic, it is true, because the singular term "ether" is allowed to refer to things that do not exist or are not part of the domain of discourse.
This fell off the back of a truck about Being.
It has it's problems, of course, but is presented here as a curiosity. Has anyone come across it before?
It introduces
E!t=df?x(x=t)
___________
Edit: A rearrangement of the title to better reflect the content. This time to include fiction.
For God and Descartes, see here.
So in our usual logic, "We detect no motion of the earth relative to the ether" is false, because the ether does not exist. In a free logic, it is true, because the singular term "ether" is allowed to refer to things that do not exist or are not part of the domain of discourse.
This fell off the back of a truck about Being.
It has it's problems, of course, but is presented here as a curiosity. Has anyone come across it before?
It introduces
E!t=df?x(x=t)
___________
Edit: A rearrangement of the title to better reflect the content. This time to include fiction.
For God and Descartes, see here.
Comments (96)
And further, since E!t=df ?x(x=t), E!t is defined using identity: "=".
So there seems to be some support here for this sort of reasoning:
Quoting Olivier5
(@Xtrix)
We get
E!(Xtrix)= ?(x)(x=Xtrix).
This by way of undermining my own point that "? Xtrix" is ill-formed.
Meinong's logic - it seems that free logic is a part of the jungle - the jungle is broken into two domains, inner, the things that exists, and outer, the things that don't - but free logic allows us to say things about stuff that doesn't exist. The puzzle is, at what cost?
Like any logical formalism, there's no question of it being right or wrong, just apt to certain purposes.
SO we have :
Classical logic
Free logic
Inclusive logic
IS that right?
:up:
So we say, as a postulate governing its interpretation, that E!x iff ?y[y=x].
Where the free logic comes in is that you can interpret ? as quantifying over only the domain of 'really existing things.' So if you say something like 'the aether exists,' and the aether is referred to by 'a,' you can put the aether in the non-existent domain, so that:
E!a comes out false, because, we interpret it as:
?x[x=a]
And this will come out false. Why? Because the quantifier ranges only over individuals in the domain of real existents, which the aether is not in (it's in the domain of non-existent things). So we find no individual x in the domain that is identical to a. So the translation of 'the aether exists' is false.
But we can still say, e.g. 'John doesn't believe in the aether,' if we interpret that as ~B(j,a), and have that come out as true – so long as we allow individual constants like 'a' to refer to things both in the domain of existents and the domain of non-existents.
The result is that we can refer to non-existent things, and say true things about them, while we cannot existentially quantify over them, meaning that it will be true to say they don't exist.
This is all from memory – anyone can correct me if they're looking at the material.
My understanding is that free logic has a split domain, one of existents and one of non-existents (although you can interpret the domains however you want – there are just two of them). The difference between them is that individual constants can refer to members of either domain, while quantifiers range only over the individuals in one domain (the domain of 'existents').
Oh wow! Banno is giving some thought to his posts. Miracles happen... So it turns out that existence is (or can be, amongst other things) a logical predicate after all...
1. Real world:
Someone called me.
Px = x is a person. Cx = x called me.
[math](\exists x)(Px \wedge Cx)[/math]
2. Fictional world: From the nonexistent book Leprechauns and their peculiar habits
Some Leprechauns eat poo.
Lx = x is a leprechaun. Ex = x eats poo.
[math](\exists x)(Lx \wedge Ex)[/math]
Something's not quite right.
Hence, I suppose, Free Logic.
At this stage there are no individuals that are not part of the domain. Introducing things that are not part of the domain leads to what has been called two domains - the things that are E! and the things that are not.
Cool.
Yeah, but there is a problem for the most common sort of philosophical arguments that try to treat existence as a predicate. Free Logic does not permit the expression of existence conditions. That is, that something exists cannot be the conclusion of an argument in free logic.
SO arguments that have such things as the existence of god, or the existence of the self, or the existence of reality as their conclusion receive no solace from free logic.
And we are left in much the same position as I have pointed out in the thread on Being: things that exist because they are the presumed individuals in the domain, and things that exist because they are the subject of a predicate.
That is, fictional species are part of the conversation, so you can talk about them in your scheme.
In free logic leprechauns would not be members of the domain of things that exist - E!. But you could still make inferences about them.
However as noted above, you could not infer their existence, even in free logic.
In classical logic, to make the inference you would have to presume the predicate "... is a leprechaun". How you understand that predicate remains moot; and one can play on that ambiguity.
This is the ambiguity apparently traded on in the Being thread.
If one supposes that all ?(x)(Lx) says is that something is a leprechaun, one need not conclude that one might meet a leprechaun walking down the street. That there are leprechauns says nothing more in this context than that we can predicate being a leprechaun to something - fictional or otherwise.
Some folk see this as problematic. Seems to me to be just an ambiguity in the use of "is". That Shamus is a leprechaun does not imply that you might meet him in the pub.
Me, too.
What?
You are claiming there’s an ambiguity to avoid existential quantification meaning exactly what it says and what everyone agrees it means. @bongo fury wasn’t trading on any ambiguity; sentences found in fiction are literally false, and that’s fine. (Lawrence Block wrote a book about fiction writing called “Telling Lies for Fun and Profit”.)
If you take away existential import there’s no way to put it back just when you want. “There is something under the bed” will just no longer mean there is something under the bed.
Also, what is “presuming a predicate”?
To simplify even further: things are always presumed to exist. Existence cannot be proven.
When people say things like "X exist", they usually mean "I cannot think through this issue without making the assumption that X exists." Or simply: "I would rather make the assumption that X exists." They are positing the existence of God, the mind, the Higgs boson or true love. And oftentimes they know perfectly well it is simply an assumption, rather than some deep ontological commitment.
Free logic, as far as I can tell, seems to have an issue with the existential quantifier because, taking the fictional statement "some unicorn ate my lunch", translating that would require a committment we can't (afford to) make viz. existence. The translation of the above example statement is [math](\exists x)(Ux \wedge Ax)[/math] where Ux = x is a unicorn and Ax = x ate my lunch.
Free logic would, in my humble opinion, open up the world of fiction - Tolkein's works, Doyle's works, etc. - to logical analysis.
You say that's not all free logic can do. It seems that you're under the impression that with free logic we can begin an analysis of a given domain of objects and then, almost miraculously, switch the topic to something outside that domain. I'm not sure I understood you correctly but if this is what you mean, what's the point, really? It appears to be something but, in fact, it is, I suspect, nothing. I'm out!
Quoting Srap Tasmaner
Well, it's false that Holmes lived at 222 Baker Street.
Yep.
Interesting, then, that introducing E! does not provide solace for those who wold do otherwise.
You seem to think this would be problematic. Why shouldn't fiction be logical?
Holmes lived at 221b Baker Street. Why shouldn't we consider this to be true, within the context of the writings of Doyle and their derivatives? Is there an argument against this?
Quoting TheMadFool
Free logic defines two domains. There's no magic here, just an attempt to make explicit the grammar around saying that something exists.
What is amusing about this is that in attempting to build a sensible account of E!(x) a string of logical difficulties arrises, allowing us to look with greater care at the problems of treating existence as a property.
To be sure, it's the problems of free logic that are fun.
If we confine ourselves to the stories of Tolkien, then we can make inferences about fictional characters without making use of free logic. So the example from the SEP article is that within the context of LOTR, Gollum hates the sun, and we can validly infer that something hates the sun*.
The two issues noted in the article are that there is no compunction on fiction to be logically consistent; and that the properties of fictional individuals are incomplete.
The first of these is apparent from watching Marvel or DC movies. There's much to be said for the usefulness of applying logic to these scripts in order to set out their inconsistencies.
The second is of use to authors, presumably, since it allows the characters they create to develop in unexpected ways. It presents problems for logicians if they erroneously treat fictional characters as complete individuals.
Free logic might be employed if one is attempting to talk across both fictional and non-fictional individuals. The example given in the SEP article is
Treated in ordinary logic this statement would be false, since Gollum does not refer to anything. But we can make use of Free logic, since here Gollum is fictional, but not Gödel, hence E!(Gödel) but ~E!(Gollum). In a positive free logic Gollum is more famous than Gödel.
Keep in mind that positive free logic is one which an individual is identical with itself, even if that individual doe snot exist: one in which (Gollum = Gollum) is true; a negative free logic is one in which (Gollum = Gollum) is false. Hence in a negative free logic one can infer from (Gollum=Gollum) that Gollum exists.
*I wonder if @Srap Tasmaner thinks this inference is invalid...? How seriously should we take the claim that all fictional statements are false?
If it's Fiction Logic within Free Logic, then aren't we committing ourselves to Epistemic Logic at this point to "conclude" anything about the domain of discourse, which is the mind of the author and/or literature or script?
So Epistemic Logic is the various logics that include predicates for belief and knowledge - yes? Where's that fit here?
But it's not true, nor even helpful, to think of fictional worlds as only existing in the mind of the author. Were this true, both Holmes and Frodo would have ceased to exist along with the minds of their respective authors.
Isn't it simply the case that the domain of discourse for LOTR includes the individuals Frodo, Gandalf and so on, and the predicates "...is a hobbit" and "...is a wizard", and that it is true within that domain that Frodo is a hobbit and false within that domain that Gandalf is a hobbit? And isn't that settled extensionally?
More aptly referred to as being stipulated or written, rather than being true. no? We have good reason to believe it is true that it is written that Holmes lived at 221b Baker Street.
If there is a real 221b Baker Street, then it is false that Holmes ever lived there. If that address is also fictional then I would say it is merely stipulated that the fictional character lives at the fictional address.
1. That a post shall accomplish something and arrive somewhere. But your post accomplishes nothing and arrives in air.
:wink:
"It is written, that's why" - Brian's mum, Life of Brian.
Well, the stipulation is that we are talking about LOTR or whatever; and the question is as much about what we might infer as what is true and what is false.
So I think it not unreasonable, given that Frodo walked into Mordor, that we might infer that something walked into Mordor. That is, that the logic of existential generalisation might be expected to hold in a fictive world.
The alternative suggestion is that it is not true that Frodo walked into Mordor, and hence that we cannot make such an inference.
It occurs to me that there are fictional creations in which such inferences do not hold. Hence, Nonsense as a genre.
Do you mean literature in which inconsistencies and contradictions abound. I'm trying to think of an example. Twain attributes some inconsistency to Fennimore Cooper's Deerslayer in the linked article. I read Deerslayer a few years ago and enjoyed it. Is poetic licence justification for inconsistency in literature?
According to that logic,
If you say three times that if you say something three times, then unsay it once it is false, then it is false.
If you say three times that if you say something three times, then unsay it once it is true, then it is true.
I guess...
Who did the original three-saying?
Quoting Lewis Carroll
To which we might add:
Does the thrice-telling make it true, or does the Bellman tell it three times because it is true? Perhaps this is the much that yet remains to be said.
Ah, and excellent point; and one that I think returns us to the incompleteness of fictional characters mentioned above and noted in the SEP article.
Given that the Bellman says other things, some of them undoubtedly true, yet this is the only one that he says three times. Hence that the Bellman says it three times is not, it seems , a necessary condition for truth. While not decisive, this perhaps mitigates against the theory that thrice-telling causes truth, and that rather thrice-telling implies truth.
This is further supported by the pivotal place of trust amongst the crew.
Perhaps; or is it that we can make such inferences as we see fit, and suit our purposes? Do we move to a paraconsistent logic, in which only statements made within the text or inferred therefrom are true, and all other statements neither true nor false?
Which raises the intriguing notion of a free, paraconsistent logic.
Sounds like a good option.
Quoting Banno
Maybe that's the salient difference between the logic of fiction and logic of fact. We say that propositions concerning what we don't know about the world, even what we cannot know about the world, must be true or false, even if we can never know it.
In the case of fiction, it would seem that propositions about what has not been told must be neither true nor false. Even if a later telling were to make them true or false, this would then be a "world" which has been altered or augmented; a "world" which was not prior to the later telling.
Imagine if there was a God (an author) and our world was like that...
But then what of spoilers - story facts kept till last? Are they neither true nor false until presented in the narrative?
If you would,
Quoting Banno
Is this correct? Or have I misunderstood?
No, no. I have a recollection of wanting to make an argument about a fictional character about 6 months ago but I couldn't figure out how because of the vexing matter of existential import. I wanted an argument that has a nonexistent entity in it but it was impossible without also saying that the nonexistent exists.
It appears, prima facie, that the difficulty lies with the implied existence of [math]\exists x[/math]. If I'm correct, free logic should have existence as a predicate. We might also need to look at the ontological aspects of categorical/sentential/predicate logic.
Excelente!
It seems story facts can't be true or false until irrevocably committed to print, because otherwise the author might change her mind. They could be true or false in the context of a draft or even the author's imagination, if nothing is written down, as long as she doesn't change her mind.
On a classical logic, E!t should be trivially true in any model, because every model has to assign to each individual constant a member of the domain. So just in virtue of naming t, you're committed to it existing in this sense. Everything exists.
On a free logic, E!t is not trivially true – it can be true on some models, and false on others, depending on whether t is in the domain of existent things. Some things don't exist (according to some models).
@Srap Tasmaner
I'm interested in your answer to a question asked previously. You claimed Quoting Srap Tasmaner
In the SEP article the following example of a logical deduction is given: within the context of LOTR, Gollum hates the sun, and we can validly infer that something hates the sun.
You have claimed that "Gollum hates the sun" is false. Hence for you presumably the inference is invalid. Is that so?
I guess if I really wanted to do this, I'd assume fiction is a type of counterfactual, so you get your extensional semantics via possible worlds. Your nonsense category will show up as impossible worlds, I guess. That doesn't solve crossover problems directly. Doesn't Kripke write about this somewhere? How there can't turn out to be a "real" Sherlock Holmes, for instance.
So here we have the best attempt to formalise existence as a predicate for individuals. And it cannot be used to infer that some particular individual exists.
In particular, the logic shows that such arguments rely on question-begging.
Of the arguments of this type, two are of particular interest:
Hence, another change of title for this thread.
But I take it you agree that the inference that in LOTR there is something that hates the sun is valid? That fictional sentences may be true within the fiction?
Maybe? It's just hard to be sure what we mean by this.
Broadly, I'm not opposed to some kind of analysis that distinguishes internal and external frames of reference, however you do that, but it's not perfectly obvious how to do that formally.
What's more, people freely cross that boundary: "But in Chapter 3, Harry said ..." That sort of thing makes me suspect the "internal" frame of reference might actually just be shorthand for the external, just a condensed manner of speaking.
But I'm not wild about that approach either. For one thing, whatever analysis we arrive at for fictional objects ought to be able to support the fact that people care about fictional objects very nearly as if they were real. (At least as far as psychology is concerned, that suggests we're using some of the same machinery for understanding fictional worlds, and their furniture, that we use to understand the real one.)
Absolutely we expect fiction to be largely logical, except when it deliberately isn't. (William Burroughs, say.) That's a chunk of Mark Twain's critique of Cooper.
But then, this isn't very first-order, is it? More as though,
Quoting Snakes Alive
You're using predicates to refer to predicates (and other formulae including individual constants why not), instead of using them to refer to (only) individuals.
Which potentially is a problem if it isn't clear, and encourages equivocation between the x and the "x", the individual and the individual constant (use and mention, as ever), resulting in the obvious multiplicity of empty constants being used to excuse obvious baloney such as
Quoting Snakes Alive
An odd response, since free logic is exactly "how to do that formally". It also allows talk of both existent and nonexistent individuals within the same conversation.
But more over, if you are to care about the characters, you might begin by talking about them. That's what free logic permits.
As with all formal systems it provides a way of understanding the grammar of such conversations, of how it is that we can say things such as "Gollum is more famous than Gödel" without such claims being simply false.
But of course it's application is far broader than that.
Obviously, only Tolkien knew about Frodo before he wrote LOTR, no?
I don't see the point in NOT invoking epistemic logic into the discussion when thinking about works of fictions originally, (at least), existing in the mind of the author, and then his book or script after he or she decides to commence it into fiction for other people.
~E!(Paris)
Paris doesn't exist in any sense?
Paris doesn't exist in {eggs, Bacon}. Hence we might introduce the predicate "...is meat", the member of which is bacon; and conclude that Paris is not a meat; and "..."is breakfast", the members of which are both eggs and bacon, and conclude that Paris is not a breakfast.
We need a broader context in order to do more with Paris.
The difficulty I see with epistemic logic is that belief is a relation between a statement and an individual. But it's not just Tolkien who believes Frodo walked into Mordor. Hence my argument that statements about Frodo continue despite the demise of the author.
I agree. The second there is a purpose or context which influences the truth conditions of E!, it should not be interpreted outside of it. E!, in what domain? Purpose or context choose the domain - therefore purpose and context choose what exists?
Yes, I think that's roughly correct.
That falls in line with the injunction to look at use rather than meaning.
Do you see a problem with that?
I see a problem with the implied claim that people choose what exists, yes!
:grin:
When you put it like that...
But our thinking doesn't make it so. Things don't appear because we so choose. All that is happening here is the choice of topic.
So consider the section on simples in Philosophical Investigations, and the argument that what is to count as a simple is what is of use in the context.We choose what we treat as existing and what we don't.
It's that the world is divided up as we intend it to be.
Quoting Banno
Which we? Particular speakers or the linguistic activity in which they are embedded?
Can you spell out the distinction between these two claims:
( 1 ) People determine what exists.
( 2 ) The linguistic community determines what exists.
Quoting fdrake
Preferably within the confines of the assumption that people determine the linguistic community. If you decide to go down the avenue that the linguistic community's language items (speech acts, utterances, writing...) are created "in contact with" reality ("always-already" interpretation), how does that contact place constraints on what exists? What stops Santa Claus existing if people talk that way?
Surely, Tolkien didn't believe in Frodo the same way we believe the cup is still on the counter after we walk out of the room, no?
I mean by this that Tolkien most probably imagined Frodo, and then all the rest followed on with his quest to destroy the ring of power.
Anyhow, epistemic logic seemingly has no issues with free logic as long as fiction logic is at play? This is my only point, and don't see how you can argue otherwise? However, if you insist, there's quite a lot of logicizing that took place during the venture of Frodo with Sam.
I'm just confused in general with these topics about Pegasus or Santa denoting nothing of epistemic import, that we rely so heavily on epistemology; yet, it goes unacknowledged when solving these linguistic dilemmas.
Roughly, the linguistic community are those participating in the language game under consideration.
So breakfast need not include Paris.
Of course not, and so far as I can see this is outside of the considerations of free logic. Whatever your point is concerning epistemic logic, it remains opaque to me.
Quoting fdrake
Quite happy with "Santa Claus exists in Rudolph the Red Nose Reindeer", seems a stretch to go from that to "Santa Claus exists", no? Former has a context, latter doesn't*. If that's the only sense of exist which matters, context decides what exists.
It's been a while since you brought out the Snark. Maybe you were conjuring them into existence. :grin:
*Edit: Well, I imagine it must have a context, but it's fuzzy and broad. If I'm thinking about whether Santa really exists as a kid, I'm thinking toward that context. And we all know what context it is. Just hard to spell it out.
Far too risky, for obvious reasons. But take care, since this is just the place for a Snark... We don't want you softly and suddenly vanishing away—
Paris = London = porridge
Quoting bongo fury
Porridge is porridge, but not "porridge", to be sure, but how would this be relevant?
p = ?x?S ?x
So, x is bound to set S.
Quoting Free Logic » 5.4 Logics of Fiction
Neat example. The statement is perfectly parsable and easy enough to understand, but not quite logical. I guess Anselmian ontology relies on such logic, just optimize/maximize x by a well-chosen ?, and even fictions come to life, "therefore Gollum exists".
Quite fun, isn't it?
But see Inexpressibility of Existence Conditions. It seems we can't conclude that Gollum exists.
In
Quoting bongo fury
"Paris", "London" and "porridge" were used, in order to mention*, in this case, nothing.
The relevance is
Quoting bongo fury
There's only one nothing, if any.
* Edit if it helps @Banno, or anyone: to mention something is to refer to it. To use a word to mention something is to use the word to refer to the thing.
I don't follow that. It's use, not mention. As for use in order to mention... I've no idea what your point might be.
I think it's correct to say that existence is not a predicate or a property to be ascribed to individuals such as God, unicorns, Abraham Lincoln, or any other individual (fictional or not). When we say that X exists, we're saying something about the concept, i.e., either the concept has an instance in reality or it doesn't. This would seem to be the case in order to make sense of the statement, "Hobbits do not exist." In other words, for a claim to have meaning it must be about something, but since Hobbits don't exist, what could the statement be about? Simply put, it's about the concept, Hobbits. Thus, existence is not something individuals possess, but is rather, a way of expressing something about concepts.
This bothers me. Not you, but others, will quickly decide that the concept "hobbit" is some sort of mental creature, and so exists in minds, and conclude that their concept of hobbit is different to your concept of hobbit, that concepts-of-hobbits are private, and all the nonsense that ensues.
But hobbits are part of a shared game, the rules of which, although set by Tolkien, are subject to modification; Hobbits are a communal exercise.
I find it hard to deny the privatization of individualistic, thought generated imagery.
Needs an edit.
My bad. One second.
Sure. You can play chess by yourself, but it is in essence a game for two players.
Hobbits are for telling other folk stories.
I agree. I don't particularly find the definition of concepts as individual things any bit useful either; I guess I just wouldn't know how to deny it.
In a practical sense, you're pretty correct.
Well, people so all sorts of strange things, maybe hobbits are running around in people's minds. :wink:
I guess you're right: it's all about concepts. Hobbits do not exist in our present reality, if by "Hobbit" you mean:
Note that this definition (from Oxford Languages) of the concept is explicit about the imaginary nature of the beast, and its inventor.
If on the other hand you mean figuratively something like "a very small adult man or woman", they do exist. If you mean instead "another species similar to modern man, but smaller", they did exist way back: various Australopithecus sp., and closer to us Homo floresiensis ("Flores Man"), actually nicknamed "the Hobbit" among paleontologists. We don't really know if he had hairy feet but the odds look good enough to me.
From the title of the thread: Whatever thinks exists.
An excerpt from a book, to be precise a thought the fictional character Smith entertains:
"This looks like a lovely spot for a family picnic," thought Smith.
[quote=René Descartes]I think, therefore I am[/quote]
Smith thinks, therefore Smith is (exists).
:chin:
That is and argument that concludes that some individual - such as the "I" in Descartes, or God in the argument from necessity - when set out in free logic finds itself presuposing what it seeks to prove.
Think of free logic as an attempt to make explicit the logical structure of such existential arguments by making explicit the first order existential predicate E!a - "a exists", where a is a proper name; so an example would be "MadFool Exists".
And what this explication found is that it cannot deduce that MadFool exists. All it can do is presuppose it, by assuming that MadFool is a part of the domain of E!x.
Put anther way, in trying to show the validity of "I think therefore I exist" it instead shows that it is circular, that "I think" already supposes that "I exist".
Descartes' argument is valid, but circular.
Now consider this in relation to your question in the "being" forum: Quoting TheMadFool
The existence of things to talk about is supposed by language.
Of course, it's not just language that does this, but anything we do - hammering a nail supposes both hammer and nail; asking a question supposes someone to ask; and so on.
I've got an issue with presuppositions with regard to free logic and the supposed circularity (question begging) in Descartes' argument and others like it.
You claim that I think, therefore I am is circular because I think, according to you, presupposes that I am.
I've always been a little confused by conditionals. Take the following two statements:
1. If it rains then the ground will become wet [classic entailment: rain makes the ground wet]
2. If you're relying on arguments to make a point then you endorse the utility of logic in making a point [presupposition: arguments presuppose logic]
Come now to Descartes' argument:
3. I think
Ergo,
4. I am
Is this conditional (I think [math]\rightarrow[/math] I am) an entailment or a presupposition? Both? Neither?
An entailment.
Quoting Banno
Didn't you say that it was a presupposition?
Perhaps it's both but then that doesn't help, does it?
To think, there's got to be a thinker. That's why Descartes' argument works. This is, as you said, an entailment.
I would treat the thinker as a necessary condition to be able to say "I think". Are all necessary conditions presuppositions.
My hunch is: all presuppositions are necessary conditions but not all necessary conditions are presuppositions.
Which isn't a problem for Descartes' purposes.