An analysis of truth and metaphysics

I'm tired, so I might not be seeing it right, but (2) looks troubling. Don't you need T("p") independent of the conditional in order to introduce the existential quantifier?

So
T("p")
_______
?"p"

might be valid.
Quoting Michael

?p: T("p") ? ?"p" (from 1, by existential introduction)

But in (1) you have ?p: T("p") ? p, not T("p").

Also, ?"p" is ill-formed, unless you are using Free logic.

Quoting Michael

T(x) ? x is true (definition)

Looks like you've defined a fixed point of some function. But I doubt that is what you mean?

Reply to Banno

According to existential introduction:

Q(a) ? ?xQ(x) (if John is bald then there exists at least one thing which is bald)

And surely:

?xQ(x) ? ?x (if there exists at least one thing which is bald then there exists at least one thing)

And so:

Q(a) ? ?a (if John is bald then John exists)

Maybe my particular symbols aren't being used quite right, but surely the logic works? In ordinary language (and providing the complete account of existential introduction) it would be:

1. proposition "p" is true if and only if p
2. if proposition "p" is true then there exists some proposition which is true (existential introduction)
3. if proposition "p" is true then there exists some proposition
4. if proposition "p" is true then proposition "p" exists

4 must follow otherwise we would have the situation where, from 2, the truth of proposition "p" would entail that there exists some other (true) proposition. Or in the case of John being bald, that John being bald entails that there exists some other (bald) thing. Which seems absurd.

Quoting jgill

Looks like you've defined a fixed point of some function. But I doubt that is what you mean?

I'm just explaining what is meant by the predicate T. I could have written the argument as:

1. ?p: "p" is true ? p

But that would require more typing.

Reply to Michael From the other thread:

Quoting Banno

2. ?p: T("p") ? ?"p" (from 1, by existential introduction)
— Michael

Again, this looks problematic. "p" is an individual variable, so ?"p" is like ?(a). You would have to move to a free logic and use ?!"p"; but that something exists cannot be the conclusion of an argument in free logic. You have to assume that p is true and that there is a sentence "p".

This is now spread over two threads. https://thephilosophyforum.com/discussion/13170/an-analysis-of-truth-and-metaphysics

Reply to Banno I'm pretty sure my comment above addresses that.

If John is bald then John exists
If the proposition "it is raining" is written in English then the proposition "it is raining" exists
If the proposition "it is raining" is true then the proposition "it is raining" exists

Reply to Michael But. ?"p" is ill-formed. "p" is an individual. So it's lie writing ?(a). Quoting Michael

If John is bald then John exists

It's "If John is bald then there is something that is bald".

Quoting Banno

It's "If John is bald then there is something that is bald".

Is there a difference between "if John is bald then there is something that is bald" and "if John is bald then there exists something that is bald"?

Reply to Michael But you have "if John is bald then John exists". That's invalid ill-formed. ?(a) is not a formula in first order logic, unless you move to free logic. Then it would be ?!(a)

?!(a) = ?(x)(a=x) (dfn)

Quoting Banno

But you have "if John is bald then John exists". That's invalid ill-formed. ?(a) is not a formula in first order logic, unless you move to free logic.

How is it ill-formed? It makes perfect sense to me:

If John is bald then something exists which is bald
If John is bald then something exists

Are you saying that the second sentence doesn't make sense? Or is false?

Reply to Michael I'm saying it's not allowed in the rules of classical logic.

God exists, Whatever thinks exists, Fiction: Free Logic might help.

Do metaphysical claims entail observables? The problem of evil, god - metaphysics - denied because his omnibenvevolence, omniscience, and omnipotence is contradicted by observation (rampant evil).

Are metaphysical claims amenable only to a priori proofs?

What theory of truth are we using for metaphysical claims? If the correspondence theory of truth then, they can be and should be verified/falsified.

Quoting Banno

I'm saying it's not allowed in the rules of classical logic.

Then what logic am I using when I say that if John is bald then John exists? Or that if the cat is on the mat then the cat exists? Because they seem like logical inferences to me. It would be strange to say that the cat is on the mat but there isn't a cat.

If your problem is with my (mis-)use of formal symbols then you can consider the argument in natural language as I started with here.

But then let's look at the argument using the complete form of existential introduction:

1. ?p: T("p") ? p (premise)
2. ?p: T("p") ? ?xT(x)
3. ?p: p ? ?xT(x)
4. ?p: ¬?xT(x) ? ¬p
5. ?p: T("¬p") ? ¬p (premise)
6. ?p: T("¬p") ? ?xT(x)
7. ?p: ¬p ? ?xT(x)
8. ?p: ?xT(x)

The main conclusions being 4 (if there are no true truth-bearers then nothing is the case) and 8 (there is at least one true truth-bearer).

And so there is still the issue that either a) truth-bearers are dependent on thought and speech and so if something is the case then something true is thought or spoken or b) truth-bearers are independent of thought and speech.

If needed we can reformulate our initial premise and derive the slightly different conclusion that for all truth-bearers there is at least one true truth-bearer:

9. ?"p": T("p") ? p
...
10: ?"p": ?xT(x)

Still working on how to properly formulate 11. Perhaps something like:

11. ?p: p ? ¬?"p"T("p") (there is at least one case where if that thing is the case then there is no true proposition that it is the case).

[quote=Banno]I'm saying it's not allowed in the rules of classical logic.[/quote]

:up: Some of us are running an old logic module; others seem to have been habituated to double-think, and still others are at ease contradicting themselves, not because they're inured to it, but for the simple reason that it makes complete sense to them.

Classical logic is 2.5k years old - time for an upgrade, oui? Paraconsistent logic comes to mind, but that's just tinkering around with the rules of logic and it, for some reason, hasn't caught on among philosophers. I wonder why?

Reply to Banno

This (and the comments by @Snakes Alive in the discussion you linked to) would suggest that ?x(x = q) is valid in first-order logic, and doesn't require free logic? So I can do T("p") ? ?x(x = "p").

I've updated my original post accordingly.

Actually, looking at this, it does appear that steps 4 and 11 (x does not exist) of my argument depend on free logic:

Free logic allows such statements to be true despite the non-referring singular term. Indeed, it allows even statements of the form ??x x=t (e.g., “the ether does not exist”) to be true, though in classical logic, which presumes that t refers to an object in the quantificational domain, they are self-contradictory.

I'm happy with this, as I would say that "the ether does not exist" is in fact true.

My one initial concern is with whether or not modus tollens applies to ?p: p ? ?x(x=q). I'll do some digging.

OK, I've done some further research and in classical logic ¬?x(x=q) isn't allowed and in free logic T(q) ? ?x(x=q) is false.

I'll strike out the free logic stuff for now as the rest is still interesting and will come back to them when I've learnt more.

On that note though, what logic would you say ordinary language uses? Because in classical logic you can't say "if the cat does not exist then the cat is not on the mat" and in free logic you can't say "if the cat is on the mat then the cat exists."

It seems to me that in ordinary language we can say both, and so an ordinary language interpretation of my argument still holds.

Quoting Michael

what logic am I using when I say that if John is bald then John exists?

"John exists" is not expressed in mere predicate logic. You need modal logic for it.

Quoting Agent Smith

Classical logic is 2.5k years old

Classical logic is about 140 years old.

Yet again you shoot your mouth off not knowing what you're talking about.

Quoting TonesInDeepFreeze

what logic am I using when I say that if John is bald then John exists?
— Michael

"John exists" is not expressed in mere predicate logic. You need modal logic for it.

Why not (with "J" for "is John" and "B" for "is Bald"),

?x (J(x) ? B(x)) => ?x (J(x))

?

Reply to bongo fury

Yes, of course, we can do that.

[You know all this; I'm writing it for benefit of those who don't:]

You can have the predicate 'is John', which is something different from just the name 'John'.

Simplest example from mathematics:

'0' is an operation symbol, defined

0 = x <-> Ay ~yex.

But we don't write:

Ex 0

And we don't write:

Ex x

They are not well formed. A quantifier is concatenated with a formula, and a mere term is not a formula.

Meanwhile, for any term T whatsoever, it's a logical theorem:

Ex x = T

which includes:

Ex x = x

To put it another way, there is not "existence predicate" in predicate logic. We go to modal logic for that.

But we could define a predicate symbol:

Mx <-> x = 0

And we can say:

ExMx

Reply to TonesInDeepFreeze Yes, I think I clarified my position [url=https://thephilosophyforum.com/discussion/comment/714164]here[url].

Reply to Michael

I added that modal logic is the main arena for this.

I think the below should avoid the need for that.

T(q) ? q is true
P(q) ? q is a proposition

1. T(q) ? p
2. T(q) ? ?xT(x)
3. ?xT(x) ? ?xP(x)
4. p ? ?xP(x)
5. ¬?xP(x) ? ¬p

Does this logic work?

If it is raining then some x is a proposition. If no x is a proposition then it is not raining. These could be seen to be problematic conclusions, as it suggests either Platonism (of propositions) or some form of antirealism.

Reply to Michael

There is one premise there:

Tq <-> p

Following that, I don't see a problem with the logic. But you use vacuous quantification with

ExTq

and

ExPq

So, though there is no mistake in the logic, I don't see any point in it.

Reply to TonesInDeepFreeze I have since edited it to ?xT(x) and ?xP(x) as I believe that's the correct application of existential introduction?

And the point is the conclusions on lines 4 and 5. It's easier to understand in ordinary language:

1. "it is raining" is true iff it is raining
2. "it is raining" being true entails that some x is true
3. Some x being true entails that some x is a proposition
4. If it is raining then some x is a proposition
5. If no x is a proposition then it is not raining

As I said above, it seems to suggest either Platonism (of propositions) or some form of antirealism. That's the point of the argument.

Reply to Michael

My mistake: The logic is not correct. Line 3 (whether original or reviesd) is a non sequitur.

1. Tq <-> p ... premise
2. Tq -> ExTx ... EG
3. ExTx -> ExPx ... non sequitur (prob you have an unstated premise in mind)
4. p -> ExPx ... sentential logic, but relies on non sequitur in step 3
5. ~ExPx -> ~p ... sentential logic, but relies on non sequitur in step 3

Quoting TonesInDeepFreeze

The logic is not correct. Line 3 (whether original or reviesd) is a non sequitur.

Line 3 is ?xT(x) ? ?xP(x).

That some x is true semantically entails that some x is a proposition, given that truth is predicated of (and only of) propositions.

Maybe it's simpler to just understand T(q) as "q is a true proposition". If some x is a true proposition then some x is a proposition.

Quoting Michael

That some x is true semantically entails that some x is a proposition, given that truth is predicated of (and only of) propositions.

Truth is semantic. My point is that you are missing the premise:

Ax(Tx -> Px)

Reply to TonesInDeepFreeze I did say that "maybe it's simpler to just understand T(q) as 'q is a true proposition'." So that gives us:

T(q) ? q is a true proposition
P(q) ? q is a proposition

1. T(q) ? p
2. T(q) ? ?xT(x)
3. ?xT(x) ? ?xP(x)
4. p ? ?xP(x)
5. ¬?xP(x) ? ¬p

3 follows from the two definitions.

1. "it is raining" is a true proposition iff it is raining
2. If "it is raining" is a true proposition then some x is a true proposition
3. If some x is a true proposition then some x is a proposition
4. If it is raining then some x is a proposition
5. If no x is a proposition then it is not raining

Quoting Michael

I did say that "maybe it's simpler to just understand T(q) as 'q is a true proposition'."

Yes, after I audited both your original and revised arguments. Of course, I have no problem with emending your argument again now.

Since "proposition" and "true proposition" are not in your argument itself, this would work:

1. Tq <-> p ... premise
2. Ax(Tx -> Px) ... premise
3. Tq -> ExTx
4. ExTx -> ExPx
4. p -> ExPx
5. ~ExPx -> ~p

That's all fine, but the more general point I mentioned is that we need to move to modal logic to have existence as a predicate.

Quoting TonesInDeepFreeze

Since "proposition" and "true proposition" are not in your argument itself

I'm not sure of the proper procedure for specifying definitions, but I did have these two (unnumbered) lines are the start:

T(q) ? q is a true proposition
P(q) ? q is a proposition

And note that I used the symbol ? (semantic entailment), not the symbol ? (material implication). Which is why I didn't think your second premise is needed.

Quoting Michael

T(q) ? q is a true proposition
P(q) ? q is a proposition

You didn't use them in the proof.

The semantic turnstile as opposed to the proof turnstile is not important in this context. You don't even need any turnstile.

Quoting TonesInDeepFreeze

The semantic turnstile as opposed to the proof turnstile is not important in this context. You don't even need any turnstile.

Maybe I don't need it but I thought it would be simpler to use it. Maybe I misunderstood what it meant.

I thought it would be enough to say "some x being a bachelor semantically entails that some x is an unmarried man".

I didn't think I'd have to say "for all x, if x is a bachelor then x is an unmarried man, and so if some x is a bachelor then some x is an unmarried man".

But if I'm wrong I'm wrong. So thanks for the correction.

And with your corrections we can then address the crux of the issue: the conclusions that if it is raining then some x is a proposition and if no x is a proposition then it is not raining. So is this Platonism (of propositions) or antirealism?

Reply to Michael

Yes, "G |= F" means G semantically entails F; and "G |- F" means G proves F.

But, due to completeness and soundness, G |= F iff G |-F.

So you don't advance any point by switching from one to the other mid-proof.

For that matter, due to the deduction theorem, you only need the implication sign, not any turnstile.

/

Depending on the context, 'proposition' stands for something different from 'sentence'. But you use 'p' for a sentence (you negate it, so it's a sentence). I don't see how one would figure out anything about platonism or anti-realism from your argument.

Quoting TonesInDeepFreeze

Depending on the context, 'proposition' stands for something different from 'sentence'. But you use 'p' for a sentence (you negate it, so it's a sentence). I don't see how one would figure out anything about platonism or anti-realism from your argument.

If a proposition is a sentence then the conclusions are:

1. if it is raining then some x is a sentence, and
2. if no x is a sentence then it is not raining

And if a sentence is an utterance then the conclusions are:

1. if it is raining then some x is an utterance, and
2. if no x is an utterance then it is not raining

This appears to connect the occurrence of rain to an utterance, suggesting antirealism. Realists would argue that there is no connection; that there is some possible world where it is raining but where nothing is uttered.

If a proposition is not a sentence, such that it's possible that some x is a proposition but no x is an utterance, then it suggests Platonism, as how else would one interpret utterance-less propositions?

Reply to Michael

'utterance' means speaking out loud. Or do you have a different sense in mind?

Quoting TonesInDeepFreeze

'utterance' means speaking out loud. Or do you have a different sense in mind?

Speaking or signing or writing. Perhaps "linguistic expression" is the more inclusive term. So the question is whether or not a proposition (or if we want to be more inclusive, "truth-bearer") is identical to a linguistic expression, or is in some sense dependent on a linguistic expression. If so then if some x is a proposition then some x is a linguistic expression, in which case if it is raining then some x is a linguistic expression and if no x is a linguistic expression then it is not raining. This seems to me to suggest antirealism.

Alternatively propositions are neither identical to nor dependent on linguistic expressions, in which case it can be that some x is a proposition even if no x is a linguistic expression. This seems to me to suggest that propositions are Platonic entities.

Quoting TonesInDeepFreeze

1. Tq <-> p ... premise

Quoting Michael

Realists would argue that there is no connection; that there is some possible world where it is raining but where nothing is uttered.

Do you mean where p without Tq?

Or where not even p, because that's an utterance?

But if uttering p is ok to describe (from outside it) the state of some utterance-free world, why not also Tq?

Quoting TonesInDeepFreeze

That's all fine, but the more general point I mentioned is that we need to move to modal logic to have existence as a predicate.

Several families of wholly non-modal logics have existence as a predicate: free logics, inclusive logics, Meinongian logics et. cetera.

Traditional modal logics that extend classical logics, like FOL or FOL=, with modal axioms, also do not treat existence like a predicate. The modal logics that treat existence like a predicate are, at least all the ones that I'm aware of, just modal extensions of some of the non-modal systems above (i.e. Barba's free modal logic).

Reply to Kuro

I should have said, "we move to modal logic of some other appropriate system more involved than mere first order logic". Of course there is no limitation on systems and semantics that may be devised.

For treatment of the existence predicate in modal logic, see the common textbook, Hughes & Cresswell.

Reply to TonesInDeepFreeze

Are you referring to the E formula from FOL= (and similar systems), such that Exists(x) =df ?y y=x?

While that's certainly an 'existence predicate', it is usually not what is really at stake in the debate of an existence predicate (i.e., it's sorta trivial). Usually, the controversial kind of existence predicate that we're interested in is the one that allows us to say ?x ¬Exists(x), aka quantify over non-existent things, whatever those are.

Quoting Kuro

Are you referring to the E formula from FOL= (and similar systems), such that Exists(x) =df ?y y=x?

Of course not. (1) AxEy y=x is a theorem, but I have never seen Ey y=x in FOL= as a definiens for Exists(x). It would be pointless. (2) My point is the opposite: FOL= does not have an existence predicate. (3) Indeed, the "existence predicate" I mean is Exists(x) as in modal logic.

Quoting TonesInDeepFreeze

(1) AxEy y=x is a theorem

Correct.

Quoting TonesInDeepFreeze

Ey y=x in FOL= as a definiens for Exists(x). It would be pointless

Correct. I was just making sure, because this formula translates to FOL= extended modal systems like FOL + S5, but it's obviously trivial and not the controversial existence predicate that logicians (or metaphysicians) are interested in.

Quoting TonesInDeepFreeze

(2) My point is the opposite: FOL= does not have an existence predicate.

I'm aware, and I agree (besides the trivial quantifier-defined one), I was simply noting that modal logic is not a prerequisite to having existence predicates in any sense: most logics with existence predicates are not modal (to this, I think you agreed)

Quoting Kuro

most logics with existence predicates are not modal

But modal logic is the more common one to study than all the others combined. (That's not an argument that modal logic is "better" or anything like that, just that it's natural enough to first turn to modal logic, as a common subject, to see what it offers, while not precluding that the number of other approaches is potentially inexhaustible too.)

Quoting TonesInDeepFreeze

But modal logic is the more common one to study than all the others combined. (That's not an argument that modal logic is "better" or anything like that, just that it's natural enough to first turn to modal logic, as a common subject, to see what it offers, while not precluding that the number of other approaches is potentially inexhaustible too.)

My issue isn't with modal logic here. I'm just unsure why you're characterizing modal logic as ones that deal with existence predicates: most modal logics are standardly extensions of FOL with K and some of the additional modal axioms, and therefore do not express nontrivial existence predicates.

Surely, certain modal logics can express existence predicates but these aren't extensions of a classical base, and by that point, there are similarly non-classical logics that express existence predicates.

So I'm just wholly confused why it is that we turn to modal logics to talk about existence predicates.

Quoting Kuro

I'm just unsure why you're characterizing modal logic as ones that deal with existence predicates

I agreed that existence predicates are handled in systems other than modal logic. And I'm not claiming that every version of modal logic in basic forms includes the advanced subject of an existence predicate.

Quoting Kuro

most modal logics are standardly extensions of FOL with K and some of the additional modal axioms, and therefore do not express nontrivial existence predicates.

But in the overall subject of modal logic, we do find a definition an existence predicate. We find that in textbooks such as Hughes & Cresswell (among the preeminent introductions to modal logic) and L.T.F. Gamut. I'm highlighting modal logic for this subject only because one is more likely to encounter a course in, or textbook on, modal logic before some of the other advanced alternative logics.

I have no interest in convincing you or anyone else not to investigate existence predicates in whatever logic systems you or anyone else wishes to study them in whatever order you or anyone else wishes to study them.

/

It's been a while since I studied this, but, if I recall correctly, Hughes & Cresswell and L.T.F. Gamut do define an existence predicate in modal logic that is an extension of classical FOL=. (I'll happily stand corrected though if I my memory is incorrect.)

Quoting TonesInDeepFreeze

It's been a while since I studied this, but, if I recall correctly, Hughes & Cresswell and L.T.F. Gamut do define an existence predicate in modal logic that is an extension of classical FOL=. (I'll happily stand corrected though if I my memory is incorrect.)

Ohh I'm actually familiar with this, I've recently read chapter 16 of Hughes & Cresswell after your recommendation. I fully understand: by relativizing formulas that lack modal quantifiers to quantify just over the actual world, we can coherently define an "existence predicate" whose extension is just those set of things that actually exist (i.e. exist in the actual world), i.e., it doesn't falsify the fact that Santa does not exist that he exists in other possible worlds, mainly because 'existence' simpliciter is relativized to the actual world. And we can generalize this such that for any world/node we can define an existential predicate just over the domain of that node.

This is moreso along the lines of the actualism vs possibilism issue and univocity of existence versus the distinct metaontological/logical debate about the nature of existence itself, though this is still nonetheless quite relevant because it is important to the related topic of modal realism. Someone like David Lewis, for instance, takes the proper existence simpliciter to be an unrestricted quantifier, quantifying over all possible worlds, so he takes "Santa exists" simpliciter to literally be true, and interprets our ordinary discourse as implicitly nested under actuality quantifiers. Of course, this will depend on our background theory of the metaphysics of modality and is highly controversial, but I digress.

I get what you're saying now. I suppose I have to make a three-fold distinction on the types of 'existence predicates' that philosophers and logicians consider:

• The trivial E formula, defining the predicate in terms of the existential quantifier. Uninteresting.
• The existential predicate defined in terms of the domain of a modal node / world
• The existential predicate simpliciter, such that ?x¬Ex

(1) is uninteresting in all regards, (2) is relevant to the disputes about existence in relation to quantification & modality, i.e. relevant to Quine, Lewis and the like, and (3) is relevant to the disputes about the existence-being distinction, i.e. those between Frege & Russell against Meinong which preceded the formal regimentation of modality.

I did not consider the possibility that the term 'existence predicate' could be in reference to (2), which makes complete sense now in retrospect. I'm familiar with the topic of (2) though, so this is certainly a fault on my end.