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

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.

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

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.

T(q) ≔ q is a true proposition
P(q) ≔ q is a proposition — Michael

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.

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

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?

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.

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. — TonesInDeepFreeze

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?

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

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

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.

1. Tq <-> p ... premise — TonesInDeepFreeze

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

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?

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. — TonesInDeepFreeze

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).

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.

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.

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

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.

(1) AxEy y=x is a theorem — TonesInDeepFreeze

Correct.

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

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.

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

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)

most logics with existence predicates are not modal — Kuro

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.)

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.) — TonesInDeepFreeze

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.

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

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.

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. — Kuro

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.)

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.) — TonesInDeepFreeze

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.