Aristotle would not have supported proving anything exists simply from logic structures alone. — Gregory

What specific theorems of modal logic do you have in mind as "proving something exists from logic structures alone"? And what passages in Aristotle do you take to be predictive that he would not countenance modern modal logic?

Modal logical leads to it's using the ontological argument. That's the best example of how ML can go to far

Is ▢ ∃(x)(a=x) well-formed? Is it a theorem of S5? — Banno

Ex a=x is a theorem of the pure predicate calculus with identity.

So it seems to me that it is the case that necessarily Ex a=x.

Am I overlooking something? (I admit that I'm rusty in modal logic.)

I searched "what had model logic proved" and there was nothing
— Gregory

https://philosophy.stackexchange.com/questions/23929/what-are-the-practical-applications-of-modal-logic

https://www.sciencedirect.com/science/article/pii/S1571066114000905 — TonesInDeepFreeze

You said you couldn't find references. I gave you ones.

Modal logical leads to it using the ontological argument. — Gregory

What system of modal logic and what specific theorem are you referring to?

Those links are about computer science

Aristotle would never take the ontological argument under consideration. But modal theorists struggle with it nevertheless

Good point.

So is ▢ ∃(x)(a=x) a rendering of "a (that individual) exists in every possible world"?

That seems problematic, since one could specify a possible world in which A did not exist.

Those links are about computer science — Gregory

You said you couldn't find mention on the Internet of anything modal logic proves. So those are links were modal logic is used for results in computer science.

And one of the links is not just about what is proved but actual use for traffic signal systems.

is ▢ ∃(x)(a=x) a rendering of "a (that individual) exists in every possible world"? — Banno

In

Ex a=x

'a' is merely a variable or constant (depending on your specification). It has no specific referent except by assignment to an individual in the domain of the model.

B applies to a theorem, not an individual. I don't think we have an answer yet. — Banno

You mean the corollary of axiom B? It does not apply to an individual, but it could apply to a proposition which has an individual as its subject, that is: “Necessarily, God exists”. If this proposition is true in some possible worlds (meaning it does not entail a contradiction), then it's possible that it's necessary that God exists, and using corollary B, the advocate of the modal ontological argument will argue that it implies that God exists, unless one can show that God exists in no possible worlds, which is to show that the proposition: “Necessarily, God exists” entails a contradiction.

Is ▢ ∃(x)(a=x) well-formed? Is it a theorem of S5? — Banno

As TonesInDeepFreeze pointed out, it seems to be well formed, since it follows logically from “the pure predicate calculus with identity”, and truths infered from pure logic are necessarily true if they are deduced from principles which are themselves necessarily true.

But it would mean something like: Necessarily, there exists/is an x (God), such that a (the greatest conceivable being/ subject of all perfections) = x.

Aristotle would never take the ontological argument under consideration. But modal theorists struggle with it nevertheless — Gregory

I asked what specific system and what specific theorem you have in mind. Apparently you don't have anything specific in formal modal logic in mind.

And I asked you what passages in Aristotle you have in mind. Again, apparently none.

Yep - thanks. You are helping me to articulate my question.

So is the implication of ▢ ∃(x)(a=x) that, in any given model, there must be an individual that exists in every possible world?

But it would mean something like: Necessarily, there exists an x (God), such that a (the greatest conceivable being/ subject of all perfections) = x. — Amalac

I don't want to go there yet - leave the good lord int he background and look at the implication of ▢ ∃(x)(a=x)...

So is the implication of ▢ ∃(x)(a=x) that, in any given model, there must be an individual that exists in every possible world? — Banno

It seems you are wondering how to formalize "there is an individual that exists in every possible world". I don't know at this moment. But I don't think it is Ex a=x.

Ex a=x really doesn't say much. It's so obviously true that translating it into English looks almost silly. Here's one way: "Whatever is named by 'a', there is an object x in the domain such that what is named by 'a' is x." That is obviously true since such an x is whatever is named by 'a'.

I've been clear but you don't follow. Aristotle had philosophy-like arguments for the first mover, something ML doesn't have. I stated right at the start that logic is used in science just as math is. But logic can't say what can exist and what can't. It is plain old logic at the end of the day, and yes for Aristotle metaphysics is not logic. This is all common knowledge

It seems you are wondering how to formalize "there is an individual that exists in every possible world'. — TonesInDeepFreeze

Well, yes, in that I don't think it can be done - it woudl not be well-formed - or it would lead to contradiction.

I guess I'm trying to formalise an apparent contradiction between "there is an individual that exists in every possible world" and "for any given individual, one can specify a possible world in which that individual does not exist".

Again, thanks for your help.

So, still, you've shown nothing specific that Aristotle wrote that would be predictive that he would reject modern modal logic.

The fact alone that Aristotle had certain arguments to prove the existence of God does not predict that he would reject modern modal logic.

But logic can't say what can exist and what can't. — Gregory

It easily says, "There exists an object that has property P if and only if it has property P".

It easily says "There does not exist an object that has property P if and only if it does not have property P".

All that comes from non logic. ML tries to rule over philosophy like logicism tried to do with math

But int he case of logic what is distinguished is coherent consistent argument against nonsense.

Individuals are members in universes. Now, suppose an individual is a member of a certain universe, of course that individual is not a member of certain other universes. So, yes, there is no individual that is a member of every universe.

On the other hand, whatever individual 'a' names by a model, that individual is a member of the universe for the model that names 'a'. But 'a' may name different individuals according to different models.

And if 'a' is in the language, then, for any model, 'a' must name some individual in the universe for the model. And syntactically in the object language we have: Ex a=x for which the derivation is trivial:

Ax x=x
a=a
Ex a=x

But logic is about contrapositives, inverse minor premises, ect. It's useful to understand how to understand one's own thinking and map the world, but the great questions of philosophy can't be answered thru finding the final modal logic idea that settles everything. Doesn't modal logic implicitly assume it can settle these questions one and for all?

All that comes from non logic. — Gregory

No, it comes from first order logic.

ML tries to rule over philosophy — Gregory

Mathematical logic is a field of study. It has no will such that it could will to rule over anything, including another field of study. Meanwhile people who work in the field of mathematical logic might have a will to rule over people in the field of philosophy. So, yes, we should have that investigated, maybe start with a Presidential commission.

Ah, so here you are using "universe" - is that not the same as "possible world"? Or is it like "domain of discourse"?

Doesn't modal logic implicitly assume it can settle these questions one and for all? — Gregory

No. That is an hilarious question.

Doesn't modal logic implicitly assume it can settle these questions one and for all? — Gregory

Well, no. Rather it allows you to recognise good and bad arguments.

Name one thing in modal logic literature that proved something in philosophy.

"universe" - is that not the same as "possible world"? Or is it like "domain of discourse"? — Banno

Yep. With first order logic, every model specifies a universe. For modal logic, we specify a set of universes.

First order logic gets its content from non logic

On the other hand, whatever individual 'a' names by a model, that individual is a member of the universe for the model that names 'a'. But 'a' may name different individuals according to different models. — TonesInDeepFreeze

So if I could describe what I think you are saying back to you in my own terms...

To build a model, we set up a bunch of possible worlds. Within that universe, "a" refers to some given individual. In some possible worlds, "a" exists, in others, "a" does not exist - but in those universes "a" refers to that very same individual.

OK so far?