It doesn't need to have a particular definition – it's just a predicate. — Snakes Alive

'x' is a variable. And 'E!' is being used as a 1-place predicate symbol. So either it is a primitive predicate symbol, in which case it would appear in the axioms, or it is a defined predicate symbol.

I haven't found a full explication of the argument in a modal logic, with all the terms defined from primitives or previously defined terms.

How does a system of modal logic talk about its own semantics? I'm not saying it can't be done, but I'd like to know how it works.
— TonesInDeepFreeze

The corollary of axiom M states that A→◇A , so systems that have axiom M do consider the actual world as one of the possible worlds, since a possible world is simply a world, real or imagined, that does involve any contradictions, and so the actual world is one of them. — Amalac

I don't see how that answers my question.

My question was a followup. The previous point is more important:

I know the definition of E!xP, but I would like to know the definition of E!x.

deleted

Motivated by the posts of Snakes Alive, I consulted a textbook so that I think I understand what he's saying, which I think is along these lines, even if not exactly:

For modal semantics, there are two different methods we may adopt for assigning denotations to the constants:

(1) For each domain of a possible world, we assign the denotations. The assignments for different domains might be different. That is the method I used earlier in this thread.

(2) We make only one assignment, so that each constant is assigned to a member of the union of the domains. Obviously, with this method, my previous remarks don't apply.

But I don't understand how (2) could work with a base of ordinary predicate logic. Suppose individual d is in the union of the domains but not in a particular domain D for a possible world w. Then if constant 'a' is assigned to d, then how could w even be a possible world? It would lack a denotation for 'a'.

You can define any predicate you like — Snakes Alive

Let 'Px' intuitively say "x exists". What is the definition of 'Px' in the form?:

Px <-> Fx where 'Fx' is a formula in a given first order language (or with a modal operator too, if you like) with only 'x' free and not including the symbol 'P'?

To make a predicate 'P' that is necessary for an individual a at a model, you just posit that the model you're working with is such that for all worlds w in the set of worlds W associated with the frame of the model, P(a) evaluates to true at w. — Snakes Alive

I was asking about a predicate "exists".

Anyway, what you mentioned is a semantical. How would we express that as a formula in the modal logic itself?

But necessity, on a Kripkean semantics, is not a matter of logical truth that generalizes over models – it's a matter of truth at all accessible worlds to some particular world, and if we have an accessibility relation on which every world is accessible from every other, then this is equivalent to truth at all worlds in that particular model. There is no impediment to supposing such a model. — Snakes Alive

I do seem to recall those particulars to be correct. But that's all semantical. I don't know how you can express such things in the modal logic itself.

you just say that E!(a), let's say, where 'a' refers to that individual, values true relative to every world W in the domain of worlds in your frame. — Snakes Alive

How do you express that as a modal formula in the object language?

f there is some individual x identical to a in the domain of individuals, then there will be at any world, — Snakes Alive

I don't quite follow you. In any domain, there is an individual named by 'a'. But the individual named by 'a' may be different in different domains.

On the other hand, you can make the domain relative to a world, such that at world w, there is an individual x identical to a, but at world w', there is no (because the domain associated with w includes a, while the domain of individuals associated with w' does not). — Snakes Alive

Yes, that is the sense I had been mentioning. If 'a' names a certain individual in one world then it is not required that 'a' names that individual in other worlds. Indeed the universes of the worlds in question could even be disjoint.

you are not forced to make existence necessary existence, but you can – you can just include a in the individual-domain of every world in your domain of worlds. — Snakes Alive

Yes. But how do you express that as a formula in the modal logic object language?

If you just have an existence predicate, E! — Snakes Alive

What systems have that predicate? Is it definable in typical modal systems? What is the definition?

A logic that banned the necessary existence of an individual would have to make some special provision for how existence is interpreted, and why you could never have a domain of worlds such that an individual exists at every world. — Snakes Alive

That I understand. Of course, there is no restriction that demands that the intersection of the set of universes is empty.

even if one accepted S5, the modal ontological argument would still have major problems, like dealing with the objection that existence is not a predicate or is a second order predicate, since that premise is required — Amalac

Ah, I admit that I am not familiar with a system that has existence as a predicate. For ordinary predicate logic, it's not a formalizable notion. I didn't know that it can be formulated in certain modal logics.

The argument would go something like this:

The actual world is one among the possible worlds — Amalac

I don't know enough about this. Isn't the above a meta-argument about the semantics for formal logic? That would be okay, but I'd like to be clear what really is afoot.

(this again follows in some systems of modal logic) — Amalac

this again follows in some systems of modal logic — Amalac

How does a system of modal logic talk about its own semantics? I'm not saying it can't be done, but I'd like to know how it works.

Now logic is about the forms of itself and can't comment on the inexperienced. — Gregory

I don't know what that means.

You admit this! — Gregory

How could I admit it when I don't even know what it means?

modal logic is just logic and much closer to programming than philosophy — Gregory

It depends on a definition of 'philosophy'. Ordinarily philosophy is regarded to include logic and formal logic. If you insist on a definition of 'philosophy' rigged to exclude formal logic, then, of course, you will have prevailed to establish, by definition, that formal logic is not part of philosophy.

...and must there be a number that exists in all universes?

The answer is "no"? — Banno

Indeed. I gave an example. And it's basic set theory. For any set S, and individual d, we have that d is not a member of S\{d}.

I think they claim that follows from the definition of God, using corollary B or corollary 5. So it's not a premise, but rather something that follows from other premises (they say). — Amalac

I'd like to see the logic, the semantics, and the proof explicitly specified.

What is the exact sentence that is proved? Is it of the form?:

Necessarily E!x x has property P [where 'P' stands for the bundle of Godlike properties].

Also, I don't know enough about the theory of definitions in modal logic, but if it is close enough to predicate logic, then a definition of an individual requires first proving an existence and uniqueness theorem. So do the proofs you mention indeed first prove there exists a unique individual with such and such properties that is then named 'God'? One can't prove that God has certain properties without first defining 'God' courtesy of an existence and uniqueness theorem. Especially, one can't just assert without proof that there does exist a unique individual having certain properties and then go on to demonstrate that that individual then has other properties for a QED.

And recall that what I mentioned is based on ordinary specifications for setting up a semantics for modal logic. I don't even know how one would formulate those specifications as formulas in the language itself to use in proof.

I admit that I am very rusty in modal logic so I might need to be corrected or qualified. But as far as I can tell, ordinarily, semantics for modal logic begins with ordinary models for propositional or predicate logic, which includes assignments for constants and variables. And in that ordinary manner, there is no object that is the member of all universes for models.

Has modal logic always fail or has it proved something which takes logic to prove? — Gregory

I don't understand the question.

logic is about proof — Gregory

One may characterize what logic is "about" in different ways. A common notion is that logic studies entailment and inference. Then proof formulates methods for inference, especially methods that correspond to entailment.

make the form of the syllogism of the understanding the basis and criterion that one might say not a single one of the metaphysical concepts could have arisen or stood on ground, if it had been subjected to the laws of logic. — Gregory

I can't parse that.

he does not allow the forms of syllogism to govern or encroach on the sphere of speculative philosophy — Gregory

Good for him! Nor does modern logic. indeed, one of the early notable aspects of modern logic is that it extends past syllogisms. And modern logic does not claim that any given logic encompasses all of reasoning. Indeed, that is why, for example, propositional logic is extended to predicate logic, and both are extended to modal logic, and extensions and alternatives to many kinds of mathematical logics and philosophical logics.

What I see as the problem with modal logic and the way many posters reason on this forum too is trying to use logic to prove something beyond itself. Proper philosophical intuition rarely considers logic as logic — Gregory

Please provide an example of modern formal modal logic used trying to prove something beyond itself. Of course, a proof may adopt premises that a reader might reject, but I'd like to know what formal modal logic proof you think is not entailed by its premises. (Various systems of modal logic are proven to be sound.)

So the member of the universe named by "a" exists in some possible worlds but not others. — Banno

Right. For example, suppose one universe is the set of even numbers, and suppose 'a' names 2. Then suppose another universe is the set of odd numbers. Then whatever 'a' names, it can't be 2.

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.
— TonesInDeepFreeze

Hmm, but isn't that what the advocates of the modal ontological argument would reject? — Amalac

I don't know. I'd have to see the specific argument formalized.

Perhaps such arguments have additional premises other than the mere proof apparatus of whatever given modal system? Of course, with certain premises, we can compel certain semantical results that are not compelled by the logic alone.

they argue that God, and God alone, is a member of all “universes” or “possible worlds” without exception. — Amalac

That strikes me as being an additional premise. Of course we can't rule out that additional premises have consequences.

First order logic gets its content from non logic — Gregory

I don't know what you intend to mean with that.

The a first order theory of course includes non-logical symbols and, if not the pure first order theory for that language, the theory includes non-logical theorems. The content expressed is given by models.

I'll give the definitions, then I'll address your formulation.

Let 'A' notate the universal quantifier.

We define 'model for a first order language' as a function F on certain symbols of the language, such that

F('A') is a non-empty set. F('A') is called 'the universe for the model F'.

For an n-place (n being 0 or greater) predicate symbol 'R', F('R') is an n-ary relation on the universe.

For an n-place (n being 0 or greater) operation symbol 'g', F('g') is an n-ary function on the universe.


So a model for a first order language specifies a "possible world".

/

We define 'model' for a first order modal language' as a tuple <W c> such that

W is a non-empty set of first order models for the language. W is the set of "possible worlds".

c is a 2-place relation on W. c is the accessibility relation.

/

To build a model, we set up a bunch of possible worlds. — Banno

For modal, a model has a non-empty set of possible worlds and an accessibility relation on the set of possible worlds.

Within that universe — Banno

For modal, there may be more than one universe.

"a" refers to some given individual. — Banno

For a given possible world, 'a' refers to a member of the universe of that possible world.

In some possible worlds, "a" exists, in others, "a" does not exist — Banno

No, 'a' is a symbol, not an individual. For a given possible world, 'a' names a member of the universe of that possible world. And for any possible world, some member of the universe of that possible world is named by 'a'.

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

I have not claimed that modal logic proves anything in philosophy. And that does not entail that modal logic does not prove anything in philosophy nor that I hold that modal logic does not prove anything in philosophy. Still, even though I am not obligated to support that which I have not claimed, one observes that the subjects of necessity and contingency are themselves philosophical topics. And, turning back around, modal logic itself is a subject of philosophy. And modal logic applied to mathematical logic has a role in the philosophy of mathematics. Meanwhile, in your readings about modal logic on the Internet, you really found nothing about application of modal logic to philosophy? You didn't look further beyond the Wikipedia article you looked at (I only guess you actually read it) - on such subjects the article mentions as knowledge, belief, temporality and morality?

It's interesting that people on the Internet have such vociferous opinions on entire fields of study that they know nothing about except skimming a Wikipedia article and a few other dubiously researched web pages and posts.

"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.

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

No. That is an hilarious question.

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.

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

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".

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'.

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.

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.

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.

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

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

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.

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.)

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?

I surmise that I am not mistaken that, ordinarily, physics uses classical mathematics, which has infinite sets and is ordinarily axiomatized by set theory. That seems salient. So, from my admittedly non-expert point of view, it would seem plausible that we might combine formal mathematics with whatever parts of physics have been, or might be, formalized. At least one example of a basic portion is, as I mentioned, in Suppes's logic book, though it is not so ambitious to undertake relativity, quantum, etc.

Here's a short excerpt from the first that is germane: " In terms of recursion theory, Bridgman’s claim can be re-interpreted such that no diverging algorithm should be allowed as legal input of any other (terminating) algorithm.One may go even further than Bridgman and assume that, since infinite entities are not operational, infinities have to be abandoned altogether." — jgill

And contrariwise in other articles. I'm sure you wouldn't want to cherry pick just gainsaying quotes.

And the article describes Bridgman's notions as stemming from Bridgman's philosophical framework. That does not preclude other frameworks, and especially doesn't entail that any form of ZFC+physics must be inconsistent. (Though the author of the article does give other vigorous arguments against set theory as a foundation for physics.)

If Aristotle is fine with you, then what particular break between Aristotle and modern modal logic do you object to?

Modal logic is not a philosophically traditional way of thinking. — Gregory

That depends on a definition of "philosophical traditional way of thinking" and its import depends on whether it is important to adhere to "philosophically traditional ways of thinking".

What do you know of Hegel's logic — Gregory

I haven't made any claims about Hegel's 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

I was explaining how necessity and contingency are used in traditional philosophy and model logic in totally different ways — Gregory

Whatever you thought you were explaining, you did it by terribly misunderstanding the modal operators in modal logic, thus giving reason to think you don't know anything about modal logic.

this is not using the word as it is used in traditional philosophy — Gregory

Do you have a problem with Aristotle's "traditional philosophy"?

Wikipedia and it's application in articles. — Gregory

Wikipedia and those articles don't mention that the operators are applied to sentences and don't refrain from saying that the operators are applied to "ideas" or noun phrases?

"The desk across the room" is a noun phrase, not a proposition. The modal operators are applied to propositions not noun phrases.

Would you please tell me what textbook in modal logic is the basis of whatever familiarity you have with the subject?

The concepts apply to propositions, or formally - sentences, including propositions about the world.

In mathematics, salient uses of modality include analysis of provability ('The Logic Of Provability' - Boolos) and for semantcs for constructive logic.