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? They would not be convinced with just “of course that individual is not a member of certain other universes” because they argue that God, and God alone, is a member of all “universes” or “possible worlds” without exception.

Or is there some significant difference between “universe” and “possible world” in the case of God?

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.

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

Aristotle did, indeed, describe the figures of syllogism just as he did with countless other forms of spirit and nature, but in his metaphysical concepts he was so far from seeking to 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. Even if, in his own way, Aristotle began much that is essentially a product of description and of understanding, the speculative concept is always what is dominant for him, and he does not allow the forms of syllogism to govern or encroach on the sphere of speculative philosophy- Hegel in "Logic"

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'.

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.

Has modal logic always fail or has it proved something which takes logic to prove? After all logic is about proof

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'. — TonesInDeepFreeze

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

Must there be an individual named by "a" that exists in every possible world? Can there be an individual that exists in every possible world?

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.

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.

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.)

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

Actually, 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).

You are right however, in that 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 (that existence is a predicate) is required (it seems to me at least) to hold the claim that the greatest conceivable being or subject of all perfections exists in all possible worlds.

The argument would go something like this:

The actual world is one among the possible worlds (this again follows in some systems of modal logic). If one admits that god exists in all possible worlds, that would imply that god exists in the actual world.

And so, if one accepts that it is possible that it is necessary that god exists in all possible worlds (meaning: in some possible worlds, necessarily God exists in all possible worlds), then it follows that in all possible worlds, god exists in all possible worlds, and therefore “god exists in all possible worlds” is true in the actual world, which is one of the possible worlds in which that statement is true, and therefore god exists in the actual world.

All this follows if one accepts system B of modal logic, from the corollary of axiom B (if the modal ontological argument is valid):

◇□X → X (If it is possible that it is necessary that X, then X is the case).

Likewise in system S5, the corollary of axiom 5:

◇□X → □X — Amalac

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.

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.

Hegel writes that way. I read and reread hundreds of pages of such. It's a striking style.

Now logic is about the forms of itself and can't comment on the inexperienced. You admit this! So then modal logic is just logic and much closer to programming than philosophy

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

The answer is "no"?

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.

...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}.

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.

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.

This depends on how you construe existence. If you just have an existence predicate, E!, then sure – 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.

If you construe existence as a matter of existential quantification over identity, as ∃x[x=a], to mean that there is an individual identical to a (the individual you're interested in), then it depends on what your quantifiers range over. In fact, on a classical Kripkean treatment, existence is always necessary existence, since if there is some individual x identical to a in the domain of individuals, then there will be at any world, since the domain of individuals and the domain of worlds are simply separate.

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). Here, 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.

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.

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.

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

You can define any predicate you like, necessary or otherwise. 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.

The standard Kripkean treatment does not, of course, allow for a primitive predicate like this to be true for all individuals (or just some individual) at all models. 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.

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.

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'.

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

I don't know what you're asking.

So do the proofs you mention indeed first prove there exists a unique individual with such and such properties that is then named 'God'? — TonesInDeepFreeze

I don't think so, they define God as having certain properties (perfections or “great making properties”) first, and then through analysis of the concept of “God”, defined as the subject of all perfections or greatest conceivable being, they argue that the proposition “God exists” is analytically true, that is: that God's non existence is as impossible as there being an object that was both round and triangular at the same time and in the same sense.

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. — TonesInDeepFreeze

That's true, here:

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. — Amalac

...I should have just said that “there is [not “exists”] an x (God), such that...” (though as I said later in the ontological argument it is argued that God's existence is analytic), since otherwise one would just assume the existence of that “x” right away, which is not what I meant to write.

That way, the existence of x is not assumed, but (supposed to be) proved from analysis of the meanings of the terms involved.

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.

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.

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

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.