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.

Your standard quantified modal logic contains an infinite number of predicate symbols, which are just assigned interpretations relative to a model. Let 'E!' be one of those symbols, and read it such that where 'a' is an individual constant, 'E!a' is true at w iff the the referent of 'a' exists at w.

Then I understand it this way: It's a primitive symbol, but there are no special logical axioms for it, and it can be interpreted differently in different models. But we are particularly interested in those interpretations in which it is interpreted as you described.

But that raises the question: What would be axioms that would entail that any model of the theory evaluates E!a as true only in the way you described?

But, even more basically, in ordinary predicate logic, for any given model for the language, the referent of a variable or constant exists in the domain for that model. That follows from the definition of 'model for the language'. So I don't understand how that can be different with interpretations of a modal language.

PS. I do find discussion about an existence predicate in Hughes & Cresswell. If explanation is too complicated for the confines of posts, I'll try to figure it out from that textbook.

Hughes & Cresswell uses just 'E', not 'E!'.

And the semantics for 'E' are fixed (in the manner described by Snakes Alive). (I would say that the semantics for 'E' is fixed in the same sense of 'fixed' when we say the semantics for '=' is fixed.)

And 'E' is not just a predicate symbol with no supporting axioms. Rather Hughes & Cresswell describes a logic system that modifies first order logic and also has axioms mentioning the special primitive predicate symbol 'E'.

Also, Hughes & Cresswell addresses the question I asked about how a model can excuse a constant from having a referent in the domain of the model. I'll understand better as I study more carefully.

Sometimes, the exclamation point signals unique existence, as opposed to just existence. It varies.

Sometimes, the exclamation point signals unique existence — Snakes Alive

Then that opens yet another can of worms. I understand what it means for there to be a unique individual having a certain property: E!xP. And in this thread I'm starting to understand an existence predicate: Ea. And I could understand writing 'E!a' to mean the same as 'Ea'. But I don't know what it would mean to say "individual a has unique existence and not merely existence". What would be the semantics for that? (And then I'd have to see what modifications for the logic would be required.)

The unique existence predicate is second-order, and serves a similar function as an existential quantifier – so it has a different syntax, and occurs alongside a variable and a formula it scopes over. It's like an existential quantifier, except only one individual in the domain is allowed to satisfy the formula.

The unique existence predicate is second-order — Snakes Alive

Where can I read more about a uniqueness predicate in a second order modal logic?

it has a different syntax, and occurs alongside a variable and a formula — Snakes Alive

I understand a uniqueness quantification symbol that is followed by a variable and then a formula: E!xP.

But I am not familiar with a syntax, even in second order logic, that has predicate symbol preceding a variable then a formula. Where can I read about that?

It's like an existential quantifier, except only one individual in the domain is allowed to satisfy the formula. — Snakes Alive

The uniqueness quantifier works that way. But where I can see a specification of the syntax and semantics of a second order modal logic that has a uniqueness predicate?

I don't mean 'second-order' in the sense of a second-order logic that quantifies over predicates. I just mean this:

Where 'v' is a variable and 'p' is a formula, E!v[p] is true at w iff there is exactly one individual x in the domain such that p is true at w on any assignment that maps v to x.

E!v[p] is true at w iff there is exactly one individual x in the domain such that p is true at w on any assignment that maps v to x — Snakes Alive

That is the ordinary uniqueness quantifier, not a predicate.

So, thus far, we have:

'E' as a predicate symbol before a term: where 't' is a term, Et.

'E' as the existential quantifier: where 'x' is a variable and P is a formula, ExP.

'E!' as a uniqueness quantifier, where 'x' is a variable and P is a formula, E!xP.

But not yet another thing that is a uniqueness predicate (whatever that would mean).

Quantifiers are (nothing but) predicates of formulae.

Quantifiers are predicates of formulae. — Snakes Alive

That's not a usage I have happened to have seen. But that doesn't in itself disqualify it.

But at least now it is clear that 'E' as a predicate is a very different kind of animal from 'E!' as a quantifier, which was not clear, except in stages, in your previous explanations. Anyway, I do appreciate your explanations in general; they have helped. Thank you.

That's right. I didn't mean to suggest that the unique existence predicate was of the same kind as the regular existence predicate (which is of individuals).

Not to say you couldn't construct such a notion, of unique existence, though – the problem with logic and trying to use it to address philosophical issues is that you can do whatever you want.

Not to say you couldn't construct such a notion, of unique existence — Snakes Alive

Sure, but I wonder what it would be.

Do you have any thoughts on my question: How can we have a method of models in which, for certain models, there are constant symbols such that the model does not assign a member of the domain of the model? It throws off the way we evaluate satisfaction and truth in models.

Do you have any thoughts on my question: How can we have a method of models in which, for certain models, there are constant symbols such that the model does not assign a member of the domain of the model? It throws off the way we evaluate satisfaction and truth in models. — TonesInDeepFreeze

Well, the point of a non-logical constant is that its value is invariant across worlds. You could, of course, have a modal logic where individual terms like constants have different denotations relative to different worlds. And you could then allow that they refer to 'nothing,' say, at worlds where the relevant individual doesn't exist. How you want to represent this formally is up to you – one old formal trick is to use a dummy object, say *, to which the value of all terms that have nothing satisfying them at the world map to. You would then need to make a semantics that deals with the dummy object – you could assign predications of it to false, or to a third undefined truth value, and so on.

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. — Snakes Alive

I'm having trouble with that. I can specify world in which Donovan doesn't exist, and speculate about the consequences. Someone in that world might say "Donovan doesn't exist", and state a truth...

I seem to be again on a different page from you.

he point of a non-logical constant is that its value is invariant across worlds. — Snakes Alive

I'm referring to constant symbols. It is not the case that the point is to have the value for a constant symbol to be invariant across models. A model assigns to each constant some member of the universe of the model. There is no requirement that all models agree on what they assign to the constant symbol. Not all models have the same universe, so it's not even possible that they all agree on what they assign to a constant symbol.

You could, of course, have a modal logic where individual terms like constants have different denotations relative to different worlds. — Snakes Alive

Yes, that is what I imagine is the default.

And you could then allow that they refer to 'nothing,' say, at worlds where the relevant individual doesn't exist. — Snakes Alive

That I don't understand. As I explained, if we fail to assign a member of the universe to a constant symbol, then the methods of evaluation for satisfaction and truth for formulas and sentences per a model falls apart.

And I don't know what 'the relevant individual' refers to in your remark. For domains with cardinality greater than 1, there is no particular individual that must be mapped to from a constant symbol. But every constant symbol must map to some member of the domain.

How you want to represent this formally is up to you – one old formal trick is to use a dummy object, say *, to which the value of all terms that have nothing satisfying them at the world map to. — Snakes Alive

That seems to me to be about a separate question.

That seems to be a semantical version of the syntactical Fregean method of "the scapegoat" for failed definite descriptions. I understand that method for handing conditional definitions of constant symbols (and can be expanded to operation symbols). This handles those definite descriptions that fail because either the existence or uniqueness condition fails. The method requires a theory in which at least one constant symbol 's' is either primitive or already defined:

If we want a definition

c = the_unique x P

but have not derived the theorem E!xp

then we revise to

(If E!xP -> c = the_unique x P) & (~E!xP -> c = s).

You would then need to make a semantics that deals with the dummy object — Snakes Alive

But the problem of failed definite descriptions needs to be dealt with syntactically first (as I did above) or we cannot ensure that definitions uphold the criteria of eliminability.

Anyway, the question I have is not what happens when a definite description fails, but rather, how do we reconcile ordinary semantics for either predicate logic or modal predicate logic with dangling non-denotating constants?

As I understand, you suggest perhaps marking any formula with such a constant as "not satisfied" or with a third truth value, or some other accommodation we would stipulate. But that would mess up the whole context of addressing the first post in this thread and its corollary questions.

Well, I'll word it slightly more carefully. Kripke's modal logic as it was initially presented as a propositional modal logic, and so it simply didn't have any terms for variables over individuals. There were just propositional variables, sentential connectives, and modal quantifiers.

In a standard quantified modal logic, however, you need both a domain of individuals and a domain of worlds. And on a standard reading of the identity relation, it is necessary by definition (Kripke made a big deal about this in NN). Thus, if a = b in w, then in all w', a = b.

Therefore, if you make existence claims as follows:

∃x[x = a], to mean 'a exists,'

Then if you have '∃' quantifying over the domain of individuals, independent of the domain of worlds, then it will have the same value at any world – either it will be necessarily true, or necessarily false.

If you want to get around this, you have to reinterpret '∃' so as only to quantify relative to a world, such that each world is associated with a sub-domain of individuals, and '∃x[x = a]' is true in w iff there is an individual x in the sub-domain associated with w that is identical to a. If you read it this way, then the formula can be contingently true, and contingently false – true in worlds whose sub-domains contain a, and false in worlds whose sub-domain does not.

I'm referring to constant symbols. It is not the case that the point is to have the value for constant symbol invariant across models. A model assigns a member of the universe of the model to a constant symbol. There is no requirement that all models agree on what they assign to the constant symbol. Not all models have the same universe, so it's not even possible that they all agree on what they assign to a constant symbol. — TonesInDeepFreeze

Across worlds, not models. A model has a set of worlds, in its frame.

Anyway, the question I have is not what happens when a definite description fails, but rather, how do we do we reconcile ordinary semantics for either predicate logic or modal predicate logic with dangling non-denotating constants? — TonesInDeepFreeze

What do you mean, 'ordinary?' Obviously in a strict sense you cannot reconcile them, since ordinary predicate logic has no notion of a constant that doesn't refer to anything. There's just a domain, and then the interpretation function maps each constant to a member of that domain.

And I don't know what 'the relevant individual' refers to in your remark. — TonesInDeepFreeze

It's whoever, intuitively, the constant is supposed to refer to. So if 'b' refers to Bob, then it might refer to Bob in all worlds where Bob exists, but to * in all worlds in which he doesn't exist.

The method requires a theory in which at least one constant symbol 's' is either primitive or already defined: — TonesInDeepFreeze

No, it just requires some distinguished object in the domain, like say *.

I can specify world in which Donovan doesn't exist — Banno

I'm not addressing philosophy of language or more advanced modal logic, but in ordinary predicate logic, 'Donovan' is a plain name (essentially, it's a constant symbol). An interpretation of the language assigns a member of the universe to the name.

That is different from a definite description. In some theories there might or not be a theorem "There exists a unique individual that wrote the song "Mellow Yellow"". If we don't have that theorem, then see my post above as to what we can do about the definite description "The unique individual who wrote "Mellow Yellow".

(I'm assuming you mean the Scottish singer and not the main character in the movie 'Donovan's Reef' nor the main character in the TV show 'Ray Donovan'.)

[EDIT: Disregard this post, except the last quote and my response to it. I think I had a misconception in this matter.]

Across worlds, not models. A model has a set of worlds, in its frame. — Snakes Alive

Right, my mistake, we're talking about modal logic. But make the correct substitution, and my remarks still pertain both to domains for models with predicate logic and to the domains for the worlds in models for modal logic.

What do you mean, 'ordinary?' — Snakes Alive

The method of semantics for first order languages as described in any textbook in mathematical logic (or as they describe methods with inessential differences). And am I wrong that also the most basic methods for modal logic in textbooks follow suit? Doesn't basic modal logic stipulate worlds recursively, based first on the ordinary predicate logic clauses and adding the clause for the modal operator?

Obviously in a strict sense you cannot reconcile them, since ordinary predicate logic has no notion of a constant that doesn't refer to anything. There's just a domain, and then the interpretation function maps each constant to a member of that domain. — Snakes Alive

Exactly. But isn't that method also used in modal predicate logic too for worlds?

So if 'b' refers to Bob, then it might refer to Bob in all worlds where Bob exists, but to * in all worlds in which he doesn't exist. — Snakes Alive

Then there has to be an object that is in the intersection of the domains of all worlds. That seems to be a big requirement. Also, there are domains that might have nothing to do with the intuitive intent.

{0} and {1}. Two domains and empty intersection between them. And a constant symbol 'c' with no intuitive referent. I don't see how what you describe is supposed to work.

[EDIT: The following is okay but perhaps no longer relevant anyway.]

No, it just requires some distinguished object in the domain, like say *. — Snakes Alive

I was referring to the syntactical side of things for definite descriptions, not to your semantical method.

AH, think I see. Thanks.

No, I don't.

That seems to imply that Donovan exists in possible world in which he does not exist.

I see now that I face an obstacle in talking about constant symbols and terms with you. That obstacle is that I am using Hughes & Cresswell, but their quantified modal system has no constants, no terms other than the variables themselves, and '=' is introduced only in a later chapter. Their language has only: universal quantifier (with existential quantifier defined), individual variables, sentential connectives, relation symbols, and the necessity operator.

Of course, that is enough, since in theories, constants and operation symbols can be defined from relation symbols.

But it makes it difficult for me to attend to the details vis-a-vis your remarks, because I lack a reference for how terms are dealt with semantically in a definitive textbook.

So, which textbook guides you the most in this subject? I can see whether I can get it cheap enough.

if a = b in w, then in all w', a = b. — Snakes Alive

I take that mean that the letters 'a' and 'b' are variables in the meta-language and not constant symbols in the object language. And with that, the above sentence seems to be the way we should understand the semantics.

∃x[x = a], to mean 'a exists,' — Snakes Alive

But there 'a' is used as constant symbol in the object language.

For me, 'Ex x=a' is just a trivial theorem of identity theory. For any term t, we have the theorem:

Ex x=t

Then if you have '∃' quantifying over the domain of individuals, independent of the domain of worlds, then it will have the same value at any world – either it will be necessarily true, or necessarily false. — Snakes Alive

As a theorem of identity theory, isn't it true in all models for a language for quantified+identity modal logic?

'∃x[x = a]' is true in w iff there is an individual x in the sub-domain associated with w that is identical to a. — Snakes Alive

I don't see how you can use 'a' as a symbol in the object language when you write 'Ex x=a', but also as a symbol in the meta-language when you talk about 'a being in a domain'. Also with 'x', but there I can reword in my own mind to make it work, while, in this particular situation, I can't do that with the way 'a' is being used, and as I don't think this is merely pedantic but rather it confuses me as to what really is being said.

I think I might have had an incorrect premise that modal semantics evaluates truth in these stages: first per world and then per model. Rather, perhaps I can answer my own questions if I dispense with that premise and view semantics as done "top-down" for the model overall. That premise disallowed me from better understanding your posts. I'm going go back to studies I had forgotten a long time ago to see whether I can correct myself now. (If you have a textbook to recommend, then it would be appreciated.)

In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction? — Banno

I don't think you even need S5 for it? Given that you can choose world elements.

W1={egg, bacon}

W2={egg}

The statement E: "At least one entity in this world is an egg"

E's w1 valuation, true.
E's w2 valuation, true.
E is true in all possible worlds, so it's necessary.

If you'd like, call the relation between W1 and W2 the breakfastibility relation. Two worlds are connected iff they consist of only breakfast goods. That accessibility relation is an equivalence relation on the set of possible worlds.

So yes, with that world set and that accessibility relation, I believe the existence of eggs is necessary.

#modal logic isn't metaphysics

That's neat. OK, I follow that answer.

The next question is, must there be an individual which exists in every possible world?

It seems not, since we can have

W1={egg, bacon}

W2={toast, coffee}

...and now for breakfast.

The next question is, must there be an individual which exists in every possible world? — Banno

In which set of possible worlds?

{egg}, {egg, bacon}

Has "egg must exist" as true, where "eggs must exist" is treated as "necessarily there is at least one egg in every possible world".

But as you noticed:

{egg}, {bacon}

doesn't have this property, so the divine mandate for egg's existence disappears.

Does an entity necessarily exist in all possible worlds? Set out a set of possible worlds and an accessibility relation and we'll talk.

If you don't set out an accessibility relation and set of possible worlds, it looks like quantifying over sets of worlds rather than worlds - ie does there exist a set of worlds in which eggs necessarily exist - yes , I just summoned it vs do eggs necessarily exist in each possible world in all sets of possible worlds - no, I just summoned one where eggs don't -.

Does an entity necessarily exist in all possible worlds? Set out a set of possible worlds and an accessibility relation and we'll talk. — fdrake

...and that's the point. Possible worlds are constructed by fiat, not discovered. Hence arguments which presume to demonstrate a being that exists in every possible world can do so only by assuming there is such a being.

Possible worlds are constructed by fiat, not discovered. — Banno

Is logic constructed by fiat? Possible worlds depend upon what is logically possible/ impossible (that's how they are defined), and it seems we don't construct “by fiat” what is and is not logically possible.

Is logic constructed by fiat? — Amalac

That's a different question. A possible world comes about as the result of a "what if..."; then we can see if that "what if..." leads to a consistent story or not. If it is inconsistent, then there can be no such possible world.

That is, logic gives us a grammar with which to judge our statements.