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Tuesday, May 31, 2011

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Bill,

Let us focus on a simple question:

Does (1) entail (2), for a given constant 'a' that refers to a contingent existent?

If your answer is that no such inferences are valid, then which inferences from (1) are valid? Are such inferences valid only when it comes to objects that exist necessarily, if they exist at all? e.g., numbers or God, etc. But what if numbers do not exist or God does not exist.

Moreover, even in the case of putatively necessary existents you have the provision 'if the object(s) in question exists, then...'. And this could be stated in a meta-linguistic form just like in the case of contingent existents: 'if 'God' refers to an existing object, then...' or 'if '2' refers to a number, then...'. Does the fact that we have such conditions renders the objects in question not necessary or the sentences of the form '...=...' contingent?

If you say that (1) does entail (2) and (2) is contingent, then you concede the objection that a necessary proposition entails a contingent one.

So it seems to me that you got a dilemma: either you concede my argument or you reject all inferences from '(x)(x=x). Opting for the later course seems to me very radical step to take.

Both sides in this discussion have made excellent points. We are at an impasse. Doesn't this merely confirm what we know already, that the predicate calculus, with its condition of applicability of a fixed domain of objects, is insufficiently rich to express notions of contingency, especially the contingent existence of objects? We are forcing a quart into a pint pot. Surely we should expect something to go wrong?

>>Let us focus on a simple question:

Does (1) entail (2), for a given constant 'a' that refers to a contingent existent?<<

Yes, of course. No one could possibly deny that. But that's not the issue. The issue is whether one can validly move from (1) to (2) without making your extralogical assumption. And my answer to that is No.

>>If you say that (1) does entail (2) and (2) is contingent, then you concede the objection that a necessary proposition entails a contingent one.<<

No. I don't know how I could explain this more clearly than I already have. The point is that you cannot get from (1) to (2) without an auxilary premise which is contingent.


David,

The simple predicate calculus (first-order predicate logic with identity) does not have the resources to express contingency, but quantified modal logic does.

We are not at an impasse. I have isolated the mistake in Peter's reasoning. He fails to see that (2) above is contingent. For if 'a' as he says refers to a contingent existent, then 'a = a' is not true in all worlds.

Dr. Vallicella,

"The point is that you cannot get from (1) to (2) without an auxilary premise which is contingent".

I do not think this is true. Every point of an argument is a proposition and not a language statement. I think you implicitly acknowledge this be writing "(1) does entail (2)". Language statements cannot imply other statements; only propositions (which are represented by statements) can. If it is indeed a proposition, it is about object a by its very nature, no auxiliary premises required. Consider

2*. Steve the Unmarried Bachelor = Steve the Unmarried Bachelor.

(2*) is not a proposition, though it is a language statement. If, arguendo, it were a proposition, it would be directed towards Steve the Unmarried Bachelor. But Steve the Unmarried Bachelor does not exist in any of the possible worlds, which would mean that (2*) is directed towards something that does not exit, which is a contradiction. An argument (1*)-(4*) parallel to (1)-(4) will therefore not work.

"For if 'a' as he says refers to a contingent existent, then 'a = a' is not true in all worlds" -- this has been long acknowledged by Peter in our discussion. His crucial point, as I understand it, is that in those worlds in which it is not true it is also neither false nor meaningless. In particular, it is not a proposition, but something less. It is possible that your differences are largely linguistic. Have you read the comment in which I summarised Peter's views as I understand them? Peter has not yet found fault with it.

Peter,

The more I think about it, the more I am certain that your view either requires there to exist something that I called proto-propositions, or for the logic to be trivalent. Do you agree? A second question: is there any object that exists in some possible world, but not in the actual world? If so, can you give an example? You deny Pegasus is an example, which is surprising to me.

Jan,

I find your comment puzzling. First of all, you shouldn't use the phrase 'language statement.' 'Statement' will do. How could a statement not be in some language or other? If you are saying that statements are distinct from the propositions they express, then that is true, but there is no need to point that out in present company.

As for (2*)I am not sure what your point is here.

Perhaps you could make your point more clearly.

Bill,

"The point is that you cannot get from (1) to (2) without an auxilary premise which is contingent."

According to this standard every inference will involve such an auxiliary assumption. You give me any inference whatsoever, and I will show you that it involves some existential or meta-linguistic assumption.

Jan,

"...is there any object that exists in some possible world, but not in the actual world? If so, can you give an example? You deny Pegasus is an example, which is surprising to me."

Sure! Consider a case where a woman is pregnant. The parents call the unborn by the name 'Pegasus'. Unfortunately, due to a tragic accident, the unborn is prematurely aborted and dies. Several years later the parents can legitimately say about their lost child:

"This year Pegasus would have been five."

The statement is true in a possible world in which there was no accident, the child was born and he lived to be five.

I object to the claim that the term 'Pegasus' could refer to some winged horse in some possible world because, following Kripke, I think that this term was introduced as a fictional name and therefore it fails to refer unless we use it to refer to some other entity (such as in the example above). So I am not committed to fixed domains throughout possible worlds; other possible worlds may contain new entities.


Peter sez: >>According to this standard every inference will involve such an auxiliary assumption. You give me any inference whatsoever, and I will show you that it involves some existential or meta-linguistic assumption.<<

Not quite. From *(x)(x = x)* one can immediately infer *~~(x)(x = x)* And that is just one case.

But I think you now see my point.

And even if you don't accept my diagnosis of the error in your reasoning, you must accept that there is an error in it. For we KNOW from elementary modal propositional logic that if p is necessary and p entails q, then q is necessary. So we know that any putative counterexample must be bogus.

Jan asks: >>is there any object that exists in some possible world, but not in the actual world?<<

Well, isn't it possible that there be one less object than there actually is, or one more, or two fewer or two more, etc.?

In my study there are two oak tables. Surely there is a possible world in which all else is the same but there is only one oak table in my study. Of course, all else can't be quite the same: the carpet would not be depressed in the places where the legs rest, etc.

Jan,

Do you agree that Socrates exists in some but not all possible worlds? Surely you do. What about the proposition *Socrates is Socrates*? If you think it exists in all possible worlds, then you need to tell me what its constituents are in those worlds in which Socrates does not exist.

Can a singular proposition, involving one or more contingent individuals, be a necessary being?

Dr. Vallicella,

Here is my argument in a nutshell:

1. Propositions are intrinsically directed towards their objects.
2. It is not the case that if p is a proposition, an auxiliary premise is required to specify the object of p. (From 1.)
3. Every point of an argument is a proposition.
Ergo:
4. It is not the case that (2) from your original argument requires an auxiliary premise to specify its object.

The second part deals with an objection that if the object of (2) is not specified, an analogical argument will prove:

4*. Steve the Unmarried Bachelor exists.

I reject this objection by saying that (2*) is not a proposition.

I am also quite sure that the argument in question pulls nothing out of the hat that has not already been there. For starters, its conclusion (4) is to be construed as a modal proposition. In particular, the argument does not prove that a exists in the actual world. It does so if and only if *a = a* is true in the actual world, that is, if a exists in the actual world. Consider an object o such that o exists in some world w but not in the actual world.

1. For any x, x = x. Ergo:
2. o = o. Ergo:
3. (Ex)(x = a). Ergo:
4. o exists.

The argument is valid and sound. Every point of the argument is a modal proposition. It does not prove o exists in the actual world. Do you and Peter agree with this?

Peter,

Thanks for the clarification. Do you agree that you are committed to holding either that proto-propositions exist, or that the logic is trivalent? If so, which of the two do you hold?

Dr. Vallicella,

The comment above had been written before I read your latest reply. Also, I forgot to thank you for correcting my linguistic mistakes. Do not hesitate to correct me again.

Dr. Vallicella,

"Well, isn't it possible that there be one less object than there actually is, or one more, or two fewer or two more, etc.?"

I most emphatically agree. I was surprised Peter did not not consider Pegasus a possible being, hence my question to him.

"What about the proposition *Socrates is Socrates*? If you think it exists in all possible worlds, then you need to tell me what its constituents are in those worlds in which Socrates does not exist."

I do not think it exists in all possible worlds. Peter's definition of a necessary proposition is:

NP*: Proposition p is necessary iff it is false in none of the possible worlds.

and not the more common:

NP: Proposition p is necessary iff it is true in all of the possible worlds.

I have refuted the conjecture that necessary propositions can imply contingent ones working under (NP). It is however not the definition that Peter holds. Peter gave some reasons to think (NP*) is reasonable. There are problems with (NP*) some of which I highlighted when I summarised Peter's views. For example, if P is an essential property of an object o, *o is P* is a necessary proposition under (NP*).

Jan,

'Pegasus' unlike 'Vulcan' is a fictional name. If you had used 'Vulcan' Peter wouldn't have balked. Note that when astronomers decided that the planet Vulcan does not exist it did not become a fictional object. The modal concepts pertain to the realm of fact, not fiction. Not sure, but I think that is Peter's point.

I'm not getting the point re: NP and NP*. For me, the existence of a proposition is not its being true; it exists whether true or false, and its existence is a precondition of its having a truth-value.

And aren't those two principles equivalent?

I'll try to get to your argument later.

I have been away in the dark industrial north so I missed this one. If it’s not too late I have two comments.

(a) Step 2 of the argument involves introducing a logically proper name ‘a’. A logically proper name is such that, to be meaningful, it must name something. But the presumption that it names something seems to smuggle in an existence claim, right at the very start. At least, if ‘a is something’ means the same as ‘a exists’. But see my second point below.

(b) Step 4 of the argument uses the verb ‘exists’. But this word is not defined in predicate calculus. Does it mean ‘a is something’, i.e. for some x, x=a? If so, the argument is valid, but only because the ‘something’ claim was already assumed in step 2. Does it mean something stronger? But then consequens est in plus, and the argument is not valid.

Welcome back from the dark industrial north.

I agree with your comments. 'a' must name something, and that thing must exist. And so existence is smuggled in at line (2).

You are right that 'exists' is not defined in predicate logic, but in the phil. commentary on the pred. calc. by such worthies as Quine -- no mean logician -- 'exists' is defined: a exists =df (Ex)(x = a). That's why the ex. quantifier is called existential. "Existence is what existential quantification expresses." (Quine)

>>If so, the argument is valid, but only because the ‘something’ claim was already assumed in step 2.<< Exactly right. But without the smuggling at line (2) the arg. is invalid.

Otherwise we have an ontological argument 'gone wild.'

Ed,

Do you agree with me that (2) is contingent? Peter thinks it necessary.

>>Do you agree with me that (2) is contingent?

According to the London version of Ockhamism,

1. we express ‘Pegasus does not exist’ as ‘nothing is identical with Pegasus’.

2. A proposition with a proper name in the grammatical subject position always implies a proposition with the proper name replaced by ‘something’.

It follows that “Pegasus is identical with Pegasus” is false, according to Londonistas. For, given that Pegasus does not exist, it follows that nothing is identical with Pegasus. But if “Pegasus is identical with Pegasus” it would follow (by principle 2) that something is identical with Pegasus, which contradicts “nothing is identical with Pegasus”.

Therefore “Pegasus is identical with Pegasus” is false. Is it necessarily false? No, because if there were such a thing as Pegasus, it would be true. And I take it as contingent that there is no such thing as Pegasus. (Actually that is debateable, but clearly a separate question).

Bill has been maintaining that the inference from (1) to (2) contains a suppressed contingent premise and therefore (2) is contingent. I have been arguing that by this standard every inference contains such a premise and challenged Bill to give me an inference that does not. He did:


"Not quite. From *(x)(x = x)* one can immediately infer *~~(x)(x = x)* And that is just one case."

Now I maintain that if the inference from (1) to (2) contains a suppressed premise of the sort Bill previously labeled (1.5), then so does the inference from the premise:

(3) (x)(x=x);

to the conclusion:

(4) ~~(x)(x=x).

The suppressed premise is something like the following:

(3.5) The symbol '~' shall always designate the negation function in every one of its occurrences as long as it is attached to a wff.

Now, (3.5) is a convention that we use regarding the symbol '~'. This convention, or one equivalent to it, is necessary in order to render the inference from (3) to (4) valid. It is a contingent convention in the sense that we could have had quite a different convention. E.g.,

(3.5*) The symbol '~' shall designate either the *affirmation function* or the *negation function*, but never both, as follows:

(i) if 'S' is a wff with no occurrences of '~' preceding it, then the first occurrence of '~' in front of 'S' shall designate the affirmative function; the second occurrence the negation function; the third occurrence the affirmative function and so on. In general, every odd occurrence of '~' in front of a wff designates the affirmative function and every even occurrence of '~' in front of a wff designates the negation function:

(ii) The *affirmative function* is defined as follows: If 'S' is a wff and 'X' is an affirmative function, then "S is true if and only if XS".

(iii) The negative function is defined as follows: If 'S' is a wff and 'X' a negative function, then "S is true if and only if XS is false".

Notice that (3) together with (3.5*) fail to entail (4) because by the convention stated in (3.5*) the formula "(~~(x)(x=x)" has the opposite truth-value from "(x)(x=x)".

Edward says; "Therefore “Pegasus is identical with Pegasus” is false."

False of which object. Not Pegasus, since Pegasus does not exist. Nor is it false of any object that is not Pegasus, since it is not the case that any other object fails to be identical with itself. Therefore, the correct conclusion is: the expression "Pegasus = Pegasus" and "It is not the case that Pegasus =Pegasus" are both not well formed, since the individual constant (name) 'Pegasus' fails to have a referent.

Of course, one could introduce into the language the term 'Pegasus' and provide a referent for it. Then both of these expressions would be wff and the first will be true in every possible world in which the object to which 'Pegasus' refers exists.

Peter >>False of which object.

There is a presumption here that for a sentence to be false, it must be false of some object. Why so? 'There are unicorns' is false, but it is not false of some object. (Although it is true of every object that it is not a unicorn, of course).

>>Nor is it false of any object that is not Pegasus

But it is true of every object that it is not Pegasus.

>>the expression "Pegasus = Pegasus" and "It is not the case that Pegasus =Pegasus" are both not well formed, since the individual constant (name) 'Pegasus' fails to have a referent

But you have said above that "Pegasus does not exist", and that it is not it false of any object that is not Pegasus. Are these sentences, which you have used in support of your argument, also not well-formed?

If I say that there is such a thing as Pegasus, that is not false "of" any object either. But surely it is false. For you might contradict me by saying "but there is no such thing as Pegasus".

Jan,

"I most emphatically agree. I was surprised Peter did not not consider Pegasus a possible being, hence my question to him."

Bill is correct in his observation in one of his reply posts to you that in my view (following Kripke) the case of Vulcan differs radically from the case of Pegasus. I do think that Vulcan could have existed, although it does not; but not Pegasus, for reasons that Bill (and I) already stated.

As for your question: "Do you agree that you are committed to holding either that proto-propositions exist, or that the logic is trivalent? If so, which of the two do you hold?"

Excuse me, but I need to go back to your post where you introduce the term 'proto-propositions' and clarify to myself what you mean by it. If proto-propositions are, or are picked out by, what is called in the jargon 'meta-linguistic', then I agree that there are such propositions and they come in when we set of the semantics of a formal language; i.e., assign extensions to the symbols.

O/w I would not even know how any inference rules can be applied. The problem arises when one mixes statements of the language with meta-linguistic statements that are about the language. e.g., expressions such as 'a' refers to Socrates are not part of the language itself; they are about the language. Hence they do not appear as premises (or anything else) when inferences among the sentences of the language are evaluated.

Jan argues:

>>1. Propositions are intrinsically directed towards their objects.
2. It is not the case that if p is a proposition, an auxiliary premise is required to specify the object of p. (From 1.)
3. Every point of an argument is a proposition.
Ergo:
4. It is not the case that (2) from your original argument requires an auxiliary premise to specify its object.<<


This is still not clear enough. How are you thinking of propositions? I need a sketch of your theory of propositions. What's the purpose of (3)? It's not (2) that requires an auxiliary premise, but the argument.

Read Edward's first comment above. He gets it.

Edward,

Your first comment was excellent and then your second was confused. You brought in 'Pegasus' which is irrelevant. I asked you whether (2) in my original argument -- 'a = a' -- is contingent in your opinion. 'a' is a log. proper name for a contingent existent. Does that fact render (2) contingent? That's what I'm asking you.

Peter,

Amazingly, you have exchanged positions with Edward. Now he gets it (in his first comment above) and you don't. Read his first comment

I think Jan commented in a nearby thread that much depends on which logic we are working in. My understanding is that, on the presupposition that 'a' names an object in the domain of discourse, the vanilla predicate calculus licenses the derivation of (2) from (1) in one step with no other premises. Otherwise, (2) is not even a wff and is unsayable. Bill says that we do need a further premise, something of the form 'Ex.x=a'. This suggests that we should look to some flavour of free logic which will allow non-referring terms. But then, with regard to wffs of the form 'a=a' where 'a' does not refer, we are faced with a choice: some true (positive FL); all without truth value (Fregean FL); none true (negative FL). It would seem that Peter is implicitly opting for positive FL, Jan for Fregean FL perhaps, and the rest of us for negative FL. To my mind we should be trying to answer the question which flavour of FL best extends to accommodate modal concepts. But this goes way beyond my competence.

David,

You write: "To my mind we should be trying to answer the question which flavour of FL best extends to accommodate modal concepts."

Exactly! I was arguing with Peter till I understood our difference is simply that we mean different things be 'necessary proposition'. For Peter, it is that p is true in every world in which p exists. For others, that it is true in all possible worlds.

Dr. Vallicella,

I certainly agree with Edward's (a). In fact, I argued along similar lines in the exchange with Peter.

You write: "'a' must name something, and that thing must exist. And so existence is smuggled in at line (2)."

Sure. If the argument is to be construed as a modal one, it suffices for a to exist in some possible world. That's the only clarification on which I would insist.

I tentatively agree with Edward's (b) too. I'm not qualified to say whether the meaning of 'a exists' and '(Ex)(x = a)' is exactly the same.

>>Your first comment was excellent and then your second was confused. You brought in 'Pegasus' which is irrelevant. I asked you whether (2) in my original argument -- 'a = a' -- is contingent in your opinion. 'a' is a log. proper name for a contingent existent. Does that fact render (2) contingent? That's what I'm asking you.

Assuming that 'a' is not logically proper, then clearly 'a = a' is contingent. 'Obama is the same person as Obama' is true. But if it were the case that no one were Obama (i.e. Obama did not exist) then it follows from what I was saying earlier that it would not be the case that Obama was the same person as Obama. For if it were, then someone (Obama) would be Obama, against what was assumed. And so it is possible that Obama is not Obama, and so (from the definition of contingency) that it is not necessary that Obama is Obama.

>>'a' is a log. proper name for a contingent existent. Does that fact render (2) contingent? That's what I'm asking you.

That is a different question, I think. I am not sure how to answer it. By a logically proper name we mean one that is meaningless in the absence of a referent. But if there is a possible world in which no one is Obama, and if 'Obama' is a l.p. name, it follows that there is a possible world where the sentence 'Obama is Obama' is meaningless (as opposed to meaningful but false). That is confusing, but it is worse. Perhaps there is a cat named 'Obama' in that possible world - so the sentence is not meaningless after all. We could attempt to escape this by the stipulation that the sentence 'Obama is Obama' is meaningless where the name is used in the same sense it is used in this world, to apply to the US president. But then if it has the same sense as in this world, it has a sense, and so is not meaningless. But now I have confused myself.

DavidB,

You basically got it right and I think I have mentioned the issue of Free Logic somewhere in one of these threads (or perhaps merely thought about it but neglected to mention it; I can't recall right now). In any event, It is absolutely and categorically clear that

(a) In standard non-Free logics (including modal logic), the inference from (1) to (2) is valid w/o any additional premises for any constant 'a' that was previously assigned a referent (as you fairly explicitly assert.) I have confirmed this with a friend and colleague who is a logician just in case I miss something.

Of course, one might complain that this assignment of a referent for 'a' at the initial stage of setting up the semantics for the language presupposes the existence of a. They will be right! But this assumption is *not* required to be stated again as an additional premise. In response one might assert that such a logic is deficient in some respects. They will be right about this too. But the validity of inferences is defined relative to a given logic and not independently of any logical system.

(b) Free logic functions differently and, like you, I am not qualified to say much about how things are set up in such logical systems. I do know that FL includes an existence predicate in addition to the existential quantifier and that vacuous names can occur because formula can express the non-existence of some object. However, I am unsure I understand how the truth-conditions are assigned to formula that include a vacuous name following the existence predicate.

Since Bill's original principle was stated in general and not relative a particular logical system, I think my example is a counterexample. Moreover, I think that some other issues lurking in the background such as whether necessary a-posteriori propositions are possible and so on.


Edward,

There are really two arguments under discussion. the first is the unmodalized argument given at the top of my post. You and I agree that this argument is valid, but only if it is presupposed that 'a' designates something that exists. But in that case the argument does not succeed in deducing a particular existent from a merely formal principle of logic, namely, the Law of Identity.

The second is an explicitly modalized version of the first. Clearly, (1) is nec. true. But what about (2), i.e. *a = a*? I take you to agree with me that it is contingent. Peter, however, thinks it necessary, but then quiote bizarrely considers (3) to be contingent.

He thus takes the move from (2) to (3) to be a counterexample to the modal principle that if p entails q and p is necessary, then q is also necessary. Since the principle is clearly true, Peter is making a mistake somewhere and the problem is to diagnose the mistake.

Peter >>Of course, one might complain that this assignment of a referent for 'a' at the initial stage of setting up the semantics for the language presupposes the existence of a. They will be right! But this assumption is *not* required to be stated again as an additional premise.

Correct, but the point was that the existence assumption was smuggled in as part of 2. No semantics was explicitly set up, but as soon as we encounter the 'a' of (2) it is implicit that 'a' names something.

Peter, you haven't addressed my question of how the falsity of 'Pegasus exists' has to be false of something.

Bill >>But what about (2), i.e. *a = a*? I take you to agree with me that it is contingent.

Only when 'a = a' is a representation of ordinary language statements. As ever, there is the difficulty of whether these formalised statements belong to a language in its own right, i.e. standard predicate calculus, or whether they simply stand in for statements of ordinary language, such as 'Obama is the same person as Obama'. As a formalisation of ordinary language, it is clearly contingent. As a statement of the predicate calculus, (2) does follow from (1), as Peter states. And (2), for the reason you state, is clearly necessary.

A Summary of My Arguments:

A Reply to Bill, Ed, and indirectly to Jan,


(I) Ed says: “There is a presumption here that for a sentence to be false, it must be false of some object. Why so? 'There are unicorns' is false, but it is not false of some object.”

The presumption is about sentences that contain *only* individual constants (plus possibly logical constants including identity). My example was ‘Pegasus=Pegasus’; this sentence contains only the individual constant ‘Pegasus’ and the identity symbol. Such sentences purport to be about individuals and not the whole domain. By contrast, your example ‘There are unicorns’ is a general statement, since it contains the existential quantifier, and it is about the whole domain. Therefore, your point is irrelevant to the case under present consideration.

(II) The rest of Ed's comments in this post confuse object language and meta-language. e.g.,

Ed says: “But you (i.e., Peter) have said above that "Pegasus does not exist", and that it is not it false of any object that is not Pegasus. Are these sentences, which you have used in support of your argument, also not well-formed?”

The sentences to which Ed refer here are part of a series of statements that belong to the meta-language; they are statements *about* the language and its domain (in first-order logic). Take for instance the truth conditions for identity statements:

(*) An identity statement; e.g., ‘(a = b)’ is true just in case the terms ‘a’ and ‘b’ refer to the same object; otherwise it is false. (‘a’ and ‘b’ are individual constants; e.g., names)

First, note that (*) is in the meta-language, not the object language, for it states the truth conditions for object-language statements such as ‘a=b’. Truth conditions for the object-language statements must be stated in some meta-language that features resources that are richer than the object language.

Second, what are the possible conditions included in the clause ‘otherwise’? One such condition is when ‘a’ and ‘b’ refer, but to different objects. Then ‘a=b’ is clearly false. What about when ‘a’ or ‘b’ or both fail to refer to anything in the domain of the language? Could this situation be included in the clause ‘otherwise’.

The answer is: not in standard first-order quantification theory! Such a situation cannot occur, for every individual constant that enters into a formula of standard first-order logic was assigned by an evaluation function V (that belongs to the meta-language) to some object or other in the domain. Statements that include an empty name cannot even occur in the well-formed formulas of the language. Therefore, the best one can say is that the statement ‘Pegasus=Pegasus’ cannot be translated into the language unless V assigns to the term ‘Pegasus’ some object in the domain.

(III) When modal operators are introduced, then the manner truth-conditions are assigned remains more or less the same except that we must consider what happens to statements in a variety of possible worlds. So we add new quantifiers to range over possible worlds. For instance, suppose ‘a’ and ‘b’ are rigid. Then Nec (a=b), if ‘a=b’ is true. Suppose that in fact ‘a=b’ is true. Then it immediately follows that ‘Nec(a=b)’ is true. What does ‘Nec(a=b)’ mean? It means that in every possible world *accessible to the actual world* (or the world in which the statement ‘Nec(a=b)’ is evaluated), ‘a=b’ is true. What about a world w* that is not accessible to the actual world by the relation of ‘accessibility’? An example of a world w* not accessible to the actual world is a world in which the object to which ‘a’ and ‘b’ refer does not exist. Hence, in such a world ‘a=b’ cannot be assessed with respect to truth-value. This could be one reason that w* is not accessible to the actual world.

So, now, what about ‘a=a’? Suppose that in the actual world, ‘a’ refers to some object o. Since ‘a=a’ is true by (*) and since ‘a’ is rigid, it immediately follows that ‘Nec(a=a)’ is true in the actual world. What does this mean? It means that in every world that is accessible to the actual world, ‘a=a’ is true. What about worlds in which o fails to exist? Those are worlds that are not accessible to the actual world; hence, in such worlds ‘a=a’ cannot be assessed as to its truth value.

(IV) In S5 the accessibility relation is universal; every world is accessible to every other world. Hence, in S5 certain modal distinctions collapse. I suspect that Bill and others think in terms of S5; I do not. Why? Because then certain unpalatable assumptions must be made; e.g., every object that exists in the actual world exists in every possible world. Those who prefer S5 are now in an uncomfortable position. For instance, Ed insisted that statements involving empty names such as ‘Pegasus = Pegasus’ is a counterexample to my argument. Perhaps it is! But they cannot simultaneously opt for S5, for in S5 Pegasus fails to exist in any possible world, if it fails to exist in the actual world. Therefore, the statement in question cannot occur as a well formed statement and, thus, cannot be assigned a truth-value. Another example is Bill who holds that there are some possible worlds in which no contingent beings exist. While such intuitions are certainly cogent as far as they go, they cannot be represented in S5, as long as some contingent beings exist in the actual world.

If we do not like a logic with such strong assumptions, then we must either allow for a model with variable domains or introduce an *existence predicate* in addition to the existential quantifier (i.e., Free-Logic). Since I am not familiar with either of these two systems, I do not know how our disputes are worked out in such systems.

(V) Suppose we consider whether 'Hesperus = Phosphorus'. According to Kripke, if it is indeed true in the actual world that Hesperus = Phosphorus, then it is going to be a necessary truth that 'Hesperus = Phosphorus', since the terms 'Hesperus' and 'Phosphorus' rigidly refer to Venus. Therefore, in a standard first-order Kripkean modal model, we *assume* that Venus exists. Of course it is also true that Venus might not have existed; i.e., there may be some possible world in which Venus does not exists. And yet, 'Nec (Hesperus = Phosphorus)' is true, (given that 'Hesperus=Phosporus' is true). So how can it be true both that

(i) 'Nec (Hesperus = Phosphorus)' is true, given that 'Hesperus = Phosphorus';

and also that

(ii) Venus; namely, the planet to which 'Hesperus' and 'Phosphorus' both refer, exists contingently?

The answer is that certain necessary propositions can be true of objects that exist contingently. The things that are true necessarily of contingent existents are their essential properties, including that they are self-identical. A Kripkean style modal logic articulates what it means to say that a contingent object features a property essentially in terms of necessity by construing such necessity in terms of truth in all possible worlds in which the object exists. The result is that some necessary propositions entail contingent consequences, such as the counter-example I gave to Bill's principle. (I mean that '(Ex)(x=a)' is contingent; not 'a=a', which is necessary as long as 'a' refers').

Now, if someone does not like this consequence, then they will have to offer an alternative to the Kripkean modal-theoretic model of essences in the case of contingent existents. What would such an alternative model be? How would one state that an object o features a property F essentially in terms of the concept of necessity? I have no clue!

(VI) Bill has maintained all along the following:

“Clearly, (1) is nec. true. But what about (2), i.e. *a = a*? I take you (i.e., Edward) to agree with me that it is contingent. Peter, however, thinks it necessary, but then quiote bizarrely considers (3) to be contingent.”

Bill also holds that there are some possible worlds in which no individual objects exist: call this world w*. Now, since (x)(x=x) is necessary, it is true in every world including w*. On the other hand, since object a is an individual object, it fails to exist in w*. I take it that Bill will say that under such circumstances ‘a=a’ is false in w*. This proves, Bill argues, that ‘a=a’ is contingent. And this holds for every object whatsoever.

But, now, is (x)(x=x) true or false in w*? Bill, of course, maintains that it must be true, since it is necessary. But, if it is true, then there must be at least one instance of the variable x for which it is true. But no such instances exist in w*. Therefore, ‘(x)(x=x)’ turns out to be false in w*. But, how can that be; isn’t ‘(x)(x=x)’ a necessary truth?

Bill undoubtedly retort that some individual objects must exist in w* because there are some necessary existents: e.g., God or the number two. So even in w* there are some instances of ‘(x)(x=x)’ that are true. One such instance would be ‘God=God’.

However, Bill cannot respond in this way, for God exists necessarily; i.e., in every possible world including w*, provided God exists in the actual world. i.e., Bill is only entitled to hold (A):

(A) If God exists in the actual world, God exists in every possible world including w*.

So, now, the argument can be restated as follows:

(1*) Nec ((x) (x=x));
(1.5*) ‘God’ refers to some existing object in the actual world;
Therefore,
(2*) God =God.

But, (1.5*) is a contingent statement. Therefore, according to Bill’s reasoning, (2*) is contingent as well. But it is not. Bill might respond by denying that (1.5*) is a contingent statement, since God is a necessary being. But such a move is also wrong. For it could turn out that God fails to exist in the actual world. If so, then one cannot conclude that God necessarily exists and therefore that God exists in every possible world including w*. What Bill wants to say is that *if God exists in the actual world*, then existence is a necessary property of God.
Therefore, under such circumstances it would turn out that God exists in every possible world, including w*, and therefore ‘God=God’ is also necessary. But he can say that only if he allows that statements such as (1.5*) are not required for inferences of the sort exemplified by (1*)-(2*). But, then, he must allow that (1)-(2) is also a valid inference without any contingent auxiliary premises.

What distinguishes necessary beings such as God (or the number two) from contingent beings such as Hesperus and Bill is not that while in the former cases identities such as ‘God=God’ are necessary, whereas in the later cases statements such as ‘Bill = Bill’ are contingent. The difference between the two cases pertains to the scope of possible worlds in which the identity statements hold. In the case of God, the statement ‘God=God’ is true in every possible world without exception, including w*. In the case of Bill, there are possible worlds in which Bill fails to exist and in such worlds the truth-value of the statement ‘Bill = Bill’ cannot be assessed.

>>The rest of Ed's comments in this post confuse object language and meta-language. e.g.,“But you (i.e., Peter) have said above that "Pegasus does not exist", and that it is not it false of any object that is not Pegasus".
<<

Emphatically not. You said quote "False of which object. Not Pegasus, since Pegasus does not exist. Nor is it false of any object that is not Pegasus, since it is not the case that any other object fails to be identical with itself. "

1. You did not enclose the name 'Pegasus' in quotes, and so I do not see how your statement, as you later claim, involved the meta-language, not the object language.

2. Perhaps you meant to say that 'Pegasus' does not refer to any object in the domain? Well I don't see how you translate your statement that it is not false of Pegasus.

3. There is the further problem I alluded to later on, that there may be an object called 'Pegasus' in the domain (perhaps I have a real horse that is called 'Pegasus'). You have to qualify your use of the name, as meaning not a real horse, but the mythical one. But then you have to explain what that meaning is.

4. And the fact remains that when I say 'Pegasus does not exist', I am not talking about the idea of Pegasus, nor the name 'Pegasus'. I am talking about Pegasus. Bill has emphasised this point many, many times.

Peter makes many interesting points above, but re-reading my post, I see no reason to change my view.

Ain't philosophy grand?

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