Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Deducing John McCain from the Principle of Identity

What, if anything, is wrong with the following argument:

   1. (x)(x = x) (Principle of Identity)
   Therefore
   2. John McCain = John McCain (From 1 by Universal Instantiation)
   Therefore
   3. (Ex)(x = John McCain) (From 2 by Existential Generalization)
   Therefore
   4. John McCain exists. (From 3 by translation into ordinary idiom)

The initial premise states that everything is identical to itself, that nothing is self-diverse. Surely this is a necessary truth, one true no matter what, or in the jargon of possible worlds: true in every (broadly logically) possible world.

(2) follows from (1) by the intuitively clear inference rule of Universal Istantiation.  Surely, if everything is self-identical, then John McCain is  self-identical. The inferential move from (2) to (3) is also quite obvious: if McCain is self-identical, then something is identical to McCain. But (3) is just a complicated way of saying that John McCain exists. So we get the surprising result that the existence of John McCain is validly deducible from an a priori knowable necessary truth  of logic!

You understand, of course, that the argument is not about John McCain: it is about any nameable entity. Supposedly, Wilhelm Traugott Krug (1770-1842) once demanded of Hegel that he deduce Herr Krug's pen. If we name that pen 'Skip,' we can then put that name in the place of 'John McCain' and run the argument as before.

There is one premise and three inferences. Does anyone have the chutzpah to deny the premise? Will anyone make bold to question inference rules U.I. and E.G.? And yet surely something has gone wrong. Intuitively, the existence of a contingent being such as McCain cannot be deduced from an a priori knowable necessary truth of logic.  For that matter, the existence of a necessary being such as God cannot be deduced from an a priori knowable necessary truth of logic.  Surely nothing concrete, not even God, is such that its existence can be derived from the Law of  Identity.

So what we have above is an ontological argument gone wild whereby the  rabbit of real existence is pulled from the empty hat of mere logic!

St. Bonaventura said that if God is God, then God exists. If such  reasoning does not work in the case of God, then a fortiori it does not work  in the case of McCain or Herr Krug's pen.

Note that (1) is necessarily true. (It doesn't just happen to be the case that each thing is self-identical.) If (2) follows immediately  from (1), (2) is also necessarily true. And if (2) is necessarily true, then (3) is necessarily true. And the same holds for (4). But surely it is not the case that, necessarily, John McCain exists. He cannot be shown to exist by the above reasoning, and he certainly cannot be shown to necessarily exist by it.

So what went wrong? By my count there are three essentially equivalent ways of diagnosing the misstep.

A. One idea is that the argument leaves the rails in the transition from (3) to (4). All that (3) says is that something is identical to John McCain. But from (3) it does not follow that John McCain exists.   For the something in question might be a nonexistent something. After all, if something is identical to Vulcan, you won't conclude that  Vulcan exists. To move validly from (3) to (4), one needs the auxiliary premise:

3.5  The domain of quantification is a domain of existents only.

Without (3.5), John McCain might be a Meinongian nonexistent object. If he were, then everything would be logically in order up to (3). But  to get from (3) to (4) one must assume that one is quantifying over existents only.

But then a point I have been hammering away  at all my philosophical life is once again thrown into relief:  The misnamed 'existential' quantifier, pace Quine, does not express existence, it presupposes existence!

B. Or one might argue that the move from (1) to (2) is invalid. Although (1) is necessarily true, (2) is not necessarily true, but  contingently true: it is not true in possible worlds in which McCain does not exist. There are such worlds since he is a contingent being. To move validly from (1) to (2) a supplementary premise is needed:

1.5 'John McCain' refers to something that exists.

(1.5) is true in some but not all worlds. With this supplementary premise on board, the argument is sound. It also loses the  'rabbit-out-of-the-hat' quality. The original argument appeared to be  deducing McCain from a logical axiom. But now we see that the argument  made explicit does no such thing. It deduces the existence of McCain  from a logical axiom plus a contingent premise which is indeed   equivalent to the conclusion.

C. Finally, one might locate the error in the move from (2) to (3). No doubt McCain = McCain, and no doubt one can infer therefrom that something is identical to McCain. But this inferential move is not existential generalization, if we are to speak accurately and nontendentiously, but particular generalization. On this diagnosis,  the mistake is to think that the particular quantifier has anything to do with existence. It does not. It does not express existence, pace Quine, it expresses the logical quantity someness.

In sum, one cannot deduce the actual existence of a contingent being from a truth of logic alone. One needs existential 'input.' It follows that there has to be more to existence than someness, more than what  the 'existential' quantifier expresses. The thin conception of existence,  therefore, cannot be right.

Now let me apply these results to what Peter Lupu has lately been arguing.   Here he argues:

(i) (x)(x=x);

(ii) a=a, for any arbitrarily chosen object a; (from (i))

(iii) (Ex)(x=a); (from (ii) by existential generalization);

Now, (i) is necessary, but (iii) is contingent. Yet (i) entails (iii) via (ii), which is also necessary. So I simply do not see how the principle (1*) which you and Jan seem to accept applies in modal logics that include quantification plus identity.

Peter thinks he has a counterexample to the principle that if p entails q, and p is necessary, then q is also necessary.  For he thinks that *(x)( x = x)*, which is necessary, entails *(Ex)(x = a)*, which is contingent.

But surely if *a = a* is necessary, i.e. true in all worlds, then *(Ex)(x = a)* is necessary as well.

The mistake in Peter's reasoning comes in with the move from *Necessarily, (x) (x = x)* to *Necessarily, a = a*.   For surely it is false that in every possible world, a = a.  After all, there are worlds in which a does not exist, and an individual cannot have a property in a world in which it doesn't exist.  One must distinguish between essential and necessary self-identity.  Every individual is essentially (as opposed to accidentally) self-identical: no individual can exist without being self-identical.  But only some individuals are necessarily self-identical, i.e, self-identical in every world.  Socrates, for example, is essentially but not necessarily self-identical: he is self-identical in every world in which he exists (but, being contingent, he doesn't exist in every world).  By contrast, God is both essentially and necessarily self-identical: he is self-identical in every world, period (because he is a necessary being).   

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Bill Vallicella

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8 responses to “Deducing John McCain from the Principle of Identity”

  1. Richard E. Hennessey Avatar
    Richard E. Hennessey

    Nice, nice, nice.
    Let me preface my comments by observing that there is some reason to believe that there are perhaps three non-identical John McCains in the one John McCain and that we find ourselves faced with another, and particularly difficult, version of the Trinitarian problem.
    Next, getting a bit more serious and making absolutely explicit what you made virtually so in your remark about “an ontological argument gone wild,” parallel arguments can be used to prove the existence of the “flying spaghetti monster” of recent notoriety and Russell’s teapot of a more vintage notoriety.
    Getting now to the points I wish to make, I’m of the view or, perhaps better, I am now thinking that the first problem arises in the first inference:

    1. (x)(x = x) (Principle of Identity)
    Therefore
    2. John McCain = John McCain (From 1 by Universal Instantiation)

    That is, it seems to me (a) that, as you put it a bit further down, “[t]he domain of quantification is a domain of existents only.” It also seems to me (b) that, as you also put it:

    To move validly from (1) to (2) a supplementary premise is needed:
    1.5 ‘John McCain’ refers to something that exists.

    Let me put your two points a bit differently. I think your first point is incorporated into the first premise in the following reformulation of the argument and the point in the second premise:

    1. For any existent x, x is identical to x [or even, x is x]
    2. John McCain is an existent.
    3. Therefore, John McCain is identical to John McCain.

    While there is much to chew through here to bring out and justify all the assumptions that may be at work in the expression of the two premises, it seems to me that the argument is quite evidently both valid and sound. The corresponding arguments regarding the “flying spaghetti monster” and Russell’s teapot would be equally evidently valid but not evidently sound; it would remain to demonstrate that their second premises are true.
    There are additional problems in the second inference, which you have sufficiently well identified in, respectively, the paragraphs beginning with “C. Finally …} and “The mistake in Peter’s reasoning…” that I feel no need to keep on opining (though I have my worries about “possible worlds”).

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  2. peterlupu Avatar
    peterlupu

    Bill,
    First let me congratulate you on an excellent (albeit, in my opinion, still wrong) post. Second, let me state that I myself am puzzled by the sort of alleged counterexamples we have been discussing. In this I join you and all those who agree with you about this point. After all, how can a necessary proposition entail a contingent one? The argument against this possibility has been well stated first by you and then again very eloquently by Jan. And yet I maintain that my argument has not been adequately answered. I shall now use Bill’s recent post to show why I think so. I will begin with some observations Bill makes at the end of his post and then focus on his solution designated by (B). I am assuming throughout (1*) or it’s variant:
    (1**) if p entails q and p is necessary, then q is necessary.
    (I) In the last paragraph Bill says:
    “The mistake in Peter’s reasoning comes in with the move from *Necessarily, (x) (x = x)* to *Necessarily, a = a*. For surely it is false that in every possible world, a = a. After all, there are worlds in which a does not exist, and an individual cannot have a property in a world in which it doesn’t exist. “
    Bill claims that my mistake is to think that ‘a=a’ is necessary. Why? Because if the object-a is a contingent being, then there are some possible worlds in which the object-a fails to exist. Therefore, Bill claims that ‘a=a’ is not necessary: it is contingent. But, now, I ask: Does anyone wishes to deny that ‘(x)(x=x)’ entails ‘a=a’ for some object-a? I doubt that. But if ‘a=a’ is contingent and ‘(x)(x=x) is necessary, then a necessary proposition entails a contingent one.
    One of the ways Bill attempts to avoid this conclusion is his option (B) above. The move is to argue that the logical truth
    (1) (x)(x=x);
    alone does not validly entail
    (2) ‘a=a’;
    Rather an additional premise is required for such an entailment. Bill suggests the following premise:
    (1.5) ‘a’ refers to something that exists;
    (I replaced his ‘John McCain with ‘a’ in order to highlight the generality of the process)
    Now (1) together with (1.5), Bill claims, validly entails (2). But since (1.5) is contingent, so is the conjunction of (1) and (1.5). Hence, we no longer have the problem of a violation of (1**); i.e., deriving a contingent consequence from a necessary truth.
    But how does Bill’s (1.5) figure into the inference from (1) to (2)? For instance, could Bill show me a step in the inference from (1) to (2) in which (1.5) plays an essential role? Which step would that be? What rule of inference would be involved in such a step? And, of course, if (1.5) is not essentially involved in the inference from (1) to (2), then it is not part of such an inference. Therefore the premises of the argument remain necessary.
    I am not aware of any inference rule which governs the behavior of a premise such as (1.5). There are no inference rules analogous to (1.5) in first-order quantification theory nor in first-order modal logic. Could Bill, or anyone else, point out to me such an inference rule? The reason no such inference rule exists is because (1.5) is a meta-linguistic expression. It belongs to the interpretation (I) component of a model; i.e., an assignment of extensions to the expressions of a language. Thus, an assignment of extensions is already determined *before* we go about stating the axioms and deriving from them theorems by means of inference rules. Any formula that contains a singular expression for which the interpretation function failed to provide an extension is not a well-formed-formula; it cannot have a truth value. Thus, Bill’s (1.5) is not a premise in any argument; it is part of the interpretation of the language. It is for this reason that there are no inference rules that govern expressions such as (1.5). Let me illustrate this point with a different case.
    Consider the following necessary truth:
    (i) (x) (Fx or ~Fx);
    One would think that (i) alone entails by UI the consequence:
    (ii) Fa v ~Fa; for some object-a.
    Is (ii) a necessary truth? Not according to Bill’s reasoning. For by a parallel reasoning to Bill’s (B), one should maintain that the inference from (i) to (ii) is invalid. We need (1.5) as an additional premise, since object-a is contingent. But, now, (ii) turns out to be a contingent truth, since it is derived from premises one of which is contingent. Surely there is something wrong with a form of reasoning that steers us to view (ii) as a contingent truth. Such examples can be multiplied.
    A most disturbing such example can be devised even in propositional logic. ‘X’, ‘Y’, ‘Z’ are sentential variables; ‘P’, ‘Q’, ‘R’ are individual sentences. Consider the following logical truth:
    (I) For every X, Y, and Z; If (X entails Y & Y entails Z), then (X entails Z));
    An instance of (I) would be:
    (II) If (P entails Q & Q entails R), then (P entails R).
    One would think that the inference from (I) to (II) is valid based on UI and that, therefore, since (I) is necessary, so is (II). But no such a result follows if we accept Bill’s reasoning. For according to this form of reasoning, the inference from (I) to (II) is invalid because we need an additional premise which tells us that ‘P’, ‘Q’, and ‘R’ express propositions. So now we need to add the following premise:
    (1.5*) ‘P’, ‘Q’, and ‘R’ express propositions.
    Now we can validly infer (II) from (I) in conjunction with (1.5*). But now we face a serious difficulty. (1.5*) is contingent because it is a contingent matter whether a given linguistic form expresses a proposition or not. And since (1.5*) is contingent, so is the conjunction of (I) and (1.5*). But then (II) is contingent as well. Yet clearly (II) ought to turn out to be a necessary truth. Any view which renders (II) contingent is simply wrong.
    Another case that illustrates the difficulties with this form of reasoning has to do with example Bill mentioned; namely, God. He contrasts God with Socrates and says: “By contrast, God is both essentially and necessarily self-identical: he is self-identical in every world, period (because he is a necessary being).”
    Bill is not entitled to say that God is a necessary being. He is only entitled to assert that God is a necessary being, *if* God exists at all. Another way of putting the matter is this: God is a necessary being, if the term ‘God’ refers to an existing being. But the if-clause is surely a contingent matter. Does the contingency of the meta-linguistic fact that the term ‘God’ refers to an existing being somehow converts the original assertion that God is a necessary being into a contingent one as well? It does not!

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  3. Bill Vallicella Avatar
    Bill Vallicella

    Richard,
    Thanks very much for the comments. At the end you express worries about possible worlds. I think one can employ ‘possible worlds’ jargon as a facon de parler without ontologically committing oneself to entities called possible worlds. One can use it simply as a graphic way of representing modal relationships in extensional terms. Thus: a contingent being is one that exists in some, but not all, possible worlds. A necessary being is one that exists in all worlds. An impossible being is one that exists in no world. To say that S. is essentially human is to say that he is human in every world in which he exists, and so on.
    To say that S. exists in a merely possible world W is to say that, had W been actual, then S. would have existed.
    We needn’t embrace the excesses of David Lewis. Indeed, if I am not mistaken, we needn’t commit ourselves at all to the existence of possible worlds, however construed.

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  4. Bill Vallicella Avatar
    Bill Vallicella

    Excellent comments, Peter. This is tricky stuff, and I don’t feel confident that I have seen to the bottom of this problematic. Forgive the pedantry, but ‘validly entail’ is a redundancy since there are no invalid entailments. An inference can be either valid or invalid, but all entailments are valid.
    >>But how does Bill’s (1.5) figure into the inference from (1) to (2)? For instance, could Bill show me a step in the inference from (1) to (2) in which (1.5) plays an essential role? Which step would that be? What rule of inference would be involved in such a step? And, of course, if (1.5) is not essentially involved in the inference from (1) to (2), then it is not part of such an inference. Therefore the premises of the argument remain necessary.<< The question is how we get from 1. (x)(x = x) to 2. a = a. I think we both agree that 'x' is an individual variable, a variable that takes individuals (not properties) as values and logically proper names as substituends. We accept the value-substituend distinction. And we agree that (1) is necessarily true. We should also agree that (1) is true in possible worlds in which there are no individuals, and that there are such worlds. We agree that 'a' is an arbitrary individual constant. The notion of an arbitrary individual constant is not entirely clear and may be a source of our puzzlement. 'McCain' is an individual constant, but not an arbitrary individual constant. So may be we need to think harder about the differences among individual variables, individual arbitrary constants and individual constants. Let W be a world in which there are no individuals. In W, (1) is true (because (1) is true in all worlds). But in W, the inference from (1) to (2) fails. For in W there are no individuals and thus nothing that could be tagged as 'a.' So how do we move validly from (1) to (2)? We need an auxiliarly premise. May be this woruld do the trick: 1.5* No world is bare of individuals Now this is a valid argument: Necessarily, for any x, x = x. Necessarily, individuals exist. Ergo Necessarily, a = a. But it is not sound because the minor is false. It is not necessary that individuals exist. Now Peter, are you assuming the truth of the minor premise? More later. It is time to BBQ a T-bone.

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  5. Philoponus Avatar
    Philoponus

    May I offer a few comments?
    The first thing to recall is that in Quantifier Logic there are not one but a series of principles of identify. In 1st order QL , where we quantify only over individual constants, we have only the Principle of Identity for Individual Constants (PIIC). But in higher order QL’s, where we introduce variables ranging over simple predicates, relations, and more complex functions, we have and use principles of identity for all of these. I assume the first premise in Peter’s is alluding to a 1st order PIIC, but this needs to be clearer. We need in particular specify the domain over which the variable x ranges.
    “McCain=McCain” is a bit of a problem. As it stands, it is not a wff in any QL language (or English) and it has no connection to the previous PIIC. We can incorporate it into a standard 1st order QL and connect it to the the previous PIIC if we stipulate that “McCain” is constant naming a thing in the domain over which we are quantifying in the previous PIIC. If this domain is appropriately specified—for example, Navy veterans who became US senators—then this premise does legitimately exemplify or instantiate PIIC for that domain. But notice, if a different domain is assumed for the PIIC, for example, natural or rational numbers, then “McCain is McCain” is just nonsense. Notice also that the things in the domain we are quantifying over need not actually exist. Suppose for example the domain of discourse is femme fatales of Euripidean tragedy. Then “Medea is Medea” or “Medea=Medea” is well formed. But from this we cannot infer that Medea exist(ed) except as a literary character in Euripides’ play of the same name.
    A clear way to put this is the “McCain=McCain” premise of Peter’s argument is actually conditional. The antecedent is “If McCain is a constant in the domain of discourse over which x ranges” and consequent “then by the previous PIIC ‘McCain = McCain’. The conclusion is then also conditional and effectively tautological: if McCain is constant naming something in the domain over x in the PIIC ranges, then something named McCain exists in that domain.
    Final point. Suppose I propose that q is the number such that q x q = 3. Is “q=q” a wff in the domain of rationals? No. Q cannot be a rational number and so is not in that domain. “q=q” is nonsense in that domain. Identity claims about things that do not exist in the domain of the bound variable are nonsense.

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  6. peterlupu Avatar
    peterlupu

    Bill,
    I agree that these issues are tricky and on more than one level. There are some technical issues and then there are some meta-logical matters. So I shall list some of these below in no particular order.
    1) I think you are right to highlight the distinction between an individual constant vs. an arbitrary individual constant. There may be some subtle distinctions as to their behavior in formal systems and some of these may be relevant to our topic.
    2) Empty Domains: Typically it is assumed that the domains are not empty. I am uncertain about the procedures when we allow empty domains, particularly in modal logic. The non-empty domain assumption proves to be significant when it comes to the manner we interpret some of the propositions we have been discussing: e.g., the case of Pegasus.
    3) Fixed vs. variable domains: In possible world semantics they distinguish between systems in which the domain is fixed throughout possible worlds vs. systems where the domains are allowed to vary from one possible world to another. There is a difference between the two systems, yet they may be in the end formally equivalent. I do not know the details, but I suspect that this one is relevant to our discussion, particularly regarding the case of Pegasus and other such non-existents.
    4) Empty Names: As I have stated before, in most systems the interpretation of the non-logical vocabulary is given in advance. In such systems individual constants are assigned in advance an object in each possible world by the interpretation on the symbols of the language; arbitrary individual constants are assigned an object as needed. So in general we do not have a problem with empty names. Of course, this technical procedure does not solve the problem of empty names in ordinary discourse (e.g., ‘Pegasus’): hence the problem of fictional objects.
    5) Necessary truths and Contingent Objects: The general question here is this: can we say that some sentences about contingent existents are necessary truths? There are several cases to consider.
    (i) Logical Form and Contingent Existents: Contingent objects exist in some worlds but not in others. This raises the following problem: how to evaluate sentences about contingent existents when the sentences have the logical-form of logical truths and, thus, appear to be necessary truths? For instance, John McCain exists in the actual world, but he fails to exist in some other possible worlds. Our problem is what to do with a sentence such as (Fa V ~Fa), where ‘a’ is an individual constant assigned to John McCain.
    We have here a tension between the contingency of the existence of such objects versus the logical-form of the sentences. On the one hand, the contingent existence of objects such as John McCain compels some to think that all sentences about them can be true at most in worlds in which the objects exist. And since these worlds are going to be a subset of all possible worlds, we naturally think of such sentences as contingent as well. These considerations might lead some to evaluate (Fa V ~Fa) as not necessary and, hence, only possibly true. On the other hand, considerations of the logical-form of such sentences compels others to view them as true in all possible worlds in which their object exists because their logical-form matches the logical form of familiar logical truths. Each consideration pulls in a different direction.
    The problem is that if we decide that a sentence such as (Fa V ~Fa) is not true in worlds in which the object-a fails to exist, then it would seem that we are committed to saying that it is false in such worlds. But evaluating such sentences as false in worlds in which John McCain does not exists makes no sense either. For suppose we say that (Fa V ~Fa) is false in a world w*, where w* is such that John McCain fails to exist in w*. Since John McCain fails to exist in w*, the sentence (Fa V ~Fa) cannot be said to be false of him; after all, he does not exist in w*. Could the sentence (Fa V ~Fa) be said to be false of some other object, say John Kerry, who does exist in w*. But how can (Fa V ~Fa) be false of John Kerry in w* (or in any other world)? First, the sentence in question is not about John Kerry, for by our stipulation ‘a’ refers to John McCain and not to John Kerry. Second, how can (F…V ~F___) be false of any object in w* (or any other world)? It can’t.
    (ii) Essential Properties and Contingent Existents: Consider the property of self-identity. Clearly, every object has this property, including a contingent existent such as John McCain. How are we to express this fact regarding John McCain? Well, we could say that (a=a). Now we ask: Is (a=a) a necessary truth or a contingent one? Well, once again we face the same predicament as in (5i) above. Since, clearly, John McCain possesses this property essentially, that is he has it in every world in which he exists, we are compelled to say that (a=a) is a necessary truth: i.e., Nec(a-a). On the other hand, since John McCain fails to exist in some possible worlds, say w*, it would seem that (a=a) cannot be true in w*. Hence, (a=a) is contingent: i.e., ~Nec(a=a). But, then (a=a) is false in w*? Now we are faced with the same dilemma we have encountered above: Which object is such that (a=a) is false of it? It cannot be John McCain, since he does not exist in w*. And it cannot be any other object such as John Kerry, for instance, for (a=a) is not about John Kerry and it would be false to say that John Kerry fails to possess the property of being self-identical.
    (iii) Rigidity and Contingent Existents: let ‘a’ and ‘b’ be rigid designators that co-refer, say to John McCain. Then (a=b) is true. But is it a necessary truth? Well, clearly, (a=b) is true in every world in which John McCain exists. I won’t go through Kripke’s arguments for this. But some on this site have been reluctant to evaluate such a sentence as outright a necessary truth because it involves the proviso ‘if true’ or ‘if such-and-such object exists’ or ‘if such-and-such a term refers’. I suppose the intuition of these commentators is that this proviso clearly signals that the sentence might actually be false in some possible worlds, perhaps because the object does not exist in such worlds or that the term fails to refer in some worlds.
    But such intuitions cannot be correct. First, the arguments considered in (5i) and (5ii) apply equally to these cases. And, secondly, it is of utmost importance to see that provisions such as ‘if true’, etc., have nothing to do with whether a given sentence is necessary or not. For instance, some of us wish to maintain that mathematical sentences are necessary truths, IF TRUE. Now, if it indeed were the case that provisions such as ‘if true’ correctly indicate that the sentences to which they are attached cannot be necessary truths, then the opponents of the necessity of mathematical truths would gain an easy victory; they could simply point out that even the proponents of mathematical necessity must employ the provision ‘if true’ in each case of a mathematical sentence. Clearly, the view that mathematical sentences are necessary, if true, cannot be defeated by such an argument.
    Consider the 5 trillion plus 1 digit expansion of Pi. Clearly, there is such a digit; call it ‘c’. Now, suppose someone speculates that c is going to turn out to be an even number. Since we don’t currently know the value of c, we do not know whether it would be even or odd. Nevertheless, we can still say that if c turns out to be even, then necessarily c is divisible by two without remainder. Are we to say that such a statement is mistaken simply because of the proviso ‘if c is even’?
    Similar considerations would pertain to mathematical existence. Suppose that we currently know that there are x number of twin-primes (x is currently a fairly large number). Suppose someone ventures to say that the x+y twin-primes have a certain property F (where y is any whole positive natural number you like). Then we can say that if x+y twin-primes exist and they have property F, then it is necessary that x+y twin-primes have property F. Of course, we currently have no idea whether x+y twin-primes even exist. Perhaps, there are only x+(y-1) twin-primes, in which case there is not going to be a twin-prime that is x+y. But if there is such a pair and it does feature F, then it must do so.
    I do not know why some commentators think that provisions such as ‘if true’, etc., deprive the sentence to which they are attached from their necessity status. Perhaps, it has something to do with focusing a bit too much on the syntax of modalities and glossing over their semantics. However, we ought to remember that the semantics of modalities involves an important concept of *accessibility* among worlds. Defining the concepts of necessity and possibility requires a careful attention to the accessibility relation among worlds.
    Roughly, the definitions go as follows, where ‘R’ is the accessibility relation among worlds: P is necessary at w* if P is true in every world w such that Rww*; P is possible at w* if P is true in some worlds w such that Rww*. Thus, formulas are necessary/possible dependent upon how one defines the accessibility relation. Some modalized formulas may be true (or valid) in some modal models because of the way the accessibility relation is defined, while the same formulas may be false in others because the accessibility relation is different. Modal models differ when the accessibility relation among worlds differs. Glossing over these subtleties may have been one of our problem throughout these discussions.
    6) De-re vs. De-dicto Modalities:
    (i) Consider the following four sentences:
    (A) Nec(x)(x=x); (A*) (x)Nec(x=x).
    (B) Nec(Ex)(x=a); (B*) (Ex)Nec(x=a).
    Here we have a mix of two different quantifiers: quantifiers that range over objects (all, some) and the quantifiers that range over possible worlds (necessary, possible). As you all know, the complexity of quantified modal logic is due to the mix between these two quantifiers. The de-re vs. de-dicto distinction is an example of such a mix.
    (Side-Note: As Kripke warns in “Speaker’s Reference and Semantic Reference”, the de-dicto/de-re distinction (or any other two-fold distinction) cannot be identified with or substitute for Russell’s scope distinctions. Unlike any two-fold distinction, Russell’s scope distinction allows for a particular quantifier, description, or modal operator to have widest scope, narrowest scope, or various intermediary scopes, when they are multiply iterated.)
    Typically (A) and (B) are de-dicto, whereas (A*) and (B*) are read de-re. The distinction is typically stated in terms of what is necessary/possible: in the case of (A) and (B) it is the sentence or proposition that is said to be necessary: e.g., (A) means that the proposition or sentence ‘(x)(x=x)’ is true in every possible world. In the case of (A*) and (B*) it is the object featuring some property or other that is said to be necessary. Thus, (A*), for instance, says that each object is such that it is self-identical in every possible world in which it exists. Similarly, (B*) says that some particular object exists such that it is necessarily self-identical.
    (ii) So your statement:
    “One must distinguish between essential and necessary self-identity. Every individual is essentially (as opposed to accidentally) self-identical: no individual can exist without being self-identical. But only some individuals are necessarily self-identical, i.e, self-identical in every world.”
    suggests that what you call “essential self-identity” is the relevant de-re case (i.e., (A*) and (B*) respectively), whereas what you call “necessary self-identity” is the de-dicto case. But I may be wrong about this interpretation of your statement.
    Regardless, we should be amiss to think that (A*) and (B*) do not make a claim of necessity. (A*), for instance, asserts that for each given object in w* (the actual world or some other world with respect to which ‘(x)Nec(x=x)’ is evaluated), it is necessarily self-identical; i.e., for each object in w* and each world w s.t. Rww*, that object is self-identical. Similarly, (B*) asserts that a particular object exists in w* such that for every w s.t. Rww*, that object is self-identical. Of course, because de-re necessity is read that way it can be used to formalize our essentialist intuitions.
    (iii) Since (A) and (A*) and (B) and (B*), respectively, are not equivalent, one must be very careful which one uses in various inferences. And perhaps our problem originates from not distinguishing carefully between these two forms of inferences. I might have been guilty of this myself.

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  7. peterlupu Avatar
    peterlupu

    Philoponus,
    We have been working with first-order; so no quantifying over properties. All quantification is over individual objects.
    Regarding “McCain=McCain”; You are right and that is why I have been using the letter ‘a’ as an individual constant to stand for the name ‘McCain’. However, so far as I can see, this has not caused any confusion on anyone’s part. You are also correct that the domain over which the individual variables range need to be specified, although I believe we all assumed that the domain contains everything that exists in the actual world, including numbers.
    “Suppose for example the domain of discourse is femme fatales of Euripidean tragedy. Then “Medea is Medea” or “Medea=Medea” is well formed. But from this we cannot infer that Medea exist(ed) except as a literary character in Euripides’ play of the same name.”
    Well, this is a different matter altogether. Two points. First, and typically, individual constants will have to be assigned an object in advance. Suppose we have ‘Medea’ as such a constant; then it will have to be assigned some object in the domain of each possible world. If the model theory employs fixed domains, then ‘Medea’ will not be assigned an object, since no such an object exists. This can create problems in the semantics, so I do not think it will be allowed or perhaps there is some technical trick to solve the problem. In a model theory employing domains that are allowed to vary, then some possible worlds may contain Medea even if the actual world does not. So ‘Medea’ may be assigned an individual in some other possible worlds, but not in the actual world.
    “…the “McCain=McCain” premise of Peter’s argument is actually conditional. The antecedent is “If McCain is a constant in the domain of discourse over which x ranges” and consequent “then by the previous PIIC ‘McCain = McCain’. The conclusion is then also conditional and effectively tautological: if McCain is constant naming something in the domain over x in the PIIC ranges, then something named McCain exists in that domain.”
    In my LOOONG post above I addressed at length arguments from provisions such as the one you mention in this passage. I do not see how such provisions render the whole of my sentence tautological”, although yours is indeed so. Consider a Kripkean case where ‘a’ and ‘b’ are rigid and co-refer; then (a=b) is necessarily true; i.e., true in every possible world in which a (or b) exists. This is far from a tautology.

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  8. Philoponus Avatar
    Philoponus

    Hi Peter,
    Suppose you and I are discussing the women that Euripides puts at the center of some of his plays, and being logicians with nothing better to do, we decide to formalize our arguments in QL. It would be a natural choice to let the women be the 1st order constants, wouldn’t it? Let m=medea, e=Electra, p=Phaedra, i= Iphigenia, etc. Now we might say “Some of these women are victims”, and it would be natural to formalize this as a 1st order existential claim, yes? But we don’t think this formalization has anything to do with the actual world in the way that, for example, “Some of Charlie’s Sheen’s women are victims” does. The difference lies in the different domains of discourse over which our quantified variables range, in one case literary characters, in the other real young women. It might be nice if QL decided to mark the difference being real people and fictional/literary characters by, for example, using italicized variables for the latter. That would block that fallacy of disparate domains (FDD), by which we conclude that “Denise Richards and Electra are both attractive women”, since each one is.
    Do literary characters like Medea populate possible worlds (PWs)? Not if PWs are maximally consistent variants of the actual world. Is Medea taller or shorter than 5 foot six? Neither! Is she 30 years old at the time of the killings or not 30? Neither! Medea has all and only the features that Euripides attributes to her. In general a work of fiction does not create a possible world. Euripides’ Medea is not someone who possibly existed. But “There is a woman named Medea at the center of Euripides’ play of that name” is unproblematic, given the understood fictional domain.
    Now suppose in the course of our discussion I say “Medea is Medea”. Is this to be formalized as “m=m’? No, that’s not what I meant. I meant Medea is unique and like no other Euripidean woman (in her will to uphold her honor). I think we’d need at least a 2nd order formulation to capture my meaning. Natural languages love to use what look like tautologies to make higher order difference claims. But however we formalize “Medea is Medea”, there is no inference to her existing or possibly existing beyond our understood domain of literary discourse. I’m not sure if you think there is and why you think there is, unless it has to do with thinking of Medea as existing in some alternative PW. By the way, for some reason every time I try to visualize Medea I get an image of Angelina Jolie. I wonder what that means—are murderous women “hot”?–especially since “Medea looks like Angelina” has to be nonsense by the FDD.

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