The Number of Zits on My Face

Hi all.

I'm currently reading a paper by Thomas Hofweber called "Innocent Statements and Their Metaphysically Loaded Counterparts". (It is available on his website for anyone that is interested.) In his paper, Hofweber is concerned with the transition between sentences like (A) "I have four zits on my face" and (B) "The number of zits on my face is four". (Hofweber uses different examples.) Hofweber claims that (A) and (B) seem truth-conditionally equivalent and that we can infer either from the other in natural language, but (B) apparently entails the existence of numbers whereas (A) does not. His paper is an attempt to determine what is going on here.

Anyway, in Section 3.1 of his paper, Hofweber draws attention to a puzzle for those who think that "The number of zits on my face" in (B) functions as a singular definite description. He notes that, in general, a sentence containing a singular definite description "the F" entails the corresponding sentence containing the indefinite description "a F". However, Hofweber says, attempting to apply this general rule to (B) yields:
(C) A number of zits on my face is four.
And (C) seems very awkward.

So, my question to you guys is: Any idea what is going on here? It certainly looks to me like "The number of zits on my face" functions as a singular definite description in (B). So why does (C) sound so strange if a sentence containing a singular definite description entails the corresponding sentence containing the indefinite description?

An Argument for Actualism

I came across an argument from van Inwagen for actualism recently. I was wondering what y'all thought about it. The idea is that to be red, in the most inclusive sense, is to be actually red. To fail to be red actually is to be red in no sense at all. Similarly, to exist in the most inclusive sense is to exist actually. To fail to exist actually is to exist in no sense at all. There is no relevant difference between redness and existence. So actualism.

The claim about redness seems right, but the corresponding claim about existence should offend possibilists. So the challenge is to note the relevant difference, if any, between redness and existence. There are clearly some moves to make here, but I was wondering what sort of move, if any, y'all found most attractive.

Kant's Cleavage II

I just came across Putnam's account of analyticity. Here it is.

S is analytic iff S is deducible from the sentences in a finite list at the top of which someone who bears the ancestral of the graduate-student relation to Carnap has printed the words 'Meaning Postulate'.

This should settle the matter once and for all.

Kripke Semantics and Strange Truths

On a standard Tarskian account of FOPL a model is taken to be an ordered pair where D is a non-empty domain of individuals. Given the non-empty constraint, there is no model on which some sentence (x)Fx expresses a vacuous truth. It is true in M iff every member of D in M (and there has to be at least one) satisfies the open formula Fx.

Now consider a Kripkean account of MFOPL. On this account a model includes a a set of world W, a domain D and a function Q from members of w to subsets of D. There is no constaint that says the subset must be non-empty. So, there are models with worlds that have an empty domain of quantification. This is wierd.

Why? Because on such a model the sentence <>(x)Fx will express a truth. Moreover, given a plausible extension of the semantics to allow for second order quantification (MSOPL) we will have the following turn out true in models that have empty world: <>(x)(F)Fx. This is very strange. Moreover, the situation becomes more strange when we realize that in the empty world it must also be true that (x)(F)~Fx. Why? Because the denial of (x)(F)~Fx is existentially commiting. But the empty world is empty. So, there are models on which the following is true <>[(x)(F)Fx & (x)(F)~Fx]. But that entails that <>[(x)(F) Fx & ~Fx]. But that can't be true. So what the hell is going on? Should there be a constraint on Q; should it be a function from worlds to non-empty subsets of D? Moreover, if that is the case, then do we have reasons from the semantics of modal discourse for the conclusion that necessarily, something exists?