There is no dead horse

Today, Bill has come up with what seems to me a confused and mistaken piece on the inadequacies of the Quinean thin theory.

First, he says, rightly, that the thin theory (or the principal version thereof) translates Something exists as

1. ∃x (x = x). 

He observes that this contains no non-logical terms and then goes on to ask

How can the extralogical and extrasyntactical fact that something exists be a matter of pure logical syntax? How can this fact be expressed by a string of merely syntactical symbols: '∃,' 'x,' '='?  

This seems to me a confusion.  What does he mean by 'pure logical syntax' or 'merely syntactical symbols'?  I would have thought a good reply would be to say that, since MPL does not regard exists as a predicate, it expresses exists through a special symbol, viz, '∃', and hence '∃' is more than 'merely syntactic', though again I don't appreciate quite what Bill means by this phrase.  My point is that existence is expressed within the core language of MPL and does not require a predicate symbol added on to the core.

Second, he introduces the universal quantification

2. ∀x (x=x),

saying that, on the thin theory, this translates Everything exists. He says that (2) is a logical truth (I agree) but then claims that (1) follows from (2) and uses this as a leg in an aporetic triad.  But there is no aporia because (1) does not follow from (2).  ∃xPx only follows from ∀xPx under the precondition that the domain is non-empty, ie, Something exists is true.

Thereafter things get more confused.  He says Everything exists cannot be translated as (2) because (2) is a necessary truth and hence its negation is necessarily false.  Yet its negation corresponds to  Something does not exist and this, according to Bill, is not necessarily false.  Now we have been here before.  To me this says There is something that does not exist.  Since I can't conceive of a world in which this could be true I regard it as necessarily false.  But Bill disagrees, I think.

Finally Bill argues that Nothing exists cannot be translated as Everything is not self-identical.  For Nothing exists is contingent whereas its translation, ∀x~(x=x), is necessarily false, he claims.  But this last claim is wrong.  ∀x~(x=x) is false when the domain is non-empty, ie, when something exists, and vacuously true when the domain is empty, ie, when nothing exists.  To see the latter, note that when the domain is empty there is no x available which could make ~(x=x) false.  Hence ∀x~(x=x) is contingent too.