On August 11th I wrote:
Suppose we acquiesce for the space of this post in QuineSpeak.
Then 'Horses exist' says no more and no less than that 'Something is a horse.' And 'Harry exists' says no more and no less than that 'Something is Harry.' But the 'is' does not have the same sense in both translations. The first is the 'is' of predication while the second is the 'is' of identity. The difference is reflected in the standard notation. The propositional function in the first case is Hx. The propositional function in the second case is x = h. Immediate juxtaposition of predicate constant and free variable [with the predicate constant coming first] is the sign for predication. '=' is the sign for identity. Different signs for different concepts. Identity is irreducible to predication which is presumably why first-order predicate logic with identity is so-called.
Those heir to the 'Fressellian' position, such as Quine and his epigoni, dare not fudge the distinction between the two senses of 'is' lately noted. That, surely, is a cardinal tenet of their brand of analysis.
So even along Quinean lines, the strict univocity of 'exist(s)' across all its uses cannot [pace van Inwagen] be upheld. It cannot be upheld across the divide that separates general from singular existentials.
But the next morning I had a doubt about what I had written. Is there an 'is' of predication in MPL (modern predicate logic)? I argued (above) that 'exist(s)' is not univocal: it does not in MPL have the same sense in 'Fs exist' and 'a exists.' The former translates as 'Something is (predicatively) an F' while the latter translates as 'Something is (identically) a.' Kicked out the front door, the equivocity returns through the back door disguised as an equivocation on 'is' as between predication and identity.
But if the 'is' in 'Grass is green' or 'Something is green' is bundled into the predicate in the Fregean manner, then it could be argued that there is no 'is' of predication in MPL distinct from the 'is' of identity and the 'is' of existence. If so, my equivocity argument above collapses, resting as it does on the unexpungeable distinction between the 'is' or identity and the 'is' of predication.
Yesterday a note from Spencer Case shows that he is on to the same (putative) difficulty with my argument:
Hey Bill, I have a professor whose pet peeve is the claim that there is an 'is' of identity and an 'is' of predication. I don't know his arguments for thinking so, but his view is that 'is' is univocal and what differs is the content of the copula. If he's right, that would be a problem for you here. Do you know more about this position than I do?
To sort this out we need to distinguish several different questions:
Q1. Is there a predicative use of 'is' in English? Yes, e.g., 'Al is fat.' This use is distinct from the existential use and the identitative use (and others that I needn't mention). So I hope Spencer's professor is not denying the plain linguistic fact that in English there is an 'is' of predication and an 'is' of identity and that they are distinct.
Q2. Must there be a separate sign for the predicative tie in a logically perspicuous artificial language such as MPL (modern predicate logic, i.e., first-order predicate logic with identity)? No. When we symbolize 'Al is fat' by Fa, there is no separate sign for the predicative tie. But there is a sign for it, namely, the immediate juxtaposition of the predicate constant and the individual constant with the predicate constant to the left of the individual constant. So we shouldn't confuse a separate or stand-alone sign with a sign. Other non-separate signs are conceivable exploiting different fonts and different colors, etc.
Q3. Must there be some sign or other for predication in a logically adequate language such as MPL? How could there fail to be? If our logical language is adequate, then it has to be able to symbolize predications such as 'Al is fat.' And note that existentials such as 'Fat cats exist' cannot be put into MPL without a sign for predication. '(∃x)(Fx & Cx)' employs non-separate signs for predication.
Q4. Is the predicative tie reducible or eliminable? No. For Frege, there is no need for a logical copula or connector to tie object a to concept F when a falls under F. The concept is "unsaturated" (ungesaettigt). Predicates and their referents (Bedeutungen) are inherently gappy or incomplete. So the predicate 'wise' would be depicted as follows: '___ wise.' What is thereby depicted is a sentential function or open sentence. A (closed) sentence results when a name is placed in the gap. The concept to which this predicate or sentential function refers is gappy in an analogous sense. Hence there is no need for for an 'is' of predication in the logical language or for an instantiation relation. Object falls under concept without the need of a tertium quid to connect them.
I would imagine that Spencer Case's professor has some such scheme in mind. One problem is that it is none too clear what could be meant by a gappy or incomplete or unsaturated entity. That a predicate should be gappy is tolerably clear, but how could the referent of a predicate be gappy given that the referent of a predicate is a single item and not the manifold of things to which the predicate applies? The idea is not that concepts exist only when instantiated, but that their instantiation does not require the services of a nexus of predication: the concept has as it were a slot in it that accepts the object without the need of a connector to hold them together. (Think of a plug and a socket: there is no need for a third thing to connect the plug to the socket: the 'female' receptacle just accepts the 'male' plug.)
There are other problems as well.
But here is the main point. Frege cannot avoid speaking of objects falling under concepts, of a's falling under F but not under G. If the notion of the unsaturatedness of concepts is defensible, then Frege can avoid speaking of a separate predicative tie that connects objects and concepts. But he cannot get on without predication and without a sign for predication.
I conclude that my original argument is sound. There is is and must be a sign for predication in any adequate logic, but it needn't be a stand-alone sign. (Nor need its referent be a stand-alone entity.) Compare '(∃x)Hx' to '(∃x)(x = h)' as translations of 'Horses exist' and 'Harry exists,' respectively. The identity sign occurs in only one of the translations, the second. And the sign for predication occurs only in the first. There is no univocity of 'exist(s)' because there is no univocity of 'is' in the translations.