## Saturday, July 23, 2011

### Do we need identity?

There is the usual vigorous debate going on at Maverick's place, this time about whether proper names can be predicated.  I have been expecting at any minute the objection that we must distinguish between the 'is' of predication and the 'is' of identity, but no sign of it yet.

Let me explain.  In natural language we say things like 'this person is Socrates' or 'that star is the planet Venus'.  We are putting a proper name ('Socrates', 'Venus') in a part of the sentence logic normally reserves for the 'predicate'.  But in modern predicate calculus (MPC) proper names cannot occur as predicates.  We owe it to Frege (as Geach says) that modern logicians accept an absolute category difference between name and predicate, so that in MPC the two types are syntactically different: small letters for proper names, and propositional functions for the predicates.  Thus 'F(a)' represents 'Socrates is running', where 'a' represents Socrates, and F( ) the function '-- is running'.

But what about 'this person is Socrates'?  Ah, that is because we must distinguish the 'is' of identity from the 'is' of predication.  We are not predicating 'Socrates' of this person, but rather the propositional function '-- is identical with Socrates'.  Thus, as Frege says (in "On Concept and Object"), a proper name like 'Venus' can never be a predicate, although it can form part of a predicate.

That is all pretty standard stuff, but is it the knock-down argument against scholastic logic that it appears to be?  I just looked again at Fred Sommers' excellent book* defending Traditional Formal Logic.  In the chapter 'Do we need identity' he asks why no logicians before Frege had appealed to the distinction between the two kinds of 'is', and argues that the distinction depends on making the category distinction between concept-word (predicate) and object-word (proper name) rather than the other way round.  Thus, it is only after we take on the odd representation of ordinary language sentences as propositional function and argument, that we are forced to make the distinction between the two forms of 'is'.  It is not that we first recognise the distinction as a fundamental principle that forces the odd syntax upon us.

Sommers argues as follows (p. 121 - I have modified his argument slightly in order to strengthen it).  When we represent ordinary proper names (say 'Venus' and 'the morning star') as logical constants, we use lower cases letters, say 'a' and 'b'.  But then the representation of 'Venus is the morning star' as 'b(a)' is ill-formed.  The lower case letter 'b' cannot appear in predicate place.  It is therefore obvious, says Sommers ironically, that it really has the form 'F(a,b), where 'F' is the grammatical predicate which represents '-- is identical with --'.  "Clearly, it is only after one has adopted the syntax that prohibits the predication of proper names that one is forced to read 'a is b' dyadically and to see in it a sign of identity".

* Fred Sommers, The Logic of Natural Language, Clarendon 1982.