Nominalistic Truth in Logic

Hilary Putnam (1971) says a nominalist logician takes the following definition for validity:

(B) "The following turns into a true sentence no matter what words or phrases of the appropriate kind one may substitute for the letters S, M, P: 'if all S are M and all M are P, then all S are P'."

As I blogged earlier, Putnam pointed out a problem with this definition as it related to
any formal language of logic. But this is not the only problem for a nominalist logician.
In addition, Putnam points out some problems with truth in nominalistic validity.

Some conception of truth must be included within any definition of validity. But for the
nominalist logic only refers to symbols within a language. So "true" and "false" must
only relate to the ink on the paper or darkened bits on a computer screen. But this makes
no sense. How can we say that these things are "true" or "false"? They just are. Putnam
reminds us that truth rather relates to what the strings of letters express. But meaning is
just the sort of thing the nominalist wants to get rid of. Putnam wants to argue then that
the conception of truth is unavailable to the nominalist.

The nominalist could try to distinguish the conception to make it more his fit in the following way:

(1) S is true

and

(2) S is true as understood by John at time t

If S is a physical object (1) makes no sense; but (2) can represent a possible relationship which may obtain. Thus a nominalist does have an answer for Putnam.

Also the nominalist can appeal to "ordinary language" as in the following statement:

(3) Jack made a true statement.

But Putnam points out that this could imply one of two things: (a) statements (non-physical entities) exist or (b) statements don't exist. If (b) is true, then nominalism is "futile" since this contradicts one of its main doctrines. If (a) is true then nominalism is false for it contradicts the very words written in the statement.

Putnam does not claim that these arguments against the nominalist are conclusive. But the nominalist is not entitled to the conception of truth without a suitable nominalist explanation. So Putnam concludes that the nominalist conception of truth in validity is unsatisfactory, "at least today."