Hilary Putnam is among the many helpful philosophers who have been informing my thought as of late. His wonderfully and compactly written Philosophy of Logic has been very helpful. Putnam begins his small book by stating some of the generally accepted principles of logic. He includes the principle of validity in the following familiar inference:
All S are M
All M are P
(Therefore) All S are P
He also includes the usual round of suspects: the laws of identity, contradiction, and excluded middle. Putnam notes that each of these general principles have been disputed by more than one contemporary philosopher. Note that this book was written in 1971; but I would imagine that this statement still holds true.
It is obvious from the first pages that Putnam wishes to show that the nominalistic view of the general rules of logic is much mistaken. In reference to the principle of valid inference he states that most logicians are likely to say:
(A) "For all classes S, M, P: if all S are M and all M are P, then all S are P."
A nominalist though is more likely to write:
(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'."
For the nominalist, the idea of "sentences" and "words" are much more "concrete" than the entities of "classes". So, based upon (B) we can see that a schema (or wff) is valid for the nominalist "just in case all substitution-instances of S in some particular formalized language L are true." Putnam believes that the logician wants to say more than this though. Validity should "mean that it is correct no matter what class-names may be substituted."
Putnam's desire to adopt (A) becomes more attractive when we see that the nominalist suggestion for validity (B) turns out to have a problem which mushrooms quickly. For if we adopt (B) we will not have one notion of validity but an infinite series of notions: validity in L1, validity in L2, validity in L3, etc. Since validity only relates to a particular formalized language L for the nominalist we must have different notions of validity for every type of language L used in logic. A most unwanted consequence.
We could avoid this maybe by stating "a schema is valid just in case all of its substitution-instances in every language L are true." But, as Putnam notes, it is highly questionable whether the notion of all possible formalized languages is any more "concrete" than the notion of "classes".
Is anybody else familiar with the nominalism-realism debate surrounding logical or mathematical entities? I would appreciate any helpful direction.