A PROOF OF NOMINALISM: AN EXERCISE IN SUCCESSFUL REDUCTION IN LOGIC

Abstract

Nominalism can be construed as maintaining that the only quantifiers we need range over are particulars (individuals) in contradistinction to second-order (and other higherorder) entities. It is shown here how to reduce all secondorder quantification to the first-order level. This is done in three stages: (1) Independence-friendly first-order logic is extended by introducing that contradictory negation need not be sentence-initial. (2) The resulting logic is given a game-theoretical interpretation. The main idea is to isolate the game G(F*) needed in interpreting a sentence S where ¬F occurs as a subformula and where F* is a substitutioninstance of F from the rest of S. (3) The hierarchy of second-order sentences is reduced step by step in the same way sigma one-one fragment is reduced to firstorder IF logic. This reduction makes both axiomatic set theory and conventional higher-order logic dispensable in the foundations of mathematics.