Quantifier-free versions of first order logic and their psychological significance (Q1187977)

This paper has a philosophical and psychological motivation. It puts forward the idea that mathematicians think by means of general objects which they denote by variables and terms involving variables, and by means of incompletely (or completely) defined objects, such as a well- ordering of the real line, a non-measurable set, etc. (the empty set, the real line, etc.). Using this idea it is shown that quantifiers in formulas can be viewed as abbreviations for expressing Skolemizations of these formulas, and the rules of proof involving quantifiers can be viewed as rules pertaining to such Skolemizations. The only new axioms needed in such a quantifier-free formalism are Skolemizations of formulas of the form \(\varphi\to\varphi\). A finitistic view of mathematics (including set theory) is founded upon this formalism. This agrees to some extent with the view of H. Poincaré, but not with his critique of set theory. A critique of the intuitionistic critique of classical mathematics is formulated.