Mathematical Facts in Light of Nonfregean Logic

Abstract

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) constructed on the grounds of second order arithmetical theories founded upon set theory. Slingshot arguments directed against facts, situations and other propositional entities are usually blocked by rejecting one of the main inference rules used in various versions of this argument. Authors distinguish two types of main inference rules used in these variants: (i) for non-propositional expressions: iota-conversion rules (ι-conv), iota-substitution rules (ι-subs), lambda-conversion principle (λ-conv); and (ii) for propositional expressions: the principle of substitutivity for logical equivalents (PSLE). Even if strategies for defending facts by the rejection of one of the mentioned rules are accepted, it may be shown that the acceptance of the requirement that mathematical facts (situations or truth-makers) differing with regard to their constituents are different facts, leads to the contradiction in meta-theories of set theoretic models for first order arithmetical theories. In the paper, a new type of slingshot argument is presented, which may be called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the former are formulated in the semantic meta-language of mathematical theories without the use of the iota-operator or the name-forming lambda-operator, whereas the latter require for their expression at least one of these non-standard term-operators. Furthermore, in hyper-slingshots PSLE is not used, whereas in conventional slingshots, PSLE plays a crucial inferential role. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots.