Naturalizing Natural Deduction

Abstract

A simplified and improved system of natural deduction for classical predicate logic is presented. The inference rules of existential instantiation EI, existential elimination (\(\exists\) E), and universal generalization UG (\(\forall\) I) are not employed in this system. Instead, a new proof-theoretic role for the quantifiers is introduced, which we call commonizing quantifiers. We argue that the commonizing quantifiers exemplify one of Frege’s basic insights by making explicit, in the context of natural deduction, the fundamental difference between the semantics of proper names and expressions of single or multiple generality. Arbitrary-object semantics, which is the best-developed philosophical justification of \(\forall\) I and \(\exists\) E and offers an explanation of the cumbersome restrictions placed on these inference rules, is discussed and criticized. It is argued that by looking at natural deduction in an intuitionistic rather than classical setting, it can be seen that the notion of an arbitrary object is not merely a harmless manner of speaking or notational variant, but a philosophical commitment that has consequences that are disguised if the discussion is framed only in terms of classical logic. In the system on offer, the notion of an arbitrary object is entirely dispensed with; in addition to considerations of simplicity, elegance, and parsimony, philosophical considerations decidedly favour the naturalized natural deduction system presented here.