A Logical Foundation of Arithmetic

The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework (ALA) that takes seriously ordinary locutions like ‘at least nFs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of ALA, and the Dedekind–Peano axioms can be derived from those definitions by logical means alone. It will also be shown that some fundamental facts about cardinal numbers expressed using singular terms of the form ‘the number of Fs’, including (a variant of) Hume’s Principle, can be derived solely from definitions.