A Logical Foundation of Arithmetic

Studia Logica 103 (1):113-144 (2015)
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Abstract

The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework that takes seriously ordinary locutions like ‘at least n Fs’, ‘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 Hume’s Principle, can be derived solely from definitions

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Joongol Kim
Sogang University

Citations of this work

What Are Quantities?Joongol Kim - 2016 - Australasian Journal of Philosophy 94 (4):792-807.
The concept horse is a concept.Ansten Klev - 2018 - Review of Symbolic Logic 11 (3):547-572.
The sortal resemblance problem.Joongol Kim - 2014 - Canadian Journal of Philosophy 44 (3-4):407-424.
The Potential in Frege’s Theorem.Will Stafford - 2023 - Review of Symbolic Logic 16 (2):553-577.

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References found in this work

Logicism and the ontological commitments of arithmetic.Harold T. Hodes - 1984 - Journal of Philosophy 81 (3):123-149.
Frege's unofficial arithmetic.Agustín Rayo - 2002 - Journal of Symbolic Logic 67 (4):1623-1638.
A Strengthening of the Caesar Problem.Joongol Kim - 2011 - Erkenntnis 75 (1):123-136.

View all 7 references / Add more references