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PLURAL ANCESTRAL LOGIC AS THE LOGIC OF ARITHMETIC

Part of: Philosophical aspects of logic and foundations General and miscellaneous specific topics General logic

Published online by Cambridge University Press:  07 February 2022

OLIVER TATTON-BROWN Open the ORCID record for OLIVER TATTON-BROWN [Opens in a new window]
OLIVER TATTON-BROWN*
Affiliation:
UNIVERSITY OF BRISTOL
*
E-mail: otattonbrownphil@gmail.com
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Abstract

Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic concepts, including finiteness, equinumerosity and addition and multiplication of cardinality—revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic.

Keywords

logicismFregeneo-Fregeanismneo-logicismarithmeticlogical knowledge

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical 03B22: Abstract deductive systems 03B65: Logic of natural languages
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 38
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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