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PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC

Published online by Cambridge University Press:  29 October 2018

MINGHUI MA  and
AHTI-VEIKKO PIETARINEN
MINGHUI MA*
Affiliation:
Department of Philosophy, Sun Yat-Sen University
AHTI-VEIKKO PIETARINEN*
Affiliation:
Tallinn University of Technology; Nazarbayev University; and Higher School of Economics, National Research University
*
*DEPARTMENT OF PHILOSOPHY INSTITUTE FOR LOGIC AND COGNITION SUN YAT-SEN UNIVERSITY, GUANGZHOU XINGANG XI ROAD 135, HAIZHU DISTRICT GUANGZHOU 510275, CHINA E-mail: [email protected]
†TALLINN UNIVERSITY OF TECHNOLOGY, TALLINN NAZARBAYEV UNIVERSITY, ASTANA NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS MOSCOW, RUSSIA E-mail: [email protected]
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Abstract

This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted by PC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to present PC as a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, in PC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection between PC and the alpha system.

Keywords

Primary 03F0306D3003G10Peircealgebra of LogicBoolean algebrasequent calculusillationconsequencealpha graphs
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 3 , September 2020 , pp. 509 - 540
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Copyright
Copyright © Association for Symbolic Logic 2018 

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