TY - JOUR

T1 - PEIRCE'S CALCULI for CLASSICAL PROPOSITIONAL LOGIC

AU - Ma, Minghui

AU - Pietarinen, Ahti Veikko

N1 - Funding Information:
§6. Acknowledgments. The work of the first author is supported by the Project Supported by Guangdong Province Higher Vocational Colleges & Schools Pearl River Scholar Funded Scheme (2017–2019). The work of the second author is supported by the Academy of Finland (1270335, The Diagrammatic Mind: Logical and Communicative Aspects of Iconicity, 2013–2017), the Estonian Research Council (PUT 1305, Abduction in the Age of Fundamental Uncertainty, 2016–2018), Nazarbayev University Social Policy Grant 2018, and the Russian Academic Excellence Project “5–100”, 2018–2020, Principle Investigator A.-V. Pietarinen. This article was submitted in August 2016. We thank the reviewer who reviewed the article in March 2018.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - 03G10

KW - 06D30

KW - 2010 Mathematics Subject Classification

KW - Primary 03F03

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U2 - 10.1017/S1755020318000187

DO - 10.1017/S1755020318000187

M3 - Article

AN - SCOPUS:85051701593

VL - 13

SP - 1

EP - 32

JO - Review of Symbolic Logic

JF - Review of Symbolic Logic

SN - 1755-0203

IS - 3

ER -