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Early Analytic Philosophy - New Perspectives on the TraditionPropositional Logic from The Principles of Mathematics to Principia Mathematica

Early Analytic Philosophy - New Perspectives on the Tradition: Propositional Logic from The... [Bertrand Russell presented three systems of propositional logic, one first in Principles of Mathematics, University Press, Cambridge, 1903 then in “The Theory of Implication”, Routledge, New York, London, pp. 14–61, 1906) and culminating with Principia Mathematica, Cambridge University Press, Cambridge, 1910. They are each based on different primitive connectives and axioms. This paper follows “Peirce’s Law” through those systems with the aim of understanding some of the notorious peculiarities of the 1910 system and so revealing some of the early history of classical propositional logic. “Peirce’s Law” is a valid formula of elementary propositional logic: [(p ⊃ q) ⊃ p] ⊃ p This sentence is not even a theorem in the 1910 system although it is one of the axioms in 1903 and is proved as a theorem in 1906. Although it is not proved in 1910, the two lemmas from the proof in 1906 occur as theorems, and Peirce’s Law could have been derived from them in a two step proof. The history of Peirce’s Law in Russell’s systems helps to reconstruct some of the history of axiomatic systems of classical propositional logic.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Early Analytic Philosophy - New Perspectives on the TraditionPropositional Logic from The Principles of Mathematics to Principia Mathematica

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References (13)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2016
ISBN
978-3-319-24212-5
Pages
213 –229
DOI
10.1007/978-3-319-24214-9_8
Publisher site
See Chapter on Publisher Site

Abstract

[Bertrand Russell presented three systems of propositional logic, one first in Principles of Mathematics, University Press, Cambridge, 1903 then in “The Theory of Implication”, Routledge, New York, London, pp. 14–61, 1906) and culminating with Principia Mathematica, Cambridge University Press, Cambridge, 1910. They are each based on different primitive connectives and axioms. This paper follows “Peirce’s Law” through those systems with the aim of understanding some of the notorious peculiarities of the 1910 system and so revealing some of the early history of classical propositional logic. “Peirce’s Law” is a valid formula of elementary propositional logic: [(p ⊃ q) ⊃ p] ⊃ p This sentence is not even a theorem in the 1910 system although it is one of the axioms in 1903 and is proved as a theorem in 1906. Although it is not proved in 1910, the two lemmas from the proof in 1906 occur as theorems, and Peirce’s Law could have been derived from them in a two step proof. The history of Peirce’s Law in Russell’s systems helps to reconstruct some of the history of axiomatic systems of classical propositional logic.]

Published: Jan 22, 2016

Keywords: Propositional Logic; Quantificational Logic; Propositional Variable; Double Negation; True Proposition

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