Skip to main content
×
×
Home

PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC

  • MINGHUI MA (a1) and AHTI-VEIKKO PIETARINEN (a2)
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.

Copyright
Corresponding author
*DEPARTMENT OF PHILOSOPHY INSTITUTE FOR LOGIC AND COGNITION SUN YAT-SEN UNIVERSITY, GUANGZHOU XINGANG XI ROAD 135, HAIZHU DISTRICT GUANGZHOU 510275, CHINA E-mail: mamh6@mail.sysu.edu.cn
TALLINN UNIVERSITY OF TECHNOLOGY, TALLINN NAZARBAYEV UNIVERSITY, ASTANA NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS MOSCOW, RUSSIA E-mail: ahti-veikko.pietarinen@ttu.ee
References
Hide All
Anellis, I. (2012). Peirce’s truth-functional analysis and the origin of the truth table. History and Philosophy of Logic, 33, 3741.
Badesa, C. (2004). The Birth of Model Theory: Löwenheim’s Theorem in the Frame of the Theory of Relatives. Princeton: Princeton University Press.
Bellucci, F. & Pietarinen, A.-V. (2016). Existential graphs as an instrument for logical analysis. Part 1: Alpha. The Review of Symbolic Logic, 9 (2), 209237. DOI 10.1017/S1755020315000362.
Bellucci, F. & Pietarinen, A.-V. (2017). From mitchell to carus: 14 years of logical graphs in the making. Transactions of the Charles S. Peirce Society, 52(4), 539575. DOI: 10.2979/trancharpeirsoc.52.4.02.
Boole, G. (1847). The Mathematical Analysis of Logic. Cambridge: Macmillan, Barclay & Macmillan.
Boole, G. (1854). An Investigation of the Laws of Thought. Cambridge: Walton & Maberly.
Brady, G. (2000). From Peirce to Skolem: A Neglected Chapter in the History of Logic. Amsterdam: Elsevier.
Brünnler, K. (2003). Deep Inference and Symmetry in Classical Proof. Ph.D. Thesis, Technische Universität Dresden.
Dipert, R. (2004). Peirce’s deductive logic: Its development, influence, and philosophical significance. In Misak, C., editor. The Cambridge Companion to Peirce. Cambridge, Mass: Cambridge University Press, pp. 257286.
Houser, N. (1985). Peirce’s Algebra of Logic and the Law of Distribution, Dissertation. University of Waterloo, Ontario.
Houser, N. (1991). Peirce and the law of distribution. In Drucker, T., editor. Perspectives on the History of Mathematical Logic. Boston: Birkhäuser, pp. 1032.
Houser, N., Roberts, D., & Van Evra, J. (editors) (1997). Studies in the Logic of Charles S. Peirce. Bloomington: Indiana University Press.
Huntington, E. V. (1904). Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society, 5, 288309.
Keynes, J. N. (1887). Studies and Exercises in Formal Logic. London: Macmillan.
Ma, M. & Pietarinen, A.-V. (2017a). Graphical sequent calculi for modal logics. Electronic Proceedings in Theoretical Computer Science, 243, 91103. 10.4204/EPTCS.243.7.
Ma, M. & Pietarinen, A.-V. (2017b). Gamma graph calculi for modal logics. Synthese, 195, 3621. https://doi.org/10.1007/s11229-017-1390-3.
Ma, M. & Pietarinen, A.-V. (2017c). Peirce’s sequent proofs of distributivity. In Ghosh, S., and Prasad, S., editors. Logic and Its Applications: 7th Indian Conference. Lecture Notes in Computer Science, Vol. 10119. Springer, pp. 168182.
Martin, R. M. (1980). Peirce’s Logic of Relations and Other Studies. Dordrecht: Foris.
Mitchell, O. H. (1883). On a new algebra of logic. In Peirce, C. S., editor. Studies in Logic, by Members of Johns Hopkins University, Boston: Little, Brown & Company, pp. 72106.
De Morgan, A. (1847). Formal Logic. London: Taylor and Walton.
Peirce, C. S. (1867). On an improvement in Boole’s calculus of logic. Proceedings of the American Academy of Arts and Sciences 7, 250261.
Peirce, C. S. (1880). On the algebra of logic. American Journal of Mathematics, 3(1), 1557. (Reprinted in Kloesel, C. J. W., editor. Writings of C. S. Peirce: A Chronological Edition, Vol. 4. Bloomington, IN: Indiana University Press, pp. 163–209.)
Peirce, C. S. (editor) (1883). Studies in Logic by Members of the Johns Hopkins University. Boston: Little, Brown, and Co.
Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notation. American Journal of Mathematics, 7(2), pp. 180196.
Peirce, C. S. (1891a). Algebra of the Copula [Version 1]. In Houser, N., and De Tienne, A., editors. Writings of Charles S. Peirce, Vol. 8 (1890–1892). Bloomington, IN: Indiana University Press, pp. 210211.
Peirce, C. S. (1891b). Algebra of the Copula [Version 2]. In Houser, N., and De Tienne, A., editors. Writings of Charles S. Peirce, Vol. 8 (1890–1892). Bloomington, IN: Indiana University Press, pp. 212216.
Peirce, C. S. (1893a). Grand Logic. Division I. Stecheology. Part I. Non Relative. Chapter VIII. The Algebra of the Copula. (R 411)
Peirce, C. S. (1893b). Grand Logic. Chapter XI. The Boolian Calculus. (R 417)
Peirce, C. S. (1893c). Grand Logic. Book II. Division I. Part 2. Logic of Relatives. Chapter XII. The Algebra of Relatives. (R 418)
Peirce, C. S. (1894). Letter to F. Russell. (R L 387)
Peirce, C. S. (1896–1897). On Logical Graphs. (R 482)
Peirce, C. S. (1897). Memoir #1 Algebra of Copula. (R 737)
Peirce, C. S. (1897–1898). On Existential Graphs (EG). (R 485)
Peirce, C. S. (1900). Letter to Christine Ladd-Franklin, November 9, 1900. (R L 237)
Peirce, C. S. (1902). Minute Logic. Chapter III. The Simplest Mathematics (Logic III). (R 430)
Peirce, C. S. (c.1902). On the Basic Rules of Logical Transformation (R 516)
Peirce, C. S. (1903a). Logical Tracts. No. 1. On Existential Graphs. (R 491)
Peirce, C. S. (1903b). Logical Tracts. No. 2. On Existential Graphs, Euler’s Diagrams, and Logical Algebra. (R 491)
Peirce, C. S. (1903c). Logical Tracts. No. 2. The Rules of Existential Graphs. (R 1589)
Peirce, C. S. (1903d). Lowell Lectures. Lecture I. (R 450, S-27)
Peirce, C. S. (1903e). Lowell Lectures. Lecture I. Graphs: A Little Account. (S-27)
Peirce, C. S. (1903f). Lowell Lectures. Lecture I. The Conventions. (S-28)
Peirce, C. S. (1903g). Lowell Lectures. Lecture IV. (R 467)
Peirce, C. S. (1904a). A Proposed Logical Notation (Notation). (R 530)
Peirce, C. S. (1904b). Letter to E. V. Huntington, February 14, 1904. (R L 210)
Peirce, C. S. (1905). A Logical Analysis of Some Demonstrations in High Arithmetic (D). (R 253)
Peirce, C. S. (1908). One, Two, Three. (R 905)
Peirce, C. S. (1910). Diversions of Definitions, July 20–23, 1910. (R 650)
Peirce, C. S. (1931–1966). The Collected Papers of Charles S. Peirce, Vol. 8. Hartshorne, C., Weiss, P., and Burk, A. W., editors. Cambridge: Harvard University Press. Cited as CP followed by volume and paragraph number.
Peirce, C. S. (1967). Manuscripts in the Houghton Library of Harvard University, as identified by Richard Robin, Annotated Catalogue of the Papers of Charles S. Peirce, Amherst: University of Massachusetts Press, 1967, and in The Peirce Papers: A supplementary catalogue, Transactions of the C. S. Peirce Society, 7, 3757. Cited as R followed by manuscript number and, when available, page number.
Peirce, C. S. (1976). The New Elements of Mathematics by Charles S. Peirce, Vol. 4. Eisele, C., editor. The Hague: Mouton. Cited as NEM followed by volume and page number.
Peirce, C. S. (1982). Writings of Charles S. Peirce: A Chronological Edition, Vols. 1-8. Kloesel, C. W., editor. Bloomington: Indiana University Press. Cited as W followed by volume number and, when available, page number.
Pietarinen, A.-V. (2006). Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication (Synthese Library 329). Dordrecht: Springer.
Pietarinen, A.-V. (2015). Exploring the beta quadrant. Synthese, 192, 941970. 10.1007/s11229-015-0677-5.
Pietarinen, A.-V. & Bellucci, F. (2014). New light on Peirce’s conceptions of retroduction, deduction and scientific reasoning. International Studies in the Philosophy of Science, 28(4), 353373. 10.1080/02698595.2014.979667.
Prior, A. N. (1958). Peirce’s axioms for propositional calculus. The Journal of Symbolic Logic, 23, 135136.
Prior, A. N. (1964). The algebra of the copula. In Moore, E. and Robin, R., editors. Studies in the Philosophy of Charles Sanders Peirce. Amherst: The University of Massachusetts Press, pp. 7994.
Roberts, D. D. (1964). The existential graphs and natural deduction. In Moore, E. and Robin, R., editors. Studies in the Philosophy of Charles Sanders Peirce. Amherst: The University of Massachusetts Press, pp. 109121.
Roberts, D. D. (1973). The Existential Graphs of Charles S. Peirce. The Hague: Mouton.
Russell, B. (1901). Sur la logique des relations avec des applications á la théorie des séries. Revue de mathématiques/Rivista di Matematiche, 7, 115148.
Schröder, E. (1890). Vorlesungen über die Algebra der Logik, Vol. 1. Leipzig: Teubner.
Sowa, J. (2006). Peirce’s contributions to the 21st century. In Schärfe, H., Hitzler, P., and Ohrstrom, P., editors. Proceedings of the 14th International Conference on Conceptual Structures. Lecture Notes in Computer Science, Vol. 4068. Berlin, Heidelberg: Springer-Verlag, pp. 5469.
Turquette, A. (1964). Peirce’s icons for deductive logic. In Moore, E. and Robin, R., editors. Studies in the Philosophy of Charles Sanders Peirce. Amherst: The University of Massachusetts Press, pp. 95108.
Valencia, V. S. (1989). Peirce’s Propositional Logic: From Algebra to Graphs. ILLI prepublication series for logic, semantics and philosophy of language LP-89-08. Amsterdam: University of Amsterdam.
Zeman, J. (1964). The Graphical Logic of Charles S. Peirce. Ph.D. dissertation, University of Chicago.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed