Abstract
Charles Peirce’s alpha system \(\mathfrak {S}_\alpha \) is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of \(\mathfrak {S}_\alpha \) is given in terms of two embedding theorems: the system \(\mathfrak {S}_\alpha \) and Brünnler’s deep inference system for classical propositional logic can be embedded into each other; and the system \(\mathfrak {S}_\alpha \) and Gentzen sequent calculus \(\mathbf {G3c}^*\) can be embedded into each other.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellucci, F., and A.-V. Pietarinen, Existential graphs as an instrument of logical analysis: part I. Alpha, The Review of Symbolic Logic 9:209–237, 2016.
Brady, G., and T. H. Trimble, A categorical interpretation of C. S. Peirce’s propositional logic Alpha, Journal of Pure and Applied Algebra 149:213–239, 2000.
Brünnler, K., Deep Inference and Symmetry in Classical Proof, Ph.D. thesis, Technische Universität Dresden, 2003.
Bruscoli, P., and A. Guglielmi, On the proof complexity of deep inference, ACM Transactions on Computational Logic 10(2):1–34, 2009.
Caterina, G., and R. Gangle, Iconicity and abduction: a categorical approach to creative hypothesis-formation in Peirce’s existential graphs, Logic Journal of IGPL 21:1028–1043, 2013.
Dau, F., Some notes on proofs with alpha graphs, in H. Schärfe, P. Hitzler, and P. Øhrstrøm (eds.), Conceptual Structures: Inspiration and Application, vol. 3874 of LNCS, Springer, Berlin, 2006, pp. 172–188.
Hammer, E., Peircean graphs for propositional logic, in G. Allwein and J. Barwise, (eds.), Logical Reasoning with Diagrams, Oxford University Press, Oxford, 1996, pp. 129–147.
Ma, M., and A.-V. Pietarinen, Peirce’s sequent proofs of distributivity, Lecture Notes in Computer Science 10119, 2017. doi:10.1007/978-3-662-54069-513.
Ma, M., and A.-V. Pietarinen, Graphical sequent calculi for modal logics, Electronic Proceedings in Theoretical Computer Science, 2017.
Negri, S., and J. von Plato, Structural Proof Theory, Cambridge University Press, Cambridge, 2001.
Norman, J., Provability in Peirce’s alpha graphs, Transactions of the Charles S. Peirce Society, 39:23–41, 2003.
Peirce, C. S., Manuscripts in the Houghton Library of Harvard University. Identified by Richard Robin, Annotated Catalogue of the Papers of Charles S. Peirce, Amherst: University of Massachusetts Press, 1967, and The Peirce Papers: A supplementary catalogue, Transactions of the C. S. Peirce Society 7:37–57, 1971. Cited as R followed by manuscript number and, when available, page number, 1967.
Peirce, C. S., and C. Ladd-Franklin, Symbolic logic or algebra of logic, in J. M. Baldwin, (ed.), Dictionary of Philosophy and Psychology, Vol. 2, Macmillan, New York, 1901.
Pietarinen, A.-V., Peirce’s diagrammatic logic in IF perspective, Lecture Notes in Artificial Intelligence 2980:97–111, 2004.
Pietarinen, A.-V., Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication, Springer, Dordrecht, 2006.
Roberts, D. D., The Existential Graphs of Charles S. Peirce, Mouton, The Hague, 1973.
Shin, S.-J., The Iconic Logic of Peirce’s Graphs, The MIT Press, Cambridge, MA, 2002.
Sowa, J., Knowledge Representation: Logical, Philosophical and Computational Foundations, Brooks/Cole, Pacific Grove, CA, 2000.
Zalamea, F., Peirce’s Logic of Continuity, Docent Press, Boston, MA, 2003.
Zeman, J., The Graphical Logic of Charles S. Peirce, Ph.D. thesis, University of Chicago, 1964.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ma, M., Pietarinen, AV. Proof Analysis of Peirce’s Alpha System of Graphs. Stud Logica 105, 625–647 (2017). https://doi.org/10.1007/s11225-016-9703-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-016-9703-y