Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T19:37:03.949Z Has data issue: false hasContentIssue false
  • English
  • Français

FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC

Published online by Cambridge University Press:  10 September 2019

MARIO PIAZZA  and
GABRIELE PULCINI
MARIO PIAZZA*
Affiliation:
Scuola Normale Superiore di Pisa
GABRIELE PULCINI*
Affiliation:
Departamento de Matemática, Universidade Nova de Lisboa
*
*SCUOLA NORMALE SUPERIORE CLASSE DI LETTERE E FILOSOFIA PISA, ITALY E-mail: mario.piazza@sns.it
†DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE NOVA DE LISBOA CAMPUS DE CAPARICA, PORTUGAL E-mail: g.pulcini@fct.unl.pt
Get access Rights & Permissions [Opens in a new window]

Abstract

This article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions.

Keywords

03B050374503B5003B42classical propositional logicproof theory and constructive mathematicsmany-valued logicslogics of knowledge and belief
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 810 - 828
Check for updates
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Avron, A. (1993). Gentzen-type systems, resolution and tableaux. Journal of Automated Reasoning, 10 (2), 265–281.CrossRefGoogle Scholar
Beall, J., Glanzberg, M., & Ripley, D. (2018). Formal Theories of Truth. Oxford: Oxford University Press.Google Scholar
Carnielli, W. A. & Pulcini, G. (2017). Cut-elimination and deductive polarization in complementary classical logic. Logic Journal of the IGPL, 25(3), 273–282.Google Scholar
Church, A. (1956). Introduction to Mathematical Logic. Princeton, NJ: Princeton University Press.Google Scholar
D’Agostino, M. (1999). Tableau methods for classical propositional logic. In D’Agostino, M., Gabbay, D. M., Hähnle, R., and Posegga, J., editors. Handbook of Tableau Methods. Dordrecht: Kluwer, pp. 45–123.CrossRefGoogle Scholar
Francez, N. (2015). Proof-Theoretic Semantics. London: College Publications.Google Scholar
Gentzen, G. (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 176–210.CrossRefGoogle Scholar
Girard, J.-Y., Taylor, P., & Lafont, Y. (1989). Proofs and Types, Vol. 7. Cambridge: Cambridge University Press.Google Scholar
Goranko, V. (1994). Refutation systems in modal logic. Studia Logica, 53(2), 299–324.CrossRefGoogle Scholar
Hughes, D. (2010). A minimal classical sequent calculus free of structural rules. Annals of Pure and Applied Logic, 161(10), 1244–1253.CrossRefGoogle Scholar
Kleene, S. C. (1967). Mathematical Logic. London: John Wiley & Sons.Google Scholar
Makinson, D. (2003). Bridges between classical and nonmonotonic logic. Logic Journal of the IGPL, 11(1), 69–96.CrossRefGoogle Scholar
Negri, S. & von Plato, J. (2001). Structural Proof Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Piazza, M. & Castellan, M. (1996). Quantales and structural rules. Journal of Logic and Computation, 6, 709–724.CrossRefGoogle Scholar
Piazza, M. & Pulcini, G. (2016). Uniqueness of axiomatic extensions of cut-free classical propositional logic. Logic Journal of the IGPL, 24(5), 708–718.CrossRefGoogle Scholar
Piecha, T. & Schroeder-Heister, P. (2016). Advances in Proof- Theoretic Semantics. Cham: Springer.CrossRefGoogle Scholar
Pulcini, G. & Varzi, A. C. (2019). Proof-Nets for Non-Theorems. Unpublished.Google Scholar
Skura, T. (2013). Refutability and Post-completeness. Available at: http://www.uni-log.org/contest2013/skura1.pdf.Google Scholar
Tiomkin, M. (1988). Proving unprovability. Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS ’88), Edinburgh, Scotland, UK, July 5–8, 1988. Los Alamitos, CA: IEEE Computer Society Press, pp. 22–26.Google Scholar
Troelstra, A. S. & Schwichtenberg, H. (2000). Basic Proof Theory, Vol. 43. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Varzi, A. C. (1990). Complementary sentential logics. Bulletin of the Section of Logic, 19(4), 112–116.Google Scholar
Wójcicki, R. (1984). Lectures on Propositional Calculi. Wroclaw: Ossolineum.Google Scholar