Logical Studies

A Gentzent-type proof system for {LP}

Document Type : Research

Author

javid jafari

Department of Philosophy, Faculity of Humanity, Tarbiat Modares University, Tehran

https://doi.org/10.30465/lsj.2025.52639.1503
Abstract
The logic {LP} is a paraconsistent logic that bears strong structural and semantic similarities to the LP logic introduced by Graham Priest. It is defined using Nmatrices, a semantic tool that plays a significant role in the study of paraconsistent logics by allowing for the analysis of contradictory statements without collapsing the entire logical system. In this paper, we first present the semantic framework of this logic and then develop a proof theory for it based on Gentzen-type sequent calculus. We show that this proof system is both sound and complete with respect to the proposed matrix semantics. Another focus of this study is the analysis of certain distinctive features of the {LP} logic, where noticeable differences from the original LP logic emerge. In particular, we examine the non-standard behavior of the conjunction operator in this logic, which functions in such a radically different way from common logics that the term “conjunction” barely seems appropriate.

Keywords

Paraconsistent logics
non-deterministic matrix
sequent calculus
proof-theory
Gentzen-type systems
Subjects
·            Arieli, O., & Avron, A. (2015). Three-Valued Paraconsistent Propositional Logics. In J.-Y. Beziau, M. Chakraborty, & S. Dutta (Eds.), New Directions in Paraconsistent Logic (pp. 91–129). Springer India.
·            Avron, A. (2005). Non-deterministic Semantics for Logical Systems. Handbook of Philosophical Logic, 16(14), 227–304.
·            Avron, A., Arieli, O., & Zamansky, A. (2018). Theory of Effective Propositional Paraconsistent Logics. Studies in Logic, College Publications.
·            Carnap, R. (1943). Formalization of Logic. Cambridge, MA: Harvard University Press.
·            Jafari, J. (2025). Non-Classical Beasts Under the Carpet. Topoi, 44, 1–8.
·            Priest, G. (1979). Logic of Paradox. Journal of Philosophical Logic, 8(1), 219–241.
Logical Studies
Volume 16, Issue 1 - Serial Number 1
September 2025
Pages 151-172

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