Labeled Sequent Calculus for Orthologic

Tomoaki Kawano

doi:10.18778/0138-0680.47.4.01

Authors

Tomoaki Kawano Tokyo Institute of Technology, School of Computing, Department of Mathematical and Computing Science

DOI:

https://doi.org/10.18778/0138-0680.47.4.01

Keywords:

quantum logic, sequent calculus, cut-elimination theorem, decidability, Kripke model

Abstract

Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL.

References

M. L. D. Chiara and R. Giuntini, Quantum Logics, Handbook of Philosophical Logic 2nd Edition 6 (2001), pp. 129–228.
Google Scholar

C. Faggian and G. Sambin, From Basic Logic to Quantum Logics with Cut-Elimination, International Journal of Theoretical Physics 37(1) (1998), pp. 31–37.
Google Scholar

G. M. Hardegree, Material Implication in Orthomodular (and Boolean) Lattices, Notre Dame Journal of Formal Logic 22(2) (1981), pp. 163–182.
Google Scholar

Z. Hou, A. Tiu and R. Gore, A Labelled Sequent Calculus for BBI: Proof Theory and Proof Search, TABLEAUX 2013 (2013), pp. 172–187.
Google Scholar

S. Negri, Proof Analysis in Modal Logic, Journal of Philosophical Logic 34 (2005), pp. 507–544.
Google Scholar

S. Negri, Proof theory for modal logic, Philosophy Compass 6(8) (2011), pp. 523–538.
Google Scholar

H. Nishimura, Sequential Method in Quantum Logic, The Journal of Symbolic Logic 45(2) (1980), pp. 339–352.
Google Scholar

H. Nishimura, https://doi.org/10.1007/BF00671616">Proof Theory for Minimal Quantum Logic I, International Journal of Theoretical Physics 33(1) (1994), pp. 103–113.
Google Scholar

H. Nishimura, Proof Theory for Minimal Quantum Logic II, International Journal of Theoretical Physics 33(7) (1994), pp. 1427–1443.
Google Scholar