Labeled Sequent Calculus for Orthologic

Tomoaki Kawano

Bulletin of the Section of Logic (2018)

  • Volume: 47, Issue: 4
  • ISSN: 0138-0680

Abstract

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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.

How to cite

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Tomoaki Kawano. "Labeled Sequent Calculus for Orthologic." Bulletin of the Section of Logic 47.4 (2018): null. <http://eudml.org/doc/295523>.

@article{TomoakiKawano2018,
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.},
author = {Tomoaki Kawano},
journal = {Bulletin of the Section of Logic},
keywords = {quantum logic; sequent calculus; cut-elimination theorem; decidability; Kripke model},
language = {eng},
number = {4},
pages = {null},
title = {Labeled Sequent Calculus for Orthologic},
url = {http://eudml.org/doc/295523},
volume = {47},
year = {2018},
}

TY - JOUR
AU - Tomoaki Kawano
TI - Labeled Sequent Calculus for Orthologic
JO - Bulletin of the Section of Logic
PY - 2018
VL - 47
IS - 4
SP - null
AB - 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.
LA - eng
KW - quantum logic; sequent calculus; cut-elimination theorem; decidability; Kripke model
UR - http://eudml.org/doc/295523
ER -

References

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  6. S. Negri, Proof theory for modal logic, Philosophy Compass 6(8) (2011), pp. 523–538. 
  7. H. Nishimura, Sequential Method in Quantum Logic, The Journal of Symbolic Logic 45(2) (1980), pp. 339–352. 
  8. H. Nishimura, Proof Theory for Minimal Quantum Logic I, International Journal of Theoretical Physics 33(1) (1994), pp. 103–113. 
  9. H. Nishimura, Proof Theory for Minimal Quantum Logic II, International Journal of Theoretical Physics 33(7) (1994), pp. 1427–1443. 

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