A Modified Subformula Property for the Modal Logic S4.2
Bulletin of the Section of Logic (2019)
- Volume: 48, Issue: 1
- ISSN: 0138-0680
Access Full Article
top
Abstract
topHow to cite
topMitio Takano. "A Modified Subformula Property for the Modal Logic S4.2." Bulletin of the Section of Logic 48.1 (2019): null. <http://eudml.org/doc/295583>.
@article{MitioTakano2019,
abstract = {The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the K5-subformulas of some formula in S which were introduced by us to show the modied subformula property for the modal logics K5 and K5D (Bull Sect Logic 30(2): 115–122, 2001). Some corollaries including the interpolation property for S4.2 follow from this. By slightly modifying the proof, the finite model property also follows.},
author = {Mitio Takano},
journal = {Bulletin of the Section of Logic},
keywords = {modal logic S4.2; sequent calculus; subformula property},
language = {eng},
number = {1},
pages = {null},
title = {A Modified Subformula Property for the Modal Logic S4.2},
url = {http://eudml.org/doc/295583},
volume = {48},
year = {2019},
}
TY - JOUR
AU - Mitio Takano
TI - A Modified Subformula Property for the Modal Logic S4.2
JO - Bulletin of the Section of Logic
PY - 2019
VL - 48
IS - 1
SP - null
AB - The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the K5-subformulas of some formula in S which were introduced by us to show the modied subformula property for the modal logics K5 and K5D (Bull Sect Logic 30(2): 115–122, 2001). Some corollaries including the interpolation property for S4.2 follow from this. By slightly modifying the proof, the finite model property also follows.
LA - eng
KW - modal logic S4.2; sequent calculus; subformula property
UR - http://eudml.org/doc/295583
ER -
References
top- M. Fitting, Subformula results in some propositional modal logics, Studia Logica 37 (1978), pp. 387–391.
- G. E. Hughes and M. J. Cresswell, A New Introduction to Modal Logic, Routledge, London and New York (1996).
- M. Takano, A modified subformula property for the modal logics K5 and K5D, Bulletin of the Section of Logic 30 (2001), pp. 115–122.
- M. Takano, A semantical analysis of cut-free calculi for modal logics, Reports on Mathematical Logic 53 (2018), pp. 43–65.
- G. Takeuti, Proof Theory, Second Edition (Studies in Logic and the Foundations of Mathematics 81), North-Holland, Amsterdam (1987).
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.