Cut-elimination for Weak Grzegorczyk Logic Go
- 160 Downloads
We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here.
KeywordsProof theory Syntactic cut-elimination Go Weak Grzegorczyk logic
Unable to display preview. Download preview PDF.
- 6.Gabelaia D. Topological, Algebraic and Spatio-Temporal Semantics for Multi-Dimensional Modal Logics, Ph.D. thesis, Department of Computer Science, King College London,2005Google Scholar
- 7.Gentzen G. The collected papers of Gerhard Gentzen, in M. E. Szabo (ed.), Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1969.Google Scholar
- 8.Goré, R., Tableau methods for modal and temporal logics, in Handbook of Tableau Methods, Kluwer, Dordrecht, 1999, pp. 297–396.Google Scholar
- 9.Goré, R., and R. Ramanayake, Valentini’s cut-elimination for provability logic resolved, in C. Areces, and R. Goldblatt (eds.), Advances in Modal Logic, Vol. 7 (Nancy, 2008), College Publications, 2008, pp. 91–111.Google Scholar
- 10.Litak T.: The non-reflexive counterpart of Grz, Bull. Sect. Logic Univ. ódź 36(3–4), 195–208 (2007)Google Scholar
- 11.Mints, G., Cut elimination for provability logic, in Collegium Logicum 2005: Cut-Elimination, 2006Google Scholar
- 13.Sasaki K.: Löb’s axiom and cut-elimination theorem. Journal of Nanzan Academic Society Mathematical Sciences and Information Engineering 1, 91–98 (2001)Google Scholar
- 14.Troelstra, A. S., Schwichtenberg H. Basic proof theory, vol. 43 of Cambridge Tracts in Theoretical Computer Science, 2nd edn., Cambridge University Press, Cambridge, 2000Google Scholar