Annals of Pure and Applied Logic 161 (9):1097-1133 (2010)

Authors
Hiroakira Ono
Japan Advanced Institute of Science and Technology
Abstract
We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit.
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1016/j.apal.2010.01.003
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 71,464
Through your library

References found in this work BETA

A Survey of Abstract Algebraic Logic.J. M. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):13 - 97.
Logics Without the Contraction Rule.Hiroakira Ono & Yuichi Komori - 1985 - Journal of Symbolic Logic 50 (1):169-201.
Algebraizable Logics.W. J. Blok & Don Pigozzi - 1989 - Advanced Reasoning Forum.

View all 21 references / Add more references

Citations of this work BETA

Order algebraizable logics.James G. Raftery - 2013 - Annals of Pure and Applied Logic 164 (3):251-283.

View all 10 citations / Add more citations

Similar books and articles

An Algebraic Approach to the Disjunction Property of Substructural Logics.Daisuke Souma - 2007 - Notre Dame Journal of Formal Logic 48 (4):489-495.
Synthesized Substructural Logics.Norihiro Kamide - 2007 - Mathematical Logic Quarterly 53 (3):219-225.
Metacompleteness of Substructural Logics.Takahiro Seki - 2012 - Studia Logica 100 (6):1175-1199.
Meeting Strength in Substructural Logics.Yde Venema - 1995 - Studia Logica 54 (1):3 - 32.
Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.
Normal Modal Substructural Logics with Strong Negation.Norihiro Kamide - 2003 - Journal of Philosophical Logic 32 (6):589-612.
Glivenko Theorems for Substructural Logics Over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Journal of Symbolic Logic 71 (4):1353 - 1384.
Modal Translations in Substructural Logics.Kosta Došen - 1992 - Journal of Philosophical Logic 21 (3):283 - 336.

Analytics

Added to PP index
2013-12-18

Total views
23 ( #495,516 of 2,520,747 )

Recent downloads (6 months)
1 ( #405,718 of 2,520,747 )

How can I increase my downloads?

Downloads

My notes