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NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES

Published online by Cambridge University Press:  15 May 2013

PETR CINTULA ,
ROSTISLAV HORČÍK  and
CARLES NOGUERA
PETR CINTULA*
Affiliation:
Institute of Computer Science, Academy of Sciences of the Czech Republic
ROSTISLAV HORČÍK*
Affiliation:
Institute of Computer Science, Academy of Sciences of the Czech Republic
CARLES NOGUERA*
Affiliation:
Artificial Intelligence Research Institute (IIIA - CSIC) and Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic
*
*INSTITUTE OF COMPUTER SCIENCE, ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 2 182 07 PRAGUE, CZECH REPUBLIC E-mail: {cintula,horcik}@cs.cas.cz
*INSTITUTE OF COMPUTER SCIENCE, ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 2 182 07 PRAGUE, CZECH REPUBLIC E-mail: {cintula,horcik}@cs.cas.cz
†INSTITUTE OF INFORMATION THEORY AND AUTOMATION ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 4 182 08 PRAGUE, CZECH REPUBLIC ARTIFICIAL INTELLIGENCE RESEARCH INSTITUTE SPANISH NATIONAL RESEARCH COUNCIL CAMPUS DE LA UNIVERSITAT AUTÒNOMA DE BARCELONA S/N,08193 BELLATERRA, CATALONIA, SPAIN E-mail: [email protected]
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Abstract

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 3 , September 2013 , pp. 394 - 423
Copyright
Copyright © Association for Symbolic Logic 2013 

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References

BIBLIOGRAPHY

Anderson, A. R., & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity. Vol. 1. Princeton, NJ: Princeton University Press.Google Scholar
Běhounek, L., & Cintula, P. (2006). Fuzzy logics as the logics of chains. Fuzzy Sets and Systems, 157, 604–610.CrossRefGoogle Scholar
Blok, W. J., & Pigozzi, D. L. (1989). Algebraizable Logics. Vol. 396. Memoirs of the American Mathematical Society. Providence, RI: American Mathematical Society. Available fromhttp://orion.math.iastate.edu/dpigozzi/.Google Scholar
Botur, M. (2011). A non-associative generalization of Hájek’s BL-algebras. Fuzzy Sets and Systems, 178, 24–37.CrossRefGoogle Scholar
Burris, S., & Sankappanavar, H. (1981). A Course in Universal Algebra. Vol. 78. Graduate Texts in Mathematics. New York, NY: Springer-Verlag.Google Scholar
Buszkowski, W., & Farulewski, M. (2009). Nonassociative Lambek calculus with additives and context free languages. In Grumberg, O., Kaminski, M., Katz, S., and Wintner, S., editors. Languages: From Formal to Natural. Essays Dedicated to Nissim Francez. Vol. 5533. Lecture Notes in Computer Science. Berlin: Springer.Google Scholar
Cignoli, R., Esteva, F., Godo, L., & Torrens, A. (2000). Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing, 4, 106–112.Google Scholar
Cintula, P., Hájek, P., & Horčík, R. (2007). Formal systems of fuzzy logic and their fragments. Annals of Pure and Applied Logic, 150, 40–65.CrossRefGoogle Scholar
Cintula, P., Hájek, P., & Noguera, C., editors (2011). Handbook of Mathematical Fuzzy Logic (in 2 volumes). Vol. 37, 38. Studies in Logic, Mathematical Logic and Foundations. London: College Publications.Google Scholar
Cintula, P., & Noguera, C. (2010). Implicational (semilinear) logics I: A new hierarchy. Archive for Mathematical Logic, 49, 417–446.CrossRefGoogle Scholar
Cintula, P., & Noguera, C. (2011). A general framework for mathematical fuzzy logic. In Cintula, P., Hájek, P., and Noguera, C., editors, Handbook of Mathematical Fuzzy Logic. Volume 1. Vol. 37. Studies in Logic, Mathematical Logic and Foundations. London: College Publications, pp. 103–207.Google Scholar
Cintula, P., & Noguera, C. (2013). The proof by cases property and its variants in structural consequence relations. Studia Logica (to appear).Google Scholar
Czelakowski, J. (2001). Protoalgebraic Logics. Vol. 10. Trends in Logic. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
Esteva, F., & Godo, L. (2001). Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124, 271–288.CrossRefGoogle Scholar
Esteva, F., Godo, L., Hájek, P., & Montagna, F. (2003). Hoops and fuzzy logic. Journal of Logic and Computation, 13, 532–555.CrossRefGoogle Scholar
Galatos, N. (2004). Equational bases for joins of residuated-lattice varieties. Studia Logica, 76, 227–240.CrossRefGoogle Scholar
Galatos, N., & Jipsen, P. (2013). Residuated frames with applications to decidability. Transactions of the American Mathematical Society, 365, 1219–1249.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Vol. 151. Studies in Logic and the Foundations of Mathematics. Amsterdam, The Netherlands: Elsevier.Google Scholar
Galatos, N., & Ono, H. (2010). Cut elimination and strong separation for substructural logics: An algebraic approach. Annals of Pure and Applied Logic, 161, 1097–1133.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50, 1–102.Google Scholar
Hájek, P. (1998). Metamathematics of Fuzzy Logic. Vol. 4 Trends in Logic. Dordrecht, The Netherlandss: Kluwer.Google Scholar
Höhle, U. (1995). Commutative, residuated l-monoids. In Höhle, U., and Klement, E. P., editors. Non-Classical Logics and Their Applications to Fuzzy Subsets, pp. 53–106.CrossRefGoogle Scholar
Horčík, R. (2011). Algebraic semantics: Semilinear FL-algebras. In Cintula, P., Hájek, P., and Noguera, C., editors. Handbook of Mathematical Fuzzy Logic—Volume 1. Vol. 37. Studies in Logic, Mathematical Logic and Foundations. London: College Publications, pp. 283–353.Google Scholar
Jenei, S., & Montagna, F. (2002). A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica, 70, 183–192.Google Scholar
Jenei, S., & Montagna, F. (2003). A proof of standard completeness for non-commutative monoidal t-norm logic. Neural Network World, 13, 481–489.Google Scholar
Lambek, J. (1958). On the calculus of syntactic types. American Mathematical Monthly, 65, 154–170.CrossRefGoogle Scholar
Lambek, J. (1961). The mathematics of sentence structure. In Jakobson, R., editor. Structure of Language and Its Mathematical Aspects. Providence, NJ: American Mathematical Society, pp. 166–178.CrossRefGoogle Scholar
Metcalfe, G., & Montagna, F. (2007). Substructural fuzzy logics. Journal of Symbolic Logic, 72, 834–864.Google Scholar
Paoli, F. (2002). Substructural Logics: A Primer. Vol. 13 Trends in Logic. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
Raftery, J. G. (2001). A characterization of varieties. Algebra Universalis, 45, 449–450.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logics. New York: Routledge.Google Scholar
Schroeder-Heister, P., & Dosen, K., editors (1994). Substructural Logics. Vol. 2. Studies in Logic and Computation. Oxford: Oxford University Press.Google Scholar
Wang, S., & Zhao, B. (2009). HpsUL is not the logic of pseudo-uninorms and their residua. Logic Journal of the Interest Group of Pure and Applied Logic, 17, 413–419.Google ScholarPubMed