Variable Sharing in Substructural Logics: an Algebraic Characterization

Guillermo Badia

doi:10.18778/0138-0680.47.2.03

Authors

Guillermo Badia Department of Knowledge-Based Mathematical Systems, Johannes Kepler Universität, Austria Author

DOI:

https://doi.org/10.18778/0138-0680.47.2.03

Keywords:

relevant logic, algebraic characterizations of logical properties, variable sharing property, substructural logics

Abstract

We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.

References

[1] A. R. Anderson and N. D. Belnap, Entailment. The Logic of Relevance and Necessity, I, Princeton University Press, 1975.

[2] A. Avron, The basic relevance criterion, talk given at the Third Workshop in Edmonton, Canada, May 2016.

[3] M. Dunn and G. Restall, Relevance Logic [in:] D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Kluwer, 2002, pp. 1–128.

[4] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, 2007.

[5] H. Kihara and H. Ono, Algebraic characterizations of variable separation properties, Reports on Mathematical Logic 43 (2008), pp. 43–63.

[6] L. L. Maksimova, Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras, Algebra i Logika 16 (1977), pp. 643–681.

[7] L. L. Maksimova, On variable separation in modal and superintuitionistic logics, Studia Logica 55 (1995), pp. 99–112.

[8] L. L. Maksimova, Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics, Logic, Algebra and Computer Science, Banach Center Publications, Vol. 46, Polish Academy of Science, Warszawa (1999), pp. 159–168.

[9] J. M. Méndez and G. Robles, A General Characterization of the Variable-Sharing Property by Means of Logical Matrices, Notre Dame Journal of Formal Logic, Vol. 53(2) (2012), pp. 223–244.

[10] D. Souma, An algebraic approach to the disjunction property of substructural logics, Notre Dame Journal of Formal Logic, Vol. 48(4) (2007), pp. 489–495.

[11] A. Wroński, Remarks on Halldén-completeness of modal and intermediate logics, Bulletin of the Section of Logic, Vol. 5/4 (1976), pp. 126–129.

[12] E. Yang, R and Relevance Principle Revisited, Journal of Philosophical Logic 42(5) (2013), pp. 767–782.