14 May 2014, Algebra|Coalgebra Seminar, Umberto Rivieccio

Abstract

One of the latest and most challenging trends of research in non-classical logic is the attempt of enriching many-valued systems with modal operators. This allows one to formalize reasoning about vague or graded properties in those contexts (e.g., epistemic, normative, computational) that require the additional expressive power of modalities. This enterprise is thus potentially relevant not only to mathematical logic, but also to philosophical logic and computer science. A very general method for introducing the (least) many-valued modal logic over a given finite residuated lattice is described in [1]. The logic is defined semantically by means of Kripke models that are many-valued in two different ways: the valuations as well as the accessibility relation among possible worlds are both many-valued. Providing complete axiomatizations for such logics, even if we enrich the propositional language with a truth-constant for every element of the given lattice, is a non-trivial problem, which has been only partially solved to date. In this presentation I report on ongoing research in this direction, focusing on the contribution that the theory of natural dualities can give to this enterprise. I show in particular that duality allows us to adapt the method used in [1] to prove completeness with respect to local modal consequence, obtaining completeness for global consequence, too (a problem that, in full generality, was left open in in [1]). Besides this, our study is also a contribution towards a better general understanding of quasivarieties of (modal) residuated lattices from a topological perspective.

(joint work with Andrew Craig - University of Johannesburg)

References

[1] F. Bou, F. Esteva, L. Godo, and R. Rodrìguez. On the minimum many- valued modal logic over a finite residuated lattice. Journal of Logic and Computation, 21(5):739–790, 2011.

For more information, see http://www.illc.uva.nl/alg-coalg/.