# Order-Based and Continuous Modal Logics

## George Metcalfe

### Mathematisches Institut, Universität Bern

Many-valued modal logics generalize the Kripke frame semantics of classical modal logic to allow a many-valued semantics at each world based on an algebra with a complete lattice reduct, where the accessibility relation may also take values in this algebra. Such logics can be designed to model modal notions such as necessity, belief, and spatio-temporal relations in the presence of uncertainty, possibility, or vagueness, and also provide a basis for defining fuzzy description logics. More generally, many-valued modal logics provide a first foray into investigating useful and computationally feasible fragments of corresponding first-order logics. The aim of my talk will be to describe recent axiomatization, decidability, and complexity results for many-valued modal logics based on algebras over (infinite) sets of real numbers. In particular, I will report on joint work with X. Caicedo, R. Rodriguez, and J. Rogger establishing decidability and complexity results for a family of order-based modal logics, using an alternative semantics admitting the finite model property. I will also present an axiomatization, obtained in joint work with D. Diaconescu and L. Schnueriger, for a many-valued modal logic equipped with the usual group operations over the real numbers that provides a first step towards solving an open axiomatization problem for a Lukasiewicz modal logic with continuous operations.

Dr. George Metcalfe is a Professor at Mathematisches Institut (MAI), Universität Bern. He received Ph.D. in Computer Science from King’s College London,

MSc in Artificial Intelligence from Edinburgh University, and BA in Mathematics and Philosophy from St. Anne’s College, Oxford. He also held positions at Vanderbilt University and was a Marie Curie Fellow at Vienna University of Technology. His research interests include Proof Theory, Non-Classical Logics, and Ordered Algebraic Structures.