Willem Blok and Modal Logic

We present our personal view on W.J. Blok's contribution to modal logic.     

On Dynamic Topological and Metric Logics

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f² (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q ⁺. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.     

Temporalising Tableaux

As a remedy for the bad computational behaviour of first-order temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining so-called monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of first-order logic like the two-variable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for first-order fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL. Some example instantiations of the framework are presented.     

The Greatest Extension of S4 into which Intuitionistic Logic is Embeddable

This paper gives a characterization of those quasi-normal extensions of the modal system S4 into which intuitionistic propositional logic Int is embeddable by the Gödel translation. It is shown that, as in the normal case, the set of quasi-normal modal companions of Int contains the greatest logic, M*, for which, however, the analog of the Blok-Esakia theorem does not hold. M* is proved to be decidable and Halldén-complete; it has the disjunction property but does not have the finite model property.