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

We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The (...) class of topological Kripke frames characterised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although tableau completeness and decidability are still open questions. (shrink)

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. (shrink)

In this paper we investigate those extensions of the bimodal provability logic ${\vec CSM}_{0}$ (alias ${\vec PRL}_{1}$ or ${\vec F}^{-})$ which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing ${\vec CSM}_{0}$ are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even (...) complete with respect to Kripke semantics. (shrink)

Almost a decade ago, the Scotistic Commission under the direction of Charles Balic, O.F.M., set out to remedy the last named condition with a new critical edition of the Opera Omnia Scoti. Innumerable difficulties and obstacles, however, have delayed the work. Although the first volume is to appear shortly, it will be many years before the complete edition will be available. Fortunately historical research is at such a stage that we are in a position to determine in main outline what (...) works are authentic, dubious or spurious. From present indications, at least, the critical edition is not expected to introduce any radical changes of doctrine, although it will make for a better and clearer reading. Consequently, a translation of some of the more important sections of the works of Scotus based upon the present editions is not without value, and will fill a very definite need until the critical editions and official versions appear. The translation which follows is intended as a step in that direction. (shrink)

In this paper we investigate those extensions of the bimodal provability logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} (alias \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec PRL}_{1}$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec F}^{-})$\end{document} which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely (...) axiomatizable subframe logics containing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics. (shrink)

Wippel and Wolter are perhaps the most respected names in metaphysical thought of the middle ages.

We construct a normal extension of K4 with the finite model property whose minimal tense extension is not complete with respect to Kripke semantics.