The evolution of completeness proofs for modal logic with respect to the possible world semantics is studied starting from an analysis of Kripke’s original proofs from 1959 and 1963. The critical reviews by Bayart and Kaplan and the emergence of Henkin-style completeness proofs are detailed. It is shown how the use of a labelled sequent system permits a direct and uniform completeness proof for...

Kripke’s professional career began as a high school student when he published his early pioneering work in logic on the semantics and completeness proofs of the normal and non-normal modal systems. Not much later, his seminal work on “Semantical Analysis of Intuitionistic Logic” appeared. Shortly after that came his founding of transfinite recursion theory with his two classic papers, “Transfin...

Abstract We prove that the existence of a measurable cardinal is equivalent to normal space whose modal logic coincides with Kripke frame isomorphic powerset two element set.

We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski’s test and Löwenheim-Skolem’s theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models K K′ are studied. —————————————————————————2000 Mathematics Subject Cla...

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semi-contin...

We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multi-agent Kripke models that contain a designated temporal relation. These operations are natural extensions of the well-known “action models” from Dynamic Epistemic Logic. Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal struc...

In this paper we investigate Kripke models, used to model knowledge or belief in a static situation, and action models, used to model communicative actions that change this knowledge or belief. The appropriate notion for structural equivalence between modal structures such as Kripke models is bisimulation: Kripke models that are bisimilar are modally equivalent. We would like to find a structur...

In the first section of this paper we show that iΠ1 ≡ W¬¬lΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear Kripke model deciding ∆0-formulas, it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM . We also show that, for each n > 1,...