This paper continues the investigations of Kripke [63]. Not only is Kripke [63] presupposed; the reader is advised to have it on hand for ready reference. The notations and terminology of that paper are used freely; in particular, P, Q, R, ... are atomic formulae, and A, B, 0, ... are arbitrary formulae built from them using the connectives A, ~, O. All propositional calculi in this paper have ...

This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does ...

In this paper we provide a simplified semantics for the logic KD45(G), i.e. the many-valued Gödel counterpart of the classical modal logic KD45. More precisely, we characterize KD45(G) as the set of valid formulae of the class of possibilistic Gödel Kripke Frames 〈W,π〉, where W is a non-empty set of worlds and π : W → [0, 1] is a normalized possibility distribution on W .

All practical digital circuits are usually a mixture of combinational and sequential logic. Flip–flops are essential to sequential logic therefore fuzzy flip–flops are considered to be among the most essential topics of fuzzy digital circuit. The concept of fuzzy digital circuit is among the most interesting applications of fuzzy sets and logic due to the fact that if there has to be an ultimat...

Let T be a first-order theory. A T -normal Kripke structure is one in which every world is a classical model of T . This paper gives a characterization of the intuitionistic theory HT of sentences intuitionistically valid (forced) in all T -normal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic (PA), the theory HPA is a proper subtheory o...

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...

This is a sequel to two previous papers, where it was shown that, for the Heyting propositional calculus H, we can give Kripke-style models whose accessibility relation R need not be a quasi-ordering relation, provided we have: x\=A &Vy(xRy=* y VA). From left to right, this is the heredity condition of standard Kripke models for H, but, since R need not be reflexive, the converse is not automat...

We prove strong completeness of the -version and the 3-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both in…nitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the …rst logic is reducible to the class of frames having twovalued accessibility relation and this logic does not enj...

The well-known Dyckoff’s 1992 calculus/procedure for intuitionistic propositional logic is considered and analyzed. It is shown that the calculus is Kripke complete and the procedure in fact works in polynomial space. Then a multi-conclusion intuitionistic calculus is introduced, obtained by adding one new rule to known calculi. A simple proof of Kripke completeness and polynomial-space decidab...

Partial Kripke structures model incomplete state spaces with unknown parts. The evaluation of temporal logic formulae on partial Kripke structures is thus based on three-valued interpretations; the additional truth value ⊥ stands for “unknown whether property true or false”. There are existing model checking algorithms as well as tools employing this three-valued interpretation. In this paper w...