All Finitely Axiomatizable Tense Logics of Linear Time Flows Are CoNP-complete

We prove that all finitely axiomatizable tense logics with temporal operators for ‘always in the future’ and ‘always in the past’ and determined by linear fows time are coNP-complete. It follows, for example, that all tense logics containing a density axiom of the form ■ⁿ⁺¹_F p → ⁿ_F p, for some n ≥ 0, are coNP-complete. Additionally, we prove coNP-completeness of all ∩-irreducible tense logics. As these classes of tense logics contain many Kripke incomplete bimodal logics, we obtain many natural examples of Kripke incomplete normal bimodal logics which are nevertheless coNP-complete.     

Two notions of compactness in Gödel logics

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well.In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, Gödel_Δ, and Gödel_～ logics.There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by Cintula and Navara, and the question of compactness (based on entailment) for Gödel logic was fully answered by Baaz and Zach (see papers [3] and [2]).In this paper we give a nearly complete answer to the problem of compactness based on both concepts for all three logics and for an arbitrary cardinality of the set of propositional variables. Finally, we show a tight correspondence between these two concepts     

Fuzzy Logic Programming and Fuzzy Control

We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program.     

A Theory of Hypermodal Logics: Mode Shifting in Modal Logic

A hypermodality is a connective whose meaning depends on where in the formula it occurs. The paper motivates the notion and shows that hypermodal logics are much more expressive than traditional modal logics. In fact we show that logics with very simple K hypermodalities are not complete for any neighbourhood frames.     

Disjunctive Quantum Logic in Dynamic Perspective

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual properties of physical systems. This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement.     

On Łukasiewicz's Four-Valued Modal Logic

ukasiewicz's four-valued modal logic is surveyed and analyzed, together with ukasiewicz's motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz's own texts, and related literature.