This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO’s) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with a survey of the duality that exists between BAO’s and relational structures, focusing on the notions of bounded morphisms, inner substructures, disjo...

In this thesis, we consider bi-intuitionistic logic and tense logic, as well as the combined bi-intuitionistic tense logic. Each of these logics contains a pair of dual connectives, for example, Rauszer’s bi-intuitionistic logic [100] contains intuitionistic implication and dual intuitionistic exclusion. The interaction between these dual connectives makes it non-trivial to develop a cut-free s...

An intuitionistic, hybrid modal logic suitable for reasoning about distribution of resources was introduced in [14, 15]. The modalities of the logic allow us to validate properties in a particular place, in some place and in all places. We give a sound and complete Kripke semantics for the logic extended with disjunctive connectives. The extended logic can be seen as an instance of Hybrid IS5. ...

Since the invention of Prolog, a programming language based on classical first-order logic, many people have tried to extend it using similiar ideas and redefine the semantics of the extended Prolog in terms of nonclassical logics [3,5,81. The success of a programming language based on nonclassical logics usually lies in the new definiton of Horn formulas and SLD-resolution-like inference rule....

Hintikka and Sandu have developed IF logic as a genuine alternative to classical first-order logic: liberalizing dependence schemas between quantifiers, IF would carry out all the ideas already underlying classical logic. But they are alternatives to Hintikka’s game-theoretic approach; one could use instead Henkin quantifiers. We will present here some arguments of both technical and philosophi...

suhrawardi’s logic of the hikmat al-ishraq is basically modal. so to understand his modal logic one first has to know the non-modal part upon which his modal logic is built. in my previous paper ‘suhrawardi on syllogisms’(3) i discussed the former in detail. the present paper is an exposition of his treatment of modal syllogisms. on the basis of some reasonable existential presuppositions and a...

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such a...

Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. T...

Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. T...