Sergio Celani

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered inAlgebraic Logic

The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions of the minimal system of normal positive modal logic

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R‐weakly Heyting algebras, the variety of T‐weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)