Adding connectives to non-classical logics
Already in 1919 Skolem studied, from an algebraic point of view, certain operations that appear afterwards from a logical point of view, for example in 1942 in a work by Moisil. This is done in the context of a logic that, more recently, has been called bi-intuitionistic. Some decades afterwards, there also appear many papers by the polish logician Rauszer on the same logic. More recently, in 2009, Priest gave a paraconsistent version of some kind of bi-intuitionistic logic. We have proved that, in fact, using a notion by Urbas, it is strictly paraconsistent. Approximately in the same tradition appears the connective ∆ of fuzzy logic. We have proved that, added to a Heyting algebra, the result is an equational class.
Another tradition was started in Russia by Novikov in the Fifties and corresponds to the notion of intuitionistic connective. These connectives are supposed to give conservative expansions and enjoy the Disjunction Property. Regarding this, we consider some problems that arise in the case of first-order intuitionistic logic for connectives suggested by Smetanich, Kuznetsov, Gabbay, and Humberstone.
Axioms may not be enough for the axiomatization, i.e. in some cases it is necessary to add a rule. From a semantical point of view, the choice is between a truth-preserving or a truth-degree-preserving consequence.
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