This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godel-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting...

In this paper, we firstly show that the $N$-dual operation of the right residual implication, which is induced by a left-conjunctive right arbitrary $vee$-distributive left semi-uninorm, is the right residual coimplication induced by its $N$-dual operation. As a dual result, the $N$-dual operation of the right residual coimplication, which is induced by a left-disjunctive right arbitrary $wedge...

in this paper, we firstly show that the $n$-dual operation of the right residual implication, which is induced by a left-conjunctive right arbitrary $vee$-distributive left semi-uninorm, is the right residual coimplication induced by its $n$-dual operation. as a dual result, the $n$-dual operation of the right residual coimplication, which is induced by a left-disjunctive right arbitrary $wedge...

We discuss the celebrated Blok-Esakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimpl...

Bi-intuitionistic propositional logic (also known as Heyting-Brouwer logic, subtractive logic) extends intuitionistic propositional logic with a connective that is dual to implication, called coimplication. It is the logic of bi-[Cartesian closed] categories and it also has a Kripke semantics. It first got the attention of C. Rauszer who studied it in a number of papers. Later, it has been inve...

The concept of duality, stating that something can or must coexist with its opposite, makes fuzzy logic seem natural, even inevitable. In fact, there are infinite degrees of uncertainty between the certainty of being and not being. This imperfection inherent to information represented in natural language has been treated mathematically using the theory of fuzzy sets. Thus, fuzzy logic introduce...

Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ` ∆ is a theorem over operators {∨,∧,¬}, then so is ∆ ` Γ, where Σ reverses the order of formulas in Σ, and exchang...

ions over incomplete types (i.e. types that do not specify the functional structure of their inhabitants completely) are meant to simulate the Π-abstractions of the λ-cube [Barendregt 1993] and the author sees fitting the Π binder into λδ architecture as a very challenging task. In particular it would be interesting to relate this extension of λδ to COC since this calculus has been fully specif...