We prove that a variety of double-Heyting algebras is a discriminator variety if and only if it is semisimple if and only if it has equationally definable principal congruences. The result also applies to the class of Heyting algebras with a dual pseudocomplement operation and to the class of regular double p-algebras.

The well-known embedding of full intuition- istic propositional calculus into the atomic polymorphic system Fat is possible due to the intriguing phenomenon of instantiation overflow. Instantiation overflow ensures that (in Fat) we can instantiate certain universal formulas by any formula of the system, not necessarily atomic. Until now only three types in Fat were identified with such property: the types that result from the Prawitz translation of the propositional connectives (?, ^, _) into Fat (or Girard's system F). Are there other types in Fat with instantiation overflow? In this paper we show that the answer is yes and we isolate a class of formulas with such property.

Some years ago, Lokhorst proposed an intuitionistic reformulation of Mally's deontic logic (1926). This reformulation was unsatisfactory, because it provided a striking theorem that Mally himself did not mention. In this paper, we present an alternative reformulation of Mally's deontic logic that does not provide this theorem.

Let ZC -  I (respectively, ZF -  I) be the theory obtained by deleting the axiom of infinity from the usual list of axioms for Zermelo set theory with choice (respectively, the usual list of axioms for Zermelo-Fraenkel set theory). In this note, we present a collection of sentences 9x'(x) for which (ZC -  I) + 9x'(x) (respectively, (ZF - I)+9x'(x)) proves the existence of an infinite set.

In this paper we will introduce N-Vietoris families and prove that homomorphic images of distributive nearlattices are dually characterized by N-Vietoris families. We also show a topological approach of the existence of the free distributive lattice extension of a distributive nearlattice.

We consider sl-semantics in which first order sentences are interpreted in potentially infinite domains. A potentially innite domain is a growing sequence of infinite  models. We prove the completeness theorem for first order logic under this semantics. Additionally we characterize the logic of such domains as having a learnable, but not recursive, set of axioms. The work is a part of author's research devoted to computationally motivated foundations of mathematics

Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional. A second theorem of Wójcicki asserts that a logic has a weakly referential semantics if and only if it is weakly self- extensional. We formulate and prove an analog of this theorem in the categorical setting. We show that a π-institution has a weakly referential semantics if and only if it is weakly self-extensional.

The logic RM3 is the 3-valued extension of the logic R-Mingle (RM). RM (and so, RM3) does not have the variable- sharing property (vsp), but RM3 (and so, RM) lacks the more "offending" paradoxes of relevance", such as A → (B → A) or A → (A → B). Thus, RM and RM3 can be useful when some relevance", but not the full vsp, is needed. Sublogics of RM3 with the vsp are well known, but this is not the case with those lacking this property. The rst aim of this paper is to dene an ample family of sublogics of RM3 without the vsp. The second one is to provide these sublogics and RM3 itself with a general Routley-Meyer semantics, that is, the semantics devised for relevant logics in the early seventies of the past century.

The present note corrects an error made by the author in answering an open problem of axiomatizing an expansion of Nelson's logic introduced by Heinrich Wansing. It also gives a correct axiomatization that answers the problem by importing some results on subintuitionistic logics presented by Greg Restall.