In this paper a simple proof of the completeness theorem of the intuitionistic predicate calculus with respect to the topological semantics is shown. From a technical point of view the proof of the completeness theorem is based on a Rasiowa-Sikirski-like theorem for the countable Heyting algebras which allows to embadd any countable Heyting algebra into a suitable topology in a such way that a ...

Abstract The variety of (pointed) residuated lattices includes a vast proportion the classes algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among outliers, one counts orthomodular and other varieties quantum algebras. We suggest ...

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (Rl-monoids) are common generalizations of BL-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain i...

We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras ...

By introducing a new operation, the exponentiation of formal languages, we can define Heyting algebras of formal languages. It turns out that some well known families of languages are closed under this exponentiation, e. g., the families of regular and of context-sensitive languages.

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into Heyting algebra topos theory. While distributivity does not always hold lattices, it algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, condition that any element transformed through corresponds to original lattice. show these equivalent, that, only ca...