An algebraic approach to intuitionistic connectives

Abstract

Abstract.It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be “completed” conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

Compatible operations in some subvarieties of the variety of weak Heyting algebras

“…The characterization for compatible functions on Heyting algebras given in Lemma 2.1 of [4] is exactly the same given in Corollary 5.…”

“…In the variety of boolean algebras the answer is no (see [11]), thus we say that it is an affine complete variety. On the other hand, the variety of Heyting algebras is not an affine complete variety (see Example 2.1 of [4], and [8]). However, it is locally affine complete in the sense that any restriction of a compatible function to a finite subset is a polynomial (see [10], [11], [13]).…”

“…The function s 0 is the successor function in the sense of [6] (see also [4], [9], [12] for the case of Heyting algebras).…”

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

“…In Boolaen algebras all compatible operations are polynomials, this is why Boolaen algebras are called affine complete. Heyting algebras are not affine complete, there are many interesting addi-tional compatible connectives, like the frontal operators, the successor, γ and G operations, studied in many papers, see [10] and [4] and [6] for more references.…”

“…[10], [4] for references. From the point of view of algebraic semantics it is natural to demand that the new connective considered as an operation on Heyting algebras should preserve congruences of the algebras, that is, the connective is compatible.…”