The logic with unsharp implication and negation

TL;DR

This paper introduces an unsharp form of implication and negation in meet-semilattices with 0, generalizing intuitionistic logic, and demonstrates that these connectives retain key properties like Modus Ponens.

Contribution

It proposes a novel unsharp approach to implication and negation in semilattices, extending intuitionistic logic to more general algebraic structures with simple axiomatizations.

Findings

Unsharp negation and implication share properties with their intuitionistic counterparts.

The connectives can be characterized by five and seven simple axioms respectively.

Examples and relationships between deductive systems, filters, and congruences are provided.

Abstract

It is well-known that intuitionistic logics can be formalized by means of Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical connective implication is considered to be the relative pseudocomplement and conjunction is the semilattice operation meet. If the Brouwerian semilattice has a bottom element 0 then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition, which is trivially satisfied in finite semilattices, and introduce the connective negation x^0 as the set of all maximal elements z satisfying x^z=0 and the connective implication x->y as the set of all maximal elements z satisfying x^z\le y. Such a negation and implication is "unsharp" since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication, respectively, still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. Several examples are presented. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.