The intuitionistic-like logic based on a poset

TL;DR

This paper generalizes intuitionistic logic from Heyting algebras to bounded posets, introducing 'unsharp' operators for logical connectives that assign sets of elements, and explores their properties and associated deductive systems.

Contribution

It extends intuitionistic logic to bounded posets without requiring pseudocomplementation, defining new 'unsharp' operators and establishing their properties and related deductive systems.

Findings

Operators for negation and implication characterized by simple poset conditions

Implication and conjunction form an adjoint pair

Deductive systems induce equivalence relations with substitution properties

Abstract

The aim of the present paper is to show that the concept of intuitionistic logic based on a Heyting algebra can be generalized in such a way that it is formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements with maximal possible truth values. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives "unsharp" or "inexact" in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence on the base set is uniquely determined by its kernel, i.e. the class containing the top element.