Free Quantification in Four-Valued and Fuzzy Bilattice-Valued Logics

TL;DR

This paper develops a novel free logic framework based on four-valued and fuzzy bilattice-valued logics, allowing for non-denoting terms and truth-value gluts, with dual-domain semantics for quantification over existing and all objects.

Contribution

It introduces a dual-domain semantics for free logic within four-valued and fuzzy bilattice-valued logics, accommodating non-denoting terms and truth-value gluts.

Findings

Dual-domain semantics separates existing and non-existing objects.

The logic accommodates truth-value gluts and gaps.

A fuzzy variant handles partial indeterminacy and inconsistency.

Abstract

We introduce a variant of free logic (i.e., a logic admitting terms with nonexistent referents) that accommodates truth-value gluts as well as gaps. Employing a suitable expansion of the Belnap-Dunn four-valued logic, we specify a dual-domain semantics for free logic, in which propositions containing non-denoting terms can be true, false, neither true nor false, or both true and false. In each model, the dual domain semantics separates existing and non-existing objects into two subdomains, making it possible to quantify either over all objects or existing objects only. We also outline a fuzzy variant of the dual-domain semantics, accommodating non-denoting terms in fuzzy contexts that can be partially indeterminate or inconsistent.