Reduction of Many-valued into Two-valued Modal Logics

TL;DR

This paper presents a method to reduce many-valued logics to two-valued multi-modal logics using higher-order Herbrand interpretations and autoreferential Kripke semantics, enabling clearer analysis and application of classical logic tools.

Contribution

It introduces a novel 2-valued reduction approach for many-valued logics via higher-order Herbrand interpretations and autoreferential Kripke semantics, applicable to logic programs and general structures.

Findings

Achieved structural reduction of many-valued logic programs

Generalized reduction to abstract many-valued logics

Developed non truth-valued modal meta-logics

Abstract

In this paper we develop a 2-valued reduction of many-valued logics, into 2-valued multi-modal logics. Such an approach is based on the contextualization of many-valued logics with the introduction of higher-order Herbrand interpretation types, where we explicitly introduce the coexistence of a set of algebraic truth values of original many-valued logic, transformed as parameters (or possible worlds), and the set of classic two logic values. This approach is close to the approach used in annotated logics, but offers the possibility of using the standard semantics based on Herbrand interpretations. Moreover, it uses the properties of the higher-order Herbrand types, as their fundamental nature is based on autoreferential Kripke semantics where the possible worlds are algebraic truth-values of original many-valued logic. This autoreferential Kripke semantics, which has the possibility of flattening higher-order Herbrand interpretations into ordinary 2-valued Herbrand interpretations, gives us a clearer insight into the relationship between many-valued and 2-valued multi-modal logics. This methodology is applied to the class of many-valued Logic Programs, where reduction is done in a structural way, based on the logic structure (logic connectives) of original many-valued logics. Following this, we generalize the reduction to general structural many-valued logics, in an abstract way, based on Suszko's informal non-constructive idea. In all cases, by using developed 2-valued reductions we obtain a kind of non truth-valued modal meta-logics, where two-valued formulae are modal sentences obtained by application of particular modal operators to original many-valued formulae.