A modal theorem-preserving translation of a class of three-valued logics of incomplete information

Abstract

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, Lukasiewicz and Nelson logics. We show that Priest logic of paradox, closely connected to Kleene's, can also be translated into our modal setting, just exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

A capacity-based framework encompassing Belnap–Dunn logic for reasoning about multisource information

“…Thus, we construct a bridge between Belnap-Dunn logic and uncertainty theories. As special cases, we recover our previous translations of Kleene logic for incomplete information [20] and Priest Logic of Paradox [21]. This translation is not a gratuitous exercise.…”

“…In the former logic, the truth-value referring to the idea of "contradiction" is eliminated, while in the latter, the truth-value referring to the idea of "ignorance" is eliminated. Belnap-Dunn logic uses all four values forming a bilattice structure, in which one partial ordering expresses relative strength of information (from no information to too much information), and the other partial ordering represents relative truth (from false to true).In recent works [20,21], we have focused attention on three-valued logics of uncertainty or inconsistency (paraconsistent logics). We were able to express them in a two-tiered propositional logic called MEL [8], which adopts the syntax of a fragment of epistemic logic, and borrows axioms of the KD logic.…”

“…The logic MEL is in the same vein, albeit using propositional logic at both levels, and a modal language.In the MEL logic, it is possible to express that an atomic proposition, or its complement, is unknown (when translating Kleene logic, for instance), or that it is contradictory (when translating Priest Logic of Paradox, for instance). The main merit of such translations in [20,21] is to lay bare the meaning of the various three-valued connectives usually defined by means of truth tables, and especially to exhibit the limited expressive power of such three-valued logics. Namely, modalities can only appear in front of literals in the translations.…”

“…Results in this section were more succinctly announced in a former conference paper [22]. Section 5 shows how Kleene and Priest logic translations in [20,21] can be retrieved by translating additional axioms or inference rules into BC. Section 6 discusses some related approaches that deal with conflicting information, including the general source processor logic of [6].…”

Weighted logics for artificial intelligence – an introductory discussion

“…However, usual epistemic logics with their relational semantics look far away from Kleene logic. Yet, a simple fragment of a KD logic, where modalities only appear in front of literals, can capture Kleene logic and other three-valued logics of partial knowledge [24]. This restriction on the expressive power (in Kleene logic, it is not possible to represent the fact that p ∨ q is unknown) is the price paid for the truth-functionality of logics of the unknown.…”