[Paper Review] Completeness results for many-valued \Lukasiewicz modal systems and relational semantics

Georges Hansoul, Bruno Teheux|ArXiv.org|Dec 19, 2006

Advanced Algebra and Logic26 citations

TL;DR

This paper introduces a family of many-valued Łukasiewicz modal logics and establishes completeness results via canonical model construction. It proves completeness for finitely-valued modal logics and an infinitary extension using a novel infinitary deduction rule, distinguishing between Kripke-completeness and n+1-frame completeness through frame definability analysis.

ABSTRACT

The paper is dedicated to the problem of adding a modality to the \Lukasiewicz many-valued logics in the purpose of obtaining completeness results for Kripke semantics. We define a class of modal many-valued logics and their corresponding Kripke models and modal many-valued algebras. Completeness results are considered through the construction of a canonical model. Completeness is obtained for modal finitely-valued logics but also for a modal many-valued system with an infinitary deduction rule. We introduce two classes of frames for the finitely-valued logics and show that they define two distinct classes of Kripke-complete logics.

Motivation & Objective

• To extend Łukasiewicz many-valued logic with a modal operator to achieve completeness under Kripke semantics.
• To define modal many-valued Kripke models and corresponding algebras (MMV-algebras) for sound and complete semantics.
• To construct a canonical model for modal many-valued logics to prove completeness, especially for finitely-valued and infinitary systems.
• To distinguish between Kripke-completeness and n+1-frame completeness by introducing n+1-frames as a new class of structures.
• To explore the correspondence between modal formulas and first-order properties on frames and n+1-frames, highlighting semantic dissimilarities.

Proposed method

• Define a many-valued Kripke model as a triple ⟨W, R, Val⟩, where W is a non-empty set of worlds, R is an accessibility relation, and Val maps propositional variables to [0,1] or to the finite set L_n.
• Introduce modal Łukasiewicz logic with connectives ⊕, ¬, □, and their standard interpretations using Łukasiewicz logic operations.
• Construct a canonical model using the Lindenbaum-Tarski algebra of formulas modulo a logic, with truth values assigned via maximal consistent sets.
• Prove that the canonical model's valuation extends to all formulas (Proposition 5.5), enabling completeness proofs.
• Introduce n+1-frames as first-order structures derived from frames by restricting valuations, enabling analysis of n+1-Kripke-completeness.
• Use an infinitary deduction rule (Inf) to derive φ from {φ⊕φ^n} for all n≥2, enabling completeness in the infinitary system.

Experimental results

Research questions

• RQ1 Can modal extensions of Łukasiewicz many-valued logic be made complete with respect to Kripke semantics using canonical models?
• RQ2 What is the role of the infinitary deduction rule (Inf) in achieving completeness for modal many-valued logics?
• RQ3 How do Kripke-completeness and n+1-Kripke-completeness differ, and what structural features of frames cause this distinction?
• RQ4 Can the canonical model construction simplify the axiomatization of finitely-valued modal Łukasiewicz logics?
• RQ5 What is the correspondence between modal formulas and first-order definable properties on frames and n+1-frames?

Key findings

• Completeness is established for finitely-valued modal Łukasiewicz logics using canonical model construction.
• An infinitary modal many-valued system with the (Inf) rule achieves completeness: Γ ⊢∞ φ if and only if φ is true in all models of Γ.
• The canonical model’s valuation extends naturally to all formulas, a key technical result (Proposition 5.5).
• The logic L₂ is not Kripke-complete, demonstrating a strict distinction between Kripke-completeness and n+1-Kripke-completeness.
• n+1-frames are introduced as a new class of structures that define a distinct class of Kripke-complete logics, showing that frame definability depends on valuation restrictions.
• The paper identifies that the infinitary rule (Inf) is necessary for completeness in the general case, as it is not known whether ⊢∞ φ is equivalent to ⊢K φ.