[Paper Review] On the relationship between fuzzy logic and four-valued relevance logic

Umberto Straccia|ArXiv.org|Oct 31, 2000

Rough Sets and Fuzzy Logic12 references3 citations

TL;DR

This paper establishes that fuzzy propositional logic, under specific dual conditions, collapses to four-valued relevance logic—where propositions take truth-values in {true, false, unknown, contradiction}—demonstrating that fuzzy entailment lies conceptually and logically between four-valued and classical two-valued entailment. The key contribution is a formal bridge between fuzzy logic and relevance logic via entailment relations.

ABSTRACT

In fuzzy propositional logic, to a proposition a partial truth in [0,1] is assigned. It is well known that under certain circumstances, fuzzy logic collapses to classical logic. In this paper, we will show that under dual conditions, fuzzy logic collapses to four-valued (relevance) logic, where propositions have truth-value true, false, unknown, or contradiction. As a consequence, fuzzy entailment may be considered as ``in between'' four-valued (relevance) entailment and classical entailment.

Motivation & Objective

• To investigate the logical relationship between fuzzy propositional logic and four-valued relevance logic.
• To determine under what conditions fuzzy logic reduces to four-valued logic.
• To clarify the position of fuzzy entailment relative to classical and four-valued entailment.
• To formalize entailment relations in fuzzy logic using meta-level constraints on truth-values.
• To show that fuzzy entailment is bounded below by four-valued entailment and above by classical entailment.

Proposed method

• The paper defines a fuzzy propositional logic where propositions are assigned truth-values in [0,1], and meta-level constraints like ⟨A ≥ n⟩ and ⟨A ≤ n⟩ are used to express bounds on truth-values.
• It introduces a meta-logic ℒf with meta-atoms that are either true (1) or false (0), representing constraints on fuzzy truth-values.
• The paper formalizes four-valued relevance logic using truth-values {true, false, unknown, contradiction}, as defined in prior work by Anderson, Belnap, and others.
• It establishes entailment relations by comparing fuzzy entailment (|≈r), four-valued entailment (⊨4), and classical entailment (⊨2), using model-theoretic semantics.
• It proves that A |≈r B holds if and only if either A ⊨4 B or ⊨2 ¬A ∧ B, showing that fuzzy entailment subsumes four-valued entailment in non-trivial cases.
• It analyzes alternative fuzzy entailment definitions (|≈a, |≈b, |≈c) and maps them to classical or four-valued entailment via meta-logical constraints.

Experimental results

Research questions

• RQ1 Under what conditions does fuzzy propositional logic collapse to four-valued relevance logic?
• RQ2 How does fuzzy entailment relate to classical and four-valued entailment in terms of logical strength?
• RQ3 Can fuzzy entailment be formally characterized as lying between four-valued and classical entailment?
• RQ4 What is the role of meta-level constraints like ⟨A ≥ n⟩ in connecting fuzzy logic to four-valued logic?
• RQ5 How do different definitions of fuzzy entailment (|≈a, |≈b, |≈c) map to classical or four-valued entailment?

Key findings

• Fuzzy entailment |≈r is logically bounded below by four-valued entailment ⊨4 and above by classical entailment ⊨2.
• For all non-trivial cases—where the premise is classically satisfiable and the conclusion is not a tautology—fuzzy entailment |≈r is equivalent to four-valued entailment ⊨4.
• The entailment A |≈r B holds if and only if either A ⊨4 B or ⊨2 ¬A ∧ B, establishing a precise logical equivalence.
• The definition |≈a corresponds exactly to classical entailment ⊨2, as |≈a B is equivalent to ⊨2 ¬A ∨ B.
• The definition |≈b corresponds to a disjunction: either B is a classical tautology or A ⊨4 B.
• The definition |≈c corresponds to a disjunctive condition over disjunctive normal forms: for each disjunct Aj of A, either ¬Aj is a tautology or Aj ⊨4 B.