Monoidal T-norm Logic

Origin. Esteva and Godo (2001). Logic of left-continuous t-norms. Weaker than BL (basic logic). Subsumed by Łukasiewicz, Gödel, Product. Foundation for fuzzy logic hierarchy.

Models. T-norm: [0,1]² → [0,1] associative, commutative, monotone, 1 identity. Left-continuous: residuated. Not necessarily continuous (unlike BL).

Formalism.

T-norm properties: Associativity: x * (y * z) = (x * y) * z Commutativity: x * y = y * x Monotonicity: x ≤ y implies x * z ≤ y * z Identity: x * 1 = x

Residuum: x * y ≤ z iff y ≤ x → z Left-continuous t-norm has residuum.

MTL axioms: (φ → ψ) → ((ψ → χ) → (φ → χ)) φ ∧ ψ → φ φ ∧ ψ → ψ ∧ φ φ ⊙ (φ → ψ) → φ ∧ ψ (φ → (ψ → χ)) → (φ ⊙ ψ → χ) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ) — prelinearity

Prelinearity: (φ → ψ) ∨ (ψ → φ) Chains: linearly ordered models.

Hierarchy: MTL ⊂ BL ⊂ Łukasiewicz MTL ⊂ IMTL (with involutive negation) MTL ⊂ SMTL (with strong conjunction)

Symbols.

Metatheory. Standard completeness: complete wrt MTL-algebras on [0,1]. Finite strong completeness. Decidable. Prelinearity essential.

Applies to. Fuzzy logic foundation. Approximate reasoning. Control systems. Vague predicates. T-norm classification.

Limitations. Not continuous t-norms only. Less intuitive than Łukasiewicz. Technical prerequisites. Specialized use.

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