Nilpotent Minimum Logic

Origin. Esteva and Godo (2001). Fuzzy logic with nilpotent minimum t-norm. Left-continuous but not continuous. Strict negation properties. Foundation for certain fuzzy reasoning.

Models. Truth values [0,1]. T-norm: min with nilpotent twist. Residuum for implication. Negation: 1-x (involutive). Combines Gödel and Łukasiewicz features.

Formalism.

Nilpotent minimum: x ⊙ y = min(x, y) if x + y > 1, else 0

Residuum: x → y = max(1-x, y) if x ≤ y = max(1-x, y) if x > y and x + y ≤ 1 = 1 if x ≤ y

Alternative definition: x → y = 1 if x ≤ y = max(1-x, y) otherwise

Negation: ¬x = x → 0 = 1 - x (Łukasiewicz-style)

Properties: Not left-continuous at (1,1). Weaker than Łukasiewicz. Stronger than Gödel in some aspects.

Axioms: MTL axioms (monoidal t-norm logic) plus: (φ → ψ) ∨ (ψ → φ) (prelinearity) ¬¬φ → φ (involution) Specific nilpotent axiom.

Comparison: Łukasiewicz: continuous, x ⊙ y = max(0, x+y-1) Gödel: min, but no negation involution NM: between these, with negation.

Symbols.

Metatheory. Standard completeness: complete wrt [0,1] NM-algebra. Decidable. Finite model property. Between Łukasiewicz and Gödel.

Applies to. Fuzzy reasoning. Approximate reasoning. Knowledge representation. Control systems. Vague predicates.

Limitations. Less studied than Łukasiewicz/Gödel. Discontinuity at boundary. Fewer applications. Specialized use cases.

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