Basic Fuzzy Logic

Origin. Hájek (1998). The logic of all continuous t-norms, complete for BL-algebras. It unifies Łukasiewicz, Gödel, and Product logic as axiomatic extensions and founds mathematical fuzzy logic.

Models. Continuous t-norms on [0,1] with their residua. BL-algebras: prelinear, divisible, bounded commutative integral residuated lattices. Standard completeness: BL is complete for the real unit interval under any continuous t-norm.

Formalism.

BL-algebra: ⟨L, ∧, ∨, *, →, 0, 1⟩ with ⟨L, ∧, ∨, 0, 1⟩ a bounded lattice,

a commutative monoid (t-norm), → its residuum: x * y ≤ z iff x ≤ y → z.

Axioms (A1–A7): (A1) (φ → ψ) → ((ψ → χ) → (φ → χ)) (A2) φ * ψ → φ (A3) φ * ψ → ψ * φ (A4) φ * (φ → ψ) → ψ * (ψ → φ) [divisibility] (A5a) (φ → (ψ → χ)) → (φ * ψ → χ) (A5b) (φ * ψ → χ) → (φ → (ψ → χ)) (A6) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ) [prelinearity] (A7) 0 → φ

Divisibility (BL = MTL + divisibility): x ∧ y = x * (x → y). Dropping it gives MTL (left-continuous t-norms); adding it forces continuity.

Mostert–Shields: Every continuous t-norm is an ordinal sum of three: Łukasiewicz xy = max(0, x+y−1), Gödel xy = min(x,y), Product x*y = x·y.

Prelinearity: (φ → ψ) ∨ (ψ → φ). The subdirectly irreducible BL-algebras are the BL-chains.

Negation: ¬φ := φ → 0. Defined, not generally involutive.

Principal extensions: BL + involution = Łukasiewicz (MV). BL + idempotence (x*x = x) = Gödel. BL + cancellativity = Product.

Standard completeness: Every BL-chain embeds into a [0,1] t-norm algebra; BL is complete for [0,1], with finite strong completeness and decidability.

Symbols.

Symbol	Unicode	Name	Meaning
*	—	Strong conjunction	Continuous t-norm
→	U+2192	Residuum	R-implication
∧, ∨	—	Lattice meet/join	Weak connectives
BL	—	Basic Logic	Hájek's system
[0,1]	—	Standard algebra	Real unit interval

Metatheory. Complete for BL-algebras and, by standard completeness, for [0,1] under continuous t-norms. BL = MTL + divisibility. BL-chains are subdirectly irreducible and embed into the standard algebra. Decidable; finitely strongly complete. Łukasiewicz, Gödel, and Product are its principal schematic extensions.

Applies to. Mathematical fuzzy logic. Approximate reasoning. Vagueness as degree. T-norm theory. Continuous-valued inference.

Limitations. Requires continuity (divisibility); left-continuous-only t-norms need MTL. Truth-functional only. No canonical modal extension. Algebraic prerequisites for the metatheory.