Fuzzy Logic

Origin. Lotfi Zadeh introduced fuzzy sets (1965) and fuzzy logic for reasoning under vagueness. Petr Hájek provided mathematical foundations (Metamathematics of Fuzzy Logic, 1998). Practical applications in control systems, AI, and decision making. Distinct from probability: fuzziness is about vague boundaries, not uncertain events.

Models. Degrees of truth for vague predicates. "John is tall" — not simply true or false, but true to degree 0.8. Fuzzy logic extends classical logic with continuous truth values [0,1]. Membership in fuzzy sets is graded. Connectives are interpreted as functions on [0,1].

Formalism.

Truth values: [0,1] instead of {0,1}

Fuzzy sets: A fuzzy set A on universe U: membership function μ_A: U → [0,1] μ_A(x) = degree to which x belongs to A

Operations on fuzzy sets:

Complement: μ_{¬A}(x) = 1 - μ_A(x)
Intersection: μ_{A∩B}(x) = T(μ_A(x), μ_B(x)) where T is a t-norm
Union: μ_{A∪B}(x) = S(μ_A(x), μ_B(x)) where S is a t-conorm

T-norms (conjunction):

Gödel: min(a,b)
Product: a · b
Łukasiewicz: max(0, a + b - 1)

Residuation (implication): a → b = sup{c : T(a,c) ≤ b}

Gödel: 1 if a ≤ b, else b
Product: min(1, b/a)
Łukasiewicz: min(1, 1 - a + b)

Fuzzy logic systems:

BL (Basic Logic): axiomatizes continuous t-norms
Łukasiewicz logic Ł: Łukasiewicz t-norm
Gödel logic G: minimum t-norm
Product logic Π: product t-norm
MTL (Monoidal T-norm Logic): left-continuous t-norms

Standard completeness: These logics are complete w.r.t. [0,1]-valued semantics with their respective t-norms.

Fuzzy rules: IF x is A AND y is B THEN z is C Compositional rule of inference: compute output fuzzy set from inputs.

Symbols.

Symbol	Unicode	Name	Meaning
μ	U+03BC	Mu / Membership	Degree of membership, μ_A: U → [0,1]
⊗	U+2297	Circled times / T-norm	Strong conjunction
⊕	U+2295	Circled plus / T-conorm	Strong disjunction
→	U+2192	Rightwards arrow / Residuum	Fuzzy implication, sup{c : T(a, c) ≤ b}
&	U+0026	Ampersand	Hájek's notation for the t-norm ⊗
∧	U+2227	Logical and	Lattice meet, min; definable as φ & (φ → ψ)
∨	U+2228	Logical or	Lattice join, max
¬	U+00AC	Not sign	Defined negation, φ → 0
Δ	U+0394	Delta	Crispness: 1 if true, 0 otherwise
≤	U+2264	Less than or equal	Order on truth degrees
·	U+00B7	Middle dot	Product t-norm, a · b
Ł	U+0141	L with stroke	Łukasiewicz logic
Π	U+03A0	Pi	Product logic
[0,1]	—	Unit interval	Truth value range
T	—	T-norm	The conjunction operation on [0,1]
S	—	T-conorm	The dual disjunction operation
BL	—	Basic Logic	Hájek's logic of continuous t-norms
MTL	—	Monoidal T-norm Logic	Logic of left-continuous t-norms
G	—	Gödel logic	The minimum t-norm

Metatheory. The mathematical core is Hájek's (1998): BL is the logic of continuous t-norms, complete for the variety of BL-algebras, and Cignoli, Esteva, Godo and Torrens (2000) proved the stronger standard completeness with respect to the algebras on [0,1] themselves. Mostert and Shields (1957) had shown that every continuous t-norm is an ordinal sum of Łukasiewicz, Gödel and product components, which is why those three extensions are not an arbitrary selection but the building blocks. Dropping divisibility to left-continuity gives Esteva and Godo's MTL (2001), standard-complete by Jenei and Montagna (2002). All of these are algebraizable in the Blok–Pigozzi sense, with MV-algebras, Gödel algebras and product algebras as the respective equivalent varieties. The propositional tautology problems are decidable and coNP-complete. First order is where the family divides: Gödel logic is recursively axiomatizable and standard-complete, whereas Scarpellini (1962) showed first-order Łukasiewicz validity over [0,1] is not recursively enumerable, the standard-model set being Π₂-complete.

Applies to. Fuzzy control (washing machines, cameras, trains). Approximate reasoning. Expert systems with linguistic variables. Decision making under vagueness. Image processing. Natural language semantics. Medical diagnosis.

Limitations. Truth-functionality is the binding constraint: the degree of a compound depends only on the degrees of its parts, so penumbral connections are lost and φ ∧ ¬φ acquires a positive degree whenever φ is intermediate. Modelling inexactness requires exact real numbers, so higher-order vagueness is untouched — the boundary between degree 0.80 and 0.81 is as sharp as the classical boundary it was meant to soften. Nothing in the idea of a degree chooses a t-norm, and the three standard choices give logics that are not comparable. Excluded middle fails in the lattice form throughout the family, non-contradiction fails as well in Łukasiewicz logic, and the deduction theorem holds only locally wherever the t-norm is not idempotent. The base systems have no truth constants and no Δ, so "φ is fully true" is not expressible without extending the language. Zadeh's original min, max and 1 − x are not a residuated triple and, in a language without truth constants, validate nothing at all, since setting every atom to ½ sends every formula to ½ — so applied fuzzy control does not run on any of these logics.