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	Membership	Degree of membership
[0,1]	—	Unit interval	Truth value range
⊗	U+2297	T-norm	Strong conjunction
⊕	U+2295	T-conorm	Strong disjunction
→	U+2192	Residuum	Fuzzy implication
&	—	Weak conjunction	min in Gödel
∧	U+2227	Lattice meet	min
∨	U+2228	Lattice join	max
Δ	U+0394	Delta	Crispness: 1 if true, 0 otherwise

Metatheory. BL and extensions are algebraizable. Standard completeness: provable iff true in all [0,1] models. Łukasiewicz logic is an MV-algebra. Deduction theorem holds with care. Decidability: propositional fuzzy logics are generally decidable (coNP-complete for Ł). First-order fuzzy logic: undecidable, but fragments are studied.

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. Membership function design is subjective. Different t-norms give different logics; no single "correct" choice. Criticism: is it better than probability for uncertainty? Vagueness vs uncertainty distinction is debated. Fuzzy control works empirically but theoretical justification sometimes questioned. Not suitable for reasoning about rare events (probability is better).