# 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).

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