Continuous Logic

Origin. Ben Yaacov, Berenstein, Henson, and Usvyatsov developed continuous logic (2000s). Model theory for metric structures. Truth values in [0,1] rather than {0,1}. Generalizes classical model theory to Banach spaces, probability algebras, etc. Part of continuous model theory program.

Models. Metric structures with approximate truth. Classical logic: discrete structures, true/false. Continuous logic: metric structures, truth in [0,1]. Metric d(x,y) replaces equality. Connectives are continuous functions. Quantifiers are sup and inf over metric balls.

Formalism.

Metric structure: M = (M, d, (fᵢ), (Pⱼ)) where:

d: M × M → [0,1] is a metric
fᵢ are uniformly continuous functions
Pⱼ are predicates M^n → [0,1]

Syntax: Terms as usual. Formulas:

P(t₁,...,tₙ) — predicate application (value in [0,1])
d(t₁,t₂) — distance
f(φ₁,...,φₙ) — continuous connective f: [0,1]^n → [0,1]
sup_x φ, inf_x φ — quantifiers

Standard connectives:

φ ∧ ψ = max(φ, ψ) (note: reversed from usual!)
φ ∨ ψ = min(φ, ψ)
¬φ = 1 - φ
φ → ψ = min(1, 1 - φ + ψ) (Łukasiewicz)
φ ·̇ ψ = max(0, φ + ψ - 1) (truncated)

Convention: 0 = true, 1 = false (distances to truth).

Approximate satisfaction: M ⊨_ε φ means φᴹ ≤ ε. Theories: sets of conditions φ ≤ r.

Symbols.

Symbol	Unicode	Name	Meaning
d	—	Metric	Distance function
sup	—	Supremum	Universal quantifier
inf	—	Infimum	Existential quantifier
∧, ∨	U+2227, U+2228	Max, Min	Continuous connectives
·̇	—	Truncated	Truncated arithmetic
⊨_ε	—	Approx satisfies	Within ε of true
[0,1]	—	Unit interval	Truth values

Metatheory. Compactness theorem holds. Löwenheim-Skolem holds (with density character). Ultraproducts generalize naturally. Types are Keisler measures. Stability theory extends (NIP, stable theories). Complete for metric structures. Omitting types theorem has metric version.

Applies to. Banach space theory. Operator algebras (C*-algebras, von Neumann algebras). Probability theory (random variables). Metric geometry. Asymptotic analysis. Approximate reasoning formalizations.

Limitations. Convention (0=true) counterintuitive. Requires uniform continuity everywhere. Not all classical results transfer. Compact metric spaces easier than general. Tool support limited. Learning curve from classical model theory. Some combinatorial arguments don't adapt.