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 = max(0, ψ - φ) (Łukasiewicz implication under the 0 = true convention)
• φ ∸ ψ = max(0, φ - ψ) (truncated subtraction)

Ben Yaacov and Usvyatsov take {¬, ∸} together with the rational constants as a basis: by Stone–Weierstrass they generate a uniformly dense set of continuous connectives.

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

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

Symbols.

Metatheory. Ben Yaacov, Berenstein, Henson and Usvyatsov (2008) set out the framework, which for normed-space structures is equivalent to Henson's earlier logic of positive bounded formulas with approximate satisfaction. Compactness holds, proved through metric ultraproducts and a continuous Łoś theorem, and downward Löwenheim–Skolem holds with density character in place of cardinality. Ben Yaacov and Pedersen (2010) gave a sound and complete proof system, but the completeness is approximate by necessity: a condition φ ≤ r is derivable only in the sense that φ ≤ r′ is derivable for every r′ > r. Type spaces are compact Hausdorff in the logic topology and carry a second, finer metric topology, and the continuous Ryll-Nardzewski theorem of Ben Yaacov and Usvyatsov makes separable categoricity equivalent to the agreement of the two on every S_n(T). Stability theory transfers, with local stability, definability of types and forking intact; infinite-dimensional Hilbert spaces and atomless probability algebras are the standard separably categorical stable examples. Classical first-order model theory is recovered as the case of {0,1}-valued predicates over the discrete metric.

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

Limitations. The signature must be bounded and uniformly continuous with a declared modulus, so unbounded metric spaces, unbounded operators and unbounded function symbols are outside the framework and can be treated only after rescaling or restriction to bounded pieces. Equality is not primitive: d is a predicate and structures are taken modulo distance zero, so nothing below the metric identification is visible. Definable sets are the zero-sets of definable predicates and are not closed under complement, so there is no Boolean algebra of definable sets and arguments that count them or take differences do not transfer. Provability is approximate — the completeness theorem yields φ ≤ r′ for every r′ > r rather than φ ≤ r — so exact conditions are never derived. Connectives are admitted only up to uniform density, so the language is closed under uniform limits rather than under arbitrary continuous functions, and a predicate that is continuous but not uniformly so is not a formula.

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