Tarskian Semantics

Origin. Tarski (1933). Model-theoretic truth definition. Recursive satisfaction. Convention T. Foundation for formal semantics.

Models. Structures interpret language. Satisfaction recursively defined. Truth = satisfaction by all assignments. Compositionality.

Formalism.

Structure: M = (D, I) D: domain (non-empty) I: interpretation of symbols

Variable assignment: σ : Var → D Maps variables to domain elements.

Term interpretation: ⟦x⟧^M_σ = σ(x) ⟦c⟧^M_σ = I(c) ⟦f(t₁,...,tₙ)⟧^M_σ = I(f)(⟦t₁⟧,...,⟦tₙ⟧)

Satisfaction (base): M, σ ⊨ P(t₁,...,tₙ) iff (⟦t₁⟧,...,⟦tₙ⟧) ∈ I(P) M, σ ⊨ t₁ = t₂ iff ⟦t₁⟧ = ⟦t₂⟧

Satisfaction (connectives): M, σ ⊨ ¬φ iff M, σ ⊭ φ M, σ ⊨ φ ∧ ψ iff M, σ ⊨ φ and M, σ ⊨ ψ M, σ ⊨ φ ∨ ψ iff M, σ ⊨ φ or M, σ ⊨ ψ M, σ ⊨ φ → ψ iff M, σ ⊭ φ or M, σ ⊨ ψ

Satisfaction (quantifiers): M, σ ⊨ ∀x.φ iff M, σ[x↦d] ⊨ φ for all d ∈ D M, σ ⊨ ∃x.φ iff M, σ[x↦d] ⊨ φ for some d ∈ D

Truth: M ⊨ φ iff M, σ ⊨ φ for all σ. Sentence true if satisfied under all assignments.

Convention T: "Snow is white" is true iff snow is white. Material adequacy condition.

Symbols.

Symbol	Unicode	Name	Meaning
⊨	U+22A8	Satisfies	Semantic truth
⟦·⟧	—	Denotation	Interpretation
σ	U+03C3	Assignment	Variable map
M	—	Model	Structure
⊭	U+22AD	Non-satisfaction	Clause for negation: M, σ ⊭ φ
∀	U+2200	Universal quantifier	Clause ranges over all d ∈ D
∃	U+2203	Existential quantifier	Clause ranges over some d ∈ D
↦	U+21A6	Assignment update	σ[x↦d], the modified assignment
∈	U+2208	Membership	Tuple in the interpretation I(P)
D	—	Domain	Non-empty universe of the structure
I	—	Interpretation	Assignment of extensions to non-logical symbols

Metatheory. The definition proceeds by recursion on formula complexity, and the coincidence lemma — M, σ ⊨ φ depends only on σ's values at the free variables of φ — is what allows truth for sentences to be defined as satisfaction by every assignment. Gödel's completeness theorem (1929) shows the semantics is exactly right for first-order derivability: ⊢ φ iff ⊨ φ, with soundness the easy direction; compactness and the Löwenheim–Skolem theorems follow from it. Over RCA₀ the completeness theorem for countable languages is equivalent to WKL₀, so the set existence needed is weak but not nothing. Satisfaction for a set-sized structure is definable in ZFC by a Δ₁ formula, absolute between transitive models, so Tarski's construction is internal to set theory; for the class-sized structure V it is not definable, by Tarski's own undefinability theorem, though Σₙ-satisfaction is definable for each fixed n. Convention T is an adequacy criterion on definitions, not a theorem of the system.

Applies to. Model theory. Formal semantics. Philosophy of language. Logic foundations.

Limitations. Domains are required non-empty and every term denotes, so ∃x(x = x) comes out valid and free logics and empty-domain reasoning are excluded without modification. Interpretation is total and bivalent: no partial predicates, no truth-value gaps, hence no built-in treatment of presupposition failure or vagueness. It is extensional, so coextensive predicates are intersubstitutable everywhere and no intensional or hyperintensional distinction survives — modality, attitudes and content must be added on top as indices or structured meanings. Truth is always truth-in-a-structure relative to an interpretation, and the definition for a language L cannot be given in L, so the stronger metalanguage is forced rather than chosen. Etchemendy (1990) argued that the resulting analysis of logical consequence, quantifying over set-sized structures, is not evidently coextensive with the modal notion it replaces.