Team Semantics

Origin. Hodges (1997), Väänänen (2007). Dependence logic. Sets of assignments. Independence-friendly logic semantics. Foundation for dependence concepts.

Models. Truth relative to teams (sets of assignments). Dependencies expressible in object language. Compositional semantics for IF logic. Captures database-like dependencies.

Formalism.

Teams: Team X = set of assignments. Assignment s: Var → Domain. X satisfies φ: M ⊨ₓ φ. Set-based satisfaction.

Atomic dependence: =(x₁,...,xₙ,y): value of y depends on x₁,...,xₙ. M ⊨ₓ =(x̄,y) iff for all s,s' ∈ X: s(x̄) = s'(x̄) implies s(y) = s'(y). Functional dependency.

Independence: x ⊥_z y: x independent of y given z. No correlation between x, y holding z fixed. Database-style independence. Key notion in dependence logic.

Downward closure: If M ⊨ₓ φ and Y ⊆ X, then M ⊨ᵧ φ. First-order fragment has this. Dependence atoms don't. Union closure instead.

Union closure: Dependence atoms: if M ⊨ₓ φ and M ⊨ᵧ φ, then M ⊨ₓ∪ᵧ φ. Complements downward closure. Characterizes dependency statements.

Expressive power: Dependence logic = Σ¹₁ (existential SO). Captures NP. More expressive than FO. Maintains compositional semantics.

Extensions: Independence logic. Inclusion logic. Exclusion logic. Team-based modal logic.

Symbols.

Symbol	Unicode	Meaning
X	—	team
=(x̄,y)	—	dependence atom
x ⊥_z y	—	conditional independence
⊨ₓ	—	satisfaction by team X

Metatheory. Team-relative truth. Dependence expressibility. Compositionality. Database connections.

Applies to. Database theory. Independence concepts. IF logic. Expressiveness studies.

Limitations. Complexity increase. Non-classical behavior. Technical sophistication. Limited tools.