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. Dependence atoms have this, and so does the whole of dependence logic — a functional dependency that holds across a team still holds across any subteam.

Union closure: If M ⊨ₓ φ and M ⊨ᵧ φ, then M ⊨_{X∪Y} φ. Dependence atoms do not have this. Take X = {s} with s(x)=0, s(y)=0 and Y = {s'} with s'(x)=0, s'(y)=1: each singleton satisfies =(x,y), the union does not. Union closure is inclusion logic's property, not dependence logic's; the two closure conditions are what divide the family, and independence atoms have neither.

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.