Past exam of the logic and set theory course of the University of Oxford / 2011 / A / 2 / c

Extend the first-order variable assignment recursively to first-order terms byFor atomic formulas, $\mathcal{A}\models t=s[v]$ exactly when $\tilde{v}(t)=\tilde{v}(s)$. Extend to the Boolean connectives by their truth tables, for example implication is false exactly when its antecedent is true and its consequent false. Quantifiers use modified first-order variable assignments:This recursion defines satisfaction for a set structure for every first-order formula.

Semantic consequence under an assignment means that for every first-order structure $\mathcal{A}$ with the relevant first-order variable assignment $v$ into its domain,More explicitly, in a fixed structure this is the satisfaction implication at $v$; logical consequence requires that implication in all structures and assignments. An assignment takes values in a particular domain, so a single set map is not silently shared between unrelated domains.

By induction on first-order formulas, their truth values depend only on the values of their free variables. A first-order sentence has none. If all formulas here are sentences, every displayed satisfaction statement, and therefore the consequence condition, is independent of $v$.