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 Incoming links: Term valuation

Past exam of the logic and set theory course of the University of Oxford / 2008 / A / 3 / a / Solution 2026-10-31

An

L

-structure consists of a nonempty domain

A

, an element

c^{A} \in A

for each constant symbol, a function

f^{A} : A^{n} \to A

for each

n

-ary function symbol, and a relation

R^{A} \subseteq A^{n}

for each

n

-ary predicate symbol. When equality is part of the logic it is interpreted as actual identity on

A

.

An assignment in a first-order structure is a function

s

from logical variables to

A

. It induces a term valuation of first-order terms recursively:

x^{A} [s] = s (x), c^{A} [s] = c^{A}, f (t_{1}, \dots, t_{n})^{A} [s] = f^{A} (t_{1}^{A} [s], \dots, t_{n}^{A} [s]) .

For an atomic formula, substitute these first-order term values into the interpreted relation, or compare them for equality. Extend satisfaction for a set structure through the truth tables of the logical connectives, and through the quantifier clauses

A, s ⊨ \forall x θ ⟺ (\forall a \in A) A, s [x \mapsto a] ⊨ θ,

A, s ⊨ \exists x θ ⟺ (\exists a \in A) A, s [x \mapsto a] ⊨ θ .

Thus the first-order structure and assignment in a first-order structure together determine truth values for all logical formulas. A first-order sentence has no free variables, so its truth does not depend on the assignment in a first-order structure.

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