12 Interpretation in FOL
Interpretations in FOL involve three components:
- specification of a domain of discourse (\(\mathbb{D}\))
- assignment of individual objects (in the \(\mathbb{D}\)) to constants
- assignment of \(n\)-tuples of objects (in the \(\mathbb{D}\)) to predicates
Thus interpretation more generally is often defined as an ordered pair:
\[\mathcal{I} = \langle \mathbb{D}, i\rangle\]
where:
- \(\mathbb{D}\) is a non-empty domain of discourse, and
- \(i\) is an interpretation function that does two things:
- assigns an object from \(\mathbb{D}\) to each constant symbol, and
- assigns a subset of \(\mathbb{D}^n\) to each \(n\)-ary predicate symbol.
12.1 The interpretation function in FOL
In FOL, the \(i\) function involves two things:
- the interpretation of constants, and
- the interpretation of predicates.
12.1.1 Interpretation of constants
The interpretation function, first, involves assigning to each constant symbol \(\gamma\) an object \(\omega\) from the domain \(\mathbb{D}\). Suppose \(\Gamma\) is the set of constants. Then:
\[i : \Gamma \to \mathbb{D}\]
And, in application:
\[i: \gamma \mapsto \omega\]
where \(\omega \in \mathbb{D}\).
12.1.2 Interpretation of predicates
The interpretation function, second, involves assigning to each predicate an extension. The extension of an \(n\)-ary predicate is the set of ordered \(n\)-tuples of objects from the domain. Now, where:
- \(\Pi^n\) is the set of all \(n\)-ary predicate symbols,
- \(\mathbb{D}^n\) is the set of all possible \(n\)-tuples of objects in the domain, and
- \(\mathcal{P}(\mathbb{D}^n)\) is the powerset of \(\mathbb{D}^n\), that is, the set of all subsets of \(\mathbb{D}^n\),
then:
\[i : \Pi^n \to \mathcal{P}(\mathbb{D}^n)\]
That is, interpretation assigns a subset of \(\mathbb{D}^n\) to each predicate. And, in application, for some predicate symbol \(\pi \in \Pi^n\):
\[i: \pi \mapsto \{ \langle \omega_1, \ldots, \omega_n\rangle, \langle \omega'_1, \ldots, \omega'_n\rangle, \ldots \}\]
where each \(\omega_i \in \mathbb{D}\).