14  Truth-functional semantics for quantifiers

In order to fully capture the nuance of FOL, we need to first introduce an independent function for FOL, and then briefly redefine our truth function.

14.1 Variable assignment function \(g\)

We begin by first defining \(g\) for the set of all variables \(\mathcal{V}\) such that:

\[g: \mathcal{V} \to \mathbb{D}\]

That is, \(g\) assigns each variable to an object in the domain. In application:

\[g: \chi \mapsto \omega\]

where \(\chi \in \mathcal{V}\) and \(\omega \in \mathbb{D}\).

14.2 The truth function \(v\) redefined

The truth function in FOL requires \(g\) in order to evaluate a sentence. (More precisely: requires \(g\) in order to evaluate a sentence in which a variable occurs, but we want \(v\) to cover all sentences in FOL.) We redefine \(v\) for FOL like this:

\[v_{i, g} : \phi \to \{1, 0\}\]

That is, the truth function returns a value of 1 or 0 for a sentence \(\phi\) on interpretation \(i\) and variable assignment \(g\). And, in application:

\[ v_{i, g}(\phi) = 1 \quad \text{or} \quad v_{i, g}(\phi) = 0\]

14.3 Truth-functional definitions

With these two amendments, we now define the truth-functional semantics for universal and existential as follows:

14.3.1 Existential (\(\exists\))

\[ v_{i,g}(\exists \chi \phi) = \begin{cases} 1 ~ \text{iff for some} ~\omega \in \mathbb{D}, v_{i, g[\chi\mapsto\omega]}(\phi)=1\\ 0 ~ \text{otherwise} \end{cases} \]

14.3.2 Universal (\(\forall\))

\[ v_{i,g}(\forall \chi \phi) = \begin{cases} 1 ~ \text{iff for every}~ \omega \in \mathbb{D}, v_{i, g[\chi\mapsto\omega]}(\phi)=1\\ 0 ~ \text{otherwise} \end{cases} \]