Prove (I,E) ∈ D

Question

Let be any term of Predicate logic.

Let I be any interpretation, with domain . Let be any environment.

How do I prove $t^{(I,)} ∈$ using structural induction?

Answer 1

Assuming that the question is:

Let $t$ be any term. Prove, using structural induction, that $t^{(I,E)} \in \mathcal D$,

where an interpretation $I$ is a function from the set of symbols of the language to the domain $\mathcal D$ or relations and functions on $\mathcal D$, and an "environment" $E$ (usually called: variable assignment function) is a function from the set of variables of the language to $\mathcal D$.

Base: the term $t$ is either a constant $c$ or a variable $x$.

For $c$, we have that $c^I \in \mathcal D$ by definition of the interpretation $I$.

For $x$, we have that $E(x) \in \mathcal D$ because $E$ is a variable assignment function, i.e. $E : \text{Var} \to \mathcal D$.

Induction: the term $t$ is $f(t_1,\ldots, t_n)$ and we assume the induction hypotheses that $t_i^{(I,E)} \in \mathcal D$, for $i=1,\ldots,n$.

Thus:

$t^{(I,E)} = [f(t_1,\ldots, t_n)]^{(I,E)}= f^I (t_1^{(I,E)},\ldots, t_n^{(I,E)})$.

But $t_i^{(I,E)} \in \mathcal D$ by hypotheses, and $f^I : \mathcal D^n \to \mathcal D$.

Thus:

$t^{(I,E)} \in \mathcal D$.