I read about structures and interpretations today. I've described them below this paragraph. Have I accurately described them? If not, what have I incorrectly described?
A structure, $\mathscr{A}$, is identical with the ordered set $\langle \mathcal{A}, \mathcal{\sigma}, \mathcal{I}\rangle$, such that
- $\mathcal{A}$ designates the domain of discourse,
- $\mathcal{\sigma}$ designates the signature, and
- $\mathcal{I}$ designates the interpretation.
The signature, $\sigma$, is identical with the ordered set $\langle \mathrm{S}_\mathrm{{func}},\mathrm{S}_\mathrm{{pred}}, ar \rangle$, such that
${\mathrm{S_\mathrm{{func}}}}$designates the set of functional relations $($for example, in predicate logic $\mathrm{S_\mathrm{{func}}}$ designates the set $\{ \land, \lor\ldots,\top, \bot\ldots \}\;)$,
$\mathrm{S_\mathrm{{pred}}}$ designates the set of predicates and non-functional relations $($for example, $\mathrm{S_\mathrm{{pred}}}$ designates the set $\{A,\ B\ldots,\ \ge\ldots \})$, and
$ar$ designates the function $f_1:\mathrm{S_\mathrm{{pred}}}\cup \mathrm{S_{\mathrm{{func}}}} \mapsto \mathbb{Z}_{\gt0}$, which ascribes arities to predicates and to the functional and non-functional relations $($for example, $f_1:\mathrm{S_\mathrm{{pred}}}\cup \mathrm{S_\mathrm{{func}}} \mapsto \mathbb{Z}_{\gt0};\; \land\mapsto2,\ \lor\mapsto2,\ \neg\mapsto1\ \ldots,A\mapsto1\ldots)$
The domain of discourse, $\mathcal{A}$, is identical with the set of all entities.$($For example, $\{a, b, c, d, e\ldots\}$$)$
The interpretation, $\mathcal{I}$, is identical with the ordered set $\langle \mathcal{W},\mathcal{R}, \mathcal{v}\rangle$
- $\mathcal{W}$ designates the set of entities the interpretation considers. $($For example, $\{a,b,c,d\})$
- $\mathcal{R}$ designates the set of all relations that obtain among the entities the interpretation will consider. $($For example $\{Ca,Db\ldots Ca\land Db\ldots \})$
- $\mathcal{v}$ designates the function $f_2:\mathcal{R}\mapsto\{\top,\bot\}$, which ascribes truth values to the elements of $\mathcal{R}$. $($For example, $f_2:\mathcal{R}\mapsto\{\top,\bot\};Ca\mapsto\top\ldots,\land_{\langle\top\top\rangle}\mapsto\top,\land_{\langle\top\bot\rangle}\mapsto\bot\ldots\})$