Substitution Theorem for Well-Formed Formulas/Corollary

Corollary to Substitution Theorem for Well-Formed Formulas

Let $\map {\mathbf A} {x_1, \ldots, x_n}$ be a WFF whose free variables are among $x_1, \ldots, x_n$.

Let $\tau_1, \ldots, \tau_n$ be closed terms, and let each $\tau_i$ be freely substitutable for $x_i$.

Let $a_i = \map {\operatorname{val}_\AA} {\tau_i}$, where $\operatorname{val}_\AA$ denotes the value of $\tau$ in $\AA$.

Then:

$\AA \models_{\mathrm{PL}} \map {\mathbf A} {\tau_1, \ldots, \tau_n}$ if and only if $\AA \models_{\mathrm{PL_A}} \mathbf A \sqbrk {a_1, \ldots, a_n}$

where $\models_{\mathrm{PL}}$ denotes model of sentence, and $\models_{\mathrm{PL_A}}$ denotes model of formula.

Proof

Applying the Substitution Theorem for Well-Formed Formulas $n$ times:

\(\ds \map {\operatorname{val}_\AA} {\map {\mathbf A} {\tau_1, \tau_2, \ldots, \tau_n} }\)

\(=\)

\(\ds \map {\operatorname{val}_\AA} {\map {\mathbf A} {x_1, \tau_2, \ldots, \tau_n} } \sqbrk {a_1}\)

\(\ds \)

\(=\)

\(\ds \map {\operatorname{val}_\AA} {\map {\mathbf A} {x_1, x_2, \ldots, \tau_n} } \sqbrk {a_1, a_2}\)

\(\ds \)

\(\vdots\)

\(\ds \)

\(\ds \)

\(=\)

\(\ds \map {\operatorname{val}_\AA} {\map {\mathbf A} {x_1, x_2, \ldots, x_n} } \sqbrk {a_1, a_2, \ldots, a_n}\)

The result follows from the definitions of model of sentence and model of formula.

$\blacksquare$

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Lemma $\mathrm{II.8.13}$