Boolean Interpretation is Well-Defined/Proof 2
Theorem
Let $\LL_0$ be the language of propositional logic.
Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation.
Then $v$ is well-defined.
Proof
This is to be done by strong induction on the length of WFFs.
By definition of $v$ being a boolean interpretation, $\map v p$ is well-defined for all $p \in \PP_0$, the vocabulary of $\LL_0$.
A WFF of length $1$ has (trivially) a unique parsing sequence.
Consequently, only a single defining rule for $v$ as a boolean interpretation applies.
So the result holds for all WFFs of length $1$.
Now, suppose the result is true for all WFFs of length $k$ or less.
Let $\mathbf A$ be a WFF of length $k+1$.
There are two possibilities:
Suppose $\mathbf A = \neg \mathbf B$ for some WFF $\mathbf B$.
Then $\mathbf B$ is of length $k$, so by the induction hypothesis has a unique value $v (\mathbf B)$ no matter what parsing sequence is used.
So as $\map v {\mathbf A} = \map {f^\neg} {\map v {\mathbf B} }$, it follows that $\mathbf A$ likewise has a unique value.
Hence the result holds for $k + 1$ in this situation.
Suppose $\mathbf A = \paren {\mathbf B * \mathbf C}$ for some WFFs $\mathbf B$ and $\mathbf C$ and some connective $*$.
By Language of Propositional Logic has Unique Parsability, $*$ must be the unique main connective.
So $\mathbf B$ and $\mathbf C$ are both WFFs shorter than $k + 1$ and therefore by the induction hypothesis have unique values $\map v {\mathbf B}$ and $\map v {\mathbf C}$.
Since $\mathbf A = \paren {\mathbf B * \mathbf C}$ for a unique main connective $*$, it follows that:
- $\map v {\mathbf A} = \map {f^*} {\map v {\mathbf B}, \map v {\mathbf C} }$
is well-defined.
Hence the result holds for $k + 1$ in this situation.
So the result follows by the Second Principle of Mathematical Induction.
$\blacksquare$
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.5$: Semantics of Propositional Logic: Theorem $1.5.1$