I am having trouble with understanding Kalmar's PL completeness proof; specifically the case of negation in the lemma it is using.
The negation case seems to be using a proof by cases to exhaust all possibilities of truth value of $\beta$. I suppose that probably works as well. But that seems to go against how I understand induction works.
We are trying to prove if $\alpha$ is $T$ under the assignment then $B_1^{'},...B_k^{'}\vdash\alpha$, so shouldn't we assume $\alpha$ to be $T$, i.e. $\lnot\beta$ to be $T$, instead of just $\beta$?
If we use this approach, then$\beta$ would be $F$. But the inductive hypothesis applies to $\beta$, so $B_1^{'},...B_k^{'}\vdash\lnot\beta$. Since we are after $B_1^{'},...B_k^{'}\vdash\alpha$, and $\alpha=\lnot\beta$, we are done.
Likewise, if $\alpha$ is $F$, then $\lnot\beta$ is $F$. That means $\beta$ is $T$, and so since the inductive hypothesis holds for $\beta$, $B_1^{'},...B_k^{'}\vdash\beta$. Now $\alpha=\lnot\beta$ is $F$, so we are after $B_1^{'},...B_k^{'}\vdash\lnot\alpha$, i.e. $\lnot\lnot\beta=\beta$. This is exactly what we showed.
