In Chiswell's Mathematical Logic, there is the following problem is posed using the language of propositions (LP):
In some systems of logic (mostly constructive systems where 'true' is taken to mean 'provable') there is a rule
if $(\Gamma\vdash(\phi\,\lor\psi))$ is a correct sequent then at least one of $(\Gamma\vdash\phi)$ and $(\Gamma\vdash\psi)$ is also correct.
By giving a counterexample to a particular instance, show that this is unacceptable as a rule for LP. [Start by giving counterexamples for both the sequents $(\vdash p_0)$ and $(\vdash(\neg p_0))$.]
In my first post on this question, it was answered that one can let $\Gamma = \{p_0\,\lor\,(\neg p_0)\}$. From there, one can deduce neither $(\Gamma\vdash p_0)$ nor $(\Gamma\vdash (\neg p_0))$.
But this does not proceed by the counterexample route that the author suggests. Namely, he seems to imply that we should (1) take an interpretation of propositional variables $p_0,\,p_1,\,\ldots$ (setting each one equal to T or F), and (2) write $\phi$ and $\psi$ as functions of these propositional variables, and (3) pick sentences for $\Gamma$ which are all T, all with the conditions that (A) $(\phi\,\lor\psi)$ is evaluated as T, yet (B) both $\phi$ and $\psi$ are evaluated as F.
This, of course, seems impossible. What, then, am I missing? It seems that taking a concrete interpretation of the propositional symbols makes the constructivist's rule irrefutable. (Note: this is supposed to be a very simple problem set, and I am an elementary logician).
