In [Mathematical Logic] by Chiswell and Hodges, within the context of natural deduction and the language of propositions LP (basically like here) it is asked to show, by counter-example that a certain 'sequent rule' is 'unacceptable'.
I suppose the proof should follow an example a few pages earlier that shows that the sequent rule $$(p_0 \to p_1 \vdash p_1)$$ is unacceptable due to the following counter-example: let both $p_0$ and $p_1$ mean $(2=3)$. The book argues that indeed if $(2=3)$ then $(2=3)$, so the left side is true, but the right side: $(2=3)$ is false. The conclusion is that we found a counter-examples and the sequent rule is unacceptable.
It is now asked to prove that this sequent rule is unacceptable:
If $$(\Gamma \vdash (\phi \lor \psi))$$ is correct (i.e has a derivation), then at least one of $$(\Gamma \vdash (\phi))$$ and $$(\Gamma \vdash (\psi))$$ is also correct.
The book hints that one should first try to give a counter-example 'for both sequents $(\vdash p_0)$ and $(\vdash \lnot p_0)$'. What could such a $p_0$ be? Does the book mean using as $p_0$ something like 'red is square'? Or some liar kind of sentence 'this sentence is false'? Or some known undecidable statement (which I doubt due to the level of this book)? Or am I totally off and misunderstand something?