I feel like a bit of an idiot, but I can't figure this question out:
Suppose any sentences $X$ and $Y$ are correctly decidable in any arbitrary system $S$, then $X \rightarrow Y$ is also correctly decidable
Using the definition that if a sentence is correctly decidable it is either provable and true or refutable and false, I have got:
There are four cases: $X$ and $Y$ both true and provable, $X$ true and provable and $Y$ false and refutable, $X$ false and refutable and $Y$ true and provable, or $X$ and $Y$ both false and refutable.
I see how this relates to the truth or falsity of $X \rightarrow Y$, via the truth table: eg. if $X$ and $Y$ are both true and provable then $X \rightarrow Y$ is true. But why would it be provable?
Any help is appreciated.
Assuming your system includes standard propositional logic, you can prove $X \to Y$ from $Y$ or from $\neg X$.