In Propositional Logic, one is often tasked with showing that some particular formula is a theorem of a given deductive system, i.e. $\emptyset \vdash \psi$. These formulas can look very simple and intuitive, e.g. $(\alpha \rightarrow \alpha)$, but the derivation can be rather cumbersome. That examples takes around 6 steps for instance, and is one of the simpler cases.
My question is immediate then: Is there any way of anticipating the correct strategy for making such derivations, and thus avoiding the lengthy 'trial and error' process of applying different axioms (and rules of inference) until something works?
Essentially, are there any 'tricks' that are unlikely to appear in notes etc. but which you are aware of the help look ahead and spot the right path?
For completeness (no pun intended), I've included the particular deductive system I am working with below, and a few indicative theorems.
$((\neg \alpha \rightarrow \alpha)\rightarrow\alpha))$
$(\neg\neg\alpha\rightarrow\alpha)$
$(\neg\alpha\rightarrow(\alpha\rightarrow\beta))$
EDIT
The above cases can be dealt with a little more tidily by applying the Deduction Theorem. Do any other results like this exist? Also, supposing the Deduction Theorem cannot be used, do strategies still exist?
Let $\mathcal{L}_0=\mathcal{L}[\{\neg, \rightarrow\}]$. Define the system $L_0$ as follows:
An axiom of $L_0$ is any formula of $\mathcal{L}_0$ of the form
- (A1) $(\alpha \rightarrow ( \beta \rightarrow \alpha))$
- (A2) $((\alpha \rightarrow ( \beta \rightarrow \gamma) \rightarrow ((\alpha \rightarrow \beta) \rightarrow ( \alpha \rightarrow \gamma)))$
- (A3) $((\neg \beta \rightarrow \neg \alpha ) \rightarrow (\alpha \rightarrow \beta))$
The only rule of inference in $L_0$ is modus ponens, i.e. from $\alpha$ and $(\alpha \rightarrow \beta)$ infer $\beta$.
