30.3 Shortcut: ∀Intro with Conditional Proof
Quantificational claims are often a universal quantifier around a conditional. For example, recall the first Aristotelian form, All Ps are Q: Ax(P(x)->Q(x)).
We can prove those claims with the rule we just learned, ∀Intro. Let’s first review how to do it the long way.
Whenever we do a proof of a universal around a conditional like this, it will always have the same pattern: there will be a subproof for the arbitrary boxed constant [a], and then a subproof for the conditional, assuming the antecedent.
In order to make this process more efficient, we will allow ourselves this shortcut: instead of making those assumptions on separate subproofs, we’ll combine them into one subproof.
So on that assumption line, we first box the constant [a], then leave one space, and then write the formula for the conditional assumption, P(a).
This shortcut makes a lot of sense: if we want to prove that all Ps are R, we don’t have to talk about a totally arbitrary object of the domain. All we have to do is talk about an arbitrary object that is P, and prove that it is also R.
Let’s see how much cleaner the proof looks this way.
Remember, this shortcut is just a convenience. If it is confusing, you never need to use it!
You can always do the proof the long way, and first prove the arrow before the universals.
Once you get used to it, though, the shortcut makes the proofs much cleaner and saves time too.