Section Progress:

10.4 vIntro

Consider this simple proof:

You know every connective has Intro and Elim rules. The Elim rules are for using a connective you already have. The Intro rules are for building a new sentence with the connective.

Here we are building a disjunction from one of the disjuncts, so this is vIntro.

If you already know P, it might seem odd to prove PvQ. Even it it seems odd, once we learn more difficult formal proofs you'll see why vIntro is important.

The key thing to understand is that no matter how odd vIntro seems, it is certainly valid. Think about the truth table for disjunction: if we know P is true, then PvQ must be true, because the truth of any disjunct makes the whole disjunction true.

Furthermore, from P you can infer PvX, for any sentence all. X could be big or small, atomic or complex. Given how v works, this will always be valid.

First let's practice. Here's an argument:

1. ~~P&~~~Q
Thus,
2. Rv~Q

Recall that with &Intro you can put the conjuncts in any order you choose.

vIntro works similarly: you can add a disjunct in front of or behind the sentence you have, just like you did to make Rv~Q.

Here's another practice problem:

1. P
Thus,
2. (~~QvP)v(S&~T)

Notice how the last line uses vIntro to add the complex sentence S&~T to what we already have. That's great: if you have any sentence, like ~~QvP, you can make a new disjunction with anything you want.

10.4 vIntro