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14.4 Know Your Premises

In this section we look at what to do for any main connective you’ll run into in a premise.

& premise: bring down the conjuncts.

We’ve already discussed conjunction: if you have a conjunction, no matter how complex, just apply &Elim and bring down each conjunct.

v premise: start proof by cases by assuming a disjunct.

Next: if your premise’s main connective is a disjunction, you’ll need to plan out vElim and do proof by cases.

But remember, planning out vElim doesn’t mean you can just bring down a disjunct in the parent proof, as if it were a conjunction. vElim requires assuming a disjunct in a new subproof.

Now you should be able to do this.

When we say “main connective of a premise,” the same idea applies to the main connective of any sentence you already have.

When we talk about what to do with the main connective of a premise, we actually mean something more general: you do the same thing for any sentence you already have, whether it is a premise, or the assumption of a subproof, or an intermediary conclusion you have proven.

For example, when you assumed P&R on line 2 above, you probably noticed that you should bring down the conjunct R with &Elim.

The strategic ideas we are giving you therefore help you throughout your proof, not just at the first step. The main connectives are always the key!

Now we’ll consider the final Boolean connective: ~.

If the main connective of a premise is negation, things get more complicated.

~~ premise: apply ~Elim.

The first possibility to consider is when you have two negations, like ~~(P&Q). Then you can do ~Elim.

But it is rare that you are given two negations like that. If you have one negation, and it isn’t paired with another one, then you won’t be able to do ~Elim.

So we need to consider what you can do in that situation. With one ~ alone, there are just three possibilities:

  1. ~ is around v, as in: ~(PvQ)
  2. ~ is around &, as in: ~(R&S)
  3. ~ is around atomic sentence, as in: ~T

Here’s what you need to know at this point. The first two possibilities, ~(PvQ) and ~(R&S) are messy, and we deal with them in the next chapter.

For now, we’ll just talk about the third possibility: ~T.

With a literal premise: look for other occurrences of the same sentence in the rest of the proof.

We treat negation around an atomic the same way we treat an atomic sentence, so we’ll cover both at once here. With a literal, you cannot apply any Elim rule. What you do is look for other occurrences of the same sentence somewhere else in the proof.

Those recurring patterns will hold the key for what to do.

For example, in the proof at the top of this section, from (P&R)v(Q&R) to R, when you got to R in the subproof on line 3, you didn’t need to worry about what Elim rule to apply to it.

Once you hit a literal, you stopped to look around, and hopefully you noticed that there was another R in the second disjunct of your premise.

Here’s another problem to practice on.

This is such an important skill, we’ll do one more practice problem. Students often get stuck doing a proof because either they don’t know where to look, or they don’t know what to do with what they’ve found.

In this section you’ve learned to look at the main connective of your premises, and you’ve learned what to do with each type of pattern you might find there.