# 14.3 Premises not = Conclusions

When you look at the main connective of a sentence, what do you do with it?

The next key idea: what you do about a main connective depends on whether it is a premise or a conclusion.

Here’s the next key idea: what you do depends on whether it is a premise or a conclusion.

When we say to look at the main connectives, we mean you have to look at the main connectives of all the premises of the argument, as well as what the conclusion is.

Elim rules are for “eliminating” a connective. They apply to premises.

Intro rules are for “introducing” a connective. They apply to conclusions.

For example, if your conclusion is P&Q, you are probably going to need to prove P somewhere in the proof, and prove Q somewhere in the proof, and then you can create the conclusion with &Intro.

But if your premise is P&Q, then things are quite different: if you have P&Q, you’re probably going to want to apply &Elim, and bring down the P and bring down the Q.

Here’s a quick practice example. You know by associativity that:

(P&Q)&~R entails P&(Q&~R)

If you pay attention to the main connectives, you’ll be able to do that proof.

In the next two sections we’ll extend this idea, by considering all the possible patterns of main connectives you might face in a premise or a conclusion.