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19.2 -> Premises: Be Patient

In the last section you learned that when you have an arrow conclusion, you know exactly what to do first. Start ->Intro proof.

An arrow premise does not tell you what to do right away. Wait!

But here is the key lesson about arrow premises: they are not like that. An arrow premise does not tell you what to do right away.

A very common mistake is to see an arrow premise and then assume the antecedent. But that’s the plan for an arrow conclusion, and you know that premises ≠ conclusions.

That is an understandable temptation, because often times we do assume the antecedent. But the reason is not just because we have an arrow premise and want do ->Elim.

It’s true that we have  an arrow premise, and we start by assuming the antecedent, but hopefully you can see why we did that: it’s because that is the antecedent of our conclusion!

This becomes quite obvious if we just change the example a little.

Both of our premises have P in the antecedent, but it’s still wrong to assume P.

One thing this illustrates is how powerful the ->Intro plan is: like we said in the previous section, an arrow conclusion always tells you what to do. Arrow conclusions are great!

What to do with an arrow premise: be patient!

The second thing it illustrates is what to actually do with an arrow premise: be patient!

In fact, you don’t do anything. You wait. But you wait with anticipation: you stay on the lookout for how to get the antecedent.

Yes, you will use the arrow premise by doing ->Elim, but that does not tell you to jump the gun and assume the antecedent.

Instead, you have to earn the antecedent. Or, to put it another way: you lie in wait for the antecedent, knowing that somewhere or other it will make an appearance.

“P! We’ve been waiting for you!”

And then you can say, “P! We’ve been waiting for you!”

Let’s see if you’ve got it.

Here’s our thought process on this proof.

We look at the conclusion: darn, it’s not an arrow. It’s an atomic, which doesn’t tell us what to do.

So we look at the premises. Premise 1: wide-scope arrow. Let’s be patient and be on the lookout for PvQ. Premise 2: wide-scope conjunction. We can bring down the conjuncts.

The first one, ~P, doesn’t look too useful, but with ~~Q we know we can get Q. Then the light bulb goes off: that can help us get PvQ, which helps us do ->Elim.

Now you know what to do with an arrow premise: be patient and stay on the lookout.