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14.6 When All Else Fails: Reductio!

In the last section, you learned that reductio is a good idea whenever your conclusion is a wide-scope negation.

When all else fails, try a reductio.

But that’s not the only time it is a good idea. Here’s some general advice: if you run out of ideas, and if you’ve looked at the main connectives and none of the normal plans work, then try reductio.

For example, consider this proof:

1. Q
2. ~(Q&~R)
Thus,
3. R

You might be thinking: no fair! We’re not learning about a premise with negation around a conjunction until Chapter 15.

That’s true. But Chapter 15 will make more sense if you understand this idea: sometimes reductio is the right idea, even if the conclusion isn’t a wide-scope negation.

Here our conclusion is R, a literal, and we know that in that situation we look around in the proof for other occurrences of it. What we notice is that R appears buried deep in the second premise, and it’s not clear yet how that will help us.

So we also look at the main connectives of our premises. Premise 1, Q, is a literal, so again we look around. It also appears in the second premise, so we can tell that the second premise is going to be key.

Premise 2 is a wide-scope negation, and it is a pattern we cannot apply ~Elim to.

So we are stuck. Looking at the main connectives hasn’t helped us make progress yet.

Often proofs are about having the right ideas, not dozens of lines of writing.

So what do we do? If you read the title of this section, you’ll know how to start. See if you can figure the rest out below.

The point of this section is to prepare you for Chapter 15. When things start to get harder, we’ll use reductio in creative ways.

When your conclusion is a main connective negation, reductio is a good idea. But we’ll need reductio in many other circumstances as well!