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19.5 Biconditional: Twice the Work

There’s a reason why we have been focusing just on the single conditional in this chapter.

The ideas for the biconditional are exactly the same. So if you understand the tricks for the conditional, you know pretty much all you need to know.

For example, when you have a biconditional premise, don’t just start assuming one side or the other. Instead, be patient!

And if you have a biconditional conclusion, the best way to proceed is always to start two subproofs to set up <->Intro.

Twice the conditional is twice the work.

Here’s the one key idea that applies to the biconditional: twice the conditional is twice the work.

For example, consider this argument.

1. ~(P<->Q)
2. ~(P&Q)

When we look at proving this, our first thought is, let’s do a reductio of that negation conclusion. That gives us a conjunction we could break up with &Elim whenever we need to.

But then we have to deal with ~(P<->Q).

Well, we know we have to build it from the inside, which will complete the reductio that we started.

The only difference is that building P<->Q takes two subproofs instead of one.

Let’s do it. Since you could do &Elim in several places, pay attention to the subproofs to see how we did it.

After dealing with single conditionals, the biconditional can feel like a lot of work. But the ideas for it are exactly the same.