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18.3 Conditional Proof

The conditional obeys transitivity: P→Q and Q→R entail P→R. But how would we prove that fact? To do it, we have to know how to prove a conditional: P→R. That’s what this chapter is about.

Now, sometimes proving a conditional is trivial. For example, you can already do this:

1. (P→R)&Q
2. P→R

But that only works because we have the conditional already hidden in the premise, so to speak. What we need to know is how to create a conditional, how to prove one anew when it’s not already hidden up our sleeves.

Conditional proof: the method of proving a conditional by temporarily assuming the antecedent and showing that the consequent follows.

That is what the method of conditional proof is for. A conditional says IF something is true, THEN something else is true. We can prove a claim like that by temporarily assuming the antecedent, and showing that the consequent follows. Given how logical validity works, that means that the consequent really must be true, if the antecedent is.

Here’s how it looks. The argument is this:

1. If Pia is guilty, then Quinn is guilty.
2. If Quinn is guilty, then Raquel is guilty.
3. If Pia is guilty, then Raquel is guilty.

We prove it this way:

Proof: Assume for conditional proof that Pia is guilty, and we want to show that Raquel is guilty. Since we know Pia is guilty (assumption), and if Pia is guilty, then Quinn is guilty (prem 1), it follows that Quinn is guilty (by MP). It then follows that Raquel is guilty (by MP with prem 3), which is what we wanted to show. Done.

Guidelines: Start a conditional proof this way: “Proof: Assume for conditional proof X, and we want to show Y”, where X is the antecedent and Y is the consequent.

You can see that conditional proof uses a temporary assumption, just like some other proof methods: proof by cases and reductio. That’s why it is helpful to declare “assume for conditional proof…”, or “assume for reductio…”. You want to make your proofs perfectly clear for your audience.

Your turn to try. Here’s an argument:

1. If Pia and Raquel are guilty, then Quinn is guilty.
2. Raquel is guilty.
3. If Pia is guilty, then Quinn is guilty.

The key to getting conditional proof right is to structure it correctly. Your temporary assumption is always identical to the antecedent of the conditional you are proving, and the thing you want to show is always the consequent.

Here’s your chance to practice.

Biconditional Proof: you just do two conditional proofs; assume the first side, and prove the second; then assume the second, and prove the first.

Since a biconditional is just two conditionals, the way we prove a biconditional is quite similar. All you do is two conditional proofs.