Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. Proving this requires assuming the premise and deriving, from that assumption, the consequent of the conditional. By proving the connection between the antecedent and the consequent, the assumption of the antecedent is justified post hoc.
For example, I claim that "if you don't leave now, you'll be late for work". I prove it with the following argument:
It takes twenty minutes to get to work.
You're supposed to start work in twenty minutes.
Assume you don't leave now.
When you do leave, you'll arrive after the time you're supposed to start.
∴ If you don't leave now, you'll be late for work.
Note that I haven't proved that you'll be late for work: I've only proven the conditional, that the consequent follows necessarily from the antecedent.
It is called ConditionalProof, because we have not proved the truth of r; we have only proved that if q is true then r is true.
We recommend that a Proof by Contradiction be one that begins with p and ~q and ends up obtaining the negation of the premise, and that a Reductio Ad Absurdum Proof be one that ends up obtaining any contradiction of a known truth.
Although RAA proofs are often easier and more convenient, a direct proof is preferred for the reason that RAA depends for its validity on the assumption that the unprovability of the negation of p is tantamount to the provability of the negation of the negation of p.
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