The assumed antecedent of a conditional proof is called the
conditional proof assumption (
CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion
necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that
if it were true it would lead to the consequent. Conditional proofs are of great importance in
mathematics. Conditional proofs exist linking several otherwise unproven
conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently. A famous network of conditional proofs is the
NP-complete class of complexity theory. There is a large number of interesting tasks (see
List of NP-complete problems), and while it is not known if a polynomial-time solution exists for any of them, it is known that if such a solution exists for some of them, one exists for all of them. Similarly, the
Riemann hypothesis has many consequences already proven. == Symbolic logic ==