# Conditional proof

A **conditional proof** is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

## Overview

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 actually 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, 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 any of them, one exists for all of them. Similarly, the Riemann hypothesis has many consequences already proven.

## Symbolic logic

As an example of a conditional proof in symbolic logic, suppose we want to prove A → C (if A, then C) from the first two premises below:

1. | A → B | ("If A, then B") |

2. | B → C | ("If B, then C") |

3. | A | (conditional proof assumption, "Suppose A is true") |

4. | B | (follows from lines 1 and 3, modus ponens; "If A then B; A, therefore B") |

5. | C | (follows from lines 2 and 4, modus ponens; "If B then C; B, therefore C") |

6. | A → C | (follows from lines 3–5, conditional proof; "If A, then C") |

## References

- Robert L. Causey,
*Logic, sets, and recursion*, Jones and Barlett, 2006. - Dov M. Gabbay, Franz Guenthner (eds.),
*Handbook of philosophical logic*, Volume 8, Springer, 2002.