Conditional proof

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

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.

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 are 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 a large number of 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")

See also

• Deduction theorem
• Logical consequence
• Propositional calculus

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.