Material implication (rule of inference)
|Rules of inference|
|Rules of replacement|
In propositional logic, material implication  is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction if and only if the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and can replace each other in logical proofs.
The material implication rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;
or in rule form:
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "";
where and are propositions expressed in some formal system.
- If it is a bear, then it can swim.
- Thus, it is not a bear or it can swim.
where is the statement "it is a bear" and is the statement "it can swim".
If it was found that the bear could not swim, written symbolically as , then both sentences are false but otherwise they are both true.