Material implication (rule of inference)

, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that either form can replace the other in logical proofs.
Where "" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given statements.

Formal notation

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 "";
or as the statement of a truth-functional tautology or theorem of propositional logic:
where and are propositions expressed in some formal system.

Partial proof

Suppose we are given that. Then, since we have by the law of excluded middle, it follows that.
Suppose, conversely, we are given. Then if is true that rules out the first disjunct, so we have. In short,. However if is false, then this entailment fails, because the first disjunct is true which puts no constraint on the second disjunct. Hence, nothing can be said about. In sum, the equivalence in the case of false is only conventional, and hence the formal proof of equivalence is only partial.
This can also be expressed with a truth table:


An example is:
Thus, the conditional fact can be converted to, which is "it is not a bear" or "it can swim",
where is the statement "it is a bear" and is the statement "it can swim".