In propositional logic, material implication[1][2] is a valid rule of replacement that allows 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-

and that either form can replace the other in logical proofs.

cannot be true either; additionally, when

may be either true or false.

" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and

To illustrate this, consider the following statements: Then, to say "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch" (

Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by contraposition).

However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch.

by the law of excluded middle[clarification needed] (i.e. either

Suppose, conversely, we are given

is true, that rules out the first disjunct, so we have

is false, then this entailment fails, because the first disjunct

is true, which puts no constraint on the second disjunct

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: we are given the conditional fact that if it is a bear, then it can swim.

Then, all 4 possibilities in the truth table are compared to that fact.

Thus, the conditional fact can be converted to

, which is "it is not a bear" or "it can swim", where