In classical propositional logic, material implication 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 and that either form can replace the other in logical proofs is 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 can be read as "(not P) or Q". To illustrate this, consider the following statements:

• : Sam ate an orange for lunch.

• : Sam ate a fruit for lunch.

Then, to say "Sam ate an orange 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.

Proof

by the law of excluded middle must not be true).

must not be true).

.

This can also be expressed with a truth table:

Example

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.

1. If it is a bear, then it can swim — T

2. If it is a bear, then it can not swim — F

3. If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.

4. If it is not a bear, then it can not swim — T (as above)

is the statement "it can swim".

Differences with implication in intuitionistic logic

Intuitionistic logic because

could be derived. However, the latter is the law of excluded middle without knowing which case applies).

References