Material implication (rule of inference)


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 not- or and that either form can replace the other in logical proofs. In other words, if is true, then must also be true, while if is true, then cannot be true either; additionally, when is not true, may be either true or false.
where "" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and can be read as " or Q". To illustrate this, consider the following statements:
Then, to say "Sam ate an orange for lunch" "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. However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit for lunch.

Proof

Suppose we are given that. Then we have by the law of excluded middle.
Subsequently, since, can be replaced by in the statement, and thus it follows that .
Suppose, conversely, we are given. Then if is true, that rules out the first disjunct, so we have. In short,.
This can also be expressed with a truth table:
PQ¬PP → Q¬P ∨ Q
TTFTT
TFFFF
FTTTT
FFTTT

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
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".

The equivalence does not hold in intuitionistic logic

Intuitionistic logic does not treat as equivalent to because
Given, one can constructively transform a proof of into a proof of. In particular, holds in intuitionistic logic. If would hold, then could be derived. However, the latter is the law of excluded middle, which is not accepted by intuitionistic logic.