Material implication (rule of inference)
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|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 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.
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.
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 P is true that rules out the first disjunct, so we have Q. In short, .
This can also be demonstrated with a truth table:
|P||Q||¬P||P→Q||¬P ∨ Q|
An example is:
- 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.