Material implication (rule of inference)
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|Rules of inference|
|Rules of replacement|
|Rules of inference|
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- or (i.e. either must be true, or must not be true) 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 (i.e. the latter logically follows from the former);
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, we have by the law of excluded middle (i.e. either must be true, or must not be true).
Subsequently, since , can be replaced by in the statement, and thus it follows that (i.e. either must be true, or must not be true).
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:
|P||Q||¬P||P→Q||¬P ∨ Q|
An example is:
- 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.
- 1st: If it is a bear, then it can swim — T
- 2nd: If it is a bear, then it can not swim — F
- 3rd: If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
- 4th: If it is not a bear, then it can not swim — T (as above)
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".