Material implication (rule of inference)

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Material implication
TypeRule of replacement
FieldPropositional calculus
StatementP implies Q is logically equivalent to not- or . Either form can replace the other in logical proofs.
Symbolic statement

In propositional logic, material implication[1][2] 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 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 not 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," and P and Q are any given logical statements. 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" 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.

Partial proof[edit]

Suppose we are given that . Then, we have by the law of excluded middle[clarification needed] (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, .[3] 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
T T F T T
T F F F F
F T T T T
F F T T T

Example[edit]

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)

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".

References[edit]

  1. ^ Patrick J. Hurley (1 January 2011). A Concise Introduction to Logic. Cengage Learning. ISBN 978-0-8400-3417-5.
  2. ^ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.
  3. ^ Math StackExchange: Equivalence of a→b and ¬ a ∨ b