Material implication (rule of inference)

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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 (i.e. either must be true, or must not be true) and that either form can replace the other in logical proofs.

Where "" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given logical statements.

Formal notation[edit]

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 "";

or as the statement of a truth-functional tautology or theorem of propositional logic:

where and are propositions expressed in some formal system.

Partial proof[edit]

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, .[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


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".


  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. CS1 maint: discouraged parameter (link)
  3. ^ Math StackExchange: Equivalence of a→b and ¬ a ∨ b