Material implication (rule of inference)

For other uses, see Material implication.
Not to be confused with Material inference.

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-P or Q and can replace each other in logical proofs.

Where "" is a metalogical symbol representing "can be replaced in a proof with."

Formal notation

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 "";

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

where and are propositions expressed in some formal system.


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.


  1. Hurley, Patrick (1991). A Concise Introduction to Logic (4th ed.). Wadsworth Publishing. pp. 364–5.
  2. Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.
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