Material implication (rule of inference)

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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 if and only if 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.

P \to Q \Leftrightarrow \neg P \or Q

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

Formal notation[edit]

The material implication rule may be written in sequent notation:

(P \to Q) \vdash (\neg P \or Q)

where \vdash is a metalogical symbol meaning that (\neg P \or Q) is a syntactic consequence of (P \to Q) in some logical system;

or in rule form:

\frac{P \to Q}{\neg P \or Q}

where the rule is that wherever an instance of "P \to Q" appears on a line of a proof, it can be replaced with "\neg P \or Q";

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

(P \to Q) \to (\neg P \or Q)

where P and Q are propositions expressed in some formal system.


If it is a bear, then it can swim.
Thus, it is not a bear or it can swim.

where P is the statement "it is a bear" and Q is the statement "it can swim".

If it was found that the bear could not swim, written symbolically as P \and \neg Q, 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.