Material implication (rule of inference)

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.

Proof

Suppose we are given that . Then, since we have by the law of excluded middle, it follows that .

Suppose, conversely, we are given . Then if P is true that rules out the first disjunct, so we have Q. In short, .[3]

This can also be demonstrated 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

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.

References

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