Material implication (rule of inference)
Transformation rules |
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Propositional calculus |
Rules of inference |
Rules of replacement |
Predicate logic |
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.
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
- ↑ Hurley, Patrick (1991). A Concise Introduction to Logic (4th ed.). Wadsworth Publishing. pp. 364–5.
- ↑ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.