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. P → Q ⇔ ¬ P ∨ Q displaystyle Pto QLeftrightarrow neg Plor Q Where " ⇔ displaystyle 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 → Q ) ⊢ ( ¬ P ∨ Q ) displaystyle (Pto Q)vdash (neg Plor Q) where ⊢ displaystyle vdash is a metalogical symbol meaning that ( ¬ P ∨ Q ) displaystyle (neg Plor Q) is a syntactic consequence of ( P → Q ) displaystyle (Pto Q) in some logical system; or in rule form: P → Q ¬ P ∨ Q displaystyle frac Pto Q neg Plor Q where the rule is that wherever an instance of " P → Q displaystyle Pto Q " appears on a line of a proof, it can be replaced with " ¬ P ∨ Q displaystyle neg Plor Q "; or as the statement of a truth-functional tautology or theorem of propositional logic: ( P → Q ) → ( ¬ P ∨ Q ) displaystyle (Pto Q)to (neg Plor Q) where P displaystyle P and Q displaystyle Q are propositions expressed in some formal system. Example[edit] An example is: If it is a bear, then it can swim. Thus, it is not a bear or it can swim. where P displaystyle P is the statement "it is a bear" and Q displaystyle Q is the statement "it can swim". If it was found that the bear could not swim, written symbolically as P ∧ ¬ Q displaystyle Pland neg Q , then both sentences are false but otherwise they are both true. References[edit] ^ 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. |