In propositional logic, material implication  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
" is a metalogical symbol representing "can be replaced in a proof with." Formal notation The material implication rule may be written in sequent notation:
( P → Q ) ⊢ ( ¬ P ∨ Q )
displaystyle (Pto Q)vdash (neg Plor Q)
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)
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.
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
P ∧ ¬ Q
displaystyle Pland neg Q
, 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.