The Info List - Material Implication (rule Of Inference)

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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)


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)



displaystyle P



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.



displaystyle P

is the statement "it is a bear" and


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.