Argument

Material implication (rule of inference)

Overview

In classical propositional logic, material implication is a valid rule of replacement that allows 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-

{\displaystyle P}

{\displaystyle Q}

and that either form can replace the other in logical proofs. In other words, if

{\displaystyle P}

is true, then

{\displaystyle Q}

must also be true, while if

{\displaystyle Q}

is not true, then

{\displaystyle P}

cannot be true either; additionally, when

{\displaystyle P}

is not true,

{\displaystyle Q}

may be either true or false.

→

⇔

∨

{\displaystyle P\to Q\Leftrightarrow \neg P\lor Q,}

where "

⇔

{\displaystyle \Leftrightarrow }

" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and

∨

{\displaystyle \neg P\lor Q}

can be read as "(not P) or Q". To illustrate this, consider the following statements:

{\displaystyle P}

: Sam ate an orange for lunch.

{\displaystyle Q}

: Sam ate a fruit for lunch.

Then, to say "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch" (

→

{\displaystyle P\to Q}

). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by contraposition). However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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The chief business of the logician is to classify arguments; for all testing clearly depends on classification. The classes of the logicians are defined by certain typical forms called syllogisms. For example, the syllogism called _Barbara_ is as follows: S is M; M is P: Hence, S is P. Or, to put words for letters— Enoch and Elijah were men; all men die: Hence, Enoch and Elijah must have died.
Charles S. Peirce, Chance, Love, and Logic
States what Material implication is.
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This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of interrelated existence as disclosed in each immediate occasion of experience. The key to the pattern means this fact:—that from a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also…
Alfred North Whitehead, Science and the modern world
States what Material implication is.
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In the proposition, “Wine is good,” it is understood with equal readiness, though for somewhat different reasons, that the assertion is not intended to be universal, but particular.(28) As is observed by Professor Bain,(29) the chief examples of Indefinite propositions occur “with names of material, which are the subjects sometimes of universal, and at other times of particular predication. ‘Food is chemically…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Material implication is.
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Philosophers who discussed Material implication (rule of inference)

C. I. Lewis

Related ideas

Place in the tradition

Part of Intermediate logic.

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