# Material conditional

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The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol ${\displaystyle \rightarrow }$ is interpreted as material implication, a formula ${\displaystyle P\rightarrow Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.[citation needed]

Quick facts: IMPLY, Definition, Truth table, Logic gate, N...
Material conditional
IMPLY
Definition${\displaystyle x\rightarrow y}$
Truth table${\displaystyle (1011)}$
Logic gate
Normal forms
Disjunctive${\displaystyle {\overline {x}}+y}$
Conjunctive${\displaystyle {\overline {x}}+y}$
Zhegalkin polynomial${\displaystyle 1\oplus x\oplus xy}$
Post's lattices
0-preservingno
1-preservingyes
Monotoneno
Affineno
Close

Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.