Conditional statement (logic): Difference between revisions

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{{Expert-subject|Mathematics|date=January 2012}}

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In [[philosophy]], [[logic]], and [[mathematics]], a '''conditional statement''' is a [[proposition]] that can be written in the form "If ''p'', then ''q''," where ''p'' and ''q'' are propositions. The proposition immediately following the word "if" is called the hypothesis (also called antecedent). The proposition immediately following the word "then" is called the conclusion (also called consequence). In the aforementioned form for conditional statements, ''p'' is the hypothesis and ''q'' is the conclusion. A conditional statement is often called simply a '''conditional''' (also called an '''implication'''). Unlike the [[material conditional]], a conditional statement need not be truth-functional.<ref>Barwise and Etchemendy 1999, p. 178-179</ref> Conditional statements are often [[symbolic logic|symbolized]] using an arrow (→) as ''p'' → ''q'' (read "''p'' implies ''q''"). The conditional statement in symbolic form is as follows:

* <math>p \rightarrow q</math>

As a proposition, a conditional statement is either [[truth|true]] or false. A conditional statement is true [[if and only if]] the conclusion is true in every case that the hypothesis is true. A conditional statement is false if and only if a [[counterexample]] to the conditional statement exists (that is, it is a [[material conditional]]). A counterexample to a conditional statement exists if and only if there is a case in which the hypothesis is true, but the conclusion is false.

Examples of conditional statements include: