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#REDIRECT [[Conditional statement]]
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In [[logic|philosophical]] and [[mathematical logic|mathematical]] logic, a '''conditional statement''' is a compound [[statement (logic)|statement]], composed of [[declarative sentence]]s or [[proposition]]s ''p'' and ''q'', that can be written in the form "if ''p'' then ''q''". In this form, ''p'' and ''q'' are placeholders for which the antecedent and consequent are substituted, (also known as the condition and consequence or hypothesis and conclusion). A conditional statement is sometimes simply called a '''conditional''' or an '''implication'''. Outside of mathematics, it is a matter of some controversy as to whether the [[truth function]] for [[material implication]] provides an adequate treatment of ‘conditional statements in English’ (a [[sentence]] in the [[indicative mood]] with a [[conditional clause]] attached, i.e., an [[indicative conditional]]; for a more technical treatment, see [[sentence (mathematical logic)|sentences in mathematical logic]]).<ref name="sep-conditionals"/><ref>Barwise and Etchemendy 1999, p. 178-179</ref> That is to say, critics argue that in some non-mathematical cases, the truth value of a compound statement, "if ''p'' then ''q''", is not adequately determined by the truth values of ''p'' and ''q''.<ref name="sep-conditionals"/> Examples of non-truth-functional statements include: "''p'' because ''q''", "''p'' before ''q''" and "it is possible that ''p''".<ref name="sep-conditionals"/> “[Of] the sixteen possible truth-functions of ''A'' and ''B'', [material implication] is the only serious candidate. First, it is uncontroversial that when ''A'' is true and ''B'' is false, "If ''A'', ''B''" is false. A basic rule of inference is [[modus ponens]]: from "If ''A'', ''B''" and ''A'', we can infer ''B''. If it were possible to have ''A'' true, ''B'' false and "If ''A'', ''B''" true, this inference would be invalid. Second, it is uncontroversial that "If ''A'', ''B''" is sometimes true when ''A'' and ''B'' are respectively (true, true), or (false, true), or (false, false)… Non-truth-functional accounts agree that "If ''A'', ''B''" is false when ''A'' is true and ''B'' is false; and they agree that the conditional is sometimes true for the other three combinations of truth-values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when ''A'' and ''B'' are both true, "If ''A'', ''B''" must be true. Some do not, demanding a further relation between the facts that ''A'' and that ''B''.”<ref name="sep-conditionals">{{cite web |first=Dorothy |last=Edgington |editor=Edward N. Zalta |year=2008 |title=Conditionals |work=The Stanford Encyclopedia of Philosophy |edition=Winter 2008 |url=http://plato.stanford.edu/archives/win2008/entries/conditionals/}}</ref>
{{quotation|The truth-functional theory of the conditional was integral to [[Gottlob Frege|Frege]]'s new logic (1879). It was taken up enthusiastically by [[Bertrand Russell|Russell]] (who called it "[[material implication]]"), [[Ludwig Wittgenstein|Wittgenstein]] in the ''[[Tractatus]]'', and the [[logical positivist]]s, and it is now found in every logic text. It is the first theory of conditionals which students encounter. Typically, it does not strike students as ''obviously'' correct. It is logic's first surprise. Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders. It is a strikingly simple theory: "If ''A'', ''B''" is false when ''A'' is true and ''B'' is false. In all other cases, "If ''A'', ''B''" is true. It is thus equivalent to "~(''A''&~''B'')" and to "~''A'' or ''B''". "''A'' ⊃ ''B''" has, by stipulation, these truth conditions.|[[Dorothy Edgington]]|The Stanford Encyclopedia of Philosophy|“Conditionals”<ref name="sep-conditionals"/>}}
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. 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 (which is to say, a conditional statement is true whenever the antecedent is false, or when the consequent and antecedent are both true).
Examples of conditional statements include:
* If I am running, then my legs are moving.
* If a person makes lots of jokes, then the person is funny.
* If the Sun is out, then it is midnight.
* If you locked your car keys in your car, then 7 + 6 = 2.
== Variations of the conditional statement ==
The conditional statement "If ''p'', then ''q''" can be expressed in many ways; among these ways include<ref>Rosen 2007, p. 6</ref><ref>Larson, Boswell, and Stiff 2001, p. 80</ref>:
# If ''p'', then ''q''. (called if-then form<ref>Larson et al. 2007, p. 79</ref>)
# If ''p'', ''q''.
# ''p'' implies ''q''.
# ''p'' only if ''q''. (called only-if form<ref>Larson, Boswell, and Stiff 2001, p.80</ref>)
# ''p'' is [[sufficient condition|sufficient]] for ''q''.
# A sufficient condition for ''q'' is ''p''.
# ''q'' if ''p''.
# ''q'' whenever ''p''.
# ''q'' when ''p''.
# ''q'' every time that ''p''.
# ''q'' is necessary for ''p''.
# A necessary condition for ''p'' is ''q''.
# ''q'' follows from ''p''.
# ''q'' unless ¬''p''.
== The converse, inverse, contrapositive, and biconditional of a conditional statement ==
The conditional statement "If ''p'', then ''q''" is related to several other conditional statements and propositions involving propositions ''p'' and ''q''.<ref>Larson et al. 2007, p. 80</ref><ref>Rosen 2007, p. 8</ref>
=== The converse ===
{{Main|Converse (logic)}}
The converse of a conditional statement is the conditional statement produced when the hypothesis and conclusion are interchanged with each other. The resulting conditional is as follows:
* <math>q \rightarrow p</math>
=== The inverse ===
{{Main|Inverse (logic)}}
The inverse of a conditional statement is the conditional statement produced when both the hypothesis and the conclusion are [[negation|negated]]. The resulting conditional is as follows:
* <math>\lnot p \rightarrow \lnot q </math>
=== The contrapositive ===
{{Main|Transposition (logic)}}
The contrapositive of a conditional statement is the conditional statement produced when the hypothesis and conclusion are interchanged with each other and then both negated. The result, which is equivalent to the original, is as follows:
* <math>\lnot q \rightarrow \lnot p </math>
=== The biconditional ===
{{Main|Logical biconditional}}
The biconditional of a conditional statement is the proposition produced out of the [[Logical conjunction|conjunction]] of the conditional statement and its converse. When written in its standard [[English language|English]] form, the hypothsis and conclusion are joined by the words "if and only if." The biconditional of a conditional statement is equivalent to the conjunction of the conditional statement and its converse. The resulting proposition is as follows:
* <math>p \leftrightarrow q </math>; or equivalently,
* <math>(p \rightarrow q) \and (q \rightarrow p) </math>
== Notes ==
{{Reflist}}
== References ==
* Barwise, Jon, and John Etchemendy. ''Language, Proof and Logic''. Stanford: CSLI (Center for the Study of Language and Information) Publications, 1999. Print.
* Larson, Ron, Laurie Boswell, and Lee Stiff. ''Geometry''. Boston: McDougal Littell, 2001. Print.
* Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. ''Geometry''. Boston: McDougal Littell, 2007. Print.
* Rosen, Kenneth H. ''Discrete Mathematics and Its Applications, Sixth Edition''. Boston: McGraw-Hill, 2007. Print.
== See also ==
* [[Material implication]]
* [[Logical implication]]
* [[Strict conditional]]
* [[Counterfactual conditional]]
* [[Indicative conditional]]
* [[Propositional logic]]
[[Category:Conditionals]]
[[Category:Mathematical relations]]
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[[Category:Philosophical logic]]
[[Category:Propositional calculus]]
[[Category:Logical connectives]]
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