|
#REDIRECT [[Conditional statement]]
<!-- Please do not remove or change this AfD message until the issue is settled -->
{{Article for deletion/dated|page=Conditional statement (logic)|timestamp=20120115042749|year=2012|month=January|day=15|substed=yes}}
<!-- For administrator use only: {{Old AfD multi|page=Conditional statement (logic)|date=15 January 2012|result='''keep'''}} -->
<!-- End of AfD message, feel free to edit beyond this point -->
{{Refimprove|date=January 2012}}
{{Expert-subject|Logic|date=January 2012}}
{{Expert-subject|Mathematics|date=January 2012}}
{{Wikify|date=January 2012}}
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. 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:
# 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]]
[[Category:Philosophy]]
[[Category:Philosophical logic]]
[[Category:Propositional calculus]]
[[Category:Logical connectives]]
[[Category:Articles created via the Article Wizard]]
| 