Conditional statement (logic)

In philosophical and mathematical logic, a conditional statement is a compound statement, composed of declarative sentences or propositions 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. Since not all conditional statements are truth-functional, not all conditional statements are material conditionals.

Conditional statements are often symbolized using an arrow (→) as p → q (read "p implies q"). The conditional statement in symbolic form is as follows:

• $p\rightarrow q$

As a proposition, a conditional statement is either 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:

1. If p, then q. (called if-then form)
2. If p, q.
3. p implies q.
4. p only if q. (called only-if form)
5. p is sufficient for q.
6. A sufficient condition for q is p.
7. q if p.
8. q whenever p.
9. q when p.
10. q every time that p.
11. q is necessary for p.
12. A necessary condition for p is q.
13. q follows from p.
14. 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.

The converse

Main article: 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:

• $q\rightarrow p$

The inverse

Main article: Inverse (logic)

The inverse of a conditional statement is the conditional statement produced when both the hypothesis and the conclusion are negated. The resulting conditional is as follows:

• $\lnot p\rightarrow \lnot q$

The contrapositive

Main article: 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:

• $\lnot q\rightarrow \lnot p$

The biconditional

Main article: Logical biconditional

The biconditional of a conditional statement is the proposition produced out of the conjunction of the conditional statement and its converse. When written in its standard 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:

• $p\leftrightarrow q$; or equivalently,
• $(p\rightarrow q)\land (q\rightarrow p)$

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