Conditional statement (logic)

Template:Wikify is deprecated. Please use a more specific cleanup template as listed in the documentation.

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. 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, or false-to-fact sentences in the subjunctive mood, i.e., a counterfactual conditional).^[1]^[2] 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.^[1] Examples of non-truth-functional statements include: "p because q", "p before q" and "it is possible that p".^[1] “[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.”^[1]

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it "material implication"), Wittgenstein in the Tractatus, and the logical positivists, 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”^[1]

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 (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^[3]^[4]:

If p, then q. (called if-then form^[5])
If p, q.
p implies q.
p only if q. (called only-if form^[6])
p is 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.^[7]^[8]

The converse

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

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

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

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,
$(p\rightarrow q)\land (q\rightarrow p)$

Notes

^ ^a ^b ^c ^d ^e Edgington, Dorothy (2008). Edward N. Zalta (ed.). "Conditionals". The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
^ Barwise and Etchemendy 1999, p. 178-179
^ Rosen 2007, p. 6
^ Larson, Boswell, and Stiff 2001, p. 80
^ Larson et al. 2007, p. 79
^ Larson, Boswell, and Stiff 2001, p.80
^ Larson et al. 2007, p. 80
^ Rosen 2007, p. 8

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.