Conditional statement (logic): Difference between revisions

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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 oftensometimes called simply a '''conditional''' (also called an '''implication'''). UnlikeOutside 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 statementclause needattached, noti.e., bean truth-functional[[indicative conditional]]).<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 "''p''."<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> 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:

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# If the Sun is out, then it is midnight.

# If you locked your car keys in your car, then 7 + 6 = 2.

{{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"/>}}

== Variations of the conditional statement ==