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 "Ifif ''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 sometimes called simply a '''conditional''' (also called 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]]).<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, "Ifif ''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> 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>