Material conditional: Difference between revisions

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{{Confusing|date=May 2013}}

The '''material conditional''' (also known as “"'''material implication'''”", “"'''material consequence'''”", or simply “"'''implication'''”", “"'''implies'''”" or “"'''conditional'''”") is a [[logical connective]] (or a [[binary operator]]) that is often symbolized by a forward arrow “→”"→". The material conditional is used to form [[statement (logic)|statements]] of the form “"''p''→''q''”" (termed a [[conditional statement]]) which is read as "if p then q" and conventionally compared to the English construction "If...then...". But unlike as the English construction may, the conditional statement “"''p''→''q''”" does not specify a causal relationship between ''p'' and ''q'' and is to be understood to mean "if ''p'' is true, then ''q'' is also true" such that the statement “"''p''→''q''”" is false only when ''p'' is true and ''q'' is false.<ref>{{cite web|title=forallx: An Introduction to Formal Logic|url=http://www.fecundity.com/codex/forallx.pdf|author=Magnus, P.D|date=January 6, 2012|publisher=Creative Commons|page=25|accessdate=28 May 2013}}</ref> The material conditional is also to be distinguished from [[logical consequence]].

The material conditional is also symbolized using:

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# <math>p \Rightarrow q</math> (Although this symbol is often used for [[logical consequence]] (i.e. logical implication) rather than for material conditional.)

With respect to the material conditionals above, ''p'' is termed the ''[[antecedent (logic)|antecedent]]'', and ''q'' the ''[[consequent]]'' of the conditional. Conditional statements may be nested such that either or both of the antecedent or the consequent may themselves be conditional statements. In the example “"(''p''→''q'')&nbsp;→&nbsp;(''r''→''s'')”" both the antecedent and the consequent are conditional statements.

In [[classical logic]] <math>p \rightarrow q</math> is [[Logical equivalence|logically equivalent]] to <math>\neg(p \and \neg q)</math> and by [[De Morgan's Law]] to <math>\neg p \or q</math>.<ref>{{cite web|title=A Modern Formal Logic Primer: Sentence Logic Volume 1|url=http://tellerprimer.ucdavis.edu/pdf/1ch4.pdf|author=Teller, Paul|date=January 10, 1989|publisher=Prentice Hall|page=54|accessdate=28 May 2013}}</ref>

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Outside of mathematics, it is a matter of some controversy as to whether the [[truth function]] for [[material implication (rule of inference)|material implication]] provides an adequate treatment of conditional statements in English (a [[sentence (mathematical logic)|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]]).<ref name="sep-conditionals"/> 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''.<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>

{{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 Logico-Philosophicus|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”"Conditionals"<ref name="sep-conditionals"/>}}

The meaning of the material conditional can sometimes be used in the [[natural language]] English "if ''condition'' then ''consequence''" construction (a kind of [[conditional sentence]]), where ''condition'' and ''consequence'' are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition ([[Protasis (linguistics)|protasis]]) and consequence ([[Consequent|apodosis]]) (see [[Connexive logic]]).{{citation needed|date=February 2012}}