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| {{Redirect|Logical conditional|other related meanings|Conditional statement (disambiguation){{!}}Conditional statement}}
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| {{Distinguish2|[[material inference]]}}
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| {{Confusing|date=May 2013}}
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| 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]].
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| The material conditional is also symbolized using:
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| # <math>p \supset q</math> (Although this symbol is confused with the superset symbol used by [[algebra of sets]].);
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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.)
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| 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'') → (''r''→''s'')" both the antecedent and the consequent are conditional statements.
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| 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]] logically equivalent 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> Whereas, in [[minimal logic]] (and therefore also intuitionistic logic) <math>p \rightarrow q</math> only [[Logical consequence|logically entails]] <math>\neg(p \and \neg q)</math>; and in [[intuitionistic logic]] (but not minimal logic) <math>\neg p \or q</math> entails <math>p \rightarrow q</math>.
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| ==Definitions of the material conditional==
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| Logicians have many different views on the nature of material implication and approaches to explain its sense.<ref>{{cite web | url=http://www.cs.cornell.edu/Info/People/gries/symposium/clarke.htm | title=A Comparison of Techniques for Introducing Material Implication | publisher=Cornell University | date=March 1996 | accessdate=March 4, 2012 | author=Clarke, Matthew C.}}</ref>
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| ===As a truth function===
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| In [[classical logic]], the compound ''p''→''q'' is logically equivalent to the negative compound: not both ''p'' and not ''q''. Thus the compound ''p''→''q'' is ''false'' [[if and only if]] both ''p'' is true and ''q'' is false. By the same stroke, ''p''→''q'' is ''true'' if and only if either ''p'' is false or ''q'' is true (or both). Thus → is a function from pairs of [[truth value]]s of the components ''p'', ''q'' to truth values of the compound ''p''→''q'', whose truth value is entirely a function of the truth values of the components. Hence, the compound ''p''→''q'' is called ''[[Truth function|truth-functional]]''. The compound ''p''→''q'' is logically equivalent also to ¬''p''∨''q'' (either not ''p'', or ''q'' (or both)), and to ¬''q'' → ¬''p'' (if not ''q'' then not ''p''). But it is not equivalent to ¬''p'' → ¬''q'', which is equivalent to ''q''→''p''.
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| ====Truth table====
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| The truth table associated with the material conditional '''not p or q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p → q''', or '''Cpq''') is as follows:
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| {|class="centered"
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| {|class="wikitable center hintergrundfarbe2"
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| ! <math>p</math> || <math>q</math> || <math>p \rightarrow q</math>
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| |-
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| !T||T
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| |T
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| |-
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| !T||F
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| |F
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| |-
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| !F||T
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| |T
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| |-
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| !F||F
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| |T
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| |}
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| |}
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| ===As a formal connective===
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| The material conditional can be considered as a symbol of a [[theory (mathematical logic)|formal theory]], taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:
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| # [[Modus ponens]];
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| # [[Conditional proof]];
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| # [[contraposition|Classical contraposition]];
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| # [[reductio ad absurdum|Classical reductio]].
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| Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various [[Formal system|logical system]]s, where somewhat different properties may be demonstrated. For example, in [[intuitionistic logic]] which rejects proofs by contraposition as valid rules of inference, {{math|(''p'' → ''q'') ⇒ ¬''p'' ∨ ''q''}} is not a propositional theorem, but [[False (logic)#False, negation and contradiction|the material conditional is used to define negation]].
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| ==Formal properties==
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| When studying logic formally, the material conditional is distinguished from the [[Logical consequence#Semantic consequence|semantic consequence]] relation <math>\models</math>. We say <math>A \models B</math> if every interpretation that makes ''A'' true also makes ''B'' true. However, there is a close relationship between the two in most logics, including [[classical logic]]. For example, the following principles hold:
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| * If <math>\Gamma\models\psi</math> then <math>\varnothing\models(\varphi_1\land\dots\land\varphi_n\rightarrow\psi)</math> for some <math>\varphi_1,\dots,\varphi_n\in\Gamma</math>. (This is a particular form of the [[deduction theorem]]. In words, it says that if Γ models ''ψ'' this means that ψ can be deduced just from some subset of the theorems in Γ.)
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| * The converse of the above
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| * Both <math>\rightarrow</math> and <math>\models</math> are [[Monotonic function|monotonic]]; i.e., if <math>\Gamma\models\psi</math> then <math>\Delta\cup\Gamma\models\psi</math>, and if <math>\varphi\rightarrow\psi</math> then <math>(\varphi\land\alpha)\rightarrow\psi</math> for any ''α'', Δ. (In terms of structural rules, this is often referred to as [[weakening]] or ''thinning''.)
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| These principles do not hold in all logics, however. Obviously they do not hold in [[non-monotonic logic]]s, nor do they hold in [[relevance logic]]s.
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| Other properties of implication (the following expressions are always true, for any logical values of variables):
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| * [[distributivity]]: <math>(s \rightarrow (p \rightarrow q)) \rightarrow ((s \rightarrow p) \rightarrow (s \rightarrow q))</math>
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| * [[transitive relation|transitivity]]: <math>(a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))</math>
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| * [[reflexive relation|reflexivity]]: <math>a \rightarrow a</math>
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| * [[total relation|totality]]: <math>(a \rightarrow b) \vee (b \rightarrow a)</math>
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| * truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.
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| * commutativity of antecedents: <math>(a \rightarrow (b \rightarrow c)) \equiv (b \rightarrow (a \rightarrow c))</math>
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| Note that <math>a \rightarrow (b \rightarrow c)</math> is [[Logical equivalence|logically equivalent]] to <math>(a \and b) \rightarrow c</math>; this property is sometimes called [[currying|un/currying]]. Because of these properties, it is convenient to adopt a [[right-associative]] notation for → where <math>a \rightarrow b \rightarrow c</math> denotes <math>a \rightarrow (b \rightarrow c)</math>.
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| Comparison of Boolean truth tables shows that <math>a \rightarrow b</math> is equivalent to <math>\neg a \or b</math>, and one is an equivalent replacement for the other in classical logic. See [[material implication (rule of inference)]].
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| == Philosophical problems with material conditional ==
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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: "''q'' because ''p''", "''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>
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| {{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"<ref name="sep-conditionals"/>}}
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| 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}}
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| The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see [[vacuous truth]]). So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement "if I have a penny in my pocket then Paris is in France" is always true, regardless of whether or not there is a penny in my pocket. These problems are known as the [[paradoxes of material implication]], though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and other natural languages) are tempted to [[equivocation|equivocate]] between the material conditional and the [[indicative conditional]], or other conditional statements, like the [[counterfactual conditional]] and the [[logical biconditional |material biconditional]]. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
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| ==See also==
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| {{col-begin}}
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| {{col-break}}
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| * [[Boolean algebra (logic)|Boolean algebra]]
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| * [[Boolean domain]]
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| * [[Boolean function]]
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| * [[Boolean logic]]
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| {{col-break}}
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| * [[Conditional quantifier]]
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| * [[Implicational propositional calculus]]
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| * [[Laws of Form]]
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| * [[Logic gate]]
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| * [[Logical graph]]
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| {{col-break}}
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| * [[Paradoxes of material implication]]
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| * [[Peirce's law]]
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| * [[Propositional logic]]
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| * [[Sole sufficient operator]]
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| {{col-end}}
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| ===Conditionals===
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| * [[Counterfactual conditional]]
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| * [[Indicative conditional]]
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| * [[Corresponding conditional]]
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| * [[Strict conditional]]
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| ==References==
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| {{Reflist}}
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| == Further reading ==
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| * Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, [[Kluwer]] Academic Publishers, [[Norwell, Massachusetts|Norwell]], MA. 2nd edition, [[Dover Publications]], [[Mineola, New York|Mineola]], NY, 2003.
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| * [[Dorothy Edgington|Edgington, Dorothy]] (2001), "Conditionals", in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic'', [[Wiley-Blackwell|Blackwell]].
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| * [[W. V. Quine|Quine, W.V.]] (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, [[Harvard University Press]], [[Cambridge]], MA.
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| * [[Robert Stalnaker|Stalnaker, Robert]], "Indicative Conditionals", ''[[Philosophia]]'', '''5''' (1975): 269–286.
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| == External links ==
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| * {{SEP|conditionals|Conditionals|Edgington, Dorothy}}
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| {{Logical connectives}}
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| [[Category:Logical connectives]]
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| [[Category:Conditionals]]
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| [[Category:Logical consequence]]
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| [[am:ጥገኛ አምክንዮ]]
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| [[ar:قضية شرطية]]
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| [[cs:Implikace]]
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| [[eo:Implico]]
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| [[fa:شرطی منطقی]]
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| [[it:Implicazione logica]]
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| [[he:אם-אז]]
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| [[mk:Материјална импликација]]
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| [[nl:Logische implicatie]]
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| [[ja:論理包含]]
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| [[no:Subjunksjon (logikk)]]
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| [[pms:Amplicassion]]
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| [[kaa:İmplikatsiya]]
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| [[sk:Implikácia]]
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| [[th:เงื่อนไขเชิงตรรกศาสตร์]]
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| [[zh:实质条件]]
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