Material conditional: Difference between revisions

Revision as of 16:38, 9 May 2006

Revision as of 15:04, 18 May 2006

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In [[propositional logic]], or logical calculus in [[mathematics]], the '''material conditional''' or the '''implies operator''' is a [[binary_relationbinary relation|binary]] [[truth-functional]] [[logical operator]] yielding the form

''If'' a ''then'' c,

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Using the horseshoe "&sup;" symbol for implication is falling out favor due to its conflict with the superset symbol <math>\supset</math> used by the [[Algebra of sets]]. A set interpretation of "<math> A \to B</math>" is "{x| A(x) is true} <math>\subseteq</math> {x| B(x) is true}".

==Symbolization==

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==Comparison with other conditional statements==

The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.

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]]. This temptation can be lessened by reading conditional statements without using the words "if" and "then". The most common way to do this is to read ''A &#8594; B'' as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true". (This [[equivalence|equivalent]] statement is captured in logical notation by <math>\neg A \vee B</math>, using negation and disjunction.)

==See also==

===Other operators===

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* [[Exclusive disjunction]]

* [[Logical conjunction]]

* [[Logical disjunction]]

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* [[Logical equality]]

* [[Logical NAND]]

* [[Logical NOR]]

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===Related topics===

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* [[Ampheck]]

* [[Boolean algebra]]

* [[Boolean ___domain]]

* [[Boolean function]]

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* [[Boolean logic]]

* [[Propositional logic]]

* [[Sole sufficient operator]]

* [[Zeroth order logic]]

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[[Category:Logic]]