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| | This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow</math>" and "<math>\Rightarrow</math>" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "<math>\rightarrow</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow</math>" being used for both concepts. | | This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow</math>" and "<math>\Rightarrow</math>" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "<math>\rightarrow</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow</math>" being used for both concepts. |
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| − | ==Symbolization==
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| − | A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, [[disjunction]], [[conjunction]], [[negation]], and (frequently) [[biconditional]]. More advanced logic books and later chapters of introductory volumes often add [[identity]], [[Existential quantification]], and [[Universal quantification]].
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| − | Different phrases used to identify the material conditional in ordinary language include ''if'', ''only if'', ''given that'', ''provided that'', ''supposing that'', ''implies'', ''even if'', and ''in case''. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "''A'' only if ''B''" is captured by the statement
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| − | ''A'' → ''B'',
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| − | but "''A'', if ''B''" is correctly captured by the statement
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| − | ''B'' → ''A''
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| − | When doing symbolization exercises, it is often required that the student give a [[scheme of abbreviation]] that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:
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| − | ''A'' → ''B'' <br>
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| − | ''A''—Kermit is a frog.<br>
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| − | ''B''—Muppets are animals.
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| − | Using the horseshoe "⊃" 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}".
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| − | ==Comparison with other conditional statements==
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| − | 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.
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| − | 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 → 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.)
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| | ==References== | | ==References== |
