Contrapositive: Difference between revisions
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Larry_Sanger (talk) Expanding article to include contrapositives in categorical logic |
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In [[predicate logic]], the <b>contrapositive</b> of the statement "''p'' implies ''q''" is "not-''q'' implies not-''p''." These are [[logical equivalence|logically equivalent]]. We can find examples in [[ordinary English]]. We might form the contrapositive of "If there is fire here, then there is oxygen here," like this: "If there is no oxygen here, then there is no fire here." |
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In [[Aristotelian logic]] (or [[categorical logic]]), moreover, [[categorical proposition]]s can have contrapositives. |
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*The contrapositive of "All S is P" is "All P is S." (These are [[A proposition|"A" propositions]].) |
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*The contrapositive of "No S is P" is "No P is S." (These are [[E proposition|"E" propositions]].) |
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*The contrapositive of "Some S is P" is "Some P is S." (These are [[I proposition|"I" propositions]].) |
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*The contrapositive of "Some S is not P" is "Some P is not S." (These are [[O proposition|"O" propositions]].) |
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So-called "E" and "I" propositions are logically equivalent to their contrapositives. For example, we can always infer from "no bachelors are women" to "no women are bachelors" (as well as the converse inference) and from "some dogs are flea-bitten animals" to "some flea-bitten animals are dogs" (and conversely). |
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However, so-called "A" and "E" propositions are ''not'' logically equivalent to their contrapositives. For example, from "all violins are musical instruments," we cannot infer "all musical instruments are violins." Similarly, from "some plants are not trees," we cannot infer "some trees are not plants." |
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Revision as of 21:01, 12 January 2002
In predicate logic, the contrapositive of the statement "p implies q" is "not-q implies not-p." These are logically equivalent. We can find examples in ordinary English. We might form the contrapositive of "If there is fire here, then there is oxygen here," like this: "If there is no oxygen here, then there is no fire here."
In Aristotelian logic (or categorical logic), moreover, categorical propositions can have contrapositives.
- The contrapositive of "All S is P" is "All P is S." (These are "A" propositions.)
- The contrapositive of "No S is P" is "No P is S." (These are "E" propositions.)
- The contrapositive of "Some S is P" is "Some P is S." (These are "I" propositions.)
- The contrapositive of "Some S is not P" is "Some P is not S." (These are "O" propositions.)
So-called "E" and "I" propositions are logically equivalent to their contrapositives. For example, we can always infer from "no bachelors are women" to "no women are bachelors" (as well as the converse inference) and from "some dogs are flea-bitten animals" to "some flea-bitten animals are dogs" (and conversely).
However, so-called "A" and "E" propositions are not logically equivalent to their contrapositives. For example, from "all violins are musical instruments," we cannot infer "all musical instruments are violins." Similarly, from "some plants are not trees," we cannot infer "some trees are not plants."