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#REDIRECT ] |
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In ], the '''contrapositive''' (or '''transposition''') of the statement "''p'' ] ''q''" is "not-''q'' implies not-''p''." A statement and its contrapositive are always ], unlike a statement's ] or its ]. |
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{{Redirect category shell| |
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One can informally convince oneself of this equivalence by examining examples from ]. Consider the statement, "If there is fire here, then there is oxygen here." The contrapositive would be, "If there is no oxygen here, then there is no fire here." If the statement and its contrapositive are indeed logically equivalent, then these sentence should either both be true or both false. But they are indeed both true. (See ].) Thus logical equivalence holds, at least in this case. |
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Note that while a statement ''is'' logically equivalent to its contrapositive (where the two statements are both negated and "swapped"), it is ''not'' logically equivalent to its ] (with the two statements "swapped", but ''not'' negated). |
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== Proofs of logical equivalence == |
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That a statement and its contrapositive are ''always'' logically equivalent can be proven rigorously using formal logic. There are two basic methods for doing this: deriving the equivalence from the axioms of the ], or with a ]. |
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=== Via the propositional calculus === |
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The logical equivalence of a statement and its contrapositive is ''not'' one of the axioms of the ], at least those defined here on ]. However, the equivalence can be proved as follows: |
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First, we can prove that p → q entails ¬q → ¬p: |
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# Suppose p → q. Assuming this, we can reason as follows: |
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## Suppose ¬q. Assuming this, we can reason as follows: |
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### Suppose p. Assuming ''this'', we can reason as follows: |
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#### q (], lines 1 and 1.1.1) |
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#### ¬q (Copying from above) |
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#### q and ¬q (]) |
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### Since this is a contradiction, then ¬p (]) |
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## Thus ¬q → ¬p (]) |
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# Thus (p → q) → (¬q → ¬p) (]) |
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A very similar proof will show that ¬q → ¬p also entails p → q. Combined, these facts show the two statements to be logically equivalent. |
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=== Via a truth table === |
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Alternatively, logical equivalence can be proved using the following ]: |
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<table frame=border rules=all> |
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<tr><th>p</th><th>q</th><th>p → q</th><th>¬p</td><th>¬q</th><th>¬q → ¬p</th></tr> |
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<tr><td>T</td><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td></tr> |
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<tr><td>T</td><td>F</td><td>F</td><td>F</td><td>T</td><td>F</td></tr> |
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<tr><td>F</td><td>T</td><td>T</td><td>T</td><td>F</td><td>T</td></tr> |
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<tr><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr> |
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</table> |
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The first two columns can be taken as given. The third follows from the first two by the truth table definition of the ]. The fourth and fifth follow from the first two by ]. The sixth follows from the fourth and fifth, again by the definition of the ]. |
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Since the third and sixth columns have the same truth values for all values of p and q, the two are logically equivalent. |
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== In Aristotelian logic == |
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In ] (or ]), moreover, ]s can have contrapositives. |
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*The contrapositive of "All S is P" is "All P is S." (These are ].) |
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*The contrapositive of "No S is P" is "No P is S." (These are ].) |
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*The contrapositive of "Some S is P" is "Some P is S." (These are ].) |
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*The contrapositive of "Some S is not P" is "Some P is not S." (These are ].) |
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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 "O" 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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] |
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