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# 4.4 Conversion Obversion And Contraposition

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Course lecture I developed over section 4.4 of Patrick Hurley\'s &quot;A Concise Introduction to Logic&quot;.

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### 4.4 Conversion Obversion And Contraposition

1. 1. 4.4 Conversion, Obversion, and Contraposition
2. 2. Overview <ul><li>Conversion </li></ul><ul><li>Obversion </li></ul><ul><li>Contraposition </li></ul>
3. 3. Conversion <ul><li>First type of operation we can apply to propositions. </li></ul><ul><li>Involves switching the subject term and the predicate term . </li></ul><ul><li>Conversion of both E and I type propositions will yield a statement that will necessarily have the same truth value. The original statement and its converse are said to be logically equivalent . </li></ul><ul><li>All S are P. Conversion  All P are S. </li></ul><ul><li> (A type) </li></ul>
4. 4. Conversion, continued <ul><li>No S are P. Conversion  No P are S. </li></ul><ul><li> (E type) </li></ul><ul><li>Some S are P. (I type) Some P are S. </li></ul>
5. 5. Conversion, continued <ul><li>Some S are not P. Conversion  Some P are not S. </li></ul><ul><li> (O type) </li></ul><ul><li>Note that the diagrams for A and O types do not match up. </li></ul><ul><ul><li>Conversion of these two types will result in a truth value that is logically undetermined. </li></ul></ul><ul><ul><li>Converting A and O types will result in a formal fallacy called an illicit conversion. </li></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul><ul><ul><ul><ul><li>All people are happy. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Therefore, all things happy are people. </li></ul></ul></ul></ul>
6. 6. Obversion <ul><li>More complicated than conversion. </li></ul><ul><li>Involves two steps: </li></ul><ul><ul><li>1) Change the quality of the statement (without changing the quantity). </li></ul></ul><ul><ul><ul><li>For A and E type statements, change the quantifier (All to no, no to all) </li></ul></ul></ul><ul><ul><ul><li>For I and O statements, change the copula (are to are not, are not to are) </li></ul></ul></ul><ul><ul><li>2) Replace the predicate term with its term complement . </li></ul></ul><ul><li>Term complement </li></ul><ul><ul><li>Word or group of words that denotes the class complement. </li></ul></ul><ul><ul><li>Dog  Non-dog, person  non-person. </li></ul></ul><ul><li>Obversion works with all four types of propositions. </li></ul><ul><ul><li>A, E, I, and O types. </li></ul></ul>
7. 7. Obversion, continued <ul><li>All S are P. No S are non-P. </li></ul><ul><li>No S are P. All S are non-P. </li></ul>
8. 8. Obversion, continued <ul><li>Some S are P. Some S are not non-P. </li></ul><ul><li>Some S are not P. Some S are non-P. </li></ul>
9. 9. Contraposition <ul><li>Similar to obversion, since it requires two steps as well: </li></ul><ul><ul><li>1) Switch the subject and predicate terms. </li></ul></ul><ul><ul><li>2) Find the term complement for each one. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>All people are happy. </li></ul></ul></ul><ul><ul><ul><li>Therefore, all non-happy things are non-people. </li></ul></ul></ul><ul><li>All S are P. All non-P are non-S. </li></ul>
10. 10. Contraposition, continued <ul><li>No S are P. No non-P are non-S. </li></ul><ul><li>Some S are P. Some non-P are non-S. </li></ul>
11. 11. Contraposition, continued <ul><li>Some S are not P. Some non-P are not non-S. </li></ul><ul><li>Notice that the E and I types do not appear the same when contraposition is applied. This means when they are contraposed, the new statement is logically undetermined. </li></ul><ul><li>But A and O type statements have the same diagram, so their truth values are logically equivalent (the same). </li></ul>
12. 12. Contraposition, continued <ul><li>Illicit contraposition </li></ul><ul><ul><li>Happens when you apply contraposition to E and I type statements, and make a claim about them (even when their truth values are logically undetermined). </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>No people are happy (No S are P). </li></ul></ul></ul><ul><ul><ul><li>Therefore, no non-happy things are non-people (No non-P are non-S). </li></ul></ul></ul><ul><ul><ul><li>Some people are happy (Some S are P). </li></ul></ul></ul><ul><ul><ul><li>Therefore, some non-happy things are non-people (Some non-P are non-S). </li></ul></ul></ul>