4.4 Conversion Obversion And Contraposition

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

1. 4.44.4
Conversion, Obversion, and Conversion, Obversion, and
ContrapositionContraposition

2. OverviewOverview
•ConversionConversion
•ObversionObversion
•ContrapositionContraposition

3. ConversionConversion
•First type of operation we can apply to propositions.First type of operation we can apply to propositions.
•Involves switching the Involves switching the subject termsubject term and the and the predicate termpredicate term..
•Conversion of both E and I type propositions will yield a statement Conversion of both E and I type propositions will yield a statement
that will necessarily have the same truth value. The original that will necessarily have the same truth value. The original
statement and its statement and its converseconverse are said to be are said to be logically equivalentlogically equivalent..
All S are P.All S are P. ConversionConversion All P are S.All P are S.
(A type)(A type)

4. Conversion, continuedConversion, continued
No S are P.No S are P. ConversionConversion No P are S.No P are S.
(E type)(E type)
Some S are P.Some S are P. (I type) (I type) Some P are S.Some P are S.

5. Conversion, continuedConversion, continued
Some S are not P.Some S are not P. ConversionConversion Some P are not S.Some P are not S.
(O type)(O type)
•Note that the diagrams for A and O types do not match up. Note that the diagrams for A and O types do not match up.
–Conversion of these two types will result in a truth value that is logically Conversion of these two types will result in a truth value that is logically
undetermined.undetermined.
–Converting A and O types will result in a formal fallacy called an illicit Converting A and O types will result in a formal fallacy called an illicit
conversion.conversion.
•Example:Example:
–All people are happy.All people are happy.
–Therefore, all things happy are people.Therefore, all things happy are people.

6. ObversionObversion
•More complicated than conversion.More complicated than conversion.
•Involves two steps:Involves two steps:
–1) Change the quality of the statement (without changing the quantity).1) Change the quality of the statement (without changing the quantity).
•For A and E type statements, change the quantifier (All to no, no to all)For A and E type statements, change the quantifier (All to no, no to all)
•For I and O statements, change the copula (are to are not, are not to are)For I and O statements, change the copula (are to are not, are not to are)
–2) Replace the predicate term with its 2) Replace the predicate term with its term complementterm complement..
•Term complementTerm complement
–Word or group of words that denotes the class complement.Word or group of words that denotes the class complement.
–Dog Dog  Non-dog, person Non-dog, person  non-person. non-person.
•Obversion works with Obversion works with all four all four types of propositions.types of propositions.
–A, E, I, and O types.A, E, I, and O types.

7. Obversion, continuedObversion, continued
All S are P.All S are P. No S are non-P. No S are non-P.
No S are P.No S are P. All S are non-P. All S are non-P.

8. Obversion, continuedObversion, continued
Some S are P.Some S are P. Some S are not non-P.Some S are not non-P.
Some S are not P.Some S are not P. Some S are non-P.Some S are non-P.

9. ContrapositionContraposition
•Similar to obversion, since it requires two steps as well:Similar to obversion, since it requires two steps as well:
–1) Switch the subject and predicate terms.1) Switch the subject and predicate terms.
–2) Find the term complement for each one.2) Find the term complement for each one.
–Example:Example:
•All people are happy.All people are happy.
•Therefore, all non-happy things are non-people.Therefore, all non-happy things are non-people.
All S are P.All S are P. All non-P are non-S.All non-P are non-S.

10. Contraposition, continuedContraposition, continued
No S are P.No S are P. No non-P are non-S.No non-P are non-S.
Some S are P.Some S are P. Some non-P are non-S.Some non-P are non-S.

11. Contraposition, continuedContraposition, continued
Some S are not P.Some S are not P. Some non-P are not non-S.Some non-P are not non-S.
•Notice that the E and I types do not appear the same when Notice that the E and I types do not appear the same when
contraposition is applied. This means when they are contraposed, contraposition is applied. This means when they are contraposed,
the new statement is logically undetermined.the new statement is logically undetermined.
•But A and O type statements have the same diagram, so their truth But A and O type statements have the same diagram, so their truth
values are logically equivalent (the same).values are logically equivalent (the same).

12. Contraposition, continuedContraposition, continued
•Illicit contrapositionIllicit contraposition
–Happens when you apply contraposition to E and I type statements, Happens when you apply contraposition to E and I type statements,
and make a claim about them (even when their truth values are and make a claim about them (even when their truth values are
logically undetermined).logically undetermined).
–Example:Example:
•No people are happy (No S are P).No people are happy (No S are P).
•Therefore, no non-happy things are non-people (No non-P are non-S).Therefore, no non-happy things are non-people (No non-P are non-S).
•Some people are happy (Some S are P).Some people are happy (Some S are P).
•Therefore, some non-happy things are non-people (Some non-P are non-S).Therefore, some non-happy things are non-people (Some non-P are non-S).