4 Categorical Propositions

1. 4 Categorical Propositions 4.4. Conversion, Obversion, and Contraposition

2. Conversion, Obversion, and Contraposition These are 3 operations that can be used to change standard form statements or change a statement into standard form. Conversion is the simplest: simply switch the subject and predicate terms. No S are P -----conversion----> No P are S

3. Conversion Compare the Venn Diagrams of each statement: No P are S No S are P S P S P For E propositions, there is no change in diagram, and so no change in truth value.

4. Conversion Compare the Venn Diagrams of each statement: All P are S All S are P S P S P For A propositions, there IS a change in diagram, and so a change in truth value.

5. Conversion Compare the Venn Diagrams of each statement: Some P are S Some S are P X X S P S P For I propositions, there is no change in diagram, and so no change in truth value.

6. Conversion Compare the Venn Diagrams of each statement: Some P are not S Some S are not P X X S P S P For O propositions, there IS a change in diagram, and so a change in truth value.

7. Conversion So, what do we know? Immediate inferences (inferences with one premise and one conclusion) are valid for the converses of E and I propositions, but not for A and O propositions. All puppies are evil. Therefore, all evil things are puppies. Since this is conversion of an A proposition, it is an illicit inference … an illicit conversion. Some puppies are evil. Therefore, some evil things are puppies. No problem.

8. Obversion Obversion (2 steps): • Change the quality • Replace Predicate term with its complement All S are P ------- obversion----->____________? No S are P -------obversion ------> _____________? Some S are P ----- obversion -----> _______________? Some S are not P ----obversion---> _____________?

9. Obversion No S are non-P All S are P All S are non-P No S are P X X X X Some S are P Some S are not non-P S S S S S S S P S P P P P P P P Some S are not P Some S are non-P

10. Contraposition Contraposition (2 steps): • Switch Subject and Predicate terms • Replace both Subject and Predicate terms with their complements All S are P ------- contraposition----->____________? No S are P -------contraposition ------> _____________? Some S are P ----- contraposition -----> _______________? Some S are not P ----contraposition---> _____________?

11. Contraposition All non-P are non-S All S are P No non-P are non-S No S are P X X X Some S are P Some non-P are non-S X S S S S S S P P P P S P P P S P Some S are not P Some non-P are not non-S

12. Method to Drawing Venn Diagrams • Place an X in each distinct area of subject class • Remove Xs based on what the proposition says using shading for universal propositions and eraser for particular ones • Remove Xs based on Boolean interpretation using your eraser

13. Helpful Wording All non-P are non-S Every single non-P is also a non-S If there were any Xs inside the left part of the S circle, the statement would allow a non-P that was an S, which is denied in the statement. If there are any Xs, they would be outside the two circles (they don’t appear because this is the Boolean interpretation) S P

14. Helpful Wording No non-P are non-S Not even one of the non-Ps is also a non-S (an X outside both circles says, here is a non-P that is also a non-S) So, to prohibit allowing an X outside those circles, that area must be shaded. P S