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# Hum 200 w4 ch5 catprops

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### Hum 200 w4 ch5 catprops

1. 1. Logic <br /> HUM 200<br /> Categorical Propositions <br />1<br />
2. 2. Objectives<br />2<br />When you complete this lesson, you will be able to:<br />Identify the four classes of categorical propositions<br />Describe the quality, quantity, and distribution of categorical propositions<br />Identify the four types of opposition<br />Apply the immediate inferences given in the Square of Opposition<br />Apply immediate inferences not directly associated with the Square of Opposition<br />Describe existential import<br />List and describe the implications of the Boolean interpretation of categorical propositions<br />Symbolize and diagram the Boolean interpretation of categorical propositions<br />
3. 3. The Theory of Deduction <br />3<br />Deductive arguments<br />Premises are claimed to provide conclusive grounds for the truth of its conclusion <br />Valid or invalid<br />Theory of deduction<br />Aims to explain the relations of premises and conclusion in valid arguments <br />Classical logic<br />Modern symbolic logic<br />
4. 4. Classes and Categorical Propositions <br />4<br />Class<br />Collection of all objects that have some specified characteristic in common <br />Relationships between classes may be:<br />Wholly included<br />Partially included<br />Excluded<br />
5. 5. Classes and Categorical Propositions, continued <br />5<br />Example categorical proposition<br />No athletes are vegetarians.<br />All football players are athletes.<br />Therefore no football players are vegetarians. <br />
6. 6. 6<br />Universal affirmative proposition (A proposition)<br />Whole of one class is included or contained in another class <br />All S is P<br />Venn diagram<br />P<br />S<br />All S is P<br />The Four Kinds of Categorical Propositions <br />
7. 7. 7<br />Universal negative proposition (E proposition) <br />The whole of one class is excluded from the whole of another class <br />No S is P<br />Venn diagram<br />P<br />S<br />No S is P<br />The Four Kinds of Categorical Propositions, continued <br />
8. 8. 8<br />Particular affirmative proposition (I proposition)<br />Two classes have some member or members in common <br />Some S is P<br />Venn diagram<br />P<br />S<br />x<br />Some S is P<br />The Four Kinds of Categorical Propositions, continued <br />
9. 9. 9<br />Particular negative propositions (O proposition)<br />At least one member of a class is excluded from the whole of another class <br />Some S is not P<br />Venn diagram<br />P<br />S<br />x<br />Some S is not P<br />The Four Kinds of Categorical Propositions, continued <br />
10. 10. Quality <br />10<br />An attribute of every categorical proposition, determined by whether the proposition affirms or denies some form of class inclusion<br />Affirmative<br />Affirms some class inclusion <br />A and I propositions<br />Negative<br />Denies class inclusion <br />E and O propositions<br />
11. 11. Quantity <br />11<br />An attribute of every categorical proposition, determined by whether the proposition refers to all members, or only some members of the class<br />Universal<br />Refers to all members of the class<br />A and E propositions<br />Particular<br />Refers only to some members of the class<br />I and O propositions<br />
12. 12. Distribution <br />12<br />Characterization of whether terms refer to all members of the class designated by that term <br />A proposition<br />Subject distributed, predicate undistributed<br />E proposition<br />Both subject and predicate distributed<br />I proposition<br />Neither subject nor predicate distributed<br />O proposition<br />Subject undistributed, predicate distributed<br />
13. 13. The Traditional Square of Opposition <br />13<br />Opposition<br />Any kind of such differing other in quality, quantity, or in both<br />Contradictories<br />Contraries<br />Subcontraries<br />Subalternation <br />
14. 14. The Traditional Square of Opposition, continued <br />14<br />Contradictories<br />One proposition is the denial or negation of the other <br />One is true, one is false<br />A and O are contradictories<br />E and I are contradictories<br />
15. 15. The Traditional Square of Opposition, continued <br />15<br />Contraries<br />If one is true, the other must be false<br />Both can be false<br />A and E are contraries<br />
16. 16. The Traditional Square of Opposition, continued <br />16<br />Subcontraries<br />They cannot both be false<br />They may both be true<br />If one is false, then the other must be true<br />I and O are subcontraries<br />
17. 17. The Traditional Square of Opposition, continued <br />17<br />Subalteration<br />Opposition between a universal proposition (superaltern) and its corresponding particular proposition (subaltern)<br />Universal proposition implies the truth of its corresponding particular proposition <br />Occurs from A to I propositions<br />Occurs from E to O propositions<br />
18. 18. The Traditional Square of Opposition, continued <br />18<br />E<br />A<br />contraries<br />(No S is P.)<br />superaltern<br />(All S is P.)<br />superaltern<br />contrad ictories<br />contradictories<br />subalternation<br />subalternation<br />subaltern<br />(Some S is not P.)<br />subaltern<br />(Some S is P.)<br />subcontraries<br />I<br />O<br />Immediate inference<br />Inference drawn from only one premise <br />
19. 19. The Traditional Square of Opposition, continued <br />19<br />Immediate inferences<br />
20. 20. Further Immediate Inferences <br />20<br />Conversion<br />Formed by interchanging the subject and predicate terms of a categorical proposition <br />
21. 21. Further Immediate Inferences, continued <br />21<br />Complement of a class<br />The collection of all things that do not belong to that class <br />Class denoted as S<br />Complement denoted as non-S<br />Double negatives<br />
22. 22. Further Immediate Inferences, continued <br />22<br />Obversion<br />Changing the quality of a proposition and replacing the predicate term by its complement <br />
23. 23. Further Immediate Inferences, continued <br />23<br />Contraposition<br />Formed by replacing the subject term of a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term <br />
24. 24. Existential Import and the Interpretation of Categorical Propositions <br />24<br />Existential import<br />Proposition asserts the existence of an object of some kind <br />Example<br />All inhabitants of Mars are blond (A proposition)<br />Some inhabitants of Mars are not blond (O proposition)<br />A and O are contradictories<br />Since Mars has no inhabitants, both statements are false, so these statements cannot be contradictories<br />
25. 25. Existential Import and the Interpretation of Categorical Propositions, continued <br />25<br />Presupposition<br />We presuppose propositions never refer to empty classes <br />Problems <br />Never able to formulate the proposition that denies the class has members <br />What we say does not suppose that there are members in the class <br />Wish to reason without making any presuppositions about existence <br />
26. 26. Existential Import and the Interpretation of Categorical Propositions, continued <br />26<br />Boolean interpretation<br />Universal propositions are not assumed to refer to classes that have members <br />I and O continue to have existential import <br />Universal propositions are the contradictories of the particular propositions <br />Universal propositions are interpreted as having no existential import <br />
27. 27. Existential Import and the Interpretation of Categorical Propositions, continued <br />27<br />Boolean interpretation<br />Universal proposition intending to assert existence is allowed, but doing so requires two propositions: one existential in force but particular, and one universal but not existential in force<br />Corresponding A and E propositions can both be true and are therefore not contraries <br />I and O propositions can both be false if the subject class is empty <br />
28. 28. Existential Import and the Interpretation of Categorical Propositions, continued <br />28<br />Boolean interpretation<br />Subalternation is not generally valid <br />Preserves some immediate inferences <br />Conversion for E and I propositions <br />Contraposition for A and O propositions <br />Obversion for any proposition<br />Transforms the traditional Square of Opposition by undoing relations along the sides of the square <br />
29. 29. Symbolism and Diagrams for Categorical Propositions <br />29<br />Boolean interpretation notation<br />Empty class: 0<br />S has no members: S = 0<br />Deny class is empty: S≠ 0<br />Product (intersection) of two classes: SP<br />No satires are poems: SP = 0<br />Some satires are poems: SP≠ 0<br />
30. 30. Symbolism and Diagrams for Categorical Propositions, continued <br />30<br />Complement of a class: S<br />All S is P: SP = 0<br />Some S is not P: SP≠ 0<br />
31. 31. Symbolism and Diagrams for Categorical Propositions, continued <br />31<br />
32. 32. 32<br />Boolean Square of Opposition<br />SP = 0<br />SP = 0<br />E<br />A<br />contrad ictories<br />contradictories<br />I<br />O<br />SP ≠ 0<br />SP≠ 0<br />Symbolism and Diagrams for Categorical Propositions, continued <br />
33. 33. 33<br />Venn diagrams of Boolean interpretation<br />S<br />S<br />x<br />S = 0<br />S≠ 0<br />P<br />S<br />SP<br />SP<br />SP<br />SP<br />Symbolism and Diagrams for Categorical Propositions, continued <br />
34. 34. 34<br />Venn diagrams of categorical propositions<br />P<br />S<br />P<br />S<br />P<br />S<br />x<br />A: All S is P<br />SP = 0<br />E: No S is P<br />SP = 0<br />I: Some S is P<br />SP≠ 0<br />P<br />S<br />x<br />O: Some S is not P<br />SP≠ 0<br />Symbolism and Diagrams for Categorical Propositions, continued <br />
35. 35. 35<br />Venn diagrams of categorical propositions<br />P<br />S<br />P<br />S<br />P<br />S<br />x<br />A: All P is S<br />PS = 0<br />E: No P is S<br />PS = 0<br />I: Some P is S<br />PS≠ 0<br />Symbolism and Diagrams for Categorical Propositions, continued <br />P<br />S<br />x<br />O: Some P is not S<br />PS≠ 0<br />
36. 36. Summary <br />36<br />Categorical propositions<br />Quality, quantity, and distribution<br />Opposition<br />Immediate inferences<br />Existential import<br />Boolean interpretation<br />Symbolism and diagrams of categorical propositions<br />