1. Logic HUM 200 Categorical Propositions 1

2. Objectives 2 When you complete this lesson, you will be able to: Identify the four classes of categorical propositions Describe the quality, quantity, and distribution of categorical propositions Identify the four types of opposition Apply the immediate inferences given in the Square of Opposition Apply immediate inferences not directly associated with the Square of Opposition Describe existential import List and describe the implications of the Boolean interpretation of categorical propositions Symbolize and diagram the Boolean interpretation of categorical propositions

3. The Theory of Deduction 3 Deductive arguments Premises are claimed to provide conclusive grounds for the truth of its conclusion Valid or invalid Theory of deduction Aims to explain the relations of premises and conclusion in valid arguments Classical logic Modern symbolic logic

4. Classes and Categorical Propositions 4 Class Collection of all objects that have some specified characteristic in common Relationships between classes may be: Wholly included Partially included Excluded

5. Classes and Categorical Propositions, continued 5 Example categorical proposition No athletes are vegetarians. All football players are athletes. Therefore no football players are vegetarians.

6. 6 Universal affirmative proposition (A proposition) Whole of one class is included or contained in another class All S is P Venn diagram P S All S is P The Four Kinds of Categorical Propositions

7. 7 Universal negative proposition (E proposition) The whole of one class is excluded from the whole of another class No S is P Venn diagram P S No S is P The Four Kinds of Categorical Propositions, continued

8. 8 Particular affirmative proposition (I proposition) Two classes have some member or members in common Some S is P Venn diagram P S x Some S is P The Four Kinds of Categorical Propositions, continued

9. 9 Particular negative propositions (O proposition) At least one member of a class is excluded from the whole of another class Some S is not P Venn diagram P S x Some S is not P The Four Kinds of Categorical Propositions, continued

10. Quality 10 An attribute of every categorical proposition, determined by whether the proposition affirms or denies some form of class inclusion Affirmative Affirms some class inclusion A and I propositions Negative Denies class inclusion E and O propositions

11. Quantity 11 An attribute of every categorical proposition, determined by whether the proposition refers to all members, or only some members of the class Universal Refers to all members of the class A and E propositions Particular Refers only to some members of the class I and O propositions

12. Distribution 12 Characterization of whether terms refer to all members of the class designated by that term A proposition Subject distributed, predicate undistributed E proposition Both subject and predicate distributed I proposition Neither subject nor predicate distributed O proposition Subject undistributed, predicate distributed

13. The Traditional Square of Opposition 13 Opposition Any kind of such differing other in quality, quantity, or in both Contradictories Contraries Subcontraries Subalternation

14. The Traditional Square of Opposition, continued 14 Contradictories One proposition is the denial or negation of the other One is true, one is false A and O are contradictories E and I are contradictories

15. The Traditional Square of Opposition, continued 15 Contraries If one is true, the other must be false Both can be false A and E are contraries

16. The Traditional Square of Opposition, continued 16 Subcontraries They cannot both be false They may both be true If one is false, then the other must be true I and O are subcontraries

17. The Traditional Square of Opposition, continued 17 Subalteration Opposition between a universal proposition (superaltern) and its corresponding particular proposition (subaltern) Universal proposition implies the truth of its corresponding particular proposition Occurs from A to I propositions Occurs from E to O propositions

18. The Traditional Square of Opposition, continued 18 E A contraries (No S is P.) superaltern (All S is P.) superaltern contrad ictories contradictories subalternation subalternation subaltern (Some S is not P.) subaltern (Some S is P.) subcontraries I O Immediate inference Inference drawn from only one premise

19. The Traditional Square of Opposition, continued 19 Immediate inferences

20. Further Immediate Inferences 20 Conversion Formed by interchanging the subject and predicate terms of a categorical proposition

21. Further Immediate Inferences, continued 21 Complement of a class The collection of all things that do not belong to that class Class denoted as S Complement denoted as non-S Double negatives

22. Further Immediate Inferences, continued 22 Obversion Changing the quality of a proposition and replacing the predicate term by its complement

23. Further Immediate Inferences, continued 23 Contraposition 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

24. Existential Import and the Interpretation of Categorical Propositions 24 Existential import Proposition asserts the existence of an object of some kind Example All inhabitants of Mars are blond (A proposition) Some inhabitants of Mars are not blond (O proposition) A and O are contradictories Since Mars has no inhabitants, both statements are false, so these statements cannot be contradictories

25. Existential Import and the Interpretation of Categorical Propositions, continued 25 Presupposition We presuppose propositions never refer to empty classes Problems Never able to formulate the proposition that denies the class has members What we say does not suppose that there are members in the class Wish to reason without making any presuppositions about existence

26. Existential Import and the Interpretation of Categorical Propositions, continued 26 Boolean interpretation Universal propositions are not assumed to refer to classes that have members I and O continue to have existential import Universal propositions are the contradictories of the particular propositions Universal propositions are interpreted as having no existential import

27. Existential Import and the Interpretation of Categorical Propositions, continued 27 Boolean interpretation 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 Corresponding A and E propositions can both be true and are therefore not contraries I and O propositions can both be false if the subject class is empty

28. Existential Import and the Interpretation of Categorical Propositions, continued 28 Boolean interpretation Subalternation is not generally valid Preserves some immediate inferences Conversion for E and I propositions Contraposition for A and O propositions Obversion for any proposition Transforms the traditional Square of Opposition by undoing relations along the sides of the square

29. Symbolism and Diagrams for Categorical Propositions 29 Boolean interpretation notation Empty class: 0 S has no members: S = 0 Deny class is empty: S≠ 0 Product (intersection) of two classes: SP No satires are poems: SP = 0 Some satires are poems: SP≠ 0

30. Symbolism and Diagrams for Categorical Propositions, continued 30 Complement of a class: S All S is P: SP = 0 Some S is not P: SP≠ 0

31. Symbolism and Diagrams for Categorical Propositions, continued 31

32. 32 Boolean Square of Opposition SP = 0 SP = 0 E A contrad ictories contradictories I O SP ≠ 0 SP≠ 0 Symbolism and Diagrams for Categorical Propositions, continued

33. 33 Venn diagrams of Boolean interpretation S S x S = 0 S≠ 0 P S SP SP SP SP Symbolism and Diagrams for Categorical Propositions, continued

34. 34 Venn diagrams of categorical propositions P S P S P S x A: All S is P SP = 0 E: No S is P SP = 0 I: Some S is P SP≠ 0 P S x O: Some S is not P SP≠ 0 Symbolism and Diagrams for Categorical Propositions, continued

35. 35 Venn diagrams of categorical propositions P S P S P S x A: All P is S PS = 0 E: No P is S PS = 0 I: Some P is S PS≠ 0 Symbolism and Diagrams for Categorical Propositions, continued P S x O: Some P is not S PS≠ 0

36. Summary 36 Categorical propositions Quality, quantity, and distribution Opposition Immediate inferences Existential import Boolean interpretation Symbolism and diagrams of categorical propositions