This document discusses categorical propositions, including their four classes (A, E, I, O), quality, quantity, distribution, and opposition. It describes the traditional square of opposition and immediate inferences. It then introduces the Boolean interpretation, which removes existential import from universal propositions and transforms the square of opposition. Finally, it presents symbolic notation and Venn diagrams for representing categorical propositions under the Boolean interpretation.

1. Logic HUM 200 Categorical Propositions 1

2. Objectives2When you complete this lesson, you will be able to:Identify the four classes of categorical propositionsDescribe the quality, quantity, and distribution of categorical propositionsIdentify the four types of oppositionApply the immediate inferences given in the Square of OppositionApply immediate inferences not directly associated with the Square of OppositionDescribe existential importList and describe the implications of the Boolean interpretation of categorical propositionsSymbolize and diagram the Boolean interpretation of categorical propositions

3. The Theory of Deduction 3Deductive argumentsPremises are claimed to provide conclusive grounds for the truth of its conclusion Valid or invalidTheory of deductionAims to explain the relations of premises and conclusion in valid arguments Classical logicModern symbolic logic

4. Classes and Categorical Propositions 4ClassCollection of all objects that have some specified characteristic in common Relationships between classes may be:Wholly includedPartially includedExcluded

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

6. 6Universal affirmative proposition (A proposition)Whole of one class is included or contained in another class All S is PVenn diagramPSAll S is PThe Four Kinds of Categorical Propositions

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

8. 8Particular affirmative proposition (I proposition)Two classes have some member or members in common Some S is PVenn diagramPSxSome S is PThe Four Kinds of Categorical Propositions, continued

9. 9Particular negative propositions (O proposition)At least one member of a class is excluded from the whole of another class Some S is not PVenn diagramPSxSome S is not PThe Four Kinds of Categorical Propositions, continued

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

11. Quantity 11An attribute of every categorical proposition, determined by whether the proposition refers to all members, or only some members of the classUniversalRefers to all members of the classA and E propositionsParticularRefers only to some members of the classI and O propositions

12. Distribution 12Characterization of whether terms refer to all members of the class designated by that term A propositionSubject distributed, predicate undistributedE propositionBoth subject and predicate distributedI propositionNeither subject nor predicate distributedO propositionSubject undistributed, predicate distributed

13. The Traditional Square of Opposition 13OppositionAny kind of such differing other in quality, quantity, or in bothContradictoriesContrariesSubcontrariesSubalternation

14. The Traditional Square of Opposition, continued 14ContradictoriesOne proposition is the denial or negation of the other One is true, one is falseA and O are contradictoriesE and I are contradictories

15. The Traditional Square of Opposition, continued 15ContrariesIf one is true, the other must be falseBoth can be falseA and E are contraries

16. The Traditional Square of Opposition, continued 16SubcontrariesThey cannot both be falseThey may both be trueIf one is false, then the other must be trueI and O are subcontraries

17. The Traditional Square of Opposition, continued 17SubalterationOpposition 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 propositionsOccurs from E to O propositions

18. The Traditional Square of Opposition, continued 18EAcontraries(No S is P.)superaltern(All S is P.)superalterncontrad ictoriescontradictoriessubalternationsubalternationsubaltern(Some S is not P.)subaltern(Some S is P.)subcontrariesIOImmediate inferenceInference drawn from only one premise

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

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

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

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

23. Further Immediate Inferences, continued 23ContrapositionFormed 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 24Existential importProposition asserts the existence of an object of some kind ExampleAll inhabitants of Mars are blond (A proposition)Some inhabitants of Mars are not blond (O proposition)A and O are contradictoriesSince Mars has no inhabitants, both statements are false, so these statements cannot be contradictories

25. Existential Import and the Interpretation of Categorical Propositions, continued 25PresuppositionWe 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 26Boolean interpretationUniversal 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 27Boolean interpretationUniversal 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 forceCorresponding 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 28Boolean interpretationSubalternation is not generally valid Preserves some immediate inferences Conversion for E and I propositions Contraposition for A and O propositions Obversion for any propositionTransforms the traditional Square of Opposition by undoing relations along the sides of the square

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

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

31. Symbolism and Diagrams for Categorical Propositions, continued 31

32. 32Boolean Square of OppositionSP = 0SP = 0EAcontrad ictoriescontradictoriesIOSP ≠ 0SP≠ 0Symbolism and Diagrams for Categorical Propositions, continued

33. 33Venn diagrams of Boolean interpretationSSxS = 0S≠ 0PSSPSPSPSPSymbolism and Diagrams for Categorical Propositions, continued

34. 34Venn diagrams of categorical propositionsPSPSPSxA: All S is PSP = 0E: No S is PSP = 0I: Some S is PSP≠ 0PSxO: Some S is not PSP≠ 0Symbolism and Diagrams for Categorical Propositions, continued

35. 35Venn diagrams of categorical propositionsPSPSPSxA: All P is SPS = 0E: No P is SPS = 0I: Some P is SPS≠ 0Symbolism and Diagrams for Categorical Propositions, continued PSxO: Some P is not SPS≠ 0

36. Summary 36Categorical propositionsQuality, quantity, and distributionOppositionImmediate inferencesExistential importBoolean interpretationSymbolism and diagrams of categorical propositions