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- 1. Categorical Propositions

4.3 (Venn Diagrams and the Modern Square of Opposition), 4.5 (The Traditional Square of Opposition) and 4.6 (Venn Diagrams and the Traditional Standpoint)

2. 4.3 Venn Diagrams and the Modern Square of Opposition

All Henrys students are achievers

All unicorns are one-horned animals.

Aristotle and Boole conflict: Two approaches to the interpretation of existence of Universal Categorical Proposition

Aristotle: assumes existence, that is things actually exist in all propositions

Boole: make no assumptions about existence, that is universal propositions have no existential import.

Aristotle and Boole agree that particular propositions make claims about actually existing things. They disagree on universal (A and E) propositions only.For Aristotle universal propositions have existential import if and only if the proposition asserts something about an actually existing thing.

3. Venn Diagrams and the Modern Square of Opposition

Aristotelian Standpoint:

All German Shepherds are dogs

No snakes are snails

All vampires are vile creatures

Boolean Standpoint:

All dogs are animals

No mammals are frogs

All squares are round objects

4. Venn Diagrams and the Modern Square of Opposition

Existential Fallacies from the Boolean Standpoint:

All S are P

Therefore, some S are P

No Some S are P

Therefore, some S are not P

Why are these arguments existential fallacies?

You cant conclude the existence of something from a premise that makes no assumptions about existence.

5. Venn Diagrams and the Modern Square of Opposition

John Venn (19th Century) created Venn Diagram system by adoption the Boolean Standpoint.

All S are P =

No members of S outside P.

No S are P =

No members of S are inside P.

6. Venn Diagrams and the Modern Square of Opposition

Some S are P =

At least one S exists

and that S is a P.

Some S are not P =

At least one S exists

and that S is not a P.

7. Venn Diagrams and the Modern Square of Opposition

8. Venn Diagrams and the Modern Square of Opposition

Modern Square of Opposition: have a contradictory relationship.

9. Venn Diagrams and the Modern Square of Opposition

Using the Modern Square of Opposition to determine the true value of an Argument:

All dogs are animals

Therefore, some dogs are not animals

10. Venn Diagrams and the Modern Square of Opposition

All dogs are animals

Therefore some dogs are animals

11. 4.5 The Traditional Square of Opposition

Aristotle: Universal propositions about actually existing things have existential import

Three new moves with traditional Square of opposition

Contrary

Subcontrary

Subalternation

12. The Traditional Square of Opposition

13. 4.5 The Traditional Square of Opposition

Immediate inference using the Traditional Square

Contradictory = opposite truth value (same as Modern Square). Relationship between A & O and E & I only.

Contrary= at least one is false, (not both true). Relationship between A & E only.

Subcontrary = at least one true, (not both false). Relationship between I & O only.

Subalternation = truth flows downward and falsity flows upward. Relationship between A & I and E & O only.

14. The Traditional Square of Opposition

Three steps for testing for immediate inference using the Traditional Square of Opposition

Determine the type of relation, that is the move that has occurred between the premise and conclusion

Using the basic relations from the traditional square of opposition, deduce the remaining truth values if possible

If the move is legal, then the argument is valid. If the move is illegal/illicit, then the argument is invalid.

15. The Traditional Square of Opposition

Three Fallacies:

Illicit contrary: invalid application of the contrary relation

Illicit subcontrary: invalid application of the subcontrary relation

Illicit subalternation: invalid application of the subalternation relation

16. The Traditional Square of Opposition

Using the Traditional Square of Opposition to make immediate inferences: Lets work our way around the square using the following propositions:

All dogs are animals (T)

Some dogs are fish (F)

Some students are not a piano (T)

17. The Traditional Square of Opposition

All dogs are animals (T)

18. The Traditional Square of Opposition

Some dogs are fish (F)

19. The Traditional Square of Opposition

Some students are not a piano (T)

20. 4.6 Venn Diagrams and the Aristotelian or Traditional Standpoint

The conflict between Aristotle and Boole is based on the whether universal proposition have existential import.

Aristotle assumes that universal propositions (A & E) that actually refer to existing things have existential import.

Boole makes no assumptions that universal propositions (A & E) refer to existing things, that is have existential import.

21. Venn Diagrams and the Aristotelian or Traditional Standpoint

22. Venn Diagrams and the Aristotelian or Traditional Standpoint

Problem Steps:

Reduce to problem to its basic form and check from the Boolean standpoint. If the problem is valid from the Boolean standpointyour doneit is also valid from the Aristotelian standpoint.

If the problem is invalid from the Boolean standpoint, then adopt the Aristotelian Standpoint and place a circled X in the un-shaded region of the premise and recheck the argument.

If the argument is conditionally valid, then check if the circled X actually represents an existing thing. If so the argument is valid from the Aristotelian Standpoint. If not, then the argument is invalid.

23. Venn Diagrams and the Aristotelian or Traditional Standpoint

Problem:

All squiresare animals

Therefore, some squires are animals

24. 4.3, 4.5 and 4.6 Summary

Aristotelian and Boolean standpoints:

Aristotle is open to existence

Boole is closed to existence

Valid arguments from the Boolean standpoint are unconditionally valid because we are not concerned with whether the terms actually refer to existing things.

Arguments from the Aristotelian Standpoint are conditionally valid because we have to determine if the subject term of premise actually refer to existing thing.

25. 4.3, 4.5 and 4.6 Summary

Existential Fallacies:

26. 4.3, 4.5 and 4.6 Summary

Modern Square of Opposition

Contradictory relationships, all other relationships are undetermined.

Traditional Square of Opposition

Contradictory

Contrary

Subcontrary

Subalternation

27. 4.3, 4.5 and 4.6 Summary

Venn Diagrams:

Boolean or Modern Standpoint

28. 4.3, 4.5 and 4.6 Summary

- Venn Diagrams:

Aristotelian or Traditional Standpoint

29. 4.3, 4.5, 4.6 Summary

Practice Problems:

Film doing problems in Logic Coach