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Euler’s circles

Euler’s circles. Some A are not B. All B are C. Some A are not C. Algorithm = a method of solution guaranteed to give the right answer. First step. Draw the diagrams that the first premise entail (p. 18) Some A are not B. B. A. A. A. B. B. A. B. C. Second step.

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Euler’s circles

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  1. Euler’s circles • Some A are not B. • All B are C. • Some A are not C. • Algorithm = a method of solution guaranteed to give the right answer

  2. First step • Draw the diagrams that the first premise entail (p. 18) • Some A are not B B A A A B B

  3. A B C Second step • Draw representation of second premise, adding to pictures of first premise e.g., draw B as a subset of C

  4. A B C shortcut • We found a diagram in which conclusion did not hold  conclusion is INVALID Conclusion = Some A are not C  NOT TRUE ABOVE!

  5. deduction • Applying logical rules to given information (premises) to see the results • E.g., Euler’s circles are the logical rules • Another type of deduction is conditional reasoning

  6. Conditional reasoning problems • If p then q • p • q • First line = first premise or premise 1 or major premise • Second line = second premise or minor premise • Third line = conclusion (is the conclusion valid or invalid?)

  7. Other parts • If p then q <- a “conditional” • On the condition that p is true, then q will also be true • p and q are “terms” also called “variables” (they vary in their values) • p <- “p is true” • q <- “q is true”

  8. more on conditional reasoning • if p then q • p • q • antecedent = part after “if” • if a person is a teacher, then they are a woman • consequent = part after “then” • Dr. Carrier is a teacher • Dr. Carrier is a woman <- VALID

  9. logical structure of problem • if p then q • p • q • always VALID conclusion

  10. another example • women are better at multitasking than are men • if one group is good at doing something and another group is not, then the first group is better • women are good at multitasking and men are not • women are better at multitasking than are men • VALID conclusion

  11. our first logical rule for CR problems • if p then q • p • q <- conclusion is VALID • affirmation of the antecedent  we are saying that the antecedent is true • a logical rule • aka, modus ponens

  12. 2nd logical rule • if a person is a teacher, then they are a woman • Dr. Carrier is a woman • Dr. Carrier is a teacher • logical structure • if p, then q • q • p  always INVALID • called AFFIRMATION OF THE CONSEQUENT

  13. 3rd logical rule • If a person is a teacher, then they are a woman. • Dr. Carrier is not a teacher • Dr. Carrier is not a woman <- INVALID • If p, then q • not p <- “p is not true” • not q <- “q is not true” • conclusion always INVALID • denying the antecedent saying that the antecedent is false

  14. 4th logical rule • if a person is a teacher, then they are a woman • Dr. Carrier is not a woman • Dr. Carrier is not a teacher <- VALID • if p then q • not q <- “q is not true” • not p <- “p is not true” • conclusion is always VALID • called DENIAL OF THE CONSEQUENT, aka modus tollens

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