# 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.

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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 A B A B A B

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

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

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

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?)

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”

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

logical structure of problem if p then q p q always VALID conclusion

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

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

2 nd 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

3 rd 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

4 th 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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