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
Second step
• Draw representation of second premise,
adding to pictures of first premise
A
B
e.g., draw B as a subset of C
C
shortcut
• We found a diagram in which conclusion
did not hold  conclusion is INVALID
A
B
Conclusion = Some A are not C  NOT TRUE ABOVE!
C
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
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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
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
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
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