1. Lesson 5.6
Deductive Proofs pp

2. Objectives: 1. To define and apply four methods of deductive proof.
2. To identify the steps required for a valid deductive proof.
3. To recognize converse and inverse fallacies.

3. Deductive reasoning can be split into two major sections: proof by direct argument and proof by contradiction. Examples of direct argument proofs include:
1. Law of Deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

4. Definition
The Law of Deduction is a method of deductive proof with the following symbolic form.

5. Definition The Law of Deduction p (assumed) q1 q2
qn (statements known to be true)
r (deduced from statements above)
pr (conclusion)

6. Definition
Modus ponens is a method of deductive proof with the following symbolic form.
Premise 1: pq
Premise 2: p
Conclusion: q

7. Definition
Modus tollens is a method of deductive proof with the following symbolic form.
Premise 1: pq
Premise 2: ~q
Conclusion: ~p

8. Definition
Transitivity is a method of deductive proof with the following symbolic form.
Premise 1: pq
Premise 2: qr
Conclusion: pr

9. EXAMPLE 2 Classify this argument.
Premise 1: If man is sinful, then he must die (Rom. 6:23).
Premise 2: Man is sinful (Rom. 3:23).
Conclusion: Man must die (second death; Rev. 21:8).
Modus Ponens; valid

10. If it rains today, the ball game will be canceled.
It rained.
the ball game was cancelled.
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

11. If we eat carrots, then our eyesight will be good.
Our eyesight is not good.
we didn’t eat our carrots.
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

12. r(pq)
~p~q
~r
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

13. A  B
B  C
C is a right angle
A is a right angle
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

14. Two fallacies result from common misuse of modus ponens and modus tollens. These fallacies are assuming the converse and assuming the inverse.

15. EXAMPLE 3 Classify the fallacy committed by this argument.
All dogs are mammals.
Fido is a mammal.
Therefore, Fido is a dog.
Assuming the converse

16. Identify the fallacy in the argument.
All US presidents are US citizens
Ken is a US citizen.
Ken is a US president.
1. Assuming the converse
2. Assuming the inverse

17. Identify the fallacy in the argument.
All whole numbers are integers.
-5 is not a whole number.
-5 is not an integer.
1. Assuming the converse
2. Assuming the inverse

18. Homework pp

19. ►A. Exercises
Suppose you want to prove the Vertical Angle Theorem using the Law of Deduction: “If two angles are vertical angles, then they are congruent.”
1. What should you assume?
Answer: Two angles are vertical angles.

20. ►A. Exercises
Suppose you want to prove the Vertical Angle Theorem using the Law of Deduction: “If two angles are vertical angles, then they are congruent.”
2. What should you derive from the assumption?
Answer: The two angles are congruent.

21. ►A. Exercises Suppose you know that 1 and 2 are vertical angles.
3. What would you conclude from the Vertical Angle Theorem?
Answer: 1  2

22. ►A. Exercises Suppose you know that 1 and 2 are vertical angles.
4. What rule of logic lets you draw the conclusion?
Answer: modus ponens

23. ►A. Exercises Suppose you know that A and B are not congruent.
5. What can you conclude from the theorem?
Answer: A and B are not vertical angles.

24. ►A. Exercises Suppose you know that A and B are not congruent.
6. What rule of logic lets you draw the conclusion?
Answer: modus tollens

25. ►A. Exercises Suppose you know that A and B are not congruent.
7. If A → B and B → C, then what?
Answer: A → C by transitivity

26. ►A. Exercises Identify the fallacy in each of the following arguments.
8. If smoke comes from a house, it must be on fire. No smoke comes from the house. Therefore, the house is not on fire.
Answer: assuming the inverse

27. ►A. Exercises Identify the fallacy in each of the following arguments.
9. If the temperature is above 100°C, then the water will boil. The water on the campfire was boiling. Thus, its temperature was above 100°C.
Answer: assuming the converse

28. ►B. Exercises
Look at the following arguments and state the form of deductive reasoning used.
13. If I break my arm, then I will go to the
hospital. I broke my arm. I will go to
the hospital.
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

29. ►B. Exercises
Look at the following arguments and state the form of deductive reasoning used.
15. If you study for your geometry test, then you will get a high score on the test. If you get a high score on the test, then you will get a good grade on your report card. If you study for your geometry test, then you will get a good grade on your report card.

30. ►B. Exercises
Look at the following arguments and state the form of deductive reasoning used.
16. Two lines intersect. Every line contains at least two points. Three distinct noncollinear points lie in exactly one plane. Therefore, two intersecting lines lie in exactly one plane.

31. ►B. Exercises
Look at the following arguments and state the form of deductive reasoning used.
17. A cat is mammal. All mammals have four-chambered hearts. A cat has a four-chambered heart.
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity

32. ►B. Exercises
Look at the following arguments and state the form of deductive reasoning used.
18. If a triangle has two congruent sides, then the triangle is an isosceles triangle. ∆ABC is not an isosceles triangle. ∆ABC does not have two congruent sides.

33. ►B. Exercises [(pq)p]  q
Give the symbolic logic form needed to prove each of the following.
19. modus ponens
[(pq)p]  q

34. ►B. Exercises [(pq)(qr)]  (pr)
Give the symbolic logic form needed to prove each of the following.
20. transitivity
[(pq)(qr)]  (pr)

35. ■ Cumulative Review
25. p: A quadrilateral is a square and only if it is both a rectangle and a rhombus.
Using s: “the quadrilateral is a square,” r: “the quadrilateral is a rectangle,” and h: “the quadrilateral is a rhombus,” which of the following expressions below best represents statement p?

36. ■ Cumulative Review 25. A. s(rh) B. (sr)h C. s(rh) D. (sr)h
E. (sr)(sh)

37. ■ Cumulative Review STATEMENTS REASONS -2x + 5  8 Given
Add prop of ineq
Mult prop of ineq
Assoc prop of mult
Mult inverse prop
Mult ident prop

38. Analytic Geometry
Graphing Lines

39. In algebra you learned the slope-intercept form of a line is y = mx + b.

40. Graph y = 2x - 3
m =
b = 2 -3

41. Find the equation of the line whose slope is 2/3 and the intercept is (0, 5).

42. Find the equation of the line whose slope is -3 and the intercept is (0, 2).

43. Find the equation of the line that passes through (3, 4) and (0, -2).

44. Find the equation of the line whose slope is 3 and passes through the point (-2, 5).