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Lesson 5.6
Deductive Proofs
pp. 189-195
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.
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
Definition
The Law of Deduction is a
method of deductive proof
with the following symbolic
form.
Definition
The Law of Deduction
p (assumed)
q1
q2
qn (statements known to be true)
r (deduced from statements above)
pr (conclusion)
Definition
Modus ponens is a method
of deductive proof with the
following symbolic form.
Premise 1: pq
Premise 2: p
Conclusion: q
Definition
Modus tollens is a method of
deductive proof with the
following symbolic form.
Premise 1: pq
Premise 2: ~q
Conclusion: ~p
Definition
Transitivity is a method of
deductive proof with the
following symbolic form.
Premise 1: pq
Premise 2: qr
Conclusion: pr
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
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
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
r(pq)
~p~q
~r
1. Law of deduction
2. Modus ponens
3. Modus tollens
4. Transitivity
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
Two fallacies result from
common misuse of modus
ponens and modus tollens.
These fallacies are assuming
the converse and assuming
the inverse.
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
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
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
Homework
pp. 193-195
►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.
►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.
►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
►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
►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.
►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
►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
►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
►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
►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
►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.
►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.
►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
►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.
►B. Exercises
Give the symbolic logic form needed to
prove each of the following.
19. modus ponens
[(pq)p]  q
►B. Exercises
Give the symbolic logic form needed to
prove each of the following.
20. transitivity
[(pq)(qr)]  (pr)
■ 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?
■ Cumulative Review
25. A.
B.
C.
D.
E.
s(rh)
(sr)h
s(rh)
(sr)h
(sr)(sh)
■ Cumulative Review
STATEMENTS
-2x + 5  8
REASONS
Given
26. -2x  3
26. Add prop of ineq
27. -½(-2x)  -½(3)
27. Mult prop of ineq
28. (-½-2)x  -½(3)
28. Assoc prop of mult
29. 1x  -3/2
29. Mult inverse prop
30. x  -3/2
30. Mult ident prop
Analytic Geometry
Graphing Lines
In algebra you learned the
slope-intercept form of a line is
y = mx + b.
Graph y = 2x - 3
m=2
b = -3
Find the equation of the line
2
whose slope is /3 and the
intercept is (0, 5).
Find the equation of the line
whose slope is -3 and the intercept
is (0, 2).
Find the equation of the line that
passes through (3, 4) and (0, -2).
Find the equation of the line
whose slope is 3 and passes
through the point (-2, 5).