2.3 Apply Deductive Reasoning

Objectives

Use the Law of Detachment

Use the Law of Syllogism

Deductive Reasoning

Recall, in previous sections we used inductive reasoning (finding examples and patterns to make a conjecture) to present arguments.

Another method is: Deductive Reasoning – a form of

argument in which facts, rules, definitions, or properties are used to reach logical conclusions (i.e. think Sherlock Holmes, Encyclopedia Brown, CSI)

Law of Detachment

Law of Detachment – is a form of deductive reasoning that is used to draw conclusions from true conditional statements (also called a direct argument)

If p q is true and p is true, then q is also true.

[(p q) ^ p] q

Basically, if the hypothesis of a true conditional is true, then the conclusion is also true.

The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.

Conclusion:

The hypothesis states that

Answer: Since the conditional is true and the hypothesis is true, the conclusion is valid.

Given:

Example 1a:

The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.

Answer: According to the hypothesis for the conditional, you must have two pairs of congruent segments. The given only has one pair of congruent segments. Therefore, the conclusion is not valid.

The hypothesis states that is a segment and

Conclusion:Given:

Example 1b:

The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a polygon is a convex quadrilateral, then the sum of the interior angles is 360.

a. Given: Conclusion: If you connect X, N, and O with segments, the figure will be a convex quadrilateral.

b. Given: ABCD is a convex quadrilateral. Conclusion: The sum of the interior angles of ABCD

is 360.

Answer: not valid

Answer: valid

Your Turn:

Law of Syllogism

Law of Syllogism – a law of logic which uses the Transitive Property of Equality,

if a = b and b = c, then a = c (also called the chain rule)

If p q and q r are true, then p r is true.

[(p q) ^ (q r)] (p r)

PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following set of statements.

(2) If Saline attends the prom, she will be voted Prom Queen.

(1) If Mark attends the prom, he will go with Saline.

Answer: This is an example of the law of syllogism. If the first statement is true, then since the second statement used the conclusion of the first as its hypothesis, then its conclusion becomes true.

Example 2a:

PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following set of statements.

(2) The Peddler Steakhouse stays open until 10 P.M.

(1) If Mel and his date eat at the Peddler Steakhouse before going to the prom, they will miss the senior march.

Answer: There is no valid conclusion. While both statements may be true, the conclusion of each statement is not used as the hypothesis of the other.

Example 2b:

Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.

a. (1) If you ride a bus, then you attend school. (2) If you ride a bus, then you go to work.

b. (1) If your alarm clock goes off in the morning, then you will get out of bed. (2) You will eat breakfast, if you get out of bed.

Answer: invalid

Answer: valid

Your Turn:

Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

(1) If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.

(3) XYZ is a right triangle.

(2) For XYZ, (XY)2 + (YZ)2 = (ZX)2.

Example 3a:

p: the sum of the squares of the two sides of a triangle is equal to the square of the third side

q: the triangle is a right triangle

By the Law of Detachment, if is true and p is true, then q is also true.

Answer: Statement (3) is a valid conclusion by the Law of Detachment

Example 3a:

(1) If Ling wants to participate in the wrestling competition, he will have to meet an extra three times a week to practice.

(3) If Ling wants to participate in the wrestling competition, he cannot take karate lessons.

(2) If Ling adds anything extra to his weekly schedule, he cannot take karate lessons.

Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

Example 3b:

p: Ling wants to participate in the wrestling competition

q: he will have to meet an extra three times a week to practice

r: he cannot take karate lessons

By the Law of Syllogism, if and are true. Then is also true.

Answer: Statement (3) is a valid conclusion by the Law of Syllogism.

Example 3b:

Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment of the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

a. (1) If a children’s movie is playing on Saturday, Janine will take her little sister Jill to the movie.

(3) If a children’s movie is playing on Saturday, Jill will get popcorn.

(2) Janine always buys Jill popcorn at the movies.

Answer: Law of Syllogism

Your Turn:

b. (1) If a polygon is a triangle, then the sum of the interior angles is 180.

(3) The sum of the interior angles of polygon GHI is 180.

(2) Polygon GHI is a triangle.

Answer: Law of Detachment

Your Turn:

Assignment

Geometry: Pgs. 90 – 93 #4 – 13, 21 –

22, 29