Welcome to Interactive Chalkboard 4 Deductive Reasoning Welcome to Interactive Chalkboard

Objectives Use the Law of Detachment Use the Law of Syllogism

Deductive Reasoning Recall, in previous sections we used inductive reasoning (finding examples 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)

If p  q is true and p is true, then q is also true. Law of Detachment Law of Detachment – a form of deductive reasoning that is used to draw conclusions from true conditional statements If p  q is true and p is true, then q is also true. [(p  q) ^ p]  q Basically, you determine the validity of the conditional before you test the validity of the conclusion.

Example 1: 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. Given: Conclusion: The hypothesis states that Answer: Since the conditional is true and the hypothesis is true, the conclusion is valid.

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) If p  q and q  r are true, then p  r is true. [(p  q) ^ (q  r)]  (p  r)

Example 2a: PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following set of statements. (2) Mark is a 17-year-old student. (1) If Salline attends the prom, she will go with Mark. 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: 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.

Your Turn: 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

Example 3b: (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.

Your Turn: 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

Practice Problems