Deductive reasoning is the application of a general statement to a specific instance.
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1 Section1.1: Deductive versus Inductive Reasoning Logic is the science of correct reasoning. Websters New World College Dictionary defines reasoning as the drawing of inferences or conclusions from known or assumed facts. Problem Solving Logic and reasoning are associated with the phrases problem solving and critical thinking. If we are faced with a problem, puzzle, or dilemma, we attempt to reason through it in hopes of arriving at a solution. Deductive Reasoning Deductive reasoning is the application of a general statement to a specific instance. A syllogism is an argument composed of two statements, or premises (the major and minor premises), followed by a conclusion. For any given set of premises, if the conclusion of an argument is guaranteed ( that is, if it is inescapable in all instances), the argument is valid. If the conclusion is not guaranteed ( that is, if there is at least one instance in which it does not follow), the argument is invalid. Example 1. All men are mortal. (major premise ) 2. Socrates is a man. (minor premise) Therefore, Socrates is mortal. (conclusion) 1
2 Deductive Reasoning and Venn Diagrams The validity of a deductive argument can be shown by use of a Venn diagram. A Venn diagram is a diagram consisting of various overlapping figures contained within a rectangle ( called the universe). Examples 2
3 ANALYZING A DEDUCTIVE ARGUMENT Example 2 Construct a Venn diagram to verify the validity of the following argument: 1. All men are mortal. 2. Socrates is a man. Therefore, Socrates is mortal. Example 3 Construct a Venn diagram to verify the validity of the following argument: 1. All doctors are men. 2. My mother is a doctor. Therefore, my mother is a man. (*) Saying that an argument is valid does not mean that the conclusion is true. If the premises of a valid argument are true, then the conclusion will also be true. 3
4 Example 4 Construct a Venn diagram to verify the validity of the following argument: 1. All professional wrestlers are actors. 2. The Rock is an actor. Therefore, The Rock is a professional wrestler. (*) Saying that an argument is invalid does not mean that the conclusion is false. In logic, validity and truth do not have the same meaning. Validity refers to the process of reasoning used to obtain a conclusion; truth refers to conformity with fact or experience. Example 5 Construct a Venn diagram to verify the validity of the following argument: 1. Some plants are poisonous. 2. Broccoli is a plant. Therefore, broccoli is poisonous. 4
5 Example 6 Construct a Venn diagram to verify the validity of the following argument: 1. No snake is warm blooded. 2. All mammals are warm- blooded. Therefore, snakes are not mammals. Inductive Reasoning Inductive reasoning involves going from a series of specific cases to a general statement. Although it may seem to follow and may in fact be true, the conclusion in an inductive argument is never guaranteed. Example 1. Joe sneezed after petting Frako s cat. 2. Joe sneezed after petting Paulettes cat. Therefore, Joe is allergic to cats. 5
6 INDUCTIVE REASONING AND PATTERN RECOGNITION Example 7 What is the next number in the sequence 1, 8, 15, 22, 29,...? 6
7 Problem Real men dont eat quiche. 2. Clint Eastwood is a real man. Therefore, Clint Eastwood doesnt eat quiche. Exercise All roads lead to Rome. 2. Route 66 is a road. Therefore, Route 66 leads to Rome. 7
8 Exercise Some animals are dangerous. 2. A tiger is an animal. Therefore, a tiger is dangerous. Exercise Some women are police officers. 2. Some police officers ride motorcycles. Therefore, some women ride motorcycles. 8
9 Exercise All squares are rectangles. 2. Some quadrilaterals are squares. Therefore, some quadrilaterals are rectangles. Exercise 22 Classify each argument as deductive or inductive. a. 1. I ate a chili dog at Joe s and got indigestion. 2. I ate a chili dog at Ruby s and got indigestion. Therefore, chili dogs give me indigestion. b. 1. All spicy foods give me indigestion. 2. Chili dogs are spicy food. Therefore, chili dogs give me indigestion. 9
10 In Exercises 23, 25, 26 and 28, fill in the blank with what is most likely to be the next number. Exercise 23 3, 8, 13, 18,... Exercise 25 0, 2, 6, 12,... Exercise 26 1, 2, 5, 10,... Exercise 28 1, 8, 27, 64,... 10
11 Exercise 33 Fill in the blanks with what are most likely to be the next letters. O, T, T, F,...,... Exercise 38 Explain the general rule or pattern used to assign the given letter to the given word. circle square trapezoid octagon rectangle i u a o... 11
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