Mat 142 Chapter 1 Logic Last updated on February 1, 2010 Dr. Firoz 1 Chapter 1. The Logic Section 1.1 Deductive versus Inductive Reasoning What is logic: The science of correct reasoning. Logic is fundamental both to critical thinking and problem solving. What is reasoning: Reasoning is defined as the drawing of inferences or conclusions from known or assumed facts. Deductive versus Inductive reasoning: The type of logic, which is known as deductive reasoning is the application of a general statement to a specific instance. Example of deductive reasoning: To solve any quadratic equation like0122xx, one can use the general formula aacbbx242 for the equation 02cbxax. On the other hand, in inductive reasoning conclusions may be probable but not guaranteed. Example of inductive reasoning: The probable answer of next term of the sequence 1, 8, 15, 22, 29 could be 36. But it is not guaranteed. One may answer that the next number in the sequence is 5, if he/she thinks the numbers are coming from Mondays in August 2005. Then next Monday is on September 5. We can only use inductive reasoning and give one or more possible answers. What is argument: The standard dictionary meaning of argument is a discussion in which there is a disagreement. Valid versus invalid argument: If the conclusion of an argument is guaranteed, the argument is valid. On the other hand if the conclusion of the argument is not guaranteed, the argument is invalid. Saying that an argument is valid does not mean that the conclusion is true: We verify the situation by an example. Consider two premises 1. All doctors are men, 2. My mother is a doctor. Then the valid argument “My mother is a man” is not a true conclusion. Saying that an argument is invalid does not mean that the conclusion is false. We verify the situation also by an example. Consider two premises 1. All professional wrestlers are actors, 2. The Rock is an actor. Then the invalid argument “the Rock is a professional wrestler”, may not be false. We will verify valid and invalid arguments and conclusions with Venn diagram. What is a Venn Diagram: A Venn diagram consists of a rectangle, representing the universal set, and various closed figures within the rectangle, each representing a set.Mat 142 Chapter 1 Logic Last updated on February 1, 2010 Dr. Firoz 2 Venn diagram and invalid arguments: To show that an argument is invalid you must construct a Venn diagram in which the premises are met yet the conclusion does not necessarily follow. Example 1. Construct a Venn diagram to determine the validity of the given argument. No snake is warm-blooded All mammals are warm-blooded Therefore, snakes are not mammals Solution: Suppose x represents snakes. The position of x in the diagram is unique and shows that the Snakes mammals argument is valid. x Worm-blooded Example 2. Construct a Venn diagram to determine the validity of the given argument. All professional wrestlers are actors The Rock is an actor Therefore, the Rock is a professional wrestler Solution: Suppose x represents Rock. Then the different position of x in the diagram shows that the argument is invalid. Professional wrestlers Professional wrestlers x x Actors Actors Here the argument is invalid. But the conclusion could be true. This example demonstrates that an invalid argument can have a true conclusion even though The Rock is a professional wrestler, the argument used obtain the conclusion is invalid. Exercise 3. a) Construct a Venn diagram and verify that the following argument is invalid. 1. (Major premise) Some plants are poisonous 2. (Minor premise) Broccoli is a plant Therefore (conclusion) Broccoli is poisonous. b) Verify that the argument in question 1 is deductive, but argument in question 2 is inductive. Section 1.2 Symbolic logic What is a statement (or proposition): A statement is a sentence that is either true or false. All logical reasoning is based on statement.Mat 142 Chapter 1 Logic Last updated on February 1, 2010 Dr. Firoz 3 Examples of statements (or propositions). 1. Apple manufactures computers. 2. A $2000 computer that is discounted 25% will cost $1500 (true) 3. A $2000 computer that is 25% discounted will cost $1000 (false) Examples which are not statements. 1. I am telling the truth (either true or false) 2. Apple manufactures the world‟s best computers (either true or false) 3. Did you go to market yesterday? (question) Symbols for logic: By tradition symbolic logic uses lowercase letters as labels for statements. The most frequently used letters are p, q, r, s, and t. Compound Statements and Logical Connectives A compound statement is a statement that contains one or more simpler statements. A compound statement can be formed by inserting the word „not‟ into a simpler statement or by joining two or more simpler statements with connective words such as „and‟, „or‟, „if …then…..‟, „only if‟, „if and only if‟ etc. The compound statement could be a negation, a conjunction, a disjunction, a conditional, or any combination thereof. Negations: The negation of a statement is the denial of the statement and is represented by the symbol ~. For example, given the statement p: it is snowing, the negation is ~p: it is not snowing. Negations of statements containing qualifiers. The words some, all, no (or none) are referred to as qualifiers. The negation of “all p are q would be some p are not q” and the negation of “some p are q would be no p are q”. One may remember the following diagram. All p are q No p are q Some p are q Some p are not q Example 4. Determine which pairs of statements are negations of each other. 1. Some of the beverages contain caffeine 2. Some of the beverages do not contain caffeine 3. None of the beverages contain caffeine 4. All of the beverages contain caffeine Solution: The negation of 1 is 3 and the negation of 3 is 1. On the other hand the negation of 2 is 4 and the negation of 4 is 2.Mat 142 Chapter 1 Logic Last updated on February
View Full Document
Unlocking...