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Week 12 3.5 Analyzing Arguments with Euler Diagrams Parts of An Student Responsibilities – Week 12 I Reading: This week: Textbook, Sections 3.5, 3.6 MAT 1160 — WEEK 12 Next week: Fallacies, Sudoku Dr. N. Van Cleave I Summarize Sections Work through Examples Spring 2010 I I Recommended exercises: I Section 3.5: evens, 2 – 32 I Section 3.6: evens, 2 – 52 N. Van Cleave, c 2010 N. Van Cleave, c 2010 3.5 Analyzing Arguments with Euler Diagrams Parts of an Arguments — Recall — A logical argument is composed of: I Two types of reasoning: inductive and deductive. 1. premises (assumptions, laws, rules, widely held ideas, or observations) and I Inductive reasoning observed patterns to solve problems. 2. conclusion I Deductive reasoning involves drawing specific conclusions from given general premises. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Valid and Invalid Arguments Euler diagrams I One method for verifying the validity of an argument is the I An argument is valid if the fact that all the premises are true visual technique based on Euler diagrams forces the conclusion to be true. I This technique is similar to Venn diagrams, in that circles are used to denote sets, with I An argument that is not valid is said to be invalid or a fallacy. I overlap indicating shared elements I disjoint circles indicating no shared elements I Deductive reasoning can be used to determine whether logical I a circle contained within another circle indicating a subset arguments are valid or invalid. I An x may be used to indicate a single element Note: valid and true are not the same — an argument can I This is like a game — if possible, we want to show the argument be valid even though the conclusion is false, as we shall see later. I is invalid ! As long as the circles and x’s do not contradict the premises, we can position them to win the game. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Example 1. Is the following argument valid? Example 2. Is the following argument valid? All dogs are animals. Fred is a dog. Animals All rainy days are cloudy. x -------------------- Today is not cloudy. Cloudy days Fred is an animal. -------------------- Dogs Today is not rainy. Rainy days Draw regions to represent the x premise. Let x represent Fred. Draw regions to represent the premise. Let x represent today. Since: the set of all animals contains the set of all dogs, and I Placing the x for today outside the cloudy days region forces it that set contains Fred I to also be outside the rainy days region. I Fred is also inside the regions for animals. Thus, if both premises are true, the conclusion that today is not Therefore, if both premises are true, the conclusion that is an Fred rainy is also true. animal must be true also. The argument is valid. The argument is valid as checked by the Euler diagram. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Example 4. Is the following argument valid? Example 3. Is the following argument valid? All expensive things are desirable. All desirable things make you feel good. All things that make you feel good make you live longer. All banana trees have green leaves Plants with -------------------------------------------------------- That plant has green leaves. green leaves All expensive things make you live longer. -------------------- make you That plant is a banana tree. that live Banana trees make you Things longer Draw regions to represent the that feel Things good premise. Let x represent that Desirable things plant. Where does the x go? Expensive things Rule: Place the x to make the argument invalid if possible. Example of a valid argument which need not have a true conclusion. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Valid or Invalid Arguments? Example 5. Is the following argument valid? 1. All boxers wear trunks. Steve Tomlin is a boxer. Some students go to the beach People who go -------------------------- for Spring Break. to the beach Steve Tomlin wears trunks. I am a student. for Spring Break ------------------------------ I go to the beach for 2. All residents of NYC love Coney Island hot dogs. Spring Break. Ann Stypuloski loves Coney Island hot dogs. ------------------------------------------------ Students Ann Stypuloski is a resident of NYC. Where does the x go? 3. All politicians lie, cheat, and steal. That man lies, cheats, and steals. Can we place the x to make the argument invalid? -------------------------------------- That man is a politician. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Each of these arguments has a true conclusion—determine if the argument is valid or invalid. 1. All contractors use cell phones. Laura Boyle does not use a cell phone. 1. All cars have tires. -------------------------------------- All tires are rubber. Laura Boyle is not a contractor. --------------------- All cars have rubber. 2. All chickens have beaks. All birds have beaks. 2. Some trucks have sound systems. ------------------------ Some trucks have gun racks. All chickens are birds. --------------------------------------------- Some trucks with sound systems have gun racks. 3. Veracruz is south of Tampico. Tampico is south of Monterrey. ------------------------------- Veracruz is south of Monterrey. N. Van Cleave, c 2010 N. Van Cleave, c 2010 Given the premises: 1. All people who drive contribute to air pollution. 1. All chickens have beaks. 2. All people who contribute to air pollution make life a little All hens are chickens. worse. ------------------------ 3. Some people who live in a suburb make life a little worse. All hens have beaks. Which of the following conclusions are valid? 2. No whole numbers are negative. -4 is negative. a) Some people who live in a suburb drive. ------------------------------ -4 is not a whole number. b) Some people who contribute to air pollution live in a suburb. c) Suburban residents never drive. d) All people who drive make life a little worse. N. Van Cleave, c 2010 N. Van Cleave, c 2010 3.6 Analyzing Arguments with Truth Tables Rewriting the Premises and Conclusion Some arguments are more easily analyzed to determine if they are Premise 1: p → q valid or invalid using Truth Tables instead of Euler Diagrams. Premise 2: p One example of such an argument is: Conclusion: q If it rains, then the squirrels hide. Thus, the argument converts to: It is raining. ((p → q) ∧ p) → q ------------------------------------- With Truth Table: The squirrels are hiding. p q ((p → q) ∧ p) → q Notice that in this case, there are no universal quantifiers such as all, some, or every, which would indicate we could use Euler T T Diagrams. T F To determine the validity of this argument, we must first identify the component statements found in the argument. They are: F T F F p = it rains / is raining q = the squirrels hide / are hiding Are the squirrels hiding? N. Van Cleave, c 2010 N. Van Cleave, c 2010 Testing Validity with Truth Tables Recall 1. Break the argument down into component statements, assigning each a letter. Direct Statement p → q 2. Rewrite the premises and conclusion symbolically. Converse q → p Inverse ∼ p →∼ q 3. Rewrite the argument as an implication with the conjunction Contrapositive ∼ q →∼ p of all the premises as the antecedent, and the conclusion as the consequent. 4. Complete a Truth Table for the resulting conditional statement. Which are equivalent? If it is a tautology, then the argument is valid; otherwise, it’s invalid. N. Van Cleave, c 2010 N. Van Cleave, c 2010 If you come home late, then you are grounded. Modus Ponens — The Law of Detachment You come home late. --------------------------------------------- Both of the prior example problems use a pattern for argument You are grounded. called modus ponens, or The Law of Detachment. p = p → q q = p ------ Premise 1: q Premise 2: Conclusion: or Associated Implication: p q ((p → q) ∧ p) → q T T T F F T Notice that all such arguments lead to tautologies, and therefore F F are valid. Are you grounded? N. Van Cleave, c 2010 N. Van Cleave, c 2010 If a knee is skinned, then it will bleed. Modus Tollens — Example The knee is skinned. -------------------------------------- If Frank sells his quota, he’ll get a bonus. It bleeds. Frank doesn’t get a bonus. ------------------------------------- p = Frank didn’t sell his quota. q = p = Premise 1: q = Premise 2: Premise 1: p → q Premise 2: ∼ q Conclusion: ∼ p Conclusion: Thus, the argument converts to: ((p → q) ∧ ∼ q) → ∼ p Associated Implication: p q p q ((p → q) ∧ ∼ q) → ∼ p T T T T T F T F F T F T F F F F (Modus Ponens)– Did the knee bleed? Did Frank sell his quota or not? N. Van Cleave, c 2010 N. Van Cleave, c 2010 Modus Tollens Modus Tollens — Example II If the bananas are ripe, I’ll make banana bread. An argument of the form: I don’t make banana bread. ------------------------------------- p → q The bananas weren’t ripe. ∼ q p = ------ ∼ p q = or Premise 1: p → q Premise 2: ∼ q Conclusion: ∼ p Thus, the argument converts to: ((p → q) ∧ ∼ q) → ∼ p ((p → q) ∧ ∼ q) → ∼ p p q ((p → q) ∧ ∼ q) → ∼ p T T is called Modus Tollens, and represents a valid argument. T F F T F F Were the bananas ripe or not? N. Van Cleave, c 2010 N. Van Cleave, c 2010 Fallacy of the Inverse — Example Fallacy of the Inverse If it rains, I’ll get wet. It doesn’t rain. An argument of the form: ------------------------------------- I don’t get wet. p → q ∼ p p = ------ ∼ q q = Premise 1: p → q Premise 2: ∼ p Conclusion: ∼ q or Thus, the argument converts to: ((p → q) ∧ ∼ p) → ∼ q ((p → q) ∧ ∼ p) → ∼ q p q ((p → q) ∧ ∼ p) → ∼ q T T T F is called the Fallacy of the Inverse, and represents an invalid F T argument.
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