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Read the various perspectives on the death penalty on pages 262-268. Evaluate each argument for validity and soundness. Cite the structural strengths and weaknesses of each argument. Discuss who has the stronger argument? Why? 

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8 How is the professor using logical argumentation in coming up with mathematical proofs? How can learning about deductive logic, such as arguments based on mathematics, help us make better-informed decisions? DEDUCTIVE WHAT’S TO COME 239 | What Is a Deductive Argument? 241 | Types of Deductive Arguments 247 | Hypothetical Syllogisms 252 | Categorical Syllogisms 257 | Translating Ordinary Arguments into Standard Form 261 | Critical Thinking Issue: Perspectives on the Death Penalty I n Sir Arthur Conan Doyle’s mystery story “Silver Blaze,” detective Sherlock Holmes uses his extraordinary powers of deductive logic to solve the mystery of the disappearance of racehorse Silver Blaze and the murder of the horse’s trainer, John Straker. His head shattered by a savage blow, Straker’s body was found a quarter mile from the King’s Pyland stables where Silver Blaze was kept. A search is carried out of the surrounding moors and of the neighboring Mapleton stables for the horse. After interviewing everyone who might have been involved and collecting all the facts, Holmes concludes that Silver Blaze is still alive and hidden in the Mapleton stables, ARGUMENTS 237 Think >> FIRST ■ What is a deductive argument? ■ What are some of the types of deductive arguments? ■ What is a syllogism, and how do we know if it is valid? even though the earlier search of the stables had failed to turn up the missing horse. “It’s this way, Watson,” [says Holmes]. “Now, supposing that [Silver Blaze] broke away during or after the tragedy, where could he have gone to? The horse is a very gregarious creature. If left to himself his instincts would have been either to return to King’s Pyland or go over to Mapleton. Why would he run wild upon the moor? He surely should have been seen by now . . . He must have gone to King’s Pyland or to Mapleton. He is not at King’s Pyland. Therefore, he is at Mapleton.”1 As it turns out, Holmes’s deduction is right. The missing racehorse is at Mapleton, the silver blaze on its nose covered over to disguise its appearance. Sherlock Holmes also solves the “murder” of the horse’s trainer through deductive logic. He learns from the stable hand that the guard dog did not bark when Silver Blaze was “stolen” from the stables. Therefore, Holmes concludes, the person who took Silver Blaze must have been familiar to the dog. This eliminated suspects who were strangers. Holmes then eliminates, one by one, the other suspects, leaving only the horse. As Holmes stated in another story: “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”2 He concludes that the horse must have accidentally killed its trainer when Straker, who was something of a scoundrel, used a surgical knife found in his possession to nick the tendons of Silver Blaze’s ham so the horse would develop a slight limp and lose the upcoming race. Holmes explains, “Once in the hollow, [Straker] had got behind the horse and had struck a light; but the creature, frightened at the sudden glare, and with the strange instinct of animals feeling that some mischief was intended, had lashed out, and the steel shoe had struck Straker full on the forehead.”3 To generations of mystery readers, Sherlock Holmes has epitomized the skilled reasoner. In this chapter we’ll learn how to evaluate deductive arguments and practice some of the strategies used by Holmes and others who are skilled in deductive argumentation. In Chapter 8 we will: • Identify the essential attributes of a deductive argument • Distinguish between validity, invalidity, and soundness in a deductive argument 238 • THiNK • Learn how to recognize and evaluate arguments by elimination, mathematical arguments, and argument from definition Some deductive arguments are more involved and may have several dependent premises and subconclusions. • Study the different types of hypothetical syllogisms, including modus ponens, modus tollens, and chain arguments Valid and Invalid Arguments • Learn how to recognize standard-form categorical syllogisms • Reevaluate categorical syllogisms using Venn diagrams • Practice putting arguments that are in ordinary language into standard form Finally, we will analyze different arguments regarding the justification of the death penalty (capital punishment). A deductive argument is valid if the form of the argument is such that the conclusion must be true if the premises are true. The form of an argument is determined by its layout or pattern of reasoning. In the above case, the form is: syllogism A deductive argument presented in the form of two supporting premises and a conclusion. valid A deductive argument where the form is such that the conclusion must be true if the premises are assumed to be true. form The pattern of reasoning in a deductive argument. All X (men) are Y (mortal), WHAT IS A DEDUCTIVE ARGUMENT? Unlike inductive arguments, in which the premises offer only support rather than proof for the conclusion, in a valid deductive argument the conclusion necessarily follows from the premises. Deductive arguments sometimes contain words or phrases such as certainly, definitely, absolutely, conclusively, must be, and it necessarily follows that. For example: Marilyn is definitely not a member of the swim team, since no freshmen are members of the swim team and Marilyn is a freshman. Deductive Reasoning and Syllogisms Deductive arguments are sometimes presented in the form of syllogisms, with two supporting premises and a conclusion. For the purpose of analysis, in this chapter the premises and conclusion of a syllogism will usually be presented on separate lines, with the conclusion last. l. 2. 3. Premise: All men are mortal. Premise: All fathers are men. Conclusion: Therefore, all fathers are mortal. Deductive arguments may also be diagrammed using the guidelines we learned on pages 112–114. In the case of a syllogism, the two premises are always dependent: 1 _____ 2 (Dependent premises) 3 (Conclusion) All Z (fathers) are X (men). Therefore, all Z (fathers) are Y (mortal). This argument is a valid form no matter what terms we use for X, Y, and Z. Because the form is valid, if we substitute different terms for men, mortal, and fathers, and the premises are still true, then the conclusion must be true, as in the following example. All cats (X) are mammals (Y). All tigers (Z) are cats (X). Therefore, all tigers (Z) are mammals (Y). A false conclusion does not necessarily mean that a deductive argument is invalid. In the two arguments we’ve examined so far, the conclusions were both true because the premises were true and the form was valid. The conclusion of a valid argument can be false only if one of the premises is false. In the following example, which uses the same form as our initial argument, we end up with a false conclusion: Hot or Not? Are deductive arguments better than inductive arguments? All men are tall people. Tom Cruise is a man. Therefore, Tom Cruise is a tall person. The conclusion in the above argument is false only because there is a false premise, not because the form of the argument is invalid. The first premise, “All men are tall people,” is false. If both premises are true and the conclusion is false, then the argument, by definition, is invalid. For example: Chapter 8 | Deductive Arguments • 239 All dogs are mammals. Some mammals are not poodles. Therefore, some poodles are not dogs. It is also possible to have an invalid argument in which the premises are true and the conclusion just happens to be true. Consider this: No seniors are freshman. All freshmen are college students. Therefore, some college students are seniors. In this argument, the premises and conclusion are true. However, the premises do not logically support the conclusion. The invalidity of a form can be demonstrated by substituting different terms for senior, freshman, and college students, and then seeing whether we can come up with an argument using this form in which the premises are true but the conclusion false, as in the following substitutions: No fish are dogs. All dogs are mammals. Therefore, some mammals are fish. Sound and Unsound Arguments An argument is sound if (1) it is valid and (2) the premises are true. The argument on page 239 about fathers being mortal is a sound argument because it is valid and the premises are true. On the other hand, although the argument about Tom Cruise on page 239 uses a valid form, it is not a sound argument because the first premise is false. Invalid arguments, because they do not meet the first criterion, are always unsound. Logic is primarily concerned with the validity of arguments. As critical thinkers, we are also interested in the soundness of our arguments and in having our premises supported by credible evidence and good reasoning. We have already discussed in previous chapters guidelines for ensuring that our premises are accurate and credible. sound A deductive argument that is valid and that has true premises. In this chapter we’ll learn how to identify the different types of deductive arguments and how to use Venn diagrams to evaluate these arguments for validity. HIGHLIGHTS DEDUCTIVE ARGUMENTS Valid Invalid Sound Unsound Unsound Valid argument: The form or layout of the argument is such that if the premises are true, then the conclusion must necessarily be true. Sound argument: The form of the argument is valid and the premises are true. ➤APPLICATION: Identify in the text an example of an argument that is (a) valid and sound, (b) valid and unsound, and (c) invalid. STOP AND ASSESS YOURSELF S E XE RC I SE 8 - 1 240 “Some mammals are fish” is an example of a false conclusion. 1. What do you mean when you say that you can prove something with certainty? Give a specific example of a proof from your everyday experience (keep it as brief as possible). What type of logic does the proof use—inductive or deductive? 2. In the story “Silver Blaze,” Sherlock Holmes tells Watson that when it comes to the art of reasoning, many people rely on opinion and unsupported assumptions. • THiNK E X E RCI SE 8- 1 c on t . The difficulty, he maintains, is to detach the framework of undeniable fact from the embellishments of hearsay and reporters. What do you think he meant by this? Explain using examples from your personal experience. 3. Using substitution, show that the form of each of the following deductive arguments is invalid. Remember: To establish invalidity, your premises must be true when you are substituting new terms for the ones in the original argument. *a. All fraternity members are men. No women are fraternity members. Therefore, no women are men. b. If it is raining, then it is cloudy. It is cloudy. Therefore, it is raining. c. No mice are humans. Some mice are rodents. Therefore, no humans are rodents. *d. Some married people are college students. All wives are married people. Therefore, some wives are college students. e. All flowers are plants. All orchids are plants. Therefore, all orchids are flowers. f. If my baby sister is a college student, then she is a high school graduate. My baby sister is not a college student. Therefore, my baby sister is not a high school graduate. 4. The following arguments are all valid arguments. Determine whether each argument is sound or unsound. *a. No mammals are birds. Some penguins are mammals. Therefore, some penguins are not birds. b. Some twins are sisters. All twins are siblings. Therefore, some siblings are sisters. c. All students are dormitory residents. No dormitory residents are birds. Therefore, no birds are students. *d. If Mexico is in South America, then Mexico is not a country bordering the United States. Mexico is in South America. Therefore, Mexico is a not country bordering the United States. e. All people living in England are citizens of the European Union. All members of the British royal family are people living in England. Therefore, all members of the British royal family are citizens of the European Union. f. All millionaires are rich people. Some Americans are not rich people. Therefore, some Americans are not millionaires. TYPES OF DEDUCTIVE ARGUMENTS There are several types of deductive arguments. In this section, we’ll be looking at three types of deductive arguments used in everyday reasoning: • • • Arguments by elimination Arguments based on mathematics Arguments from definition Arguments by Elimination An argument by elimination rules out different possibilities until only one possibility remains. In the introduction to this chapter, we saw Sherlock Holmes using an arguargument by elimination A ment by elimination. He deductive argument that rules out different possibilities until only one reasoned that Silver Blaze remains. had to be at one of the two stables. Since it wasn’t at King’s Pyland, it must be at Mapleton. In “Thinking Outside the Box: Bo Dietl, Top Cop” on page 242, we profile Chapter 8 | Deductive Arguments • 241 a New York City detective who is skilled in this type of deductive reasoning. Like detectives, physicians are trained in this type of deductive logic. In diagnosing an illness, a physician starts by doing a physical examination and, often, by ordering tests. If the examination and test results eliminate the most common explanations of the symptoms, then the physician moves on to check out less obvious possibilities until the mystery is solved. Indeed, Dr. Joseph Bell, one of Sir Arthur Conan Doyle’s professors at the University of Edinburgh Medical School, was the inspiration for the character Sherlock Holmes. Arguments by elimination are frequently used in everyday life. For instance, suppose it is the first day of the semester and you arrive on campus with 10 minutes to spare. You check your schedule and see that your first class, Introduction to Psychology, is in Winthrop Hall. However, on your schedule the room number is smudged and you can’t read Thinking Outside the Box B O DI E T L , Top Cop DI SC U SSI O N Q UE STI ONS Bo Dietl is a modern Sherlock Holmes. Born in Queens, New York, in 1950, Dietl wanted a job where he could make a real difference in people’s lives. When he learned about the test to get into the police academy, he decided to give it a try. One of the most highly decorated detectives in the history of the New York Police Department, Dietl investigated numerous high-profile murders and other felonies, obtaining evidence through research, interviews, and other investigative techniques. He attributes much of his success in solving more than 1,500 felonies to what he calls his “sixth sense—a nontangible feeling good detectives use in solving cases.”* One of the most famous crimes he solved was the 1981 rape and torture of a Catholic nun in an East Harlem convent. Dietl concluded from the evidence that the crime was a burglary gone wrong, rather than a sex crime, thus narrowing his search to people with burglary records. He also knew, from interviewing witnesses, that one of the men was probably tall and that the other had a limp. Days later he received a tip that the two men who committed the crime lived somewhere on 125th Street in Harlem. However, there were hundreds of buildings and thousands of people living on this street. He began the process of elimination by going to the local hangouts and tenements, knocking on doors, giving a brief description of the suspects, and asking questions. He also passed out hundreds of business cards. His efforts paid off, and the two suspects were apprehended and arrested. The 1998 movie One Tough Cop is based on Dietl’s autobiography of the same name. 242 • THiNK 1. Discuss how Dietl’s method of solving the murder of the nun in the East Harlem convent demonstrates deductive reasoning using an argument by elimination. 2. In Chapter 2 we learned that much of reasoning is unconscious and automatic and that scientists and mathematicians, as well as great detectives, often resolve complex problems without any conscious deliberation. However, to develop this ability, they have spent years consciously resolving problems and mentally rehearsing solutions. Think of a type of problem in your life that you find easy to resolve with little or no conscious deliberation. Discuss what factors, such as your familiarity and experience with the problem, contributed to your ease of resolution. *Conversation with Bo Dietl on August 8, 2005. it. What do you do? It would take too long to get a new schedule. Instead, you head over to Winthrop Hall and check out the building directory. It lists twelve room numbers. Nine of them are faculty offices, so you eliminate those nine. The remaining three are classrooms A, B, and C. You go into classroom A and ask some students what class it is. They tell you that it’s English Literature. You proceed to classroom B and repeat the process; it turns out to be a course in Business Statistics. When you get to classroom C, you just go inside and take a seat. How do you know this is the correct classroom? Through the use of an argument by elimination. Assuming that your premises are true (that your psychology course is being taught somewhere in Winthrop Hall), the third classroom by necessity must be your classroom. My class is either in room A, B, or C. My class is not in room A. My class is not in room B. Therefore, my class must be in room C. In the previous example, there were three alternatives. If there are only two alternatives, the argument is referred to as a disjunctive syllogism. A disjunctive syllogism takes one of two forms: Either A or B. Either A or B. Not A. Not B. Therefore, B. Therefore, A. A mouse locates the prize at the end of the maze through the deductive process of elimination. In determining the whereabouts of Silver Blaze, Sherlock Holmes used a disjunctive syllogism: Either Silver Blaze is at King’s Pyland or Silver Blaze is at Mapleton. Silver Blaze is not at King’s Pyland. Therefore, Silver Blaze is at Mapleton. Here is syllogism: another example of a disjunctive Either you finished cleaning your room or you’re staying in tonight. You are not staying in tonight. Therefore, you finished cleaning your room. In a disjunctive syllogism, the two alternatives presented in the first premise—clean your room or stay in tonight—must be the only two possibilities. If there are other possible alternatives that have not been stated, then the argument commits the fallacy of false dilemma. For example: Either we fight the war on terrorism in Iraq, or we’ll have to fight the terrorists here in America on our own soil. We’re fighting the war on terrorism in Iraq. Therefore, we won’t have to fight the terrorists here in America on our own soil. In this argument, the two alternatives in the first premise do not exhaust all possible alternatives. Perhaps disjunctive syllogism A type of deductive argument by elimination in we could go after individwhich the premises present only two ual terrorist cells instead of alternatives. attacking a country or even argument based on negotiate a truce with some mathematics A deductive argument of the organizations or govin which the conclusion depends on a ernments associated with mathematical calculation. terrorist activities. Because the argument commits the fallacy of false dilemma, it is not a sound argument. Arguments Based on Mathematics In an argument based on mathematics, the conclusion depends on a mathematical or geometrical calculation. For example: My dormitory room is rectangular in shape. One side measures 11 feet and the side adjacent to it measures 14 feet in length. Therefore, my room is 154 square feet. You can also draw conclusions about your new roommate, Chris, even before you meet, using this type of deductive reasoning ...
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