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SCHEDULE – Spring ‘06 June 15, Test 5 (Ch 10.5-11.5)- FINAL
Are You Logical? Why is it when ducks fly in a “v” shape that one side of the “v” is always longer than the other side? There are more ducks on one side.
1.1 What is Logic? Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experience. We read them in books and newspapers, hear them on television, and formulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.
Proposition – An assertion that something is (or is not) the case; all propositions are either True or False. Statement– The meaning of a declarative sentence at a particular time; in Logic, the word “statement” is basically interchangeable with the word “proposition.” The following sentences are statements: Aluminum is attacked by hydrochloric acid. Broccoli is a good source of vitamin A. Argentina is located in North America. Napoleon prevailed at Waterloo. Rembrandt was a painter, & Shelley was a poet. The first two simple statements are true, the second two false. The last one, acompoundstatement, expresses two statements, both of which are true. Truth and falsity are called the two possibletruth valuesof a statement. 1.2 Propositions
. Logic is the “hard-wiring” of our brains. Both math and grammar are based on Logic (and not the other way around). You are logical (despite what other people tell you). The fact that you made it to this class from work or home means that you are capable of logical deduction. But don’t get too excited yet. Even dogs and cats are capable of doing some of the things we will be doing this semester. But they cannot do the parts that are uniquely human logic. Some birds have the mental powers of a 5 year-old, so being “bird-brained” is not necessarily a bad thing.
. Unlike statements, many sentences cannot be said to be either true or false. Questions, proposals, suggestions, commands, and exclamations usually cannotbe determined to be true or false, and so are not usually classified as statements. The following sentences are not statements: In Logic, the only “meaningful” statements are declarative statements, and they can have a truth value.
. Simple proposition - A statement making only one assertion. • Compound proposition • A statement containing two • or more simple statements As noted above, declarative statements used in Logic can be either simple or compound. The most common compound statements in Logic are Disjunctive Statements and Conditional Statements. Conditional (or Hypothetical Proposition – A type of compound statement; it is false when the antecedent is true and the consequent is false. Disjunctive (or Alternative) proposition - A type of compound statement; if true, at least one of the component statements is true IF (1) you make a 100 on the last test, THEN (2) I’ll give you an “A” for the course. (1) You can have an apple, OR (2) you can have an orange.
Conditional Statements If……………………..then
. CONDITIONAL STATEMENTS (non-argumentative alone; no claim): antecedent consequent consequent antecedent If ____________, then ____________. ____________, if____________. Regardless of the location of the “if,” the antecedent comes after the “if.” sufficient condition: when the occurrence of A is ALL that is needed for the occurrence of B necessary condition: A cannot occur without the occurrence of B A B If X is a drake, then X is a male. If X is a drake, then X is a duck. Being a tiger is a sufficient condition for being an animal. Being an animal is a sufficient condition for being a tiger. Drinking a pint of gin is a necessary condition for being drunk. Having water present is a necessary condition for plant life. Getting into a fistfight is a sufficient condition for someone's being injured. Having one's eyes open is a sufficient condition for watching television. Sufficient condition; A can stand alone Necessary condition; A needs B true false false true true false sufficiansa
. Fill in the blanks with ‘‘necessary’’ or ‘‘sufficient’’ to make the following statements true. After the blanks have been filled in, express the result in terms of conditional statements. 1. Being a tiger is a ___________ condition for being an animal. 2. Being an animal is a ___________condition for being a tiger. 3. Drinking water is a _____________ condition for quenching one’s thirst. 4. Having a racquet is a _____________ condition for playing tennis. 5. Pulling the cork is a __________ condition for drinking an expensive bottle of wine. 6. Stepping on a cat’s tail is a __________ condition for making the cat yowl. 7. Burning leaves is a ___________ condition for producing smoke. 8. Paying attention is a ____________ condition for understanding a lecture. 9. Taking a swim in the North Sea is a _________ condition for cooling off. 10. Opening a door is a __________ condition for crossing the threshold.
. Propositional statements are the building blocks of which arguments are made. From propositional statements that are supposed to be providing facts, we can make an inference (or draw a conclusion) from those supposed “facts.” Inference– A process of linking statements by affirming one particular statement on the basis of one or more other supposedly factual statements. Argument – A structured group of statements reflecting an inference. Nice definitions, huh?
1.3 – Arguments . But here is the next big question: WHAT THE HECK IS AN ARGUMENT?
An argument in Logic is NOT a disagreement with raised voices. An argument, as it occurs in Logic, is a group of statements of which one or more (the premises) claim to provide support for (or reasons to believe) the one (the conclusion) that follows from them.
. AN ARGUMENT: 1) at least one statement must claim to present evidence 2) there must be a claim that the evidence implies something 3) an argument is NOT just a contradiction (see movie ) In Chapter 2, we’ll look at some things that are NOT arguments, and we’ll look at the confusing argument v. explanation problem. ARGUMENTS prove THATEXPLANATIONS show WHY Some other non-argumentative forms (lacking an inferential claim): warnings pieces of advice statements of opinion reports loosely assorted statements illustrations (can be argumentative) expository passages (sometimes argumentative)
Premise – A statement used in an argument which provides the supposed “evidence” needed in order to support some other statement which we are to supposed believe follows from the premise (or premises) Conclusion – The statement in an argument that the other “factual” statement (or statements) claim to support and provide evidence
. Premise indicator words: since (not temporal) as indicated by because for in that may be inferred from as given that seeing that for the reason that inasmuch as owing to Conclusion indicator words: therefore wherefore accordingly we may conclude entails that hence thus* consequently we may infer it must be that for this reason it follows that implies that as a result *Note: "Because" and "thus" can be used in both explanations and arguments.
. Sometimes and argument has no indicator words: The space program deserves increased expenditures in the years ahead. Not only does the national defense depend on it, but the program will more than pay for itself in terms of technology. Furthermore, at current funding levels the program cannot fulfill its anticipated potential. Similar to the thesis statement in a paragraph in English, if there is no conclusion indicator, odds are high that it’s the first sentence. The conclusion is always listed after the premises: P1: The national defense is dependent upon the space program. P2: The space program will more than pay for itself with technological spinoffs. P3: At current funding levels the program cannot fulfill its potential. C: The space program deserves increased expenditures.
. Dachshunds are ideal dogs for small children, as they are already stretched and pulled to such a length that the child cannot do much harm one way or the other. Cats can think circles around dogs! My cat regularly used to close and lock the door to my neighbor’s doghouse, trapping their sleeping Doberman inside. Try telling a cat what to do, or putting a leash on him—he’ll glare at you and say, ‘‘I don’t think so. You should have gotten a dog.’’ (below) A not-so-smart cat
Identify the premise(s) and conclusion in the following passages A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed. • Premise: A well regulated militia being necessary to the security of a free state. Conclusion: The right of the people to keep and bear arms shall not be infringed. • Premise: The right of the people to keep and bear arms shall not be infringed. Conclusion: A well regulated militia being necessary to the security of a free state.
. Of all our passions and appetites, the love of power is the most imperious and unsociable nature, since the pride of one man requires the submission of the multitude. Premise: Of all our passions and appetites, the love of power is the most imperious and unsociable nature. Conclusion: The pride of one man requires the submission of the multitude. Premise: The pride of one man requires the submission of the multitude. Conclusion: Of all our passions and appetites, the love of power is the most imperious and unsociable nature.
. Unquestionably, no more important goal exists in medical research today than the development of an AIDS vaccine. Last year (1998) AIDS, caused by HIV (the Human Immunodeficiency Virus) was the infectious disease that killed the most people around the world, and the epidemic is not abating. • Premise: In 1998, AIDS was the infectious disease that killed the most people around the world. Premise: The epidemic is not abating. Conclusion: No more important goal exists in medical research today than the development of an AIDS vaccine. Premise: In 1998, AIDS was the infectious disease that killed the most people around the world. Premise: No more important goal exists in medical research today than the development of an AIDS vaccine. Conclusion: The epidemic is not abating.
1.4 – Deductive v. Inductive Arguments Evaluating the strength of an inferential claim: 1) occurrence of indicator words 2) nature of the link between premise(s) and conclusion 3) form of argumentation by the arguer Deductive: premises are claimed to provide NECESSARY support Inductive: premises are claimed to prove only PROBABLE support ….IF the premises are assumed to be true
. Inductive indicator wordsDeductive indicator words probable/improbable necessarily plausible/implausible certainly likely/unlikely definitely reasonable to conclude absolutely Note 1: "Must" can be used with either probability (induction) or necessity (deduction). Note 2: "It certainly follows that" can be used with either also, since the phrase is often used rhetorically with induction without asserting necessity. In other words, both “must” and “it certainly follows” should be first to considered as part of an inductive argument that the arguer is trying to make sound more certain that it actually is. On the other hand, it still could be a deductive argument.
. INDUCTIVE (probably the case): The vast majority of saleswomenare extroverts. Elizabethis a saleswoman. Therefore, Elizabethis an extrovert. DEDUCTIVE (necessarily the case): All saleswomenare extroverts. Elizabeth is a saleswoman. Therefore, Elizabethis an extrovert.
Without indicator words to determine induction or deduction, it is necessary to take into account the FORM of the argument. DEDUCTIVE: 1) Argument based on mathematics -- if the conclusion depends on some arithmetic or geometric computation or measurement (but not statistics), it is deductive (eg., 2 + 2 = 4) -- if it is statistical, it is probably inductive 2) Argument from definition -- conclusion is claimed to depend upon the definition of some word or phrase used in the premise or conclusion
. 3) Categorical Syllogism: each statement begins with one of the words "all,“ "no," or "some"; almost always deductive All lasersare optical devices. Some lasersare surgical devices. Therefore, some optical devicesare surgical instruments. 4) Hypothetical Syllogism: has a conditional statement for one or both of its premises; usually deductive but occasionally inductive If X, then YA B If Y, then Z B C Therefore, if X, then Z.A C
. 5) Disjunctive Syllogism: has "either/or" statements for one of its premises; usually deductive Either breach of contract is a crime or it is not punishable by the state. Breach of contract is not a crime. Therefore, it is not punishable by the state. Generally, with DEDUCTIVE arguments, the premise(s) begins with the familiar and conclusion moves to a subject that less is known about. Recap: These forms are generally Deductive arguments: from mathematics from definition hypothetical syllogism categorical syllogism disjunctive syllogism
. INDUCTIVE (probability & predictions) forms: 1) Argument from analogy (probablistic at best) --depends on the existence of a similarity of two things or states of affair; a condition which affects a better-known thing is by analogy demonstrated to affect similarly something less-known Because Christina's Jaguar is a great handling car, it stands to reason that Angela's new Jaguar will also handle well. 2) Inductive generalization (often statistics) --knowledge concerning a selected sample is used to make a claim about a larger group with similar characteristics Because four out of five times that I go to Burger King they get my order wrong, it stands to reason that BK generally gets most orders wrong. Even if it’s true, it’s still inductive.
. 3) Argument from Authority --the conclusion rests upon a statement made by some presumed authority or witness --A witness my testify to observing a murder, but because he might be lying or simply mistaken, the argument is probablistic 4) Argument based on signs --a certain sign is ascribed to symbolize the knowledge of a thing or situation --a road sign might indicate that there is a steep hill ahead, leading one to argue that there is indeed a steep hill within a mile or so. But the sign could be in the wrong location or in error about the slope of the hill, so the conclusion is only probable. Dancing Cow Ahead Is that probable?
. 5) Casual inference --argument proceeds from knowledge of a cause to knowledge of an effect, or visa-versa; lacking certainty to the conclusion, it is inductive --the knowledge that a bottle of wine had been left in the freezer overnight might make someone conclude that it had frozen (however, wine will NOT freeze overnight--usually--because of the alcohol content, and, thus, the conclusion would be incorrect) inductive arguments: predictions arguments from analogy inductive generalizations arguments from authority arguments based on signs causal inferences deductive arguments: arguments based on mathematics arguments from definition categorical syllogisms hypothetical syllogisms disjunctive syllogisms
. SCIENTIFIC AEGUMENTS: can be either deductive or inductive, depending upon the circumstances 1) discovery--usually inductive and generalized based on empirical evidence (premises); less known and unknown conclusions 2) application--deductive with reservations; application of previous laws and mathematical equations to something new (but because sometimes a law works in one instance, this might be inductive with the possibility that the same law might not work in the future or in a different theory) A PARTICULAR STATEMENT is one that makes a claim about one or more particular members of a class A GENERAL STATEMENT makes a claim about all the members of a class There are no hard and fast rules about particular and general when determining induction and deduction, but we generally see……..
. Deductive moving from particular to general: One is a prime number. Three is a prime number. Five is a prime number. Seven is a prime number. Therefore, all odd numbers between 0 and 8 are prime numbers. Deductive moving from particular to particular: Gabriel is a wolf. Gabriel has a tail. Therefore, Gabriel's tail is the tail of a wolf. Inductive moving from the general to the particular: All emeralds previously found have been green. Therefore, the next emerald to be found will be green. Recap in difference between induction and deduction: 1) differences in the occurrences of indicator words 2) differences in the nature of the link between premise and conclusion 3) differences in the form of argumentation that the arguer uses
How “reasonable” is “Beyond a Reasonable Doubt”? You are on the jury for this court case. It is a well-known fact that Bob hates Bill, and Bill has turned up dead. In court you learn that: (1) there was a smoked Camel unfiltered cigarette lying next to Bill’s body, and Bob smokes Camel unfiltered cigarettes (2) there was an impression of a man’s size 11 Wilson tennis shoe next to Bill’s body, and Bob wears Wilson tennis shoes – size 11
. (3) Bill was killed with a .38 revolver; Bob owns a .38 revolver, but he says he can’t find, it so it is missing Now, add up the evidence. How reasonable would it be for you, as a member of the jury, to conclude that Bob had killed Bill? .Bob and cigs………Bob’s shoes…………..Bob’s missing gun
. Sherlock Holmes always said that he “deduced” who the criminal was, but he did not “deduce” it, rather he “induced” it. He added up the particular information and drew a general conclusion. Deductive Inductive
. PRACTICE: Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation). 1. Because triangle A is congruent with triangle B, and triangle A is isosceles, it follows that triangle B is isosceles. 2. The plaque on the leaning tower of Pisa says that Galileo performed experiments there with falling objects. It must be the case that Galileo did indeed perform those experiments there.
. 3.The rainfall in Seattle has been more than 15 inches every year for the past thirty years. Therefore, the rainfall next year will probably be more than 15 inches. 4. No e-mail messages are eloquent creations. Some love letters are eloquent creations. Therefore, some love letters are not e-mail messages. 5. Amoco, Exxon, and Texaco are all listed on the New York Stock Exchange. It must be the case that all major American oil companies are listed on the New York Stock Exchange. 6. The longer a pendulum is, the longer it takes to swing. Therefore, when the pendulum of a clock is lengthened, the clock slows down.
. 7. Paying off terrorists in exchange for hostages is not a wise policy, since such action will only lead them to take more hostages in the future. 8. The Matterhorn is higher than Mount Whitney, and Mount Whitney is higher than Mount Rainier. The obvious conclusion is that the Matterhorn is higher than Mount Rainier. 9. Although both front and rear doors were found open after the burglary, there were pry marks around the lock on the rear door and deposits of mud near the threshold. It must be the case that the thief entered through the rear door and left through the front. 10. The Encylopaedia Britannica has an article on symbiosis. The Encyclopedia Americana, like the Britannica, is an excellent reference work. Therefore, the Americana probably also has an article on symbiosis.
. 11. Cholesterol is endogenous with humans. Therefore, it is manufactured inside the human body. 12. Either classical culture originated in Greece, or it originated in Egypt. Classical culture did not originate in Egypt. Therefore, classical culture originated in Greece. 13. World-renowned physicist Stephen Hawking says that the condition of the universe at the instant of the Big Bang was more highly ordered than it is today. In view of Hawking’s stature in the scientific community, we should conclude that this description of the universe is correct. 14. If Alexander the Great died from typhoid fever, then he became infected in India. Alexander the Great did die from typhoid fever. Therefore, he became infected in India.
. 25. The Simpson incident had shown me that a dog was kept in the stables, and yet, though someone had been in and had fetched out a horse, he had not barked enough to arouse the two lads in the loft. Obviously the midnight visitor was someone whom the dog knew well. (A. Conan Doyle, Memoirs ofSherlock Holmes) D’oh!! Wrong “Simpson incident”!!!! Sorry!
. 15. It seems likely that young people will be at war with old people in another 15 or 20 years. You can see it coming in the numbers. In 1900 only 1 percent of the population was older than 75. Today 4 percent of all Americans are more than 75 years old, and in a few years it’s going to be 5 percent . . . 13 million people. 26. Eternity is simultaneously whole. But time has a before and an after. Therefore time and eternity are not the same thing. (Thomas Aquinas, Summa Theologica) 27. Ordinary things that we encounter every day are electrically neutral. Therefore, since negatively charged electrons are a part of everything, positively charged particles must also exist in all matter. (James E. Brady and Gerard E. Humiston, General Chemistry) 30. Because the moon moves relative to the earth so that it returns to the same position overhead after about 25 hours, there are two high and two low tides at any point every 25 hours.
. Answer ‘‘true’’ or ‘‘false’’ to the following statements: 1. In an inductive argument, it is intended that the conclusion contain more information than the premises. 2. In a deductive argument, the conclusion is not supposed to contain more information than the premises. 3. The form of argumentation the arguer uses may allow one to determine whether an argument is inductive or deductive. 4. The actual strength of the link between premises and conclusion may allow one to determine whether an argument is inductive or deductive. 5. A geometrical proof is an example of an inductive argument. 6. Most arguments based on statistical reasoning are deductive. 7. If the conclusion of an argument follows merely from the definition of a word used in a premise, the argument is deductive.