Tip #15: The end branches of a probability tree must total to 1, which is equal to the aggregate of all individual probabilities.
Tip #16: Weighted ranking is a tool for finding solutions using a weighted average. To calculate weighted average, we multiply each event by its a.s.sociated weight and total the results. In the case of probabilities, we multiply each event by its respective probability and total the results.
Tip #17: Utility a.n.a.lysis takes into account desirability of outcomes, which may be different from monetary payoffs.
Tip #18: For the purposes of hypothesis testing, the minimum requirement for causal inference is evaluation using a atwo-waya table.
Chapter 4: a.n.a.lyzing Arguments.
Tip #19: Evidence + a.s.sumption = Conclusion. The a.s.sumption is the glue that holds the evidence to the conclusion.
Tip #20: There are effectively two ways to attack an argument: attack the evidence or attack the a.s.sumption(s).
Tip #21: The five most common critical reasoning errors people make include: comparing aapples with oranges,a overgeneralizing on the basis of small samples, ignoring relevant evidence, confusing cause and effect, and failing to antic.i.p.ate bottlenecks when plans are put into action.
Tip #22: Watch for ascope shifts,a which occur when one term is subst.i.tuted for another as an argument unfolds.
Tip #23: Changes in definition destroy the ability to make valid comparisons Tip #24: If a situation involves a asurvey,a check to see if the survey is based on a sample which is both quant.i.tatively and qualitatively representative.
Tip #25: Representativeness a.s.sumptions are based on the idea that some smaller factor is representative of the larger whole.
Tip #26: Broadly speaking, representativeness a.s.sumptions occur anytime that we argue from the particular to the general.
Tip #27: Check to see if evidence has been handpicked to support a claim being made. Otherwise we may fall victim to aproof by selected instances.a Tip #28: aEvidence omitteda may hold the key to determining an argumentas validity.
Tip #29: Correlation does not equal causation.
Tip #30: Cause-and-effect a.s.sumptions are grounded in the idea that because one event follows another in time, the first of the two events is the cause and the second is the effect.
Tip #31: When tackling cause-and-effect scenarios, think first in terms of alternative causal explanations. If the argument states that A is causing B, then check to see that another cause, namely C, is not instead causing B.
Tip #32: A more complex form of alternative explanation occurs when two effects result from a single cause. Thus, if an argument suggests that A is causing B, consider the possibility that another cause, namely C, could be causing both A and B.
Tip #33: If A is thought to be causing B, the idea that B is causing A is called reverse causation and casts serious doubt on the notion that A is really causing B.
Tip #34: Test the opposite scenario a" if you hear that a full moon causes the crime rate to rise, always ask what the crime rate is like when the moon is not full.
Tip #35: Theory may be divorced from practice. Plans may not equal completed action. Do not a.s.sume that plans will be implemented without a hitch.
Tip #36: aCana does not equal awill.a The ability to do something should not imply application of that ability, whether due to choice or neglect.
Tip #37: One way to uncover implementation a.s.sumptions is to antic.i.p.ate bottlenecks.
Tip #38: An argument may depend on the a.s.sumption that a person or organization is aware of a pre-existing fact, situation, or condition.
Tip #39: Searching for something does not guarantee that weall recognize it once weave found it. The ability to accurately identify that which we are seeking to find may be a key a.s.sumption.
Tip #40: Always look for potentially vague terms in an argument and ask for or seek clarification.
Chapter 5: Mastering Logic.
Tip #41: Be careful not to reverse the flow of aIf a thena statements. aIf A, then Ba is not the same as aIf B, then A.a This is known in logic as the fallacy of affirming the consequent.
Tip #42: The statement aIf A, then Ba leads to the logical inference aIf not B, then not A.a This is known in logic as the contrapositive.
Tip #43: The statement aIf A, then Ba does not equal aIf not A, then not B.a This is known in logic as the fallacy of denying the antecedent.
Tip #44: One way to think about an aIf a thena statement in the form of aIf A, then Ba is that just because A leads to B does not mean that C, D, or E could not also lead to B.
Tip #45: Necessary conditions are not the same as sufficient conditions. The statement aA person needs water to remain healthya does not mean that water alone is enough to keep a person healthy. Water is a necessary but not sufficient condition for someone to remain healthy.
Tip #46: The statement aIf A, then Ba does not equal aOnly As are Bs.a For example, the statement aIf one wants to make a good salad, one should use tomatoesa should not be interpreted to mean that only tomatoes are necessary to make a good salad.
Tip #47: The statement aEvery A is a Ba does not equal aOnly As are Bs.a For example, the statement aEvery cat is a mammala should not be interpreted to mean that aonly cats are mammals.a Tip #48: aAlla statements imply inclusivity; asomea statements imply crossover; anoa statements imply exclusivity.
Tip #49: Whereas asomea statements imply reciprocality, amosta statements do not necessarily imply reciprocality.
Tip #50: The word aunsurpa.s.sed,a both technically and legally, does not mean abest.a It is possible that all products, compet.i.tors, etc., could be tied at number one and yet all could claim to be unsurpa.s.sed.
APPENDIX II a" FALLACIOUS REASONING A traditional approach to argumentation involves studying common types of fallacious reasoning. This method is consistent with what a person might encounter in an introductory logic course in college or university.
It is valuable to recognize fallacious types of reasoning and defend against them. One benefit of studying fallacious reasoning is to avoid unknowingly constructing fallacious arguments. aFeelinga that an argument is fallacious is one thing; being able to specifically identify why something is fallacious is quite another. This difference is a.n.a.logous to feeling sick but not knowing why, as opposed to seeing a doctor and finding out exactly what the reason is.
The fallacies presented in this section are divided into four major groupings, including: (1) fallacies based on language; (2) fallacies based on abada evidence; (3) fallacies based on flawed a.s.sumptions; and (4) fallacies found in deductive logic. Each fallacy is accompanied by a brief description of the fallacy and at least one key example to ill.u.s.trate it.
1) Fallacies Based on Language Equivocation The fallacy of equivocation occurs when someone uses a word or phrase that has two different meanings depending on its context.
aGambling should be legalized everywhere because it is something we canat avoid. It is part of life itself. People gamble every time they get behind the wheels of their cars or say their wedding vows.a The word agamblinga is used in two different contexts, creating ambiguity or equivocation.
Equivocation can even occur in the revered domain of syllogistic logic.
aAll lemons are yellow.
This car is a lemon.
Therefore, this car is yellow.a Since lemon has two different meanings a" the first referring to a fruit and the second meaning ainferiora or aof cheap qualitya a" a case of fallacious reasoning arises: the conclusion does not follow logically from the premises.
Distinction Without a Difference The fallacy of distinction without a difference occurs when a person argues that his or her position is different from a previous one by means of careful distinction of language. However, even though the words chosen are different, they are in substance identical to the ones already used.
aI didnat lie. I just stretched the truth a bit.a What is the difference between lying and stretching the truth?
2) Fallacies Based on aBada Evidence Hasty Generalization The fallacy of hasty generalization occurs when a conclusion is reached based on too small a sample size or on a sample that is unrepresentative.
aI have been to Phoenix three times and each time it has rained. Phoenix sure does get a lot of rain.a Circular Reasoning The fallacy of circular reasoning occurs when a conclusion is based on a premise (a piece of evidence) which in turn is based on the conclusion.
aMr. Weebleas desk is always messy because heas incompetent. A messy desk is the sign of a cluttered mind, and this tells us something: he simply canat get the job done.a The argument above effectively states: aincompetence leads to a messy desk and a messy desk leads to work not getting done (i.e., to incompetence).a Fallacy of Negative Proof The fallacy of negative proof occurs when a claim is deemed true because it is not proven false, or false because it is not proven true. For example, the fact that no one has found evidence that Atlantis (the fabled lost continent) existed does not prove that Atlantis did not exist.
aBecause no intern in our office has ever complained about his or her salary, we can be confident that interns in our office are satisfied with their wages.a Ad Hominem The ad hominem fallacy consists in attacking the person, often in a personal or abusive way, rather than attacking the argument or claim being made.
aHow can you let Sheila give you marital advice? Donat you know she spent some time in jail for mail fraud?a Poisoning the Well The fallacy of poisoning the well consists of rejecting an opponentas argument because of the opponentas background, particularly as it relates to nationality, race, gender, or geography.
aHow can what you say be credible? Youare from Sydney, and of course you think Sydney is a better city than Melbourne.a Tu Quoque The tu quoque fallacy consists of refusing to accept an opponentas stance or position because the opponent is guilty of these very same charges.
Father: aYou shouldnat drink, son. Drinking hurts your liver, and being drunk and stupid is no way to go through life.a Son: aDad, isnat that a gin and tonic in your hand?a Even though the boyas father is drinking, it doesnat mean the fatheras point is without merit.
Red Herring The red herring fallacy consists in attempting to hide a weakness in an argument by drawing attention away from the real issue and emphasizing a side issue.
(Boss to employee): aDonat tell me about low wages. When I was your age I was making only a hundred dollars a week.a The fact that this employeeas boss made less when she or he was young is irrelevant to a concerned employee seeking a wage increase.
a.s.signing Irrelevant Goals or Functions The fallacy of a.s.signing irrelevant goals or functions consists of rejecting a policy or course of action because it fails to satisfy certain goals or objectives that it was never intended to achieve.
Peter: aDo you really think that studying logic is going to help us solve the worldas problems?a Tiffany: aNo, probably not.a Peter: aThen why are you wasting your time studying it?a Wishful Thinking The fallacy of wishful thinking occurs when one a.s.sumes that just because one wishes something to be true (or false), it will therefore be the case.
aForget all about our teamas rather lackl.u.s.ter performance this season. Our football team is going to defeat the defending champions in the first round of the playoffs. All of our players believe in themselves and are determined to pull off an upset victory.a Fallacy of Tradition The fallacy of tradition consists in trying to persuade someone based on history, heritage, or reverence rather than relying on the strength of the evidence.
aTim, you canat be serious about going to Annapolis! Your family has been and always will be Army a" your father, brother, grandfather, and uncle. Itas West Point for you, my friend.a Appeal to Public Opinion The fallacy of appeal to public opinion occurs when an idea or position is accepted because a great number of people also accept or support it.
aIam going to vote ayesa on the proposed tax bill amendment. According to a recent poll, more than two-thirds of the registered voters under the age of twenty-five support it.a Appeal to Pity The fallacy of appeal to pity consists of using sympathy to persuade rather than relying on the strength of the evidence presented.
(Sheila to John) aYou have to donate to the orphanage. It is bad enough that children will never know who their real parents are, let alone being deprived of the basic necessities of life.a 3) Fallacies Based on Flawed a.s.sumptions False Alternatives The fallacy of false alternatives involves a.s.suming that one of two alternatives is correct. These alternatives are usually extreme and fail to acknowledge middle ground. The fallacy of false alternatives is also referred to as the aeither-ora fallacy or ablack-and-whitea fallacy.
aIf youare not for free trade, you must be for protectionism.a Fallacy of the Golden Mean The fallacy of the golden mean consists in believing that the middle option is best because it is the point between two extremes. This is also called the fallacy of moderation.
aJunior high school teachers believe students should follow a fixed course curriculum. The parents of junior high school students believe that students should be allowed to pick their own courses. The best solution is to combine the wishes of teachers and parents.a Fallacy of Composition The fallacy of composition consists in believing that what is true for the parts must therefore be true for the whole.
aBrad is a fine young man. Janet is a fine young woman. Theyall make a wonderful couple.a As marriage is deemed to be more than the sum of its parts by virtue of the coupleas relationship in the marital whole, the result might be something more or less than the apartsa added together.
Fallacy of Division The fallacy of division involves believing that what is true for the whole is also true for the individual parts. For example, just because the New Zealand rugby team is a great team, this does not mean all players on the New Zealand rugby team are great players.
aBecause a car is a heavy object, all components that go into making a car are heavy in weight.a Fallacy of the Continuum The fallacy of the continuum is the result of believing that small or incremental differences can be ignored because they are inconsequential on a larger scale.
aTry building your vocabulary by studying one new word each day. Take a medium-sized dictionary and start at the beginning. Learn one word one day and another the next. Eventually, you will have gone cover to cover, and, more importantly, you will have learned almost every important word in the English language. How many people can boast of that?a Incorrect Attack on a Generalization The fallacy of incorrect attack on a generalization consists in believing that a generalization is open to attack because a single exception can be cited. For example, a theatre sign stating that those under eighteen years of age should not enter a theatre does not mean that parents carrying their seven-month-old baby should not be permitted to enter.
Student #1: aIt is a well-known fact that smoking shortens your life expectancy.a Student #2: aYeah, well then how do you explain the fact that my great grandfather smokes a pack a day and heas rockina into his nineties?a Finding an exception to a generalization does not undermine the generalization a" a generalization is merely a generalization.
Distortion The fallacy of distortion consists in twisting an opponentas point of view or claim, thereby making it easier to attack.
Proponent: aThe only way to increase education in developing countries is to have materials, and this means textbooks.a Opponent: aWhat youare saying is that you couldnat care less about depleting our forests to provide paper for, G.o.d knows, how many more textbooks.a Faulty a.n.a.logy The fallacy of faulty a.n.a.logy consists in believing that because two things are alike (or unlike) in one or more aspects they are alike (or unlike) in one or more other aspects.
aWhen it comes to artificial lures, the Rappala artificial minnow has worked like a charm for me. Iave used it every day this summer to catch small mouth ba.s.s and canat wait to try it when going trout fishing in the fall.a Cause and Effect (Causal Oversimplification) The fallacy of causal oversimplification occurs when one a.s.sumes that a particular cause is responsible for a given effect.
aI have heard that rich people work hard. I am going to work hard and get rich.a Hard work will contribute to someoneas becoming rich, but it is probably only one of several contributing factors. It is wrong to a.s.sume that hard work alone is a sufficient condition for becoming rich.
Domino Fallacy The domino fallacy consists in a.s.suming that because one event may cause another to occur it will also cause a series of future events to occur. This fallacy is also referred to as the chain reaction fallacy.
aI donat see a problem with giving the homeless a free meal, but a free meal leads to free clothing, and if weare not careful, free accommodation. Pretty soon weall be giving them a guaranteed annual salary as well.a Gambleras Fallacy The gambleras fallacy consists in concluding that a future eventas chance or probability is altered based on a previous eventas outcome, even though the two are independent events with absolutely no influence on one another.
(Parents conversing with their doctor): aBecause we already have three boys, the odds greatly favor our having a baby girl.a The result of one birth has absolutely no influence (almost no influence, biologically speaking) on the gender of future offspring. The odds are still 50-50.
Fallacy of False Precision The fallacy of false precision consists in making a claim with a level of mathematical precision that is likely impossible to obtain.
aOne out of every four people in Shakespeareas day did not actually like Shakespeareas plays.a It is unlikely that such a precise figure could be obtained, particularly during that time period.
4) Fallacies Found in Deductive Logic Fallacy of Affirming the Consequent The fallacy of affirming the consequent a.s.sumes that in an aIf A, then Ba scenario, the opposite, aIf B, then A,a is also true. This is sometimes called the fallacy of false conversion.
aWhenever I go on vacation, I feel relaxed. I feel relaxed, so I must be on vacation.a Maybe the pub sells a lot of beer all week long, all year round.
Fallacy of Denying the Antecedent The fallacy of denying the antecedent a.s.sumes that in an aIf A, then Ba scenario, it is also true that aIf not A, then not B.a aWhenever it rains the ground gets wet. It didnat rain last night so the ground canat be wet.a For all we know, the sprinkler system may have been turned on, causing the ground to be wet.
For a review of both of the preceding fallacies, refer to the discussion of aIf a thena type statements in chapter 5.
APPENDIX III a" AVOIDING IMPROPER INFERENCES Developing the mindset to master critical reasoning requires knowing where to adraw the linea between inferable and non-inferable statements. First, letas contrast the terms inference and a.s.sumption. An a.s.sumption is an unstated premise which is an integral component of any argument a" it is the missing link needed to make an argument work. An inference is a logical extension a" a logically inferable statement based on an argument, statement, or pa.s.sage.
In everyday life we make loose inferences. We make inferences based on what is amost probably true.a For example, the statement, aItas been raining a lot lately,a begs the inference aumbrella sales are up.a The statement, aThe weatheras getting colder,a begs the inference that aice cream sales are down.a But these are not logically inferable. What is alogically inferablea is a much firmer, tighter noose than what is amost probably true.a Take the following argument: aBuy our product. Itas the cheapest in the market.a The conclusion is abuy our product.a The evidence is aitas the cheapest in the market.a The a.s.sumption is aprice is the dominant factor in a purchasing decision.a The statement aother products in the market are more expensivea is not an a.s.sumption but an inference a" a logically inferable statement based on the argument.
Problem 40: Little Italy Antonio is a well-known Italian connoisseur operating the Little Italy restaurant in Devon City. Not only do three out of every four food and beverage consultants recommend his Italian restaurant but a survey by the cityas Food and Beverage a.s.sociation shows that his restaurant is unsurpa.s.sed by any other Italian restaurant in the city. The proof is in the eating. Antonioas customers prefer his style of Italian cooking by a ratio of 2 to 1.
Which of the statements below are logically inferable based on the brief pa.s.sage about the Little Italy restaurant above?
Tip #50: The word aunsurpa.s.sed,a both technically and legally, does not mean abest.a It is possible that all products, compet.i.tors, etc., could be tied at number one and yet all could claim to be unsurpa.s.sed.
1. Antonioas restaurant Little Italy is regarded by the cityas Food and Beverage a.s.sociation as the best Italian restaurant in the city.
a" True a" False 2. Antonio enjoys preparing Italian food.
a" True a" False 3. Food and beverage consultants recommend Antonioas restaurant more than they do other Italian restaurants in the city.
a" True a" False 4. Antonioas customers prefer his style of Italian cooking to that of other comparable Italian restaurants.
a" True a" False 5. Spaghetti is one of the restaurantas most requested dishes.
a" True a" False 6. Antonio uses high-quality ingredients in the dishes served in his restaurant.
a" True a" False 7. The Little Italy restaurant is a profitable business.
a" True a" False 8. If Antonio moved to another city, he would also be able to establish himself as a well-known Italian connoisseur.