# Reasoning and Fallacies

This chapter reviews logical rules that produce valid arguments and common rule violations that lead to fallacies. Understanding fallacies helps us to avoid committing them and to recognize fallacious arguments made by others.

Reasoning can be inductive or deductive. Deductive reasoning is what we call "logic" informally. It is a way of thinking mathematically about all kinds of things: Given a set of assumptions (premises), what must then be true? In contrast, inductive reasoning attempts to generalize from experience (data) to new situations: How strong is the evidence that something is true or false about the world? Inductive reasoning is inherently uncertain. Deductive reasoning—if logical—is as certain as mathematics can be. Much of the meat of Statistics, covered in other chapters, concerns inductive reasoning. Exceptional care is needed to draw reliable conclusions by inductive reasoning. And good inductive reasoning requires correct deductive reasoning, the subject of this chapter.

Deductive reasoning that is mathematically correct (logical) is valid. Deductive reasoning that is incorrect (logically faulty, illogical) is fallacious. Reasoning can be valid even if the assumptions on which it is based are false. If reasoning is valid and based on true premises, it is sound.

Many deductive and inductive arguments rely on statistical evidence. Even the best statistical evidence can lead to wrong conclusions if it is used in a fallacious argument. The difficulty many people have understanding statistics makes statistics especially effective "red herrings" to distract the listener. Moreover, statistics can give fallacious arguments an undeserved air of scientific precision.

Fallacies have been studied at least since classical times. This chapter gives a loose taxonomy of fallacies. Validity, soundness and formal fallacies (which result from not following valid rules of logic or misapplying valid rules) are covered in more detail in covers reasoning about categories of things. The post hoc ergo propter hoc fallacy is addressed in more detail in The regression fallacy gets special attention in The principle of insufficient reason, mentioned in is related to the appeal to ignorance fallacy. The base rate fallacy and the Prosecutor's fallacy are mentioned in Nonresponse bias and sampling design, studied in is related to the hasty generalization fallacy. Simpson's paradox, a version of the composition fallacy, is covered in

## Valid rules of reasoning

### Arguments

An argument is a sequence of statements, one of which is called the conclusion. The other statements are premises (assumptions). The argument presents the premises—collectively— as evidence that the conclusion is true. For instance, the following is an argument:

If A is true then B is true. A is true. Therefore, B is true.

The conclusion is that B is true. The premises are If A is true then B is true and A is true. The premises support the conclusion that B is true. The word "therefore" is not part of the conclusion: It is a signal that the statement after it is the conclusion. The words thus, hence, so, and the phrases it follows that, we see that, and so on, also flag conclusions. The words suppose, let, given, assume, and so on, flag premises.

A concrete argument of the form just given might be:

If it is sunny, I will wear sandals. It is sunny. Therefore, I will wear sandals.

Here, A is "it is sunny" and B is "I will wear sandals."

We usually omit the words "is true." So, for example, the previous argument would be written

If A then B. A. Therefore, B.

The statement not A means A is false. give a more mathematical treatment of logic. This chapter gives only a few valid rules of reasoning (and some fallacies to avoid).

### Validity and Soundness

An argument is valid if the conclusion must be true whenever the premises are true. In other words, an argument is valid if the truth of its premises guarantees the truth of its conclusion. Stating the conclusion explicitly is in some sense redundant, because the conclusion follows from the premises: it serves to draw our attention to the fact that that particular statement is one (of many) that must be true if the premises are true. An argument that is not valid is invalid or fallacious.

If an argument is valid and its premises are true, the argument is sound. If an argument is not sound it is unsound. An argument can be valid even if its premises are false—but such an argument is unsound. For instance, the following argument is valid but unsound:

Cheese more than a billion years old is stale. The Moon is made of cheese. The Moon is more than a billion years old. Therefore, the Moon is stale cheese.

If all three premises were true, the conclusion would have to be true. The argument is valid despite the fact that the Moon is not made of cheese, but the argument is unsound—because the Moon is not made of cheese. discusses validity and soundness in more detail.

The logical form of the argument just above is (roughly):

For any x, if x is A and x is B then x is C. y is A. y is B. Therefore, y is C.

Here, A is "made of cheese," B is "more than a billion years old." and C is "stale." The symbol x is a free variable that can stand for anything; the symbol y stands for the Moon. Note that this example uses A, B, and C to represent properties of objects (categories, see rather than to represent statements (whole sentences), as above.

Some Valid Rules of Reasoning

• A or not A. (Law of the excluded middle)
• Not (A and not A).
• A. Therefore, A or B.
• A. B. Therefore, A and B.
• A and B. Therefore, A.
• Not A. Therefore, not (A and B).
• A or B. Not A. Therefore, B. (denying the disjunct)
• Not (A and B). Therefore, (not A) or (not B). (de Morgan)
• Not (A or B). Therefore, (not A) and (not B). (de Morgan)
• If A then B. A. Therefore, B. (affirming the precedent, modus ponendo ponens, "affirming by affirming")
• If A then B. Not B. Therefore, not A. (denying the consequent, modus tollendo tollens, "denying by denying")

### Formal Fallacies

Formal fallacies are errors that result from misapplying or not following these rules. For instance, consider the argument:

If A then B. B. Therefore, A.

A concrete example of this might be:

If it is sunny, I will wear sandals. I will wear sandals. Therefore, it is sunny.

(I encourage you to "plug in" values in abstract expressions to get plain-language examples—not only in this chapter, but anytime you encounter a mathematical expression. That can really clarify the math.) This is a fallacy known as affirming the consequent. (In the conditional If A then B, A is called the antecedent and B is called the consequent. To affirm something is to assert that it is true; to deny something is to assert that it is false.) The premises say that if A is true, B must also be true. It does not follow that if B is true, A must also be true. To draw the conclusion that A is true, we need an additional premise: If B then A. That premise, together with the other two premises, would allow us to conclude that A is true.

More generally, consider proving something from the premise If A then B and an additional premise: that A is true, that A is false, that B is true, or that B is false.

If the additional premise is that the antecedent A is true, we are affirming the antecedent, which allows us to reach the logically valid conclusion that B is also true. If the additional premise is that the antecedent A is false, we are denying the antecedent, which does not allow us to conclude anything about B.

If the additional premise is that the consequent B is true, we are affirming the consequent, which does not allow us to conclude anything about A. If the additional premise is that the consequent B is false, we are denying the consequent, which allows us to reach the logically valid conclusion that A is also false.

There are countless fallacies; some, such as affirming the consequent and denying the antecedent are so common they have names. Some do not.

Non sequitur is the name of another common type of formal fallacy. For instance, consider the argument:

If A then B. A. Therefore, C.

This is a fallacy known as non sequitur, which is Latin for "does not follow." The conclusion does not follow from the two premises. The premises guarantee that B is true. They say nothing about C. The conclusion B follows from the premises but the conclusion C does not. There is a missing premise: If B then C. We will distinguish between two kinds of non sequitur; see the box below.

Common Formal Fallacies

• A or B. Therefore, A.

• A or B. A. Therefore, not B. (affirming the disjunct)

• Not both A and B are true. Not A. Therefore, B. (denying the conjunct)

• If A then B. B. Therefore, A. (affirming the consequent)

• If A then B. Not A. Therefore, not B. (denying the antecedent)

• If A then B. C. Therefore, B. (non sequitur of evidence)

• If A then B. Not C. Therefore, not A. (non sequitur of evidence)

• If A then B. A. Therefore, C. (non sequitur of relevance)

• If A then B. Not B. Therefore, not C. (non sequitur of relevance)

The following exercises check your understanding of the correct application of rules of reasoning, valid and sound arguments, and formal and informal fallacies. Reload the page to get more examples.

## Informal Fallacies

A fallacy is an argument in which the premises do not justify the conclusion as a matter of logic. An argument can be fallacious for many reasons. The argument might mis-apply a legitimate rule of logic. Or it might omit a crucial premise or misconstrue a premise. Or it might misconstrue the conclusion.

For instance, consider the argument: Mary says X is true. Mary does Y. Anybody who does Y is a bad person. Therefore, X is false.

That argument is fallacious: It is a non sequitur of relevance because the conclusion that X is false does not follow from the two premises ("Mary does Y" and "Anybody who does Y is a bad person"). The form of the argument is: If A then B. A. Therefore C. To get to the conclusion that X is false, we would need an additional premise, namely, that everything a bad person says is false, that is, the premise if B then C. If we added that premise, the argument would be valid—but not sound, because it is not true that everything bad people say is false.

Because this fallacy has, at its heart, a non sequitur of relevance, we call it a fallacy of relevance. Instead of establishing the conclusion it claims (that X is false), it establishes a different conclusion (that Mary is bad) and ignores the difference. This variant of a fallacy of relevance is very common. It has a name, ad hominem, which means "at the person" in Latin. Instead of addressing Mary's argument that X is true, it attacks Mary herself. Ad hominem arguments are discussed below.

The tacit premise that everything that comes of something bad is bad—and its opposite, that only good comes of good—are genetic fallacies. Inappropriate appeal to authority, discussed below, is another genetic fallacy.

Consider the argument: All Ys are Zs. Mary says X is a Y. Therefore, X is a Z.

That argument is fallacious: It is a non sequitur of evidence because the conclusion that X is a Z does not follow from the two premises ("All Ys are Zs" and "Mary says X is a Y"). The form of the argument is: If A then B. C. Therefore B. To get to the conclusion that X is a Z, we would need an additional premise, namely, that if Mary says X is a Y, X is in fact a Y: the premise if C then A. In words, that extra premise is "if Mary says X is a Y, then X is a Y." If we added that premise, the argument would be valid. But it would not be sound unless it is impossible for Mary to be mistaken that X is a Y. Because this fallacy has, at its heart, a non sequitur of evidence, we call it a fallacy of evidence. The fallacy consists in treating one of the stated premises (Mary says X is a Y) as if it were a different premise (X is a Y). This particular kind of fallacy of evidence is common: It is an (inappropriate) appeal to authority. There are more examples and discussion of fallacies of evidence and fallacies of relevance below.

To sum up, many (if not all) informal fallacies are of this form: The argument is a non sequitur of relevance or a non sequitur of evidence. Non sequiturs of relevance can be made valid by adding a premise that says the real conclusion implies the desired conclusion. Non sequiturs of evidence can be made valid by adding a premise that says one of the given premises implies a necessary but missing premise. In both cases, the missing premise is false, so the patched argument would be valid but not sound.

Fallacies of Relevance and Fallacies of Evidence

A fallacy of relevance commits a non sequitur of relevance: It establishes a conclusion, but not the desired conclusion. An extra (and false) premise is needed for the actual conclusion to imply the desired conclusion. A canonical form of a fallacy of relevance is:

If A then B. A. Therefore, C.—together with the real-world fact that B does not imply C.

Alternatively, a fallacy of relevance is:

If A then B. Not B. Therefore, not C.—together with the real-world fact that not A does not imply not C.

Conversely, a fallacy of evidence commits a non sequitur of evidence: It does not establish any conclusion. An extra (and false) premise is needed for one of the stated premises to imply a premise that can be used to reach the desired conclusion. A canonical form for a fallacy of evidence is:

If A then B. C. Therefore B.—together with the real-world fact that C does not imply A.

Alternatively, a fallacy of evidence is:

If A then B. Not C. Therefore, not A.—together with the real-world fact that not C does not imply not B.

### Fallacies of Relevance

Logicians often distinguish among kinds of relevance. A piece of evidence is positively relevant to some assertion if it adds weight to the assertion. It is negatively relevant if it takes weight away from the assertion. Some evidence is irrelevant to a given assertion.

For instance, consider the assertion "it's hot outside." The observation that passers-by are sweating would be positively relevant to the assertion: it supports the assertion that the weather is hot. The observation that passers-by are wearing parkas would be negatively relevant: it is evidence that the weather is not hot. The observation that passers-by are listening to mp3 players would be irrelevant.

Many superficially persuasive arguments in fact ride on irrelevant observations. Here are some examples.

Nancy claims the death penalty is a good thing. But Nancy once set fire to a vacant warehouse. Nancy is evil. Therefore, the death penalty is a bad thing.

This argument does not address Nancy's argument, it just says she must be wrong (about everything) because she is evil. Whether Nancy is good or evil is irrelevant: It has no bearing on whether her argument is sound.

This is a fallacy of relevance: It establishes that Nancy is bad, then equates being bad and never being right. In symbols, the argument is If A then B. A. Therefore C. (If somebody sets fire to a vacant warehouse, that person is evil. Nancy set fire to a vacant warehouse. Therefore, Nancy's opinion about the death penalty is wrong.)

Ad hominem is Latin for "towards the person." An ad hominem argument attacks the person making the claim, rather than the person's reasoning. A variant of the ad hominem argument is "guilt by association."

Bob claims the death penalty is a good thing. But Bob's family business manufactures caskets. Bob benefits when people die, so his motives are suspect. Therefore, the death penalty is a bad thing.

This argument does not address Bob's argument, it addresses Bob's motives. His motives are irrelevant: They have nothing to do with whether his argument for the death penalty is sound.

This is related to an ad hominem argument. It, too, addresses the person, not the person's argument. However, rather than condemning Bob as evil, it impugns his motives in arguing for this particular conclusion.

Amy says people shouldn't smoke cigarettes in public because cigarette smoke has a strong odor. But Amy wears strong perfume all the time. Amy is clearly a hypocrite. Therefore, smoking in public is fine.

This argument does not engage Amy's argument: It attacks her for the (in)consistency of her opinions in this matter and in some other matter. Whether Amy wears strong fragrances has nothing to do with whether her argument against smoking is sound.

The abstract form of this argument is also a non sequitur: If A then B. A. Therefore C. (In words: If you complain about strong smells and wear strong fragrances, you are a hypocrite. Amy complains about strong smells and wears strong perfume; therefore, her opinion about smoking is incorrect.)

Tu quoque is Latin for "you also." It related to ad hominem arguments: it addresses the person rather than the person's argument. But instead of generally condemning the other party, it says that his or her claim in the matter at issue is hypocritical because it is inconsistent with something else the person has done or said. We are supposed to conclude that he or she must therefore be wrong on this particular point.

Yes, I hit Billy. But Sally hit him first.

This argument claims it is fine to do something wrong because somebody else did something wrong. The argument is of the form: If A then B. A. Therefore C. (In words: If Sally hit Billy, it's OK for Billy to hit Sally. Sally hit Billy. Therefore, it's OK for me to hit Billy.)

Generally, the two-wrongs-make-a-right argument says that the justified wrong happened after the exculpatory wrong, or was less severe. For instance, Sally hit Billy first, or Sally hit Billy harder than I did, or Sally pulled a knife on Billy.

On the other hand, it might be quite reasonable to argue, "yes, I hit Billy. But he was beating me with a baseball bat—I acted in self defense. " In that case, the first "wrong" might justify hitting Billy, which otherwise would be wrong.

If you don't give me your lunch money, my big brother will beat you up. You don't want to be beaten up, do you? Therefore, you should give me your lunch money.

Ad baculum is Latin for "to the stick." It is essentially the argument "might makes right."

Not all arguments of the form If you do A then B will happen. B is bad. Therefore, don't do A are ad baculum arguments. It depends in part on whether B is a real or imposed consequence of A. For instance, If you cheat on your exam, you will feel guilty about it for the rest of your life; therefore, you should not cheat is not an ad baculum argument. But If you cheat on your exam, I will turn you in to the Student Conduct Office and have you expelled; therefore, you should not cheat is an ad baculum argument. (Either way, don't cheat on your exam!)

Yes, I downloaded music illegally—but my girlfriend left me and I lost my job so I was broke and I couldn't afford to buy music and I was so sad that I was broke and that my girlfriend was gone that I really had to listen to 100 variations of She caught the Katy.

This argument justifies an action not by claiming that it is correct, but by an appeal to pity: extenuating circumstances of a sort.

Ad misericordium is Latin for "to pity." It is an appeal to compassion rather than to reason. Another example:

Yes, I failed the final. But I need to get an A in the class or I [won't get into Business school] / [will lose my scholarship] / [will violate my academic probation] / [will lose my 4.0 GPA]. You have to give me an A!

Millions of people share copyrighted mp3 files and videos online. Therefore, sharing copyrighted music and videos is fine.

This "bandwagon" argument claims that something is moral because it is common. Common and correct are not the same. Whether a practice is widespread has little bearing on whether it is legal or moral. That many people believe something is true does not make it true.

Ad populum is Latin for "to the people." It equates the popularity of an idea with the truth of the idea: Everybody can't be wrong. Few teenagers have not made ad populum arguments: "But Mom, everybody is doing it!"

Bob: Sleeping a full 12 hours once in a while is a healthy pleasure.

Samantha: If everybody slept 12 hours all the time, nothing would ever get done; the reduction in productivity would drive the country into bankruptcy. Therefore, nobody should sleep for 12 hours.

Samantha attacked a different claim from the one Bob made: She attacked the assertion that it is good for everybody to sleep 12 hours every day. Bob only claimed that is was good once in a while.

This argument is also a non sequitur of relevance: If A then B. A. Therefore C. (In words: If an action would have bad consequences if everyone did it all the time, then that action should not be performed by everyone all the time. Sleeping 12 hours would have bad consequences if everyone did it all the time. Therefore, nobody should ever do it.)

A straw man argument replaces the original claim with one that is more vulnerable, attacks that more vulnerable claim, then pretends to have refuted the original.

Art: Teacher salaries should be increased to attract better teachers.

Bette: Lengthening the school day would also improve student learning outcomes. Therefore, teacher salaries should remain the same.

Art argues that increasing teacher salaries would attract better teachers. Bette does not address his argument: She simply argues that there are other ways of improving student learning outcomes. Art did not even use student learning outcomes as a reason for increasing teacher salaries. Even if Bette is correct that lengthening the school day would improve learning outcomes, her argument is sideways to Art's: It is a distraction, not a refutation.

A red herring argument distracts the listener from the real topic. Red herring arguments are very common in political discourse.

All men should have the right to vote. Sally is not a man. Therefore, Sally should not necessarily have the right to vote.

This is an example of equivocation, a fallacy facilitated by the fact that a word can have more than one meaning. This argument uses the word man in two different ways. In the first premise, the word means human while in the second, it means male. Generally, equivocation is considered a fallacy of relevance, but this example fits our definition of a fallacy of evidence.

The logical form of this argument is If A then B. Not C. Therefore, B is not necessarily true. That argument is a formal fallacy. There is a missing premise that equates one of the premises given (Sally is not male) with a different premise not given (Sally is not human). That is, if not C then not A. That (false) premise relates evidence given to evidence not given, so this is a fallacy of evidence according to our definition. The fact that the same word can mean "human" and "male" hides the formal fallacy.

Another common structure for fallacies that involve equivocation is: All P1s are Qs. X is a P2. Therefore, X is a Q. The equivocation is that the same word is used to refer to P1 and P2, which hides the fact that P1 and P2 are not the same.

Here is an example of equivocation hiding a fallacy of relevance:

If you are a Swiss citizen living in the U.S., you are an alien (foreigner). Birgitte is a Swiss citizen living in California. Therefore, Birgitte is an Alien (from another planet).

The structure of this example is For any x, if x is A then x is B. y is A. Therefore, y is C. The missing (and false) premise is that all aliens are Aliens (For any x, if x is B then x is C), which would relate the valid conclusion (Birgitte is an alien) to the desired conclusion (Birgitte is an Alien). Thus this equivocation fallacy is a fallacy of relevance.

The straw man, red herring and equivocation fallacies all change the subject: they argue for (or against) something that is sideways to the original claim but easy to confuse with the original claim. Ad hominem arguments also change the subject—from whether the speaker is right to whether the speaker is "good."

There is no circumstance that justifies killing another person. The death penalty involves killing another person. Therefore, even if someone commits a brutal murder, he should not be put to death.

This argument begs the question. It assumes what it purports to prove, namely, that there is no circumstance that justifies killing. "No circumstance" already precludes "commits brutal murder."

The form of the argument is A. Therefore, A. That is indeed logically valid—it just isn't much of an argument. Where this "fallacy" gets legs is when the premise and the conclusion use different words to say the same thing, creating the illusion that the conclusion is different from the assumption. Here is another example:

Jack is overweight. Therefore, Jack is fat.

Petitio principii is Latin for "attack the beginning." The premise assumes the truth of the conclusion. A circular argument is a variant of begging the question.

Common Fallacies of Relevance

• Attacking the motive
• Look who's talking (tu quoque)
• Two wrongs make a right
• Appeal to force (ad baculum)
• Appeal to pity (ad misericordium)
• Straw man
• Red herring
• Equivocation (semantic fallacy)

• Begging the question (petitio principii)

### Fallacies of Evidence

This section gives examples of fallacies of evidence. Recall that such fallacies are of the form: If A then B. C. Therefore B. There is a missing premise, namely, If C then A. Without that premise, the argument is not much of an argument at all: The premises cannot be combined to support a conclusion other than the premises themselves (which would be a circular argument: C. Therefore, C.). is of this form.

Another form for a fallacy of evidence is: If A then B. Not C. Therefore Not A. Again, there is a missing premise, namely, If not C then not B. And without that premise, the argument is no argument at all. is of this form.

Yet another fallacy is of the form: A. Therefore, B. For instance:

Eat your brussels sprouts. There are children starving in Africa.

(In this argument, the premise A is "there are children starving in Africa," and the conclusion B is "you should eat your brussels sprouts.") This could be classified as a fallacy of evidence: There's a missing premise, If A then B: "If children are starving in Africa, then I should eat my brussels sprouts." Without that premise, the statements are not much of an argument.

The examples of fallacies of evidence given below differ in how C is related to A and B.

All animals with rabies go crazy. Jessie says my cat has rabies. Therefore, my cat will go crazy.

This argument is fallacious. The form of the argument is If A then B. C. Therefore, B. There is a missing premise, namely, that if Jessie says my cat has rabies, then my cat has rabies (If C then A.). That premise relates a stated premise (Jessie says my cat has rabies—the evidence given) to an unstated premise (my cat has rabies—the evidence required to be able to use the premise that animals with rabies go crazy). Thus, this is a fallacy of evidence.

If we add that missing premise, the argument might or might not be sound, depending on whether Jessie could be mistaken. The argument is clearly stronger if Jessie is a veterinarian who tested my cat for rabies than if Jessie is a 5-year-old child who lives next door. Any time we take somebody's say-so as evidence, we are making an appeal to authority. That person might or might not be "an authority."

Our legal system has elaborate rules governing evidence. A witness can testify about what he or she saw or heard or has personal knowledge of. If I say "Jane told me she saw Frank hotwire the car," all other things being equal, that could be used in court as evidence that Jane told me something, but not as evidence that Frank hotwired the car—it is hearsay because I am reporting something I heard about, not something I witnessed directly. I could not appeal to Jane's authority for evidence about Frank's actions. There is a big difference between "I heard it" and "I heard of it."

The fallacy of appealing to authority is called argumentum ad verecundiam, argument to veneration (respect). A more blatant example is:

Professor Stark says 1+1=3. Professor Stark has a Ph.D. He is a learned professor of statistics at one of the world's best universities. He has published many scholarly articles in refereed journals, lectured in many countries, and written a textbook about Statistics. His former students hold positions at top universities, in government research agencies, and in the private sector. He has consulted for many top law firms and Fortune 100 companies. He has been qualified as an expert in Federal court and has testified to Congress. Therefore, 1+1=3.

To study user satisfaction with our software product, we sent out 5,000 questionnaires to registered users. Only 100 users filled out and mailed back their questionnaires. Since more than 4,900 of the 5,000 users surveyed did not complain, the vast majority of users are satisfied with the software.

This argument is fallacious: It treats "no evidence of dissatisfaction" as if it were "evidence of satisfaction." Lack of evidence that a statement is false is not evidence that the statement is true. Nor is lack of evidence that a statement is true evidence that the statement is false.

This is an example of nonresponse in a survey. Nonresponse generally leads to nonresponse bias: people who return the survey are generally different from those who do not. Moreover, even if everyone surveyed responds, the opinions expressed in a survey do not tend to be representative of the opinions of the population surveyed (all users, in this case) unless the group that is administered the survey (the 5,000 users who were sent questionnaires) is selected at random. discusses sample surveys, survey design, nonresponse, and nonresponse bias.

Either you support the war in Iraq, or you don't support our soldiers who risk their lives for our country. You do support our soldiers. Therefore, you support the war.

This argument is valid but not sound: It starts with a premise that is an artificial "either-or,"—a false dichotomy. It is possible to support our soldiers and still to oppose the war in Iraq, so the first premise is false.

The same false dichotomy could have been disguised in slightly different language: The first premise could have been written "if you don't support the war in Iraq, you don't support our soldiers who risk their lives for our country." That is because the premise if A then B is logically equivalent to the premise not A or B.

False dichotomies show up in question form as well: "So, if you didn't get that money by embezzling it, did you rob someone at gunpoint?" If a lawyer asked a witness that question in court, you would expect the opposing attorney to say, "Objection: compound."

Did you know that the Sun goes around the Earth?

This statement presupposes that the Sun goes around the Earth— it is a loaded question. Classical examples of loaded qustions include, "Have you stopped beating your wife?" and "Does your mother know you are an alcoholic?" The word "stop" presupposes that something has already started; the word "know" presupposes that something is true.

After nearly eight years of the Bush administration, the stock market had the largest drop since the Great Depression. The Republican government ruined the economy.

It might be true that Bush administration policies led to the stock market crash. And it might be true that there would have been a comparable "market correction" under a Democratic administration. The fact that the crash occurred late in the Bush administration is not in itself proof that it was caused by Republican government. The crash also occurred shortly after the introduction of the iPhone. Does it follow that the iPhone caused the stock market crash?

This argument is an example of questionable cause. In particular, it is an example of the post hoc ergo propter hoc (after this, therefore because of this) fallacy.

Moreover, even if actions of the Bush government contributed to the stock market crash, it would be an oversimplification to pin everything on the government: an example of the oversimplified cause fallacy. For instance, mortgage banks that gave "sub-prime" loans to borrowers who were not creditworthy might also have played a role.

In addition to post hoc ergo propter hoc, the cum hoc ergo propter hoc (with this, therefore because of this) fallacy is common in misapplications of statistics: data show that two phenomena are associated—tend to occur together or to rise and fall together—and the arguer mistakenly concludes that one of them must therefore cause the other. These fallacies are discussed in more depth in

Giving coincidences special causal import is another example of the questionable cause fallacy. For instance:

Dogs bark before earthquakes. Therefore, dogs can sense that an earthquake is coming.

Dogs do bark before earthquakes. And during earthquakes. And after earthquakes. And between earthquakes.

Marijuana must remain illegal. If it were legalized, cocaine would soon follow. And if cocaine were legalized, then opium would be, and eventually heroin, too. Before you know it, everybody would be on drugs, from nursing infants to great-great-grandparents. The only babies born would be crack babies. People would be dying by the tens of millions from AIDS transmitted by sharing hypodermic needles. There would be rampant prostitution so drug-crazed women and men could pay for their habits. Construction workers would fall off buildings, right and left. Nobody would be able to drive safely—highway fatalities would claim millions of lives a year. The police and military would all become addicted, so there would be no law enforcement or national security. Doctors would be so whacked-out that they couldn't treat patients. The economy would collapse. Within five years, the U.S. would be a third-world country. Eventually, nobody would live past age 20—if they managed to survive infancy.

This argument is of the form:

If A then B. If B then C. If C then D, etc. Eventually, Z. You don't want Z, do you? So, you must prevent A.

While each step in this progression is possible, it is by no means inevitable: The conditional statements are really "if A then possibly B", etc., but they are asserted as "if A then necessarily B," and so on. The implicit argument is that since there is no "bright line" demarcating where to stop the progression, the progression will not stop. That is fallacious. It is also a resort to scare tactics—an appeal to emotion rather than to reason. There is no law of Nature that says government could not draw the line at any point in the progression. Currently, the line is drawn at alcohol, although during the Prohibition, the line was drawn on the other side of alcohol. The line on tobacco is being re-drawn to impose more restrictions: no smoking on airplanes, in restaurants, etc. Lines can be drawn anywhere.

The adage "give them an inch and they'll take a mile" is in this family. Surely, "they" want a mile—or a hundred—in the first place. It's up to us to draw the line where we see fit: at zero, at a millimeter, at an inch, or at a megaparsec.

Not every argument of the form "If A then B. If B then C. If C then D, etc. Eventually, Z" is a slippery-slope fallacy. If each of the conditionals (the if-then statements) is valid, this is a perfectly valid argument. For instance, this argument has no fallacy (assuming the bomb is big): If I push the detonator, the bomb will go off. If the bomb goes off, the building will collapse. If the building collapses, people will be injured and killed. I don't want that to happen, so I shouldn't push the detonator.

I interviewed 20 students in the lunch line at noon today. Nineteen were hungry. Therefore, most students are hungry.

This argument is of the form: Some x are (sometimes) A. Therefore, most x are (always) A. The data do not support such a generalization: They were taken at a particular time and place of my choosing. To draw reliable generalizations from the sample to other times, places, and students requires a sample that is drawn using proper sampling techniques, discussed in This sample has built-in bias by virtue of where and when it was taken. It is called a convenience sample because it simply selected students who were readily available. At another time, the same students might not be hungry. At noon, students not in the lunch line might not be hungry.

Hasty generalizations occur in Statistics when the sample from which we generalize is not representative of the larger group ("population") to which we would like to generalize. This tends to happen when the method used to draw the sample is not a good one—leading to large bias—or because the sample is small, so that the luck of the draw has a big effect on the accuracy. To ensure that a sample is representative generally requires using some method of random selection. These issues are discussed in Another example:

I want to estimate the fraction of free "adult" (commercial pornography) websites that are hosted in the United States. I run a search on a popular search engine for "free porn" and find a link to a list of 25,000 free porn websites, with user ratings. I look up whether each of those websites is hosted in the U.S. or elsewhere. Seventy percent are hosted in the U.S. Therefore, the majority of free porn websites are hosted in the U.S.

This is a hasty generalization. Even though the list is large, there is no reason it should be representative of the population of all free porn websites, with respect to the country they are hosted in or any other variable. The list is a sample of convenience. It was found by running a query in English on a U.S. search engine. It is just a list somebody made and that I happened to find.

To try to get a better sample, I think of more search terms. I run searches on a popular search engine for "free porn," "free sex videos," "naked girls," "hot sex," "free intercourse videos," and 15 other terms. For each search, I record the first 100 links to free porn websites that the search engine returns, producing a list of 2,000 websites. I look up whether each of those websites is hosted in the U.S. Seventy percent are; therefore, the majority of free porn websites are hosted in the U.S.

This is still a hasty generalization from a sample of convenience. There is no reason the first 100 websites returned by a search engine in response to each of 20 queries I make up should be representative of free porn websites, with respect to host country or any other characteristic—and there are many reasons for it to be unrepresentative: More popular websites tend to be returned closer to the top of the list of results for each search. Queries in English tend to return links in the U.S. U.S. search engines tend to return links in the U.S. The search engine databases are not an exhaustive list of all websites. Other queries intended to retrieve porn websites would get different results. There are countless more reasons.

Countries are like families. Government is like the parents, and citizens are like the children. "Spare the rod, spoil the child." Therefore, our legal system must impose harsh penalties for legal infractions: "tough love."

This argument is of the form: x is similar to y in some regards. Therefore, everything that is true for x is true for y. Another example:

Physicists are like physicians. Both study calculus as undergraduates; both have many years of education, culminating in advanced degrees; both know lots of science; both tend to have above average intelligence; both tend to be arrogant; both even start with the same seven letters, "physici." Physicians treat illness. Therefore, physicists treat illness.

Weak analogies often arise in some forms of sampling, such as quota sampling and convenience sampling. A sample can resemble the population from which it is drawn in many ways, and yet be unrepresentative of the population with respect to the property we care about. The Hite report example in shows that a sample can match the demographics of the population from which it is drawn quite accurately, but still be unlike the population with respect to opinions and experiences. One of the chief merits of random sampling is that the resulting samples tend to be representative of the population with respect to all properties; moreover, we can quantify the extent to which a random sample is likely to be unrepresentative, and those differences tend to be smaller the larger the sample size. For other ways of drawing samples, the samples are generally unrepresentative even when they are large.

"Nobody goes there anymore. That place is too crowded."—Yogi Berra.

This isn't an argument because there is no conclusion, only premises, but that's not the point. These two premises amount to: A. Not A. They contradict each other. If one of the premises is true, the other must be false: It is logically impossible for both to be true. Hence, any argument that stems from them cannot be sound—since, even if it is valid, it will have a false premise. Generally, the fallacy of inconsistency occurs whenever the premises cannot all be true—as a matter of logic. Yogi Berra was a master of inconsistency. I don't know how many of the following he actually said, but I've seen these quotes attributed to him:

"Baseball is 90% physical. The other half is mental." "Half the lies they tell about me aren't true." "I never said most of the things I said." "If the world was perfect, it wouldn't be." "It was impossible to get a conversation going, everybody was talking too much." "It gets late early out there." "I wish I had an answer to that because I'm tired of answering that question."

Common Fallacies of Evidence

• Inappropriate appeal to authority
• Appeal to ignorance
• False dichotomy
• Questionable cause
• Slippery slope
• Hasty generalization
• Weak analogy
• Inconsistency

The following exercises check your ability to recognize informal fallacies.

## Summary

An argument consists of a sequence of statements. One is the conclusion; the rest are premises. The premises are given as evidence that the conclusion is true. If the conclusion must be true if the premises were true, the argument is valid. A valid argument is sound if its premises are true. Valid arguments result from applying correct rules of reasoning. Examples of correct rules of reasoning include:

• A or not A.
• Not (A and not A).
• A. B. Therefore, A and B.
• A. Therefore, A or B.
• A and B. Therefore, A.
• A or B. Not A. Therefore, B.
• Not A. Therefore, not (A and B).
• Not (A and B). Therefore, (not A) or (not B).
• Not (A or B). Therefore, (not A) and (not B).
• If A then B. A. Therefore, B.
• If A then B. Not B. Therefore, not A.

Using incorrect rules of reasoning or misapplying correct rules results in a formal fallacy. There are many common formal fallacies. One is the non sequitur. In a non sequitur, a necessary premise is missing. If the missing premise relates one of the stated premises to a different premise (i.e., ties evidence given to evidence not given), the fallacy is a non sequitur of evidence. If the missing premise relates a valid conclusion but unstated conclusion to the stated conclusion (i.e., ties the conclusion given to a conclusion not given), the fallacy is a non sequitur of relevance. There are other formal fallacies as well. Examples of formal fallacies include:

• A or B. A. Therefore, not B.
• Not (A and B). Not A. Therefore, B.
• If A then B. B. Therefore, A.
• If A then B. not A. Therefore, not B.
• If A then B. C. Therefore, B.
• If A then B. A. Therefore, C.

In addition to formal fallacies, there are informal fallacies. Although the categorization is not strict, informal fallacies generally fall into two groups: fallacies of relevance and fallacies of evidence. Fallacies of relevance have non sequiturs of relevance at their core; fallacies of evidence have non sequiturs of evidence. Examples of the former include ad hominem and other genetic fallacies, appeals to emotion (fear, pity), the straw man, the red herring, and arguments that beg the question. Examples of the latter include appeals to authority, slippery slope, hasty generalizations, weak analogies, post hoc ergo propter hoc, and cum hoc ergo propter hoc. Equivocation can hide a fallacy of relevance or a fallacy of evidence. Inconsistency can produce a valid argument, but never a sound argument.

Statistics can be (and often is) misused to produce both kinds of informal fallacy. Be alert to the structure of arguments to avoid being deceived and to avoid deception.

## Key Terms

• affirming the consequent
• anecdotal fallacy
• antecedent
• appeal to authority
• compound question
• consequent
• cum hoc ergo propter hoc
• equivocation
• evidential fallacy
• false dichotomy
• formal fallacy
• genetic fallacy
• hasty generalization
• informal fallacy
• modus ponendo ponens
• modus tollendo tollens
• non sequitur of evidence
• non sequitur of relevance
• post hoc ergo propter hoc
• premise
• presupposition
• quantificational fallacy
• red herring
• semantic fallacy
• slippery slope
• sound
• straw man
• syllogism
• tautology
• valid
• weak analogy