LOGIC

 

From Great Issues in Philosophy, by James Fieser

Home: www.utm.edu/staff/jfieser/120

Copyright, updated 4/1/2016

 

CONTENTS

 

A. What is an Argument?

Propositions and Non-Propositional Utterances

Premise and Conclusion Indicators

Argument Diagrams

B. Informal Fallacies

Fallacies of Relevance

Other Common Fallacies

C. Propositional Statements

Complex Propositions and Logical Connectives

Nested Logical Connectives

D. Propositional Logic

Valid Argument Forms

Fallacious Argument Forms

Sound and Unsound Arguments

E. Inductive Logic

Inductive vs. Deductive Arguments

Inductive Probability

Inductive Argument Forms

 

In ancient Greece, a group of traveling teachers called Sophists had the reputation of being able to argue for any point, no matter how absurd. One Sophist offered this argument:

 

(1) Fido is Joe’s dog.

(2) Fido is a mother.

(3) Therefore, Fido is Joe’s mother.

 

As bizarre as this argument is, at least some parts of it seem compelling, specifically the first two statements. We can easily assume that Fido is Joe’s dog, and that Fido is has had puppies and is thus a mother. The strange part about this argument, though, is that the third statement appears to follow from the first two. Something has gone wrong here, since Joe, who is a human being, was clearly not given birth by a dog. Thus, the above argument is a bad one, even though we might not yet be able to identify exactly where the problem lies. Here, by contrast, is an example of a good argument:

 

(1) If David Hume was a bachelor, then he was unmarried.

(2) David Hume was a Bachelor.

(3) Therefore, David Hume was unmarried.

 

The first two statements are true, and, on top of that, the third statement seems to follow with compelling necessity. The goal of logic is to help us understand what makes arguments like the Fido one bad, and other arguments like the Hume one good.

            The study of logic is different than the study of other areas of philosophy. Philosophy of mind or ethics, for example, involve an unfolding story of competing theories where one philosophical school attempts to outdo or improve upon the views of rivals. Logic, by contrast, is an analytic skill that requires mastery of abstract argument structures. Learning logic often involves looking at many examples and trying practice exercises, similar to how we learn other analytic subjects like math or computer programming. This chapter presents basic themes covered in typical introduction to logic text books, with accompanying practice exercises and solutions.

 

A. WHAT IS AN ARGUMENT?

 

Whether an argument is good or bad, there are a few basic components that all arguments have, namely that they consist of premises and conclusions. In both of the examples above, statements 1 and 2 are the premises; statement 3 in both of the examples is the conclusion. Here are the basic definitions of these concepts:

 

Premise: a statement which is used as evidence for a conclusion.

Conclusion: a statement which is supported by at least one premise.

Argument: at least one premise accompanied with a conclusion.

 

Propositions and Non-Propositional Utterances

It helps to think of an argument as a series of factual statements where the first ones provide evidence for the final one. Each line in an argument must be a “proposition”, that is, a true or false statement about the world. Consider again the above “David Hume” argument. In premise one, it is either true or false that “If David Hume was a bachelor, then he was unmarried.” In this case, this is a true statement. In premise two, it is also either true or false that “David Hume was a Bachelor.” Again, this is also true. Finally, the conclusion “David Hume was unmarried” is either true or false, which, again, is also true.  Although all three of these statements are true, false statements in arguments also count as propositions. Similarly, even bizarre and wildly implausible statements can be propositions as long as they are either true or false, such as the following:

 

“Fido is Joe’s mother”

 

“Joe is 20 feet tall”

 

“Gramps just swallowed his teeth”

 

            While the concept of a proposition might seem straight forward enough—an either true or false statement about the world—it is easy to confuse propositions with other kinds of verbal expressions. Accordingly, we must distinguish between two notions:

 

Proposition: an either true or false statement about the world, such as, “The door is brown,” “David Hume was unmarried,” or “Fido is Joe’s mother”.

 

Non-propositional Utterance: a verbal expression that conveys meaning, but is not a true or false statement about the world. Non-propositional utterances include questions (“who am I”), commands (“get that porcupine out of my face!”), expressions of feelings (“three cheers for old glory!”).

 

Taken literally, questions, commands and expressions of feelings are not true or false statements about the world. It makes no sense to ask whether it is true or false whether “who am I” or whether it is true or false “get that porcupine out of my face”. However, sometimes even non-propositional utterances like these may be rewritten to express propositions. Consider the following:

 

“Keep your flesh-eating zombie out of my yard!”

 

This is strictly speaking a command, but its essential meaning can be preserved in this proposition:

 

“You better keep your flesh-eating zombie out of my yard!”

 

This revised statement is either true or false. More precisely, it is the phrase “you better” which is the true or false component of this statement. That is, it is either true or false that “you better keep your flesh-eating zombies out of my yard.”

 

Premise and Conclusion Indicators

An initial challenge with understanding arguments concerns simply identifying which sentences are premises and which are conclusions. Sometimes, for example, the conclusion might appear first in an argument, such as the following:

 

A trumpet is not a stringed instrument since a trumpet doesn’t have any strings.

 

In this argument, the word “since” is a premise indicator, that is, it is a clue word that tells us that a premise follows it. Laid out formally, the argument is this:

 

premise (1): a trumpet doesn’t have any strings.

concl.    (2): a trumpet is not a stringed instrument.

 

There are many premise indicators, and the more common ones are these:

 

Since

For

Because

Given that

For the reason that

In view of the fact that

 

Just as there are premise indicators, there are also conclusion indicators, which tell us that a conclusion follows, such as the following:

 

A guitar has strings, therefore a guitar is a stringed instrument.

 

In this argument, the word “therefore” is the conclusion indicator, and, more formally, the argument is this:

 

premise (1):  a guitar has strings.

concl.    (2): a guitar is a stringed instrument.

 

While the word “therefore” is the most common conclusion indicator, there are others such as these:

 

Therefore

Thus

Hence

So

Accordingly

For this reason

Consequently

It follows that

 

The above arguments contain only one premise each, but a typical argument often contains two or more premises. Regardless of the number of premises, we still look for premise and conclusion indicators to help identify which is which. Take, for example, this one:

 

The pipa is a musical instrument from China, and it has strings; for this reason it is a stringed musical instrument.

 

The conclusion indicator here is “for this reason” and, laid out formally, the argument is this:

 

premise (1): the pipa is an object that has strings.

premise (2): the pipa is musical instrument from China.

concl.    (3): it is a stringed musical instrument.

 

Argument Diagrams

Once we are able to identify the premises and conclusions in a given argument, the next step is to see how the premises lead to the conclusions, and there are different ways that they can do that. It is sometimes helpful to diagram argument structures using arrows and plus signs to reveal an argument’s structure. Take again the above argument:

 

premise (1): the pipa is an object that has strings.

premise (2): the pipa is musical instrument from China.

concl.    (3): it is a stringed musical instrument.

 

The argument diagram of this is as follows:

 

1+2 |→ 3

 

This tells us that both premise 1 and 2 must be taken together to produce the conclusion; each premise independently will not do that. This is a joint inference. Premise one by itself only tells us that some strange thing called a “pipa” has strings, and for all we know it may just be a clothesline. Premise two by itself tells us that the pipa is in fact a musical instrument, but it says nothing about it having strings. Both pieces of information are needed to lead to the conclusion.

            Other times each premise in an argument might lead independently to the conclusion, without the assistance of the other premise, as in the following:

 

premise (1): a typical trumpet is made of brass.

premise (2): the Harvard Dictionary of Music classifies the trumpet as a brass instrument.

concl.    (3): the trumpet is a brass instrument.

 

Using an arrow diagram, the structure of the argument is this:

 

1 |→ 3 and 2 |→ 3

 

This tells us that premise 1 by itself leads to the conclusion, and also that premise 2 by itself leads to the same conclusion. This is an independent inference. We have here two distinct arguments for the same conclusion, each of which stands independently of the other. The benefit of independent arguments like this is that sometimes it is more effective when arguing to offer separate arguments for the same conclusion, just in case one of the arguments may not be compelling. Throughout this chapter, though, our focus will be on arguments like the earlier pipa one, where premises are taken together to produce a conclusion.

 

B. INFORMAL FALLACIES

 

Recall again our opening argument by an ancient Greek sophist:

 

(1) Fido is Joe’s dog.

(2) Fido is a mother.

(3) Therefore, Fido is Joe’s mother.

 

There is some logical trickery going on here, and it contains what logicians call a fallacy, that is, an error of reasoning that makes an argument flawed. To help expose the logical tricks used by the Sophists, the ancient Greek philosopher Aristotle wrote a book titled Fallacies of the Sophists, which catalogs around a dozen fallacious patterns of reasoning. Today we call them informal fallacies since they occur within the context of normal language, and do not require a formal or abstract analysis of argument patterns. Although the list of informal fallacies has changed since Aristotle, acquaintance with the fallacies is still an effective way of detecting bad argumentation. Some discussions of informal fallacies list as many as 200 different types. We will look at thirteen of the more important ones. For many of these fallacies, there will be exceptions where use of that argument pattern will be valid. Thus, the informal fallacies should be seen as rough guidelines, and not absolute rules.

 

Fallacies of Relevance

A first group of fallacies are called fallacies of relevance, where the premises of the argument are irrelevant for establishing the conclusion. What counts as relevant is a matter of degree: some premises are wildly irrelevant to the conclusion, and others only mildly so. Many of the fallacies have Latin names that trace back to the middle ages, and these are given in parentheses.

            Argument against the Person (argumentum ad hominem): attacking a person’s character instead of the content of that person’s argument. For example, “Heidegger was a poor philosopher since he was a member of the Nazi party.” “Bob is an alcoholic, so don’t take his investment advice too seriously.” “Of course Jones would argue for gun control, after all, Jones is a Democrat.” In these examples, the specific features of someone’s private life may not have any bearing on whether he is a person is a good philosopher, has sound investment advice or has a good argument for gun control.

            Argument from Ignorance (argumentum ad ignorantiam): concluding that something is true since you can’t prove it is false. For example, “Zeus must exist, since no one can demonstrate that he does not exist.” A famous example of this fallacy is from Senator Joseph McCarthy who accused a certain person of communist connections with the following argument: “I do not have much information on this except the general statement of the agency that there is nothing in the files to disprove his Communist connections.” In these cases, it is unreasonable to insist that someone be able to prove a negative when it would require virtually complete knowledge about the world.

            Appeal to Pity (argumentum ad misericordiam): appealing to a person’s unfortunate circumstance as a way of getting someone to accept a conclusion. For example, “you need to pass me in this course since I’ll lose my scholarship if you don’t.” “I implore you to find Mrs. Bobbit not guilty of assaulting her husband, since her home life was so traumatic.” “Please don’t arrest me, I have a wife and kids to support.” “Yes, I murdered my parents, but take pity on me for now I’m an orphan.”

            Appeal to the Masses (argumentum ad populum): going along with the crowd in support of a conclusion. For example, “Gee mom, all the guys in school carry guns.” “I need to get a Facebook account since everyone I know has one.” In these cases, mere popular opinion is not a good indicator of what we should do or believe.

            Appeal to Authority (argumentum ad verecundiam): appealing to a popular figure who is not an authority in that area. For example, “Einstein believed in God, so God must exist.” “Bart Simpson likes Butterfinger candy bars, so they must be good.” While it may be proper to cite Einstein as an authority in physics, questions of religion are outside his area of expertise. And Bart Simpson, if he counts as an authority at all, is only an expert in causing mischief, not in culinary matters.

            Irrelevant Conclusion (non sequitur): drawing a conclusion which does not follow from the evidence. Technically speaking, all of the above fallacies of relevance involve the drawing of an irrelevant conclusion. This specific fallacy of “irrelevant conclusion,” though, is a very general one, and when an argument does not fit any of the more specific patterns of irrelevance above. For example, “My business went under last year, hence the U.S. president should be impeached.” “My shoe string broke; I guess that means it’s time to buy a new car.”

 

Other Common Fallacies

Fallacies of relevance are just one type of erroneous argumentation. There are other informal fallacies that are often classed together under their own special headings. The ones below are among the more common of these.

            False Cause (post hoc ergo procter hoc): inferring a causal connection based on mere correlation. For example, “the number of stroke victims in hospitals is directly proportional to the number of tar bubbles on the road; thus, tar bubbles cause strokes.” “Successful people have expensive clothing; hence the best way to become a success is to buy expensive clothing.”

            Circular Reasoning: implicitly using your conclusion as a premise. For example, “God must exist since the Bible says that God exists, and the Bible is true because God wrote it.” “It is impossible to talk without using words, since words are necessary for talking.”

            Equivocation: an argument which is based on two definitions of one word. For example, “Good steaks are rare these days, so you shouldn’t order yours well done.” “Jones is a poor man, and he loses whenever he plays poker; therefore, Jones is a poor loser.” “You don’t find cars like yours in these parts, so don’t let your car out of your sight.”

            Composition: assuming that the whole must have the properties of its parts. For example, “Each part of this machine is light, therefore the whole machine is light.” “A bus uses more gas than a car, therefore all busses combined use more gas than all cars combined.”

            Division: assuming that the parts of a whole must have the properties of the whole. For example, “This whole machine is heavy, therefore each part of this machine is heavy.” “This corporation is important, hence each worker in this corporation is important.”

            Red Herring: introducing an irrelevant or secondary subject and thereby diverting attention from the main subject. For example, “seat belts in cars do not really increase safety, and, besides, it’s my business, not the government’s, how I choose to sit in my car.” In this example, the real issue is seatbelt safety, and the diversion is government intrusion on liberty. “Women should have the freedom to choose to have an abortion; restricting this freedom is just another instance of male oppression of women.” In this example, the real issue freedom of choice, and the diversion is male oppression of women.

            Straw Man: distorting an opposing view so that it is easy to refute. For example, “vote against gun control, since gun control advocates believe that no one should own any type of fire arm.” “The pro-life position on abortion is wrong since pro lifers believe a woman would have to bring her fetus to term even when her life is in danger.”

 

C. PROPOSITIONAL STATEMENTS

 

Learning the informal fallacies may help you detect specific types of logical errors, but this will not guarantee that you can construct error-free arguments of your own. A different approach to argumentation, called propositional logic (also sentential logic), assists in constructing arguments which fit valid argument forms. Consider again this earlier example:

 

(1) If David Hume was a bachelor, then he was unmarried.

(2) David Hume was a bachelor.

(3) Therefore, David Hume was unmarried.

 

The logical structure of this argument is this:

 

(1) if P then Q

(2) P

(3) therefore, Q

 

The arguments of propositional logic have a special grammar to them, and they often consist of splicing together simple propositions into longer ones. Consider premise one above: “If David Hume was a bachelor, then he was unmarried”. This contains two simple propositions that we have abbreviated with the letters P and Q. Here, P stands for the simple proposition that “David Hume was a bachelor”, and P stands for the simple proposition that “He was unmarried.” P and Q each express a single, simple idea that is either true or false. However, the entirety of premise one expresses a complex idea that is formed by combining these two simple ideas. In propositional logic, the letters P, Q, R, etc., are commonly used to designate simple propositions. However, we could just as easily use Greek letters or geometrical shapes as abbreviations.

 

Complex Propositions and Logical Connectives

In the above “David Hume” example, premise one is a complex proposition that splices together the simple propositions P and Q in an “if-then” statement. Within the system of propositional logic, there are four and only four basic logical connectives that are used to construct complex propositions from simple ones:

 

P and Q

P or Q

if P then Q

not P

 

For example, the complex proposition, “Bob will be happy and Beth will be sad” may be abbreviated, “P and Q”, where P stands for “Bob will be happy” and Q stands for “Beth will be sad.” Each of the above four logical connectives have special names and have specific meanings, which we will next examine.

            Conjunction: “P and Q” An example of a conjunction is “Bob is rich and Joe is poor.” This can be abbreviated “P and Q” where “P” stands for the simple proposition “Bob is rich” and “Q” stands for the simple proposition “Joe is poor.” The “P” and “Q” elements of a conjunction are referred to as conjuncts. The P’s and Q’s of conjunctions can be switched around and mean the same thing; for example, the statement “Bob is rich and Joe is poor” means the same thing as “Joe is poor and Bob is rich.” In ordinary conversation we use conjunctions regularly, but often in a disguised form. It is important to see through these disguises and translate concealed conjunctions into standard propositional form (standard form being “P and Q”). Here is a list of some typical disguises:

 

P, but Q

P, although Q

P; Q

P, besides Q

P, however Q

P, whereas Q

 

            Disjunction: “P or Q”. An example of a disjunction is, “Beth pawned her wedding ring or Beth sold blood.” This can be abbreviated as “P or Q” where “P” stands for the simple proposition “Beth pawned her wedding ring,” and “Q” stands for the simple proposition, “Beth sold blood.” The “P” and “Q” elements of a disjunction are each referred to as disjuncts. Like conjunctions, the two disjuncts in a disjunction may also be switched around and mean the same thing. Disjunctions are more complicated than they first appear since in ordinary conversation the word “or” can be used in two distinct ways. First, the word or is used inclusively in the above example since Beth could have pawned her ring, or sold blood, or both of these. Second, in ordinary language, the word “or” can be used exclusively as in the statement, “Mary is dead or Mary is alive,” where Mary cannot be both dead and alive at the same time. Although the word “or” can be either inclusive or exclusive in ordinary language, in logic, however, it is used only inclusively. There is a more complex way of logically expressing the notion of “or” in an exclusive sense, which will be discussed further down.

            Negation: “not P”. An example of negation is, “it is not the case that Fido just chased a raccoon into Walmart.” This can be abbreviated “not P” where “P” stands for the simple proposition “Fido just chased a raccoon into Walmart.” A sentence which has a negative word in it, such as “not,” “never,” or “none,” may often (but not always) be translated into a negated proposition. For example, consider the sentence, “I knew that Bill was not really a communist.” Since this sentence is an assertion about my knowledge, it does not translate into a negation.

            Conditional: “If P then Q”. An example of a conditional proposition is, “if you eat of the forbidden fruit, then you will surely die.” This can be abbreviated “if P then Q” where P stands for the simple proposition “you eat of the forbidden fruit,” and Q stands for “you will surely die.” The P part of a conditional is referred to as the antecedent (sometimes also called the “sufficient condition”), and the Q part is called the consequent (sometimes also called the “necessary condition”). An important feature about conditionals is that if the P’s and Q’s are switched around, the meaning of the sentence changes. This is precisely why the P and Q components of conditionals have separate names, unlike with conjunctions and disjunctions whose parts can be switched and retain the same meaning. For example, compare the above example to this: “if you die, then you will have eaten of the forbidden fruit.” Clearly, the two sentences do not mean the same thing. Assume that everyone who eats the forbidden fruit subsequently dies; still, not everyone who dies will have eaten of the forbidden fruit, such as someone who dies in a skydiving accident. Some typical disguised conditionals are,

 

If P, it follows that Q

P implies Q

P entails Q

Whenever P, Q

P, therefore Q

Q follows from P

Q, since P

 

Nested Logical Connectives

Complex propositions often contain a number of logical connectives nested within each other. Consider the following: “I will not cry and roll around on the floor if you simply give me that piece of candy.” This proposition contains a negation, conjunction, and a conditional. Translated into standard form the proposition reads, “If you will simply give me that piece of candy, then it is not the case that (I will cry and I will roll around on the floor). Abbreviated this would be as follows:

 

if P then not (Q and R)

 

where,

 

P = you simply give me that piece of candy

Q = I will cry

R = I will roll around on the floor

 

The benefit of nesting logical connectives is that it is possible to put even quite complicated propositions into standard form. As noted above, an or (that is, disjunction) in logic is inclusive. Yet, by nesting a number of logical connectives, it is possible to put exclusive or’s into standard form without violating any rules. Take again the example “Mary is dead or Mary is alive.” The intent of this proposition is that Mary cannot be both dead and alive, hence the “or” here is used exclusively. To be in proper logical form, this sentence must be translated into, “(Mary is dead or Mary is alive) and it is not the case that (Mary is dead and Mary is alive). Abbreviated this says,

 

(P or Q) and not (P and Q)

 

where,

 

P = Mary is dead

Q = Mary is alive

 

It is important to recognize when a nested proposition is formed properly—or “well-formed” as logicians call it. For example, the following statements are not well-formed:

 

P not (if and)

(P Q) not R

or P then Q if

P if Q and R or S not

 

In each of these statements, the sub-components are just randomly strung together, and each whole statement is essentially gibberish. By contrast, the following propositions are well-formed:

 

(P and Q) or R

If P then (Q and R)

not (P or R)

(P and Q) or (R and S)

 

In these cases, (a) all of the sub-components within the parentheses fit the basic pattern of one of the four logical connectives, and (b) the whole proposition also fits the basic pattern of one of the four logical connectives. For example, in the first proposition, “P and Q” fits the pattern of a conjunction, and the whole proposition fits the pattern of a disjunction.

 

D. PROPOSITIONAL LOGIC

 

So far we have seen that propositional statements may be simple, complex, and nested. Any of these may be used as elements of an argument. Propositional logic is a system of logic that builds arguments from propositional statements.

 

Valid Argument Forms

As noted earlier, many arguments are bad ones, and when constructing logical arguments in propositional logic, our goal is to make good ones. The first step in forming a good argument is to follow a valid argument form, and for our purposes we will define a valid argument as follows:

 

Valid Argument: an argument which fits a valid argument form (such as modus ponens below).

 

There are an infinite number of valid argument forms, but we will be interested in just four commonly used ones:

 

Modus Ponens

 

premise (1) if P then Q

premise (2) P

concl.    (3) therefore, Q

 

(1) If the president pushes the button, then a nuclear bomb will explode.

(2) He pushed the button.

(3) Therefore, a nuclear bomb will explode.

 

Modus Tollens

 

premise (1) if P then Q

premise (2) not Q

concl.    (3) therefore, not P

 

(1) If Bob climbed up the police radio tower then he would have been arrested.

(2) It is not the case that Bob was arrested.

(3) Therefore, it is not the case that Bob climbed up the police radio tower.

 

Disjunctive Syllogism

 

premise (1) P or Q

premise (2) not P

concl.    (3) therefore, Q

 

(1) Joe will eat a stick of butter or Joe will eat a tub of lard.

(2) It is not the case that Joe will eat a stick of butter.

(3) Therefore, Joe will eat a tub of lard.

 

Hypothetical Syllogism

 

premise (1) if P then Q

premise (2) if Q then R

concl.    (3) therefore, if P then R

 

(1) If you bribe the officer then he will tear up the ticket.

(2) If he tears up the ticket then you won’t pay a fine.

(3) Therefore, if you bribe the officer then you won’t pay a fine.

 

Fallacious Argument Forms

All four of these valid argument forms have a strong intuitive appeal. Unfortunately, it is easy to make a tiny mistake when constructing these arguments, which can radically alter its validity. This happens so often that, based on such mistakes, logicians have introduce a group of fallacious argument forms, or formal fallacies (in contrast to the informal fallacies discussed earlier). These arguments are so similar to genuine valid argument forms that they can be frequently mistaken for the real thing. We will consider three of these.

 

Fallacious Modus Ponens: fallacy of affirming the consequent

 

premise (1) if P then Q

premise (2) Q

concl.    (3) therefore, P

 

(1) If the president pushes the button, then a nuclear bomb with explode.

(2) A nuclear bomb exploded.

(3) Therefore, the president pushed the button.

 

Intuitively, we can see that this argument fails since, even if a nuclear bomb did explode, it wouldn’t necessarily be because the president pushed the button. The Whitehouse janitor, for example, might have accidentally pushed the button. For comparison, here is the genuine version of modus ponens:

 

premise (1) if P then Q

premise (2) P

concl.    (3) therefore, Q

 

The error with the fallacious form occurs in premise two by affirming the consequent, rather by affirming the antecedent is required in the genuine version of modus ponens.

 

Fallacious Modus Tollens: fallacy of denying the antecedent

 

premise (1) if P then Q

premise (2) not P

concl.    (3) therefore, not Q

 

(1) If Bob climbed up the police radio tower then he would have been arrested.

(2) It is not the case that Bob climbed up the police radio tower.

(3) Therefore, it is not the case that Bob was arrested.

 

This argument fails since Bob might be arrested for any number of reasons, such as tap dancing on top of a patrol car. Climbing the police radio tower is just one reason for someone to be arrested. Here again is the genuine version of modus tollens:

 

premise (1) if P then Q

premise (2) not Q

concl.    (3) therefore, not P

 

The mistake with the fallacy occurs in premise two by denying the antecedent, rather than by denying the consequent as is required in the genuine version of modus tollens.

 

Fallacious Disjunctive Syllogism: fallacy of asserting an alternative

 

premise (1) P or Q

premise (2) P

concl.    (3) therefore, not Q

 

(1) Joe will eat a stick of butter or Joe will eat a tub of lard.

(2) Joe will eat a stick of butter.

(3) Therefore, it is not the case that Joe will eat a tub of lard.

 

The fallacy in this case rests on forgetting that the “or” in logic is inclusive. Even if Joe does eat a stick of butter, he could also eat a tub of lard. Here, by comparison, is the genuine version of disjunctive syllogism:

 

premise (1) P or Q

premise (2) not P

concl.    (3) therefore, Q

 

The error with this fallacy is by asserting one of the disjuncts in premise two, rather than denying a disjunct.

 

Sound and Unsound Arguments

 So far we have seen that the first step in forming a good argument within propositional logic is that it must be valid. However, more is needed. A good argument must be valid and have all true premises. The combined requirements of validity plus truth is called soundness:

 

Sound Argument: an argument which (a) follows a valid argument form, and (b) has only true premises.

 

The David Hume argument presented at the outset is an example of a sound argument since it is valid and has only true premises.

 

(1) If David Hume was a bachelor, then he was unmarried.

(2) David Hume was a Bachelor.

(3) Therefore, David Hume was unmarried.

 

First, the argument is valid since it follows a valid modus ponens argument form. Second, both premises are true. Premise 1 is true by definition since a bachelor is defined as an unmarried man. Premise 2 is true because it is a fact of history: David Hume in fact was a bachelor his entire life. Since this argument is valid and has only true premises, then it is thereby a sound argument.

            A few examples will illustrate why a sound argument must be both valid and have all true premises. The argument below follows a valid form but does not have true premises:

 

(1) Uncle George is a golfing shoe or Uncle George is a tennis shoe.

(2) It is not the case that Uncle George is a golfing shoe.

(3) Therefore, Uncle George is a tennis shoe.

 

The above argument is valid since it follows the form of disjunctive syllogism. But, premise 1 is obviously false: Uncle George is a human being and thus is not any type of shoe. The argument, then, is unsound because premise 1 is false. There are also some arguments that have true premises, but do not follow a valid argument form, and are likewise unsound. For example:

 

(1) If the President was the pilot of Air Force One, then he could fly in the presidential plane.

(2) The President can fly in the presidential plane.

(3) Therefore, the President is the pilot of Air Force One.

 

Premises 1 and 2 in this argument are true. With premise 1, “Air Force One” is simply the nickname of the president’s personal plane; whoever is the pilot of Air Force One will thereby be flying in the presidential plane. Premise 2 is true since flying in the presidential plane is a provision of the president’s job. However, even though both premises are true, the argument follows the fallacious argument form of fallacious modus ponens, so this argument is also unsound. In short, since soundness entails both validity and true premises, there are two ways that an argument can be unsound: (1) it will be invalid, or (2) it will have at least one false premise.

 

E. INDUCTIVE LOGIC

 

Inductive vs. Deductive Arguments

All four of the above valid argument forms in propositional logic are classified as deductive, as defined here:

 

Deductive argument: an argument whose conclusion follows necessarily from its basic premises.

 

Deduction was the main message of the concept of soundness: if you have a valid argument with all true premises, then the conclusion follows with necessity. In a sense, the conclusion in its full force is already built into the combination of premises, and all that a deductive argument does is extract the conclusion from those premises. To “deduce” means to take some facts and see what necessarily follows from them.

            The argument forms of propositional logic are just one type of deductive logical system, and, in fact, it is only within the past 100 years that it has become the dominant deductive system within the field of logic. Prior to that, the main deductive approach was categorical syllogistic logic, a system first created by Aristotle over 2,000 years ago. Briefly, here is the classic example of a categorical syllogism:

 

1. All human beings are mortal things.

2. Socrates is a human being.

3. Therefore, Socrates is a mortal thing.

 

The categorical syllogism, as the name implies, is all about categories of things, and how some categories are contained within other categories. Think of it like a series of boxes that are contained within each other. In this case, there is a large box labeled “Mortal Things”, a smaller box labeled “Human Beings” and an even smaller box labeled “Socrates”. Using  this box metaphor, the above argument is this:

 

1. The box labeled “Human Beings” is inside the box labeled “Mortal Things”.

2. The box labeled “Socrates” is inside the box labeled “Human Beings”.

3. Therefore, the box labeled “Socrates” is inside the box labeled “Mortal Things.”

 

The conclusion states the obvious: the box containing “Socrates” is within the box of “Mortal Things” – and this is so because the Socrates box is in the “Human Beings” box, and that box is within the larger box of “Mortal Things”. The conclusion follows from the two premises with necessity.

            The larger point here is that both propositional logic and syllogistic logic are deductive systems since their conclusions necessarily follow from their premises. But inductive logic takes a completely different approach: the conclusion is likely to be true, but it doesn’t follow from the premises with absolute necessity. Induction is about probability, not necessity. In contrast with deductive argumentation, the definition of an inductive argument is this:

 

Inductive Argument: an argument in which the premises provide reasons supporting the probable truth of the conclusion.

 

Here is an example of a simple type of inductive argument:

 

1. Rock 1 falls to the ground when I open my hand.

2. Rock 2 falls to the ground when I open my hand.

3. Therefore, all rocks similar to 1 and 2 will probably fall to the ground when I open my hand.

 

The first two premises here are observations about how two rocks behave in a similar fashion. Strictly speaking, all that this tells us is how only those two rocks behave. But the conclusion moves well beyond those two rocks and makes the sweeping claim that all other rocks similar to those two will behave similarly. The conclusion here does not follow with necessity from the premises, since, for all we know, the next rock that we pick up that is similar to rocks 1 and 2 will levitate to the sky. That is why the conclusion contains the important word “probably”, which warns us that there is no guarantee that the next rock that is similar to rocks 1 and 2 will behave the same way.

            Induction is most often connected with scientific experimentation. Scientists will look at a comparatively small group of cases and conclude that this is how things work in all similar cases. A test group of 100 bald men are given a drug, and they all grow hair. The scientists then conclude that this drug will probably promote hair growth on all bald men. This is no guarantee that it will work on every bald man, but the evidence does support the probable truth of the conclusion.

 

Inductive Probability

With deductive arguments, validity is the key concept: an argument is either valid or invalid, and there is no in between. With induction, however, the concept of validity makes no sense: induction is all about the probability of a conclusion, not its necessity. Thus, in place of validity, inductive arguments use a different standard, namely, inductive probability, which is the degree to which a conclusion is probable given the truth of the premises. There are degrees of inductive probability based on the relative strength or weakness of the inference. There is no sharp line between strong and weak inductive reasoning, but for convenience we will use the following four degrees of strength:

 

Inductively very strong: probability is close to certain.

Example: “Every living person that we know of breathes air; therefore Joe probably breathes air.”

Inductively strong: probability is high.

Example: “Regular cigarette smoking reduces a person’s life expectancy by seven years on average; therefore, Joe, who smokes regularly, will probably die seven years earlier than he would otherwise.”

Inductively weak: probability is low.

Example: “Some people watch reruns of the Andy Griffith Show; therefore, Joe probably watches reruns of the Andy Griffith Show.”

Inductively very weak: probability is close to non-existent.

Example: “I once saw a guy balance a chair on top of his head; therefore Joe can probably balance a chair on top of his head.”

 

Depending on the type of inductive argument used, there are several factors that determine an argument’s inductive strength. But the critical point is whether the conclusions move too far beyond the data in the premises. For example, if every person we know of can balance a chair on top of their heads, then it is reasonable to conclude that a random person named Joe can do this too. But if very few people can balance a chair on top of their heads, then it is improper to conclude that Joe can do this.

 

Inductive Arguments Forms

There are several types of inductive arguments, but we will consider here just three commonly used forms. For each of these forms there are specific fallacies that often occur, and we will also look that these.

 

Statistical Syllogism: drawing a conclusion about an individual based on the population as a whole. Below is the formula for this argument, and example of it, and the fallacy associated with it:

 

premise (1) n percentage of a population has attribute A.

premise (2) x is a member of that population.

concl.    (3) Therefore, there is an n probability that x has A.

 

(1) 36% of Americans ages 18-24 have tattoos.

(2) Joe is an American within that age range.

(3) Therefore, there is a low probability that Joe has a tattoo.

 

Fallacy of small proportion: a conclusion is too strong to be supported by the small population proportion (or percentage) with the attribute.

 

As an example of the fallacy of small proportion, suppose in the above illustration we concluded that “There is a very strong probability that Joe has a tattoo.” This new conclusion would be too strong since the population proportion is only 36%.

 

Statistical Induction: drawing a conclusion about a population based on a sample. Here are the details of this:

 

premise (1) n percent of a sample has attribute A.

concl.    (2) Therefore, n percent of a population probably has attribute A.

 

(1) 27% of 1033 randomly surveyed adults believe that God helps decide who wins sporting events.

(2) Therefore, 27% of the population probably believes that God helps decide who wins sporting events.

 

Fallacy of small sample: a conclusion is too strong to be supported by a small sample number.

 

Fallacy of biased sample: a conclusion is too strong to be supported by a nonrandom sampling technique.

 

With statistical induction, there are two distinct fallacies that are often committed. First, with the fallacy of small sample, the issue rests on what an appropriate size sample should be for a study, and this differs based on the field of research. For example, with psychological studies, such as testing the reaction time of elderly people, 30 is an acceptable number. With national surveys, such as the one above about sporting events, 1,000 is acceptable, which will have a margin of error of 3%. As an example of the fallacy of small sample, if in the above illustration we only randomly surveyed 50 people rather than 1033 people, then the margin of error would be too high to reliably support the conclusion. With the second fallacy, the fallacy of biased sample, the issue is not about the number of people sampled, but the nonrandom nature of the sample. As an example of the fallacy of biased sample, suppose in the above illustration the sample group was restricted to 1033 members of an organization called “The Association of Christian Sports Fans.” Since this would not be a random sample of the whole population, we could thus not reliably draw a conclusion about the views of the whole population.

 

Argument from Analogy: drawing a conclusion about one individual based on its similarities with another individual. The following are the details:

 

premise (1) Objects x and y each have attributes A, B and C.

premise (2) Object x has an additional attribute D.

concl.    (3) Therefore, object y probably also has attribute D.

 

(1) Humans and chimpanzees each have pain receptors, neurological pain pathways within their brains, and natural pain killers.

(2) Humans consciously experience pain.

(3) Therefore, chimpanzees probably also consciously experience pain.

 

Fallacy of false analogy: comparing two items that have trivial points in common, but differ from each other in more significant ways.

 

An example of the fallacy of false analogy is the following:

 

(1) Humans and store manikins both have the human form, stand upright, and wear clothes.

(2) Humans consciously experience pain.

(3) Therefore, store manikins probably also consciously experience pain.

 

While both premises 1 and 2 are both true, this argument commits the fallacy of false analogy since the three attributes in premise 1 (human form, standing upright, and wearing clothes) are trivial attributes that are not relevant to the psychological ability to consciously experience pain.

 

IN-CLASS EXERCISES

 

Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions.

 

(1) I have a very refined sense of smell.

(2) Not here, Bob!

(3) I think I’m going to sell little Joey into slavery.

 

Instructions: Identify the premises and conclusions in the following arguments, and identify any premise and conclusion indicators.

 

(4) English is the best language since it’s the only one that I speak.

(5) Bob likes to argue all the time, and for that reason he would make a good lawyer.

(6) In view of the fact that Joe cheated on his taxes, we consequently cannot appoint him to the ethics committee.

 

Instructions: Diagram the following arguments. First number each statement, then use plus signs and arrows to designate the argument structure as either a joint inference or an independent inference.

 

(7) Joe has no friends since the only people he knows are on Facebook, and those aren’t real friends.

(8) Bob was voted most popular student in class, and Bob is always seen with lots of people around him. Thus, Bob has many friends.

(9) Joe and Bob aren’t friends because each says that he can’t stand the other, and each angrily insults the other when they pass in the hall.

 

Instructions: Identify the informal fallacy in each of the following.

 

(10) “The Dead Milkmen” is a rock band. Most people who were once milkmen in the U.S. are now dead. Yikes! That’s one big rock band!

(11) Hey, forget about Beth, she’s nothing special. Is there anything special about her kidneys, tonsils, or small intestine? She’s only a collection of those things.

(12) Of course the Major thinks that the Army offers good career opportunities. He’s an Army man himself.

(13) I think Beth will go out with you. I haven’t heard anything which suggests that she wouldn’t.

(14) We have a good faculty here at Preppy State University. Therefore, Dr. Joseph Drunkard, who teaches here, is a good faculty member.

 

Instructions: In each of the following identify the logical connective being used and translate the proposition into standard form.

 

(15) Father Joe’s marriage to Beth implies that he first leaves the priesthood.

(16) I was accepted at Yale University, but I’d much rather attend Thrift Community College.

(17) Bob’s name does not appear on Santa’s “nice” list.

 

Instructions: Determine which of the following are well-formed nested propositions.

 

(18) if P then (Q or R)

(19) (P and Q) not

(20) not (P or Q)

(21) P and (if Q then R)

 

Instructions: Translate the following premises and conclusions into standard form and decide which valid argument form or fallacious argument form is being used.

 

(22) If the band “Satan’s Pitchfork” performs in town, they will play “Hell, Sweet Hell.” If they perform “Hell, Sweet Hell” then dudes will stage dive. Therefore, if they perform, dudes will stage dive.

(23) Either Bob will go bankrupt, or I will. Bob will go bankrupt. Therefore, I will not.

(24) If Joe flunks out of college, then his brother Bob will inherit the family business. Joe will not flunk out of college. Therefore, Bob will not inherit the family business.

 

Instructions: Make up a valid argument that leads to the conclusion given. Use the rule indicated in parentheses. You will need to invent some simple proposition to make your premises complete.

 

(25) Polly wants a cracker. (disjunctive syllogism)

(26) If you insult Beth’s mother, you will go to the hospital. (hypothetical syllogism)

(27) Joe will fail his exam. (modus ponens)

(28) Thrift Community College is a good school. (modus tollens)

 

Instructions: Are the following arguments valid, invalid, sound, or unsound?

 

(29) If Fido is a Dalmatian, then Fido would have lots of spots

It is not the case that Fido is a Dalmatian

Therefore, it is not the case that Fido has lots of spots

 

(30) If Joseph Stalin had U.S. citizenship, then he would have been born in the U.S.

It is not the case that Joseph Stalin was born in the U.S.

Therefore, it is not the case that Joseph Stalin had U.S. citizenship.

 

Instructions: The following test your understanding of soundness.

 

(31) Can a valid argument have a false conclusion?

(32) Can a sound argument have a false conclusion?

 

Instructions: What is the inductive strength of each of the following (that is, very strong, strong, weak, very weak)?

 

(33) Some notable guitarist have died in their 20s. Joe is a notable guitarist. Therefore, Joe will probably die in his 20s.

(34) College dropouts make $1 million less during their careers than college graduates. Joe is a college dropout. Therefore, Joe will probably make around $1 million less during his career than an average college graduate.

(35) College students with seven or more body piercings have a significantly higher rate of deviant behavior than students with no piercings. Joe has only one body piercing. Therefore Joe has only a slightly higher rate of deviant behavior than students with no piercings.

 

Instructions: For each of the following, indicate the inductive argument form that is followed, and whether it commits any inductive fallacy.

 

(36) Joe and Bob live in the same town, listen to the same music, and like the same sports teams. Joe is Presbyterian. Therefore, Bob is probably also Presbyterian.

(37) 60% of college students in the U.S. are women. Preppy State University is a U.S. College. Therefore, there is a very high probability that the next student who walks out of Preppy State’s student center will be a woman.

(38) 100% of 20 randomly surveyed adults in the small town of Hornbeak, Tennessee shop at Walmart. Therefore, 100% of Americans shop at Walmart.

 

ANSWERS TO IN-CLASS EXERCISES

 

(1) This is a proposition as written.

(2) This is not a proposition. It can be rewritten as a proposition as follows: “I request that you do not do that here Bob.” The true/false component of the revised version is “I request that. . . .”

(3) This is a proposition as written. The true or false component is “I think. . . .” 

(4) The conclusion is “English is the best language”, the premise is “it’s the only one that I speak,” and “since” is a premise indicator.

(5) The premise is “Bob likes to argue all the time,” the conclusion is “he would make a good lawyer,” and “for that reason” is a premise indicator.

(6) The premise is “Joe cheated on his taxes,” the conclusion is “we cannot appoint him to the ethics committee,” “in view of the fact that” is a premise indicator, and “consequently” is a conclusion indicator.

(7) The numbered argument and diagram are as follows:

 

1 [Joe has no friends] since 2 [the only people he knows are on Facebook], and 3[those aren’t real friends].

 

2 + 3 |→ 1

 

(8) The numbered argument and diagram are as follows:

 

1 [Bob was voted most popular student in class], and 2 [Bob is always seen with lots of people around him]. Thus, 3 [Bob has many friends].

 

1 |→ 3 and 2 |→ 3

 

(9) The numbered argument and diagram are as follows:

 

1 [Joe and Bob aren’t friends] because 2 [each says that he can’t stand the other], and 3 [each angrily insults the other when they pass in the hall].

 

2 |→ 1 and 3 |→ 1

 

(10) Equivocation. This argument assumes two meanings of the phrase “Dead Milkmen.”

(11) Composition. This argument assumes that a property of each part (i.e., “nothing special”) applies to the whole (i.e., Beth herself).

(12) Argument against the person. This argument attacks the personal attribute of the Army Major, without examining the content of the Major’s argument.

(13) Argument from ignorance. This argument concludes that something is true (i.e., that Beth will go out with you) since it isn’t proven false.

(14) Division. This argument assumes that a property of the whole university (i.e., “good faculty”) applies to each member (i.e., Dr. Joseph Drunkard).

(15) This is a conditional (if-then). The clue term is “implies”, and the proposition translates “IF Father Joe marries Beth THEN he must first leave the priesthood.”

(16) This is a conjunction (and). The clue term is “but”, and the proposition translates “I was accepted at Yale University AND I’d much rather attend Thrift Community College.”

(17) This is a negation. The clue term is “not”, and the proposition translates “It is not the case that Bob’s name appears on Santa’s ‘nice’ list.”

(18) This is well formed as written.

(19) This is not well formed as written since (a) a logical connective needs to precede “not” and (b) a proposition needs to follow (not).

(20) This is well formed as written.

(21) This is well formed as written.

(22) This is a valid hypothetical syllogism argument. Here it is translated into proper form:

 

IF the band “Satan’s Pitchfork” performs in town, THEN they will play “Hell, Sweet Hell.”

IF they perform “Hell, Sweet Hell” THEN dudes will stage dive.

Therefore, IF the band “Satan’s Pitchfork” performs in town, THEN dudes will stage dive.

 

 (23) This is an invalid fallacious disjunctive syllogism argument. Here it is translated into proper form:

 

Bob will go bankrupt, OR I will go bankrupt.

Bob will go bankrupt.

Therefore, I will not go bankrupt.

 

(24) This is an invalid fallacious modus tollens argument. Here it is translated into proper form:

 

IF Joe flunks out of college, THEN his brother Bob will inherit the family business.

IT IS NOT THE CASE THAT Joe will flunk out of college.

Therefore, IT IS NOT THE CASE THAT his brother Bob will inherit the family business.

 

(25) A possible created proposition is “Polly wants a hamburger.” The argument, then, is this:

 

Polly wants a hamburger OR Polly wants a cracker.

IT IS NOT THE CASE THAT Polly wants a hamburger.

Therefore, Polly wants a cracker.

 

(26) A possible created proposition is “Beth will hit you.” The argument, then, is this:

 

IF you insult Beth’s mother, THEN Beth will hit you.

IF Beth hits you, THEN you will go to the hospital.

Therefore, IF you insult Beth’s mother, THEN you will go to the hospital.

 

(27) A possible created proposition is “Joe parties all night.” The argument, then, is this:

 

IF Joe parties all night, THEN Joe will fail his exam.

Joe partied all night.

Therefore, Joe will fail his exam.

 

(28) A possible created proposition is “the faculty members are drunks”. To make sure a negation appears in the conclusion, yet preserves the original meaning of stipulated conclusion, the conclusion should be reworded as follows: “It is not the case that Thrift Community College is a bad school.” The argument, then, is this:

 

IF Thrift Community College is a bad school. THEN the faculty members are drunks.

IT IS NOT THE CASE THAT the faculty members are drunks.

Therefore, IT IS NOT THE CASE THAT Thrift Community College is a bad school.

 

(29) This is an invalid fallacious modus tollens argument, since premise two denies the antecedent instead of the consequent. It is, thus, both invalid and unsound.

(30) This is a valid modus tollens argument. However, premise 1 is false since U.S. citizens don’t have to be born in the U.S. (i.e., they can be naturalized). Thus, it is valid, but unsound for having a false premise.

(31) A valid argument can have a false conclusion, such as the earlier argument that concludes “Uncle George is a tennis shoe.” Validity only concerns the form of an argument, not the truth of the content.

(32) A sound argument cannot have a false conclusion. Validity is a truth-preserving mechanism; thus, if a valid argument has true premises (and thus is sound), then the truth carries over into the conclusion.

(33) The inductive probability is very weak since the conclusion is much stronger than what the premises indicate. Of the hundreds of notable guitarists, the early death of “some” of them implies almost nothing about the probability of Joe’s early death.

(34) The inductive probability of this argument is strong since the conclusion is consistent with the premises. However, it does not rise to the level of  “very strong” since $1 million is only the average difference, and for specific dropouts it might be much higher or lower than this.

(35) The inductive probability of this argument is very weak since the conclusion goes far beyond what the premises state. Premise 1 speaks only about the increased deviance of students with seven or more body piercings. It does not indicate where the threshold for deviancy begins, and for all we know the deviancy rate for one piercing is the same as for no piercings.

(36) This is an argument from analogy that commits the fallacy of false analogy. The attributes of living in the same town, listening to the same music, and having the same parents are trivial and not clearly connected with the attribute of being lazy.

(37) This is an argument from statistical syllogism. It commits the fallacy of small proportion since the clause “very high probability” in the conclusion is much stronger than the 60% indicated in the first premise.

(38) This is an argument from statistical induction that commits both the fallacy of small sample and biased sample. A sample of only 20 people is too small and will produce a high margin of error. And, a sample from only a small town in Tennessee is not a random sample of the whole American population.

 

TAKE HOME EXERCISES (These are the “Questions for Review” to put in the blue book; there are no “questions for analysis” for this chapter).

 

Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions.

 

(1) I think that the Feds are watching me.

(2) Put down the porcupine and come out with your hands up.

(3) I have contacted the spirits of the dead.

 

Instructions: Identify the premises and conclusions in the following arguments, and identify any premise and conclusion indicators.

 

(4) In view of the fact that I almost never wash my hair, the odds are pretty slim that I’ll get a date.

(5) I’ll do whatever the guy in front of me wants given that he is holding a gun.

(6) There is an ever-increasing population of flesh-eating zombies; it follows that the apocalypse has begun.

 

Instructions: Diagram the following arguments. First number each statement, then use plus signs and arrows to connect the numbers in the appropriate argument structure.

 

(7) My life is over since I lost my cell phone, and my life depends upon that device.

(8) I won’t play follow the leader with you since you have no leadership skills, and, even if you did, you’d just lead me right off a cliff.

 

Instructions: Identify the informal fallacy in each of the following.

 

(9) Defenders of capital punishment are wrong since they think that even shoplifters should be executed.

(10) John has diabetes, which he probably got from drinking water since he drinks more water than anyone I’ve ever seen.

(11) The sign said “fine for littering”. Since it was fine, I littered.

 

Instructions: Put the following in standard propositional form by replacing the clue word with the correct logical connective (for example, “d however e” translates into “d and e”).

 

(12) x follows from y

(13) a although b

(14) r implies s

 

Instructions: Which of the following are not well formed propositions?

 

(15) (P or Q) not (R and S)

(16) if (P and Q)

(17) (P or Q) if and (R then S)

 

(18) What is the name of the following argument form or fallacy?

 

if A then B

not B

therefore, not A

 

(19) What is the name of the following argument form or fallacy?

 

X or Y

not X

therefore, Y

 

(20) Make up a modus tollens which leads to the following conclusion: “Therefore, the taxman is not my friend.”

 

(21) Is the following argument sound or unsound? If unsound, why?

 

If I am at Graceland then I can see Elvis memorabilia

It is not the case that I am at Graceland

Therefore, it is not the case that I can see Elvis memorabilia

 

Instructions: What is the inductive strength of the following (that is, very strong, strong, weak, very weak)?

 

(22) A sixteen-year-old girl was struck by lightning and the next day she won the lottery. Joe was struck by lightning today. Therefore, Joe will probably win the lottery tomorrow.

 

Instructions: For each of the following, indicate the inductive argument form that is followed, and whether it commits any inductive fallacy.

 

(23) 21% of Americans believe a UFO crashed at Roswell in 1947. Therefore, there is a 21% chance that Joe believes a UFO crashed at Roswell in 1947.

 

(24) 4% of 1,247 registered American voters believe that shape-shifting reptilian people control our world by taking on human form and gaining power. Therefore, there is a 4% chance that Joe, who is an American voter, believes that shape-shifting reptilian people control our world by taking on human form and gaining power.