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(/) <( e::z f- <( 17. Today I met someone in an Internet chat room. But there is no chance that we could have a suc­ cessful long-term relationship, for no people who meet in internet chat rooms can have successful long-term relationships. *18. Some Texans are tall. Billy Bob is a Texan. Therefore, Billy Bob is tall. 19. No sharks are friendly. Hammerheads are sharks. Therefore, no hammerheads are friendly. *20. Columbus was either Spanish or Italian. He was not Spanish. Therefore, he was Italian. 21. Lady Gaga has no problems. She is a famous singer, and no famous singer has problems. 22. All famous politicians are celebrities. Some governors are famous politicians; therefore, some gover­ nors are celebrities. *23. Some comedians are Canadians. Mike Meyers is a comedian. So Mike Meyers is probably a Canadian. 24. Lake Michigan most likely carries commercial shipping, since it's one of the Great Lakes, and the other Great Lakes carry commercial shipping. *25. No hip-hop artist is a fan of harmonica music. Since Zoltan is a fan of harmonica music, it follows that he is not a hip-hop artist. VIII. All the arguments below have missing premises and may be counted as either deductive or inductive, depending on what missing premises are put in. For each argument, provide the missing premise that would make it (a) deductive, or (b) inductive. Some flexibility in wording is allowed! 1. People waste a huge amount of time surfing the web. It follows that the web is not such a great invention. SAMPLE ANSWER: 1 a: No invention that allows people to waste a huge amount of time is great. 1 b: Many inventions that allow people to waste a huge amount of time are not great. 2. Ellen is a sophisticated artist, hence she listens to jazz. 3. Digsby was fired. After all, he had been spending all day surfing the web. 4. Latino purchasing power is approaching billions of dollars in the United States. Therefore, there will be better employment opportunities for talented Latinos. 5. Air Canada is an airline. Therefore, Air Canada charges a baggage fee to passengers who check bags. 6. The British red squirrel is a rodent. Consequently, the British red squirrel is an endangered species. 7. The galaxies are flying outward. This suggests that the Milky Way Galaxy will spin apart. 8. Mount Everest is a tall mountain. Therefore, Mount Everest is difficult to climb. 9. President Calvin Coolidge was a fiscal conservative. So he was not a gambler. 10. The NAFTA treaty regulates North American commercial relations. Therefore, the NAFTA treaty is unpopular with opponents of free trade. IX. YOUR OWN THINKING LAB 1. Consider the claim 'Ray has at least one sibling.' Write two arguments for it, one deductive (i.e., pro­ viding conclusive reasons) and the other inductive (i.e., providing nonconclusive reasons). 2. Write an argument with a missing premise, and then identify that premise. 3. Consider the claim 'There is life after death.' Write an argument for it and another one against it. Discuss whether these arguments are conclusive or nonconclusive. 4.5 Norm and Argument What Is a Normative Argument? We've seen that all arguments fall into either one of the other of two classes: they're either deductive or inductive. From a different perspective, both deductive and inductive arguments could be classified as being either normative or non-normative. The examples we've discussed in this book up to now have nearly all been made up entirely of statements that assert or deny some facts (or putative facts) about the world, such as 'Toronto is the largest city in Ontario,' 'Mercury is heavier than water,' and 'Jerry Seinfeld is a comedian.' Statements of this sort fall under the category of non-normative. But some other expressions go beyond facts to assess in­ dividuals, actions, and things, or to say what an individual ought to do (or ought not to do) or how things should be (or not be). For example, 'You ought to keep your promises,' 'Reggae music is cool,' 'Hitler was evil,' and 'Elena deserves credit for her hard work.' Expressions of this latter type are used to make normative judgments, which figure in a sort of reasoning that we'll call normative reasoning. When we make a normative judgment and offer reasons intended to support it, the result is a normative argument. These are arguments for the conclusion that something has a certain value, such as being good or bad, right or wrong, just or unjust, beautiful or ugly, and the like. Also, arguments for the conclusion that something is permissible (may be done), obligatory (ought to be done), or forbidden (should not �e done) may be classified as being normative arguments. Consider 14 1. One ought to obey one's parents. 2. My parents told me not to go to the party on Friday night. 3. I ought not to go the party on Friday night. The conclusion of (14) is a normative judgment, since it represents a certain action (going to the party on Friday night) as being forbidden. By doing so, it directs or guides the arguer's be­ havior in a certain way-namely, away from the Friday-night party. That the conclusion is a normative judgment here is sufficient to make argument (14) normative. In addition, (14)'s premise 1 belongs to the category of general normative judgments, sometimes also called 'principles,' because they state rules that are supposed to apply not just to one person, but to anyone. We may distinguish between normative judgments that express a generalization or rule and those-like (14)'s conclusion-that are particular sentences used to make claims about individual persons, things, events, and so on. The distinctions we have in mind here are summarized in Box 6. 1-­ z w :'? ::J (9 a: <( 0 z <( :'? a: 0 z "'. '3" ml 1- z w � :::, (.') a: <( a: Cf) 0 in LL >–

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We'll classify them accordingly as aesthetic, legal, prudential, or moral judgments. Whenever any such normative judgment is the conclusion of an argument, we'll say that the argument itself is aesthetic, legal, prudential, or moral, as the case may be. A normative judgment is aesthetic just in case it expresses an evaluation or norm involving a matter of taste such as that some piece of art is beautiful or ugly, a dish is tasty or inedible, or that we ought to admire good music. Aesthetic judgments could be either particu­ lar ('Beyonce's recordings are superior art,' 'Frank Lloyd Wright's designs are overrated,' 'The Parliament buildings in Ottawa are a majestic sight') or general ('White socks don't go well with black shoes,' 'You ought to watch Law and Order'). The conclusion of a legal argument features a normative judgment involving a legal mat­ ter: something that's said to be a duty or obligation according to the law, or to be permitted to do or forbidden to do by statute-for example, that drivers ought not to tear up a parking ticket or are permitted to turn right on red (except in New York City!) and that adults have a duty to file an income tax return. Legal normative judgments could have a conditional form, as in 'If a person is called for jury service, that person must show up,' and 'When a person is sworn as a witness in court, that person is obligated to tell the truth.' The conclusion of a prudential argument makes a claim about what it would be in your own self-interest to do, such as 'You ought to be especially nice to your rich Aunt Gertrude,' 'It's not in your interest to antagonize your boss,' 'People should look out for themselves first!' and 'Don't cheat your business associates if you don't want them to cheat you.' The conclusion of a moral argument is a moral judgment. Judgments of this sort make a claim about what is good or bad, just or unjust, and what ought (or ought not) to be done, not because it's sanctioned by the law, but because, as the case may be, it deserves praise or blame- for example, 'Lying is wrong,' 'You ought to help the earthquake survivors,' 'Matthew's behavior was dishonest,' and 'The firefighters showed great courage on 9/11.' The upshot, then, is that when normative judgments of any of these four types occur in the conclusion of an argument, the argument is itself normative. And it's by paying attention to the type of normative judgment in the conclusion that we tell which type of normative argument it is: aesthetic, legal, prudential, or moral. Missing Normative Premises Earlier in this chapter, we saw that when arguments are presented in everyday language, they sometimes have missing premises that need to be restored if the argument is to be reconstructed in a way that respects the principles of faithfulness and charity. One especially common way in which important premises may be left out is a pattern that sometimes occurs in normative arguments. In fact, such arguments often have normative judgments, not only in their conclusions, but also in at least one premise, and it's that premise that is sometimes left out. What we shall call normative general premises, such as 'Keeping promises is right,' 'Slavery is unjust,' or 'One ought to obey the law,' are judgments that may seem to the arguer too obvi­ ous to need repeating, and so they may get left out. Here are some examples of normative ar­ guments in which the normative general premises are in place. As you read them, try imagin­ ing what they'd sound like with that crucial premise left out. 19 Legal argument: 1. Driving faster than 55 miles per hour on the Taconic Parkway is forbidden by law. � NORMATIVE GENERAL PREMISE 2. Yesterday I drove faster than 55 miles per hour on the Taconic Parkway. 3. Yesterday I did what I ought not to do, according to the law. 20 Aesthetic argument: 1. Music that consists of only a random collection of honks, bleats, and screeches is worthless noise. � NORMATIVE GENERAL PREMISE 2. Professor Murgatroyd's 'Second Symphony' consists of only a random collection of honks, bleats, and screeches. 3. Professor Murgatroyd's 'Second Symphony' is worthless noise. r­ z w 2 ::J (9 a: < 0 z < 2 a: 0 z "'. s;- DIii 1- z w 2 ::) CJ a: –

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CHAPTER

Evaluating Deductive
Arguments

In this chapter, you’ll look more closely at deductive reasoning, focusing first on the
concept of validity and then on related topics, including

The difference between valid and invalid arguments.

Some alternative ways of talking about validity.

The relation between validity and argument form.

How to represent propositional and categorical argument forms.

Soundness as an evaluative standard.

Deductive cogency as an evaluative standard.

The practical implications (or ‘cash value’) of validity, soundness, and cogency.

94

5.1 Validity

Sometimes people use ‘valid’ to mean ‘true’ or ‘reasonable’ and ‘invalid’ to mean ‘false’ or
‘unreasonable.’ But these are not what ‘valid’ and ‘invalid’ mean in logical thinking. A deductive
argument is valid if and only if its premises necessitate or entail its conclusion, where ‘entail­
ment’ is defined as in Box 1.

As we’ve seen, a deductive argument is one in which the conclusion is supposed to follow neces­
sarily from the premises-so that if the premises were all true, the conclusion would be, too.
Since a valid argument’s premises, if true, determine that the conclusion is true, valid arguments
can also be said to be truth-preserving. Any argument that fails to be truth-preserving would be
one whose premises could be true and its conclusion false at once. Such an argument is, by defi­
nition, invalid: its premises do not entail its conclusion. Note that we’re introducing here some
different expressions that all mean the same thing. To say that an argument is valid is equivalent
to saying that its premises entail its conclusion. And both of these are equivalent to saying that
the argument is truth-preserving, and that its conclusion follows necessarily from its premise or
premises. The upshot of all this is:

In a valid argument, it makes no logical sense to accept the premises and reject
the conclusion.

Once you accept a valid argument’s premises, were you to reject its conclusion (i.e., think that it
is false), that would be contradictory or nonsensical. Contradictory statements cannot have the
same truth value: if one is true, the other must be false. Consider this valid argument:

1 If the Ohio River is in North America, then it is not in Europe. The Ohio River is in
North America; therefore it is not in Europe.

You cannot accept both that if the Ohio River is in North America, then it is not in Europe and also
that it is in North America and at the same time reject that the Ohio River is not in Europe. That
would be contradictory, thus making no logical sense.

Validity is one of the standards used to evaluate deductive arguments. Whether an argu­
ment is valid or not is never a matter of degree, but instead one of all or nothing. An argument
cannot be ‘sort of valid.’ It’s either valid or it’s not. Furthermore, there is a simple test to deter­
mine the validity of an argument. As you read it, ask yourself, ‘Could the conclusion be false
with all the premises true at once?’ If so, the argument flunks the test: it’s invalid. But if not,

then you may accept it as valid. Let’s consider some examples. Suppose we ventured to predict
what next summer in Baltimore will be like. We might say,

BOX 1 ■ ENTAILMENT

There is entailment in an argument if and only if the truth of the argument’s premises guarantees
the truth of its conclusion-in the sense that, if the premises are all true, the conclusion cannot be
false. Such an argument is valid and truth-preserving.

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Since beavers are nocturnal, we may infer that badgers, weasels, and wolverines are, too, for all of these animals are small, fur-bearing mammals found in the upper Midwest. 9. Seven-year-old Jason has contracted chicken pox. This occurred only a week after his three younger sisters, Gwendolyn, Samantha, and Hermione, were stricken with chicken pox. Consequently, Jason caught the chicken pox from his sisters. *10. Bart Simpson cannot run for governor of California because Bart Simpson is a cartoon character, and no cartoon characters are citizens of California. Only citizens of California are eligible to run for governor of California. 11. For as long as records have been kept, every winter there has been some rain in Vancouver. Therefore, next winter there will be some rain in Vancouver. 12. Since Venus Williams and Serena Williams are star tennis players, and Venus and Serena are sisters , we may infer that at least two members of the Williams family are athletes; for all tennis stars are athletes. *13. Since this is a freshman-level course, it is an easy course, for all freshman-level courses are easy. 14. It is unlikely that Joe will be a senator. Most senators are people who win public debates, and so far Joe has lost every one. 15. If my computer keeps crashing, then it must have picked up a virus somehow. Therefore, it must have a virus, because it keeps crashing! *16. The Washington Redskins is a football team that has thousands of enthusiastic fans. The same is true of the Denver Broncos, the New York Jets, the Minnesota Vikings, and the Dallas Cowboys. It follows that all American professional football teams have thousands of fans. 17. Since no health-conscious people are sedentary couch potatoes, no marathon runners are seden­ tary couch potatoes, for all marathon runners are health-conscious people. 18. The value of stocks is now falling every day. Whenever this happens, stocks are not a good invest­ ment. Thus stocks are not a good investment now. *19. For us, the options tonight are either to watch a movie at home or go out for dinner. We won't watch a movie at home. Thus we' ll go out for dinner. 20. Sally is always happy, because she is a singer, and many singers are always happy. 21. Nat is not a spy. All spies have espionage training, and he has never had such training. *22. JJ's won't get the support of the Chamber of Commerce, for the Chamber of Commerce usually supports only local firms, and JJ's is from out of state. 23. The Ethiopian city of Addis Ababa is a center of African culture. All cities that are centers of African culture are large cities. Hence, Addis Ababa is a large city. 24. Either Syria will stop supporting Lebanon or it wants a war with Israel. But clearly Syria does not want a war with Israel, so Syria will stop supporting Lebanon. *25. Many undergraduates in the United States receive some form of financial aid. Since Jane is a college undergraduate, she has financial aid. 26. An inspector at a Sony computer factory found that, out of the many computers she inspected, none had defects. She concluded: 'At this factory, no computer is defective.' 27. Since it's a Friday, Atkins will not be home until late tonight. Most Fridays, Atkins makes a stop at Miller's Bar and Grill on the way home for a beer or two and never leaves Miller's before 11 :00 p.m. *28. Simon Peterson is a cardinal. Since no cardinals are Protestants, Peterson is not a Protestant. 29. Most people who have a tooth extracted without an anesthetic are in pain. I' ll have one extracted without an anesthetic later today. Therefore, I' ll be in pain. 30. Mr. Abernathy must be at least sixty-five years old, since no one can be receiving Social Security payments unless he is sixty-five years old or older, and Mr. Abernathy gets a Social Security check in the mail every month. Ill. For each of the above arguments that are valid, construct an argument of your own that follows the same pattern. In this exercise, premises need not be true. IV. Determine whether the following types of argument are logically possible or impossible. For each that's logically possible, give an example. 1. A valid argument whose premises are true and conclusion false. SAMPLE ANSWER: Logically impossible. 2. An invalid argument whose premises are true and conclusion false. *3. An invalid argument whose premises are true and conclusion true. 4. An invalid argument form that cannot have true premises and a false conclusion. *5. A valid argument whose premises are false and conclusion false. 6. A valid argument whose premises are false and conclusion true. 7. 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<� :::) :::) _J 0 ;; a: w< (13) is an instance of a hypothetical syllogism, for it has the form 13' 1. If I, then C 2. If C, then A 3. If I, then A And, as you can prove for yourself, (14) and (15) below illustrate the two versions of disjunctive syllogism in Box 4, while (16) illustrates contraposition: 14 1. Either American Dennis Tito or South African Mark Shuttleworth was the first space tourist. 2. South African Mark Shuttleworth was not the first space tourist. 3. American Dennis Tito was the first space tourist. 15 1. Either American Dennis Tito or South African Mark Shuttleworth was the first space tourist. 2. American Dennis Tito was not the first space tourist. 3. South African Mark Shuttleworth was the first space tourist. 16 1. If Persia was a mighty kingdom, then Lydia was a mighty kingdom. 2. If Lydia was not a mighty kingdom, then Persia was not a mighty kingdom. All these arguments are substitution instances (or simply, instances) of one or another of the argument forms in Box 4, which are all valid. This means that in any argument that is an instance of one of these forms, there is entailment, no matter what actual statements the symbols stand for. That is, no actual arguments of the forms listed in Box 4 above could have true premises and a false conclusion. There are many such forms, but again, we'll examine this topic at greater length in Chapter 12. SUGGESTION: In this section, there are a number of valid argument forms. For quick reference and to gain familiarity, construct a card with these forms. Write down on one side those where validity hinges on relations among propositions, and on the other side those where validity hinges on relations among terms. Categorical Argument Forms Many arguments are clearly valid, even though they don't fit into any form of propositional logic. Consider 17 1. All dentists have clean teeth. 2. Dr. Chang is a dentist. 3. Dr. Chang has clean teeth. (17) is plainly valid, for if its premises are true, then its conclusion must be true. Now suppose we replace its parts by letter symbols, treating the argument as if it were an instance of an ar­ gument form in propositional logic. We would then get this form: 17' 1. D 2. C 3. E But (17') is an invalid form, since there are counterexamples to it: that is, arguments of the same form with true premises and a false conclusion. Here is one, 18 1. Whales are mammals. 2. California is the most populous state in the United States. 3. The Earth is flat So to take (17), a valid argument, to have an invalid argument form such as (17') would be in­ correct. What's needed is a different system, one where letter symbols do not stand for propo­ sitions. In other words, (17') is too coarse-grained to serve as the correct argument form of (17), where the entailment hinges on relations among certain expressions within the propositions that make up that argument, rather than on relations among the propositions themselves that constitute premises and conclusion. In (17) the entailment depends on relations among terms such as 'all,' 'Dr. Chang,' 'dentist' and 'clean teeth.' A more fine-grained representation is needed for arguments such as (17). We shall repre­ sent their forms by adopting the following conventions: 1. Use 'to be' in present tense as the main verb in each premise and conclusion. 2. Make explicit any logical expressions, such as 'all,' 'some,' and 'no.' 3. Replace expressions such as 'dentist' and 'clean teeth' with capital letters. 4. Replace expressions for specific things or individuals, such as 'Dr. Chang,' 'Fido,' 'I,' and 'that chair' with lowercase letters. In this language, the logical form of (17) is similar to that of 19 1. All soda companies are businesses that prosper. 2. Pepsi is a soda company. 3. Pepsi is a business that prospers. (19)'s argument form is 19' 1. All A are B. 2. c is an A. 3. cis a B In (19'), 'A' stand for the term 'soda companies,' 'B' for 'businesses that prosper,' and 'c' for 'Pepsi.' We can also represent in this language arguments such as 20 1. All ophthalmologists are doctors. 2. Some ophthalmologists are short. 3. Some doctors are short. >­
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<( � 🙂 🙂 _J CJ :; a: w <( - (20) is a plainly valid argument: it is a substitution instance of a valid categorical argument form. Another instance of the same form is 21 1. All red squirrels are rodents. 2. Some red squirrels are wild animals. 3. Some rodents are wild animals. The argument form of both (20) and (21) is, 20' 1. All A are B 2. Some A are C 3. Some B are C Here 'A' stand for 'red squirrels' (or 'ophthalmologists'), 'B' for 'rodents' (or 'doctors'), and 'C' for 'wild animals' ( or 'short'). Let's now recall a point made at the beginning of this section: that another way to under­ stand validity is to say that whether an argument is valid or not is simply a matter of whether it has a valid form. Consider 22 1. No Peloponnesians are Euboeans. 2. All Spartans are Peloponnesians. 3. No Spartans are Euboeans. Even someone who knew nothing at all about Greek geography could nevertheless see that the argument is valid, because it is an instance of the valid form number 3 in Box 5. No argument with this form could have true premises and a false conclusion. Similarly, the following argu­ ment is valid even though its premises are false. Why? Simply because it has valid form num­ ber 3 in Box 5. BOX 5 ■ SOME VALID CATEGORICAL ARGUMENT FORMS 1 2 All A are B Some A are B No Bare C All A are C No Care A Some Care B 3 4 NoAareB All A are B All Care A All Care A No Care B All Care B 5 6 All A are B All A are B All Bare C Some A are not C All A are C Some B are not C 23 1. All apples are oranges. 2. All bananas are apples. 3. All bananas are oranges. Validity, then, is entirely a matter of argument form. The same could be said for the other ex­ amples above. This brings us to another important point: for each form that is valid, all of the arguments that have it will be valid. Similarly, for each invalid form, all of the arguments that have it will be invalid. Propositional or Categorical? ■ When you see certain connections between propositions, such as 'Either . . . or . . . ' and 'If ... then . . . ,' the argument is probably better reconstructed as propositional. ■ On the other hand, when you see in the premises certain words indicating quantity, such as 'All,' 'No,' and 'Some,' the argument is probably better reconstructed as categorical. The Cash Value of Validity Logical thinking has goals, such as learning, understanding, and solving problems. Each of these requires argument analysis and sometimes refutation, the process by which a given argument is shown to fail. But, far from being among logical thinking's primary goals, refuta­ tion is a result of argument analysis unavoidable in some cases. Achieving logical thinking's primary goals greatly depends on charitable and faithful reconstruction of arguments. For those that are deductive, charity recommends making them as strong as possible, maximizing the truth of their premises and conclusion and the validity of their forms-while faithfulness recommends trying to capture the arguer's intentions. In all of this, logical thinkers strive to capture the form of an argument correctly, adding missing premises when needed. Once they have properly reconstructed an argument, they then move on to evaluate it, keeping in mind rules such as ■ Do not criticize/accept an argument by focusing solely on its conclusion. ■ Direct each objection to the argument form, or to a clearly identifiable premise. ■ Use the evaluative criteria offered here. ■ Do not make unsubstantial criticisms, such as 'that is a matter of opinion.' Any challenge to validity is a challenge to the argument form. If the premises of an argument with a certain form could be true and its conclusion false, then the argument is invalid because it has an invalid form. Yet finding an argument invalid is not a conclusive reason to reject it, since it could still be a good inductive argument (more on this in Chapter 6). Once an argu­ ment is found valid, logical thinkers should then check whether its premises are true, a topic we'll take up later in this chapter. >­
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<{ � :::i :::i _J CJ ;; a: w <{ *2. Alaska is a large state in the United States. No large state in the United States is densely populated. Thus, Alaska is not densely populated. (a, L, D) 3. All rattlesnakes are snakes. No snakes are friendly pets. Therefore, no rattlesnakes are friendly pets. (R, S, P) 4. Some mathematicians are logicians. No logicians are space travelers. Thus some mathematicians are not space travelers. (M, L, 1) *5. All Afghans are peace-loving people. Some peace-loving people are French. Therefore, some Afghans are French. (A, P, F) 6. All residents of the Vatican are religious believers. No religious believer is an atheist. Therefore, no resident of the Vatican is an atheist. 0/, B, A) 7. Kate is not a tourist. All Niagara Falls visitors are tourists. It follows that Kate is not a Niagara Falls visitor. (k, T, \/'i *8. No Marine Corps drill instructors are sympathetic friends. So Sergeant Osberg is not a sympathetic friend, since he is a Marine Corps drill instructor. (I, F, o) IX. For each of the following arguments, determine whether it is propositional or categorical. 1. All living creatures need liquid water. My cat is a living creature. Thus my cat needs liquid water. SAMPLE ANSWER: Categorical argument 2. There is no extraterrestrial intelligence. After all, if there were extraterrestrial intelligence, we should have evidence of it by now. But we don't have it. *3. No desert is humid. The Atacama is a desert. Therefore, the Atacama is not humid. 4. Doctors are exposed to agents that cause ailments. Jane is a doctor. Hence, she is exposed to agents that cause ailments. *5. If the Orinoco crocodile is a rodent, then the Chinese alligator is a rodent. But the Chinese alligator is not a rodent. Therefore, the Orinoco crocodile is not a rodent. 6. Euripides enjoyed tragedy. After all, all fifth-century Greeks enjoyed tragedy, and Euripides was a fifth-century Greek. *7. All hibernating mammals slow their breathing in the winter. Since all black bears are hibernating mammals, therefore all black bears slow their breathing in the winter. 8. If one has poor health, one goes to the doctor. If one goes to the doctor, one spends money. Thus, if one has poor health, one spends money. *9. Chris will take summer courses this year, because either he takes them or he'll wait until next fall for graduation, and he won't wait that long. 1 O. All diesel engines produce exhaust gases. All school buses have diesel engines. Thus all school buses produce exhaust gases. *11. The universe can act as a magnifying lens, since if relativity theory is correct, the universe can act as a magnifying lens. And relativity theory is correct. 12. All people who drink a glass of warm milk before bedtime are sound sleepers. Given that Beth always drinks a glass of warm milk before bedtime, she is a sound sleeper. *13. All chameleons are lizards that change their color. All lizards that change their color are scary crea­ tures. So all chameleons are scary creatures. 14. If crocodiles wallow in mud holes, then they are rarely killed by predators. Crocodiles do wallow in mud holes. Thus they are rarely killed by predators. *15. No Ohio farmer grows papayas, for no northern farmer grows papayas, and Ohio farmers are north­ ern farmers. X. The arguments below are either propositional or categorical. Indicate which is which and give the argument form. 1. If the defendant's car was used in the robbery, then the car was at the scene of the crime. But it was not at the scene of the crime. Thus the defendant's car was not used in the robbery. (D, C) SAMPLE ANSWER: Propositional If D, then C NQ1_Q Not D 2. If these snakes are cobras, then they're poisonous. Therefore, if these snakes are not poisonous, then they are not cobras. (C, P) 3. If offenses against the innocent are punished, then we have a fair system of justice. If we have a fair system of justice, then the guilty are treated as they deseNe. So if offenses against the innocent are punished, then the guilty are treated as they deseNe. (0, J, G) *4. Since all computers are mechanical devices, no computers are things that can think, for no things that can think are mechanical devices. (C, D, T) 5. Archie doesn't eat chicken, for Archie is a vegan, and if he is a vegan, then he doesn't eat chicken. (A, C) *6. If Mississippi does allow gay marriage, then its laws governing marriage are liberal. In fact, its laws governing marriage are not liberal. So, Mississippi does not allow gay marriage. (M, L) 7. Either doctors favor the new health program or the uninsured suffer. But doctors do not favor the new health program. Hence the uninsured suffer. (N, U) 8. All accountants are good at math. Greg is not an accountant. Therefore, he is not good at math. (A, M, g) *9. If a flower is an orchid, then it is a tropical flower. Therefore, if it is not a tropical flower, then it is not an orchid. (0, F) 10. Since no tropical country has blizzards and Venezuela is a tropical country, Venezuela doesn't have blizzards. (C, B, v) >­
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<( � 🙂 🙂 _J ('.) :; a: w <( 11111B *11. All babies are infants. Some babies are good at crawling. Therefore, some infants are good at crawl­ ing. (B, I, C) 12. All carrots are vegetables full of vitamin A All vegetables full of vitamin A are foods good for your eye­ sight. So carrots are foods good for your eyesight. (C, V, F) *13. No schoolchildren are college graduates. All college graduates have a college diploma. Therefore, no schoolchildren have a college diploma. (S, G, D) 14. If penguins are birds, then they are likely to have feathers. Since it is the case that penguins are birds, we must conclude that they are likely to have feathers. (B, F) *15. No planet is a star. Hence, Venus is not a star, since Venus is a planet. (P, S, v) 16. Anne is Mario's wife. Thus Mario is not a bachelor. For if Anne is his wife, then he is not a bachelor. (A, B) 17. No professional gambler is good at saving money. Since Nathan is a professional gambler, we may infer that he is not good at saving money. (G, M, n) *18. If oxygen is the lightest element, then oxygen is lighter than hydrogen. But oxygen is not lighter than hydrogen. Therefore, oxygen is not the lightest element. (0, H) 19. If Winston Churchill was English, then he was not Brazilian. But if he was not Brazilian, then he was not South American. Thus if Winston Churchill was English, then he was not South American. (E, B, S) *20. Melissa will either pledge Gamma Phi or she will not join a sorority at all this year. Accordingly, she will not join a sorority at all, since she will not pledge Gamma Phi. (M, J) XI. For each of the above arguments that is propositional, give the name of its form (answers to 3, 6, 9, 18, and 20 in the back of the book). SAMPLE ANSWER: 1. Modus to/lens XII. Indicate whether the following statements are true or false. 1. A valid argument cannot have a false conclusion. SAMPLE ANSWER: False 2. A valid argument cannot have a false premise. *3. A valid argument cannot have true premises and a false conclusion. 4. Invalid arguments always have true premises and false conclusions. *5. A valid argument could have a counterexample. 6. All valid argument forms are truth-preserving. *7. An invalid argument could never have a true conclusion. 8. An invalid argument could never have true premises. *9. If there is entailment in an argument, then that argument is truth-preserving. 10. An invalid argument could have no counterexample. XIII.The following categorical arguments are invalid. After symbolizing their forms accordingly, show invalidity in each case with a counterexample. (Tip: Use the same counterexample for arguments exemplifying the same invalid form. When the given argument plainly has true premises and a false conclusion, you can simply point that out in lieu of counterexample.) 1. All female college students are students. Some students are smokers. Therefore, some female col­ lege students are smokers. SAMPLE ANSWER: All Fare D Some o are M Some Fare M. Counterexample: F, D, and M stand for 'fish,' 'animal,' and 'mammal.' [All fish are animals. Some animals are mammals. Thus some fish are mammals.] 2. All giraffes are mute. That animal is mute. Thus that animal is a giraffe. *3. Most American citizens are permitted to vote in the United States. Michael is not permitted to vote in the United States. So, Michael is not an American citizen. 4. Roses are flowers. Some flowers are daffodils. Thus roses are daffodils. *5. No SUVs are easy to park. Some SUVs are speedy vehicles. Hence, no speedy vehicles are easy to park. 6. Some days are rainy days. Some days are sunny days. Therefore, some rainy days are sunny days. *7. Fido is a dog. Some dogs bark. Therefore, Fido barks. 8. Most Mexicans speak Spanish. Some non-Mexicans speak Spanish. Therefore, some non-Mexicans are Mexicans. 9. All intellectuals support stem-cell research. Barbra Streisand supports stem-cell research. Therefore, Barbra Streisand is an intellectual. 10. No desktop computer is light. My computer is not light. Hence, my computer is not a desktop. XIV. YOUR OWN THINKING LAB 1. Explain in your own words the relation between 'invalidity' and 'counterexample.' 2. Explain in your own words the claim that validity is a matter of argument form. 3. Give two arguments of your own for each of the following valid argument forms: modus ponens, modus to/lens, hypothetical syllogism, disjunctive syllogism, and contraposition. 4. Give a counterexample to the following argument: Horses are domestic animals. Dobbin is a domes­ tic animal. Therefore, Dobbin is a horse. 5. For each of the following argument forms, construct an argument with true premises on a topic of your choice that illustrates that form: 1. All A are B AIIB areC AIIA areC >–
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<� :J :J _J 0 ;; a: w< 11111 2. All A are B Some A are not C Some B are not C 3. No As are Bs All Care A No Care B 4. All A are B Some A are C Some B are C 5. AIIA areB c is not a B c is not an A 6. AIIA areB c is an A c is a B 6. For each of the following argument forms, construct an argument with true premises on a topic of your choice that illustrates that form: modus ponens, modus to/lens, contraposition, hypothetical syllogism, and disjunctive syllogism. 5.2 Soundness Must we then always accept the conclusions of valid arguments? No, for there may still be something wrong with them (as is clear in some of the examples above). To evaluate an argu­ ment, validity is the first criterion we use, but not the only one. After we have decided that an argument is valid, we must also determine whether it is sound, bearing in mind that An argument is sound if and only if it is valid and all of its premises are true. Thus consider some arguments given earlier: 22 1. No Peloponnesians are Euboeans. 2. All Spartans are Peloponnesians. 3. No Spartans are Euboeans. 4 1. All dogs are fish. 2. All fish are mammals. 3. All dogs are mammals. 5 1. All Democrats are vegetarians. 2. All vegetarians are Republicans. 3. All Democrats are Republicans. BOX 7 ■ SOUND ARGUMENT 1. An argument is sound if and only if it is valid and all of its premises are true. 2. An argument is unsound if it lacks either validity or true premises, or both. 3. Unsoundness is a reason to reject an argument even if it's valid. 4. The conclusion of a sound argument is true. 5. Given (4), a sound argument's conclusion cannot be rejected without saying something false. Argument (22) is sound. But (4) and (5) are unsound. This is because if an argument lacks either validity or true premises (or both), then it is unsound. The problem with (4) and (s) is that their premises are false, thus rendering the arguments unsound, even though, as we have seen, both are valid. Important things to remember are, first, that if even one of an argument's premises is false, then the argument is unsound, whether it's valid or not. Second, a premise counts as true only if there is no controversy about whether it's true. Third, since validity is a necessary condition of soundness, an argument can also be unsound because its form is invalid. For example, 26 1. Any city that is the capital of a country is a center of political power. 2. Chicago is a center of political power. 3. Chicago is the capital of a country. Here both premises are true, yet the argument is unsound because it is invalid. Validity and true premises, then, are the necessary conditions for soundness. The ques­ tion of whether an argument's premises are in fact true or false is another matter entirely (which cannot be answered by logic alone). Most such answers belong rather to the sciences, or to the investigations of historians, geographers, and other fact finders. To be sure, a good logical thinker will want to get her facts straight! But this is where she must head for the library or the laboratory and consider the evidence for the premises of arguments that purport to be sound. BOX 8 ■ SOUNDNESS DEDUCTIVE ARGUMENT SoUND UNSOUND Valid with all true premises Invalid At least one false premises Invalid and at least one false premises (/) (/) w z 0 z ::J 0 (/) N l() w >

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<:� ::J ::J _J CD ;; er: w <: The Cash Value of Soundness Why, then, is soundness so important? Why is soundness a desirable characteristic in argu­ ments? Because if one is aware of an argument's soundness, then not only is one fully justified in accepting its conclusion, one has to accept it! As defined above, all valid arguments are truth-preserving-if their premises are true, their conclusions must be true. If an argument is valid and also actual!}, has true premises, it is sound; and that means that the truth of the prem­ ises transfers to the conclusion. Thus there is no denying the conclusion of any such argument without saying somethingfalse. Soundness has a practical impact or worth because whenever a given deductive argument meets this standard, its conclusion is guaranteed to be true. In fact, the cash value of sound­ ness is twofold: When an argument is sound, then ■ Given its validity, you can't assert its premises and deny its conclusion without saying something contradictory. ■ Given its validity and its true premises, you can't deny its conclusion without saying something false. What about unsoundness? What is its cash value for logical thinkers? An unsound argument fails to guarantee the truth of its conclusion. If an argument is valid but unsound, that means that it has at least one false premise. The realization of this is sufficient reason to reject that argument. If an argument is unsound but has all true premises, that means that its argument form is invalid: as we'll see in Chapter 6, some such arguments are better considered inductive and evaluated according to standards other than soundness. 5.3 Cogency Validity and soundness are not the only standards used to evaluate deductive arguments. There is also deductive cogency or persuasiveness, a standard that is met when a proposed argument has BOX 9 ■ THREE CONDITIONS FOR COGENCY 1. Recognizable validity. 2. Acceptable premises. 3. Premises that are more clearly acceptable than the conclusion. Given 1, the validity of a cogent argument should be clear to the logical thinker evaluating the argument. Given 2 and 3, the premises of a cogent argument should provide the logical thinker with good reasons to accept its conclusion. Note that this falls short of requiring that the cogent argument be sound: that is, an unsound argument could be cogent, provided that the thinker recognizes its validity and takes the premises to provide good reasons for its con­ clusion, even when, unknown to the thinker, at least one of its premises happens to be false. Consider a thinker who seems to have seen Ingrid at the library and reasons as follows: 27 1. Ingrid is at the library. 2. If Ingrid is at the library, then she is not at the cafeteria. 3. Ingrid is not at the cafeteria. Since he just saw Ingrid at the library, he seems to have evidence that premise 1 is true. 2 is also true, since nobody could be in two different places at the same time. From 1 and 2 together, conclusion 3 follows validly by modus ponens. So the argument is deductively cogent: it is recognizably valid and has acceptable premises that support the conclusion. But suppose that in fact, unknown to the thinker, it wasn't Ingrid at all he saw in the library but her identical twin, Greta. In this situation, the argument is unsound-but still perfectly cogent! On the other hand, there can sometimes be sound arguments that are non-cogent. For having recognizable validity and true premises might not be enough for an argument to be persuasive. Consider: 28 1. The Earth is not flat and is not the center of the universe. 2. The Earth is not the center of the universe (28) is plainly valid, since if its premise is true, its conclusion cannot be false. Moreover, since its premise is in fact true, the argument is also sound. But anyone who reasonably doubted (28)'s conclusion would not be persuaded to accept it. (28) fails to meet condition 3 above: namely, hav­ ing a premise that is more clearly acceptable than the conclusion. Imagine this argument being offered in the Middle Ages, when all available evidence pointed to the falsity of its conclusion. Even though people at that time would have rejected premise 1, unknown to them that premise was true-and the argument sound. Thus even sound arguments can fail to be cogent when their premises fail to be more acceptable than the conclusion they are offered to support. BOX 10 ■ SECTION SUMMARY A deductive argument meeting the three conditions of cogency in Box 9 is one whose premises give the logical thinker good reasons to accept its conclusion. The Cash Value of Cogency Anyone who recognizes the validity of an argument and finds its premises to provide good reasons for its conclusion cannot rationally reject the argument. Such an argument may be said to be 'rationally compelling' (or simply 'compelling'). If the thinker were to reject the argument, that would be irrational: it would make no logical sense. Since argument (28) cannot persuade thinkers to accept its conclusion on the basis of its premises, the argument is not co­ gent (i.e., not rationally compelling). Logical thinkers should be on guard for such arguments and strive to avoid them altogether. In Chapter 8 we'll discuss a pattern of mistake in arguing that affects the cogency of some valid, and even sound, arguments. >–
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<( � 🙂 🙂 __J CJ ;;: a: l1J <( Exercises 1 . What does it mean to say that a deductively cogent argument is rationally compelling? 2. Explain the difference between soundness and cogency. 3. Given a certain standard for deductive arguments, it is contradictory to assert the premises of an argument that satisfies it and yet at the same time deny the conclusion. What standard is that? 4. What's the effect of denying the conclusion of sound argument? 5. What's the effect of denying the conclusion of an argument that one recognizes as cogent? 6. Must the conclusion of an invalid argument always be rejected? Explain your answer. XVI. Which of the following statements are true, and which are false? 1 . An unsound argument could have a valid form SAMPLE ANSWER: True •2. A sound argument could have all false premises. 3. A sound argument could have one false premise. *4. A sound argument could be invalid. 5. An unsound argument could have a true conclusion. *6. A sound argument could have a false conclusion. 7. A sound argument could have true premises and true conclusion. *8. A sound argument could have a true conclusion. XVII. What's the matter with the following arguments? Explain. 1. An argument whose premises entail its conclusion is valid. Hence, one should accept the conclusion of any valid argument. SAMPLE ANSWER: Such an argument could be non-cogent, or have false premises and thus be unsound. •2. Only sound arguments guarantee the truth of their conclusions. Thus entailment and therefore valid­ ity are of no importance. 3. Logic books make too much fuss about soundness. After all, unsound arguments may also have true conclusions. *4. Validity doesn't matter in science, for science values truth, and there is no relation between validity and truth. XVIII. Indicate whether the following scenarios are logically possible or impossible: 1. An unsound argument where there is entailment SAMPLE ANSWER: Logically possible (a valid argument with at least one false premise) 2. An unsound argument with a false conclusion 3. A valid argument where there is no entailment *4. A sound argument that is not truth-preserving 5. An argument that is an instance of a valid form *6. An invalid argument with true premises and a false conclusion 7. A sound argument with false premises and a true conclusion *8. An unsound argument with false premises and a false conclusion 9. A sound argument where there is no entailment *10. A cogent argument that is not rationally persuasive 11 . A cogent argument that is invalid *12. A cogent argument that is unsound XIX. YOUR OWN THINKING LAB 1. For any possible arguments in the previous exercise, provide an example of your own. 2. Give two examples of your own to illustrate the following: modus ponens, contraposition, and dis­ junctive syllogism. 3. Explain why your examples above are valid. 4. Explain each of the following claims: A. Denying the conclusion of a cogent argument is irrational. B. Asserting the premises while denying the conclusion of a valid argument is contradictory. C. Some valid arguments might not be cogent. D. Some unsound arguments might not be cogent. E. Some unsound arguments might be cogent. F. A deductively valid argument might yet be clearly unsound. 5. Illustrate each of the statements above with an example, supplying the context when needed. ■ Writing Project Find two arguments on the issue of immigration policy, one supporting tighter restrictions on undocumented aliens, the other opposing them. The sources, which should be identified in your work, may be biogs or articles from a website, a newspaper, or a magazine. First, recon­ struct the arguments as deductive, and then discuss whether they are sound. The simpler the argument, the easier it is to determine whether it is sound. A length of 700 words should be adequate. f-- 0 w ---, 0 a: 0... (9 z t:: a: w >

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w <( IEI ■ Chapter Summary Propositional argument: the relation of inference hinges on relations among the propo­ sitions expressed by its premises and conclusion. Categorical argument: the relation of inference hinges on relations among the terms within its premises and conclusion. Argument form: the symbolic pattern of the logical relations in an argument. Counterexample to an argument: another argument of the same form with clearly true premises and a false conclusion. It proves that the original argument has an invalid form. Substitution instance of an argument form: an actual argument exemplifying that form. Some valid forms of propositional arguments: modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, and contraposition. Validity, soundness, and cogency: standards for evaluating deductive arguments. VALIDITY Definition An argument is valid if and on!}> if it has entailment (its premises necessitate its conclusion).

Cash Value

■ It is not possible that the argument’s premises are true and its conclusion false.
■ The conclusion could be false, if at least one of the premises is false.
■ It is contradictory to accept a valid argument’s premises and reject its conclusion.

SOUNDNESS

Definition

An argument is sound if and on!)> if it is valid and all its premises are true.

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■ The argument’s conclusion is true: to deny it is to say something false.
■ A logical thinker who recognizes an argument as sound must accept its conclusion.

COGENCY

Definition

An argument is cogent if and on!)> if it is recognizab!J valid and has acceptable premises which
are more acceptable than the conclusion they attempt to support.

Cash Value

■ Any argument that satisfies these conditions is rationally compelling, in the sense that
it would move the thinker to accept its conclusion (provided she accepts its premises
and works out the entailment).

■ It would be irrational for the thinker to reject the conclusion of that argument.

■ Key Words

Validity

Entailment

Truth-preserving argument

Argument form

Propositional argument

Categorical argument

Soundness

Counterexample

Substitution instance

Cogency

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::<'.'. CHAPTER Analyzing Inductive Arguments This chapter looks more closely at inductive reasoning. Among its topics are: The nature of inductive arguments. Universal and non-universal generalizations. Identifying types of inductive argument: enumerative induction, statistical syllogism, ■ causal argument, and analogy. Two standards for evaluating inductive arguments: reliability and strength. Mill's methods for establishing causal connections between events: agreement and difference, and concomitant variation. 122 6.1 Reconstructing Inductive Arguments Since we have already dwelt at some length on deductive arguments, in this chapter we turn to inductive ones, which are crucial to ordinary and scientific reasoning. As we have seen, an argument is either deductive or inductive, depending whether the premises guarantee the truth of the conclusion. If they do, the argument is deductive; if not, it's inductive. There are a number of related tests that may help in recognizing an inductive argument. First, in the case of any such argument, ask yourself Could the premises of the argument be asserted and the conclusion denied without logical contradiction? ■ If yes, the argument is inductive. ■ If no, the argument is deductive. Let's consider some examples-first, a simple deductive argument: 1 Pam is energetic and athletic. Therefore, Pam is athletic. The first test recommends trying to see what happens when (1)'s premises are asserted and its conclusion denied. The test yields 2 Pam is energetic and athletic. But Pam is not athletic. (2) is contradictory: there is no logically possible scenario in which the statements that make up (2) could all be true or all false at once. In light of such a result, argument (1) above is deductive. By contrast, consider 3 1. Pam is athletic. 2. Most of those who are athletic don't eat junk food. 3. Pam doesn't eat junk food. Argument (3)'s premises could all be asserted and its conclusion denied without contradiction. After all, there are possible scenarios in which these premises are true and the conclusion false­ for example, a scenario in which Pam is athletic, and most athletic people do not eat junk food, but Pam does eat junk food. Thus (3) is inductive. Similarly, (4) and (s) are inductive, given that their premise could be asserted and their conclusion denied without contradiction: 4 1. Many horses are friendly. 2. Mr. Ed is a horse. 3. Mr. Ed is friendly. 5 Housing prices will continue to go down, for we are having a recession and usually housing prices go down in recessions. w >
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But this feature makes inductive arguments ideally suited for scientific reasoning in fields such as physics and biology, where scientists often make causal connections or reach general conclusions on the basis of only a sample of observed cases. The observation that a great number of metals expand under heat plays a role in the scientists' conclusion that all metals do so-as does research on the habits of people with lung disease in their concluding that smoking increases the risk of contracting such ailments. But both conclusions add something that was not among the scientists' premises. Another distinctive feature of inductive arguments is that newly acquired evidence could always make a difference in the degree of support for their conclusions, strengthening it in some cases, weakening it in others. Consider 8 1. 98% of State College students are involved in politics. 2. Heather is a State College student. 3. Heather is involved in politics. Argument (8) is inductive. Its premises, if true, would provide some support for its conclusion. New evidence to the effect that Heather is indifferent to politics, however, could undermine that support. Once that evidence is added, the argument then is 9 1. 98% of State College students are involved in politics. 2. Heather is a State College student. 3. Heather never votes. 4. Heather is involved in politics. A quick comparison of (8) and (9) shows that in the latter, support for the claim that Heather is involved in politics has been undermined by the addition of premise 3. The features of inductive arguments so far reviewed suggest that there is no entailment in them: their premises, even in cases where they succeed in supporting their conclusions, could never necessitate them. That is, no inductive argument is truth-preserving. Although an inductive argument may in fact have true premises and a true conclusion, what makes the argument inductive is that an argument of the same form could have true premises and a false conclusion-which, again, is the same as saying that the premises of an inductive argument do not entail its conclusion. Yet, as we shall see in this chapter, the lack of entailment in inductive arguments does not mean that they cannot offer support for their conclusions. In fact, they often make their conclusions probably true, or reasonable to believe, by providing evidence for them, even though their premises always fall short of necessitating their conclusions. This is why it is common to refer to the premises of inductive arguments as 'evidence.' At the same time, since the conclusions of such arguments may be supported but are never completely proved true by the premises, they have the status of conjectures and are often called 'hypotheses.' Given these features, inductive arguments are plausibility arguments. That is, although the evidence that any such argument may provide for its hypothesis never entails that hypothesis, when successful, they can make it plausible. To say that a claim is plausible is to say that it is likely to be true, probably true, or at least reasonable to accept. We shall look closely at the standard for successful induction once we have examined some common types of inductive argument. Before leaving this section, however, it is important that you know the answers to the questions in Box 1. BOX 1 ■ INDUCTIVE ARGUMENTS What sort of argument counts as inductive? ■ Any argument whose premises may provide evidence for its conclusion or hypothesis but do not guarantee it. How does one determine whether an argument is inductive or not? By checking ■ whether it would be possible for an argument with the same form to have true premises and a false conclusion. ■ whether one can assert its premises and deny its conclusion without contradiction. ■ whether the conclusion adds information not contained in the premises. If in any case the answer is Yes, then the argument is inductive. 6.2 Some Types of Inductive Argument Enumerative Induction Of the four types of inductive argument discussed in this chapter, we'll begin with enumerative induction. An enumerative induction always has a universal conclusion to the effect that all things of a certain kind have (or lack) a certain feature. This conclusion is drawn from evidence that some things of that kind have (or lack) that feature. The conclusion of any such argument, often called an 'inductive generalization,' is a universal generalization: UJ >
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<( z <( A universal generalization Consider ■ is a statement asserting that all of the members of a certain class have (or don't have) a certain feature. ■ may be expressed by a great number of different patterns of sentence. Some standard patterns are 'All ... are ... ,' 'Every ... is ... ,' 'No ... is ... .' 1 0 Roses blossom in the summer. Unless more information is provided here, (10) should be read as saying 'All roses blossom in the summer,' which illustrates the pattern 'All A are B.' To support (10) with an enumerative induction, we may adopt one or the other of two equiv­ alent strategies. First, offer a single premise to the effect that, for example, many roses have been observed to blossom in the summer. That would be a non-universal generalization: A non-universal generalization ■ is a statement asserting that some, perhaps many, of the members of a class have (don 't have) a certain feature. ■ may be expressed by a great number of different patterns of sentence. Some standard patterns are 'Most ... are ... ,' 'A few ... are ... ,' 'Many ... are ... ,' 'n percent of .. . are ... '(where n percent is less than 100 percent), 'Some ... are ... ,' and 'Some .. . are not. .. .' According to this strategy, the argument would run: 11 1. Many roses have been observed to blossom in the summer. 2. All roses blossom in the summer. Why is conclusion 2 a universal generalization? Because it asserts that all things of a certain kind (roses) have a certain feature (blossoming in the summer). Here are other such general­ izations common in science and everyday life: 12 Every metal expands when heated. 13 Any potato has vitamin C. 14 Each body falls with constant acceleration. 15 All bodies attract each other in proportion to their masses and in inverse proportion to the square of the distance between them. 16 No emeralds are blue. 17 No seawater quenches thirst. 18 No mules are fertile. Following the above strategy, we could attempt to support these generalizations by enumerative induction. Clearly, scientists could not have observed all metals in order to conclude (12), so the premise for (12) must be a non-universal generalization saying, for example, that many metals so far observed expand when heated. Similar enumerative inductions support the other universal generalizations in our list. Each such enumerative induction would have a premise that would be a non-universal generalization to the effect that things of the relevant kind have (12 through 15) or do not have (16 through 18) a certain feature. An alternative, yet equivalent, strategy to support these universal generalizations by enumerative induction would have specific statements as premises. A specific statement is a statement about an individual thing or person. For example: 'Benjamin Franklin founded the University of Pennsylvania,' 'That oak is infested,' 'Mary's cap is waterproof,' and 'The UN is in session.' If we wish to use this strategy to support the conclusion that roses blossom in the summer, our argument may run: 19 1. Rose 1 has been observed to blossom in the summer. 2. Rose 2 has been observed to blossom in the summer. 3. Rose 3 has been observed to blossom in the summer ... 4. Rose number n, has been observed to blossom in the summer. 5. All roses blossom in the summer. When n is a large number (say, billions) of individual roses, the universal generalization in con­ clusion 5 would be supported by the argument's premises, each of which is a specific statement about individual roses found to blossom in the summer. This strategy is equivalent to the one used in (n) above, given that (19)'s premises spell out what (n)'s premise summarizes. Similar to (n) is 20 1. Every raven so far observed has been black. 2. Ravens are black. Argument (2o)'s conclusion is a universal generalization ascribing a certain feature (blackness) to all ravens. Like other inductive arguments, this makes an inferential leap: from a number of ravens having a certain feature, it draws the conclusion that all ravens have that feature. Its premise, if true, supports the claim that a great number of ravens have that feature, but it does not guarantee that all ravens do. After all, nobody can observe all past, present, and future ravens! Argument (2o)'s conclusion, then, goes beyond the information given in its premise. Inductions of this sort run along the lines of (21). 21 1. A number, n, of A have been observed to be B 2. All A are B Clearly, any argument with this form could have a true premise and a false conclusion, since it is always possible that some unobserved A lacks the feature of being a B. This could happen even in cases where n turns out to be a very large number. Note that if n were taken to involve all cases, the argument would be deductive. w >
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<( z <( Causal Argument It is a matter of well-documented observation that whenever a flame comes in contact with combustible substances, this is invariably followed by a fire. Given that evidence, we may safely conclude that Jim lighting a match this morning near the gas caused the fire that erupted immediately after. Here we reason from an observed effect (the fire) to a possible cause that we may, or may not, have observed (Jim lighting a match this morning near the gas). Parallel causal reasoning is at work when only the effects of an event have been observed and we infer from them their likely cause-as is not uncommon in crime investi­ gations. Other times, facts have been observed pertaining to the cause of an event and these are then used in causal reasoning to predict possible effects, as in recent medical research that has revealed certain genes likely to be responsible for a type of mental illness mani­ festing as a social pathology. Here the genes appear to be a likely cause, in the sense that their presence is necessary (though not sufficient) for developing the social pathology. After all, not everyone with the genes will develop the illness: other factors, including environ­ mental ones, would also be needed. In the explosion case, Jim's lighting of a match this morning near the gas was sufficient but not necessary for the explosion to occur: in the described circumstances, an action of that sort would invariably cause an explosion, but other types of action could also cause an explosion. Knowledge of the causal relations between events is instrumentally valuable for us, because the control of nature is essential for human survival and flourishing. From a prudential point of view, we wish to promote those causes that have good effects while preventing those that have bad effects. Knowing that droughts were causally related to failed crops spurred the early development of irrigation systems by engineers, and farmers in antiquity. Similarly, our prospect of learning about causal connections between certain micro­ organisms and illnesses has triggered medical research that has resulted in our being able to prevent or contain infections and deadly diseases such as malaria and polio. So it is not an exaggeration to say that much of our everyday lives and scientific progress depend greatly on our being able to make causal connections between things and events. We take some phenomena (things and occurrences of things) to be the effects or results of other phenomena, which are their causes, and reason accordingly, ascribing causal relationships to new phenomena that we encounter. Reasoning about how certain events stand in cause/effect relations with other events takes the form of causal arguments: A causal argument makes the claim that two or more things or events are causally related in any of these ways: 1. Y results from Z. 2. Y causes Z. 3. Y and Z are the cause or the effect of another thing X The reasoning underwriting causal arguments is fundamental to both commonsense and scientific knowledge. It is at work when, if presented with some empirical evidence of state of affairs E, we set out to discover how E came to be. This requires determining which state of affairs C is linked to E-as its sufficient cause, its necessary cause, or its necessary and sufficient cause- for the word 'cause' can be used to mean a number of different relationships. When used to talk about a phenomenon that is always enough to bring about a certain outcome all by itself, it means sufficient cause, as illustrated by this causal argument: 32 1. There was a power blackout in my neighborhood yesterday. 2. My computer malfunctioned yesterday. 3. Yesterday's blackout was responsible for the malfunctioning of my computer. In (32), yesterday's blackout is taken to be the sufficient cause of the malfunctioning of the computer-just as overcooking one's dinner is sufficient for spoiling it. But the blackout isn't a necessary cause of the computer malfunctioning, because in the absence of a power blackout, the computer could still malfunction because of some other condition, such as rough handling, obstructed ventilation, and defective parts. Other times, an event C is the necessary cause of another event E, which is to say that E cannot occur in the absence of C. Since AIDS cannot occur in the absence of HN, this is the sense of 'cause' at work in the claim that 33 HN causes AIDS. There is also a sense of 'cause' that denotes a condition that is both necessary and sufficient to bring about a certain effect: that would be the case of a cause that's enough all by itself to cause something to happen and also necessary, in the sense that the effect could not have happened without it-as when we say that 34 Having the genome of a cat causes Fluffy the kitten to grow up to be a cat. Here, having a certain genetic code is both a necessary cause for the animal to be a cat (rather than, say, a monkey) and a sufficient cause, since it will always produce the same result. Note, however, that causal claims could be a generalization such as (33), or a particular statement such as (34) and the conclusion of (32). But when arguers make particular causal claims, their claims often rest on implicit generalizations. In the case of (32), it's a missing premise that could be reconstructed as being either a universal or a non-universal generalization. If the former, the argument would be deductive; if the latter, inductive. As an inductive argument, it could run this way: 32' 1. Power blackouts are often the cause of computer malfunctions. 2. There was a power blackout in my neighborhood yesterday. 3. My computer malfunctioned yesterday. 4. Yesterday's blackout was responsible for my computer's malfunction. (32')'s conclusion has the status of a hypothesis, which would be well supported, provided its premises are true. Because, as reconstructed here, (32' )'s premises don't guarantee its conclusion, the argument is inductive. w >
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<( z <( BOX 3 ■ THREE MEANINGS OF 'CAUSE' 1. Sufficient cause: C is a sufficient cause of E if and only if C always produces E. 2. Necessary cause: C is a necessary cause of E if and only if E cannot occur in the absence of C. 3. Necessary and sufficient cause: C is a necessary and sufficient cause of E if and only if C always is the sole cause of E. The methods of agreement and difference, and of concomitant variation. In his System of Logic (1843), John Stuart Mill (1806-1873) made use of ordinary intuitions in an attempt to establish generalizations about cause-and-effect relations. According to those intuitions, whenever something occurs, it is often possible to narrow the range of acceptable hypotheses about its likely cause-or about its effect-by eliminating plainly irrelevant factors until at last we find the hypothesis most likely to be the actual cause (or effect) of the occurrence. Of the five methods to establish generalizations about causal relationships proposed by Mill, we'll here consider two: the so-called method of agreement and difference and the method of concomitant variation. The method of agreement and difference The method of agreement and difference rests on the following basic principles: 1. Agreement: What different occurrences of a certain phenomenon have in common is probably its cause. 2. Difference: Factors that are present only when some observed phenomenon occurs are probably its cause. Suppose a coach wants to find out why Mick, Jim, and Ted, three of his best players, often perform poorly on Friday afternoons. After collecting some data about what each player does before the game, the coach reasons along these lines: 35 1. Mick,Jim, and Ted have been performing poorly on Friday afternoons. 2. Going to late parties on Thursday is the one and only thing that all three do when and only when they perform poorly. 3. Going to late parties on Thursday likely causes their poor game performance. The coach's reasoning here illustrates 'agreement,' since it runs roughly alone these lines: 36 1. X has occurred several times. 2. Y is the one and only other thing that precedes all occurrences of X. 3. Y causes X. But to make a more precise cause-effect claim, the coach should also use the method of difference: first, he should compare the players' performance when they've been going to late parties and when they haven't, and then, if they perform poorly only in the former cases, he should conclude that that difference also points to late-evening party-going as the likely cause of their poor performance. In fact, although the methods of agreement and difference are independent, they are usually employed jointly for the sake of greater precision. The method of concomitant variation The method of concomitant variation rests on the following principles: 1. When variations of one sort are highly correlated with variations of another, one is likely to be the cause of the other, or they may both be caused by something else. 2. When variations in one phenomenon are highly correlated with variations in another phenomenon, one of the two is likely to be the cause of the other, or they may both be caused by some third factor. Suppose now someone asks the coach why being fit matters for the members of a team. He may safely invoke empirical evidence to argue that there is a causal relationship between a player's being fit and his or her performance: 37 1. The more fit the players are, the better their performance. 2. Probably, being fit causes their better performance, or their better performance causes their being fit, or something else causes both their better performance and their being fit. The underlying reasoning is roughly 38 1. X varies in a certain way if and only if Yvaries in a certain way. 2. Y causes X, or X causes Y, or some Z causes both X and Y. Analogy Analogy is a type of inductive argument whereby a certain conclusion about individuals, qualities, or classes is drawn on the basis of some similarities with other individuals, qualities, or classes. Here is an example of an analogy whose conclusion about a certain vehicle rests on this vehicle having some things in common with other similar vehicles: 39 1. Mary's vehicle, a 2007 SUV, is expensive to run. 2. Jane's vehicle is a 2007 SUV and is expensive to run. 3. Simon's vehicle is a 2007 SUV and is expensive to run. 4. Peter's vehicle is a 2007 SUV. 5. Peter's vehicle is expensive to run. In (39), the arguer attempts to make her conclusion reasonable by analogy: Peter's vehicle shares two features with Mary's, Jane's, and Simon's: being a 2007 model and an SUV. This provides some reason to think that it may also have in common a third feature, that of being expensive to run. 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<{ z <{ 39' 1. m is A, B, and C 2. j is A, B, and C 3. s is A, B, and C 4. pis A and B 5. p is C Any argument along these lines would fall short of being deductive (i.e., of entailing its conclusion). Yet if its premises are true, they might provide good evidence for it. Analogies can make their conclusions plausible, provided that they meet the standards for good inductive arguments discussed below. Among the specific factors that matter for the success of analogies are those presented in Box 4. Now consider 40 Extensive research on polar bears and hippos has shown that they have a great number of relevant features in common with large animals that live in the wild. These animals are also listed as endangered species. So polar bears and hippos might disappear. The pattern of reasoning underlying this analogy is 1. Polar bears and hippos have a number of relevant things in common with species x, y, and z. 2. Species x, y. and z also have feature f (being an endangered species). 3. Polar bears and hippos probably have feature f. If polar bears and hippos do in fact share a number of features with threatened species, and such features are truly relevant to the conclusion of this argument, then (40) can be said to succeed in rendering its conclusion plausible. BOX 4 ■ ANALOGY Whether an analogy succeeds or not depends on 1. The number of things and the number of features held to be analogous. ■ Greater numbers here would make an analogy stronger. 2. The degree of similarities and dissimilarities among those things. ■ More of the former and less of the latter would make an analogy stronger. 3. The relevance of ascribed features to the hypothesis. ■ Greater relevance would make an analogy stronger. 4. The boldness of the hypothesis with respect to the evidence. ■ Modesty in the hypothesis would make an analogy stronger. Exercises 1 . Discuss three features of inductive argument that distinguish them from deductive arguments. 2. What's the problem with asserting the premises of an inductive argument while denying its conclusion? 3. Why are the premises and conclusion of an inductive argument called 'evidence' and 'hypothesis' , respectively? 4. What does it mean to say that a hypothesis is 'plausible'? 5. What is an enumerative induction? 6. What's the difference between universal and non-universal generalizations? How can a universal generalization be proved false? 7. Describe the structure of a statistical syllogism. 8. Describe the structure of a causal argument. 9. Why is the word 'cause' ambiguous? 1 0. Describe the structure of an analogy. II. Determine whether the following arguments are deductive or inductive. 1. Many whales observed in this region are white mammals. Therefore, any whale in this region is a white mammal. SAMPLE ANSWER: Inductive argument 2. Triangles have exactly three internal angles. Rectangles have exactly four internal angles. Therefore, rectangles are not triangles. *3. If all magnolias have a scent, then the magnolias in the vase have a scent. But they don't. It follows that it isn't true that all magnolias have a scent. 4. Buying a house is a good investment. After all, that's exactly what statistics have shown for the last ten years. 5. All samples of river water so far tested have been polluted. Thus all river water is polluted. *6. The Crusades were bloody, for most medieval wars were bloody, and the Crusades were medieval wars. 7. Surely the Earth is not flat. If it were flat, then Magellan could not have circumnavigated it. But he did! 8. Jane is a dentist and has clean teeth. Bruce is a dentist and has clean teeth. Therefore, all dentists have clean teeth. *9. Cars are mechanical devices. No mechanical devices are easy to fix. Thus no car is easy to fix. 10. Many medical doctors care about their patients. Tom is a medical doctor. Thus he cares about his patients. 11. Mary doesn't like being denied a salary increase, for she is a state worker, and no state worker likes that. *12. To be an ophthalmologist is to be an eye specialist MD. My new neighbors are eye specialist MDs, so they are ophthalmologists. w >
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When an inductive argument is reliable, it has a form that makes its conclusion plausible

provided that its premises are true. Consider

41 1. 99 percent of guitar players also play other musical instruments.
2. Phong is a guitar player.
3. Phong also plays other musical instruments.

This inductive argument seems pretty reliable: its form is such that, if its premises were true,
its conclusion would be plausible. Compare (42), which is itself less reliable than (41) but more
reliable than (43):

42 1. 59 percent of guitar players also play other musical instruments.
2. Phong is a guitar player.
3. Phong also plays other musical instruments.

43 1. 39 percent of guitar players also play other musical instruments.
2. Phong is a guitar player.
3. Phong also plays other musical instruments.

Inductive reliability is, then, a matter of degree. An inductive argument of (44)’s form is more
reliable than (45):

44 1. 59 percent of A are B
2. p is an A
3. p is a B

45 1. 39 percent of A are B
2. p is an A

3. p is a B

The cash value of inductive reliability for logical thinkers can be better appreciated by comparing
it to the cash value of deductive validity. Each of these concerns argument form, as well as the sup­
port an argument’s premises may give its conclusion, provided they are true. In the case of a valid
argument, if its premises are true, its conclusion must be true. In that of a reliable argument, if its
premises are true, its conclusion is likely to be true. As we saw in Chapter 5, a valid deductive
argument is truth-preserving. By contrast, a reliable inductive argument is not. Even so, inductive
reliability is one of the two desirable features that ordinary and scientific arguments should have.

Inductive Strength

Strength is another desirable feature for inductive arguments; thus we may use it to
evaluate such arguments. An inductive argument is strong just in case it meets the
conditions listed in Box 5.

BOX 5 ■ STRONG INDUCTIVE ARGUMENT

An inductive argument is strong if and only if

1. It is reliable.
2. It has all true premises.

When an inductive argument is strong, it is reasonable to accept its conclusion. That is, it
is reasonable to think that the conclusion is true. We may think of this standard in terms of
competition: given the structure of an inductive argument, rival conclusions are always
logically possible. Imagine a case where a professor in Biology 100 has just received an email
from one of her new students, whose name is Robin Mackenzie. She is trying to decide
whether she should begin her reply, ‘Dear Mr. Mackenzie’ or ‘Dear Ms. Mackenzie.’ Let’s
assume that it is true that 80 percent of the students in Biology 100 are women and reason
through the steps of this inductive argument:

46 1. 80 percent of the students in Biology 100 are women.
2. Robin is a student in Biology 100.

3. Robin is a woman.

Since (46) is an inductive argument, the conclusion, statement 3, may in fact fail to be true,
even if both premises are true. After all, a person named ‘Robin’ could be a man. Even so, given
the evidence provided by the premises, it seems that conclusion 3 is more plausible than the
other competing hypothesis (i.e., that Robin is a man). But imagine a different scenario:
suppose that we knew that 80 percent of the students in Biology 100 were men. Then, among
then the competing hypotheses, the conclusion that is most likely to be true on the basis of
that information is that Robin is a man. The argument now is

47 1. 80 percent of the students in Biology 100 are men.
2. Robin is a student in Biology 100.

3. Robin is a man.

We may alternatively define inductive strength in this way:

An inductive argument is strong if and only if its hypothesis is the one that has the greatest
probability of being true on the basis of the evidence.

In the same way that inductive reliability can be contrasted with deductive validity,
inductive strength can be contrasted with deductive soundness. For one thing, the latter does
not come in degrees, since it depends on validity and truth, neither of which is itself a matter
of degree (there’s no such thing as a ‘sort of true’ premise or a ‘sort of valid’ argument). Hence,
just as any given deductive argument is either valid or invalid, so, too, it’s either sound or
unsound. On the other hand, inductive strength does come in degrees, for it depends in part
on reliability, which is a matter of degree. What about the cash value of these standards? When
an argument is deductively sound, its conclusion is true-and must be accepted by any logical
thinker who recognizes the argument’s soundness. But the conclusion of any inductively
strong argument can be, at most, probably true-and thus reasonable to accept by a logical

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<( z <( 111m thinker who recognizes the argument's strength. For each of the two criteria by which we assess inductive arguments, then, we may summarize its cash value as follows: Inductive Reliability's Cash Value ■ If an argument has a good share of reliability, then it would be reasonable to accept its conclusion, provided that its premises are true. Inductive Strength's Cash Value ■ If an argument has a good share of inductive strength, then it's reasonable to accept its conclusion, since it has a reliable form and its premises are true. What, then, of inductive arguments that fail by one or the other of these two criteria? No such argument could provide good reasons for their conclusions. Exercises 1. What are the two standards for evaluating an inductive argument? Define each. 2. Does inductive reliability depend on the form of an argument? What about strength? 3. What question should we ask to determine whether an inductive argument is reliable? 4. Assuming that an inductive argument is reliable, when would it be strong? 5. Does the cash value of deductive validity differ from that of inductive reliability? Explain. 6. What factors are relevant to determining whether an enumerative induction is reliable? 7. What factors are relevant to determining whether an enumerative induction is strong or weak? 8. What factors are relevant to determining whether an analogy is reliable? VII. Identify whether the arguments below are enumerative inductions, analogies, causal arguments, or statistical syllogisms, and determine which are reliable and which are not. For any argument whose reliability cannot be determined, explain why not. 1. There is consensus among experts that heavy drinking is linked to liver disease. Therefore, heavy drinking leads to liver disease. SAMPLE ANSWER: Causal argument, reliable 2. Millions of fish so far observed have all been cold-blooded animals. Thus all fish are cold-blooded animals. *3. Most South American coffee beans are dark. Brazilian coffee beans are South American coffee beans. Hence, Brazilian coffee beans are dark. 4. Nancy lives downtown and pays a high rent. Bob lives downtown and pays a high rent. Pam pays a high rent. Thus Pam probably lives downtown. 5. 40 percent of college students sleep less than eight hours a day. Peter is a college student. Therefore, Peter sleeps less than eight hours a day. *6. Every pizza eater I have met liked mozzarella. Thus pizza eaters like mozzarella. 7. Betty's pet is carnivorous, and so are Lois's, Brenda's, and John's. It follows that all pets are carnivorous. 8. Senegal is an African nation and has a forest. Nigeria is an African nation and has a forest. Since Egypt is also an African nation, it probably has a forest. *9. Caffeine is related to poor memory. All recent studies have shown that people can improve their memory by reducing their daily consumption of caffeinated drinks. 10. Among families that have lived in Spring Valley for more than ten years, nearly 90 percent say they like it there. My family will soon move to Spring Valley. So my family will like it there, too. 11. Since their discovery, microorganisms have been observed to be present in all infections. Thus microorganisms are responsible for infections. *12. Mike sells junk food, for he owns a fast-food restaurant, and that's what most fast-food restaurants sell. 13. Frank Sinatra sang in a 1950s movie wearing a tuxedo. Sammy Davis Jr., Peter Lawford, and Joey Bishop were all in tuxedos in that movie with Frank, and they made up 90 percent of the male actors cast in it. Since Dean Martin also sang in the same movie, he must have worn a tuxedo. 14. From 1951 to 2001, Sir Richard Doll documented the mortality rate of British male doctors born between 1900 and 1930. 81 percent of nonsmokers lived to at least age seventy, but only 58 percent of smokers lived to that age. Cigarette smoking stood out in Doll's findings as the only major factor distinguishing these two groups of doctors. Thus the shorter survival rate in the second group was a result of smoking. *15. Chase is a bank and makes home-finance loans. Citibank is a bank and makes home-finance loans. Wells Fargo is a bank and also makes home-finance loans. This suggests that all U.S. banks make home finance loans. 16. After an extensive study involving major research universities, scientists discovered that poison ivy grew there at 2.5 times its normal rate when they pumped carbon dioxide through pipes into a pine forest. Their work suggests that atmospheric carbon dioxide is at least partially responsible for the higher growth rate of poison ivy. 17. I' ll be accepted. Let's not forget that 98 percent of applicants with my qualifications get accepted. *18. Given that T ina Turner is a famous singer and has an insurance policy on her legs, Queen Latifah probably has one, too. 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All leopards so far observed have been carnivorous 2. All leopards are carnivorous When a representative sample of leopards has been observed to be carnivorous, the conclusion of this inductive argument is well supported. Similarly, if we've observed a representative sample of leopards and found them all to be wild animals, we would be justified in drawing the general conclusion that leopards are wild animals on the basis of those observations. Our inductive argument would be, 2 1. Every leopard observed so far has been a wild animal 2. All leopards are wild animals But for any such inductive conclusion to be justified, the conditions listed in Box 3 must be met. If either of those two conditions, or both, is unfulfilled, then the argument commits the fallacy of hasty generalization and therefore fails. Hasty generalization is the mistake of trying to draw a conclusion about all things of a certain kind having a certain feature on the basis of having observed too small a sample of the things that allegedly have it, or a sample that is neither comprehensive nor randomly selected. Suppose a team of naturalists were to observe 500,000 leopards, which all turn out to be wild animals. Yet they were all observed in India, during the first week of August, at a time when these animals were about to eat. The sample seems large enough, and the observers might therefore draw the conclusion that 3 All leopards are wild animals. But they would be committing a hasty generalization, since leopards are also found in other parts of the world. And they are found at other times of the year, and in other situations. Clearly, the sample lacks comprehensiveness and randomness. In this case, argument (2) would fail to provide a good reason for its conclusion. On the other hand, suppose the naturalists directly observed patterns of wild behavior among leopards in all parts of the world where such animals are found, at different times of the year, and in many different situations. Yet the sample now consists of only thirty-seven leopards. Do the naturalists have better grounds for concluding (3) above? No, because although the comprehensiveness and randomness criteria are now met, the sample is too small. The charge of hasty generalization would similarly apply in this scenario. It is, however, not only naturalists and other scientists who will need to beware of this sort of blunder. Logical thinkers will want to be on guard for hasty generalization in many everyday situations. Among these is the familiar mistake of stereotyping people. Suppose someone from the Midwest visits California for the first time. He becomes acquainted with three native Californians, and it happens that all three practice yoga. 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<{ ...J 3� UJ LL �z 0 <{ (/) 0 infer that she caught the chicken pox from her sisters. Given what we know about how infectious diseases are transmitted, this inductive conclusion seems supported. But not all causal arguments are strong. When either of the two types of error listed in Box s occurs, a fallacy of false cause has been committed. False cause is the mistake of arguing that there is a significant causal connection between two phenomena, when in fact the connection is either minimal or nonexistent. BOX 5 ■ HOW TO AVOID THE FALLACY OF FALSE CAUSE Causal arguments can fail in two basic ways: ■ The argument concludes that there is a cause-effect connection between two phenomena where there is none at all. ■ The argument mistakenly identifies some phenomenon as a sufficient (or determining) cause, when in fact it's only a contributory cause (i.e., one among many) of some observed effect. Let's consider three different ways the fallacy of false cause may occur. One is: Post hoc ergo propter hoc ('after this, therefore because of this'): The fallacy of concluding that some earlier event is the cause of some later event, when the two are in fact not causally related. The inclination to commit this fallacy in everyday life rests on the fact that, when we see two events constantly conjoined-so that they are always observed to occur together, first the one, then the other-it may eventually seem natural to assume that the earlier is the cause of the latter. But it is not difficult to imagine cases of precisely this sort where an imputation of causal connec­ tion would be absurd. Suppose we saw a bus passing the courthouse in the square just before the clock in the tower struck 9:00 a.m., and we then continued to see the exact same sequence of events day after day. Do we at last want to say that it's the bus's passing the courthouse that causes the clock to strike 9:00 a.m.? Of course not! And yet, in our experience, the two events have been constantly conjoined: the clock's striking has always been preceded by the bus's passing. Clearly it would be preposterous to argue, in that case, that, from the evidence of constant conjunction between the bus's passing and the clock's striking 9:00, it follows that the former causes the latter. But equally absurd arguments are in fact sometimes heard in everyday life. Suppose that Hector and Barbara are not getting along, and one of their friends ventures to explain the source of the problem: 8 1. Hector was born under the sign of Capricorn. 2. Barbara was born under Pisces. 3. Capricorns and Pisces are not compatible. 4. Their recent difficulties are owing to their having incompatible zodiac signs. Argument (8) fails to support its conclusion, since it claims a causal connection for which the argument gives no good evidence-nor, in this case, should we expect good evidence to be forthcoming. After all, there's no good reason to think that configurations of stars and other celestial events really do affect the courses of our lives, and whatever the cause of this couple's troubles may be, it's probably traceable to something else. Argument (8) is a fallacy of post hoc ergo propter hoc, a form of false cause, for it assumes a cause-effect relation between being born on a day when celestial bodies have a certain configuration (which determines a certain zodiac sign) and subsequently growing up to develop certain personality traits. But there is no reason to think that these two sequential events are in fact causally related. Another way false cause may occur is Non causa pro causa (roughly, what is not the cause is mistaken for the cause): A fallacy in which the error is not an imputation of causality in a temporal sequence of events (as in post hoc ergo propter hoc, where an earlier event is wrongly thought to be the cause of a later one), but rather the simple mistake of misidentifying some event contemporaneous with another as its cause, when in reality it's not. One form of this error occurs when cause and effect are confused. An early nineteenth-century study of British agriculture noted that, of farmers surveyed, all the hard-working and industri­ ous ones owned at least one cow, while all the lazy ones owned no cows. This led the researchers to conclude that productivity could be improved overall and habits of industry encouraged in the lazy farmers by simply giving them each of those farmers a cow! Now, plainly there is something wrong in this reasoning. But what? It seems to rest on an extended argument along these lines: 9 1. All of the observed industrious farmers are cow owners. 2. None of the observed lazy farmers is a cow owner. 3. All and only cow-owning farmers are industrious. 4. There is a positive correlation between cow-owning and industriousness. 5. It's cow-owning that causes industrious farmers to be industrious. If we grant, for the sake of discussion, that the sample of British farmers in the study was large enough, and that it was also comprehensive and randomly selected, then premises 1 and 2 support conclusion 3, and its restatement, conclusion 4. But s's claim about cause and effect fails to be supported! It's industriousness that is probably the cause of cow-owning, and not the other way around. By confusing cause with effect, (9) commits non causa pro causa. Finally, there is a version of false cause in which the source of the mistake is something rather different from what we've seen so far: Oversimplified cause: The fallacy of overstating the causal connection between two events that do have some causal link. 0 (.') (/) f­ z w � :::, (.') 0: –
<( __J � <( LlJ u.. �z 0 <( (/) 0 Suppose a vice president, campaigning for reelection, says, 10 1. At the beginning of this administration's term, the national economy was sluggish. 2. At the end of this administration's term, the national economy is booming. 3. White House economic policies do have an effect on the nation's economy. 4. The improvement in the economy is due to this administration's policies. (10) fails to support its conclusion. Let's assume that the premises are true. Even then, the causal relation asserted in premise 3 is merely one of contributory cause-in effect, one causal factor among others-which amounts to a rather weak sense of 'cause.' But 4 grandly asserts as the conclusion something much more than that: namely, that the actions taken by the incum­ bent administration are a sufficient cause of the improvement in the economy. Now, surely this is an exaggeration. The campaigning vice president commits a fallacy of oversimplified cause by taking full credit for the nation's economic turnaround, thereby overstating the sense in which his administration's policies 'caused' it. Of course, many politicians are quite prepared to take credit for anything good that happens while they're in office. But proving that it was due entirely to their efforts is something else again. The logical thinker should be on guard for this and any of the other versions of false cause as representing different ways in which a causal argument may fail. BOX 6 ■ THREE TYPES OF CAUSAL FALLACY Post Hoc Ergo Propter Hoc Appeal to Ignorance FALLACY OF FALSE CAUSE Non Causa Pro Causa Oversimplified Cause Another fallacy of failed induction is the appeal to ignorance (or ad ignorantiam): an argu­ ment that commits this fallacy concludes either that some statement is true because it has never been proved false, or that it is false because it has never been proved true. More generally, The fallacy of appeal to ignorance is committed by any argument in which the conclusion that something is (or isn't) the case is supposedly supported by appeal to the lack of evidence to the contrary. Suppose someone reasons, 11 1. It has never been proved that God doesn't exist. 2. We can confidently assert that God exists. (11) commits the fallacy of appeal to ignorance, but so does (12): 12 1. It has never been proved that God exists. 2. We can confidently assert that God doesn't exist. Similarly, a believer in 'extrasensory perception' might argue, 13 1. No one has ever been able to prove that ESP doesn't exist. 2. It's reasonable to believe that there is ESP. Clearly, the only reason offered by (13) to support its conclusion is the absence of contrary evi­ dence. But from that premise, all that can be supported is that we don't know what to say about ESP! The conclusion given-that "it's reasonable to believe that there is ESP"-is far too strong to be supported by such a flimsy premise. Reasoning along similar lines could also be used to demonstrate the failure of (11) and (12). BOX 7 ■ HOW TO AVOID THE FALLACY OF APPEAL TO IGNORANCE ■ An argument whose premises merely invoke the lack of evidence against a certain conclusion commits the fallacy of appeal to ignorance. Such premises are bad reasons for the conclusion they attempt to support, and the argument therefore fails. ■ Why? Because the mere lack of negative evidence does not in itself constitute positive evidence for anything! It justifies nothing more than an attitude of non belief (i.e., neutrality) toward the conclusion. We must, however, add a note of caution. Suppose that the attempt to prove some claim has occasioned rigorous scientific investigation, and that these efforts have repeatedly turned up no evidence in support of the claim. Furthermore, suppose that the claim doesn't serve the purpose of explaining anything. In that case, it is not a fallacy to reject that claim out of hand. Here we have to proceed case by case. Consider the claim, 14 There are witches. Although there is of course a long history of claims that witches exist, all efforts to prove those claims have so far failed for lack of evidence. Furthermore, the concept of a witch has no serious explanatory function in any scientific theory: the existence of witches doesn't explain anything that happens in the natural world. These considerations suggest that it is not a fallacy to conclude, 15 Probably there are no witches. Inductive conclusions of this sort are rendered plausible by the absence of reliable empirical evidence after thorough investigation, and must not be confused with the fallacy of appeal to ignorance. 0 C)

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If we want to have well-founded beliefs about the French Revolution, the Ming Dynasty, or the presidency of Theodore Roosevelt, we should look to writers whose books are not "self-published" or published by vanity presses (where the authors pay to get their books into print). We should look for historians who are held in high regard by peers in their fields and whose work has been favorably reviewed. Although none of these criteria guarantee expertise, they make it vastly more likely. Similarly, for beliefs about nature, it goes without saying that respected journals in the natural sciences are generally dependable sources of information, unlike supermarket tabloids that describe miracle cures for cancer and 'evidence' of mental telepathy. For scientists, too, being the author of respected, mainstream scholarship and having the favorable regard of fellow scholars are usually the marks of credibility as genuine experts. For logical thinkers, then, an important competence is the ability to tell the difference between real experts and bogus ones, since it is often on that distinction that the difference between legitimate appeals to authority and the fallacy of appeal to unqualified authority turns. BOX 9 ■ HOW TO AVOID FALLACIOUS APPEALS TO AUTHORITY To avoid fallacious appeal to authority, keep in mind the way it differs from appeals to authority that aren't fallacious. The difference hinges on whether the authority cited in support of a claim ■ does indeed have sufficient expertise in the relevant field; and ■ is expressing a view well represented (perhaps the prevailing one) among experts on the topic. Exercises 1 . What is a fallacy? 2. What's the point of studying fallacies, as far as logical thinking is concerned? 3. What is the fallacy of hasty generalization? 4. Are all generalizations to be avoided? 5. What is stereotyping? And how is this related to hasty generalization? 6. What is the fallacy of weak analogy? 7. The fallacy of false cause has at least three different forms. Identify the kind of mistake each makes, and explain why they are all mistakes in causal reasoning. 8. What is the fallacy of appeal to ignorance? 9. When a fallacy of appeal to unqualified authority is committed, who commits it? Is it the arguer or the bogus authority? 1 o. What is the difference between the legitimate use of appeal to authority and the fallacy of appeal to unqualified authority? II. Each of the following arguments commits one of the fallacies of failed induction discussed in this chapter. Identify the fallacy. 1. I'm an Aquarius, so I love doing lots of projects at once. SAMPLE ANSWER: False cause 2. Some people can cure heart disease by meditation. I know because the coach of my son's soccer team told me. *3. Wage and price controls will not work as a means of controlling the rate of inflation. After all, no econ­ omist has ever been able to give conclusive proof that such controls are effective against inflation. 4. Most HIV patients are young. Thus youth causes HIV. 5. Yogi Berra, an Italian American, was one of the greatest baseball players of all time. Other all-time greats of baseball include Joe DiMaggio, Mike Piazza, and Roy Campanella. So, no doubt about it, Italian Americans are great baseball players. *6. Last week, when Notre Dame won the game, the coach was wearing his green tie. So their victory must have been due to the coach's choice of necktie, since this nearly always happens when he wears that tie. *7. Foreign wars are good for a nation, just as exercise is good for the body. In the same way that exercise keeps the body fit, foreign wars keep a nation fit as a society. 8. According to recent polls of registered voters, the state of Massachusetts has a large percentage of voters who are political liberals. This suggests that all states have a large percentage of voters who are political liberals. *9. The chances for stability in the Middle East will continue to improve. Popular singer Britney Spears has recently said that that is what she expects to happen. 10. Dallas and Houston are North American cities, and one can drive from the one to the other in only a few hours. Montreal and Los Angeles are also North American cities. Thus one can drive from Montreal to Los Angeles in only a few hours. 11. Some years ago, after not having seen my best friend from Duckwood High School for several years, we met for lunch and were surprised to find that our clothes and hairstyles were the same! The only possible explanation for this is that we both went to Duckwood High. *12. Some regular churchgoers believe that taxpayers' dollars should not be used to fund laboratories that carry out tests on animals for medical research. 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CHAPTER

Avoiding Ungrounded
Assumptions

In this chapter you’ll learn about the fallacies of presumption and some related
logical and philosophical issues. The topics include

■ Circular reasoning: When is it vicious? When is it benign?

■ The fallacies of begging the question and begging the question against.

The concept of burden of proof.

■ The fallacy of complex question.

■ The fallacy of false alternatives.

■ The fallacy of accident.

166

8.1 Fallacies of Presumption

Now we’re ready to look at some fallacies that can be grouped together because arguments

committing them take for granted something that is in fact debatable. Such arguments rest on
presumptions, which are strong assumptions or background beliefs taken for granted.

Generally, there is nothing wrong with presumptions: arguments commonly rest on implicit

beliefs that create no fallacy of presumption at all. But when an argument takes for granted a

belief that is in fact debatable, it commits a fallacy of presumption. The unsupported belief at

work in such fallacious arguments may at first seem innocent or even acceptable, though in

reality it is neither. The patterns of mistake illustrated by arguments that rest on debatable
presumptions include the five types of fallacy listed in Box 1.

BOX 1 ■ SOME FALLACIES OF PRESUMPTION

I FALLACIES OF I PRESUMPTION

BEGGING THE BEGGING THE

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8.2 Begging the Question

In Chapter s we saw that the premises of valid arguments could be true yet fall short of count­

ing as persuasive reasons for their conclusion. That would be the case with any sound argument

that failed to be cogent. As a result, no such argument can move a rational thinker to accept its

conclusion, even when the validity of the argument may be obvious to the thinker. Why?

Imagine that we intend to convince you rationally to accept a certain claim-say, that

1 We care about logical thinking.

We offer you this reason as a premise:

2 It is not the case that we don’t care about logical thinking.

The argument is

3 1. It is not the case that we don’t care about logical thinking.
2. We care about logical thinking.

(3) is valid, and we may assume that it has a true premise. Yet it lacks cogency, since it doesn’t

offer reasons that could persuade a logical thinker of the truth of its conclusion if that thinker

is skeptical about that very conclusion. Philosophers call this ‘circular reasoning.’ (3) is affected

by a degree of circularity that may be considered ‘vicious,’ since it would make any argument

that has it fail-by contrast with ‘benign circularity,’ which, as we’ll see, is the tolerable degree

of circularity that affects many deductive arguments.

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<( <( - As just noted, any valid argument has some degree of circularity, due either to its form, the concepts involved, or both. The following two arguments are affected by circularity hinging on argument form: 8 1. Today it's cloudy and breezy. 2. Today it's breezy. 9 1. The Pope is in Rome. 2. The Pope is in Rome. Clearly, the circularity afflicting these arguments hinges on their forms-which are 8 1. C and B 2. B 9 1. E 2. E Any argument with either of these forms is valid, for if its premise is true, then its conclusion must also be. But in each case the premise is not more acceptable than the conclusion it attempts to support. As a result, no one could come to accept the conclusion on the basis of having worked out the validity of the argument and found the premise acceptable. Yet circularity may also be conceptual, hinging on the meaning or concepts involved. Consider 10 1. If Kobe Bryant is a basketball player, then he plays basketball. 2. Kobe Bryant is a basketball player. 3. Kobe Bryant plays basketball. 11 1. Marianne was once abducted by alien beings from outer space. 2. There are alien beings from outer space. In each argument, the premises presuppose the conclusion they are supposed to support, given the concepts involved. In (10), no logical thinker who doubts that Kobe Bryant plays basketball could come to accept that on the basis of the argument's premises. In (11), no logical thinker who doubts the existence of alien beings from outer space could be persuaded that there are such beings by the argument's premise. Each argument begs the question, thereby failing to be cogent. As in (8) and (9) above, these arguments too are affected by circularity that renders them fallacious. Whether formal or conceptual, circularity comes in degrees: too much of it causes an argument to beg the question. Benign Circularity But circularity does not always make an argument question-begging. Compare 1 2 1. If the mind is the brain, then the mind is organic matter. 2. If the mind is organic matter, then it perishes with the body. 3. If the mind is the brain, then it perishes with the body. BOX 3 ■ FORMAL AND CONCEPTUAL CIRCULARITY Two TYPES OF CIRCULARITY Here the argument form is 12' 1. If M, then 0 2. If 0, then B 3. If M, then B Formal It hinges on argument form Conceptual It hinges on concepts involved In an argument with this form, there is some formal circularity, since the propositions represented as M and B appear not only in the conclusion, but also in the premises. Yet (12) does not beg the question, because finding its premises acceptable and recognizing the argument's validity could provide reasons to move logical thinkers to accept its conclusion. Anyone who accepts the argument's premises and works out the entailment thereby possesses a compelling reason to accept its conclusion. By contrast with viciously circular arguments, coming to accept (12)'s conclusion on the basis of its premises amounts to a cognitive achievement. Let's now compare some conceptually circular arguments such as 13 1. Salsa is music for dancers. 2. Salsa is music for those who dance. 14 1. Andrew is a bachelor. 2. Andrew is unmarried. 15 1. She has drawn an isosceles triangle. 2. She has drawn a triangle. All three of these arguments are valid: if their premises are true, their conclusions must be true, as well. Yet under ordinary circumstances, each begs the question, for in each case acceptance of its premise requires a previous acceptance of the conclusion. No logical thinker who disputed the conclusion could be compelled to accept it on the basis of the argument's premise and recognition of the argument's validity. But consider 16 1. The Moon orbits the Earth. 2. The Moon is a large celestial body. 3. Any large celestial body that orbits a planet is a satellite. 4. The Moon is a satellite. z 0 1- (/) w :::J a w I 1- (') z (') (') w OJ N 00 0 IJj 0 z ::::) 0 a: CJ (/) zz ::::) 0 CJ i= z 0.. - � e ::::i 0 (/) > (/)
<( <( Although in (16) there is some conceptual circularity between the concepts 'satellite' and 'large celestial body that orbits a planet,' this does not make the argument question-begging. For a logical thinker who lacked some basic astronomical knowledge and initially doubted claim 4 could be persuaded to accept it on the basis of deducing it from 1, 2, and 3, provided that she were led to recognize the acceptability of those premises and the validity of the argument. An important point to keep in mind is that Logical circularity, whether formal or conceptual, comes in degrees. Some valid arguments have more circularity than others. The more logically circular an argument is, the more its conclusion follows trivially from its premises and is likely to beg the question. The Burden of Proof It is not uncommon to find the expression 'burden of proof' in dialectical contexts, which are situations involving deliberation among two or more parties, such as a debate, controversy, or deliberation on a disputed question between opposing sides defending incompatible claims. 'Burden of proof' refers to the obligation to take a turn in offering reasons, which, at any given stage of the deliberation, is on one side or the other (except for the paradoxical situations discussed below). A deliberation commonly follows this pattern: one party, C, makes a claim. The other party, 0, replies by raising some objections to it. If these objections are adequate, the burden of proof is now on C, who must get rid of (or 'discharge') it by offering reasons for her claim. If she comes up with a sound or strong argument that outweighs O's argument, the burden of proof then switches to 0, who must try to discharge it by offering the appropriate arguments. It may happen, however, that the reasons on both sides appear equally strong. As a result, there would then be a dialectical impasse, or standoff in the deliberation. No progress can be made until new reasons are offered to resolve the conflict. Except for these situations, however, we may expect that the burden of proof will, at any given stage of a deliberation, be on either the one side or the other. As the deliberation progresses, it will likely switch from the one side to the other more than once, always falling on the participant whose claim is more in need of support. BOX 4 ■ WHERE IS THE BURDEN OF PROOF? In the following debate,® shows the burden of proof andQan impasse. 1. A rejects a claim made by!!, which is a commonly held belie£®A 2. A defends her rejection with an argument that begs the question against!!-® A 3. A recasts her argument so that it now seems cogent.®!! 4. !! offers an argument that turns out to be clearly invalid.®!! 5. !!'s argument is modified and now seems as cogent as A's.Q 6. A provides further strong evidence in support of her view.®!! 7. !! replies by offering weak evidence for his view.®!! 8. !! offers further evidence which is equally strong as A's.Q Commonsense beliefs, which are ordinary beliefs based on observation, memory, and inference, enjoy a privileged standing with respect to the burden of proof. Whoever challenges them has, at least initially, the burden of proof. For example, the belief that the Earth has existed for more than five minutes belongs to common sense. If someone challenges it, the burden of proof is on the challenger, who must now offer adequate reasons against that commonsense belief. But that advantage can be overridden by a strong argument if available. Knowing where the burden of proof is at any given stage of a debate has this cash value: ■ If you know that the burden of proof is on you, you know you must discharge it by offering an adequate argument in support of your claim. ■ If you know that the burden of proof is on the other side, you can rest until a sound objection to your view has been offered. ■ If you know that you are defending a claim that is part of common sense, then you also know that the burden of proof is on any challenger. Finally, note that some deliberation goes on 'internally'-for example, when a person reflects upon which of two opposite theories is correct. If, in the course of inner delibera­ tion, a thinker is fair-minded, then the burden of proof will tend to shift from one position she is considering to an opposite view, following the same general considerations outlined above. BOX 5 ■ RATIONAL DELIBERATION EITHER ON ONE SIDE THE BURDEN OF PROOF IS ON THE OTHER SIDE, OR 8.3 Begging the Question Against THERE IS AN IMPASSE A common mistake that undermines argument is committed by failing to discharge the burden of proof. Suppose that we assert 'Not P' (i.e., that Pis false), but someone else, Melinda, has just offered us some good reason for thinking that P is true. BOX 6 ■ HOW TO AVOID BEGGING THE QUESTION AGAINST Don't include any controversial statement among your premises without first offering adequate reasons for it. I­ (/) z <( (') <( z 0 I­ (/) LiJ =i a LiJ I I- (') z (') (') LiJ (l] "'. co 11111 0 UJ 0 z :::) 0 0: 0 (f) zz :::) 0 0 f'.= z a. -� e ::::i 0 (f) > (f) <( <( The burden of proof is now on us, and we must discharge it by offering an adequate argument against P. The failure to do so-by assuming that P is false, without offering a reason for this-commits the fallacy of begging the question against Melinda. For it amounts to implicitly reasoning in either of these viciously circular forms: 17 1. P is false 2. P is false Or similarly, 18 1. NotP 2. Not P The fallacy of begging the question against (your opponent) is often committed in everyday arguments on controversial topics. For example, when someone maintains 19 1. In abortion, the fetus is intentionally killed. 2. A fetus is an innocent person. 3. Intentionally killing an innocent person is always murder. 4. Abortion is always murder. Although 1 seems unobjectionable, 2 and 3 are controversial premises that cannot be employed unless good reasons have already been offered to back them up. Premise 2 begs the question against the view that a fetus is not a person-a view that can be supported in a number of ways (as most parties to the current popular debate over the morality of abortion now recognize). Begging the question against can be difficult to detect, for it involves presupposing the truth of premises that, although controversial, are sometimes inadvertently taken for granted. To avoid this fallacy, always abide by the rule in Box 6 above. BOX 7 ■ SECTION SUMMARY 1. When an argument begs the question, at least one premise assumes the conclusion being argued for. 2. When an argument begs the question against, at least one premise assumes something that is in need of support. Exercises 1 . What do all fallacies of presumption have in common? 2. What does it mean for an argument to be 'circular'? Is all circularity bad? 2. Define non-cogency in relation to begging the question. 3. What's wrong with a question begging argument? 4. What is the fallacy of begging the question against? How does it differ from begging the question? 5. Against whom is the question begged in any argument that begs the question against? 6. Could the conclusion of a question-begging argument be true? Explain. 7. What is meant by burden of proof? How do commonsense beliefs matter to it? 8. Where is the burden of proof at each stage of a deliberation? II. Each of the following arguments begs the question. Explain why. 1. Dylan is a brother. Therefore, Dylan has a sibling. SAMPLE ANSWER: The logical thinker who rejects the conclusion would reject that Dylan is a brother. 2. Capital punishment is cruel, for it is cruel and unusual punishment. And it's demeaning to the society that inflicts it. *3. The mind is different from the body. Hence, the mind and the body are not the same. 4. Mount Aconcagua and Mount Whitney are both tall mountains. But Mount Aconcagua is taller than Mount Whitney. Consequently, Mount Whitney is shorter than Mount Aconcagua. *5. Demons are supernatural beings. Supernatural beings are only fictional. Therefore, demons do not exist. 6. Dorothy is a historian. For, she is a historian and art collector. *7. Since Aaron is a hunter, he is someone who hunts. 8. The U.S. president and the British prime minister both oppose the treaty. Hence, it's false that both leaders do not oppose the treaty. *9. If a plane figure is a circle, then it is not a rectangle. Therefore, if the figure is a rectangle, then it is not a circle. 10. The first witness is not trustworthy, since he is not reliable. Ill. [Note: This exercise is somewhat more challenging.] For each of the above arguments, determine whether the circularity is formal, conceptual, or both. IV. For each of the following arguments, determine whether it would, under normal circumstances, beg the question, beg the question against, or do both. 1. Whoever is less productive should have lower wages. Women are less productive than men. Hence, women should have lower wages. SAMPLE ANSWER: Begs the question against 2. Euthanasia is murder and is wrong. So, euthanasia is wrong. *3. Fido is a puppy. Therefore, Fido is a young dog. 4. A woman has an absolute right to control her own body. And if a woman has an absolute right to control her own body, then abortion is morally permissible. Therefore, abortion is morally permissible. *5. Since the Democrats won the '08 presidential election, it is simply false that they didn't win. 6. Derek Jeter has an insurance policy on his cars, for it is not the case that his cars lack such a policy. *7. The fetus is an unborn baby. Therefore, it is not the case that the fetus is not an unborn baby. 8. Anyone who is an idealist is also a loser. Thus idealists are losers. 9. Vladimir is a bachelor. Therefore, Vladimir is unmarried. f--­ (/) z <( CJ
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<( <( 11&1 *10. Infanticide is always morally wrong. So, infanticide is never morally right. 11. If there is intelligent life elsewhere in the universe, then life on Earth is not unique. But life on Earth is unique. Hence, there is no intelligent life elsewhere in the universe. 12. The right to life is God's will. Therefore, the right to life is the will of Divine Providence. *13. Given atheism, God doesn't exist. But it is not the case that He doesn't exist, so atheism is mistaken. 14. Priscilla got a B on her philosophy paper this semester. Therefore, she turned in a philosophy paper this semester. 15. There is life after death. Therefore, there is an afterlife. *16. Since no person should be denied freedom, and Bruno is a person, it follows that Bruno is entitled to freedom. 17. Magdalene is a sister. Therefore, she is a female. 18. Since capital punishment is murder, capital punishment is wrongful killing. *19. Northfield is not far from Minneapolis. Thus Northfield is close to Minneapolis. 20. Socialism is an unjust system of government. Unjust systems of government must be abolished. Therefore, socialism must be abolished. V. Determine whether the following arguments are possible or impossible. 1. An argument that is cogent for a logical thinker but not rationally compelling. SAMPLE ANSWER: Impossible 2. A valid argument that is non-cogent. 3. A sound argument that is non-cogent. *4. A question-begging argument that is not circular. 5. A circular argument that is not fallacious. *6. A cogent argument that begs the question against. 7. A sound argument that is cogent. *8. A question-begging argument that is sound. 9. A question-begging argument that is rationally compelling. *1 0. The burden of proof being on the side that has offered the most cogent argument. VI. In the deliberation described below, determine where the burden of proof lies at each stage: if on Carolyn, write 'C'; if on Karl, write 'K'; and if there is a dialectical impasse, write 'I.' 1. C rejects a commonsense belief held by K. SAMPLE ANSWER: C 2. C defends her rejection with an argument that begs the question against K. 3. C recasts her argument in a way that makes it clearly unsound. *4. C offers a new argument that turns out be invalid. 5. C's argument undergoes another recast that makes it cogent. *6. K advances a valid yet question-begging argument against C. 7. K offers a non-question-begging argument with clearly false premises. *8. K recasts his argument so that it is now as cogent as C's. VII. In the following deliberation, either Sor O has the burden of proof. Identify which has it at any given stage in the deliberation, and mark dialectical impasses. Explain your choice. 1. S makes a claim that challenges a commonly held belief. SAMPLE ANSWER: Burden of proof on S. When commonsense beliefs are at issue, the burden of proof is on the challenger. 2. S attempts to support her claim by offering an inductively weak argument. *3. S recasts her argument so that it is now clearly valid but unsound. 4. S recasts her argument again so that it is now sound but question-begging. *5. S's argument undergoes another recast that makes it deductively cogent. 6. 0 responds with a valid argument that begs the question against S. *7. 0 recasts her argument so that it is now non-question-begging but plainly unsound. 8. 0 recasts her argument once more so that it is now as cogent as S's argument. VIII. YOUR OWN THINKING LAB *1. Consider the following argument: "Marriage can be only between two persons of different sexes. Therefore, gay couples cannot be married." What's the matter with this argument? 2. Provide an argument that both begs the question and begs the question against. 3. Provide an argument that begs the question without begging the question against. 4. Provide an argument that begs the question against without begging the question. *5. Discuss the conditions an argument must meet to be deductively cogent. *6. An argument that is invalid always falls short of being rationally compelling, but could such an argument be cogent? Must its conclusion be rejected? Explain your answers. 7. Imagine a debate in which two rival claims are equally well supported by observational data. Of the two, one agrees with common sense, the other doesn't. Does this make a difference? Where is the burden of proof? Explain. 8. Discuss what's wrong with an argument that begs the question. 9. Discuss what's wrong with an argument that begs the question against. I­ (/) z <( 0 (/)
<( <( - *10. Suppose you are engaged in a rational debate. Your opponent has just offered a seemingly adequate argument for a claim that you wish to reject. Where is the burden of proof? What does 'burden of proof' mean? 8.4 Complex Question Another fallacy of presumption is complex question, which is a pattern of mistake in asking a question that can be answered only by yes or no, but which assumes either 1. that there is only one question when there are in fact two or more, each with its own answer, or 2. that some claim is true when in fact it is either false or, at the very least, doubtful. Whenever this fallacy is committed, the question being asked is unfair because it has an unjustified assumption embedded within it, in one or the other of these two ways. An example of 1 would be a question one can imagine being asked of a presidential candidate: 20 If elected, would you continue the best traditions of your party and promote wasteful spending on welfare programs that only encourage laziness? Yes or no? Clearly, this is not one question but two. The candidate may indeed want to continue the best traditions of his party but also have no intention of promoting "wasteful spending on welfare programs that only encourage laziness." But the interlocutor is demanding a yes or no on the whole query at once and not allowing him to divide the question. An example of 2 would be a question that is implicitly critical of the person being queried. A classic case of this is that of a man who's asked, 21 Have you stopped beating your wife? Here a 'yes' is just as bad as a 'no,' because it seems to follow from this question that the addressee was engaged in wife beating. Questions of this sort are unfair, since the person queried will convict himself with either answer. (Note, however, that context does matter. If a man is actually known to be a wife beater, then posing (21) to him would not commit a fallacy.) Consider another example: 'Tyler is a high school student who plays in a punk-rock band. He has multiple piercings and tattoos but has never used drugs of any kind. One evening he has a date with Dahlia to go to the movies. But when he arrives to pick her up, he meets her father, who regards him with suspicion and says, 22 Before you take my daughter to the movies, I must ask you this: do you intend to conceal from me your history of marijuana use? Now, what is the correct answer to this? Obviously, 'Tyler doesn't want to answer 'yes.' But if he answers 'no,' then that is equally to admit to marijuana use (something he's innocent of). Either answer will convict him. Notice, however, that that is only because the question itself is unfair. It assumes-without any supporting evidence-that the young man has used marijuana! It's not difficult to see the mistake here. But how is this an argument? First, the question asks whether or not Tyler intends to conceal his history of marijuana use. If he does, then he has a history of marijuana use. And if he doesn't, then he also has a history of marijuana use. Assuming that he either does or doesn't, it follows that he has a history of marijuana use. But there is a problem with these premises, since they rest on an assumption that is false­ namely, that the person queried ('Tyler) does have a history of marijuana use. Yet not all arguments that commit the complex-question fallacy are intended specifically to trap an individual. Some consist simply in questions phrased so that any answer a respondent gives to them must necessarily endorse an unsupported assumption built into the question itself. Suppose that a politician, in a speech, asks, 23 Does my opponent agree with the president's disastrous economic policy which is now leading our nation to ruin? Because the question assumes (without anything in the context making it plausible) that the president's economic policy is 'disastrous,' and that it is 'leading our nation to ruin,' anyone who responds to (23), either in the affirmative or in the negative, will be implicitly endorsing those views! Again, a fallacy of complex question has been committed, in this case by the politician. To a complex question, it seems, any answer is a wrong answer. But that is only because there is something wrong with the question itself. It is phrased so that it assumes something not yet supported. BOX 8 ■ HOW TO AVOID COMPLEX QUESTION Beware of any yes/no question presupposing that, if the answer is yes, a questionable proposition P (for which no argument has been offered) follows, and if you answer is no, P also follows. 8.5 False Alternatives False alternatives is a defect in reasoning that might affect an argument containing a disjunction as premise. A disjunction is a compound proposition with two members or 'disjuncts.' An exclusive disjunction has the form 24 Either P or Q.(but not both). Here P and Q represent propositions standing as exclusive alternatives, because if one is true, the other is false and vice versa. For example, 25 Either the groundhog hibernates during the winter or it continues in a state of animation. This is an exclusive disjunction, since it presupposes that exactly one of the alternatives is true. By contrast, consider the inclusive disjunction 26 Apples that are either too small or too ripe are discarded. (j) w >

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<{ <{ El XI. The following arguments are instances of begging the question, begging the question against, complex question, false alternatives, or accident. Determine which is which. (Note that in some cases an argument could both beg the question and beg the question against.) 1. Jane has to come to work. She may be sick, as she says, but she is needed in the office. Whenever an employee is needed in the office, she must show up. SAMPLE ANSWER: Accident 2. Women should not be deployed for military service, because no woman should serve in the military. 3. We've had record hot weather for two weeks. In a heat wave as bad as this, there are only two options: either one spends all day complaining about it, or one shuts up and goes about one's work. So we'll have to go on with our work, since complaining is not an option for us. *4. Is MacKenzie still forcing his employees to use obsolete technology in his office? 5. Everyone should get some strenuous physical exercise every day, like running a mile before breakfast. So, my Uncle Olaf, who is ninety-seven years old, ought to run a mile every day before breakfast. *6. To be wealthy, you have to be either a Wall Street financier or a drug dealer. You are wealthy, but you are not a Wall Street financier. It follows that you are a drug dealer. 7. Have they given up yet on casting a short blond actor, Daniel Craig, as James Bond? 8. Sarah had better marry Dombrowsky. For either she marries him or ends up single. *9. In general, any average person's statistical chance of suffering a gunshot wound is minimal. So I'm not worried about my friend Al, who just joined the police force. 10. Did Melissa manage to get along with Justin? 11. Was he able to stay out of trouble during his last visit? *12. Murderers don't have a right to life. Since Joe is a murderer, he doesn't have a right to life. 13. Certainly there is life after death, since there are people who have lived previous lives and have memories of those earlier selves long ago. 14. The law clearly states that if citizens fail to pay their taxes, they'll be prosecuted. So my four-year-old cousin Egbert should be prosecuted! After all, I happen to know that he paid no taxes last year. *15. Do you support Senator Krank's ridiculous school appropriations bill, which would bankrupt our state government? 16. Martial arts are either taekwondo or jujitsu. Yoshizuki is trained in the martial arts, but he doesn't practice taekwondo. Therefore, he practices jujitsu. *17. It will surely be in the interest of the United States to abolish tariffs on commerce with nations south of the Rio Grande, for free trade with Latin America can only be in the interest of the United States. 18. Do you really want to pass up your chance of a lifetime to invest in Swampwood Estates, Florida's most exclusive and luxurious new residential neighborhood? 19. Were you sober last weekend? *20. I can say whatever I want about my neighbor, O'Connor. Whether it's true or not, I can say it, and no one can stop me! After all, the First Amendment guarantees freedom of speech in the United States. 21. If there is no substantial economic growth in our country this year, then either there will be disruptive social upheaval or the military will overthrow the government. We're at a point now where there could only be minimal economic growth. Therefore, we can expect a military overthrow of the government to happen soon. *22. Lakeesha's donation of her prize money to AIDS research was selfless. It follows that her action was not selfish. 23. Either the United States invades Mexico or drugs cartels continue to destabilize Mexican society. Since the United States won't intervene, it follows that drugs cartels continue to destabilize Mexican society. 24. Since media literacy is a proven tool against crime, it could be used to reform convicted murderers. *25. Abigail has been reporting intractable insomnia since 1999. A warm glass of milk before going to bed should end the problem for her. After all, it helps others to sleep well. XII. YOUR OWN THINKING LAB *1. 'If a principle has proved to be generally true or reliable, it's probably true all of the time.' Should we agree with this rule? Explain your answer. 2. Suppose I find in an argument some premise that itself can be accepted only if the conclusion has already been accepted. What's wrong with the argument? Explain the fallacy it commits. 3. Provide two circular arguments, one that begs the question and one that doesn't. What's wrong with those arguments? *4. Ask a complex question and explain why it is a fallacy. ■ Writing Project The principle that we should be tolerant of the beliefs and actions of others is highly regarded in our culture. Find at least three examples of current beliefs or actions that you think fall beyond the reach of that principle and write a short essay entitled "Beyond Tolerance." In each case, first construct an argument where that principle is used to contend that some episode (of hate speech, Holocaust denial, terrorism, etc.) should be tolerated. Then explain why the argument fails by showing that it commits the fallacy of accident. ■ Chapter Summary Fallacies of Presumption: They make an argument fail in virtue of some unwarranted assumption built into its premises. The argument seems OK only when the assumption is made. They include: 1. Begging the Question. The argument features at least one premise that itself depends on the conclusion's being true, so that it can be accepted only if one has already accepted the conclusion. 2. Begging the Question Against. The argument features at least one controversial premise that is assumed to be true but not argued for. Note: when a claim is controversial, an argument that commits this fallacy is no help in discharging the burden of proof. >­
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< < - 3. Complex Question. The question either has an unsupported assumption built into it or is a conflation of two or more different questions. 4. False Alternatives. The argument features a premise with a disjunction, mistakenly taking it to be either exclusive, when in fact both disjuncts could be true, or exhaustive, when in fact there is a third alternative. 5. Accident. The argument assumes that some principle generally applicable is applicable also in the anomalous case, when in fact it isn't. ■ Key Words Presumption Begging the question Begging the question against Burden of proof Commonsense belief Vicious circularity Formal circularity Conceptual circularity Complex question False alternatives Accident Benign circularity CHAPTER From Unclear Language to Unclear Reasoning This chapter considers some common forms of unclarity in language, and the ways in which they lead to unclarity in reasoning. Its topics will include Three types of linguistic unclarity that may lead to fallacies: vagueness, ambiguity, and confused predication. ■ The heap paradox. ■ The fallacy of slippery slope. The fallacy of equivocation. ■ The fallacy of amphiboly. ■ The fallacy of composition. The fallacy of division. 187 w 00 �z 0z zo <( (fJ _J <( a: w <( a: w a: _J <( Ow z _J :::> 0
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9.1 Unclear Language and Argument Failure

Vagueness, ambiguity, and confused predication are three different sources of unclear
language. Each may lead to argument failure, and we shall find, rooted in these defects, several
types of informal fallacy as well as a type of puzzling argument. When an expression is vague
to a significant degree, it is unclear whether it applies to certain things. For instance, it’s
unclear whether ‘rich’ applies to Betty, who has $900,000 in her bank account. She’s certainly
doing well, but she’s not even a millionaire, much less a billionaire! The problem is that ‘rich’ is
a vague word: for some cases, it’s not clear what (or who) counts as being ‘rich.’ By contrast,
when an expression is ambiguous to a significant degree, it has more than one meaning and
reference, and it is unclear which one is intended by its user. For example, it is unclear whether
“challenging arguments” means either the act of disputing some arguments or complex
arguments that are difficult to follow. Roughly, the reference of an expression is what the
expression applies to, while its meaning is its content. Consider

1 The sum of 1+1

2 The smallest even number.

Both (1) and (2) may be used to refer to the same thing, since they both apply to the same
number-namely, the number 2. Yet (1) and (2) don’t have the same content, which is equivalent
to saying that they don’t have the same meaning, for

MEANING = CONTENT

Since reference and meaning belong to the semantic dimension of a language, vagueness and
ambiguity are two different forms of semantic unclarity. Each may undermine an argument by
affecting some of the terms or concepts that make up its premises and conclusion.

Confused predication, on the other hand, also amounts to semantic unclarity, but it can
arise only at the level of relations between statements in an argument. That is, confused predi­
cation is a fallacy involving a certain error committed in using some predicate, or expression
that attributes some feature or quality to a thing- for example, ‘occupying 60 percent of the
surface of the Earth’ in the conclusion of this argument:

3 Since oceans occupy 60 percent of the surface of the Earth and the Mediterranean is
an ocean, therefore the Mediterranean occupies 60 percent of the surface of the
Earth.

While ‘occupying 60 percent of the surface of the Earth’ might be truly predicated of all oceans
taken collectively, it obviously fails to be true of the Mediterranean Sea. The confusion in (3) is
a common type of mistake that stems from an erroneous inference involving a predicate (we’ll
have more to say about predicates later in this chapter).

Linguistic unclarity rooted in any of these phenomena (confused predication, vagueness,
or ambiguity) can render an argument fallacious. Yet before we examine common ways
in which this may happen, we must ask why such mistakes matter to logical thinking at all.

BOX 1 ■ SOME FALLACIES OF UNCLEAR LANGUAGE AND

A PARADOX

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Heap

I paradox
Vagueness

Slippery

Slope

Equivocation
UNCLEAR

Ambiguity
LANGUAGE

Amphiboly

Composition
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Predication
Division

Two millennia ago, Greek philosophers pointed out that unclarity in language is a sign of
unclarity in reasoning. Today we’d say much the same. Assuming that speakers are sincere,

what they say is what they believe. And since beliefs are the building blocks of their reasoning,
it is then quite likely that any unclarity in what they say results from unclarity in how they rea­

son (for more on this topic, see Chapters 2. and 3).

9.2 Semantic Unclarity

Vagueness and ambiguity are forms of semantic unclarity that may affect linguistic expressions
of different kinds, as well as the logical relations between them. When an expression is vague, it
is unclear whether or not certain cases fall within its reference. When an expression is ambiguous,

it is unclear which of its possible meanings is the one intended by the speaker. Suppose some­

one says

4 She got the cup.

Furthermore, this is said in a room where there are several women, without pointing to any­
one in particular. In this context, it is unclear to whom the word ‘she’ applies. At the same time,
the term ‘cup’ is ambiguous, since it may equally mean and refer to either ‘bowl-shaped drink­

ing vessel’ or ‘sports trophy.’ Furthermore, if we assume that it is used to refer to a drinking
vessel, it is unclear just how wide the range of its application may be. Does it apply, for exam­

ple, to coffee mugs? What about beer tankards? These seem borderline cases about which ‘cup’
neither definitely applies nor definitely fails to apply. Hence, ‘cup’ is not only ambiguous, but
also to some degree vague.

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BOX 2 ■ VAGUENESS AND AMBIGUITY

■ When an expression is vague, there are borderline cases where it is unclear whether the
expression applies.

■ When an expression is ambiguous, it has more than one meaning and sometimes more than
one reference.

Vagueness and ambiguity are, however, also found at a higher level in the statements that

make up arguments. Here the worst-case scenario is one where a defect of either type renders

an argument misleading. In any such argument, although its conclusion might at first appear

acceptable on the basis of the argument’s premises, a closer look could show that in fact it’s

not. The premises actually provide no support for it.

Logical thinkers should be alert for misleading arguments and try, through careful, case­

by-case scrutiny, to unmask fallacies lurking behind vague or ambiguous language.

These two forms of semantic unclarity are unavoidable features of many everyday

arguments. Such arguments are, after all, cast in a natural language, which, unlike a formal

language, is rich in semantic connotations. For example, suppose that a college instructor, on

the day of an examination, receives this phone message on her answering machine:

5 This is Mary. I was at the bank during the test, so I’d like to take the makeup.

Unable to recognize the voice, and aware of several financial institutions as well as a river

nearby, the instructor cannot make much of (5). For one thing, of the several students

named ‘Mary’ who missed the exam, it is unclear who the caller in (s) is. Furthermore, of the

two meanings of ‘bank’ possible in (5), either ‘financial institution’ or ‘side of a river,’ it is

unclear which one is intended. Suppose the student who left the message later sends a note

from the local Citibank branch attesting that, on the date of the exam, she, Mary McDonald,

had to go there to refinance her mortgage. Putting two and two together, the instructor

reasons that

6 Mary McDonald was the student who reported her absence. She can prove she was at

the local Citibank branch the day of the exam. Thus she qualifies for the makeup.

No ambiguity remains now: a look at contextual information has eliminated the semantic

unclarity in (s) above.

Yet sometimes semantic unclarity bearing on the soundness or strength of an argument

persists even after we have engaged in a charitable and faithful reconstruction of the argu­

ment. In that case, we must reject the argument on the ground that its premises provide no

support for its conclusion, even though they might at first appear to support it. As we shall

presently see in detail, each of these two types of semantic unclarity can render an argument

misleading.

BOX 3 ■ HOW TO AVOID AMBIGUITY AND VAGUENESS

Ambiguity and vagueness are a matter of degree. Although they affect most expressions in
natural languages (which is in part why symbolic logic has developed formal languages to study
logical relations such as inference), the fog they raise can often be thinned by looking at the context

that is, other linguistic expressions surrounding the affected ones, and factors in the arguer’s
environment. When we are engaged in argument reconstruction, the principles of charity and
faithfulness recommend that we check the context, when available, to gain semantic clarity.

9.3 Vagueness

Vagueness is at the root of some philosophically interesting puzzling arguments and also of
many fallacious ones. Later in this section, we’ll examine some cases of each. But first, let’s
consider a shortcoming common to all arguments affected by vagueness: indeterminacy.

When either the premise or conclusion of an argument is significantly vague, that

statement is indeterminate: neither determinately true nor determinately false. Such

indeterminacy undermines the argument as a whole.

This is because, as you may recall, to be deductively sound or inductively strong, an argu­
ment must have premises that are determinately true. Without that, it counts as neither.
Consider this argument:

7 1. Tall buildings in Chicago are in danger of terrorist attacks.
2. The 30-story Nussbaum Building in Chicago is a tall building.
3. The 30-story Nussbaum Building in Chicago is in danger of terrorist attacks.

This argument seems valid, since if its premises are true, its conclusion cannot be false. At the
same time, it also seems unsound, for soundness requires determinately true premises, and
premise 2 suffers from a significant degree of vagueness: putting aside the problem that tall­
ness is relative, although a 100-story building is clearly tall (even by Chicago standards) and a

2-story building clearly not tall, it is unclear whether a 30-story building is tall in Chicago. No
contextual information is available to reduce the vagueness of premise 2, which results from
the two facts described in Box 4. The problem is that there is no determinate point or cutoff
between tall Chicago buildings and Chicago buildings that are not tall.

BOX 4 ■ WHAT’S WRONG WITH ARGUMENT 7?

1. It uses the expression ‘tall,’ which has no clear cutoff point between the cases to which it
determinately applies and those to which it determinately does not apply.

2. The 30-story Nussbaum Building is among the borderline cases of things about which it is
indeterminate whether that word applies or not. It is neither determinately tall nor
determinately not tall.

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(.’) (.’) <( z :J - CJ z zo <( (j) _J <( a: w <( a: w a: _J <( Ow z _J :J 0 �z 0 :J a: 0 LL f- When a statement has a vague term applied to a borderline case, that statement is neither determinately true nor determinately false. Try this yourself: run another series with, for example, 'cold,' beginning with the determinately true statement, 'A temperature of zero degrees Fahrenheit is cold,' and continuing to a point where you 'cannot draw the line.' Is it 47 degrees? 48 degrees? 50? Again, any cutoff point in the series would be rather arbitrary. Keep in mind, however, that vague terms may have non-vague occurrences. Compare 8 The 30-story Nussbaum Building is tall. 9 The 100-story John Hancock Building is tall. 10 The one-story Exxon Station on Route 10 is tall. While (8) is indeterminate, (9) seems determinately true and (10) determinately false. BOX 5 ■ SUMMARY OF VAGUENESS When a term is vague, ■ It is indeterminate whether it applies or not to certain borderline cases. ■ There is no cutoff between the cases to which it determinately applies and those to which it determinately does not. When a statement is vague, it is neither determinately true nor determinately false. The Heap Paradox Every bit as puzzling to us today as it was to the philosophers of ancient Greece who discov­ ered it is the heap paradox, also called 'argument from the heap' or 'sorites' (from the Greek, soros, 'a heap'). The argument begins with obviously true premises, but, because they contain a vague term, ends with an obviously false conclusion: 11 1. One grain of sand is not a heap. 2. If 1 grain of sand is not a heap, then 2 grains of sand are not a heap. 3. If 2 grains of sand are not a heap, then 3 grains of sand are not a heap. 4. If 3 grains of sand are not a heap, then 4 grains of sand are not a heap. 5. If 4 grains of sand are not a heap, then ... 6. A large number (say, a million) grains of sand are not a heap. Given (u), no matter how many grains of sand there are, they never make up a heap. Something has gone wrong in (u), but since it is difficult to tell what, (11) is a puzzle or para­ dox. After all, it seems that, A. The argument is valid. B. Its premises are true. C. Its conclusion is false. D. But a valid argument can't have true premises and a false conclusion. Like other heap arguments, (11) then creates a paradox, for D is true by definition of 'valid argument.' Therefore, A, B, and C cannot all be true, but it is difficult to say which of them is false. A paradox is a puzzle without apparent solution involving claims that cannot all be true at once, even though each seems independently true. Standardly, a paradox may be dealt with in one or the other of two ways: it may be solved or it may be dissolved. To solve a paradox, at least one of its claims must be shown false. To dissolve it, it has to be shown that the claims are not really inconsistent. Until we do either the one or the other, the paradox remains. Since antiquity, the heap paradox has resisted many attempts of both kinds, all of which have turned out to be flawed in one way or another. Let's now use another vague word, 'child,' to run a simplified heap paradox. 12 1. A 3-year-old is a child. 2. If a 3-year-old is a child, then a 4-year-old is a child. 3. If a 4-year-old is a child, then ... 4. A 90-year-old is a child. Again, the argument seems valid, its premises true, and its conclusion false. Premise 2 suggest a chain of premises such as 13 If a 4-year-old is a child, then a 5-year-old is a child. 14 If a 5-year-old is a child, then a 6-year-old is a child. The series eventually reaches borderline cases such as a 14-year-old or a 15-year-old, about whom to the term 'child' neither clearly applies nor doesn't apply. There is no cutoff point between these and the previous cases, to which the word clearly applies. Or between these and the following case, to which the word clearly doesn't apply: 15 A no-year-old person is a child. The unclarity affecting (12), then, is owing to the vagueness of the word 'child.' More needs to be said about what goes wrong in the heap argument, but on the basis of its puzzling aspects, it has all the marks of a paradox. The above arguments run into the heap paradox because they feature words such as 'heap' and 'child,' which are affected by vagueness. (/) (/) w z w :J 0 <( C? Ol ml w C) C) <{ z ::J - C) z zo <{ (/) ....J <{ a: w <{ a: w a: ....J <{ Ow z ....J ::J 0 �z 0 ::J a: 0 lJ.. I- IDI The Slippery-Slope Fallacy By contrast with the heap paradox, we can tell what has gone wrong in arguments that commit the fallacy of concern here: A slippery-slope argument proceeds from a premise about a harmless scenario to one or more premises about apparently similar scenarios that are taken to have unwelcome consequences, either flouting well-accepted rules or leading to disaster. The argument would commit a fallacy just in case there is no good reason to think ■ That the scenarios in question are analogous in the way assumed in the argument, or ■ That the chain of events envisioned will in fact happen as assumed in the argument. Thus arguments that commit this fallacy begin with a premise that seems clearly true and move through a continuum of cases to a conclusion that appears to be the unavoidable result of sound reasoning. Yet close scrutiny often shows that it isn't sound. Imagine two arguments on opposite sides of a City Council debate on whether to enact legislation requiring registration of handguns. 16 Council Member Robinson argues, "If we pass a law requiring registration of handguns, that will lead inevitably to other laws requiring registration of all firearms, including hunting weapons. And that will then mean that the government will have a list of all the gun owners. But if the government has such a list, the next inevitable step is the confiscation of all weapons by the government. From this, it is but a small step to dictatorship and the end of freedom." 16' Council Member Richardson replies, "If we fail to pass a law requiring the registra­ tion of handguns, these guns will become easier and easier to obtain. And this will mean that criminals and psychopaths of every description will have their pick of dangerous firearms, including assault weapons. If that happens, crime will increase exponentially and our cities will become lawless battlefields. Ultimately, all social order will break down. Armed thugs will run roughshod over the rights of citizens as our nation descends into anarchy." The thing to notice in this debate is that both of these arguments commit the slippery-slope fallacy. They both begin by issuing a warning about an apparently innocent first step and then predict a succession of worsening conditions, leading ultimately to disaster, if the initial step is taken (Robinson) or not taken (Richardson). But is there really any good reason to think that the hopeless slide to catastrophe envisioned in either of these two very different arguments really would happen? Of course, such things could happen. But we're not justified in believing it would on the basis of the "reasons" presented here. Both of these council members are offering arguments that are little more that speculative fearmongering and hype. They plainly commit slippery slope. In a different variant of the slippery-slope argument, an argument may fail because there is no good reason to think that an assumed similarity between premises actually obtains. Suppose you arrive five minutes late to a wedding without making much of it. The conven­ tional rule is that arriving, say, sixty minutes late to a wedding is a serious breach of etiquette and therefore not acceptable. Someone thinks that your five-minute delay is not acceptable, because allowing it is not significantly different from allowing a sixty-minute delay, which would in effect overthrow that rule altogether. This example of slippery-slope fallacy runs, 17 Whatever justifies arriving five minutes late to a wedding would justify arriving six minutes late, seven minutes late, ... and even sixty minutes late! Thus accepting a justification for arriving five minutes late would amount to overthrowing an important social convention. Clearly there is a kind of reasoning by analogy here, since consistency requires that we treat like cases alike, ascribing the same qualities to each pair of relevantly similar cases in the series (six minutes late is not much different from seven minutes late, which is itself not much different from eight minutes late, etc.). But the background assumption seems to be that a sequence of small differences can never amount to a substantial difference between any two points in the sequence. And that's plainly false. Small differences can sometimes add up to a big difference in the end. Furthermore, even in the comparison of two similar cases, it may turn out that some predicates are true of one without being true of the other. For example, on some highways the law stipulates a speed limit of 70 miles per hour. Now, there is no significant difference in speed between 70 miles per hour and 71 miles per hour; but because of the law, driving at 70 miles per hour on those highways is legal, while driving at 71 is technically illegal. So the predicate 'legal' truly applies in one case but not in the other, even though they are otherwise not substantially different. We may conclude that any argument committing the slippery-slope fallacy rests on this false principle: What is true of A is also true of Z, provided there is a series of cases B, C, ... , Y between A and Z that differ from each other only minimally. BOX 6 ■ HOW TO AVOID THE SLIPPERY-SLOPE FALLACY Reject the principle fueling a slippery-slope argument, for that something is true of some given case doesn't guarantee that it's likewise true of any other similar case. Although it is reasonable that similar cases share many predicates, small differences in a series of cases can add up to a big difference between the initial case and the one featured in the slippery-slope argument's conclusion. The slippery-slope arguer fails to take this into account. 9.4 Ambiguity Vagueness must be distinguished from ambiguity. As we have seen, a word or phrase is vague if its reference is indeterminate, so that it is unclear whether or not it applies to a certain case. But ambiguity is a different kind of semantic unclarity that is also apt to cause havoc in argu­ ments, and therefore is equally likely to mislead. A word is ambiguous if it has more than one meaning and a given context makes unclear which meaning is intended. >­
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<( 0.. MU• 10.1 Fallacies of Relevance Another source of error in reasoning that can cause an argument to be misleading is the failure of premises to be relevant to the conclusion they are offered to support. Even if a premise is plainly true, if it is also irrelevant to the conclusion it is supposedly backing up, then it cannot count as a reason for it, and the argument fails. Arguments that are fallacious by virtue of having irrelevant premises often rely on distractions that draw attention away from what truly matters for the conclusions at hand, and thus are sometimes employed as rhetorical tricks by artful persuaders who aim to influence us by psychologically effective but logically defective means. There are several types of informal fallacy that manifest this form of error, often known as 'fallacies of relevance.' We'll consider six of them here. BOX 1 ■ FALLACIES OF RELEVANCE I FALLACIES OF I . RELEVANCE . APPEAL TO APPEAL TO PITY FORCE .----� _-_--._-_-_-_-_,-' AD HOMINEM BESIDE THE jsTRA� MANI POINT . 10.2 Appeal to Pity One type of fallacy of relevance is the appeal to pity (also called ad misericordiam). An argument commits the fallacy of appeal to pity if and only if its premises attempt to arouse feelings of sympathy as a means of supporting its conclusion. Consider, for example, an argument that was once made on behalf of clemency for Rudolf Hess, a close associate of Hitler arrested in Britain during World War II and later sentenced to life imprisonment for war crimes. In 1982, when Hess was old and in poor health, some people argued that he should be freed from prison. The argument went this way: 1 1. Hess has already spent more than forty years in prison. 2. He is in his eighties now and his health is failing. 3. This elderly man should be permitted to spend his last years with his family. 4. Hess should be granted clemency. But Hess's age and failing health were irrelevant to the real issue: his guilt as one of the founders of a regime that had terrorized Europe. Many Russians, whose country had suffered millions of deaths at the hands of the German invaders, recognized this argument as an appeal to pity and objected vigorously. As a result, Hess's sentence was never commuted and he died in prison. A similar argument was offered recently by the mother of a sea pirate, who begged the president of the United States for leniency in her son's case on the grounds that he was "lured into piracy by older friends." According to a report in the Associated Press, the pirate himself expressed contrition. "I am very, very sorry about what we did," he said through an interpreter. "All of this was about the problems in Somalia." But even if we do feel sorry for him, in view of his wretched existence in a war-torn, lawless land, that is hardly enough to justify the murder of innocent merchant seamen on foreign-flag ships. The argument is plainly an appeal to pity. It's worth noting, however, that it's not only on behalf of scoundrels and criminals that people resort to the appeal to pity. We find it in everyday life in many guises, including some uses we may (wrongly) think free of this fallacy-for example, when a student argues, 2 You gave me a B in this course, but . . . can't you give me an A? If I don't have an A, then it'll mean that my grade average will fall, and I won't be able to get into law school! And I've been working hard all semester. The argument in fact is: 2' 1. I've been working hard in this course. 2. Any grade below an A would adversely affect my chances for law school. 3. I should get an A in this course. This argument commits the fallacy of appeal to pity. But not because of premise 1: plainly, how hard the student has been working is not relevant to its conclusion, but that is the fallacy known as 'beside the point' (more on this later). What's making the argument count as an appeal to pity is premise 2: that premise shows that the argument attempts to support its conclusion by making the professor feel sorry for the student. It might succeed in doing that, but it fails to make its conclusion rationally acceptable. More generally, an appeal to pity is a fallacious argument trading on the fact that feeling sorry for someone is often psychologically motivating. Yet that is not a good reason for the argument's conclusion. Logical thinkers would want to be able to recognize and avoid this fallacy. For some tips on this, see Box 2. , BOX 2 ■ HOW TO AVOID APPEAL TO PITY 1. An argument whose premises attempt to provoke feelings of sympathy that might move an audience to accept its conclusion commits the fallacy of appeal to pity. 2. Any such argument should be rejected, since it provides no reason relevant to its conclusion­ that is, it provides no rational support for it. 10.3 Appeal to Force Another informal fallacy trading on feelings, though in an entirely different way, is the appeal to force (sometimes called ad baculum, literally, 'to the stick'). w 0 a: 0 LL 0 f­ __J <{ w 0.. 0.. <{ (') 0 1- z w
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One of the most notorious examples of emotively charged images in the history of political advertising, the commercial was widely denounced as tasteless, prompting Democrats to withdraw it. BOX 4 ■ HOW TO AVOID APPEAL TO EMOTION 1. Be on guard for arguments that attempt, through the use of emotively charged words or images, to elicit a strong psychological response conducive to the acceptance of its conclusion. 2. Any such argument commits the fallacy of appeal to emotion and should be rejected. Why? Because its premises offer only "reasons" that are irrelevant, in the way suggested in (1), to the argument's conclusion. No such argument can provide rational support for its conclusion. The Bandwagon Appeal Some forms of emotional appeal are intended to take advantage of common feelings that seem to be part of human nature, such as the desire not to miss out on the latest trends-for example, when books are marketed as 'best sellers' or a film is touted as 'the Number One Hit Movie of the Summer!' This so-called bandwagon appeal exploits our desire to join in with the common experiences of others and not be left out. But the reasons offered for buying the book or seeing the movie merely note their popular appeal, not their quality. A best seller might be only a shallow entertainment, a hit movie little more than a television sitcom. That they're widely sought does nothing to support the claim that they're worth seeking. Appeal to Vanity Appeal to vanity (sometimes called 'snob appeal') is another of the varieties of ad populum­ this time trying to exploit people's unspoken fears about self-esteem. When a car is advertised as in (7), the advertiser attempts to persuade prospective buyers to buy the car by making an appeal to their vanity. 7 Not for everyone-this is the car that tells the world who you are! In another example of this argumentation tactic, Virgin Atlantic Airways has decided to attract customers to its premium-class service by calling it, not 'first class,' but 'Upper Class.' Can you see what is going on here? 10.5 Ad Hominem Another way arguments can fail because of irrelevant premises is the very common fallacy of ad hominem (literally, 'to the man'), which has less to do with emotion than with personal attack. It is sometimes called 'argument against the person,' but we'll call it by its Latin name, since that has now come to be familiar in everyday usage. An argument commits the fallacy of ad hominem if and only if it attempts to discredit someone's-or some group's-argument, point of view, or achievement by means of personal attack. That is, the fallacious ad hominem rests on some personal consideration strictly irrelevant to the matter at hand, which is intended to undermine someone's credibility, as a means of indirectly attacking the person's position or argument. The problem with such an ad hominem, of course, is that in this way the question of the real merit of that person's position is evaded. Instead, the ad hominem offers only a cheap shot aimed at the person herself. Before turning to some specific arguments of this sort, notice that they all fail to support their conclusions-yet they can be recognized easily and avoided in the way suggested in Box 5. Examples of ad hominem are, unfortunately, easy to find-sometimes committed by people you'd not expect to be committing fallacies. Planned Parenthood recently ran a series of advertisements on buses and subways that featured a photo of several grumpy-looking men in suits. Across the photo was mounted the ad copy, which read, "79% of abortion opponents are men. 100% of them will never be pregnant." We may smile at this rhetorically clever juxtaposition of image and slogan, but, make no mistake, this is an ad hominem against male opponents of abortion. Instead of focusing on what those men's objections to abortion may be, the effect of the ad is simply to dismiss the objections as men's views. But the views of men­ on abortion or any other topic-cannot be legitimately rejected solely on the basis of their provenance (that they are "men's views"). Rather, the question is: Are these views well supported? It's not whose views they are that matters, but do the proponents have good or bad arguments for their claims? Suppose a new political scandal erupts in Washington. Senator Dunster has been caught using public funds to pay for expensive luxury vacations for himself and his family, and another legislator, Senator Brewster, has taken to the Senate floor to denounce this impropriety. But Dunster is a Harvard man and cannot resist pointing out that Brewster's college days were spent at Yale. In a speech, Dunster loudly responds, 8 These charges are all false! And these unfounded accusations are coming from exactly the place we would expect. Apparently Senator Brewster, like all Yalies, cannot resist the temptation to besmirch the reputation of a Harvard man! Here Senator Dunster's argument is an ad hominem that attempts to discredit Brewster's statements, not by speaking to their content (the accusations of impropriety), but by pointing to Brewster's personal background-the fact that he is a Yale graduate. Its clear assumptions are that all Yalies are naturally prejudiced against Harvard graduates, and that that is why Brewster is saying these things! But Dunster's argument simply engages in personal attack: � w z � 0 I 0 <( "'. 0 r­ z <( > w
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<( 0.. it introduces an irrelevant consideration that has no power to actually discredit the opponent's claim (though it may appear to do so). The thing to keep in mind, again, is that it's not who says it that makes a claim well supported or not, but rather whether there are in fact good reasons to back it up. Those reasons should be judged on their own merits: either they provide some support for the claim, or they don't. In our example, we would of course need to hear Senator Brewster 's argument­ presumably citing facts in support of the conclusion that Dunster had behaved inappropriately-in order to determine this. BOX 5 ■ HOW TO AVOID A FALLACIOUS AD HOMINEM 1. Beware of any argument that appeals to some personal facts (or alleged facts) that are irrelevant to its conclusion. 2. Any such argument commits the fallacy of ad hominem and should be rejected, for its premises are irrelevant to its conclusion-that is, they are offered as a means of attempting to discredit an argument or point of view by discrediting the person who presents it. The Abusive Ad Hominem Sometimes ad hominem arguments attack a person's character. Suppose a moviegoer announces, 9 I have no desire to see Woody Allen's latest movie. I'm sure it's worthless, and I wouldn't waste my money on it-not after what I know about him now! He betrayed Mia Farrow and broke her heart when he became romantically involved with Mia's adopted daughter, Soon-Yi Previn. So his movies are without artistic merit, as far as I'm concerned. Now, (9) plainly commits the fallacy of ad hominem, since it seeks to discredit Woody Allen as a film director not by invoking evidence that his movies are artistically questionable, but by a personal attack that refers to his relationship to Soon-Yi Previn (whom he later married). But this ad hominem is of a more abusive sort, since it attacks Allen's character-he is denounced on moral grounds as a 'betrayer,' which is, of course, a term of contempt. But, whatever we may think of Allen's personal qualities, does any of that prove that his films are bad? Isn't all of that simply irrelevant to an assessment of his art? Tu Quoque Finally, the fallacy of ad hominem is also committed when one tries to refute someone's point of view by calling attention to the person's hypocrisy regarding that very point of view. This is sometimes called 'tu quoque' (literally, 'you also'). For example, consider how Thomas Jefferson's writings must have sounded to the British in his day. Jefferson famously wrote, in the Declaration of Independence, "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness." But one can easily imagine how this must have been received in conservative circles in Britain in 1776. Tories certainly regarded this lofty language as risible political rhetoric, since they knew very well that Jefferson was himself a prominent slave holder. In London, Dr. Samuel Johnson scoffed, "How is it that we hear the loudest yelps for 'liberty' among the drivers of Negroes?" Johnson's remark could be expanded into an extended argument that looks like this: 10 1. Jefferson claims that all men are created equal and have rights to liberty. 2. But Jefferson himself is a slave owner. 3. He preaches lofty principles for others that he does not practice himself. 4. Jefferson's claims about liberty and equality are false. Yet if any did actually offer such an argument, it would have committed the fallacy of tu quoque, a form of ad hominem. The imagined argument, after all, tries to bring a personal matter-Jefferson's real-life hypocrisy about race and human nature-into the discussion to cast doubt on his assertions about human equality and rights. Now, it is of course true that the Sage of Monticello did not permit his own black slaves to enjoy the very liberty and equality he so forcefully advocated for himself and his fellow white men. But did that personal failure go any way at all toward showing that Jefferson's claims about liberty and equality were false? Naturally, we all think that people should not be hypocrites. People should practice what they preach. Yet if someone fails to heed this moral maxim, and we point out his hypocrisy, we have not thereby proved that what he preaches is false. In fact, we are only indulging in a form of ad hominem, a tu quoque. Nonfallacious Ad Hominem Before we leave the discussion of ad hominem, there remains one important clarification that should be added. Some uses of argument against the person are not fallacious, for there are contexts in which such an argument may be in order. In public life, for instance, the moral character of a politician may be a highly relevant issue to raise during a campaign, since we do very reasonably expect our elected leaders to be trustworthy. In the second example given above, Senator Brewster's speech calling Senator Dunster's personal rectitude into question amounts to a kind of personal attack, but it commits no fallacy (as does Dunster's reply), since conduct that is unethical (or illegal!) would not be irrelevant to an assessment of a person's fitness to serve as a senator. Brewster's remarks, then, could justifiably be seen as an ad hominem argument but not a fallacious one, for they commit no fallacy of irrelevant premises. Similarly, in the Anglo-American system of justice, which employs an adversarial model in court-with attorneys on opposing sides each presenting an argument for their client's case and trying to undermine their opponent's position-some of what happens in the courtroom may appear to be ad hominem. Here, after all, attorneys might try to discredit a witness by presenting evidence about his personal life. But in fact this does not amount to a fallacious ad hominem at all, since in the courtroom, the reliability of a witness is not irrelevant. Given that the purpose of a witness just is to give testimony, it is highly relevant to know whether the person can be believed or not. Thus an attorney does not commit a fallacy of ad hominem when she appeals to relevant personal matters in an attempt to discredit the claims made by a witness. An attorney's job is to defend her client's interest by aggressively pressing his case, and part of that may include presenting facts about a witness's background and personal life in an effort to undermine his credibility. This is a kind of personal attack, but it commits no fallacy. � w z � 0 I 0
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<( Q. WU• Logical thinkers must bear in mind that courtroom procedure is a specialized subject in the law, and that we're not attempting here to venture into its complexities. When one is called to serve on a jury, one should follow the instructions of the judge. The important thing to notice now, however, is simply that there can be some uses of ad hominem that are not fallacious, and that it is the context that determines when this is so. 10.6 Beside the Point An argument might commit a fallacy of relevance by offering premises that simply have little or nothing to do with its conclusion. Maybe they support some conclusion, but they don't support the one given by the argument. When this happens, the argument commits a beside­ the-point fallacy (also known as ignoratio elenchi). An argument commits the fallacy of beside the point if and only if its premises fail to support its conclusion by failing to be logically related to its conclusion, though they may support some other conclusion. Faced with an argument of this sort, we may at first find ourselves unable to identify the source of the confusion. For example, imagine that opponents of cruelty to animals introduce legislation to ban the mistreatment of chickens, pigs, and cows in certain 'factory farms.' But suppose the corporations who own the farms respond, 11 These farms are not cruel to animals. After all, the farms provide the food that most consumers want, and they do so in a manner that is cost-effective; moreover, these poultry, pork, and beef products are nourishing and contribute to the overall health of American families. The odd thing about (11) is that nothing in its premises contributes toward providing support for the conclusion, 'These farms are not cruel to animals.' Perhaps the premises support some conclusion. But they don't support that one, since they offer no reason to think that the factory farms in question are not cruel. As a result, (11) commits the beside the point fallacy. Here's another example that does so as well. Early in Barack Obama's administration, a state dinner at the White House was attended by a local couple who had not been invited and had no authorization to enter the White House. They were, in effect, party crashers. Threatened with prosecution under federal law for having breached White House security, they responded that they should not be prosecuted because they had "made a sacrifice in time BOX 6 ■ HOW TO AVOID THE BESIDE-THE-POINT FALLACY Logical thinkers should be on guard for 1. Arguments whose premises are simply irrelevant to proving the conclusion. 2. Any such argument is defective, even if nothing else is wrong with it; it commits a beside-the-point fallacy and should be rejected. and money to get ready for the party." Now, let us suppose that it's true that they had made such a sacrifice. Even so, how is that relevant to their claim that they do not deserve prosecution for breaking the law? The proposed "reason" why they should not be prosecuted (namely, the alleged "sacrifice in time and money") is not a reason that supports the conclusion. This argument is plainly an instance of the beside-the-point fallacy. Yet another example of this type of mistake was inadvertently provided by a radio listener who responded to a BBC program predicting a crisis of overpopulation in the United Kingdom by 2051. "We can meet this challenge," the listener confidently asserted, "because we all stood together as one people when we were fighting the Nazis." But there is more than one problem in this argument, not least of them the fact that none of the Britons who fought the Germans in World War II are likely to be alive in 2051. So, whatever the coping skills of those who prevailed in Britain's Finest Hour, their application in the envisaged crisis to come at mid­ century seems unlikely. Moreover, it is not at all clear how a nation's possessing the military skills necessary to defeat Hitler proves anything at all about their ability to overcome an entirely different sort of problem in the foreseen population crisis. Thus the argument is only a beside-the-point fallacy. Its premise, though manifestly true, provides no support for the conclusion. 12 1. We all stood together as one people when we were fighting the Nazis. 2. We can meet the coming challenge of overpopulation. 10. 7 Straw Man Finally, let us consider a type of informal fallacy committed by any argument where the view of an opponent is misrepresented so that it becomes vulnerable to certain objections. The distorted view may consist of a statement or a group of related statements (i.e., a position or a theory). Typically ignored in such distortions are charity and faithfulness, the principles of argument reconstruction discussed in Chapter 4. Given the principle of charity, interpreting someone else's view requires that we maximize the truth of each of its parts (in the case of an argument, premises and conclusion) and the strength of the logical relation between them. Given the principle of faithfulness, such interpretation requires that we strive for maximum fidelity to the author's intentions. It is precisely the lack of charity, faithfulness, or both, in the interpretation of the views of others with whom the arguer disagrees that results in straw man. An argument commits the fallacy of straw man if and only if its premises attempt to undermine some view through misrepresenting what that view actually is. Situations where this type of informal fallacy often occurs include deliberations, such as de­ bates and controversies. Straw man is (regrettably) a common tactic in public life, often heard in the rhetoric of political campaigns. Typically, the straw-man argument ascribes to an oppo­ nent some views that are in fact a distortion of his actual views. These misrepresentations may be extreme, irresponsible, or even silly views that are easy to defeat. The opponent's position, then, becomes a 'straw figure' that can be easily blown away. But to refute that position is of z
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For each of the following arguments, identify the fallacy of relevance it commits. 1. CBS News is trying to make people believe that there are unsafe working conditions in this factory. But I tell you this: anyone who plans to continue working for me should not talk to reporters. SAMPLE ANSWER: Appeal to force *2. I deserve the highest grade, Professor Arroyo, because I studied harder than anyone else. 3. You cannot say that divorce is immoral. After all, you yourself are divorced. *4. A Princeton student found guilty of plagiarism admitted that the work was not her own but argued that the university ought not to penalize her for this infraction, since she had been 'under enormous pressure at the time, having to meet a deadline for her senior thesis with only one day left to write the paper.' -The New York Times, May 7, 1982 5. If Einstein's theory is right, then everything is relative. But 9-11 really happened, and that's a fact. So not everything is relative. Therefore Einstein's theory is wrong. 6. We needn't take seriously what the Reverend Brimstone says when he tells us that people should always be honest in their dealings with others. Just yesterday the Billy Brimstone Evangelistic Association was found guilty of soliciting funds for missionary work and then using them to buy the Reverend Brimstone a new Cadillac. *7. Everywhere, people are increasingly getting rid of their iPods and instead listening to music on their cell phones. That's the way to listen to music on the move! So, if you're up-to-date and in touch with the latest things, you'll get rid of your iPod and use the phone to listen to music. 8. I know you're the coach of this baseball team, and you're entitled to your opinion. But I'm the owner of this ball club, and you work for me. If you really want Scooter Wilensky to play third base, you can put him there. Of course, I can always find another coach. 9. Everybody visits the Art Institute of Chicago. Therefore, you should, too. *10. It's true that Knut Hamsun, the early twentieth-century Norwegian novelist, won the Nobel Prize for Literature, but as far as I am concerned his works are worthless. Anyone who collaborated with the Nazis, as Hamsun did during World War II, was not capable of producing works with literary merit. 11. A protestor demonstrating against the new president said, 'A recount of the ballots is needed in this presidential election. If not, we will blockade airports and highways, we' ll take over embassies, and we' ll bring traffic to a halt all over the country.' 12. People often point out that Richard Wagner, the nineteenth-century German composer of operas, wrote some of the most beautiful and powerful music ever written . But I say all of his music is worthless junk and should never be performed! It's well known that Wagner was a raving anti-Semite. And decades after his death, his biggest fan turned out to be Hitler. *13. In Britain, the president of the Royal Society has suggested that scientific research on how to protect the environment should be supported by 'carbon taxes,' levied on countries producing the most air and water pollution. But this is nonsense, for his own country would be near the top of the list, and he himself drives a pollution-producing car! 14. The governor shouldn' t be blamed for his staff members lying under oath to the grand jury. After all, he was under tremendous pressure at the time. 15. Humans are capable of creativity. Therefore, creativity is a value. *16. Many contemporary physicists accept Heisenberg's indeterminacy principle, which implies that everything is indeterminate. But this cannot possibly be correct, as shown by the fact that mathematical truths are determinate. 17. Professor Nathan's history of the Catholic Church is a classic. But she is a Protestant, so we cannot expect her treatment of Catholicism to be fair. 18. In his dialogue Meno, Plato describes an exchange between Socrates and Anytus, a powerful and influential Athenian politician. Socrates suggests that the reason why the sons of prominent Athenian families often turn out badly is that their parents do not know how to educate them. To this, Anytus replies, 'Socrates, I think that you are too ready to speak evil of men; and, if you will take my advice, I would advise you to be careful. Perhaps there is no city in which it is not easier to do men harm than to do them good, and this is certainly the case at Athens, as I believe you know.' *19. Over two million people die in the United States every year. Therefore, the United States is a dangerous place to visit, and we should take our vacation elsewhere. 20. Reverend Armstrong urges us not to support the war, saying that violence is barbaric in all forms and only breeds more violence in return. 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11.1 Argument as a Relation between Propositions

In this chapter and the next, we’ll return to a topic briefly addressed in Chapter 5: propositional
arguments. Here we’ll have a close look at propositions, the building blocks of propositional
arguments. Consider

1 1. If the Earth is a planet, then it moves.
2. If the Earth does not move, then it is not a planet.

(1) is a propositional argument because it consists entirely in the relation between the proposi­
tions that make it up. Its premise and conclusion are compound propositions, which result
from logical connections established between two simple propositions: ‘The Earth is a planet’
and ‘The Earth moves.’ The connections ‘if … then … ‘ and ‘not’ are among the five types of
truth-functional connectives (or simply ‘connectives’) that we’ll study here-namely,

Truth-Functional Connectives

negation*
conjunction
disjunction
conditional
biconditional

Standard English Expression

notP
PandQ
either Por Q
if P,then Q
P if and only if Q

* As we’ll see, negation is called a ‘connective’ by courtesy.

Here we are using capital letters such as ‘P,’ •�• and ‘R’ as symbols or “dummies” for any propo­
sition. We’ll use other capital letters from ‘A’ to ‘O’ to translate propositions in English into
symbols, reserving P through W to represent non-specific propositions. Whenever possible,
we’ll pick the first letter of a word inside the proposition that we are to represent in symbols,
preferably a noun if available. For example, ‘If the Earth is a planet, then it moves’ may be
represented as ‘If E, then M’-where

E

M

The Earth is a planet
The Earth moves

We’ll resort to the same chosen symbol every time the proposition it symbolizes occurs again.
And if we have already used a certain letter to stand for a different proposition, then a letter of
another word, preferably a noun, in the proposition in question will serve. The argument form
of example (1) may now be represented by replacing each proposition occurring in its premise
and conclusion with a propositional symbol in this way, while momentarily retaining the con­
nective ‘if … then … ‘ in English. The resulting translation is

1 ‘ 1. If E, then M
2. If not M, then not E

Let’s now consider the following arguments with an eye toward translating their propositions
into symbols:

2 1. Ottawa is the capital of Canada.
2. It is not the case that Ottawa is not the capital of Canada.

3 1. Either Fido is in the house or he’s at the vet.
2. Fido is not in the house.
3. Fido is at the vet.

4 1. Jane works at the post office and Bob at the supermarket.
2. Bob works at the supermarket.

5 1. TV is amusing if and only if it features good comedies.
2. TV does not feature good comedies.
3. TV is not amusing.

Once we have translated the propositions into symbols, we obtain

2′ 1. 0

2. It is not the case that not 0

3′ 1. Either For E

2. NotF

3. E

4′ 1. J and B
2. B

5′ 1. A if and only if C
2. Not C
3. NotA

Although (2 1 ) through (s’) feature connectives, not all propositional argument forms do:
(6) doesn’t.

6 1. p

2. p

In ( 6), the propositional symbol ‘P’ stands for exactly the same proposition in the premise and in the

conclusion. Known as ‘identity,’ any argument with this form would of course be valid, since if its

premise were true, its conclusion could not be false. But this is not our present concern. Rather, in

this section we’ve considered propositional arguments and discovered that their premises and
conclusions often feature truth-functional connectives. So let’s now look more closely at these.

11.2 Simple and Compound Propositions

Any proposition that has at least one truth-functional connective is compound; otherwise, it is
simple. Consider

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Here is the picture that will emerge: BOX 1 ■ TRUTH-FUNCTIONAL CONNECTIVES Connective In English In Symbols Symbol's Name negation notP ~P tilde conjunction PandQ P•Q dot disjunction PorQ PvQ wedge conditional if Pthen Q P:J Q horseshoe bi conditional P if and only if Q P=Q triple bar Before turning to each of these connectives, notice that there is always one connective governing a compound proposition, called 'main connective.' By identifying the main connec­ tive, we determine what kind of compound proposition a given proposition is: a conjunction, a negation, a disjunction, etc. Obviously, in cases where a compound proposition contains more than one connective, it is crucial to be able to determine which connective is the main one. Negation Negation is a truth-functional connective standardly expressed in English by 'not,' and symbolized by'-', the tilde. Negation can affect one proposition by itself. Even so, we'll refer to it as a 'connective' by courtesy. In ordinary English, the expression for a negation may occur in any part of a statement. When a negation is added to a simple proposition, that proposition becomes compound. (10), which may be represented as (10') exemplifies this: 1 0 Russell Crowe is not an actor. 10' ~C Here the simple proposition that has become compound by adding a negation is 'Russell Crow is an actor.' In (10'), we've used the tilde to represent negation, and C for the simple proposition affected by it. When possible, we'll use the first letter of an important word occurring in the proposition we wish to represent in symbolic notation. Propositions affected by negation could also be themselves compound. For example, 11 It is false that both Mars and Jupiter have water. 12 It is not the case that Mary is not at the library. To represent propositions that are negations, the symbol for negation always precedes what is negated. (12) is the negation of 'Mary is not at the library,' which is already a negation. So we have a double negation: the negation of a proposition that's itself a negation, which we can represent by the propositional formula 12'~ ~ L Since the two negations cancel each other out, (12') is logically the same as 12" L Any proposition or propositional formula affected by a negation is a compound proposition. The 'truth-value rule' that defines negation, and can be used to determine the truth value of a proposition (or propositional formula) that's affected by that connective, is: A negation is true whenever the negated proposition is false. A negation is false whenever the negated proposition is true. When a proposition is the logical negation of another, the two could not both have the same truth value: where 'P' is true,'~ P' is false; where 'P' is false,'~ P' is true. For example, (11) above, which is true, is the negation of 'both Mars and Jupiter have water,' which is false. But (14) below is not the negation of (13), since both propositions are false. 13 All orthodontists are tall. 14 No orthodontists are tall. Now consider these: 15 Some orthodontists are not tall. 16 Some orthodontists are tall. (15) is the negation of (13), and (16) is the negation of (14), for those pairs could not have the same truth value. But propositions that are logically the same would have the same truth value. For example, if (17) is true, (18) is also true. 17 Lincoln was assassinated. 18 It is not the case that Lincoln was not assassinated. (18) is a case of double negation: it is the negation of 'Lincoln was not assassinated.' Notice that propositions featuring expressions such as 'it is not true that,' 'it is false that,' 'it never happened that' are commonly negations-as are some propositions containing prefixes such as 'in-,' 'un-,' and 'non-.' For example, 0 z :::::, 0 0. � 0 0 0 z (/) <( z �Q 0. I­ � in in 2i'. NO ,..: a: � 0. (/) z 0 f­ [/) 0 0.. 0 er: 0.. 0 z :J 0 0.. ::;;; 0 0 19 My right to vote is inalienable. Here 'inalienable' means 'not alienable.' (19) is logically the same as 19' My right to vote is not alienable. Similarly, since 'unmarried' means 'not married,' (20) and (20') are also logically the same: 20 Condoleezza Rice is unmarried. 20' It is not the case that Condoleezza Rice is married. But (21) is not a negation: 21 Unmarried couples are also eligible for the prize. Here 'unmarried' is not being used to deny the whole proposition. It affects only the word 'couples.' Finally, notice that although verbs such as 'miss,' 'violate,' 'fail,' and the like have a negative meaning, they need not be taken to express negations. Conjunction Conjunction is a compound proposition created by a truth-functional connective standardly expressed in English by 'and,' and in symbols by '• ', the dot. The connective for conjunction is always placed between two propositions, each of which called a 'conjunct.' Conjuncts may themselves be simple or compound propositions. Let's consider the conjunctions of some simple propositions: 22 Mount Everest is in Tibet and Mont Blanc is in France. 23 Mars and Jupiter have water. In symbols, these are 22' E • B 23' M •J Recall (11) above: 11 It is false that both Mars and Jupiter have water. The formula that represents this proposition is (11'), which has parentheses to indicate that both M and J are under the scope of the negation. 11 '-(M • J) We'll have more to say on the use of parentheses and other punctuation signs later. Now let's consider why conjunction is a truth-functional connective: because it determines the truth value of the compound proposition affected by it, given the values of its members and this truth-value rule: A conjunction is true if and only if its conjuncts are both true. Otherwise, a conjunction is false. (22) is true since both its conjuncts are in fact true. But if one conjunct is false and the other true, or both are false, then a conjunction is false. Thus (23) is false, since for all we know, both of its conjuncts are false. The following are also false: 24 Mount Everest is in Tibet and Mont Blanc is not in France. 25 Mount Everest is not in Tibet and Mont Blanc is not in France. Since Mont Blanc is in France, the second disjunct in (24) is false, which makes the conjunction false. In a conjunction, then, falsity is like an infection: if there's any at all, it corrupts the whole compound. (Logical thinkers who are contemplating a career in politics should keep this in mind!) In (25)1 both conjuncts are false, since each is the negation of a true proposition. In symbols: 24'E•-B 25'-E •-B Note also that, like (23), many conjunctions in ordinary language are abbreviated. For instance, 26 Rottweilers and Dobermans are fierce dogs. This is just a shortened way of saying 27 Rottweilers are fierce dogs and Dobermans are fierce dogs. Yet (28) is not short for a conjunction of two simple propositions, but is rather a single propo­ sition about a certain relation between some such dogs. 28 Some Rottweilers and Dobermans are barking at each other. Another thing to notice is that conjunction, as a truth-functional connective, is commutative­ that is, the order of the conjuncts doesn't affect the truth value of the compound. Assuming that (26) is true, the facts that make it true are exactly the same as those that make 'Dobermans are fierce dogs and Rottweilers are fierce dogs' true, which are also the same that make (27) true. However, we must be careful about this, since sometimes order matters. When it does, the conjunction is not a truth-functional connective: for example, 29 He took off his shoes and got into bed. The facts that make (29) true do not seem to be the same as those that make (30) true: 30 He got into bed and took off his shoes. The order of events, and therefore of the conjuncts, does matter in these non-truth-functional conjunctions-as it also does in (31) and (32). 0 z :::i 0 Q. 2 0 0 0 z (/) <( z �Q Q. 1- 2 U) -o (/) Q. (\J 0 ,...: a: � Q. (/) z 0 C- U) 0 0.. 0 a: 0.. 0 z :::, 0 0.. � 0 0 El 31 He saw her and said 'hello.' 32 He said 'hello' and saw her. Finally, note that besides 'and,' there are a number of English expressions for conjunction, including 'but,' 'however,' 'also', 'moreover,' 'yet,' 'while,' 'nevertheless,' 'even though,' and 'although.' Disjunction Disjunction, also a commutative connective, is a type of compound proposition created by the truth-functional connective standardly expressed in English by 'or,' and in symbols by 'v', the wedge. In representing a disjunction, the connective is placed between two propositions called 'disjuncts,' which may themselves be simple or compound propositions. Here are two disjunc­ tions, first in English and then in symbols: 33 Rome is in Italy or Rome is in Finland. 33' I v F 34 Rome is not in Italy or Paris is not in France. 34'-I v-F (33) and (34) are disjunctions and thus compound propositions. Disjunction is a truth­ functional connective because it determines the truth value of the compound proposition it creates on the basis of the values of its members and this truth-value rule: A disjunction is false if and only if its disjuncts are both false. Otherwise, a disjunction is true. Given the above rule, at least one of the disjuncts must be true for the disjunction to be true. So (33) is true, but (34) is false. (35) is also false, for both its disjuncts, both of them compound propositions, are false: 35 Either snow tires are useful in the tropics and air conditioners are popular in Iceland, or it is not the case that Penguins thrive in cold temperatures. 35' (S • A) v ~ P Clearly, the conjunction (S • A) is false because both conjuncts are false, and - P is false because it is the negation of P, which is true. Since both disjuncts in (35) are false, given the truth-value rule for disjunction, (35) is false. In addition to 'or,' disjunction can be expressed by 'either ... or ... ' and 'unless,' and other locutions of our language. It is also sometimes found embedded in a negation in 'neither ... nor ... '(where negation is the main connective). Thus these are also disjunctions: 36 She is the director of the project, unless the catalog is wrong. 36' Either she is the director of the project, or the catalog is wrong. (37) is a shortened version of (37'): 37 Neither the CIA nor the FBI tolerates terrorists. 37' Neither the CIA tolerates terrorists nor the FBI tolerates terrorists. Since 'neither ... nor ... ' is a common way to express the negation of a disjunction, (37) is log­ ically the same as (or equivalent to) 38 It is false that either the CIA tolerates terrorists or the FBI tolerates terrorists. Thus both (37) and (38) may be symbolized as the negation of a disjunction: 38'-(CvF ) Note that here the main connective is negation, not disjunction. Furthermore, (37) and (38) are logically equivalent to (39), which may be symbolized as 39 The CIA doesn't tolerate terrorists and the FBI doesn't tolerate terrorists. 39'- C • -F Finally, a truth-functional disjunction may be inclusive, when both disjuncts could be true ('either P or Qor both'), or exclusive, when only one could be ('either P or Qbut not both'). This book focuses on inclusive disjunction, whose truth-value rule is given above. Material Conditional Material conditional, a type of compound proposition also called 'material implication' or simply 'conditional,' is created by a truth-functional logical connective, standardly expressed in English by 'if ... then ... ,' and in symbols by '::)', the horseshoe. For example, 40 If Maria is a practicing attorney, then she has passed the bar exam. A conditional has two members: the proposition standardly preceded by 'if' is its antecedent, and the one that follows 'then,' its consequent. The conditional is a truth-functional connective because the value of the compound proposition it creates is determined by the truth value of the antecedent and consequent, together with this truth-value rule: A material conditional is false if and only if its antecedent is true and its consequent false. Otherwise, it is true. Thus any conditional with a true consequent is true, and any conditional with a false antecedent is true. The two propositions in a conditional, which may themselves be either simple or compound, stand in a hypothetical relationship, where neither antecedent nor consequent is being asserted independently. Does (40) assert that Maria is a practicing attorney? No. Does it 0 z ::::, 0 a. L 0 0 0 z (/) <( z ::j Q a. I­ L in -o (/) a. NO � er: � a.. (/) z 0 1- if) 0 Q. 0 a: Q. 0 z ::J 0 Q. � 0 0 claim that she has passed the bar exam? No. Rather, in any conditional, 'If P, then �• P and Q stand in a hypothetical relationship such that P's being true implies that Qis also true. To chal­ lenge a conditional, one has to show that its antecedent is true and its consequent false at once. Notice that sometimes the 'then' that often introduces the consequent of a conditional sen­ tence may be left out. Moreover, besides 'if ... then ... ,' many other linguistic expressions can be used in English to introduce one or the other part of a conditional sentence. Such expressions may precede that sentence's consequent, its antecedent, or both-as shown in the examples below, where double underlines mark the antecedent and single underlines the consequent: Maria has passed the bar exam, provided she is a practicing attorney. Supposing that Maria is a practicing attorney. she has passed the bar exam. On the assumption that Maria is a practicing attorney. she has passed the bar exam. Maria is a practicing attorney only if she has passed the bar exam. That Maria is a practicing attorney implies that she has passed the bar exam. We'll now translate these conditional sentences into our symbolic language, using 'M' to stand for 'Maria is a practicing attorney' and 'E' for 'Maria has passed the bar exam.' Our formula rep­ resenting any of these propositions has 'M' for the antecedent and 'E' for the consequent. It lists 'M' first, then the horseshoe symbol, and 'E' last: 40' M :J E Here the rule is: To translate a conditional sentence into the symbolic language, we must list its antecedent first and its consequent last, whether or not these two parts occur in the English sentence in that order. Let's now translate the conditionals below into the symbolic language using this glossary: N = The United States is a superpower I = China is a superpower C = China has agents operating in other countries 0 = The United States has agents operating in other countries 41 If China is a superpower, then China and the United States have agents operating in other countries. 41' I :J (C • 0) 42 It is not the case that if the United States has agents operating in other countries, then it is a superpower. 42' - (0 ::J N) 43 China has agents operating in other countries provided that the United States and China are superpowers. 43'(N • I) ::JC 44 If the United States doesn't have agents operating in other countries, then it is not a superpower. 44'- 0::J-N 45 That China has agents operating in other countries implies that either it is a super­ power or the United States is not a superpower. 45' C ::J (Iv -N) 46 If either the United States or China has agents operating in other countries, then neither the United States nor China is a superpower. 46' (0 v C) ::J -(Nv I) 47 If the United States is not a superpower, then it either has or doesn't have agents operating in other countries. 47' - N ::J (0 v -0) Note that 'P unless Q'. could also be translated as 'if not P, then Q' Thus 'China is a member of the UN unless it rejects the UN Charter' is equivalent to 'If China is not a member of the UN, then it rejects the UN Charter.' Necessary and Sufficient Conditions. In any material conditional, the antecedent expresses a sufficient condition for the consequent, and the consequent a necessary condition for the antecedent. Thus another way of saying 'If P, then Q'. is to say that P is sufficient for � and Q is necessary for P. A necessary condition of some proposition P's being true is some state of affairs without which P could not be true, but which is not enough all by itself to make P true. In (40), Maria's having passed the bar exam is a necessary condition of her being a practicing attorney (she could not be a practicing attorney if she had not passed it, though merely having passed doesn't guarantee that she's practicing). A sufficient condition of some proposition Q'.s being true is some state of affairs that is enough all by itself to make Q true, but which may not be the only way to make Q true. In (40 ), Maria's being a practicing attorney is sufficient for her having passed the bar exam (in the sense that the former guarantees the latter). In a material conditional ■ Its consequent is a necessary (but not sufficient) condition for the truth of its antecedent. ■ Its antecedent is a sufficient (though not a necessary) condition for the truth of its consequent. 0 z 🙂 0 CL � 0 () 0 z (/) <{ z �Q CL I­ � in -o (/) CL NO ,..: a: � CL fl€■ (/) z 0 t: (/) 0 Q_ 0 a: 0 z :::) 0 Q_ � 0 0 Material Biconditional A material biconditional is a type of compound proposition, also called 'material equiva­ lence,' or simply 'biconditional,' created by the truth-functional connective standardly expressed in English by 'if and only if,' and in symbols by '!!!!', the triple bar. Some other English expressions for the biconditional connective are 'just in case,' 'is equivalent to,' 'when and only when,' and the abbreviation 'iff.' Each of the two members of a biconditional could be either simple or compound. Here is a biconditional, in both English and symbols, made up of simple propositions: 48 Dr. Baxter is the college's president if and only if she is the college's chief executive officer. 48' B= 0 The truth value of the compound proposition the biconditional creates is determined by the truth value of its members, together with this truth-value rule: A material biconditional is true whenever its members have the same truth value­ that is, they are either both true or both false. Otherwise, a biconditional is false. Given this rule, for a biconditional proposition to be true, the propositions making it up must have the same truth value-that is, be both true or both false. When a biconditional's members have different truth values, the biconditional is false. (49) through (51) are false, for each features propositions with different truth values. 49 The Himalayas are a chain of mountains if and only if the Pope is the leader of the Anglican Church. 50 London is in England just in case Boston is in Bosnia. 51 Parrots are mammals if and only if cats are mammals. By contrast, the following biconditionals are all true because in each case its members have the same truth value: 52 Lincoln was assassinated if and only if Kennedy was assassinated. 53 Beijing is the capital of France just in case Bill Gates is poor. 54 That oaks are trees and tigers are felines are logically equivalent. In any biconditional, each member is both a necessary and a sufficient condition of the other. Thus in (48), Baxter's being the college's CEO is both a necessary and sufficient condition for her being the college's president, and her being the college's president is both a necessary and sufficient condition for her being the college's CEO. So a biconditional can be understood as a conjunction of two conditionals. Thus we can represent (52) in either of these ways: 52' L=K 52" (L :::l K) • (K :::l L) (5211 ) is the conjunction of two conditionals whose antecedent and consequent imply each other. That is why the material equivalence relation is called a 'biconditional,' and, obviously, this connective is commutative. BOX 2 ■ SUMMARY: COMPOUND PROPOSITIONS ■ Any proposition that is affected by a truth-functional connective is compound. Otherwise, it is simple. ■ The truth value of a compound proposition is determined by factoring in: (1) the truth values of its members, and (2) the truth-value rules associated with each connective affecting that proposition. ■ Negation is the only connective that can affect a single proposition. Exercises 1 . What is a compound proposition? 2. What are the five logical connectives? And what does it mean to say that they are truth-functional? 3. Besides 'and,' what are some other words used to express a conjunction? 4. Besides 'either ... or ... ,' what are some other words used to express a disjunction? 5. Besides 'if ... then ... ,' what are some other words used to express a conditional? 6. Besides 'if and only if,' what are some other words used to express a biconditional? 7. In a material conditional, which part is understood to present a necessary condition of the other? Which part is understood to present a sufficient condition of the other? 8. How could the biconditional be rephrased using other truth-functional connectives? II. For each of the following propositions, determine whether or not its main connective is a negation. Indicate double negation whenever appropriate. 1. Either London's air pollution is not at dangerous levels or San Francisco's isn't. SAMPLE ANSWER: Not a negation 2. It is false that London's air pollution is at dangerous levels. *3. San Francisco's air pollution is unhealthy. 4. It is not the case that Mexico City's air pollution is not harmful. 5. Non-dangerous levels of air pollution are rare in big cities. 6. Dangerous levels of air pollution are illegal. *7. Dangerous levels of air pollution violate the Kyoto Protocol. 8. It is not the case that dangerous levels of air pollution violate the Kyoto Protocol. 9. Dangerous levels of air pollution are not illegal. *10. Cleveland's air quality now reaches non-dangerous levels of pollution. 0 z ::) 0 Cl. � 0 0 z Cf) <( z ::J Q Cl. I­ � U) U) � NO .,..:. a: �o.. mll Cf) z 0 0 0.. 0 a: 0.. 0 z :J 0 0.. � 0 0 111m Ill. For each of the following propositions, determine whether or not its main connective is a conjunction. 1. Mexico City's air pollution is not harmful, but Houston's is. SAMPLE ANSWER: Conjunction 2. Dangerous levels of air pollution are illegal and unhealthy. 3. Chicago's air is polluted; however, Washington's is worse. *4. Rome's air is as unpolluted as Cleveland's. 5. In Toronto, air pollution is a fact of life; moreover, people are resigned to it in the summer. 6. New York's polluted air is often blown out to sea by westerly winds. *7. The Kyoto Protocol mandates steps to reduce air pollution, but the United States has not complied. 8. London's air pollution is not at dangerous levels; however, that's not the case in San Francisco. *9. Either Vancouver has low levels of air pollution or Montreal has dangerous levels of air pollution. 10. It is not the case that Canada is not a signatory of the Kyoto Protocol. IV. For each of the following propositions, determine whether or not its main connective is a disjunction. 1. Neither China nor North Korea is a signatory of the Kyoto Protocol. SAMPLE ANSWER: Not a disjunction *2. China and North Korea are not signatories of the Kyoto Protocol. 3. Either the United States complies with the Kyoto Protocol or it doesn't. 4. It is not the case that Mexico City's air pollution is either harmful or unhealthy. 5. Mexico City's air pollution is neither harmful nor unhealthy. *6. New York's polluted air blows either out to sea or north to Canada. 7. Dangerous levels of air pollution violate health laws as well as the Kyoto Protocol. *8. Dangerous levels of air pollution violate either the Kyoto Protocol or internal regulations. *9. San Francisco's air pollution is at dangerous levels unless there is fresh air blowing from the sea. 10. It is false that neither China nor North Korea is a signatory of the Kyoto Protocol. V. For each of the following propositions, determine whether or not its main connective is a material conditional. 1. If the United States and China sign the Kyoto Protocol, then the biggest polluters agree to comply. SAMPLE ANSWER: Conditional 2. That London's air pollution is not at dangerous levels implies that London is complying with the Kyoto Protocol. *3. Either Montreal has dangerous levels of air pollution or Rome does. 4. Mexico City's air is not harmful provided that Houston's air is healthy. *5. Chicago's air is unhealthy only if it has dangerous levels of pollutants . 6. Washington's air pollution is not a fact of life unless people are resigned to it. 7. That Canada has signed the Kyoto Protocol implies that Canada is willing to comply. *8. It is not the case that if London has dangerous levels of air pollution, the United Kingdom has not signed the Kyoto Protocol. 9. Either Mexico City has air pollution or if Houston has it, so does Vancouver. *10. That China has not signed the Kyoto Protocol implies that neither Canada nor the United Kingdom has signed it. VI. For each proposition in Exercise V that is a conditional, mark its antecedent with double underline and its consequent with single underline (*4, *7, and *10). 1. SAMPLE ANSWER: If The USA and China sign the Kyoto Protocol, then the biggest polluters agree to � VII. For each of the following propositions, determine whether or not its main connective is a material biconditional. 1. Only if Chicago has dangerous levels of air pollutants is its air unhealthy. SAMPLE ANSWER: Not a biconditional *2. China has signed the Kyoto Protocol if and only if North Korea has. 3. Washington's air pollution is a fact of life just in case people are resigned to it. 4. If London's air pollution is not at dangerous levels, the United Kingdom has signed the Kyoto Protocol. *5. Montreal has dangerous levels of air pollution if Rome does. *6. London's air pollution is at dangerous levels if and only if its air is unhealthy. 7. It is false both that Houston's air is harmful and that it is unhealthy when and only when it reaches dangerous levels of pollution. *8. Chicago's air is unhealthy just in case it has pollutants that are either dangerous or otherwise unhealthy. 9. Dangerous levels of air pollution violate the Kyoto Protocol if and only if they violate UN environmen­ tal regulations. 10. New York's air does not reach dangerous levels of pollution only if it is either blown out to sea by westerly winds or dispersed by thunderstorms . VIII. YOUR OWN THINKING LAB In each of the following, a proposition is taken either to be or not to be a condition that's necessary, sufficient , or both for the truth of another proposition. Provide the correct representation of each using the propositional symbols in parentheses and connectives as needed. 0 z :::, 0 0.. � 0 0 0 z rJ) <( z :'.:j Q 0.. f­ � U) -o (f) 0.. NO ...: a: � 0.. (/) z 0 1- (/) 0 0.. 0 a: 0.. 0 z ::J 0 0.. L 0 0 11111m 1. 'The potato has nutrients' (0) is necessary and sufficient for 'The potato is nutritious' (N). SAMPLE ANSWER: (0 ::> N) · (N ::J 0) or 0 = N

2. ‘John hunts’ (J) is necessary for ‘John is a hunter’ (H).

*3. ‘This figure is an isosceles triangle’ (I) is a sufficient for ‘This figure is a triangle’ (F).

4. ‘Fluffy is a cat’ (C) is not a sufficient condition for ‘Fluffy is a feline’ (F).

5. ‘Mary is a sister’ (A) is necessary and sufficient for ‘Mary is a female sibling’ (F).

*6. ‘Laurence is not British’ (B) is not necessary for ‘Laurence is not European’ (E).

11.3 Propositional Formulas for
Compound Propositions

Punctuation Signs

As we have seen in some examples above, parentheses, brackets, and braces can be used to remove
ambiguity in formulas by indicating the scope of their logical connectives. When a compound
proposition is joined to a simple proposition or to another compound proposition by a logical
connective, parentheses are the first recourse for determining the scope of occurring connectives
if necessary. When the compound proposition is more complex, brackets may be needed, and for
even more complex compound propositions, braces. Thus parentheses are introduced first, then
brackets, and finally braces. For examples illustrating their correct use, see Box 3.

The compound proposition (P • Q} :::> R is a conditional, while P • (Q:::> R) is a conjunction.
Without brackets, the proposition (P • Q} :::> R v ~ S is ambiguous, since it is unclear which con­
nective is its main connective: it admits of two different interpretations, one as a conditional,
the other as a disjunction. Finally, the main connective in ~ {[(P • Q} :::> R] v ~ S} is the negation
in the far left of this formula, which affects the whole formula. Compare ~ [(P • Q} :::> R] v ~ S.

Now, without braces, the scope of that negation is the conditional marked by brackets, and the
whole formula is not a negation but a disjunction.

Well-Formed Formulas

A formula representing a proposition, whether simple or compound, is well formed when it is
acceptable within the symbolic notation that we are now using. To determine whether a com­
pound formula is well formed, the scope (or range) of its truth-functional connectives matters.

BOX 3 ■ PUNCTUATION SIGNS

parentheses ‘()’as in: (P • Q) :::> R

brackets ‘[]’as in [(P • Q) ::) R] v ~ S

braces ‘{}’as in: ~ {[(P • Q) ::) R] v ~ S}

Within the scope of negation falls the simple or compound proposition that follows it.
Negation is the only connective that has a single formula, simple or compound, within its
scope. For the other connectives, each has two formulas (simple or compound) within its
scope. Well-formed formulas (WFFs) often require punctuation signs to mark the scope of
their connectives.

Recall (52 11

) above, (L :::J .K) • (K:::, L), which is a well-formed formula with two conditionals
set inside parentheses to eliminate ambiguity: parentheses are needed here to indicate that
the compound proposition is a conjunction of two conditionals.

A different arrangement of punctuation signs could yield a different proposition even
when all propositional symbols remain the same, as shown by L :::, [K • (K :::, L)]. This is a
conditional featuring a simple proposition as antecedent and a compound consequent
that’s a conjunction of a simple proposition with another conditional (one made up of
simple propositions). If that conditional is false, we can say that by introducing negation
and braces in this way: – {L :::J [K • (K :::J L)]}. These are all WFFs, but the formulas in Box 4
are not.

BOX 4 ■ SOME FORMULAS NOT WELL FORMED

P-Q

P­

PvQ•P

Symbolizing Compound Propositions

We’ll now have a closer look at some compound propositions. But first, consider

55 Fox News is on television.

Since there is no connective here, (55) is a simple proposition that we may symbolize as

55′ F

By contrast, (56), symbolized below as (561), has a negation and is therefore compound.

56 CBS News is not on television.

56′-C

Now consider (57), an abbreviated version of the longer proposition (57′):

57 Fox News is on television but CBS News is not.

57’ Fox News is on television but CBS News is not on television.

Either way, we have a compound proposition featuring two connectives: conjunction and
negation. The main connective, however, is the conjunction, whose scope is the entire
compound proposition. The scope of negation is only the second proposition. The underlying
principle for determining this is

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Exercises

1 . Why are punctuation signs part of the symbolic notation for propositions?

2. What’s the scope of a negation?

3. When Pis false and Q is true, what’s the value of P :::i Q?

4. Define P”” Q using only material conditional and conjunction.

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� <( � f- EDI (/) z 0 i== in 0 n. 0 a: n. 0 z :::i 0 n. � 0 0 IED of the rule given in Box s to each row in the column on the left yields the results, recorded in the column inside the box on the truth table's right-hand side. They are as follows: First row: -Pis false when Pis true Second row: -Pis true when Pis false Other Truth-Functional Connectives. To define the other truth-functional connectives, first we keep in mind that they involve two propositions, each of which could be T (true) or F (false). This determines the number of Ts and Fs that we'll write on the column on the left-hand side of the truth table. To calculate the total number of Ts and Fs in each column, we use the for­ mula 2° , where 2 stands for the two truth values a proposition may have (true or false) and n for the number of propositions of different types that occur in the formula to which we'd apply the procedure. In the definition of negation, there is only one proposition, so n is 1 and the for­ mula 2 ° produces two truth values: one T and one F. But, for conjunction, disjunction, condi­ tional, and biconditional, each definition features two propositions, represented by P and Q. So the formula is 2 2 and yields four places for the truth values of occurring propositions: two Ts and two Fs. In the first column on the left, we assign half Ts followed by half Fs: that is, two Ts, and two Fs (see truth tables below). In the other column on the same side of the truth table, we assign four values in this way: T, F, T, F. On the top right-hand side of the table, have the for­ mula whose truth value we wish to determine, and below it, the truth value resulting from the application of the corresponding truth-value rule to each row of assigned values on the left. The final result is marked by putting it inside a box and is obtained by applying the truth value rule of each occurring connective to the values on the left-hand side of the truth table. Let's now construct truth tables for each of the remaining types of compound propositions. Conjunction. To define this connective with a truth table, keep in mind that BOX 6 ■ TRUTH-VALUE RULE FOR CONJUNCTION A conjunction is true if and only if its conjuncts are both true. Otherwise, a conjunction is false. The truth table has two columns on the left, each of which assigns four truth values, two Ts and two Fs, to each of its conjuncts. Its four horizontal rows are obtained by calculating the possi­ ble combinations of those values while applying the rule in Box 6. The result, in a box on the right, shows that a conjunction is true just in case the two conjuncts are true. PQ TT T F FT F F Disjunction. To define this connective with a truth table, keep in mind that A disjunction is true if and only if at least one disjunct is true. Otherwise, a disjunction is false. The truth table for disjunction has two columns on the left, each assigning four truth values, two Ts and two Fs, to each of the disjuncts. Its four horizontal rows are the result of reading the possible combinations of those values while applying the rule in Box 7. The final result, inside the box in the right-hand column, amounts to a definition of disjunction: it shows that a disjunction is true just in case at least one of its disjuncts is true. Equivalently, it defines disjunction as a compound proposition that is false just in case both of its disjuncts are false. p TT T F FT F F PvQ Material Conditional. To define this connective with a truth table, keep in mind that BOX 8 ■ TRUTH-VALUE RULE FOR MATERIAL CONDITIONAL A material conditional is false if and only if its antecedent is true and its consequent false. Otherwise, a conditional is true. As before, it has two columns on the left, each assigning four truth values, two Ts and two Fs, to its antecedent and its consequent. Its four horizontal rows are obtained by calculating the possible combinations of those values and applying the rule in Box 8. The result, inside the box on the right-hand side, amounts to a definition of the material conditional. It shows that it is true in all cases except when its antecedent is true and its consequent false. p TT T F FT F F Material Biconditional. To define this connective with a truth table, keep in mind that BOX 9 ■ TRUTH-VALUE RULE FOR MATERIAL BICONDITIONAL A material biconditional is true if and only if both members have the same truth value. Otherwise, a biconditional is false. I 1- :::J a: I I- � (/) UJ >
i=
0
UJ
z
z
0
0

0
z
z
u..
UJ (/) 0 UJ

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� <( � I- Cf) z 0 1- u.i 0 Q.. 0 a: Q.. 0 z ::i 0 Q.. � 0 0 11111 This truth table has two columns on the left, each assigning four truth values (two Ts and two Fs) to each simple proposition in the formula. Its four horizontal rows are obtained by calculating the possible combinations of those values and applying the rule in Box 9. The truth table's result appears inside the box on the right, and the truth table amounts to a definition of the material biconditional. It shows that it is true just in case its two members have exactly the same truth value: that is, they are either both true or both false. PQ TT T F FT FF P=Q The five truth tables we've now constructed provide truth-functional definitions for each of the five logical connectives. We can now use a similar procedure to determine the truth values of other compound propositions. 11.5 Truth Tables for Compound Propositions The truth value of compound propositions can be determined with truth tables. To construct a truth table for a compound proposition, first identify the simple propositions in the formula whose value you wish to check. Once you have written them down on the top of the left-hand side of the truth table in the order that they appear in the formula, assign Ts and Fs in the way outlined above, using 2n to calculate the total number of rows. For example, the truth table for F • - C is: 62 F C --+----- T T T F FT F F � F T F T The formula on the right-hand side of this truth table is a conjunction of F, whose values we read in the first column on the left, and - C, whose values we need to determine first. We do this by applying the truth-value rule for negation to each row in the second column on the left. Once we determine - C's values, we enter them under the tilde on the right. We then determine the truth value of the conjunction by applying the truth-value rule for conjunction to F's values (available on the left-hand side of the truth table) and- C's values (under the tilde). We enter the values thus obtained under the dot, marking the resulting column with a box. This column under the main connective is the most important one, because it provides infor­ mation about the truth values of the compound proposition F • - C. It tells us that this com­ pound proposition is true only when 'F' and '- C' are true (as shown in the second horizontal row). On all other assignments of values, that proposition is false. Now let's construct a truth table for 58' (H • M) ::) - B The truth table for (58') is: 63 HMB (H • M) ::J -B TTT T F F TT F T T T TFT F T F TFF F T T PTT F T F FT F F T T FFT F T F FF F F T T On its left-hand side, this truth table shows all possible combinations of truth values for the three members of the compound proposition represented by the formula on its right-hand side. That formula, whose truth value we want to determine, has three different simple propositions symbolized by H, M, and B. As before, to calculate the number of rows needed, we use the algorithm 2n, here 23, which reveals that eight rows are needed. Accordingly, we assign Ts and Fs to the three columns on the left-beginning with the one farthest to the left (the one under H), which has the top half Ts and the bottom half Fs-and continue to divide that pattern in half as we move across to each of the two other columns to the right (under M and B). This convention guarantees that we do get all possible combinations of truth values. On the top line, it's all Ts, on the bottom line, it's all Fs, and in between are all other possible arrangements. Once we have entered these values, we look at the compound proposition formula on the top right. It is a conditional, so the main connective is ':::>‘, under which we place the final

result (inside the box). But we can determine the possible truth values of the conditional only
after we first find the possible truth values of the antecedent, H • M, and the consequent, – B.

Those truth values make up the column under’•’ and the column under’-‘. The final step con­

sists in applying the rule for the truth value of the conditional to those two columns.

BOX 10 ■ TRUTH TABLES FOR COMPOUND PROPOSITIONS

As we’ve seen, in a truth table, the number of truth values assigned to each simple proposition on
the left-hand side depends on how many different propositions occur in the formula at the top of
the right-hand side, whose truth value we wish to determine. For any simple proposition there are
only two possible truth values (true and false); therefore, for a compound proposition such as – P,
only two rows are needed. But with more propositions, the number of truth values would increase
according to the formula 2n : with two, it’s four lines; with three, it’s eight lines; with four, it’s
sixteen lines; and so on. In the case of (62), then, we need four lines. And, just to make sure that
we get all possible combinations of truth values, we’ll adopt this convention: in the column under
whatever letter symbol is farthest to the left, we put T in the top half of the rows and F in the
bottom half; and in the column under the other letter symbol to the right of that, we put a
sequence of alternating Ts and Fs.

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<( (/J rz IO �E a: (/) rO Q. l{) 0 ,..: a: � Q. (j) z 0 E 0 Q_ 0 a: Q_ 0 z :::, 0 Q_ � 0 0 - 11.6 Logically Necessary and Logically Contingent Propositions Contingencies What, then, have we learned about the truth values of the compound propositions on the right-hand side of truth tables (62) and (63)? Just this: that each is neither necessarily true nor necessarily false, but instead sometimes true and sometimes false, depending on the truth values of the component simple propositions and the logical connectives. Propositions that yield such truth values are contingencies. A compound proposition is a contingency if its truth table displays at least one T and at least one F in the column under the main connective. In (63), there is at least one T and at least one F under the ':::i'-and in (62), under the'•'. In light of those results, each of these compound propositions is a contingency. Contradictions Contradictions are compound propositions that are always false, simply by virtue of their form (and regardless of the actual truth values of their component simple propositions). In a truth table for a contradiction, the column under the main connective symbol is all Fs. Consider 64 B =-B Since (64) contains no proposition other than B, which occurs twice, the algorithm 21 yields two places for truth values, one for T and the other for F. Accordingly, the truth table runs: 65 B B = -B ---+---- T F f"FlF l!_JT This truth table reveals (64) to be a contradiction. Tautologies Some propositions are tautologies: they are always true, simply by virtue of their form (and regardless of the actual truth values of their component propositions). The truth table of a tautology would have all Ts under the formula's main connective. The negation of (64) above is a tautology, which reads, 66 -(B = - B) The truth table for this proposition shows all Ts under the formula's main connective: 67 B -(B = - B) --+-=---- ; w F F FT (67) gives the truth value of (66), thus confirming that it is a tautology. Among well-known tautologies in logic are the so-called principles of excluded middle, P v - P, and non-contradiction, -(P • -P). For further practice, check that these are tautologies by constructing a truth table for each. Keep in mind that BOX 11 ■ CONTRADICTIONS, TAUTOLOGIES, AND NEGATION The negation of a contradiction is a tautology, and the negation of a tautology is a contradiction. Exercises 1 . How are truth tables used to define the five propositional connectives? 2. In a truth table for a compound proposition, how do we know how many horizontal rows are required? What is the rationale for this? 3. In a truth table for a compound proposition, which column is the most important? And what does that column tell us? 4. What is a tautology? 5. What is a contradiction? 6. What is a contingency? XX. For each of the following formulas, construct a truth table to determine whether it is a contingency, tautology, or contradiction. 1. W:J-K SAMPLE ANSWER: Contingency WK W :J -K TT TF FT FF �F T F T 2. (L v N) vA *3. B :J (M :J B) 4. -J • (G vN) *5. - [(A • B) :J (B • A)) 6. D v (-M :J -0) 7. -[-(A • B) =(-Av -B)] *8. (-A v-B) :J (B •A) 9. (F :J -N) • -(F :J -N) *10. - A""+ K v -H) 11. (0 v M) :J (M V 0) *12. -[(-A • H) v -(H :J -1)] 13. (E • -G) :J G 14.A "'-A 15. - (A = B) "' -L *16. - {� • (8 • C)] = [(A • B) • Cl} 17. - {[(A • B) v (-B :J A)) :J B} *18. (A • B)..,, (B • A) 19. (-B :J A)"" [(B v -0) :J CJ *20. (A= B) = [(A :J B) • (B :J A)] (/) z 0 l- o ui zo – 0
a: a:
0

*2. Earth is not the center of the universe or our planet is special. (E, 0)

3. Either our planet is special or it isn’t. (0)

*4. Earth is not the center of the universe just in case there is something special about our planet. (E, 0)

5. There is something special about our planet: however, Earth is not the center of the universe. (0, E)

6. It is false that either our planet is special or it isn’t. (0)

•7. If Earth is the center of the universe and there is something special about our planet, then there is

something special about our planet. (E, 0)

8. It is not the case that human life has value if and only if human life has value. (H)

9. Human life has a purpose, but it is not the case that it has value. (L, H)

10. Human life has value only if it has a purpose. (H, L)

*11. Human life has value and a purpose if and only if it is not false that human life does have value and a

purpose. (H, L)

*12. It is not the case that both Earth is the center of the universe and there is something special about

our planet just in case it is false that human life has value and a purpose. (E, 0, H, L)

13. Neither is Earth the center of the universe nor is there something special about our planet. (E, 0)

*14. Neither is Earth the center of the universe nor is there something special about our planet if and only

if both Earth is not the center of the universe and it is not the case that there is something special

about our planet. (E, 0)

15. Either human life has both value and a purpose or if it is false that there is something special about

our planet, then Earth is the center of the universe. (H, L, 0, E)

XXII. YOUR OWN THINKING LAB

Write down ordinary English sentences for each of the formulas below following this glossary: ‘F’ = Fred

is at the library; ‘M’ = Mary is at the library; ‘L’ = The library is open; ‘I’ = I have Internet access; ‘E’ =

The essay is due on Thursday.

SAMPLE ANSWER: 1. Mary is at the library but the library is not open.

2. F .. (L • M)

*3. F ss (L v – M)

4. (L • I) :::, (F v M)

*5. E”” (L:::, I)

■ Writing Project

6. (E • L) :::, (M v F)

*7. – [- F:::, (- L v M)]

8. (M • F) “” (E • L)

Find five compound propositions in English where the words used to translate the logical con­
nectives don’t accord with the truth-value rules for those connectives. Write a short piece
arguing that the connectives in your examples are not truth-functional. Suggestions: look for

conjunctions that are not commutative, or ‘if … then … ‘ sentences where the consequent
appears not to be a necessary condition of the antecedent (e.g., ‘If I have money for bus fare
then I’ll take the bus’).

BOX 12 ■ SYMBOLIC NOTATION FOR PROPOSITIONS

Propositional Letters

From A to O for specific
propositions

From P to W for
unspecific ones

■ Chapter Summary

Connectives

~ negation
• conjuction
v disjunction
:::, Conditional
“‘ Biconditional

Punctuation Signs

( ) parentheses
I ] square brackets
{ } braces

Propositions: the building blocks of propositional arguments. Each proposition is either true

or false, and either simple or compound.

Compound proposition: any proposition whose truth value is in part determined by the

truth-value rule of one of the truth-functional connectives. It falls into one or another of

these categories:

1. Tautology. Always true, by virtue of its form. Its truth table shows only Ts under the main
connective.

2. Contradiction. Always false, by virtue of its form. Its truth table shows only Fs under the
main connective.

3. Contingency. Neither always true nor always false. Its truth table shows at least one T and

one F under the main connective.

Five Connectives and their truth-value rules:

1. Negation. True if and only if the proposition denied is false.

2. Conjunction. True if and only if its conjuncts are both true.

>­
a:
<( � � 🙂 CJ) a: w f- 0.. <( I () Cl) z 0 0 0 0 z 0 0 3. Disjunction. True if and only if at least one disjunct is true. 4. Conditional. False if and only if its antecedent is true and its consequent false. 5. Biconditional. True if and only if both members have the same truth value. Truth Tables for the Connectives Negation ; I� Conjunction Disjunction p Q p . Q PQ TT � TT TF TF FT FT F F FF ■ Key Words Compound proposition Negation Conjunction Disjunction Material conditional Antecedent P vQ � Conditional Biconditional p Q TT T F FT F F p 🙂 Q PQ PaaaQ � TT T F FT F F Consequent Material biconditional Tautology Contradiction Contingency � CHAPTER Checking the Validity of Propositional Arguments In this chapter you'll learn some ways to determine whether propositional arguments are valid or invalid. The topics will include ■ The use of truth tables in checking argument forms for validity. ■ Some standard valid argument forms in propositional logic: modus ponens, modus to/lens, contraposition, hypothetical syllogism, and disjunctive syllogism. ■ The formal fallacies of affirming the consequent, denying the antecedent, and affirming a disjunct. ■ An introduction to proofs of validity. 261 Cf) LL f­ Q Z >– �
t:: :::i
g CJ
__J a:
<( <( > __J
w <( IZ f- 0 CJ � z­ - Cf) � 0 0 0... WO I a: 0 0... 12.1 Checking Validity with Truth Tables As we have seen, truth tables provide a procedure for determining whether a compound proposition is a tautology, contradiction, or contingency. Moreover, they generate that outcome in a mechanical way, applying certain rules that yield a result in a finite number of steps. But they have another use that we'll explore at length here: they allow us to determine mechanically whether an argument is valid or not. Consider, for example, this argument: 1 1. Either buffalo are prairie animals or coyotes are. 2. Buffalo are prairie animals. 3. Coyotes are not prairie animals. To determine whether (1) is valid or not requires that we first obtain its argument form. We translate (1)'s premises and conclusion into our standard symbolic notation and obtain: 1' 1.B vC 2.B 3. -C Our next step is to transform this vertical listing of premises and conclusion into a horizontal one, using commas to separate premises and writing the symbol ':. ', which reads 'therefore,' in front of the conclusion. This gives us 1' B v C, B ... -C We can now test this argument form for validity with a truth table. We enter the formula at the top of the truth table on its right-hand side, and each different simple proposition that occurs in that formula on the left-hand side. Next, we assign truth values to those simple propositions, following the algorithm 2 n , which for the formula under consideration is 2 2 (since the simple propositions occurring in it are two, B and C). Once this is done, we focus on the smaller formulas that represent the argument's premises and conclusion, calculating their truth values one at a time. These calculations are performed in the standard way described in Chapter 11. In the final step, we check (in a way we'll presently explain) to see whether the argument is valid or not. Our truth table for checking the validity of (1) above looks like this: 2 B C TT T F FT FF B v C, B :. -C In (2), a value has been calculated for each formula representing the argument's premises and conclusion. How? By reasoning as follows: the first premise is a disjunction, so its truth value is calculated by applying the truth-value rule for disjunction to B and C, whose values have been assigned on the left-hand side. The first column on the right, placed under the wedge, shows the result of this calculation. Since the second premise, B, is a simple proposition, we cannot calculate its values by using any of the truth-value rules for connectives. So we assign to B the same values that we have assigned it in the first column on the left-hand side of the table. That is, we simply transfer those values to the second column on the right-hand side of the table (this step can be omitted, since B's values are readily available in the first column on the left, and could be read di­ rectly from there). We then proceed to calculate the truth value of -C by applying the truth-value rule for negation to the values of C, which are displayed in the second column on the left. We write down the results of this calculation under the tilde, as shown by the truth table's third col­ umn on the right. We're ready now to check whether the formula on its top right-hand side is a valid argument form. To determine this, we scan horizontally each row displaying the values of B v C, B, and - C (ignoring all vertical columns). We are looking for a row where the premises B v C and B are both true and the conclusion - C false. And we do find precisely that in the first row. This shows the argument represented by the formula to be invalid, since If a truth table devised to test the validity of an argument displays at least one row where premises are all true and conclusion false, that proves the invalidity of the argument form tested. BOX 1 ■ WHAT DO TRUTH TABLES HAVE TO DO WITH VALIDITY? The relation between validity and truth tables is simply this: 1. If it is possible for an argument to have all of its premises true and its conclusion false at once-that is, if this occurs on one or more horizontal rows in its truth table-then the form is invalid (as is any argument with that form). 2. But if this is not possible-that is, if its truth table shows no such row-then the form is valid. (Recall that if an argument's form is such that it is possible for all of its premises to be true and its conclusion false at once, then its premises do not entail its conclusion.) The first row (indicated by an arrow) in the above truth table demonstrates the invalidity of the form being tested (see rationale in Box 1). In this way we show that (1) is invalid. Similar truth tables could be constructed to demonstrate the invalidity of any argument of the same form. For example, 3 Either the media foster public awareness, or public opinion leads to public policy. Since the media foster public awareness, it is not the case that public opinion leads to public policy. Since this argument has the same form as (1) above, the results of any correct truth table for checking its validity would be exactly the same as those displayed in (2). (You should construct such a truth table for your own practice.) Let's now use a truth table to check the validity of another argument: 4 1. If Sally voted in the presidential election, then she is a citizen. 2. Sally is not a citizen. 3. Sally did not vote in the presidential election. I I- >–
1-
0
_J
�
(9 (/)
� w
� _J
0 Ul
w <( I 1- 0 I � ::J "" a: � I- EDI (/) LL 1- 0 Z w >– 2
‘= ::)
0(9
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<( <( > _J
w <( Iz I- 0 0 i== z­- (/) �o 0 0... WO I a: 0 0... 111m This has the form 4' M :J C,-C :. -M First, notice that there are only two different simple propositions in this argument form, M and C, each of which occurs twice. Thus we need only four assignments of values (two Ts and two Fs) on the left-hand side of the truth table, and four horizontal rows. The next step is to calculate the truth value of (4')'s premises and conclusion. Each of these is a compound propo­ sition, for which we'll write the truth value in the column under its connective symbol: in the first premise, under'::>‘, and in the second premise and conclusion under’-‘. In this argument

there are no simple propositions; to test for validity, therefore, we scan only the rows in

columns under the connectives: in the premises, these are the columns under the horseshoe

‘:::,’ and the tilde’-‘, and in the conclusion, it’s the column under the tilde’-‘. We’re looking for
a row in which all the premises are true and the conclusion false, which would indicate inva­

lidity. But the scan shows that there is no such row in this truth table.

5 MC

T T

T F

F T

F F

The absence of such a row means that (4′), and therefore also (4), is valid. This test proves

validity because the truth table gives an exhaustive list of all possible combinations of truth

values of the premises and conclusion, and no horizontal row shows that the former can be

true and the latter false at once. Thus in all arguments with (4′)’s form, the premises entail the

conclusion. Consider these arguments:

6 If Professor Tina Hare is at the University of Liverpool, then she works in England.

Professor Tina Hare doesn’t work in England. Thus Professor Tina Hare is not at the

University of Liverpool.

7 If the Earth is not a planet, then Mars is not a planet. But Mars is a planet. Hence, the

Earth is a planet.

For your own practice, construct a truth table for these arguments to check their validity. You’ll

see that their final result will be exactly like that in (s) above.

Let’s try one more argument, this time more complex.

8 Since France is not a member of the union, it follows that Britain is not. For if France

is not a member, then either the Netherlands is or Britain is.

We can reconstruct (8) as

8’ 1. France is not a member of the union.

2. If France is not a member of the union, then either the Netherlands is a member

of the union or Britain is a member of the union.

3. Britain is not a member of the union.

which has the form

8″ -F, -F :J (N v B) :. -B

To test (8″) for validity, we first note that since three different simple propositions occur in it,
the truth table will need eight horizontal lines. Once we write down all possible combinations
of truth values for these simple propositions on the left-hand side of the truth table, we then
calculate the truth values of premises and conclusion and enter the results under each
connective symbol on the right-hand side. Here is the truth table, with the rows showing the
argument’s invalidity indicated by an arrow:

9 F NB -F, -F 🙂 (N v B) :. -B
TTT F F T T F
TTF F F T T T
TFT F F T T F
T FF F F T F T
FTT T T T T F +-

FT F T T T T T
FF T T T T T F
F F F I T F F I

The more complex formula on (9)’s right-hand side is the one representing the argument’s
second premise: it’s got three connectives in it. How do we determine which is the most
important? We do this by reading carefully and looking at the parentheses: they tell us that it is
the horseshoe placed between – F and (N v B). But in order to determine the truth values in the
column under the horseshoe, we first have to know the possible truth values of its antecedent, -F,
and its consequent, (N v B). Once we have the value of – F, which can be obtained by applying
the rule for the truth value of negation to F on the left-hand side of the truth table, we enter
those values under – F, the first premise of the argument (so they don’t need to be written twice
if desired). The value of (N v B) can be obtained by applying to the values of N and B the rule for
the truth value of the disjunction on the left-hand side of the truth table. To calculate the value
of – B, we proceed in a manner similar to that in which we calculated the values of – F. Once
this is done, then, ignoring all the other columns, we scan each horizontal row showing the
truth value for each premise and conclusion on the right-hand side of the truth table. We ask
ourselves: is there any horizontal row in which both premises are true and the conclusion false?
And the answer is Yes! It happens twice: on rows 5 and 7. Thus the argument form (8″) has been
proved invalid, and so any argument that has that argument form, such as (8) above, is invalid.

BOX 2 ■ HOW TO CHECK VALIDITY WITH TRUTH TABLES

■ When we use a truth table to check an argument’s validity, we first write the formula captur­
ing the argument’s form at the top on the right.

■ Each different type of proposition that occurs in that formula goes at the top on the left.

■ The rows under the formula itself offer an exhaustive list of possible combinations of truth
values for premises and conclusion.

■ To decide whether an argument form is valid or not, we scan each row under the formula.

■ Any row showing that there is a configuration of truth values in which premises are true and
the conclusion false proves that the argument form is invalid.

■ If there is no such row, then the argument form is valid.

:r:
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0

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� UJ
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o en
UJ <( I� o:r: � � ::J <'i a: �� (j) LL f­ Q Z UJ >– ::?:
I::: :::) 0 (9
::i a::
<( <( > _J

UJ <( IZ f- 0 0 � z­- (j) ::.:: 0 0 0.. wo I a:: 0 0.. IIIEI Exercises 1 . How do we construct a truth table to check the validity of a propositional argument? 2. How do we tell by a truth table whether an argument is valid or not? 3. Why is the truth-table technique a mechanical procedure for checking the validity of an argument? 4. If at least one row of a truth table for an argument shows all false premises and a true conclusion, is that relevant to determining the argument's validity? II. Use truth tables to determine whether each of these formulas represents a valid or an invalid argument. 1. H :::i -K , -K :. H SAMPLE ANSWER: Invalid HK H :::i -K, -K :. H TT F F F T F T T T FT T F F F F T T .I. 2. L v N, -N:.L 3. R :J (M :J R) :. M v R *4. J, J v N :. -N 5. -(B,., A) :J -L :. L :J (B ""A) 6. D v (-M :J -D) :. M :J D 7. -(G • E), -G :. E • -G *8. -C v -B, -(B • A) :. A v C 9. (F :J -N) • -(F :J -N) :. -N *10. -B, -(-K"" -H) :. K :J -H 11. (D v M) :::l (M v D) :. M v D 12. - [(-A• H) v (H :J -B)] :. -A• -H *13. K •(-Ev 0), -E :J -K :. 0 *14. E :J A, --A :. -Ev -A 15. A :J B :. -(-B :J -A) 16. -M v 0, M :. 0 17. A• B :. -A 18.C:JD:.D:JC 19. C :J D :. -C v D 20. C :J D, -C :. -D 21. A"" B :. (A :J B) • (B :J A) *22. H • (-1 v J), J :J -H :.J *23. - 0, A :::i B :. -0 • B 24. C :J D, D :. C 25. C :. C v D Ill. Translate each of the following arguments into symbolic notation. Then construct a truth table to determine whether it is valid or invalid. 1. Henry's running for mayor implies that Bart will not move to Cleveland, for if Henry runs for mayor then Jill will resign, and Jill's resigning implies that Bart will not move to Cleveland. (H, J, B) SAMPLE ANSWER: Valid HJ 8 H :::> J, J :::> -8 :. H :::> -8

TTT ,= FF FF

TTF T TT TT

TFT F T F F F

T FF F TT TT

FTT T F F TF

FTF T TT TT

FF T T T F TF

F F F .I .I T ..I T

2. If Quebec is a part of Canada, then some Canadians are voters. If Ontario is a part of Canada, then some

Canadians are voters. Hence, if Quebec is a part of Canada, then Ontario is a part of Canada. (8, C, 0)

*3. Algeria will not intervene politically if and only if Britain will not send economic aid. Thus Algeria will in­

tervene politically unless France will not veto the treaty, for Britain will not send economic aid only if

France will veto the treaty. (A, B, F)

4. Neither Detroit nor Ann Arbor has cold weather in February. If Michigan sometimes has snow in

winter, then either Detroit or Ann Arbor has cold weather in February. Therefore, it is not the case that

Michigan sometimes has snow in winter. (D, A, M)

5. Either the examinations in this course are too easy or the students are extremely bright. In fact, the

students are extremely bright. From this it follows that the examinations in this course are not too

easy. (E, B)

*6. If John is a member of the Elks lodge, then either Sam used to work in Texas or Timothy is a police

officer. But it is not the case that Sam used to work in Texas, and Timothy is not a police officer.

Therefore, John is not a member of the Elks lodge. (J, A. I)

*7. Both antelopes and Rotarians are found in North America. But Rotarians are found in North America

if and only if French police rarely drink gin. It follows that if it is not the case that French police rarely

drink gin, then antelopes are not found in North America. (A, 0, F)

8. Dogs are not always loyal. For rattlesnakes are always to be avoided unless either dogs are always

loyal or cats sometimes behave strangely. (D, A. C)

9. If either Romans are not fast drivers or Nigeria does have a large population, then it is not the case that

both Nigeria does have a large population and Argentineans are coffee drinkers. Hence, Romans are

fast drivers, for Argentineans are coffee drinkers only if Nigeria does not have a large population. (F, N, A)

*10. We may infer that mandolins are easy to play but French horns are difficult instruments. For man­

dolins are easy to play if and only if either didgeridoos are played only by men or French horns are

difficult instruments. But if French horns being difficult instruments implies that didgeridoos are not

played only by men, then it is not the case that mandolins are easy to play. (M, F, D)

11. If both Ellen is good at math and Mary is good at writing, then Cecil is a pest. It follows that Mary is

good at writing. For either Cecil is not a pest unless Mary is not good at writing, or both Ellen is not

good at math and Cecil is a pest. But Cecil is a pest if and only if Ellen is good at math. (E, M, C)

I
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UJ <( I 1- 0 I � ::J N a: � I- Cf) LL 1--­ Q Z w >– 2
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:::::i a:
-1
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0 a:
<( (J) 02 z a: �o (J) l.l.. LU 1- 2 Z 0 LU (J)2 :::::, (\J (9 Na: � <( (/) u..r- 0 di >– �
‘= :J
0 CJ
:J a:
<( <( > ..J w <( IZ r- 0 CJ i== z­- (/) -::t: 0 0 a.. wo I a: 0 a.. - (14)'s form mirrors that of contraposition, since it is 14' 1. p ::J Q 2. -Q ::J -P (14') helps us to see that (14) is a valid argument given that it has a valid argument form. Why is that form valid? For similar reasons modus tollens is: since in the premise Q is the consequent of a material conditional, it is a necessary condition for the truth of the conditional's antecedent P. Thus if Q is false, then P must be false, too. The validity of contraposition is shown by this truth table: 15 P Q TT T F FT FF p :::> Q :. -Q :::>-P

�

F

�

F
T FF
F TT

T TT

Hypothetical Syllogism

Hypothetical syllogism is labeled this way because it has two premises (that’s the ‘syllogism’ part) and
because its premises (as well as its conclusion) are hypothetical or conditional statements. Consider

16 1. If Elaine is a newspaper reporter, then she is a journalist.

2. If Elaine is a journalist, then she knows how to write.

3. If Elaine is a newspaper reporter, then she knows how to write.

This argument has the form

16 1

1. p ::J Q

2. Q ::JR

3. p ::J R

(161) allows us to see that (16) mirrors hypothetical syllogism, which is a valid argument form.
A closer look at this form reveals that premise i’s consequent is premise 2’s antecedent, and

premise i’s antecedent together with premise 2’s consequent are, respectively, the antecedent and
consequent of the conclusion. Obviously, since the antecedent of a conditional expresses a sufficient
condition for the truth of its consequent, when P is a sufficient condition for Q, and Q a sufficient
condition for R, it follows that P is a sufficient condition for R (16) is a substitution instance of this
form and is therefore valid. The following truth table shows the validity of hypothetical syllogism:

17 p Q R P::JQ Q::JR :. p ::J R

TTT T T T

TTF T F F

TFT F T T

T FF F T F

FTT T T T

FT F T F T

FF T T T T

F F F .I I I.

Disjunctive Syllogism
Finally, in our sample of valid argument forms, there is one that does not use conditionals at
all: disjunctive syllogism. The form is labeled this way because it has two premises (that’s the
‘syllogism’ part) and because one of its premises is a disjunction. Here, one premise presents a
disjunction, and the other denies one of the two disjuncts, from which the affirmation of the
other disjunct then follows as the conclusion. For example,

18 1. Either my car was towed away by the police or it was stolen.

2. My car was not towed away by the police.

3. My car was stolen.

(18) is a substitution instance of disjunctive syllogism, and as such may be correctly represented
in one of the two possible arrangements for the premises of that argument form, depending on
which disjunct is denied:

18a 1. p V Q
2.-P

3.Q

18b 1. p V Q
2.-Q
3. p

In the case of (18), since the negation affects the first disjunct of the disjunctive premise, the correct
representation is (18a). But the principle underlying either version of disjunctive syllogism is: given
the truth-functional definition of inclusive disjunction, if a premise that is an inclusive disjunction
is true but one of its disjuncts false, it follows that the other disjunct must be true. Thus any
argument mirroring (18a) or (18b) is valid-as demonstrated by this truth table:

19 P Q PvQ, -P :. Q

TT T F T

T F T F F

FT T T T

F F F T F

More Complex Instances of Valid Forms

When we set about trying to analyze propositional arguments, it’s immensely helpful to be
able to recognize these five basic valid argument forms, because any time you find an
argument that has one, you thereby know that it’s valid! No further procedure is required. For
the argument to be valid, it is enough that the general form of the argument’s premises and
conclusion mirror that of a valid form. This means that the premises and conclusion of a valid
argument could feature connectives other than those featured in the valid form mirrored by
that argument. That’s fine, provided that main connectives are exactly the same. Let’s make a
list including this and other considerations to keep in mind when deciding about the form of
propositional arguments:

#1. The order of the premises does not matter for an argument to have the form of a
modus ponens, modus tollens, hypothetical syllogism, or disjunctive syllogism.

#2. The English expression for a connective may be other than the standard one.

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2. B :::> J

3. E :::> J

Now consider the following arguments, together with their correct representations:

24 1. Costa Rica is a peaceful country and doesn’t have an army.

2. Costa Rica is a peaceful country and doesn’t have an army only if it doesn’t have
public unrest.

3. Costa Rica doesn’t have public unrest.

24′ 1. C • -A

2. (C • -A) :::> -N

3. -N

25 1. Joey was either tried in Europe or extradited to the United States.

2. That Joey was either tried in Europe or extradited to the United States implies that
his defense failed and he is not free.

3. Joey’s defense failed and he is not free.

25′ 1. J v E

2. (J v E) :::> (D • -F)
3. D • -F

If we focus strictly on the main connective in premises and conclusions, then it’s clear that
both arguments turn out to be substitution instances of modus ponens. This is so because each
consists of a conditional premise (which happens to come second) and another premise that
asserts the antecedent of that conditional (which happens to come first). Neither the order of
these premises nor the fact that they themselves are compound propositions made up of several
connectives affects the status of the arguments as instances of modus ponens.

Exercises

1 . Explain the validity of modus ponens by reference to the necessary and sufficient conditions in a

material conditional.

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<( <( > _J w <( IZ I- 0 0 � z­- (/) � 0 0 Cl. WO I er: 0 Cl. Mil• 2. Which sense of disjunction is required for disjunctive syllogism to be valid? Which of the valid forms employs disjunction? 3. Suppose the order of the premises in a valid propositional argument is changed. Does that affect the validity of the argument? 4. When you have established that an argument is a formal fallacy, what have you discovered about that argument? V. The following formulas are instances of modus ponens, modus to/lens, contra­ position, hypothetical syllogism, or disjunctive syllogism. Say which is which. 1. A:) -B A -B SAMPLE ANSWER: Modus ponens 2. (KvN)vA -A KvN *3. L:) -M B:) L B:) -M 4. -(F • H) A:) (F • H) -A *5. -E:) -D -E -D *6. (A v LP (B • C) -(B • C) 🙂 -(A v L) 7. -C v -A A -C *8. (A • -F) 🙂 -G G -(A• -F) 9. J:) A A:) -C J:) -C 10. -H 🙂 -(Ev A) (Ev A):) H 11. -B = C ( -B ""C):) -A -A *12.A v(G vF) -(G VF) A 13. A :J 0 -0:J-A 14. C vO :£_ 0 15. (G v F) :J A (A :J 0):J (G v F) (A :J 0) :J A VI. The following arguments are instances of the valid forms discussed above. Symbolize each and identify its form. 1 . Wynton Marsalis is an authority on music, for he is a famous jazz trumpeter who is equally well known as a performer of classical music. But if he is a famous jazz trumpeter who is equally well known as a performer of classical music, then Wynton Marsalis is an authority on music. (F, A) SAMPLE ANSWER: F :J A, F .. A Modus ponens 2. Ernie is a liar or Ronald is not a liar. It is not the case that Ronald is not a liar. Therefore, Ernie is a liar. (E, L) *3. If Staten Islanders are not Mets fans, then Manhattan's being full of fast talkers implies that Queens is not the home of sober taxpayers. Thus if it is not the case that Manhattan's being full of fast talk­ ers implies that Queens is not the home of sober taxpayers, then it is not the case that Staten Islanders are not Mets fans. (I, M, H) 4. Penelope is not a registered Democrat. For Penelope is a registered Democrat only if she is eligible to vote in the United States. But she is not eligible to vote in the United States. (D, E) *5. If Democrats are always compassionate, then Republicans are always honest. For if Democrats are always compassionate, then they sometimes vote for candidates who are moderates. But if they sometimes vote for candidates who are moderates, then Republicans are always honest. (D, M, H) *6. If Emma is a true pacifist, then she is not a supporter of war. Emma is a true pacifist. It follows that she is not a supporter of war. (E, A) 7. If this cheese was not made in Switzerland, then it's not real Emmentaler. Therefore, if it is real Emmentaler, then it was made in Switzerland. (C, E) *8. Either gulls sometimes fly inland or hyenas are not dangerous. But hyenas are dangerous. So, gulls sometimes fly inland. (G, H) 9. If both Enriquez enters the race and Warshawsky resigns, then Bosworth will win the election. But if Bosworth will win the election, then Mendes will not win the election. Thus if both Enriquez enters the race and Warshawsky resigns, then Mendes will not win the election. (E, A, B, M) *10. Microbes are not creating chronic diseases such as diabetes, multiple sclerosis, and even schizo­ phrenia. Hospitals need to improve their cleaning practices only if it is the case that microbes are 0 _J 0 a: <( U) 0 � z a: �o U) LL w f­ � z ow U) � ::J N (') N a: � <( (/) LL t­ Q Z >– ‘.2
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w <( IZ t- 0 (.') j:: z­_(/) �o 0 0.. WO I a: 0 0.. creating many chronic diseases such as diabetes, multiple sclerosis, and even schizophrenia. It follows that hospitals need not improve their cleaning practices. (M, H) 11 . California farmers grow either vegetables that thrive in warm weather or citrus fruits and bananas. Since they don't grow citrus fruits and bananas, they must grow vegetables that thrive in warm weather. (A, C, B) 12. Steve's attacker was not a great white shark. An attack of the sort he suffered last week must be by either a great white shark or by a shark of another type that felt threatened in the presence of a swimmer unknowingly wading into its feeding area. Therefore, Steve was attacked by a shark of another type that felt threatened in the presence of a swimmer unknowingly wading into its feeding area. (G, A) 13. Calcium is good for healthy bones. Either vitamin D is good for healthy bones or calcium is not good. Therefore, vitamin D is good for healthy bones. (C, D) 14. If herons wade either in mud holes or lagoons, then they catch bacterial infections. But they don't catch bacterial infections. Thus herons wade in neither mud holes nor lagoons. (H, L, I) 15. If she has a tune stuck in her head, she is either happy or annoyed. Therefore, if she is neither happy nor annoyed, then she doesn't have a tune stuck in her head. (H, A, N) VII. YOUR OWN THINKING LAB 1. Construct an argument of your own for each of the argument forms listed in exercise M- 2. Construct truth tables for each of the argument forms listed in exercise (VI). 12.3 Some Standard Invalid Argument Forms Already we have seen that arguments may have defects of various kinds that cause them to fail. Types of defects that undermine arguments constitute the so-called fallacies, among which, as we have already seen at some length, the informal fallacies figure prominently. Now we must consider their analogues in propositional logic, which include some of the formal fallacies. All formal fallacies have in common that they occur in an argument that has a superficial similarity to some valid form but departs from that form in some specifiable way. They are there­ fore instances of failed deductive arguments. Recall that an argument is invalid if it is possible that an argument with the same form could have true premises and a false conclusion. To prove the invalidity of an argument, then, it is enough to find a single case of an argument with exactly the same logical form whose premises are true and conclusion false. Consider the following argument: 26 1. If the messenger came, then the bell rang about noon. 2. The bell rang about noon. 3. The messenger came. This argument is invalid because it is possible for its premises to be true and its conclusion false. Even if the premises and conclusion all happen to be true in a certain case, there are other scenarios in which arguments with an identical form could have true premises and BOX 3 ■ INVALID ARGUMENT FORMS ['-___ s_o_M_ E_ F_O_R_ M_A _L_F _A_L_LA_C_I_E _s ___ ] Either P orQ p AFFIRMING A DISJUCT Either P or Q Q Therefore not Q Therefore not P AFFIRMING THE CONSEQUENT if P, then Q Q Therefor P DENYING THE ANTECEDENT if P, then Q Not P Therefore not Q a false conclusion. Suppose that the messenger didn't come, but the bell did ring about noon, though it was a neighbor who rang it. In this scenario, (26)'s premises are true and its conclusion false. Thus the scenario amounts to a counterexample that shows the invalidity of (26). It is often possible to find real-life counterexamples that prove the invalidity of certain arguments. Yet we could do without such counterexamples, since to show that an argument is invalid, it is sufficient to describe a 'possible world' (which may or may not be the actual world-it's simply a scenario involving no internal contradiction) where an argument with the same form would have true premises and a false conclusion. Thus the invalidity of an argument can be proved in the way just shown: one tries to describe a scenario where the premises of the argument in question are true and its conclusion is false. If such a scenario is not forthcoming, we may first extract the argument form-which, in the case of (26), is 26' 1. p:) Q 2.Q 3. p Then we try to find an example of an argument with the same form that in some possible scenario would have true premises and a false conclusion. For example, 27 1. If Barack Obama is a Republican, then he is a member of a political party. 2. Barack Obama is a member of a political party. 3. Barack Obama is a Republican. (27) shows that, in a scenario where the possible world is the actual world, an argument with the same form as (27) has true premises and a false conclusion. By the definition of inva­ lidity, (27) is invalid. At the same time, it amounts to a counterexample to any argument with the same form. 0 __J
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w <( IZ f- 0 ('J i= z­_(f) ::,::: 0 0 CL wo I a: 0 CL IElil Affirming a Disjunct Another formal fallacy is affirming a disjunct: Affirming a disjunct is the fallacy committed by any argument featuring a premise that is an inclusive disjunction, another premise affirming one of the disjuncts, and a conclusion denying the other disjunct. Affirming a disjunct is an invalid form because, as we saw earlier in our discussion of the truth-functional connectives, 'or' is to be understood in the inclusive sense (i.e., either P or Q or both)-not the exclusive sense (i.e., either P or Q but not both). The inclusive disjunction is true in all cases except where both disjuncts are false. Thus assuming that a certain inclusive disjunction is true, denying one disjunct (which amounts to saying that it is false) entails that the other disjunct must be true. But affirming one of its disjuncts (which amounts to saying that it is true) does not entail the denial of the other-that is, does not entail that the other is false. (In the case of an exclusive disjunction, what we are calling 'affirming a disjunct' would not be a fallacy.) Consider the following example: 32 1. Either my car was towed away by the police or was stolen. 2. My car was in fact towed away by the police. 3. My car was not stolen. Is there any way this conclusion could be false if both premises were true? Yes! A possible scenario is that thieves came in the night and broke into my car, then drove it to an illegal park­ ing space, from which the police towed it! If that were the case, then both of (32)'s premises would be true and its conclusion false at once. Thus the conclusion does not follow necessarily from the premises-it is not entailed by them. So the argument is invalid. But the thing to notice is that (32) instantiates version (a) of the invalid form affirming a disjunct. This fallacy is committed by any argument of one of these forms: 32' a l. p V Q 2.P 3. -Q b 1.PvQ 2. Q 3. -P Since, here, the 'either ... or ... ' connective in (32) is inclusive, to affirm one of the two alternatives does not entail a denial of the other. In any case where the disjunction is inclusive, an argument with either of (32')'s forms is invalid. The invalidity of affirming a disjunct is clearly shown by this truth table: 33 P Q PvQ, p :. -Q TT T T F T F T T T FT T F F F F F F T We have identified three invalid argument forms that correspond to three types of formal fallacy. Whenever you find an argument that has one, a truth table is not required. All you need to do to prove invalidity is simply to show that the argument has one of these forms: affirming the con­ sequent, denying the antecedent, or affirming a disjunct. If you can keep separate in your mind these three invalid forms and the five valid forms discussed earlier, you should find it much easier to distinguish valid and invalid propositional arguments. BOX 6 ■ HOW TO AVOID AFFIRMING A DISJUNCT Note that in a disjunctive syllogism, a premise denies one of the disjuncts of the other premise, and the conclusion asserts the other. ■ Thus, watch out for any argument that appears to be a disjunctive syllogism but it is not, since one of its premises asserts a disjunct of the other premise, while its conclusion denies the other. Exercises YIII Review Ouestjons 1 . How does the type of disjunction at work in disjunctive syllogism bear on the fallacy of affirming a disjunct? 2. How does affirming a disjunct differ from disjunctive syllogism? 3. How does affirming the consequent differ from modus ponens? 4. How does denying the antecedent differ from modus to/lens? 5. What's the cash value of recognizing that an argument commits a formal fallacy? IX. Some of the following arguments commit formal fallacies, and some don't. Indicate which do and which don't, identifying formal fallacies and valid argument forms by name. 1. If the defendant's 2007 Toyota sedan was used as the getaway car in the robbery, then it was not in the mechanic's garage with a cracked engine block on the date of the crime. But it was in the me­ chanic's garage with a cracked engine block on that date! From this it follows that the defendant's 2007 Toyota sedan was not used as the getaway car in the robbery. SAMPLE ANSWER: Modus to/lens. Valid. 2. If this car has faulty brakes, then it's dangerous to drive. But this car does not have faulty brakes. Therefore it's not dangerous to drive. *3. If our public officials take bribes, then there is corruption in our government. But if the mayor and several City Council members were paid to support the appropriations bill, then our public officials take bribes. So, if the mayor and several City Council members were paid to support the appropria­ tions bill, then there is corruption in our government. 4. Barry is a union member, for he will not cross the picket line. And if he were a union member, then he would not cross the picket line. *5. Ireland does not allow abortion. Either Ireland allows abortion or Ireland is a conservative country. Hence, Ireland is a conservative country. 0 _J 0 a: <( (/) 0� z a: �o (/) LL w 1- � z 0W (/)� :::>
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w <( IZ I- 0 (') i= z-- (J) � 0 0 a. UJ 0 I a: 0 a. ED 3. For each of the propositional arguments below, give its form and standard name (if any), use a truth table to decide whether it is valid or not, and propose an argument of your own with exactly the same form. A. B. C. D. E. F. G. H. I. J. K. L. M. If the cold front is here, then we don't go to the beach. Thus if we go the beach, then the cold front is not here. Either the small apples or the ripe ones are on sale. The ripe apples are on sale. Therefore, the small apples are not on sale. She is at Lalo's if her class is over. She is at Lalo's. Therefore, her class is over. I don't see my glasses there. If I don't see them there, then they are not there. Hence, they are not there. If Ptolemy was right, then the Sun and planets orbit the Earth. But it is not the case that the Sun and planets orbit the Earth. Therefore, Ptolemy was not right. The ring is made of either gold or silver. In fact, it is not made of silver. Therefore, it is made of gold. If the pool doesn't have chlorine, then it is not safe to swim in it. Since it is not safe to swim in it, it follows that the pool doesn't have chlorine. Irving is either a bachelor or he is a Dodgers fan. He is not a Dodgers fan. Therefore, he is a bachelor. If magnets cure rheumatism, then there is a market for them. But since it is not the case that magnets cure rheumatism, there isn't a market for them. There is a storm outside. If there is a storm outside, I'd better stay indoors. So, I'd better stay indoors. If Mary knows Juan, then she knows Jennifer. She knows Jennifer. Therefore she knows Juan. Tokyo is the capital of either Japan or Bangladesh. Tokyo is not the capital of Japan. So Tokyo is the capital of Bangladesh. Customer Services handles complaints about merchandise that is either damaged or imperfect. Customer Services handles complaints about merchandise that is damaged. Therefore, Customer Services doesn't handle complaints about merchandise that is imperfect. N. If the 'Big Bang' theory is not wrong, then the universe is expanding. The 'Big Bang' theory is not wrong. Therefore the universe is expanding. 0. Either students who got As or those who have missed no class are eligible for the prize. Students who have missed no class are eligible for the prize. So students who got As are not eligible for the prize. 12.4 A Simplified Approach to Proofs of Validity Some valid argument forms such as those discussed above are often used as basic rules of inference in the so-called proofs of validity. This is a procedure designed to show the steps by which the conclusion of a valid propositional argument follows from its premises. In constructing a proof for an argument, we assume that it is in fact valid, and we try to show that it is. Before we can proceed to construct some such proofs, we'll add other basic valid argument forms and rules of replacement to our list so that we can have enough rules of inference to prove the conclusions of a great number of valid propositional arguments. The Basic Rules In constructing our proofs of validity, then, we'll need some valid argument forms and some logical equivalences between compound propositions. The former will serve as rules of inference, which will permit us to draw a conclusion from a premise or premises. The latter will serve as rules of replacement, which will permit us to substitute one expression for another that is logically equivalent to it. Our list of rules includes the following: Basic Rules of Inference 1. Modus Ponens (MP) 2. Modus Tollens (MT) 3. Hypothetical Syllogism (HS) 4. Disjunctive Syllogism (DS) 5. Simplification (Simp) 6. Conjunction (Conj) 7. Addition (Add) Basic Rules of Replacement 8. Contraposition (Contr) 9. Double Negation (DN) 10. De Morgan's Theorem (DeM) 11. Commutation (Com) 12. Definition of Material Conditional (Cond) 13. Definitions of Material Biconditional (Bicond) What Is a Proof of Validity? p :::> Q,P :. Q

p :::> Q, -Q :. -P

P :::> Q, Q :::> R :. P :J R

P V Q,-P :. Q

p. Q … p

P, Q :. p • Q

p ;. p V Q

(P :::> Q)=(-Q :::> -P)

P=–P

-(P • Q)=(-P v -Q)
-(P v Q) = (-P • -Q)

(P V Q) “”(Q VP)
(P • Q)=(Q • P)

(P :::> Q) = (-P v Q)

(P “” Q) = [(P :::> Q) • (Q :::> P)]
(P”” Q) = [(P • Q) v (-P • -Q)]

Proofs of validity may be formal or informal. In a formal proof, the relation of entailment is taken
to obtain strictly between certain well-formed formulas of a system of logic that need have no
interpretation in a natural language (such as English, Portuguese, Mandarin). Furthermore, the
basic rules of inference and replacement used in formal proofs are such that they could be used
to prove the conclusion of any valid propositional argument from its premises. On the other
hand, in the informal proofs proposed here, entailment is taken to be a relation that obtains
between certain propositions that are expressible in a natural language. When a proof is offered
as involving only formulas, it is assumed in the informal approach that these have an
interpretation in a natural language. Moreover, the basic rules offered in our informal approach
fall short of allowing proofs of validity for any valid propositional argument.

I
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Such rules are 'basic' in the sense of being accepted without a proof. (Since any proof at all within this system would assume at least some of them, there are basic rules that cannot be proved within the system.) How to Construct a Proof of Validity Let's now put our basic rules to work and demonstrate the validity of the following argument: 34 Both Alice and Caroline will graduate next year. But if Caroline will graduate next year, then Giselle will win a scholarship if and only if Alice will graduate next year. So, either Giselle will win a scholarship if and only if Alice will graduate next year, or Helen will be valedictorian. First, we translate the argument into the symbolic notation as follows: 34' A• C, C:::) (G = A):.(G = A) v H We can now prove that this argument's conclusion, (G = A) v H, follows from its premises. How? By showing that such a conclusion can be deduced from (34')'s premises by applying to them only basic rules of inference and replacement. Our proof, whose four steps (numbered 3 1 4 1 5, and 6) aim at deducing the intended conclusion from (34')'s premises, runs 3411 1.A•C 2. C:::) (G =A) 3. C • A 4.C 5. G=A 6.( G=A) v H :.( G=A) vH from 1 by Com from 3 by Simp from 2 and 4 by MP from 5 by Add In line 3 1 we deduce C•A by applying commutation (see Com in the rules above) to premise 1. Any time we deduce a formula, we justify what we've done on the right-hand side of the proof. In this example, the justification includes expressions such as 'from,' 'and,' and 'by' that we'll later omit ('from') or replace by punctuation marks ('and' and 'by'). Note that a proof 's justification requires two things: (a) that we state the premise number to which a certain rule was applied (if more than one, we write down the premises' numbers in the order in which the rule was applied to them), and (b) that we state the name of the rule applied. After justify­ ing how a formula was deduced from the premise/s of an argument, that formula can be counted as a new premise listed with its own line number. Since C • A in line 3 has been deduced from the argument's premises, it is now a premise that can be used in further steps of the proof. In fact, it is used in line 4 to deduce C in the way indicated on the right-hand side of that line. Premises 2 and 4 allow us to deduce G = A in line 5, which follows from them by modus ponens (MP). In line 6, addition (Add) allows us to deduce the formula that proves (34)'s validity: namely, the conclusion of that argument. We have thus shown that its conclusion follows from its premises, and we have done so by showing that it can be obtained by applying only basic rules of inference and replacement to those premises. Thus (34) has been proved valid. Proofs vs. Truth Tables As we've seen, in the case of truth tables, the truth values of an argument's premises and conclusion are assigned according to rules associated with the truth-functional connectives involved in that argument. Although here we've defined only five such connectives, their total number is in fact sixteen. This is a fixed number. By contrast, the actual number of valid argument forms and logically equivalent expressions that could be used to construct proofs of validity may vary from one deductive system to another. Furthermore, the proof procedure allows for no fixed number of steps to correctly deduce an argument's conclusion from its premises: it often depends on which premises and basic rules we decide to use. Since in these respects proofs permit a certain degree of flexibility, it is sometimes possible, within a single system of basic rules, to construct more than one correct proof to demonstrate the validity of a certain argument. That is, unlike a truth table, a proof is not a mechanical procedure that always yields a result in the same way in a fixed number of steps. Moreover, it might happen that, in constructing a proof for a certain valid argument, we err in our assessment of its validity. We might simply "fail to see" at the moment that certain rules can be put at the service of deducing that argument's conclusion from its premises and mistakenly conclude that the argument is invalid. That's why we say that, for any valid argument, one could 'in principle' construct a proof of its validity. It must be admitted, however, that proofs do have one big advantage over truth tables: namely, that the latter tend to be very long and unwieldy when an argument features propositions of many different types. Proofs face no such problem. Exercises 1 . In what does the method of proof consist? 2. Do proofs offer any advantage over truth tables? 3. What is a rule of inference? 4. How are rules of inference used in a proof? 5. What are rules of replacement? 6. In this section, a distinction has been drawn between a formal and an informal approach to proofs. What is that distinction? I 0 <( a: >–
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5. (D :::> C) :::> D

6. ~(D :::> C) v D

7. D v ~(D :::> C)

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3.1 • G

4. –(I• G)

5. ~(~H v L)

6. ~~H • -L

7. H • ~L

8. ~L • H

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w <( IZ f- 0 C) � z­- (/) � 0 0 [l_ wo I a: 0 [l_ MIH 13. 1. ~A :J ( I v -0) 2. ~A• ~I 3. -A 4.1 v ~D 5. ~I• -A 6. ~I 7. -D 14. 1. E • [~I• (~D v I)] 2. [ ~I • (~D v I)] • E 3. ~I • (~D v I) 4. ~I 5. (-0 v I) • ~I 6. -D v I 7. ~D 8. -D v F 15. 1. E 2. Ev D 3. (Ev D) v C 4. C v (Ev D) /:. ~D I:. -D VF !:. C v (Ev D) XIV. Translate each of the following arguments into symbolic notation using the propositional symbols within parentheses and construct a correct proof of validity for it. 1. The Bensons and the Nelsons will be at the party. But if the Nelsons are at the party, then the Finnegans will not be there. The Finnegans will be at the party only if the Bensons will not be there. It follows that the Finnegans will not be at the party. (B, N, F) SAMPLE ANSWER: 1. B • N 2. N :J -F 3. F :J -B 4. N • B 5.N 6. -F !:. ~F 1 Com 4Simp 2, 5 MP 2. If elephants are mammals, then they are not warm-blooded creatures. It is not the case that elephants are not warm-blooded creatures. From this we may infer that either elephants are not mammals or they are not warm-blooded creatures. (E, C) 3. Either municipal bonds will not continue to be a good investment or stocks will be a wise choice for the small investor at the present time. Municipal bonds will continue to be a good investment. Therefore, stocks will be a wise choice for the small investor at the present time. (M, C) *4. If Romania establishes a democracy, then Bulgaria will, too. Either Mongolia will not remain inde­ pendent or Romania will not establish a democracy. Bulgaria will not establish a democracy, but Romania will. Thus Mongolia will not remain independent. (D, B, M) 5. Zoe will not resign next week. For Keith will serve on the committee, and either Zoe will not resign next week or Oliver will. But if Zoe does resign next week, then Keith will not serve on the committee. (K, E, 0) *6. Honduras will support the treaty, but it is clear that either Russia will not support it or Japan will support it. Japan supporting the treaty implies that Honduras will not support it. Therefore, Japan will not support the treaty. (H, I, J) 7. If Macedonians and the Danes were polytheists, then most ancient Europeans also were. The Romans' not being polytheists implies that both the Macedonians and the Danes were polytheists. It follows that if most ancient Europeans were not polytheists, then the Romans were polytheists. (M, D, E, 0) *8. Railroads are safe investments, but oil companies are not. It follows that oil companies are not safe investments but public utilities are, because railroads are safe investments only if public utilities are, too. (I, C, B) 9. Dramatists are not opinionated or historians are not disputatious. For if dramatists are opinionated, then musicians are not good at math. But musicians are good at math. (D, M, H) *10. Sicily is an island. Besides, if Italy is the home of famous soccer players, then Egypt is not the birthplace of Caesar. In addition, if Italy is not the home of famous soccer players, then Norway being full of tourists implies that Egypt is not the birthplace of Caesar. It follows that Sicily is an island, and Egypt's being the birthplace of Caesar implies that if Norway is full of tourists, then Egypt is not the birthplace of Caesar. (I, H, E, N) ■ Writing Project Provide a hypothetical syllogism for the conclusion that if globalization is promoted, products will be cheaper. Then offer a modus ponens for the conclusion that globalization entails fewer American jobs. Compare the relative strength of these two arguments by discussing the support for their premises. At the end of this discussion, reconstruct both arguments, marking in each case premises and conclusion. ■ Chapter Summary Procedures for determining whether an argument is valid: 1. Truth Tables: a mechanical technique that shows an argument form to be invalid if there is a row where all premises are true and the conclusion false. Otherwise, it is valid. 2. Proofs: a nonmechanical technique that shows an argument to be valid if its conclusion can be deduced by applying only valid rules of inference and replacement to the argument's premises. Some valid forms. When an argument has any of these forms, it is valid: 1. Modus ponens 2. Modus tollens 3. Hypothetical Syllogism 4. Disjunctive Syllogism 5. Contraposition p :J Q,P :. Q p :J Q, -Q :. -P P :J Q, Q :J R :. P :J R P vQ,-P :. Q p :J Q :. -Q :J -P -Q :J -P :. p :J Q >­
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w <( IZ I- 0 ('.) i= z­- (/) ::.:'. 0 0 0... WO I er: 0 0... Some invalid forms. When an argument has any of these forms, it commits a formal fallacy and is invalid: 1. Affirming the Consequent 2. Denying the Antecedent 3. Affirming a Disjunct ■ Key Words Truth table for arguments Modus ponens Modus tollens Contraposition Hypothetical syllogism Disjunctive syllogism p:) Q 'Q :. p p 🙂 Q '-P :. -Q p V Q 'p ,', -Q Formal fallacy Affirming the consequent Denying the antecedent Affirming a disjunct Counterexample Proof of validity CHAPTER Categorical Propositions and Immediate Inferences In this chapter you'll read about logical relations between categorical propositions, which are the building blocks of syllogistic arguments. The topics include Standard categorical propositions and the class relationships they represent. Non-standard categorical propositions and their translation into standard categorical propositions. How to represent categorical propositions in Venn diagrams and in traditional logic. The Square of Opposition, both traditional and modern versions. The problem of existential import. Other immediate inferences from categorical propositions. 293 (f) (f) Zw Qo f- z if) w 0 a: 0... w Q LL a: � 0... w _J f­ <( ::i: 00 -w a: 2 0 ""' 0 == w f- 0 <( z 0 <( 11&1 13.1 What Is a Categorical Proposition? Categorical Propositions Categorical propositions are propositions that represent relations of inclusion or exclusion between classes of things, such as 1 All philosophers are wise persons. 2 No philosophers are wise persons. Or between partial classes, such as 3 Some philosophers are wise persons. Or between partial classes and whole classes, such as 4 Some philosophers are not wise persons. The relationships between classes that matter for categorical propositions are, then, these four: ■ Whole inclusion of one class inside another ■ Mutual, total exclusion between two classes ■ Partial inclusion, whereby part of one class is included inside another. ■ Partial exclusion, whereby part of one class is wholly excluded from another In the above examples of categorical propositions, 'philosophers' is the subject term and 'wise persons' the predicate term. These terms are the logical, rather than syntactical, subject and pred­ icate of a categorical proposition. Each of them denotes a class of entities: that made up by all and only the entities to which the term applies. Thus 'philosophers' denotes the class of philoso­ phers and 'wise persons' the class of persons who are wise. Categorical propositions (1) through (4) illustrate four ways in which the class of philosophers and the class of wise persons can stand in relationships of inclusion or exclusion. Each of these relationships may be represented in one of the following ways: 1 ' All philosophers are wise persons. 2' No philosophers are wise persons. 3' Some philosophers are wise persons. 4' Some philosophers are not wise persons. Philosophers X In traditional logic, first developed in antiquity by Aristotle (384-322 BCE), the standard notation to represent the logical form of categorical propositions is to use 'S' as a symbol for any subject term, and 'P' for any predicate term. In that notation, then, the logical form of the above categorical propositions is, respectively, 1. All Sare P 2. No Sare P 3. Some S are P 4. Some S are not P In traditional logic, only statements that can be shown to have these logical forms qualify as expressing a categorical proposition. Any such proposition always represents one of the four relationships between classes mentioned above, which can now be described by using the symbols 'S' and 'P,' which stand for the classes denoted by the proposition's subject and predicate. Those relationships are as in Box 1. But we can also represent them by circle diagrams, in which case we'd have _J < 0 er: 0 0 w � 0 < C'-· f?
NO
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term (‘U.S. citizens’) and the other for the predicate (‘voters’).

p

Since, according to (10), all members of the class denoted by its subject term are members of the

class denoted by its predicate term, the crescent-shaped part of S that has no members (i.e., that

representing U.S. citizens who are not voters) has been shaded out in the diagram. With the

Venn-diagram technique, shading a space means that that space is empty. Thus, in the above

diagram, S non-P is shaded out, to represent that there is nothing that is S that is non-P. This is

consistent with reading (10) as saying that the subclass of U.S. citizens who are not voters is an

empty subclass-or, equivalently, that there are no U.S. citizens who are not voters.

On the previous page, (1o)’s translation is provided, first, in the algebraic notation

introduced by the English mathematician George Boole (1815-1864), which reads, ‘S non-P

equals o,’ and then in the notation of traditional logic, reading ‘All S are P.’ What both say is

captured by the Venn diagram in the box: namely, that the subclass of S non-P (represented by

the shaded portion of the diagram) is empty.

Now let’s look at (11), an instance of the universal negative.

11 No U.S. citizens are voters.

p

Boolean Notation:

SP=o

E Proposition:

No Sare P

Since (11) is a universal proposition, its Venn diagram shows an empty subclass that has been

shaded out: the football-shaped center area, the intersection of’ S’ and ‘P,’ which represents the

U.S. citizens who are voters. The diagram thus captures that (11) denies that there are any such

voters: in other words, asserting (11) amounts to saying that the class of voting U.S. citizens has

no members. To the left of the diagram, (n)’s Boolean notation ‘S P = o’ tells us that the sub­

class ‘S P’ is empty. Immediately below, we find (n)’s notation and type in traditional logic.

Keep in mind that, for any universal categorical proposition (whether affirmative or negative),

there will be a part of the circles shaded out, to indicate that that part has no members.

Next, consider the particular affirmative

12 Some U.S. citizens are voters.

Boolean Notation:

SP=#=o

I Proposition:

Some Sare P

p

This time, no universal claim is being made, but rather a particular one: a claim about part of a

class. As a result, the diagram shows no shading at all, but an ‘x’ instead, in the area where there

are some members. Since ‘some’ logically amounts to ‘at least one,’ (12) is equivalent to

12′ There is at least one U.S. citizen who is a voter.

Putting an ‘x’ in the football-shaped center space indicates that that space, ‘SP,’ is not empty­

in effect, that it has some members (at least one). To the left of the diagram, we find (12)’s

Boolean translation’ SP=#= o,’ which tells us that the subclass ‘SP’ (i.e., the football-shaped area

in the center) is not empty-together with its type and notation in traditional logic.

Finally, what about a particular negative? Consider

13 Some U.S. citizens are not voters.

p

Boolean Notation:

SP=#=o

0 Proposition:

Some S are not P

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The two intersecting circles represent the two classes of things related in a categorical
proposition-the one on the left, the class denoted by its subject, the one on the right, the
class denoted by its predicate. The circles also determine four subclasses that we may
identify with the spaces drawn. The space in the center, where they overlap, represents the
subclass of things that are both Sand P at once (i.e., the subclass of things that are simulta­
neously members of both classes), which is indicated by the notation ‘SP.’ The crescent­
shaped space on the left represents the subclass of things that are S but not P, where the

negation is indicated by a bar over the symbol ‘P.’ The crescent-shaped space on the right
represents the subclass of things that are P but not S, where the negation is indicated by a
bar over the symbol ‘S.’ The space outside the two interlocking circles represents the class

of things that are neither S nor P. As we have seen, with these spaces we can use the Venn­
diagram technique to represent the class inclusion and exclusion relationships described in
each of the four standard categorical propositions. To see how this works, let’s start with a

concrete example. Consider the four categorical propositions that may be constructed out
of ‘U.S. citizens’ as the subject term and ‘voters’ as the predicate term. All four relationships
of inclusion and exclusion between the class of U.S. citizens and the class of voters, as rep­

resented in those propositions, are captured in the Venn diagram in Box 3. There we may

BOX 3 ■ VENN DIAGRAMS FOR CATEGORICAL

PROPOSITIONS

U.S. citizens
(S)

Nonvoting
Non-U.S. citizens

identify the following subclasses: (1) U.S. citizens who are voters, (2) U.S. citizens who are
not voters, (3) voters who are not U.S. citizens (which would include, for instance, those who
vote in other countries), and (4) non-U.S. citizen who are non-voters (which would include,
for instance, not only current citizens of other countries who do not vote, but also Henry
VIII, Julius Caesar, and even things like the Eiffel Tower, the Magna Carta, and the Grand
Canyon-in fact everything we can think of belongs to one or the other of these four possi­
ble subclasses).

For each categorical proposition, then, there is a Venn diagram that shows the relation­
ship of inclusion or exclusion that it involves. The bottom line is:

■ The areas displayed by a Venn diagram relevant to representing a categorical proposition
are three: those inside each intersecting circle and their intersection itself.

■ A Venn diagram for an A or E proposition shows a shaded area where there are no
members. No ‘x’ occurs in this diagram.

■ A Venn diagram for an I or O proposition shows an ‘x’ in the area where there are
members. No area is shaded in this diagram.

Exercises

1. In the previous section, Venn diagrams were used to represent categorical propositions. Explain
how this technique works.

2. What do Venn diagrams for universal propositions have in common? What about those for particu­
lar propositions?

3. What does it mean when spaces are shaded out in Venn diagrams for categorical propositions?
And what’s the meaning of an ‘x’ placed in one of the circles?

4. What do the two circles stand for in a Venn diagram for a categorical proposition?

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__J !;;: <( - (.) 0 - LU 0: 2 0 2 0_ LU f- 0 <( z 0 <( IX. Determine whether each Boolean notation for the diagram on the right is correct. If it isn't, provide the correct one. 1.SP=O SAMPLE ANSWER: Incorrect. It should be: p S P,¢.0 2. S P¢c 0 p *3. SP ¢c 0 s p 4. SP� 0 p X. For each categorical proposition below, first identify its letter name and traditional notation. Then select the correct Venn diagram and Boolean notation for it from the following menu: Venn Diagram 1 p 2 p 3 s p 4 p _J 0

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<( � 00 -w a: � 0 � (.'J - w ...... 0 <( z 0 <( (1) SP= 0 (2) SP= O Boolean Notation (3) SP :it: 0 (4) S P :it= 0 1. No Sumo wrestlers are men who wear small-size shirts. SAMPLE ANSWER: E proposition; no Sare P. Venn Diagram 2, Boolean notation 1. 2. Some sports cars are very expensive machines. *3. All llamas are bad-tempered animals. 4. Some grocers are not members of the Rotary Club. 5. All waterfalls are places people in kayaks should avoid. *6. No spiders are insects. 7. Some advertisers are artful deceivers. 8. Some cowboys are not rodeo riders. *9. No atheists are churchgoers. 10. All oranges are citrus fruits. 11 . Some rivers that do not flow northward are not South American rivers. *12. If a number is even, then it is not odd. 13. There are marathon runners who eat fried chicken. 14. Some accountants who are graduates of Ohio State are not owners of bicycles. *15. Not all oils are good for you. 16. Some reference works are books that are not in the library. 17. If an architect is well known, then that architect has good taste. *18. Nothing written by superstitious people is a reliable source. 19. Chiropractors who do not have a serious degree exist. *20. Some resorts that are not in the Caribbean are popular tourist destinations. 13.3 The Square of Opposition The Traditional Square of Opposition Categorical propositions of the above four types were traditionally thought to bear logi­ cal relations to each other that enable us to draw certain immediate inferences. These are single-premise arguments, to some of which we'll turn now. We'll first look at the imme­ diate inferences represented in the Traditional Square of Opposition, a figure that looks like this: BOX 4 11 TRADITIONAL SQUARE OF OPPOSITION (All Sare P) (No Sare P) A CONTRARIETY E s s u u B B A A L L T T E E R R N N A A T T I I 0 0 N N SUBCONTRARIETY 0 (Some S are P) (Some S are not P) The relations represented in the Traditional Square of Opposition, which involve two categorical propositions at a time, are as follows: Relation Contradiction Contrariety Subcontrariety Subalternation Established Between A and O; E and I AandE I and 0 A and I; E and O Let's now take up each of these relations in turn. Name of Related Propositions Contradictories Contraries Sub contraries Superalterns: A and E Subalterns: I and 0 Contradiction. Propositions of the types in diagonally opposite corners of the Square are contradictories. Propositions that stand in the relation of contradiction cannot have the same truth value: if one is true, then the other is false, and vice versa. Thus A and O propositions will always have opposite truth values if their subjects and predicates are the same, as will proposi­ tions of types E and I. Thus if (1) is true, (4) is false: 1 All philosophers are wise persons. 4 Some philosophers are not wise persons. On the other hand, if (1) is false, then that's logically the same as saying that (4) is true. Similarly, if it's true that 3 Some philosophers are wise persons, (that is, there is at least one philosopher who is a wise person), then it is false that 2 No philosophers are wise persons. z 0 (j) 0 n.. n.. 0 LL 0 w 0: a (j) w f- (j) (j) Zw Qo f- z And conversely, if (3) is false, then (2) is true. When we infer the truth value of a proposition from that of its contradictory, as we've been doing here, we make a valid immediate inference: a single-premise argument whose conclusion must be true if its premise also is. But contra­ diction is only one sort of valid immediate inference according to traditional logicians; as we shall see next, there are others. BOX 5 ■ CONTRADICTION in� O w A and O ¢ <::i Contradictories n_ LL � � E and I ¢ <::i Contradictories n. w ....J f- < ::!'. 00 - w Contrariety, Subcontrariety, and Subalternation. The Traditional Square of Opposition also a: � g � includes the following logical relationships between propositions, which are valid given � o certain assumptions that we'll discuss presently: < z O< - A and E ¢ Contrariety I and 0 ¢ Subcontrariety A and I ¢ Subalternation Eand 0 ¢ Subalternation Contrary propositions cannot both be true at once, but can both be false. For instance, by con­ trariety, if (14) is true we can infer that (15) is false: 14 All bankers are cautious investors. 15 No bankers are cautious investors. That's because these categorical propositions cannot both be true. Yet they could both be false (as in fact they are). But contrariety differs from subcontrariety, and neither of these is the same as contradiction. Subcontrary propositions can both be true at once but cannot both be false. By subcontrariety, if (16) is false, then (17) is true: 16 Some students are vegetarians. 17 Some students are not vegetarians. These categorical propositions cannot both be false but could both be true (as in fact they are). Finally, there is the relationship of subalternation, which is a little more complex, since the correct inference of truth values varies depending on whether we go from the universal proposition to the corresponding particular, or the other way around. Logically speaking, to say that an A proposition and the corresponding I proposition are in the relation of subalternation is to say that if the A proposition is true, then the I proposition must be true, as well, but also that if the I proposition is false, then the A must be false. And similarly, for an E proposition and the corresponding 0, to say that they are in the relation of subalternation means that if the E proposition is true, then necessarily the O proposition is true, but also that if the O is false, then the E must be false as well. In either case, the universal proposition is called 'superaltern,' and the particular of the same quality 'subaltern.' So Subalternation is a logical relation between: A and I (A superaltern, I subaltern) E and O (E superaltern, 0 subaltern) In this relation: Truth transmits downward (from superaltern to subaltern) Falsity transmits upward (from subaltern to superaltern) Let's reason by subalternation as traditional logicians would. Suppose it's true that 18 All trombone players are musicians. Then it must also be true that 19 Some trombone players are musicians. This suggests that truth transmits downward. At the same time, since it is false that some trombone players are not musicians, it follows that it is also false that no trombone players are musicians-and this suggests that falsity transmits upward. But a false superaltern such as (14) clearly fails to entail a false subaltern, since that some bankers are cautious investors is true. 14 All bankers are cautious investors. And a true subaltern such as (17) fails to entail a true superaltern, since that no students are vegetarians is false. 17 Some students are not vegetarians. Truth-Value Rules and the Traditional Square of Opposition Let's now summarize all rela­ tionships represented in the Traditional Square of Opposition, together with the rules to be used for drawing immediate inferences from it: Contradiction: Contradictory propositions cannot have the same truth value (if one is true, the other must be false, and vice versa). Contrariety: Contrary propositions cannot both be true at once, but can both be false. Subcontrariety: Subcontrary propositions cannot both be false at once, but can both be true. Subalternationfrom the superaltern to subaltern (i.e., from the universal proposition to the particu­ lar proposition of the same quality): If the superaltern is true, then the subaltern must be true. If the superaltern is false, then the subaltern is undetermined Subalternationfrom the subaltern to superaltern (i.e., from the particular proposition to the universal proposition of the same quality): If the subaltern is true, then the superaltern is undetermined. If the subaltern is false, then the superaltern must be false. z 0 1- (j) 0 0... 0... 0 LI.. 0 w a:
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<( ::; 00 -w 0: ::?: 0 "<;: 0 � w f-- 0 <( z 0 <( Given the relationships of contradiction, contrariety, subcontrariety, and subalternation repre­ sented in the Traditional Square of Opposition, then assuming the truth values listed on the left, we can infer the values listed on the right. If A is true ¢ E is false, 0 is false, and I is true. If A is false ¢ E is undetermined, 0 is true, and I is undetermined. If Eis true ¢ A is false, I is false, and O is true. If Eis false ¢ A is undetermined, I is true, and O is undetermined. If I is true ¢ A is undetermined, E is false, and O is undetermined. If I is false ¢ A is false, E is true, and O is true. If O is true ¢ A is false, E is undetermined, and I is undetermined. If O is false ¢ A is true, E is false, and I is true. Existential Import Although inferences by contrariety, subcontrariety, and subalternation are all licensed as valid by the Traditional Square of Opposition, our ability to draw such inferences is undermined by a significant difference between universal propositions, on the one hand, and particular propositions, on the other: namely, that the latter (I and 0) have existential import while the former (A and E) do not. That is, I and O propositions implicitly assume the existence of the entities denoted by their subject terms. Since 'some' is logically the same as 'at least one,' therefore an I proposition such as (20) is logically equivalent to (20'): 20 Some cats are felines. 20' There is at least one cat that is a feline. Note that 'there is at least one cat ... ' amounts to 'cats exist.' Similarly, an O proposition such as (21) is logically the same as (21'), which likewise presupposes that some cats exist: 21 Some cats are not felines. 21 ' There is at least one cat that is not a feline. On the other hand, A and E propositions are logically the same as conditionals: (22) is equiva­ lent to (22') and (23) to (23'). 22 All cats are felines. 22' If anything is a cat, then it is a feline. 23 No cats are felines. 23' If anything is a cat, then it is not a feline. Understood in this way, a universal categorical proposition doesn't have existential import, since it is equivalent to a conditional, a compound proposition that is false if and only if its antecedent is true and its consequent false. So (22') would be false if and only if there are cats but they are not felines, as would (23') if there are cats but they are felines. If cats did not exist, the antecedents of these conditionals would be false, and those conditionals true (independent of the truth value of their consequents). Thus the inference by contrariety is undermined: given this understanding of universal propositions, contrary propositions could both be true in cases where their subjects are empty (i.e., have no referents). Consider (24), which is equivalent to (241): 24 All unicorns are shy creatures. 24' If anything is a unicorn, then it is a shy creature. Since nothing is a unicorn, (24')'s antecedent is false, and the whole conditional therefore true. Now consider its contrary, (25), which is equivalent to (25'): 25 No unicorns are shy creatures. 25' If anything is a unicorn, then it is not a shy creature. Here again, since nothing is a unicorn, (25')'s antecedent is false, and the whole conditional therefore true. Clearly, then, (24) and (25) could both be true! It follows that, unless we assume that the subject term of a true universal proposition is non-empty, we cannot infer that its con­ trary is false. Now, what about subcontrariety? This involves I and O propositions-which, in the modern understanding, do have existential import. Although, given the Traditional Square of Opposition, subcontraries cannot both be false, in the modern understanding they can. Consider now 26 Some unicorns are shy creatures. This is equivalent to 26' There are unicorns and they are shy creatures. Thus understood, (26) is false, since there are no unicorns. Compare 27 Some unicorns are not shy creatures. This is equivalent to 27' There are unicorns and they are not shy creatures. Since there are no unicorns, (27) turns out to be false as well. Thus (26) and (27) could both be false at once. It follows that we cannot draw valid inferences by subcontrariety. Finally, consider subalternation. From what we have just seen, this relation also begins to look suspicious. How can one validly infer, for example, from an A proposition that has no existential import, an I proposition that does? Of course, I-from-A and O-from-E inferences might seem unproblematic at first, whenever the things denoted by their subject terms exist­ for example, trombone players, accountants, and tigers. But when we're talking about entities whose existence is questionable, inference by subalternation leads to absurdities, such as 28 1. All unicorns are shy creatures. 2. Some unicorns are shy creatures. Since the conclusion in (28) is equivalent to (26') above, the argument appears to have "proved" that unicorns exist! This attempt to draw a conclusion by subaltemation fails because it ignores the fact that the premise has no existential import, while the conclusion (its subaltern) does have it. z 0 r (J) 0 Cl. Cl. 0 LL 0 llJ a: <( :::, 0 (J) llJ I r C') C') (/) (/) z lJJ Qo I- z - lJJ (/) a: 0 lJJ Q_ u.. Oz a: - Q_ UJ _J I­ <{ � 00 - UJ a: � 0 � CJ - UJ I- 0 <{ z 0 <{ IIE1D The Modern Square of Opposition Some qualifications of the allowable valid inferences according to the Square of Opposition are needed to restrict the range of valid inferences involving categorical propositions. As shown in Box 6, the Modern Square modifies the traditional one so that it leaves out the relationships of subalternation, contrariety, and subcontrariety, retaining only contradiction as a relation sanctioning valid immediate inferences. Contradiction holds between A and O and between E and I propositions, which are in opposite corners of the Square, marked by the two diagonals, as shown in Box 6. From this Modern Square, we can see two things about a proposition and the negation of its contradictory. First, they are logically equivalent: if the proposition in one corner is true, then the negation of its contradictory must be true; and if the proposition in one corner is false, then the negation of its contradictory must be false. Second, they entail each other: any BOX 6 ■ MODERN SQUARE OF OPPOSITION (All Sare P) A (Some S are P) (No Sare P) E 0 (Some S are not P) inference from a proposition to the negation of its contradictory preserves truth value and is therefore valid. Here, then, is a complete list of the equivalences (and entailment relations) between a proposition of one of the four standard types and the negation of its contradictory sanctioned by the Modern Square of Opposition: 1. A= not 0 2. E= not I 3. I= not E 4. O= not A So, given (1), if 'All oranges are citrus fruits' is true, then 'It is not the case that some oranges are not citrus fruits' must be true; and vice versa. But given (4), if 'Some oranges are not citrus fruits' is true, then 'All oranges are citrus fruits' must be false while 'It is not the case that all oranges are citrus fruits' must be true. You should try, as an exercise, to run an example for each of these equivalences. The bottom line is that for the listed propositions, each pair have the same truth value: if one is true, the other must also be true; and if one is false, the other must likewise be false. The former yields validity, the two combined logical equivalence. Venn diagrams are consistent with the modern view of the Square of Opposition. After all, it is only for particular propositions that we're required to use an 'x' to indicate where there are members of the subject class (if they exist at all). Universal proposi­ tions never require us to indicate where there are members, but only where there aren't any (i.e., by shading). BOX 7 ■ LOGICAL EQUIVALENCE AND VALIDITY Logical Equivalence When two propositions are logically equivalent, if one is true, then the other is also true; and if one is false, then the other must be false as well. This is because the conditions under which they are true or false are the same. Thus logically equivalent propositions have the same truth values: they are either both true or both false. As a result, one of them could be substituted for the other while preserving the truth value of the larger expression in which they occur, provided that nei­ ther occurs in a special context that could not allow such substitutions. For example, a proposi­ tion 'P' is logically equivalent to 'It is not the case that not P'; therefore, one can be replaced by the other while preserving the truth value of the larger expression in which one of them occurs, pro­ vided that, for instance, the expression does not occur inside quotation marks. Validity When two propositions are logically equivalent, if one is true, the other is true as well. This satisfies the definition of entailment or valid argument: logically equivalent propositions entail each other. Any argument from one to the other is valid. Exercises 1 . What is an immediate inference? 2. Which immediate inferences are valid according to the Traditional Square of Opposition, and which according to the Modern Square of Opposition? Support your answer with examples. 3. Subalternation works differently depending on whether it is an inference from superaltern to sub­ altern or vice versa. Explain. 4. What does it mean to say that certain propositions have existential import? Which categorical propositions have it, according to the modern interpretation of the Square of Opposition? z 0 1- (/) 0 Q_ Q_ 0 LL 0 w a: <{ :::i 0 (/) w I 1- (") (") E1III (/) (/) z LlJ Qo I- z -w (/) a: 0 LlJ a. u.. Oz a: - a. LlJ .J I­ <( � 00 - LlJ a: � o""' 0:::: I- 0 <( z 0 <( XII. For each of the following, first name the type of the proposition related to it by contrariety or subcontrariety, as the case may be, and state that proposition. Then assume that the proposition given is true and determine the truth value of its contrary or subcontrary. 1. All Icelanders are believers in elves. SAMPLE ANSWER: E. Contrary. No Icelanders are believers in elves. False. 2. No epidemics are dangerous. *3. Some humans are not mortal. 4. No riverboat gamblers are honest men . *5. All labor unions are organizations dominated by politicians. 6. Some conservatory gardens are not places open to the public. *7. Some lions are harmless. 8. No used-car dealers are people who can be trusted. *9. Some bats are not nocturnal creatures. 10. Some historians are interested in the past. XIII. For each of the propositions above, assume that it is false and determine the truth value of its contrary or subcontrary. (*4, *6, *10) SAMPLE ANSWER: 1. E. Contrary. Undetermined. XIV. For each of the following, give the letter name of its contradictory and state that proposition. 1. All bankers are fiscal conservatives. SAMPLE ANSWER: 0. Some bankers are not fiscal conservatives. *2. No Democrats are opponents of legalized abortion. 3. Some SUVs are vehicles that get good gas mileage. *4. All professional athletes are highly paid sports heroes. 5. Some tropical parrots are not birds that are noisy and talkative. *6. Some chipmunks are shy rodents. 7. No captains of industry are cheerful taxpayers. *8. Some cartographers are amateur musicians. 9. All anarchists are opponents of civil authority. *10. Some airlines are not profitable corporations. XV. First suppose each categorical proposition listed in the previous exercise is true. What could you then know about the truth value of its contradictory? Second, suppose each proposition in the list is false. What could you then know about the truth value of its contradictory? XVI. For each of the following, first name the type of the proposition related to it by subalternation and state that proposition. Then assume that the proposition given is true and determine the truth value of its superaltern or subaltern. 1. Some westerns are not good movies. SAMPLE ANSWER: E. Superaltern. No westerns are good movies. Undetermined. 2. Some string quartets are works by modern composers. *3. No butterflies are vertebrates. 4. No parakeets are philosophy majors. *5. Some comets are not frequent celestial events. 6. All Internal Revenue agents are hard workers. *7. Some porcupines are not nocturnal animals. 8. Some Rotarians are pharmacists. *9. No extraterrestrials are Republicans. 10. All amoebas are primitive creatures. XVII. For each of the propositions above, assume that it is false and determine the truth value of its superaltern or subaltern. (*4, *8, *10) SAMPLE ANSWER: 1. E. Superaltern. False. XVIII. For each proposition below, first give the letter names of all propositions related to it according to the Traditional Square of Opposition, specify those relationships, and state those propositions. Then, assuming that each proposition listed below is true, what would be the truth values of the given propositions? (Tip for in-class correction: Move clockwise through the relations in the Square.) 1. All tables are pieces of furniture. SAMPLE ANSWER: E. Contrary. No tables are pieces of furniture. False. 0. Contradictory. Some tables are not pieces of furniture. False. I. Subaltern. Some tables are pieces of furniture. True. 2. Some griffins are mythological beasts. 3. No liars are reliable sources. *4. Some bassoonists are anarchists. 5. All trombone players are musicians. z 0 f­ (/) 0 a.. a.. 0 0 w a: <( ::, 0 (/) w I f- C") C") (/) (/) z LI.J Qo 1-- z (/) LI.J 0 a: 0.. LI.J 0 LL a: � 0.. LI.J ....J 1-­ <( ::; 00 - LI.J a: � 0� C)_ LI.J 1-- 0 <( z 0 <( 1111m *6. No Americans are people who care about global warming. 7. All white horses are horses. *8. All acts of cheating are acts that are wrong. 9. Some cyclists are not welcome in the Tour de France. *10. Some things are things that are observable with the naked eye. XIX. Assuming that the propositions listed in the previous exercise are false, what is the truth value of each proposition related to them by the Traditional Square of Opposition? (*3, *5, *7) SAMPLE ANSWER: E. Contrary. No tables are pieces of furniture. Undetermined. 0. Contradictory. Some tables are not pieces of furniture. True. I. Subaltern. Some tables are pieces of furniture. Undetermined. XX. YOUR OWN THINKING LAB 1. Assuming that the propositions listed in (XVIII) above are true, use the Modern Square of Opposition to draw a valid inference from each of them. SAMPLE ANSWER: All tables are pieces of furniture. It is false that some tables are not pieces of furniture. 2. Consider propositions such as 'No centaur is a Freemason,' 'All hobbits live underground,' and 'Some Cyclops are nearsighted.' What's the matter with them according to modern logicians? Explain. *3. Determine which logical relation among those represented in the Traditional Square of Opposition holds between premise and conclusion in each of the following arguments. Is the argument valid according to the Modern Square of Opposition? Discuss. A. All automobiles that are purchased from used-car dealers are good investments. Therefore, some automobiles that are purchased from used-car dealers are good investments. SAMPLE ANSWER: Subalternation. Invalid by the Modern Square. B. Some residents of New York are dentists. Therefore, it is not true that no resident of New York is a dentist. *C. No boa constrictors are animals that are easy to carry on a bicycle. Therefore, it is false that boa constrictors are animals that are easy to carry on a bicycle. D. Some motorcycles that are made in Europe are not vehicles that are inexpensive to repair. Therefore, it is not the case that all motorcycles that are made in Europe are vehicles that are inex­ pensive to repair. *E. It is false that some restaurants located in bus stations are places where one is likely to be poisoned. Therefore, some restaurants located in bus stations are not places where one is likely to be poisoned. F. It is not the case that some politicians are not anarchists. Therefore, no politicians are anarchists. *G. No pacifists are war supporters. Therefore, it is not true that some pacifists are war supporters. 13.4 Other Immediate Inferences We'll now turn to three more types of immediate inference that can be validly drawn from categorical propositions: conversion, obversion, and contraposition. In some cases, conversion and contraposition allow an inference from a universal to a particular proposition, but the validity of those inferences requires the assumption that the subject terms in the universal premises do not refer to empty classes such as mermaids and square circles. Conversion Conversion allows us to infer, from a categorical proposition called the 'convertend,' another proposition called its 'converse' by switching the former's subject and predicate terms while retaining its original quantity and quality. Thus from an E proposition such as 29 No SUV is a sports car, we can infer by conversion 29' No sports car is an SUV. Here the convertend's subject and predicate terms have been switched, but its quantity and quality remain the same: universal negative. The inference from (29) to (29') is valid: if (29) is true, then (29') must be true as well (and vice versa). Similarly, by conversion, an I proposition yields an I converse when the subject and predicates terms of the convertend are switched. For example, the converse of (30) is (30'): 30 Some Republicans are journalists. 30' Some journalists are Republicans. If (30) is true, then (30') must also be true and vice versa-so the inference is valid and the two propositions are logically equivalent. For A propositions, however, an inference by conversion in this straightforward way would not be valid. For, clearly, (31 1 ) does not follow from (31): 31 All pigs are mammals. 31 ' All mammals are pigs. Rather, an A proposition can be validly converted only 'by limitation-for (311 ') does follow from (31) 31" Some mammals are pigs. In such a case of conversion by limitation, the convertend's quantity has been limited in the converse: the valid converse of an A proposition is an I proposition where the subject and predicate terms have been switched and the universal quantifier 'all' replaced by the non­ universal quantifier 'some.' Finally, note that in the case of O propositions, there is no valid conversion at all. If we tried to convert the true proposition (32), we'd get the false proposition (32'). 32 Some precious stones are not emeralds. 32 Some emeralds are not precious stones. (f) w 0 z w a: w u.. z w 0 w 2 2 a: w I l­ o Ellllll (j) (j) Zw Qo f- z -w (/) cc Ow Q.. LL Oz cc - a.. w _j � <( - 00 -w cc � 0� C) - w f- 0 <( z 0 <( This proves the invalidity of the inference from (32) to (32'). For any O proposition, an immediate inference by 'conversion' commits the fallacy of illicit conversion, and the same fallacy is com­ mitted when an A proposition is inferred by 'conversion' from another A proposition. To sum up, here are the rules for conversion: BOX 8 ■ CONVERSION Convertend Converse Inference A All Sare P Some Pare S (Valid by limitation only) E No Sare P No Pare S VALID I Some Sare P Some Pare S VALID 0 Some S are not P (No valid conversion) Obversion A categorical proposition's obverse is inferred by changing the proposition's quality (i.e., from affirmative to negative, or negative to affirmative) and adding to its predicate the prefix 'non.' The proposition deduced by obversion is called the 'obverse,' and that from which it was deduced, the 'obvertend.' The inference is valid across the board. Thus from the A proposition (33) it follows by obversion (33'): 33 All eagles are birds. 33' No eagles are non-birds. From the E proposition (34), obversion yields (341): 34 No cell phones are elephants. 34' All cell phones are non-elephants. The obverse of I proposition (35) is (35'): 35 Some Californians are surfers. 35' Some Californians are not non-surfers. The obverse of O proposition (36) is (36'): 36 Some epidemics are not catastrophes. 36' Some epidemics are non-catastrophes. In each of these, the obvertend's predicate has been replaced in the obverse proposition by the predicate for its class complement, which is the class made up of everything outside of the class in question. For instance, for the class of senators, the class complement is the class of non-senators, which includes mayors, doctors, bricklayers, airplanes, butterflies, planets, postage stamps, inert gases, and so forth .. . in fact, everything that is not a senator. The class complement of the class of horses is non-horses, a similarly vast and diverse class of things. For the class of diseases, the class complement is non-diseases. And so on. The expression that denotes any such complement is a term complement. BOX 9 ■ EQUIVALENCES AND NON-EQUIVALENCES BY CONVERSION All Sare P. All Pare S. s p NoSareP. NoPareS. p Some Sare P. Some PareS. p Some S are Some P are �� �£ p p p Unlike conversion, obversion is a valid immediate inference for all four types of categorical proposition. For each of the four pairs of categorical propositions listed below, an immediate inference from obvertend to obverse would be valid: if the obvertend is true, the obverse would be true too. The following table summarizes how to draw such inferences correctly: BOX 10 ■ OBVERSION Obvertend Obverse Inference A All Sare P No Sare non-P VALID E No Sare P All Sare non-P VALID I Some Sare P Some Sare not non-P VALID 0 Some S are not P Some Sare non-P VALID w 0 z w a: w LL � w 0 w � � a: w I r- 0 (") � - (J) (J) Zw Qo f- z U)W 0 a: Q_ w 0 LL a: � Q_ w _J f- <( � 00 -w a: 2 0 2 0_ w f- 0 <( z 0 <( BOX 11 ■ EQUIVALENCES BY OBVERSION All Sare P. No Sare non-P. s p No Sare P. All Sare non-P. p Some Sare P. Some Sare not non-P. p Some S are Some Sare not P. non-P. Contraposition s p p F Contraposition allows us to infer a conclusion, the contrapositive, from another proposition by preserving the latter's quality and quantity while switching its subject and predicate terms, each preceded by the prefix 'non.' Thus the contrapositive of (37) is (37'): 37 All croissants are pastries. 37' All non-pastries are non-croissants. Given contraposition, an A proposition of the form 'All S are P' is logically equivalent to another A proposition of the form 'All non-P are non-S.' Recall that whenever two propositions are logically equivalent, they have exactly the same truth value: if (37) is true, (37') is also true, and if (37) is false, (37') must be false. And, as noted in Box 7 in the previous section, whenever two propositions are logically equivalent, we may infer the one from the other: any such inference would be valid. To visualize this relationship between (37) and (37'), you may want to have a look at the corresponding Venn diagrams in Box 12 (think of' S' in the diagram as standing for 'croissants' and 'P' for 'pastries'). BOX 12 ■ A'S EQUIVALENT AND E'S NON-EQUIVALENT CONTRAPOSITIVES All Sare P. s No Sare P. p p All non-Pare non-S. No non-P are non-S. p The contrapositive of an I proposition is another proposition of exactly the same quality and quantity (that is, another I proposition), where the subject and predicate terms have been switched and prefixed by 'non.' The contrapositive of (38) is (38'): 38 Some croissants are pastries. 38' Some non-pastries are non-croissants. But (38) and (38') are not logically equivalent, as can be seen in the corresponding Venn diagram in Box 12. Thus any inference drawn from one to the other by contraposition would be invalid, an instance of the fallacy of illicit contraposition. With E propositions there is also a danger of committing the fallacy of illicit contraposition. But the fallacy can be avoided by limiting the quantity of the original E proposition in its contra­ positive. That is, an E proposition's valid contrapositive is an O proposition in which subject and predicate have been switched and pre-fixed by 'non.' Thus consider 39 No leopards are reptiles. The correct contrapositive, one that limits the quantity of (39) while preserving its quality, is (39'), which is also true. 39' Some non-reptiles are not non-leopards. (40) is inferred from (39) by contraposition without limitation, which makes the inference invalid. Cf) w 0 z w cc w LL z w 0 w 2 2 cc w I 1-- 0 ­
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CHAPTER

Categorical
Syllogisms

Here you’ll read more about traditional logic. This chapter is entirely devoted to
syllogistic arguments. It first explains what categorical syllogisms are and then
examines two methods of checking them for validity. The topics include

Recognizing categorical syllogisms.

How to determine the form of a syllogism on the basis of its mood and figure.

Testing syllogism forms for validity using Venn diagrams.

Distribution of terms.

Testing syllogism forms for validity using traditional logic’s rules of validity.

Some patterns of failed syllogism.

330

14.1 What Is a Categorical Syllogism?

Beginning in antiquity with Aristotelian logic, and continuing for many centuries in other

schools of logic, a number of methods have been proposed for analyzing deductive arguments
of the sort we have broadly called ‘syllogistic.’ A syllogism is a deductive argument with two

premises. A categorical syllogism is a syllogism made up entirely of categorical propositions.

Thus there are several different kinds of syllogistic argument, some of which were considered

in Chapters s and 12. In this chapter, we’ll look closely at categorical syllogisms, which, for our

purposes here, we’ll refer to simply as ‘syllogisms.’ For example,

1 1. All rectangles are polygons.

2. All squares are rectangles.

3. All squares are polygons.

Argument (1) is a syllogism, since it has two premises and a conclusion, all of which are cate­

gorical propositions. A closer look at (1)’s premises and conclusion reveals that it has exactly

three terms in the position of subject or predicate: ‘rectangle,’ ‘polygon,’ and ‘square.’ Each of

these denotes a category (or class) of things, and these categories are related in such a way that

the argument’s conclusion follows validly from its premises. According to that conclusion, the

class of squares is wholly included in the class of polygons, which must be true provided that

(1)’s premises are true. This is a valid deductive argument: its conclusion is entailed by its

premises. But other syllogisms might be invalid. When a syllogism meets the deductive stan­

dard of validity, entailment hinges on relations among the terms of three different types that

occur as subject or predicate of the categorical propositions that make up the syllogism. Since

the validity of an argument depends on its having a valid form, several methods have been

proposed for determining when syllogisms have such forms. But before turning to these, more

needs to be said about the structure of standard syllogisms.

The Terms of a Syllogism

A standard syllogism consists of three categorical propositions, two of which function as

premises and one as a conclusion. Each of these has a subject term and a predicate term

denoting two classes of things, with the proposition as a whole representing a certain relation

of exclusion or inclusion among the classes denoted by its subject and predicate terms. Our

inspection of each of the categorical propositions making up (1) above showed that its

component propositions feature subject and predicate terms of three different types: namely,

‘polygon,’ ‘square,’ and ‘rectangle.’ In fact, this is something all standard syllogisms have in

common, since they all feature terms of three different types: the so-called major, minor, and

middle terms. The major term is the predicate of the conclusion. The minor term is the
subject of the conclusion, and the middle term is the term that occurs only in the premises.

Consider (1) again,

1 1. All rectangles are polygons.

2. All squares are rectangles.

3. All squares are polygons.

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<( a: 0 CJ LI.J I­ <( IED By looking at the predicate and subject of (1)'s conclusion, we can identify this syllogism's major and minor terms respectively: 'polygons' (the conclusion's predicate) and 'squares' (its subject). Notice that each of these terms occurs also in the premises, but that does not bear on their status as major and minor terms, which is determined solely by their functions as predicate and subject of the conclusion. But in (1), there is also the term 'rectangles,' which occurs in the subject and predicate positions in the premises. It is the 'middle term,' so called because its function is to mediate between the two premises-to connect them, so that they're both talking about the same thing. In any syllogism, the middle term occurs in both premises but not in the conclusion. Another thing to notice is this: that although each of the three terms of argument (1) is a single word, this is not so in all syllogisms-since sometimes phrases can function as subject and predicate of a categorical proposition. Let's now identify the major, minor, and middle terms in 2 1. No military officers are pacifists. 2.. All lieutenant colonels are military officers. 3. No lieutenant colonels are pacifists. By using the rule just suggested, we can determine that the major term here is 'pacifists,' the minor term 'lieutenant colonels,' and middle term 'military officers.' BOX 1 ■ A SYLLOGISM'S TERMS The important thing to keep in mind is that in order to identify the three words or phrases that are to count as the terms of a syllogism, we look first to the syllogism's conclusion. The major term is whatever word or phrase turns up in the predicate place (i.e., after the copula) in the conclusion. The minor term is whatever word or phrase turns up in the subject place (i.e., between the quantifier and the copula) in the conclusion. And the middle term is the term that does not occur in the conclusion at all but occurs in both premises-whether it be a single word, as in (1), or a more complex expression, as in (2.). The Premises of a Syllogism The conclusion of (1) above is the proposition I 3. All squares are polygons. In the notation of traditional logic, this is symbolized as 3'.AIISareP It is common practice to represent the minor and major terms of a syllogism as 'S' and 'P' respectively, and its middle term as 'M.' We'll adopt that practice and represent any syllogism by replacing its three terms by those symbols, keeping logical words such as quantifiers and negation. In the case of (1) above, we thus obtain 1 ' 1 All M are P 2 All Sare M 3 All Sare P In a standard syllogism, the minor and the major terms occur in different premises. That con­ taining the major term is the 'major premise.' Since (1)'s major term is 'polygons,' its major premise is 1 . All rectangles are polygons. In symbols this becomes 1'. All MareP The premise that contains the minor term is the minor premise. Since (1)'s minor term is 'squares,' its minor premise is 2. All squares are rectangles. In symbols this becomes 2'. All Sare M You may have noticed that, in both examples of syllogism considered thus far, each has been arranged with its major premise first, its minor premise second, and its conclusion last. This is standard order for a reconstructed syllogism. Although in ordinary speech and writing a syllogism's premises and conclusion might be jumbled in any order whatsoever, when we reconstruct it, its premises must be put into standard order (this will become especially important later). We can now determine which premise is which in (1) above: 1 ' 1. All M are P 2. All Sare M 3. All Sare P ¢:i MAJOR PREMISE ¢:i MINOR PREMISE Recognizing Syllogisms However jumbled they may be in their real-life occurrences, syllogisms can be recognized by first identifying their conclusions. Once we've identified the conclusion of a putative syllogism, we can check whether it is indeed a syllogism: the conclusion's predicate gives us the major term, its subject the minor term. Once we've identified these terms, we can then look at the argument's premises and ask: Which premise contains the major term? (That's the major premise.) Which contains the minor term? (That's the minor premise.) After listing these in the standard order, as premises 1 and 2 respectively, and replacing its relevant terms _J -< 0 a: 0 ('J LU 0 -< � ('• f- � < Cl) I- 3: g � _J -–
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