COLLEGE AND UNIVERSITY NOTES: Critical Thinking and Problem Solving

Critical Thinking and Problem Solving

Pokhara University

Seventh Semester

Answer the question based on the information given.
Question No. : 1
Many business offices are located in buildings having 2-8 floors. If a building has more than 3 floors, it has a lift. If the above statements are true, which of the following must be true? And explain.

A. 2nd floors do not have lifts

B. 7th floors have lifts

C. Only floors above the 3rd floors have lifts

D. All floors may be reached by lifts

What Is an Analogy?

An analogy is something that shows how two things are alike, but with the ultimate goal of making a point about this comparison.

The purpose of an analogy is not merely to show, but also to explain. For this reason, an analogy is more complex than a simile or a metaphor, which aim only to show without explaining. (Similes and metaphors can be used to make an analogy, but usually analogies have additional information to get their point across.)

Attempt all Questions

1. List four importance of critical thinking.
2. Write four advantages of qualitative approach of decision making.
3. State the characteristics of directive decision making style.
4. Define fuzzy logic.
5. Distinguish between linear creativity and lateral creativity.
6. List the techniques for improving group decision making.
7. What is sensitivity analysis?
8. List four abilities of a creative person.
9. Give the concept of utility function in one sentence.
10. Write four benefits of brainstorming.

11. What is fallacious reasoning? Explain any four common fallacious with example.
12. What is the importance of decision making process? Explain the different approaches of decision theories.
13. What do you mean by judgmental biases? Describe any four types of judgmental biases and their corrective procedures.
14. The research department of a company has recommended to the marketing department to launch a product of three different types. The marketing manager has to decide one of the types of the product to be launched under the following estimated payoffs for various level of sales:

Types of productEstimated level of sales (units)15000100005000A301010B40155C55203

What will be the marketing manager?s decision if a) Maximin b) Maximax c) Laplace and d) regret criteria are applied?

15. Describe briefly the relationship between problem solving and decision making. Explain the process of problem identification and formulation.
16. Chandan had to decide whether or not to drill a well on his farm. In his village, only 40% of the wells drilled were successful at 200 feet of depth. Some of the villagers who did not get water at 200 feet. Drilled further up to 250 feet byt only 20% struck water at 250 feet. Cost of drilling is Rs.50 per foot. He estimated that he would pay Rs. 18000 during a five year period in the present value terms, if he continues to buy water from the neighbor rather than go for the well which would have a life of five years. He has three decisions to make: a) should he drill upon 200 feet and b) if no water is found at 200 feet, should he drill up to 250 feet? c) Should he continue to buy water from his neighbor? Draw a decision tree and give best solution to him.

Group ?C? – Comprehensive Answer Questions: [2 X 10 = 20]

17. A retailer stocks bunches of fresh-cut flowers. He has an uncertain demand; the best estimates available follow:

Demand (bunches)20254060Probability0.100.300.500.10

The retailer buys these for $6 a bunch and sells then for $10.

a. Set up the payoff table.
b. If he stocks 40 every day. What will his expected profit per day be?
c. What quantity should he buy every day to maximize expected profits?
d. What is the expected value of perfect information for him?
e. Construct an opportunity loss table and fine EOL.

18. Consider a game in which each player selects one of three colored poker chips: red, white or blue. The players must select a chip without knowing the color of the chip selected by other player. The player the reveal their chips. Payoffs to player A in dollars are follows:

Player APlayer BRed b1White b2Blue b3Red a10-12White a254-3Blue a323-4

a. Determine the optimal strategies that should be adopted by players A and B.
b. What is the value of the game?

Introduction to Logic and Critical Thinking

Version 2.0

Matthew J. Van Cleave

vancleave@mac.com

Introduction to Logic and Critical Thinking by Matthew J. Van Cleave is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Table of contents

Preface

Chapter 1: Reconstructing and analyzing arguments

1.1 What is an argument?

1.2 Identifying arguments

1.3 Arguments vs. explanations

1.4 More complex argument structures

1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form

1.6 Validity

1.7 Soundness

1.8 Deductive vs. inductive arguments

1.9 Arguments with missing premises

1.10 Assuring, guarding, and discounting

1.11 Evaluative language

1.12 Evaluating a real-life argument

Chapter 2: Formal methods of evaluating arguments

2.1 What is a formal method of evaluation and why do we need them? 2.2 Propositional logic and the four basic truth functional connectives 2.3 Negation and disjunction

2.4 Using parentheses to translate complex sentences

2.5 “Not both” and “neither nor”

2.6 The truth table test of validity

2.7 Conditionals

2.8 “Unless”

2.9 Material equivalence

2.10 Tautologies, contradictions, and contingent statements 2.11 Proofs and the 8 valid forms of inference

2.12 How to construct proofs

2.13 Short review of propositional logic

2.14 Categorical logic

2.15 The Venn test of validity for immediate categorical inferences 2.16 Universal statements and existential commitment

2.17 Venn validity for categorical syllogisms

Chapter 3: Evaluating inductive arguments and probabilistic and statistical fallacies

3.1 Inductive arguments and statistical generalizations

3.2 Inference to the best explanation and the seven explanatory virtues

3.3 Analogical arguments

3.4 Causal arguments

3.5 Probability

3.6 The conjunction fallacy

3.7 The base rate fallacy

3.8 The small numbers fallacy

3.9 Regression to the mean fallacy

3.10 Gambler’s fallacy

Chapter 4: Informal fallacies

4.1 Formal vs. informal fallacies

4.1.1 Composition fallacy

4.1.2 Division fallacy

4.1.3 Begging the question fallacy

4.1.4 False dichotomy

4.1.5 Equivocation

4.2 Slippery slope fallacies

4.2.1 Conceptual slippery slope

4.2.2 Causal slippery slope

4.3 Fallacies of relevance

4.3.1 Ad hominem

4.3.2 Straw man

4.3.3 Tu quoque

4.3.4 Genetic

4.3.5 Appeal to consequences

4.3.6 Appeal to authority

Answers to exercises

Glossary/Index

Preface

This is an introductory textbook in logic and critical thinking. The goal of the textbook is to provide the reader with a set of tools and skills that will enable them to identify and evaluate arguments. The book is intended for an introductory course that covers both formal and informal logic. As such, it is not a formal logic textbook, but is closer to what one would find marketed as a “critical thinking textbook.” The formal logic in chapter 2 is intended to give an elementary introduction to formal logic. Specifically, chapter 2 introduces several different formal methods for determining whether an argument is valid or invalid (truth tables, proofs, Venn diagrams). I contrast these formal methods with the informal method of determining validity introduced in chapter 1. What I take to be the central theoretical lesson with respect to the formal logic is simply that of understanding the difference between formal and informal methods of evaluating an argument’s validity. I believe there are also practical benefits of learning the formal logic. First and foremost, once one has internalized some of the valid forms of argument, it is easy to impose these structures on arguments one encounters. The ability to do this can be of use in evaluating an argumentative passage, especially when the argument concerns a topic with which one is not very familiar (such as on the GRE or LSAT).

However, what I take to be of far more practical importance is the skill of being able to reconstruct and evaluate arguments. This skill is addressed in chapter 1, where the central ideas are that of using the principle of charity to put arguments into standard form and of using the informal test of validity to evaluate those arguments. Since the ability to reconstruct and evaluate arguments is a skill, one must practice in order to acquire it. The exercises in each section are intended to give students some practice, but in order to really master the skill, one must practice much, much more than simply completing the exercises in the text. It makes about as much sense to say that one could become a critical thinker by reading a critical thinking textbook as that one could become fluent in French by reading a French textbook. Logic and critical thinking, like learning a foreign language, takes practice because it is a skill.

While chapters 1 and 2 mainly concern deductive arguments, chapter 3 addresses inductive arguments, including probabilistic and statistical fallacies. In a world in which information is commonly couched within probabilistic and statistical frameworks, understanding these basic concepts, as well as some of the common mistakes is essential for understanding our world. I have tried to

Preface

write chapter 3 with an eye towards this understanding. As with all the chapters, I have tried to walk the fine line between being succinct without sacrificing depth.

Chapter 4 picks out what I take to be some of the most common fallacies, both formal and informal. In my experience, many critical thinking textbooks end up making the fallacies sound obvious; one is often left wondering how anyone could commit such a fallacy. In my discussion of the fallacies I have tried to correct this not only in the particular examples I use in the text and exercises, but also by discussing what makes a particular fallacy seductive.

I have used numerous different textbooks over the years that I have been teaching logic and critical thinking courses. Some of them were very good; others were not. Although this textbook is my attempt to improve on what I’ve encountered, I am indebted to a number of textbooks that have shaped how I teach logic and critical thinking. In particular, Sinnott-Armstrong and Fogelin’s Understanding Arguments: An Introduction to Informal Logic and Copi and Cohen’s Introduction to Logic have influenced how I present the material here (although this may not be obvious). My interest in better motivating the seductiveness of the fallacies is influenced by Daniel Kahneman’s work in psychology (for which he won the Nobel Prize in economics in 2002).

This textbook is an “open textbook” that is licensed under the Creative Commons Attribution 4.0 license (CC BY 4.0). Anyone can take this work and alter it for their own purposes as long as they give appropriate credit to me, noting whether or not you have altered the text. (If you would like to alter the text but have come across this textbook in PDF format, please do not hesitate to email me at vancleave@mac.com and I will send you the text in a file format that is easier to manipulate.) Many colleges and universities have undertaken initiatives to reduce the cost of textbooks. I see this as an issue of access to education and even an issue of justice. Some studies have shown that without access to the textbook, a student’s performance in the class will suffer. Many students lack access to a textbook simply because they do not buy it in the first place since there are more pressing things to pay for (rent, food, child care, etc.) and because the cost of some textbooks is prohibitive. Moreover, both professors and students are beholden to publishers who profit from selling textbooks (professors, because the content of the course is set by the author of the textbook, and perhaps market forces, rather than by the professor herself; students, because they must buy the newest editions of increasingly expensive

Preface

textbooks). If education is necessary for securing certain basic human rights (as philosophers like Martha Nussbaum have argued), then lack of access to education is itself an issue of justice. Providing high quality, low-cost textbooks is one, small part of making higher education more affordable and thus more equitable and just. This open textbook is a contribution towards that end.

Matthew J. Van Cleave

January 4, 2016

iii

Chapter 1: Reconstructing and analyzing arguments

1.1 What is an argument?

This is an introductory textbook in logic and critical thinking. Both logic and critical thinking centrally involve the analysis and assessment of arguments. “Argument” is a word that has multiple distinct meanings, so it is important to be clear from the start about the sense of the word that is relevant to the study of logic. In one sense of the word, an argument is a heated exchange of differing views as in the following:

Sally: Abortion is morally wrong and those who think otherwise are seeking to justify murder!

Bob: Abortion is not morally wrong and those who think so are right-wing bigots who are seeking to impose their narrow-minded views on all the rest of us!

Sally and Bob are having an argument in this exchange. That is, they are each expressing conflicting views in a heated manner. However, that is not the sense of “argument” with which logic is concerned. Logic concerns a different sense of the word “argument.” An argument, in this sense, is a reason for thinking that a statement, claim or idea is true. For example:

Sally: Abortion is morally wrong because it is wrong to take the life of an innocent human being, and a fetus is an innocent human being.

In this example Sally has given an argument against the moral permissibility of abortion. That is, she has given us a reason for thinking that abortion is morally wrong. The conclusion of the argument is the first four words, “abortion is morally wrong.” But whereas in the first example Sally was simply asserting that abortion is wrong (and then trying to put down those who support it), in this example she is offering a reason for why abortion is wrong.

We can (and should) be more precise about our definition of an argument. But before we can do that, we need to introduce some further terminology that we will use in our definition. As I’ve already noted, the conclusion of Sally’s argument is that abortion is morally wrong. But the reason for thinking the conclusion is true is what we call the premise. So we have two parts of an argument: the premise and the conclusion. Typically, a conclusion will be supported by two or more premises. Both premises and conclusions are

statements. A statement is a type of sentence that can be true or false and

Chapter 1: Reconstructing and analyzing arguments

corresponds to the grammatical category of a “declarative sentence.” For example, the sentence,

The Nile is a river in northeastern Africa

is a statement. Why? Because it makes sense to inquire whether it is true or false. (In this case, it happens to be true.) But a sentence is still a statement even if it is false. For example, the sentence,

The Yangtze is a river in Japan

is still a statement; it is just a false statement (the Yangtze River is in China). In contrast, none of the following sentences are statements:

Please help yourself to more casserole

Don’t tell your mother about the surprise

Do you like Vietnamese pho?

The reason that none of these sentences are statements is that it doesn’t make sense to ask whether those sentences are true or false (rather, they are requests or commands, and questions, respectively).

So, to reiterate: all arguments are composed of premises and conclusions, which are both types of statements. The premises of the argument provide a reason for thinking that the conclusion is true. And arguments typically involve more than one premise. A standard way of capturing the structure of an argument is by numbering the premises and conclusion. For example, recall Sally’s argument against abortion:

Abortion is morally wrong because it is wrong to take the life of an innocent human being, and a fetus is an innocent human being.

We could capture the structure of that argument like this:

1. It is morally wrong to take the life of an innocent human being 2. A fetus is an innocent human being

3. Therefore, abortion is morally wrong

Chapter 1: Reconstructing and analyzing arguments

By convention, the last numbered statement (also denoted by the “therefore”) is the conclusion and the earlier numbered statements are the premises. This is what we will call standard argument form. We can now give a more precise definition of an argument. An argument is a set of statements, some of which (the premises) attempt to provide a reason for thinking that some other statement (the conclusion) is true. Although arguments are typically given in order to convince or persuade someone of the conclusion, the argument itself is independent of one’s attempt to use it to convince or persuade. For example, I have just given you this argument not in an attempt to convince you that abortion is morally wrong, but as an illustration of what an argument is. Later on in this chapter and in this book we will learn some techniques of evaluating arguments, but for now the goal is to learn to identify an argument, including its premises and conclusion(s). It is important to be able to identify arguments and understand their structure, whether or not you agree with conclusion of the argument. In the next section I will provide some techniques for being able to identify arguments.

Exercise 1: Which of the following sentences are statements and which are not?

1. No one understands me but you.

2. Alligators are on average larger than crocodiles.

3. Is an alligator a reptile or a mammal?

4. An alligator is either a reptile or a mammal.

5. Don’t let any reptiles into the house.

6. You may kill any reptile you see in the house.

7. East Africans are not the best distance runners.

8. Obama is not a Democrat.

9. Some humans have wings.

10. Some things with wings cannot fly.

11. Was Obama born in Kenya or Hawaii?

12. Oh no! A grizzly bear!

13. Meet me in St. Louis.

14. We met in St. Louis yesterday.

15. I do not want to meet a grizzly bear in the wild.

Chapter 1: Reconstructing and analyzing arguments

1.2 Identifying arguments

The best way to identify whether an argument is present is to ask whether there is a statement that someone is trying to establish as true by basing it on some other statement. If so, then there is an argument present. If not, then there isn’t. Another thing that can help in identifying arguments is knowing certain key words or phrases that are premise indicators or conclusion indicators. For example, recall Sally’s abortion argument:

Abortion is morally wrong because it is wrong to take the life of an innocent human being, and a fetus is an innocent human being.

The word “because” here is a premise indicator. That is, “because” indicates that what follows is a reason for thinking that abortion is morally wrong. Here is another example:

I know that the student plagiarized since I found the exact same sentences on a website and the website was published more than a year before the student wrote the paper.

In this example, the word “since” is a premise indicator because what follows it is a statement that is clearly intended to be a reason for thinking that the student plagiarized (i.e., a premise). Notice that in these two cases, the premise indicators “because” and “since” are interchangeable: I could have used “because” in place of “since” or “since” in the place of “because” and the meaning of the sentences would have been the same. In addition to premise indicators, there are also conclusion indicators. Conclusion indicators mark that what follows is the conclusion of an argument. For example,

Bob-the-arsonist has been dead for a year, so Bob-the-arsonist didn’t set the fire at the East Lansing Starbucks last week.

In this example, the word “so” is a conclusion indicator because what follows it is a statement that someone is trying to establish as true (i.e., a conclusion). Here is another example of a conclusion indicator:

A poll administered by Gallup (a respected polling company) showed candidate x to be substantially behind candidate y with only a week left before the vote, therefore candidate y will probably not win the election.

Chapter 1: Reconstructing and analyzing arguments

In this example, the word “therefore” is a conclusion indicator because what follows it is a statement that someone is trying to establish as true (i.e., a conclusion). As before, in both of these cases the conclusion indicators “so” and “therefore” are interchangeable: I could have used “so” in place of “therefore” or “therefore” in the place of “so” and the meaning of the sentences would have been the same.

Table 1 contains a list of some common premise and conclusion indicators:

Premise indicators	Conclusion indicators
since	therefore
because	so
for	hence
as	thus
given that	implies that
seeing that	consequently
for the reason that	it follows that
is shown by the fact that	we may conclude that

Although these words and phrases can be used to identify the premises and conclusions of arguments, they are not failsafe methods of doing so. Just because a sentence contains them does not mean that you are dealing with an argument. This can easily be shown by examples like these:

I have been running competitively since 1999.

I am so happy to have finally finished that class.

Although “since” can function as a premise indicator and although “so” can function as a conclusion indicator, neither one is doing so here. This shows that you can’t simply mindlessly use occurrences of these words in sentences to show that there is an argument being made. Rather, we have to rely on our understanding of the English sentence in order to determine whether an argument is being made or not. Thus, the best way to determine whether an argument is present is by asking the question: Is there a statement that someone is trying to establish as true or explain why it is true by basing it on some other statement? If so, then there is an argument present. If not, then there isn’t. Notice that if we apply this method to the above examples, we will

Chapter 1: Reconstructing and analyzing arguments

see that there is no argument present because there is no statement that someone is trying to establish as true by basing it on some other statement. For example, the sentence “I have been running competitively since 1999” just contains one statement, not two. But arguments always require at least two separate statements—one premise and one conclusion, so it cannot possibly be an argument.

Another way of explaining why these occurrences of “so” and “since” do not indicate that an argument is present is by noting that both premise indicators and conclusion indicators are, grammatically, conjunctions. A grammatical conjunction is a word that connects two separate statements. So, if a word or term is truly being used as a premise or conclusion indicator, it must connect two separate statements. Thus, if “since” were really functioning as a premise indicator in the above example then what followed it would be a statement. But “1999” is not a statement at all. Likewise, in the second example “so” is not being used as a conclusion indicator because it is not conjoining two separate statements. Rather, it is being used to modify the extent of “happy.” In contrast, if I were to say “Tom was sleeping, so he couldn’t have answered the phone,” then “so” is being used as a conclusion indicator. In this case, there are clearly two separate statements (“Tom was sleeping” and “Tom couldn’t have answered the phone”) and one is being used as the basis for thinking that the other is true.

If there is any doubt about whether a word is truly a premise/conclusion indicator or not, you can use the substitution test. Simply substitute another word or phrase from the list of premise indicators or conclusion indicators and see if the resulting sentence still makes sense. If it does, then you are probably dealing with an argument. If it doesn’t, then you probably aren’t. For example, we can substitute “it follows that” for “so” in the Bob-the-arsonist example:

Bob-the-arsonist has been dead for a year, it follows that Bob-the-arsonist didn’t set the fire at the East Lansing Starbucks last week.

However, we cannot substitute “because” for “so” in the so-happy-I-finished that-class example:

I am because happy to have finally finished that class.

Chapter 1: Reconstructing and analyzing arguments

Obviously, in the latter case the substitution of one conclusion indicator for another makes the sentence meaningless, which means that the “so” that occurred originally wasn’t functioning as a conclusion indicator.

Exercise 2: Which of the following are arguments? If it is an argument, identify the conclusion of the argument.

1. The woman in the hat is not a witch since witches have long noses and she doesn’t have a long nose.

2. I have been wrangling cattle since before you were old enough to tie your own shoes.

3. Albert is angry with me so he probably won’t be willing to help me wash the dishes.

4. First I washed the dishes and then I dried them.

5. If the road wasn’t icy, the car wouldn’t have slid off the turn. 6. Albert isn’t a fireman and he isn’t a fisherman either.

7. Are you seeing that rhinoceros over there? It is huge!

8. The fact that obesity has become a problem in the U.S. is shown by the fact that obesity rates have risen significantly over the past four decades. 9. Bob showed me a graph with the rising obesity rates and I was very surprised to see how much they’ve risen.

10.Albert isn’t a fireman because Albert is a Greyhound, which is a kind of dog, and dogs can’t be firemen.

11.Charlie and Violet are dogs and since dogs don’t sweat, it is obvious that Charlie and Violet don’t sweat.

12.The reason I forgot to lock the door is that I was distracted by the clown riding a unicycle down our street while singing Lynyrd Skynyrd’s “Simple Man.”

13.What Bob told you is not the real reason that he missed his plane to Denver.

14.Samsung stole some of Apple’s patents for their smartphones, so Apple stole some of Samsung’s patents back in retaliation.

15.No one who has ever gotten frostbite while climbing K2 has survived to tell about it, therefore no one ever will.

Chapter 1: Reconstructing and analyzing arguments

1.3 Arguments vs. explanations

So far I have defined arguments in terms of premises and conclusions, where the premises are supposed to provide a reason (support, evidence) for accepting the conclusion. Many times the goal of giving an argument is simply to establish that the conclusion is true. For example, when I am trying to convince someone that obesity rates are rising in the U.S. I may cite evidence such as studies from the Center for Disease Control (CDC) and the National Institute of Health (NIH). The studies I cite would function as premises for the conclusion that obesity rates are rising. For example:

We know that obesity is on the rise in the U.S. because multiple studies carried out by the CDC and NIH have consistently shown a rise in obesity over the last four decades.

We could put this simple argument into standard form like this:

1. Multiple studies by the CDC and NIH have consistently shown a rise in obesity over the last four decades.

2. Therefore, obesity is on the rise in the U.S.

The standard form argument clearly distinguishes the premise from the conclusion and shows how the conclusion is supposed to be supported by the evidence offered in the premise. Again, the goal of this simple argument would be to convince someone that the conclusion is true. However, sometimes we already know that a statement or claim is true and we are trying to establish why it is true rather than that it is true. An argument that attempts to show why its conclusion is true is an explanation. Contrast the previous example with the following:

The reason that the rate of obesity is on the rise in the U.S. is that the foods we most often consume over the past four decades have increasingly contained high levels of sugar and low levels of dietary fiber. Since eating foods high in sugar and low in fiber triggers the insulin system to start storing those calories as fat, it follows that people who consume foods high in sugar and low in fiber will tend to store more of the calories consumed as fat.

Chapter 1: Reconstructing and analyzing arguments

This passage gives an explanation for why obesity is on the rise in the U.S. Unlike the earlier example, here it is taken for granted that obesity is on the rise in the U.S. That is the claim whose truth we are trying to explain. We can put the obesity explanation into standard form just like any other argument. In order to do this, I will make some paraphrases of the premises and conclusion of the argument (for more on how to do this, see section 1.5 below).

1. Over the past four decades, Americans have increasingly consumed foods high in sugar and low in fiber.

2. Consuming foods high in sugar and low in fat triggers the insulin system to start storing those calories as fat.

3. When people store more calories as fat, they tend to become obese. 4. Therefore, the rate of obesity is on the rise in the U.S.

Notice that in this explanation the premises (1-3) attempt to give a reason for why the conclusion is true, rather than a reason for thinking that the conclusion is true. That is, in an explanation we assume that what we are trying to explain (i.e., the conclusion) is true. In this case, the premises are supposed to show why we should expect or predict that the conclusion is true. Explanations often give us an understanding of why the conclusion is true. We can think of explanations as a type of argument, we just have to distinguish two different types of argument: those that attempt to establish that their conclusion is true

(arguments), and those that attempt to establish why their conclusion is true (explanations).

Exercise 3: Which of the following is an explanation and which is an argument? Identify the main conclusion of each argument or explanation. (Remember if the premise(s) seems to be establishing that the conclusion is true, it is an argument, but if the premise(s) seems to be establishing why the conclusion is true, it is an explanation.)

1. Wanda rode the bus today because her car was in the shop. 2. Since Wanda doesn’t have enough money in her bank account, she has not yet picked up her car from the shop.

3. Either Bob or Henry rode the bus to work today. But it wasn’t Henry because I saw him riding his bike to work. Therefore, it was Bob. 4. It can’t be snowing right now since it only snows when it is 32 degrees or below and right now it is 40 degrees.

Chapter 1: Reconstructing and analyzing arguments

5. The reason some people with schizophrenia hear voices in their head is that the cognitive mechanism that monitors their own self-talk is malfunctioning and they attribute their own self-talk to some external source.

6. Fracking should be allowed because, although it does involve some environmental risk, it reduces our dependence on foreign oil and there is much greater harm to the environment due to foreign oil drilling than there is due to fracking.

7. Wanda could not have ridden the bus today because today is a city wide holiday and the bus service is not operating.

8. The Tigers lost their star pitcher due to injury over the weekend, therefore the Tigers will not win their game against the Pirates. 9. No one living in Pompeii could have escaped before the lava from Mt. Vesuvius hit. The reason is simple: the lava was flowing too fast and there was nowhere to go to escape it in time.

10.The reason people’s allergies worsen when they move to Cincinnati is that the pollen count in Cincinnati is higher than almost anywhere else in the surrounding area.

1.4 More complex argument structures

So far we have seen that an argument consists of a premise (typically more than one) and a conclusion. However, very often arguments and explanations have a more complex structure than just a few premises that directly support the conclusion. For example, consider the following argument:

No one living in Pompeii could have survived the eruption of Mt. Vesuvius. The reason is simple: the lava was flowing too fast and there was nowhere to go to escape it in time. Therefore, this account of the eruption, which claims to have been written by an eyewitness living in Pompeii, was not actually written by an eyewitness.

The main conclusion of this argument—the statement that depends on other statements as evidence but doesn’t itself provide any evidence for any other statement—is:

A. This account of the eruption of Mt. Vesuvius was not actually written by an eyewitness.

Chapter 1: Reconstructing and analyzing arguments

However, the argument’s structure is more complex than simply having a couple of premises that provide evidence directly for the conclusion. Rather, some statement provides evidence directly for the main conclusion, but that statement itself is supported by another statement. To determine the structure of an argument, we must determine which statements support which. We can use our premise and conclusion indicators to help with this. For example, the passage contains the phrase, “the reason is…” which is a premise indicator, and it also contains the conclusion indicator, “therefore.” That conclusion indicator helps us to identify the main conclusion, but the more important thing to see is that statement A does not itself provide evidence or support for any of the other statements in the argument, which is the clearest reason why statement A is the main conclusion of the argument. The next question we must answer is: which statement most directly supports A? What most directly supports A is:

B. No one living in Pompeii could have survived the eruption of Mt. Vesuvius.

However, there is also a reason offered in support of B. That reason is that:

C. The lava from Mt. Vesuvius was flowing too fast and there was nowhere for someone living in Pompeii to go in order to escape it in time.

So the main conclusion (A) is directly supported by B, and B is supported by C. Since B acts as a premise for the main conclusion but is also itself the conclusion of further premises, we refer to B as an intermediate conclusion. The important thing to recognize here is that one and the same statement can act as both a premise and a conclusion. Statement B is a premise that supports the main conclusion (A), but it is also itself a conclusion that follows from C. Here is how we would put this complex argument into standard form (using numbers this time, as we always do when putting an argument into standard form):

1. The lava from Mt. Vesuvius was flowing too fast and there was nowhere for someone living in Pompeii to go in order to escape it in time.

2. Therefore, no one living in Pompeii could have survived the eruption of Mt. Vesuvius. (from 1)

Chapter 1: Reconstructing and analyzing arguments

3. Therefore, this account of the eruption of Mt. Vesuvius was not actually written by an eyewitness. (from 2)

Notice that at the end of statement 2 I have written in parentheses “from 1” (and likewise at the end of statement 3 I have written “from 2”). This is a shorthand way of saying: “this statement follows from statement 1.” We will use this convention as a way of keeping track of the structure of the argument. It may also help to think about the structure of an argument spatially, as figure 1

shows:

The main argument here (from 2 to 3) contains a subargument, in this case the argument from 1 to 2. In general, the main argument is simply the argument whose premises directly support the main conclusion, whereas a subargument is an argument that provides indirect support for the main conclusion by supporting one of the premises of the main argument. You can always add further subarguments to the overall structure of an argument by providing evidence that supports one of the unsupported premises.

Another type of structure that arguments can have is when two or more premises provide direct but independent support for the conclusion. Here is an example of an argument with that structure:

I know that Wanda rode her bike to work today because when she arrived at work she had her right pant leg rolled up (which cyclists do in order to

Chapter 1: Reconstructing and analyzing arguments

keep their pants legs from getting caught in the chain). Moreover, our coworker, Bob, who works in accounting, saw her riding towards work at 7:45 am.

The conclusion of this argument is “Wanda rode her bike to work today” and there are two premises that provide independent support for it: the fact that Wanda had her pant leg cuffed and the fact that Bob saw her riding her bike. Here is the argument in standard form:

1. Wanda arrived at work with her right pant leg rolled up.

2. Cyclists often roll up their right pant leg.

3. Bob saw Wanda riding her bike towards work at 7:45.

4. Therefore, Wanda rode her bike to work today. (from 1-2, 3 independently)

Again, notice that next to statement 4 of the argument I have written the premises from which that conclusion follows. In this case, in order to avoid any ambiguity, I have noted that the support for the conclusion comes independently from statements 1 and 2, on the one hand, and from statement 3, on the other hand. It is important to point out that an argument or subargument can be supported by one or more premises. We see this in the present argument since the conclusion (4) is supported jointly by 1 and 2, and singly by 3. As before, we can represent the structure of this argument spatially, as figure 2 shows:

There are endless different argument structures that can be generated from these few simple patterns. At this point, it is important to understand that arguments can have these different structures and that some arguments will be longer and more complex than others. Determining the structure of very

Chapter 1: Reconstructing and analyzing arguments

complex arguments is a skill that takes some time to master. Even so, it may help to remember that any argument structure ultimately traces back to some combination of these.

Exercise 4: Write the following arguments in standard form and show how the argument is structured using a diagram like the ones I have used in this section.

1. There is nothing wrong with prostitution because there is nothing wrong with consensual sexual and economic interactions between adults. Moreover, since there’s no difference between a man who goes on a blind date with a woman, buys her dinner and then has sex with her and a man who simply pays a woman for sex, that is another reason for why there is nothing wrong with prostitution.

2. Prostitution is wrong because it involves women who have typically been sexually abused as children. We know that most of these women have been abused from multiple surveys done with women who have worked in prostitution and that show a high percentage of self-reported sexual abuse as children.

3. There was someone in this cabin recently because there was warm water in the tea kettle and because there was wood still smoldering in the fireplace. But the person couldn’t have been Tim because Tim has been with me the whole time. Therefore, there must be someone else in these woods.

4. It is possible to be blind and yet run in the Olympic Games since Marla Runyan did it at the 2000 Sydney Olympics.

5. The train was late because it had to take a longer, alternate route since the bridge was out.

6. Israel is not safe if Iran gets nuclear missiles since Iran has threatened multiple times to destroy Israel and if Iran had nuclear missiles it would be able to carry out this threat. Moreover, since Iran has been developing enriched uranium, they have the key component needed for nuclear weapons—every other part of the process of building a nuclear weapon is simple compared to that. Therefore, Israel is not safe.

7. Since all professional hockey players are missing front teeth and Martin is a professional hockey player, it follows that Martin is missing front teeth. And since almost all professional athletes who are missing their front teeth have false teeth, it follows that Martin probably has false teeth.

Chapter 1: Reconstructing and analyzing arguments

8. Anyone who eats the crab rangoon at China Food restaurant will probably have stomach troubles afterward. It has happened to me every time, which is why it will probably happen to other people as well. Since Bob ate the crab rangoon at China Food restaurant, he will probably have stomach troubles afterward.

9. Albert and Caroline like to go for runs in the afternoon in Hyde Park. Since Albert never runs alone, we know that any time Albert is running, Caroline is running too. But since Albert looks like he has just run (since he is panting hard), it follows that Caroline must have ran

too.

10.Just because Jeremy’s prints were on the gun that killed Tim and the gun was registered to Jeremy, it doesn’t follow that Jeremy killed Tim since Jeremy’s prints would certainly be on his own gun and someone else could have stolen Jeremy’s gun and used it to kill Tim.

1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form

Although sometimes we can just lift the premises and conclusion verbatim from the argument, we cannot always do this. Paraphrases of premises or conclusions are sometimes needed in order to make the standard form argument as clear as possible. A paraphrase is the use of different words to capture the same idea in a clearer way. There will always be multiple ways of paraphrasing premises and conclusions and this means that there will never be just one way of putting an argument into standard form. In order to paraphrase well, you will have to rely on your understanding of English to come up with what you think is the best way of capturing the essence of the argument. Again, typically there is no single right way to do this, although there are certainly better and worse ways of doing it. For example, consider the following argument:

Just because Jeremy’s prints were on the gun that killed Tim and the gun was registered to Jeremy, it doesn’t follow that Jeremy killed Tim since Jeremy’s prints would certainly be on his own gun and someone else could have stolen Jeremy’s gun and used it to kill Tim.

What is the conclusion of this argument? (Think about it before reading on.) Here is one way of paraphrasing the conclusion:

Chapter 1: Reconstructing and analyzing arguments

The fact that Jeremy’s prints were on the gun that killed Tim and the gun was registered to Jeremy doesn’t mean that Jeremy killed Tim.

This statement seems to capture the essence of the main conclusion in the above argument. The premises of the argument would be:

1. Jeremy’s prints would be expected to be on a gun that was registered to him

2. Someone could have stolen Jeremy’s gun and then used it to kill Tim

Notice that while I have paraphrased the first premise, I have left the second premise almost exactly as it appeared in the original paragraph. As I’ve said, paraphrases are needed in order to try to make the standard form argument as clear as possible and this is what I’ve tried to do in capturing premise 1 as well as the conclusion of this argument. So here is the reconstructed argument in standard form:

1. Jeremy’s prints would be expected to be on a gun that was registered to him

2. Someone could have stolen Jeremy’s gun and then used it to kill Tim 3. Therefore, the fact that Jeremy’s prints were on the gun that killed Tim and the gun was registered to Jeremy doesn’t mean that Jeremy killed Tim. (from 1-2)

However, as I have just noted, there is more than one way of paraphrasing the premises and conclusion of the argument. To illustrate this, I will give a second way that one could accurately capture this argument in standard form. Here is another way of expressing the conclusion:

We do not know that Jeremy killed Tim.

That is clearly what the above argument is trying to ultimately establish and it is a much simpler (in some ways) conclusion than my first way of paraphrasing the conclusion. However, it also takes more liberties in interpreting the argument than my original paraphrase. For example, in the original argument there is no occurrence of the word “know.” That is something that I am introducing in my own paraphrase. That is a totally legitimate thing to do, as long as introducing new terminology helps us to clearly express the essence of the premise or

Chapter 1: Reconstructing and analyzing arguments

conclusion that we’re trying to paraphrase.¹ Since my second paraphrase of the conclusion differs from my first paraphrase, you can expect that my premises will differ also. So how shall I paraphrase the premises that support this conclusion? Here is another way of paraphrasing the premises and putting the argument into standard form:

1. Tim was killed by a gun that was registered to Jeremy and had Jeremy’s prints on it.

2. It is possible that Jeremy’s gun was stolen from him.

3. If Jeremy’s gun was stolen from him, then Jeremy could not have killed Tim.

4. Therefore, we do not know that Jeremy killed Tim. (from 1-3)

Notice that this standard form argument has more premises than my first reconstruction of the standard form argument (which consisted of only three statements). I have taken quite a few liberties in interpreting and paraphrasing this argument, but what I have tried to do is to get down to the most essential logic of the original argument. The paraphrases of the premises I have used are quite different from the wording that occurs in the original paragraph. I have introduced phrases such as “it is possible that” as well as conditional statements (if…then statements), such as premise 3. Nonetheless, this reconstruction seems to get at the essence of the logic of the original argument. As long as your paraphrases help you to do that, they are good paraphrases. Being able to reconstruct arguments like this takes many years of practice in order to do it well, and much of the material that we will learn later in the text will help you to better understand how to capture an argument in standard form, but for now it is important to recognize that there is never only one way of correctly capturing the standard form of an argument. And the reason for this is that there are multiple, equally good, ways of paraphrasing the premises and conclusion of an argument.

¹How do we know that a paraphrase is accurate? Unfortunately, there is no simple way to answer this question. The only answer is that you must rely on your mastery and understanding of English in order to determine for yourself whether the paraphrase is a good one or not. This is one of those kinds of skills that is difficult to teach, apart from just improving one’s mastery of the English language.

Chapter 1: Reconstructing and analyzing arguments

1.6. Validity

So far we have discussed what arguments are and how to determine their structure, including how to reconstruct arguments in standard form. But we have not yet discussed what makes an argument good or bad. The central concept that you will learn in logic is the concept of validity. Validity relates to

how well the premises support the conclusion, and it is the golden standard that every argument should aim for. A valid argument is an argument whose conclusion cannot possibly be false, assuming that the premises are true. Another way of putting this is as a conditional statement: A valid argument is an argument in which if the premises are true, the conclusion must be true. Here is an example of a valid argument:

1. Violet is a dog

2. Therefore, Violet is a mammal (from 1)

You might wonder whether it is true that Violet is a dog (maybe she’s a lizard or a buffalo—we have no way of knowing from the information given). But, for the purposes of validity, it doesn’t matter whether premise 1 is actually true or false. All that matters for validity is whether the conclusion follows from the premise. And we can see that the conclusion, Violet is a mammal, does seem to follow from the premise, Violet is a dog. That is, given the truth of the premise, the conclusion has to be true. This argument is clearly valid since if we assume that “Violet is a dog” is true, then, since all dogs are mammals, it follows that “Violet is a mammal” must also be true. As we’ve just seen, whether or not an argument is valid has nothing to do with whether the premises of the argument are actually true or not. We can illustrate this with another example, where the premises are clearly false:

1. Everyone born in France can speak French

2. Barack Obama was born in France

3. Therefore, Barack Obama can speak French (from 1-2)

This is a valid argument. Why? Because when we assume the truth of the premises (everyone born in France can speak French, Barack Obama was born in France) the conclusion (Barack Obama can speak French) must be true. Notice that this is so even though none of these statements is actually true. Not everyone born in France can speak French (think about people who were born there but then moved somewhere else where they didn’t speak French and

Chapter 1: Reconstructing and analyzing arguments

never learned it) and Obama was not born in France, but it is also false that Obama can speak French. So we have a valid argument even though neither the premises nor the conclusion is actually true. That may sound strange, but if you understand the concept of validity, it is not strange at all. Remember: validity describes the relationship between the premises and conclusion, and it means that the premises imply the conclusion, whether or not that conclusion is true. In order to better understand the concept of validity, let’s look at an example of an invalid argument:

1. George was President of the United States

2. Therefore, George was elected President of the United States (from 1)

This argument is invalid because it is possible for the premise to be true and yet the conclusion false. Here is a counterexample to the argument. Gerald Ford was President of the United States but he was never elected president, since Ford Replaced Richard Nixon when Nixon resigned in the wake of the Watergate scandal.² So it doesn’t follow that just because someone is President of the United States that they were elected President of the United States. In other words, it is possible for the premise of the argument to be true and yet the conclusion false. And this means that the argument is invalid. If an argument is invalid it will always be possible to construct a counterexample to show that it is invalid (as I have done with the Gerald Ford scenario). A counterexample is simply a description of a scenario in which the premises of the argument are all true while the conclusion of the argument is false. If you can construct a counterexample to an argument, the argument is invalid.

In order to determine whether an argument is valid or invalid we can use what I’ll call the informal test of validity. To apply the informal test of validity ask yourself whether you can imagine a world in which all the premises are true and yet the conclusion is false. If you can imagine such a world, then the argument is invalid. If you cannot imagine such a world, then the argument is valid. Notice: it is possible to imagine a world where the premises are true even if the premises aren’t, as a matter of actual fact, true. This is why it doesn’t matter for validity whether the premises (or conclusion) of the argument are actually true. It will help to better understand the concept of validity by applying the informal test of validity to some sample arguments.

²As it happens, Ford wasn’t elected Vice President either since he was confirmed by the Senate, under the twenty fifth amendment, after Spiro Agnew resigned. So Ford wasn’t ever elected by the Electoral College—as either Vice President or President.

Chapter 1: Reconstructing and analyzing arguments

1. Joan jumped out of an airplane without a parachute

2. Therefore, Joan fell to her death (from 1)

To apply the informal test of validity we have to ask whether it is possible to imagine a scenario in which the premise is true and yet the conclusion is false (if so, the argument is invalid). So, can we imagine a world in which someone jumped out of an airplane without a parachute and yet did not fall to her death? (Think about it carefully before reading on.) As we will see, applying the informal test of validity takes some creativity, but it seems clearly possible that Joan could jump out of an airplane without a parachute and not die—she could be perfectly fine, in fact. All we have to imagine is that the airplane was not operating and in fact was on the ground when Joan jumped out of it. If that were the case, it would be a) true that Joan jumped out of an airplane without a parachute and yet b) false that Joan fell to her death. Thus, since it is possible to imagine a scenario in which the premise is true and yet the conclusion is false, the argument is invalid. Let’s slightly change the argument, this time making it clear that the plane is flying:

1. Joan jumped out of an airplane traveling 300 mph at a height of 10,000 ft without a parachute

2. Joan fell to her death (from 1)

Is this argument valid? You might think so since you might think that anyone who did such a thing would surely die. But is it possible to not die in the scenario described by the premise? If you think about it, you’ll realize that there are lots of ways someone could survive. For example, maybe someone else who was wearing a parachute jumped out of the plane after them, caught them and attached the parachute-less person to them, and then pulled the ripcord and they both landed on the ground safe and sound. Or maybe Joan was performing a stunt and landed in a giant net that had been set up for that purpose. Or maybe she was just one of those people who, although they did fall to the ground, happened to survive (it has happened before). All of these scenarios are consistent with the information in the first premise being true and also consistent with the conclusion being false. Thus, again, any of these counterexamples show that this argument is invalid. Notice that it is also possible that the scenario described in the premises ends with Joan falling to her death. But that doesn’t matter because all we want to know is whether it is possible that she doesn’t. And if it is possible, what we have shown is that the

Chapter 1: Reconstructing and analyzing arguments

conclusion does not logically follow from the premise alone. That is, the conclusion doesn’t have to be true, even if we grant that the premise is. And that means that the argument is not valid (i.e., it is invalid).

Let’s switch examples and consider a different argument.

1. A person can be President of the United States only if they were born in the United States.

2. Obama is President of the United States.

3. Kenya is not in the United States.

4. Therefore, Obama was not born in Kenya (from 1-3)

In order to apply the informal test of validity, we have to ask whether we can imagine a scenario in which the premises are both true and yet the conclusion is false. So, we have to imagine a scenario in which premises 1, 2, and 3 are true and yet the conclusion (“Obama was not born in Kenya”) is false. Can you imagine such a scenario? You cannot. The reason is that if you are imagining that it is a) true that a person can be President of the United States only if they were born in the United States, b) true that Obama is president and c) true that Kenya is not in the U.S., then it must be true that Obama was not born in Kenya. Thus we know that on the assumption of the truth of the premises, the conclusion must be true. And that means the argument is valid. In this example, however, premises 1, 2, and 3 are not only assumed to be true but are actually true. However, as we have already seen, the validity of an argument does not depend on its premises actually being true. Here is another example of a valid argument to illustrate that point.

1. A person can be President of the United States only if they were born in Kenya

2. Obama is President of the United States

3. Therefore, Obama was born in Kenya (from 1-2)

Clearly, the first premise of this argument is false. But if we were to imagine a scenario in which it is true and in which premise 2 is also true, then the conclusion (“Obama was born in Kenya”) must be true. And this means that the argument is valid. We cannot imagine a scenario in which the premises of the argument are true and yet the conclusion is false. The important point to recognize here—a point I’ve been trying to reiterate throughout this section—is that the validity of the argument does not depend on whether or not the

Chapter 1: Reconstructing and analyzing arguments

premises (or conclusion) are actually true. Rather, validity depends only on the logical relationship between the premises and the conclusion. The actual truth of the premises is, of course, important to the quality of the argument, since if the premises of the argument are false, then the argument doesn’t provide any reason for accepting the conclusion. In the next section we will address this topic.

Exercise 5: Determine whether or not the following arguments are valid by using the informal test of validity. If the argument is invalid, provide a counterexample.

1. Katie is a human being. Therefore, Katie is smarter than a chimpanzee.

2. Bob is a fireman. Therefore, Bob has put out fires.

3. Gerald is a mathematics professor. Therefore, Gerald knows how to teach mathematics.

4. Monica is a French teacher. Therefore, Monica knows how to teach French.

5. Bob is taller than Susan. Susan is taller than Frankie. Therefore, Bob is taller than Frankie.

6. Craig loves Linda. Linda loves Monique. Therefore, Craig loves Monique.

7. Orel Hershizer is a Christian. Therefore, Orel Hershizer communicates with God.

8. All Muslims pray to Allah. Muhammad is a Muslim. Therefore, Muhammad prays to Allah.

9. Some protozoa are predators. No protozoa are animals. Therefore, some predators are not animals.

10.Charlie only barks when he hears a burglar outside. Charlie is barking. Therefore, there must be a burglar outside.

1.7 Soundness

A good argument is not only valid, but also sound. Soundness is defined in terms of validity, so since we have already defined validity, we can now rely on it to define soundness. A sound argument is a valid argument that has all true premises. That means that the conclusion of a sound argument will always be true. Why? Because if an argument is valid, the premises transmit truth to the

Chapter 1: Reconstructing and analyzing arguments

conclusion on the assumption of the truth of the premises. But if the premises are actually true, as they are in a sound argument, then since all sound arguments are valid, we know that the conclusion of a sound argument is true. Compare the last two Obama examples from the previous section. While the first argument was sound, the second argument was not sound, although it was valid. The relationship between soundness and validity is easy to specify: all sound arguments are valid arguments, but not all valid arguments are sound arguments.

Although soundness is what any argument should aim for, we will not be talking much about soundness in this book. The reason for this is that the only difference between a valid argument and a sound argument is that a sound argument has all true premises. But how do we determine whether the premises of an argument are actually true? Well, there are lots of ways to do that, including using Google to look up an answer, studying the relevant subjects in school, consulting experts on the relevant topics, and so on. But none of these activities have anything to do with logic, per se. The relevant disciplines to consult if you want to know whether a particular statement is true is almost never logic! For example, logic has nothing to say regarding whether or not protozoa are animals or whether there are predators that aren’t in the animal kingdom. In order to learn whether those statements are true, we’d have to consult biology, not logic. Since this is a logic textbook, however, it is best to leave the question of what is empirically true or false to the relevant disciplines that study those topics. And that is why the issue of soundness, while crucial for any good argument, is outside the purview of logic.

1.8 Deductive vs. Inductive arguments

The concepts of validity and soundness that we have introduced apply only to the class of what are called “deductive arguments”. A deductive argument is an argument whose conclusion is supposed to follow from its premises with absolute certainty, thus leaving no possibility that the conclusion doesn’t follow from the premises. For a deductive argument to fail to do this is for it to fail as a deductive argument. In contrast, an inductive argument is an argument whose conclusion is supposed to follow from its premises with a high level of probability, which means that although it is possible that the conclusion doesn’t follow from its premises, it is unlikely that this is the case. Here is an example of an inductive argument:

Chapter 1: Reconstructing and analyzing arguments

Tweets is a healthy, normally functioning bird and since most healthy, normally functioning birds fly, Tweets probably flies.

Notice that the conclusion, Tweets probably flies, contains the word “probably.” This is a clear indicator that the argument is supposed to be inductive, not deductive. Here is the argument in standard form:

1. Tweets is a healthy, normally functioning bird

2. Most healthy, normally functioning birds fly

3. Therefore, Tweets probably flies

Given the information provided by the premises, the conclusion does seem to be well supported. That is, the premises do give us a strong reason for accepting the conclusion. This is true even though we can imagine a scenario in which the premises are true and yet the conclusion is false. For example,

suppose that we added the following premise:

Tweets is 6 ft tall and can run 30 mph.

Were we to add that premise, the conclusion would no longer be supported by the premises, since any bird that is 6 ft tall and can run 30 mph, is not a kind of bird that can fly. That information leads us to believe that Tweets is an ostrich or emu, which are not kinds of birds that can fly. As this example shows, inductive arguments are defeasible arguments since by adding further information or premises to the argument, we can overturn (defeat) the verdict that the conclusion is well-supported by the premises. Inductive arguments whose premises give us a strong, even if defeasible, reason for accepting the conclusion are called, unsurprisingly, strong inductive arguments. In contrast, an inductive argument that does not provide a strong reason for accepting the conclusion are called weak inductive arguments.

Whereas strong inductive arguments are defeasible, valid deductive arguments aren’t. Suppose that instead of saying that most birds fly, premise 2 said that all birds fly.

1. Tweets is a healthy, normally function bird.

2. All healthy, normally functioning birds can fly.

3. Therefore, Tweets can fly.

Chapter 1: Reconstructing and analyzing arguments

This is a valid argument and since it is a valid argument, there are no further premises that we could add that could overturn the argument’s validity. (True, premise 2 is false, but as we’ve seen that is irrelevant to determining whether an argument is valid.) Even if we were to add the premise that Tweets is 6 ft tall and can run 30 mph, it doesn’t overturn the validity of the argument. As soon as we use the universal generalization, “all healthy, normally function birds can fly,” then when we assume that premise is true and add that Tweets is a healthy, normally functioning bird, it has to follow from those premises that Tweets can fly. This is true even if we add that Tweets is 6 ft tall because then what we have to imagine (in applying our informal test of validity) is a world in which all birds, including those that are 6 ft tall and can run 30 mph, can fly.

Although inductive arguments are an important class of argument that are commonly used every day in many contexts, logic texts tend not to spend as much time with them since we have no agreed upon standard of evaluating them. In contrast, there is an agreed upon standard of evaluation of deductive arguments. We have already seen what that is; it is the concept of validity. In chapter 2 we will learn some precise, formal methods of evaluating deductive arguments. There are no such agreed upon formal methods of evaluation for inductive arguments. This is an area of ongoing research in philosophy. In chapter 3 we will revisit inductive arguments and consider some ways to evaluate inductive arguments.

1.9 Arguments with missing premises

Quite often, an argument will not explicitly state a premise that we can see is needed in order for the argument to be valid. In such a case, we can supply the premise(s) needed in order so make the argument valid. Making missing premises explicit is a central part of reconstructing arguments in standard form. We have already dealt in part with this in the section on paraphrasing, but now that we have introduced the concept of validity, we have a useful tool for knowing when to supply missing premises in our reconstruction of an argument. In some cases, the missing premise will be fairly obvious, as in the following:

Gary is a convicted sex-offender, so Gary is not allowed to work with children.

Chapter 1: Reconstructing and analyzing arguments

The premise and conclusion of this argument are straightforward:

1. Gary is a convicted sex-offender

2. Therefore, Gary is not allowed to work with children (from 1)

However, as stated, the argument is invalid. (Before reading on, see if you can provide a counterexample for this argument. That is, come up with an imaginary scenario in which the premise is true and yet the conclusion is false.) Here is just one counterexample (there could be many): Gary is a convicted sex-offender but the country in which he lives does not restrict convicted sex-offenders from working with children. I don’t know whether there are any such countries, although I suspect there are (and it doesn’t matter for the purpose of validity whether there are or aren’t). In any case, it seems clear that this argument is relying upon a premise that isn’t explicitly stated. We can and should state that premise explicitly in our reconstruction of the standard form argument. But what is the argument’s missing premise? The obvious one is that no sex offenders are allowed to work with children, but we could also use a weaker statement like this one:

Where Gary lives, no convicted sex-offenders are allowed to work with children.

It should be obvious why this is a “weaker” statement. It is weaker because it is not so universal in scope, which means that it is easier for the statement to be made true. By relativizing the statement that sex-offenders are not allowed to work with children to the place where Gary lives, we leave open the possibility that other places in the world don’t have this same restriction. So even if there are other places in the world where convicted sex-offenders are allowed to work with children, our statements could still be true since in this place (the place where Gary lives) they aren’t. (For more on strong and weak statements, see section 1.10). So here is the argument in standard form:

1. Gary is a convicted sex-offender.

2. Where Gary lives, no convicted sex-offenders are allowed to work with children.

3. Therefore, Gary is not allowed to work with children. (from 1-2)

This argument is now valid: there is no way for the conclusion to be false, assuming the truth of the premises. This was a fairly simple example where the

Chapter 1: Reconstructing and analyzing arguments

missing premise needed to make the argument valid was relatively easy to see. As we can see from this example, a missing premise is a premise that the argument needs in order to be as strong as possible. Typically, this means supplying the statement(s) that are needed to make the argument valid. But in addition to making the argument valid, we want to make the argument plausible. This is called “the principle of charity.” The principle of charity states that when reconstructing an argument, you should try to make that argument (whether inductive or deductive) as strong as possible. When it comes to supplying missing premises, this means supplying the most plausible premises needed in order to make the argument either valid (for deductive arguments) or inductively strong (for inductive arguments).

Although in the last example figuring out the missing premise was relatively easy to do, it is not always so easy. Here is an argument whose missing premises are not as easy to determine:

Since children who are raised by gay couples often have psychological and emotional problems, the state should discourage gay couples from raising children.

The conclusion of this argument, that the state should not allow gay marriage, is apparently supported by a single premise, which should be recognizable from the occurrence of the premise indicator, “since.” Thus, our initial reconstruction of the standard form argument looks like this:

1. Children who are raised by gay couples often have psychological and emotional problems.

2. Therefore, the state should discourage gay couples from raising children.

However, as it stands, this argument is invalid because it depends on certain missing premises. The conclusion of this argument is a normative statement— a statement about whether something ought to be true, relative to some standard of evaluation. Normative statements can be contrasted with descriptive statements, which are simply factual claims about what is true. For example, “Russia does not allow gay couples to raise children” is a descriptive statement. That is, it is simply a claim about what is in fact the case in Russia today. In contrast, “Russia should not allow gay couples to raise children” is a normative statement since it is not a claim about what is true, but what ought to

Chapter 1: Reconstructing and analyzing arguments

be true, relative to some standard of evaluation (for example, a moral or legal standard). An important idea within philosophy, which is often traced back to the Scottish philosopher David Hume (1711-1776), is that statements about what ought to be the case (i.e., normative statements) can never be derived from statements about what is the case (i.e., descriptive statements). This is known within philosophy as the is-ought gap. The problem with the above argument is that it attempts to infer a normative statement from a purely descriptive statement, violating the is-ought gap. We can see the problem by constructing a counterexample. Suppose that in society x it is true that children raised by gay couples have psychological problems. However, suppose that in that society people do not accept that the state should do what it can to decrease harm to children. In this case, the conclusion, that the state should discourage gay couples from raising children, does not follow. Thus, we can see that the argument depends on a missing or assumed premise that is not explicitly stated. That missing premise must be a normative statement, in order that we can infer the conclusion, which is also a normative statement. There is an important general lesson here: Many times an argument with a normative conclusion will depend on a normative premise which is not explicitly stated. The missing normative premise of this particular argument seems to be something like this:

The state should always do what it can to decrease harm to children.

Notice that this is a normative statement, which is indicated by the use of the word “should.” There are many other words that can be used to capture normative statements such as: good, bad, and ought. Thus, we can reconstruct the argument, filling in the missing normative premise like this:

1. Children who are raised by gay couples often have psychological and emotional problems.

2. The state should always do what it can to decrease harm to children. 3. Therefore, the state should discourage gay couples from raising children. (from 1-2)

However, although the argument is now in better shape, it is still invalid because it is still possible for the premises to be true and yet the conclusion false. In order to show this, we just have to imagine a scenario in which both the premises are true and yet the conclusion is false. Here is one counterexample to the argument (there are many). Suppose that while it is true that children of gay couples often have psychological and emotional problems, the rate of

Chapter 1: Reconstructing and analyzing arguments

psychological problems in children raised by gay couples is actually lower than in children raised by heterosexual couples. In this case, even if it were true that the state should always do what it can to decrease harm to children, it does not follow that the state should discourage gay couples from raising children. In fact, in the scenario I’ve described, just the opposite would seem to follow: the state should discourage heterosexual couples from raising children.

But even if we suppose that the rate of psychological problems in children of gay couples is higher than in children of heterosexual couples, the conclusion still doesn’t seem to follow. For example, it could be that the reason that children of gay couples have higher rates of psychological problems is that in a society that is not yet accepting of gay couples, children of gay couples will face more teasing, bullying and general lack of acceptance than children of heterosexual couples. If this were true, then the harm to these children isn’t so much due to the fact that their parents are gay as it is to the fact that their community does not accept them. In that case, the state should not necessarily discourage gay couples from raising children. Here is an analogy: At one point in our country’s history (if not still today) it is plausible that the children of black Americans suffered more psychologically and emotionally than the children of white Americans. But for the government to discourage black Americans from raising children would have been unjust, since it is likely that if there was a

higher incidence of psychological and emotional problems in black Americans, then it was due to unjust and unequal conditions, not to the black parents, per se. So, to return to our example, the state should only discourage gay couples from raising children if they know that the higher incidence of psychological problems in children of gay couples isn’t the result of any kind of injustice, but is due to the simple fact that the parents are gay.

Thus, one way of making the argument (at least closer to) valid would be to add the following two missing premises:

A. The rate of psychological problems in children of gay couples is higher than in children of heterosexual couples.

B. The higher incidence of psychological problems in children of gay couples is not due to any kind of injustice in society, but to the fact that the parents are gay.

So the reconstructed standard form argument would look like this:

Chapter 1: Reconstructing and analyzing arguments

1. Children who are raised by gay couples often have psychological and emotional problems.

2. The rate of psychological problems in children of gay couples is higher than in children of heterosexual couples.

3. The higher incidence of psychological problems in children of gay couples is not due to any kind of injustice in society, but to the fact that the parents are gay.

4. The state should always do what it can to decrease harm to children. 5. Therefore, the state should discourage gay couples from raising children. (from 1-4)

In this argument, premises 2-4 are the missing or assumed premises. Their addition makes the argument much stronger, but making them explicit enables us to clearly see what assumptions the argument relies on in order for the argument to be valid. This is useful since we can now clearly see which premises of the argument we may challenge as false. Arguably, premise 4 is false, since the state shouldn’t always do what it can to decrease harm to children. Rather, it should only do so as long as such an action didn’t violate other rights that the state has to protect or create larger harms elsewhere.

The important lesson from this example is that supplying the missing premises of an argument is not always a simple matter. In the example above, I have used the principle of charity to supply missing premises. Mastering this skill is truly an art (rather than a science) since there is never just one correct way of doing it (cf. section 1.5) and because it requires a lot of skilled practice.

Exercise 6: Supply the missing premise or premises needed in order to make the following arguments valid. Try to make the premises as plausible as possible while making the argument valid (which is to apply the principle of charity).

1. Ed rides horses. Therefore, Ed is a cowboy.

2. Tom was driving over the speed limit. Therefore, Tom was doing something wrong.

3. If it is raining then the ground is wet. Therefore, the ground must be wet.

4. All elves drink Guinness, which is why Olaf drinks Guinness. 5. Mark didn’t invite me to homecoming. Instead, he invited his friend Alexia. So he must like Alexia more than me.

Chapter 1: Reconstructing and analyzing arguments

6. The watch must be broken because every time I have looked at it, the hands have been in the same place.

7. Olaf drank too much Guinness and fell out of his second story apartment window. Therefore, drinking too much Guinness caused Olaf to injure himself.

8. Mark jumped into the air. Therefore, Mark landed back on the ground.

9. In 2009 in the United States, the net worth of the median white household was $113,149 a year, whereas the net worth of the median black household was $5,677. Therefore, as of 2009, the United States was still a racist nation.

10.The temperature of the water is 212 degrees Fahrenheit. Therefore, the water is boiling.

11.Capital punishment sometimes takes innocent lives, such as the lives of individuals who were later found to be not guilty. Therefore, we should not allow capital punishment.

12.Allowing immigrants to migrate to the U.S. will take working class jobs away from working class folks. Therefore, we should not allow immigrants to migrate to the U.S.

13.Prostitution is a fair economic exchange between two consenting adults. Therefore, prostitution should be allowed.

14.Colleges are more interested in making money off of their football athletes than in educating them. Therefore, college football ought to be banned.

15.Edward received an F in college Algebra. Therefore, Edward should have studied more.

1.10 Assuring, guarding and discounting

As we have seen, arguments often have complex structures including subarguments (recall that a subargument is an argument for one of the premises of the main argument). But in practice people do not always give further reasons or argument in support of every statement they make. Sometimes they use certain rhetorical devices to cut the argument short, or to hint at a further argument without actually stating it. There are three common strategies for doing this:

Chapter 1: Reconstructing and analyzing arguments

Assuring: informing someone that there are further reasons although one is not giving them now

Guarding: weakening one’s claims so that it is harder to show that the claims are false

Discounting: anticipating objections that might be raised to one’s claim or argument as a way of dismissing those objections.³

We will discuss these in order, starting with assuring. Why would we want to assure our audience? Presumably when we make a claim that isn’t obvious and that the audience may not be inclined to believe. For example, if I am trying to convince you that the United States is one of the leading producers of CO2

emissions, then I might cite certain authorities such as the Intergovernmental Panel on Climate Change (IPCC) as saying so. This is one way of assuring our audience: by citing authorities. There are many ways to cite authorities, some examples of which are these:

Dentists agree that…

Recent studies have shown…

It has been established that…

Another way of assuring is to comment on the strength of one’s own convictions. The rhetorical effect is that by commenting on how sure you are that something is true, you imply, without saying, that there must be very strong reasons for what you believe—assuming that the audience believes you are a reasonable person, of course. Here are some ways of commenting on the strength of one’s beliefs:

I’m certain that…

I’m sure that…

I can assure you that…

³This characterization and discussion draws heavily on chapter 3, pp. 48-53 of Sinnott Armstrong and Fogelin’s Understanding Arguments, 9^thedition (Cengage Learning).

Chapter 1: Reconstructing and analyzing arguments

Over the years, I have become convinced that…

I would bet a million dollars that…

Yet another way of assuring one’s audience is to make an audience member feel that it would be stupid, odd, or strange to deny the claim one is making. One common way to do this is by implying that every sensible person would agree with the claim. Here are some examples:

Everyone with any sense agrees that…

Of course, no one will deny that…

There is no question that…

No one with any sense would deny that…

Another common way of doing this is by implying that no sensible person would agree with a claim that we are trying to establish as false:

It is no longer held that…

No intelligent person would ever maintain that…

You would have to live under a rock to think that…

Assurances are not necessarily illegitimate, since the person may be right and may in fact have good arguments to back up the claims, but the assurances are not themselves arguments and a critical thinker will always regard them as somewhat suspect. This is especially so when the claim isn’t obviously true.

Next, we will turn to guarding. Guarding involves weakening a claim so that it is easier to make that claim true. Here is a simple contrast that will make the point. Consider the following claims:

A. All U.S. Presidents were monogamous

B. Almost all U.S. Presidents were monogamous

C. Most U.S. Presidents were monogamous

Chapter 1: Reconstructing and analyzing arguments

D. Many U.S. Presidents were monogamous

E. Some U.S. Presidents were monogamous

The weakest of these claims is E, whereas the strongest is A and each claims descending from A-E is increasingly weaker. It doesn’t take very much for E to be true: there just has to be at least one U.S. President who was monogamous. In contrast, A is much less likely than E to be true because it require every U.S. President to have been monogamous. One way of thinking about this is that any time A is true, it is also true that B-E is true, but B-E could be true without A being true. That is what it means for a claim to be stronger or weaker. A weak claim is more likely to be true whereas a strong claim is less likely to be true. E is much more likely to be true than A. Likewise, D is somewhat more likely to be true than C, and so on.

So, guarding involves taking a stronger claim and making it weaker so there is less room to object to the claim. We can also guard a claim by introducing a probability clause such as, “it is possible that…” and “it is arguable that…” or by reducing our level of commitment to the claim, such as moving from “I know that x” to “I believe that x.” One common use of guarding is in reconstructing arguments with missing premises using the principle of charity (section 1.9). For example, if an argument is that “Tom works for Merrill Lynch, so Tom has a college degree,” the most charitable reconstruction of this argument would fill in the missing premise with “most people who work for Merrill Lynch have college degrees” rather than “everyone who works for Merrill Lynch has a college degree.” Here we have created a more charitable (plausible) premise by weakening the claim from “all” to “most,” which as we have seen is a kind of guarding.

Finally, we will consider discounting. Discounting involves acknowledging an objection to the claim or argument that one is making, while dismissing that same objection. The rhetorical force of discounting is to make it seem as though the argument has taken account of the objections—especially the ones that might be salient in a person’s mind. The simplest and most common way of discounting is by using the “A but B” locution. Contrast the following two claims:

A. The worker was inefficient, but honest.

B. The worker was honest, but inefficient.

Chapter 1: Reconstructing and analyzing arguments

Although each statement asserts the same facts, A seems to be recommending the worker, whereas B doesn’t. We can imagine A continuing: “And so the manager decided to keep her on the team.” We can imagine B continuing: “Which is why the manager decided to let her go.” This is what we can call the “A but B” locution. The “A but B” locution is a form of discounting that introduces what will be dismissed or overridden first and then follows it by what is supposed to be the more important consideration. By introducing the claim to be dismissed, we are discounting that claim. There are many other words that can be used as discounting words instead of using “but.” Table 2 below gives a partial list of words and phrases that commonly function as discounting terms.

although	even if	but	nevertheless
though	while	however	nonetheless
even though	whereas	yet	still

Exercise 7: Which rhetorical techniques (assuring, guarding, discounting) are being using in the following passages?

1. Although drilling for oil in Alaska will disrupt some wildlife, it is better than having to depend on foreign oil, which has the tendency to draw us into foreign conflicts that we would otherwise not be involved in.

2. Let there be no doubt: the entity that carried out this attack is a known terrorist organization, whose attacks have a characteristic style—a style that is seen in this attack today.

3. Privatizing the water utilities in Detroit was an unprecedented move that has garnered a lot of criticism. Nonetheless, it is helping Detroit to recover from bankruptcy.

4. Most pediatricians agree that the single most important factor in childhood obesity is eating sugary, processed foods, which have become all too common in our day and age.

5. Although not every case of AIDS is caused by HIV, it is arguable that most are.

6. Abraham Lincoln was probably our greatest president since he helped keep together a nation on the brink of splintering into two.

7. No one with any sense would support Obamacare.

8. Even if universal healthcare is expensive, it is still the just thing to do.

Chapter 1: Reconstructing and analyzing arguments

9. While our country has made significant strides in overcoming explicit racist policies, the wide disparity of wealth, prestige and influence that characterize white and black Americans shows that we are still implicitly a racist country.

10.Recent studies have show that there is no direct link between vaccines and autism.

1.11 Evaluative language

Yet another rhetorical technique that is commonly encountered in argumentation is the use of evaluative language to influence one’s audience to accept the conclusion one is arguing for. Evaluative language can be contrasted with descriptive language. Whereas descriptive language simply describes a state of affairs, without passing judgment (positive or negative) on that state of affairs, evaluative language is used to pass some sort of judgment, positive or negative, on something. Contrast the following two statements:

Bob is tall.

Bob is good.

“Tall” is a descriptive term since being tall is, in itself, neither a good nor bad thing. Rather, it is a purely descriptive term that does not pass any sort of judgment, positive or negative, on the fact that Bob is tall. In contrast, “good” is a purely evaluative term, which means that the only thing the word does is make an evaluation (in this case, a positive evaluation) and doesn’t carry any descriptive content. “Good,” “bad,” “right,” and “wrong” are examples of purely evaluative terms. The interesting kinds of terms are those that are both descriptive and evaluative. For example:

Bob is nosy.

“Nosy” is a negatively evaluative term since to call someone nosy is to make a negative evaluation of them—or at least of that aspect of them. But it also implies a descriptive content, such as that Bob is curious about other people’s affairs. We could re-describe Bob’s nosiness using purely descriptive language:

Bob is very curious about other people’s affairs.

Chapter 1: Reconstructing and analyzing arguments

Notice that while the phrase “very curious about other people’s affairs” does capture the descriptive sense of “nosy,” it doesn’t capture the evaluative sense of nosy, since it doesn’t carry with it the negative connotation that “nosy” does.

Evaluative language is rife in our society, perhaps especially so in political discourse. This isn’t surprising since by using evaluative language to describe certain persons, actions, or events we can influence how people understand and interpret the world. If you can get a person to think of someone or some state of affairs in terms of a positively or negatively evaluative term, chances are you will be able to influence their evaluation of that person or state of affairs. That is one of the rhetorical uses of evaluative language. Compare, for example,

Bob is a rebel.

Bob is a freedom fighter.

Whereas “rebel” tends to be a negatively evaluative term, “freedom fighter,” at least for many Americans, tends to be a positively evaluative term. Both words, however, have the same descriptive content, namely, that Bob is someone who has risen in armed resistance to an existing government. The difference is that whereas “rebel” makes a negative evaluation, “freedom fighter” makes a positive evaluation. Table 3 below gives a small sampling of some evaluative terms.

beautiful	dangerous wasteful		sneaky	cute
murder	prudent	courageous timid		nosy
sloppy	smart	capable	insane	curt

English contains an interesting mechanism for turning positively evaluative terms into negative evaluative ones. All you have to do is put the word “too” before a positively evaluative terms and it will all of a sudden take on a negative connotation. Compare the following:

John is honest.

John is too honest.

Chapter 1: Reconstructing and analyzing arguments

Whereas “honest” is a positively evaluative term, “too honest” is a negatively evaluative term. When someone describes John as “too honest,” we can easily imagine that person going on to describe how John’s honesty is actually a liability or negative trait. Not so when he is simply described as honest. Since the word “too” indicates an excess, and to say that something is an excess is to make a criticism, we can see why the word “too” changes the valence of an evaluation from positive to negative.

Evaluative language provides a good illustration of the difference between logic, which is concerned with the analysis and evaluation of arguments, and rhetoric, which is concerned with persuasion more generally. There are many ways that humans can be caused to believe things besides through rational argumentation. In fact, sometimes these other persuasive techniques are much more effective. (Consider advertising techniques in the 1950s, which more often tried to used argument and evidence to convince consumers to buy products, compared to advertising today, which rarely uses argument and evidence.) In any case, evaluative language—especially the use of hybrid terms that have both descriptive and evaluative aspects—can lead people to subtly accept a claim without ever arguing for it. As an analogy for how this could work in conversation, consider the concept of what philosophers⁴have called “presupposition.” If is say something like

Even Jane could pass

I have asserted that Jane could pass the course. But I have also presupposed that Jane is not a very good student (or not very smart) by using the word “even.” If I were to say “nuh-uh,” this would naturally be taken as rejecting the claim that Jane could pass (i.e., I would be saying that she couldn’t pass). And if I were to agree, I would naturally be taken as agreeing that she could pass. But notice that there isn’t any simply yes/no way to disagree with the presupposition that Jane isn’t a smart/good student. Since presuppositions are more difficult to challenge, they can end up influencing what people in the conversation are taking for granted and in this way presupposition can influence what people accept as true without any argument or evidence. Of course, a person could explicitly challenge the implicit presupposition that Jane isn’t smart or a good student, but that takes extra effort and many times people don’t realize that a presupposition has just slipped into a conversation.

⁴For example, see David Lewis’s “Scorekeeping in a Language Game” (1979).

Chapter 1: Reconstructing and analyzing arguments

I suggest that hybrid evaluative/descriptive terms can work as a kind of presupposition. If I describe someone as an “insurgent,” for example, I am saying something both descriptive—person who has risen in armed resistance against an existing government—and negatively evaluative since the connotation of the term “insurgent” (as compared to “freedom fighter”) has come to be that of someone doing something bad or negative. In using the term “insurgent” no one has explicitly claimed that the individual/group in question is bad, but because the term has (for us) a negative connotation it can lead us to be more receptive to accepting (implicitly) claims such as that the person/group is bad or is doing something bad/harmful.

Thus, like assuring and discounting (section 1.10), evaluative language is a rhetorical technique. As such, it is more concerned with non-rational persuasion than it is with giving reasons. Non-rational persuasion is ubiquitous in our society today, not the least of which because advertising is ubiquitous and advertising today almost always uses non-rational persuasion. Think of the last time you saw some commercial present evidence for why you should buy their product (i.e., never) and you will realize how pervasive this kind of rhetoric is. Philosophy has a complicated relationship with rhetoric—a relationship that stretches back to Ancient Greece. Socrates disliked those, such as the Sophists, who promised to teach people how to effectively persuade someone of something, regardless of whether that thing was true. Although some people might claim that there is no essential difference between giving reasons for accepting a conclusion and trying to persuade by any means, most philosophers, including the author of this text, think otherwise. If we define rhetoric as the art of persuasion, then although argumentation is a kind of rhetoric (since it is a way of persuading), not all rhetoric is argumentation. The essential difference, as already hinted at, is that argumentation attempts to persuade by giving reasons whereas rhetoric attempts to persuade by any means, including non-rational means. If I tell you over and over again (in creative and subliminal ways) to drink Beer x because Beer x is the best beer, then I may very well make you think that Beer x is the best beer, but I have not thereby given you a reason to accept that Beer x is the best beer. Thinking of it rationally, the mere fact that I’ve told you lots of times that Beer x is the best beer gives you no good reason for believing that Beer x is in fact the best beer.

The rhetorical devices surveyed in the last two sections may be effective ways of persuading people, but they are not the same thing as offering an argument.

Chapter 1: Reconstructing and analyzing arguments

And if we attempt to see them as arguments, they turn out to be pretty poor arguments. One of the many things that psychologists study is how we are persuaded to believe or do things. As an empirical science, psychology attempts to describe and explain the way things are, in this case, the processes that lead us to believe or act as we do. Logic, in contrast, is not an empirical science. Logic is not trying to tell us how we do think, but what good thinking is and, thus, how we ought think. The study of logic is the study of the nature of arguments and, importantly, of what distinguishes a good argument from a bad one. “Good” and “bad” are what philosophers call normative concepts

because they involve standards of evaluation.⁵ Since logic concerns what makes something a good argument, logic is sometimes referred to as a normative science. They key standard of evaluation of arguments that we have seen so far is that of validity. In chapter 2 we will consider some more precise, formal methods of understanding validity. Other “normative sciences” include ethics (the study of what a good life is and how we ought to live) and epistemology (the study of what we have good reason to believe).

1.12 Analyzing a real-life argument

In this section I will analyze a real-life argument—an excerpt from President Obama’s September 10, 2013 speech on Syria. I will use the concepts and techniques that have been introduced in this chapter to analyze and evaluate Obama’s argument. It is important to realize that regardless of one’s views—

whether one agrees with Obama or not—one can still analyze the structure of the argument and even evaluate it by applying the informal test of validity to the reconstructed argument in standard form. I will present the excerpt of Obama’s speech and then set to work analyzing the argument it contains. In addition to creating the excerpt, the only addition I have made to the speech is numbering each paragraph with Roman numerals for ease of referring to specific places in my analysis of the argument.

I. My fellow Americans, tonight I want to talk to you about Syria, why it matters and where we go from here. Over the past two years, what began as a series of peaceful protests against the repressive regime of Bashar al-Assad has turned into a brutal civil war. Over a hundred thousand people have been killed. Millions have fled the country. In that time, America has worked with allies to provide humanitarian support, to help the moderate opposition and to shape a political settlement.

⁵We encountered normative concepts when discussing normative statements in section 1.9.

Chapter 1: Reconstructing and analyzing arguments

II. But I have resisted calls for military action because we cannot resolve someone else's civil war through force, particularly after a decade of war in Iraq and Afghanistan.

III. The situation profoundly changed, though, on Aug. 21st, when Assad's government gassed to death over a thousand people, including hundreds of children. The images from this massacre are sickening, men, women, children lying in rows, killed by poison gas, others foaming at the mouth, gasping for breath, a father clutching his dead children, imploring them to get up and walk. On that terrible night, the world saw in gruesome detail the terrible nature of chemical weapons and why the overwhelming majority of humanity has declared them off limits, a crime against humanity and a violation of the laws of war.

IV. This was not always the case. In World War I, American GIs were among the many thousands killed by deadly gas in the trenches of Europe. In World War II, the Nazis used gas to inflict the horror of the Holocaust. Because these weapons can kill on a mass scale, with no distinction between soldier and infant, the civilized world has spent a century working to ban them. And in 1997, the United States Senate overwhelmingly approved an international agreement prohibiting the use of chemical weapons, now joined by 189 governments that represent 98 percent of humanity.

V. On Aug. 21st, these basic rules were violated, along with our sense of common humanity.

VI. No one disputes that chemical weapons were used in Syria. The world saw thousands of videos, cellphone pictures and social media accounts from the attack. And humanitarian organizations told stories of hospitals packed with people who had symptoms of poison gas.

VII. Moreover, we know the Assad regime was responsible. In the days leading up to Aug. 21st, we know that Assad's chemical weapons personnel prepared for an attack near an area where they mix sarin gas. They distributed gas masks to their troops. Then they fired rockets from a regime-controlled area into 11 neighborhoods that the regime has been trying to wipe clear of opposition forces.

VIII. Shortly after those rockets landed, the gas spread, and hospitals filled with the dying and the wounded. We know senior figures in Assad's military machine reviewed the results of the attack. And the regime increased their shelling of the same neighborhoods in the days that followed. We've also studied samples of blood and hair from people at the site that tested positive for sarin.

IX. When dictators commit atrocities, they depend upon the world to look the other way until those horrifying pictures fade from memory. But these things happened. The facts cannot be denied.

X. The question now is what the United States of America and the international community is prepared to do about it, because what happened to those people, to

Chapter 1: Reconstructing and analyzing arguments

those children, is not only a violation of international law, it's also a danger to our security.

XI. Let me explain why. If we fail to act, the Assad regime will see no reason to stop using chemical weapons.

XII. As the ban against these weapons erodes, other tyrants will have no reason to think twice about acquiring poison gas and using them. Over time our troops would again face the prospect of chemical warfare on the battlefield, and it could be easier for terrorist organizations to obtain these weapons and to use them to attack civilians.

XIII. If fighting spills beyond Syria's borders, these weapons could threaten allies like Turkey, Jordan and Israel.

XIV. And a failure to stand against the use of chemical weapons would weaken prohibitions against other weapons of mass destruction and embolden Assad's ally, Iran, which must decide whether to ignore international law by building a nuclear weapon or to take a more peaceful path.

XV. This is not a world we should accept. This is what's at stake. And that is why, after careful deliberation, I determined that it is in the national security interests of the United States to respond to the Assad regime's use of chemical weapons through a targeted military strike. The purpose of this strike would be to deter Assad from using chemical weapons, to degrade his regime's ability to use them and to make clear to the world that we will not tolerate their use. That's my judgment as commander in chief.

The first question to ask yourself is: What is the main point or conclusion of this speech? What conclusion is Obama trying to argue for? This is no simple question and in fact requires a good level of reading comprehension in order to answer it correctly. One of the things to look for is conclusion or premise indicators (section 1.2). There are numerous conclusion indicators in the speech, which is why you cannot simply mindlessly look for them and then assume the first one you find is the conclusion. Rather, you must rely on your comprehension of the speech to truly find the main conclusion. If you carefully read the speech, it is clear that Obama is trying to convince the American public of the necessity of taking military action against the Assad regime in Syria. So the conclusion is going to have to have something to do with that. One clear statement of what looks like a main conclusion comes in paragraph 15 where Obama says:

And that is why, after careful deliberation, I determined that it is in the national security interests of the United States to respond to the Assad regime's use of chemical weapons through a targeted military strike.

Chapter 1: Reconstructing and analyzing arguments

The phrase, “that is why,” is a conclusion indicator which introduces the main conclusion. Here is my paraphrase of that conclusion:

Main conclusion: It is in the national security interests of the United States to respond to Assad’s use of chemical weapons with military force.

Before Obama argues for this main conclusion, however, he gives an argument for the claim that Assad did use chemical weapons on his own civilians. This is what is happening in paragraphs 1-9 of the speech. The reasons he gives for how we know that Assad used chemical weapons include:

• images of the destruction of women and children (paragraph VI) • humanitarian organizations’ stories of hospitals full of civilians suffering from symptoms of exposure to chemical weapons (paragraph VI) • knowledge that Assad’s chemical weapons experts were at a site where sarin gas is mixed just a few days before the attack (paragraph VII) • the fact that Assad distributed gas masks to his troops (paragraph VII) • the fact that Assad’s forces fired rockets into neighborhoods where there were opposition forces (paragraph VII)

• senior military officers in Assad’s regime reviewed results of the attack (paragraph VIII)

• the fact that sarin was found in blood and hair samples from people at the site of the attack (paragraph VIII)

These premises do indeed provide support for the conclusion that Assad used chemical weapons on civilians, but it is probably best to see this argument as a strong inductive argument, rather than a deductive argument. The evidence strongly supports, but does not compel, the conclusion that Assad was responsible. For example, even if all these facts were true, it could be that some other entity was trying to set Assad up. Thus, this first subargument should be taken as a strong inductive argument (assuming the premises are true, of course), since the truth of the premises would increase the probability that the conclusion is true, but not make the conclusion absolutely certain.

Although Obama does give an argument for the claim that Assad carried out chemical weapon attacks on civilians, that is simply an assumption of the main argument. Moreover, although the conclusion of the main argument is the one I have indicated above, I think there is another, intermediate conclusion that

Chapter 1: Reconstructing and analyzing arguments

Obama argues for more directly and that is that if we don’t respond to Assad’s use of chemical weapons, then our own national security will be put at risk. We can clearly see this conclusion stated in paragraph 10. Moreover, the very next phrase in paragraph 11 is a premise indicator, “let me explain why.” Obama goes on to offer reasons for why failing to respond to Assad’s use of chemical weapons would be a danger to our national security. Thus, the conclusion Obama argues more directly for is:

Intermediate conclusion: A failure to respond to Assad’s use of chemical weapons is a threat to our national security.

So, if that is the conclusion that Obama argues for most directly, what are the premises that support it? Obama gives several in paragraphs 11-14:

A. If we don’t respond to Assad’s use of chemical weapons, then Assad’s regime will continue using them with impunity. (paragraph 11) B. If Assad’s regime uses chemical weapons with impunity, this will effectively erode the ban on them. (implicit in paragraph 12)

C. If the ban on chemical weapons erodes, then other tyrants will be more likely to attain and use them. (paragraph 12)

D. If other tyrants attain and use chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield (paragraph 12) E. If we don’t respond to Assad’s use of chemical weapons and if fighting spills beyond Syrian borders, our allies could face these chemical weapons. (paragraph 13)

F. If Assad’s regime uses chemical weapons with impunity, it will weaken prohibitions on other weapons of mass destruction. (paragraph 14) G. If prohibitions on other weapons of mass destruction are weakened, this will embolden Assad’s ally, Iran, to develop a nuclear program. (paragraph 14)

I have tried to make explicit each step of the reasoning, much of which Obama makes explicit himself (e.g., premises A-D). The main threats to national security that failing to respond to Assad would engender, according to Obama, are that U.S. troops and U.S. allies could be put in danger of facing chemical weapons and that Iran would be emboldened to develop a nuclear program. There is a missing premise that is being relied upon for these premises to validly imply the conclusion. Here is a hint as to what that missing premise is: Are all of these things truly a threat to national security? For example, how is Iran having a

Chapter 1: Reconstructing and analyzing arguments

nuclear program a threat to our national security? It seems there must be an implicit premise—not yet stated—that is to the effect that all of these things are threats to national security. Here is one way of construing that missing premise:

Missing premise 1: An increased likelihood of U.S. troops or allies facing chemical weapons on the battlefield or Iran becoming emboldened to develop a nuclear program are all threats to U.S. national security interests.

We can also make explicit within the standard form argument other intermediate conclusions that follow from the stated premises. Although we don’t have to do this, it can be a helpful thing to do when an argument contains multiple premises. For example, we could explicitly state the conclusion that follows from the four conditional statements that are the first four premises:

1. If we don’t respond to Assad’s use of chemical weapons, then Assad’s regime will continue using them with impunity.

2. If Assad’s regime uses chemical weapons with impunity, this will effectively erode the ban on them.

3. If the ban on chemical weapons erodes, then other tyrants will be more likely to attain and use them.

4. If other tyrants attain and use chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield.

5. Therefore, if we don’t respond to Assad’s use of chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield. (from 1-4)

Premise 5 is an intermediate conclusion that makes explicit what follows from premises 1-4 (which I have represented using parentheses after that intermediate conclusion). We can do the same thing with the inference that follows from premises, 1, 7, and 8 (i.e., line 9). If we add in our missing premises

then we have a reconstructed argument for what I earlier called the “intermediate conclusion” (i.e., the one that Obama most directly argues for):

1. If we don’t respond to Assad’s use of chemical weapons, then Assad’s regime will continue using them with impunity.

2. If Assad’s regime uses chemical weapons with impunity, this will effectively erode the ban on them.

Chapter 1: Reconstructing and analyzing arguments

3. If the ban on chemical weapons erodes, then other tyrants will be more likely to attain and use them.

4. If other tyrants attain and use chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield.

5. Therefore, if we don’t respond to Assad’s use of chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield. (from 1-4)

6. If we don’t respond to Assad’s use of chemical weapons and if fighting spills beyond Syrian borders, our allies could face these chemical weapons.

7. If Assad’s regime uses chemical weapons with impunity, it will weaken prohibitions on other weapons of mass destruction.

8. If prohibitions on other weapons of mass destruction are weakened, this will embolden Assad’s ally, Iran, to develop a nuclear program. 9. Therefore, if we don’t respond to Assad’s use of chemical weapons, this will embolden Assad’s ally, Iran, to develop a nuclear program. (from 1, 7-8)

10. An increased likelihood of U.S. troops or allies facing chemical weapons on the battlefield or Iran becoming emboldened to develop a nuclear program are threats to U.S. national security interests.

11.Therefore, a failure to respond to Assad’s use of chemical weapons is a threat to our national security. (from 5, 6, 9, 10)

As always, in this standard form argument I’ve listed in parentheses after the relevant statements which statements those statements follow from. The only thing now missing is how we get from this intermediate conclusion to what I earlier called the main conclusion. The main conclusion (i.e., that it is in national security interests to respond to Assad with military force) might be thought to follow directly. But it doesn’t. It seems that Obama is relying on yet another unstated assumption. Consider: even if it is true that we should respond to a threat to our national security, it doesn’t follow that we should respond with military force. For example, maybe we could respond with certain kinds of economic sanctions that would force the country to submit to our will. Furthermore, maybe there are some security threats such that responding to them with military force would only create further, and worse, security threats. Presumably we wouldn’t want our response to a security threat to create even bigger security threats. For these reasons, we can see that Obama’s argument, if it is to be valid, also relies on missing premises such as these:

Chapter 1: Reconstructing and analyzing arguments

Missing premise 2: The only way that the United States can adequately respond to the security threat that Assad poses is by military force.

Missing premise 3: It is in the national security interests of the United States to respond adequately to any national security threat.

These are big assumptions and they may very well turn out to be mistaken. Nevertheless, it is important to see that the main conclusion Obama argues for depends on these missing premises—premises that he never explicitly states in his argument. So here is the final, reconstructed argument in standard form. I have italicized each missing premise or intermediate conclusion that I have added but that wasn’t explicitly stated in Obama’s argument.

1. If we don’t respond to Assad’s use of chemical weapons, then Assad’s regime will continue using them with impunity.

2. If Assad’s regime uses chemical weapons with impunity, this will effectively erode the ban on them.

3. If the ban on chemical weapons erodes, then other tyrants will be more likely to attain and use them.

4. If other tyrants attain and use chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield.

5. Therefore, if we don’t respond to Assad’s use of chemical weapons, U.S. troops will be more likely to face chemical weapons on the battlefield. (from 1-4)

6. If we don’t respond to Assad’s use of chemical weapons and if fighting spills beyond Syrian borders, our allies could face these chemical weapons.

7. If Assad’s regime uses chemical weapons with impunity, it will weaken prohibitions on other weapons of mass destruction.

8. If prohibitions on other weapons of mass destruction are weakened, this will embolden Assad’s ally, Iran, to develop a nuclear program. 9. Therefore, if we don’t respond to Assad’s use of chemical weapons, this will embolden Assad’s ally, Iran, to develop a nuclear program. (from 1, 7-8)

11.Therefore, a failure to respond to Assad’s use of chemical weapons is a threat to our national security. (from 5, 6, 9, 10)

Chapter 1: Reconstructing and analyzing arguments

12. The only way that the United States can adequately respond to the security threat that Assad poses is by military force.

13. It is in the national security interests of the United States to respond adequately to any national security threat.

14. Therefore, it is in the national security interests of the United States to respond to Assad’s use of chemical weapons with military force. (from 11-13)

In addition to showing the structure of the argument by use of parentheses which show which statements follow from which, we can also diagram the arguments spatially as we did in section 1.4 like this:

This is just another way of representing what I have already represented in the standard form argument, using parentheses to describe the structure. As is perhaps even clearer in the spatial representation of the argument’s structure, this argument is complex in that it has numerous subarguments. So while statement 11 is a premise of the main argument for the main conclusion (statement 14), statement 11 is also itself a conclusion of a subargument whose premises are statements 5, 6, 9, and 10. And although statement 9 is a premise in that argument, it itself is a conclusion of yet another subargument whose premises are statements 1, 7 and 8. Almost any interesting argument will be complex in this way, with further subarguments in support of the premises of the main argument.

Chapter 1: Reconstructing and analyzing arguments

This chapter has provided you the tools to be able to reconstruct arguments like these. As we have seen, there is much to consider in reconstructing a complex argument. As with any skill, a true mastery of it requires lots of practice. In many ways, this is a skill that is more like an art than a science. The next chapter will introduce you to some basic formal logic, which is perhaps more like a science than an art.

Chapter 2: Formal methods of evaluating arguments

2.1 What are formal methods of evaluation and why do we need them?

In chapter 1 we introduced the concept of validity and the informal test of validity. According to that test, in order to determine whether an argument is valid we ask whether we can imagine a scenario where the premises are true and yet the conclusion is false. If we can, then the argument is invalid; if we can’t then the argument is valid. The informal test relies on our ability to imagine certain kinds of scenarios as well as our understanding of the statements involved in the argument. Because not everyone has the same powers of imagination or the same understanding, this informal test of validity is neither precise nor objective. For example, while one person may be able to imagine a scenario in which the premises of an argument are true while the conclusion is false, another person may be unable to imagine such a scenario. As a result, the argument will be classified as invalid by the first individual, but valid by the second individual. That is a problem because we would like our standard of evaluation of arguments (i.e., validity) to be as precise and objective as possible, and it seems that our informal test of validity is neither. It isn’t precise because the concept of being able to imagine x is not precise—what counts as imagining x is not something that can be clearly specified. What are the precise success conditions for having imagined a scenario where the premises are true and the conclusion is false? But the informal test of validity also isn’t objective since it is possible that two different people who applied the imagination test correctly could come to two different conclusions about whether the argument is valid. As I noted before, this is partly because people’s understanding of the statements differ and partly because people have different powers of imagination.

The goal of a formal method of evaluation is to eliminate any imprecision or lack of objectivity in evaluating arguments. As we will see by the end of this chapter, logicians have devised a number of formal techniques that accomplish this goal for certain classes of arguments. What all of these formal techniques have in common is that you can apply them without really having to understand the meanings of the concepts used in the argument. Furthermore, you can apply the formal techniques without having to utilize imagination at all. Thus, the formal techniques we will survey in this chapter help address the lack of precision and objectivity inherent in the informal test of validity. In general, a formal method of evaluation is a method of evaluation of arguments that does not require one to understand the meaning of the statements involved in the argument. Although at this point this may sound like gibberish, after we have

Chapter 2: Formal methods of evaluating arguments

introduced the formal methods, you will understand what it means to evaluate an argument without knowing what the statements of the argument mean. By the end of this chapter, if not before, you will understand what it means to evaluate an argument by its form, rather than its content.

However, I will give you a sense of what a formal method of evaluation is in a very simple case right now, to give you a foretaste of what we will be doing in this chapter. Suppose I tell you:

It is sunny and warm today.

This statement is a conjunction because it is a complex statement that is asserting two things:

It is sunny today.

It is warm today.

These two statements are conjoined with an “and.” So the conjunction is really two statements that are conjoined by the “and.” Thus, if I have told you that it is both sunny and warm today, it follows logically that it is sunny today. Here is that simple argument in standard form:

1. It is sunny today and it is warm today.

2. Therefore, it is sunny today. (from 1)

This is a valid inference that passes the informal test of validity. But we can also see that the form of the inference is perfectly general because it would work equally well for any conjunction, not just this one. This inference has a particular form that we could state using placeholders for the statements, “it is sunny today” and “it is warm today”:

1. A and B

2. Therefore, A

We can see that any argument that had this form would be a valid argument. For example, consider the statement:

Kant was a deontologist and a Pietist.

Chapter 2: Formal methods of evaluating arguments

That statement is a conjunction of two statements that we can capture explicitly in the first premise of the following argument:

1. Kant was a deontologist and Kant was a Pietist.

2. Therefore, Kant was a deontologist. (from 1)

Regardless of whether you know what the statements in the first premise mean, we can still see that the inference is valid because the inference has the same form that I just pointed out above. Thus, you may not know what “Kant” is (one of the most famous German philosophers of the Enlightenment) or what a “deontologist” or “Pietist” is, but you can still see that since these are statements that form a conjunction, and since the inference made has a particular form that is valid, this particular inference is valid. That is what it means for an argument to be valid in virtue of its form. In the next section we will delve into formal logic, which will involve learning a certain kind of language. Don’t worry: it won’t be as hard as your French or Spanish class.

2.2 Propositional logic and the four basic truth functional connectives

Propositional logic (also called “sentential logic”) is the area of formal logic that deals with the logical relationships between propositions. A proposition is simply what I called in section 1.1 a statement.¹ Some examples of propositions are:

Snow is white

Snow is cold

Tom is an astronaut

The floor has been mopped

The dishes have been washed

¹Some philosophers would claim that a proposition is not the same as a statement, but the reasons for doing so are not relevant to what we’ll be doing in this chapter. Thus, for our purposes, we can treat a proposition as the same thing as a statement.

Chapter 2: Formal methods of evaluating arguments

We can also connect propositions together using certain English words, such as “and” like this:

The floor has been mopped and the dishes have been washed.

This proposition is called a complex proposition because it contains the connective “and” which connects two separate propositions. In contrast, “the floor has been mopped” and “the dishes have been washed” are what are called atomic propositions. Atomic propositions are those that do not contain any truth-functional connectives. The word “and” in this complex proposition is a truth-functional connective. A truth-functional connective is a way of connecting propositions such that the truth value of the resulting complex proposition can be determined by the truth value of the propositions that compose it. Suppose that the floor has not been mopped but the dishes have been washed. In that case, if I assert the conjunction, “the floor has been mopped and the dishes have been washed,” then I have asserted something that is false. The reason is that a conjunction, like the one above, is only true when each conjunct (i.e., each statement that is conjoined by the “and”) is true. If either one of the conjuncts is false, then the whole conjunction is false. This should be pretty obvious. If Bob and Sally split chores and Bob’s chore was to both vacuum and dust whereas Sally’s chore was to both mop and do the dishes, then if Sally said she mopped the floor and did the dishes when in reality she only did the dishes (but did not mop the floor), then Bob could rightly complain that it isn’t true that Sally both mopped the floor and did the dishes! What this shows is that conjunctions are true only if both conjuncts are true. This is true of all conjunctions. The conjunction above has a certain form—the same form as any conjunction. We can represent that form using placeholders— lowercase letters like p and q to stand for any statement whatsoever. Thus, we represent the form of a conjunction like this:

p and q

Any conjunction has this same form. For example, the complex proposition, “it is sunny and hot today,” has this same form which we can see by writing the conjunction this way:

It is sunny today and it is hot today.

Chapter 2: Formal methods of evaluating arguments

Although we could write the conjunction that way, it is more natural in English to conjoin the adjectives “sunny” and “hot” to get “it is sunny and hot today.” Nevertheless, these two sentences mean the same thing (it’s just that one sounds more natural in English than the other). In any case, we can see that “it is sunny today” is the proposition in the “p” place of the form of the conjunction, whereas “it is hot today” is the proposition in the “q” place of the form of the conjunction. As before, this conjunction is true only if both conjuncts are true. For example, suppose that it is a sunny but bitterly cold winter’s day. In that case, while it is true that it is sunny today, it is false that it is hot today—in which case the conjunction is false. If someone were to assert that it is sunny and hot today in those circumstances, you would tell them that isn’t true. Conversely, if it were a cloudy but hot and humid summer’s day, the conjunction would still be false. The only way the statement would be true is if both conjuncts were true.

In the formal language that we are developing in this chapter, we will represent conjunctions using a symbol called the “dot,” which looks like this: “⋅” Using this symbol, here is how we will represent a conjunction in symbolic notation:

p ⋅ q

In the following sections we will introduce four basic truth-functional connectives, each of which have their own symbol and meaning. The four basic truth-functional connectives are: conjunction, disjunction, negation, and conditional. In the remainder of this section, we will discuss only conjunction.

As we’ve seen, a conjunction conjoins two separate propositions to form a complex proposition. The conjunction is true if and only if both conjuncts are true. We can represent this information using what is called a truth table. Truth tables represent how the truth value of a complex proposition depends on the truth values of the propositions that compose it. Here is the truth table for conjunction:

p	q	p ⋅ q
T	T	T
T	F	F
F	T	F
F	F	F

Chapter 2: Formal methods of evaluating arguments

Here is how to understand this truth table. The header row lists the atomic propositions, p and q, that the conjunction is composed of, as well as the conjunction itself, p ⋅ q. Each of the following four rows represents a possible scenario regarding the truth of each conjunct, and there are only four possible scenarios: either p and q could both be true (as in row 1), p and q could both be false (as in row 4), p could be true while q is false (row 2), or p could be false while q is true (row 3). The final column (the truth values under the conjunction, p ⋅ q) represents how the truth value of the conjunction depends on the truth value of each conjunct (p and q). As we have seen, a conjunction is true if and only if both conjuncts are true. This is what the truth table represents. Since there is only one row (one possible scenario) in which both p and q are true (i.e., row 1), that is the only circumstance in which the conjunction is true. Since in every other row at least one of the conjuncts is false, the conjunction is false in the remaining three scenarios.

At this point, some students will start to lose a handle on what we are doing with truth tables. Often, this is because one thinks the concept is much more complicated than it actually is. (For some, this may stem, in part, from a math phobia that is triggered by the use of symbolic notation.) But a truth table is actually a very simple idea: it is simply a representation of the meaning of a truth-functional operator. When I say that a conjunction is true only if both conjuncts are true, that is just what the table is representing. There is nothing more to it than that. (Later on in this chapter we will use truth tables to prove whether an argument is valid or invalid. Understanding that will require more subtlety, but what I have so far introduced is not complicated at all.)

There is more than one way to represent conjunctions in English besides the English word “and.” Below are some common English words and phrases that commonly function as truth-functional conjunctions.

but	yet	also	although
however	moreover	nevertheless	still

It is important to point out that many times English conjunctions carry more information than simply that the two propositions are true (which is the only information carried by our symbolic connective, the dot). We can see this with English conjunctions like “but” and “however” which have a contrastive sense. If I were to say, “Bob voted, but Caroline didn’t,” then I am contrasting what

Chapter 2: Formal methods of evaluating arguments

Bob and Caroline did. Nevertheless, I am still asserting two independent propositions. Another kind of information that English conjunctions represent but the dot connective doesn’t is temporal information. For example, in the conjunction:

Bob brushed his teeth and got into bed

There is clearly a temporal implication that Bob brushed his teeth first and then got into bed. It might sound strange to say:

Bob got into bed and brushed his teeth

since this would seem to imply that Bob brushed his teeth while in bed. But each of these conjunctions would be represented in the same way by our dot connective, since the dot connective does not care about the temporal aspects of things. If we were to represent “Bob got into bed” with the capital letter A and “Bob brushed his teeth” with the capital letter B, then both of these propositions would be represented exactly the same, namely, like this:

A ⋅ B

Sometimes a conjunction can be represented in English with just a comma or semicolon, like this:

While Bob vacuumed the floor, Sally washed the dishes.

Bob vacuumed the floor; Sally washed the dishes.

Both of these are conjunctions that are represented in the same way. You should see that both of them have the form, p ⋅ q.

Not every conjunction is a truth-function conjunction. We can see this by considering a proposition like the following:

Maya and Alice are married.

If this were a truth-functional proposition, then we should be able to identify the two, independent propositions involved. But we cannot. What would those propositions be? You might think two propositions would be these:

Chapter 2: Formal methods of evaluating arguments

Maya is married

Alice is married

But that cannot be right since the fact that Maya is married and that Alice is married is not the same as saying that Maya and Alice are married to each other, which is clearly the implication of the original sentence. Furthermore, if you tried to add “to each other” to each proposition, it would no longer make sense:

Maya is married to each other

Alice is married to each other

Perhaps we could say that the two conjuncts are “Maya is married to Alice” and “Alice is married to Maya,” but the truth values of those two conjuncts are not independent of each other since if Maya is married to Alice it must also be true that Alice is married to Maya. In contrast, the following is an example of a truth

functional conjunction:

Maya and Alice are women.

Unlike the previous example, in this case we can clearly identify two propositions whose truth values are independent of each other:

Maya is a woman

Alice is a woman

Whether or not Maya is a woman is an issue that is totally independent of whether Alice is a woman (and vice versa). That is, the fact that Maya is a woman tells us nothing about whether Alice is a woman. In contrast, the fact that Maya is married to Alice implies that Alice is married to Maya. So the way to determine whether or not a conjunction is truth-functional is to ask whether it is formed from two propositions whose truth is independent of each other. If there are two propositions whose truth is independent of each other, then the conjunction is truth-functional; if there are not two propositions whose truth is independent of each other, the conjunction is not truth-functional.

Chapter 2: Formal methods of evaluating arguments

Exercise 8: Identify which of the following sentences are truth-functional conjunctions. If the sentence is a truth-functional conjunction, identify the two conjuncts (by writing them down).

1. Jack and Jill are coworkers.

2. Tom is a fireman and a father.

3. Ringo Starr and John Lennon were bandmates.

4. Lucy loves steak and onion sandwiches.

5. Cameron Dias has had several relationships, although she has never married.

6. Bob and Sally kissed.

7. A person who plays both mandolin and guitar is a multi instrumentalist.

8. No one has ever contracted rabies and lived.

9. Jack and Jill are cowboys.

10.Josiah is Amish; nevertheless, he is also a drug dealer.

11.The Tigers are the best baseball team in the state, but they are not as good as the Yankees.

12.Bob went to the beach to enjoy some rest and relaxation.

13.Lauren isn’t the fastest runner on the team; still, she is fast enough to have made it to the national championship.

14.The ring is beautiful, but expensive.

15.It is sad, but true that many Americans do not know where their next meal will come from.

2.3. Negation and disjunction

In this section we will introduce the second and third truth-functional connectives: negation and disjunction. We will start with negation, since it is the easier of the two to grasp. Negation is the truth-functional operator that switches the truth value of a proposition from false to true or from true to false. For example, if the statement “dogs are mammals” is true (which it is), then we can make that statement false by adding a negation. In English, the negation is most naturally added just before the noun phrase that follows the linking verb

like this:

Dogs are not mammals.

Chapter 2: Formal methods of evaluating arguments

But another way of adding the negation is with the phrase, “it is not the case that” like this:

It is not the case that dogs are mammals.

Either of these English sentences expresses the same proposition, which is simply the negation of the atomic proposition, “dogs are mammals.” Of course, that proposition is false since it is true that dogs are mammals. Just as we can make a true statement false by negating it, we can also make a false statement true by adding a negation. For example, the statement, “Cincinnati is the capital of Ohio” is false. But we can make that statement true by adding a negation:

Cincinnati is not the capital of Ohio

There are many different ways of expressing negations in English. Here are a few ways of expressing the previous proposition in different ways in English:

Cincinnati isn’t the capital of Ohio

It’s not true that Cincinnati is the capital of Ohio

It is not the case that Cincinnati is the capital of Ohio

Each of these English sentences express the same true proposition, which is simply the negation of the atomic proposition, “Cincinnati is the capital of Ohio.” Since that statement is false, its negation is true.

There is one respect in which negation differs from the other three truth functional connectives that we will introduce in this chapter. Unlike the other three, negation does not connect two different propositions. Nonetheless, we call it a truth-functional connective because although it doesn’t actually connect two different propositions, it does change the truth value of propositions in a truth-functional way. That is, if we know the truth value of the proposition we are negating, then we know the truth value of the resulting negated proposition. We can represent this information in the truth table for negation. In the following table, the symbol we will use to represent negation is called the “tilde” (~). (You can find the tilde on the upper left-hand side of your keyboard.)

Chapter 2: Formal methods of evaluating arguments

p	~p
T	F
F	T

This truth table represents the meaning of the truth-functional connective, negation, which is represented by the tilde in our symbolic language. The header row of the table represents some proposition p (which could be any proposition) and the negation of that proposition, ~p. What the table says is simply that if a proposition is true, then the negation of that proposition is false (as in the first row of the table); and if a proposition is false, then the negation of that proposition is true (as in the second row of the table).

As we have seen, it is easy to form sentences in our symbolic language using the tilde. All we have to do is add a tilde to left-hand side of an existing sentence. For example, we could represent the statement “Cincinnati is the capital of Ohio” using the capital letter C, which is called a constant. In propositional logic, a constant is a capital letter that represents an atomic proposition. In that case, we could represent the statement “Cincinnati is not the capital of Ohio” like this:

Likewise, we could represent the statement “Toledo is the capital of Ohio” using the constant T. In that case, we could represent the statement “Toledo is not the capital of Ohio” like this:

We could also create a sentence that is a conjunction of these two negations, like this:

~C ⋅ ~T

Can you figure out what this complex proposition says? (Think about it; you should be able to figure it out given your understanding of the truth-functional connectives, negation and conjunction.) The propositions says (literally): “Cincinnati is not the capital of Ohio and Toledo is not the capital of Ohio.” In later sections we will learn how to form complex propositions using various

Chapter 2: Formal methods of evaluating arguments

combinations of each of the four truth-functional connectives. Before we can do that, however, we need to introduce our next truth-functional connective, disjunction.

The English word that most commonly functions as disjunction is the word “or.” It is also common that the “or” is preceded by an “either” earlier in the sentence, like this:

Either Charlie or Violet tracked mud through the house.

What this sentence asserts is that one or the other (and possibly both) of these individuals tracked mud through the house. Thus, it is composed out of the following two atomic propositions:

Charlie tracked mud through the house

Violet tracked mud through the house

If the fact is that Charlie tracked mud through the house, the statement is true. If the fact is that Violet tracked mud through the house, the statement is also true. This statement is only false if in fact neither Charlie nor Violet tracked mud through the house. This statement would also be true even if it was both Charlie and Violet who tracked mud through the house. Another example of a disjunction that has this same pattern can be seen in the “click it or ticket” campaign of the National Highway Traffic Safety Administration. Think about what the slogan means. What the campaign slogan is saying is:

Either buckle your seatbelt or get a ticket

This is a kind of warning: buckle your seatbelt or you’ll get a ticket. Think about the conditions under which this statement would be true. There are only four different scenarios:

Chapter 2: Formal methods of evaluating arguments

Your seatbelt is buckled	You do not get a ticket	True
Your seatbelt is not buckled	You get a ticket	True
Your seatbelt is buckled	You get a ticket	True
Your seatbelt is not buckled	You do not get a ticket	False

The first and second scenarios (rows 1 and 2) are pretty straightforwardly true, according to the “click it or ticket” statement. But suppose that your seatbelt is buckled, is it still possible to get a ticket (as in the third scenario—row 3)? Of course it is! That is, the statement allows that it could both be true that your seatbelt is buckled and true that you get a ticket. How so? (Think about it for a second and you’ll probably realize the answer.) Suppose that your seatbelt is buckled but your are speeding, or your tail light is out, or you are driving under the influence of alcohol. In any of those cases, you would get a ticket even if you were wearing your seatbelt. So the disjunction, click it or ticket, clearly allows the statement to be true even when both of the disjuncts (the statements that form the disjunction) are true. The only way the disjunction would be shown to be false is if (when pulled over) you were not wearing your seatbelt and yet did not get a ticket. Thus, the only way for the disjunction to be false is when both of the disjuncts are false.

These examples reveal a pattern: a disjunction is a truth-functional statement that is true in every instance except where both of the disjuncts are false. In our symbolic language, the symbol we will use to represent a disjunction is called a “wedge” (v). (You can simply use a lowercase “v” to write the wedge.) Here is the truth table for disjunction:

p	q	p v q
T	T	T
T	F	T
F	T	T
F	F	F

As before, the header of this truth table represents two propositions (first two columns) and their disjunction (last column). The following four rows represent the conditions under which the disjunction is true. As we have seen, the disjunction is true when at least one of its disjuncts is true, including when they are both true (the first three rows). A disjunction is false only if both disjuncts are false (last row).

Chapter 2: Formal methods of evaluating arguments

As we have defined it, the wedge (v) is what is called an “inclusive or.” An inclusive or is a disjunction that is true even when both disjuncts are true. However, sometimes a disjunction clearly implies that the statement is true only if either one or the other of the disjuncts is true, but not both. For example, suppose that you know that Bob placed either first or second in the race because you remember seeing a picture of him in the paper where he was standing on a podium (and you know that only the top two runners in the race get to stand on the podium). Although you can’t remember which place he was, you know that:

Bob placed either first or second in the race.

This is a disjunction that is built out of two different atomic propositions: Bob placed first in the race

Bob placed second in the race

Although it sounds awkward to write it this way in English, we could simply connect each atomic statement with an “or”:

Bob placed first in the race or Bob placed second in the race.

That sentence makes explicit the fact that this statement is a disjunction of two separate statements. However, it is also clear that in this case the disjunction would not be true if all the disjuncts were true, because it is not possible for all the disjuncts to be true, since Bob cannot have placed both first and second. Thus, it is clear in a case such as this, that the “or” is meant as what is called an “exclusive or.” An exclusive or is a disjunction that is true only if one or the other, but not both, of its disjuncts is true. When you believe the best interpretation of a disjunction is as an exclusive or, there are ways to represent that using a combination of the disjunction, conjunction and negation. The reason we interpret the wedge as an inclusive or rather than an exclusive or is that while we can build an exclusive or out of a combination of an inclusive or and other truth-functional connectives (as I’ve just pointed out), there is no way to build an inclusive or out of the exclusive or and other truth-functional connectives. We will see how to represent an exclusive or in section 2.5.

Chapter 2: Formal methods of evaluating arguments

Exercise 9: Translate the following English sentences into our formal language using conjunction (the dot), negation (the tilde), or disjunction (the wedge). Use the suggested constants to stand for the atomic propositions.

1. Either Bob will mop or Tom will mop. (B = Bob will mop; T = Tom will mop)

2. It is not sunny today. (S = it is sunny today)

3. It is not the case that Bob is a burglar. (B = Bob is a burglar) 4. Harry is arriving either tonight or tomorrow night. (A = Harry is arriving tonight; B = Harry is arriving tomorrow night)

5. Gareth does not like his name. (G = Gareth likes his name) 6. Either it will not rain on Monday or it will not rain on Tuesday. (M = It will rain on Monday; T = It will rain on Tuesday)

7. Tom does not like cheesecake. (T = Tom likes cheesecake) 8. Bob would like to have both a large cat and a small dog as a pet. (C = Bob would like to have a large cat as a pet; D = Bob would like to have a small dog as a pet)

9. Bob Saget is not actually very funny. (B = Bob Saget is very funny) 10.Albert Einstein did not believe in God. (A = Albert Einstein believed in God)

2.4 Using parentheses to translate complex sentences

We have seen how to translate certain simple sentences into our symbolic language using the dot, wedge, and tilde. The process of translation starts with determining what the atomic propositions of the sentence are and then using the truth functional connectives to form the compound proposition. Sometimes this will be fairly straightforward and easy to figure out—especially if there is only one truth-functional operator used in the English sentence. However, many sentences will contain more than one truth-functional operator. Here is an example:

Bob will not go to class but will play video games.

What are the atomic propositions contained in this English sentence? Clearly, the sentence is asserting two things:

Chapter 2: Formal methods of evaluating arguments

Bob will not go to class

Bob will play video games

The first statement is not an atomic proposition, since it contains a negation, “not.” But the second statement is atomic since it does not contain any truth functional connectives. So if the first statement is a negation, what is the non negated, atomic statement? It is this:

Bob will go to class

I will use the constant C to represent this atomic proposition and G to represent the proposition, “Bob will play video games.” Now that we have identified our two atomic propositions, how can we build our complex sentence using only those atomic propositions and the truth-functional connectives? Let’s start with the statement “Bob will not go to class.” Since we have defined the constant “C” as “Bob will go to class” then we can easily represent the statement “Bob will not go to class” using a negation, like this:

The original sentence asserts that, but it is also asserts that Bob will play video games. That is, it is asserting both of these statements. That means we will be connecting “~C” with “G” with the dot operator. Since we have already assigned “G” to the statement “Bob will play video games,” the resulting translation should look like this:

~C ⋅ G

Although sometimes we can translate sentences into our symbolic language without the use of parentheses (as we did in the previous example), many times a translation will require the use of parentheses. For example:

Bob will not both go to class and play video games.

Notice that whereas the earlier sentence asserted that Bob will not go to class, this sentence does not. Rather, it asserts that Bob will not do both things (i.e., go to class and play video games), but only one or the other (and possibly neither). That is, this sentence does not tell us for sure that Bob will/won’t go to

Chapter 2: Formal methods of evaluating arguments

class or that he will/won’t play video games, but only that he won’t do both of these things. Using the same translations as before, how would we translate this sentence? It should be clear that we cannot use the same translation as before since these two sentences are not saying the same thing. Thus, we cannot use the translation:

~C ⋅ G

since that translation says for sure that Bob will not go to class and that he will play video games. Thus, our translation must be different. Here is how to translate the sentence:

~(C ⋅ G)

I have here introduced some new symbols, the parentheses. Parentheses are using in formal logic to show groupings. In this case, the parentheses represent that the conjunction, “C ⋅ G,” is grouped together and the negation ranges over that whole conjunction rather than just the first conjuct (as was the case with the previous translation). When using multiple operators, you must learn to distinguish which operator is the main operator. The main operator of a sentence is the one that ranges over (influences) the whole sentence. In this case, the main operator is the negation, since it influences the truth value of all the rest of the sentence. In contrast, in the previous example (~C ⋅ G), the main operator was the conjunction rather than the negation since it influences both parts of sentence (i.e., both the “~C” and the “G”). We can see the need for parentheses in distinguishing these two different translations. Without the use of parentheses, we would have no way to distinguish these two sentences, which clearly have different meanings.

Here is a different example where we must utilize parentheses:

Noelle will either feed the dogs or clean her room, but she will not do the dishes.

Can you tell how many atomic propositions this sentence contains? It contains three atomic propositions which are:

Noelle will feed the dogs (F)

Chapter 2: Formal methods of evaluating arguments

Noelle will clean her room (C)

Noelle will do the dishes (D)

What I’ve written in parentheses to the right of the statement is the constant that I’ll use to represent these atomic statements in my symbolic translation. Notice that the sentence is definitely not asserting that each of these statements is true. Rather, what we have to do is use these atomic propositions to capture the meaning of the original English sentence using only our truth-functional operators. In this sentence we will actually use all three truth-functional operators (disjunction, conjunction, negation). Let’s start with negation, as that one is relatively easy. Given how we have represented the atomic proposition, D, to say that Noelle will not do the dishes is simply the negation of D:

Now consider the first part of the sentence: Noelle will either feed the dogs or clean her room. You should see the “either…or” there and recognize it as a disjunction, which we represent with the wedge, like this:

F v C

Now, how are these two compound propositions, “~D” and “F v C” themselves connected? There is one word in the sentence that tips you off—the “but.” As we saw earlier, “but” is a common way of representing a conjunction in English. Thus, we have to conjoin the disjunction (F v C) and the negation (~D). You might think that we could simply conjoin the two propositions like this:

F v C ⋅ ~D

However, that translation would not be correct, because it is not what we call a well-formed formula. A well-formed formula is a sentence in our symbolic language that has exactly one interpretation or meaning. However, the translation we have given is ambiguous between two different meanings. It could mean that (Noelle will feed the dogs) or (Noelle will clean her room and not do the dishes). That statement would be true if Noelle fed the dogs and also did the dishes. We can represent this possibility symbolically, using parentheses like this:

Chapter 2: Formal methods of evaluating arguments

F v (C ⋅ ~D)

The point of the parentheses is to group the main parts of the sentence together. In this case, we are grouping the “C ⋅ ~D” together and leaving the “F” by itself. The result is that those groupings are connected by a disjunction, which is the main operator of the sentence. In this case, there are only two groupings: “F” on the one hand, and “C ⋅ ~D” on the other hand.

But the original sentence could also mean that (Noelle will feed the dogs or clean her room) and (Noelle will not wash the dishes). In contrast with our earlier interpretation, this interpretation would be false if Noelle fed the dogs and did the dishes, since this interpretation asserts that Noelle will not do the dishes (as part of a conjunction). Here is how we would represent this interpretation symbolically:

(F v C) ⋅ ~D

Notice that this interpretation, unlike the last one, groups the “F v C” together and leaves the “~D” by itself. These two grouping are then connected by a conjunction, which is the main operator of this complex sentence.

The fact that our initial attempt at the translation (without using parentheses) yielded an ambiguous sentence shows the need for parentheses to disambiguate the different possibilities. Since our formal language aims at eliminating all ambiguity, we must choose one of the two groupings as the translation of our original English sentence. So, which grouping accurately captures the original sentence? It is the second translation that accurately captures the meaning of the original English sentence. That sentence clearly asserts that Noelle will not do the dishes and that is what our second translation says. In contrast, the first translation is a sentence that could be true even if Noelle did do the dishes. Given our understanding of the original English sentence, it should not be true under those circumstances since it clearly asserts that Noelle will not do the dishes.

Let’s move to a different example. Consider the sentence:

Either both Bob and Karen are washing the dishes or Sally and Tom are. This sentence contains four atomic propositions:

Chapter 2: Formal methods of evaluating arguments

Bob is washing the dishes (B)

Karen is washing the dishes (K)

Sally is washing the dishes (S)

Tom is washing the dishes (T)

As before, I’ve written the constants than I’ll use to stand for each atomic proposition to the right of each atomic proposition. You can use any letter you’d like when coming up with your own translations, as long as each atomic proposition uses a different capital letter. (I typically try to pick letters that are distinctive of each sentence, such as picking “B” for “Bob”.) So how can we use the truth functional operators to connect these atomic propositions together to yield a sentence that captures the meaning of the original English sentence? Clearly B and K are being grouped together with the conjunction “and” and S and T are also being grouped together with the conjunction “and” as well:

(B ⋅ K)

(S ⋅ T)

Furthermore, the main operator of the sentence is a disjunction, which you should be tipped off to by the phrase “either…or.” Thus, the correct translation of the sentence is:

(B ⋅ K) v (S ⋅ T)

The main operator of this sentence is the disjunction (the wedge). Again, it is the main operator because it groups together the two main sentence groupings.

Let’s finish this section with one final example. Consider the sentence:

Tom will not wash the dishes and will not help prepare dinner; however, he will vacuum the floor or cut the grass.

This sentence contains four atomic propositions:

Chapter 2: Formal methods of evaluating arguments

Tom will wash the dishes (W)

Tom will help prepare dinner (P)

Tom will vacuum the floor (V)

Tom will cut the grass (C)

It is clear from the English (because of the “not”) that we need to negate both W and P. It is also clear from the English (because of the “and”) that W and P are grouped together. Thus, the first part of the translation should be:

(~W ⋅ ~P)

It is also clear that the last part of the sentence (following the semicolon) is a grouping of V and C and that those two propositions are connected by a disjunction (because of the word “or”):

(V v C)

Finally, these two grouping are connected by a conjunction (because of the “however,” which is a word the often functions as a conjunction). Thus, the correct translation of the sentence is:

(~W ⋅ ~P) ⋅ (V v C)

As we have seen in this section, translating sentences from English into our symbolic language is a process that can be captured as a series of steps:

Step 1: Determine what the atomic propositions are.

Step 2: Pick a unique constant to stand for each atomic proposition. Step 3: If the sentence contains more than two atomic propositions, determine which atomic propositions are grouped together and which truth-functional operator connects them.

Step 4: Determine what the main operator of the sentence is (i.e., which truth functional operator connects the groups of atomic statements together).

Step 5: Once your translation is complete, read it back and see if it accurately captures what the original English sentence conveys. If not,

Chapter 2: Formal methods of evaluating arguments

see if another way of grouping the parts together better captures what the original sentence conveys.

Try using these steps to create your own translations of the sentences in exercise 10 below.

Exercise 10: Translate the following English sentences into our symbolic language using any of the three truth functional operators (i.e., conjunction, negation, and disjunction). Use the constants at the end of each sentence to represent the atomic propositions they are obviously meant for. After you have translated the sentence, identify which truth functional connective is the main operator of the sentence. (Note: not every sentence requires parentheses; a sentence requires parentheses only if it contains more than two atomic propositions.)

1. Bob does not know how to fly an airplane or pilot a ship, but he does know how to ride a motorcycle. (A, S, M)

2. Tom does not know how to swim or how to ride a horse. (S, H) 3. Theresa writes poems, not novels. (P, N)

4. Bob does not like Sally or Felicia, but he does like Alice. (S, F, A) 5. Cricket is not widely played in the United States, but both football and baseball are. (C, F, B)

6. Tom and Linda are friends, but Tom and Susan aren’t—although Linda and Susan are. (T, S, L)

7. Lansing is east of Grand Rapids but west of Detroit. (E, W) 8. Either Tom or Linda brought David home after his surgery; but it wasn’t Steve. (T, L, S)

9. Next year, Steve will be living in either Boulder or Flagstaff, but not Phoenix or Denver. (B, F, P, D)

10.Henry VII of England was married to Anne Boleyn and Jane Seymour, but he only executed Anne Boleyn. (A, J, E)

11.Henry VII of England executed either Anne Boleyn and Jane Boleyn or Thomas Cromwell and Thomas More. (A, J, C, M)

12.Children should be seen, but not heard. (S, H)

2.5 “Not both” and “neither nor”

Chapter 2: Formal methods of evaluating arguments

Two common English phrases that can sometimes cause confusion are “not both” and “neither nor.” These two phrases have different meanings and thus are translated with different symbolic logic sentences. Let’s look at an example of each.

Carla will not have both cake and ice cream.

Carla will have neither cake nor ice cream.

The first sentence uses the phrase “not both” and the second “neither nor.” One way of figuring out what a sentence means (and thus how to translate it) is by asking the question: What scenarios does this sentence rule out? Let’s apply this to the “not both” statement (which we first saw back in the beginning of section 2.4). There are four possible scenarios, and the statement would be true in every one except the first scenario:

Carla has cake	Carla has ice cream	False
Carla has cake	Carla does not have ice cream	True
Carla does not have cake	Carla has ice cream	True
Carla does not have cake	Carla does not have ice cream	True

To say that Carla will not have both cake and ice cream allows that she can have one or the other (just not both). It also allows that she can have neither (as in the fourth scenario). So the way to think about the “not both” locution is as a negation of a conjunction, since the conjunction is the only scenario that cannot be true if the statement is true. If we use the constant “C” to represent the atomic sentence, “Carla has cake,” and “I” to represent “Carla has ice cream,” then the resulting symbolic translation would be:

~(C ⋅ I)

Thus, in general, statements of the form “not both p and q” will be translated as the negation of a conjunction:

~(p ⋅ q)

Note that the main operator of the statement is the negation. The negation applies to everything inside the parentheses—i.e., to the conjunction. This is very different from the following sentence (without parentheses):

Chapter 2: Formal methods of evaluating arguments

~p ⋅ q

The main operator of this statement is the conjunction and the left conjunct of the conjunction is a negation. In contrast with the “not both” form, this statement asserts that p is not true, while q is true. For example, using our previous example of Carla and the cake, the sentence

~C ⋅ I

would assert that Carla will not have cake and will have ice cream. This is a very different statement from ~(C ⋅ I) which, as we have seen, allows the possibility that Carla will have cake but not ice cream. Thus, again we see the importance of parentheses in our symbolic language.

Earlier (in section 2.3) we made the distinction between what I called an “exclusive or” and an “inclusive or” and I claimed that although we interpret the wedge (v) as an inclusive or, we can represent the exclusive or symbolically as well. Since we now know how to translate the “not both,” I can show you how to translate a statement that contains an exclusive or. Recall our example:

Bob placed either first or second in the race.

As we saw, this disjunction contains the two disjuncts, “Bob placed first in the race” (F) and “Bob placed second in the race” (S). Using the wedge, we get:

F v S

However, since the wedge is interpreted as an inclusive or, this statement would allow that Bob got both first and second in the race, which is not possible. So we need to be able to say that although Bob placed either first or second, he did not place both first and second. But that is just the “not both” locution. So, to be absolutely clear, we are asserting two things:

Bob placed either first or second.

and

Bob did not place both first and second.

COLLEGE AND UNIVERSITY NOTES

Monday, December 5, 2022

Critical Thinking and Problem Solving

No comments:

Post a Comment