Critical Thinking and Informal Logic

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Critical Thinking Timothy A. Crews-Anderson

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ISBN 978-1-84760-046-2

Critical Thinking and Informal Logic Timothy A. Crews-Anderson

Philosophy Insights. Tirril: Humanities-Ebooks, 2007

Contents A Note on the Author Acknowledgements Introduction: Thinking Skilfully Chapter 1: The Basics 1.1 Propositions 1.2 Arguments and Inferences 1.3 Deductive and Inductive Reasoning 1.4 Form versus Content

Chapter 2: Deductive Reasoning 2.1 Introduction to Deductive Logic 2.2 Syllogisms

Chapter 3: Inductive Reasoning 3.1 Introduction to Inductive Reasoning 3.2 Evaluation of Inductive Arguments 3.3 Arguments from Analogy 3.4 Statistical Reasoning

Chapter 4: Rhetorical Devices and Informal Fallacies 4.1 Rhetorical Devices 4.2 Informal Fallacies

Chapter 5: Complex Arguments 5.1 Analysis versus Interpretation 5.2 Evaluating Complex Arguments

Chapter 6: Bringing It All Together 6.1 Good Habits 6.2 Composing an Argument 6.3 Information Literacy

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Appendix A: Categorical Logic A.1 Categorical Propositions A.2 The Square of Opposition and Immediate Inferences A.3 Categorical Syllogisms

Appendix B: Resources B.1 Textbooks B.2 Other Resources

A Note on the Author Tim Crews-Anderson took his BA at the Florida State University and his MA at Georgia State University, where he has also taught as a Visiting Instructor. He is currently pursuing his PhD at the University of Illinois at Chicago.

Acknowledgements I would like first to thank Dr. Sandra Dwyer both for her guidance and for the many opportunities that she has made available to me. Additionally, I would like to extend my gratitude to Dr. George Rainbolt and Dr. Melissa Merritt. Finally, I would like to thank Victoria Anne Crews-Anderson, my wife, for her tireless support and encouragement.

Introduction: Thinking Skilfully Try for a moment to imagine a field of human endeavour that does not require reasoning. This is likely to be impossible as there is little that human beings do that does not involve our ability to think. Many philosophers over the years have postulated that if there is anything essential to the human being, it is the capacity for reason. Indeed, thought is so fundamental to the human experience that it is only rarely that one is without it. The operations of the mind come so naturally and so easily that one scarcely realizes that they are going on. It is perhaps for this reason that the notion of thinking skilfully may seem strange.

The goal of this book is to consider thought as an activity, as an act or series of acts that a person deliberately, intentionally and wilfully undertakes. In simpler terms, the present purpose is to think about thinking. Close attention will be paid to the process of thought with the aim of evaluating reasoning and cultivating good thinking skills. This project falls within the confines of logic, which is the branch of philosophy that studies the reasoning process and seeks to understand the differences between good and bad reasoning.

Human beings naturally reason well when they take the time to do so, and if their attention is properly directed, they are, in the vast majority of cases, capable of great insight. If there is one fundamental difference between a person who is an incisive critical thinker and one who is not, it is that the critical thinker takes the process of thinking seriously, consciously attends to that process and asks the right questions. The focus, therefore, will be on the concerns that one should keep in the forefront of the mind and on which questions should be asked. It is also worth noting that this book does not introduce much that the average person does not already do. Everyone has carried out all of the described reasoning activities countless times. What the book offers is a careful cataloguing of the various types of reasoning with a discussion of which of them are reliable and which are not. It is, in a sense, a guided tour of the human capacity for reasoning along with instructions for its use.

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With most human activities the development of skills does not come without practice, and it is no different with thinking. It is probably unreasonable to expect them to come easily, but it is almost certain that time and effort will pay a hefty dividend.

Considering the length of this book, its scope is perhaps somewhat larger than it should be. The idea is to present the basics of critical thinking and informal logic and to point the way towards further study, so it is perhaps best to consider the book as a primer. It will provide a solid introduction to the fundamental concepts and considerations as well as links to other resources on the web and in print.

Chapter 1: The Basics Before delving into definitions of the basic elements of critical thinking, a brief overview of the structure of these elements will perhaps be helpful. The discussion offered in this book will focus primarily on the argument, which itself consists of at least two propositions. All arguments are either deductive or inductive, and an understanding of this distinction is required for criticism. The success of deductive arguments is evaluated in terms of what philosophers call validity and soundness, while inductive arguments are rated along a spectrum from weak to strong. Many of these words are familiar to most, but it is likely that they are associated with a number of different possible meanings. To remove these potential ambiguities, precise definitions will be introduced. The importance of a thorough familiarity with these concepts cannot be overstated.

1.1 Propositions The precise nature of propositions is a matter of some philosophical debate, but for present purposes, it will suffice to define a proposition as a claim or assertion that affirms or denies that something is the case. All propositions are either true or false, and no proposition can be both true and false. Furthermore, they are the only sort of thing that can properly be called true or false. Put simply, propositions are the sole bearers of truth and falsehood, and as will become clear shortly, this feature is of crucial importance for identifying them in ordinary language. Here are some examples of propositions.

All triangles have three sides.

Either George W. Bush won the U.S. election, or John Kerry won it.

People ought not to lie.

There is some debate among philosophers about the nature of â&#x20AC;&#x153;oughtâ&#x20AC;? claims. Some deny that such statements are propositions, claiming rather that they are either non-propositional commands or expressions of emotional attitudes. Click here for an overview of the work of A. J. Ayer, a proponent of this view.

Critical Thinking 11 If today is Wednesday, then tomorrow is Thursday.

All circles are squares.

The majority of propositions that one encounters come in the form of a declarative sentence, but it is important to note that a proposition is not identical to the sentence that expresses it. A proposition is that to which a declarative sentence refers. For this reason, multiple sentences may express or refer to the same proposition.

George W. Bush won the U.S. election.

The U.S. election was won by George W. Bush.

George W. Bush was the winner of the U.S. election.

It is perhaps helpful to think of a declarative sentence as pointing to a kind of abstract object. It is this sort of object that philosophers have termed a proposition.

Non-propositional language: exclamations, commands, and questions There are many uses of language that do not express propositions. As noted above, all propositions are either true or false, so whether a particular phrase or sentence can be considered to be true or false will determine if it indeed expresses a proposition. Consider the following.

What are we doing here?

To decide whether this question expresses a proposition, simply ask whether it makes sense to say that, “what are we doing here?” is either true or false. Clearly it does not, and just as clearly this sentence does not express a proposition.

Another tactic is to add, “It is the case that…” in front of the phrase in question and consider whether this new construction makes sense. If it does, then it is very likely that the phrase expresses a proposition. If it does not, then it is likely a non-propositional use of language. Consider.

It is the case that, “what are we doing here.”

This is, of course, nonsensical. By contrast, consider the following propositions.

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It is the case that, “all triangles have three sides.”

It is the case that, “either George W. Bush won the U.S. election, or John Kerry won it.”

It is the case that, “people ought not to lie.”

It is the case that, “if today is Wednesday, then tomorrow is Thursday.”

The concern here is not whether the constructed sentence is true or false but whether it makes sense. Consider a false proposition.

It is the case that, “all circles are squares.”

Obviously this claim is false, but it does make sense, and thus the phrase “all circles are squares” expresses a proposition.

Other non-propositional uses of language include commands and exclamations.

Go to bed.

Oh, no!

A quick consideration of each of these constructions reveals that they are neither true nor false and thus do not express propositions. Note that the following do not make sense.

It is the case that, “go to bed.”

It is the case that, “oh, no!”

This provides the basis for a general rule. Questions, commands, and exclamations do not express propositions. They are non-propositional uses of language.

Questions of interpretation: rhetorical questions as assertions and commands as normative claims It should not come as a surprise that there are exceptions to this rule. In some cases certain questions or commands can be interpreted as actually expressing propositional language.

A rhetorical question is a question that is meant to have an obvious answer and can be interpreted and reformulated as a declarative sentence that expresses a proposition. Consider the meaning of the question “who won the 2007 election?” in the following.

Critical Thinking 13 Marie: I think it is clear that France is moving to the right politically.

Thomas: I’m not so sure.

Marie: And who won the 2007 election?

Thomas: It’s a good point, but I am still not entirely convinced.

In this dialogue, Marie is not really wondering who won the French election in 2007. She is, rather, asking a question that is entirely rhetorical. The meaning of this phrase should be interpreted as “Nicolas Sarkozy won the 2007 French election,” which is clearly a propositional use of language.

Like rhetorical questions, it is sometimes possible to interpret and reformulate a command as a proposition. Consider.

Don’t lie.

In some contexts, this command is simply a non-propositional command that is neither true nor false. In others, however, it can be interpreted as a normative claim that contains “ought” or “should.” If this is the case, then it can be reformulated as one of the following declarative sentences, both of which express a proposition.

You should not lie.

You ought not to lie.

Unfortunately, the correct interpretation of language is seldom a straightforward task. It requires constant and careful thought about what the author or speaker actually means. Rhetorical questions and commands that express normative claims are fairly common, and when critically analyzing an argument, it is important to recognize them and to reformulate them as clear and concise declarative sentences.

Simple and compound propositions A simple proposition makes only one claim or assertion, and a compound proposition contains two or more simple propositions. There are three distinct types of compound propositions, and it is important both to recognize them and to understand the conditions under which they are true or false (truth conditions).

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Conjunctive propositions The first type is a conjunctive proposition, which consists of two or more simple propositions that are connected by the conjunctions “and” or “but.”

While these two words carry somewhat different connotations, they have the same truth value, so for present purposes, they will be treated as synonyms. Here are some examples of conjunctive propositions.

Nicolas Sarkozy won the French election, and Ségolène Royal lost.

All triangles have three sides, and all circles are squares, but all squares have four sides.

A conjunctive proposition asserts each of its simple parts, so such propositions are true just in case all of the simple propositions that it contains are themselves true. If one or more of the constitutive simple propositions is false, then the entire conjunctive proposition is false. The first example above is true because both simple propositions are, in fact, true claims. With respect to the second, while it is the case that all triangles have three sides and that all squares have four, it is not the case that all circles are squares. Thus, the second example is false.

Disjunctive propositions The second type of compound proposition is a disjunctive proposition. Such propositions contain two or more propositions that are connected by “or.” There are actually two possible meanings of the word “or,” and for reasons that will become clear later, it is important to carefully distinguish between the two. The “inclusive or” means “one or the other or both” while the “exclusive or” means “one or the other but not both.” In this book, the inclusive meaning will be assumed, and any use of the exclusive will be noted. Following are some examples of disjunctive propositions.

Either George W. Bush won the US election, or John Kerry won it.

Either George W. Bush is the US President, or Nicolas Sarkozy is the French President.

All circles are squares, or all triangles have three sides.

A disjunctive proposition is false if and only if all of the constitutive simple propositions are false. If one or more of them are true, then the entire disjunctive proposition is true. Consider now an implication of these truth conditions.

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Assume that the following disjunctive proposition is true.

All gadgets are widgets, or all widgets are sprockets.

Based merely on the knowledge that this proposition is true, is it possible to infer that either of the simple propositions is true? Indeed, while it can be said for sure that one of them is true, it is impossible to say for sure which one. There are several possibilities.

“All gadgets are widgets” is true, and “all widgets are sprockets” is false.

“All gadgets are widgets” is false, and “all widgets are sprockets” is true.

“All gadgets are widgets” is true, and “all widgets are sprockets” is true.

This example illustrates the fact that unlike a conjunctive proposition, a disjunctive proposition does not, in fact, assert each of its constitutive propositions; rather, it asserts only that at least one of them is true without revealing which one.

Hypothetical propositions The third and final type of proposition is the hypothetical. A hypothetical proposition consists of two propositions connected by “if…then.”

If you look into the abyss, then the abyss will change you.

A couple of terms are helpful for discussing hypothetical propositions. The antecedent is the constitutive proposition that is governed by “if.” The consequent is the constitutive proposition that is governed by “then.” A hypothetical proposition is false if and only if its antecedent is true and its consequent is false. Otherwise, it is true. These conditions can generate some strange results.

Consider the following true propositions.

If the sun revolves around the earth, then the sum of two and two is four.

If the sun revolves around the earth, then the sum of two and two is five.

If the earth revolves around the sun, then the sum of two and two is four.

In none of these examples is it the case that the antecedent is true and the consequent false, thus all of the examples are true. This is straightforward enough with respect to the third example as it is indeed the case that “the earth revolves

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around the sun” and that “the sum of two and two is four,” but it may seem somewhat strange to say that the first two examples are true. While counterintuitive, a hypothetical proposition with a false antecedent is always true (albeit trivially so) regardless of the truth or falsehood of its consequent.

One might wonder why this is so, and this is indeed a good question. It may help to keep in mind the precise assertion that hypothetical propositions actually make. Unlike conjunctive and disjunctive propositions, they do not make a claim about the truth of either or both of their constitutive propositions. They do, however, affirm a relationship of some sort between them. The precise nature of this relationship depends greatly on the context of the proposition, but in general terms, it can be said that a hypothetical proposition makes the claim that the truth of the antecedent is sufficient for the truth of the consequent.

With this in mind consider the structure of a false hypothetical proposition.

Since the antecedent is true and consequent is false, the claim that the truth of the antecedent suffices for the truth of the consequent is demonstrated to be false.

Moreover, since the claim being made about the consequent depends entirely on the antecedent being true, a false antecedent has no bearing whatsoever on the truth of the consequent. Thus, a false antecedent means a true hypothetical proposition regardless of the truth or falsehood of the consequent.

On a final note, considering the above examples, it may seem strange to think that the behaviour of the sun and earth is, in any way, sufficient for the truth of basic arithmetic. A likely source of this discomfort is the assumption that the relationship being claimed by these propositions is one of cause and effect.

While this is certainly one possible meaning of the affirmed relationship, it is not the only one. For this reason it is not safe to assume the precise nature of the relationship without carefully considering the context in which the proposition occurs. The only universal way to show that any hypothetical proposition is false is to show that the antecedent is true while the consequent is false.

These are the three basic types of compound propositions. Things can, however, be quite a bit more complex. The constitutive propositions of a compound proposition can themselves be compound. This can make for propositions that are quite monstrous for lack of a better word. Here are some examples.

Genghis Kahn was Mongolian, but if Julius Caesar was Roman, then Napoleon Bonaparte was German, and Alexander the Great was Greek.

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If there is a God, and He is all-powerful, all-loving, and all-knowing, then there should not be pain and suffering, and if there is pain and suffering, then God does not exist, or is not all-powerful, all-loving, and all-knowing.

Determining the truth values of complicated propositions requires specific rules of precedence similar to the order of operations in arithmetic. This procedure requires precise interpretation and symbolization, and for this reason, it will not be introduced here. For those who are interested, any of the introductory texts to symbolic logic mentioned in Appendix B will explicitly deal with this question.

Considering propositions Only rarely can philosophy, logic or critical thinking skills help someone determine whether a proposition is actually true or false. The sorts of propositions that are commonly encountered in the course of daily life will most likely require evidence that cannot be supplied by a philosopher. That being said, it is crucial to thinking skilfully to consider carefully what it would take to prove that a given proposition is true. What burden must be met if a compelling case is to be made, and how might someone meet that burden? Consider a claim that one can easily imagine hearing on the evening news.

Divorce rates in the United States have decreased substantially over the last three years.

Think for a moment about what kind of information would be necessary to support a claim like this. For this particular proposition, demographic studies and reliable statistics would be required. Other propositions will call for different kinds of support. Here are some more examples.

All triangles have three sides.

God exists.

U.S. President Harry Truman used atomic weapons on Japanese cities in order to frighten the Soviets.

The first example requires nothing more than a proper understanding of the definition of a triangle, thus little more needs to be said to show that it is true. The second example, however, is far more problematic as it is very likely impossi-

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ble to know for sure whether it is true. Indeed, there are many propositions for which this is the case. The third is also a difficult claim to prove. What kinds of evidence and information would be required to support it? Clearly, it would be extremely difficult to remove all doubt, but it might very well be possible to offer strong supporting evidence for or against it. Such evidence would likely include historical documents, eyewitness accounts of Truman’s decision-making process, etc.

Coming to at least a cursory understanding of the burden of proof that a proponent of a claim must meet is absolutely fundamental to a good critical analysis. As the discussion moves from propositions to arguments, it will be very helpful to keep this in mind.

For the sake of concision, propositions will often be represented as capital letters throughout this text. P is the default designation for a proposition, Q is a proposition that is not P, R is a proposition that is neither P nor Q, and so on. To express the negation of a proposition, a “not” will simply be placed in front of the letter.

If “P” is “the cat is on the mat,” then “Not P” is simply “the cat is not on the mat” or “it is not the case that the cat is on the mat.”

Keep in mind that the negation of a denial is a possibility.

If P is “people ought not to lie,” then “Not P” would be “it is not the case that people ought not to lie” or simply “people ought to lie.”

Finally, the negation of a particular proposition always bears the opposite truth value of that proposition, so if “P” is true, then “Not P” is false and vice versa.

1.2 Arguments and Inferences The English word “argument” has many meanings, and when most people hear it, they think of a (perhaps heated) disagreement. It is very likely that they do not immediately think of philosophy or critical thinking. Perhaps they should however, because the argument, or rather a particular kind of argument, is the philosopher’s bread and butter. In philosophy, an argument is a group of two or more propositions that express an inference. An inference, in turn, is a mental

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process of linking propositions by offering support to one proposition on the basis of one more other propositions. The conclusion of an argument is that single proposition which is supported by other propositions, and a premise is a proposition that provides a basis of support for the conclusion. Consider the following argument.

P1: If you look into the abyss, then the abyss will change you.

P2: The abyss did not change you.

C: Therefore, you did not look in to the abyss.

In this example, P1 and P2 are premises, and C is the conclusion. All three propositions taken together are the argument. The inference, however, does not lie here on the screen. It is a mental activity, and as such, it occurs in the mind. Indeed, the inference is an instance of the reasoning activity for which human beings are so famous. Recognizing the differences between good inferences, those in which the premises provide adequate support for the conclusion, and bad ones, those in which the premises are inadequate to this task, is the essence of critical thinking and the primary focus of logic.

1.3 Deductive and Inductive Reasoning A deductive argument is one in which it is claimed that the premises provide a guarantee of the truth of the conclusion. In a deductive argument, the premises are intended to provide support for the conclusion that is so certain that, if the premises are true, it would be impossible for the conclusion to be false. Because successful deductive arguments are those in which the conclusion is completely guaranteed by the premises, the conclusion must be contained within the premises. The conclusion cannot go beyond what the premises implicitly assert. For this reason, deductive arguments are usually found in inferences that follow from definitions, mathematics and rules of formal logic. The following are examples of deductive arguments.

There are 32 books on the top-shelf of the bookcase, and 12 on the lower shelf of the bookcase. There are no books anywhere else in the bookcase. Therefore, there are 44 books in the bookcase.

Melbourne is either in Victoria or New South Wales. If Melbourne is in Victoria, then

Critical Thinkingâ&#x20AC;&#x192; 20 Melbourne is in Australia, and if Melbourne is in New South Wales, then Melbourne is in Australia. Therefore, Melbourne is in Australia.

An inductive argument is an argument in which it is claimed that the premises provide reasons supporting the probable truth of the conclusion. In an inductive argument, the premises are intended only to be so strong that, if they are true, then it is unlikely that the conclusion is false. They can appeal to any consideration that might be thought relevant to the probability of the truth of the conclusion. Inductive arguments, therefore, can take very wide ranging forms, including arguments dealing with statistical data, generalisations from past experience, appeals to signs, evidence or authority, and causal relationships.

Please note that while causal reasoning is a complex and important part of any robust discussion of induction, there is simply no way to do it justice within the scope of this book. Click here for an overview of Millâ&#x20AC;&#x2122;s methods for identifying causal relationships. The following are examples of inductive arguments.

On every single morning in human history the sun has appeared to rise. Therefore, the sun will appear to rise tomorrow morning.

It has snowed in Massachusetts every December in recorded history. Therefore, it will snow in Massachusetts this coming December.

The real difference between the deductive and inductive arguments comes from the sort of relation the author or expositor of the argument takes there to be between the premises and the conclusion. If the author of the argument believes that the truth of the premises definitely establishes the truth of the conclusion due to definition, logical entailment or mathematical necessity, then the argument is deductive. If the author of the argument does not think that the truth of the premises definitely establishes the truth of the conclusion, but nonetheless believes that their truth provides good reason to believe that the conclusion is true, then the argument is inductive. It is usually fairly easy to decide whether an argument is deductive or inductive, but because it requires insight into what the arguer is thinking, it can occasionally be difficult or controversial.

1.4 Form versus Content Put simply, the form of an argument is its logical structure or the manner in

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which the premises offer support for the conclusion. Since the form describes the relationship between the premises and the conclusion, it cannot be true or false (remember that only propositions can be true or false). The content of an argument is the group of actual propositions that comprise the argument. It is with respect to content alone that one may consider truth and falsehood. Consider the following arguments.

If the sun shines, then the tarmac will be hot.

The sun is shining.

Therefore, the tarmac is hot.

If John runs home, then Jane will see him.

John ran home.

Therefore, Jane saw him.

If elephants can fly, then rocks can float in water.

Elephants can fly.

Therefore, rocks can float in water.

If P, then Q. P. Therefore, Q

Notice what is similar and what is different about these four arguments. What is similar about these arguments is their form. What is different about these arguments is their content. In the last example, the actual propositions have been replaced with capital letters. This makes the point that it is possible to consider the form of argument without any reference to actual propositions or content.

This is important because the form of an argument is the most direct expression of the inference. Generally speaking, the study of logic primarily concerns itself with evaluations of the form of arguments and the reasoning process that they involve.

Chapter 2: Deductive Reasoning 2.1 Introduction to Deductive Logic Validity Recall that a deductive argument is one in which it is claimed that the premises guarantee the conclusion with absolute certainty. Consider that there are only two possible outcomes of such a claim. Either the claim is successful, in which case the premises do indeed guarantee the conclusion, or the claim is a failure, in which case the premises do not succeed in guaranteeing the conclusion.

A deductive argument in which the claim that the premises guarantee the conclusion succeeds is said to be valid. If all of the premises of a valid argument are true, then it is impossible for the conclusion to be false. Thus, a valid argument is a deductive argument such that if its premises are true, then the inference to which the argument refers necessitates that the conclusion must also be true. A deductive argument in which the premises do not guarantee the conclusion is invalid. In such an argument, the inference that the argument expresses fails to necessitate the conclusion. All deductive arguments are either valid or invalid, and none can be both. Since the question of whether or not a deductive argument is valid or invalid is a question that concerns only the form of the argument and not the content, it is extremely important to keep the form-content distinction in the forefront of the mind. Consider the following examples.

A. All plants are living things.

All trees are plants.

Therefore, all trees are living things.

B. All x’s are z’s.

All y’s are x’s.

Therefore, all y’s are z’s

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C. All trees are plants.

All trees are living things.

Therefore, all living things are plants.

D. All x’s are y’s.

All x’s are z’s.

Therefore, all z’s are y’s.

Closely consider arguments A and C. At first glance, they may look similar, but there are subtle and crucial differences. It should be clear that in A, the premises succeed in guaranteeing the conclusion and that the inference to which the argument refers is valid. While in C, the premises fail to guarantee the conclusion.

Notice that B expresses the form of A without reference to any content, and D expresses the form of C also without reference to any content.

B represents a valid argument form, and any argument with this form is always valid regardless of what content may actually be in a particular argument and regardless of whether that content ends up being true or false. C, on the other hand, expresses an invalid argument form, and any argument structured in this way is invalid irrespective of the truth content of the constitutive propositions.

It is of fundamental importance to ignore the truth or falsehood of the constitutive propositions of an argument until the validity of its form is established.

It is entirely possible for an invalid argument to have true premises and a true conclusion. Consider this example.

All trees are plants.

All trees are living things.

Therefore, all plants are living things.

The real question is whether it is possible to infer the truth of the conclusion from the truth of the premises, and the actual truth or falsehood of the premises and conclusion does nothing to resolve this issue. If one were to consider only the truth of the propositions that comprise this argument, one could easily be lead astray. While all of these propositions are indeed true, it is not possible to infer

These examples are categorical syllogisms. An overview of determining the validity of this sort of argument is available in Appendix A: Categorical Logic.

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the conclusion merely from these premises. If oneâ&#x20AC;&#x2122;s knowledge is limited only to what the premises assert, then an object that is a plant but neither a tree nor a living thing is possible. It might be tempting to suggest that any plant would be a living thing, and as a matter of fact such a suggestion would actually be true, but remember that this fact is not in the premises. The question, again, is whether the conclusion can be inferred from these premises and these premises alone.

The existence of an object that is a plant and not a living thing would allow the two propositions that constitute the premises of this argument to be true while making the conclusion false, and that is all that is necessary to show that one can not infer this conclusion from these premises.

As mentioned above, the general purpose of logic as a branch of philosophy is the evaluation of reasoning. Deductive logic, however, specifically deals with the determination of the validity of deductive arguments. There are a number of different systems for making these determinations, including sentential or propositional logic, predicate logic and modal logic. These are, of course, well beyond the present scope, so only a brief discussion will be possible here. For those who are interested in the basics of a deductive system, please find an overview of Categorical Logic in Appendix A. Counterexamples The most direct way to make a determination of the validity of an argument is simply to pay close attention to the inference that an argument expresses. The basic human capacity for reasoning will often suffice for at least the relatively simple argument forms. The basic test for validity is as follows.

1. Ignore the actual truth or falsehood of the premises and conclusion of the argument.

2. Assume for the sake of testing, that all of the premises are true.

3. Consider whether the inference that is expressed by the argument necessitates that the conclusion must be true. If so, the argument is valid. If not, the argument is invalid.

Another method is to produce what logicians call a counterexample, which is an interpretation, instance, object or scenario in which the premises of an argument are true and the conclusion is false. From the definition of validity, it should be clear that if there is a counterexample to a particular argument, then that argument is invalid. Remember that a valid argument is a deductive argu-

Critical Thinking 25

ment such that if its premises are true, then its conclusion must be true. So, if one can show that it is possible for the premises of an argument to be true while its conclusion is nonetheless false, then one has proven that said argument is not valid. Consider the invalid argument C from above.

C. All trees are plants.

All trees are living things.

Therefore, all living things are plants.

To identify a counterexample for this argument, simply identify a scenario or object that allows the premises to be true while making the conclusion false.

Obviously, any object that is a living thing and not a plant, an animal for instance, would serve as a counterexample to the argument.

Even without any particular content, it is possible to identify a counterexample for an argument like this. Consider example D from above, which is of course merely an expression of the form of the argument C. D. All x’s are y’s.

All x’s are z’s.

Therefore, all z’s are y’s.

To identify a counterexample, simply imagine an object that allows the premises to be true but makes the conclusion false. The best place to start is with the conclusion. What sort of object would make the claim that “all z’s are y’s” false?

The obvious answer is an object (call it “o”) that is a z but not a y. Let o be an object that is a z but is neither an x nor a y. Since o is not an x and also not a y, it can still be true that, “all x’s are y’s.” Although o is not an x, o can still be a z. This is because while it is true that “all x’s are z’s,” it is possible that there are some z’s that are not x’s (nowhere does it say that all z’s are x’s). Thus, o is a z but need not be a y. This is an interpretation of the argument in which the premises are true and the conclusion false, and the argument form is proven to be invalid.

While it is the case that the presentation of a counterexample will indeed show unequivocally that an argument is invalid, the fact that one is unable to come up with such a scenario does not suffice to show that an argument is valid.

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This is because there are two possible reasons for such a failure. The first, of course, is the possibility that the argument is valid. A valid argument has no counterexample. The second possibility, however, is that there is a counterexample, but no one has yet been able to identify it. From the fact that there is no known counterexample, one can not justify a claim that an argument is valid.

Proving an argument to be valid requires one of the aforementioned systems of logic. Fortunately for the relatively simple argument forms that one most commonly encounters, careful attention to the inference being expressed and the basic human capacity to reason will suffice to determine whether an argument is valid.

Soundness Once the task of determining that a particular argument is valid has been accomplished, attention may be turned to whether the propositions that comprise its content are actually true or false. If it is determined that a deductive argument is invalid, then the argument is unsound, and no further analysis is necessary.

An argument with an invalid form fails regardless of whether the premises and conclusion are actually true.

An argument is sound just in case it is deductive, valid and contains only premises that are true. If it turns out that a valid deductive argument has at least one false premise, then it is unsound. Whether an argument is sound is a question that concerns both its form and content, and it is with respect to soundness alone that the truth and falsehood of particular propositions need be considered.

Consider some examples.

A. All plants are living things.

All trees are plants.

Therefore, all trees are living things.

B. All plants are rocks.

All lions are plants.

Therefore, all lions are rocks.

C. All xâ&#x20AC;&#x2122;s are yâ&#x20AC;&#x2122;s.

Critical Thinking 27 All z’s are x’s.

Therefore, all z’s are y’s

It should be clear that both A and B are arguments of the same form (C) and that this form is valid. The premises in A are clearly true, thus A is sound. The premises in B, however, are clearly false, and so B is unsound.

2.2 Syllogisms Many of the arguments that one will encounter are what logicians call syllogisms, so a discussion of four of the most common syllogistic forms will be valuable. In general, a syllogism is a deductive argument that contains exactly two premises.

Disjunctive syllogisms A disjunctive syllogism is a deductive argument with exactly two premises, one of which is a disjunctive proposition. Here is an example.

A. Either Mary is with her mother, or she is with a doctor.

Mary is not with a doctor.

Therefore, Mary is with her mother.

As should be clear, this argument is valid. Now consider the following.

B. Either Mary is with her mother, or she is with a doctor.

Mary is with a doctor.

Therefore, Mary is not with her mother.

While this argument might appear valid at first glance, it is not, and here is why.

Remember that by assumption, “or” should be defined as inclusive, thus the first premise in B is true if any of the following three possibilities obtain. (1) Mary is with her mother, (2) Mary is with a doctor, or (3) Mary is with her mother and with a doctor. It is entirely possible that Mary is with two people or that Mary’s mother is a doctor. This third option is a counterexample to B. Assuming the inclusive definition of “or”, a disjunctive syllogism is valid if

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and only if it has one of these two forms.

P or Q Not P Therefore, Q P or Q Not Q Therefore, P

It is invalid if it has one of these.

P or Q P Therefore, not Q P or Q Q Therefore, not P

If the exclusive â&#x20AC;&#x153;orâ&#x20AC;? is assumed, then, of course, all four forms are valid.

Pure hypothetical syllogisms A pure hypothetical syllogism is a deductive argument in which both of the premises and the conclusion are hypothetical propositions. Here is an example.

If the cat is on the mat, then the dog is on the rug.

If the dog is on the rug, then the bird is in the cage.

Therefore, if the cat is on the mat, then the bird is in the cage.

A pure hypothetical syllogism is valid if it has the following form.

If P, then Q. If Q, then R. Therefore, If P, then R

It is worth noting that this sort of syllogism may be trivially sound. With the truth conditions of the hypothetical proposition in mind, consider that if P, Q and R are all false, the argument is sound while saying nothing about the locations of the cat, the dog, or the bird.

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Modus ponens, modus tollens, the fallacy of denying the antecedent and the fallacy of affirming the consequent Another type of hypothetical syllogism, which is sometimes called a mixed hypothetical syllogism, is a deductive argument with exactly two premises, only one of which is a hypothetical proposition. This is the most frequently encountered type of argument, so much so, that the four most common forms of mixed hypothetical syllogisms have been given names. The valid forms are modus ponens and modus tollens, and the invalid forms are the fallacy of denying the antecedent and the fallacy of affirming the consequent.

Modus ponens, which means “the affirming mode,” is expressed by the following.

If P, then Q P Therefore, Q

Modus tollens¸ or “the denying mode,” is an argument with the following form.

If P, then Q Not Q Therefore, not P

Any argument in the form of modus ponens or modus tollens is always valid.

While this may seem simple enough, it is very easy to confuse these valid forms with invalid forms that are very similar. Consider now the fallacious mixed hypothetical syllogisms.

An argument commits the fallacy of denying the antecedent and is always invalid if it has this form.

If P, then Q Not P Therefore not Q

An argument commits the similar fallacy of affirming the consequent and is likewise invalid if it has the following form.

Critical Thinkingâ&#x20AC;&#x192; 30 If P, then Q Q Therefore, P

To make this a bit clearer, it will be helpful to consider some examples. Here is instance of modus ponens.

If Mary is visiting [P], then there are eggs and toast for dinner [Q].

Mary is visiting [P].

Therefore, there are eggs and toast for dinner [Q].

Since this argument is valid, there is, of course, no counterexample to it. As mentioned above, this lack of a counterexample does not suffice to show the argument to be valid. Without a comprehensive system of deduction, the only way to show that it is valid is to identify it as an example of an argument in the form modus ponens, but it will be helpful to step through the reasoning process.

To test for validity, first assume for the sake of testing that all of the premises are true. Second, consider under what conditions the premises can actually be true. Recall that a hypothetical proposition is false if and only if the antecedent is true and the consequent false; otherwise, it is true. Since it is assumed that the second premise is true, it can be assumed that P is true. If it is the case that P is true, the only interpretation that would allow the first premise to be true is if Q is also true. Thus, if the premises are true, the conclusion must also be true, and the argument form is valid.

Here is a very similar argument in modus tollens.

If Mary is visiting [P], then there are eggs and toast for dinner [Q].

There are not eggs and toast for dinner [Not Q].

Therefore, Mary is not visiting [Not P].

Again this argument form is valid, so there is no counterexample, but here is the reasoning process. Assuming that the premises are true, from the second premise it is established from the fact that Not Q is true that Q is false. Since Q is false, the only way for the first premise to be true is if P is also false. If P were true, then the first premise would be false. Since P is false, it must be the case that Not P (the conclusion) is true, and the argument form is valid.

Now consider the two invalid forms. Here is the fallacy of denying the

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antecedent.

If Mary is visiting [P], then there are eggs and toast for dinner [Q].

Mary is not visiting [Not P].

Therefore, there are not eggs and toast for dinner [Not Q].

It is not too difficult to identify a counterexample for this example. While Mary is not visiting, someone else decided to make eggs and toast for dinner. Assuming that the premises are both true, it is known from the second premise that Not P is true, and thus that P is false. Since P is false, the hypothetical first premise is true regardless of whether Q is true or false. Hence, it cannot be inferred that Not Q is true merely from these premises, and the argument form is proven to be invalid.

Here is an instance of the fallacy of affirming the consequent.

If Mary is visiting [P], then there are eggs and toast for dinner [Q].

There are eggs and toast for dinner [Q].

Therefore, Mary is visiting [P].

The same counterexample that was identified for the fallacy of denying the antecedent will serve as such for this argument as well. Someone else could have decided to make eggs and toast for dinner. Supposing again that the premises are true, it is affirmed that Q is true, but given this fact the first premise is true regardless of whether P is true or false. If P is false, then both premises are true while the conclusion is false, and the argument form is shown to be invalid.

Hopefully, these examples have served to illuminate the basic of deductive reasoning. From the standpoint of critical thinking, it is always a good practice to consider the validity of a deductive argument before considering the truth of the premises. Determining the truth of the content of an argument is often a somewhat difficult and time-consuming undertaking if it is possible at all. Furthermore, even if this truth can be established, it can turn out to be deceptive if the inference is flawed (See Chapter 5: Rhetorical Devices and Informal Fallacies). With practice determining the validity of the commonly encountered deductive argumentative forms is usually quick and easy. Once it has been determined that a particular deductive argumentâ&#x20AC;&#x2122;s form is invalid, it has been conclusively refuted, and no further effort or time is required. It is only in the case that a deductive argument is valid, that time should be spent considering the truth of the premises.

Chapter 3: Inductive Reasoning 3.1 Introduction to Inductive Reasoning As was mentioned in the introduction, in a very real sense this book does not introduce much that the average person does not already do. Nowhere is that more true than with inductive reasoning. Recall that an inductive argument is an argument in which it is claimed that the premises provide reasons that support the probable truth of the conclusion. It turns out that this sort of reasoning comprises the vast majority of human thinking and provides the support for most beliefs. Indeed, any belief that comes as a result of experience is supported by inductive reasoning and thus is a matter of probability. An explication of the reasons for this would require a more in-depth voyage into philosophy than is possible here, but for those who are interested, follow these links to a discussion of the problem of induction and the philosophy of David Hume.

For current purposes, it should suffice to say that an inductive argument expresses an inference in which the conclusion goes beyond what is implicit in the premises. Contrast this with a valid deductive argument in which the conclusion can be inferred merely by unpacking what is already stated in the premises.

For this reason, while a valid deductive argument gives absolute certainty that its conclusion is true (assuming that the premises are true), an inductive argument can at best only suggest the likelihood that its conclusion is true.

3.2 Evaluation of Inductive Arguments In inductive arguments, it is claimed only that the conclusion is probably true, so evaluating them is not as simple as it is for a deductive argument. The success of an inductive inference must be measured across a spectrum and not in the eithervalid-or-invalid system that works for deduction. Another way to think of this is that while successful and unsuccessful deductive argument forms are different

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in kind, more or less successful inductive arguments differ in degree.

Strength, weakness and cogency A weak inductive argument is one that contains premises that provide a less powerful basis for accepting its conclusion than might be desired while a strong inductive argument contains premises that provide a more forceful case. It is important to keep in mind that inductive inferences are not simply either weak or strong; rather, they lie somewhere along a scale that extends from completely unfounded inferences on the weak side to arguments that approach the absolute certainty of a deductive argument on the strong. A cogent inductive argument is simply an argument in which the premises lend enough support to the conclusion that if they are true, then it is more likely than not that the conclusion is true. Unfortunately, determining the strength and cogency of inductive inferences is often difficult and controversial. Nonetheless, there are some relatively simple rules for estimating the strength of most of the more common types of inductive arguments. Please note that for the purposes of this book, the technical terms “valid,” “invalid,” “sound” and “unsound” will be used to refer only to deductive arguments, and “weak,” “strong” and “cogent” will be used only to denote inductive arguments.

3.3 Arguments from Analogy The word “analogy” has its origins in the Greek term analogos, which means “proportionate.” In inductive logic, an argument from an analogy or an analogical argument is an argument in which it is inferred from the fact that two or more objects, situations or instances are similar in certain ways to a conclusion that they are also similar in other ways. This definition is somewhat cumbersome, so consider the following the example, which will hopefully make things clearer.

Note that no inductive argument can ever achieve absolute necessity and certainty in the way that deductive inferences do. If an argument actually does achieve complete certainty, then it is, by definition, deductive.

Analogies may be drawn between all types of things, situations, objects, etc. For brevity, the term “instance” will be used to express all of these possibilities.

Critical Thinking 34 Tom’s laptop was manufactured by Widgets, Inc., and it is very dependable.

Mary’s new laptop was also manufactured by Widgets, Inc. Therefore, Mary’s new laptop will be very dependable as well.

In this argument, it is being inferred from the fact that Tom and Mary’s laptops are similar in one way, namely the manufacturer, that they will also be similar in another way i.e., very dependable.

It is important to pause here and reflect on just how common inferences like this actually are. How does one decide how much time to allot for the morning commute to work or school? This decision is made through an analogical inference in which it is inferred from the fact that since tomorrow morning’s trip into school or the office is similar to previous trips in certain ways (distance, traffic conditions, route, etc.) that it will be similar in another way as well (time spent in transit). Suppose that Padma is listening to a story being related by Frederick, who she knows to be a habitual liar. On what basis does she question the veracity of the story she is hearing? This again is an analogical reasoning process in which Padma infers from the fact that the current story is similar in some ways to certain other stories that she has heard (being told by Frederick) to the conclusion that it will be similar in another (not true). These examples are fairly straightforward, but just about any scenario in which experience is used as a guide can be interpreted in terms of an analogical argument. This sort of reasoning is, then, truly fundamental to human thought and life.

Evaluation of arguments from analogy Analogical inferences are ubiquitous, and for the most part people seem naturally to make fairly good evaluations of them. There are likely good evolutionary reasons for this. Making accurate predictions from past experience on the basis of analogy provides an obvious adaptive advantage. There is, however, always room for improvement, and it is usually the case that merely making explicit what people actually do when they evaluate these sorts of inferences will suffice to significantly enhance one’s ability to assess them. Thus, the strategy will be to walk through the reasoning process in several different variations of a common sort of analogical inference in order to illustrate the criteria for evaluating them.

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Identifying the elements of the analogy Reconsider the aforementioned question of how much time one allots for the morning commute. Here is the basic inference that might be made in such a situation and a good starting point.

A. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

Therefore, today the trip will take about 40 minutes.

To begin, it is important to identify the elements of the analogy by reformulating the definition of analogical arguments into the terms of this example. This argument expresses an inference from the fact that Abdul’s commute today is similar to his commutes last week in one way (it is by bus) to the notion that it will be similar in another way (it will take about 40 minutes).

First criterion: relevance The conclusion certainly seems reasonable; that is, it is not completely unfounded, but more information could make it weaker or stronger. In this example, what additional information would change the strength of the reasoning? The weather and traffic conditions along the bus’ route would obviously influence one’s thinking. If, for instance, a rare snowstorm had struck the city the night before or of perhaps there is traffic-snarling construction along the route, the trip will likely take longer than it did last week. Once the elements of the analogy are correctly identified, careful thought should be given to whether the given information is relevant to the question at hand.

In precise terms, one must determine whether the manner in which the instances in the premises and conclusion are known to be similar is relevant to the manner in which they are being inferred to be similar. In the present example, consider whether the manner in which Abdul’s commutes today and last week are similar as stated by the premises (they are by bus) is relevant to the way that they are being inferred to be similar in the conclusion (40 minutes in duration).

Is the mode of transportation relevant to the amount of time that a trip will take?

Certainly it is, and so this inference is stronger than it would be if the mode of

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transportation was not relevant to trip time.

This is the first criterion for evaluating analogical arguments. The information given in the premises offers support to the conclusion or affects the strength of the inference only if it is relevant to the conclusion. The subsequent criteria for evaluating analogical arguments will involve adding information to the basic argument in this example in order to ascertain how this additional information strengthens or weakens the inference. Before it is possible to evaluate how such information may change the strength of the argument, it is first necessary to consider whether it is relevant. It is of crucial importance to consider carefully what sorts of factors or information are likely to affect the probability of the conclusion being true. Consider.

B. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

Today there is a brand new decal for Widgets Computers on the side of the bus.

Therefore, today the trip will take about 40 minutes.

C. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

30 minutes prior to Abdulâ&#x20AC;&#x2122;s departure this morning, a flying saucer with a 10 kilometre diameter suddenly appeared over the city.

Therefore, today the trip will take about 40 minutes.

New decals on the side of Abdulâ&#x20AC;&#x2122;s bus are unlikely to affect transit time, but the sudden appearance of a giant flying saucer over the city very well might! Keep in mind that the question of whether specific information is relevant to the question at hand is often a matter of controversy, and disagreement about the relevance of the information in the premises will almost always translate into disagreement about the strength of the inference.

Second criterion: number of instances in the premises Consider now the following variation of the Abdul example.

Critical Thinkingâ&#x20AC;&#x192; 37 D. Abdul has taken the bus to school every day for the last five years, and the trip has always taken about 40 minutes.

Abdul will take the bus to school today.

Therefore, the trip will take about 40 minutes.

It should be immediately apparent that the inference that is expressed in D is stronger than the one that is expressed in A. More relevant information has been made available, and so the reasons for believing that Abdulâ&#x20AC;&#x2122;s commute today will take about 40 minutes carries more force. This illustrates a second criterion for evaluating analogical arguments, which is that the greater the number of instances that are known (from the premises) to have a similarity that is relevant to the inferred similarity between the instances in the premises and the instance in the conclusion, the stronger the inference (and the less this number, the weaker the inference).

Third criterion: variety of instances in the premises While simply adding to the number of instances in the premises about which the relevant information is known increases the strength of the argument, additional information about the instances in the premises can either strengthen or weaken it. Determining the effect that supplemental information will have requires an analysis not only the relevance of the new information but also careful attention to the question of which elements of the analogy the new data concerns.

Consider now an example that gives the same number of instances in the premises as are in A but offers some additional data as well.

E. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

On Monday and Tuesday during his trip, it was sunny and warm.

On Wednesday during his trip, it snowed.

On Thursday during his trip, it rained.

And, on Friday during his trip, the roads along the way were covered with ice.

Abdul will take the bus today.

Therefore, the trip will take about 40 minutes.

The new information in this version of the argument is limited to those instances that are only in the premises, namely the five morning bus trips that Abdul took

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last week. Notice that this additional data introduces differences or variety among these instances. The force of this additional variety is to make the case that certain factors that one might otherwise think would be relevant to the conclusion are in fact not. This serves to accentuate the importance of the similarity that is known to exist between all of the instances in the premises and that in the conclusion, and thus it increases the strength of the inference. In this example, one might reasonably suspect that weather conditions would be relevant to duration of Abdul’s commute by bus, but this new information suggests that in these particular cases they are not. The third criterion for evaluating analogical arguments is that the greater the dissimilarities among those instances that are known (from the premises) to have a similarity that is relevant to the inferred similarity between the instances in the premises and the instance in the conclusion, the stronger the inference (and the less these dissimilarities, the weaker the inference).

Fourth criterion: disanalogies Consider again example C. C. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

30 minutes prior to Abdul’s departure this morning, a flying saucer with a 10 kilometre diameter appeared over the city.

Therefore, today the trip will take about 40 minutes.

Given this new information, it seems very likely that today Abdul’s commute will take a bit longer than the usual 40 minutes. The reasoning here is fairly straightforward. The sudden appearance of what could only be visitors from an alien civilization would almost certainly affect traffic conditions during the morning rush hour. In this example like E, additional data has been provided that introduces differences among the elements of the analogy. Unlike E, however, the additional information reveals differences between the instance in the conclusion on the one hand and all of the instances that are only in the premises on the other. Such dissimilarities are called disanalogies. The effect of this kind of information should be fairly obvious, thus the fourth criterion for evaluating

Critical Thinkingâ&#x20AC;&#x192; 39

analogical arguments is that the greater the relevant dissimilarities between the instances that are only in the premises and the instance in the conclusion, the weaker the inference (and the greater the similarities, the stronger the inference).

Note that in B, a disanalogy is indeed introduced (this morning there is a new decal on the side of the bus). It would not seem that this sort of thing is likely to affect the duration of the bus trip. Since the advertisement on the side of Abdulâ&#x20AC;&#x2122;s bus is probably not relevant, this dissimilarity does not weaken the inference.

Fifth criterion: Modesty of the Conclusion The final consideration when evaluating analogical inferences is the boldness of the assertion that the conclusion makes. Consider the following.

F. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

Therefore, today the trip will definitely take precisely 40 minutes.

All of the relevant information is the same as in A, but in this example, the conclusion is significantly less modest, and the inference is clearly weaker. This is because the premises are being asked to do more work than if the conclusion were less bold. This is the final fifth criterion for evaluating analogical arguments: the bolder the claim made by the conclusion, the weaker the inference (and the more modest the claim, the stronger the inference). Consider one final example.

G. Every day last week Abdul took the bus to school, and each day the trip took about 40 minutes.

Abdul will take the bus to school today.

Therefore, it is reasonable to think that today the trip will take about 40 minutes.

This inference is clearly stronger than the one expressed in F, and the only difference is the modesty of the conclusion.

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3.4 Statistical Reasoning In this age of demographics, reference is often made to the approval ratings that politicians enjoy (or suffer), to opinion polls on subjects of all kinds, and to various statistical claims about everything from risk factors for various diseases to preferences for television programmes. The statistical mathematics involved in this sort of reasoning is somewhat complex, so the current discussion will not delve into them too deeply. However, it is of great importance to be in a position at least to interpret properly these claims both in terms of critical thinking in general and in the more practical business of understanding what precisely is being reported in the media. The basic inference upon which these sorts of claims are based is the statistical generalization.

Statistical generalizations In general terms, a statistical generalization is an inductive inference in which an inference is made from the fact that a certain percentage of a portion of a group bears a particular property to a conclusion that the same percentage of the entire group bears the property. Consider the following example of an explicitly stated statistical generalization.

65% of adult citizens of France polled approve of the job that Nicolas Sarkozy is doing as their president.

Therefore, 65% of all adult citizens of France approve of the job that Nicolas Sarkozy is doing as their president.

It is very seldom that claims like these are stated in this fashion. Usually a news report will state simply that Sarkozy is enjoying a 65% approval rating. The more precise reporting outlets will add the qualifier that this number has a margin of error of plus or minus (Âą) 3% or something comparable. A proper interpretation of these claims requires a brief discussion of the basic elements of this sort of argument.

In a statistical generalization, the population is the group of instances in the conclusion or that group of which a certain percentage is being inferred to bear the property in question. Note that the population need not be a collection of people. Statistical generalizations can be made about virtually any sort of

Critical Thinkingâ&#x20AC;&#x192; 41

thing, including but not limited to objects, situations, events, plants, animals, etc. Alternately, the population may sometimes be referred to as the target or target population because it is that group that is the target of the inference. The sample is the group of instances in the premises or that group of which a certain percentage is known to bear the property in question. The sample may also be called the sample population. Finally, the relevant property is simply that property that is of interest in the inference. In the example above the population is all adult citizens of France. The sample is those adult citizens of France that have been polled, and the relevant property is approval of Sarkozy as president.

Sample randomness and bias Contrary to common sentiment, these sorts of inferences can be strong given that there is a sufficient amount data that has been properly gathered. The primary concerns in any evaluation of a statistical generalization are the randomness and size of the sample. A sample is random if each member of the population has an equal chance of being selected to be a member of the sample. In terms of the present example, if each adult citizen of France had an equal chance of being polled, then the sample is random. The randomness of a sample is greatly determined by the method that the data collectors use. If, for example, the people polled were solicited to fill out an online questionnaire available only on the website of a conservative newspaper, then the sample of the poll would hardly be random; the obvious reason is that socialists who do not visit a conservative website would be much less likely to be polled. Very likely, such a method for selecting a sample would generate artificially high results for Sarkozyâ&#x20AC;&#x2122;s approval rating. Likewise, if solicitations for participation in the poll were to be found only on a socialist website, then the sample would be skewed in the opposition direction.

The extent to which a sample is not truly random is the extent to which it suffers from what is termed sample bias. It should be clear at this point, that the more random the sample in a statistical generalization, the stronger the inference, and the more biased the sample, the weaker it is. A bias in the selection of the sample is a way to influence the results, but it need not be intentional to count as such. Indeed, it is impossible to be completely certain that all sample bias has

Critical Thinking 42

been avoided. Assuming that the method of collecting data is well designed and correctly executed so as to minimize bias, there is still a significant possibility that sample bias will creep in by chance.

In terms of the current example, assume that the method of selecting those who are to be polled about their approval of Sarkozy is designed so as to ensure as much as possible that every adult citizen of France has an equal chance of being chosen to answer the poll questions. Even if this method is properly implemented, it is still possible entirely by chance that more supporters (or detractors) of Sarkozy will be chosen for the sample than is reflected in the population. The only way to reduce the probability of this happening is to increase the sample size.

Sample size Another fundamental concern when considering the strength of statistical reasoning is the size of the sample. In general, the greater the size of a properly selected sample, the greater the chance that the percentage of sample members that bear the relevant property will be the same as the percentage of the population that bear the property. Interestingly and perhaps counterintuitively, the only consideration that is relevant here is the size of the sample. The size of the population is irrelevant. Accordingly, given that a sample is indeed random and of a particular size, the strength of the generalization is no different whether the population is 10,000 or 10 billion. The strength of the generalization depends entirely on the randomness and size of sample and not, as one might suspect, on the ratio of the sample to the population.

Margin of error and levels of confidence The likelihood of unintentional bias creeping into a properly designed and executed sample selection method is expressed by the margin of error, which is a mathematical function of the size of the sample. It is the square root of the reciprocal of the number of individuals in the sample. Thus, if 1,600 citizens are polled for their approval of Sarkozy, the margin of error is the square root of 1/1600, which is 1/40 or 2.5%. One often sees results reported in this way, “Sarkozy’s approval rating is at 65%, ±2.5%.” A more precise interpretation, however, would be that the truth (the percentage of the population that actually

Critical Thinking 43

approves of Sarkozy) lies somewhere in a range from 62.5% to 67.5%, which is a range from the result of the poll minus the margin of error to the result of the poll plus the margin of error. The actual result of the poll taken of the sample is informative only as the centre of the range of possibilities for the actual true percentage in the population.

This is especially important when considering pre-election polls. Consider an all too common example in which a poll of 1600 likely voters is conducted to determine which of two candidates will win an upcoming election. Candidate A receives 49% of the likely vote and Candidate B receives 51%. Based on these results, one might be led to believe that Candidate B is more likely to defeat Candidate A than these statistics indicate. What these results actually indicate is that the likely votes for Candidate A lie somewhere between 46.5% and 51.5% and those for Candidate B between 48.5% and 53.5%. It is entirely possible that Candidate A could win the election without this statistical claim being in any way wrong!

It is worthwhile also to mention why most published opinion polls have a margin of error in the vicinity of ±3%. Consider the following table which shows the relationship between margin of error and sample size.

Sample Size 25 100 400 1600 6400

Margin of Error ±20% ±10% ±5% ±2.5% ±1.25%

This illustrates a simple rule. In order to reduce the margin of error by half, one must quadruple the sample size. This means that as the sample size increases, the change in the margin of error diminishes. For polling organizations that have to pay someone to carry out each survey, there is little value in a sample size over about 1,500 or so.

Increasing the sample adds cost without generating much in the way of increased precision. A margin of error of ±3% is small enough to generate fairly reliable results while keeping the cost of conducting the survey at a reasonable level.

Another consideration for statistical inferences is the level of confidence.

Calculating this factor is somewhat more complicated in that it requires an understanding of the probability of the polling apparatus generating false results. In the

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Sarkozy example, this would involve consideration of the likelihood that a particular respondent might intentionally lie about their approval. In terms of a medical study in which researchers are trying to determine what percentage of the general population has a particular gene, the determination of level of confidence would require knowledge of the accuracy of the medical test for said gene. The greater the possibility of the testing procedure producing inaccurate results, the lower the level of confidence in the conclusion of the statistical generalization. For most opinion polls, the level of confidence is around 95%, and this is considered the standard for a significant statistical study.

A level of confidence of 95% in a survey means that if the survey were carried out 100 times, the truth would fall within the margin of error 95 times. This may seem like a very high level of confidence, and in a certain sense, it is. Nonetheless, that 5% should not be ignored. The implications are that five times out of a hundred (or one time out of 20) the truth will not be in the margin of error and the results will be completely wrong. If one considers that thousands of opinion polls are conducted every year, there are likely quite a few that are not at all informative.

Biased questions In addition to the aforementioned sample bias, one other factor has the potential to affect seriously the results, particularly of opinion polling. This takes the form of biased questions in the study. It is relatively easy to influence a person’s responses to a survey by the manner in which the questions are phrased. A prime example of this has occurred in the United States over the question of repealing a tax on the inheritance of estates worth more than a certain amount of money. The proponents of the tax call it an “estate tax” while those who wish to abolish it call it a “death tax.” Not surprisingly, when polls are conducted that ask whether one supports the abolition of the “death tax” the results are significantly higher in favour than if same question is phrased with “estate tax” instead of “death tax.” To maximize the reliability of a poll, then, questions should be formulated so as not to unduly influence or bias the responses, at least to the extent that this is possible. Note that although many unscrupulous parties have used biased questions to generate statistical results that are favourable to their ends, biased questions need not be intentional. It is not always easy to avoid this mistake when constructing a poll.

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Interpreting a statistical generalization Despite their somewhat poor reputation, statistics can and do provide valuable and reliable information about a great variety of things from medical risk to political opinion polls to market research. While simple at its basic levels, statistics can become very complicated very quickly as one examines their inner workings. Such an exploration would require formal training in statistical mathematics. For the majority of statistical claims to which one is exposed in the course of daily life, however, a basic knowledge of how to interpret them will suffice to improve significantly the incisiveness of oneâ&#x20AC;&#x2122;s critical thought. When presented with a statistical generalization in the form of the results of an opinion poll like the Sarkozy example, it will likely come in a form like this.

Nicolas Sarkozy has an approval rating of 65%.

Margin of error Âą3%

Often, but not always, the margin of error will be given (usually in smaller print). Assuming that these results were generated by a reliable polling organization i.e., one that uses unbiased sample selection methods and questions, they can reasonably be assumed to have a level of confidence of 95%. Given this level of confidence, this example should be interpreted as.

There is a 95% chance that the percentage of adult French citizens who approve of Nicolas Sarkozy is somewhere between 62% and 68%.

Note that the number 65% is relevant only as the centre of the range of the likely possibilities.

Most of the major and respectable polling organizations make their methodologies and questions available to the public. A quick internet search should provide this information.

Chapter 4: Rhetorical Devices and Informal Fallacies Not surprisingly, people who wish to convince others to adopt their positions do not always rely solely on attempts to establish sound and cogent arguments.

People are often more interested in convincing rather than seeking the truth of the matter. Advertisers, advocates, politicians, political pundits, etc. (the list truly goes on and on) have developed quite an arsenal for levelling arguments or colouring the language of argumentation in such a way as to make the reasons for accepting their point of view seem to be much more compelling than they actually are. This need not be intentional; that is, a personâ&#x20AC;&#x2122;s intent need not be deception to make use of these highly persuasive tactics. Certainly there are many cases in which a devious type will deliberately set out to deceive, but there also many instances in which people in good faith unwittingly make use of them as well. Those who have an interest in identifying attempts at deception as well as those who are interested in compellingly arguing their points in a reasonable manner should take note of what follows.

4.1 Rhetorical Devices A rhetorical device, in general, is a use of language that makes a position seem to be more compelling than it would otherwise be by generating an emotional response. This sort of tactic is especially insidious as it often much easier to manipulate emotions than it is to appeal to reason. Heightened emotions cloud the thinking and often have the effect of masking the fact that emotions have been manipulated in the first place. It is crucial to insightful and critical thought that such manipulation be identified and that a position be dispassionately considered on its merits rather than on emotional responses that have been generated by its proponents or opponents. The following is a brief overview of some rhetorical devices. This list is by no means exhaustive.

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Euphemisms and dysphemisms A person’s emotional attitude towards something can often and easily be influenced by the term that is used to refer to it. A euphemism is a replacement term that is introduced in order to make something seem less offensive or negative than it would otherwise be. The paradigmatic example of a euphemism is the replacement of the term “civilian casualties” with “collateral damage.” By contrast, a dysphemism is a replacement term that is introduced to make something seem more offensive or negative then it would otherwise be. By way of example, consider the difference between “causing civilian casualties” and “killing babies.”

Prejudicial rhetoric (comparison, definition and explanation) In general, prejudicial rhetoric is a use of language that creates a preconception in the audience. Such rhetoric often comes in the form of comparisons.

Consider the difference between these two examples.

John: Padma is like a bull in a china shop. She doesn’t let the opinions or feelings of others affect her.

Tom: Padma is as resolute and steady as a rock. She doesn’t let the opinions or feelings of others affect her.

Both Tom and John could be speaking about the same woman, but while Tom’s language reveals a level of respect for her, John is likely expressing a negative attitude. Another common place to find prejudicial rhetoric is in definitions.

Social welfare programs are a safety net for people who are down on their luck.

Social welfare programs are an institutionalized racket whereby those who work for a living are forced to support those who refuse to take care of themselves.

One’s attitude could be greatly affected by the emotional tone of these definitions. Explanations are fertile ground for such uses of language as well.

He is a conservative because he cares more about profits than he does about people.

Critical Thinkingâ&#x20AC;&#x192; 48 The reason for her lack of faith is a deep-seated depravity and lack of a moral compass.

Derision and hyperbole Another common rhetorical device is the use of derision, which is to set up an opposing position to ridicule without offering a countering argument. This can often be accomplished with nothing more than a sneer or a dismissive joke.

Ha! You believe that we should vote for him!? Well, Iâ&#x20AC;&#x2122;ve got a bridge to sell you.

Note that no reason has been given as to why it might not be a good idea to vote for him. The implication, of course, is that the listener must be gullible even to consider it. The use of hyperbole, an overly excessive or demonstrative exaggeration, may also be used to colour a particular position in an emotional tone that is either positive or negative.

Her election to Parliament would be the best thing ever to happen to this country!

Her election to Parliament would be the worst thing ever to happen to this country!

While it may, in fact, be significant that a particular person be elected to the legislative body, it is very unlikely that it would be the best (or worst) thing to have ever happened. These are clear instances of exaggeration.

Complex questions The complex question, also known as a loaded question, is a question that is formulated so as to make a presumption that is unproven or unjustified. Any direct answer to such a question tacitly accepts the presumption. Consider the classic example of a complex question.

Has Tom stopped beating his wife?

Note that since this is formulated as a yes-or-no question, any possible direct answer to it necessarily accepts the assumption that Tom beats (or has beaten) his wife. There is no response if Tom has never beaten his wife. Here are some more examples.

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Have the liberals in this country finally stopped supporting the terrorists?

Is it true that he has finally embraced his own self-avowed principles?

Regardless of how these questions are answered, an unpalatable assumption is accepted.

Innuendo One of the more insidious rhetorical devices is innuendo, which is a use of language in which a charge is indirectly implied without being explicitly stated.

The user of innuendo thereby makes a potentially ugly claim while maintaining the appearance of not having done so.

I’m not saying that you’re a drunk, but I did notice you’ve replaced all of the bottles in your liquor cabinet.

I’m not saying that she’s a violent criminal, but her opponents in the election should watch their backs.

In both of these instances, it is clear that the speaker is in fact implying what he is explicitly denying.

The use of emotionally-laden language is very common. Indeed, it is probably safe to say that the vast majority of claims that are made in everyday contexts contain some sort of emotional content. This is only natural if one considers that human beings, if not rational in every instance, are almost always emotional. It is important to realize that the fact that a speaker or writer makes use of these sorts of devices does not in itself indicate that the position being espoused is false or that the person making the claim is being disingenuous. Someone may be passionate about what he honestly feels to be the truth, and this is not at all a bad thing. The crucial concern from the standpoint of critical thought, however, is not to let emotions unduly affect the interpretation and analysis of a particular claim.

4.2 Informal Fallacies In a broad sense, a fallacy is any argument that involves faulty reasoning, so it

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is correct to say that any argument that is neither valid nor cogent is a fallacy.

The current discussion will, however, focus on a narrower notion of an informal fallacy, which is an argument that while appearing at first glance to express a compelling inference actually expresses a flawed one. These are arguments that are psychologically deceptive in that they appear to make use of valid or cogent reasoning when, in fact, they do not.

Ad hominem Perhaps the most common informal fallacy is the argument ad hominem, or argument against the person. This is an argument that relies on claims about a person taking a position rather than against the position itself. A likely reason that such arguments are encountered so often is that they are extremely effective. People are inclined to think falsely that a person’s character or circumstance actually has an effect on the truth of the claims that she might make. While a person’s character may very well influence whether a listener believes a particular claim, it is unlikely that a speaker’s character will actually change the truth value of a proposition. Consider the following scenario.

Abdul has been in a closed room with no contact with the outside world for two weeks.

Mary, who Abdul knows from personal experience is a habitual liar, enters the room and reports to Abdul that it is raining outside.

It is likely that Abdul will not believe that it is raining outside because he does not trust Mary. However, it is either true or false that it is raining outside, and the fact that Mary is a liar has absolutely no effect on the weather.

Keep in mind that there are some claims for which the person’s character or circumstance may affect truth. These are usually claims that are actually about the character or circumstance of a person. Consider that if Mary claims that she is an honest person, her character is relevant to whether her statement is true.

Indeed, the fact that she is a habitual liar makes the claim a false one, but in the vast majority of cases, a person’s character or circumstance is completely irrelevant to the question of the soundness or cogency of her reasoning or the truth of her claims. One must resist efforts to argue for or against a position by pointing

The identification of these fallacies requires consideration of their content. It is for this reason that they are termed “informal.”

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to alleged shortcomings of its proponents or opponents and concentrate on good interpretation and analysis of the position itself.

There are several common types of these arguments that are worth noting.

The first of these is the abusive ad hominem, which attempts to refute a position by insulting the character or intelligence of its proponent(s). Here is an example.

Her foreign policy plans are idiotic. Don’t you know that she got bad grades in history when she was at university?

Even if it is the case that this particular woman did poorly in history class while she was studying at university, her foreign policy plans may be fine. At any rate, the plans should be considered on their own merits.

Another type is the circumstantial ad hominem, which assumes that a person’s circumstances should preclude them from taking a certain position.

Importantly, no argument is actually offered against the position in question.

Consider.

Mary opposes affirmative action in all its forms, but since she is an African-American, her position is untenable.

Mary’s personal circumstances are completely irrelevant when considering the merits of her position. The truth of her position and her right to espouse it are unrelated to her ethnicity. A variation of the circumstantial ad hominem is the tu quoque fallacy. “Tu quoque” is Latin for “you, too,” and this fallacy is committed when it is pointed out that the proponent of an argument is hypocritical.

Tony Blair is always saying that we should be concerned about global warming, but he’s constantly flying around the world in greenhouse-gas-spewing jumbo jets, so I don’t worry about it.

If Blair’s argument for global warming turns out to be cogent, the fact that he is contributing to the problem is not relevant.

Arguments ad hominem need not be used in a negative way. It is possible not only to attack a particular position by pointing to the proponents but also to defend it.

Ms. Willkie is the most honest person to hold this office. Her arguments for going to war cannot possibly be flawed.

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In order to make a case that an argument ad hominem has been made, one must identify the claim about the person taking a position and show that the character or circumstances of the person is irrelevant to the truth of the claim.

Equivocation A fallacy of equivocation is an argument that plays on the ambiguity of a particular term. The meaning of a term is ambiguous if it can reasonably be interpreted in two or more ways. A fallacy of equivocation is committed when two different interpretations of a term are used at different points in the argument.

Consider the following.

A half a loaf of bread is better than nothing.

Nothing is better than good health.

Therefore, a half a loaf of bread is better than good health.

A quick glimpse of this argument should raise alarm bells if for no other reason than the fact that it seems valid, the premises appear to be true and the conclusion is absurd. Indeed, there is a problem here with the term “nothing.” In the first premise, “nothing” means something like “nothing to eat.” The first proposition might be formulated more precisely by saying that having a half a loaf of bread to eat is better than going hungry. In the second premise, however, “nothing” means that there is nothing at all more valuable to a person than her health.

“Nothing” in this sense references a much broader group of things that include anything that a human being might value.

Note that if one assumes that “nothing” in both premises means “nothing to eat,” then the second premise is false, but if the assumption is the interpretation that makes the second premise true, the first premise is false. Certainly a half a loaf of bread is not better than anything else a human being might value. Here is another well-known example of equivocation.

Happiness is the end of life.

The end of life is death.

Therefore, happiness is death.

Again, there is obviously a problem with this argument. The equivocation is on

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the word “end.” In the first premise, “end” means “goal” or “purpose” while in the second, it means “termination.”

In order to prove (or at least make a case) that a fallacy of equivocation has been committed, it is necessary to identify the term that is used twice, and then show that the definition which is appropriate for one of its uses would not be appropriate for the other use.

Amphiboly and Accent Amphiboly refers to a fallacy in which a loose or awkward combination of words causes ambiguity in the meaning of a sentence. They are similar to equivocation in that they play on ambiguity, but they involve different interpretations of a statement or sentence rather than merely a term. Here is an example.

This morning, Abdul saw a ten kilometre-wide flying saucer in his nightclothes.

Therefore, large flying saucers sometimes wear human clothes.

The inference in this argument assumes one possible interpretation of the premise in which it is apparently being claimed that Abdul saw a flying saucer that was somehow wearing his nightclothes. This interpretation is possible based on the loose language in the sentence. More precise language would, of course, solve this problem. The more plausible interpretation is that Abdul, while still wearing his nightclothes this morning, saw a large flying saucer.

Here are some other examples of poorly-constructed statements that admit of multiple interpretations.

God told the president that if he started a war, there would be a great victory.

It is not clear, of course, which side will enjoy the victory.

Dog for sale. Will eat anything. Especially fond of children.

Would children be safe around this animal?

Used cars for sale. Why go elsewhere to be cheated? Come here first!

This advertisement is perhaps not as enticing as it could be.

The fallacy of accent is a specific type of amphiboly in which ambiguity in

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the meaning of a statement arises because of a shift in emphasis from one word or phrase in the statement to another. Consider.

The flight attendant indicated that it was necessary to fasten your seatbelt.

So, she wouldn’t approve if you fastened mine.

In this example, it is plausible to assume that the flight attendant meant that it was now necessary for the passengers to fasten their seatbelts. The conclusion drawn from this statement, however, assumes a different interpretation of the statement and exploits an ambiguity that arises from a shift of emphasis to the word “your.” Note that the meaning of a sentence can be significantly changed simply by shifting the accent to different words.

Fasten your seatbelt.

Assuming a more or less neutral intonation, this command instructs one to fasten a seatbelt.

Fasten YOUR seatbelt.

When an accent is placed on “your,” the meaning changes so as to indicate that you shouldn’t fasten anyone else’s seatbelt.

FASTEN your seatbelt.

This means that one should fasten the seatbelt instead of doing something else with it.

Fasten your SEATBELT.

Here of course, the suggestion seems to be that one is attempting to fasten something other than a seatbelt.

Fallacies of accent are often used in advertisements when a particular word or phrase is graphically accentuated in order to draw one’s attention away from qualifiers or caveats that while visible are de-emphasized. Similarly to equivocation, the burden of proof for a charge that a fallacy of amphiboly or accent has been committed is to identify the statement that admits of multiple possible interpretations and show that the interpretation that is being assumed in one part

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of the argument is not appropriate for another part of the argument.

Composition and division A fallacy of composition is an argument in which it is mistakenly inferred from the fact that the parts of a whole (or individuals of a type of thing) bear a certain property that the whole (or the group of a type of thing taken as a collective whole) also bears that property. Some examples will hopefully make this definition a bit clearer.

The players on this football team are among the best that have ever played the game.

Therefore, this football team is one of the best teams to have ever played the game.

The problem with this argument is that while the players are good, it may be the case that they do not play well together and thus constitute a poor team.

An elephant will eat several times more food in a day than a human being.

Therefore, all elephants taken as a group consume more food in a day than all humans taken as group.

Clearly, this argument fails to take into account the fact that there are many, many more human beings in the world than there are elephants, and so while individual elephants may indeed eat more than individual humans, it is not the case that the collective group of elephants eats more than the collective group of human beings.

An argument that expresses an inference that moves in the opposite direction is also problematic. A fallacy of division is an argument in which it is mistakenly inferred from the fact that a whole (or the group of a type of thing taken as a collective whole) bears a certain property that the parts of that whole (or individuals of a type of thing) also bear that property. Consider some examples.

Acme automobiles are very expensive.

Therefore, each of the parts that are used to build it is very expensive.

There may, of course, be some parts of these automobiles that are quite inexpensive.

Because of its complexity and organisation, it is clear that the Universe has a purpose.

Since every human being is part of the Universe, each of us must have a purpose as well.

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Even if the Universe has a purpose, it does not follow that every human being does as well.

To make a case that someone has committed a fallacy of composition or division, it is necessary to show that it does not follow from the fact that the parts of a whole bear a property that the whole bears the property or vice versa.

Red herring A red herring is a fallacy in which a distraction is introduced into the discourse in order to lead an opponent or audience away from the issue at hand. This distraction can come in the form of a mischaracterization of an opponent’s position (straw man) or merely an interruption that ends discussion altogether.

Interviewer: Your opponent has argued for immigration reform. Do you agree with her position?

Candidate: I think the more important question confronting this great nation is the question of terrorism. Let me tell you how I plan to defeat it.

The more interesting the red herring, the more effective this tactic becomes. The burden of proof for showing that such a fallacy has been committed is simply to point out that the committer has changed the subject.

Straw man A straw man is a fallacious argument in which the opponent’s position is mischaracterized in such a way that it is easier to refute or dismiss. The constructor of the straw man proceeds to tear it down, thus appearing to have defeated the opposing position. It is a type of red herring because the mischaracterization has a distracting effect. Importantly, the opponent’s real position remains untouched.

This sort of fallacy is quite common in political contexts.

Mary: We must not betray the principles of justice and democracy. Suspected terrorists must be granted basic rights as well as legal representation and access to a fair court.

Tom: Mary is advocating the release of known terrorists. We cannot afford to allow our enemies to move freely in our society.

In this example, Tom has mischaracterized Mary’s position in such a way as to

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make it appear absurd.

A straw man fallacy can be identified by explicitly stating the actual position and showing that it is different from the mischaracterization.

Ignoratio elenchi The fallacy of ignoratio elenchi or irrelevant conclusion is any argument for which the conclusion is not directly relevant to the question at hand. Typically, the committer will offer a very compelling argument for a conclusion that is subtly different from the real issue, thus appearing to have provided good reasons for his position. Consider the following example.

Mary: We must not betray the principles of justice and democracy. Suspected terrorists must be granted basic rights as well as legal representation and access to a fair court.

Tom: It is difficult to understand why my opponent would even consider extending the rights to legal representation and court hearings to the terrorist suspects. The threat that terrorism poses to our way of life is profound. A rigorous and well-considered response is of crucial importance if we are to survive and emerge victorious.

The sleight of hand is subtle. While the issue at hand is whether terrorist suspects should be afforded basic rights, Tom offers an argument for a rigorous response to terrorism. It is perfectly possible that Mary would agree with Tom’s conclusion, although Tom’s commission of ignoratio elenchi makes it appear that Mary is opposed to a “rigorous and well-considered response.” Tom fails to show that there is anything wrong or unwise with Mary’s position. Like the red herring and the straw man, the proper response to an instance of ignoratio elenchi is to point out that the committer has argued for a position that is not directly relevant to the question at hand.

False dilemma A false dilemma or false dichotomy is committed when an opponent is presented with a certain number of choices as if these choices exhaust the possibilities when, in fact, they do not. The force of this fallacy is to constrain artificially and unreasonably an opponent’s options. Here are some examples.

Abdul: The proposition is either true or false, so you must either accept it or deny it.

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This example is especially deceiving because it is true that all propositions are either true or false. Nevertheless, one is not necessarily forced to either accept or deny a proposition. One may withhold judgement, neither accepting nor denying it.

Tom: You are either with us, or you are with the terrorists.

Unfortunately, this example is all too real. The tacit argument is to compel the audience to join up on Tomâ&#x20AC;&#x2122;s side, for few would wish to join the terrorists. It is likely, however, that it is possible to disagree with Tomâ&#x20AC;&#x2122;s position and to reject terrorism. To make a case that a false dilemma has been presented, it is necessary to point out the fact the choices that have been presented do not constitute all of the available options.

Hasty generalization A hasty generalization is an inductive argument in which one makes a fallacious inference from a relatively small number of cases to a generalization about a class of instances. This type of argument gains its deceptive strength from the fact that the presented evidence may be relevant but is nonetheless insufficient to establish the conclusion, making the argument seem stronger than it is. This fallacy is a common tool, particularly of the racist or xenophobe.

Look at those three Martians over there! They are drunk and fighting.

Martians are violent and impulsive.

While it may indeed be true that these three Martians are intoxicated and fighting, it cannot be inferred from three instances that all Martians are likely to engage in this sort of behaviour. To make a case that this fallacy has been committed, one must show that the number of instances in the premises do not warrant the inference to the general class of which those instances are a member.

Appeal to inappropriate authority A fallacious appeal to inappropriate authority is either (1) an argument in which the conclusion is based on the judgment of someone who is not actually an authority on the issue at hand or (2) an argument which is based on the judg-

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ment of a genuine authority on the issue at hand but concerns an issue about which there is disagreement among experts in the field. Given the complexity of contemporary life, it is very difficult to ascertain whether a particular person or group is a reliable authority on a particular issue and even more problematic to determine if there is significant disagreement among experts in a particular field.

The best policy is always to be somewhat suspicious of appeals to authority. If one is not in a position to evaluate an authority, then one is not in a position to evaluate the strength of an argument that appeals to that authority. Consider a couple of examples of the first type.

Tom Jones, a respected actor who plays the brilliant cardiologist Dr. John Smith in the film Emergency, recommends Drug X for improving the overall health of the heart.

Therefore, it would be wise to take Drug X.

This example is fairly obvious as it is extremely unlikely that an actor is a genuine authority on the health of the heart, but the next example is perhaps a little more deceptive.

Dr. John Rawls, a respected professor at Harvard University, recommended Drug X for improving the overall health of the heart.

Therefore, it would be wise to take Drug X.

The fact that John Rawls was both a doctor and a professor at Harvard might lead one to believe that he is a genuine authority on the subject. However, it turns out that John Rawls was a professor of philosophy and by no means an expert on cardiology. Consider now an example of the second type.

Dr. John Rawls, a respected professor of political philosophy at Harvard University, argued that the practical philosophy of the 18th century philosopher Immanuel Kant provides a plausible foundation for democracy in multi-cultural societies.

Therefore, it would be wise to look to Kant to solve some of societyâ&#x20AC;&#x2122;s political problems.

The argument seems strong enough. John Rawls is indeed a world-renowned political philosopher and a genuine expert on the issue at hand. There is, however, significant controversy and disagreement on this subject among the experts in this field, thus this argument is not as strong as it appears. Showing that an argument commits this particular fallacy will require that a case be made that the authority is not legitimate or that there is disagreement among the experts.

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It is important to note here that disputing the legitimacy of an authority does not constitute an argument ad hominem. So long as the authority’s expertise is relevant to evaluating the strength of the argument, no fallacy is committed. It is, of course, possible to point to some fact about a person that is irrelevant to her status as a legitimate authority in order to undermine fallaciously her credibility.

In this case, the argument would indeed be ad hominem Argument from ignorance An argument from ignorance is one in which it is inferred from the fact that a proposition has not or cannot be proven true that it is false or from the fact that it has not or cannot be proven false that it is true. A noteworthy example of this fallacy may be found in a popular film.

Han Solo (to Luke Skywalker): Kid, I’ve flown from one side of the galaxy to the other. I’ve seen a lot of strange stuff, but I’ve never seen anything that could make me believe there’s one all-powerful “Force” controlling everything.

The tacit conclusion, of course, is that there is no “Force.” It does not follow, however, from the fact that it has never been proven to exist that it does not.

Other common examples include the following.

Cigarettes do not cause cancer. No one has ever been able to prove conclusively that they do.

The reason for this inability has to do with the limitations of scientific knowledge (follow these links to a discussion of the problem of induction and the philosophy of David Hume) and the impermissibility of experimentation involving people. It is clear that this difficulty does not provide a basis for the claim that cigarettes do not cause cancer. To meet the burden for showing that an argument from ignorance has been made, one must show that the premise shows only that there is ignorance about the truth or falsehood of the conclusion and not that the conclusion is, in fact, true or false.

This list is by no means exhaustive. There are many, many ways in which bad argumentations and unwarranted inferences can deceive. Those who are interested in further reading on the subject will find additional resources in Appendix B.

From Star Wars Episode IV: A New Hope by George Lucas in accordance with the fair use doctrine

Chapter 5: Complex Arguments Now that the basics of argumentation and its evaluative analysis are in hand, attention may be turned to a discussion of how arguments are often encountered in the real world. Attempts to persuade are as common as advertisements on television. They may be found in political speeches, letters to the editor, essays and countless other media. Seldom simple and straightforward, they are often confusing, deceptive and complicated.

5.1 Analysis versus Interpretation When confronted with a complex argumentative passage there are two fundamental concerns. The first of these to be discussed was the question of analysis.

Is the argument deductive or inductive? Is it valid, sound, or cogent? The first four chapters have dealt exclusively with this question, but before an analysis can be even be attempted, one must know precisely what the argument is. This is the question of interpretation, which concerns itself with several broad issues.

What exactly is the speaker or author trying to say? Is an argument actually being made? What is the point that the proponent is trying to make i.e., what position is being espoused? What are the premises and conclusion of the argument? Coming to terms with these questions requires some understanding of the intent and thinking process of the speaker or author, and unfortunately, this is not always an easy undertaking. Interpretation is a skill, and as such, practice will be necessary to develop it.

The best way to approach questions of interpretation is simply to read or listen with them in mind. The human mind is actually very good at comprehending language, and quite often all that is required for an insightful interpretation is simply paying close and explicit attention to the task. There are, of course, many tricks of the trade, some of which will be discussed shortly, but to reiterate, the single most important difference between one who is an insightful critical

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thinker and one who is not is that the critical thinker takes the process of thinking seriously, consciously attends to that process and asks the right questions.

Arguments versus explanations An important distinction should be mentioned here. The language of argumentation (see the section on conclusion and premise indicators below) is somewhat similar to that of explanation, and it is possible to become confused between the two. In general, when an argument is deployed, the intent is to give reasons (premises) to support a claim (conclusion) that is controversial or that is not yet accepted by the reader. The thinking process involved is an inference that moves from the premises to the conclusion. An explanation, however, works in the opposite direction. The purpose of this use of language is to give an account of a how or why it is that a more-or-less widely accepted or uncontroversial claim is the case. The thinking process moves from a generally accepted claim (that which is to be explained) to the account (the explanation). Determining whether a particular piece contains an argument or an explanation will require consideration of the status of the particular claim in question. Is it a matter of common knowledge, or does work need to be done to establish that it is the case? Consider the following.

On Earth the sky is blue because blue light is scattered by the Earth’s atmosphere while others colours of light pass through the gases unobstructed and so are not visible.

This is clearly an explanation of why the Earth’s sky is blue. The basis of this determination is that it is widely known and ostensibly uncontroversial that the sky is blue. By contrast consider the following example of an argument.

Planet X orbits a star that is very similar to the sun at a distance that is almost exactly the same as that of the earth. The planet also has an atmosphere that is very similar in composition to the Earth’s. Hence, on the surface of Planet X, the sky is blue.

Here the conclusion is that the sky of Planet X is blue. The reasons for thinking this are that Planet X is similar to the Earth in the ways that are relevant to the colour of the sky. The important factor is that since no one has ever stood on the surface of Planet X, the colour of the sky on Planet X is not widely known and may reasonably be considered to be a matter of controversy.

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In general, it is fairly rare that someone will become confused as to whether a particular passage is an argument or an explanation. Most people are fairly good at recognizing when an attempt to persuade is being made. The fundamental question to ask is whether the force of the language is to convince one to accept a particular position or to explain why something that is already accepted is the case.

The principle of charity A further word on interpretation involves what philosophers call the principle of charity, according to which one is in some sense obligated to offer the strongest possible interpretation of an opposing argument. This may seem somewhat counterintuitive. Why, after all, should one try to make an opponentâ&#x20AC;&#x2122;s position stronger? There are several compelling reasons for doing just that. The first of these is simply to save time. Suppose that an opponent offers an objection to a weak interpretation of an argument. In response to such an objection, a defender of the position can simply reformulate the argument in a stronger way. If, however, an opponent interprets the argument in the strongest possible form from the outset, such an option is not available to the defender, who must then offer a more substantive defence of the position. A second reason to do this is to reduce the chances of committing a straw man fallacy (see 4.2).

The best reason for operating under a principle of charity, however, is to maximize the force of an objection. Consider how in competitive sports a victory can be soured by the fact that the defeated team had some of their best players out of the match with injuries. It can always be said that the winning team won only because the defeated team was playing at a disadvantage. The ideal scenario is to defeat opponents when they are at their strongest. In a similar way, if an opponent interprets the argument in the best possible light (perhaps in an even stronger form than the proponent originally stated it!) and grants as many of the premises as possible, a devastating objection is all the more damaging.

5.2 Evaluating Complex Arguments In the real world, arguments are often embedded in speeches or essays that may or may not be clearly worded. Crucial premises or even the conclusion may be

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tacitly implied and never explicitly stated. An argument that contains one or more unstated propositions is called an enthymeme, and any unstated proposition is called an enthymematic proposition. It is quite common for arguments to contain an enthymematic conclusion or unstated premises. Additionally, the bulk of the piece may be support for the premises in the main argument and so not directly relevant to its conclusion. The argument that is presented may not express a sound or cogent inference. Indeed, its proponentâ&#x20AC;&#x2122;s intent may be to deceive or coerce rather than to offer a compelling argument. All of these factors along with the other vagaries of human discourse serve to make interpretation a difficult and challenging endeavour. It is, however, a necessary skill for critical thought. What follows is a step-by-step guide for evaluating complex argumentative pieces.

Identifying the issue or question at hand Incontrovertibly, the first step in evaluating a complex argument is to identify the issue or question at hand. Assuming that the argument is part of a written essay, the best way to do this is to give the piece a thorough and careful initial reading. At this point, one should refrain from any attempt to interpret or analyze the argument that might be contained, opting instead for a solid synoptic grasp of the scope and subject matter of the piece. It is important at this point to give the author the benefit of the doubt on controversial claims. Remember that the point here is merely to figure out what precisely she is trying to say, so it is important to take the piece at face value. This acceptance should, of course, be provisional. Final acceptance should be kept separate from this interpretive exercise in order to avoid being deceived by bad reasoning. Any final judgment should be withheld until a thorough analysis as been made.

After this first reading, one should carefully consider the direction that the piece has taken. Towards what position or goal is the argument targeted? What thoughts or opinions are left in the mind? What does the author want the reader to believe? Ideally, the author will explicitly state her position. If so, the question with respect to which she has taken a position is the de facto issue at hand.

Of course, it is entirely possible that the argumentation that she offers does not actually support this particular claim. If the force of the reasons that are offered

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seem to lead somewhere besides this explicitly stated conclusion, there is quite possibly a serious problem with the argument. There is also the possibility that the author has not explicitly stated her stand on the issue that is discussed in the piece. If this is the case, then it is probably safe to say that the issue at hand is the target of the reasoning that the language of the piece expresses. The context in which the piece is being presented should also be considered. For example, if the author is attempting to refute an argument in another article or essay, then the subject matter of the original piece is the issue at hand. In such cases, it is very important to be on guard against subtle re-interpretations of the opposing position.

Correctly identifying the issue at hand requires a holistic grasp of the content, scope and context of a piece, and there is no simple method to accomplish this task. Its importance cannot be overstated because it is a crucial step to recognizing the conclusion of the main argument. Any evaluative analysis will rely on how well the premises of this main argument support its conclusion, so failure to identify properly the issue at hand will make a critical evaluation impossible.

Recognizing arguments and their elements: premise and conclusion indicators Recognizing that an argument is being made is not usually difficult. The best clue is the context in which language is being used. If a person is attempting to persuade others to adopt a particular position or to accept a particular claim by offering reasons, then she is very likely making an argument. What can be somewhat more difficult is the correct identification of the elements of an argument.

The second step in evaluating a complex argument is to identify the conclusion of the authorâ&#x20AC;&#x2122;s main argument. This step and those that follow will require multiple close readings of the piece. The conclusion of the main argument should be the position that the author takes with respect to the issue at hand, thus while the issue at hand is the subject about which she has written or the question that she has set out to answer, her conclusion is the stand she takes on this issue or the answer that she offers. For the sake of simplicity in the analysis process, the conclusion should be formulated as one proposition. This proposition may, of course, be a complex proposition (conjunctive, disjunctive or hypothetical), and the argumentative burden will change accordingly.

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Identification of the conclusion of the main argument requires a clear grasp of the intent and reasoning process of the author, so there is no way to accomplish this task without careful reading and consideration. There are a number of phrases and words that often indicate that a conclusion is about to be stated.

So Therefore Thus Hence It can be inferred that From these facts, it can be concluded that This implies that This entails that It follows that

Obviously this list is not exhaustive, and the purpose of giving it is to show what kinds of language may used to state the conclusion. Keep in mind that in a complex argumentative passage, it is very likely that the author will provide support for the premises of her main argument. This support will necessarily come in the form of an argument, so the presence of conclusion-indicating language does not necessarily point to the main conclusion. The only reliable way to make this identification is to come to an understanding of the authorâ&#x20AC;&#x2122;s reasoning process.

Before moving to the premises, it is a good idea to take a close look at the conclusion and to consider what kind of evidence would be required to show that it is true. Does it make a statistical claim? If so, then one should anticipate a statistical generalization. If it is a conjunctive proposition, then there should be support for each of the individual propositions that constitute it. A hypothetical proposition will require evidence of the existence of some sort of relationship between its antecedent and its consequent. A working idea of the burden of proof that the author will have to meet in order to establish the conclusion will allow a critical reader to anticipate the sort of argument and supporting evidence that should be presented. This will aid in the recognition of the main premises as well as in the evaluation of the overall argument.

The third step in evaluating a complex argument is to identify the premises of the authorâ&#x20AC;&#x2122;s main argument. This is likely to be the most difficult part of the process as these premises may be enthymematic or presented in a haphazard way. They may be mixed in with claims that are meant to offer them support,

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so there may actually be a large number of propositions in a passage, not all of which are premises of the main argument. The important thing to remember is to look for those propositions that offer direct support to the conclusion. This reveals why it is necessary to identify the conclusion of the main argument first.

It is impossible to identify those propositions that offer direct support for the conclusion if one does not know that the conclusion is. There are many phrases and words that provide clues to the presence of a premise.

Because For Since As From the fact that For the following reason This follows from It may be implied from the fact that

Again, this list is not exhaustive, and the presence of this kind of language indicates only that a reason for believing some claim is being given. It does not reveal whether the claim for which support is being offered is the conclusion of the main argument. Much of the challenge of formulating a correct interpretation arises when one attempts to identify the elements of the main argument and to separate them from the support that is being offered to the premises of the main argument.

The analysis of the main argument Once the constitutive propositions of the main argument have been identified, one may move to the fourth step in evaluating a complex argument, which is an analysis of the form of the main argument. At this point, one should assume that the premises are true. The reason for this is that a determination of the actual truth of the premise is often very time-consuming, and if the form of the argument has serious problems (deductive invalidity, inductive weakness, or fallacious reasoning) then it wonâ&#x20AC;&#x2122;t be necessary to carry out such a determination.

The first question is whether the argument is deductive or inductive? An answer to this will again require consideration of the intent of the author. Is she claiming to have proven her conclusion with deductive certainty? If so, is the

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argument valid? Is she claiming only to show inductively that the conclusion is likely to be true? If this is the case, is the argument a strong one? Note that if an argument that has been interpreted as deductive turns out to be invalid, the principle of charity will likely require the reader to reconsider the argument with the assumption that it is inductive. An invalid deductive inference may very well be interpreted as a strong inductive one. Note that if an argument expresses an inference that is exceedingly weak, then one need go no further in this procedure. Even if the premises are actually true, they do not provide compelling support for the conclusion, so consideration of their truth is a waste of time.

A further concern at this point is to consider whether the terms of the argument are being used consistently wherever they appear. Does a word or phrase mean the same thing in one part of the argument as it does in another? If not, then there is good possibly that a fallacy of equivocation has occurred. Also consider whether the propositions that constitute the argument have multiple reasonable interpretations. Is the language loose or awkward? Is the emphasis in the sentence in the appropriate place? Has a fallacy of amphiboly or accent been committed?

In addition to attending to the relationship between the premises and the conclusion in the main argument, it is also important to examine the relationship between the main argument and the issue at hand. It is possible that the conclusion of the main argument is subtly different from the position that the author has ostensibly taken with respect to the issue at hand, or the premises may actually support a proposition that is subtly different from the stated conclusion. It is especially important to keep these possibilities in mind when the main argument is an enthymeme. If an argument is to be compelling, the premises must offer support for the conclusion, and the conclusion must be relevant to the issue at hand.

Support for the premises Once the validity or cogency of the main argument and its relevance to the issue at hand have been established, attention may be turned to the actual truth of the premises. Chances are that the bulk of the essay will be evidence for these propositions. In practice, this support will come in the form of subsidiary arguments,

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which, for simplicity, will be termed sub-arguments. Critical evaluation of these sub-arguments should follow the same steps as the evaluation of the main one.

First, identify the conclusion of the sub-argument, which is, of course, a premise of the main argument. Consider the sort of evidence that would be required to show this proposition to be true, and identify the premises of the sub-argument.

Once satisfied that these premises are indeed relevant to the conclusion of the sub-argument, consider its validity or cogency. This process should be repeated for each premise for which the author lends support. It is always possible that she will not offer any support for one or more premises. This need not be a problem if these propositions are uncontroversial or a matter of common knowledge.

But if they are not, if they need support that is lacking, this is a problem for the argument.

In more complex pieces, another level of argumentation may be offered as evidence for the premises of the sub-arguments. Since this could obviously go on forever, any argument must eventually rest on propositions for which no support is offered. These will usually be grounded in an appeal to authority, empirical data, common sense, definitions or some combination of these. It is important to keep in mind that none of these should be taken completely at face value.

Appeals to authority can be inappropriate, empirical data can be misleading, common sense can be wrong, and definitions can be incorrect or misapplied.

The final evaluation Once the sub-arguments have been properly interpreted and analysed, a final evaluation may be made of the main argument. The purpose of this is to make a determination of how compelling the complete argument is when all things are considered. Is the inference valid or at least cogent? Are there good reasons for accepting the truth of the premises? Are they as well-supported as they could be? All told does the argument offer a compelling case for the acceptance of its conclusion? As should be clear, this process can require a lot of time and effort, so it should be easy to understand why it is always a good idea to evaluate the inference that is expressed by the main argument before considering the truth of the premises.

A summary of this step-by-step process is as follows.

Critical Thinkingâ&#x20AC;&#x192; 70 1. Identify the issue at hand.

2. Identify the conclusion of the main argument 2. Consider the burden of proof of the conclusion of the main argument.

3. Identify the premises of the main argument.

4. Conduct an evaluative analysis of the inference expressed in the main argument and the relevance of the main argument to the issue at hand.

5. Repeat steps 1 through 5 for each of the subsidiary arguments that lend support to the main premises.

6. Make a final evaluation

Paraphrasing and diagramming Paraphrasing and diagramming are helpful tactics for successfully navigating the above steps. In the course of identifying the constitutive elements of the main argument and sub-arguments, it is very useful to reformulate the propositions in clear and concise language on a note page. This is paraphrasing, and in addition to making the argument less cumbersome, it may facilitate the discovery of equivocation, amphiboly or accent fallacies. It is of fundamental importance when paraphrasing an argument to maintain in the interpretation all of the nuance and meaning of the original language. If an analysis is conducted on an inaccurate formulation of an argument, then the argument in question has not been analyzed!

Once an argument has been precisely and accurately interpreted and paraphrased, it is also helpful to map graphically the inferences that the argument expresses. This is especially useful when evaluating large and complex arguments because it will make it easier to follow the authorâ&#x20AC;&#x2122;s reasoning process and will also aid in identifying propositions that lack support. A suggested procedure for diagramming a complex argument is to label the premises of the main argument P1, P2,â&#x20AC;ŚPN and its conclusion C. The premises of the sub-argument for a main premise (P1) can then be designated with subscripted letters P1a, P1b, and so on. Once the propositions are so denoted, simply draw the inferential relationships between them in a way that indicates whether they are deductive or inductive.

Given a deductive main argument with two premises, a good way to indicate this would look like this.

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Given an inductive main argument, use the following.

Given these basic graphical tools, it is fairly easy to diagram even complex arguments.

For those who prefer to avoid pencil and paper, there are argument-mapping software packages available online. A good program is available for download from Austhink (click to go the site).

Chapter 6: Bringing It All Together The development of robust critical thinking skills will require more than a mere mastery of the set of definitions and tactics here presented. In addition to the cultivation of these interpretative and analytical tools, a thinker must remember to use them. When presented with an attempt to persuade, one must be aware of it and actively and consciously engaged in the thinking process. This can only truly be accomplished by forming what might best be called good mental habits. This is not to say that one should walk through life in a constant state of distrust and suspicion, but it is important to be cognizant of one’s reasoning processes and those of others. If, with these basic skills in hand, close attention is paid to the arguments that are presented in daily life, the sheer volume of poor and fallacious reasoning (not to mention its unwarranted success) is genuinely staggering.

Critical thinking has more to offer than just an improvement in one’s ability to evaluate arguments. It will also aid in the development of a well-considered and consistent world-view, in the composition of rationally compelling arguments, and in improvement in handling information.

6.1 Good Habits Develop an awareness of one’s epistemic status with respect to beliefs Epistemology is the study of knowledge, so “epistemic” means of or having to do with knowledge. One’s epistemic status with respect to a particular proposition or belief is, then, how certain one is (or can be) that the proposition is true or false. It is important to develop and maintain an awareness of one’s epistemic status with respect to particular beliefs. Such awareness requires that close atten-

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tion be paid to the reasons that one has for a belief. When these reasons are identified and examined, are they rationally compelling? Do they rationally obligate one to accept the belief in question, or are they entirely arbitrary? Do the reasons rely on certain assumptions that might be controversial? Consider that the great majority of human beliefs are held without this sort of examination.

The development of a habit of awareness of epistemic status will greatly increase the insightfulness of one’s argumentative criticism, improve information literacy, and strengthen the force of one’s own arguments. Indeed, such mental discipline is necessary for the quick recognition of argumentative language or attempts to persuade, for the realization that claims being made are not well supported, and for insightful examination of one’s own system of beliefs.

This is certainly not an easy habit to develop. Furthermore, even if one remembers to engage critical thinking skills, it can be difficult to ascertain one’s epistemic status with respect to a particular claim. Keep in mind, however, that a realization that one’s epistemic status is unknown counts, in a very real sense, as awareness of one’s epistemic status. It is good to know that one does not know.

This insight will often illuminate the fact that further research or investigation is necessary and may even provide direction for such inquiry.

Seek consistency in beliefs A fundamental principle in philosophy holds that a proposition cannot be both true and false at the same time and from the same point of view. Critical thinking will often uncover contradictions in a system of beliefs. This sort of thing can occur in a variety of ways, but it perhaps most commonly happens when the implications of a belief are examined. When it is realized that that a belief or one of its implications directly contradicts another belief or one of its implications, a rational obligation to resolve the conflict arises. There are really only two ways this sort of tension can be rectified. One can somehow show that there really is no contradiction, or one can abandon one of the contradictory beliefs. This can be a burdensome task, especially if both beliefs are fundamental. The rejection of a long-held fundamental belief, aside from being psychologically difficult, will often entail a major revision in one’s entire world-view.

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Identify dogmatic beliefs When considering the reasons that a particular belief is held either by oneself or by another, it is often a good idea also to consider what kind of evidence it would take to change the belief. If it turns out that there is no possible evidence that would suffice to change the belief, it is likely that the belief is held dogmatically. A dogmatic belief is a belief that is taken to be absolutely true. In practice, if a belief is held dogmatically, then any possible evidence that could be presented to show that it is incorrect would simply be reinterpreted or denied.

By way of example, suppose that a person, who believes a proposition P dogmatically, is presented with direct sensory evidence that P is false. The person would be forced either to reinterpret the significance of the evidence or to deny that the senses are reliable in this specific case. There are several reasons why it is important to identify beliefs that are held in this way.

Dogmatic beliefs are often a matter of religion. Please note that the author intends no negative evaluation of either religious faith or dogmatic beliefs. Such beliefs may very well be true, and if they are, there will certainly be no conclusive evidence to show that they are false. In the interests of constructive and respectful discourse, it is important to realize when a discussion is entering potentially sacred territory.

Once it is established that a belief is genuinely dogmatic and that there is no possible evidence that could be produced to change it, there is often little more to be said on the issue. If any further discussion is to be productive, a mutual agreement will need to be reached with respect to how this belief is to be handled. Can everyone agree to assume it? Can the discussion be altered to avoid reference to this particular belief, or is the questioning of the belief necessary for the issue at hand? Realizing and appropriately resolving a conflict in dogmatic beliefs will save time and maximize the possibility of productive interaction. Nothing in this should be interpreted as suggesting that dogmatic beliefs preclude the possibility of productive discourse, but their presence will shape the boundaries of the discussion.

In the examination of oneâ&#x20AC;&#x2122;s own beliefs, it may be discovered that a belief has been held in way that has unduly shielded it from scrutiny. While not genuinely dogmatic, such a belief, if not grounded in good reasons, may unnecessarily

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prevent an incisive investigation as well as the development of a more coherent belief system.

Follow reason Philosophers traditionally believe that the purpose of skillful reasoning and argumentation is to eventually arrive at the truth. Their goal is not (or should not be) merely to win an argument for the sake of winning. In principle, a philosopher should welcome a conclusive refutation of her argument because it provides the valuable service of eliminating a false possibility. The discovery of what is not true is a step down the road to the truth. In this spirit, one should not shy away from the ever-present possibility that one is wrong.

It is not uncommon, of course, for even the most insightful thinkers to resist clear evidence that they are wrong. The reasons for this range from pride to stubbornness. Nevertheless, it is important to keep open the possibility that an opponent’s position is correct until it has been conclusively refuted. For the purposes of critical thought, maintaining objectivity with respect to one’s certainty in the truth of a belief is crucial for identifying potential weakness in the position, which, in turn, is useful for the development of compelling supporting arguments. Furthermore, keeping an open mind about the possibility of one’s being wrong is necessary for cultivating a philosophically informed worldview.

6.2 Composing an Argument While many of the considerations that come in to play when evaluating the arguments of others will also play an important role in developing arguments to support one’s own position, there is, unfortunately, no sure-fire process for composing a conclusive argument. What follows is a general strategy for the development of a rationally compelling argument that will focus primarily on the sort of things that one should keep in mind.

Preparation The first step in the development of a compelling argument is to come to a good understanding of what precisely is being claimed. The best way to do this is to

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formulate the position as a clear and concise proposition. This will be the conclusion of the argument.

The audience With the conclusion in hand, attention should be turned to the intended audience of the nascent argument. Ideally, an argument should appeal to the sorts of reasons that as many people as possible would accept, but it is entirely possible that an argument is intended for a more limited audience. If, for example, the audience consists of believers of a particular religion, it is unlikely that the sacred text of a different religion would provide compelling evidence. The crucial concern is to identify the kinds of evidence that the audience can reasonable be expected to accept, and this will require a determination of who precisely the audience is.

The burden for the philosopher is to develop arguments that appeal only to reason, and indeed evidence that requires only that the audience be rational will have the widest possible application. However, particular groups may share certain assumptions or articles of faith that may count as compelling reasons for members of those groups but not for others. Once the audience has been identified, one should dismiss any evidence that it is not reasonable to expect that group of people to accept.

Developing the position At this point, it is often helpful to brainstorm on the pros and cons of the position. Make a list of all of the evidence that supports the position as well as all of the reasons to think that it might be wrong. Be sure to distinguish carefully between that evidence that offers direct support for the conclusion and evidence that lends support to what will eventually be the premises of the main argument.

Consider the inferences that one must make if this evidence is to be relevant. Are they valid or at least cogent? Take time to develop the more compelling evidence.

Make lists of respected information sources that may provide documentation.

Being as objective as possible and keeping open the possibility that the conclusion is false, weigh the evidence in support of it against the reasons to think that it is false. Carefully consider how certain it is possible to be that the propo-

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sition is true. Is it more likely than not to be true, is it just one of several viable possibilities, or are there good reasons for thinking that it is wrong? It is always possible that a belief will not survive this process.

Consider whether the supporting reasons offer conclusive support for the conclusion or merely suggest that it is true. Supporting reasons that are genuinely conclusive might very well provide the material for a deductive argument, but if the force of the evidence is only sufficient to show the likelihood of the conclusion being true, then an inductive argument should probably be developed.

Formulating the argument After careful consideration of the reasons that can be offered in direct support of the conclusion, it is a good idea to formulate them as clear and concise propositions. These are the premises of the main argument. The goal is to reproduce oneâ&#x20AC;&#x2122;s reasoning process in the mind of the audience, and this will only be successful if the structure of the argument accurately and clearly expresses that inference.

Once the main argument has been formulated, the inference that it expresses should be rigorously analyzed for validity or cogency. At this point, one should be as objective as possible in determining how compelling the inference is. If the inference is intended to be deductive turns out to be invalid, can it be reformulated as an inductive argument? If it is intended as an inductive argument, can it be strengthened by making the conclusion more modest? If it is discovered that the inference is not compelling, more development will be necessary.

When satisfied that the main argument clearly and accurately expresses the most compelling possible inference, attention may be turned to the supporting evidence for the main premises. One should follow the same procedure for the development and formulation of the sub-arguments as for the main argument. A compelling case that the main premises are true should involve either a valid or cogent inference from the testimony of a respected and legitimate authority, common knowledge, empirical data, or definitions. It is important to trace all of the supporting claims for the main premises back to one of these grounds or something similar. Additionally, any possible documentation (particularly for those grounds other than common knowledge) should be collected and referenced accordingly.

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Before moving on, one should have written in notes a clear and concise set of propositions that express the inferences that support the conclusion. Each of these propositions should be supported by referenced documentation, common knowledge or further propositions that express an inference.

Anticipate and answer objections Once the main argument and its supporting sub-arguments have been clearly formulated, one should return to the list of reasons to think that the conclusion is wrong in order to develop arguments against the main position. When this is complete, responses to these anticipated objections can be composed. It is worth noting that it is very important to formulate these objections objectively and in as compelling a manner as possible, so that when they are answered, little doubt will be left that one’s position has been adequately defended.

Write an essay Putting one’s argument in essay form is crucial for making it accessible to others, and while writing and composition are not the subject of this book, a brief discussion of the basics of an argumentative essay is appropriate. The essential concern when communicating an argument is to make the question of interpretation as easy as possible for the reader to answer. This will require that careful attention be paid to organization and clarity of language. There is no best way to organize an essay, so what follows is meant merely as a suggestion.

The issue at hand should be made apparent to the reader as quickly and clearly as possible. This should be the topic of the very first sentence in the essay. Indeed, one should not be afraid to jump right in. Here is an example of a first sentence that explicitly states the issue at hand.

There has been much debate in recent months on the question of whether the Government should adopt the Reform Act.

The second concern is to make one’s position clear as soon as possible. This very likely should be in the second sentence. Here is an example of a second sentence that explicitly states the author’s position with respect to the issue at hand.

Critical Thinking 79 Proponents have offered plausible arguments for this revolutionary piece of legislation, but there are compelling reasons to think that its passage would not be in the best interests of the country.

Next, a brief summary of the reasons that the author has for her position should be offered. These should relate the premises of the main argument, and ideally they should be presented in such a way that it is clear whether the inference that they express should be interpreted as deductive or inductive. The following is an example of language that introduces a summary of the main reasons for the author’s position and also indicates that the inference should be interpreted as an inductive one.

While perhaps not conclusive, the following reasons strongly suggest that this is so.

The idea is to state concisely the main argument in the introductory paragraph.

The body of the essay should first offer a systematic and well-documented survey of the supporting evidence for the truth of the main premises. Second, it should present the anticipated objections against the main conclusion and the responses to them. Finally, a brief recapitulation of the main argument should be offered to conclude the essay.

6.3 Information Literacy The period from the early 1980s to the present day is commonly referred to as the Information Age. The reasons for this moniker are obvious. Technology has accelerated the flow of information and made massive volumes of data available literally at mankind’s fingertips. The Internet is a technological innovation that will arguably be as profound in its effects on human life as the wheel or perhaps the mastery of fire. As the quantity and speed of information increases, so also does the importance of its skilful management. This set of skills has been termed information literacy, which the American Library Association characterizes in the following way. “To be information literate, a person must be able to recognize when information is needed and have the ability to locate, evaluate, and use effectively the needed information.”

There is obviously much more to this than can be adequately covered in the

http://www.ala.org/ala/acrl/acrlpubs/whitepapers/presidential.htm

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current context, and it is doubtful that critical thought can be practically applied without serious attention being paid to it. The purpose of the following overview is to show that critical thinking skills both require and aid in the development of information literacy.

Recognizing when information is needed It has never been easier to find and retrieve information. Entire libraries are being made available online, and searching is as easy as typing a phrase into a text field. A more fundamental concern, however, is recognizing when additional information or documentation is needed to make a point. The development of an awareness of oneâ&#x20AC;&#x2122;s epistemic status with respect to beliefs is the single most important consideration in this recognition. Attention should also be paid to what it is reasonable to expect others to accept. Is a claim a matter of common knowledge? If the answer to this question is no, then it is likely that supporting information will be necessary to bolster the strength of the claim. Empirical data should be sourced, and authorities should be referenced.

Evaluating sources of information The best way to ensure the viability of particular information is independent confirmation. If a particular piece of information can be verified by multiple and separate reliable sources, then the chances of it being true are improved. Another issue of great importance is, then, the correct evaluation of sources of information. Is a referenced authority a legitimate one? Is there controversy in the field on the issue at hand? Is the source of empirical data reliable?

Unfortunately, there is no infallible way to determine whether a particular source is trustworthy. In general, the most important thing is not merely to assume that it is. Some knowledge of any biases that the source may have is also important. Perhaps most importantly, one should have a good understanding of how a particular source came to have the information. The Internet is to a very great extent self-referencing. In other words, many information sources

Note that since a bias is potentially relevant to the reliability of the information that a source provides, it is not necessarily an ad hominem to consider a sourceâ&#x20AC;&#x2122;s personal circumstances in this context.

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on the Internet also get their information from the Internet. This, in itself is not a problem, but if the goal is to confirm independently a particular piece of information, this can lead to confusion. Two separate websites may seem to verify independently the information, but it could be that both websites actually got the information from the same source. In fact, this is very common. There are relatively few information-gathering organizations that many, many people reference. Information from one of these sources can quickly propagate through the internet and appear to be coming from multiple independent sources. The point here is that simply checking information on several websites is not likely to count as independent confirmation. A rigorous attempt to verify the reliability of information will require tracking the information back to its original source.

Fortunately, most respectable websites provide references for their claims, so this process can usually be carried out fairly quickly.

An investigation into the past performance of a sourceâ&#x20AC;&#x2122;s information might also be a good idea. It is important to mention that the fact that a source has on occasion been wrong does not, in itself, mean that it is not reliable. Human beings are known for their fallibility, and it is inevitable that a person or organization will make mistakes. The important thing to consider isnâ&#x20AC;&#x2122;t that mistakes are made; rather, it is how often they are made, and perhaps more importantly how forthright the source is in correcting them. The concerns, then, are that the source be competent and in good faith.

Another important consideration to evaluating the reliability of a particular source is to have a good understanding of its nature and how it functions. Is the source a public relations firm for a particular industry? Is it a non-governmental organization with a particular purpose, or perhaps a news outlet with a particular bias? The important thing is come to an understanding of the manner in which information is collected and released by the source. What is the method that the source uses to gather information, and does the source have any reason to filter its output? Answering these questions requires careful thought and research into the background and design of the source.

An exciting and growing trend on the Internet is sites for which users provide the content. Blogs, wikis, social networks, and video sharing sites are becoming

This term indicates a site for which a community works together in a collaborative way to add and edit content. The well-known site wikipedia.org is the prime example.

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ubiquitous, and it is unquestionable that they are referenced more and more as sources of information. The question of their reliability, however, is somewhat more difficult to answer. The website wikipedia.org will be taken as an example for the purposes of developing a good understanding of the nature and functionality of a source. According to the site, Wikipedia is a multilingual, web-based, free content encyclopedia project. Wikipedia is written collaboratively by volunteers from all around the world. With rare exceptions, its articles can be edited by anyone with access to the Internet, simply by clicking the edit this page link.

The result of this design has been the development of an enormous repository of information in a very short amount of time. As of July 26, 2007, there were over 1.9 million articles in English alone. This is truly astonishing when one keeps in mind that the site first came online in 2001.

But is Wikipedia a reliable source of information? It is crucial to keep in mind that the result of a search on the site is literally what the last person to edit it wrote. Is that anonymous person a legitimate expert? Does he or she have any relevant biases or axes to grind? Quite simply there is no way to know, and so there is really no way to evaluate any particular article. In general, it is reasonable to think that as a whole, Wikipedia tends strongly towards accurate information, and it is probably safe to say that for any given search the results will be more or less correct. It should, however, be treated with a certain degree of scepticism. The most responsible way to use it would seem to be as a general guide for the direction that one might take in researching a topic and not as an authoritative source. A similar examination should be conducted before any source is considered reliable.

Plagiarism Plagiarism is the presentation of the work of another as oneâ&#x20AC;&#x2122;s own. This is a cardinal sin for a critical thinker, and the importance of avoiding it cannot be overstated. Any time use is made of the work or ideas of another whether as a direct quotation or paraphrasing, a properly documented reference must be provided.

http://en.wikipedia.org/wiki/Wikipedia:About

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Summing Up The current work is by design an overview of the fundamental concepts and considerations that constitute critical thinking. While it has hopefully provided the reader with much to think about, it should be viewed only as the beginning of a process of developing these skills through further study and practice.

Appendix A: Categorical Logic The following is a very brief and cursory survey of a system of deductive logic that was first described by Aristotle. It is as a good example of a more-or-less complete system of deductive logic, and it is for this purpose that it is included here. Those who are interested in a more in-depth examination of the history and details of the system please click here.

A.1 Categorical Propositions A categorical proposition is a proposition that makes an assertion about the relation of membership between groups, classes, or categories of objects. A basic element of a categorical proposition is its terms. A term is a word or phrase that describes or designates some group of objects. This group of objects can also be called a class or category. In general, the group that a term indicates consists of all the objects in the Universe that are described by the term. For example, the term “cats” indicates or “picks out” all of the objects in the Universe that might truthfully be described as a cat. Note also that a class or category might be empty if the term that denotes it describes something that does not exist.

For present purposes, the term that comes first in the proposition is the subject term, and the one that comes second is the predicate term. These are traditionally abbreviated as “S” and “P.” Consider that there are only four possible relations of membership between any two groups.

Every member of one group of objects is also a member of a second group of objects.

All cats are mammals.

None of the members of one group of objects are members of a second group of objects.

No rocks are mammals.

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One or more members of one group of objects are also members of a second group of objects.

Some blades are swords.

One or more members of one group of objects are not members of a second group of objects.

Some cats are not pets.

There are thus four different kinds of categorical propositions, and using the above abbreviations, these four types can be expressed formally without using specific terms and without referencing actual groups.

All S are P. No S are P. Some S are P. Some S are not P.

Each type of categorical proposition has a title and an abbreviation of it own.

They are as follows.

“All S are P” is a universal affirmative and is abbreviated as an “A proposition.”

“No S are P” is a universal negative and is abbreviated as an “E proposition.”

“Some S are P” is a particular affirmative and is abbreviated as an “I proposition.”

“Some S are not P” is a particular negative and is abbreviated as an “O proposition.”

Quantity, quality and distribution Note that there are two universal propositions (A and E), two particular propositions (I and O), two affirmative propositions (A and I) and two negative propositions (E and O). Whether a proposition is universal or particular is a question of its quantity, and whether it is affirmative or negative is a matter of its quality.

The quantity of an A proposition, for example, is universal while its quality is affirmative. Consider the following table.

Universal Particular

Affirmative A: All S are P. I: Some S are P.

Negative E: No S are P. O: Some S are not P

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While quantity and quality are relatively straightforward, the notion of distribution is somewhat more difficult. A categorical proposition distributes a term if it makes reference to all of the members of the class that is picked out by that term. This is, perhaps, somewhat abstract, but consider an A proposition.

All cats are mammals.

Note that this proposition is informative about all of the members of the class that is indicated by the subject term. Based solely on this proposition, it can be said that something is known about every single cat, namely that they are all mammals. Consider now the predicate term. Does this proposition provide information about all mammals, or is it possible that there are some mammals, some members of the class denoted by the predicate term, that are not mentioned in this proposition? It is hopefully clear that while making reference to all of the cats, this proposition does not necessarily say anything about all of the mammals. Thus, A propositions (all A propositions) distribute the subject term while leaving the predicate term undistributed.

E propositions make reference to all of the members of both classes in that they assert that all members of the subject class are not members of the predicate class and that all members of the predicate class are not members of the subject class. E propositions therefore distribute both terms.

I propositions assert only that some member (or members) of the subject class is among the members of the predicate, the rest of which are not referenced, so I propositions distribute neither term.

The distribution of the O proposition is slightly more difficult to grasp. Based on the distributions of the first three types, one might expect that the O proposition distributes the predicate term only, and this is, indeed, correct. An O proposition provides information about only some member (or members) of the subject class and so does not distribute the subject term. The assertion that it makes about this member or these members of the subject term does reference all of the members of the predicate class. It asserts that none of the members of the predicate class are the particular member (or members) of the subject class that the proposition references.

It may also be helpful to note that the quality of a proposition determines whether the predicate term is distributed. If the proposition is affirmative, the

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predicate term is not distributed. If the proposition is negative, the predicate term is distributed. The quantity of a proposition determines whether the subject term is distributed. If the proposition is universal, the subject term is distributed, but if the proposition is particular, the subject term is undistributed.

For the purposes of determining the deductive validity of categorical syllogisms (see A.3), it is only necessary to remember the distribution patterns of the four types of propositions. Memorizing the following table will suffice.

A: All S are P. E: No S are P. I: Some S are P. O: Some S are not P.

Distributes S only Distributes both terms Distributes neither term Distributes P only

A.2 The Square of Opposition and Immediate Inferences

Consideration of the claims being made by the various types of categorical propositions will reveal that if the same terms are assumed, the truth or falsehood of one proposition will determine the truth value of at least some of the others.

Consider that if “all cats are mammals” is true, it must be the case that “no cats are mammals” is false. There are many such relationships among the four kinds of categorical propositions, and these relationships exhibit a symmetrical pattern. This can be clearly seen in the square of opposition.

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Contradiction, contrary, subcontrary and subalternation Any two propositions (P and Q) are contradictory if and only if (1) the truth of P necessitates the falsehood of Q, (2) the truth of Q necessitates the falsehood of P, (3) the falsehood of P necessitates the truth of Q and (4) the falsehood of Q necessitates the truth of P. Put another way, P and Q are contradictory if and only if both cannot be true at the same time and both cannot be false at the same time. Assuming identical subject and predicate terms, an A proposition and an O proposition are contradictories as are an E proposition and an I proposition.

These propositions lie diagonally from one another on the square.

Any two propositions (P and Q) are contrary if and only if (1) the truth of P necessitates the falsehood of Q and (2) the truth of Q necessitates the falsehood of P. In other words, P and Q are contradictions if and only if both cannot be true at the same time. It is possible that they both be false at the same time, so the falsehood of one does not allow a determination of the truth value of the other. Assuming identical subject and predicate terms, an A proposition and an O proposition are contraries. These propositions lie on the top corners of the square.

Any two propositions (P and Q) are subcontrary if and only if (1) the falsehood of P necessitates the truth of Q and (2) the falsehood of Q necessitates the truth of P. P and Q are subcontrary if and only if they both cannot be false at the same time. It is, however, possible that they both be true at the same time, so the truth of one does not allow a determination of the truth value of the other.

Assuming identical subject and predicate terms, an I proposition and an O proposition are subcontraries. These propositions lie on the bottom corners of the square.

Subalternation describes the relationship between two categorical propositions with the same quality. From the fact that two categorical propositions have the same quality and identical subject and predicate terms, the following inferences can be made.

1. The truth of the A proposition entails the truth of the I proposition.

2. The truth of the E proposition entails the truth of the O proposition.

3. The falsehood of the I proposition entails the falsehood of the A proposition.

4. The falsehood of the O proposition entails the falsehood of the E proposition.

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However, given the falsehood of an A proposition it cannot be inferred that the I proposition with identical terms is either true or false. Nor does the falsehood of an E proposition entail the truth or falsehood of the O proposition with the same terms. Similarly, the truth of an I proposition does not entail the truth or falsehood of the A proposition, nor does the truth of the O proposition necessitate the truth or falsehood of the E proposition. On the square of oppositions, the rules of thumb for subalternation is that the “truth moves down” and “falsehood moves up.”

The following chart catalogues all of the possible valid inferences from the truth value of a categorical proposition to the truth value of the other types of categorical propositions with identical terms.

Given that A: “All S is P” is Given that E: “No S is P” is Given that I: “Some S is P” is Given that O: “Some S is not P” is

True.

False.

True.

False.

True.

False.

True.

False.

A: “All S is P” E: “No S is P” I: “Some S is O: “Some S is must be must be P” must be not P” must be T F T F F U U T F T F T U F T U U F T U F T F T F U U T T F T F

It is important to note that some of these inferences assume a certain interpretation of categorical assertions. Some interpretations deny that these sorts of inferences are possible. For a more detailed discussion of the square of opposition, click here.

More immediate inferences In addition to those on the square of oppositions, there are several other valid inferences that may be made. These inferences require an understanding of the notion of a class complement. The complement of a class is a group that contains all of the objects in the Universe that are not a member of that class. Recall that a class is a group of all the objects that are denoted by a term, so for example the class designated by the term “cats” consists of all the objects in the Universe

“U” indicates that the truth value is undetermined.

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that may properly be called cats. The complement of the class designated by “cats” is the class that contains all of the objects in the Universe that are not properly called cats. This class is designated by the term “non-cats,” which includes everything from a hydrogen atom in the Andromeda galaxy to the reader of this book to the President of the Republic of France, indeed everything that is not a cat. Note that the “non” in this term is part of the term and does not constitute a negative for the purposes of determining the quality of a categorical proposition.

Consider the following I proposition.

Some dogs are non-cats.

While it may be tempting to identify this as an O proposition, it is crucial for the determination of the validity of categorical syllogisms and for the following immediate inferences that this be resisted. “Non-cats” is a term that designates the class complement of the class designated by the term “cats.” It does not make the proposition a negative one. Note also that complement of the class designated by the term “non-X” is the class designated by the term “X,” thus the complement of “non-cats” is “cats.”

Obversion, conversion and contraposition The obverse of a categorical proposition is obtained by (1) changing the quality of the proposition and (2) replacing the predicate term with the complement of the predicate term. Consider the following examples.

An A proposition: All cats are mammals.

The obverse of which is: No cats are non-mammals An E proposition: No cats are dogs.

The obverse of which is: All cats are non-dogs.

An I proposition: Some cats are pets.

The obverse of which is: Some cats are not non-pets.

An O proposition: Some cats are not pets.

The obverse of which is: Some cats are non-pets.

The obverse of a categorical proposition P may be inferred to have the same truth value as P, and this is the case for all four types of categorical proposi-

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tions. Thus, it can be said that obversion is valid for all four types of categorical propositions.

The converse of a categorical proposition is obtained by (1) replacing the subject term with the predicate term and (2) replacing the predicate term with the subject term. Consider the following valid conversions.

An E proposition: No dogs are cats.

The valid converse of which is: No cats are dogs.

An I proposition: Some cats are pets.

The valid converse of which is: Some pets are cats.

Conversion is valid only for E propositions and I propositions, hence the truth value of the converse of a proposition Q can be inferred to be the same as the truth value of Q, given that Q is an E or I proposition. This inference cannot be validly made for A and O propositions. Consider the following invalid conversions.

An A proposition: All cats are mammals.

The invalid converse of which is: All mammals are cats An O proposition: Some people are not men.

The invalid converse of which is: Some men are not people.

The contrapositive of a categorical proposition is obtained by (1) replacing the subject term with the complement of the predicate term and (2) replacing the predicate term with the complement of the subject term. Consider the following valid contrapositives.

An A proposition: All cats are mammals.

The valid contrapositive of which is: All non-mammals are non-cats.

An O proposition: Some people are not men.

The valid contrapositive of which is: Some non-men are not non-people.

Contraposition is valid for A and O propositions only. Consider the following invalid contrapositions.

An E proposition: No dogs are cats.

The invalid contrapositive of which is: No non-cats are non-dogs.

An I proposition: Some non-dogs are non-cats.

The invalid contrapositive of which is: Some cats are dogs.

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By limitation Although direct conversion is invalid for A propositions, it is possible to infer by subalternation from the truth of an A proposition to the truth of an I proposition with identical terms. Conversion of the I proposition would then be valid.

Given the true A proposition: All cats are mammals.

By subalternation, it also true that: Some cats are mammals.

By conversion, it is also true that: Some mammals are cats.

This process is called conversion by limitation. Keep in mind that since valid subalternation is sensitive to truth value (“truth moves down” and “falsehood moves up”), conversion by limitation is valid only in the case of a true A proposition.

Similarly, the contraposition by limitation of a true E proposition can be validly inferred.

Given a true E proposition: No cats are dogs.

By subalternation, it is also true that: Some cats are not dogs.

By contraposition: Some non-dogs are not non-cats.

A.3 Categorical Syllogisms The following is a system of deductive logic that will allow one to determine the validity of any categorical syllogism. As indicated in 2.2, a syllogism is a deductive argument with exactly two premises. A categorical syllogism is a syllogism in which both of the premises and the conclusion are categorical propositions.

Each of the terms in a categorical syllogism is given a designation, which may be determined by examining the conclusion. The major term of a categorical syllogism is the predicate term of the conclusion of the syllogism, and the major premise is the premise that contains the major term. The minor term of a categorical syllogism is the subject term of the conclusion of the syllogism, and the minor premise is the premise that contains the minor term. The middle term of a categorical syllogism is the term that occurs exactly once in each of the two premises and not in the conclusion.

A categorical syllogism is in standard form if its major premise is given first and the minor premise is given second. Consider the following syllogisms, the first of which is in standard form and the second of which is not.

Critical Thinking 93 All cats are mammals.

Some cats are pets.

Some pets are mammals.

Some cats are pets.

All cats are mammals.

Some pets are mammals.

The form in which an argument appears i.e., the order in which the premises are given does not affect the validity of the argument, and it is not a necessary consideration in what follows.

Four rules to determine the validity of a categorical syllogism It is always possible to determine the validity of a categorical syllogism. If the syllogism is in accordance with all four of the following rules, then it must be valid. If it fails to satisfy any one or more of these rules, it must be invalid.

The first rule of a valid categorical syllogism is that it must contain exactly three terms. It is absolutely crucial for this system of deductive logic to generate reliable results that there be precisely three terms in the syllogism, so one must be on guard against equivocation. It is possible that though the terms may sound or appear to be the same, that they actually designate different classes. Consider the following.

All things that are read are things that contain written language.

All apples are things that are red.

Thus, all apples are things that contain written language.

When reading this argument on the screen, it is obvious that there are four terms, (1) things that are read, (2) things that contain written language, (3) apples and (4) things that are red. However, because “red” and “read” are homophones, it is possible that a listener could fail to realize that an equivocation has occurred.

In the context of categorical logic, this is called the fallacy of four terms. Any categorical syllogism that contains more than three terms is invalid.

The second rule of a valid categorical syllogism is that the number of negative propositions in the conclusion must be the same as the number of negative propositions in the premises. Since the conclusion must either be affirmative or negative, the number of negative propositions in the conclusion can only be one

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or zero. If the conclusion is negative, exactly one premise must also be negative. If the conclusion is affirmative, then both premises must be affirmative. If both premises are negative, the syllogism is invalid because there cannot be two negative conclusions. Consider the following syllogisms, the first of which is in accordance with this rule and the second of which is not.

Some dogs are pets.

No cats are dogs.

Therefore, some pets are not cats.

No cats are dogs.

Some dogs are pets.

Therefore, some cats are pets.

The third rule of a valid categorical syllogism is that if a term is distributed in the conclusion, it must also be distributed in the premise in which it occurs.

The question of whether a proposition distributes a term may be resolved by referencing this table.

A: All S are P. E: No S are P. I: Some S are P. O: Some S are not P.

Distributes S only Distributes both terms Distributes neither term Distributes P only

Consider the following syllogism.

All people are rational beings.

All women are people.

Therefore, all women are rational beings.

Since the conclusion is an A proposition and A propositions distribute their subject terms, the term “women” must also be distributed in the conclusion. In this example, the premise that contains “women” is an A proposition with “women” as its subject term, thus the minor premise distributes the term “women.” This syllogism is therefore in accordance with the second rule. Consider now a syllogism that fails with respect to the second rule.

All men are people.

All people are rational beings.

Therefore, all rational beings are men.

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Here the term “rational beings” is distributed by the conclusion, but since “rational beings” is not distributed in the second premise, this syllogism is invalid. Note that this rule only requires that if a term is distributed in the conclusion that it also be distributed in the premise in which it occurs. An argument in which a term is distributed in a premise but not in the conclusion would be in accordance with this rule.

The fourth rule of a valid categorical syllogism is that the middle term must be distributed at least once. Consider the following.

All men are people.

All rational beings are people.

Therefore, all rational beings are men.

In this example, the middle term is “people,” which is the predicate term of both premises. Since an A proposition does not distribute its predicate term, the middle term is not distributed at all in the premises of this syllogism. It is thus invalid.

If a syllogism is in accordance with all four of these rules, then it is always valid. If, however, the syllogism fails with respect to one or more of them, then it is always invalid.

Appendix B: Resources B.1 Textbooks Jago, Mark. Formal Logic. Philosophy Insights. Tirril: Humanities-Ebooks, 2007. This is a concise, inexpensive and informative introduction to formal symbolic logic and a good starting point for further study.

Klenk, Virginia. Understanding Symbolic Logic. Prentice Hall, 2007. This is the standard introduction to Symbolic Logic. It is clearly written and comprehensive, and concentrates on the method known as “natural deduction.” It also briefly outlines the “tree method.” It is difficult, and will take a serious commitment to master.

Jeffrey, Richard C. Formal Logic: Its Scope and Limits. Hackett Publishing Company, 2004. This is another introductory text for Symbolic Logic, one that focuses exclusively on the “tree method.”

Garson, James W. Model Logic for Philosophers. Cambridge University Press, 2006. This is an introduction to modal logic.

B.2 Other Resources General Resources Rationale argument mapping software from austhink.org The Stanford Encyclopedia of Philosophy. This is a comprehensive and evergrowing collection of definitive articles on various issues in philosophy.

The Internet Encyclopedia of Philosophy. This is a similar internet project.

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Rhetorical Devices and Informal Fallacies A discussion of informal fallacies at austhink.org More on fallacies from Drury University Still more in informal fallacies from Lander University Information Literacy A comprehensive guide to information literacy from the Association of College and Research Libraries A similar resource from informationliteracy.org in the UK

Humanities Insights The following Insights are available or forthcoming at: http://www.humanities-ebooks.co.uk/ Genre FictionSightlines Octavia E Butler: Xenogenesis / Lilith’s Brood Reginal Hill: On Beulah’s Height Ian McDonald: Chaga / Evolution’s Store Walter Mosley: Devil in a Blue Dress Tamora Pierce: The Immortals

History Insights The British Empire: Pomp, Power and Postcolonialism The Holocaust: Events, Motives, Legacy Methodism and Society Southern Africa

Literature Insights (by author) Chatwin: In Patagonia Conrad: The Secret Agent Eliot, George: Silas Marner Eliot, T S: ‘The Love Song of J Alfred Prufrock’ and The Waste Land Faulkner: The Sound and the Fury Gaskell: Mary Barton Hardy: Tess of the d'Urbervilles Hopkins: Selected Poems Lawrence: The Rainbow Lawrence: Sons and Lovers Lawrence: Women in Love Shakespeare: Hamlet Shakespeare: Henry IV Shakespeare: Richard II Shakespeare: Richard III Shakespeare: The Tempest Shelley: Frankenstein Wordsworth: Lyrical Ballads

Literature Insights (general) English Renaissance Drama: Theatre and Theatres in Shakespeare’s Time Fields of Agony: English Poetry and the First World War

Philosophy Insights American Pragmatism Business Ethics Ethics Existentialism Formal Logic Heidegger Informal Logic and Critical Thinking Islamic Philosophy Marxism Meta-Ethics Philosophy of History Philosophy of Language Philosophy of Mind Philosophy of Sport Sartre: Existentialism and Humanism Wittgenstein

General Titles An Inroduction to Feminist Theory An Introduction to Critical Theory An Introduction to Rhetorical Terms

Commissioned Titles Include Aesthetics Austen: Pride and Prejudice Blake: Songs of Innocence & Experience and The Marriage of Heaven & Hell’ Eliot: Four Quartets Fielding: Tom Jones Heaney: Selected Poems Hughes: Selected Poems Lawrence: Selected Poems Mental Causation Toni Morrison: Beloved Plato Plato’s Republic Renaissance Philosophy