A Formal Introduction to Critical Thinking 3e

Preface

This text, as the title suggests, is a formal introduction to critical thinking.  There are a number of approaches to teaching Critical Thinking.  Most of them tend toward informal approaches, in which students spend a lot of time reading newspaper articles and watching video clips and are instructed to evaluate the argumentation in them.  I believe that this is the wrong approach to teaching Critical Thinking.  The analogy I use to explain my approach to my students is that this is like sending soldiers to war without ever having gone through basic training.  I ask my students what the result of this might be, and they are quick to respond that these soldiers would likely die very quickly, because they don’t know what they are doing.  In order to succeed on the battlefield, a person needs physical training and education in the ways of war.  I believe that formal logic is to a person’s mind or reason what basic training is to a person’s body.  It is training in and the strengthening of the mental muscles that one needs to be a successful reasoner.

I use war as an analogy not by accident.  I believe that the rational stakes in our world are high, and that people are involved in an intense battle for their beliefs.  There are a number of people who would be happy to overcome us rationally and control and dominate us by convincing us to believe what they want us to believe, whether it is true or false.  The primary enemies in this battle are advertisers, politicians, and religious leaders.  This is not to say that there is anything wrong with politics or religion (I will have nothing positive to say about advertising); just that these are the primary areas in our lives in which people try to convince us to believe what they want us to believe, and in which people often resort to bad arguments to convince us.  The only adequate preparation for withstanding these attempts to control us is to have a rigorous training in the fundamental concepts of logic and reason, which is Deductive or Formal Logic.  Hence this text, of which about 80% deals with deductive logic.  With good basic training in the rules of rational engagement, I believe students will be more successful students of life.

At the same time, since this text is designed for a one-semester Critical Thinking course, I have simplified the formal instruction somewhat.  In Categorical Logic, I have used only Venn Diagrams, since they are faster and more efficient tools for analyzing arguments than the complex rules of mood, figure, distribution, etc.  In Sentential Logic, I have limited the operators to the Negation, Conjunction, Disjunction, and Conditional, since the Biconditional can be formed from these and is an uncommon operator.  In Natural Deduction, I have pared the set of derivation rules down to a small set, which seems to me sufficient to do the most common derivations. 

In the end, I hope to have fashioned a worthwhile introduction to Critical Thinking, which avoids the excesses of informality and lack of rigor on the one end and of overly difficult technical logic on the other.

Chapter 1: Introductory Concepts

1.1 – Critical Thinking, Logic and Arguments

Critical Thinking, as we will use it in this course, is a branch of the philosophical discipline of Logic.  Logic is the study of argument.  Another way of saying this is that logic is the study of human reasoning.  The characteristic that is most often associated with how humans differ from other animals is that humans reason, while other animals don’t.¹ Reason is the capacity to make decisions or come to conclusions based on the evaluation of evidence, or based on argument.  So logic is a very important study; it is the study of what makes us distinctive creatures in the world.

Arguments

When defining terms, it is not very helpful to define a term with other terms that are themselves not well defined.  This is the case with defining logic as the study of argument.  So it will help to provide a definition of argument.  Informally, when we think of an argument, we think of a group of people, who disagree about something, voicing their opinions about what they think is the right or wrong way of viewing the issue.  In addition to just voicing their opinions however, people in an argument are usually trying to convince the other people that their opinion is the correct one.  The attempt to persuade or convince is the essential feature of arguments.  To begin, we can define an argument informally as a rhetorical situation in which someone is trying to persuade or convince someone else that his or her opinion is right or wrong.  Rhetorical situation refers to any situation involving communication.  We can define argument more specifically, based on this informal definition.  The formal definition of argument (the ones used by logicians) is, a set of statements that contains at least one premise, one conclusion, and an inference. This definition is VERY important.  It will guide everything we study in this book.

Statements

We now, though, have another definition that includes some undefined terms (statement, premise, conclusion, inference), so we should define those terms as well.  There are several definitions for statement.  The two simplest ones are a sentence that is either true or false and a sentence that says something about the world.  For instance, the sentence, The sky is blue is a statement.  It says something about the world, about the color of the sky, and it is either true or false (in this case it is true).  Statements may be more complex than this one.  For instance, The governor of California asked the state government to pass a resolution declaring stem cell research legitimate and establishing a fund to promote stem cell research.  This sentence still says something about the world (an action of the governor’s) and it is either true or false (either the governor asked the government to do this or he didn’t).  All statements have these two characteristics.

Not all sentences are statements; statements are a subgroup of sentences.  There are 3 types of sentences that are not statements.  Questions are sentences, but not statements.  Are you going to get out of bed? is a proper sentence, but it doesn’t say anything about the world and it is neither true nor false (the answer to the question will be yes or no, but the question itself is neither true nor false).  Commands are also sentences but not statements.  Get out of bed! is a is a proper sentence, but it doesn’t say anything about the world and it is neither true nor false.  It is just a command.  Finally, expressions of emotion, sometimes referred to as expletives are sentences but not statements.  If I stub my toe and yell Ouch! I have expressed a sentence, a complete thought, but I have not said anything about the world and my expression is neither true nor false.

One important clarification about statements must be made.  Although statements are either true or false, it is not necessary that we know whether a sentence is true or false to call it a statement.  For instance, There is life on other planets is a statement, even though we don’t know whether it is true or false.  It is either true or false; either there is life on other planets or not; the state of our knowledge doesn’t affect that fact.  Similarly, There is a large animal living in Loch Ness in Scotland is a statement, even though we don’t know whether it is true or false.

Premises and Conclusions

An argument is composed of statements.  There are two types of statements in an argument that have a special relationship to each other.  The premises of an argument are statements that give support or evidence for another statement.  In an argument between people, they are the facts or information the arguer would present to try to convince the other person(s) of whatever the arguer is trying to convince them of.  The conclusion of an argument is the statement that the premises give support or evidence for, the thing the arguer is trying to convince the other(s) is true. For instance, in this argument;

All college students are humans.

Bob is a college student.

Therefore, Bob is a human.

The italicized statements, All students are humans and Bob is a student, are the premises.  They are taken to give support or evidence for the statement, Bob is a human, which is the conclusion.  An argument must have AT LEAST ONE premise (if no information is being presented, there can't be an attempt to convince anyone of anything!).  But there is no limit to the number of premises in an argument (except an infinite number of premises).  Technically, an argument can have only ONE conclusion.  If some evidence leads to more than one conclusion, then either it is not enough evidence, or there is more than one argument.

Inference

The other part of an argument is called the inference.  I will give the formal definition of inference, then explain what it meant.  An inference is a relationship between statements in which the truth of one (or more) statements affects the probability of the truth of another statement.  The statement(s) whose truth is affecting the probability is (are) the premise(s).  The statement whose probability is being affected is the conclusion.  In an argument, if the premises are true, then their truth affects the probability that the conclusion is true as well.  In a good argument, then, the truth of the premises will establish that the conclusion is probable.  In turn,    a good argument will (ideally) motivate or influence the listener to believe that the conclusion is true as well.  This is a deep insight into reason and human nature.  This is what it means to say that humans are rational: we are naturally convinced by arguments in which the premises support the conclusion.  We can’t help but be persuaded by good arguments.²

The term infer is a Latin term, meaning to carry in.  This definition is helpful in understanding the sense of infer in arguments – the premises infer or carry in the conclusion.  You can’t let the premises in (as true) without letting the conclusion in (as true) along with them.  Thus, the premises have a certain sort of relationship to the conclusion; namely, a support relationship, or an inferential relationship.

Arguments are divided into categories depending on the type and strength of the support relationship between the premises and the conclusion.  First, the premises can support the conclusion in an absolute way; that is, the truth of the premises guarantees the truth of the conclusion, absolutely.  For instance, consider the argument already presented:

All students are humans.

Bob is a student

Therefore, Bob is a human.

In this argument, IF the premises are true (which we will assume for the moment), the conclusion MUST be true; it cannot be false.  So we say the support relationship here is absolute.  Arguments in which the support relationship is absolute are referred to as DEDUCTIVE arguments.

Now consider the argument:

Most people who drive Jaguar cars are rich.

Bob drives a Jaguar car.

Therefore, Bob is rich.

In this argument, even if we take the premises to be true, there is no attempt to demonstrate absolutely that Bob is rich.  The attempt is only to make the reader think it is probable that Bob is rich (the key here is the word most in the first premise).  There is an attempt at persuasion here, but not in an absolute sense.  Arguments in which the premises only give probabilistic support for the conclusion (that is, they make the conclusion probably, not necessarily) are called INDUCTIVE arguments.  Inductive arguments that make the conclusion very probable are better (more convincing) arguments, while inductive arguments that don’t make the conclusion very probable are less convincing and worse arguments.

There is one more possible (but not very interesting) support relationship.  That is, the premise could have nothing at all to do with the conclusion, in which case they don’t really have a support relationship (that is, their truth has no effect on the probability of the truth of the conclusion.  For instance,

The sky is blue.

Therefore, birds fly.

Although this text has the form of an argument, the premise has no support relationship with the conclusion, so this text is not properly an argument at all.

The different support relationships can be illustrated like this:

Strength of Support Relationships

Identifying Arguments

Types of arguments

Arguments, then, fall into two categories.  If the premises support the conclusion absolutely (the conclusion MUST be true if the premises are true), the argument is