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Introduction

1.1 Logic

Logic¹ is the analysis and appraisal of arguments. Here we’ll examine reasoning on philosophical areas (like God, free will, and morality) and on other areas (like backpacking, water pollution, and football). Logic is a useful tool to clarify and evaluate reasoning, whether on deeper questions or on everyday topics.

¹ Key terms (like “logic”) are introduced in bold. Learn each key term and its definition.

Why study logic? First, logic builds our minds. Logic develops analytical skills essential in law, politics, journalism, education, medicine, business, science, math, computer science, and most other areas. The exercises in this book are designed to help us think more clearly (so people can better understand what we’re saying) and logically (so we can better support our conclusions).

Second, logic deepens our understanding of philosophy – which can be defined as reasoning about the ultimate questions of life. Philosophers ask questions like “Why accept or reject free will?” or “Can one prove or disprove God’s existence?” or “How can one justify a moral belief?” Logic gives tools to deal with such questions. If you’ve studied philosophy, you’ll likely recognize some of the philosophical reasoning in this book. If you haven’t studied philosophy, you’ll find this book a good introduction to the subject. In either case, you’ll get better at recognizing, understanding, and appraising philosophical reasoning.

Finally, logic can be fun. Logic will challenge your thinking in new ways and will likely fascinate you. Most people find logic enjoyable.

1.2 Valid arguments

I begin my basic logic course with a multiple-choice test. The test has ten problems; each gives information and asks what conclusion necessarily follows. The problems are fairly easy, but most students get about half wrong.² 0002

² Http://www.harryhiker.com/logic.htm has my pretest in an interactive format. I suggest that you try it. I developed this test to help a psychologist friend test the idea that males are more logical than females; both groups, of course, did equally well on the problems.

Here’s a problem that almost everyone gets right:

· If you overslept, you’ll be late.

· You aren’t late.

Therefore

· (a) You did oversleep.

· (b) You didn’t oversleep. ⇐ correct

· (c) You’re late.

· (d) None of these follows.

With this next one, many wrongly pick answer “(b)”:

· If you overslept, you’ll be late.

· You didn’t oversleep.

Therefore

· (a) You’re late.

· (b) You aren’t late.

· (c) You did oversleep.

· (d) None of these follows. ⇐ correct

Here “You aren’t late” doesn’t necessary follow, since you might be late for another reason; maybe your car didn’t start.¹ The pretest shows that untrained logical intuitions are often unreliable. But logical intuitions can be developed; yours will likely improve as you work through this book. You’ll also learn techniques for testing arguments.

¹ These two arguments were taken from Matthew Lipman’s fifth-grade logic textbook: Harry Stottlemeier’s Discovery (Caldwell, NJ: Universal Diversified Services, 1974).

In logic, an argument is a set of statements consisting of premises (supporting evidence) and a conclusion (based on this evidence). Arguments put reasoning into words. Here’s an example (“∴” is for “therefore”):

Valid argument

· If you overslept, you’ll be late.

· You aren’t late.

· ∴ You didn’t oversleep.

An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. “Valid” doesn’t say that the premises are true, but only that the conclusion follows from them: if the premises were all true, then the conclusion would have to be true. Here we implicitly assume that there’s no shift in the meaning or reference of the terms; hence we must use “overslept,” “late,” and “you” the same way throughout the argument.²

² It’s convenient to allow arguments with zero premises; such arguments (like “∴ x = x”) are valid if and only if the conclusion is a necessary truth (couldn’t have been false).

Our argument is valid because of its logical form: how it arranges logical notions like “if-then” and content like “You overslept.” We can display the form using words or symbols for logical notions and letters for content phrases:

· If you overslept, you’ll be late.

· You aren’t late.

· ∴ You didn’t oversleep.

· If A then B Valid

· Not-B

· ∴ Not-A

Our argument is valid because its form is correct. Replacing “A” and “B” with other content yields another valid argument of the same form: 0003

· If you’re in France, you’re in Europe.

· You aren’t in Europe.

· ∴ You aren’t in France.

· If A then B Valid

· Not-B

· ∴ Not-A

Logic studies forms of reasoning. The content can deal with anything – backpacking, math, cooking, physics, ethics, or whatever. When you learn logic, you’re learning tools of reasoning that can be applied to any subject.

Consider our invalid example:

· If you overslept, you’ll be late.

· You didn’t oversleep.

· ∴ You aren’t late.

· If A then B Invalid

· Not-A

· ∴ Not-B

Here the second premise denies the first part of the if-then; this makes it invalid. Intuitively, you might be late for some other reason – just as, in this similar argument, you might be in Europe because you’re in Italy:

· If you’re in France, you’re in Europe.

· You aren’t in France.

· ∴ You aren’t in Europe.

· If A then B Invalid

· Not-A

· ∴ Not-B

1.3 Sound arguments

Logicians distinguish valid arguments from sound arguments:

An argument is valid if it would be contradictory to have the premises all true and conclusion false.

An argument is sound if it’s valid and every premise is true.

Calling an argument “valid” says nothing about whether its premises are true. But calling it “sound” says that it’s valid (the conclusion follows from the premises) and has all premises true. Here’s a sound argument:

Valid and true premises

· If you’re reading this, you aren’t illiterate.

· You’re reading this.

· ∴ You aren’t illiterate.

When we try to prove a conclusion, we try to give a sound argument: valid and true premises. With these two things, we have a sound argument and our conclusion has to be true.

An argument could be unsound in either of two ways: (1) it might have a false premise or (2) its conclusion might not follow from the premises: 0004

First premise false

· All logicians are millionaires.

· Gensler is a logician.

· ∴ Gensler is a millionaire.

Conclusion doesn’t follow

· All millionaires eat well.

· Gensler eats well.

· ∴ Gensler is a millionaire.

When we criticize an opponent’s argument, we try to show that it’s unsound. We try to show that one of the premises is false or that the conclusion doesn’t follow. If the argument has a false premise or is invalid, then our opponent hasn’t proved the conclusion. But the conclusion still might be true – and our opponent might later discover a better argument for it. To show a view to be false, we must do more than just refute an argument for it; we must give an argument that shows the view to be false.

Besides asking whether premises are true, we can ask how certain they are, to ourselves or to others. We’d like our premises to be certain and obvious to everyone. We usually have to settle for less; our premises are often educated guesses or personal convictions. Our arguments are only as strong as their premises. This suggests a third strategy for criticizing an argument; we could try to show that one or more of the premises are very uncertain.

Here’s another example of an argument. In fall 2008, before Barack Obama was elected US president, he was ahead in the polls. But some thought he’d be defeated by the “Bradley effect,” whereby many whites say they’ll vote for a black candidate but in fact don’t. Barack’s wife Michelle, in an interview with Larry King, argued that there wouldn’t be a Bradley effect:

· Barack Obama is the Democratic nominee.

· If there’s going to be a Bradley effect, then Barack wouldn’t be the nominee [because the effect would have shown up in the primaries].

· ∴ There isn’t going to be a Bradley effect.

Once she gives this argument, we can’t just say “Well, my opinion is that there will be a Bradley effect.” Instead, we have to respond to her reasoning. It’s clearly valid – the conclusion follows from the premises. Are the premises true? The first premise was undeniable. To dispute the second premise, we’d have to argue that the Bradley effect would appear in the final election but not in the primaries. So this argument changes the discussion. (By the way, there was no Bradley effect when Obama was elected president a month later.)

Logic, while not itself resolving substantive issues, gives us intellectual tools to reason better about such issues. It can help us to be more aware of reasoning, to express reasoning clearly, to determine whether a conclusion follows from the premises, and to focus on key premises to defend or criticize.

Logicians call statements true or false (not valid or invalid). And they call arguments valid or invalid (not true or false). While this is conventional usage, it pains a logician’s ears to hear “invalid statement” or “false argument.”0005

Our arguments so far have been deductive. With inductive arguments, the conclusion is only claimed to follow with probability (not with necessity):

Deductively valid

· All who live in France live in Europe.

· Pierre lives in France.

· ∴ Pierre lives in Europe.

Inductively strong

· Most who live in France speak French.

· Pierre lives in France.

· This is all we know about the matter.

· ∴ Pierre speaks French (probably).

The first argument has a tight connection between premises and conclusion; it would be impossible for the premises to all be true but the conclusion false. The second has a looser premise–conclusion connection. Relative to the premises, the conclusion is only a good guess; it’s likely true but could be false (perhaps Pierre is the son of the Polish ambassador and speaks no French).

1.4 The plan of this book

This book starts simply and doesn’t presume any previous study of logic. Its four parts cover a range of topics, from basic to rather advanced:

· Chapters 2 to 5 cover syllogistic logic (an ancient branch of logic that focuses on “all,” “no,” and “some”), meaning and definitions, informal fallacies, and inductive reasoning.

· Chapters 6 to 9 cover classical symbolic logic, including propositional logic (about “if-then,” “and,” “or,” and “not”) and quantificational logic (which adds “all,” “no,” and “some”). Each chapter here builds on previous ones.

· Chapters 10 to 14 cover advanced symbolic systems of philosophical interest: modal logic (about “necessary” and “possible”), deontic logic (about “ought” and “permissible”), belief logic (about consistent believing and willing), and a formalized ethical theory (featuring the golden rule). Each chapter here presumes the previous symbolic ones (except that Chapter 10 depends only on 6 and 7, and Chapter 11 isn’t required for 12 to 14).

· Chapters 15 to 18 cover metalogic (analyzing logical systems), history of logic, deviant logics, and philosophy of logic (further philosophical issues). These all assume Chapter 6.

Chapters 2–8 and 10 are for basic logic courses, while other chapters are more advanced. Since this book is so comprehensive, it has much more material than can be covered in one semester.

Logic requires careful reading, and sometimes rereading. Since most ideas build on previous ideas, you need to keep up with readings and problems. The companion LogiCola software (see Preface) is a great help.