Introduction to Logic

Harry J Gensler

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781317436102
• Edition: 3
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Introduction to Logic is clear and concise, uses interesting examples (many philosophical in nature), and has easy-to-use proof methods. Its key features, retained in this Third Edition, include:

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• simpler ways to test arguments, including an innovative proof method and the star test for syllogisms;

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• a wide scope of materials, suiting it for introductory or intermediate courses;

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• engaging examples, from philosophy and everyday life;

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• useful for self-study and preparation for standardized tests, like the LSAT;

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• a reasonable price (a third the cost of some competitors); and

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• exercises that correspond to the free LogiCola instructional program.

This Third Edition:

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• improves explanations, especially on areas that students find difficult;

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• has a fuller explanation of traditional Copi proofs and of truth trees; and

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• updates the companion LogiCola software, which now is touch friendly (for use on Windows tablets and touch monitors), installs more easily on Windows and Macintosh, and adds exercises on Copi proofs and on truth trees. You can still install LogiCola for free (from http://www.harryhiker.com/lc or http://www.routledge.com/cw/gensler).

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1
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...