Forallx - An Introduction to Formal Logic

P.D. Magnus

Forallx - An Introduction to Formal Logic

ISBN 978-80-272-2658-0

Produced by Studium Publishing, 2018

Studium Publishing, 2018. No claim to the material licensed as Creative Commons CC BY-SA 3.0.

Forallx, An Introduction to Formal Logic, by P.D. Magnus, available on-line at http://www.fecundity.com/logic

is offered under a Creative Commons license (Attribution-ShareAlike 3.0), https://creativecommons.org/licenses/by-sa/3.0/

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© 2005-2014 by P.D. Magnus. Some rights reserved

The author would like to thank the people who made this project possible. Notable among these are Cristyn Magnus, who read many early drafts; Aaron Schiller, who was an early adopter and provided considerable, helpful feedback; and Bin Kang, Craig Erb, Nathan Carter, Wes McMichael, Selva Samuel, Dave Krueger, Brandon Lee, Toan Tran, and the students of Introduction to Logic, who detected various errors in previous versions of the book.

Table of Contents

1 What is logic?

Arguments

Sentences

Two ways that arguments can go wrong

Deductive validity

Other logical notions

Formal languages

Practice Exercises

2 Sentential logic

Sentence letters

Connectives

Other symbolization

Sentences of SL

Practice Exercises

3 Truth tables

Truth-functional connectives

Complete truth tables

Using truth tables

Partial truth tables

Practice Exercises

4 Quantified logic

From sentences to predicates

Building blocks of QL

Quantifiers

Translating to QL

Sentences of QL

Identity

Practice Exercises

5 Formal semantics

Semantics for SL

Interpretations and models in QL

Semantics for identity

Working with models

Truth in QL

Practice Exercises

6 Proofs

Basic rules for SL

Derived rules

Rules of replacement

Rules for quantifiers

Rules for identity

Proof strategy

Proof-theoretic concepts

Proofs and models

Soundness and completeness

Practice Exercises

Other symbolic notation

Solutions to selected exercises

Quick Reference

Chapter 1

What is logic?

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Logic is the business of evaluating arguments, sorting good ones from bad ones. In everyday language, we sometimes use the word ‘argument’ to refer to belligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you.

In logic, we are not interested in the teeth-gnashing, hair-pulling kind of argument. A logical argument is structured to give someone a reason to believe some conclusion. Here is one such argument:

(1) It is raining heavily.

(2) If you do not take an umbrella, you will get soaked.

.˙. You should take an umbrella.

The three dots on the third line of the argument mean ‘Therefore’ and they indicate that the final sentence is the conclusion of the argument. The other sentences are premises of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion.

This chapter discusses some basic logical notions that apply to arguments in a natural language like English. It is important to begin with a clear understanding of what arguments are and of what it means for an argument to be valid. Later we will translate arguments from English into a formal language. We want formal validity, as defined in the formal language, to have at least some of the important features of natural-language validity.

Arguments

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When people mean to give arguments, they typically often use words like ‘therefore’ and ‘because.’ When analyzing an argument, the first thing to do is to separate the premises from the conclusion. Words like these are a clue to what the argument is supposed to be, especially if— in the argument as given— the conclusion comes at the beginning or in the middle of the argument.

premise indicators: since, because, given that

conclusion indicators: therefore, hence, thus, then, so

To be perfectly general, we can define an ARGUMENT as a series of sentences. The sentences at the beginning of the series are premises. The final sentence in the series is the conclusion. If the premises are true and the argument is a good one, then you have a reason to accept the conclusion.

Notice that this definition is quite general. Consider this example:

There is coffee in the coffee pot.

There is a dragon playing bassoon on the armoire.

.˙. Salvador Dali was a poker player.

It may seem odd to call this an argument, but that is because it would be a terrible argument. The two premises have nothing at all to do with the conclusion. Nevertheless, given our definition, it still counts as an argument— albeit a bad one.

Sentences

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In logic, we are only interested in sentences that can figure as a premise or conclusion of an argument. So we will say that a SENTENCE is something that can be true or false.

You should not confuse the idea of a sentence that can be true or false with the difference between fact and opinion. Often, sentences in logic will express things that would count as facts— such as ‘Kierkegaard was a hunchback’ or ‘Kierkegaard liked almonds.’ They can also express things that you might think of as matters of opinion— such as, ‘Almonds are yummy.’

Also, there are things that would count as ‘sentences’ in a linguistics or grammar course that we will not count as sentences in logic.

Questions In a grammar class, ‘Are you sleepy yet?’ would count as an interrogative sentence. Although you might be sleepy or you might be alert, the question itself is neither true nor false. For this reason, questions will not count as sentences in logic. Suppose you answer the question: ‘I am not sleepy.’ This is either true or false, and so it is a sentence in the logical sense. Generally, questions will not count as sentences, but answers will.

‘What is this course about?’ is not a sentence. ‘No one knows what this course is about’ is a sentence.

Imperatives Commands are often phrased as imperatives like ‘Wake up!’, ‘Sit up straight’, and so on. In a grammar class, these would count as imperative sentences. Although it might be good for you to sit up straight or it might not, the command is neither true nor false. Note, however, that commands are not always phrased as imperatives. ‘You will respect my authority’ is either true or false— either you will or you will not— and so it counts as a sentence in the logical sense.

Exclamations ‘Ouch!’ is sometimes called an exclamatory sentence, but it is neither true nor false. We will treat ‘Ouch, I hurt my toe!’ as meaning the same thing as ‘I hurt my toe.’ The ‘ouch’ does not add anything that could be true or false.

Two ways that arguments can go wrong

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Consider the argument that you should take an umbrella (on p. 5, above). If premise (1) is false— if it is sunny outside— then the argument gives you no reason to carry an umbrella. Even if it is raining outside, you might not need an umbrella. You might wear a rain pancho or keep to covered walkways. In these cases, premise (2) would be false, since you could go out without an umbrella and still avoid getting soaked.

Suppose for a moment that both the premises are true. You do not own a rain pancho. You need to go places where there are no covered walkways. Now does the argument show you that you should take an umbrella? Not necessarily. Perhaps you enjoy walking in the rain, and you would like to get soaked. In that case, even though the premises were true, the conclusion would be false.

For any argument, there are two ways that it could be weak. First, one or more of the premises might be false. An argument gives you a reason to believe its conclusion only if you believe its premises. Second, the premises might fail to support the conclusion. Even if the premises were true, the form of the argument might be weak. The example we just considered is weak in both ways.

When an argument is weak in the second way, there is something wrong with the logical form of the argument: Premises of the kind given do not necessarily lead to a conclusion of the kind given. We will be interested primarily in the logical form of arguments.

Consider another example:

You are reading this book.

This is a logic book.

.˙. You are a logic student.

This is not a terrible argument. Most people who read this book are logic students. Yet, it is possible for someone besides a logic student to read this book. If your roommate picked up the book and thumbed through it, they would not immediately become a logic student. So the premises of this argument, even though they are true, do not guarantee the truth of the conclusion. Its logical form is less than perfect.

An argument that had no weakness of the second kind would have perfect logical form. If its premises were true, then its conclusion would necessarily be true. We call such an argument ‘deductively valid’ or just ‘valid.’

Even though we might count the argument above as a good argument in some sense, it is not valid; that is, it is ‘invalid.’ One important task of logic is to sort valid arguments from invalid arguments.

Deductive validity

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An argument is deductively VALID if and only if it is impossible for the premises to be true and the conclusion false.

The crucial thing about a valid argument is that it is impossible for the premises to be true at the same time that the conclusion is false. Consider this example:

Oranges are either fruits or musical instruments.

Oranges are not fruits.

.˙. Oranges are musical instruments.

The conclusion of this argument is ridiculous. Nevertheless, it follows validly from the premises. This is a valid argument. If both premises were true, then the conclusion would necessarily be true.

This shows that a deductively valid argument does not need to have true premises or a true conclusion. Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider this example:

London is in England.

Beijing is in China.

.˙. Paris is in France.

The premises and conclusion of this argument are, as a matter of fact, all true. This is a terrible argument, however, because the premises have nothing to do with the conclusion. Imagine what would happen if Paris declared independence from the rest of France. Then the conclusion would be false, even though the premises would both still be true. Thus, it is logically possible for the premises of this argument to be true and the conclusion false. The argument is invalid.

The important thing to remember is that validity is not about the actual truth or falsity of the sentences in the argument. Instead, it is about the form of the argument: The truth of the premises is incompatible with the falsity of the conclusion.

Inductive arguments

There can be good arguments which nevertheless fail to be deductively valid. Consider this one:

In January 1997, it rained in San Diego.

In January 1998, it rained in San Diego.

In January 1999, it rained in San Diego.

.˙. It rains every January in San Diego.

This is an INDUCTIVE argument, because it generalizes from many cases to a conclusion about all cases.

Certainly, the argument could be made stronger by adding additional premises: In January 2000, it rained in San Diego. In January 2001. . . and so on. Regardless of how many premises we add, however, the argument will still not be deductively valid. It is possible, although unlikely, that it will fail to rain next January in San Diego. Moreover, we know that the weather can be fickle. No amount of evidence should convince us that it rains there every January. Who is to say that some year will not be a freakish year in which there is no rain in January in San Diego; even a single counter-example is enough to make the conclusion of the argument false.

Inductive arguments, even good inductive arguments, are not deductively valid. We will not be interested in inductive arguments in this book.

Other logical notions

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In addition to deductive validity, we will be interested in some other logical concepts.

Truth-values

True or false is said to be the TRUTH-VALUE of a sentence. We defined sentences as things that could be true or false; we could have said instead that sentences are things that can have truth-values.

Logical truth

In considering arguments formally, we care about what would be true if the premises were true. Generally, we are not concerned with the actual truth value of any particular sentences— whether they are actually true or false. Yet there are some sentences that must be true, just as a matter of logic.

Consider these sentences:

1. It is raining.

2. Either it is raining, or it is not.

3. It is both raining and not raining.

In order to know if sentence 1 is true, you would need to look outside or check the weather channel. Logically speaking, it might be either true or false. Sentences like this are called contingent sentences.

Sentence 2 is different. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or not. This sentence is logically true; it is true merely as a matter of logic, regardless of what the world is actually like. A logically true sentence is called a TAUTOLOGY .

You do not need to check the weather to know about sentence 3, either. It must be false, simply as