Elementary Logic

Brian Garrett

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317547495
• Edition: 1
• Topic: Philosophie
• Subtopic: Histoire et théorie de la philosophie

Elementary Logic explains what logic is, how it is done, and why it can be exciting. The book covers the central part of logic that all students have to learn: propositional logic. It aims to provide a crystal-clear introduction to what is often regarded as the most technically difficult area in philosophy. The book opens with an explanation of what logic is and how it is constructed. Subsequent chapters take the reader step-by-step through all aspects of elementary logic. Throughout, ideas are explained simply and directly, with the chapters packed with overviews, illustrative examples, and summaries. Each chapter builds on previous explanation and example, with the final chapters presenting more advanced methods. After a discussion of meta-logic and logical systems, the book closes with an exploration of how paradoxes can exist in the world of logic. Elementary Logic's clarity and engagement make it ideal for any reader studying logic for the first time.

1 OVERVIEW

The aim of this book is to introduce students to the ideas and techniques of symbolic logic. Logic is the study of arguments. After working through this book the reader should be in a position to identify and evaluate a wide range of arguments.

Once an argument has been identified, we need to determine whether it is a good argument or a bad one. By ‘good argument’ we mean a valid argument; by ‘bad argument’ we mean an invalid argument. Our primary method for determining validity will be natural deduction proofs, but we also use the (simpler but more cumbersome) method of truth-trees. In addition, we briefly show how truth-tables can also be used to test for validity.

Elementary logic studies arguments, and, in doing so, it studies the logical or inferential properties of the so-called logical connectives: ‘and’, ‘if … then …’, ‘or’, ‘not’ and ‘if and only if’. We use these logical words much of the time, even if we might find it hard to say what they mean. In logic, however, these key words have a clear and explicit meaning.

SOME KEY TERMS AND IDEAS

Premises and conclusion

In elementary logic, the premises and conclusion of an argument are all declarative sentences; that is, they are sentences that are either true or false. There are only two truth-values and each declarative sentence has one and only one of them. ‘The cat is on the mat’, ‘no one loves Raymond’ and ‘all bachelors are bald’ are examples of declarative sentences.

‘Is the cat on the mat?’, ‘Put the cat on the mat!’, ‘Don’t park there’ and ‘I pronounce you man and wife’ are examples of non-declarative sentences. It would be odd to respond to utterances of any of these four sentences with ‘That’s true!’ (or ‘That’s false!’). In these latter cases, the utterer is not stating or asserting things to be a certain way (so that things would either be that way or not).

Logical connectives

As noted, we are concerned with five logical connectives: ‘and’, ‘if … then …’, ‘or’, ‘not’ and ‘if and only if’. These connectives are used to form new sentences. Thus, from the sentences ‘Bill is bald’ and ‘Fred is fat’ we can use ‘and’ to form the sentence ‘Bill is bald and Fred is fat’. We can use ‘if … then …’ to form the sentence ‘if Bill is bald then Fred is fat’. And so on with the other connectives. (Note that all the connectives except ‘not’ take two sentences in order to form a new sentence. ‘Not’ takes only one.)

The logical connectives are thus sentence-forming operators. In addition, they are truth-functional operators. That is, the truth-value of a new sentence formed using one (or more) of our connectives is fixed entirely by the truth-values of the original sentence or sentences. For example, if it is true that Bill is bald, but false that Fred is fat, then the truth-value of ‘Bill is bald and Fred is fat’ is immediately fixed (as false). The truth-functionality of the five connectives allows their meaning to be displayed by their complete truth-tables (see Chapter 2).

The meaning of the logical connectives is captured not only by their truth-tables, but also by the rules of inference associated with each connective. Thus, for example, ‘and’ is associated with the rules: from ‘A’ and ‘B’ infer the conjunction ‘A and B’; from ‘A and B’ infer ‘A’ (or infer ‘B’).

There is some debate in philosophical logic over whether the fundamental meaning of the connectives is given by truth-tables or by inference rules. This issue is discussed in Chapter 13.

Argument

An argument typically consists of one or more premises and a conclusion. (I say typically because some arguments have no premises. This sounds odd, but see Chapter 8.) Words like ‘so’, ‘thus’, ‘therefore’, ‘hence, ‘whence’, ‘accordingly’ and ‘consequently’ indicate that what comes next is the conclusion.

Here are some arguments:

(A) Bill is in Paris and Bill is happy; so
Bill is in Paris.

(B) If Bill is in Paris, Bill is in France;
Bill is not in France; thus
Bill is not in Paris.

(C) Bill is either in France or in Germany;
Bill is not in France; hence
Bill is in Germany.

Validity

Arguments (A)–(C) are valid. Logic is the study of valid argument. Detecting and displaying validity is the goal of logic. Our central method for demonstrating the validity of an argument will be that of natural deduction proof.

A successful proof reveals that a particular valid argument is valid, and why. In a valid proof the conclusion is shown to follow from the premises by self-evidently valid (hence, truth-preserving) rules of inference.

Since rules of inference are sensitive only to the shape or form of lines in a proof, we can say that an argument is valid just if it is truth-preserving (i.e. never leads from truths to falsehoods) in virtue of its pattern or form.

Arguably, the notion of truth-preservation implicitly involves the notion of form. That is, an argument counts as truth-preserving if and only if no other argument of the same form ever leads from true premises to a false conclusion.

We should say a little more about the notions of form and rule of inference.

Form

What do we mean by logical form, and why is it important? We can understand the notion of form by considering the following two arguments:

(D) If Mary loves Bill, she will marry him;
Mary loves Bill; so
Mary will marry him.

(E) If Ingrid hates Lars, she will divorce him;
Ingrid hates Lars; so
She will divorce him.

Arguments (D) and (E) are both valid. In each case, given the premises, the conclusion must follow. In each case, it is impossible for the premises to be true and the conclusion false.

But the explanation of the validity of (D) is not different from the explanation of the validity of (E). If you look at (D) and (E), you can see that they have a common form or shape. They are both valid in virtue of having that common form. (We shall assume throughout the book that each argument has one and only one form.)

We can represent the common form thus:

(MP) If P then Q;
P; so
Q

The italicized letters P and Q are variables. That is, they do not stand for particular sentences, but represent the positions that can be occupied by any pair of sentences. This simple argument form – called modus ponens – is clearly valid. English arguments of this form – such as (D) or (E) – are valid in virtue of being of that form.

The point generalizes. Any valid argument we encounter in this book is valid because it is an instance of a valid form, a form that can be shared by a potential infinity of other arguments.

Rule of inference

The notions of form and rule of inference are connected. A rule of inference, which allows the move from one or more lines in a proof to a new line, is responsive only to the form or shape of the individual lines.

We can now see the connection between the notion of a form and the notion of a rule of inference. A rule of inference is sensitive only to shapes. For example, the rule of inference we shall use to show (D) and (E) to be valid, known as the rule of Arrow Out (→O), says that from two lines of the shape ‘if P then Q’ and ‘P’, you may create a new line ‘Q’. (This is the topic of Chapter 3.)

This rule validates (D) and (E) (and any other natural-language argument of the same form). The content is irrelevant. It is irrelevant that one set of premises concerns Mary and marriage, the other Ingrid and divorce. All that matters is the form, not the content.

Just as a particular argument is valid in virtue of having a valid form, so the rules of inference we employ in our proof system must also be valid, that is, truth-preserving. The rule of →O, for example, is clearly truth-preserving. No application of it will lead from truth to falsity. All the inference rules we use in this book are self-evidently valid.

The concept of validity thus applies to natural-language arguments such as (A)–(E), to forms of argument, to rules of inference, and to sequents. (A sequent, as we shall see in Chapter 3, is an argument translated fully into our symbolic language.)

Validity and form

The aim of logic is to show that a valid argument is valid, and why. A valid argument is one in which the conclusion follows from the premises by valid rules of inference. Rules of inference are responsive only to the form or shape of lines in an argument or proof. There is, there...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Overview
• 2 Logical connectives and truth-tables
• 3 Conditional
• 4 Conjunction
• 5 Conditional proof
• 6 Solutions to selected exercises, I
• 7 Negation
• 8 Disjunction
• 9 Biconditional
• 10 Solutions to selected exercises, II
• 11 Derived rules
• 12 Truth-trees
• 13 Logical reflections
• 14 Logic and paradoxes
• Glossary
• Further reading
• References
• Index