Truth-Functional Logic

J. A. Faris

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781000735536
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Originally published in 1962. This book gives an account of the concepts and methods of a basic part of logic. In chapter I elementary ideas, including those of truth-functional argument and truth-functional validity, are explained. Chapter II begins with a more comprehensive account of truth-functionality; the leading characteristics of the most important monadic and dyadic truth-functions are described, and the different notations in use are set forth. The main part of the book describes and explains three different methods of testing truth-functional aguments and agument forms for validity: the truthtable method, the deductive method and the method of normal forms; for the benefit mainly of readers who have not acquired in one way or another a general facility in the manipulation of symbols some of the procedures have been described in rather more detail than is common in texts of this kind. In the final chapter the author discusses and rejects the view, based largely on the so called paradoxes of material implication, that truth-functional logic is not applicable in any really important way to arguments of ordinary discourse.

Chapter One
INTRODUCTION

1. Sentences, propositions, arguments and the point of view of logic. Logic begins as the study from a particular point of view of certain types of argument. A person who makes an inference, i.e. does a piece of reasoning, and expresses it in some way, whether privately or publicly, may be said to use an argument. An argument has two essential parts: a set of one or more premisses and a conclusion, and it is said to be from its premisses to its conclusion. In this book we shall take it that the basic elements of an argument, i.e. the premisses and the conclusion, are propositions. We begin by saying something about how this term is to be understood. Consider the following sentence:

The population of the world is increasing.

This sentence may be used to make a statement or express a proposition: namely the proposition that the population of the world is increasing. Nevertheless although the proposition is expressed by the sentence it is not identical with the sentence. For the sentence could be translated into another language: if this were done we should have a different sentence but one which expressed the same proposition.

Propositions have two important properties. One of these has already emerged: a proposition is expressed by an indicative sentence but is not identical with a sentence, and any particular proposition may be expressed equally well by any one of a number of different sentences. The other property is that every proposition is either true or false.

The distinction between proposition and sentence, though of theoretical importance, will not be prominent in this book. On the contrary we will adopt for the sake of convenience two practices which will confine it to the background: in the first place in our illustrations we will normally use the same sentence to express the same proposition; in the second place when we wish to refer to a proposition which we express by means of a sentence S instead of using the phrase:

the proposition expressed by the sentence S

we shall say simply

the proposition S.

The second of these phrases is to be understood as an abbreviation for the first.

The other property we have referred to will, however, be constantly before us. It should be mentioned perhaps that it is sometimes questioned whether the property of being true or false is one that is in fact possessed by every proposition: it is suggested that there may be some propositions which are neither true nor false. However, although systems of logic, and indeed systems of truth-functional logic, have been devised which might be applicable if this were so, these must be taken to be outside our present field of study. It is a fundamental postulate of the system of logic which we are to expound that every proposition to which it applies is either true or false. We therefore regard this property of being either true or false as an essential property of propositions as we are using the term.

Let us now have an example of an argument. The following passage occurs in one of Bishop Berkeley’s dialogues.¹

¹ First Dialogue between Hylas and Philonous.

‘Because intense heat is nothing else but a particular kind of painful sensation; and pain cannot exist but in a perceiving being; it follows that no intense heat can really exist in an unperceiving corporeal substance.’

Berkeley is here presenting an argument which we may regard as having two premisses, namely the propositions:

Intense heat is nothing else but a particular kind of painful sensation,

and

Pain cannot exist but in a perceiving being.

The conclusion is the proposition:

No intense heat can really exist in an unperceiving corporeal substance.

In any argument the set of premisses is put forward as a reason for accepting the conclusion and in any presentation of an argument there is some indication of this relationship which enables us to identify premisses and conclusion respectively. In the present example the word because marks the premisses and the expression it follows that the conclusion. The word therefore is of course often used to mark the conclusion of an argument. The conclusion immediately follows the therefore; any propositions immediately preceding it are premisses. We will henceforward use this as our standard method of distinguishing premisses and conclusion. Premisses and conclusions will usually be numbered for ease of reference and an argument will sometimes be labelled with a capital letter and a number. This is exemplified in the following argument, which we call Al, in which the premisses are numbered (1) and (2) and the conclusion is numbered (3). It is obvious that Al is very closely related to Berkeley’s argument, though it might perhaps be disputed whether the two are identical:

(Al) (1) Every instance of intense heat is a painful sensation.

(2) No painful sensation is a thing capable of existing in an unperceiving being.

Therefore (3) No instance of intense heat is a thing capable of existing in an unperceiving being.

We began by saying that initially logic is the study of arguments from a certain point of view. We must now explain what this point of view is. Let us consider the argument Al which is set out above. Al is a philosophical argument; that is to say, it is of interest primarily to philosophers. If we study this argument as philosophers our interest will be to decide whether it provides us with sufficient reason for believing in its conclusion. To answer this question affirmatively we must convince ourselves that two conditions are satisfied: (i) that the premisses are both true; (ii) that the conclusion follows from the premisses. The argument provides us with good reason for accepting its conclusion as true if, but only if, it satisfies both conditions. The philosopher then will be interested in the two questions: whether condition (i) is satisfied and whether condition (ii) is satisfied. The logician as such, on the other hand, in studying this argument is interested in the second of these questions only, the question, that is, of whether or not the conclusion follows from the premisses. Another way of saying the same thing is to say that the logician is interested, not in the question of whether the conclusion of an argument is true, but rather in the question of whether the conclusion must be true if the premisses are. Yet another way is to say that the logician is interested not in whether the conclusion is true but in whether the argument is valid. These are various alternative ways of indicating roughly the point of view of the logician as such. In the last of them we have used the term valid. The concept of validity is of fundamental importance in the study of logic and we must now attempt to give a systematic account of it. To do this we must first introduce and explain the notion of a form of argument.

2. Argument forms; validity. Let us look again at the argument Al and compare with it another argument which we shall call A2.

(A2) (4) Every visitor is a person now present.

(5) No person now present is a prizewinner.

Therefore (6) No visitor is a prizewinner.

These two arguments are obviously different from one another in at least one respect. They are different in respect of what they are about. Al is about intense heat, painful sensations and things capable of existing in an unperceiving being; A2 is about visitors, persons now present, and prizewinners. Yet in another respect they are similar to one another. The respect in which Al and A2 are similar is that they have the same form. The form of these two arguments may be expressed as follows and called A.

What is meant by saying that Al and A2 have the same form A is this: if in the blank spaces in A marked a, b and c we write respectively:

intense heat, painful sensations, things capable of existing in an unperceiving being,

then we have the argument Al; whereas, on the other hand, if we write:

visitor, person now present, prizewinner,

then we have the argument A2.

If two arguments are of the same form they may be said to exemplify that form or to be exemplifications of it. We should regard each form as having a set of exemplification rules which tell us what may and what may not be put in the blank spaces or gaps. We need to know this in order to know what is to count as an exemplification of a given form. In each case two rules at least are required. One rule, which we may call the type rule, governs the type of expression which is to be inserted in the blank spaces; the other rule, here called the distribution rule, governs the way in which expressions of the appropriate type or types may be distributed over the different spaces. For the form A the two rules are:

(Type rule) Only a general term, i.e. a general noun or nominal phrase may be put in any space.

(Distribution rule) Where two spaces are marked by the same letter they must be filled in in the same way; spaces marked by different letters may be filled in either in the same way or in different ways.

There is a generally accepted convention that this distribution rule which has just been stated for A applies to all forms. Hence a distribution rule is rarely stated explicitly. The type rule is often conveyed in the guise of a description. The space-labelling letters (a, b, c in our example) are known as variables; when we fill in a space we are said to be making a substitution for or replacing a variable. Sometimes a writer, after setting out a form which contains variables x, y and z, may say: ‘where x, y and z are general term variables’ or ‘where x, y, z are proper name variables’ or again ‘where x, y and z are general terms’ or ‘where x, y and z are proper names’. Any such phrase really serves the purpose of giving the type rule for the form in question. Very often however the type rule is omitted altogether. There is a general principle that an argument form must be filled in in such a way that the result has for premisses and conclusion significant propositions expressed by properly constructed sentences. With many forms this principle alone is sufficient to determine what sort of insertion is legitimate; apart from this it is often obvious from the context what type rule is intended.

The form A would more usually be written:

(A) (1) Every a is a b.

(2) No b is a c.

Therefore (3) No a is a c.

An argument is not an exemplification of a form if it cannot be obtained when proper substitutions are made for the variables in the form. For example the following argument is not an exemplification of the form A:

(Bl) (1) Every visitor is a competitor in the last race.

Therefore (2) Every competitor in the last race is a visitor.

To see that this is so it is sufficient to notice...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Contents
• Chapter 1. Introduction
• Chapter 2. Truth-functions
• Chapter 3. The Truth-Table Method
• Chapter 4. The Deductive Method
• Chapter 5: Part I: Normal Forms
• List of Abbreviations
• Index of Definitions