Quantification Theory

J. A. Faris

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781000735529
• Edition: 1
• Topic: Filosofia
• Subtopic: Storia e teoria della filosofia

Originally published in 1964. This book is concerned with general arguments, by which is meant broadly arguments that rely for their force on the ideas expressed by all, every, any, some, none and other kindred words or phrases. A main object of quantificational logic is to provide methods for evaluating general arguments. To evaluate a general argument by these methods we must first express it in a standard form. Quantificational form is dealt with in chapter one and in part of chapter three; in the remainder of the book an account is given of methods by which arguments when formulated quantificationally may be tested for validity or invalidity. Some attention is also paid to the logic of identity and of definite descriptions. Throughout the book an attempt has been made to give a clear explanation of the concepts involved and the symbols used; in particular a step-by-step and partly mechanical method is developed for translating complicated statements of ordinary discourse into the appropriate quantificational formulae. Some elementary knowledge of truth-functional logic is presupposed.

CHAPTER ONE

1. Introduction. Quantification theory is concerned fundamentally with general arguments. We begin by explaining what we mean by this term. In much of our reasoning we make use of the ideas expressed by the words or phrases: all, every, each, any, some, at least one, not all, not one, none. Let us refer to these expressions and to others which are similar in meaning and use as generalizes. A distinctively general argument may be defined as any argument the force of which depends on the meaning and use of one or more generalizers. For example the following simple argument the force of which depends on the generalizers every and some is a distinctively general argument.

Every applicant will be employed; some refugees will be applicants; therefore some refugees will be employed.

Distinctively general arguments form the main part of the field of quantificational logic. However, it would not be satisfactory to define quantificational logic as being concerned solely with distinctively general arguments; so we must try to state the position more exactly. In doing so it will be necessary to refer to truth-functional logic. We assume in the present book, which is indeed written more or less as a companion volume to my earlier monograph Truth-Functional Logic,¹ a reasonable acquaintance with this subject.

It is desirable to regard quantificational logic as including truth-functional logic. It follows of course that it is applicable to any arguments to which truth-functional logic is applicable. Let us refer to any argument to which truth-functional logic is applicable as a truth-functional argument¹ and let us say that an argument is a general argument if and only if it is either distinctively general or truth-functional. We may now say that the field of quantificational logic is the class of general arguments. Our purpose is to show how truth-functional logic may be extended to form a system, quantificational logic, which is applicable to any general argument.

¹ Truth-Functional Logic will be referred to hereafter as TFL.

There are of course other systems of logic which deal with distinctively general arguments. One such system is the traditional syllogistic logic; another is the Boolean algebra of classes. However, neither of these is fully comprehensive. To get a true comparison we need to consider what the scope of each of these systems would become if it were supplemented by truth-functional logic. We shall say that a system of logic is adequate to a certain field if it provides non-intuitive methods by which we may attempt to test the validity of any argument within the field. Now the traditional syllogistic, even when supplemented by truth-functional logic, is adequate only to a small section of the field of distinctively general arguments. Similarly a supplemented Boolean algebra of classes is adequate only to a part of the field. The interest of quantificational logic, as compared with these earlier and related disciplines, is that it is adequate to the whole field of general, including distinctively general, arguments.

2. Direct and indirect evaluation. We shall speak of evaluating an argument. By this we mean determining whether an argument is valid or invalid. Any particular system of formal logic provides methods of evaluating arguments of a certain kind. In general the methods of a system of formal logic apply directly only to arguments which have a certain standard form. We can make use of these methods to evaluate an argument Ζ only if Ζ is already in the standard form or if we are able to find some argument Z′ which is in the standard form and is equivalent to Ζ in the sense that the premisses of Z′ can be true if and only if the premisses of Ζ are true and the conclusion of Z′ can be true if and only if the conclusion of Ζ is true. When two arguments Ζ and Z′ are equivalent in this sense we may say that Ζ is expressible in standard form as Z′. If an argument Ζ is in the form which is standard for a particular system of logic the methods of that system may be applied to it directly. If Ζ is not in the standard form the methods of the system in question may yet be used indirectly, provided that Ζ can be expressed as Z′ where Z′ is in the standard form: the methods of the system are in this case applied directly to Z′ and from the result the evaluation of Ζ itself immediately follows.

¹ The scope of truth-functional logic is discussed in TFL, chapter V.

Quantificational logic is a particular system of formal logic the methods of which are applicable directly to arguments of a certain standard form which will here be called quantificational form. All general arguments can be expressed in quantificational form and may thus be evaluated by quantificational logic. The evaluation is indirect in the sense explained in the last paragraph. To evaluate a general argument G we first express it as an argument Q in quantificational form; we then use quantificational methods to evaluate Q and when this has been done we are able immediately to evaluate G. It can thus be seen that to be able to use quantificational logic to evaluate general arguments we need to be able to do two things; first, to express general arguments in quantificational form; second, to apply the methods of quantificational logic. What we are here calling quantification theory may accordingly be regarded as having two main elements:

(1) The study of quantificational form and of the way in which general arguments may be expressed in quantificational form.

(2) Quantificational logic proper, an account of the methods by which arguments in quantificational form may be evaluated.

The study of quantificational form will be our main concern in the rest of this chapter and in chapter three, sections 4 and 5; and broadly speaking quantificational logic proper will be dealt with in the remaining parts of the book.

We proceed now to give a preliminary account of quantificational form and an explanation of how general arguments may be expressed in this form.

3. Quantifier-matrix form of singular statements. Any statement¹ which is premiss or conclusion of a general argument and is not truth-functionally compound is either a general statement or a non-general statement. A general statement is normally expressed in quantificational form by means of a formula consisting of two main elements, a quantifier and a matrix. A non-general statement is not normally expressed in a quantifier-matrix form. However, although this is so, it will be convenient to begin here by taking a simple non-general statement and showing in stages how it may be transformed into a statement with the distinctively quantifier-matrix structure. The quantificational expression of general statements can then be explained as a natural development from the quantificational expression of non-general statements.

¹ The word ‘statement’ is used throughout the present book in the sense in which ‘proposition’ is used in TFL. I hope that what this sense is will become sufficiently clear in the contexts. I may add that I do not now regard what is said about ‘proposition’ on page 2, lines 9 to 16, of TFL as giving a satisfactory account of the use of the word in that book.

Consider the following statement:

(1) Nero succeeded Claudius.

In some contexts it would be natural to express this in this form:

(2) Of Nero it is true that he succeeded Claudius, which we may regard as consisting of two parts: a prefix, Of Nero it is true that and a matrix he succeeded Claudius. Let us now substitute a different prefix, Nero is such that. This gives us:

(3) Nero is such that he succeeded Claudius.

(3) might normally be taken to differ slightly in meaning from (1) and (2); it might be supposed not merely to mean that Nero succeeded Claudius but also to carry the implication that he was the sort of person who might have been expected to succeed Claudius and perhaps even the implication that he still exists. We stipulate that in what follows the phrase is such that is to be understood as being entirely without any additional implications of this sort. The prefix Nero is such that is to be understood as having exactly the same sense as Of Nero it is true that. Thus (1), (2) and (3) all have the same meaning.

We now transform (3) in a different sort of way. Using a construction found in verse:

Lars Porsena of Clusium By the Nine Gods he swore‒and common enough in colloquial speech, though frowned upon in literary prose, let us instead of (3) write:

(4) Nero he is such that he succeeded Claudius, w...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Contents
• Abbreviations
• Chapter One
• Chapter Two
• Chapter Three
• Chapter Four
• Index of Definitions