The Structure of Aristotelian Logic

James Wilkinson Miller

• Publisher: Routledge
• Year: 2015
• Print ISBN: 9781138942424
• eBook ISBN: 9781317375425
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Originally published in 1938. This compact treatise is a complete treatment of Aristotle's logic as containing negative terms. It begins with defining Aristotelian logic as a subject-predicate logic confining itself to the four forms of categorical proposition known as the A, E, I and O forms. It assigns conventional meanings to these categorical forms such that subalternation holds. It continues to discuss the development of the logic since the time of its founder and address traditional logic as it existed in the twentieth century. The primary consideration of the book is the inclusion of negative terms - obversion, contraposition etc. – within traditional logic by addressing three questions, of systematization, the rules, and the interpretation.

CHAPTER I

POSTULATES FOR ARISTOTELIAN LOGIC¹

The task of the first four chapters is to formulate traditional logic as a deductive system. The present chapter will lay the foundation for this systematization; that is, it will give the undefined ideas, the postulates, and the definitions of the system.

We must begin with two preliminary matters, however. First, it will be instructive to note the partial systematization which appears in the introductory manuals of logic. Second, we must consider the general character which our systematization will have.

10. Reduction as Deduction. The standard textbooks of logic, following the procedure of Aristotle himself, give a partial systematization of traditional logic. This appears in the customary chapters on “the reduction of the imperfect figures of the syllogism.” A close scrutiny of “reduction” will show that it is nothing more nor less than an incomplete application of the postulational method to Aristotelian logic. Reduction of one form of syllogism to another, is the same as deduction of the former from the latter. In the textbooks all of the valid forms of syllogism in the second, third, and fourth figures are “reduced” to the valid forms in the first figure; that is simply another way of stating that from the valid forms of the first figure the other valid forms are deduced. In short, the textbooks take the valid forms of the first figure as axioms or postulates, and from them deduce the others as theorems.

This matter of reduction may be carried even further, as some of the textbooks suggest. For we need not postulate all six valid forms of the first figure. The weakened forms of Barbara and Celarent may be reduced “directly” to Barbara and Celarent themselves. Also, Darii and Ferio may be reduced “indirectly” to Celarent. Barbara and Celarent are thus sufficient to yield all the other forms, as Aristotle himself was aware; and this is what is meant by saying that the whole theory of the syllogism is covered by the dictum de omni et nullo. Finally, by means of obversion, Celarent may be reduced directly to Barbara. Therefore, from Barbara all of the other twenty-three valid forms of the syllogism may be deduced, our list of postulates is cut down to one, and the whole theory of the syllogism is covered by the dictum de omni.

Certain misapprehensions about the nature of this reduction of all forms of valid syllogisms to Barbara need to be guarded against.

In the first place, it might be assumed that Barbara acquires a certain priority or preeminence as a result of this reduction. In one sense this is true; in that particular systematization Barbara is prior to the others. But Barbara need not be made prior. Other syllogistic forms will serve quite as well. With the exception of the forms which have both premises universal but the conclusion particular, any one of the valid forms may replace Barbara as postulate. There are accordingly sixteen syllogistic forms from any one of which all the valid forms may be deduced. Another reason why Barbara—and the other syllogisms of the first figure—are supposed to be preëminent is that they are held to be more “natural” and “self-evident” than syllogisms in the other figures. This, however, is an epistemological or perhaps psychological matter, and need not concern us.

Another misapprehension to be noted is the supposition that the dictum de omni et nullo is more fundamental than Barbara and Celarent themselves. The dictum may be taken in either of two ways. First, it may be regarded as merely stating in verbal form what Barbara and Celarent state in quasi-symbolic form; in that case it is just the same thing in a different medium of expression, and can claim no greater profundity. Second, it may be considered as the interpretation of the abstract forms Barbara and Celarent; but how an interpretation can be considered deductively more fundamental than the abstract system to which it applies is not clear.

A third misapprehension would be the supposition that because only one form of the syllogism need be assumed, nothing else need be assumed in order to deduce all of the valid forms of the syllogism. Now the deduction of the other valid syllogisms from Barbara—or from any other postulated syllogistic form—makes use of simple conversion, conversion by limitation, the relation of contradiction on the square of opposition, and obversion. Unless these principles of eduction and opposition could all be deduced from our syllogistic axiom—and they cannot—further assumptions are required. In affirming that Barbara yields all the other forms of the syllogism we intend the tacit proviso “on certain assumptions about immediate inference and opposition.”

The ordinary presentation of traditional logic thus employs the postulational method, but the resulting systematization is a partial one only. It is incomplete in the following respects.

First, a system ought to state what its undefined or primitive ideas are and ought to define all the other ideas of the system in terms of these primitive ones. To lay down postulates and derive theorems is not sufficient. The ideas of the system should also be set forth, some as assumed and some as defined. But the systematization given in the textbooks is entirely lacking in this respect.

Second, not only the forms of the syllogism but also the forms of eduction and the principles of the square of opposition ought to be systematized. They are part of traditional logic; moreover, they are required for the deduction of the forms of the syllogism. Each one of them should appear either as a postulate or as a theorem. Here also the ordinary account of traditional logic is defective.

Third, the textbooks fail to treat postulationally the principles of invalidity. By a principle of invalidity I mean, for example, such a proposition as It is false that for all values of s, m, and p, All m is p and All m is s implies All s is p. In a complete systematization of traditional logic that proposition would appear either as a postulate or as a theorem.

11. General Character of our Systematization. But one further preliminary need detain us. Before we begin the task of systematizing traditional logic, we must consider the general character which our systematization is to have.

We propose to formulate traditional logic as a deductive system. But there are two species of deductive systems, and we must explain to which species our system belongs.¹

One type of deductive system may be called the logistic type. The distinguishing feature of that kind of system is the fact that the general principles of logic, by means of which the theorems are deduced from the postulates, are themselves contained in the postulates of the system, and that the general ideas of logic, by means of which the postulates, definitions, and theorems are stated, are themselves assumed ideas of the system or are definable in terms of the assumed ideas without the use of any further ideas. Examples of this kind of system are (1) Whitehead and Russell’s system of logic set forth in their Principia Mathematica and (2) C. I. Lewis’s “System of Strict Implication.”

The other and more common type of system may be called non-logistic. In this type of system, though the theorems are deduced rigorously from the postulates, this deduction is effected by means of general principles of logic which are not themselves contained in the postulates. These principles are outside the system, so to speak. By means of these principles which are external to the system, the system is generated. Moreover, in systems of this type, certain ideas of logic, which are not part of the system itself, are employed to state the posutlates, definitions, and theorems of the system. Most of the modern “sets of postulates” are of this sort; for example, Professor Huntington’s sets of postulates for Boolean algebra (the modern logic of classes), and his postulates for geometry.

Now our system will be of the second rather than the first type; it will be non-logistic. Indeed it must be so. For the logistic method can be used only in the case of a subject matter of a peculiarly fundamental nature. The fact that we shall thus use principles and ideas of logic which are outside our system is not a logical defect; it is simply a characteristic which belongs inevitably to many systems.

The ideas of logic which are not part of the system but which we shall use to state our postulates, definitions, and theorems are the following: “implication,” “equivalence,” “negation,” and “conjunction.” When we say that one proposition, p, “implies” another, q, we me...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Copyright Page
• Table of Contents
• PREFACE
• INTRODUCTION
• CHAPTER I. POSTULATES FOR ARISTOTELIAN LOGIC
• CHAPTER II. IMMEDIATE INFERENCE
• CHAPTER III. MEDIATE INFERENCE
• CHAPTER IV. COMPLETION OF THE SYSTEM
• CHAPTER V. DISTRIBUTION, QUALITY, AND QUANTITY
• CHAPTER VI. THE RULES OF ARISTOTELIAN LOGIC
• CHAPTER VII. INTERPRETATIONS OF ARISTOTELIAN LOGIC
• CONCLUSION
• INDEX