The Problems of Logic

Andrew Paul Ushenko

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781000737110
• Edition: 1
• Topic: Filosofia
• Subtopic: Storia e teoria della filosofia

Originally published in 1941. Professor Ushenko treats of current problems in technical Logic, involving Symbolic Logic to a marked extent. He deprecates the tendency, in influential quarters, to regard Logic as a branch of Mathematics and advances the intuitionalist theory of Logic. This involves criticism of Carnap, Russell, Wittgenstein, Broad and Whitehead, with additional discussions on Kant and Hegel. The author believes that the union of Philosophy and Logic is a natural one, and that an exclusively mathematical treatment cannot give an adequate account of Logic. A fundamental characteristic of Logic is comprehensiveness, which brings out the affinity between logic and philosophy, for to be comprehensive is the aim of philosophical ambition.

Chapter I

THE NATURE OF LOGIC

§ 1. INTRODUCTION

Ideas about the nature of logic, when they do not develop from a working experience of problems within logic, are likely to be “empty and jejeune”, and it might appear sensible to begin by merely saying that logic is a theory concerned with such and such problems, viz. the problems that contemporary logicians are trying to solve, and then proceed with a discussion of these special problems. But as things happen to stand now this would be an extremely dangerous, if not a disastrous, course to take. For the particular logical problems of to-day are specialized to the extent of being highly technical points of symbolic calculus, and if they should become the only concern of logic, logic would become a science, more precisely a branch of mathematics. This is, of course, just what the majority of contemporary logicians, who are at their best mathematicians, believe logic must be. They like to remind us that philosophers were responsible, as was recognized by Kant himself in a famous passage (which, however, was not intended to be used for condemning logic), for the century-long stagnation of formal logic, and that to stir it up to its present state of feverish activity (carried on by a yearly increasing number of adepts), mathematicians, Boole, Peirce, De Morgan, Peano, Schröder, Russell, etc., had to take the matter in their hands. From this point of view logic, to be alive, must be symbolic or mathematical.

I have called the activity of symbolic logicians “feverish”, but I do not know whether this should imply the delicate question whether it signifies the momentum or the inertia of logical research. We must not forget that the results of a similar activity of mediaeval logicians have been scored under the name of scholasticism throughout the subsequent ages. However, I am willing to admit that in the main symbolic logic is an improvement upon the Aristotelean tradition. And it is certain that in making a definition of logic we must take account of the development of technical exercises in symbolic calculus. Even more, we must do justice to the interests of to-day and put the technical aspect of logic in the foreground. But I still take exception to the “greed” of the mathematical logician when he identifies the technical aspect with the whole of logic or when he decides that because of the prominence of mathematicians in all recent developments of logic it is time to have the latter transferred from the competence of philosophy to the departments of mathematics. This brings a vital issue before both philosophy and logic, and because of it a discussion of the status of logic cannot be merely academic. Philosophy has suffered many amputations. The most recent of those, the segregation of psychology into an independent science, has proved to have been a procedure of very doubtful value. Hence my concern is not simply a personal grudge of a philosopher against the success of mathematicians; I feel that an act of violence against the natural union of philosophy and logic is about to be perpetrated. And I shall argue the natural character of their union. I believe that an exclusively mathematical treatment cannot give an adequate account of logic.

A fundamental characteristic of logic is comprehensiveness. This brings out the affinity between logic and philosophy, for to be comprehensive is the aim of philosophical ambition. Philosophers are so interested in categories, such as substance, relation, and the like, because these are comprehensive concepts the application of which not only transcends the confines of special sciences and arts, but is not even hampered by the barriers between human endeavours and processes of raw nature. Logical forms are, even more than philosophical categories, independent of the variations in their subject-matter or content: they have a claim to universal application. From this point of view it is very difficult to construe logic as a science, for mere sciences are always specialized. In fact scientific success is an outcome of specialization. This is particularly true of sciences which, like mathematics, take the form of postulational systems.* A postulational system is, of course, worked out as an abstract structure and can be contrasted with its interpretations which are taken from various fields of actuality. Nevertheless the construction of different postulational systems is not only guided by the natural segregation of the fields of interpretation into distinct groups, but it must be made flexible enough to allow for the incessant differentiation and specialization of sub-groupings within the already established groups. The postulational method is adopted by the scientist because of the ease with which it responds to the increasing specialization; it responds by a mere addition of new postulates to the original set. For example, the postulates for serial order are, to begin with, tools for grouping together such various things as natural numbers in their succession of increasing magnitude, Kings of England in their chronological succession, and so on. But the practical importance of this set of postulates is the fact that with an addition of a new postulate it can single out a special type of series, for instance, the type of dense series which are interpretable, among other things, as fractions in their order of magnitude. The adjustment of postulational systems to the persevering desire for specialization is so striking that one can venture to conjecture that postulational treatment is essentially a method of differentiation. And this conjecture has a further confirmation whenever postulational development must perforce be carried out without the aid of interpretation. The splitting of geometry into euclidean and non-euclidean varieties is, of course the memorable instance. If so, it is only natural to expect that a thorough application of the postulational treatment to logic would run contrary to its essential claim to comprehensiveness. Indeed a differentiation of logic in the hand of mathematicians is exactly what has happened. First, there are different systems of mathematical logic.* Even if they were translatable into one another, they would not be comprehensive systems because their inter-translation cannot be performed within either of them, but presupposes a neutral medium, such as English, with a non-mathematical logic embedded in it. Secondly, there are alternative mathematical logics which cannot even be reduced to one another. The discovery of alternative logics may be a positive achievement, but it, certainly, means that classical logic, as an interpretation of the two-valued alternative system, has rivals and must relinquish its monopoly upon the valid forms of thought. Of course, the relationships among alternative logics are not yet made entirely clear. But, again, the very fact that problems about them are raised proves that these alternative systems are not comprehensive. Their relationships must be established within a neutral medium with a non-mathematical logic embedded in it.

* It is likely that sciences must always take a postulational form as soon as they reach a high degree of exactitude. For example, the procedure of physical theory, according to Mr. R. B. Lindsay, “is to start from certain intuitive and logically indefinable concepts and from these to build more elaborate concepts by purely postulational methods…. It is not denied, to be sure, that ideas gained from experience and in particular laboratory operations enter into this process. Inevitably, however, there is and always will be an element of arbitrary choice in the construction of concepts: we build those which we believe are going to be useful to us in physical description, even if they do not always correspond closely to raw experience…. Having constructed the concepts and assigned symbols to them, we continue the process of theory building by assuming certain mathematical relations among the symbols….” Cf. “Operationalism in Physics”, Philosophy of Science, October 1937, p. 459.

* For example, there is the Boolean algebra of logic, the system of the Principia, Curry’s Combinatory logic, Church’s system, etc.

I do not for a moment doubt that forms of deduction can be exhibited in full by the postulational method. But formal logic is not the whole of logic. There is besides the theory of logic. This is the medium in which an introduction as well as a discussion of logical forms and of symbolic notation must take place. And it must, at least in part, consist of non-symbolic and non-formal statements.* For symbols have to be explained, the various modes of their combinations noted, and the rules for the derivation of certain complexes of symbols (theorems) from others (postulates) expressed. Furthermore, the given symbolic system of logic must be compared, as we have already seen, with other symbolic logics, whether reducible to one another or alternative. Of course, these facts cannot be altogether denied even by the advocates of the thoroughly mathematicized logic. The disagreement comes through a twist of interpretation which they can give to the facts. They recognize the difference between formal logic and the theory of logic, but, instead of taking it as a distinction between an exhibition and a description of the forms of actual discourse, they treat it as an actual separation of systems. Accordingly the theory of logic is said to be not a part of logic at all but a separate system which under the name of metalogic or syntax-language is then contrasted with the symbolic postulational system (which is the object of metalogical discussion) to be known as the object-logic or the object-language or, simply, as logic. This separation between metalogic and logic does not require, on the other hand, that the former should be given in terms of ordinary non-formal language. It is suggested that dependence on ordinary language and its intuitive logic in the matter of exposition of symbolism is a contingency of human civilization, since one can conceive of other beings, such as Martians, who would have a native ability to think along the lines of a symbolic calculus. Furthermore, after a sufficiently resourceful symbolic language had been introduced one can even formulate within it most of its own syntax, much as the syntax of English is formulated in English.* This does not mean, however, that the separation between syntax and its object can be dispensed with. The syntax-language is always more comprehensive than the object-language, for, as the work of Gödel, Carnap, Tarski, and others has shown, some of the metalogical statements must remain untranslatable into the expressions of the object-language. At the same time, although more comprehensive than formal logic, metalogic cannot play the part of a comprehensive system, because metalogic is itself an object for analysis and therefore presupposes a still more comprehensive meta-metalogic, which in its turn is the object of a meta-meta-metalogic, and so on ad infinitum. This awkward assumption of an unending hierarchy of logics is, to my mind, a distorted analogue of the claim to comprehensiveness in intuitive logic, necessarily distorted by the mathematical treatment.

* Cf. W. E. Johnson, Logic, v. 2, ch. iii, pp. 2 and 3.

* It is sometimes said that the formulation of the syntax within the object-language is inconsistent with their separation. This is not so. The formulas of the object-language, regardless of what they “express”, remain distinct from their syntactical interpretation.

But let us defer criticism to a context in which a closer examination of metalogic is possible. In this Introduction the need is for a mere illustration of the striking divergence of the two contemporary views on the status of logic. These opposing views will be referred to as the intuitional and the postulational theories, and their respective adherents as the intuitionalists and the postulationalists.* To further the cause of the intuitional theory is the main purpose of this essay.

§ 2. LOGIC AND METALOGIC

The intuitional theory is not hostile to the use of symbols in logic. But while the postulationalists segregate the symbols in use into an isolated calculus or language, the object-logic, the intuitionalists take these same symbols as a part of English (or of whatever other common language the logician happens to speak), a sort of shorthand intended to avoid recourse to technical terms or a construction of cumbersome statements. Take, for example, the formula:

(1) pV ~ p.

* The intuitional theory of logic must not be confused with the intuitionism of the intuitionist Brouwer, in which the two-valued logic is co-extensive with the field of finite aggregates. The postulational theory is a designation which applies both to the formalism of Hilbert’s school and to the linguistic theory of logic of logical positivism.

To the intuitionalist this is a symbolic expression of the traditional principle of the excluded middle and an abbreviation of what can be put in words as:

(1′) A proposition is either true or false.

Such symbols as “wedge” and “curl” are obviously used in (1) as ideographical for the logical constants “either-or” and “it is false that”, which are phrases in plain English. As to the variables p, q, etc., they do not function as abbreviations directly, but by representing any of the indefinite number of English propositions they secure brevity of expression even to a greater extent than the symbols for logical constants. The fact that in effect the variables are merely devices of abbreviation must be plain since, as the comparison of (1) and (1′) illustrates, the same logical principles which are given as symbolic formulas can be expressed at a greater length by means of such technical terms as “proposition” without making any use of variables.

In the postulational treatment formula (1) is merely a complex of marks written down in a combination which is allowed for by the rules of the symbolic calculus within its context. A “postulate” or a “theorem” are just technical names for such a combination where it can be used within the system without restriction. A “variable” like “p” is then an element-mark in a symbolic formation for which other marks can be substituted under specified conditions. The connection which a postulational system of logic can have with propositions in actual discourse is established outside the system by interpretation. A mark like “p” by itself does not represent anything, therefore it does not represent a proposition, but we can, if we choose, interpret it as a proposition. We might, however, interpret it as “house” or as any other word provided one condition is satisfied: an interpretation of a “postulate” or a “theorem” in terms of English words must describe a situation which, conventionally or otherwise, is assertible as a fact.

The postulationalist divorce of a symbolic language from its interpretation in English or from any other syntax-language is supposed to give a twofold gain in the generality of treatment. First, it leaves the possibility of important interpretations of symbolic formulas other than in terms of propositions. Second, it provides for a construction of alternative systems which after they are interpreted in English would give non-aristotelean modes of argument. In mathematics a similar claim to a twofold advantage of the postulational approach has been, no doubt, well founded.* But in symbolic logic the possibility of new interpretations remains so far merely theoretical. As to non-aristotelean logics, although they might have been in...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Contents
• Chapter I. The Nature of Logic
• Chapter II. The Paradoxes of Logic
• Chapter III. Consistency and the Decision-Problem
• Chapter IV. Conceptual Reference
• Chapter V. Logic and Reality
• Chapter VI. The Existence of Propositions
• Index