George Novack
An Introduction to the Logic Of Marxism: LECTURE II

The Limitations of Formal Logic

In the first lecture we dealt with three questions.

1. What is logic? We defined logic as the science of the thought process in its connections with all other processes in the universe. We learned that there were two main systems of logic: formal logic and dialectics.

2. What is formal logic? We stated that formal logic was thinking dominated by the laws of identity, of contradiction, and of the excluded middle. We pointed out that these three fundamental laws of formal logic have a material content and an objective basis; that they are explicit formulations of the instinctive logic of common sense; that they constitute the prevailing rules of thought in the bourgeois world.

3. What are the relations between formal logic and dialectics? These two systems of logic grew out of and correspond to two different stages in the development of the science of thought. Formal logic preceded dialectics in the historical evolution of logic as it usually does in the intellectual development of individuals. Then dialectics arose out of the criticism of formal logic, overthrew and replaced it as its revolutionary opponent, successor and superior.

In this second lecture we propose to uncover the limitations of formal logic and indicate how dialectics necessarily emerges from a critical examination of its fundamental ideas. Now that we have grasped what the basic laws of formal logic are, what they reflect in reality, why they are necessary and valuable instruments of thought, we must proceed a step further and find out what the laws of formal logic are not: what features of reality they neglect and distort, and where their usefulness ends and their uselessness begins.

This next step in our investigation will not produce purely negative results or culminate in a sceptical denial or dismissal of all logic. It will on the contrary lead to the most positive results. As the deficiencies of formal logic are exposed, there will be simultaneously disclosed the necessity and the main characteristics of the new logical ideas destined to replace them. Thus in the very process of dissecting elementary logic and separating the valid elements in it from the false, we shall be laying the basis for dialectical logic. The acts of criticism and creation, negation and affirmation, go hand in hand as two sides of the same process.

This dual movement of destruction and creation occurs not only in the evolution of logic but in all processes. Every leap forward, every creative act, involves the destruction of outgrown and intolerably restrictive conditions. In order to be born the chicken must peck through and smash the eggshell which had sheltered and nourished it in its embryonic stage. So, in order to obtain room for its freer and further development, the science of logic had to break through and smash the petrified shell of formal logic.

Formal logic starts from the proposition that A is always equal to A. We know that this law of identity contains some measure of truth, since it serves as an indispensable function in all scientific thought and is constantly used by all of us in everyday activity. But how true is this law? Is it always a thoroughly reliable guide through the complicated processes of reality? That is the question.

We prove that any proposition is true or false by going to objective reality and seeing in practice whether or not, and to what degree, the concrete content asserted in the proposition is exemplified. If the content corresponding to the assertion can be produced in reality, then the proposition has truth in it; if it cannot be produced, it is untrue.

Now what do we find when we go to reality and look for evidence of the truth of the proposition: A equals A? We discover that nothing in reality corresponds perfectly to the content of this proposition. On the contrary, we find that the opposite of this axiom is far closer to the truth.

Wherever we encounter some really existing thing and examine its character, we find that A is never equal to A. Says Trotsky: “. . . if we observe these two letters under a lens, they are quite different from each other. But, one can object, the question is not of the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar — a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is this true — all bodies change uninterruptedly in size, weight, colour, etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at any given moment.”

Aside from the extremely dubious practical value of this ‘axiom,’ it does not withstand theoretical criticism either. How should we really conceive the word ‘moment’? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that ‘moment’ to inevitable changes. Or is the ‘moment’ a purely mathematical abstraction, that is, a zero of time? But everything exists in time; . . . time is consequently a fundamental element of existence. Thus the axiom ‘A is equal to A’ signifies that a thing is equal to itself if it does not change, that is, if it does not exist.” (In Defence of Marxism, p. 49.)

Driven into this corner, some defenders of formal logic try to extricate themselves by saying: While it is true that the laws of formal logic never can be applied with absolute exactitude to any existing things, that does not nullify the worth of these regulating principles. Although they do not directly and wholly correspond to reality, these ideal generalisations are true “in themselves” without reference to reality and therefore serve to direct thinking along the right lines. This position does not remove the contradiction; it accentuates it. If, as they contend, the law of identity remains wholly true only so long as it is not applied, then it follows that the moment it is applied to any real thing, it becomes the source of error.

As Trotsky remarks: “The axiom ‘A’ is equal to ‘A’ appears on one hand to be the point of departure for all our knowledge, on the other hand the point of departure for all the errors in our knowledge.” (In Defence of Marxism, p. 49.) How can one and the same law be both the source of knowledge and the source of error? This contradiction can be explained by the fact that the law of identity has a two-sided character. It is in itself both true and false. It holds true of things insofar as they can be regarded as fixed and immutable, or insofar as the amount of change in them can be disregarded or discounted as negligible. That is to say: the law of identity gives correct results only within certain limits. These limits are given by the essential characteristics exhibited by the actual development of the object in question, on the one hand, and by the practical purpose in view, on the other.

Once these specific limits have been transgressed, the law of identity no longer suffices and turns into a source of error. The farther beyond these limits the process of development goes, the farther from the truth does the law of identity take us. Other laws must then be invoked and employed in order to correct the errors emanating from this rudimentary law and to cope with the new and more complex state of affairs.

Let us give some examples. From Albany to New York the Hudson River is clearly equal to itself and to no other body of water. A always equals A. But beyond these limits it becomes increasingly difficult to distinguish the Hudson River from other bodies of water. At its mouth beyond New York harbour the Hudson loses its identity and becomes more and more one with the Atlantic Ocean. At its source the Hudson disintegrates into separate streams and springs, which, although they go to make up the Hudson, nevertheless have each a specific identity and material existence of their own, different from the river itself. Thus at both ends of its course the identity of the Hudson River tends to disappear and to pass over into non-identity.

A similar loss of identity occurs constantly along the course of the river. The spatial identity of the river is usually defined and maintained by the banks between which it flows. But, as the river becomes higher or lower, or as erosion takes place, these banks change. Rains and floods change existing limits permanently or provisionally for miles at a stretch. Even where the river remains spatially the same, it never contains the same water. Every drop is different. Thus the Hudson River keeps changing its identity all the time.

Or let us take the example of the dollar cited by Trotsky. We ordinarily assume, and act correctly upon the assumption, that a dollar bill is a dollar. A equals A. But we are beginning to realise that nowadays a dollar is no longer the same dollar it was. It is becoming less and less of a dollar in value. The 1942 dollar can buy only three-quarters as much as the 1929 dollar bought. (In 1963 the dollar was worth 40.8 cents in 1939 terms.)

It looks like the same dollar — the law of identity is still applicable — but at the same time the dollar is beginning to alter its identity by diminishing in value.

In 1923 the German people found that the mark, which since 1875 had been equal to 23 cents in gold, had, as a result of inflation, become equal to zero, was valueless. A, which for almost half a century had been equal to A, had suddenly become equal to non-A! In the course of the inflationary process, A had turned into its opposite.

The certificate of value had no value.

“Every worker knows that it is impossible to make two completely equal objects. In the elaboration of bearing brass into cone bearings, a certain deviation is allowed for the cones which should not, however, go beyond certain limits (this is called tolerance). By observing the norms of tolerance, the cones are considered as being equal. (“ A” is equal to “A.” ) When the tolerance is exceeded the quantity goes over into quality; in other words the cone bearings become inferior or completely worthless.

Our scientific thinking is only a part of our general practice, including techniques. For concepts there also exists ‘tolerance,’ which is established not by formal logic issuing from the axiom ‘A’ is equal to ‘A,’ but by dialectical logic issuing from the axiom that everything is always changing. ‘Common sense’ is characterised by the fact that it systematically exceeds dialectical ‘tolerance’.” (In Defence of Marxism, p. 50.)

In the machine shop, grades of tolerance usually range from one one-hundredth to one ten-thousandth of an inch, depending upon the class of work to be done. It is the same with brainwork and the concepts which are its tools. Where the permissible margin of error is considerable, the laws of formal logic suffice; but when finer tolerances are demanded, new tools must be created and used. In the field of intellectual production, these tools are the ideas of dialectical logic.

The law of identity can exceed dialectical tolerance in two opposite directions. Just as tolerances usually have not one but two limits, a maximum and a minimum, so the law of identity continually exceeds dialectical tolerance by becoming either more or less valid. If, for example, as a result of deflation, a dollar doubles in value, then A is no longer equal to A but is greater than A. If, during inflation, the dollar dwindles to half its value, A again is not equal to A, but to less than A. In either case the law of identity is no longer strictly true, but becomes more and more untrue, according to the amount and specific character of the change in value. Instead of A equalling A, we now have A equalling either 2A or 1/2 A.

Notice that we started, quite correctly, with the law of identity. We had A and nothing more.

And then we inevitably come to this contradiction: it is true that A equals A; it is likewise true that A does not equal A. In addition to equalling A, it equals 2A and 1/2 A.

This gives us a clue to the true nature of A. A is not the simple fellow, the fixed, unchangeable category the formal logicians make him out to be. That is only one of the appearances of A. In reality A is extremely complex and contradictory. It is not only A but also at the same time something else. That makes A very elusive and slippery. We can never quite catch hold of A because the minute we try to pin A down, it begins to change into something more or less different.

What then, you may ask in exasperation, is A, if it’s not simply and solely A? The dialectical answer is that A is both A and non-A. If you take A as simply A and nothing more, as the formal logician does, you see only one side of A and not its other side, its negative side. A, taken by itself as simply A and nothing more, is an abstraction that can never be fully realised or found in actuality. It is a useful abstraction so long as you understand its limits and do not take it or, better, mistake it, for the full and final truth about any given thing. This elementary law of identity holds good for most of the ordinary acts of everyday life and thinking, but it must be replaced by more deep-going and complex laws where more complicated and long-drawn-out processes are involved.

Any machinist should easily comprehend why this law of thought can have only a limited value. Isn’t this true of all tools and machines? Each is useful only under certain conditions and for certain definite operations: a saw for cutting, a lathe for turning, a boring mill for boring. At each stage in the process of industrial production, the workers run up against the intrinsic limitations of each and every tool and machine tool. They overcome the limitations of the tools at their disposal in two ways: either by using a different tool or else by combining different tools in the same continuous process of production. Operations on a turret lathe provide an excellent example of this.

Thinking is essentially a process of intellectual production — and the limitations of the tools of thought can be overcome in the same manner. Whenever we strike a snag with the law of identity, we either have to resort to a different logical law or else we have to combine old laws in new ways in order to get at the truth. Here is where dialectical logic comes in. Just as we bring in a more developed machine or set of machines in industrial production, so, when we want more correct and exact results in intellectual production, we apply the more developed ideas of dialectics.

If we return now to our original abstract equation, A equals A, we observe that it has developed in a very contradictory fashion. A has differentiated itself. In other words, A is always changing — and changing in different directions. A is always becoming more or less itself, always approaching or receding from itself.

Now there comes a point, in this process of realising or losing its identity, at which A becomes something other than the self it started with. If we subtract enough from A or add enough to A, it changes its specific quality and turns into something else, into a new quality. At this critical point where A loses its identity, the law of identity, which has hitherto retained some validity, becomes utterly false.

The Hudson River loses its identity and becomes part of the Atlantic Ocean; the German mark becomes no longer a mark but a slip of printed paper; the cone bearing, instead of being an integral part of a machine, turns into worthless scrap metal. In algebraic terms, A becomes -A. In dialectical language, quantitative changes destroy the old and bring about a new quality. “To determine at the right moment the critical point where quantity changes into quality is one of the most important and difficult tasks in all the spheres of knowledge including sociology.” (In Defence of Marxism, p. 50.)

One of the central problems of the science of logic consists in recognising and formulating this law. We have to understand how and why at a certain point quantitative changes give rise to new qualities, and vice versa.

We arrive then at this conclusion. While the law of identity correctly reflects certain features of reality, it either distorts or fails to reflect others. Moreover, the aspects which it falsifies and cannot express are far more pervasive and fundamental than those it more faithfully depicts. Intermixed with its particles of fact, this elementary generalisation of logic contains a serious infusion of fiction. As a result this instrument of truth becomes in turn a generator of error.

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