It is an astonishing fact that the basic laws of formal logic worked out by Aristotle have remained fundamentally unchanged for over two thousand years. In this period, we have witnessed a continuous process of change in all spheres of science, technology and human thought. And yet scientists have been content to continue to use essentially the same methodological tools that were used by the mediaeval Schoolmen in the days when science was still on the level of alchemy.

Given the central role played by formal logic in Western thought, it is surprising how little attention is paid to its real content, meaning and history. It is normally taken as something given, self-evident, and fixed for all time. Or it is presented as a convenient convention which reasonable people agree upon, in order to facilitate thought and discourse, rather as people in polite social circles agree upon good table manners. The idea is put forward that the laws of logic are entirely artificial constructions, made up by logicians, in the belief that they will have some application in some field of thought, where they will reveal some truth or other. But why should the laws of logic have any bearing upon anything, if they are only abstract constructions, the arbitrary imaginings of the brain?

On this idea, Trotsky commented ironically:

“To say that people have come to an agreement about the syllogism is almost like saying, or more correctly it is exactly the same as saying, that people came to an agreement to have nostrils in their noses. The syllogism is no less an objective product of organic development, i.e., the biological, anthropological, and social development of humanity than are our various organs, among them our organ of smell.” In reality, formal logic is ultimately derived from experience, just as any other way of thinking. From their experience, humans draw certain conclusions, which they apply in their daily life. This applies even to animals, though at a different level.

“The chicken knows that grain is in general useful, necessary, and tasty. It recognises a given piece of grain as that grain – of the wheat – with which it is acquainted and hence draws a logical conclusion by means of its beak. The syllogism of Aristotle is only an articulated expression of those elementary mental conclusions which we observe at every step among animals.”[55]

Trotsky once said that the relationship between formal logic and dialectics was similar to the relationship between lower and higher mathematics. The one does not deny the other and continues to be valid within certain limits. Likewise, Newton’s laws, which were dominant for a hundred years, were shown to be false in the world of subatomic particles. More correctly, the old mechanistic physics, which was criticised by Engels, was shown to be one-sided and of limited application. Later Trotsky would write:

The dialectic is neither fiction nor mysticism, but a science of the forms of our thinking insofar as it is not limited to the daily problems of life but attempts to arrive at an understanding of more complicated and drawn-out processes.[56]

The most common method of formal logic is that of deduction, which attempts to establish the truth of its conclusions by meeting two distinct conditions: a) the conclusion must really flow from the premises; and b) the premises themselves must be true. If both conditions are met, the argument is said to be valid. This is all very comforting. We are here in the familiar and reassuring realm of common sense. “True or false?” “Yes or no?” Our feet are firmly on the ground. We appear to be in possession of “the truth, the whole truth, and nothing but the truth”. There is not a lot more to be said. Or is there?

Strictly speaking, from the standpoint of formal logic, it is a matter of indifference whether the premises are true or false. As long as the conclusions can be correctly drawn from its premises, the inference is said to be deductively valid. The important thing is to distinguish between valid and invalid inferences. Thus, from the standpoint of formal logic, the following assertion is deductively valid: All scientists have two heads. Einstein was a scientist. Therefore, Einstein had two heads. The validity of the inference does not depend upon the subject matter in the slightest. In this way, the form is elevated above the content.

In practice, of course, any mode of reasoning that did not demonstrate the truth of its premises would be worse than useless. The premises must be shown to be true. But this leads us into a contradiction. The process of validating one set of premises automatically raises a new set of questions, which in turn need to be validated. As Hegel points out, every premise gives rise to a new syllogism, and so on ad infinitum. So that what appeared to be very simple turns out to be extremely complex, and contradictory.

The biggest contradiction of all lies in the fundamental premises of formal logic itself. While demanding that everything else under the sun must justify itself in the High Court of the Syllogism, logic becomes utterly confused when asked to justify its own presuppositions. It suddenly loses all its critical faculties, and resorts to appeals to belief, common sense, the “obvious”, or, the final philosophical get-out clause – a priori. The fact is that the so-called axioms of logic are unproved formulas. These are taken as the starting point, from which all further formulae (theorems) are deduced, exactly as in classical geometry, where the starting point is provided by Euclid’s principles. They are assumed to be correct, without any proof whatsoever, i.e., we just have to take them on trust.

But what if the basic axioms of formal logic turn out to be false? Then we would be in just the same position as when we gave poor Mr. Einstein an additional head. Is it conceivable that the eternal laws of logic might be flawed? Let us examine the matter more closely. The basic laws of formal logic are:

1) The law of identity (“A” = “A”).
2) The law of contradiction (“A” does not equal “not-A”).
3) The law of the excluded middle (“A” does not equal “B”).

These laws, at first sight, seem eminently sensible. How could anyone quarrel with them? Yet closer analysis shows that these laws are full of problems and contradictions of a philosophical nature. In his Science of Logic, Hegel provides an exhaustive analysis of the Law of Identity, showing it to be one-sided and, therefore, incorrect.

Firstly, let us note that the appearance of a necessary chain of reasoning, in which one step follows from another, is entirely illusory. The law of contradiction merely restates the law of identity in a negative form. The same is true of the law of the excluded middle. All we have is a repetition of the first line in different ways. The whole thing stands or falls on the basis of the law of identity (“A” = “A”). At first sight this is incontrovertible, and, indeed, the source of all rational thought. It is the Holy of Holies of Logic, and not to be called into question. Yet called into question it was, and by one of the greatest minds of all time.

There is a story by Hans-Christian Andersen called The Emperor’s New Suit of Clothes, in which a rather foolish emperor is sold a new suit by a swindler, which is supposed to be very beautiful, but invisible. The gullible emperor goes about in his fine new suit, which everyone agrees is exquisite, until one day a little boy points out that the emperor is, in fact, stark naked. Hegel performed a comparable service to philosophy in his critique of formal logic. Its defenders have never forgiven him for it.

The so-called law of identity is, in fact, a tautology. Paradoxically, in traditional logic, this was always regarded as one of the most glaring mistakes that can be committed in defining a concept. It is a logically untenable definition which merely repeats in other words what is already contained in the part to be defined. Let us put this more concretely. A teacher asks his pupil what a cat is, and the pupil proudly informs him that a cat is – a cat. Such an answer would not be considered very intelligent. After all, a sentence is generally intended to say something, and this sentence tells us nothing at all. Yet this not very bright scholar’s definition of a feline quadruped is a perfect expression of the law of identity in all its glory. The young person concerned would immediately be sent to the bottom of the class. Yet for over two thousand years, the most learned professors have been content to treat it as the most profound philosophical truth.

All that the law of identity tells us about something is that it is. We do not get a single step further. We remain on the level of the most general and empty abstraction. For we learn nothing about the concrete reality of the object under consideration, its properties and relationships. A cat is a cat; I am myself; you are you; human nature is human nature; things are as they are. The emptiness of such assertions stands out in all its uncouthness. It is the consummate expression of one-sided, formalistic, dogmatic thinking.

Is the law of identity invalid, then? Not entirely. It has its applications, but these are far more limited in scope than what one might think. The laws of formal logic can be useful in clarifying certain concepts, analysing, labelling, cataloguing, defining. It has the merit of neatness and tidiness. This has its place. For normal, simple, everyday phenomena, it holds good. But when dealing with more complex phenomena, involving movement, sudden leaps, qualitative changes, it becomes wholly inadequate, and, in fact, breaks down completely.

The following extract by Trotsky brilliantly sums up Hegel’s line of argument in relation to the law of identity:

I will here attempt to sketch the substance of the problem in a very concise form. The Aristotelian logic of the simple syllogism starts from the proposition that ‘A’ is equal to ‘A’. This postulate is accepted as an axiom for a multitude of practical human actions and elementary generalisations. But in reality ‘A’ is not equal to ‘A’. This is easy to prove 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; and existence itself is an uninterrupted process of transformation; 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.

At first glance it could seem that these ‘subtleties’ are useless. In reality they are of decisive significance. The axiom ‘A’ is equal to ‘A’ appears on one hand to be the point of departure for all the errors in our knowledge. To make use of the axiom ‘A’ is equal to ‘A’ with impunity is possible only within certain limits. When quantitative changes in ‘A’ are negligible for the task at hand then we can presume that ‘A’ is equal to ‘A’. This is, for example, the manner in which a buyer and a seller consider a pound of sugar. We consider the temperature of the sun likewise. Until recently we considered the buying power of the dollar in the same way. But quantitative changes beyond certain limits become converted into qualitative. A pound of sugar subjected to the action of water or kerosene ceases to be a pound of sugar. A dollar in the embrace of a president ceases to be a dollar. 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…

Dialectical thinking is related to vulgar thinking in the same way that a motion picture is related to a still photograph. The motion picture does not outlaw the still photograph but combines a series of them according to the laws of motion. Dialectics does not deny the syllogism, but teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality. Hegel in his Logic established a series of laws: change of quantity into quality, development through contradictions, conflict of content and form, interruption of continuity, change of possibility into inevitability, etc., which are just as important for theoretical thought as is the simple syllogism for more elementary tasks.[57]

Similarly with the law of the excluded middle, which asserts that it is necessary either to assert or deny, that a thing must be either black or white, either alive or dead, either “A” or “B”. It cannot be both at the same time. For normal everyday purposes, we can take this to be true. Indeed, without such assumptions, clear and consistent thought would be impossible. Moreover, what appear to be insignificant errors in theory sooner or later make themselves felt in practice, often with disastrous results. In the same way, a hairline crack in the wing of a jumbo jet may seem insignificant, and, indeed, at low speeds may pass unnoticed. At very high speeds, however, this tiny error can provoke a catastrophe. In Anti-Dühring, Engels explains the deficiencies of the so-called law of the excluded middle:

“To the metaphysician, things and their mental images, ideas, are isolated, to be considered one after the other and apart from each other, fixed, rigid objects of investigation given once for all. He thinks in absolutely unmediated antitheses. ‘His communication is “yea, yea; nay, nay”; for whatsoever is more than these cometh of evil.’ For him a thing either exists or does not exist; a thing cannot at the same time be itself and something else. Positive and negative absolutely exclude one another; cause and effect stand in a rigid antithesis one to the other.

“At first sight this way of thinking seems to us most plausible because it is that of so-called sound common sense. Yet sound common sense, respectable fellow that he is in the homely realm of his own four walls, has very wonderful adventures directly he ventures out into the wide world of research. The metaphysical mode of thought, justifiable and even necessary as it is in a number of domains whose extent varies according to the nature of the object, invariably bumps into a limit sooner or later, beyond which it becomes one-sided, restricted, abstract, lost in insoluble contradictions, because in the presence of individual things it forgets their connections; because in the presence of their existence it forgets their coming into being and passing away; because in their state of rest it forgets their motion. It cannot see the wood for the trees…

“In like manner, every organic being is every moment the same and not the same; every moment it assimilates matter supplied from without and gets rid of other matter; every moment some cells of its body die and others build themselves anew; in a longer or shorter time the matter of its body is completely renewed and is replaced by other molecules of matter, so that every organic being is always itself, and yet something other than itself.”[58]

The relationship between dialectics and formal logic can be compared to the relationship between quantum mechanics and classical mechanics. They do not contradict but complement each other. The laws of classical mechanics still hold good for an immense number of operations. However, they cannot be adequately applied to the world of subatomic particles, involving infinitesimally small quantities and tremendous velocities. Similarly, Einstein did not replace Newton, but merely exposed the limits beyond which Newton’s system did not work.

Formal logic (which has acquired the force of popular prejudice in the form of “common sense”) equally holds good for a whole series of everyday experiences. However, the laws of formal logic, which set out from an essentially static view of things, inevitably break down when dealing with more complex, changing and contradictory phenomena. To use the language of chaos theory, the “linear” equations of formal logic cannot cope with the turbulent processes which can be observed throughout nature, society and history. Only the dialectical method will suffice for this purpose.

  1. Trotsky, L. Writings, 1939-40, pp. 399 and 400.

  2. Trotsky, L. In Defence of Marxism, p. 65, our emphasis.

  3. Ibid pp. 63-6.

  4. Engels, F. Anti-Dühring, pp. 26-7.