Philosophy and Ideology. Z. A. Jordan 1963

Prior to a closer examination of the arguments in support of the Marxist-Leninist views on formal logic and the principle of non-contradiction certain terminological distinctions must be made.

We often use the term ‘logic’ as an abbreviated form of the expression ‘formal logic’, but this is not by any means always the case. ‘Logic’ is sometimes used as the collective name denoting a historically developed subject of teaching consisting of a great variety of topics such as elements of formal logic, the theory of knowledge, methodology, psychology, sometimes metaphysics and recently also syntax and semantics. The only principle unifying all these various topics is the fact that they are taught by philosophers who hold the chair of logic at universities. ‘Logic’ probably has this broad meaning in all languages, and it is certainly used in this sense in Poland^{[623]}.

While this usage of the term ‘logic’ could be justified by some precedents, its widespread adoption at the present time reflects the high repute that logic now enjoys, even among thinkers who are not actually interested in formal logic. The usage is, however, misleading and inappropriate. It is misleading because it makes logic and philosophy indistinguishable. It is inappropriate because it tends to blur the important difference between the formal approach, peculiar to logic, and the non-formal analytic procedures applied in philosophy. It is simply untrue to say that a philosopher and a logician do or try to do the same thing. Deductive reasoning plays a considerable role in philosophical investigations. These investigations pursue, however, an objective entirely foreign to a logician. To make use of the rules of valid inference and to study them for their own sake are two basically different activities.

To define ‘traditional logic’ is perhaps more difficult than to say what is the object of logic in the broad sense of this term. A modern logician often applies the term ‘traditional logic’ to all systems of the past that were not conceived and presented on the pattern set by Principia Mathematica. This might be sufficient for practical purposes, but strictly speaking it is incorrect or at least misleading, for we would have to consider some parts of ancient logic to be a part of modern logic.

In the past the view was common that what we now call ‘traditional logic’ dealt with the forms of thought. Since Descartes already knew that our thoughts have no extension, it is clear that what was intended to be a definition was in fact a metaphor, which was not always harmless. It did lead sometimes to the so-called psychologism in logic, and psychologism was a mark of the decay of logic in modern philosophy.

The above definition of traditional logic can be accepted provided that ‘forms of thought’ are interpreted to mean ‘forms of inference’. What seems to be the most general characteristic of logical investigations and what differentiates them from other studies on the ‘forms of thought’ is exactly the concern of logic with the rules or forms of inference. This concern is an important indication of whether some examinations belong to the field of logic, helps to establish its origin and the various periods of its development. Just for this reason, Bocheński is inclined to speak of Zeno’s or Plato’s logic, but not of that of the Pythagoreans, and his periodisation of the history of logic is based on the varying techniques of investigating rules of inference applied in different times ^{[624]}’.

While some logicians prefer to consider modern formal logic as an entirely new discipline, which differs from the Aristotelian class logic as much as ‘the railroad from the oxcart’, others recognise in traditional logic the direct progenitor of modern formal logic with the same subject-matter, broadly speaking. The latter agree, however, that traditional logic differs from modern logic in three important respects. As a rule, traditional logic was not formal; its subject-matter was only a fragment of that investigated by modern formal logic; and the methods of traditional logic, compared with those applied at present, were at a prescientific stage of development. For all these reasons traditional logic could have been of little use, if any at all, either to philosophy or to science, to mathematics in particular. Many great philosophers of the past testified to this fact. On the other hand, the usefulness of modern formal logic is widely recognised by representatives of the empirical and deductive sciences. It provides a method which enables us to obtain knowledge that commands assent of all qualified persons.

Since the discussion of the relation of dialectics to formal logic requires a sharp distinction between the terms ‘logic’, ‘traditional logic’ and ‘modern formal logic’, the meaning of the last of these terms must be at least delimited. For this purpose Alonzo Church’s initial definition might provide a starting point. According to Church, formal logic is concerned with the analysis of propositions and of proofs with attention paid to the form and in abstraction from the matter of content^{[625]}. ‘Form’ in this context means as much as ‘structure’, and ‘structure’ can in turn be explained to mean ‘arrangement’ or ‘order’ in which things are found or put together. We can now restate Church’s definition and say that formal logic is concerned with the relations that are revealed among objects of any kind when only their form is considered. The theory of the A-E-I-O relations in the field of universal terms, i.e. the logic of Aristotle as rediscovered by Łukasiewicz, Frege’s, Russel’s and Whitehead’s, or Łukasiewicz’s propositional calculus provide the simplest examples of what is meant by a system of modern formal logic.

Whenever Engels, Lenin, and Plekhanov discussed logic, the term ‘logic’ had the above indicated broad meaning, the various ingredients – metaphysics, the theory of knowledge, and methodology in particular – entering into it in different proportions. It is a common theme of Anti-Dhring, Dialectics of Nature, Philosophical Notebooks, and Dialectic and Logic that – in Marxian or Marxist-Leninist materialism – logic, the theory of knowledge, and dialectics either converge or become identical. There is no need of three words, Lenin wrote, for what is a single discipline. Engels called this single discipline the ‘science of thought’. It is ‘like every other, a historical science’, it assumes different contents at different times. It must be, therefore, distinguished from the ‘philistine’ conception of logic which tries to establish the ‘laws of thought’ as ‘eternal truth’. Thus, logic in the broad sense, either dialectical or non-dialectical is not a new kind of logic in the strict sense, different from traditional or formal logic, but something not to be compared with the latter^{[626]}.

Engels, Lenin, and Plekhanov also used the term ‘formal logic’, but what they then had in mind had nothing in common with modern formal logic. They spoke of formal logic, but referred in fact to traditional logic. Engels drew his knowledge of ‘formal logic’ from Hegel Wissenschaft der Logik and Enzyklop"die der philosophischen Wissenschaften; Lenin derived it from the same sources, to which some histories of philosophy and some of Aristotle’s works should be added. Plekhanov was the most advanced of the three, since apart from Engels’ and Lenin’s sources he also studied sberweg’s System der Logik (first published in 1857) and *Tredelenburg Logische Untersuchungen* (first published in 1870).

The idea of traditional logic which we find in the writings of the founders of Marxism-Leninism was mainly inspired by Hegel and had the imprint of his conception of logic. Hegel’s objection to traditional logic was that it was infused with the wrong kind of metaphysics. He set about correcting this error and by his enrichments he accomplished the complete divorce of logic from science. Engels, Lenin, and Plekhanov admired Hegel for this achievement and saw in him a great reformer of logic instead of its grave-digger. They had not the slightest inkling of another reforming trend initiated by George Boole and Augustus De Morgan about the middle of the last century, which was gathering strength at the time when Lenin and Plekhanov were philosophically most active. The direction in which Engels, Lenin, and Plekhanov wished traditional logic to develop was just the opposite of the one it has actually taken to resume its progress and to be transformed into modern formal logic.

The pronouncements of Engels, Lenin, and Plekhanov, which are still quoted with awe and respect to support the present-day views on the relation of dialectics to logic, as a rule confuse different meanings in which the term ‘logic’ is used. On the one hand, they compare dialectical logic in the broad meaning with nondialectical logic in the narrower and more technical sense of this term. On the other, ‘formal logic’, to which their pronouncements refer, means in fact ‘traditional logic’, to be sharply distinguished from ‘modern formal logic’. Consequently, what Engels, Lenin, and Plekhanov said about the relation of dialectical and non-dialectical logic in the past is not only open to objection, but also irrelevant to present-day problems. Whatever the relation of dialectics to formal logic might be, the pronouncements of Engels, Lenin, and Plekhanov provide no help to its elucidation and evaluation.

This remains unnoticed by Marxist-Leninists in general, those in Poland in particular. The same terminological confusions, of which Engels, Lenin, and Plekhanov were guilty, are very apparent in their own writings whenever they discuss logical matters. ‘Formal logic’ remains for them an ambiguous term, to denote both traditional and modern formal logic. Their comments on the former interfuse with those on the latter and it is hardly ever possible to decide which of them they have in mind. The considerable advance beyond traditional logic achieved in modern formal logic is largely ignored and little appreciated. While Engels could not know that traditional logic was not formal, because at that time formal logic did not yet exist, and Lenin could have become acquainted with it, had he wished and tried hard, there is no such excuse for a contemporary Marxist-Leninist philosopher.

The task of presenting and defining the Marxist-Leninist views on the ‘dogma of traditional logic’, the principle of non-contradiction, was undertaken by Schaff^{[627]}. He set about it with an apparent effort to dissociate these views from their Hegelian anti-rationalistic tradition and to accommodate them to what he considered to be the requirements of scientific thinking.

Three different stages might be distinguished in Schaff’s argument intended to justify the contention that the ontological principle of non-contradiction is false and that its logical and semantical formulations must be, therefore, rejected. The first tried to establish the epistemological superiority of dialectics over formal logic. The second discussed the foundations of formal logic and placed their dialectical revision within a more general trend of thought, the aim of which is to clarify the foundations of logic. The actual criticism of the ontological principle of non-contradiction followed last. Schaff did not distinguish these three stages and in his argument they were interwoven with each other. It appears, however, that his argument becomes somewhat clearer if the three stages in question are kept separate.

Two main considerations were put forward by Schaff to support the thesis concerning the superiority of dialectics over formal logic. The first of them emphasised that the cognitive value of logical laws is nil. Thus, the law of identity, to which he gave the form A = A, but which he wants to be read: for all A (A is A), is either false or trivial. It is false if it implies that nothing changes; it is trivial if it means no more than it says. A law that does not add anything to our knowledge is an empty tautology, as was rightly pointed out already by Hegel. Finally, if the law of identity is interpreted to mean that the meaning of any term should be kept invariable throughout the inference in which it is used, the law of identity is reduced to a mere linguistic rule^{[628]}.

The conclusion to be drawn from this analysis is familiar and goes back to Engels and Lenin. A theory like dialectics that enables the examination of how concepts and propositions change and are transformed in the actual process of thinking, which reflects the ever changing and moving phenomena and things of Nature, is richer and more significant than that concerned with mere forms of thought or rules of inference. ‘Dialectics’, wrote Lenin, ‘is living many-sided knowledge, with the number of sides eternally increasing, with an infinite number of shadings of every sort of approach and approximation to reality’. On the other hand, ‘the old logic’ is sterile, deserving only contempt and ridicule^{[629]}. The ‘merely formal logic’ is content with ‘enumerating the forms of motion of thought, i.e. the various forms of judgement and conclusion, and placing them side by side without any connection’. Ordinary logic, Engels argued, is common to men and the higher animals, dialectical thought is ‘only possible for man, and for him only at a comparatively high stage of development’ ^{[630]}. It is, therefore, clear that the former should conform to the latter, and not conversely. Dialectics is superior to formal logic, it is the most important mode of thinking for presentday science.

The second argument in favour of dialectics is also based on extra-logical considerations. Briefly, it says that dialectics does and logic does not reflect the regularities in the external world, or, more specifically, that the former lays bare the source of motion in Nature, the source being the struggle of the opposites, without resorting to supernatural factors, which by implication the latter is unable to do. Thus dialectics solves one of the most difficult problems of science and establishes its theoretical superiority over formal logic^{[631]}.

Both Schaff’s arguments display terminological confusions of the various meanings of the term ‘logic’. In the first of them ‘dialectics’ means the same as ‘logic’ in the broad meaning of this term and is compared with ‘logic’ in the sense of traditional logic, the science of the ‘forms of thought’, the ‘childish play of solving jig-saw puzzles’, as it was once described by Lenin. The latter is ‘static’, while dialectics is ‘dynamic’. With the tacit assumption that only a ‘moving thought’ can comprehend and adequately describe change and motion the conclusion is reached that traditional logic and its mode of thought are inapplicable to physical and social reality.

Hegel, Engels, Lenin, and Schaff seem to be essentially in agreement that well defined concepts are an inadequate instrument to describe what continually changes, what ‘simultaneously is itself and something else’. From this follows that the ‘abstract and refined conceptual framework’, worked out by bourgeois philosophy, is merely a subtle ideological weapon that conceals the rejection of the materialist tradition for the sake of idealism. On the other hand, a truly scientific procedure makes use of ‘dialectically elastic concepts’ and thus obtains a theoretically correct and adequate knowledge. The demand for the abandonment of scientific procedure is made in the name of science. ‘Science’ stands here for something quite different from what is commonly called by this name and from what it is to a man who studies science in its own right^{[632]}.

The objection that the truths of traditional logic are trivial is familiar and to some extent true. Many laws of traditional and formal logic do seem to convey very little knowledge about reality. They are, however, an indispensable condition for speaking with precision about anything at all, including dialectics. The much ridiculed law of identity is an example in point. It is always risky to use a term in two different meanings and it is a serious error to do it in an inference. This danger is by no means always easy to see; Marxist-Leninist writings provide some instances of this. Moreover, the laws of logic which say only little about reality constitute an integral part of an immense, abundant, and diversified system of theorems or rules. There is no reason to underestimate either the intricacies or significance of logical calculi. Even the simplest of them – the propositional and functional calculus – includes theorems neither self-evident nor trivial. Every mathematical proof may be considered as a substitution of one or more laws of the propositional and functional calculus. The theory of relations has its own intrinsic interest and many applications, neither trite nor uninformative. Aristotle was aware of the existence of relations but he did not realise that their logic cannot be accommodated within the theory of classes. While some laws of logic may give the impression of triviality, formal logic includes many theories of great simplicity combined with a high degree of abstraction, which makes them difficult to discover and easy to comprehend once the discovery has been accomplished. It also includes numerous difficult and complicated theorems, derived from the fundamental ones and providing important knowledge about reality, which to obtain otherwise would be extremely hard, if not impossible.

The second argument in favour of dialectics displays terminological confusions even to a higher degree than the first. In the indicated passages dialectics is called the ‘logic of contradiction’. But logic means there not only the theory of knowledge, psychology, or methodology, as was the case in the previous argument, but also metaphysics. In the second argument the term ‘logic’ is clearly used in its broadest sense, in which practically anything might be called a problem of logic.

Let us assume that dialectics does solve the metaphysical problem of the origin of motion. It does not follow therefrom that the dialectical conception of logic is theoretically superior to traditional or formal logic. The problem of the origin of motion is not a logical one and whatever the flaws of traditional or formal logic might be, the fact that it disregards it is not one of these flaws. The objection is pointless and irrelevant, a classical example of Hegelian logic or metaphysics -from which it was borrowed by Lenin^{[633]} – since for Hegel logic and metaphysics were the same thing.

It is confusing to argue that dialectics is superior to formal logic since it can claim the solution of the problem of the origin of motion, and that it is, therefore, entitled to make and unmake the laws of logic. On this basis one could hold the view that biology is superior to physics, because the former does and the latter does not investigate organic matter. While some reasons can be produced to support the view concerning the superiority of biology over physics, for instance, that biology assumes the laws of physics and supplements them by some others of its own, ‘superiority’ would then carry a different meaning from that suggested with respect to the relation between dialectics and formal logic. In particular, it would not imply that physics should change and make its laws conform to what the biologist finds to be the ‘laws of movement’ of organic matter. A logically posterior science cannot dictate what the laws of a logically prior science should be, but has to accept them as they are and, if need be, to try to do its best by them. Schaff’s argument about the superiority of dialectics over formal logic seems to suggest that exactly the opposite course is the correct one.

Although the suggestion that formal logic should conform to dialectics, and not the other way round, conflicts with the established scientific procedure, it is in keeping with the fundamental premisses of Marxist-Leninist philosophy. The ‘dialectics of the brain’ is according to Engels only the reflection of the forms of motion in the real world; the basic ‘laws of motion’ are always the same irrespective of whether they refer to the motion of physical bodies, human society or thought^{[634]}. On this assumption it might not appear bizarre that the laws of logic should be deduced from those of dialectics, which is the ‘science of the general laws of motion’.

This particular assumption of Marxist-Leninist philosophy provides an example of the usefulness of the law of identity which the dialectical philosophers have declared to be trivial and dispensable. The term ‘motion’ has been taken over from physics, where it has a precise and well-defined meaning; it can be applied outside physics, in logic, history, or sociology, only metaphorically. If the dialecticians use the term ‘motion’ in the sense accepted in physics, they should explain what the expression ‘change of place’ means with respect to mental or social phenomena. On the other hand, if they prefer to apply the term ‘motion’ to any kind of change, and not specifically to that of place, they should be aware of the fact that the term in question loses its precise sense, ceases to be a scientific term and acquires the ambiguity peculiar to the words loosely used in colloquial speech^{[635]}. Marxist-Leninists do not adopt either of these courses. ‘Motion’ does not have for them its clear meaning familiar from physics, and is still considered to possess all the precision of the scientific term a physicist may justly associate with it. This is clearly the error that results from the failure to keep the meaning of the terms used invariable.

The metaphorical usage of the term ‘motion’ applied equally to the changes of physical bodies, human society, history, or thought, is harmless as long as the metaphor is recognised for what it is. It becomes the source of misunderstandings of the crudest kind, when the metaphor is taken to be literally true. No metaphor can provide the basis for precise investigations. For when we speak of the movement of a physical body we wish to say that its position relative to some frame of reference is changing. Now, no thought or social change has a position whose movement can be observed and described in terms of their spatial-temporal co-ordinates. To be applied to human society or thought in some other than metaphorical sense, the concept of motion would have to be appropriately defined, and we do not find such a definition in the writings of Marxist-Leninists. If we try this course ourselves, we are at once faced by the difficulty that the kind of change we have in mind when it concerns human society or thought is a structural and not a spatial change. What relation there might be between these two kinds of change, it is not easy to see.

The same must be said of the ‘laws of motion’. This expression has a welldefined sense in physics, but in logic, psychology or sociology it can be at most a figure of speech. When Marx spoke of the ‘natural laws of movement’ which a society may discover in its development, he made a similar use of a picture language as when he referred to the ‘Furies of private interest’. As the term ‘motion’ has in logic and sociology a metaphorical meaning, we could only speak of the ‘laws of metaphorical motion’, but this expression simply does not make sense^{[636]}.

The argument that the laws of logic have practically no cognitive value, that they are responsible for the distortions of our view of reality in the manner made familiar by Zeno’s paradoxes, or that they are useless in acquiring new knowledge, is to justify the contention that formal logic, being an inferior discipline, should be subordinated to dialectics. Dialectics, forcing its way beyond the narrow horizon and sterile ground of formal logic, provides the reasons for deciding in each case which laws of logic should be accepted and which rejected or restricted in their application. Generally speaking, formal logic applies to things which do not change or are at rest, and loses its relevance whenever change and motion are involved. Since things constantly change or are in motion and remain unchanged or at rest only occasionally, dialectics has an incomparably wider application than formal logic and contains a more comprehensive view of the world. Although they are in a certain sense complementary, they should not be treated on the same footing, dialectics being hierarchically superior to formal logic^{[637]}.

The thesis of ‘logical dualism’ suggests that an Eleatic and a Heraclitean world exist side by side, with formal logic applying to the former and dialectics to the latter. The division results from a misconception of what logic is and what it can do. For obvious reasons formal logic is not responsible for keeping Zeno’s ‘arrow in flight’ poised motionless in the air or for considering ‘things as static and lifeless, each one by itself, alongside of and after each other’ ^{[638]}. To use Zeno’s argument as evidence for the distortions which formal logic apparently imposes on change and motion, as Marxist-Leninists do, is to mistake a particular application of the principles of identity, non-contradiction, and the excluded middle, with these principles themselves.
There is nothing in the principle of identity that would limit its application to what does not change or move. In the formula: (A) (A is A), or better: (p).p⊃p, every suitable expression can be substituted for the variable. ‘If a body is in motion, then a body is in motion’ is as good an example of the identity law as ‘a table is a table’. The law of identity neither confirms nor denies change or motion. Logic is not a science about bodies in motion or at rest; it is altogether indifferent to such matters and applies equally to both. An inference is a substitution of a logical formula, formally valid if the formula is valid, but its truth and falsehood, its correspondence with reality, is determined by extra-logical considerations, which do not belong to logic and for which logic cannot be responsible.

The same applies to the principle of non-contradiction and of the excluded middle. The sentence ‘A moving body either remains or does not remain in the position x at the time t’ is a correct substitution of the law of the excluded middle. Whether it is true or false, is a matter which does not belong to formal logic; it is a question of physics. If by means of logical formulae we come to the conclusion that motion is a series of stationary states, not our logic but our physics should be blamed. Since the question of whether we dismiss motion as an illusion or accept it as real must be decided on extra-logical grounds; since, moreover, formal logic applies both to stationary and moving bodies, both changing and unchanging, there is no reason to accept two logics, a formal and a dialectical logic, based on different principles^{[639]}.

If logic is subordinated to dialectics, the accepted order of priority is reversed. For it is assumed, as a rule, that formal logic provides the basis for all science and that it logically precedes any other discipline. This claim is made for several related reasons. Formal logic is the simplest instance of a discipline constructed in accordance with certain intelligible principles stated beforehand and strictly adhered to. These principles are of such generality, clarity and precision that they serve as a universal model of theory construction. Consequently, in any deductive theory logic is presupposed. Logical terms are applied without definition and logical laws made use of in proofs without their validity being examined.

Deductive method and deductive reasoning are not restricted to deductive sciences. Logical argument plays an important and, according to some methodologists, an all-important role also in the empirical science^{[640]}. Only deductive reasoning makes possible the discovery of what a hypothesis implies and a hypothesis cannot be critically examined, to be disconfirmed or confirmed by the failure if its refutation, unless all its implications, including its remote and not easily perceptible consequences, are clearly realised. It is logical reasoning that leads our steps from the level of theory to the level of facts and helps to discover whether the latter confirm the former. Logic is presupposed as much by the deductive as by the empirical sciences.

In general, no argument whether in science or in everyday life can do without logical concepts and every valid inference is made in accordance with some logical laws. The use of logical concepts and laws might not be consciously made, but this does not invalidate their fundamental role. There are probably few men who are aware all the time of all the logical laws they apply in their inferences, but everybody resorts to them from time to time to verify a proof or to discover the source of errors. A Marxist-Leninist is no exception to this rule and the recognition of this fact ultimately led Schaff also to the rejection of his view on the superiority of dialectics over formal logic.

Formal logic can be described as a study of forms on which valid inferences are based. A proof is a sequence of propositions and to ascertain whether the conclusion is arrived at by valid inferences we do not need to and, for certain reasons, should not consider the content of the propositions in question. To study these structures in a precise manner we must investigate various types of propositions and relations among them in abstraction from their content, i.e. to restrict ourselves to the investigation of propositional forms.

Marxist-Leninists recognised that formal logic is solely concerned with the ‘formal aspect of the processes of thought’ and they also somewhat grudgingly conceded that this is a reason for its success. The formalism of formal logic has achieved its highest perfection in mathematical logic, by which they understood, in agreement with the view more and more widely accepted in Poland, various logical methods applied in the investigations of the structure and foundations of mathematics. The achievements of mathematical logic, in Poland and elsewhere, could not have been denied. The fact that mathematical logic has grown out of formal logic was, for instance, for Schaff an important factor in the philosophical evaluation of formal logic itself^{[641]}.

The formal character of modern logic sharply differentiates it from dialectical logic, which, unlike the former, considers also content. This, Marxist-Leninists thought, was not necessarily a handicap and did not imply that dialectical logic was a discipline inferior to formal logic. There are logical systems that are formal without being formalised, the abstraction from the content varies and displays a variety of degrees. To paraphrase Plekhanov’s dictum, paying to formal logic the homage which is its due leaves a dialectician free to pay homage also to dialectics. Formalism, which is desirable in one branch of knowledge, may be less or not at all desirable in another field. Since ‘dialectical logic’ means here ‘logic in the broad sense’, his opinion is basically true.

Although logic to be formal must abstract from content, this does not release it from the obligation of testing the correspondence of its formulae with the observable relations and matters of fact. Formal logic, Schaff argued, would be a futile game and not a serious scientific investigation if it did not pass this test successfully. His moderation in this respect was persuasive in a manner which a more extreme militancy would have failed to obtain^{[642]}.

Marxist-Leninists were neither the first nor the only philosophers who demanded some kind of verification of logical insights concealed behind its symbolical garment. Before the war Leśniewski was the logician who was probably most acutely aware of the problems involved and Lukasiewicz never forgot them though he perhaps failed to pay them enough attention^{[643]}. Among philosophers the phenomenologist Ingarden was the most incisive and by far the most competent critic of formal logic and its philosophical claims. He urged incessantly that the relation between reality and logical assumptions should be carefully studied. After the war Ajdukiewicz discussed the problem of the relations between formal logic and experience and indicated various possible approaches^{[644]}. Schaff’s demand was, therefore, nothing new, but none of the Marxist-Leninist opinions on formal logic gained a greater number of supporters among logicians and philosophers than the view concerning the necessity and urgency of examining the objective validity of formal logic. It might be said that the support for it was practically unanimous^{[645]}.

It is, however, one thing to request the test of formal logic by observable phenomena and quite a different matter to say exactly how it should be done. The verification of formal logic confronts us with problems of considerable difficulty. Schaff’s ideas on the subject were simplified in the extreme. What he conceived to be a verification differed essentially from the logician’s view on the matter and though they seemed to speak of the same problem they really spoke of two different things^{[646]}.

Schaff did not approach the verification problem of logic on its own ground and felt that the issue can be satisfactorily resolved by examining the interpretations of logical formulae in ordinary language. As a matter of fact, he did not see any difficulty at all in answering the question as to whether logic applies to reality. He followed in Lenin’s footsteps and in the simplest, most common expressions, like the sentence ‘Fido is a dog’ or ‘John is a man’, saw a convincing proof that reality involves contradictions sensu stricto and that in general does not adhere to the laws of logic. These sentences apparently reveal that the ‘singular is the general’, that the opposites are identical, and that everything simultaneously is itself and something else. What is revealed by the analysis in the linguistic expressions, is abundantly confirmed by natural science in the realm of facts. ‘Objective nature’ also shows us the transformation of the singular into the general, the interfusion and transition of the opposites^{[647]}.

The error which these views involve, results from the confusion of logical types and of identity with class membership or class inclusion. ‘The singular is general’ is a perfectly true sentence if it is understood to mean “’being singular’ is a general characteristic.” Otherwise it is a meaningless sentence-like composition of words. No contradiction is involved in the sentence ‘John is a man’. John remaining identical with himself may be a member of the class of men, of husbands, soldiers, or whatever it might be, without helping to establish the dialectical law of contradiction. The same thing might be a member of different classes and no contradiction arises therefrom unless the classes are considered to be identical or the inclusion relation is taken to be that of identity, for which, however, there is no warrant.

While a fallacious logical analysis of language is the apparent source of the fanciful conclusions, a more fundamental error is hidden a little deeper. To test formal logic by observable phenomena it is not enough to find an interpretation of the former in ordinary language and to examine how it corresponds to the external world. For various reasons, of which only one will be briefly mentioned, this course encounters insuperable difficulties. Ordinary speech is too vague and imprecise to be reduced to simple elements and to fit unambiguously logical formulae, in order that their correspondence with reality may be directly ascertained. Between the interpretation and the experienced reality there lies always a screen of vague meanings inherent in the expressions of ordinary language and this screen prevents seeing clearly what the language only dimly conveys. This should be borne in mind when at the third stage of his proof Schaff tries to show that in the light of his interpretation the principle of noncontradiction does not apply to things or phenomena of Nature.

Simplified ideas about the manner in which formal logic might be shown to correspond or not to correspond to observable matters of fact are not independent of what Marxist-Leninists believed to be the foundations of formal logic. In their opinion formal logic is always based on the principles of identity, of the excluded middle and of non-contradiction^{[648]}. By testing these principles we can, therefore, test by one stroke the whole of formal logic. Only many-valued logics include certain deviations from this rule. Since, however, they still remain purely abstract constructions they can be left aside for the moment.

By saying that the three above-mentioned principles constitute the base of formal logic, Marxist-Leninists meant that in any formal system they are either assumed as axioms or follow from axioms, the latter having been so chosen that the three principles can be derived from them. Moreover, the three principles constitute the base of formal logic also in this sense that they are used as metalogical theorems. Thus, for instance, the so-called principle of bivalence corresponds to the logical principles of non-contradiction and of the excluded middle. The three logical principles are ‘organically interrelated’; they are either materially or inferentially equivalent. They must either be accepted together or rejected together^{[649]}.

Formal logic accepts the three principles, Schaff argued, because they are considered to be self-evident and thus particularly well suited to constitute the base for the whole of formal logic. Besides this explanation, for which there is no evidence, Schaff mentioned another reason closer, though in a misleading fashion, to the prevailing views about the nature of logical and mathematical theorems. In his view, formalism fosters the tendency towards a-priorism, and a-priorism protects our preconceived ideas from the verification by experience. The three principles are considered by their supporters to be a priori truths (and thus universally valid). This last view is emphatically and quite rightly rejected by Marxist-Leninists^{[650]}.

Dialectics was supposedly not alone in its attempt to revise the foundations of logic. Seen against the background of the tendencies apparent in modern logic, dialectics, with its rejection of the law of non-contradiction, is nothing extraordinary. The logicians themselves have become aware that the old logic requires ‘some generalisation and extension’. Aristotle considered the principle of non-contradiction as the ‘firmest of all opinions’, but was not convinced that it was an axiom immune from revision. Medieval philosophy killed what was strong and creative in Aristotle’s thought; it replaced his moderation with ‘absolute conceptions’, which dominate the traditional way of thinking and its inclination to see phenomena as static and at rest. Only Lukasiewicz revealed what was Aristotle’s authentic view on the principle of non-contradiction and threw doubt on its validity^{[651]}.

Schaff suggested that in many-valued logics a distinct step towards the revision of the traditional approach to the principle of non-contradiction has been made. Many-valued and intuitionist logics reject altogether the principle of the excluded middle, and consequently change their ‘attitude to the principle of noncontradiction’. Thus, the restricted validity of traditional logic has been put into relief by investigations which, starting from different points of departure, have reached the same destination. The rejection of the principle of bivalence brings dialectics and formal logic closer together. In Schaff’s opinion, we witness a revision of the fundamental laws of logic. A new viewpoint on the foundation problem of logic, to which dialectics contributes its share, is being formed^{[652]}.

The defence of the dialectical revision of logic was elaborate but ineffective, for most of its arguments rest on errors or misunderstandings. They are so numerous that not all of them can be dealt with. The exposure of the more fundamental ones is sufficient to make the whole case for dialectics collapse.

To begin with, there is the claim that the principles of identity, of the excluded middle, and of non-contradiction are inferentially equivalent, that they constitute the base of all logic, that they have this role ascribed to them because of their being considered as a priori laws of thought, and that logic built on such assumptions is an unsound discipline, in need of a critical examination. This claim would have been justified, if its premisses were correct. This, however, is not the case.

While in the so-called classical calculi the principles of identity, of the excluded middle, and of non-contradiction are inferentially equivalent, we face a different position on the ground of philosophical logic, which is that chosen by Schaff for his critical appraisal of logic. We cannot say, as Schaff did, that they are ‘various formulations of one and the same fundamental principle, namely that of noncontradiction’. If someone rejects the formula

∼ (φχ.∼φχ)

he may still accept that

φχ.∼ φχ.⊃. φχ.∼ φχ,

that is, recognise the law of identity. This was the case with thinkers like Heraclitus, Nicholas of Cusa, Hegel, and Engels who all rejected the principle of non-contradiction. Moreover, if we reject the principle of identity, as Kratylos did, all statements become false. For from the negation of p ⊃ p we can infer ∼ p. The rejection of the principle of non-contradiction implies that all statements are true. From p and ∼ p we can infer an arbitrary q. The two principles mean different things and carry different implications. They are not various formulations of the same law, the rejection of one of them does not imply the rejection of the other^{[653]}.

Nor are the principles of non-contradiction and of the excluded middle as closely bound up as Schaff suggested. To state that a certain object, e.g. a mathematical entity, is not contradictory, is not equivalent to the statement that it is determinate and that the principle of the excluded middle applies to it. This is exactly what the intuitionists claim in their dispute with the formalists over the sense to be associated with the existence of mathematical entities. We can, without inconsistency, respect the principle of non-contradiction and question the validity of the principle of the excluded middle. What is more, we can actually indicate the method of constructing a non-contradictory entity, e.g., a number which is indeterminate in the sense that we are unable to show that it is either rational or non-rational, either real or non-real^{[654]}.

The opinion that in modern investigations on the foundations of logic and mathematics there is a return to Kant’s a-priorism is not shared by Schaff alone^{[655]}. It does not seem to be a case of an intentional distortion of the opponent’s views in order to assail him with a seemingly devastating effectiveness. It appears to be a genuine misunderstanding, however difficult it might be to comprehend. The school to which the return to Kant’s a-priorism is ascribed is, of course, logical empiricism with its division of all meaningful statements into empirical and analytic. To the latter group belong all statements which are true or false by virtue of their composition alone. They owe this distinction to the fact that they do not say anything about reality and are derived from arbitrarily laid down axioms and definitions by well-defined rules of transformation. Their origin explains their name – analytic or tautological statements. According to the theory of logical empiricism, logical and mathematical theorems are tautologies in the above indicated sense^{[656]}.

Whatever else we may say about this theory, it is clear that a tautology in the sense of logical empiricism is neither synthetic nor a priori in the Kantian sense. It is close to Kant’s analytic sentences, which, like tautologies, are discovered by pure thought, without being identical with them. Kant’s analytical sentences must have a subject and a predicate, and their analycity depends thereon. Analytic statements in the sense of logical empiricism are only seldom subjectpredicate sentences. Most of them are functorial propositions, that is to say, they are propositions in which propositional functors occur and their analycity is closely bound up with this characteristic. Moreover, they are not extricated, as it were, from the mind itself, as are Kant’s analytical sentences, but are transformations of more or less arbitrary axioms carried out by means of more or less arbitrary rules. Their truth and falsehood is recognisable by inspection in view of the fact that the criteria of logical truth refer to the notational features of statements. The more formalistic a logical system is, the less reason there is for the revival of Kant’s division and the less room for a priori truths in his sense.

The view that the principles of identity, of the excluded middle and of noncontradiction constitute the base of logical systems is erroneous. It is true that they follow from the axioms in any complete calculus, which shows that they are logical laws as much as but not more than any other logical theorem proved in this calculus. Upon the publication of Principia Mathematica this view was universally accepted, for there is no reason for differentiating various logical laws once they are presented in axiomatic form. The choice of axioms is prompted by many considerations, one of them being that of the completeness of the system, and this objective is achieved when any meaningful expression within this system can be either proved or disproved. So far as the role of the principles of the excluded middle and of non-contradiction in the propositional calculus are concerned, it has been found, and caused much astonishment, that in spite of their fame they are only rarely used in proofs. Other less famous theorems are endowed with a much greater inferential power than the two principles in question.

History of logic has something to say on how the three principles have acquired the reputation of being the ‘supreme laws of thought’, a reputation they still seem to enjoy with some conservative philosophers. The law of identity is independent of all the other theses of Aristotle’s syllogistic. If it is to be included in the system, it must be accepted axiomatically. Moreover, if a proposition in which the predicate is contained in the subject (an analytic proposition in the Kantian sense) is accepted as self-evident, the same, and a fortiori, applies to the law of identity. In the latter case the predicate is not only contained in but is also identical with the subject.

So far as the principles of the excluded middle and of non-contradiction are concerned, they play a considerable role in discussions and deductions in which the reductio ad absurdum provides a powerful instrument of inference. This seems to be the reason why Aristotle accorded to the latter an exalted position. Modern research in the history of logic suggests, however, that even Aristotle ceased to ascribe to the principle of non-contradiction its elevated rank when he realised that his syllogistic is independent of it^{[657]}.

The laws of identity, of non-contradiction, and of the excluded middle do not seem to have acquired the reputation of the ‘supreme laws of thought’ until the views of Leibniz, Kant, and Fichte became known. The reasons which prompted these thinkers to exalt them were mostly of an extra-logical nature and sprang from epistemological and metaphysical considerations. The ‘supreme laws of thought’ cannot be found in the English nineteenth century textbooks of logic, in J. S. Mill System of Logic, W. S. Jevons’ Principles of Science, J. N. Keynes’ Formal Logic but they do appear in sberweg System der Logik and Wundt Logik^{[658]}. It should be remembered that sberweg’s textbook was Plekhanov’s and Schaff’s source from which they both drew their knowledge of logic.

Czeżowski suggested that in the absence of modern formal logic the three principles under discussion were fulfilling the function of a substitute for the non-existent propositional calculus. They seem to continue to fulfil this function in informal discussions. In such discussions the law of non-contradiction plays the role of the rejection rule which protects us from the acceptance of a pair of expressions from which any arbitrary expression may be derived. If that were possible, it would be no use to argue about anything.

There remains the question of the relation between the principle of bivalence and the notion of consistency of a formalised system on the one hand, the principles of the excluded middle and non-contradiction on the other. Schaff quoted Łukasiewicz as his authority to support the claim that on the ground of modern formal logic the principle of bivalence has restored the two principles to their elevated position of the ‘supreme laws of thought’ ^{[659]}. He failed to notice, however, that with the principle of bivalence we move to a more abstract level, that of metalogic. Łukasiewicz did not overlook this fact. He made it clear at that time that the principle of bivalence corresponds to the metalogical (and not logical) principles of non-contradiction and of the excluded middle, but he soon abandoned this view and later investigations confirmed that this was the right course to take^{[660]}. The logical relations in various many-valued systems turned out to be more varied than it at first appeared and their study threw light also on the relation between the principles of bivalence, of non-contradiction, and of the excluded middle.

The formulation of the principles of bivalence and of the excluded middle makes it clear that they should be kept apart. While the first of these principles states that every proposition has one and only one of the two possible truth-values, the second lays down that of two contradictory propositions one must be true. The principle of bivalence is a metatheorem or a metaprinciple, characteristic of the so-called classical logical systems. In a bivalent system, in which negation of a false proposition is a true proposition, the negation of a true proposition is a false proposition and the alternative of p and q is true, if at least one of its arguments is true, the law of the excluded middle holds. In other words, the principle of the excluded middle follows from the law of bivalence and the matrices of negation and alternative. The difference between the two principles becomes still clearer if the three-valued propositional calculus is considered. In this calculus the laws of the excluded middle and of non-contradiction might or might not be valid and this depends on how the matrices are determined. The rejection of the bivalence principle does not necessarily imply that the laws of the excluded middle and of non-contradiction must be rejected too. It follows that the principle and the laws in question cannot be equivalent^{[661]}.

The concept of consistency and that of non-contradiction are concepts of different orders or levels. The latter is a logical, the former a metalogical concept. The law of non-contradiction may not be a theorem of a consistent theory. On the other hand, since an inconsistent theory includes all its meaningful expressions, the theorem of non-contradiction is a theorem of an inconsistent system. The sharp distinction between the metalogical principles of bivalence and consistency and the logical laws of the excluded middle and non-contradiction were firmly established at the time Schaff set out to revise the foundations of logic^{[662]}.

The consistency of a formal system might be defined without resorting to semantical concepts, that is to say, it might be given a definition syntactical in character, making no use of the metalogical principle of non-contradiction. But the requirement of consistency is prompted by semantical considerations; its objective is to exclude from the system pairs of propositions which cannot be true together. It offers the same kind of guarantee with respect to the whole system that outside it is provided by the law of non-contradiction. A deductive system which includes a pair of contradictory propositions would include any proposition whatsoever. All contradictory systems are, therefore, identical and their study would be of no interest. They would not help us in the search for truth by establishing valid forms of inference and by discovering the invalid ones. While it can be argued whether consistency is both the sufficient and necessary condition of the truth of logical and mathematical theories, the fact that it is a necessary condition is not questioned by any qualified person. To suggest, as Marxist-Leninists did, that the main obstacle to the acceptance of the dialectical principle of contradiction is force of habit and tradition, does not testify to the awareness of the issues involved, both in formal and non-formal reasoning^{[663]}.

The semantical principle of non-contradiction is indispensable in classical and non-classical logic, in formal and non-formal theories, it is a guidepost without which no advance of rational thought could be made and no argument about anything whatsoever could be carried on. In this sense Łukasiewicz called the semantical principle of non-contradiction an absolute principle^{[664]}. Its indispensability includes also the discourse in which a Marxist-Leninist states that he does not accept its validity. He cannot say what he does unless he assumes that his statement and its denial cannot be true together. Only on this assumption is his statement meaningful and can its truth and falsehood be examined. What a Marxist-Leninist denies in his theory he has to accept on the metatheoretical level, unless he speaks like an oracle, neither understanding himself nor expecting others to understand what he is saying.

To support the dialectical revision of formal logic Schaff referred to some modern developments in logic, which, as he thought, spoke in favour of dialectics. What Schaff said on this matter might be reduced to two main claims. First, that many-valued logics lead to the rejection or restriction of the principle of non-contradiction. Second, that many-valued and intuitionist logics deny the principle of the excluded middle and this denial affects in some way or other also the validity of the principle of non-contradiction.

The first of these claims needs a few introductory comments. We do not pass from the bivalued to the three- or many-valued propositional calculus by rejecting the principle of non-contradiction and of the excluded middle, but by rejecting the principle of bivalence, which states that every proposition is either true or false. For this purpose we accept the assumption that besides true and false propositions there are – to confine ourselves to the three-valued calculus also propositions which are neither true nor false. In other words, instead of two we accept three truth-values which any statement may have, and thus we replace the current and intuitive true-false dichotomy by the trichotomy of true, false, and tertium. Every statement has one and only one truth-value, it is either true or false or tertium. Tertium non datur is replaced by quartum non datur.

The question arises whether we can construct a three-valued propositional calculus which is consistent, that is to say, in which no meaningful propositional formula ? is both asserted and rejected. As is well known, this question has been answered affirmatively.

Three- and bivalued calculi are incomparable in view of the fact that different principles are assumed as their basis, and, consequently, their truth matrices are different. They both include only identically true formulae (tautologies), i.e. formulae which are true for all permissible substitutions. Not every formula valid in one system is, however, valid in the other, and conversely. Thus, there are formulae of the three-valued calculus which have no analog in the bivalued system and bivalued formulae which are no longer identically true formulae in the three-valued calculus. In particular, in Łukasiewicz’s three-valued system the law of the excluded middle and of non-contradiction

p V ∼ p, ∼ (p. ∼ p)

are no longer valid. If ‘p’ assumes the value of truth or falsehood their value is truth. But if ‘p’ is tertium, their value is tertium. Not being tautologies, the two laws do not belong to the three-valued propositional calculus. It would be an error, however, to infer that their negations are theorems or that two formulae like ?, and ∼ may be true together. This is excluded by the consistency of the system.

Schaff’s first claim is not, therefore, justified. The three-valued calculus does not make the contention of contradictory propositions being both true any more acceptable than it would have been without it. The fact that the law of noncontradiction is not a valid theorem of Łukasiewicz’s calculus is a consequence of the trichotomy of truth-values and of the manner in which the truth matrices have been determined. Our inferences are bound to follow a different, though consistent, pattern if besides true and false propositions those which are neither true nor false are also accepted. Schaff’s error seems to result from the confusion of the statement ‘? is neither true nor false’ with ‘? and ∼? are true together.

Schaff assumed that the laws of the excluded middle and of non-contradiction are always materially or inferentially equivalent. This appears to have led him to the contention that many-valued logics, including intuitionist logic, throw doubt upon the universal validity of the law of non-contradiction, the reason being that the law of the excluded middle is not one of their theorems. It is true to say that in the two-valued propositional calculus the two laws are inferentially equivalent (with respect to De Morgan laws), and that if we reject one of them, we have also to reject the other. In many-valued logics this is no longer the case. Generally speaking, the position depends on how the truth matrices are determined and they might be determined in various ways without the risk of inconsistency. Thus, in Łukasiewicz’s three-valued calculus neither the law of noncontradiction, nor that of the excluded middle are valid. On the other hand, in intuitionist logic the law of non-contradiction is a valid theorem and the law of the excluded middle is not.

It is misleading to say that intuitionist logic altogether rejects the law of the excluded middle. An intuitionist does not question its validity outside mathematics at all. For him as much as for anybody else two statements like ‘this body is in motion’ and ‘this body is not in motion’ cannot be both true. Intuitionist logic, wrote Heyting, concerns only mathematical propositions^{[665]}.

Not all objects, but only those which Brouwer called ‘mental mathematical constructions’ require a novel treatment. Upon being formalised, the formal system, of which intuitionist mental mathematical constructions provide a model, proves to have peculiar characteristics. Intuitionist logic is a part of mathematics. It grows out of the latter and is not something logically prior by means of which mathematics is constructed and which serves as its foundation. Moreover, within mathematics the principle remains valid as long as we deal with finite sets. The difficulty begins when we pass to the infinite sets or sequences of numbers. According to the intuitionists in these cases the principle is no longer admissible as a basis for existential inferences.

The principle of the excluded middle: (p) . p V ∼ p, as it is understood by the intuitionists, demands a general method of solving every problem. To assert this principle means for them that they are able to prove by an actual construction either ‘p’ or ‘∼ p’. Since such a general method of construction does not exist, they refrain from asserting the principle of the excluded middle. A ‘mathematical theorem’, in the sense in which an intuitionist uses this term, refers to an empirical fact, that is, to the fact that a certain construction has been successful.

The issue between the intuitionists and other mathematicians does not, in fact, concern the validity of the principle of the excluded middle but the problem of mathematical existence, of what is really meant when the mathematician says ‘there is such x that φx’ and the range of the variable x is an infinite set. Let us assume that we try to prove that there exists a number x that fulfils a certain condition (φx). Let us further assume that we have found the proof which shows that the assumption ‘there is no such x’ results in a contradiction. A mathematician of the logistic or formalistic school of thought would then consider that by the law of the excluded middle he has proved the existence of a number such that φx. But an intuitionist would reject this conclusion. The negation of a general statement is an insufficient basis for proving the validity of an existential particular statement. To prove the latter we must be able to construct what in this particular statement is asserted to exist. An indirect proof does not allow us, as a rule, actually to construct a number x such that φx, and the truth of a mathematical theorem is determined by the law that enables the construction of the entities to which the theorem refers^{[666]}.

Intuitionist logic assumes, therefore, that besides true and false theorems there exist those unprovable, and thus rejects the principle of bivalence (which a formalist accepts, thus rejecting unprovable theorems). In other words, intuitionist logic accepts besides truth and falsehood at least one other truthvalue, different from truth and falsehood^{[667]}. This third value of the intuitionists is different from Łukasiewicz’s tertium. While in Łukasiewicz’s system the negation of tertium has tertium as its truth-value, the negation of the intuitionist’s third value gives falsehood. On this account the law of non-contradiction is not valid in Łukasiewicz’s system and is valid in intuitionist logic. Łukasiewicz’s statements, whose value is tertium, can be described as statements which are neither true nor false; in intuitionist logic to say that a ‘statement is unprovable’ means the same as ‘it is not false but it cannot be proved to be true’. In both systems the law of the excluded middle is not valid, but the lack of validity results from different considerations. Since an unprovable statement of intuitionist logic is not false, it is absurd to assume that the denial of the law of the excluded middle is true. While p V ∼ p is unprovable, the so-called law of the absurdity of the absurdity of the principle of the excluded middle

∼∼ (p V ∼ p)

is a valid theorem of intuitionist logic.

It turns out that Schaff’s second claim is also based on error. Intuitionist logic does not deny the validity, as it were, of the elementary law of the excluded middle, in which dialectics is interested. Intuitionist logic considers the law as a theorem unprovable for certain classes of mathematical entities, but in these cases it assumes that its denial is absurd, and dialectics is concerned exactly with this denial. In fact, intuitionist logic is as much incompatible with dialectics as is the classic conception of logic that assumes the validity of tertium non datur without exception.

The discussion of the problems which have arisen from non-classical logic and from support of the claims of dialectics with which the principles adopted in the non-classical calculi allegedly provided dialectics has an air of unreality. A Marxist-Leninist should have refused to draw any advantages from the development of many-valued logics, even if there were any advantages to be drawn therefrom. In his opinion, any deviation from the principle of bivalence is incompatible with Marxist-Leninist philosophy and its theory of ‘objective truth’. The theory of ‘objective truth’ is essentially the familiar one, known as the ‘correspondence theory of truth’. Marxist-Leninists felt that this theory was perfectly capable of dealing with the problem of contingent propositions, which in Poland played a considerable role in the discovery and development of manyvalued logics. A satisfactory solution of the difficulty presented by the contingent propositions would dispose of one of the most important reasons for the abandonment of the dichotomy of truth and falsehood. To achieve this purpose Schaff, in particular, adopted some of Kotarbiński’s views concerning the truth value of statements in the future tense and, with slight modifications, incorporated them into what he called the ‘materialist theory of truth’.

Kotarbiński distinguished two kinds of statements in the future tense, the one consisting of statements either true or false and the other of those neither true nor false. The statement ‘Caesar crossed the Rubicon’ is neither true nor false at any time preceding the moment of his crossing the Rubicon. The statement ‘Halley’s comet passed through the orbit of the Earth’ is true at any time also prior to the time of its passing through the orbit of the Earth. The second statement is always true because the statement concerning the cause of the event is always true, its sufficient and necessary conditions are known. Statements in the future tense are either true or false if the events which they refer to are determined by the laws of Nature and neither true nor false otherwise^{[668]}.

Schaff accepted Kotarbifiski’s distinction and also his explanation why the first kind of statements in the future tense are either true or false. He disagreed, however, with Kotarbiński’s opinion concerning the second kind of statements and followed Chwistek instead^{[669]}. Chwistek felt that Łukasiewicz’s views concerning the three-valued calculus were ‘naive’. What Łukasiewicz wished to achieve by means of his three-valued calculus can be accommodated within the framework of two-valued logic by means of the calculus of probability. It is not quite clear whether Chwistek denied the existence of ‘true’ contingent propositions in Aristotle’s sense. It cannot be doubted, however, that in his opinion whatever the status of these propositions might be, they do not require a different logic from that based on the dichotomy of truth and falsehood. This was also the view which Kotarbiński finally adopted under Leśniewski’s influence.

Schaff argued as follows. If truth consists in the conformity of thought to reality, statements about the future which are predictions based on uniform regularities of Nature are either true or false as much as any other proposition. This view, which Schaff did not adequately explain, seems to imply that if a future event is predetermined by a cause or causes existing today – and this applies to events which are particular cases of a uniform regularity of Nature – the conformity of thought to reality can also be ascertained in advance. Predictions are verified in future but are true or false at present in so far as they do or do not correspond to what is the case, the case being determined by uniform regularities of events.

On the other hand, a statement about a future contingent event, e.g. ‘John will be in the Soviet Union a year hence’, is no prediction in the proper sense of this term; it is not a statement about a predetermined event and its truth or falsehood cannot be ascertained in the manner indicated above. It does not follow therefrom that it is neither true nor false. Such a statement cannot be qualified in terms of truth and falsehood, but only in those of probability. What is undecided at present, what may or may not happen, is not true or false, but probable, the probability being associated with the ‘prediction’ and not with the event to which it refers. The question whether it is true or false cannot be asked at all, and the answer, if given, is meaningless. A statement is true, if it corresponds to the facts of the case, and false, if it does not. Neither of these possibilities applies to a statement about a future event, since there are yet no ‘facts of the case’. With one term of the correspondence relation missing, no correspondence relation can exist and be qualified as true or false^{[670]}. Schaff would agree with what Professor Ryle said on this matter. As the adjectives ‘deceased’ and ‘extinct’ cannot be applied to people and mastodons while they exist, so ‘true’ and ‘false’ cannot be used with respect to statements in the future tense. In a way ‘true’ and ‘false’ are obituary or valedictory epithets^{[671]}.

Thinkers who consider the principle of bivalence to be an absolute truth may dispose of the counter-examples, intended to show that it is too narrow to account for the great variety of formal structures embedded in the discourse, in three different ways. They can say that every statement is either true or false irrespective of whether we are able to find out which is the case. Propositions in the future tense referring to matters not yet determined constitute no exception to this rule. This is the most common argument against many-valued logics, close to the views of Aristotle himself, represented by Leśniewski in Poland before the war and more recently by Greniewski^{[672]}.

Second, the supporters of the principle of bivalence can contend that the examples of statements neither true nor false are not genuine statements, but statement forms with free variables of time, place, and so forth. They must be bound before they can become statements. As soon as this is done, they turn out to be either true or false^{[673]}.

The third way of dealing with statements apparently neither true nor false – and this one deals specifically with statements in the future tense – is to point out that the question whether some of these statements are true or false is not an ‘askable’ one, since it is inappropriate to the circumstances and in this sense absurd. The sentence “’p’ is neither true nor false” assumes that the question whether “’p’ is false” can be resolved. This assumption is wrong and must be rejected with respect to some sentences in the future tense. We cannot discuss what a man yet unborn does or fails to do. Timeless shadows of events which have not yet been successful ‘in the competition for actuality’ cannot be made into ‘surrogate contemporary things’ ^{[674]}. This is the position which Schaff took. Statements referring to contingent events are right or wrong guesses. They can be dealt with, as Chwistek suggested, by means of the calculus of probability. The calculus of probability can be developed within the framework of two-valued logic and only two-valued logic is, in Schaff’s opinion, compatible with the Marxist-Leninist conception of truth.

Schaff’s position is not, however, consistent. While he rejects statements which are neither true nor false, he accepts those which are contradictory, i.e. statements both true and false. Moreover, he also accepts the existence of a class of statements which admit different degrees of truth and, being either more or less true than some other statements, are partly true and partly false. The latter are not, as he expressly stated, probability statements to be accommodated within two-valued logic. He seems to argue that the principle of bivalence both is and is not adequate to describe the structure of the universe^{[675]}.

Although one can be uncertain what Schaff’s views on the principle of bivalence were, they were understood to mean a firm adherence to the dichotomy of truth and falsehood. The suggestion made earlier by a supporter of Marxism-Leninism that Marxism-Leninism should adopt the principle of the polyvalence of truth and falsehood, was never seriously considered^{[676]}. The implications of such a step could not be easily foreseen and the idea of many-valued logic appeared not only bold, but also subversive with respect to the naive and uncritical conception of ‘objective truth’. Moreover, some sound and good reasons against manyvalued logics could be given. Statements which for specific reasons cannot be qualified as true or false provide a flimsy ground for the rejection of the principle of bivalence. In modal logic we encounter some stronger but not decisive reasons for questioning this principle^{[677]}. Other examples of statements which might be interpreted in terms of three-valued logic find an adequate explanation also on the basis of the dichotomy of truth and falsehood. The principle of bivalence has served science well and is universally accepted in scientific investigations. No consideration important enough can be found to justify its rejection^{[678]}.

Marxism-Leninism, as interpreted in Poland, seems to agree with those who say: the fact that many-valued logics can be constructed does not entail that they are significant. They are purely abstract systems with no connection with reality. Consequently, many-valued logics might be didactically useful in reminding that no laws, be it even those of logic and mathematics, are immune from revision, but they have no other use^{[679]}. This implies that nothing whatever that manyvalued logics might have to say about the principle of non-contradiction is of any relevance to the issue of its validity. Marxist-Leninists failed to notice this implication. They should have insisted that the validity of the principle of noncontradiction is ultimately an ontological and not a formal problem to be solved in accordance with the ‘empirical evidence’ which might be brought in support of the claims of dialectics.