Schaff put forward three arguments to support the dialectical thesis that the ontological principle of non-contradiction must be rejected. He considered them to be empirical and decisive.

Nature, Schaff argued referring to Engels, is the test of dialectics. Modern natural science, and physics in particular, has furnished a vast body of evidence which reveals contradictions inherent in natural phenomena. An ‘objective contradiction’ is something more than a unity of the opposites; the latter does not always involve a contradiction sensu stricto. Thus, an atom is a unity of positive and negative electric charges, a magnetic needle – of two opposite poles, the motion of a planet on its orbit – of the centrifugal and centripetal forces, society – of mutually exclusive class interests. In each case the unity of the opposites is real and ‘objective’, but a statement referring to it does not entail two contradictory propositions being both true. This is the case with respect to an ‘objective contradiction’. An electron or a photon is both a particle and a wave; mass is and is not an attribute of matter. Since a contradiction sensu stricto is inherent in the nature of any particle of matter, an adequate description of a particle entails the truth of the conjunction

(∃φ) . φx. ∼ φx^{[680]}.

The second argument is stated very succinctly. It does not offer any proof, follows closely in Engels’ footsteps, and is satisfied with awakening some intuitions at the back of the reader’s mind, without examining their meaning and their logical implications. Any process of quantitative or qualitative change involves contradiction, any changing object is itself and not itself at the same time. Engels made the metaphysical mode of thought responsible for thinking in ‘discontinuous antitheses’; Schaff put the blame on traditional logic. Logic encourages conceiving things as fixed and isolated objects instead of as events, which, while they change, pass through contradictory stages and are transformed into their opposites. We are referred to Hegel for examples and reminded that already the ancients knew that summum ius est summa iniuria^{[681]}.

Ajdukiewicz tried to extricate from this almost inarticulate argument some formal pattern of inference of which it makes use. Let us consider, he suggested, any process of change, for instance, a ripening fruit or a rising column of mercury in a thermometer. Let us then consider two distinct stages of the change, the green and the ripe fruit or the height of the mercury column as it indicates 20 and 21 centigrades. Let A and B stand for these states at different times. We often assume that if an object x passed from the state A to the state B there must have been a time t at which x was neither A nor B. This common assumption, which seems to be in agreement with our everyday experience, might be formulated as the principle of continuity. This principle requires that in any sufficiently small interval of time the rate of change occurring during this interval becomes arbitrarily small. In other words, a continuous change is a compact series of ‘states’; no two ‘states’ are consecutive; between any two, however close to each other they might be, there are always others (this is a crude idea of continuity, but it is good enough for this purpose). The principle of continuity excludes the possibility that there should exist such an instant t during the time interval of the change from the state A to the state B that at the time t the object x is A and at every moment later than t the object x is B. It seems to follow that in the time interval of the change from A to B there must be contained such a sub-interval that within it the object x is neither A nor B.

If this premiss is accepted, a dialectical philosopher might infer his contention in the following manner. Let us consider the change of water (A) into steam (B). Water which is no longer in the liquid state might be described to be in the state non-A. In accordance with the principle of continuity there must have been such a time-interval when the considered portion of water was neither A nor nonA. But ‘x is non-A’ means the same as ‘x is not A’, and to say ‘it is not true that x is not A’ is equivalent to ‘x is A’. It follows from the premiss

∼ (x is A) . ∼ (x is non-A)

that

∼ (x is A) . (x is A).

The principle of continuity leads to the conclusion that during a certain timeinterval a changing object has contradictory attributes^{[682]}.

Schaff’s third and final argument for the rejection of the ontological principle of non-contradiction is Zeno’s time-honoured argument against motion. It was Hegel who in modern times revived Zeno’s paradoxes and turned them to his own advantage, namely to prove that motion is a ‘living contradiction’. Engels adopted Hegel’s use of Zeno’s arguments and Plekhanov made it prominent in his own presentation of dialectics. He also re-established the chain of ideas leading from Zeno through Hegel to Engels and passed it on to the Marxist-Leninist philosophy of our own times. Motion, wrote Plekhanov, is a ‘contradiction in action’, the moving body presents itself as an ‘irrefutable argument’ in favour of the ‘logic of contradiction’ ^{[683]}.

Bertrand Russell observed a long time ago that Zeno’s paradoxes have inspired practically all the theories concerned with the concepts of time, space, infinity, and continuity, and that they have acted as a powerful stimulus to the development of these concepts from Zeno’s times to our own day. Marxist-Leninist philosophy is not interested in the theoretical aspect of Zeno’s paradoxes. The interest of Marxist-Leninist philosophy in Zeno’s arguments has been invariably limited to a single point, namely, to their usefulness in showing that contradictions sensu stricto exist ‘objectively’ in the phenomena of Nature.

Of all Zeno’s arguments against motion Schaff considered only one, that of the ‘arrow in flight’. In his opinion, it is not a paradox but a flawless argument^{[684]}.

It proves either that from the assumption ‘the arrow is in flight’ follows the conclusion ‘the arrow is at rest’, or that a moving body is at the same time in motion and at rest, it is at a given point and it is not there. In both cases the premiss that something moves entails a contradiction. We thus face the following alternative: either to deny the fact that motion is real, as Zeno did, or to accept Zeno’s argument and the reality of motion. The latter course implies the conclusion that there are contradictions sensu stricto in Nature. Something must be given up, either the reality of motion or the laws of traditional logic. A Marxist-Leninist accepts the evidence of experience and rejects the laws of traditional logic ‘Motion is a contradiction, a unity of contradictions’. Thus, to use Engels’ words, he recognises that contradictions are ‘objectively present in things and processes themselves’, and, as it were, assume a ‘corporeal form’ ^{[685]}.

Engels himself was probably all the more convinced that we actually cannot describe motion consistently, because in his opinion the differential and integral calculus is based on the ‘contradiction that in certain circumstances straight lines and curves are identical’ and yet the calculus produces ‘correct results’ ^{[686]}. This was no longer true when Anti-Dhring was published. The classical theory of functions of a real variable, which we learn today in elementary calculus -the definitions of limit and continuity, of derivative and definite integral, of convergence and continuous function – is fundamentally that of Cauchy published in the ‘twenties of the last century. The exculpatory comment that Marx and Engels were misled by the mathematics of their time cannot be maintained^{[687]}. If he wished, Engels could have known that motion may be described as a continuous function of time and that the concept of continuous function does not involve any contradiction.

Lenin was aware of this fact but dismissed it as a standard objection of Hegel’s ‘metaphysical’ opponents. A mathematical analysis of motion describes the ‘result of motion and not motion itself’; it disregards the ‘possibility’ (origin) of motion; and it constructs motion as a sum of stationary states, to escape the logical but not the dialectical contradiction. The latter is thus concealed at a deeper level, without being resolved.

Lenin’s comment delighted some stubborn Polish adherents of ‘real contradiction’ and prompted them to add that the consistency of mechanics is relative, dependent on the consistency of the analysis. The analysis managed to ‘overcome’ the contradiction inherent in the concept of the infinitesimal to fall into new ones, arising from the concepts of class, infinite class and continuum. Thus Zeno’s paradoxes are reproduced at a higher level of abstraction. Mechanics provides a conceptual tool by means of which a consistent description of motion can be given, but this description is ‘banal’, phenomenal and dependent on arbitrary assumptions. Real motion, its source and cause, remain as elusive and inexplicable as before^{[688]}.

Once the dialectical implications of the ‘arrow in flight’ are accepted, certain conclusions become inevitable. If the fundamental laws of formal logic require that there should be no contradiction in motion and motion does exhibit contradictory attributes, formal logic is shown to be a theory too narrow to account for the phenomena of Nature. The logic of ‘either-or’ must make room for a more comprehensive one that includes Plekhanov’s formula ‘yes is no and no is yes’. There are, therefore, two logics: formal logic which excludes contradictions and is concerned with things at rest, and dialectical logic which is a ‘logic of contradiction’ and applies to things in motion. The former is a special and marginal case of the latter^{[689]}.

We should accept, Schaff argued, the thesis of logical dualism formulated by Plekhanov in Dialectic and Logic. Plekhanov’s thesis that dialectics does not suppress formal logic but ‘merely deprives the laws of formal logic of the absolute value which metaphysicians have ascribed to them’, is consistent with the teaching of Marx and Engels, basically true and to be accepted with some minor modifications^{[690]}. The modifications are as follows.

The so-called classics of Marxism-Leninism did not subscribe to the view that every asserted proposition can be justifiably denied. Engels emphasised that consistency of thought helps to ‘get over defective knowledge’ and spoke disapprovingly of ‘losing one’s way in insoluble contradictions’ ^{[691]}. The opinion that may be found in some Soviet textbooks and publications to the effect that ‘formal logic does not make sense’ can cause nothing but astonishment. There can be no question of dialectical logic ousting and supplanting formal logic altogether. Only the absolute monarchy of the latter must be abolished, and some kind of dual power, with priority accorded to dialectics, installed in its place^{[692]}.

If instead of the validity of the laws of formal logic being restricted, the laws of logic were altogether rejected, dialectics itself would not be immune from contradiction. Both the asserted and the rejected theorems of dialectics would then be valid and true. Dialectical logic recognises the objective existence of ‘concrete contradictions’, i.e. of objects which have contradictory attributes, but dialectics itself is not self-contradictory. Dialectics does not allow the acceptance of every pair of contradictory propositions, since this would invalidate ‘correct thinking’. Dialectics being a ‘logic of contradiction’ remains a consistent system. It rejects theses which deny its own theorems and does not include theorems contradictory to each other^{[693]}.

Dialectical logic returns to formal logic the services which the latter renders to dialectics. Dialectics guards formal logic against the dangers of formalism and of losing touch with reality by abandoning itself excessively to abstractions. Dialectical logic formulates the important methodological rule that prescribes to ‘split the unity’ which conceals the opposites inherent in every object, to reveal and to get hold of them by means of appropriate procedures. ‘To split the unity and to learn its contradictory factors’, wrote Lenin, ‘is the essence of dialectics’. This methodological rule is of particular importance in the social sciences, since it helps to lay bare the internal contradictions of bourgeois society and to guide to action conducive to the social development which results from the struggle and unity of the opposites. This is probably the main obstacle, Schaff suggested, to the acceptance of dialectical logic and accounts for the fact that the purely scientific dispute concerning the relation of formal logic to dialectics has degenerated into a distinctly political quarrel^{[694]}.

Thus, the relation between formal logic and dialectics turns out to be itself dialectic. On the one hand, they are interrelated and supplement each other. On the other, they are mutually exclusive and inconsistent. Marxist-Leninists did not seem to be able to decide which of the two possibilities applies and to be inclined to the view that both should be accepted. The urge for consistency, induced by the critical examination of Marxism-Leninism carried out by nonMarxist philosophers, is occasionally apparent but is ultimately overruled by extra-logical considerations^{[695]}.

The attempts at the reconciliation of formal logic and dialectics conceived as a ‘logic of contradiction’ was logically untenable, but in practice it had its advantages. The thesis of logical dualism has provided some protection for and made some concessions to the rights of formal and mathematical logic. It set restrictive limits to their study and applications, but within these limits logic could exist and survive.

Logical dualism prompted the extension of the so-called dialectical method to various fields of philosophical inquiry to the exclusion of any other method and reserved for dialectics a large place in the natural and social sciences, which led to something like total annihilation of some of them. On the other hand, logical dualism repudiated the intention of suppressing formal logic altogether and was explicitly committed to the recognition of its limited but real importance for scientific thinking.

The protection of logical dualism, which formal logic enjoyed, was reinforced by the results of the Soviet discussions on the subject-matter of formal logic and its relation to dialectics^{[696]}, and by Stalin’s pronouncements, known under the collective title Marxism and Problems of Linguistics, first published in Pravda in June-August, 1950. Before Stalin’s intervention the attitude of Marxist-Leninist philosophy to formal logic was, even in Poland, ambiguous and by no means provided a clear and safe assurance that Marxism-Leninism would recognise a further need for it. Upon Stalin’s pronouncement, formal logic, like mathematics and grammar, could no longer be considered as a ‘superstructure on the basis’, the doubt as to its usefulness and the desirability of its continued development was precluded and could not be voiced by a faithful Party member^{[697]}.

To return to the discussion of the arguments in favour of the ‘logic of contradiction’, it should be observed that the first argument for the rejection of the ontological principle of non-contradiction has concerned matters highly controversial since the late ‘twenties. Many theories have been formulated to explain them. They constitute problems to be solved by research and new explanations in the field of physics, for which a speculative short-cut is no substitute.

Schaff was quickly corrected in two essential respects. It is not true to say that an electron has simultaneously the characteristics of a particle and a wave, thus revealing contradictory attributes (in the logical sense of this term). No observation justifies this contention. Electrons do behave as if they were particles when they move in Wilson’s cloud chamber, and as if they were waves when they pass through a crystal. These are, however, two different phenomena, the time and conditions of their occurrence are different. We cannot speak of the simultaneous but only of the successive duality of electrons. The successive duality of electrons is not inconsistent with the contradictory characteristics of atomic particles; however, it does not imply them. If it did, we would be unable to understand what we were saying^{[698]}. For we know from the propositional calculus that if p implies q ≡ ∼ q, p must be a falsehood

p ⊃ . q ≡ ∼ q : ⊃ ∼ p.

There are grounds to believe that the conditions under which the discussed phenomena occur (the recording apparatus) and the phenomena themselves are interrelated. This admits of various interpretations. None of them is a sufficient basis for implying that a statement describing the electron trail in the cloud chamber (the particle characteristic of electrons) and a statement describing the diffraction and interference pattern of a beam of electrons (the wave characteristic of electrons) should be considered as a conjunction of contradictory statements. There exists the puzzling problem of accounting for the waveparticle duality of electrons, but this is a question of physics and not of philosophy^{[699]}.

Moreover, the physicists were not on the side of the interpretation in terms of the ‘logic of contradiction’. Both abroad and at home a great many of them were warning against reifying physical models of atomic particles and against the conclusion, drawn from their usefulness in describing the results of experiments, that something analogous, corresponding to the models, actually exists in Nature. What we know about the micro-objects makes it necessary to renounce the notion that they have a unique and precisely definable conceptual model or that their objective existence constitutes a definable whole, similar to that of large-scale phenomena, simultaneously and unambiguously accessible to observation. Those who refuse to accept the finality of this interpretation and the absolute validity of the indeterminacy principle would not consider for a moment an arbitrary assumption as a permissible way out of the difficulty^{[700]}.

Schaff was also corrected on the point concerning mass and matter and the transformation of mass and energy, governed by Einstein’s equation, which in his opinion provided another striking instance of an objective contradiction in a ‘corporeal form’. Mass and matter should not be identified. Like energy, mass is an attribute of matter, which can take the form of a particle or that of a field. In the former case physics speaks of mass as an attribute of matter, in the latter of its energy. Einstein’s equation establishes the relation between two attributes of matter, or between the two forms that matter can take. There is again the problem of the field-particle duality of matter that so far defies a satisfactory solution and indicates our inadequate knowledge of facts. The problem involved belongs to physics and cannot be explained away by philosophical speculations^{[701]}.

The second argument, namely that every process of change involves the simultaneous emergence of contradictory attributes in the changing object, fared little better in discussions. Ajdukiewicz provided an intelligible formal pattern of inference which the argument seems to follow. When this was done, it became clear that it is unsound.

It should first be observed that the conclusion of the argument, that is to say, the statement to the effect that a changing object must at a certain instant display contradictory characteristics

∼ (x is A) . (x is A)

is never confirmed by experience. Observation reveals that a changing object becomes different but it never verifies the contention that it exhibits contradictory attributes. This contention is an extrapolation; it refers to a possible state of affairs, which, being unconfirmed by experience and absurd in its implications, should be rejected^{[702]}.

Furthermore, there seems to be an error in the inference, by which the conjunction

∼ (x is A) . (x is A)

is reached. The characteristic ‘non-A’ occurs in the inference as if it were a singular term, which is not in fact the case. Since ‘non-A’ denotes all states different from A, it is a general term. The principle of continuity is applicable to a process of change from a definite state A to a definite state B, when both ‘A’ and ‘B’ are singular terms. In the case when ‘B’ (‘non-A’) is the name of a class of states, the principle of continuity does not apply. The inference, therefore, by which the conclusion

∼ (x is A) . (x is A)

was arrived at, is not valid^{[703]}.

Finally, and generally, Ajdukiewicz drew attention to the fact that our inability to decide in a particular case which of the incompatible predicates should be applied to a given object does not imply that both can be truly applied to this object, or that contradictory characteristics inhere in it at the same time. Sometimes our ignorance and sometimes the ambiguity of our language appear to suggest that two contradictory statements are both true. If we disregard these ambiguities we may easily reach the conclusion that Socrates who is turning old and bald is both old and young, both bald and not bald. This does not confirm the existence of contradictions in the external world, but indicates that the user of language has failed to specify the rules for the use of expressions, or to follow these rules, or to define the expressions themselves in relation to other expressions with a common ‘fringe’. It is the use, or rather the misuse, of language that makes contradictions and inconsistencies possible. They serve as a warning that the meaning of the expressions has not been fixed and that it must be defined before it is descriptively applied. Limitations in the applicability of descriptive terms invalidate reasonings by means of which the existence of contradictions is inferred from the occurrence of ambiguous terms in our speech. A descriptive term which does not distinguish an object from what it is not, is more or less useless, and a sentence in which it occurs purports to say something though in fact it does not say anything at all^{[704]}.

Ajdukiewicz’s explanations of elementary logic and logic of language had also some unintended implications, which might have impressed Marxist-Leninists. Ajdukiewicz showed that the emergence of contradictory attributes in the changing objects depends on the continuity of change. But for a Marxist-Leninist philosopher change is both a continuous and a discontinuous process, or a continuous process with ‘breaks in continuity’. The principle of continuity explains the appearance of contradictory attributes; it also clashes with and undermines the ‘theory of leaps’, and that some time ago was pronounced to be a heresy^{[705]}. The ‘theory of leaps’ explains by itself the emergence of new qualities and does not require the existence of ‘corporeal contradictions. In other words, one does not need to assume that every process of change involves a contradiction. It is enough to retain the Hegelian principle of the object being transformed into its opposite by the action of those polar forces which determine its existence. Instead of the contradictions, the polarity of the opposites provides a sufficient basis for the explanation of the spontaneous change and self-development of Nature. While the latter speculation is as arbitrary as the former, it is at least not incompatible with the fundamental requirements of logic and somewhat reduces the area of arbitrary and oracular thought.

Thus, however, the contradiction allegedly inherent in mechanical motion remains as the only mainstay left to support the contention that the law of noncontradiction is a relic of the metaphysical mode of thought, to be abandoned at the dialectical stage of the development of mankind. The question as to whether Zeno’s analysis of the ‘arrow in flight’ can be maintained acquires crucial importance.

Ajdukiewicz subjected this matter to a thorough examination. His discussion of Zeno’s paradoxes, in many respects novel, deserves close attention in view of the role his examination has played in the evolution of Marxist-Leninist philosophy in Poland^{[706]}.

Ajdukiewicz examined two possible interpretations of the flying arrow argument. Schaff seems to have used both of them in different places or in support of each other. The first of these interpretations starts with the assumption that the arrow is in flight and reaches the conclusion that if it is in flight then it is at rest. Since

(p) : p ⊃ ∼ p . ⊃ ∼ p,

the assumption must be rejected as self-contradictory. The argument proceeds by the following stages.

If the arrow is in flight during an interval T, then for every instant t of the interval T there is a position x in which the arrow is to be found (A). If there is a position in which the arrow is to be found at every instant t of its flight (T), then the arrow remains at rest throughout its flight (B). This implication makes use of the definition of rest by which a body is considered to be at rest during the interval T if there is a position in which it is to be found at every instant t of the interval T. From (A) and (B) follows the conclusion: if the arrow is in flight throughout T, then the arrow is at rest throughout T. The assumption that the arrow is in flight implies that the arrow is not in flight.

The inference seems to be a substitution of the following theorem of the propositional calculus

p ⊃ q . q ⊃ r : ⊃ . p ⊃ r.

This, however, is not the case. The sentences to be substituted for q in p ⊃ q and in q ⊃ r are not the same but two different propositions. The consequent of the first premiss is, ‘for every instant t of the interval T there is a position x in which the arrow is to be found’, and the antecedent of the second premiss runs, ‘there is a position in which the arrow is to be found at every instant t of its flight (T)’. Symbolically expressed, the consequent in p ⊃ q has the form

(t) (∃x) φxt (q1),

and the antecedent in q ⊃ r

(∃x) (t) φxt (q2).

The inference under discussion is, therefore, a substitution of a different theorem of the propositional calculus, namely of

p ⊃ q . s ⊃ r : ⊃ . p ⊃ r,

which is a valid formula if and only if q ⊃ s. But this is not so in our case, since q1 does not imply q2 (though q2 does imply q1.). The rules which govern the use of quantifiers do not allow the inference

(t) [(∃x) φxt] . ⊃ . (∃x) [(t) φxt] (C).

A different example might make it clearer. The premiss ‘for every x there exists such a y that x

The second, more common interpretation can be reduced to the following inference. If a body is in a definite position x at an instant t, then this body is at rest in x at the instant t (D). If the arrow in flight is in a definite position x at every instant of its flight, then the arrow is at rest at every instant of its flight (E). If the arrow in flight is at rest at every instant of its flight, then the arrow is at rest throughout its flight (F)^{[708]}.

The fallacy of Zeno’s argument is twofold. The first depends on the ambiguity of the connective ‘is’. The connective ‘is’ in the expression ‘the arrow is in the position x’ may mean the same as ‘the arrow remains at x’ (a). The connective ‘is’ may be also used, however, in the most general sense which does not specify the kind of relation between the arrow and its position, to mean ‘it passes x’ or ‘leaves x behind’ or ‘reaches x’ (b). If in the premiss (D) the first ‘is’ is used in the sense (b), the consequent of the premiss does not follow from the antecedent and the whole inference is destroyed. If, however, the connective ‘is’ in the antecedent of the premiss (D) has the meaning (a), it becomes a false statement, because a moving body certainly does not remain anywhere. Thus, there is no such meaning of ‘is’ in which the premiss (D) is true and the whole inference valid^{[709]}.

With the above distinction of meanings that the connective ‘is’ may have, Plekhanov’s problem does not offer any difficulty. If we disagree with the thesis that ‘a body in motion is at a given point, and at the same time it is not there’, Plekhanov wrote, we will be ‘forced to proclaim with Zeno that motion is merely an illusion of the senses’ ^{[710]}. This is exactly the contention which made of the ‘arrow in flight’ the main argument for the existence of contradictions in the phenomena of Nature. The antecedent of Plekhanov’s thesis is a perfectly true, though trivial proposition, if the first ‘is’ carries the meaning (b), and the second the meaning (a). We can agree with it without committing ourselves to the existence of ‘corporeal contradictions’. On the other hand, if ‘is’ is differently interpreted, Plekhanov’s antecedent becomes a false statement. We can disagree with it without being forced to admit that motion is illusory.

In order to know whether a body is in motion or at rest at an instant t, we must always consider what happens to this body at the instant earlier and later than t, i.e. we must consider a time interval that contains t ^{[711]}. Only then can we give exact definitions of what is motion, continuous motion, and rest. Zeno’s argument goes astray at this point. If the arrow is where it is at a given instant, it does not follow therefrom that the arrow is then at rest. We cannot know whether it is in flight or at rest without considering its position at earlier and later instants. Where it is now does not presume where it was before or where it will be after, i.e. whether it is or is not in flight. There are obvious excuses for Zeno’s paralogism; no physical definition of rest and motion was available at his time. Zeno cannot be blamed either for having made the assumption that there are consecutive instants and points. There are no such excuses for errors of the same kind when they are committed today^{[712]}.

To show that the alternative ‘either motion is illusory or it involves contradiction’ is invalid, since both its constituents are false, it is sufficient to apply the modern concept of mathematical continuity to Zeno’s paradoxes. Motion consists in the occupation by the moving body B of different places at different times. When throughout an interval, however short it might be, different times are correlated with different places, B is in motion. Similarly, B is at rest, when throughout an arbitrary interval different times are correlated with the same place. This prompted Bertrand Russell to say that Weierstrass ‘by strictly banishing all infinitesimal, has at last shown . . . . . . that the arrow, at every moment of its flight, is truly at rest’ ^{[713]}. By ‘being at rest’ Russell understands occupying a place equal to itself’. On this understanding Zeno was right in pointing out that an arrow in flight is at each instant where it is, irrespective of whether it does or does not fly. When the modern concept of continuity is applied, this does not imply that the arrow in flight is not at different places at different times throughout the interval of its flight.