Dimitris Moustakas uses formal logic to confront Zeno’s paradoxes in a deconstruction of Marxian “dialectical logic.”

Richard Artschweger, Zeno’s Paradox, 2004

Within Marxist circles, among both activists and academics, it is widely agreed that Marxism’s distinct identity as a scientific theory comes from its unique method, understood as ‘dialectical’.¹ Interpretations of this term can vary depending on the source, yet there are some consistent themes seen in every rendition of ‘dialectics’. One such theme is the notion that dialectics, to some degree, resists formal logic, urging instead the adoption of a more evolved form of logic termed “dialectical logic”. One might counter by arguing that:

a) the nature of dialectical logic remains unclear,

b) the rules governing it are unspecified, and

c) how to correctly apply it is yet undefined.

While I am open to raising such points, my objective in this piece is more narrowly focused. I aim to unpack just one aspect of the argument made by ‘dialectical’ thinkers as to why dialectical logic surpasses any form of formal logic. This aspect revolves around the issue of contradiction.

In many forms of logic (with paraconsistent logics being exceptions), encountering a contradiction (i.e., asserting a statement and its negation) signifies a serious problem. Dialectical logic, however, maintains that contradictions are feasible and should not be viewed with alarm. This stance largely arises from the idea that, according to dialectical logic, understanding movement and change is impossible within the bounds of formal logic. But does this claim hold up under scrutiny?

In this article, my objective is to lay out the arguments which purport that formal logic is incapable of grappling with change/movement. I will critically evaluate these arguments in an effort to demonstrate their flawed reasoning. It is not my intent here to declare dialectical logic and dialectics as entirely fallacious. Nonetheless – I am of the opinion that the compatibility of change with formal logic, as has been argued by several Marxists, points to some possible inconsistencies within dialectics. This, however, will be a topic for further discussion.

Contradictions in Formal Logic

To avoid any linguistic confusion, I will begin by defining what ‘contradiction’ signifies within the context of formal logic. This is crucial, as an accurate understanding of the term ‘contradiction’ is key to discussing its (non-)existence.

Suppose we have a proposition – this could be anything from “today is sunny” or “Bill is tall”, to “If he is German, then his position on the Palestinian issue is irrelevant”. For simplicity, let’s represent the proposition as p. Its negation is symbolized as ~p, and in formal logic, we understand the truth value of p to be the inverse of the truth value of ~p. Here’s an important note: the negation of a statement is not p, and not some other proposition q, that deviates from not p. This distinction is crucial, as one might incorrectly infer that “it’s raining today” is the negation of “it’s sunny today”. The correct negation of “it’s sunny today” is actually “it’s not sunny today”. “Today is sunny” and “today is not sunny” are different statements, and there are instances where it’s not raining, yet it isn’t sunny either (e.g., it could be cloudy or snowing). While “raining” and “sunny” may seem contrary or opposite, one is not the negation of the other.

This allows us to establish the principle of non-contradiction. Assuming a proposition p, the statements p and ~p cannot both be true simultaneously. Referring to our previous example, it cannot be both sunny and not sunny today. It may be rainy and sunny, or other ‘contrary’ conditions may occur, but this doesn’t constitute a contradiction in the formal-logical sense. Another principle, derivable from the principle of non-contradiction (and vice versa), is the principle of the excluded middle. This asserts that either p or not-p must be true, meaning at least one of them has to hold. The principle of non-contradiction implies that at most one of these conditions can be true. Consequently, we deduce that exactly one of the statements – p or not-p – must be true.

Now we must discuss why the law of contradiction holds such significance in formal logic. Let’s assume that both p and not-p are true, and let q represent any given sentence. As p is true, the inclusive statement p or q must also be true. However, we’ve also assumed that not-p is true, leading us to conclude that q must also be true. This process implies that, by assuming a contradiction, we can substantiate any proposition conceivable. Every statement, from “ice melts at -90 degrees Celsius” to “this website is called Cracked”, or “Marx did not do economic theory”, can be proven true. This outcome is termed the principle of explosion, and can be concisely summarised as ‘from contradiction, anything follows’. Given standard, and arguably self-evident, rules of inference, we can derive any number of meaningless statements if even a single contradiction is permitted. For logicians, and indeed anyone interested in making inferences, this presents a strong incentive to refute the validity of contradictory statements.

This amount of formal logic should suffice for our present discussion. We are now equipped to comprehend what is implied when a dialectician asserts the existence of real contradictions, necessitating the rejection of formal logic. One might assign any meaning to ‘contradiction’, even if it does not make any sense, but when discussing the incompatibility between dialectical and formal logic, the aforementioned definition of contradiction is what should be considered.

Dialectical Workhorse

The question might arise: Why emphasize Zeno’s Paradox? Are there no other potent arguments to ‘prove’ the existence of real contradictions? While other arguments may indeed exist, and I’ll briefly touch upon them as well, it’s safe to say that Zeno’s ‘proof’ – asserting the impossibility of movement – is recurrently invoked as justification for the acceptance of real contradictions among a wide range of Marxist thinkers. To preempt any potential counterarguments, let’s consider some quotes that underline this consensus.

Engels, in Anti-Dühring, asserts that “so long as we consider things as static and lifeless, each one by itself, alongside and after each other, it is true that we do not run up against any contradictions in them… Motion itself is a contradiction: even simple mechanical change of position can only come about through a body being at one and the same moment of time both in one place and in another place, being in one and the same place and also not in it. And the continuous origination and simultaneous solution of this contradiction is precisely what motion is.” Lukács, in Ontology of Social Being, states “when Zeno said that a flying arrow is at rest, he expressed certain dialectical contradictions in the relationship of space, time and motion that were similarly fruitful and paradoxical”. Additionally, to address potential criticism that I haven’t incorporated views from “non-dogmatic” thinkers, I reference Ilyenkov’s Dialectics of Abstract and Concrete, where he informs us that:

The ban in its Aristotelian formulation applies, as has long been established, to the proposition expressing the famous paradox of Zeno concerning the flying arrow. That is why all logicians endeavouring to raise the Aristotelian ban to an absolute, have for two thousand years made attempts, as persistent as they have been unsuccessful, to present this paradox as the result of errors in the expression of facts. They run the risk of spending another two thousand years of vain effort, for Zeno expressed in the only possible (and therefore the only correct) form an extremely typical case of the dialectical contradiction contained in any fact of transition, motion, change, or transformation.

I could go on quoting similar statements from an array of dialectical Marxism proponents, but the aforementioned should suffice in demonstrating that my target is not arbitrarily chosen. To establish that Zeno’s paradoxes do not necessitate the inevitability of contradictions in studying change or movement significantly undermines the traditional argument for the existence of real contradictions.

Let’s begin by examining the first of the two paradoxes discussed in this article – “the paradox of dichotomy”. Suppose we have a body, C, moving from position M at time t, to position M’ at time t’. The movement’s duration, T=t’-t, and the distance traversed, l=M’-M, are both finite. However, for the body to cover the distance l, it must pass through all points between M’ and M. These points are infinite as we can segment this interval by first travelling l/2, then l/4, l/8 and so on, essentially traversing half the distance between the current point and the destination M’ each time. To move from any one point in this infinite sequence to the next, C will take a non-zero amount of time. This duration may continually decrease, but it will never reach zero. Zeno argues that since the body must traverse an infinite number of points, and each traversal takes a non-zero amount of time, the total time taken to move from M to M’ must be infinite. Hence, these two statements must concurrently hold:

1. After a finite amount of time, the body must be at a different point (based on our assumption of movement).
2. The body cannot reach a different point within a finite interval, as argued above.

Thus, Zeno concludes, movement is impossible.

Another paradox, known as “Achilles and the Tortoise”, bears considerable similarity to the dichotomy paradox and will thus be omitted here. Instead, we’ll discuss the “Arrow Paradox”. This paradox posits that for motion to occur, an object must change the position it occupies. However, at any given instant, an arrow in flight is not moving. It cannot move elsewhere, as no time passes for it to do so, and it cannot move to where it already is. Consequently, if it’s neither moving to where it is, nor where it isn’t, it must be at rest at every moment of its flight. Zeno thus argues that the flying arrow is always at rest, leading to the conclusion that, since a body at rest cannot be moving, motion is impossible.

What About Those Derivatives?

Zeno, as we have noted, suggested that motion is impossible based on the arguments of the first paradox discussed. Despite this assertion blatantly defying our everyday experiences, Zeno chose to dismiss experience as an unreliable source of knowledge, favoring apriorism (rationalism) instead. This philosophy holds that reason alone is the most dependable source of knowledge. The consequence of this position is a dissonance between “truth of reason” and empirical evidence.

Several other philosophers of antiquity adopted Zeno’s radical apriorism. However, there were also philosophers who did accept the validity of the paradox but were reluctant to completely dismiss their sensory experiences of motion. This led them to the only viable alternative: rejecting the principles of “reason”. They posited that since motion, evidenced through experience, contradicts the “truth of reason”, we should accept this contradiction and proceed. Consequently, they proposed discarding the law of contradiction, which posits that two contrary statements cannot both be true. They reasoned that while this law may be applicable to static entities, it doesn’t apply to dynamic, changing, moving reality. Hence, they posited that contradiction is intrinsic to motion, and since motion is observable and undeniable, contradiction must also be accepted as a fact. As Ajdukiewicz² presented this dialectical position, “let us abandon the law of contradiction which may be valid with respect to something that is motionless and fossilized but does not apply to live, moving and changeable reality!”

However, simply because an ancient Greek philosopher proclaimed something doesn’t necessarily make it correct. Indeed, it would represent a conflict between experience and the law of non-contradiction if the assumption that a body in motion implied both that the body would be at a different position after a finite time and that the body could not be in a different position relative to its starting point. But we now know (and we’ve known it for a couple of centuries!) that Zeno’s argument that an infinite sum of finite quantities has to sum up to infinity is false. In fact, Zeno’s sum, T/2+T/4+T/8+T/1…, is what’s known as a geometric series and equals T. This sum’s divergence was critical to Zeno’s paradox and hence, the entire argument collapses.

One might be tempted to afford Zeno some historical leeway, considering that the tools used to identify the fallacy are closely tied to the development of various calculus concepts in recent centuries. However, proofs of the aforementioned or similar results for other geometric series can be found in Euclid’s Elements, in the works of Archimedes, and among medieval mathematicians. Despite these refutations, some might still argue that these are attempts to disprove the argument using advancements in mathematical knowledge. Regrettably, even Aristotle provided counterarguments to Zeno’s crucial assumption. Based predominantly on philosophical intuition, he noted that Zeno assumed a finite distance could be divided into infinitely many parts (e.g., l/2, l/4, l/8, etc.) but refused to accept that a finite time-interval could be divided into infinitely many parts of definitive, non-zero length (e.g., T/2, T/4, T/8, etc.). If a finite distance can consist of infinitely many parts, Aristotle contended, there’s no reason to reject the idea that a finite time-interval can also consist of infinitely many parts. Aristotle’s remark practically hints at the mathematical solution, debunks Zeno’s argument, and underscores the inconsistency in treating distance and time differently. In other words, the likes of Hegel, Engels, Lenin, Lukacs, Ilyenkov and others had no solid ground on which to claim that Zeno’s paradox demonstrated the existence of real contradictions and that formal logic was incapable of dealing with motion.

Arrow Paradox: A Warm-Up

After revealing the shortcomings of the argument for contradictions based on the Dichotomy Paradox, we can now turn our attention to the Arrow Paradox, which was outlined earlier. Zeno’s argument is perhaps easily understood in this context, but for the sake of completeness, I would like to present an alternative, less-frequently-considered interpretation of the paradox.

This interpretation can be broken down into the following series of propositions:

1. If an arrow flies for T time, then for every point in time, t, during that period, there is a position x such that the arrow is at position x.
2. If there exists a position x such that for every point in time t the arrow is at position x, then the arrow is at rest for the whole period T.
3. If the arrow flies for T period, then the arrow is at rest for the period T, which is a contradiction. Hence, the original assumption cannot be the case, as the arrow cannot move.

At first glance, this interpretation appears to demonstrate the familiar logical structure of: (I) leads to (II), which leads to (III), therefore (I) leads to (III). However, upon closer examination, it becomes apparent that this is not accurate. In reality, the logical structure of the argument is: (I) leads to (II), and (III) leads to (IV), therefore (I) leads to (IV). This is not a valid inference, as (II) does not imply (III).

In detail, the consequent of proposition 1 (II) is ‘for every point in time there is a position x such that the arrow is at position x” but the premise of proposition 2 (III) is “there exists a position x such that for every point in time (t) the arrow is at position (x)”. It is clear that (II) and (III) are different and, if one reads the above proposition using the quantifiers ‘for every’ and ‘exists’ we could name this kind of fallacious reasoning as an inadmissible interchange of quantifiers.

Not Another Plekhanov Bashing

In the brief previous section, one might argue that the interpretation I dismissed was not the popular one among those who see change as synonymous with contradiction. While I’m willing to accept this, I will, in this more extended section, thoroughly explore the prevalent interpretation of the Arrow Paradox. I aim to show that this interpretation is both A) fallacious and B) in opposition to our best physical theories. I consider that Point A alone is enough to make my point, but I will also briefly discuss Point B to target the (supposedly large) Marxist demographic that subscribes to some form of scientific realism and naturalism.³

This interpretation flows straight out of what was presented in the second section:

1. For every point in time, t, the arrow is at a particular point in space
2. Hence, the arrow is at rest for every t.
3. Therefore, the arrow is at rest for the whole duration of its movement.

Numerous philosophers throughout history have interpreted the Arrow Paradox in this way, although they have disagreed with its conclusions. The likes of Diogenes, Aristotle, Aquinas, Bergson, and Russell come to mind. In this discourse, we will concentrate solely on Adolf Reinach’s phenomenological critique, as suggested by Ajdukiewicz. This focus is due to Reinach’s critique embodying the essence of the “change implies contradiction” viewpoint, and its relevance to Plekhanov’s consideration of the alleged paradox.

The question arises – haven’t we had enough of Plekhanovian critiques in Marxist philosophy? Isn’t it time we move past this tendency? I believe we haven’t reached that point yet. Plekhanov’s politics and interpretation of Marxism continue to be subjects of debate. The discourse around Plekhanov’s “determinism” and “vulgar Marxism” has grown somewhat tiresome. Therefore, in this article, I wish to present an innovative argument: Plekhanov’s mistake wasn’t due to his “determinism” or any related issue, but because of his allegiance to dialectics. More controversially, Plekhanov’s interpretation would have improved had he rejected dialectics and dialectical logic.

But let’s not get straight to the fun part! First, we have to present Reinach’s criticism of Zeno’s Paradox. Reinach offers a nuanced perspective on motion by proposing four different types of contact between an object and its location in a moment of time: passing through, reaching, leaving, or staying at the location. This reflects the various stages of motion, with the object leaving its starting point, moving through several locations, and ultimately reaching and staying at its final destination. Pulling a Bill Clinton, Reinach argues that the validity of Zeno’s Paradox heavily relies on the interpretation of “is”. If “is” is taken broadly, meaning any type of contact, then it’s true that an object in motion is at a certain place in each moment of its motion. But this doesn’t mean the object is at rest at each moment, which is the conclusion Zeno wants to reach. If “is” is taken to mean “stays”, the initial statement becomes false, as an object in motion does not stay stationary in every moment of its motion. Essentially, Reinach provides a linguistic solution to the paradox, highlighting the nuanced meanings of “is”. Given these nuances, he debunks the assumptions behind Zeno’s Paradox, showing the argument’s fallibility depending on the interpretation of “is”. From Reinach’s perspective, no interpretation of “is” would render Zeno’s premise true and the argument valid.

The point of the above paragraph is to try to understand what Plekhanov tried to do when he sought to resolve Zeno’s Paradox by challenging the validity of either the law of contradiction or the law of the excluded middle when it comes to motion and change. Plekhanov rejects Zeno’s premise that a body in motion is in a particular place in each moment of its motion, but also refutes the counterclaim that a body in motion is not in a particular place in every moment of motion. Instead, he suggests that a body in motion both is and is not in a definite place. This can be understood as saying that a body in motion is in a definite place (in the broadest sense of “is”, not specifying the type of contact), while, at the same, it is not in that place in the sense of “staying in” it. Hence, we can interpret the claim that a body in motion is and is not in a definite place as saying that the body in every moment of motion has contact with a particular place but does not remain there. Interpreted in this way, Plekhanov’s position doesn’t imply that motion itself is contradictory; rather, it highlights the need for distinguishing different meanings of “is”. By introducing dialectics, Plekhanov only introduced a linguistic confusion to an already suspicious argument.

Reinach seems to be quite useful for putting into a coherent series of sentences the Hegelian/Marxist reading of the Arrow Paradox. While that can be satisfying, we must point out that his elaboration of the notion of motion leaves a lot to be desired. I mean, what more can you expect from a phenomenologist behind vaguely pointing towards some intuitions and hoping for the best? We’re still lacking a definition of when a body is in motion and when it is at rest; for that, we will have to take a page out of physics.

Let us begin with the following definitions:⁴

1. A body C is in motion at time t if and only if there exists a time interval (t_1, t_2) such that t_1 < t <t_2 and for every two moments in that time interval the body is at a different place.
2. A body C is at rest at time t if and only if the body is not in motion.

It is obvious that these definitions satisfy one of the most basic facts we know: motion and rest are mutually exclusive. They also point out another crucial aspect of motion – that to say that a body is in motion, we have to know its position across time. More explicitly, motion is defined as the change of position of a body across time.

Another definition, taken directly from our best scientific theory of motion, says that a body C is in motion if the first derivative of position with respect to time is non-zero while it is at rest if said derivative is equal to zero. This definition uses the infamous (in unscientific Hegelian circles) notion of instantaneous velocity. This completely uncontroversial concept is somehow a casus belli for Hegelians, and I opt to include this, as I wanted to point out to the scientific-realist Marxists the difficulty of their position when they want to keep both contradictions and their realism at the same time. In any case, let me point out that the two different definitions are co-extensive with regard to what they define in motion and in rest.

In both cases, the definition of motion or rest necessarily involves understanding a body’s state in the moments preceding and following the given instant. Zeno’s premise, however, posits that an arrow in flight is at each instant in some fixed position, from which he concludes that an arrow in flight is at rest at each moment of its flight. Relative to the two definitions provided above we notice a fundamental difference in Zeno’s idea of motion: it abstracts away from what happens to the arrow before or after a given instant.

But any declaration of a body being in motion or at rest at an instant is tied to its immediate past or future state. Therefore, it is impossible to infer a body’s state (either rest or motion) at an instant just by knowing its position at that instant, without considering its position before or after that moment. Zeno’s argument erroneously draws this conclusion, and therein lies its fallacy.

When Zeno claims that an arrow in flight is in a fixed position at each instant of its flight, he uses the term ‘instant’ to denote a point in time without any duration. At such a time-point, it indeed makes sense to say that a body is in a specific position. However, when Zeno concludes that the arrow is at rest at each instant because it is at a definite position at each instant, he seems to redefine ‘instant’ to mean ‘a very short time-interval’—a brief period with some non-zero duration. In this context, if a body is continuously at a specific location during this brief interval, it can be reasonably considered at rest; but when ‘instant’ refers to a time-point, being at a specific position at that instant doesn’t mean that the body is at rest! This is because the state of motion or rest at a time-point is dependent on the body’s positions before and after that instant, rather than its position at that isolated instant. The contradiction that Zeno “proves” is dissolved.

Picking Off the Remains

While the above covers our discussion of the Zeno Paradox, there are other, less prevalent, arguments for the incompatibility of change/movement and formal logic. I will not be able to cover every such argument; however, I would like to argue against one that seems in contrast to other dialectical pronouncements – but more on that later.

The proposed argument would go something like this: when a body changes state from A to B, there is a point in time that the body is no longer in state A but has not yet become state B. Or, more succinctly, the body is neither in state A nor in state B. One can create examples like that, like moving from point x to point y implies that there are times such that someone is neither in x nor in y. Numerous natural phenomena can be described like that, so I don’t think it is needed to elaborate on them. The idea behind this is that for every change between two states, there is time such that the body is passing from A to B and hence, is in neither state. Let us call this idea the “postulate of transition”. This postulate is justified by a more general principle, the “principle of continuity” – that every change has to occur in small, gradual steps and not in jumps. The Leninists reading this are already shouting at the screen, but just let me cook for a bit!

According to this principle, for any change to occur from state A to state B, there must be a period where the subject is transitioning from A to B. In other words, the subject should not exist solely in state A or state B at any point in time, but rather it must pass through a transitional phase. Therefore, if change were to occur in a single moment, it would contradict the principle of continuity, as the shift from state A to state B could not become infinitesimally small. The principle of continuity, then, implies a “postulate of transition”, which not only aligns with our empirical observations of change but also preserves the logical consistency of continuity.

It is not hard to imagine how the postulate of transition can be used to attack the law of non-contradiction. Let us visualise the process of ice melting, i.e., changing from a state of solid to a state of liquid. According to the state postulate, there exists some time-point t such that the body is neither an ice code nor liquid water. We do not have a paradox yet, however, unless these are the only three possible states (they are not!). But we create a peculiar situation if we say that the ice cube is changing from solid (A) to non-solid (not-A). Due to our postulate, there must be a time that this body is neither A nor not-A. However, to say that x is an object which is in state non-A amounts to saying that x is not a state A. In other words, to say x is an object which is not in state non-A amounts to saying that x is not not in state A, i.e. it is in state A. Therefore, at the time a body is in the transition from state A to non-A, it is both not A and is A, hence it has contradictory properties. And we can get our contradiction, solely from the process of change.

I think that I have not presented a weak version of the argument just so I can take it down. So, with that in mind, let’s try and see if there are any weak points. Something that I think stands out is the analogy between the “A change to B” and “A change to non-A“. I have the support of experience when I say that when I travel from my home to a football stadium there are times where I am at neither of those places. But what is the common-sense empirical evidence for accepting that there are cases when something is neither in state A nor in state non-A? I am hard-pressed to find anything. Unless one opts to go the quantum mechanics way (a very dangerous and cranky way), I do not believe that a good example can be found. And even then, one would have to justify his interpretation of the linear combination of classical states etc. – something that is far more controversial that what we are talking about. To base a theory of dialectics on quantum mechanics would put the theory immediately in huge strains. But if we have no good empirical reason to accept this use of the postulate of transition, i.e., for changes from A to non-A, then what we would be doing is accepting one unjustifiable conjecture of “proof” just to save our belief to real contradictions. By my standards, that seems a huge cost and a very glaring hole to only get the equally unsupported Hegelian result. This is not a trade-off one should accept easily.

The postulate of transition has also in its defense the fact that it can be derived from the principle of continuity. But the principle of continuity is not some untouchable axiom; rather, it is more accurately considered an extrapolation based on experiential observations. In fact, its applicability isn’t universally acknowledged within modern scientific paradigms. Dynamical systems, for example, display a number of non-continuous properties. Numerous examples can be found – but, given that the discussion is happening within Marxist theory, I believe equally important is that according to Lenin, and various philosophers following him, dialectics is characterized by its acknowledgments of leaps, of sudden breaks (supposedly) unlike formal logic. There is an obvious contradiction here; one cannot attack formal logic with the argument in this section and with the argument about the leaps and the breaks. And, to cover all the bases, if one wants to make the preposterous claim that he is happy to take two contradictory positions, then he has the problem that he has assumed what needs to be shown, i.e., the existence of real contradictions. So, this line of defense, which I hope no one takes, is just a vicious circle.

I will conclude this section by making an argument for something a bit more general. I claim that even if we accept that the principle of continuity holds, even for a particular situation and not in general, then the special version of the postulate of transition from state A to non-A does not follow.

Let us define, for the sake of argument, a ‘transition’ as an intermediate state during which an object is neither in state A nor in non-A. The principle of continuity demands that any change should occur without jumps. If we were to move from state A to state B at time t, this would imply a discontinuity or a ‘jump’ from A to B, which contradicts the principle of continuity.

The key point of the argument is when considering the change from state A to a non-A state. Here, “non-A” doesn’t refer to a specific state, but to any state different from A. The difference between state A and non-A, thus, isn’t a fixed value but could be any value based on the specific non-A state we are considering. However, given the principle of continuity, we notice that the set of possible states cannot have a quantised structure. If it were so, the difference between A and any non-A could not be arbitrarily small and that would contradict the principle of continuity, which allows for arbitrarily small changes.

If we assume that the states are continuous, i.e., there are no minimum differences between two states, then the change from state A to non-A could potentially be arbitrarily small. In such a scenario, moving from state A to non-A at a given time t would not violate the principle of continuity because we could potentially find a state A’ so close to A that the ‘jump’ is virtually nonexistent. But we just showed that the principle of continuity does not necessarily support the postulate of transition when considering changes from a definite state to an unspecified one. Therefore, one shouldn’t invoke the principle of continuity to argue that change implies contradiction.

What is Left?

Have I dealt with any possible objections or arguments for the existence of real contradictions? Certainly not. I can think of cases where I have come up against appeals to vagueness, for example, as an argument against formal logic. Vague sentences refer to sentences where we do not know their truth value due to the indeterminacy of their statements or of the concepts used. A classic example would be “X is bald” where X may have hair in his head but not enough to proclaim with confidence that “X is not bald”. Is there a certain number of hairs above regarding which we can call a person not bald? Maybe there is, maybe there isn’t. But this does not seem appealing enough to me. Contradictions are not limits of knowledge and to regard them as such would be a very counter-intuitive use of the word “contradiction”. Besides, in analytical philosophy,⁵ one can find convincing arguments that deflate the whole problem of vagueness. I will not elaborate further on this, even though what I’ve said is not adequate to cover this complex topic, but I am pretty sure this would not be a ripe field for objections.

Another tactic that I’ve encountered is to point towards surprising or counterintuitive results in scientific fields. These do not need to be literal contradictions; they merely need to challenge the prevailing scientific consensus. For example, Ilyenkov cites the “rotating disk paradox” to argue for the existence of real contradictions.⁶ The rotating disk paradox, a thought experiment, disrupts our understanding of space and time through a spinning disk and its measurements. According to relativity theory, a rotating disk’s circumference should contract due to time dilation effects, resulting in a shorter measured distance. However, observers on the rotating disk would not notice this contraction, thereby creating a paradox. Ilyenkov’s examples of contradictions are the rotating disk paradox and Zeno’s motion paradoxes. As discussed earlier, neither holds water when it comes to real contradictions. The rotating disk paradox only seems paradoxical if one denies that spacetime’s geometry is non-Euclidean. Once this assumption is discarded, the paradox disappears, leaving only a perceived contradiction based on an observer’s perspective. But unless one wants to proclaim the existence of contradictions because he does not correctly estimate the distance of a car when he’s looking at it through his own car’s mirror (or when submerging a stick in water), then that move is blocked too. And in the case that this move is made by someone, I do not think that we should really pay any attention to it.

Another type of “contradiction” that arises in Marxism is the rhetorical one. A prominent example of this is the supposed contradiction between socialised production and capitalist appropriation. I wholeheartedly support socialisation, but that does not render the above statement a contradiction. These two concepts can coexist harmoniously; a “paleomarxist” might even argue that capitalist appropriation is a necessary precursor to the emergence of socialised production. The only “contradiction” lies between our shared socialist ideals and the reality of capitalism. Needless to say, these sorts of examples provide shaky grounds to refute our best current scientific understanding.

I have also noticed that occasionally, the existence of contradictions is inferred from phenomena, natural or social, that have complex dynamics. In particular, phenomena that can be decomposed to several subcomponents, with each having its own unique dynamic, are presented as an argument that the “totality”, i.e., the phenomena are contradictory. I can follow the point, but the conclusion drawn seems to me as confused. Are there such phenomena? Without a doubt. Has our best scientific theories dealt with such phenomena successfully? Definitely. But our best scientific theories are all based on formal logic. So, in the formal-logical sense of the term, there are no real contradictions that can be inferred from the existence of such phenomena. To put a Marx quote on this point,⁷

The market must, therefore, be continually extended, so that its interrelations and the conditions regulating them assume more and more the form of a natural law working independently of the producer, and become ever more uncontrollable. This internal contradiction seeks to resolve itself through expansion of the outlying field of production. But the more productiveness develops, the more it finds itself at variance with the narrow basis on which the conditions of consumption rest. It is no contradiction at all on this self-contradictory basis that there should be an excess of capital simultaneously with a growing surplus of population. For while a combination of these two would, indeed, increase the mass of produced surplus-value, it would at the same time intensify the contradiction between the conditions under which this surplus-value is produced and those under which it is realised.

It is clear that Marx did not use the term contradiction in this passage in the formal sense of the term, but in reference to different dynamics within a phenomenon. But this is neither distinct in Marxist theory nor does it need any movement away from formal logic to model it. In fact, formal logic gives us the best tools to understand such dynamics. The use of dialectics in such cases is redundant.

Finally, I would like to very briefly comment on Ilyenkov’s attempt to show the existence of real contradictions. He, of course, also subscribes to the view that Zeno’s paradoxes are lethal for formal logic. The incoherence of this view was presented above. He also tries to claim that, actually, it’s self-movement that requires contradiction and not movement in general. However, there is no argument to be made against formal logic from this ground, as nothing prevents ‘self-movement’ in formal logic. It’s just making up a guy to get mad at. But I would like to quickly reply to another of Ilyenkov’s arguments – Prof. David Bakhurst summarizes Ilyenkov’s argument in the following way:

He argues that Marx’s derivation of value turns on the identification of an objective contradiction (Ilyenkov 1957: 67-9, commenting on Marx 1867: 131-55). Value is something a commodity has only relative to some other commodity. When a certain amount of commodity A is taken to be equivalent to (i.e., exchangeable for) a certain amount of commodity B (e.g., 5A = SB), then A measures its value relative to B. A is said to be in the “relative form” of value, whereas B, playing the role of A‘s equivalent, is in the “equivalent form.” Now, the relative and equivalent forms of value are mutually exclusive. No commodity may be in both forms simultaneously for, if it were, it could serve as the measure of its own value, and that is impossible. But each commodity is simultaneously in both forms because the terms of the equation can be reversed: When A is in the relative form with respect to B it simultaneously plays the role of B’s equivalent (and vice versa). Thus the value of the commodity is a consequence of the simultaneous existence of mutually exclusive forms, a living contradiction. Ilyenkov claims that it is by tracing the resolution of this contradiction that Marx derives other crucial economic categories.

Marx’s presentation of this in Capital is really great,⁸ but there is no reasonable way to take from Marx’s definition of an equivalence relation that a real contradiction exists. Value is like length; it is a relational property. The value of a commodity is measured only in relation to some other commodity. Consequently, when the value of A is determined relative to B, the value of B is determined relative to A. But there’s nothing paradoxical here, only the point that there is no Archimedian point from which to measure value. The contradiction amounts to saying that “My tv is on the left relative to my bookshelf, the bookshelf is on the right relative to my tv.” and concluding, as Ilyenkov would want us to do, that this amounts to my tv being in two different places at the same time. Ilyenkov’s desired conclusion does not follow from his argument.

A Hope

Have I answered to all arguments that purport to show that “change implies contradiction”? It’s highly unlikely. My focus was primarily on debunking the two primary drivers of this claim, Zeno’s paradoxes of motion, arguing that they do not provide valid grounds for the existence of real contradictions. From a scientific-realist perspective, these paradoxes pose no significant threat to formal logic. Yet, I’ve laid out a case that I believe can stand firm, independent of the metaphysical commitments tied to such a realist viewpoint. Additionally, I’ve lightly touched upon other arguments put forth by ‘dialectical’ Marxists against formal logic. Despite this, it’s safe to assume that dialectical thinking will continue to dominate discussions on scientific socialism. But changing that wasn’t my primary objective. Instead, my goal was to propose that if dialectical logic aims to be coherent, it should first discard any reference to real contradictions and align with the principles that formal logic imposes on theories.⁹ I suspect many readers might be wondering, “What is dialectical logic without real contradictions?” I genuinely don’t know. I even question the existence of dialectical logic. However, those who advocate this viewpoint must uphold higher standards if they intend to maintain its predominance in a theory that labels itself “scientific.” Anything less would be an admission of defeat.

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1. Analytical Marxists of course disagree with this claim, but there are a minority within the broader theory.
2. Ajdukiewicz, K. (1978), Change and Contradiction (1948) The Scientific World-Perspective and Other Essays, 1931–1963, 192-208. It needs to be noted that Ajdukiewicz positioned himself against the dialectics and part of the discussion in this article follows his arguments in this paper.
3. Naturalism in the Quineian use of the term.
4. In the following, we will use “is” in the broadest sense of the term, i.e., by saying that a body is in contact with a position, but not specifying the kind of contact as Reinach does.
5. Williamson T., Vagueness.
6. Dialectics of the Abstract & the Concrete in Marx’s Capital, Chapter 5.
7. Capital Vol 3, Chapter 15
8. Ulrich Krause, in his Money and Abstract Labour: On the Analytical Foundations of Political Economy was the first, at least as far as I know, to provide a formal presentation of Marx’s definition of exchange as an equivalence relation between commodities. Stathis Psillos, in Value and Abstraction in Marx’s Capital has more recently made a similar point from a philosophical point of view.
9. It is important to note that Elster, in Logic and Society: Contradictions and Possible Worlds, offers an alternative definition of what constitutes a ‘contradiction’ in Marxist theory. I am totally behind his definition actually, but his approach is not only not in opposition to formal logic but is actually based on modal logic, a type of logic that would fall under the ‘formal logic’ for a Hegelian. There’s no dialectical structure there; it is a non-starter for a dialectical Marxist.