On Quantum Theory and Contradictory Claims/Beliefs

In the SD

introduction

Quantum theory is full of oddities, and many even see it as logical contradictions. The uncertainty principle states that there are pairs of physical quantities for which definite values ​​cannot be attributed to the same body at the same time. For example, a certain body cannot be attributed a definite position and a definite velocity at the same time. If it has a definite position, it does not have a definite velocity, and vice versa. The same goes for energy and time, and so on. This is a strange principle and is difficult to grasp, but on the surface there is no logical contradiction in it. There is no logical necessity for a body to have a position and velocity at the same time (I will dispute this below). On the surface, this is just a difficulty but not a contradiction, similar to the reality of four dimensions, which is very difficult for us to grasp and certainly imagine, but there is no contradiction in such a reality. The inability to imagine this is probably based on the limitations of our perception. In contrast, the principle of complementarity (= completion), which is attributed to the Danish physicist Niels Bohr, seems like a real logical contradiction. This principle states that any body can be both a particle and a wave. And more generally, it is possible for there to be two contradictory descriptions of reality that are both true. Here we have already gone one step further, beyond the uncertainty principle, since here we are making a claim about reality that contains a logical contradiction. A particle is not a wave and a wave is not a particle. The claim that a particle is a wave, or that some object is both a wave and a particle, seems similar to the claim that in reality (or even in the world of concepts or in Plato’s world of ideas) there exists a married bachelor, or a round triangle.

Various physicists and philosophers propose to explain the oddities of quantum theory on the basis of the logical problems it contains. They claim that quantum theory is based on a different logic than the one we are familiar with. [1] This logic does not satisfy some of the logical principles included in classical logic. There are several proposals for such logics, including proposals that it does not satisfy the law of contradiction (that it is impossible for a statement X and its opposite to be both true at the same time) or the law of the excluded middle (that either a statement X is true or it is false, and there is no third possibility). The prevailing approach proposes a logic that dispenses with the distributive property that exists in ordinary logic (what is called in Hebrew the ‘law of distributivity’): P*(QUR) = (P*Q) U (P*R). [2]

Interpretations of this type are collectively called “quantum logic,” and as a rule they assume that the claim that quantum theory is based on a different logic is an explanation for its oddities and the contradictions we find in it.

From this, many thinkers conclude that the laws of logic or the logical constraints that we are so accustomed to and see no way out of, are not really so draconian and absolute. According to these claims, quantum theory allows us to make contradictory claims and we need not be alarmed by this.

This conclusion has obvious applications in theology, since various descriptions of God inherently lead to contradictions, and we seemingly have a simple way to deal with such contradictions: God is above logic, and thus, for example, He is both present in the entire world and only outside it, He knows in advance what we will choose and we have free choice, He is both perfect and profitable, and so on. Many believe that these claims are not so far-fetched, since even our physics (especially quantum theory) does not always obey logic. There is an assumption here that the laws of logic are not fundamentally different from the laws of physics, and just as God can perform miracles, that is, act against the laws of nature, He can also act in a way that contradicts the laws of logic.

In this article, I would like to examine this fundamental claim, and focus on the logical interpretations of quantum theory. To clarify matters, I will state my fundamental claim here: an interpretation based on a different logic is nothing but nonsense (meaningless discourse), both in relation to quantum theory and in relation to philosophy and theology. These are just meaningless combinations of words. My position is that there is no possibility of making a claim that does not obey the laws of logic, neither in a theological context nor in general. [3]

The Judeo-Christian Dialectic Regarding Attributing Contradictions to Divinity

In the writings of some of the first, one can find strong arguments against attributing logical contradictions to God. As surprising as it may be, their words indicate that He too is subject to the laws of logic. Here I will give one striking example from the words of Maimonides in Moreh Hanebuchim 3:55: [4]

A nature that is not subject to the action of an agent has a permanent existence, is not subject to the action of an agent, and cannot change at all, and because of this, God cannot be described as capable of it, and no one among the scholars disputes this at all, and only someone who does not understand the rationality will deny this.

Here Maimonides states that logical contradictions should not be attributed to God. He sees this as an agreed-upon matter, and anyone who disagrees with him is “not one of the learned” and does not understand reason.

But immediately afterwards he adds that there is still room for disagreement on such matters:

And indeed, the point of disagreement among all scholars is the allusion to one type of imaginary thing, because some scholars say that it is the plague of the impossible, which God cannot be described as capable of, and others say that it is the plague of the possible, which God’s ability to create is dependent on when He wills.

Such a dispute does not concern the attribution of contradictions to God. This is not possible by all accounts. One can only disagree on the question of whether this is indeed a logical contradiction (“logically impossible”) or merely an incomprehensible matter (or something that contradicts the laws of nature. “physically impossible”). [5] But if we have come to the conclusion that this is an impossible (logical contradiction), then God, too, “cannot be described in terms of His ability.”

He then provides examples of attributing contradictions to God (such as He can make a square whose diagonal is equal to its side) and later also mentions issues about which there is disagreement among philosophers as to whether they belong to the logical or physical plane. In this context, one can add the question of divine knowledge and our freedom of choice (see Rambam, p. 5, reply to the 5th chapter, and also in columns 299-303 on my website) and other questions that will be mentioned later in the article.

In contrast, in Christian thought, the view that God is not subject to the laws of logic is very widespread and that logical contradictions can certainly be attributed to Him. A clear expression of this is found in the doctrine of the “unity of opposites” or “overlap of opposites” (Coincidentia Oppositorum) of Nicholas of Cusa, a fifteenth-century Catholic philosopher, theologian, and mathematician. Tertullian, one of the Church Fathers (in the second-third century AD), also went even further and stated: “I believe because it is absurd” (Credo quia absurdum). Note, not “although it is absurd” but “because it is absurd.” He sees the essence of belief in the absurd. Not only is belief in absurdities possible, but faith relates primarily to absurdities. Apparently, in his eyes, something that is not absurd does not belong to the realm of faith but to knowledge, science, or logical thinking. This concept can also be found in the existential philosophy of the Danish Captain Kierkegaard from the nineteenth century, for in his doctrine of the “Knight of Faith” (the figure of the perfect believer represented by Abraham, especially in the form of a contract), one is required to bend his logic and his ethics in favor of the absurdity that faith demands. For him, too, the essence of the meaning of faith is living (and not just believing) in the absurd.

In short, I will say that in recent generations this idea has entered Jewish thought more strongly, and many speak of the unity of opposites in Hasidic literature, in the works of Rabbi Kook and others. [6] As far as I am concerned, the fact that the origin of any idea is Christian or that it was created under Christian influence does not disqualify it. An idea or approach should be examined for its substance, whatever its origin. I oppose the idea of ​​the unity of opposites, regardless of its origin, simply because it is meaningless talk. I will now show that there really is no such idea, and that whoever raises it only moves his lips but says nothing.

An explanation of the impossibility of contradictions, or: What are the ‘laws of logic’?

What confuses people when they come to discuss logical contradictions in divinity is the assumption that from God’s omnipotence and His being the source of all reality and all the laws that govern it, it follows that it is impossible for any system of laws to limit and obligate Him, meaning that He would be subject to it. From this seemingly follows the conclusion that the laws of logic cannot limit Him either, and therefore it is also possible to attribute to Him abilities or attributes that include logical contradictions.

At the heart of this argument is a reference to the laws of logic as if there were a system of laws similar to the laws of physics, or the laws of the state. But this is a mistake. A triangle is not round not because there is a law that prohibits it from being so, but because it simply is not so. If it were round, it would not be a triangle. The non-existence of a round triangle is not the result of any law, and therefore no one legislated for it. It is the result of the definition of the concepts themselves. The term ‘laws of logic’ is a confusing and unsuccessful conceptual mix-up. It stems from the fact that from Aristotle to the present day, logic has become an independent field of study and research, and like every field there are fundamental laws. But people ignore the major difference: in all other fields there are laws that are the product of enactment, and at their core is an authoritative legislator. Even the laws of nature are like that, where here the legislator is God Himself. And hence they could also have been different (depending on the decision of the ‘legislator’). The same goes for the laws of the state or the laws of some guild, they are the product of enactment and could have been different. All of these are systems of ‘laws’ in the conventional sense. But the ‘laws of logic’ were not enacted by anyone, since they could not have been different. And hence, although God cannot deviate from them either, they do not ‘obligate’ Him and are not forced upon Him. Simply because there is nothing that can be forced upon Him. He who cannot overcome the laws of nature is not omnipotent because he is forced to do something. But he who cannot ‘overcome the laws of logic’ is not incapable, if only because there is no such thing as “overcoming the laws of logic.”

From this you can understand that it is not correct to say that a circular triangle does not exist. A circular triangle is a meaningless concept, and therefore I cannot say anything about it. Neither that it exists nor that it does not exist, neither that it is beautiful or generous, nor that it is a rare creature. Any sentence containing the phrase “circular triangle” cannot be true, but neither can it be false. If a sentence contains a hidden concept, it becomes meaningless. [7] When I say that a circular triangle does not exist, I do not intend to assert any claim of fact, but only to say that this concept is undefined.

It is impossible to say that God can create a wall that stops any shell and a shell that penetrates any wall, not because there is any lack of ability in Him, but because there is no logical possibility for the simultaneous existence of such a wall and a shell. This is a logical contradiction. The same is true of the statement that He cannot make a circular triangle. Therefore, the statement that He cannot make a wall that stops any shell and a shell that penetrates any wall, or that He cannot make a circular triangle, is not a lack of ability in Him. When we say that someone is omnipotent, this means the ability to do everything that is defined (or everything that can be imagined). But what is not defined does not have ability, and when we say that God cannot create such a wall and a shell, or a circular triangle, we do not mean “cannot” in the sense of lacking ability. The concept of ability itself is irrelevant with regard to such situations or concepts. A more precise translation could be that the sentence “God cannot make a circular triangle” means: “The claim ‘God can make a circular triangle’ is meaningless.” This is not an argument about God and his abilities, but about the combination of the words ’round triangle’.

Example: The stone that God cannot lift

From this we can also understand why the banal sophism about the stone that God cannot lift (the omnipotence paradox) is based on the same error in understanding the laws of logic. People wonder whether God can create a stone that he himself cannot lift. The argument is that if he can do this then there is a stone that he cannot lift and therefore he is not omnipotent. If he cannot do this then again he is not omnipotent. The conclusion is that the claim “God is omnipotent” is incorrect (because it leads to a logical contradiction).

I will present the error in this argument on two levels. First, it should be noted that this is an attack on the concept of omnipotence itself, regardless of God. The claim is that this concept is meaningless (its normal content leads to a logical contradiction). But if that is the case, then at most one can argue that the sentence “God is omnipotent” is meaningless, because we are using an empty or contradictory concept. But this is an argument about us, not about him. This does not mean that God is lacking, but only that the concept ‘omnipotent’ cannot be found in our language.

Second, think of the debate between the believer and the atheist as a debate between two people: Reuben, the believer, believes that God is omnipotent, and Simon, the atheist, makes an argument that attacks him. He says to Reuben: In your opinion, if God is omnipotent, can He create a stone that He Himself cannot lift? Simon’s assumption is that Reuben can answer him either yes or no, and there is no third possibility (this is the law of the excluded middle). But whether he answers yes or no, the conclusion is that God is not omnipotent, which means that Reuben is wrong. This is what is called in logic a ‘dilemma argument’.

But surprisingly, here both answers are incorrect. Not because there is a third possibility (the law of the unavoidable third is also true here, of course). The reason is that there is no such stone. [8] It is like asking whether God can create a circular triangle. Both the positive and negative answers are irrelevant here. Reuben, who assumes that God is omnipotent, cannot understand the concept of ‘a stone that God cannot lift,’ because in his view it translates to the concept of ‘a stone that the omnipotent cannot lift.’ Therefore, from his point of view, it is like a circular triangle. Simon the atheist can of course understand this concept because he assumes that God is not omnipotent. In his view, there is no contradiction in the concept of ‘a stone that God cannot lift.’ But a logical attack is supposed to start from the assumptions of the person being attacked (Reuben) and show that they lead to a contradiction. Proving that God is not omnipotent on the basis of Simon’s preliminary assumption that he is not omnipotent is simply assuming the desired thing. But as we have seen, if we start from Reuven’s assumptions, this attack is meaningless and incomprehensible. Reuven’s answer to Simon would be: Please explain to me the concept of ‘a stone that the Almighty cannot lift,’ and then I can try to answer you whether God can create such a stone or not. Simon, of course, will never be able to explain this concept to Reuven. He will be able to explain to him the concept of ‘a stone that God cannot lift,’ but that is only on his assumption that God is not omnipotent. Even Simon himself cannot understand the concept of ‘a stone that the Almighty cannot lift.’ In short, his question is meaningless.

The difference between the two answers I have given here is that the first answer holds that the concept of ‘all power’ is simply undefined. In this sense, the believer was indeed wrong when he used it, although this is not a death blow to his faith. He just needs to correct his language. In contrast, the second answer is willing to accept the concept of ‘all power’ as a defined and non-contradictory concept, and still rejects the logical attack on the believer. This shows us that the atheist is simply confused. In fact, he sees logical contradiction as something that should be within the ability of the omnipotent God, but this is a mistake. A logical contradiction is not within the scope of His abilities, not because there is some lack in His abilities, but because arguments with logical contradictions are meaningless. As we saw above, the claim “God cannot create a stone that He Himself cannot lift” is interpreted as follows: “The claim ‘God can create a stone that He Himself cannot lift’ is meaningless.”

Consider, for example, the well-known children’s story, “Puss in Boots.” In this story, the cat arrives at the palace of the terrible wizard, wondering in his ears whether he can turn himself into a mouse, and when the wizard does so, the cat devours him and thus gets rid of him. And in the parable, consider the following question: Can God turn himself into a human being? If so, then he can be shot in the head and killed. One could perhaps argue that he can turn into a human being, but when he is shot he will not die. But then he has not really become a human being (since a person who is shot dies). The correct answer is that he cannot, of course, turn himself into an ordinary human being. A being whose existence is necessary and who is omnipotent cannot bring about a situation in which he himself will become extinct. To the same extent, he probably cannot make himself imperfect or limit his abilities. But all these “shortcomings” do not constitute an impairment of his entire ability, since these are logical contradictions. God is also incapable of logical contradictions. By saying “God cannot become a man,” I actually meant to say: “The claim ‘God can become a man’ is meaningless.”

Interim conclusion: What do we do with contradictions?

The fundamental question we are dealing with here is what we should do when we reach two logical conclusions that contradict each other. Note that if claim X and claim Y contradict each other, this in itself is not problematic. We should simply choose one of them and reject the other. At most, we will be in a dilemma about which one to choose and which to give up, but this is not a contradiction but a doubt. The problem arises when each of these two claims seems very logical to us and we tend to adopt both. In such a situation we are in intellectual confusion. There are thinkers or people who in such a situation choose to declare the ‘unity of opposites’ and see this as a solution to the problematic I described. They claim that there are things that are beyond reason or logic, especially if it is about God.

But as Rudolf Otto wrote, in the introduction to the English edition of his book, The Sacred , the ‘unity of opposites’ is the refuge of the lazy. When a person is faced with a contradiction of the kind I have described, if he cannot find a solution to it or is too lazy to look for it, it is very easy for him to declare the unity of opposites and see this as a kind of solution. This sounds very profound and mysterious, and supposedly saves him from an embarrassing situation (a contradiction without a solution). It also allows him to hold both logical claims at the same time without giving up either of them.

But as we saw above, this is actually nonsense. When we are faced with two contradictory claims, we have one of two options: either to adopt one of them and reject the other, or to show that we were mistaken and that there is actually no logical contradiction here. Sometimes it is difficult to show this, and therefore some choose instead to declare the ‘unity of opposites’ and thus exempt themselves from the logical effort. But if there is no solution, we must reject one of them. This can perhaps be attributed to our short-sightedness. But declaring that God is above logic or that we can hold two contradictory claims without being bound by logic is nothing more than an expression of intellectual laziness, since it is a collection of meaningless words.

And what about quantum theory?

This conclusion brings us to quantum theory. There we seemingly encounter scientific claims that involve a logical contradiction. If an object can be both a particle and a wave (the principle of complementarity), then logical contradictions are seemingly meaningful. According to this way of thinking, quantum theory shows us that the ‘laws of logic’ are also laws in the ordinary sense. From here the possibility of attributing contradictions to God arises again, and of course in particular the omnipotence paradox (the stone paradox) also resurfaces.

But as we saw above, this is not possible. Contradictory claims are meaningless, and the fact that they are asserted within some scientific framework (such as quantum theory) cannot change this. I will put it another way. Quantum theorists are also not exempt from eliminating contradictions in their doctrine. Their use of the unity of opposites is meaningless nonsense. It must be understood that quantum theory itself is based on mathematical tools and observational tools developed on the basis of ordinary logic, that is, logic in which the law of contradiction and the third law of avoidance hold. If ordinary logic is incorrect, then it is also not possible to use everything we have arrived at by negation. A proof by negation shows that assumption X leads to a contradiction and therefore it is necessarily not true that X. But if there is a third possibility, then this proof naturally fails. The same is true for the distributive property of our logic. Furthermore, the measuring tools we use in the laboratory, and through which we discovered quantum theory itself, were also developed on the basis of the principles of ordinary logic (since they are not products of quantum theory but are the basis on which it itself was built). Therefore, it is clear that in quantum theory, once we have proven that X is not true, we can conclude that ‘not X’ is true. If this were not the case, the findings of quantum theory would have no meaning, since within its framework it would be possible to adopt a claim and its refutation.

More generally, (classical) logic teaches us that if we adopt a set of claims that includes a logical contradiction, then any conclusion can be drawn from it. This is why mathematicians work so hard to prove consistency (=the absence of contradiction) in any logical and mathematical system. If quantum theory really had any contradiction, then any claim could be drawn from it, in particular a claim and its converse. We could deduce that the electron is a particle with 0 charge or that it is a particle with a different charge, and we could also adopt both of these claims together. Furthermore, it would also follow from this that quantum theory is devoid of scientific value and scientific content. If any conclusion can be drawn from any scientific theory, then it has no prediction of what will happen in a future experiment. But then it is also impossible to subject it to refutation tests (to conduct an experiment that will test whether its predictions come true or not), and according to the accepted definition in the philosophy of science (following Karl Popper), this was not a scientific theory.

The conclusion is that, as we have seen, the ‘laws of logic’ are not laws in the usual sense. It is not possible to deviate from them (to break the law), and claims that contradict them are meaningless. The laws of logic are the basis on which our thought is based, and therefore it is impossible to measure the laws of logic in the laboratory, and it is not possible to deduce any laws of logic from empirical scientific findings. This inference itself would be made using (ordinary) logic. Therefore, neither thinking nor observation can show us that our logic is wrong. If we return to quantum theory, we must understand that it is a branch of physics, and as such is a scientific field. In contrast, the laws of logic do not belong to science but constitute an a priori basis (prior to observation) for scientific thinking and thinking in general. Therefore, there is no possibility of deriving other laws of logic on the basis of quantum theory.

In the margins of my remarks, I will only note that more precise descriptions of quantum logic do not treat it as an explanation of quantum theory, nor as an alternative logic. They present it as a formal system that allows us to present the relationships between various claims in quantum theory. For example, we find at the beginning of Wikipedia in English (about ‘quantum logic’):

Quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory .

This logical structure has nothing to do with logic as a description of our thinking. It is a technical tool for presenting the (strange) findings of quantum theory, nothing more.

The conclusion is that if observations in quantum theory yield contradictory claims, that is, findings that contradict logic, the explanation must be found elsewhere, it must be formulated under that well-known and familiar logic. The meaning of these things is that if we do indeed have logical pathologies in quantum theory, such as a claim and its opposite, then we must take one of two paths: either understand that the two claims cannot be true at the same time, and then adopt one side and reject the other, or show that there is no logical contradiction here. And it is certainly not possible to rely on quantum theory to philosophically substantiate claims that contain a contradiction, whether they deal with God or not. Such claims are meaningless. Now let’s look at some pathologies in quantum theory, and see how this picture is expressed in them.

The two-crack experiment [9]

As Richard Feynman, Nobel Prize winner in physics and one of the researchers who participated in the Los Alamos nuclear project, used to say, the best way to explain quantum theory is through the two-slit experiment. This experiment has an interesting and very volatile history. It dates back to Newton’s time, when there was a debate among physicists about the nature of light. Some claimed that it was composed of particles (like Newton) and some saw it as a wave (the Fresnel-Huygens theory). In 1801, 74 years after Newton’s death, Thomas Young performed the first two-slit experiment to settle this issue. In the experimental setup that Young used, there is a clear difference between a wave and a particle beam, and precisely because of this, it was useful in settling the dispute about the nature of light. This is precisely why this experimental setup will be found incredibly useful in the next century in resolving a similar dispute, this time regarding the nature of particles like the electron (whether they have a wave or particle nature).

To understand this, let’s start with the simpler case of a single crack. The drawing below depicts a source (in the shape of a cylinder on the right) that sends a beam of particles or a wave (the thick dotted line) in the direction of a partition containing a crack. Behind the partition is a screen (shown by a double line), essentially a type of photographic film that is sensitive to the impact of the wave or particles.

Figure 1: Single crack experimental setup

The graph on the left of the drawing depicts the experimental results (what is seen on the screen) for a single crack. Its height (on the y-axis) represents the amount of particles or wave intensity received at each point on the screen (each point on the x-axis of the graph represents its corresponding point on the screen).

It can be seen that the largest amount is received exactly in front of the crack, and on both sides of it the amount decreases (this represents some of the light or particles that have deviated slightly to the side). This picture is true for both a particle beam and a wave beam. Young’s idea was to distinguish between a particle beam and a wave by means of a similar experiment, except that a partition with two cracks is placed in it. To understand this, we must recognize something that distinguishes wave phenomena, interference. When two particle beams move in space, the total number of particles at each point is the sum of the numbers of particles from the two beams at that point. In contrast, when two waves rotate in space, the total field (wave) intensity at each point is not a simple sum of the intensities of the two waves at the point in question. The intensity at each point in space will rise and fall as a result of the effect of ‘interference’ between the waves. [10] For example, it is possible for two high-intensity waves to sum to zero, since one is positive and the other is negative. In some places the intensity of the two waves cancel each other out, and in other places they reinforce each other (then the overall intensity is greater than the sum of the intensities).

Now consider the experimental setup proposed by Young, which looks similar to the one we saw above, but this time there are two slits in the partition. If the source on the right emits a beam of particles (such as electrons, tennis balls, or elephants), some of them will pass through each of the two slits. Around each slit, the number of particles will be distributed as in the graph of the single slit (see the graph in Figure 1). So, in an experiment with two slits, at any given point on the screen the number of particles that will reach it will be the sum of the particles that came from the right slit and the left slit. Therefore, the image on the screen will look like the sum of the images of the two single slits, meaning we will get a graph with two equal peaks, as depicted in Figure 2:

Figure 2: Two-slit experiment with particle beam

In contrast, when the source sends a wavy beam, the interference phenomenon will cause the image to look completely different, as depicted in Figure 3: [11]

Figure 3: Two-crack experiment with a wavy beam

In the wavy case depicted in Figure 3, [12] it is precisely between the two slits (where the particle image almost vanishes. See Figure 2) that the maximum intensity appears. On either side of the center there are side lobes whose peaks go up and down. Thus, the two-slit experiment provides us with a sharp distinction between a particle beam and a wave beam.

In Young’s experiment, conducted at the beginning of the 19th century, the two slits were tested in relation to a light wave, and the result was unequivocal: a wave image was obtained (Figure 3). Thus the debate between Newton and Huygens (who had long since died) was settled, and it was determined that light is a wave phenomenon.

Over a hundred years later, at the beginning of the twentieth century, the situation began to reverse. Various pieces of evidence and arguments began to accumulate in favor of the picture that sees particles as waves. In 1924, a French prince named Louis de Broglie, in his doctoral thesis, proposed viewing the electron as a wave and even offered a mathematical description of its wave function. Now a debate began that was the opposite of the one that took place between Newton and Huygens: do objects that we are accustomed to seeing as particles (for example, the electron) actually have a wave nature? To examine this question, two-slit experiments were performed again, but this time with an electron beam. The expectation was that the image would look like Figure 2 above, since electrons are supposedly particles. The results were amazing. It turned out that the resulting image was similar to Figure 3. It followed that electrons also actually have a wave nature. If so, electrons behave like waves of light, and in essence the difference between a particle and a wave is seemingly erased. It was not clear how a beam of particles could interfere with itself? As I mentioned, interference is a property of waves, not particles.

At some point, it was hypothesized that the interference was a result of the fact that it was a beam of electrons, and not a single electron. Different electrons interfere with each other, and from their meeting a kind of interference image is created. The advantage of physics is that it is an empirical science, so to test this suggestion, the two-slit experiment was repeated when the beam of electrons was emitted from the source at a very slow and sparse rate. In such a situation, only one electron hit the screen at a time, and therefore interference between two electrons could not occur. To our amazement, it turned out that a wave image was still obtained (as in Figure 3). In other words, the single electron is also a wave, since it interferes with itself. The amazing conclusion is that the electron is not a tiny particle located in a specific place (a kind of ball), as was thought until then, but rather a wave that is present in all space, just like light (we, of course, still remember Fresnel-Huygens and Young).

If the electron does indeed interfere with itself, this means that the single electron is actually a wave that passes through both slits. Therefore, the two parts of the electron wave (the wave of the single electron) arrive at the screen from different places and create an interference pattern (Figure 3). This already seems unbearable. A tennis ball only passes through one of the slits at a time. How is it possible that if there are two slits in the screen, the particle passes through both slits at the same time?

To examine this question, we must examine each time which of the two slits our electron passes through, or whether it passes through both. If so, we of course return to the laboratory. Now conduct the experiment again, but this time place a detector next to one of the slits (let’s call it slit A). When the electron passes through the slit, the detector detects it, and it reports to us that the electron passed through slit A. If the detector did not report anything, the conclusion is that the electron passed through slit B. Repeat the experiment with the two slits in the presence of the detector, and here the astonishment was already complete. The detectors did show us which slit the electron passed through, but this time a particle image was obtained on the screen, namely the graph in Figure 2. The phenomenon of the electron’s interference with itself (Figure 3) disappeared. The electron ceased to be a wave, and began to behave like a cultured particle (like a tiny tennis ball). Occasionally the detector showed that the electron passed through slit A and then there was a buildup in front of that slit, and other times the detector showed that it passed through slit B and then the buildup was in front of slit B. In a particle beam, the image in Figure 2 is obtained.

Just to clarify, with tennis balls the situation is of course different. If we conduct an experiment with two cracks in tennis balls, with or without a detector, the result will always be a particle (Figure 2). The conclusion is that a tennis ball is not a quantum particle. But the electron, at least as long as there is no detector near one of the cracks, behaves like a wave. Therefore, the electron is a quantum particle. When there is a detector checking it, it goes back to behaving like a decent particle (tennis ball), but when no one is looking at it, it really goes wild. Note that even when there is a detector near the crack, the electron is still not completely a tennis ball. Even if we shoot the electron from the source towards crack A, unlike a tennis ball, there is some chance that the electron will still pass through crack B. When there is a detector, the single electron will not pass through both cracks, but you cannot know for sure which one it will pass through. Only after the detector has measured its position does it begin to behave like a regular classical particle (meaning that if it passed through crack A, it will of course hit the screen opposite that crack and not on the other side). But as long as it is not actually measured, there is a chance that the detector will detect it at A and there is also a chance that it will be detected at B.

Before I continue, I will just comment on the wonders of the historical pendulum, and say that after quantum theory was formulated, it became clear that light is not entirely a wave either. There are situations where it behaves as a collection of particles (photons), and even there it depends on whether or not a photon detector is placed near one of the slits. Surprisingly, after about two hundred years, Newton returned from the dead.

The accepted interpretation

The interpretation of the strange findings of quantum theory is not agreed upon to this day. According to the accepted interpretation that developed after these experiments (called the “Copenhagen interpretation,” after the Danish physicist Niels Bohr and his school), the distinction between particle and wave collapsed. The electron is both a particle and a wave, and this depends on the experimental system that measures it. This is a particular expression of Niels Bohr’s complementarity principle. It later became clear that we have no way of measuring the velocity and position of an electron simultaneously, and this is the uncertainty principle, which also falls under the complementarity principle. This is another expression of the fact that the electron is both a particle (a body that has a defined position) and a wave (a body without a defined position). When its position is measured, it becomes a particle, and when its position is not measured (but velocity can be measured), then it is a wave.

The conclusion that physicists have drawn from this confusing picture is that as long as it is not looked at (a detector is not placed) the electron does indeed pass through both slits. This means that the electron is in a state known as a superposition, which is a sum of pure particle states. It is essentially a sum of two particles, one passing through slit A and the other through slit B. By the way, if we were to put three slits in, it would be a sum of three such particles, and so on. To put it more precisely, the electron is one particle of course, but the state of the electron is a sum of two particle states (for us, this is already a wave state). But that’s only as long as we don’t put in a detector. When we put in the detector, the electron ‘collapses’ into one of the particle states, and somehow decides to become a classical particle that passes through A or a classical particle that passes through B. The decision to which of the two particle states it will collapse into is a kind of lottery, and quantum theory can also describe to us the chances of these two outcomes. They are described by the ‘wave function’ that describes the electron (or photon).

We denote the superposition state between two particle states as follows:

Where state [A> is a trajectory in which the particle behaves like a point particle passing through slit A , and the probability of receiving it (if a detector is placed near the slit) is defined by , and state [B> is a trajectory in which the particle behaves like a point particle passing through slit B , and the probability of receiving it (if a detector is placed near the slit) is defined by the coefficient . The general state is the sum, or superposition, of these two states. If we place a detector, the particle will either pass only through slit A (which will happen with probability ) or only through slit B (which will happen with probability ).

Two notes

In the margins of my remarks, two more comments that are not important to our discussion, but are necessary to complete the picture:

• On large scales, when we are dealing with objects from our ordinary world (a pigeon, a chair, a person, a bacterium, a living cell, or a ping-pong ball), quantum effects usually disappear. The reason for this is that even if our particles are quantum, a large system containing masses of particles (every macroscopic body is one), its state is the sum of the states of the particles. The sum of lots of random outcomes gives a simple average result (this is the ‘law of large numbers’ in probability). This is an effect called quantum decoherence (or dephasing).
• In the past, there have been suggestions (which are still cited today) that the person measuring influences the results of the experiment. People have drawn far-reaching conclusions from this about telekinesis (a non-physical effect of the mind on distant objects and people), about the nature of reality (Kant’s distinction between the thing as it is in itself (the noumena) and the thing as it appears to us (the phenomenon), and more. I will not go into all these implications here, but I will say that it is now generally accepted that none of this is true. The two-slit experiment was performed when the results measured by the detector were immediately erased, that is, there was no human consciousness observing them, and yet it turned out that the very act of placing the detector caused the wave function to collapse into a particle state. [13] From this it seems that the source of the strangeness is not necessarily human consciousness but the very act of measurement. [14]

Contradiction and compromise

If we return to focusing on the small, quantum scales, on the surface there seems to be a logical contradiction here. A particle is a body with certain properties (we will refer to it for simplicity as a point of matter located in some defined place), while a wave is an entity with other (opposite) properties, and in particular it does not have a defined location. In quantum theory, an entity with a pure wave nature (such as a photon with a defined wavelength) is spread out over the entire world. So who is the electron (and also the photon), one or the other? It cannot be both.

The answer I will offer here to this question is that the electron is neither a particle nor a wave. It is a different entity, which can exist in two different types of states: a particle state or a wave state. According to this interpretation, a state of superposition does not mean that the particle has passed through both slits (this is a logical contradiction), but rather that its wave function consists of two functions, each of which describes the state of a particle passing through one of the slits. In everyday language, we can say that part of the particle passes through each slit.

So, there is no contradiction here, but a complexity that is difficult to grasp. It is not a particle passing through the two cracks, which is a paradoxical situation (logically contradictory), but an entity whose state is described as a superposition between two particle states. There is no contradiction in this. A particle and a wave are not entities but states of entities. But what exactly is the being whose states are these? What does it look like? What are its true properties? We have no way of knowing or describing it. You see that we have passed from a logical contradiction to a difficulty in description (as in the four-dimensional example given above).

Schrödinger’s Cat and Kiddushin Not Dedicated to a Lioness

In the thought experiment known as “Schrödinger’s Cat,” a cat is placed in a closed box with a sealed test tube of poison. The stopper of the test tube is controlled by a quantum process and its state is described by the sum of two simple (classical) states, as in the equation above. However, here the state [A> is that the test tube is open and the state B>] is that it is closed. But if the test tube is open, the poison spreads throughout the box and kills the cat, meaning the cat is dead. And if the test tube is sealed, then the cat is alive. The meaning of a superposition state is that the cat is in a state that is the sum of a live state and a dead state. When we open the box and measure the state of the cat, we will find a live cat or a dead cat, but not both. But before opening it, it was in a state of both live and dead. As we have seen, it is not accurate to say that the cat is both alive and dead, but rather that its quantum state (its wave function) consists of two classical states: this is considered a paradox because such a state is already more difficult to digest. Cats are classical macroscopic creatures, and we thought that at least we understand such creatures. They are either alive or they are dead. Unlike a photon or an electron that none of us have ever seen, a cat is an object that we think we know, and it cannot be alive and dead at the same time.

Various solutions to this paradox can be proposed, but in all of them we are not left with a logical contradiction. The first possibility is that even with a cat, as long as we have not observed it, it is indeed a different entity than what we imagine. The image of the cat that we are familiar with is the product of a ‘measurement’ (looking into the eyes), and here there is only one of two possible outcomes: alive or dead. The measurement collapses the wave function of the cat into one of these two states. But life and death are states of the cat, and therefore a living cat and a dead cat are not two types of cats but two different states of an entity that is not a cat in the sense that we are familiar with. Such an entity can indeed be found in a superposition of alive and dead.

There is also another possibility. The poison that spreads in the box acts on cells or mechanisms in the cat’s body, thus causing its death. But cells or particles in the cat are small entities for which quantum theory is relevant. They may indeed be in a state of superposition, but this does not mean that the cat as a whole is in a state of superposition. The cat’s state is a sum of the states of its cells and particles, and here a single macroscopic result is obtained (this is the decoherence effect I mentioned above), alive or dead. When we measure it, we will find out which of the two. Either way, what is important for our purpose is that this is not a logical contradiction.

There is also a halachic example that depicts a completely identical situation, a kiddushin that is not dedicated to Levia (see it on my website in column 303). Think of a situation where Reuben has two daughters, Rachel and Leah. Now Shimon comes to him and gives him a penny and tells him that he is sanctifying one of his two daughters with this penny (without specifying which one of them). A problem arises here, since if he is married to Rachel, then Leah is his wife’s sister, with whom marital relations are forbidden to him as a matter of shame. And the same is true vice versa, if Leah is the one he is married to, then Rachel is his wife’s sister, with whom he is forbidden. Because there is no way of knowing which of them is his wife, it is permissible for him to have marital relations with either of them. These are kiddushin that are not dedicated to Levia, and there is a dispute as to whether they are valid or not. In any case, he is prohibited from having marital relations with both of them, the question is whether they are both his wives and he must give them both a satisfactory divorce, or whether they are not married to him at all.

The commentators discuss the nature of this situation. From the Talmud it seems that this is a state of doubt, since we do not know which of the two is his wife. But a closer look reveals that this is not a normal state of doubt. In normal doubt, there is one correct answer, but we do not know it. Think of a man who sent a messenger to consecrate one of his two daughters. The messenger arrived at his father and consecrated one of them, say Leah, and now the father and the messenger are dead. No one knows which of the two is his wife. This is a state of normal doubt, since there is one correct answer (God knows that it was Leah who was consecrated to him), and only in our case does doubt arise due to partial information. Doubt is a lack of information. But in consecrations that are not dedicated to Leah, there is no correct answer at all, not even theoretically. Even God Himself does not know which of the two is consecrated, since neither of them is defined as his wife. We can say that this is a state of ambiguity (which is a feature of reality itself) and not of doubt (which is a feature of a lack of information in man).

In our terminology, we can say that kiddushin that are not dedicated to Leah is a state of quantum doubt, since both answers are correct. It is not a doubt whether Rachel is his wife or Leah is his wife. Ostensibly, they are both his wives. But this cannot be because a person cannot be married to two sisters. It is more correct to describe this as a state of quantum superposition: his wife is a third entity, and this entity is a composite of two ordinary entities: Rachel and Leah. This is a sum of two states, one in which he is married to Rachel and the other in which he is married to Leah. But as we have seen in quantum theory, the correct description is that his wife is a sum of two states, and not that both Rachel and Leah are his wives. Just as it is not correct to say that the particle passed through both slits together, but rather that the state of the entity (electron) is a sum of two particle states, one in which the particle passes through slit A and the other through slit B.

The Arrow in Flight and the Uncertainty Principle

I have already mentioned that one of the fundamental principles of quantum theory is the uncertainty principle, which states that there are pairs of quantities that cannot be attributed simultaneously to the same body. Thus, for example, it is impossible for any body to have an exact velocity and an exact position at the same time. There is a reflection of the complementarity principle here, since an exact velocity is attributed to an electron when it is in a wave state and an exact position exists for it only when it is in a particle state. The uncertainty principle also sets a measure for the levels of uncertainty in these two properties, but I will not address this here.

In my article “Zenon’s Cross and Modern Physics” ( Iyun Mo, 1999, p. 425), I discussed the connection between the uncertainty principle and Zeno’s paradox of the arrow in flight. I will address this briefly here, because there too there is a solution to the contradiction in quantum theory, and again one can see there the claim that the unity of opposites is the refuge of the lazy.

The Greek philosopher Zeno of Ilia (a fifth-century BCE student of Parmenides) presented several paradoxes that challenge the concepts of motion and time, and wanted to argue that motion is nothing more than a fiction. It exists only in our perception but cannot exist in reality itself. One of these paradoxes is the paradox of the arrow in flight, and I will present a popular formulation of it here. If we observe an arrow in flight, at every (discrete) moment that we observe it, we find that it is standing in a different place. So at what moment exactly does it change its position? How does it move from one place to another? After all, if it is standing, then it does not move at any moment, and how is it possible that we nevertheless see it in a different place at every moment?

First, I will ask whether this is a logical contradiction or a physical difficulty? On the surface, it seems to be a logical contradiction. The particle is both stationary and moving. Its speed at that moment is both 0 and non-zero. If so, this is a difficulty that cannot be lived with and a solution must be sought. The solutions proposed for this difficulty are based, among other things, on the infinitesimal calculus (the timeline is a continuum. There are really no isolated points in time) and on the uncertainty principle in quantum theory (according to which it is impossible to speak of the position and speed of the arrow at the same time). In my above article, I explained that neither of these can constitute a solution. Here I am just saying that it is essentially the same argument that I presented above: it is impossible to live with a logical contradiction. The infinitesimal calculus offers us a description of reality free of contradictions, but as I explained in my above article, this is tantamount to adopting a language in which it is forbidden to express the paradox. This is not a real solution to it.

The uncertainty principle does apparently offer a solution. We have seen that it is not possible to attribute an exact speed and position to any body at the same time. Therefore, if the arrow is located in a specific place, it has no speed and vice versa. This undermines the formulation of the paradox I presented above. But the other side of the coin is that this paradox shows why the uncertainty principle itself is problematic, and in fact contradictory. We have a point body and theoretically there is no reason to talk about its location at some point and the speed it has (speed is the rate of change of place). What does it mean that it has no speed? Does it not change its position? After all, our eyes see that it does. And what about the position? What’s more, an arrow is a macroscopic object, and therefore there is no reason to attribute a position and speed to it at the same time. We have seen that quantum theory is irrelevant on these scales. As I noted above, the fact that the contradiction also appears in the framework of a scientific discussion is not a solution for it. A scientific theory cannot contain contradictions either. If there is a contradiction in quantum theory, it is just another reason to seek an appropriate and logical solution to it. In my article there, I proposed a reversal of the approach: to seek a logical solution to the paradox, and through it to explain the strangeness of the uncertainty principle in quantum theory.

The root of the problem in this paradox lies in the lack of distinction between two seemingly similar concepts: standing and being. When we say that a body is standing at place X at time T, this is not equivalent to saying that it is at place X at time T. In other words, a body can be at some place at some time at speed 0 (in which case we say that it is ‘standing’ at that place) or it can be there at some other speed (in which case we say that it is ‘being’ there). When we say that a body is at a place, this does not necessarily mean that it is standing. It can be there in motion. When we say that a body is standing at place X at time T, this means that it is there at speed 0. Now you can see that the paradox simply disappears. To the question of at what moment the body is moving, the answer is at every moment. There is no contradiction between the statement that at every moment it is (but not standing) at another place, and the statement that at those very moments it is also moving.

What is confusing here is that it is impossible to say that at every moment the body changes its position. Changing position takes some time, and cannot be done in a single moment of time. In other words, the claim that a body is in two places at the same moment is logically contradictory and not just physically impossible. Here we are not talking about infinite speed but about two contradictory claims (like the particle passing through both slits). If so, a body cannot change its position at one discrete moment of time, but it can be in motion at a discrete moment of time. What confuses us about this matter is that velocity (being in motion) is not a change of position. Velocity is a quantity that also exists at one discrete moment, and it is the potential for a change of position. But a change of position requires a segment of time, and a body also has velocity at one discrete moment of time. The meaning of saying that a body has velocity at a certain moment is that it has the potential to change position at the next moment, and therefore it will probably change position. [15] In my article there, I explained this problem in terms of a concept derived from the infinitesimal calculus used to define speed in mechanics. There, speed is defined through position differences divided by the time it takes to traverse them, and when the speed is not constant, very small time segments must be taken (which tend to zero). But as I explained there, this is only a computational (operational) definition of speed (i.e., this is the way to calculate speed) but not a definition of the concept of speed itself. The conclusion is that the arrow-in-flight paradox is the result of conceptual confusion and nothing more.

Later in the article, I explained the uncertainty principle in quantum theory in light of this distinction. The reason why we cannot attribute both velocity and position to the same body at the same time is because measuring velocity requires a different observation than measuring position. Measuring position is essentially taking a photograph of the body at any given moment (with a camera with an exposure time that is one discrete moment in time). When observing the world with such a static camera, there is no possibility of seeing movement or speeds (as we saw in the arrow paradox). A camera is blind to movement. In contrast, measuring velocity requires me to film the body, [16] and this necessarily requires tracking it over a period of time. This cannot be done over a discrete moment in time. The film is blind to positions (because only the movements are seen. Quite the opposite of a camera). In any case, these are two forms of observation that exclude each other, and therefore it is impossible to measure speed and position simultaneously: either I observe the body (and the world) through a camera or I observe it through a film. [17]

I showed there that the meaning of the division between the two points of view I described (the camera and the video camera) is what is called in quantum theory the ‘picture of place’ and the ‘picture of momentum’ (the velocity). In the picture of place, all of physical reality is described in terms of the spatial position of the particles at any moment. There is no possibility of talking about velocities there. In contrast, in the picture of momentum, all of reality is described only in terms of the velocities of the particles at any moment, and then there is no possibility of talking about positions. This is exactly parallel to our camera and video camera. In quantum theory too, these are two pictures that subtract from each other, and it is not possible to describe the body in both at the same time. A person must choose a point of view or a picture. It is true that it is possible to switch from one to the other, but not to use both simultaneously.

Distinguishing between two points of view is not a contradiction.

This means that, as we saw with the complementarity principle, so it is with the uncertainty principle. Here too, it is not a contradiction but a complex picture. In the background, there are two different perspectives that exclude each other, but here there are not two truths that contradict each other at the same time. Therefore, the uncertainty principle does not express a logical contradiction, but at most a situation that is difficult for us to imagine. The explanation I have proposed here in light of the arrow paradox even brings this a little closer to our reason and imagination. The bottom line is that quantum theory does not contain contradictions or deviations from conventional logic, and in any case does not create the possibility of making contradictory claims or holding contradictory beliefs.

Now I will present three applications that sharpen the difference between holding in contradiction and two different perspectives. The first is a well-known parable about a debate between two people describing an elephant. Reuben claims that it is an animal with two legs far apart, with one eye. Shimon argues that it is an animal with two legs close together and two eyes. These are seemingly two contradictory descriptions, but in fact they are looking at the same elephant from two different perspectives. Reuben sees it from the side, so its legs are far apart and it has only one eye (this is what is seen from the side). Whereas Shimon sees it from the front, so he sees two legs close together (the two front ones) and two eyes. The descriptions are seemingly contradictory, but in fact they are descriptions of the same object from two different perspectives. Each of the descriptions is partial, and the complete description is the combination of both.

A second application can be seen in the Sage article “These and These are the Words of the Living God.” Originally, this was said regarding the disputes between the Bible and the Bible (Eruvin 13b), and in the issue of Gittitin 6b regarding a concubine in the hill country. As is known, the commentators apply this to all disputes in the Talmud and perhaps even after it. The puzzlement over this article is well-known, and the clear source that presents it is the Ritva in his additions to the Eruvin there, who writes:

These and these are the words of the living God. The rabbis of France, z”l, asked how it is possible that both are the words of the living God, one prohibiting and the other permitting, and they explained that when Moses ascended to receive the Torah, he was shown 49 faces for prohibition and 9 faces for permission for every matter, and he asked God, blessed be He, about this, and He said that this should be handed down to the sages of Israel in every generation and there should be a decision like theirs, and it is correct according to the sermon and in the way of truth there is a reason and a secret in the matter.

Here too, it seems to be a logical contradiction. But in truth, this difficulty disappears if we look at the parallel issue, which is the only one in the Talmud in which an explanation of this rule is also given, in Gittin 6b:

As it is written, “And his concubine committed adultery with him,” Rabbi Abiathar said, “A fly found her.” Rabbi Jonathan said, “Nima found her.” And Ashkhaiah. Rabbi Abiathar told Elijah, “May the Holy One, blessed be He, engage in a concubine on the hill.” And what did he say? Abiathar said to him, “My son, thus says Jonathan, my son, thus says the Lord, the Almighty.” And who is there to say, “Come up to heaven, O God?” These and these are the words of the living God. A fly found her, and he did not observe, but he observed and observed.

The conditions differ as to what exactly the man found for his mistress. What provoked his anger? One says he found a nymph and the other says he found a fly for her. The prophet Elijah reveals that God himself raised both possibilities, but this is not a doubt but a double assertion: He found both a fly and a gnat for her, and only the two together created his cupid. [18] Each side only grasped a partial picture, and the full picture is the combination of the two. Although this is a legendary issue, it seems from within that this is the explanation for the phrase “These and these are the words of the living God” also in its halakhic context. And indeed, further up on the same page in the issue of Eruvin, the possibility of finding many correct reasons against a versed halakhic law is presented, such as the fact that there are reasons to purify the insect, even though the Torah itself states that it is impure. The explanation for this is that there are indeed reasons to purify it and there are also reasons to defile it. The bottom line is that it is impure, because the reasons for impurity outweigh the reasons for purity. But this does not mean that the reasons for purity are not correct. The reasons in all directions are correct, and the halakhic law is the weighted result of all of them. Here again, there is A view is made up of different perspectives, each of which is only partial. The apparent contradiction (the creature is both impure and pure) is not really a contradiction but a complexity created by the combination of conflicting perspectives. [19]

The third application appears in my article “What is ‘chalot’? Halacha, Logic and Adherence to It”, Tzohar 2, 2006. There I cited Rabbi Shimon Shkop’s words regarding a woman who divorces on a condition. His argument is that until the condition is met or not met, the woman’s status is both a husband’s wife and a divorced woman. This is essentially a quantum superposition state, as I described it above. If she meets the condition – the wave function ‘collapses’ to the state of a divorced woman, and if not – it ‘collapses’ to the state of a husband’s wife. There I explain how it is possible for two contradictory states, a husband’s wife and a divorced woman, to be combined. On the face of it, this is impossible, after all, if she is a husband’s wife she is not divorced, and vice versa. There I argued that there are two chalots on the woman, a chalot of a husband’s wife and a chalot of a divorced woman. The legal situation created is a combination of both, and in such a situation there is no contradiction. I likened it to a situation where there is salt and sugar in a dish, which is of course completely possible. It is not possible for the dish to be both salty (completely) and sweet (completely). The contradiction exists in the implications, but the mere existence of the two aspects is not a contradiction. ‘Chalot’ is a type of entity, and therefore the existence of two contradictory chalots on the same woman at the same time is not a problem (like salt and sugar in the same dish). The halachic implications, of course, cannot be contradictory, and indeed they always lean towards one of the sides (I explain there which of the two).

In other words, it can be said that in this woman’s situation there is one reason to treat her as a divorcee and another reason to see her as a man’s wife. As we saw above, the existence of reasons that lead in contradictory directions is not a state of contradiction. Here too, it is not a contradiction but rather two aspects (with contradictory characteristics) that can both exist simultaneously.

Implications and Examples: Some Types of Contradictions

The meaning of the picture I have described here is that there is no place for contradictory claims, neither in faith, nor in quantum theory, nor in general. When we encounter a contradiction, we must choose one side and abandon the other, or alternatively show that it is not a contradiction, and then and only then can we hold on to both. Relying on another logic (quantum, for example), or using expressions like the ‘unity of opposites’ are not helpful because they say nothing. If we are dealing with logical contradictions, there is no place to unite them, and if not, then this must be shown, thereby resolving the apparent contradiction. People who fail to show that there is no logical contradiction between the claims and yet still want to hold on to both, prefer, probably due to intellectual laziness, to speak in the above-mentioned empty terms.

One reason for this confusion is the reckless use that many make of the concept of ‘contradiction.’ In most cases where the ‘unity of opposites’ is discussed, it is not a logical contradiction but at most a matter that is difficult to understand. In cases where we do not have an explicit explanation and yet it seems to us that both sides are correct and neither should be given up, we must at least show that there is no logical contradiction between them but rather that it is a difficulty, or a softer ‘contradiction’ (physically avoidable, or just a matter that is difficult to imagine, such as four dimensions, etc.).

I cannot go into more detail here about the distinction between the types of contradictions (logical contradictions versus physical contradictions or difficulty imagining), but to complete the picture I will only add that in the articles “Is belief in logical contradictions possible?” (appears on my website), I define another type of contradiction, using Kantian terminology: There are situations in which the contradiction between the two claims is neither logical (analytical) nor observational-scientific, but rather a priori. The contradiction arises because of an a priori principle (such as the principle of causality), but this does not mean that the combination of the two claims is meaningless. An analytical contradiction is empty talk, as I explained in this article, but an a priori contradiction has a meaning of its own and therefore should not be ruled out, at least in relation to God. To conclude the article, I will now give some examples of such reckless uses of the concept of ‘contradiction’, and the meaning of the picture I proposed here.

The first example is the words of Chazal about the red cow. Chazal use the law of the red cow to speak about contradictions that we cannot understand. For example, in Pesikta Derab Kahana 4, at the beginning of Parashat Para, we find (see also Bamidbar Rabbah 19:1):

Tamman Taninan: Bright as impure grain, a flower in its purest form. Who did so, who commanded so, who decreed so, not the One and Only of the world?

Tamman Taninan: If a woman dies and gives birth inside her womb, and an animal reaches out its hand and touches it, the animal is unclean for seven days, and the woman is pure until the birth comes out.

He who dies in a pure house, and leaves it, is unclean. Who did this, who commanded this, who decreed this, not one, not the only one in the world?

And they said, “All who are occupied with the cow from beginning to end are defiled with their garments, for it itself purifies the impure.” Rather, the Holy One, blessed be He, said: I have enacted a statute and decreed a decree, and I have no power to transgress my decree – “This is the law of the Torah which the Lord commanded to be said” (Num. 19:2).

All of these are examples of contradictions (some of them are cited in another midrash as Korach’s mocking difficulties with Moses, with many more examples. See, for example, Yerushalmi Sanhedrin, chapter 10). In the case of leprosy, if a person’s flesh has a spot the size of a grain of sand, it defiles, but if it has spread throughout the body, he is pure. The second example is a midwife who becomes impure from touching a dead fetus, as opposed to the woman who carries it in her womb, who is pure until it comes out. The third example is when there is a dead person in the house, who is pure, and he becomes impure when the dead person comes out of it. A red cow is a fourth example, since it defiles the pure who deal with it, but its whole point is that it purifies the impure. All of these examples seem like logical contradictions, and therefore the midrash attributes them to God, the One and Only of the Universe. Ostensibly, this is a literal statement of the ‘unity of opposites’ that assumes that God is not subject to the constraints of logic.

But a closer look reveals that this is not a logical contradiction. And is there a contradiction between the claim that a cow defiles the pure and the claim that it purifies the impure? A contradiction would only exist if we were to say that it both purifies the impure and does not purify them. Purifying the impure and defilement of the pure is an incomprehensible matter, but not a logical contradiction. And so it is with all the other examples cited in the aforementioned midrash. Our treatment of such cases as if there were contradictions here is reckless. If these were indeed logical contradictions, then we could not adopt both sides of the contradiction simultaneously. One of them was wrong. When the matter is not understood, but not contradictory, there is no reason to hold to both sides of the ‘contradiction’ together. After that, one can even try to understand the idea, but understanding here is not a condition for adopting the two laws in question. It can be said that in soft contradictions such as these, they leave us with a question but not with a difficulty or a contradiction.

The Polish logician, Jan Lukaszewicz, developed three-valued logic, that is, logic in which each statement can have three truth values ​​(true, false, and something third). He showed that it is possible to present a consistent logic that is not binary (i.e., does not fulfill the law of the excluded middle), and some saw this as a challenge to classical (binary) logic. Many authors, when discussing contradictions in Adam’s doctrine, use three-valued logic to explain this. Supposedly, there is justification for a contradictory doctrine here. There were even those who wanted to use this logic to offer a solution to paradoxes. In their view, a paradoxical statement is neither true nor false, but this should not bother us because the third truth value can be attached to it (T – true, F – false, P – paradox).

But as I explained above, such logic cannot be an explanation for anything. Even Lukashevich himself, when he built his logic, used ordinary logic. His logic does not replace ordinary logic but rather constitutes a formal structure that can serve us as a useful tool in certain contexts (as we saw above regarding quantum logic). For example, some also refer to probability as a different type of logic, since in probability theory, claims can take on a sequence of values ​​(a number between 0 and 1). Does probability undermine our ordinary logic? Of course not. It is a tool for dealing with situations of uncertainty, but not a change in the principles of logic. Therefore, it is clear that dependencies in three-valued logic cannot be an explanation for a contradictory proposition or paradoxes.

In Feuer’s book (above note 1), Lukashevich himself is quoted, who said that his motivation for developing three-valued logic was the feeling that the determinism imposed on us by classical binary logic (true or false) freezes thought. His friends and colleagues used his logic as a basis for understanding quantum theory, which shattered the determinism that had been accepted until that time. You can see in Feuer how many ideas were based on and hung on this logic, without any injustice. Contrary to the words of its creators and the thinkers who came after them, such logic cannot form the basis for any idea on earth, but perhaps be a source of inspiration for thinking that goes outside the box (but not the logical box, of course). There is no possibility of departing from accepted logic, neither from the law of contradiction nor from the law of the third rule. Thus, for example, in the development of Lukashevich’s logic, it was also possible to use proofs by negation (which are based on the law of the excluded third).

It is no wonder that authors such as Benjamin Ish Shalom, in his book Rabbi Kook Between Rationalism and Mysticism (above, note 5), propose to explain in this way some of Rabbi Kook’s basic ideas, which seem contradictory to us. In several places (see note 71 to the first chapter and note 133 to the third chapter) he links Rabbi Kook’s positions to the ideas of the unity of opposites on the logical level, and even mentions in this context Lukashevich’s three-valued logic. Similarly, Avi Sagi, in his book These and These (Hillel Ben Chaim Library, Kibbutz Hameuhad, 1996. See there, chapter 7 of part three), refers to the Talmudic statement “These and these are the words of the living God” as an expression of the unity of opposites, which, according to Rabbi Tzadok HaCohen of Lublin, is not “bound by the laws of contradiction” like human thinking. The same is true for philosophical and kabbalistic issues, such as the contrast between the image of “surrounding all the universe” (transcendence) and the image of “filling the universe” (immanence).

Meir Monitz, in his article (above, note 5), begins with quotes from the writings of Rabbi Kook that speak of the unity of opposites in God, contrary to the law of contradiction in human thought. He also cites from the aforementioned book by Benjamin Ish Shalom, Four Ways to Understand Contradictions in Rabbi Kook’s Teaching, the fourth of which is the boldest of all, which speaks of accepting the unity of opposites (he mixes opposites in nature, thought, and logic) without resolving the paradox. His claim is that the concept of divinity requires us to enter into logical contradictions, on the issues of freedom and necessity, finitude and infinity, completeness and completion, and so on. As part of his “logical” explanation, he cites from Hugo Bergman in his book Introduction to the Theory of Logic – The Theoretical Science of Order (Bialik Institute, Jerusalem, 1953), which raises several claims against the Law of the Avoided Third and the Law of Contradiction. I will not be able to enter into a critical examination of these claims here (some of which I have addressed here, such as the Arrow Paradox), but a priori it is clear that all of his claims there cannot hold water. They themselves are based on our classical binary logic, and what That said, outside of it, it is simply meaningless.

In most of these places, there are not really logical contradictions, and therefore, instead of clinging to vague statements about the ‘unity of opposites’ or three-valued logic, it would be better to look for an explanation and show that there is no logical contradiction. Without such an explanation, there is indeed a logical contradiction (this is the case with the problem of completeness and completion. See it on my website, in columns 170, 278, 519, and more). In such a situation, there is no use waving around concepts like the ‘unity of opposites’ and/or different logics. One of the rays of the dilemma is wrong and should be abandoned. However, if we have found an explanation, then there is no need for the unity of opposites or a change in the principles of logic, because there is no contradiction to be resolved here.

I will add that in many cases the problem is interpretive. An author speaks of the ‘unity of opposites’ but does not intend to make a claim on the logical level (as if there were a different logic) but only to use this metaphor to sharpen the complex picture he is drawing. His intention is to suggest that we should not think that if there is a contradiction we must immediately abandon one of its sides. On the contrary, we should seek a solution and an explanation and show that there is no contradiction. In these cases the ‘unity of opposites’ is merely a metaphor. At the beginning of his article, Monitz cites six such quotations (see more sources in his note 2) from the writings of Rav Kook. This can be seen in each of them. Here I will use as an example the quote from the book of the Rabbi, Arpali Tahar , p. 13, where he writes:

One should not be alarmed by the grouping of great opposites as is famously said, because what appears to many as disparate and opposite things is only because of the smallness of their minds and the narrowness of their outlook, which sees only a very small part of the supreme perfection. And even this part is in a very distorted form. But those who have enlightened knowledge, their thoughts spread to different places and large spaces. And they grasp the treasures of goodness in every place. And they unite everything together in a peaceful unity.

Is it necessary to interpret this as a logical thesis? I could equally explain that his intention is to point out the short-sightedness of some people who, due to their narrow-mindedness, fail to understand that there is no contradiction here, in contrast to the “people of enlightened knowledge” who do find an explanation and resolve the contradiction.

It is important to understand that such an interpretation of his words is fundamentally different from the logical interpretation offered by Monitz, since according to the logical interpretation there is no point in seeking an explanation, since there is no explanation accessible to people who are as deficient in logic as we are. Only God, the Blessed One, is not subject to the constraints of logic. According to this interpretation, it is not clear who the “people of enlightened opinion” that Rav Kook is talking about are. After all, there are no people who can transcend logic, like God Himself. Therefore, the more plausible interpretation of the Rabbi’s words is precisely the one I have proposed, according to which his intention is to encourage us here to seek a solution, and not to be content with a superficial observation, to declare the unity of opposites, and to sit quietly. If we encounter a contradiction, we must seek an explanation and show that there is no real contradiction here (and thus be among the “people of enlightened opinion”).

I could now go through all the apparent contradictions mentioned in the above sources and ‘explained’ through various logics or hinged on the ‘unity of opposites’, and show why they all fall into one of these two categories: a minority of them are logical contradictions, in which case we must reject one of the claims. And the rest, most of them as a whole, are imaginary contradictions, for which there is no need for a unity of opposites. I did this above with regard to the theory of quantum mechanics, which is one of the more difficult cases, and also with regard to the Talmudic maxim “These and these are the words of the living God.” One can learn from these two cases for the other easier examples. But for our purposes, this is enough, and there is no room here to go into the other examples in detail.

[1] See, for example, the book by Louis S. Feuer, Einstein and His Contemporaries , translated by Gad Levy, Sifrat Afakim, Am Oved, Tel Aviv 1979, in the chapter “The Logical Revolution Against Determinism”, pp. 174-181.

[2] In the formula above, the symbol * represents ‘also’, and the symbol U represents ‘or’.

[3] See on this subject Israel Netanel Rubin’s book, What God Cannot Do: The Problem of God’s Subjection to the Laws of Logic and Mathematics in Jewish Philosophy and Theology , Reuven Mass, Jerusalem 2016. These matters have also been discussed in several places on my website, in responsa and articles (see, for example, columns 549-550 and many more).

[4] The words are almost copied in the Rashba’s responsa, Ch. 1, Ch. 9 and Ch. 18, and also in Ch. 1, Ch. 10, Ch. 11.

[5] On this distinction, see Yehudit Ronen’s article, “Everything is predictable and the authority is given,” in the collection Between Religion and Morality , Daniel Statman and Avi Sagi (eds.), Bar-Ilan University, 1994, pp. 35-43. These matters were also discussed in the fourth book in the Talmudic Logic series , The Logic of Time in the Talmud (pp. 50 ff.).

[6] See, for example, Meir Monitz’s article, “The Logical Foundation for the Unity of Opposites in the Teachings of Rabbi Kook,” Elyon Shvut 143-144, 1995. You can also find references in Benjamin Ish Shalom’s book, Rabbi Kook: Between Rationalism and Mysticism , Am Oved, second printing, 1990 (a second edition has also been published, Resling 2019). Monitz proposes a ‘logical foundation’ for the unity of opposites in his article (in my opinion, this is an oxymoron), but Ish Shalom also links this approach to the three-valued logic of the Polish logician Lukaszwicz. I will comment on this below.

[7] In analytic philosophy there is debate about this claim, but only because the term ‘interpretation’ is not agreed upon. For our purposes here, this is not important.

[8] Analytic philosophers ask whether the current king of France is bald. If we examine the group of bald people, we will not find him there, but if we examine the group of hairy people, we will not find him there either. This is because there is no current king of France. Applying the third law of non-existence to a non-existent object is misleading.

[9] See about it on my website in column 302.

[10] The waves are summed by the sum of the values ​​of the function that describes the wave, but the intensities are the squares of the values ​​of the function. Therefore, it is clear that the sum of the intensities of the two waves (the sum of the squares of the values ​​of the two functions) is not equal to the intensity of the wave that is composed of them (the square of the sum of the values ​​of the function).

[11] Here and below I present a very simple and schematic picture. The actual results of a two-slit experiment are more complex. There are many lobes between the slits, and their distances and heights depend on the wavelength, the distance and angle of the beam source, and the distance between the slits. The description here is schematic, and for our purposes here it is sufficient.

[12] For the intensity distribution to be as described here in Graph 3, a certain distance is required between the two slits, which depends on the wavelength of the wave beam. I do not need these details here so as not to complicate the description.

[13] See for example here: RHS Carpenter, Andrew J. Anderson (2006). “The death of Schrödinger’s cat and of consciousness based quantum wave-function collapse”, Annales de la Fondation Louis de Broglie , 31 (1).

[14] Although according to this it is not clear what the definition of measurement is at all, since the detector itself is just a physical object. Without a measurer there is no measurement. To this puzzle, as far as I know, there is no agreed-upon solution to this day.

[15] Alternatively, it will get stuck in a wall that does not allow it to move, and then its energy will come out in another form, such as dissipating heat into the environment.

[16] I explained there that this is an ideal film. Our films build the film from successive static shots, meaning that it involves continuous use of the camera. In our case, this is the perspective of a camera, and in this perspective we accept classical physics, in which there is no uncertainty between position and velocity.

[17] I cannot go into detail here about whether this is a limitation of ours (we cannot determine position and velocity simultaneously), i.e. an epistemic claim, or whether it is an ontological principle, i.e. a principle that deals with reality itself. This is an old debate among quantum theorists, and today the tendency is that it is an ontological principle. The explanation I have given here seems to be an epistemic principle, but it is certainly not necessary. The distinction between a camera and a video camera does not lie in our limitation but rather indicates something in the nature of these quantities (position and velocity) themselves.

[18] That’s how it is, as I understand it, and not that the Nima alone brought about his capitulation.

[19] See this in columns 248-247 on my website, and in the articles “Is Halacha Pluralistic?”, Ha’ain 2017.

Link to the article in Word file format for those interested .