On Quantum Theory and Contradictory Claims/Beliefs

With God's help

Introduction

Quantum theory is full of oddities, and many even see in it logical contradictions. The uncertainty principle states that there are pairs of physical quantities whose definite values cannot be ascribed to the same object simultaneously. Thus, for example, one cannot ascribe to a given object 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 is true of energy and time, and so on. This is a strange principle and difficult to grasp, but on its face it contains no logical contradiction. There is no logical necessity that an object have both position and velocity simultaneously (below I will challenge this). Seemingly, this is only a difficulty, not a contradiction, similar to the reality of four dimensions, which is very hard for us to grasp and certainly to imagine, but such a reality involves no contradiction. Our inability to imagine it is apparently rooted in the limitations of our perception. By contrast, the principle of complementarity (= completion), attributed to the Danish physicist Niels Bohr, appears to be a genuine logical contradiction. This principle states that an object can be both a particle and a wave. More generally, there can be a situation in which there are two contradictory descriptions of reality and both are true. Here we have already taken one step further, beyond the uncertainty principle, because here we are making about reality a claim 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 wave-like and particulate, resembles the claim that there exists in reality (or even in the world of concepts or in Plato's world of ideas) a married bachelor or a circular triangle.

Various physicists and philosophers propose to explain the oddities of quantum theory on the basis of the logical problems found in it. They claim that quantum theory is based on a logic different from the one familiar to us.[1] This logic does not satisfy some of the logical principles included in classical logic. There are several proposals for such logics, including proposals according to which it does not satisfy the law of contradiction (that it cannot be the case that proposition X and its negation are both true simultaneously) or the law of the excluded middle (that either proposition X is true or it is not true, with no third possibility). The prevailing approach proposes a logic that gives up the property of distributivity found in ordinary logic (what is called in Hebrew the 'law of distribution'): P*(Q U R) = (P*Q) U (P*R).[2]

Interpretations of this kind are collectively called 'quantum logic,' and in general they assume that the claim that quantum theory is based on a different logic constitutes an explanation of its oddities and of the contradictions we find in it.

From here it is a short road, in the eyes of many thinkers, to conclude that the laws of logic or the logical constraints to which we are so accustomed, and from which we see no way to escape, are not really so draconian and absolute. According to these claims, quantum theory allows us to make contradictory claims, and there is no reason to be alarmed by that.

This conclusion has obvious applications in theology, because different descriptions of God naturally lead to contradictions, and seemingly we have a simple way to deal with such contradictions: God is above logic, and so, for example, He is both present throughout the whole world and only outside it, He both knows in advance what we will choose and we nevertheless have free choice, He is both perfect and in the process of becoming perfected, and so forth. Many believe that these claims are not so far-fetched, for after all 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, so too He can act in a way that contradicts the laws of logic.

In this article I would like to examine this fundamental claim, focusing on the logical interpretations of quantum theory. To sharpen the point, I will state my principal claim already here: an interpretation based on a different logic is nothing but nonsense (meaningless discourse), both with respect to quantum theory and with respect to philosophy and theology. These are merely meaningless strings of words. My position is that there is no possibility whatsoever of making a claim that does not obey the laws of logic, not in a theological context and not at all.[3]

The Jewish-Christian dialectic regarding the attribution of contradictions to God

In the writings of several of the medieval authorities (Rishonim), one can find unequivocal claims against attributing logical contradictions to God. Surprising as this may be, their words imply that even He is subject to the laws of logic. Here I will bring one prominent example from Maimonides in Guide of the Perplexed, III:15:[4]

What is impossible has a fixed and permanent nature; it is not the product of an agent's action, and it cannot change at all. Therefore God is not described as having power over it. No one among people of inquiry disputes this at all, and only one who does not understand intelligibles is ignorant of it.

Here Maimonides states that one must not attribute logical contradictions to God. He sees this as an agreed matter, and one who disagrees with it, in his view, 'is not among the people of inquiry' and does not understand intelligibles.

But immediately afterward he adds that there is nevertheless room for disputes in such matters:

However, the point of dispute among all thinkers concerns one type of imagined thing: some thinkers say that it belongs to the class of impossibilities, over which God is not described as having power, while others say that it belongs to the class of possibilities, whose existence God's power can bring about whenever He wills.

Such a dispute does not concern attributing contradictions to God. That is impossible according to all views. One can dispute only the question whether this is indeed a logical contradiction ('logically impossible') or merely something not understood (or something that contradicts the laws of nature, that is, 'physically impossible').[5] But if we have reached the conclusion that it belongs to the realm of impossibilities (a logical contradiction), then even God 'is not described as having power over it.'

Later he gives examples of attributing contradictions to God (for example, that He can make a square whose diagonal equals its side), and he also mentions issues regarding which philosophers dispute whether they belong to the logical plane or the physical one. One may add in this context the question of divine foreknowledge and our free choice (see Maimonides, Laws of Repentance 5:5, and also columns 299-303 on my site), along with other questions that will be mentioned later in this article.

By contrast, in Christian thought the conception is very widespread that God is not subject to the laws of logic, and that one may certainly attribute even logical contradictions to Him. A clear expression of this is found in the doctrine of the 'unity of opposites' or 'coincidence of opposites' (Coincidentia Oppositorum) of Nicholas of Cusa, a Catholic philosopher, theologian, and mathematician of the fifteenth century. Tertullian as well, one of the Church Fathers (in the second-third century CE), went even further and stated: 'I believe because it is absurd' (Credo quia absurdum). Note carefully: not 'despite the fact that it is absurd,' but 'because it is absurd.' He sees the essence of faith in the absurd. Not only is belief in absurdities possible; faith relates primarily to absurdities. Presumably, in his view, something that is not absurd does not belong to the plane of faith but to that of knowledge, science, or rational thought. This conception can also be found in the existential philosophy of the nineteenth-century Danish thinker Søren Kierkegaard, since in his doctrine the 'knight of faith' (the figure of the perfect believer, represented by Abraham our patriarch, especially in the binding of Isaac) is required to sacrifice his reason and his ethics for the sake of the absurd demanded by faith. For him as well, the essence of the meaning of faith is life (not merely belief) in the absurd.

Briefly, I will say that in recent generations this idea has entered Jewish thought as well with much greater force, and many speak of the unity of opposites in Hasidic literature, in Rav Kook, and elsewhere.[6] In my view, the fact that a given idea has a Christian source or was formed under Christian influence does not invalidate it. An idea or approach must be examined on its own merits, whatever its source. I oppose the idea of the unity of opposites, regardless of its origin, simply because it is meaningless verbiage. I will now show that there is really no such idea, and whoever advances it merely moves his lips without saying anything.

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 God is the assumption that from God's omnipotence, and from His being the source of all reality and all the laws that govern it, it follows that it cannot be that any system of laws should limit and obligate Him, that is, that He should be subject to it. From this, seemingly, follows the conclusion that the laws of logic also cannot limit Him, and therefore one may attribute to Him even abilities or properties that include logical contradictions.

At the base of this argument lies a treatment of the laws of logic as though we were dealing with 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 circular not because there is a law forbidding it to be so, but because it simply is not so. If it were circular, it would not be a triangle. The nonexistence of a circular triangle is not the result of any law, and therefore no one legislated it. It is a consequence of the definitions of the concepts themselves. The term 'laws of logic' is a confusing and unfortunate conceptual muddle. It arises because from Aristotle until today logic has become an independent field of study and research, and like every field it too has foundational laws. But people ignore the great difference: in all the other fields there are laws that are the product of legislation, grounded in an authoritative legislator. Even the laws of nature are of this kind, where the legislator is God Himself. And from this it follows that they could have been otherwise (depending on the decision of the 'legislator'). The same is true of the laws of the state or the rules of some guild; they are the product of legislation and could have been otherwise. All these are systems of 'laws' in the ordinary sense. But the 'laws of logic' were legislated by no one, because they could not have been otherwise. And from this it follows that although even God cannot deviate from them, they do not 'bind' Him and are not imposed on Him. Quite simply, because there is nothing there to be imposed on Him. One who cannot overcome the laws of nature is not omnipotent because he is compelled to do something. But one who cannot 'overcome the laws of logic' is not lacking in ability, if only because there is no such thing as 'overcoming the laws of logic.'

From here 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. Not that it exists and not that it does not exist; not that it is beautiful or broad-hearted, and not that it is a rare creature. Any sentence that contains the phrase 'circular triangle' cannot be true, but it cannot be false either. If a sentence includes a contradictory concept, it thereby becomes meaningless.[7] When I say that a circular triangle does not exist, I do not mean to assert any factual claim, but only to say that this concept is undefined.

One cannot say that God can create a wall that stops every shell, and also a shell that penetrates every wall, not because there is some deficiency in His power, but because there is no logical possibility of the simultaneous existence of such a wall and such 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 create a wall that stops every shell and a shell that penetrates every wall, or that He cannot create a circular triangle, does not imply any lack in His powers. When one says that someone is omnipotent, the meaning is the ability to do everything that is defined (or everything that can be imagined). But the category of ability does not apply to what is undefined, and when one says that God cannot create such a wall and shell, or a circular triangle, one does not mean 'cannot' in the sense of a lack of ability. The very concept of ability is irrelevant with regard to such states or concepts. In more precise translation one can say that the sentence 'God cannot make a circular triangle' means: 'The claim "God can make a circular triangle" is meaningless.' This is not a claim about God and His abilities but about the phrase 'circular triangle.'

An example: the stone that God cannot lift

From here we can also understand why the hackneyed sophism about the stone that God cannot lift (the omnipotence paradox) is based on the same mistake in understanding the laws of logic. People wonder whether God can create a stone that He Himself cannot lift. The claim is that if He can do this, then there is a stone He cannot lift, and therefore He is not omnipotent. And if He cannot do this, then again He is not omnipotent. The conclusion is that the claim 'God is omnipotent' is not true (because it leads to a logical contradiction).

I will present the mistake in this argument on two levels. First, one should note that this is a challenge to the concept of omnipotence itself, irrespective of God. The claim is that this concept is meaningless (its ordinary content leads to a logical contradiction). But if so, then at most one can claim that the sentence 'God is omnipotent' is meaningless, because we are using an empty or contradictory concept. But this is a claim about us, not about Him. It does not mean that God is deficient, only that the concept 'omnipotent' cannot be found in our language.

Second, think of the dispute between the believer and the atheist as a dispute between two people: Reuven the believer holds that God is omnipotent, and Shimon the atheist raises an argument attacking him. He says to Reuven: according to your view, that God is omnipotent, can He create a stone that He Himself cannot lift? Shimon's assumption is that Reuven can answer 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, that is, that Reuven is mistaken. This is what in logic is called a 'dilemma argument.'

But astonishingly, here both answers are incorrect. Not because there is a third possibility (the law of the excluded middle is valid here as well, of course). The reason is that there is no such stone.[8] This is like asking whether God can create a circular triangle. Both the affirmative and the negative answer are irrelevant here. Reuven, who assumes that God is omnipotent, cannot understand the concept 'a stone that God cannot lift,' because in his view it is translated into the concept 'a stone that the omnipotent cannot lift.' Therefore, from his point of view, it is like a circular triangle. Shimon the atheist, of course, can understand this concept because he assumes that God is not omnipotent. In his view there is no contradiction in the concept 'a stone that God cannot lift.' But a logical attack is supposed to proceed from the attacked party's premises (Reuven's) and show that they lead to a contradiction. To prove that God is not omnipotent on the basis of Shimon's prior assumption that He is not omnipotent is simply begging the question. But as we saw, if one proceeds from Reuven's premises, this attack is devoid of meaning and significance. Reuven's answer to Shimon will be: please explain to me the concept 'a stone that the omnipotent cannot lift,' and then I can try to answer you whether God can create such a stone or not. Shimon, of course, will never be able to explain this concept to Reuven. He can explain to him the concept 'a stone that God cannot lift,' but only on his own assumption that God is not omnipotent. Shimon himself cannot understand the concept 'a stone that the omnipotent cannot lift' either. 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 'omnipotence' is simply undefined. In this sense the believer indeed erred when he used it, although this is not a death blow to his faith. He merely needs to correct his language. By contrast, the second answer is prepared to accept the concept of 'omnipotence' 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 the logical contradiction as something that ought to be within the power of an omnipotent God, but this is a mistake. A logical contradiction does not belong to the domain of His powers, not because there is some deficiency in His abilities, but because claims containing logical contradictions are meaningless. As we saw above, the claim 'God cannot create a stone that He Himself cannot lift' is interpreted as: 'The claim "God can create a stone that He Himself cannot lift" is meaningless.'

Think, for example, of the well-known children's story, 'Puss in Boots.' In that story the cat comes to the palace of the terrible sorcerer and wonders aloud whether he can turn himself into a mouse; once the sorcerer does so, the cat devours him and thus gets rid of him. Now in the moral of the story, think about the following question: can God turn Himself into a human being? If so, then one can shoot Him in the head and kill Him. One might perhaps claim that He can turn into a human being, but when He is shot He will not die. But then He has not truly become a human being (for a human being who is shot dies). The correct answer is that, of course, He cannot turn Himself into an ordinary human being. A being whose existence is necessary and who is omnipotent cannot bring about a state in which He Himself ceases to exist. By the same token, He probably also cannot turn Himself into something imperfect or limit His own abilities. But all these 'deficiencies' do not constitute any injury to His omnipotence, because these are logical contradictions. Even God cannot do logical contradictions. When I say, 'God cannot turn into a human being,' what I really mean is: 'The claim "God can turn into a human being" is meaningless.'

An interim conclusion: what does one do with contradictions?

The fundamental question with which we are dealing here is what we should do when we arrive at two rational conclusions that contradict one another. Notice that if proposition X and proposition Y contradict one another, that in itself is not problematic. We simply have to choose one of them and reject the other. At most we are faced with a dilemma about which to choose and which to give up, but that is not a contradiction but a doubt. The problem arises when each of these two propositions seems very rational to us, and we tend to adopt both. In such a case we are in an intellectual quandary. There are thinkers or people who, in such a situation, choose to proclaim a 'unity of opposites' and see in that a solution to the difficulty I have described. They claim that there are things that are above reason or above logic, especially when God is involved.

But as Rudolf Otto wrote in the preface to the English edition of his book, The Holy, 'the unity of opposites' is the lazy person's refuge. When a person stands before a contradiction of the sort I described, if he does not find a solution or is too lazy to look for one, it is very easy for him to proclaim a unity of opposites and see in that some kind of solution. It sounds very deep and mysterious, and seemingly rescues him from an embarrassing situation (a contradiction without a solution). It also allows him to hold on to both rational claims together without giving up either one.

But as we saw above, in fact this is nonsense. When we have before us two claims that contradict one another, we have one of two possibilities: either to adopt one of them and reject the other, or to show that we made a mistake and that there is really no logical contradiction here. Sometimes it is difficult to show that, and therefore some choose instead to proclaim a 'unity of opposites,' thus sparing themselves the logical effort. But if there is no solution, we must reject one of them. One can perhaps attribute this to the shortness of our understanding. But declaring that God is above logic, or that we can hold two contradictory claims without being bound by logic, is nothing but an expression of intellectual laziness, because it is merely a collection of meaningless words.

And what about quantum theory?

This conclusion brings us to quantum theory. Seemingly, there we encounter scientific claims that include a logical contradiction. If some 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 once again arises of attributing contradictions to God, and of course the omnipotence paradox (the paradox of the stone) also returns.

But as we saw above, that cannot be. Contradictory claims are meaningless, and the fact that they are asserted within some scientific framework (such as quantum theory) cannot change that. Let me formulate it differently. Those engaged in quantum theory are not exempt from removing contradictions from their doctrine either. For them too, using a unity of opposites is meaningless nonsense. One must understand that quantum theory itself is based on mathematical tools and observational tools developed on the basis of ordinary logic, that is, a logic in which the law of contradiction and the law of the excluded middle hold. If ordinary logic is not correct, then there is no possibility of using anything we reached by way of reductio ad absurdum. A proof by contradiction shows that assumption X leads to a contradiction, and therefore X must be false. But if there is a third possibility, then such a proof obviously collapses. The same applies to the distributive property of our logic. Moreover, the measuring devices we use in the laboratory, by means of which we discovered quantum theory itself, were developed on the basis of the principles of ordinary logic (for they are not products of quantum theory; rather, they are the basis on which it itself was built). Therefore it is clear that even in quantum theory, once we have proved that X is not true, one may infer from that that 'not X' is true. If this were not the case, the findings of quantum theory would have no meaning at all, since within it one could adopt a proposition and its negation alike.

More generally, classical logic teaches us that if we adopt a system of propositions that contains a logical contradiction, then any conclusion whatsoever can be derived from it. This is why mathematicians work so hard to prove consistency (= absence of contradiction) in every logical and mathematical system. If there really were some contradiction in quantum theory, then one could derive from it any proposition whatever, and in particular a proposition and its negation. We could derive that the electron is a particle with charge 0, or that it is a particle with some other charge, and we could also adopt both claims together. Moreover, it would follow from this that quantum theory is devoid of scientific value and scientific content. If any conclusion can be derived from a scientific theory, then it has no predictive power concerning what will happen in a future experiment. But then it also cannot be subjected to tests of falsification (an experiment that examines whether its predictions are fulfilled or not), and according to the accepted definition in the philosophy of science (following Karl Popper), it would not be a scientific theory.

The conclusion is that, as we saw, the 'laws of logic' are not laws in the ordinary sense. There is no possibility of deviating from them (of 'breaking the law'), and claims that stand in opposition to them are meaningless. The laws of logic are the basis on which our thinking is founded, and therefore one cannot measure the laws of logic in a laboratory, nor can one infer any laws of logic from empirical scientific findings. That very inference would itself be carried out by means of logic (the ordinary kind). Therefore neither thought nor observation can show us that our logic is mistaken. If we return to quantum theory, we must understand that it is a branch of physics, and as such it is a scientific field. By contrast, the laws of logic do not belong to science, but constitute an a priori basis (prior to observation) for scientific thinking and for thought in general. Therefore there is no possibility whatsoever of deriving different laws of logic on the basis of quantum theory.

As a marginal note, I will only remark that more precise descriptions of quantum logic indeed 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 relations among different propositions in quantum theory. Thus, for example, one finds at the beginning of the English Wikipedia entry '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, and nothing more.

The conclusion is that if the observations of quantum theory yield contradictory propositions, that is, findings that stand in opposition to logic, the explanation must be found elsewhere; it must be formulated under that same known and familiar logic. The meaning of this is that if there really are logical pathologies before us in quantum theory, such as a proposition and its negation, then we must take one of two paths: either understand that nevertheless the two claims cannot both be true together, and then adopt one side and reject the other, or show that there is no logical contradiction here. And certainly one cannot rely on quantum theory in order to provide a philosophical basis for claims that contain a contradiction, whether they concern God or not. Such claims are meaningless. Let us now examine several pathologies in quantum theory, and see how this picture is expressed in them.

The double-slit experiment[9]

As Richard Feynman, Nobel laureate in physics and one of the researchers who participated in the nuclear project (at Los Alamos), used to say, the best way to explain quantum theory is by means of the double-slit experiment. This experiment has a very interesting and highly changeable history. It begins already in Newton's time, when a dispute took place among physicists about the nature of light. Some argued that it was composed of particles (like Newton), while others saw it as a wave (the Fresnel-Huygens theory). In 1801, 74 years after Newton's death, Thomas Young performed the first double-slit experiment in order to decide this question. In the experimental setup Young used, there is a clear difference between a wave and a beam of particles, and precisely for that reason it was useful in deciding the dispute about the nature of light. This is exactly why this experimental arrangement would also prove wonderfully useful in the next century for deciding a similar dispute, this time regarding the nature of particles such as the electron (whether they have a wave-like nature or a particulate one).

To understand this, let us begin with the simpler case of a single slit. In the drawing below, a source (in the shape of a cylinder on the right) sends a beam of particles or a wave (the thick dashed line) toward a partition with a slit in it. Behind the partition there is a screen (depicted by a double line), which is in effect a kind of photographic film sensitive to the impact of the wave or the particles.

Figure 1: The experimental setup of a single slit

The graph on the left of the drawing describes the results of the experiment (what one sees on the screen) in the case of a single slit. Its height (on the y-axis) represents the quantity of particles or the intensity of the wave that is received at each point on the screen (each point on the x-axis of the graph represents the corresponding point on the screen).

One can see that the greatest quantity is received exactly opposite the slit, and on both sides of it the quantity decreases progressively (this represents part of the light or the particles that deviated somewhat to the side). This picture is correct both for a beam of particles and for a wave beam. Young's idea was to distinguish between a beam of particles and a wave by means of a similar experiment, except that here one places a partition with two slits in it. To understand this, we need to know something that characterizes wave phenomena, namely interference. When two beams of particles move through space, the total number of particles at each point is the arithmetic sum of the numbers of particles from the two beams at that point. By contrast, when two waves move through space, the total field intensity (the wave) 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 rises and falls as a result of an 'interference' effect between the waves.[10] Thus, for example, it is possible for two waves of high intensity to sum to zero, because one is positive and the other negative. In certain places the intensities of the two waves cancel one another out, and in other places they reinforce one another (in which case the total intensity is greater than the sum of the intensities).

Now think about the experimental setup proposed by Young, which looks similar to what we saw above, except that now the partition has two slits. 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 quantity of particles will be distributed as in the graph of the single slit (see the graph in Figure 1). Thus, in the two-slit experiment, at every given point on the screen the number of particles that reach it will be the sum of the particles that came from the right slit and from the left slit. Therefore the picture on the screen will look like the sum of the pictures from the two single slits, that is, we will obtain a graph with two equal peaks, as depicted in Figure 2:

Figure 2: The double-slit experiment with a beam of particles

By contrast, when the source sends a wave beam, the phenomenon of interference will cause the picture to look completely different, as depicted in Figure 3:[11]

Figure 3: The double-slit experiment with a wave beam

In the wave case depicted in Figure 3,[12] specifically between the two slits (at the place where the particle picture almost vanishes; see Figure 2), the maximal intensity appears. On both sides of the center there are side lobes whose peaks gradually decrease. Thus, the double-slit experiment gives us a sharp distinction between a beam of particles and a wave beam.

In Young's experiment, which as stated was carried out at the beginning of the nineteenth century, the double-slit experiment was performed with a light wave, and the result obtained was unequivocal: a wave pattern was obtained there (Figure 3). Thus the dispute between Newton and Huygens (both long since dead) was decided, and it was established that light is a wave phenomenon.

More than a hundred years later, at the beginning of the twentieth century, the situation began to reverse itself. Various pieces of evidence and arguments then 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 dissertation, proposed seeing the electron as a wave and even proposed a mathematical description of its wave function. At this point a reverse dispute began from that which had taken place between Newton and Huygens: are objects that we had become accustomed to seeing as particles (for example, the electron) actually wave-like in nature? In order to examine this question, double-slit experiments were once again carried out, but this time with an electron beam. The expectation was that the picture would look like Figure 2 above, since apparently electrons are particles. The results were astonishing. It turned out that the picture obtained resembled Figure 3. From this it emerged that electrons too are in fact wave-like in nature. If so, electrons behave like light waves, and the difference between a particle and a wave apparently disappears. It was unclear how a beam of particles could interfere with itself. As I noted, interference is a property of waves and not of particles.

At some stage the hypothesis arose that the interference was the result of the fact that we were dealing with a beam of electrons, and not with a single electron. Different electrons interfere with one another, and from their encounter a kind of interference pattern is formed. The advantage of physics is that it is an empirical science, and therefore, in order to test this proposal, the double-slit experiment was repeated when the electron beam was fired from the source at a very slow and sparse rate. In such a situation, only one electron hit the screen each time, and therefore no interference could be formed between two electrons. Astonishingly, it turned out that a wave pattern was still obtained (as in Figure 3). That is, the single electron too is a wave, since it interferes with itself. The astonishing conclusion is that the electron is not a tiny particle located in some specific place (a sort of little ball), as had been thought until then, but rather a wave spread throughout all space, exactly like light (and we of course still remember Fresnel-Huygens and Young).

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

To test this question, we must examine each time through which of the two slits our electron passes, or perhaps whether it passes through both. So, of course, we return to the laboratory. The experiment was now carried out again, but this time a detector was placed near one of the slits (let us call it slit A). When the electron passes through the slit, the detector picks it up and reports to us that the electron passed through slit A. If the detector reported nothing, the conclusion is that the electron passed through slit B. The double-slit experiment was repeated in the presence of the detector, and here the astonishment was complete. The detectors indeed showed us through which slit the electron passed, but on the screen this time a particle picture was obtained, that is, the graph in Figure 2. The phenomenon of the electron's interference with itself (Figure 3) disappeared. The electron stopped being a wave and returned to behaving like a civilized particle (like a tiny tennis ball). From time to time the detector showed that the electron had passed through slit A, and then there was accumulation opposite that slit; at other times the detector showed that it had passed through slit B, and then the accumulation was opposite slit B. With a beam of particles, the picture of Figure 2 is obtained.

Just to sharpen the point: with tennis balls the situation is of course different. If we perform a double-slit experiment with tennis balls, whether with a detector or without one, the result will always be particulate (Figure 2). The conclusion is that a tennis ball is not a quantum particle. But the electron, at least so long as there is no detector near one of the slits, behaves like a wave. Therefore the electron is a quantum particle. When there is a detector examining it, it returns to behaving like a respectable classical particle (like a tennis ball), but when nobody is looking at it, it really runs wild. Note that even when there is a detector near the slit, the electron is still not completely a tennis-ball-like little sphere. Even if we fire the electron from the source toward slit A, unlike a tennis ball there is some chance that the electron will nevertheless pass through slit B. When there is a detector, the single electron indeed will not pass through both slits, but you cannot know with certainty through which of them it will pass. Only after the detector has measured its position does it begin to behave like an ordinary classical particle (that is, if it passed through slit A it will of course hit the screen opposite that slit and not on the other side). But as long as it has not actually been measured, there is a chance that the detector will discover it at A and also a chance that it will be discovered at B.

Before I continue, I will just remark on the wonders of the historical pendulum, and note that after quantum theory was formulated it became clear that light too is not entirely a wave. There are situations in which it behaves as a collection of particles (photons), and there too it depends on whether one places photon detectors near one of the slits or not. To everyone's surprise, after about two hundred years Newton rose from the dead.

The accepted interpretation

The interpretation of the strange findings of quantum theory remains disputed to this day. According to the accepted interpretation that developed after these experiments (called the 'Copenhagen interpretation,' following 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 principle of complementarity. It later became clear that we have no way to measure the velocity and position of an electron simultaneously, and this is the uncertainty principle, which also falls under the principle of complementarity. This is another expression of the fact that the electron is both a particle (a body that has a definite position) and a wave (a body without a definite position). When one measures its position, one turns it into a particle; and when one does not measure position (but one may measure velocity), then it is a wave.

The conclusion the physicists drew from this confusing picture is that so long as nobody looks at it (no detector is placed), the electron indeed passes through both slits together. This means that the electron is in a state called superposition, which is a sum of pure particle states. It is essentially the sum of two particles, one passing through slit A and the other through slit B. By the way, if we were to place three slits, it would be the sum of three such particles, and so on. More precisely, the electron is of course one particle, but the state of the electron is a sum over two particle states (which for us is already a wave state). But that is only so long as we have not placed a detector. When one places 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 into which of the two particle states it will collapse is a kind of lottery, and quantum theory can even describe for us the probabilities of these two outcomes. They are described by the 'wave function' that describes the electron (or the photon).

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

When the state [A> is a path in which the particle behaves like a point particle passing through slit A, and the probability of obtaining it (if one places a detector near the slit) is defined by   , and the state [B> is a state in which the particle behaves like a point particle passing through slit  B, and the probability of obtaining it (if one places a detector 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 pass either only through slit A (and this will happen with probability ) or only through slit B (and this will happen with probability ).

Two remarks

As a marginal note, two further remarks that are not important for our discussion, but are necessary for completing the picture:

• At large scales, when we are dealing with objects from our ordinary world (a dove, a chair, a person, a bacterium, a living cell, or a ping-pong ball), the quantum effects usually disappear. The reason is that even if our particles are quantum, a large system containing masses of particles (every macroscopic body is such) has a state that is the sum of the states of the particles. A sum of many random results yields a simple average result (this is the 'law of large numbers' in probability). This is the effect called quantum decoherence (or dephasing).
• In the past, proposals were raised (and are quoted to this day) according to which the human measurer affects the results of the experiment. People drew from this far-reaching conclusions about telekinesis (a non-physical influence of the spirit on distant objects and persons), about the nature of reality (Kant's distinction between the thing as it is in itself [the noumenon] and the thing as it appears to us [the phenomenon]), and more. I will not enter here into all these implications, but will only say that today all this is generally regarded as incorrect. The double-slit experiment was carried out when the results measured by the detector were immediately erased, that is, there was no human consciousness that observed them, and nevertheless it turned out that the very placement of the detector caused the collapse of the wave function into a particle state.[13] From here it appears that the source of the weirdness is not necessarily human consciousness but the very act of measurement itself.[14]

The contradiction and its resolution

If we return and focus on the small, quantum scales, then on its face there seems to be a logical contradiction here. A particle is a body with certain properties (for simplicity we will relate to it as a material point located at some defined place), whereas a wave is an entity with different (indeed opposite) properties, and in particular it has no defined position. In quantum theory, an entity of purely wave-like character (such as a photon with a defined wavelength) is spread over the whole world. So what exactly is the electron (and the photon as well) – this or that? It cannot be both.

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

If so, there is no contradiction here, but rather a complexity that is hard to grasp. This is not a particle that passes through both slits, which would be a paradoxical state (logically contradictory), but an entity whose state is described as a superposition between two particle states. There is no contradiction in that at all. Particle and wave are not entities but states of entities. But what exactly is the being whose states these are? What does it look like? What are its true properties? We have no way of knowing or describing that. You see that we have moved from a logical contradiction to a difficulty of description (as in the example of four dimensions mentioned above).

Schrödinger's cat and betrothal that does not permit intercourse

In the well-known thought experiment called 'Schrödinger's cat,' one places a cat in a closed box with a sealed poison vial. The stopper of the vial is controlled by a quantum process, and its state is described by a sum of two simple (classical) states, as in the equation above. But here the state [A> is that the vial is open, and the state B>] is that it is closed. But if the vial is open, the poison spreads in the box and kills the cat, that is, the cat is dead. And if the vial 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 an alive state and a dead state. When we open the box and measure the state of the cat, we will discover a live cat or a dead cat, but not both. But before the opening it was in a state of alive and dead together. As we saw, it is not accurate to say that the cat is both alive and dead, but rather that its quantum state (its wave function) is composed of two classical states: <alive] + <dead]. This is considered a paradox because such a state is even harder to digest. Cats are classical macroscopic creatures, and we thought that at least such creatures we understand. They are either alive or dead. Unlike a photon or an electron, which none of us has ever seen, a cat is an object that seems to us familiar, and it cannot be alive and dead simultaneously.

One can propose different solutions to this paradox, but in all of them we are not left with a logical contradiction. One first possibility is that even with regard to a cat, so long as we have not observed it, we are dealing with an entity different from what we imagine. The picture of the cat familiar to us is the product of 'measurement' (looking with our eyes), and here there is only one of two possible outcomes: alive or dead. The measurement collapses the cat's wave function into one of these two states. But life and death are states of the cat, and therefore a live 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 familiar to us. Such an entity can indeed be 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 body of the cat and thereby brings about its death. But cells or particles in the cat are small entities for which quantum theory is relevant. It may be that they really are in a state of superposition, but that does not mean that the cat as a whole is in a state of superposition. The state of the cat is the sum over the states of its cells and particles, and here one macroscopic result is obtained (this is the decoherence effect I mentioned above), alive or dead. When we measure, we discover which of the two it is. Either way, what matters for our purposes is that there is no logical contradiction here.

There is also a halakhic example that depicts a completely identical state: betrothal that does not permit intercourse (see on this my site, column 303). Think of a situation in which Reuven has two daughters, Rachel and Leah. Now Shimon comes to him and gives him a perutah and tells him that with this perutah he betroths one of his two daughters (without specifying which one). A problem arises here, because if he is married to Rachel, then Leah is his wife's sister, with whom marital relations are forbidden to him as incest. And the same in the reverse direction: if Leah is the one married to him, then Rachel is his wife's sister and forbidden to him. Because there is no way of knowing which of them is his wife, he is forbidden out of doubt to have marital relations with either one. These are betrothals that do not permit intercourse, and there is a dispute whether they take effect or not. In any case he is forbidden to have marital relations with either of them; the question is whether both of them are his wives and he must give both of them a bill of divorce out of doubt, or whether they are not married to him at all.

The commentators discuss the nature of this state. From the Talmud it appears that this is a state of doubt, for we do not know which of the two is his wife. But a closer look reveals that this is not an ordinary state of doubt. In an ordinary doubt there is one correct answer, but we do not know it. Think of a man who sent an agent to betroth for him one of the two daughters. The agent came to the father and betrothed one of them, say Leah, and now the father and the agent have died. No one knows which of the two is his wife. This is a normal state of doubt, because there is one correct answer (God knows that Leah is the one who was betrothed to him), and only for us has doubt arisen because of partial information. Doubt is a lack of information. But in betrothal that does not permit intercourse there is no correct answer at all, not even theoretically. Even God Himself does not know which of the two is betrothed, because neither of them is defined as his wife. One may say that this is a state of vagueness (which is a property of reality itself) and not of doubt (which is a property of the human lack of information).

In our terminology, one can say that betrothal that does not permit intercourse is a state of quantum doubt, since both answers are true. It is not a doubt whether Rachel is his wife or Leah is his wife. Seemingly, both are his wives. But that cannot be, because a man cannot be married to two sisters. More accurately, this should be described 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. It 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 saw in quantum theory, the correct description is that his wife is the sum of two states, and not that both Rachel and Leah are his wives. Exactly as it is not correct to say that the particle passed through both slits together, but that the state of the entity (the electron) is the 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 ascribed simultaneously to the same object. Thus, for example, it is impossible for an object to have both an exact velocity and an exact position at the same time. There is here a reflection of the principle of complementarity, because an exact velocity is attributed to the electron when it is in a wave state, while 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 characteristics, but I will not address that here.

In my article 'Zeno's Arrow and Modern Physics' (Iyyun 46, 1998, p. 425), I discussed the connection between the uncertainty principle and Zeno's paradox of the arrow in flight. I will address it here briefly, 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 Elea (the disciple of Parmenides, from the fifth century BCE) presented several paradoxes that challenge the concepts of motion and time, and he wished to argue that motion is nothing but 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 here a popular formulation of it. If we observe a flying arrow, then at every (discrete) moment at which we observe it, we find that it is standing in a different place. So at which exact moment does it change its position? How does it pass from place to place? After all, if it is standing still, then it is not moving at any moment; how is it possible that nevertheless at every moment we see it in a different place?

First, I will ask whether this is a logical contradiction or a physical difficulty. On its face, it appears to be a logical contradiction. The particle is both standing still and moving. Its velocity at that moment is both 0 and non-zero. If so, this is a difficulty with which one cannot live, and one must seek a solution. The solutions proposed to this difficulty rely, among other things, on infinitesimal calculus (the axis of time is continuous; there are not really isolated points of time) and on the uncertainty principle in quantum theory (according to which one cannot speak of the position and velocity of the arrow simultaneously). In the above-mentioned article I explained that neither of these can serve as a solution. Here I will only say that this is really the same claim I presented above: there is no possibility of living with a logical contradiction. Infinitesimal calculus offers us a description of reality free of contradiction, but as I explained in that article, this is comparable to adopting a language in which it would be forbidden to express the paradox. That is not a real solution to it.

The uncertainty principle seemingly does offer a solution. We saw that according to it one cannot ascribe to any object both exact velocity and exact position simultaneously. If so, if the arrow is located in a defined place, it has no velocity, and vice versa. This pulls the rug out from under 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 object, and theoretically there is no obstacle to talking about its position at some point and about the velocity it has (velocity is the rate of change of place). What does it mean that it has no velocity? Does it not change its position? Our eyes see that it does. And what about position? All the more so since an arrow is a macroscopic object, and therefore there is no obstacle to ascribing to it position and velocity simultaneously. We saw that quantum theory is not relevant on these scales. As I noted above, the fact that the contradiction appears within a scientific discussion is not a solution. A scientific theory too cannot contain contradictions. If there is a contradiction in quantum theory, that is only one more reason to seek for it a fitting and rational resolution. In that article I proposed reversing the direction of the inquiry: to seek a rational solution to the paradox, and by means of it to explain the oddity of the uncertainty principle in quantum theory.

The root of the problem in this paradox lies in a failure to distinguish between two concepts that seem similar: standing still and being located. When one says that the body is standing at place X at time T, this is not equivalent to the statement that it is located at place X at time T. In other words, a body can be located at some place at a given time with velocity 0 (and then we say that it is 'standing' in that place), or be located there with some other velocity (and then we say that it is 'located' there). When one says that the body is located somewhere, this does not necessarily mean that it is standing still. It can be located there in motion. When one says that the body is standing in place X at time T, the meaning is that it is located there with velocity 0. Now you can see that the paradox simply evaporates. To the question at what moment the body moves, the answer is: at every single moment. There is no contradiction between the statement that at every moment it is located (but not standing still) in a different place, and the statement that in those very same moments it is also moving.

What confuses us here is that one cannot say that at every moment the body changes its place. A change of place takes some amount of time, and cannot be carried out in a single discrete moment of time. In other words, the claim that a body is in two places at the same moment is logically contradictory and not merely physically impossible. Here we are not speaking of infinite velocity, but of two contradictory claims (like the particle that passes through both slits). Thus, a body cannot change its place in one single discrete moment of time, but it can indeed be in motion at a discrete moment of time. What confuses us in this matter is that velocity (being in motion) is not change of place. Velocity is a quantity that exists even in a single discrete moment, and it is the potential for a change of place. But a change of place requires an interval of time, whereas velocity exists for a body even in a single discrete moment of time. The meaning of the statement that some body has a velocity at a given moment is that it has the potential for a change of place, and therefore it will presumably change its place in the next moment.[15] In that article I explained this problematic point in terms of the concept of the derivative from infinitesimal calculus, which is used to define velocity in mechanics. There velocity is defined through differences of position divided by the time it takes to traverse them, and when the velocity is not constant one must take very small intervals of time (that approach zero). But as I explained there, this is only a computational (operational) definition of velocity (that is, this is the way to calculate velocity), and not a definition of the concept of velocity itself. The conclusion is that the paradox of the arrow in flight is the product of conceptual confusion and nothing more.

Later in the article I explained, in light of this distinction, the uncertainty principle in quantum theory. The reason we have no possibility of ascribing to the same body velocity and position simultaneously is that measuring velocity requires a different kind of observation from measuring position. Measuring position is essentially taking a photograph of the body at each moment (with a camera whose exposure time is one discrete moment of time). When one observes the world with such a static camera, there is no possibility whatsoever of seeing motion or velocities (as we saw in the paradox of the arrow). A camera is blind to motion. By contrast, measuring velocity requires me to film the body,[16] and this necessarily requires following it over an interval of time. That cannot be done over a discrete moment of time. The movie camera is blind to positions (because one sees only the motions – just the opposite of a camera). In any case, these are two ways of looking that exclude one another, and therefore one cannot fix velocity and position simultaneously: either I observe the body (and the world) through a camera or I observe it through a movie camera.[17]

I showed there that the meaning of the division between the two points of view I described (the camera and the movie camera) is what is called in quantum theory the 'position picture' and the 'momentum picture' (velocity). In the position picture one describes all physical reality in terms of the spatial position of the particles at every moment. There is no possibility there of speaking about velocities. By contrast, in the momentum picture one describes all physical reality only in terms of the velocities of the particles at every moment, and then there is no possibility of speaking about positions. This is precisely parallel to our camera and movie camera. In quantum theory too these are two pictures that exclude one another, and one cannot describe the body in both simultaneously. A person must choose a point of view or picture. It is indeed possible to pass from one to the other, but not to use both simultaneously.

A distinction between two perspectives is not a contradiction

The meaning of this is that just as we saw with respect to the principle of complementarity, so too with respect to the uncertainty principle. Here too we are not dealing with a contradiction, but with a complex picture. In the background there are two different perspectives that exclude one another, but there are not here two contradictory truths simultaneously. Therefore the uncertainty principle does not express a logical contradiction, but at most a state that is difficult for us to imagine. The explanation I proposed here in light of the paradox of the arrow even brings it a bit closer to our reason and imagination. In the final analysis, the conclusion is that quantum theory does not contain contradictions or deviations from accepted logic, and consequently it does not create any possibility of making contradictory claims or holding contradictory beliefs.

I will now bring three applications that sharpen the difference between holding a contradiction and holding two different perspectives. The first is a well-known parable about a dispute between two people describing an elephant. Reuven claims that it is an animal with two legs far from one another and with one eye. Shimon claims against him that it is a creature with two legs close to one another and with two eyes. Seemingly these are two contradictory descriptions, but in fact they are views of the same elephant from two different perspectives. Reuven sees it from the side, and therefore its legs are far from one another and it has only one eye (that is what one sees from the side). Shimon, by contrast, sees it from the front, and therefore he sees two adjacent legs (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 points of view. Each description is partial, and the full description is the combination of the two.

A second application may be seen in the saying of the Sages, 'both these and those are the words of the living God.' In its original context this was said regarding disputes between Beit Shammai and Beit Hillel (Eruvin 13b), and in the sugya in Gittin 6b regarding the concubine at Gibeah. As is known, the commentators apply this to disputes throughout the Talmud, and perhaps also afterward. The perplexity over this statement is well known, and the clearest source that presents it is the Ritva in his novellae to Eruvin there, who writes:

'Both these and those are the words of the living God.' The sages of France of blessed memory asked: how can both be the words of the living God, when one forbids and the other permits? They answered that when Moses ascended on high to receive the Torah, he was shown for every matter forty-nine arguments for prohibition and forty-nine arguments for permission. He asked the Holy One, blessed be He, about this, and He said that the matter would be entrusted to the sages of Israel in every generation, and the ruling would be decided according to them. This is correct on the homiletical level, and in truth there is a reason and a secret in the matter.

Here too, on its face, we seem to be dealing with a logical contradiction. But in truth this difficulty disappears if we look at the parallel sugya, the only place in the Talmud where an explanation of this rule is also brought, in Gittin 6b:

As it is written, 'and his concubine played the harlot against him.' Rabbi Evyatar said: he found a fly with her. Rabbi Yonatan said: he found a hair with her. Rabbi Evyatar later met Elijah and said to him, 'What is the Holy One, blessed be He, doing?' He replied, 'He is occupied with the story of the concubine at Gibeah.' 'And what is He saying?' He said to him: 'My son Evyatar says thus, and My son Yonatan says thus.' He said to him: 'Heaven forbid! Can there be doubt before Heaven?' He replied: 'Both these and those are the words of the living God: he found a fly and was not upset; he found a hair and was upset.'

The tannaim disagree over the question of what exactly that man found with the concubine. What aroused his anger? One says he found a hair and the other says he found a fly. Elijah the prophet reveals that God Himself advanced both possibilities, but this is not doubt; it is a double determination: he found both a fly and a hair, and only both together created his displeasure.[18] Each of the sides grasped only a partial picture, and the full picture is the combination of both. True, this is an aggadic sugya, but from it it appears that this is the explanation of the saying 'both these and those are the words of the living God' in its halakhic contexts as well. And indeed, higher on that same page in the sugya in Eruvin, the possibility is brought of finding many valid reasons against an accepted legal ruling, such as 150 reasons to declare the creeping creature pure, although the Torah itself rules that it is impure. The explanation is that there are indeed 150 reasons to declare it pure and there are also 150 reasons to declare it impure. In the bottom line it is impure, because the reasons for impurity outweigh the reasons for purity. But that does not mean that the reasons for purity are not correct. The reasons in every direction are correct, and the halakhic ruling is the weighted result of all of them. Again we have here a complex view from different perspectives, each of which is only partial. The apparent contradiction (the creeping creature is both impure and pure) is not really a contradiction but a complexity created by the combination of opposing perspectives.[19]

The third application appears in my article 'What Is "Halut"? Jewish Law, Logic, and Cleaving to God,' Tzohar 2, 2000. There I brought the words of Rabbi Shimon Shkop regarding a woman who is divorced conditionally. His claim is that until fulfillment or non-fulfillment of the condition, the woman's state is that of both a married woman and a divorced woman together. This is in fact a state of quantum superposition, as I described above. If she fulfills the condition, the wave function 'collapses' to the state of a divorced woman, and if not, it 'collapses' to the state of a married woman. I explain there how a combination of two contradictory states, a married woman and a divorced woman, can exist together. Seemingly this is impossible, for if she is a married woman she is not divorced, and vice versa. My claim there is that there are two legal statuses resting upon the woman, the status of a married woman and the status of a divorced woman. The legal state created is a composition of both, and in such a state there is no contradiction whatsoever. I compared this to a situation in which there is salt and sugar in a dish, which is of course entirely possible. It is impossible for the dish to be both completely salty and completely sweet. The contradiction exists in the implications, but the very existence of both aspects is not a contradiction. 'Halut' is a kind of entity, and therefore the existence of two contradictory legal statuses upon the same woman at the same time presents no problem (like salt and sugar in the same dish). The halakhic implications, of course, cannot be contradictory, and indeed they always incline to one side (I explain there to which of the two).

In another formulation, one can say that in the state of this woman there is one reason to treat her as divorced and another reason to see her as a married woman. As we saw above, the existence of reasons that lead in opposite directions is not a state of contradiction. Here too we are not dealing with a contradiction but with two aspects (with contradictory characteristics), both of which can exist in parallel.

Implications and examples: several kinds of contradictions

The meaning of the picture I have described here is that there is no place at all for contradictory claims, not in faith, not in quantum theory, and not anywhere at all. When we encounter a contradiction, we must choose one of the sides and abandon the other, or alternatively show that this is not a contradiction – and only then can one hold both. Reliance on another logic (quantum logic, for example), or the use of expressions such as 'the unity of opposites,' is not helpful because these expressions say nothing. If these are logical opposites, there is no place to unite them; and if not, then that must be shown, thereby dissolving the apparent contradiction. People who do not succeed in showing that there is no logical contradiction between the claims, and yet still wish to hold both, apparently prefer – probably out of intellectual laziness – to speak in the empty terms mentioned above.

One reason for this confusion is the hasty use many make of the concept 'contradiction.' In the overwhelming majority of the cases in which people speak of the 'unity of opposites,' this is not a logical contradiction but at most something that is difficult to understand. In cases where we have no 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 a difficulty, or a 'softer contradiction' (physically impossible, or merely something hard to imagine, such as four dimensions, and the like).

I cannot enter here into greater detail regarding the distinction between the types of contradictions (logical impossibilities as opposed to physical impossibilities or merely something hard to imagine), but to complete the picture I will only add that in my article 'Is Belief in Logical Contradictions Possible?' (available on my site), I define another kind of contradiction, using Kantian terminology: there are situations in which the contradiction between two propositions is neither logical (analytic) nor observational-scientific, but a priori. The contradiction arises because of an a priori principle (such as the principle of causality), but that does not mean that the combination of the two propositions is devoid of meaning. An analytic contradiction is empty verbiage, as I explained in this article, but an a priori contradiction has meaning in itself, and therefore it should not be ruled out, at least with respect to God. To conclude the article, I will now bring several examples of such hasty uses of the concept 'contradiction,' and of the significance of the picture I have proposed here.

The first example is the words of the Sages about the red heifer. The Sages use the law of the red heifer in order to speak about contradictions that we cannot understand. Thus, for example, in Pesiqta de-Rav Kahana 4, in the proem to Parashat Parah, we find (see also Numbers Rabbah 19:1):

There we learned: A bright spot the size of a bean renders one impure; if it spread over the whole of him, he is pure. Who did this, who commanded this, who decreed this, if not the Unique One of the world?

There we learned: If a woman's fetus died within her womb and the midwife put out her hand and touched it, the midwife is impure with seven-day impurity, while the woman is pure until the fetus emerges.

A corpse in the house is pure; once it comes out of it, it is impure. Who did this, who commanded this, who decreed this, if not the One, the Unique One of the world?

And there too we learned: All who are involved with the red heifer from beginning to end render garments impure, while it itself purifies the impure. So the Holy One, blessed be He, said: I have enacted a statute and decreed a decree, and you are not permitted to transgress My decree – 'This is the statute of the Torah that the Lord has commanded, saying' (Numbers 19:2).

All these are examples of contradictions (some of them are brought in another midrash as Korah's scoffing questions to Moses our teacher, together with several additional examples. See, for example, Jerusalem Talmud Sanhedrin chapter 10). In the impurity of leprosy, if a person's flesh has a bright spot the size of a bean, it renders him impure, but if it spread over the whole body he is pure. The second example is a midwife who becomes impure by touching the dead fetus, whereas the woman carrying it in her womb is pure until it emerges. The third example is that when there is a corpse in the house, the house is pure, and it becomes impure when the corpse leaves it. The red heifer is a fourth example, because it renders impure the pure people who are occupied with it, while its whole purpose is to purify the impure. All these examples appear like logical contradictions, and therefore the midrash attributes them to God, the unique One of the world. Seemingly this is truly an assertion of the 'unity of opposites,' which assumes that God is not subject to the constraints of logic.

But on closer inspection one discovers that we are not dealing here with a logical contradiction. Is there really a contradiction between the claim that the red heifer renders the pure impure and the claim that it purifies the impure? A contradiction would exist only if we were to say that it both purifies the impure and does not purify them. To purify the impure and render the pure impure is something not understood, but not a logical contradiction. And so it is with respect to all the other examples brought in the above midrash. Our treatment of such cases as if they were contradictions is hasty. If these really were logical contradictions, then we could not adopt both sides of the contradiction simultaneously. One of them would be mistaken. When the matter is not understood, but is not contradictory, there is no obstacle to holding both sides of the 'contradiction' together. One can then perhaps even try to understand the idea, but understanding here is not a condition for adopting both of the laws in question. One may say that in such soft contradictions, they leave us with a question, but not with a difficulty or contradiction.

The Polish logician Jan Łukasiewicz developed a three-valued logic, that is, a logic in which every proposition can have three truth values (true, false, and something third). He showed that one can present a consistent logic that is not binary (that is, one that does not satisfy the law of the excluded middle), and there were those who saw in this a challenge to classical (binary) logic. Many writers, when discussing contradictions in a person's doctrine, use three-valued logic to explain them. As though there were here a justification for a contradictory doctrine. There were even those who wanted to use this logic in order to offer a solution to paradoxes. According to their view, a paradoxical proposition is neither true nor false, but this is not supposed to trouble us because one can attach to it the third truth value (T – true, F – false, P – paradox).

But as I explained above, such a logic cannot be an explanation of anything. Even Łukasiewicz himself, when he built his logic, used ordinary logic. His logic does not replace ordinary logic; rather, it is a formal structure that can serve us as a useful tool in certain contexts (as we saw above regarding quantum logic). Thus, for example, some also relate to probability as a different kind of logic, because in probability theory propositions can receive a continuum of values (a number between 0 and 1). Does probability undermine our ordinary logic? Certainly not. It is a tool for dealing with states of uncertainty, but not a change in the principles of logic. Therefore it is clear that reliance on three-valued logic cannot provide an explanation for a contradictory doctrine or for paradoxes.

In Feuer's book (above, note 1), he cites Łukasiewicz himself, 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 there in Feuer how several ideas were based on and hung from this logic, though it had done no wrong. Contrary to the words of its creators and also of the thinkers who came after them, such a logic cannot serve as the basis for any idea on earth, except perhaps as a source of inspiration for thinking outside the box (but not the logical box, of course). There is no possibility whatsoever of deviating from accepted logic, neither from the law of contradiction nor from the law of the excluded middle. Thus, for example, even in the development of Łukasiewicz's logic one could use proofs by contradiction (which rely on the law of the excluded middle).

It is no wonder that writers such as Benjamin Ish-Shalom, in his book Rabbi Kook: Between Rationalism and Mysticism (above, note 5), propose to explain in this way several of Rav Kook's basic ideas, which appear to us contradictory. In several places (see note 71 to the first chapter and note 133 to the third chapter) he connects Rav Kook's positions to ideas of the unity of opposites on the logical plane, and even mentions in this context Łukasiewicz's three-valued logic. The same is true of Avi Sagi, in his book These and Those (Hillel ben Hayyim Library, Hakibbutz Hameuchad, 1996; see there chapter 7 of the third part), who relates to the Talmudic statement 'both these and those are the words of the living God' as an expression of a unity of opposites which, according to Rabbi Tzadok ha-Kohen of Lublin, is not 'bound by the laws of contradiction' as human thought is. The same applies to philosophical and kabbalistic issues such as the contrast between an image of 'surrounding all worlds' (transcendence) and an image of 'filling all worlds' (immanence).

Meir Munitz, in his article (above, note 5), opens with citations from the writings of Rav Kook that speak of the unity of opposites in divinity, in contrast to the law of contradiction in human thought. He also brings there, from Benjamin Ish-Shalom's above-mentioned book, four ways of understanding the contradictions in Rav Kook's doctrine, the fourth being the boldest of all, which speaks about accepting a unity of opposites (he mixes opposites in nature, in thought, and in logic) without resolving the paradox. His claim is that a conception of divinity obligates us to enter logical contradictions, in matters of freedom and necessity, finitude and infinity, perfection and becoming perfect, and the like. As part of his 'logical' explanation he cites Hugo Bergmann's book Introduction to Logic – The Theoretical Science of Order (Bialik Institute, Jerusalem, 1953), which raises several arguments against the law of the excluded middle and the law of contradiction. I cannot enter here into a critical discussion of those arguments (some of them I addressed here, such as the paradox of the arrow), but a priori it is clear that none of his claims there can hold water. They themselves are asserted on the basis of our classical binary logic, and what is said outside it is simply meaningless.

In the overwhelming majority of these places, there is not really a logical contradiction at all, and therefore instead of relying on vague statements about 'the unity of opposites' or on three-valued logic, it would have been better to seek an explanation and show that this is not a logical contradiction. Without such an explanation, then indeed we are dealing with a logical contradiction (this is the case, for example, with the problem of perfection and becoming perfect; see on my site, columns 170, 278, 519, and others). In such a situation there is no point whatsoever in waving around concepts such as 'the unity of opposites' and/or different logics. One horn of the dilemma is wrong and must be abandoned. But if we have found an explanation, then there is no need for the unity of opposites or for a change in the principles of logic, because there is no contradiction here that needs resolving.

I would add that in many cases the problem is interpretive. Some author speaks about 'the unity of opposites,' but does not mean to assert a claim on the logical plane (as though there were a different logic), but only to use this metaphor in order 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, one must seek a solution and an explanation, and show that this is not a contradiction. In such cases, 'the unity of opposites' is only a metaphor. At the beginning of his article, Munitz brings six such citations (see also the sources in his note 2) from the writings of Rav Kook. This can be seen in each of them. Here I will take as an example the quotation from Rav Kook's Arpelei Tohar, p. 95, where he writes:

One should not be alarmed by a conjunction of great opposites, as is commonly thought, for everything that appears to the many as divided and opposed is so only because of the smallness of their intellect and the narrowness of their perspective, since they see only a very tiny part of the supreme perfection. And even that part is in a very distorted form. But people of clear understanding let their thought spread to different places and wide expanses, grasp the treasures of good in every place, and unite everything together in a complete unity.

Is it really necessary to interpret this as a logical thesis? To the same extent I could explain that his intention is to point to the limited capacity of certain people who, because of the narrowness of their perspective, fail to understand that there is no contradiction here – unlike the 'people of clarified understanding,' who do find an explanation and resolve the contradiction.

It is important to understand that such an interpretation of his words differs essentially from the logical interpretation proposed by Munitz, because according to the logical interpretation there is no point in seeking an explanation, since no explanation is accessible to us human beings, fashioned as we are by logic. Only God is not subject to the constraints of logic. According to this interpretation, it is not clear who the 'people of clarified understanding' of whom Rav Kook speaks are. After all, there are no human beings who can rise above logic like God Himself. Therefore the more plausible interpretation of Rav Kook's words is precisely the one I have proposed: that he intends here to spur us to seek a solution, and not to suffice with a superficial glance, declare a unity of opposites, and then sit quietly. If we encounter a contradiction, we must seek an explanation and show that there is no genuine contradiction here (and thereby be among the 'people of clarified understanding').

At this point I could go through all the apparent contradictions mentioned in the above sources that were 'explained' by means of different logics or hung 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, almost all of them, are only apparent contradictions, and for them there is no need for a unity of opposites. I did this above with regard to quantum theory, which is one of the more difficult cases, and also with regard to the Talmudic dictum 'both these and those are the words of the living God.' From these two cases one can learn to the rest of the easier examples. But for our purposes this is enough, and this is not the place to enter the rest of the examples in detail.

[1] See, for example, Lewis S. Feuer's book, Einstein and His Contemporaries, translated by Gad Levi, Afakim Library, Am Oved, Tel Aviv 1979, in the chapter 'The Logical Revolution Against Determinism,' pp. 174-181.

[2] In the formula above, the symbol * represents 'and,' and the symbol U represents 'or.'

[3] See on this matter the book by Israel Netanel Rubin, 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. The matter has also been discussed in several places on my site, Responsa and Articles (see, for example, columns 549-550 and many more).

[4] These words are almost copied in Rashba, Responsa, part I, siman 9 and siman 418, and likewise there in part IV, siman 234.

[5] On this distinction, see Judith Ronen's article, 'Everything Is Foreseen and Permission Is Given,' in the anthology Between Religion and Morality, Daniel Statman and Avi Sagi (eds.), Bar-Ilan University, 1994, pp. 35-43. The matter was also discussed in the fourth book in the series Talmudic Logic, The Logic of Time in the Talmud (p. 50 and onward).

[6] See, for example, Meir Munitz's article, 'The Logical Foundation of the Unity of Opposites in the Thought of Rav Kook,' Alon Shevut 143-144, 1995. You can also find discussions in Benjamin Ish-Shalom's book, Rabbi Kook: Between Rationalism and Mysticism, Am Oved, second printing, 1990 (a second edition also appeared, Resling 2019). Munitz proposes in his article a 'logical foundation' for the unity of opposites (in my eyes this is an oxymoron), but Ish-Shalom too hangs this approach on the three-valued logic of the Polish logician Łukasiewicz. I will comment on this below.

[7] In analytic philosophy there is a dispute over this claim, but only because the term 'meaning' is not agreed upon. For our purposes here that is not important.

[8] Analytic philosophers ask whether the current king of France is bald. If we examine the set of bald people, we will not find him there, but even if we examine the set of hairy people he will not be found there. And that is because France has no current king. Use of the law of the excluded middle with respect to a non-existent object is misleading.

[9] See about this on my site in column 302.

[10] The waves combine by summing the values of the function that describes the wave, but the intensities are the squares of the function values. 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 composed of both (the square of the sum of the function values).

[11] Here and below I present a very simple and schematic picture. The actual results of the double-slit experiment are more complex. There are many lobes between the slits, and the distances between them and their 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 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, depending on the wavelength of the wave beam. I do not need those details here, so as not to complicate the description.

[13] See, for example, here: R. H. S. 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] Admittedly, according to this it is not clear what the definition of measurement is at all, since the detector in itself is just a physical object. Without a measurer there is no measurement. As far as I know, this riddle still has no agreed-upon solution to this day.

[15] Alternatively, it will strike a wall that does not allow it to move, and then its energy will emerge in another form, such as the dispersal of heat into the surroundings.

[16] I explained there that this is an ideal movie camera. Our movie cameras construct the film from successive still photographs, that is, this is a constant use of a camera. For our purposes, that is a camera-like way of looking, and in that mode one gets classical physics, in which there is no uncertainty between position and velocity.

[17] I cannot enter here in detail into the question whether this is our limitation (we cannot determine position and velocity simultaneously), that is, an epistemic claim, or whether this is an ontological principle, that is, a principle that deals with reality itself. This is an old dispute among researchers of quantum theory, and today the tendency is to see it as an ontological principle. The explanation I gave here seems at first glance like an epistemic principle, but that is by no means necessary. The distinction between a camera and a movie camera is not rooted in our limitation, but points to something about the nature of these quantities (position and velocity) themselves.

[18] That is the plain sense, to my understanding, and not that the hair alone caused his displeasure.

[19] See on this in columns 248-247 on my site, and in my article 'Is Jewish Law Pluralistic?', HaMa'ayan, 2007.

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