Gate Twelve: The Synthetic A Priori in a New Sense

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This is an AI-generated English translation of a chapter from the book Two Wagons and a Hot Air Balloon (שתי עגלות וכדור פורח) by Rabbi Michael Avraham. Translated by OpenAI’s GPT-5.4 model with high reasoning effort. Read the original Hebrew (PDF).

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From the book Two Wagons and a Hot Air Balloon by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

The Synthetic A Priori in a New Sense¹

This gate contains six chapters:

Chapter One: Living in Paradox
Chapter Two: The Meaning of Negation: Between the Logical and the Analytic
Chapter Three: The Logical-Philosophical Significance of the Distinction Between the Logical and the Analytic
Chapter Four: Torah Perspectives
Chapter Five: Human Thought and Logic
Chapter Six: Back to the Synthetic A Priori

Introduction

The Debate over Analytic Thinking

Up to this point, we have presented analytic thinking as a form of thought accepted even by those who hold the synthetic (hearing-based) position. The difference between these two positions lay only in the question whether this is the only way to justify various claims, or whether, in addition to this mode of thought, one also recognizes the synthetic mode as a legitimate form of justification and cognition.

Accordingly, logical consistency was presented as a universal necessary condition for accepting propositions or theories, though not necessarily as a sufficient condition. In the synthetic approach, testing for consistency is not the only mode of thought; therefore it is a necessary condition for certainty, but not a sufficient one. There may be logically valid arguments whose conclusions still will not be accepted as true, because their premises are not true, and proponents of the synthetic position dispute even fundamental premises. Conversely, there may be arguments that are not analytic—meaning, in the usual sense, not purely logical—whose conclusions are nevertheless acceptable.² In any event, even for advocates of the synthetic approach, it is impossible for logical contradictions to be accepted as truths.

Thus, according to the picture presented so far, logical inconsistency is a sufficient condition for rejecting an argument, but not a necessary one; there are other ways to reject it as well. In this gate I would like to raise, cautiously, considerations that support the claim that some principles of analytic thought do not even constitute a sufficient condition for rejecting arguments. In other words, I will raise the possibility of accepting an argument that, in certain respects, stands in opposition to them. The reader is invited to formulate his own position regarding these proposals in light of what will be presented here.

In other words, in this gate I will point to an additional debate. Beyond the debate over the legitimacy of using synthetic thought, there is also a debate over the very classification itself: what counts as analytic thought and what counts as synthetic thought. We will try here to identify the difference between those who hold an analytic position and those who hold a synthetic one with respect to the very definition of a given mode of thought as analytic.

A Positive Characterization of the Synthetic Position

In the previous gate we began to characterize the synthetic position as an alternative to the analytic position. We already pointed out that there is an inherent vagueness in the synthetic position, since its essential component—synthetic thought—is, by its nature, not a formal mode of thought, and therefore it cannot be defined and characterized with precision.

One implication of the proposals that will arise in the present gate is a certain advance toward a more precise characterization, and one that is positive in nature, of synthetic thought—and through it, of the synthetic position as well. We will attempt here to circumvent the vagueness inherent in syntheticity by dealing with a domain that, on the one hand, still belongs to the logical, and can therefore be treated more precisely, and yet, on the other hand, as we shall see, no longer lies within the analytic. That is, the domain we will attempt to characterize is synthetic—non-analytic—yet still logical.

Back to the Synthetic A Priori

In Kantian terms, one could say that the domain mentioned above is a priori and not analytic—that is, synthetic a priori. This formulation implies that there is here a new possibility of distinguishing between the a priori and the analytic, that is, of constructing synthetic a priori propositions—a distinction Kant attempted to make but, as we saw, unsuccessfully.

It is therefore no surprise that this gate brings us back to the synthetic a priori. The argument presented here will uncover a family of synthetic a priori propositions without recourse to transcendental methods, and therefore they will bear a meaning different from the Kantian one.

For this reason, there seems to be here a genuine alternative to the analytic and the transcendental, which many regard as the only two possibilities for justifying synthetic a priori propositions (see Gate Seven). This, then, completes the presentation of the synthetic alternative as the only possible form for acquiring knowledge of the world—synthetic knowledge—even though the tools are a priori and do not employ Kantian methods. This is the essence of the synthetic alternative that this book seeks to present.

Chapter One: Living in Paradox

Introduction

Religious faith is often associated with thinking that is not rational. The symbol of this widespread association is Tertullian’s well-known saying:[^49] “I believe because it is absurd.” A common approach, especially in religious existentialist philosophy—Kierkegaard and his followers—and particularly in the Christian world, expands this relation and says that religious life is life in paradox.³ Here there is already a link between religious faith and anti-reason, with contradiction, and not merely with non-rationality.

In another context, we encounter the paradoxical in the following passage from Goethe’s Poetry and Truth—a work that dealt extensively with demons and demonization—a passage that describes the encounter with the demonic:[^51]

…It is something that reveals itself only in contradictions, and therefore cannot be grasped in a concept, much less expressed in words. It is not divine, for it behaves without reason; nor human, being devoid of understanding; nor satanic, for its action is beneficent; but neither is it angelic, for one frequently notices in it a kind of malicious delight… It seems that all the limitations that confine us do not exist for it… It seems that only in the impossible does it feel at home, and it rejects with contempt everything that is possible.

A sharper and more explicit reference to logical paradoxicality in religious cognition is found in Nicholas of Cusa, who deals extensively and directly with this point. A characteristic passage of this approach appears in his book The Coincidence of Opposites:[^52]

Thus I saw how great the need is that I enter into darkness in order to receive the union of opposites beyond the capacity of reason to grasp, and there seek the truth, where I encounter the impossible [emphasis mine].

And beyond that, beyond even the highest summit of the intellect, when I have already attained what is unknown to every other intellect, in the place where every judgment tells me that it is furthest from the truth—there, my God, You are.

It should be noted that the first paragraph appears to deal with anti-logical planes, whereas the second seems more concerned with non-logical ones. I will note briefly that it is surely no accident that the sources cited here—Tertullian, Kierkegaard, Goethe, and Cusanus—are all Christian. In Jewish thought such a conception is not widespread, though in recent generations it has penetrated there as well.⁴ Even in Christianity, the intention is usually not paradoxes in their logical sense, but rather the numinous—the sublime, the exalted—and that which cannot be grasped in human concepts.⁵

The question we want to address here is whether one can indeed live in paradox. In the terms of the quotation above from Cusanus, which certainly deals with logical paradoxicality, we may formulate the question as follows: Is a life possible that is not merely “beyond the capacity of reason to grasp” (non-logical), but also constitutes an “encounter with the impossible” (anti-logical)?

The believer is also a human being, and as such possesses human thought subject to accepted logic. Human beings, apparently, cannot believe in paradoxes. More than that, some claim that paradoxical propositions—in the logical sense—have no meaning at all, and if so, it is obvious that one cannot believe them. At most one can utter them, but that movement of the lips cannot be accompanied by cognitive processes such as understanding and thinking, or by any layer of meaning whatsoever—at least not at the level of the paradoxical proposition as a whole. If someone says he believes in the existence of a round triangle, is he really asserting anything? Apparently we must adopt, at least in such contexts, Wittgenstein’s recommendation at the end of the Tractatus and simply remain silent.

The Problem of Knowledge and Choice

In order to make the discussion that follows more concrete, I will accompany it with reference to one of the oldest and best-known problems in religious philosophy: the problem of divine foreknowledge and free will (hereafter: “knowledge and choice”). This example will continue to accompany the discussion later on as well, but it should be remembered that it serves here only as an illustration. Our current concern is not to resolve this specific issue, but to clarify the very possibility of a unity of opposites.

Much ink has already been spilled over the problem of knowledge and choice, and many claim that it is only a pseudo-problem. In the following note I will attempt to survey briefly the main responses to this problem.

Note 28: The Problem of Knowledge and Choice

In the context of the problem of knowledge and choice, there is more than one basic problem, and one can point to at least two distinct questions:

How does God know the future before it happens—at least with respect to human actions, which are not deterministically derived from the present state, since human beings possess free choice? In such a situation, the information, apparently, does not yet exist before the act is performed.
Granted that He does know it, how can human beings nevertheless possess free choice? Apparently, the existence of free choice means that both possibilities are open before the chooser. If a person chooses the opposite of what was known to God, it would turn out, so to speak, that God did not know correctly—something that is obviously absurd in the extreme.

The various answers that have been proposed to the problem often mix together different aspects of these two questions, and frequently do not answer them in any relevant way. A concise summary of the main types of answers is as follows.

There are three components to the problem: knowledge, choice, and time. The problem arises from the fact that the knowledge exists before the choice. Knowledge after the choice raises no difficulty. Maimonides’ answer in Mishneh Torah, Laws of Repentance 5:5, is that God’s knowledge is not like our knowledge. This approach concerns the concept of knowledge. Rabbi Hasdai Crescas, in Light of the Lord, answers that in truth we have no free choice at all, only the illusion of choice.⁶ This is a line of thought that answers the problem by altering the concept of choice. In addition, many ground the answer in the claim that God is not subject to the rule of time. This changes the conception of the third component of the problem, time.

Each of these answers is problematic, though for different reasons. The answer that grounds God’s ability to know a future act in His not being subject to time seems to answer the first question—how God gains knowledge of the future—but not the second question: how we can still have choice, given that He indeed possesses this information. Crescas’s answer in effect claims that choice exists only in one’s attitude toward what happens, whereas the events themselves are not under human control at all. But then the problem has not been solved. A person’s attitude to what happens to him is also an event—a mental-spiritual event. One can equally ask whether God knows such an event in advance. If He does, and if we wish to preserve belief in any kind of free will, then we have returned to the original problem.

Maimonides’ answer is also problematic. Maimonides determined that God’s knowledge is unlike human knowledge, but he did not explain what this means with respect to “knowledge” in its human sense. If such knowledge exists in God, then the problem has not been solved. If He does not possess knowledge in that sense, then in effect we have given up one horn of the dilemma—God’s foreknowledge. Beyond that, if it really is another kind of knowledge, then when we say that God knows what will happen in the future, we are apparently merely moving our lips without saying anything at all. It seems that when we say that God possesses knowledge, we intend the concept “knowledge” in its ordinary human meaning.⁷

There are attempts to say that there is no conceptual contradiction at all between foreknowledge and free will. What such a claim probably means is that this is a physical contradiction, not a logical one—that the inability to know the future does not conceptually contradict the concept of time, but rather describes an inability that stems from the limitations of the laws of nature. It is clear that God is not subject to those limitations, but this answer addresses only the first question, and it is doubtful whether it helps at all with the second. Clearly, if this is a real contradiction—logical and not physical—the problem remains intact, and that is indeed how many thinkers, Jewish and otherwise, understood it.⁸ Later in this gate I will attempt to propose a third concept of logical contradiction, one that is not a physical contradiction, but also not an analytic one.

In Rabbi Chaim ibn Attar’s Or HaChaim on the Torah, on Genesis 6:5, the possibility is raised that God could have known the future, but renounced this ability, and in that way left us with freedom.⁹ Here, apparently, a philosophical mistake is hidden. The problem of knowledge and choice does not depend on whether God actually knows the future. It is enough that the information exists, or that He has the actual ability to know it, for choice to become impossible.¹⁰ Or HaChaim solves the problem of omnipotence, but not the problem of knowledge and choice. The very formulation of the problem as a contradiction between knowledge and choice is misleading. If the information exists, then even if God does not use it, or renounces actually knowing it, choice is already impossible.¹¹

It may be that Rabbi Chaim ibn Attar means to say that if the knowledge is not present in God, then the situation cannot be described in the words, “the information exists, but God does not know it.” Rather, one must say that the information does not exist at all. But then he too, in practice, joins the claim that the problem has no resolution, and that if God were to use this ability, there truly would be no choice. By the very fact that God gave us choice, He renounced His ability to know the future, and indeed the information does not exist in the world. The alternative possibility before God was to create a deterministic world. If so, we have returned to an approach that gives up one horn of the dilemma, as we shall also see from Maimonides’ words.¹²

In the introduction to Rabbi Isaiah Horowitz’s Shenei Luhot HaBerit, in the section called Beit HaBechirah, this is stated explicitly: God does not actually know the future that will occur, but only what is likely to occur. See the discussion there regarding how such a claim is reconciled with prophecy.

From all of this discussion it is clear that when one tries to translate the various solutions into human language—that is, into concepts of “what am I to believe?”—one is forced to give up one horn of the dilemma. Usually the tendency is to give up the doctrine of God’s foreknowledge, not the doctrine of free will. It should be noted that this does not diminish God’s omnipotence, just as there is no diminishment of His power when we say that He cannot create a round triangle. God’s inability to create logical impossibilities stems from the limits of human language and thought. The statement that God is not subject to these limits cannot be uttered in our language. It is in fact a statement about us, not about Him. Still, in light of what will be clarified later in the chapter, this inability seems less significant. It is only an inability to create analytic contradictions, not logical contradictions as such.

A common approach holds that God’s knowledge of the future really is incompatible with the free choice of any created being whatsoever. Some claim that the problem is only apparent, and I do not intend here to address those views. If the problem is indeed real—and for the purpose of discussion we will assume that it is, as also emerges from the note above—then apparently it is impossible for any person to believe simultaneously in God’s knowledge of the future and in the existence of free choice.

The Argument of the “Unity of Opposites”

In relation to this question, and even more so with respect to questions concerning the attributes of God, the believer sometimes enlists to his aid the method—if one may call it that—of Nicholas of Cusa, in the aforementioned work, who argued for the possibility of a unity of opposites in religious thought. Such a person will say that because God is an infinite being, He cannot be grasped in the categories of the finite human intellect, and therefore His description can, and perhaps even must, contain logical contradictions. These contradictions, proponents of this method will argue, stem from the finite character of our human perception.

This kind of argument assumes two premises, one explicit and one implicit:

The explicit premise: God is a being that can be characterized by contradictions.¹³
The implicit premise: the statement of a person who claims to believe two contradictory beliefs together is meaningful; that is, it is accompanied by some cognitive process.

The second premise is critical if the unity of opposites is to provide a solution to any theological or philosophical problem. If a person cannot believe both in God’s knowledge of the future and simultaneously in his own ability to choose freely, then the claim that God is an exceptional being who tolerates contradictions is irrelevant. The discussion concerns human beliefs about God—that is, the human concept of God—and not God as He is in Himself. Every human being, unlike God, is a creature with a logical mind and all its limitations. Therefore one cannot, apparently, demand that a person believe that God knows the future and at the same time that human beings possess free choice. By the same token, one cannot sincerely and seriously believe someone who claims that these are indeed his beliefs.¹⁴

The Difficulty

Let us expand a bit here in order to clarify the difficulty involved in believing logical contradictions. When a person believes that some entity possesses property A and non-A simultaneously, the main question is not how he arrived at that belief, but whether that belief has any meaning or cognitive content. Apparently, the whole content of the belief that the entity possesses the property non-A is that it does not possess the property A. The conjunction of those two beliefs, then, is completely meaningless.

My assumption is that faith is an alternative route to certain cognitions—as we saw at the end of the previous gate—but those cognitions are still supposed to stand up to accepted human tests. If not, then should such a believer wish to infer some conclusion from his belief, he would be able to infer anything whatsoever from it. As is well known from elementary logic, from the premise “A and non-A,” any conclusion can be derived. If so, such a belief has no cognitive standing at all.

As I noted above, the problem is not the kind of entity under discussion—the bearer of the property—nor the way in which the believer arrived at his belief. The problem lies in the cognition of the believer himself: is it possible that the movement of the lips of such a believer, who says such things, is genuinely accompanied by corresponding cognitive activity in his awareness?

Someone may come along and claim that there are mystical experiences that can be accompanied by such feelings. My purpose here is not to examine the psychological possibility of mystical experiences, nor to argue that faith is not such experiences but something else—though that is indeed my understanding. I will simply assume here, for the sake of discussion, that faith is a cognitive process, and ask myself whether, even in this sense of faith, contradictory contents can exist within religious consciousness, which is human by its very nature.

Several Kinds of Contradiction That We Will Not Address

It is important to sharpen another point in this context, one that becomes clearer in light of the last few paragraphs. In the present discussion we are concerned only with logical contradictions. “Physical” contradictions—those that violate the laws of nature rather than the laws of logic—do not pertain to our question, because there is no doubt that propositions that include physical contradictions are meaningful, even if they are usually false. Their utterance is certainly accompanied by a parallel cognitive occurrence. To believe that God split the Red Sea against the laws of nature is a claim with entirely clear meaning. The question whether it is true or not depends, among other things, on the first premise above—that God is infinite, or omnipotent. The second premise—that the proposition is meaningful—manifestly holds for propositions that include physical contradictions, and therefore they are not the subject of the present inquiry. This point will become clearer and sharper as the discussion proceeds.¹⁵

For the same reason, we will not deal here with statements that speak of a unity of opposites in relation to phenomena of historical or ideological dialectic. A prominent example is the attempt to describe communist materialist dialectic, or the historical dialectic of Hegel, or that of Rabbi Kook, as examples of a logic of the unity of opposites.¹⁶ Such uses of the term “logic of the unity of opposites” are purely borrowed usages, because they do not involve genuine logical contradictions, but opposing ideologies or opposing historical processes—at most philosophically opposed, but not logically so.

For example, if someone says that in our world there is, on the one hand, a historical-ideological trend toward strengthening the status of the individual, and on the other hand, at the same time, a strong collective conception is also developing, and perhaps adds that both trends are important and will merge into a perfect synthesis at the end of the process, it is clear that there is no logical contradiction here in the strict sense. Contradictions of this dialectical sort do not require changes at the logical level of thought, but at most at its historical-ideological level.¹⁷ As stated, in this example—as in most other cases involving history or ideology and the like—the concepts of “unity of opposites” or “departure from accepted logic” are used only metaphorically. In the terminology we proposed above, such contradictions are at most “physical,” if that, and certainly not logical contradictions.

For the same reason, we will also ignore cases in which people speak of a unity of opposites, or of changing logic, when in fact the logical contradiction does not exist at all, but we simply fail to articulate explicitly why. The easy solution adopted in such cases is to speak of a logic of the “unity of opposites,” but that is only a cover for the fact that the speaker does have a clear meaning in mind for the conjunction of the two concepts, but is unable to explain it explicitly.¹⁸

Common justifications for claims of “unity of opposites” employ expressions such as “a grasp by means above reason” and the like, but they do not explain how the results of such a grasp can exist within human thought and constitute objects of belief. Such explanations are, at best, attempts to ground the first premise above. But, as noted, the difficulty does not lie in the possibility of obtaining such cognitions, but in the question whether beliefs that include logical contradictions are themselves meaningful—that is, in premise 2. The main question is whether the results of such a “supra-rational grasp” can dwell in our minds as information. Such statements usually do not explain how such an anti-logical perception is possible; they merely claim that it exists. Such an explanation would have to be accompanied by a theory of meaning underlying it, a theory itself explained in terms clear even to ordinary mortals—those who are not yet aware of the paradox-detecting powers supposedly hidden deep within them, perhaps too deep.

Very often these statements serve as a somewhat lazy substitute for the effort of thought required in a substantive engagement with a given problem. It is very easy to say that with respect to God one can believe in a unity of opposites, and imagine that by this we have solved all the antinomies connected with the conception of God. This rather easily frees us from continuing to deal with those questions. As I argued above, the central question is whether such a proposition really says anything. Otto himself, in his introduction to the first English edition of his book in 1923, explicitly distances himself from the anti-rationalist tendencies that were beginning to take over at that time, and writes:[^68]

The non-rational today serves as a favorite topic for all those too lazy to think, or too quick to evade the wearisome duty of clarifying their ideas and grounding their beliefs on the foundations of coherent thought… And not only must philosophical discussion of the non-rational itself be rational, but religious faith too is aided by conceptual expression, for only through it is it fixed as “faith”… as distinct from mere feeling.

To summarize: all of these explanations are of no help to us here. Strange as it may sound, what we are looking for is a logical foundation for the unity of logical opposites. That is, we wish to ask the seemingly absurd question: at the logical level, can one offer a justification that grounds the possibility of a meaningful belief in contradictory or paradoxical expressions at the logical level?[^^69]

The Genuine Unity of Opposites

The answer, surprising as it sounds, is that there is such a logical foundation. In order to advance toward clarifying it, I will argue that the term “logical contradiction” includes within it two different notions: “analytic contradiction” and “a priori contradiction.” I will further argue that, in accordance with the requirements of meaning presented above, a unity of opposites can exist only when the opposites are a priori, but not when they are analytic.

One can intuitively sense the force of this distinction by paying attention to the example that has accompanied us throughout this discussion: the question of knowledge and choice. The contradiction between these two concepts is clearly a priori, for it is hardly plausible that it was generated by empirical observation of some reality. There is nothing in reality that could teach us that foreknowledge contradicts the possibility of free choice. On the other hand, as I will argue below, this contradiction is not analytic. One can see this if one asks whether the terms “divine knowledge” and “free choice” have independent meanings. One can certainly understand each of these terms without grasping the other. If so, the contradiction between them is not analytic but synthetic. It does not derive from the very meaning of these terms, but from some relation between them.¹⁹

If that is so, then in order to construct a solution of this kind to the problem of the unity of opposites—that is, to find a meaningful unity between opposites—we must separate the analytic from the a priori. In other words, we must identify a sector of synthetic a priori propositions, and on its basis establish a parallel sector of synthetic a priori contradictions.

The result of this process of making sense of the unity of opposites will be a new meaning for the term “synthetic a priori.” We will present here synthetic a priori propositions of a new kind—propositions that are usually classified as analytic. The significance of this new sense will be a different view of the relation between the analytic and the a priori. This issue will be discussed briefly below.

Chapter Two: The Meaning of Negation: Between the Logical and the Analytic

Three Ways of Addressing the Problem of Knowledge and Choice

In logical literature there are various challenges to the “law of excluded middle.”²⁰ In logic, this is the name of the law that states that there are only two possibilities: either P is true or not-P is true, and there is no third possibility. It is difficult to find any challenge that genuinely raises significant doubts about the validity of this law. In any event, it is quite clear that challenges to the “law of non-contradiction”—the law stating that it cannot be the case that both P and not-P are true simultaneously—are rarer, and certainly less convincing.

In light of what has been said here, there are apparently two possible ways of relating to the contradiction between knowledge and choice:[^72]

The first possibility is to say that this is not a logical contradiction, but at most a physical one, and the human intellect too is not bound by contradictions of this kind. In other words, the inability to know the future is not a logical problem but a human limitation, one that can in principle be overcome. Propositions that contain statements of this kind are therefore certainly meaningful.²¹ Just as there is meaning to the proposition whose subject is a stone moving upward from the ground by its own power, even though such a phenomenon violates the laws of nature, so too there is meaning in speaking of God’s knowledge of the future, even if such knowledge is beyond our capacity. According to this view, one may certainly say that God is not subject to the law of nature that rules out reversal of the time axis, just as He is not subject to other laws of nature—for example, when He performs miracles. According to this approach, just as God can circumvent the laws of gravity when He wishes to perform miracles, thus He can also know the future.²² Such an approach characterizes positions that see no problem in simultaneous belief in knowledge and choice.

The second possibility is to say that this is indeed a logical contradiction, and therefore one must give up one of the two claims and adopt only the other.²³ It is clear that a claim of the second kind is not a claim about God Himself, but about the believing human being, who cannot formulate or think a contradictory proposition as a principle in which he believes.

Our proposal here is that there is a third possibility: this is indeed a contradiction, and not a physical contradiction, but it is also not a purely logical one. The claim is that there is meaning in believing simultaneously in two principles that contradict one another in this intermediate sense. In other words, we will try to challenge the very classification of a contradiction of this sort, even if it exists, as a problem on the analytic plane. This is apparently an intermediate case between the analytic and the synthetic. The significance of this is that “logical” and “analytic” are not coextensive concepts. The logical is a weaker concept than the analytic.

The Operation of Negation: The Basis of the Distinction Between the Logical and the Analytic

Let us illustrate this as follows. If the set A consists of all items that possess the property P, then all items that do not possess the property P are not included in that set. The claim that a given item, a, which is included in A, possesses the property P, is a purely analytic claim. It is a deduction that infers from the general rule to the particular case. In order to recognize the truth of this claim, we need only examine the set A carefully, analyze it, and sharpen what is already known to us. Since we know that all members of A possess the property P, it is clear that the item a, which is one of them, also possesses that property. In other words, in saying that all members of the set A possess the property P, we have already said implicitly that a is such as well. This is an analytic consideration.

Now let us consider the item b, which does not possess the property P. We can claim this if we examine all the members of the set A and verify that it is not among them. But if we wish to say that the item b possesses the property Q—where Q is the property not-P—it seems that we will not be able to infer this from looking at the set A, but only from looking at the set B, which contains all those that possess Q. This is, in fact, the complement set of A—the set that contains all elements not included in A. In other words, from examining A one cannot infer who the members of the set B, its complement, are by means of a purely analytic process. The reason is that this is not merely a focused examination of the set A and the extraction of information from what is already known to us, as in the usual analytic process of learning from the general rule to the particular case.

We already explained in the first gate that analyticity means analysis. An analytic claim is one that arises from analyzing knowledge that is already in our possession. But in order to know the members of the complement set, we need additional knowledge beyond knowing who the members of the set A are and what their characteristics are. This is not merely an analysis of our knowledge about the set A. It is a look beyond that set. For example, one must know who all the items relevant to this property are—the items with respect to which the attribution or denial of the property P is applicable²⁴—that is, what the entire space is, A+B. Only once one knows all of them can one say, in an analytic process, that if b is not in A, then it is included in its complement, that is, in B. Before we are equipped with such a broader kind of knowledge, we can say only, in a weaker and more passive way, that it is not true that b is included in A.

Most logical operations, or logical operators, are “analytic,” meaning that they deal only with analyzing existing knowledge. The operator of “intersection,” for example, represents the logical operation of finding the common part of two sets—the set of all elements that belong to both. See, for example, Note 27. To do this we must examine the two sets carefully, but in no way do we need to go beyond this known domain—that is, beyond the scope of the two sets in question—in order to know the result of the operation of intersection.

Analysis is a decomposition of existing knowledge, whereas synthesis is the use of additional information and its joining to existing knowledge. Therefore the operation of intersection can be defined as an analytic logical operation.

From the description above it appears that, in this sense, negation is not a simple—or pure—analytic operation. Negation is an inference that in certain respects resembles synthetic inferences. In a consideration of this sort, it is not enough merely to analyze the information already in our possession, and therefore it resembles a synthetic consideration. In order to carry out the logical operation of negation, we must go beyond the knowledge we already possess, and therefore it is not a purely analytic operation.

The conclusion is that negation is indeed a logical operation, but it is not an analytic one. The distinction between the logical and the analytic begins to emerge through the analysis of the operation of negation.

Two Kinds of Logical Contradiction

In light of all this, we can now also distinguish between two types of logical contradiction, beyond all the other types mentioned above that do not concern us here. The statement that a belongs to the set A and yet does not possess the property P counts as a logical-analytic contradiction. But the statement that a belongs both to the set A and to its complement B is not such a contradiction. This is apparently a contradiction that is not analytic, for we saw that in order to determine its truth or falsity, it is not enough merely to examine the set A.²⁵

To summarize: the operation of negating P is essentially different from the operation of contemplating the meaning of P. The latter is an analytic operation upon the subject P, one that looks at what is included in it, or in the set of items of which it is a property. The statement about the individual was already actually included in the general contemplation.²⁶ By contrast, the former operation—negation—is a quasi-synthetic one, because it goes beyond the negated subject, P, and considers what is not included in it, namely the complement set.

The Concept of an Opposite

In order to clarify the picture further, let us discuss for a moment the concept of an “opposite.”

There is an ancient philosophical question: does non-being not exist, or does non-being in some sense exist? That is, is the opposite of being a merely abstract notion, or does it represent some kind of existence?[^^79] If it represents some form of existence, then it is not merely the negation of being, but something with independent meaning of its own. As we have seen, the transition from being to non-being is not analytic, but synthetic. It is a transition to another entity located outside the entity under discussion. By contrast, analytic negation has no independent existence; it is merely the absence of what is negated.

Closely related discussions appear in philosophical literature regarding light and darkness. Is darkness merely the absence of light, or is it a real entity? One might similarly discuss additional pairs of opposites, such as good and evil, cold and heat, rest and motion, and so on. Some formulations ask, with regard to such pairs, which one truly exists and which is merely the absence of something. There is an assumption here that only one of the two opposites can truly exist. An interesting example of such a discussion appears in the well-known ethical work Ways of the Righteous,²⁷ which writes in its introduction as follows:

There is a trait that must be used in most places, and a trait that must be used only a little. This is like preparing a dish that requires vegetables, meat, water, salt, and spices: from each of these one must take the proper measure—of this, a little; of that, a lot. If one uses too little meat, the food will be thin; if one uses too much salt, it will be inedible from its saltiness. So too with all the ingredients: if one takes too little of what requires much, and too much of what requires little, the dish will be ruined. But one who is skilled, and takes from each the proper weight, then the dish will be pleasant and sweet to those who eat it. So too with character traits: there are traits of which one must take a large measure, such as humility and shame, and the like, and there are traits of which one must take only a little, such as pride, brazenness, and cruelty. Therefore a person should weigh on the scales of understanding and take from each trait its proper measure…

The analogy drawn here—from seasoning food to balancing the traits of the soul—requires explanation. In the matter of food, it is clear that adding sugar is not the same as reducing salt, even though there is some offsetting effect between them. But with respect to character traits, it seems that increasing pride is like decreasing humility, increasing shame is like decreasing brazenness, and vice versa. If so, then in the realm of traits there is no room for balancing opposite traits, but only for the correct dosage of each one. Put differently: why does he state that one should take a great deal of humility, and in addition reduce pride? Are not these two steps identical? Apparently it would have been enough to say that one should settle for as little pride as possible. From his words it appears as if having little pride and much humility are two independent actions, like adding much sugar and little salt to a dish.

It should be noted that in the world of concrete objects, as we assumed above, there do not seem to be true opposites. Salt is not the opposite of sugar, but merely a different substance with properties opposed in certain respects. It is difficult to think of two objects whose properties are exactly opposite. In abstract concepts, such as character traits, this is easier. This stems from the fact that inversion is an operation on the properties of objects—their form—and not on the objects themselves, their matter. Even with tastes, such as saltiness and sweetness, one can perhaps think of a genuine opposite.

Theoretically, it seems possible that there could also be opposite objects—objects all of whose properties are exactly opposite. Such pairs are very rare, if they exist at all. Apparently, two objects with a whole set of mutually opposite properties would have to be generated with some sort of relation between them. It is unlikely that such a situation would arise by chance.

Such a relation between two objects appears, seemingly, between a particle and an antiparticle in modern physics. There the two particles really do possess exactly opposite properties, and indeed they are usually produced under the constraint that the total of all their properties—their “charges,” in the standard physics terminology—be zero. Here we see an additional characteristic of opposites: there must be a defined operation of summation between them, such that their sum yields zero.²⁸ Clearly this requires that they be of the same kind in some sense, so that a relation can be created between them. This is completely obvious: two totally unrelated objects are not opposites. A bird is not the opposite of a table. A predatory bird is the opposite of a non-predatory bird.

In various mathematical theories, the opposite of an element—its inverse element—is defined as an element whose sum with that element yields the result zero. The zero element is the element that, when added to any other element, leaves that other element unchanged. In the banal and simple case of the integers, one may write:

1 + (-1) = 0.

Because of this property, one may regard 1 and -1 as opposites. The sum of 0 with any other number yields that same number, and therefore 0 is the zero element. Among elementary particles, one may define “summation” as their unification into one entity, whose properties are the sum of the properties—the “charges”—of the two particles.

We may now ask whether light and darkness, when combined, cancel each other out. It seems that combining light and darkness—turning on a lamp in a dark room—produces light. From this it is quite clear that darkness functions as a zero element, not as a true opposite of light. In other words, darkness is the absence of light, not its genuine opposite. Their relation is like 1 and 0. The opposite of light is presumably the phenomenon that would be produced in our consciousness by an anti-photon—the antiparticle of the photon, which is what produces the phenomenon of light in our consciousness.²⁹

By contrast, with respect to heat and cold, when we combine them—for example, by pouring a hot liquid into a cold liquid—we obtain an intermediate temperature, that is, lukewarm liquid. Therefore we will say that in the world of concepts, “heat” and “cold” are opposites that cancel one another out like 1 and -1.³⁰

With respect to heat and cold, one can imagine intensifying the cold independently of weakening the heat—for example, if we keep the amount of hot liquid the same and add more cold liquid. This is not the case with light and darkness, where it seems impossible to separate the two operations so that they remain independent. This stems from the fact that these are two different kinds of opposites: heat and cold are opposites like 1 and -1, whereas light and darkness are opposites like 1 and 0. Each of the opposites of the first type has separate meaning that stands on its own. Opposites of the second type have no separate meaning; one is only the nullification of the other. It has no content of its own, and therefore one cannot relate to each of them independently of the other. Darkness is the absence of light, whereas heat is not merely the absence of cold; it has an independent meaning.

According to this, we can also understand the statement of the author of Ways of the Righteous in the quotation above. Pride and humility are indeed opposites, but of the type of 1 and -1. Therefore one can work on each such trait independently: to limit pride, and in addition, independently, to increase—or not increase—humility. I do not intend here to discuss this issue in itself; I cited it only as an example relevant to the matter of opposites under discussion.³¹

Thus, the two concepts of opposite presented here sharpen the distinction between the two kinds of contradiction discussed above. When the two sides of the contradiction relate to one another like opposites of the 1-and-0 type, believing both together has no meaning at all, for the whole meaning of A is simply that it is not B. Therefore the statement, “I believe that A and B,” has no meaning. By contrast, if A and B relate to one another like 1 and -1, then each has independent meaning, though their properties are opposite. In such a case we have a contradiction that is logical but not analytic, since principle A has meaning in itself, beyond the fact that it is the opposite of B. In such a case one can say, “I believe in A and B together.” This statement has meaningful content. It should be noted that such a statement is usually false, but our claim here is that it is not meaningless. Exactly as we saw with respect to a physical contradiction: a claim that includes one is usually false, but belief in it still has meaning.

Returning to the Problem of Knowledge and Choice

If we apply the conclusion we have reached to the problem of knowledge and choice, we will say that the principle of free choice—or the concept of free choice—has a clear meaning even without any reference to the concept of God’s knowledge, and vice versa. If so, it is incorrect to say that the whole meaning of one concept or principle is simply that it is the opposite of the other. In such a case one can say, “I believe in both principles,” and that statement will have meaning and significance.

True, foreknowledge and free choice are generally two beliefs that do not accord with one another, but one may say that with respect to God matters are different. This is exactly as we saw regarding physical contradictions—for example, a stone rising against the force of gravity, without any force pushing it. Belief in such things is generally false, but it is certainly meaningful.

For this reason, one may attribute such phenomena to divine action, since God can overcome technical barriers. Only a lack of meaning obligates us to refrain from believing a contradiction, because of principle 1 above—that even if we declare belief in two contradictory principles, we do not truly believe cognitively in that conjunction. If the conjunction has meaning, there is nothing to prevent us from attributing it to God, for He, in itself, has no limitations, and the limitation of meaning that pertains to us does not arise in such a case.

A Generalization to Other Contradictions

Clearly, this argument will also hold with respect to other contradictions. Every contradiction contains belief in two principles that contradict one another. The contradiction between them is expressed thereby that one is the “negation” of the other—either as mere absence or as a genuine contrary—or the “opposite” of the other—either as a zero-like opposite or as an active contrary.

If the relation of opposition is like 1 and 0, then one cannot believe it, even with respect to God—not because He is limited, but because this belief has no cognitive meaning for us, the believers. By contrast, if the relation is like 1 and -1, then there is cognitive meaning to belief in both contradictory principles, and therefore one may attribute them together to some entity, and in particular to God.

Chapter Three: The Logical-Philosophical Significance of the Distinction Between the Logical and the Analytic

Do We Have a Three-Valued Logic Here?

According to the presentation offered here, one might think that we obtain a three-valued logic. That is, “not-X” can be interpreted in more than one sense. If we apply the argument presented here to the world of logical truth-values, we obtain that for every true proposition, its opposite—its negation—may be false, or something else in addition, which can be marked as 0. There have indeed been such proposals in the history of logic, but as noted, they do not seem persuasive.

One might argue that the true negation of 1 is the composite value “0 or -1,” and thus escape the problem. But this does not help us in assigning truth-values to claims. The negation of a true proposition, according to this, is “false or 0.” It is true that, according to what we have said, saying that a proposition is “not true” still uses an ordinary truth-value within conventional two-valued logic. But the statement that a proposition is “false” no longer does so. If we wish to continue using ordinary two-valued logic, we must use the concepts “true” and “not true.” The use of “false” in place of the negation “not true” cannot remain two-valued. Between truth and falsehood there is an additional intermediate concept.

This seems to be a general phenomenon: if one describes the opposite of some concept X using the words “not-X,” then one still remains within the domain of conventional logic, that is, there are only two possibilities. But if one chooses a special word to describe the opposite, and instead of “not-X” calls it “Y,” then one is in trouble. For example, it is clear that the opposite of “light” is “not light,” and the opposite of “cold” is “not cold.” But the customary opposites—“darkness” and “heat”—are not necessarily one-to-one opposites. With darkness, we saw that perhaps this is indeed the case, but with cold it is not, because here there is an intermediate state: lukewarm.

We saw a similar phenomenon above when we distinguished between the claim “the item a does not belong to the set A” and the claim “the item a belongs to its complement set, B.” When the opposite of a certain property has independent meaning—reflected also in the fact that it has its own separate word in the language, and not merely the expression “not-X”—then it may be that we are dealing with an opposite that is not complete. Negation requires something additional beyond contemplation of the negated thing itself, and therefore it is an operation located somewhere between the analytic and the synthetic, as explained above.

One might say that this phenomenon exists only where there really are three states. In such a case, when we say that we are not in a certain state, two possibilities still remain open to us. If that is indeed the case, then there is nothing here that stands in contradiction to conventional logic. But even so, we have encountered something more. We saw that, in any case, at least at the logical level of relation to a proposition, there are three states. Even when in reality there are only two states—as with light and darkness—at the logical level one can relate to the concept “not light” not merely as a synonym for darkness. True, at the level of reality there is no such phenomenon—that is, no real object characterized by “not-light”—but at the logical level there is such a third concept, one not equivalent to the concept of “darkness.”

In order to escape the problem presented here, someone might claim that one should use only concepts like “not-X” and not concepts with positive content. The problem is that such usage completely sterilizes the ability to use negation in philosophical arguments, as well as in others. If only such usage were permitted, we could derive nothing from the fact that X is true, or exists, or from any other statement about X. A philosophically substantive inference would want to derive from this that the meaningful opposite of X does not exist, not merely that it is not true that X exists. But that is already a quasi-synthetic—or a priori—operation, and not an analytic one. In other words, even if one can save the honor of logic in this way, one loses its meaning entirely, along with the ability to use it for meaningful inferences.

Such a solution is a conventionalist one, typically adopted by the analytic thinker, and he pays for it with the heavy price of loss of meaning. As we have already seen more than once—see, for example, Gate Seven—the analytic thinker chooses terminology that is not itself problematic and that seemingly solves the problems. But the only meaning of such a “solution” is to prevent the problems from even being raised in the language being used, instead of truly solving them.

Again the Problem of Knowledge and Choice: The Law of Non-Contradiction and the Law of Excluded Middle

Let us now return to the question of knowledge and choice and similar cases presented at the beginning of the gate. There I argued against common theological solutions to various paradoxes that say, “God is not subject to the laws of logic.” At the end of the previous chapter we saw that such solutions can be stated with respect to pairs of concepts that are opposites of the type of 1 and -1—that is, concepts that have independent meaning beyond their opposition to one another. God’s knowledge of the future is a concept we understand clearly with no dependence on the question of free choice, and it can therefore stand on its own. One may perhaps believe simultaneously in two concepts that are opposites of this sort.

But this cannot be said of an opposite of the type of 1 and 0, for the whole meaning of the 0 is simply that it is not 1. It has no independent meaning beyond that. One cannot believe that God knows, and at the same time that it is not true that He knows; or that every person has free choice, and at the same time that he does not. By contrast, one can believe—at least at the conceptual-analytic level—that He knows and at the same time that every person has free choice. The concept “free choice” is not an opposite of the 1-and-0 type relative to the concept “knowledge of the future.” It is a concept with independent content, and not merely the negative content of absence. At the end of the previous chapter we generalized this proposal to other logical contradictions in which the relation between the two sides is like the relation between knowledge and choice—that is, a relation of 1 and -1.

It should be noted that in the last paragraph we moved from a claim in the context of the law of excluded middle to a claim in the context of the law of non-contradiction. We claim that if these are not really opposites in the full sense of the term, and if they have independent meaning beyond being opposites of one another—or if there is a third state beyond these two opposite states—then perhaps the law of non-contradiction also does not hold with respect to them. That is, it may be possible to believe simultaneously in both of these opposites, even within human reason. This is a more far-reaching claim, and this is not the place to develop it. It may be that such simultaneous belief is itself the third state. That state is composed of the simultaneous conjunction, which apparently is excluded—according to the law of excluded middle—but in fact is not so, in light of the claims made here.

The A Priori: Between the Analytic and the Logical

We have seen here that saying God’s knowledge does not logically contradict free choice is indeed due to the fact that He is not subject to the laws of logic—human knowledge does indeed contradict choice—but an additional condition is needed if we are to demand that someone believe this: it is necessary that in our own minds too, those two concepts be able to live together. This is actually an intermediate case between analytic contradiction and physical contradiction, which we distinguished above. It is a “soft” analytic contradiction.

A sentence such as “God can create a round triangle” seems more problematic, although even here there are apparently two opposites with independent meaning. The difficulty arises because the concept of a triangle is the subject in that sentence, and the term “round” is its description. There is no triangular entity that bears the attribute “round.” This point too still requires further study.

To summarize, we have tried to argue here that there is an intermediate concept between the analytic and the synthetic, or between analytic contradiction and physical contradiction. We called this type of contradiction a logical contradiction that is not analytic. An analytic contradiction is logical, but the converse is not necessarily true. The example of such a logical operation is the operation of negation, at least the negation of the 1-and–1 type.

The significance of this claim is that there is an intermediate domain between the analytic and the physical that also belongs to the domain of logic. In propositions belonging to this domain we can recognize them a priori—that is, they are a priori—yet they are not analytic. In other words, these are synthetic a priori propositions.

Chapter Four: Torah Perspectives

Introduction

In this chapter I would like to discuss the negative attributes of God, and then present a note dealing with the relation between positive and negative commandments in halakha (Jewish law). These are both examples of the implications of the distinction between the different kinds of negation developed in the previous chapters.

The Doctrine of Negative Attributes

Maimonides, like a substantial portion of the medieval sages, argues that the infinite God cannot be described by ordinary human attributes.³² Expressions that appear in Scripture as divine attributes are interpreted by them as negative attributes. To say that God is “gracious” means to say that He is not “not-gracious.” Maimonides elaborates at length in explaining how negative attributes are not empty—that is, they do add some understanding of divinity, and therefore there is value in dealing with them.

Apparently, however, the more difficult question is the reverse one. Maimonides deals with the question of why a negative attribute has any meaning at all, but the real problem is what, if anything, the difference is between negative and positive attributes. It is seemingly obvious that both have meaning. Is the claim that God is not “not-gracious” not logically equivalent to the claim that He is gracious? If that is so, then certainly there is value in dealing with such attributes, but it is not at all clear what, if anything, we have gained from shifting to a negative description of God.³³

Therefore it seems that there is an implicit assumption here that the negation of “not-gracious” is not equivalent to saying that He is gracious. There are still other concepts that belong to the negation “not-gracious.” This situation seems highly problematic. In the previous chapter we saw that “gracious” and “not-gracious” are certainly ordinary, univocal logical opposites, and only when there is a word with independent content can one speak of independence between opposites. Here we see that even explicit negation is not merely the absence of what is negated.

It seems, then, that we must understand the claims of medieval thinkers regarding negative descriptions of God as implicitly assuming an intermediate state between “gracious” and “not-gracious.” In any event, we have here another example—this time a very common and familiar one for anyone who deals with medieval theological thought—of a different kind of negation, a kind resembling the relation between 1 and -1, between which there is an intermediate state, 0, unlike the relation between 1 and 0, which leaves no room for any additional middle state.

Let us conclude this part of the discussion with an interesting halakhic example that presents the problematic nature of the concept of negation as discussed here.

Note 29: Positive and Negative Commandments

The halakhic-Torah system is fundamentally made up of a collection of commandments that appear in various forms in the Torah. There is a standard division of them into two central kinds: positive commandments and negative commandments. Positive commandments, such as putting on tefillin (phylacteries), express a demand for a positive act that the Torah expects us to perform.³⁴ Negative commandments express a demand that we not perform a certain action.

Therefore, in general, to fulfill a positive commandment one must perform an action, and violating the will of the Torah as expressed in a positive commandment is done by not acting—in standard halakhic language, through passive non-action. With negative commandments the situation is reversed: fulfilling such a commandment generally consists in non-action, and violating it is done by an undesired active deed—in halakhic language, by active commission.

For example, with the positive commandment to put on tefillin, fulfillment is achieved by the act of putting them on, and violation by non-action, that is, by not putting them on. By contrast, with a negative commandment such as the prohibition against desecrating the Sabbath, fulfillment is achieved by refraining from action—that is, by not performing prohibited labor on the Sabbath—and violation occurs by doing the prohibited labor.

There are also exceptional examples. There are positive commandments fulfilled through non-action, and negative commandments fulfilled through action. For example, one who eats human flesh, according to Maimonides, violates a positive commandment; see Mishneh Torah, Laws of Forbidden Foods, beginning of chapter 2.³⁵ This positive commandment requires one not to eat human flesh and the like—that is, the requirement is one of non-action. Therefore violating this commandment, although by definition it is a positive commandment, is done by action—namely, eating. The commandments of resting on the Sabbath and in the seventh year, the Sabbatical year, are further examples in which the Torah positively commands us to rest—that is, not to perform prohibited labors. A counterexample on the side of a negative commandment that requires action in order to fulfill it is the commandment, “You shall not bring bloodguilt upon your house,” meaning that a person must ensure that there are no hazards in his home. For example, one must build a parapet around one’s roof so that no one will fall from it. Here, in order to avoid the prohibition, one must perform an action—build a parapet—and one who fails to act violates the prohibition.³⁶

Two questions arise here:

Why define a given commandment as positive or negative if its content is not necessarily action or inaction? Ordinarily we would understand the definition of a positive commandment as deriving from the fact that it requires action, and vice versa for a negative commandment. If, as we saw above, this is not a necessary characterization, what then is the meaning of one commandment being a positive commandment and another a negative one? Is there some additional, more essential significance that causes a commandment to be defined as positive or negative?[^^90]
At the more basic logical level: what is the difference between formulating a commandment as positive or as negative? Even the initial characterization of positive and negative commandments is not clear here. Apparently, every divine desire can be formulated in both ways, and the meaning would be identical. To say, “You must do X,” has exactly the same logical meaning as saying, “You must not fail to do X.” If that is so, why do we distinguish at all between positive and negative commandments? Every positive commandment is also a negative one, and vice versa. When the Torah says one must put on tefillin, one can equally understand this as a prohibition against not putting them on, which seems logically equivalent to the earlier formulation.³⁷

To sharpen the question even more, and also to point toward an answer, let us now discuss the words of Nahmanides in his novellae on the Babylonian Talmud, Kiddushin 34a. There is a halakhic principle that women are exempt from positive commandments that are time-bound. For example, women are exempt from the commandment to sit in a sukkah, a festival booth, during the festival of Sukkot, since this commandment applies only during that festival—that is, it is time-bound. With negative commandments, by contrast, women are obligated even when they are time-bound.

Nahmanides states that there are negative commandments whose entire purpose is to strengthen and support a positive commandment. For example, the negative commandment “You shall not bring bloodguilt upon your house” is intended to support the commandment obligating one to build a parapet around a dangerous roof—the positive commandment of making a parapet. In such negative commandments, Nahmanides writes there, if they are time-bound, women will be exempt just as they are from positive commandments. Here we see clearly that these commandments are in effect positive commandments formulated as negative ones.

If so, two further questions arise:

Why formulate them in this way?
In what sense does this formulation support or strengthen the positive commandment that it is meant to support? What difference does it make to a person who is hesitating whether to violate a positive commandment that there is also an additional negative commandment here? Are two transgressions really essentially different from one? A person prepared to commit one transgression will be prepared to commit two. Beyond that, it requires explanation why the support is specifically in the form of a negative commandment rather than an additional positive one, or vice versa.³⁸

It appears that the meaning is as follows. There is a clear difference between the positive formulation of a positive commandment—for example, to put on tefillin—and the negative formulation as a negative commandment forbidding a Jew not to put on tefillin. In the first case the desire is positive: the Torah wants tefillin to be put on. In the second case the Torah merely does not want the negative state, namely the state in which tefillin are not put on. The question is where the center of gravity of the Torah’s will lies: is the failure to put them on a negative state, and therefore one must put on tefillin, or is putting them on a positive act, and therefore it is forbidden not to put them on? The practical expression of these two desires is apparently identical, but at the level of will they are two utterly different determinations. A positive commandment determines what the Torah’s will is with respect to a positive state; a negative commandment determines the opposite. The opposite formulation could lead to the same practical results, but the divine will expressed in it is entirely different. A negative commandment determines a state that the Torah does not want, and in order not to arrive at it one must refrain from doing a certain act.

If the Torah had formulated a negative commandment forbidding one not to put on tefillin, then if I failed to put them on I would be in direct violation of the Torah’s command, because I would be in the very state it defines as negative. Such a command would leave me no option and would force me to put on tefillin, even though in this definition putting on tefillin would not constitute a value in itself—it would merely be avoidance of not putting them on. By contrast, a Torah command that one should put on tefillin—a positive commandment—actually leaves me the option of not putting them on, since in doing so I am not performing something blatantly negative. The state without tefillin is not bad in itself in God’s eyes. If I do not put on tefillin in such a case, I am not directly violating the Torah’s command; rather, I am failing to reach a state that would have been more desirable from its perspective.³⁹

It now becomes clear why these two formulations, which appear to be equivalent on the practical plane, are not equivalent at all on the ideational plane. It is completely obvious that if there were a prohibition against not putting on tefillin, that would be a far stronger incentive to put them on than the mere existence of the positive commandment to do so. This is not a quantitative matter of adding one more prohibition, but a qualitative difference that determines that by not putting on tefillin I am acting directly against the will of God. This is in effect a “stick” and a “carrot.” One can induce the performance of an act by placing a “carrot,” namely by determining that a certain state is positive. Such a carrot draws the person forward to perform the act—that is the positive commandment. By contrast, one can induce the act also by striking with a “stick,” which compels him to get up and do it and does not let him simply sit still and refrain. That is the negative commandment.

According to this, what Nahmanides wrote becomes clear: the negative commandment “You shall not bring bloodguilt upon your house” can indeed support the positive commandment “You shall make a parapet for your roof.” The formulation as a negative commandment is intended precisely for that supportive purpose, since the value is clearly the positive state in which a parapet is made and the house is therefore safe. The aim of adding the prohibition is to add a “stick” to the “carrot,” namely to determine that one who does not build a parapet is not in a neutral state with respect to God’s will, but is colliding with it directly. It now becomes clear why this indeed constitutes an incentive to build the parapet.

Up to this point it seems that there really is a difference between a positive commandment in its positive form and a negative commandment expressing disapproval of the opposite state. But the problem has not disappeared; it has only been swept under the rug. We showed that the two formulations have different meanings. If so, it is unclear how this fact can be reconciled with the apparent logical equivalence between the two. How can two formulations that are logically equivalent have different meanings?

We are forced to conclude that these are not logically equivalent propositions. But is the proposition “I want you to do X” really not logically equivalent to the proposition “I do not want you not to do X”? Apparently this is just double negation, which should bring us back to the first proposition.

Anyone with a little training in logic immediately notices that the true negation of “I want you to do X” is “It is not true that I want you to do X.” This state also includes the possibility that I have no desire at all concerning you—a zero state, “I do not care whether you do X.” Therefore we have here another example of negation of the 1-and–1 type, which allows an intermediate state. As we have seen, this differs from negation of the 1-and-0 type. Still more clearly, a further negation of “It is not true that I want you to do X” is not the command, “It is not true that I want you to do not-X,” but rather: “It is not true that it is not true that I want you to do X.”⁴⁰

It should be noted that even a formulation that apparently uses direct negation—“not-X” or “not want”—can be understood in both ways, that is, according to both kinds of negation. In any event, on the practical plane there may be a situation in which these two determinations are equivalent, meaning that they have the same practical content. Such a situation is created in a reality in which there is no practical intermediate state between “I want you to build a parapet” and “I do not want you not to build a parapet,” even if logically there is a third state. In the case of a house with a dangerous roof that requires a parapet, apparently there is no practical difference between the two formulations, since both determine that I must build one. Yet there is still a feeling that these are two different formulations. One may attribute this to the fact that there is a difference between them in a situation where the house already has a parapet, or where I am unable to build one.⁴¹ But intuition indicates that even in a case where there is no difference in the practical consequences between the two determinations, there is nevertheless a difference at the level of value.

In some sense, logic does not exhaust the forms of human thought and expression. Here I mean to say not only that it is not a necessary condition, but that it is not even a sufficient one. That is, even arguments that appear to have solid logical structure may not be accepted by human reason. Reason clearly sees that there is a difference between these two formulations, even where this difference has no practical implications. One of them is a “carrot” in favor of the act X, and the other is a “stick” against the act “not-X.” One approaches the required act from within it—an analytic approach—and the other by negating the opposite that stands outside it—a quasi-synthetic approach.

If we are indeed correct in this distinction, then the essential difference between positive and negative commandments becomes quite clear. A positive commandment says that there is a state desirable to God. A negative commandment says that another state is undesirable. From this it follows that, in principle, one cannot violate a positive commandment by active commission, or a negative commandment by passive non-action. Every violation of a negative commandment is a direct act—essentially an active commission—against the Torah’s will, even if in the real world this is done by not acting. The fact that a transgression is active rather than passive is determined by the character of the collision with the Torah’s will, not by the way in which it is carried out in the physical world. In this sense, a violation of a positive commandment never has the character of direct frontal collision, whereas violation of a negative commandment is always a direct action against the Torah’s will. That is the difference in formulation between these two kinds.⁴² To conclude, let us note that there are views among the medieval authorities according to which violation of a positive commandment is always through passive non-action, and vice versa; they do not accept the exceptions mentioned above.

Chapter Five: Human Thought and Logic

Further Deviations from Logic

Beyond the analysis of the operation of negation that we saw above, there are additional examples of ways in which thought deviates from ordinary logical rules; some of them were discussed above in Gate Eight. When my son asks me, “From what point is it afternoon?” I do not have a clear answer for him. Is 2:00 p.m. already afternoon, or does it still belong to noon? 4:00 p.m. is already clearly afternoon. The same is true of all everyday concepts. The typical paradox that describes this phenomenon—the sorites, or heap paradox—was presented at the end of Gate Eight. I will mention it again here only to complete the picture:

A collection of two pebbles is not a heap.
If there is a collection of stones that is not a heap, adding one stone cannot change its status into that of a heap.
A collection of a million pebbles is called a heap.

It is perfectly obvious that these three premises cannot coexist. One cannot accept all of them together. On the other hand, none of them seems doubtful or easy to reject. The required conclusion is that the concept “heap” is not a concept that can be defined mathematically. In Chapter Four we mentioned that this is true of almost all concepts of ordinary language. Some know this paradox as the paradox of baldness, which asks in a similar way: from what minimal number of hairs is a person called bald? And one may continue in this way with many everyday concepts. The analytic thinker will try to change ordinary language in order to solve the problem, but thereby he merely circumvents it rather than solves it, as we explained there. This is another example of the fact that human thought is not always subject to the laws of logic; that is, not only are those laws insufficient, they are also not necessary.

It is interesting to examine whether the use of such concepts violates the law of non-contradiction or only the law of excluded middle. Apparently the clear tension is only with the law of excluded middle, since there is a vague state that is neither “heap” nor “not-heap.” True, one might think of this state as both “heap” and “not-heap,” but that seems more artificial. One should note, however, that in terms of the structure of the argument, analytic thought breaks down here in the strongest sense, as defined in this chapter. There is a contradiction among three premises, all of which are considered true. From this aspect there is also a collision with the law of non-contradiction.⁴³

There are still further examples of apparent contradictions between human thought and logic. In the appendix we will address what Yuval Steinitz calls “the start-up problem.” There too, it seems, there is a contradiction with accepted logic, but we will see there that this is only a pseudo-contradiction. There is a clear error in the argument.

I would also like to refer the reader here to the dialogue of Yonatan Ben-Dov with David Graves in the book Questions about God, mentioned above. He points to the same issue: that even logically contradictory claims can have meaning. That is, even if they cannot be defined as true, they may still be meaningful.⁴⁴ If so, it seems that we need not remain silent with respect to such claims, contrary to Wittgenstein’s famous recommendation—the analytic positivist—at the end of his Tractatus.

Summary

There is a clear sense that logic does not capture the totality of thought and intellectual activity. Up to the present chapter we saw that there is thought that is not purely logical—that is, analytic—but also synthetic; in fact, we saw that all thought is of this sort, meaning that it is composed of dimensions that are not purely logical. Here we have tried to convey a feeling that is very difficult to formulate explicitly—and the reasons are clear: formal formulation is a logical act—that human thought can in some sense operate even against logic, and not merely beyond it.

Put differently: from the first half of this statement it follows that philosophical problems are not necessarily logical problems, and therefore their solution is not necessarily on the logical plane. From the second half it follows that logical problems are not necessarily philosophical problems, and perhaps therefore do not require a solution at all.

In Closing: Religion and Paradoxes

Let us conclude the discussion with a quotation from Nicholas of Cusa’s The Coincidence of Opposites, cited in that same dialogue in Questions about God:

Thus I saw how great the need is that I enter into darkness in order to receive the union of opposites beyond the capacity of reason to grasp, and there seek the truth, in the place where I encounter the impossible.

And beyond that, beyond even the highest summit of the intellect, when I have already attained what is unknown to every other intellect, in the place where every judgment tells me that it is furthest from the truth—there, my God, You are.

It is usually accepted that the religious person, in light of his faith, is also prepared to accept thought that appears paradoxical and non-rational. In contrast, the claim presented in this gate is that such thought may have an independent grounding. It is not as paradoxical as it appears, and therefore it has a place in any rational system of thought, not only a religious one—provided that one does not flee into forced analytic shelters.

Chapter Six: Back to the Synthetic A Priori

Introduction

As noted in the introduction to this gate, the essence of the synthetic alternative is to offer an alternative path to the transcendental one—Kant’s path—which, as we saw, is nothing but a thin disguise for analyticity. We propose an alternative way to arrive at and define synthetic a priori propositions. Let us now examine the implication of everything said in this gate for the existence of such propositions.⁴⁵

If we examine the type of contradiction that was called here a logical contradiction that is not analytic—or an a priori contradiction—we find that it is a contradiction between two concepts with independent meanings. The contradiction between “light” and “darkness” is of this kind. By contrast, the contradiction between “light” and “not-light” is analytic, since the whole meaning of “not-light” is simply that it is not “light.”

Thus, contradictions belonging to this intermediate domain are characterized by the fact that their conjunction has meaning in language, but one cannot imagine the actual existence of that conjunction. This is unlike physical contradictions, which also have meaning, but whose impossible existence can still be imagined, even visually.

With respect to the problem of knowledge and choice, we saw that the concept of “divine knowledge” does not analytically contradict the concept of “free choice,” since each of those concepts has an independent meaning. If someone does not know the concept of God, and certainly not God’s knowledge, he can still understand the concept of free choice, and vice versa.

Above we formulated the problem of the unity of opposites as consisting of two different problems:

Is the conjunction of the opposites possible in reality?
Is the paradoxical claim that joins opposites meaningful?

We said that one may perhaps accept the claim that there is no obstacle to attributing contradictory properties to the divine being, since He is not subject to human laws. But the second problem—that this conjunction has no meaning in the human mind and therefore provides no solution to our problem—is not solved by the unity of opposites.

If we examine this claim with respect to contradictions in the intermediate domain, we will find that there is meaning to such a conjunction of concepts, and therefore the second problem is solved. The first problem, as noted, may not exist even with respect to analytic contradictions, and certainly not with respect to contradictions of the intermediate type. Our conclusion, then, is that one may attribute contradictions of this kind to God. Only belief in analytic contradictions about God is impossible, because such belief has no meaning in our language and in our human thought.

If we now return once more to the example of the problem of knowledge and choice and ask ourselves why we see a contradiction there, it is clear that this is not an analytic contradiction, since the concepts have independent meaning. On the other hand, it is clear that the problem created by believing the conjunction of divine foreknowledge and free choice is not located on the physical plane. It is not derived from any observation, scientific or otherwise; that is, it is not a posteriori.

Therefore, the determination that divine foreknowledge and free choice are incompatible is, on the one hand, synthetic—that is, not analytic—and, on the other hand, a priori—that is, not a posteriori. This determination is a synthetic a priori proposition. In terms of the kinds of contradiction presented in the previous chapters: the contradiction between free choice and God’s knowledge of the future is an a priori contradiction—synthetic a priori—and not an analytic one.

Thus, the intermediate domain of which we have spoken throughout this gate—the domain situated between the physical and the analytic, while still included within the logical—is in fact the a priori domain. More precisely: the synthetic a priori domain. We have here, almost inadvertently, discovered a new kind of synthetic a priori proposition.

We must now note that the justification for adopting the proposition that free choice and divine knowledge are incompatible is not based on transcendental reasoning. I know of no such reasoning that leads to this complex proposition. The real rationale for it is contemplation of the concepts that make up this conjunction, and of the relation between them. As we recall, Kant and Hume held that relations, unlike objects, are not observable, and therefore that only the transcendental path is open to us for justifying propositions of this kind. The synthetic alternative is that such propositions can be recognized by cognition with the “mind’s eye,” that is, by using the synthetic part of the intellect—hearing-based reasoning—to observe the concepts involved in the problem.

Our certainty in the results of such observation—or hearing—is based on reasoning similar to the reasoning an analytic thinker would give for his certainty in the results of sensory observation of objects in the material world. What I have observed is certain in my eyes, and this requires no further justification. The same is true of intuitive “observation”—or, more precisely, “listening”—to concepts, relations, and ideas.

Every reader can examine his own position on this issue by examining his own thought. He should ask why he perceives divine knowledge and free choice as contradictory principles. Is this the result of observation? In all likelihood, for no reader is that the case. Is it the result of analyzing the concepts? Apparently that too is not the method by which one recognizes this contradiction, since their meanings do not clash directly—that is, not by virtue of the meaning of each one considered separately. If so, then here is a synthetic a priori proposition in which you believe.

If we generalize what has been said here, there are synthetic a priori propositions that we accept as self-evident. The reason is not transcendental reasoning but “listening” to the concepts that compose those propositions. The claim of a contradiction between knowledge and choice, for one who believes in both, is synthetic a priori. So too with all religious “paradoxes.” Paradoxes that are truly analytic contain no cognitive content, and therefore they cannot be believed, even if someone declares that he believes them.

In previous gates we explained that the only reasonable basis for accepting synthetic a priori propositions is a religious basis. Here we saw a clear example of this. Religious “paradoxes” provide another example of the connection we find between a religious—or at least theistic—worldview and trust in a cognizing, synthetic-a-priori part of reason. Those who regard all these paradoxes—including the synthetic a priori ones—as something one cannot live with thereby express an implicit analytic position, which, as noted, is also connected with atheism.

Summary of the Discussion in This Gate

In this gate we distinguished between different concepts of negation, and through them defined a type of synthetic a priori proposition: a priori paradoxes.

Our conclusion is based on the argument concerning the life of a religious person within a framework of paradoxical beliefs. Such belief can be explained only if we accept that there is a type of contradiction that is not analytic—that is, it is meaningful—yet it belongs to the domain of logic and not to that of physics. From this it emerged that the religious-synthetic thinker is subject to analytic rules, but not necessarily to all logical-a-priori rules.⁴⁶

If so, by this route one can also infer a broader conclusion: there is an intermediate domain between the analytic and the physical—at least in religious thought—which we identified with the synthetic a priori. According to the interpretation proposed here, whenever we speak of living in paradox, or whenever we use contradictory or vague concepts, ones that seem contrary to accepted logic, we are dealing with a state of living within an a priori contradiction—a synthetic a priori contradiction—and not an analytic one. These states themselves are the evidence for the distinction we proposed between the domains.

In Gates Seven and Eight we rejected Kant’s distinction between the analytic and the a priori in light of his own arguments. Here we have reestablished that distinction. We separated the analytic in its stricter sense—according to which negation is not an analytic operation—from the a priori, for all forms of negation are certainly a priori operations.

It seems that synthetic a priori propositions, according to this definition, would be analytic propositions in Kant’s sense. For example, the proposition “There is no light in this dark room” is quasi-synthetic—non-analytic—because it includes an operation of negation, but it is certainly a priori. Kant would apparently classify such a proposition as analytic, like the well-known “All bachelors are unmarried.”

It seems to me that if there is any meaningful distinction at all between the analytic and the a priori—that is, the synthetic a priori—its meaning is the distinction we have made in this gate.⁴⁷

Footnotes

For expansion and elaboration of what was said in this gate, see my article “Is Belief in Logical Contradictions Possible?,” submitted to Iyyun in 5762. ↩
In fact, as we have seen, no truth is accepted as the result of a logical argument. Every addition of information is made synthetically. Of course, whoever accepts the premises will also accept the conclusions, but the focus of disagreements usually lies at the level of the premises. Adopting the conclusion of a logical argument already happens when we accept the premises that lead to it. Accepting the truth of the premises is not a formal logical process, as we saw in the previous gate and also in Mill’s challenge presented in the eighth gate. ↩
Both Kierkegaard and Rudolf Otto deal extensively with the irrational and the paradoxical, but often—mainly in Otto—the reference is to the numinous (the sublime, the exalted) and not to the paradoxical. See, for example, Otto’s The Idea of the Holy, translated by Miriam Ron, Carmel, Jerusalem, 1999, p. 68, where he discusses the meaning of the irrational. It is clear that he does not mean paradoxicality at the logical level (this is made explicit, for example, in note 42 on p. 71 there), but rather the possibility of relating to that which cannot be related to in human concepts, to the sublime (see, for example, p. 70 there, and Joseph Ben-Shlomo’s afterword to Otto’s book, p. 195). In Kierkegaard, by contrast, there are several places where it is clear that he means a genuine paradox. ↩
An overwhelming majority of the early Jewish thinkers rejected the unity of opposites discussed below. Statements to the effect that contradictory attributes cannot be ascribed to the Holy One, blessed be He, are very common in medieval rabbinic literature. See, for example, Rashba, responsum 4:234, where this issue is discussed directly at the principled level. For additional sources see Aviad Billar’s article in Alon Shevut – Graduates, issue 7, p. 139, note 26. That article also discusses, among other things, Rabbi Kook’s attitude toward this idea, and shows an interesting development in the matter: his later thought specifically uses the unity of opposites intensively, even on logical planes. ↩
Many do not sufficiently distinguish between the different meanings of paradoxicality in the religious context: is the issue the incomprehensible (or the indescribable), or a logical paradox? Very often, references are brought from thinkers speaking of paradox in its numinous sense, and a description of the problem—and even different responses to it—is offered as though a logical paradox were involved. See also later in the chapter when I discuss physical contradictions and metaphorical uses of the term “paradox.” The question of in which places various religious thinkers intend logical paradoxicality requires separate “empirical” research. In my estimation, the results of such a study may be rather surprising. ↩
See, for example, Simhah B. Urbach’s Pillars of Jewish Thought, Department for Torah Education and Culture in the Diaspora of the World Zionist Organization, Jerusalem 5737, vol. 3. ↩
If Maimonides means to say that God’s lack of subjection to the concepts of time enables Him to know the future, then he is answering a different question (question 1 above), as explained earlier. If so, in any event, there is no answer here to the second question. ↩
See Judith Ronen’s article, “Everything Is Foreseen Yet Permission Is Granted,” in the collection Between Religion and Morality, edited by Daniel Statman and Avi Sagi, Bar-Ilan University, 5754, p. 35. This article also appears in issue 2 of Higayon. In this article Ronen distinguishes between two kinds of contradiction: the logical and the physical. This distinction may also be viewed as the distinction between an analytical and a synthetic contradiction (or, more precisely, between a priori and a posteriori; see the first gate). In her article Ronen also distinguishes between necessity that I voluntarily choose a certain act and the necessity of my choice. This distinction is meant to answer the second question and to say that knowledge does not conceptually contradict choice. In my view, however, there still seems to be a contradiction. If it is known in advance that I will do something, then it is impossible that I do otherwise. See the appendix for a discussion of this question. See there also a discussion of whether the characteristics of time are physical or logical characteristics. ↩
This very assertion—that the omnipotent is able to diminish His own power—is highly problematic. Consider the famous children’s tale Puss in Boots, in which the cat wants to overcome the terrible sorcerer and asks him to turn himself into a mouse; when he does so, the cat devours him. Thus the terrible sorcerer is eliminated by a simple cat. Apparently, a terrible sorcerer cannot turn himself into an ordinary mouse. He can assume the form of a mouse, but not become mortal. Let us imagine that God, as it were, turns Himself into an ordinary human being, since He is omnipotent, and then some person comes and shoots Him and kills Him. Can one thus “eliminate” the one whose existence is necessary? Hardly. Apparently, any being can limit itself only in things that are not of its essence—that is, it can put on and shed forms from the contingent part of its form, not from the essential part of it (see the discussion of this in the second gate, regarding Putnam’s claim about the possibility of changing a concept). Omnipotence itself, or the necessity of existence itself, even the omnipotent cannot shed. We will see below that this is not necessarily considered a limitation on His power. It seems that this is the point over which the mystics disagreed: whether the “contraction” is literal. In Kabbalah, the creation of the world is described as a contraction of God’s infinite presence (the Infinite Light) in a certain place, making the formation of finite entities possible. Some views understand that the contraction is not literal, because otherwise God would not be infinite, since He would be limited. Others hold that the contraction is literal, since the infinite can also limit itself. Some interpretations separate the two questions. The contraction is performed in the Infinite Light, which is not God Himself but His initial manifestation, and this is not the place to expand. See Shalom Rosenberg’s article “The Kabbalistic Doctrine in The Soul of Life,” appearing in Shanah be-Shanah, 5758, p. 357, and Tamar Ross’s article “Two Interpretations of the Doctrine of Contraction: Rabbi Hayyim of Volozhin and Rabbi Shneur Zalman of Liadi,” Jerusalem Studies in Jewish Thought, vol. 1, issue 2, 5742, p. 153. ↩
See a beautiful presentation of this claim in Richard Taylor’s Metaphysics (mentioned in note 21 in the seventh gate), in the chapter on fatalism, and also in the discussion of logical determinism. See there as well the story of Osmo. ↩
See also the brief discussion of logical determinism in the appendix. ↩
As we saw earlier, Maimonides states that God’s knowledge is not like our knowledge. This is an answer to question 1: how He attains the information. As for question 2, it seems that the only way to understand his answer is that knowledge of the type familiar to us is in truth not possessed by Him. If so, Maimonides too answers this question by giving up the principle that establishes God’s knowledge of the future. ↩
I am not entering here into the details of the problem. This assumption is also insufficient, since the existence of free choice is an assumption about human beings, not about God. For the sake of what follows I shall ignore this problematic aspect and assume that this statement, if it has any meaning at all, can solve the problem. ↩
All this is according to our working assumption here, that there is indeed a logical contradiction between knowledge and choice. ↩
For a discussion of these two kinds of contradiction, in the context of the problem of knowledge and choice, see Judith Ronen’s article in Between Religion and Morality, Avi Sagi and Daniel Statman (eds.), Bar-Ilan University, 5754, p. 35. As stated, for the purposes of the discussion that follows I shall assume the position that there is a logical contradiction between knowledge and choice, not a physical contradiction, although I am not sure I agree with that determination. ↩
See Benjamin Ish-Shalom’s book Rabbi Kook between Rationalism and Mysticism, Am Oved, second printing, 1990. In note 71 to the first chapter and note 133 to the third chapter he links Rabbi Kook’s positions to the unity of opposites on the logical level, and even cites Łukasiewicz’s three-valued logic for that purpose. ↩
The most problematic example in this regard is the principle of complementarity in quantum physics, where it apparently seems that there is a unity of opposites on the logical plane; indeed, there is an approach called “quantum logic” that claims quantum thought reflects a different theory of logic. For my part, I do not agree with this approach even within the quantum domain itself (see my article “What Is Legal Effect?”). For the linking of ideological and historical “paradoxes” to the physical and logical level, see Meir Monitz’s article “The Logical Basis for the Unity of Opposites in Rabbi Kook’s Teaching,” Alon Shevut, Gush Etzion, Nisan 5755, p. 112. An extreme expression of such logical reductionism is found in Lewis S. Feuer’s book Einstein and His Generation, Am Oved, 5739, pp. 175–76, where he describes Łukasiewicz’s three-valued logic as the basis of an anti-deterministic mood of the period, which itself served as fertile ground for the growth of quantum theory (the whole point of that book is a historical reductionism that I do not accept on the principled level; see the fifth gate). The appeal to Łukasiewicz’s theory as an explanation of phenomena of logical absurdity is very widespread, and it is important to make clear that this provides no explanation at all. There one finds a formal description of a logical system with three truth-values, but no explanation of the logic behind such “thinking.” It is clear that the understanding of Łukasiewicz’s system is itself carried out (in the metalanguage) in terms of conventional two-valued logic. There is no new “logic” here, only a formal description of a possible formal system. Its domain of application is a matter to be determined on the basis of understanding in ordinary logic. Therefore, as stated, citing Łukasiewicz adds no explanatory value for absurdities, whose main point lies in semantics rather than formal logic. ↩
See, for example, my article in Tzohar, vol. 2, Tel Aviv, Winter 5760, which brings such an example, and my discussion there of Daniel Weil’s article—one of the proponents of quantum logic—cited there. ↩
There is an important point that should be clarified here. The well-known philosopher Quine argued that understanding concepts is based on a network model rather than an atomistic model. That is, concepts are woven into one another, and their understanding is holistic; the understanding of all of them develops together. One cannot understand a concept in itself, without its relations to its surroundings. It is important to note here that I am assuming an atomistic epistemological conception of concepts, not a network conception. The term “knowledge” and the term “choice” are each examined in themselves. The question whether there is a contradiction between them is asked only after a separate examination of the meaning of each one independently. It seems to me that the Kantian assumption I make below—that the analytical is not a priori, that is, that there is a synthetic-a priori sector—implicitly assumes an atomistic conception of concepts. In the “network” conception (a molecular conception) it is very difficult to distinguish between analytical and synthetic. All synthetic relations enter the picture through connections in the a priori network and in effect become analytical. If so, the Kantian assumption of a distinction between the a priori and the analytical fits very well with the assumption of conceptual atomism, and therefore we do not have two assumptions that are completely independent; it is reasonable to assume one if one assumes the other. If so, this is not an ad hoc assumption for apologetic purposes, but an assumption required by the whole argument. ↩
See, for example, Bergmann’s Introduction to Logic, chapter 3, section 21. ↩
There are even various proposals made by physicists for travel to the future or the past, and according to their view this is certainly not even a physical problem. In my opinion, these proposals stem from conceptual confusion, and this is not the place to discuss it. ↩
However, this direction too apparently answers only question 1 and not question 2 in note 28. ↩
We saw in the note above a solution in this direction proposed by Rabbi Isaiah Horowitz in his book Two Tablets of the Covenant, in the introduction, in the section “The Chosen House.” See Rabbi Shmuel Ingber’s article in The Weekly Page of the Department of Basic Studies at Bar-Ilan University, no. 58, on the portion of Va’era, and the note above. ↩
For example, virtue does not belong to the domain of application of the adjective “red.” The sentence “virtue is triangular” is neither true nor false. It is simply not a sentence at all, but a collection of words lacking compositional meaning. ↩
Descriptions through properties and through membership in sets parallel the two forms of definition discussed above in note 23: definition through extension and definition through intension. Here, if the argument in the text is correct, we see that these two descriptions are not quite identical. This sheds new light on the relation between these two forms of definition. ↩
See Mill’s challenge to the process of deduction, described in the seventh gate. ↩
An ancient Jewish ethical work. It was probably composed in the fifteenth century. The author is unknown. Corrected and edited by Gabriel Zeloshinsky, Feldheim, Jerusalem, 5751. ↩
It is not clear how existence itself is annihilated. It seems to me that one aspect of two such particles cannot be annihilated, namely their very existence. Both exist in the same sense. It would not be correct to say that one exists and the other anti-exists. If so, it is not clear to me how annihilation occurs. It seems to me that one should say that in fact both such particles continue to exist even after the interaction, but their properties have been annihilated. They are now “naked” objects without properties. See the second gate, in the discussion of the ontological proof, where we pointed out that existence is not a property—that is, it does not relate to the form of the object but to its matter. Since opposites exist only in the world of properties, such a concept has no opposite in the sense later to be defined as (-1), but only as 0. One might perhaps speak of a lack of existence, but not of non-existence (or anti-existence). According to this, even objects whose properties are exact opposites of each other are not opposites of each other. In my article “What Is Legal Effect?,” Tzohar issue 2, I show that opposition is a relation that exists between properties and not between objects. Leibniz (the author of the principle of the identity of indiscernibles) apparently did not accept this determination. ↩
See the discussion at the beginning of the second gate (around the quotation from Russell) on the relation between the photon, or electromagnetic field, in the world, and the phenomenon of light, which exists only in our consciousness. Admittedly, the statement in the body of the text does not seem necessary to me. It is not clear that the effect of an anti-photon on our consciousness would yield the phenomenon opposite to that of light. It is plausible that the consciousness’s concepts of opposites do not stand in a one-to-one relation to the concepts of opposites in the physical world. It should be noted that this discussion is purely theoretical. As far as is known, there is no anti-photon in the world. ↩
Here too it should be noted that in the world of physics there are no “heat particles,” and therefore there are also no anti-particles of “cold.” In thermodynamics these two phenomena are described as one phenomenon that changes quantitatively (higher and lower temperatures). In our consciousness they are two opposite phenomena (apparently because there is a zero state defined by body temperature: what is above it is called hot, and what is below it is called cold). This is another example of concepts of opposition in consciousness that are not derived from concepts of opposition in the description of physical reality. See the previous note. ↩
For a discussion of these two kinds of opposites, see, for example: Knowing Heart by Moshe Hayyim Luzzatto, section 38; Maimonides’ Guide of the Perplexed, part 1, chapter 73, seventh introduction, and part 3, chapter 10; Maimonides’ Treatise on Logic, chapter 11 (which contradicts what he says in the Guide); Rabbi Moses Isserles’ The Doctrine of the Burnt Offering, part 3, chapter 9; Rabbi Moshe Avigdor Amiel’s The Principles for the Study of Halakha, principles 12–13; Maharal of Prague’s The Mighty Acts of the Lord, chapter 5; Rabbi Menachem Mendel Kasher’s Unraveling Mysteries, introduction to chapter 6; and more. Usually these sources addressed the question as a general one: whether an opposite is this or that. I do not know of any proposal for a criterion to distinguish between these two kinds of opposites, even among those who recognize that both exist. The claim that opposition is a relation between properties and not between objects is discussed in my article “What Is Legal Effect?,” in issue 2 of Tzohar. ↩
There are, however, differences among the various approaches. Some distinguish between attributes of action, which may be positive, and attributes of essence, which are only negative. For the purposes of the discussion here I assume the view that all divine attributes are negative. ↩
Beyond all this, one must understand what the definition of a negative attribute is at all. Apparently, every attribute for which a special word is dedicated in a language is a positive attribute, whereas a description by “not X” is a negative attribute. But this seems a somewhat arbitrary criterion. Why should we not create a “positive” word for “not gracious” and use it in the language? Would that make the description into a positive attribute? There is an assumption here that every attribute that says something in language—that is, one for which a word of its own has been assigned—is a positive attribute. An opposite always contains within itself the negation of an intelligible and defined content, not another intelligible and defined content. This point too seems related to our argument here, but this is not the place to expand on it. ↩
In the Torah world it is customary to say that the Torah instructs us or expects us to do things. It is not only the giver of the Torah who can command; the Torah itself can do so. This is an interesting terminology that points to a conception of the Torah as an entity that exists for itself (of course it too is subject to the Creator). This reflects the statement, to which we will return below, that at the basis of halakha (Jewish law) there stands a metaphysical system, unlike civil systems of law, at whose basis stand, at most, values and norms. The metaphysical system that stands at the basis of halakha and Torah is the “commander” here. ↩
There it is, however, a “prohibition inferred from a positive command,” and only in halakhic terms is it a positive command. What exactly this determination means is open to discussion, but this is not the place. ↩
I do not wish to enter here into the details of the various halakhic approaches concerning the obligation of the commandment to build a parapet and the commandment “Do not bring bloodguilt into your house.” Here they are presented as two aspects of the same mitzvah (commandment), and that is apparently how Ramban understood them, as we will discuss shortly. There are also other approaches in halakha. ↩
The second question may be entirely incomprehensible to someone accustomed to Talmud study, because this distinction has already been very deeply internalized in him: saying “I want X” is altogether different from saying “I do not want not-X.” My purpose here is only to point out and sharpen this correct point, which sometimes is not even clear to the learners themselves (if only they would pause for a moment and think about the meaning of this principle). ↩
There are sometimes also cases in which one prohibition really does support another, and in Talmudic terminology: “so that one transgresses two prohibitions by it.” Every such case indeed requires separate discussion. The simple understanding cannot be correct, for the reasons listed in the body of our discussion. ↩
Ramban, in his commentary on the Ten Commandments in the portion of Yitro, discusses why the rule that a positive commandment overrides a negative commandment when the two conflict does not contradict the principles that establish the priority of negative commandments (for example, that in order to avoid transgressing them we are obliged to spend all our money). Ramban states that transgressing a positive commandment is lighter than transgressing a negative commandment, but fulfilling a positive commandment is more important than fulfilling a negative commandment. We see here that fulfilling any commandment (positive or negative) does not receive exactly the opposite value of not fulfilling it. In the notes to the ninth gate we cited Ramban’s words in another context. ↩
It is worth noting here a common formulation of the liar paradox (this is the original formulation cited from the New Testament), which fails precisely at this point of performing negation in a logically imprecise way. The formulation is as follows: a Cretan says, “All Cretans are liars.” Here there is ostensibly a paradox. If all of them are liars, then the speaker too, who is also a resident of Crete, is a liar. But if so, then this statement is false, and the residents of Crete are truth-tellers. But if so, then the statement is true, and so on ad infinitum. The mistake here lies in applying the logical negation operator. One can say that this statement is indeed false. But that does not mean that all the residents of Crete are truth-tellers; it means only that it is not true that all the residents of Crete are liars. If so, one can say that this man is indeed among the liars, and therefore the statement he uttered is indeed false. The falsehood of the statement follows from the fact that there is a group of residents of Crete (which of course does not include this man himself) who are not liars. The question whether this is indeed the meaning of the sentence that was said depends on the principle known as the “principle of charity.” According to this principle, one gives a sentence the meaning it can bear, not necessarily what the speaker intended to convey through it, and this is not the place to discuss it. It is also worth noting that the formulation “A says: A is a liar” is not paradoxical for the same reason. The simple meaning of this sentence is that all sentences uttered by A are false. It may be incorrect because there are sentences (other than the present one) that are in fact true. A simple paradoxical formulation is: “Sentence A: Sentence A is false.” This is the formulation we reached at the end of the proof of Gödel’s theorem in the ninth gate. There is also a formulation through a pair of sentences, which we used in the notes to the discussion of Russell’s theory of types in the eighth gate. ↩
By this possibility I mean that if the formulation is positive—“Build a parapet!”—then when there is already a parapet on the house and it is no longer necessary to build one, the commandment of parapet has still not been fulfilled, since one did not perform an act of building. But if the commandment were that there not be a house without a parapet, then in such a situation he does not have such a house, and the commandment has therefore been fulfilled. The same applies to a situation in which he cannot build a parapet for some reason. If the formulation is positive, then in the end he has not fulfilled the commandment even if it is not his fault. But in a negative formulation, it is clear that he has not committed a transgression. Similarly, in the Jerusalem Talmud, tractate Gittin, there appears a rule that states: “Coercion is not as though one acted.” That is, if a person was coerced and therefore did not perform a certain act, although ordinarily a person is exempt for transgressions committed under coercion, here it is clear that he will not be considered as though he did perform it. If a person was coerced and therefore did not fulfill a positive commandment, he may not be punished, but that does not mean that the commandment was fulfilled. By contrast, one who transgresses a prohibition under coercion—see note 16—according to some views this state is considered as though no transgression at all was committed (at least not by him). ↩
This discussion seems connected to the formalism called in modern philosophy “deontic logic,” which deals with the logic of concepts of duty, and moral concepts generally. See, for example, Abraham Meidan’s article “Deontic Logics and Possible Worlds,” at the end of the collection The Just and the Unjust, and the references there. This is not the place to expand on it. ↩
All the claimed independence among the three basic logical laws—the law of identity (everything is identical to itself), the law of non-contradiction (it cannot be that X and not-X), and the law of the excluded middle (either X is true or “not-X” is true; there is no third possibility)—seems to me somewhat dubious. Problems raised against one of them can often be solved by a variation of another. For example, if one does not accept the law of non-contradiction, then it follows that there is a third state in which a given sentence is both true and not true, and in that way we have also broken the law of the excluded middle. The reverse independence seems to me correct, but this is not the place to expand on it. Another point to note is that we are speaking here about a conflict with the law of non-contradiction. At first glance this itself is an absurd situation: if indeed the law of non-contradiction is not correct, then the conflict poses no problem, for it is forbidden only by the law of non-contradiction itself, which forbids logical conflicts. It seems that there is here a move to the metalanguage of logic itself. We are speaking of contradiction on the plane from which the laws of logic are examined (including the law of non-contradiction itself). This is a departure beyond the most basic system itself: logic. Again we have here an example of a departure from analytical thinking. See on this the ninth gate. ↩
There there is a reference to his article with Ilai Alon, “A Language for the Description of God.” See David Graves and Ilai Alon, “A Language for the Description of God,” International Journal for Philosophy of Religion 36 (1994): 169–186, and the relevant references there. ↩
I refer the reader interested in greater detail to my article “Is Belief in Logical Contradictions Possible?,” since the discussion requires recourse to fine nuances that I cannot enter into here. ↩
It seems to me that this characterizes primarily the religious person and not necessarily every synthetic person. This is what all those who say that religious life is a life in paradox mean. This is a stronger kind of syntheticity. It may be, however, that the materialist dialectic of the communist school is also an example of “religious” belief in paradox. Communists too seem willing to live with a priori contradictions, even if not analytical ones. Already in the third gate we saw that communism has many religious characteristics. I refer the reader to my notes at the beginning of the gate regarding ideological or historical “paradoxes,” which do not constitute a real problem on the logical plane. ↩
In fact, as we have seen, one may also say that even in the theory proposed here there is no real distinction between the analytical and the a priori, for what we called here the synthetic-a priori is actually a posteriori. A synthetic-a priori proposition is the result of observation of the concepts involved in it, and as such it is an a posteriori proposition. This question is, in many respects, semantic, and we already noted this in the eleventh gate. ↩

← Gate Eleven: The Synthetic PositionContentsGate Thirteen: The Synthetic Position in the Beit Midrash →

The Synthetic A Priori in a New Sense1

Introduction

The Debate over Analytic Thinking

A Positive Characterization of the Synthetic Position

Back to the Synthetic A Priori

Chapter One: Living in Paradox

Introduction

The Problem of Knowledge and Choice

Note 28: The Problem of Knowledge and Choice

The Argument of the “Unity of Opposites”

The Difficulty

Several Kinds of Contradiction That We Will Not Address

The Genuine Unity of Opposites

Chapter Two: The Meaning of Negation: Between the Logical and the Analytic

Three Ways of Addressing the Problem of Knowledge and Choice

The Operation of Negation: The Basis of the Distinction Between the Logical and the Analytic

Two Kinds of Logical Contradiction

The Concept of an Opposite

Returning to the Problem of Knowledge and Choice

A Generalization to Other Contradictions

Chapter Three: The Logical-Philosophical Significance of the Distinction Between the Logical and the Analytic

Do We Have a Three-Valued Logic Here?

Again the Problem of Knowledge and Choice: The Law of Non-Contradiction and the Law of Excluded Middle

The A Priori: Between the Analytic and the Logical

Chapter Four: Torah Perspectives

Introduction

The Doctrine of Negative Attributes

Note 29: Positive and Negative Commandments

Chapter Five: Human Thought and Logic

Further Deviations from Logic

Summary

In Closing: Religion and Paradoxes

Chapter Six: Back to the Synthetic A Priori

Introduction

Summary of the Discussion in This Gate

Footnotes

שתף

The Synthetic A Priori in a New Sense¹