Is Belief in Logical Contradictions Possible?[1]

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A New Meaning for the Synthetic A Priori

Introduction

Religious faith is often associated with thinking that is not logical. The symbol of this widespread association is Tertullian's famous saying: "I believe because it is not logical." A prevalent approach, especially in religious existential philosophy (Kierkegaard and his followers), and particularly in the Christian world, expands this relation and says that religious life, by its very essence, is a life in paradox.[2] Here there is already a link between religious faith and anti-logic, contradiction, and not merely illogicality.

In most discussions of the problem of logic in religious thought, the intention is not paradoxes in their logical sense, but the numinous and the incomprehensible in human concepts. A clearer and sharper treatment of logical paradoxicality in religious cognition is found in Nicolaus Cusanus, who deals extensively, and directly, with this point, as is expressed in his book The Unity of Opposites.

In this article I wish to discuss the idea of the unity of opposites at its logical level, which is the sharpest and most problematic. The discussion will also yield a treatment of paradoxicality in weaker senses. The question I wish to address in the present article is: can one in fact live in paradox? That is, are there possible lives that are not merely non-logical, but anti-logical as well?

My assumption is that the believer is also a human being, and as such possesses human thought that is subject to accepted logic. Human beings, apparently, cannot believe in paradoxes. More than that, there are those who claim that paradoxical propositions (at least in the severe logical sense) have no meaning at all, and if so it is clear that they cannot be believed. At most, one can utter them verbally, but this movement of the lips cannot be accompanied by cognitive processes, such as understanding and thought, or by any layer of meaning. Does a person who says that he believes in the existence of a round triangle actually claim anything? Apparently we must adopt, at least in these contexts, Wittgenstein's famous recommendation at the end of the Tractatus, and remain silent.

In order for the discussion below to be more concrete, I will accompany it with reference to one of the oldest and best-known antinomies in religious philosophy, the problem of divine foreknowledge and free will (hereafter: "knowledge and choice"). This example will accompany the discussion later as well, but it should be remembered that it serves only as an illustration, and our present concern is not the clarification of this topic, but the clarification of the principled possibility of a unity of opposites.

A common approach holds that God's knowledge of what is going to occur is incompatible with the free choice of some created being. There are those who claim that the problem is illusory, and I do not intend here to address those positions. If the problem is indeed real, as I will assume for the sake of the discussion, then apparently it is impossible for any human being to believe in God's knowledge of the future and at the same time also in the existence of free will.

In addressing this question, and even more so with respect to questions touching on the attributes of God, the believer sometimes recruits to his aid the "method," if it may be called that, of Nicolaus Cusanus, who argued for the possibility of a unity of opposites in religious thought. Such a person will say that since God is an infinite being, He cannot be grasped in the categories of finite human intellect, and therefore His description can, and perhaps even must, contain logical contradictions as well. These contradictions, proponents of the above argument will claim, stem from the limited character of human apprehension.

This kind of argument assumes two premises, one openly and the other implicitly: 1. The explicit premise: God is a being that can be characterized by contradictions.[3] 2. The hidden premise: the statement of the person who claims to believe in two contradictory beliefs at once is meaningful, that is, it is accompanied by a cognitive process.

The second premise is critical if the unity of opposites is to constitute a solution to some theological or philosophical problem. If a person cannot believe in God's knowledge of the future and at the same time in his own possibility of choosing freely, then the statement that God is an exceptional being who tolerates contradictions has no relevance. The discussion concerns the human being's beliefs about God, not God as He is in Himself.

I will try here to elaborate a bit, and to clarify the problematic nature of belief in logical contradictions. When a person believes that some being possesses property A and also not-A simultaneously, the main question is not how he arrived at this belief, but whether this belief has meaning. Apparently the entire content of the belief that the being is characterized by non-A is that it is not characterized by A. A simultaneous conjunction of these two beliefs is meaningless. My assumption is that faith, and it may be based on revelation or inner illumination, is perhaps an alternative way of arriving at certain cognitions, but those cognitions are supposed to stand up to accepted human tests. For example, if that believer wishes to derive some conclusion from his contradictory belief, he will be able to derive anything whatever, since from the premise A and not-A any conclusion whatsoever can be derived. If so, a belief of this kind has no cognitive status.

One may argue that there are mystical experiences that can be accompanied by such feelings. In what I say here I do not wish to examine the psychological possibility of any mystical experiences, nor to argue that faith is not such experiences but something else (although that is indeed my understanding). I will simply assume here, for purposes of the discussion, that faith is a cognitive process, and ask myself whether even in this sense of faith there can be contradictory contents in religious cognition.

Here is the place to return and sharpen another point, which already arose above. In the present discussion we are dealing solely with logical contradictions. "Physical" contradictions (= those that violate the laws of nature, and not the laws of logic) do not bear on the question before us, since there is no doubt that propositions containing physical contradictions are meaningful (even if in our world they are not true), that is, uttering them 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 perfectly clear meaning. The question whether it is true or not depends, among other things, on premise 1 above (that God is infinite, or omnipotent). Premise 2 (that this proposition is meaningful) plainly holds for propositions that include physical contradictions, and therefore they are not the concern of the present inquiry. This point will be clarified and sharpened in the course of the discussion below.[4]

For the same reason, I will not later address statements that resemble expressions of a unity of opposites with respect to phenomena of historical or ideological dialectic. A prominent example of this is attempts to describe the communist doctrine of materialist dialectic, or the historical dialectic of Hegel's school, or that of Rabbi Kook, as examples of a logic of the unity of opposites.[5] These uses of the term "logic of the unity of opposites" are merely figurative, since they do not involve actual logical contradictions, but opposed ideologies or opposed historical processes (at most philosophically, but not logically). For example, if someone says that in our world there is a historical-ideological trend toward strengthening the status of the individual on the one hand, while on the other hand a strong conception of the collective is simultaneously developing within it, and perhaps even adds that both trends are important, and that they will merge into a perfect synthesis at the end of the process, it is clear that there is no logical contradiction here in the severe sense. Dialectical contradictions of this kind do not require changes at the logical level of thought, but at most at its historical-ideological level. As stated, in most of these cases the concept "unity of opposites" can serve only in a figurative sense.[6]

For the same reason, in what follows I will also ignore cases in which people speak of a unity of opposites, or of a change in logic, when in fact what is meant is that the logical contradiction does not exist at all, but that we do not succeed in formulating explicitly why. The easy solution adopted in such cases is to speak of a logic of the "unity of opposites," but this is merely a cover for the fact that the speaker does have a clear meaning for the conjunction of the two concepts, but is unable to make it explicit.[7]

Common justifications for statements of "unity of opposites" use expressions like "grasping by means above the intellect," and the like, but these do not explain how the results of such a grasp exist within human thought and constitute an object of faith. These explanations are attempts to ground premise 1 above, if anything. But, as stated, the problem does not lie in the possibility of attaining such items of knowledge, but in the question whether beliefs that include logical contradictions are themselves meaningful (premise 2).[8]

Otto, in his introduction to the first English edition of his book in 1923, writes:[9]

The non-rational today serves as a favorite subject for all who are too lazy to think, or too quick to shirk the wearisome duty of clarifying their ideas and founding their beliefs on the bases of coherent thought… And not only must philosophical discussion of the non-rational itself be rational, but religious faith itself too is aided by conceptual expression, for only through it is it determined as "faith"… as opposed to mere feeling.

If we summarize, we are seeking, strange as this may sound, a logical foundation for a unity of logical opposites. That is, we wish to ask the seemingly absurd question whether, at the logical level, one can offer a justification that grounds the possibility of meaningful belief in logically contradictory expressions.

My answer, surprising as it may sound, is that there is such a logical foundation. To advance toward clarifying this foundation, we must shake some of the dust off the "analytic" and the "a priori" in Kant's teaching. My main claim will be 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 demands of meaning presented above (premise 2), a unity of opposites is possible only when they are a priori, and not when they are analytic.

One can intuitively appreciate the force of this distinction if one notices the example accompanying us throughout the discussion, the question of knowledge and choice. The contradiction between these two concepts is clearly a priori, since it is not plausible that it arose from empirical observation of some reality. On the other hand, it is clear that this contradiction is not analytic. This can be seen if we try to ask ourselves whether the term "divine knowledge" and the term "free will" have independent meanings. Without doubt, each of these terms can be understood even without recourse to the other. If so, the contradiction between them does not derive from the very meaning of these terms, but from some relation between them. We will say that this contradiction is synthetic and not analytic.[10] If so, in order to establish a solution of this kind to the problem of the unity of opposites, one must separate the analytic from the a priori, that is, find a sector of synthetic a priori propositions, and establish on its basis a parallel sector of synthetic a priori contradictions.

The structure of the article is as follows: in the next chapter I will briefly present the common (Kantian) treatment of the a priori and the analytic. Chapter C will deal with the logical negation operator, and distinguish between two different meanings of it. In chapter D I will propose a distinction between two types of contradiction that are based on the two types of negation defined in the previous chapter. In this chapter I will also explain the proposed solution to the problem of the unity of opposites. Chapter E will present the doctrine of negative divine attributes as a philosophical precedent for this distinction. Chapter F will clarify that in fact a new way of viewing the synthetic a priori underlies the whole discussion, and in chapter G I will conclude.

B. Analytic, A Priori, and Synthetic A Priori

The two divisions, a priori-a posteriori and analytic-synthetic, by their very nature appear at first glance to be independent. The division between the a priori and the a posteriori lies on the epistemological plane. By contrast, the division between the analytic and the synthetic lies in thought (in the structure of the proposition itself, or in the definition of the concepts that appear in it).

Since these two divisions belong to different domains, epistemology and thought, we would expect four independent groups of propositions to arise: a priori-analytic, a priori-synthetic, a posteriori-analytic, and a posteriori-synthetic.[11] Nevertheless, before Kant (and also after him) the opinion prevailed that there are only two groups of propositions: analytic (= a priori) and synthetic (= a posteriori). The reason for the identification, in a simplified description, is that the analytic proposition of course does not need experience in order to verify it, and therefore the analytic is a priori. On the other hand, the a priori cannot arise from any source other than analysis of the concept that serves as the subject of the proposition, for from what other non-empirical source could we validate it? Therefore we infer in the opposite direction as well that the a priori is analytic. The general conclusion is that the a priori is identical with the analytic (a priori = analytic).[12] A simple logical consideration shows that these two entailments are also equivalent to the following identification: synthetic = a posteriori.

Already from this concise description it is clear that the second side of the identification is the more problematic. The analytic is undoubtedly narrower than the a priori. An analytic proposition does not require experience, but an a priori proposition might perhaps be obtained as the result of another, non-analytic method.

The pre-Kantian view held that there is no "margin" here, that is, that these are overlapping categories. Following Hume's challenges to the rational grounding of the possibility of scientific understanding, Kant argued that there are synthetic a priori propositions, that is, that the a priori consists of two sectors: the analytic, which is the familiar part, and the synthetic a priori, which is Kant's innovation.

In what follows I will not need the Kantian argument, which has serious defects, and will assume only the very existence of synthetic a priori propositions; that is, I will assume Kant's separation between the analytic and the a priori without any commitment to his arguments (see below in chapter F for a different argument). As emerges from Hume's arguments, the existence of such propositions must be accepted by anyone who regards scientific understanding as a rational understanding of the world, and therefore it is not directly dependent on accepting the particular argument Kant proposed for them.

C. Is Negation an Analytic Operation

There are several basic logical operators that are used in most logical theories, such as conjunction, disjunction, and negation. These operators act on concepts, where by means of them we create a complex concept, or on propositions, in the same way. The logical operations on concepts can also be formulated as operations between sets (parallel to Venn diagrams).

There is a relation between these two planes of use. A logical operator that acts on concepts creates a new concept out of the meanings of the given concepts. That same operator, when it acts on the extension-sets of the corresponding concepts, yields the extension-set of the complex concept. For example, the operation of conjunction between the concepts "Jewish state" and "democratic state" yields the complex concept "a Jewish and democratic state." The operation of intersection between the set of democratic states and the set of Jewish states yields the extension-set of the complex concept, that is, the set of Jewish-democratic states.[13]

Let us now define the concept "analytic operation." An "analytic operation" is a logical operation such that in order to understand its result, we need know only the given concepts (or sets), and apply logical tools to them, and nothing more. For example, someone who knows the set of democratic states, or the characteristics of the concept "democratic state," and likewise the concept "Jewish state," will be able to know and understand on the basis of that knowledge alone also the concept/set "Jewish and democratic state." No additional knowledge is required for this.

The basic logical operations are apparently analytic operations. The example of the concept "Jewish and democratic state" illustrates the analyticity of the conjunction operation. The same can be done with disjunction. I now wish to examine the nature of the negation operation, which, as we shall see, is exceptional in relation to the other logical operations.[14]

One exceptional property of the negation operation is that it is a unary operator, that is, a logical operator that acts on only one datum (a set or a concept). The other operators listed above are binary, that is, they define an operation on, or between, two data.

It seems to me that the negation operation is exceptional also in that it is not an analytic operation, in the sense we defined above. By this I mean to argue that in order to understand the concept/set that results from applying the negation operator to another concept/set, it is not enough to know the negated concept/set and the nature of the logical negation operation alone. The operation parallel to logical negation in set theory is taking the complement. It should be noted that this is not a full parallel, since the complement is a set with independent content, and not merely the negation of the set in question. In order to understand the result of the operation of taking the complement, we must know the entire relevant logical space (the union of the two sets), and then negation becomes logical "subtraction."

Let us try to illustrate this in the following way: if the set A is composed of all particulars possessing the property P (A is the extension-set of the property P), then all particulars that do not possess property P are not included in this set. The claim that a certain particular a, which is included in the set A, possesses the property P is a purely analytic claim. This is a deduction that learns from the general to the particular. In order to see the correctness of this claim, we need only look carefully "inside the entrails" of the set A, that is, analyze and sharpen what is already known to us. Since we know that all the members of A possess property P, it is clear that the particular a, which is one of them, also possesses this property. In other words: in saying that all members of the set A possess the property P, we have in fact already said, though implicitly, that a is such as well. This is a clear analytic (= analyzing) consideration.[15]

We now want to discuss the set B, which is the extension-set of the property not-P (= Q). We ask ourselves about the particular b whether it possesses the property P or not. We can infer this by looking through the entire set A and verifying that it is not a member of it. If that is the situation, the obvious conclusion is that it does not possess the property P. But if we wish to say that the particular b possesses the property Q (which is the property not-P), it turns out that we cannot infer this from looking at the set A, for it may be that this element lies outside the relevant logical space, but only from looking at the set B, which contains all bearers of property Q. This is in fact the complement set of A (the one that contains all the elements in the relevant space that are not included in A). In other words, by looking at A one cannot infer who the members of the set B (its complement) are by a purely analytic process. This is not merely focused inspection of the set A and extraction of information from what we already know about it alone, as in the ordinary analytic process of learning from the general to the particular.

To know the members of the complement set, we require knowledge beyond knowing who the members of the set A itself are, and what their characteristics are. This is not analysis of the knowledge about the set A that we have in hand alone, but looking beyond this set. For example, we must know who all the relevant particulars are (those to whom the claim ascribing property P can be affirmed or denied), that is, what the whole space is (A+B). After one knows them all, one can say analytically that if b is not found in the set A, then it is included in its complement, that is, in B. Before we are equipped with more general knowledge of this kind, we can say only that it is not true that it is included in A, but we cannot positively determine that it is included in B.

Up to this point we have discussed the possibility of knowing whether a given particular is included in the extension-set of the complementary concept. For the same reason it is also clear that from acquaintance with the set A and the property of its members (P), one cannot positively determine the properties of the members of B (that is, positively understand what Q is), beyond understanding that they are members of the complement set of A. The meaning of the concept "darkness," or the property "dark" (Q), cannot be understood solely on the basis of understanding the concept "light," or the property "illuminated" (P). That is, the non-analyticity of negation is a feature of it both when it acts on meanings of concepts and when it acts on their extension-sets.[16]

From this description it appears that negation is not a simple (or pure) analytic operation. Analysis is the analysis of existing knowledge, whereas synthesis is the use of additional information and its conjunction with the existing knowledge. Here this is a transformation between sets that in certain respects resembles synthetic inference. In a consideration of this sort, analysis of the information in our possession alone is insufficient, and therefore it resembles a synthetic consideration. Our conclusion, up to this point, is that negation is a synthetic operation.

Of course, what we have seen up to this point is not sufficient to justify the argument above. My aim is to show that negation is a synthetic a priori operation. Up to this point we have seen, at most, that negation is not an analytic operation, that is, that it is a synthetic operation. The question is whether this is an a priori or a posteriori operation. By the term "a priori operation" I mean to say that one can know the properties of the set that is the result of the logical operation, in the case of negation this is the complement set (B), from the properties of the given set (A), in an a priori way (without the addition of experience). If indeed one can do so, then the negation operation can rightly be called a "synthetic a priori operation," in the terminology defined above.

From the very fact that the negation operation is classified as one of the logical operations, it will be difficult to understand it as an a posteriori operation. The negation of a concept is a logical activity and not a physical one, and logic deals not with the ways experience operates, but with the ways the intellect thinks, with the a priori. To clarify more fully the a priority of the negation operation, and as a basis for a better understanding of the "unity of opposites," let us look a bit at the subject of opposites.

Ancient philosophers wrestled with the question whether light exists and darkness is the absence of light, or whether the reverse is true. One can ask the same question with respect to every pair of opposites, and there are even formulations that discuss the same question with respect to the concepts of "being" and "non-being" themselves (they ask: is the "non-being" not, or does it exist).

A fundamental question in this context is why it is assumed at all that only one of these concepts exists, while the second is an absence. Why should we not also consider the possibility that both of them exist? It seems to me that the explanation for this assumption lies in the fact that light and darkness relate to one another not as opposites of contrariety (1 as opposed to -1) but as opposites of privation (1 and 0). When we shine light into a dark room, we get light in exactly the quantity we introduced there, and not dim light. There is no offsetting between light and darkness that lowers the level of light. If we relate to such a state as an "addition" of light to darkness, it is clear that we interpret the relation between them as 1 as opposed to 0, since: 1=1+0.

By contrast, the pair cold and heat relate to one another like 1 to (-1), since adding hot water to cold water will yield water of average temperature (0 in the algebraic terms used here). We see that cold and heat are a pair of concepts whose addition offsets their intensity, and therefore they do not relate to one another as existence and absence but as opposed properties.[17]

We thus learn that there are two types of inverse relation between properties: a relation of privation, and a relation of opposition. A relation of privation, or of existence and absence, denotes a relation similar to the mathematical relation between 1 and 0. A relation of opposition denotes a relation similar to the mathematical relation between 1 and (-1). The negation operation is interpreted, in different contexts, in both senses: privative negation or oppositional negation.[18]

The question relevant to our concern is how we determine a relation of inversion, or opposition, between two phenomena: is this an a priori determination, or an a posteriori one? At first glance the determination that light is the opposite of darkness is empirical, and therefore a posteriori. And straightforwardly, the same would seem to hold for cold and heat.

It seems to me that this determination is incorrect. Hume raised the problematic nature of inferring a causal relation between two events from empirical observation. Observation can yield only the fact that there is usually a relation of temporal precedence between them; the relation of causal production is our conclusion. In similar fashion we can argue with respect to the relation of inversion, as also with respect to many other relations.[19] Treating light as the opposite of darkness, and vice versa, cannot be based on observations. Empirical observation can only show that usually, when there is no light, there is darkness, but not define this pair as a pair of opposites. The determination that they are opposites is made by us, not by the power of observation. The same applies to cold and heat. If so, it seems that the determination that a given pair of concepts are opposites is not derived from experience: it is a priori and not a posteriori.

Anyone who wishes to insist on defining inversion in a technical way, that is, that whenever there is no darkness there is light, may perhaps escape the problem, exactly like the formal definitions of causality as temporal precedence that were adopted because of similar difficulties. But it is clear that the essence of these concepts cannot be based on observations. This is precisely the reason that led Kant to his innovation concerning the synthetic a priori. He wanted to preserve the essence of the synthetic a priori concepts of causality, and not remain at the narrow technical-formal level that Hume defended. Our conclusion at this point is that negation is an a priori operation.

In the last paragraphs we saw that negation is an a priori operation. This seems obvious with respect to oppositional negation, but at first glance one might understand the relation of privative inversion as analytic (and not merely a priori). In privative negation, all the content of the opposite concept is that it is the absence of the first concept. It has no independent content, and therefore in order to understand it, it is enough to know its opposite (see below in the discussion of the theories of negative attributes, where we will see that these theories apparently do not accept the analyticity of privative negation).[20]

It seems to me that this is nothing but an illusion. For example, we saw that light and darkness are opposites of the privative kind (1 and 0). Is it really enough to understand what light is in order to know what darkness is? "Absence of light" is an expression devoid of positive content, and it is clear that it can be understood only on the basis of understanding the concept of light. But "darkness" is a positive manifestation of that negating concept. It seems to me that it cannot be understood only on the basis of understanding the concept of light. As for the relation of negation that exists between them, we have already seen above that it cannot be inferred analytically from their concepts (from David Hume's argument).

If so, only the relation between the concept "light" and the concept "not-light" can be considered analytic in the terminology we proposed above.[21] As for "light" and "darkness," while their physical relation is indeed one of privation, their conceptual-logical relation is one of opposition. The concept "darkness" has positive and independent cognitive content, beyond its being the absence of light.

A fine example of a two-stage process of emptying out a concept and then pouring positive content into that negation is found in the following quotation from Claus Harms:[22]

Thesis no. 37: I know a term in the religious lexicon that reason grasps only halfway: the term "holiday." "To observe a holiday" means, in the language of reason, "not to work," and the like.[23] But when the word is inflected into "festivity," it immediately slips from reason as a strange word, exalted above its understanding. The same applies to the words "sanctification" and "blessing."

Our conclusion is that negation is an a priori operation in its essence. Earlier we saw that negation is a non-analytic, that is, synthetic, operation. If we combine these two intermediate conclusions, we obtain the final conclusion of this chapter: the negation operation is a synthetic a priori operation.

• A Logical Basis for the "Unity of Opposites": Two Types of Contradiction

It follows from what we have said so far that there are two types of logical operations: "analytic operations" and "synthetic a priori operations." Negation is a synthetic a priori logical operation, and not an analytic one. The basis for this distinction was a distinction between two negation operations: privative negation and oppositional negation. Although it seems that privative negation itself is an operation analytic in essence, the giving of positive content (or a positive interpretation) to the result of negation is a synthetic operation.

A statement that includes an analytic contradiction, such as: "A certain being is illuminated and not illuminated at once," is a statement utterly devoid of meaning, since all the content of "not illuminated" is a privative negation of "illuminated." It has no positive content beyond the absence of light. If so, in such a case it is reasonable that a statement claiming belief of this kind cannot be accompanied by cognitive processes. By contrast, a statement that includes an a priori contradiction that is not analytic, that is, a synthetic a priori contradiction, such as: "A certain being is both illuminated and dark," definitely has meaning on the logical plane (although such a statement is usually false). A synthetic a priori contradiction between two concepts describes a situation in which each has an independent meaning, without dependence on the meaning of the opposite concept, and therefore their conjunction also has meaning. A being illuminated and dark simultaneously is a description with sense at the logical level, although it does not accord with the laws of physics known to us (a posteriori), and even contradicts our a priori understanding. This is indeed an a priori contradiction, but not an analytic one.

It is important to sharpen here that for Kant a proposition such as "The illuminated being is not dark" (like the famous "bachelor is unmarried") would be considered an analytic proposition, whereas here it is classified as synthetic a priori. Likewise the contradiction expressed in the proposition "This illuminated being is dark," which would be considered by Kant a meaningless proposition, or analytically impossible, will here be classified as a priori impossible.

The difference derives from the fact that the concept "illuminated" and the concept "not illuminated" do not have independent meaning. One cannot understand the one without the other, and in order to understand the one no additional assumption is required beyond understanding the other. By contrast, the concept "illuminated" and the concept "dark" do have meanings independent of one another. Establishing an oppositional relation between "illuminated" and "dark" is the result of an additional, a priori, assumption, but it is not an analytic result of understanding the concepts in themselves.

Let us now return to the question of knowledge and choice that was presented as an example at the beginning of our discussion. We saw that in order for an argument about the exceptional nature of the divine being to constitute an adequate solution to the paradox, two premises must hold: 1. The divine being tolerates contradictory descriptions. 2. These contradictory descriptions have logical meaning for the person who believes in them.

In light of our discussion above, it follows that such solutions may perhaps be stated with respect to pairs of concepts that stand in opposition to one another, since the simultaneous conjunction of such a pair has meaning on the logical level. God's knowledge of the future is a concept that we understand clearly without any dependence on the question of free choice, and the concept of choice is fully understandable even without understanding foreknowledge. Therefore each of these concepts can stand before itself. The contradiction between them will be derived from a complex consideration, one of whose links will include a step of oppositional negation. A connection between two conceptual worlds, whether a connection of compatibility or of contradiction, requires a synthetic "leap." Privative negation cannot create a new conceptual world, but only negate the present one. In order to connect the conceptual world of divine knowledge with the conceptual world of free choice, which differs from it, we must "leap" from the one to the other. Such a step will be made in the link of the argument that has the character of oppositional negation, or some other synthetic character equivalent to such negation.

The conclusion is that when a person declares his simultaneous belief in two principles (concepts) whose contradiction is synthetic a priori, there is a cognitive process that accompanies this statement. When there is an analytic contradiction between the concepts, that is, when one of them has no meaning beyond being the privative opposite of the second, one cannot declare belief in both of them together.

If we return to the problematic nature of the method of the unity of opposites, then simultaneous belief in two claims that contradict one another in a synthetic-a-priori way has clear meaning on the logical level (premise 2 holds for them). True, such a belief is usually false, since after all there is an (a priori) opposition between those claims, but here premise 1 enters, concerning the infinity of God, which allows a priori contradictions in His description. The source of this premise is religious faith. That is, faith, with its various sources, can provide a basis for premise 1, but without premise 2 the solution-argument has no logical basis even for a believer, and this holds only for synthetic a priori contradictions.[24]

In fact, we have created here an additional intermediate sector that separates physical contradictions, which are obviously possessed of cognitive content, from analytic contradictions, which obviously are not. The main claim is that synthetic a priori contradictions certainly do possess such content, despite being on the a priori (logical), and not physical, plane.

It should be noted that both analytic contradictions and synthetic a priori contradictions can be called "logical contradictions," since both the synthetic a priori and the analytic are a priori. On the other hand, the synthetic a priori contradiction has something in common with the a posteriori (physical) contradiction: both are non-analytic. This means that the contradiction between the pairs of concepts involved in them does not arise from the very definitions of the concepts, but because of the addition of some further principle: a priori in our case, and a posteriori in the case of physical contradictions. In the example of knowledge and choice, this principle is connected to the nature of time, and to the inability to leap over it. It is this principle that creates the contradiction between knowledge and choice, and not the meanings of the concepts as such. This principle is like a kind of "law of nature," which must be added beyond the meaning of the concepts themselves in order to infer that there is a contradiction here, except that in this case the principle is a priori and not a posteriori.

The simultaneous cognitive existence of the pairs of concepts in such cases is made possible by the ability to give up that additional principle that creates the synthetic contradiction between them (this principle is abandoned when the subject of the discussion is a divine being. This is premise 1 in theories of the unity of opposites). Such a process is impossible with respect to an analytic contradiction, since no additional principle is involved there, and the contradiction is derived from the very meaning of the concepts. Consequently, in cases of analytic contradictions there is nothing to give up, and premise 2 in Cusanus's argument does not hold, even when the issue is a divine being.

Perhaps the argument here can be considered from a somewhat different direction. There is a feeling that all contradictions constitute a logical problem at the same level, for in the end they are all formalized in the form: P and not-P, and therefore one can derive from every such contradiction any logical conclusion. In other words, contrary to my claim here, it seems at first glance that all types of logical contradiction lack cognitive value.

This feeling is mistaken because the formalization itself is the result of a synthetic process of thought. Let us discuss, for example, a physical contradiction, as in a situation where I believe that a log is in the fire and yet it is not burned (like the miracle of the bush that burns and is not consumed, in the book of Exodus). If the logical formalization of the proposition "The log is in the fire" is P, then the formalization of the proposition "The log is not burned" is by no means not-P, for there is no logical connection between the two claims. It is simply a proposition whose formalization is Q. In order to reach the formalization of P and not-P, we must perform an inference that assumes the physical contradiction between those claims. Therefore, the formal formalization of this claim is P and Q, and only with the addition of a further principle stating that this is impossible according to the laws of nature do we reach the contradictory formalization: P and not-P. Therefore this is a synthetic procedure.

With respect also to synthetic a priori contradictions, the structure is similar. The formalization of "I believe in divine foreknowledge" is not the formal inverse of "I believe in free will." There is no direct formal logical connection between these two claims. Here too, similar to the physical contradiction, in order to create a contradiction between these two beliefs I must adopt an additional (synthetic) principle, and this time an a priori one (the essence of time), which enables me to formalize these two beliefs as P and not-P.

If so, the criterion I propose for contradictions to which one cannot apply the principle of the unity of opposites is that their formalization be P and not-P, without any additional synthetic consideration. Recourse to some additional principle on the way to the contradictory formalization enables us to claim that with respect to the divine being this additional principle is not valid (premise 1, whose basis is religious faith), and thereby to adopt both contradictory positions simultaneously.

E. A Philosophical Precedent: The Doctrine of Negative Divine Attributes

There is another aspect in religious thought that hints at such a logic in the description of God, and therefore it can serve as a philosophical precedent for the argument presented up to this point. The discussion of it will be brief and not exhaustive, since this is not the place to elaborate on it. Among medieval thinkers (Jewish and non-Jewish), the approach was widespread that God cannot be described in human attributes, and therefore the use of human descriptions with respect to Him has only negative meaning. This is the doctrine called "the doctrine of negative attributes." Maimonides held that God's attributes of action are not negative, but there are thinkers who regarded all of His attributes as negative. According to this approach, when I say "God is gracious," my intention is to say "It is not true that God is not gracious."

Many have dealt with the question whether description in this form can add to our knowledge of God (Maimonides himself already dealt with this in the Guide of the Perplexed), and with other problematic aspects of these doctrines. Clearly one cannot say that the intention is only to claim that God should not be described at all in human terms, that is, that they are irrelevant to Him, for if that were so, one could just as well say that God is "not gracious," a statement whose meaning, according to this doctrine, is: it is not true to say that God is gracious. Clearly there is a higher truth-value to the statement that He is gracious than to the statement that He is not gracious, that is, the attribute "gracious" does describe God in some sense.

There are two important questions in this context that relate to our concern, and to the best of my knowledge they are discussed less: a. Is the statement "It is not true that God is not gracious" not equivalent to the statement "God is gracious"? If so, what theological gain is obtained from the doctrine of negative attributes? b. What defines an attribute as negative? Why is "not gracious" a more negative attribute than the attribute "gracious," and therefore a proposition containing it constitutes a legitimate description of God? Is it because no special word has been designated for it in the language, and it is described by means of a negation operator? I could define the attribute "not gracious" as "fanun," and then it would become positive, while the attribute "not-fanun" (which means "gracious") would become a negative attribute, and describing God by means of it would ostensibly be legitimate.

It seems that implicitly there are in the doctrine of negative attributes two premises that touch on the discussion here: a. The negation of "not gracious" is not equivalent to the attribute "gracious." This reminds us of the oppositional inversion discussed above. There too we saw that two negating concepts are not merely the analytic absence of one another. There is meaning to the statement that it is not true that God is not gracious, and this is not equivalent to the statement that God is gracious. True, there we saw this only with respect to pairs of concepts with independent definitions (X and Y), and not as in the present case, where the pair is a privative inversion (X and "not-X").

b. The additional premise is that an attribute for which there is a word in the language expresses independent content. Defining not-X as an independent attribute Y is an arbitrary logical operation. Usually, when there is an independent word for some attribute and it is used by speakers of the language, one cannot reduce that attribute to the privation of another attribute. In such a case the relation between them is always a relation of opposition and not of privation. Therefore "gracious" has positive content, whereas "not-gracious" has no positive content whatsoever; it is only the privation of the attribute "gracious."[25] The doctrine of negative attributes determines that one may use in describing God only attribute-words that have no positive content.

This premise solves the problem only if we add to it another point. The doctrine of negative attributes claims that the statement "God is gracious" is not analytically equivalent to the statement "It is not true that He is not gracious," that is, that the equivalence between them is a priori, but not analytic. We see here a conception according to which even privative negation is a non-analytic operation.

F. A New Meaning for the Synthetic A Priori

The distinction between logical and physical necessity is accepted in the philosophical literature. This distinction, in the terms of modal logic, says that physical necessity is a necessity only in our world, whereas logical necessity is a necessity in every possible world. Likewise, logical impossibility is impossible in every possible world, whereas physical impossibility is impossible only in our world.

The question is what place is occupied by the distinction I have drawn here between analytic necessity and synthetic-a-priori necessity. Analytic necessity is certainly a necessity in every possible world, and in this it parallels accepted logical necessity. The question is what the status is of a priori necessity, or more precisely synthetic-a-priori necessity. This is a weaker necessity, but its weakness is not expressed in terms of possible worlds. It seems to me that knowledge and choice are contradictory in every possible world, but not with respect to every possible being. This seems to express a weaker intensity of contradiction, and not really a different kind of contradiction.

The hierarchy among these propositions is not expressed in the terms of possible worlds. To try to define this, perhaps one can think of a different way, from Kant's, of looking at the relation between the analytic and the a priori. In chapter C I presented them, as Kant does, as orthogonal concepts, that is, as two independent axes of reference: the epistemological axis stretched from the a priori to the a posteriori, and the axis of thought-language stretched from the analytic to the synthetic. Because of the independence of the axes, it seemed at first glance (= a priori?) that four different categories of propositions had to arise here, like a table of two cells by two cells. The fact that in practice only two such categories are created (in pre-Kantian philosophy), or three (in Kantian philosophy), is a fact of nature (= not a priori) that stems from the structure of human cognition and thought. We as human beings have no way to process information except either in the form of analytic thought or in the form of learning from experience. In another possible world there may be creatures for whom there would be four different categories of propositions.

Kripke[26] takes this way of looking to an extreme and argues, precisely because of the assumption that these are two perpendicular axes, that even in our world there really are all four of these categories. For example, a proposition like 2+2=4 is for Kripke analytic-a-posteriori,[27] and not a priori, since an ordinary person learns it from experience, by means of empirical demonstrations of combining objects. Kripke is correct at the technical level, but there is a sense that this is still only a technical argument, since the child does not know this from experience. Experience seems only to help him actualize the conceptual analysis. Therefore it seems that in terms of the level of certainty there is here an analytic proposition, or at least an a priori one. In fact Ayer already notes this point,[28] but in effect he "throws out the baby with the bathwater," that is, he refuses to distinguish between the analytic and the a priori at all, and therefore his approach is not relevant to the present argument.[29]

The point hidden here is that perhaps one can look at these terms in a linear fashion, rather than in the accepted two-dimensional one. There are not two perpendicular axes here, an epistemological and a logical axis, but one axis only, namely the axis of certainty, or necessity. The analytic proposition is a completely necessary proposition, and therefore it stands highest on this axis. Its opposite has no meaning at all. The opposite of an analytic proposition will be an analytic contradiction. An analytic proposition is of course also a priori, but this adds nothing to it.

Second in this linear hierarchy is the synthetic-a-priori proposition. It is necessary; we also cannot imagine a situation in which it would not be true, but at the purely semantic level there is meaning to a statement that contradicts it. Therefore one can say that there is here a priori necessity, but not analytic necessity. Its negation is a synthetic-a-priori contradiction. That is, the conjunction "synthetic a priori" does not mean an intersection between two concepts that belong to perpendicular axes (logical and epistemological), but rather its real meaning is in fact "a priori," and the addition of "synthetic" comes only to say that we are speaking of an a priority that is not analytic. Lowest in the hierarchy is the a posteriori proposition, which is necessary only in our world (by virtue of the laws of nature that prevail in it).[30]

It seems, then, that there is a linear relation between the degree of necessity of three kinds of propositions, and only these are possible at the essential level (as Kant held): the analytic proposition (which is always a priori), the synthetic-a-priori proposition, and the a posteriori proposition (which is always synthetic). The same hierarchy also exists among contradictions, as I showed above.

It seems to me that such a way of looking presupposes that the distinction between epistemology and thought (= the two perpendicular axes according to the accepted way of viewing the matter) is not as sharp as is commonly thought. The Kantian explanation, which relies on transcendentality, does not seem plausible to me, as it does not to many others. A more plausible explanation, to which I can only allude here, is the adoption of an essentialist approach to concepts. Thinking, according to this approach, is "contemplation" of concepts, that is, a kind of epistemic process. The "objects" at which we contemplate are not tangible objects but concepts. A process of analytic thought is an analysis of a concept that stands before my cognition, and not an internal, introspective process of thought. If this is indeed the case, then thought and cognition are not two sharply distinct processes, and therefore it is possible to place all the above kinds of propositions in a linear relation on one shared axis.[31]

G. Summary

At the beginning of this article I mentioned that it is commonly thought that the religious person, in light of his faith, is also prepared to accept thinking that appears paradoxical and non-rational. At the outset I briefly hinted at the fact that usually, when people speak of paradoxicality in this context, they mean paradox only in a borrowed sense. Even a believer does not so quickly arrive at paradoxical statements. Many times, when we find statements about logical paradoxes, what is intended is historical or physical paradoxes, that is, a posteriori ones, or at most a priori paradoxes, and certainly not analytic paradoxes. Thinking of this sort, despite its apparent paradoxicality, can have a logical grounding.

Such thinking has a place in any rational system of thought, not necessarily a religious one. Ignoring it generally reflects disregard for the principled possibility of synthetic-a-priori propositions, something like what Ayer proposes in his above-mentioned article. That very approach prevents the possibility of rational scientific understanding, because of the problems raised by Hume. Religious "mysticism" and scientific thinking are both based on synthetic-a-priori propositions, and anyone who seeks the differences between them (which certainly do exist, although they are not as sharp as is commonly thought) must look elsewhere.

[1] I thank the members of the Department of Philosophy at Ben-Gurion University who invited me to lecture before them on this subject, and especially Dalia Drai. Quite a number of their useful comments are embedded in the discussion below. I also thank the anonymous reviewer who drew my attention to several important points and necessary clarifications.

[2] Kierkegaard, like Rudolf Otto, deals extensively with the non-rational and the paradoxical, but often, especially in Otto, what is meant is the numinous (= the sublime, the exalted) and not the paradoxical. See, for example, Otto, The Idea of the Holy, Karmel, Jerusalem, 1999, translated by Miriam Ron, p. 68, where he discusses the meaning of the non-rational, and clearly does not mean paradoxicality at the logical level (this is explicit, for example, in note 42 on p. 71). For this reason it is also clear that his ideograms, which constitute a mode of relating to descriptions of God, are not meant to solve logical paradoxes, but to make possible a relation to what cannot be related to in human concepts, to the sublime (see, for example, ibid., p. 70, and Joseph Ben-Shlomo's afterword to Otto's above-mentioned book, p. 195). With Kierkegaard, by contrast, there are several places where it is clear that he means a real paradox, and apparently this is also the case with Cusanus. Many do not distinguish between these two contexts; see below on this point.

[3] I do not enter here into the details of the problem. This premise too is not sufficient, since the existence of free will is a premise about the human being and not about God. For the sake of what follows I will ignore this problematic aspect, and assume that this statement, if it is indeed meaningful, can solve the problem.

[4] For a discussion of these two types 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 purpose of the discussion that follows I will assume the position that there is a logical contradiction, and not a physical contradiction, between knowledge and choice. I am not sure that I agree with this determination, but that is not important at the moment. Below I will clarify this point further.

[5] 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 a unity of opposites on the logical plane, and even brings Lukasiewicz's three-valued logic for this purpose.

[6] The most problematic example in this regard is the principle of complementarity in quantum physics, where at first glance there appears to be a unity of opposites on the logical plane, and indeed there is an approach called "quantum logic," which claims that quantum thought reflects a different theory of logic. I do not agree with this approach even in the scientific domain (see my article in Tzohar that is cited in the next note).

An expression of such logical reductionism is found in the above-mentioned notes of Ish-Shalom, and in Lewis S. Feuer's book, Einstein and His Generation, Am Oved, 5739, pp. 175-6, where he describes Lukasiewicz's three-valued logic as a basis for the anti-deterministic temper of the age, which itself served as fertile ground for the emergence of quantum theory.

Recourse to Lukasiewicz's doctrine as an explanation for phenomena of logical absurdity is very widespread, and it is important to clarify that there is no explanation in this at all. In Lukasiewicz there is a formal description of a logical system with three truth-values, but there is no explanation there of the logic behind such "thinking." There is no doubt that understanding Lukasiewicz's logical system is itself carried out (in the metalanguage) in the terms of conventional (two-valued) logic. There is no new "logic" here, but rather a formal description of a possible formal system. Its domain of applicability is a matter for determination that stems from understanding within ordinary logic. Therefore, as stated, citing Lukasiewicz adds no philosophical value to the explanation of absurdities, whose essence lies in semantics and not in formal logic. Of course, the correlation and historical influence may certainly be correct. In that sense Feuer has something that Ish-Shalom does not.

[7] See, for example, my article in Tzohar, vol. 2, Tel Aviv, winter 5760, which brings such an example, together with my response there to the article by Daniel Weil, one of the quantum logicians, cited there.

[8] Similarly, in the problem of knowledge and choice there is sometimes a failure to distinguish between two different questions: 1. How does God manage to attain knowledge about a future event that has not yet occurred at all? This information does not yet exist. 2. Even assuming that He is omnipotent, and therefore can attain information even if it does not exist (?), how is my future free choice compatible with the fact that all the information is in His possession in the present?

[9] See Ben-Shlomo's afterword to Otto's book, p. 187.

[10] It is important to note here that I am assuming an atomistic epistemological conception of concepts, and not a network conception such as Quine's. The term "knowledge" and the term "choice" are each examined on their own. The question whether there is a contradiction between them is asked only after separate examination of the meaning of each by itself.

It seems to me that the Kantian assumption that I will adopt below, namely that the analytic does not exhaust the a priori, that is, that there is a synthetic-a-priori sector, implicitly assumes an atomistic conception of concepts. In a "network" conception (a molecular conception), it is very difficult to distinguish between analytic and synthetic. All the synthetic relations enter the picture through connections in the a priori network, and in effect become analytic.

If so, Kant's assumption of a distinction between the a priori and the analytic fits well with an assumption about conceptual atomism, and therefore these are not two assumptions that are completely independent. It is reasonable to assume the one if one assumes the other. I intend to argue that this is not an ad hoc move.

[11] Kripke distinguishes between the necessary and the a priori; see, for example, his book Names and Necessity, University Publishing Enterprises, 1994 (see also the introduction of the translator and commentator Avishai Raveh). The "necessary," according to him, belongs to the metaphysical domain, whereas the "a priori" belongs to epistemology. Kripke argues that since these two divisions belong to two different domains, it is clear that propositions of all four types exist. I intend here a similar argument, but the argument here concerns the distinction between the a priori and the analytic.

I should note that Adi Tzemach also argues for the non-identity of the three concepts: the necessary, the a priori, and the analytic. See his article in Iyyun, issue 36, 5747, p. 168. There, however, there is illustration and not argument, and therefore what follows here can contribute an additional layer to understanding the picture.

[12] The concepts are of course different. My intention here is to say that the set of analytic propositions and the set of a priori propositions are identical.

[13] Among the characteristics of these concepts one must specifically perform a union in order to obtain the characteristics of the resulting concept. The more characteristics a concept has, the fewer states there are in its extension-set. It is interesting to note that according to the interpretation of Professor Aharon Barak, president of the Supreme Court, the result of this logical operation in the example before us yields only the characteristics of the concept "democratic state."

[14] As for the negation operator, see for example:

1965. S. Clarke, Jr., `Negating the Subject`, Philosophical Studies 43, 1983; P. T. Geach, `Assertion`, The Philosophical Review 74, 1965.

[15] I clarify here that I do not intend to claim that the property P is analytic for every member of the set A, but that the fact of any such element's possessing the property P is derived analytically from the fact of its being a member of the set of bearers of the property in question.

[16] There is a connection between the two exceptional properties of the negation operator. The fact that it acts on one datum and not on two necessarily leads to the result of applying the negation operator being located outside the given set, and therefore it cannot be understood from within that set alone.

[17] It should be noted that inversion is a relation between properties of objects and not between the objects themselves. Salty is the opposite of sweet, but salt is not the opposite of sugar. Therefore it is clear that the discussion here concerns the properties "illuminated" and "dark," and not the entities light and darkness. In the text I sometimes refer to properties as entities for the sake of simplicity, although this way of speaking is not precise. See my above-mentioned article in Tzohar.

[18] For further discussion of these two types of negation, see, for example, Maimonides' Guide of the Perplexed, Part I, chapter 73, seventh premise, and Part III, chapter 10. Also Torat Ha-Olah, by Rabbi Moshe Isserles (Rema), Part III, chapter 9. For additional references, see Mefane'ach Tzefunot, by Rabbi Menachem Mendel Kasher, at the beginning of chapter 5.

It should be noted that in set theory there are no parallels to these two operations. Oppositional negation is taking a complement, but privative negation is related to the (very problematic) concept of the "empty set." The emptiness of such a set is universal. In set theory there is no emptiness of a particular type. Darkness, by contrast, is the absence of light, but not total absence.

[19] Relations are not facts, and therefore as a rule they are not empirically learnable. There is admittedly room to discuss certain relations, such as _ is the father of _, and the like, which perhaps are learnable from experience alone. In any event, the very concept of relation, like the concepts "causality" or "opposition," is certainly not learned empirically.

[20] It should be noted that this distinction, even if it is correct, is not very significant with respect to the present discussion. Anyone who wishes to ground logical negation only in the first type deprives it of all its philosophical meanings. A philosophical argument by way of negation always uses the second type of negation (opposition), for otherwise nothing constructive can be inferred from such arguments. Every transition from one concept to another (even if it is the opposite of the first) is made only through an operation of opposition. Arguments that make do with a logic containing only negation of privation will be devoid of philosophical value. The principles that arise from them will be, at most, of the type "it is not true that P," and not of the type "it is true that Q." Thus nothing can be proven, but at most denied, and in fact this is a sharp expression of the emptiness of the analytic. By contrast, a logic that also allows negation of the second type, negation of opposition, is not empty because it is not analytic. It is a priori, but not analytic. The analytic is indeed empty, but the a priori is not.

[21] The positive properties of darkness (as opposed to those of light) may perhaps even be a posteriori (one cannot derive them a priori from the properties of light). The claim that darkness is the absence of light is an a priori determination.

[22] See Otto's above-mentioned book, p. 70.

[23] In Hebrew this claim may not seem self-evident, although this is indeed the meaning of the word "holiday" for most users of it. In English, as in some other languages, the words are synonymous: vacation = HOLIDAY (= "holy day," in literal translation).

[24] I note here briefly that in fact the argument in the last two chapters begins with the relation of inversion to the law of the excluded middle, that is, it presents opposites that "do not satisfy" the law of the excluded middle (there is a state 0 between them), and ends with an apparent violation of the law of non-contradiction. Opposites of this type (= contraries) do generally contradict one another, but a situation can arise, for example with respect to God, who is an infinite being, in which I will be prepared to ignore the contradiction that exists between them. This is not really a breaking of the law of non-contradiction, which according to our discussion here deals only with analytic contradictions, but of the "law of a priori contradiction," which indeed does not hold for beings such as God.

[25] The above proposal to define "fanun" is artificial, and therefore also does not exist. According to the premise here, every word in a language has positive meaning. Even if we were ostensibly to define "fanun" as "not gracious," this would only be an appearance. In fact, such a definition would reflect some positive insight that is merely stated in negative language.

[26] In his above-mentioned book.

[27] Kant considered these propositions synthetic-a-priori propositions. Exactly the opposite of Kripke.

[28] My thanks to Robert Albin of Ben-Gurion University for this comment.

[29] It seems to me that this article presents a collection of claims that can serve as significant counterexamples to Ayer's argument, that is, to support the existence of a synthetic-a-priori sector.

[30] If I had defined the axis as the axis of truths and not the axis of necessities, there would be a fourth level here (lower): a true (a posteriori) proposition that is not necessary at all.

[31] For fuller detail, see my book Two Carts and a Balloon, Beit-El, 5762.