Is belief in logical contradictions possible?[1]

2019

A new meaning for synthetic-a priori

introduction

Religious belief is often associated with illogical thinking. The symbol of this common connection is Tertullian's famous sentence: "I believe because it is illogical." A widespread approach, especially in religious existential philosophy (Kierkegaard and his followers), and especially in the Christian world, extends this relationship and says that religious life is, by its very nature, a life in paradox.[2] Here there is already a link between religious faith and anti-reason, contradictions, and not just illogicality.

In most references to the problem of logic in religious thought, the intention is not to paradoxes in their logical sense, but to the nominative and the incomprehensible in human terms. A clearer and sharper reference to logical paradoxes in religious cognition is found in Nicolaus Cusanus, who deals extensively and directly with this point, as expressed in his book Unity of opposites.

In this article I would like to discuss the idea of the unity of opposites at its logical level, which is the sharpest and most problematic. From these considerations, a reference to paradoxes in weaker senses will also arise. The question I would like to address in the present article is: is it indeed possible to live a paradox, that is, is it possible to live a life that is not only illogical, but also anti-logical.

My assumption is that the person who believes is also a person, and as such has human thinking that is subject to accepted logic. Humans apparently cannot believe in paradoxes. Moreover, some argue that paradoxical sentences (at least in the strict logical sense) have no meaning at all, and if so, it is clear that they cannot be believed. At most, they can be expressed with the mouth, but this movement of the lips cannot be accompanied by cognitive processes, such as understanding and thinking, or by any level of meaning. Is a person who says that he believes in the existence of a circular triangle actually claiming anything? Ostensibly, we should adopt, at least in these contexts, Wittgenstein's well-known recommendation at the end of the Tractatus, and remain silent.

In order for the discussion to be more concrete, I will accompany it with a reference to one of the oldest and best-known antinomies in religious philosophy, the problem of divine knowledge and free choice (hereinafter 'knowledge and choice'). This example will accompany the discussion further on, but it should be remembered that it serves as a demonstration only, and our current concern is not to investigate this issue, but to investigate the principled possibility of a unity of opposites.

A common view is that God's knowledge of the future is incompatible with the free choice of any created being. There are those who argue that the problem is imaginary, and I do not intend to address these positions here. If the problem is real, as I will assume for the sake of discussion, then it is seemingly impossible for any person to believe in God's knowledge of the future and at the same time in the existence of free choice.

In addressing this question, and even more so in relation to questions concerning the titles of God, the believer sometimes enlists the help of the "method," if one may call it that, of Nicholas of Cusa, who argued for the possibility of a unity of opposites in religious thought. Such a person would say that since God is an infinite entity, He cannot be conceived in the categories of the finite human mind, and therefore His description can, and perhaps even must, contain logical contradictions. These contradictions, those who advocate the above argument would argue, stem from the limited nature of human perception.

This type of argument assumes two assumptions, one explicit and the other implicit: 1. The explicit assumption: God is an entity that can be characterized by contradictions.[3]  2. The implicit assumption: The statement of the person who claims to believe in two contradictory beliefs is meaningful, that is, accompanied by a cognitive process.

The second assumption is critical for the unity of opposites to be a solution to any theological or philosophical problem. If a person cannot believe in God's knowledge of the future and at the same time in his ability to choose freely, there is no relevance to the statement that God is an exceptional entity that suffers contradictions. The discussion is about man's belief in God, not in God as He is to Himself.

I will try to expand a little here, and try to clarify the problematic nature of believing in logical contradictions. When a person believes that an object has both the property A and not-A at the same time, then the main question is not how he arrives at this belief, but whether this belief has any meaning. Ostensibly, the entire content of the belief that the object is characterized by the property not-A is that it is not characterized by the property A. A simultaneous combination of these two beliefs is meaningless. My assumption is that belief, and it can be based on revelation or inner enlightenment, is perhaps an alternative form of reaching some knowledge, but those knowledge are supposed to meet accepted human tests. For example, if that believer wants to draw any conclusion from his contradictory belief, he can draw anything, since from the assumption: A and not-A, any conclusion can be derived. If so, such a belief has no cognitive status.

It can be argued that there are mystical experiences that can be accompanied by such feelings. In my remarks here I do not wish to examine the psychological possibility of any mystical experiences, nor to claim that faith is not such experiences but something else (although this 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 the existence of contradictory contents in religious cognition is possible.

Here is the place to return to and clarify another point, which has already been raised above. In the present discussion we are dealing exclusively with logical contradictions. 'Physical' contradictions (= such as those that contradict the laws of nature, and not the laws of logic) do not concern the question before us, since there is no doubt that sentences that include physical contradictions are meaningful (even if in our world they are not true), meaning that their utterance is certainly accompanied by a corresponding cognitive event. Believing that God parted the Red Sea against the laws of nature is a claim with a completely clear meaning. The question of whether it is true or not depends, among other things, on assumption 1 above (God is infinite, or omnipotent). Assumption 2 (that this sentence is meaningful) clearly holds for sentences that include physical contradictions, and therefore they are not the subject of the clarification here. This point will be clarified and clarified in the course of the discussion that follows.[4]

For the same reason, I will not address further statements that mention expressions of the unity of opposites in relation to the phenomena of historical or ideological dialectics. A prominent example of this is attempts to describe the communist Mishnah regarding materialist dialectics, or the historical dialectics of Hegel's school, or that of Rabbi Kook, as examples of the logic of the unity of opposites.[5] These uses of the term 'logic of unity of opposites' are merely borrowed, since they are not real logical contradictions, but opposing ideologies, or opposing historical moves (at most philosophically, but not logically). For example, if someone were to say that there is a historical-ideological trend in our world of strengthening the status of the individual on the one hand, and at the same time a strong perception of the collective is also developing in it, and perhaps he would also add 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 strict sense. Dialectical contradictions of this kind do not require changes at the logical level of thinking, but at most at its historical-ideological level. As mentioned, in most of these cases the concept of 'unity of opposites' can be used only in a borrowed way.[6]

For the same reason, I will also ignore in what follows cases in which we speak of a unity of opposites, or of a change in logic, when in fact we mean that the logical contradiction does not exist at all, but we are unable to formulate explicitly why. The easy solution we adopt in these cases is to speak of a logic of the 'unity of opposites', but this is only a cover for the fact that the speaker has a clear meaning for the combination of the two concepts, but is unable to express it explicitly.[7]

The accepted justifications for the statements of the 'unity of opposites' use phrases such as 'perception by means beyond reason', and the like, but none of these explain how the results of such a perception exist in human thought, and constitute an object of belief. These explanations are attempts to substantiate assumption 1 above, if at all. However, as stated, the problem lies not in the possibility of obtaining this information, but in the question of whether beliefs that include logical contradictions are in themselves meaningful (assumption 2).[8]

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

The irrational today serves as a favorite subject for all those too lazy to think, or those quick to evade the tedious duty of clarifying their ideas and founding their beliefs on the foundations of coherent thought...and not only must philosophical discussion of the irrational be rational, but even religious belief itself is aided by conceptual expression, because only through it is it determined as 'faith'...as opposed to mere emotion.

To summarize, we are looking, as strange as it may sound, Logical basis for the unity of logical oppositesThat is, we would like to ask the seemingly absurd question: whether, at the logical level, it is possible to offer a reason that would substantiate the possibility of the existence of a belief that has meaning for logically contradictory expressions.

My answer, as surprising as it may sound, is that there is such a logical foundation. In order to proceed to the clarification of this foundation, we must shake the dust off the 'analytic' and the 'a priori' in Kant's doctrine a little. My main argument will be that the term 'logical contradiction' includes two different terms: 'analytic contradiction' and 'a priori contradiction.' I will further argue that, in accordance with the requirements of meaning presented above (Proposition 2), there is a unity of contradictions only when they are a priori, and not when they are analytic.

The power of this distinction can be intuitively seen if we pay attention to the example that accompanies us in the discussion, the question of knowledge and choice. The contradiction between these two concepts is clearly a priori, since it is unlikely that it arose from empirical observation of any reality. On the other hand, it is clear that this contradiction is not analytical. We can see this if we try to ask ourselves whether the terms 'divine knowledge' and 'free choice' have independent meanings. Without a doubt, each of these terms can be understood without needing the other term. 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 analytical.[10] So, in order to provide this type of solution to the problem of the unity of contradictions, one must separate the analytic from the a priori, that is, find a sector of synthetic-a priori theorems, and posit 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 conventional (Kantian) relation to the a priori and the analytic. Chapter 3 will deal with the logical negation operator, distinguishing between its two different meanings. In Chapter 4, 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 5 will present the theory of the negation of the divine titles as a philosophical precedent for this distinction. Chapter 6 will clarify that in fact a new form of relation to the synthetic-a priori is at the foundation of things, and in Chapter 7 I will conclude.

B. Analytical, a priori and synthetic-a priori

The two divisions: a priori-a posteriori and analytic-synthetic, by their very nature seem seemingly independent. The division between the a priori and the a posteriori is found on the epistemological level. In contrast, the division between the analytic and the synthetic is found in thinking (in the structure of the sentence itself, or in the definition of the concepts that appear in it).

Since these two divisions belong to different fields: epistemology and thinking, we would expect four independent groups of sentences to be formed: a priori-analytic, a priori-synthetic, a posteriori-analytic, and a posteriori-synthetic.[11] However, before Kant (and after him) the opinion prevailed that there were only two groups of sentences: analytic (=a priori) and synthetic (=a posteriori). The reason for the identification, in a simplistic description, is that the analytic sentence obviously does not need experience to verify it, and therefore the analytic is a priori. On the other hand, the a priori cannot derive from any other source than the analysis of the concept that serves as the subject of the sentence, otherwise we can confirm it from some other non-experimental source. Therefore, we also conclude in the opposite direction that the a priori is analytic. The general conclusion is that the a priori is the same as the analytic (a priori=analytic).[12] A simple logical consideration shows that these two derivations are also equivalent to the following identification: synthetic=a posteriori.

It is clear from this concise description that the other side of identification is the more problematic. The analytical is undoubtedly narrower than the a priori. An analytical theorem does not require experience, but an a priori theorem can perhaps be obtained as a result of another, non-analytical method.

The pre-Kantian view held that there were no 'margins' here, that is, these were overlapping categories. Following Hume's challenges to the rational foundation for the possibility of scientific understanding, Kant argued that there are synthetic-a priori sentences, 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 reasoning, which has serious flaws, and I will assume only the very existence of synthetic-a priori sentences, that is, I will assume Kant's separation between the analytic and the a priori without any commitment to his reasoning (see below in Chapter 6, Different Reasoning). As emerges from Hume's arguments, the existence of such sentences must be acceptable to anyone who perceives scientific understanding as a rational understanding of the world, and therefore is not directly conditional on accepting the particular reasoning that Kant proposed for them.

C. Is negation an analytic operation?

There are some basic logical operators used in most logical theories, such as conjunction, disjunction, and negation. These operators operate on concepts, where we use them to create a complex concept, or on sentences, in the same way. The logical operations on concepts can also be formulated as operations between sets (similar to Venn diagrams).

There is a relationship between these two levels of use. A logical operator that operates on concepts creates a new concept from the meanings of the given concepts. The same operator that operates on the scope groups of the corresponding concepts will give the scope group of the complex concept. For example, the conjunction operation between the concepts 'Jewish state' and 'democratic state' will give the complex concept 'Jewish and democratic state'. The intersection operation between the group of democratic states and the group of Jewish states will give the scope group of the complex concept, that is, the group of Jewish-democratic states.[13]

We will now define the concept of 'analytical operation'. An 'analytical operation' is a logical operation whose result requires us to know only the given concepts (or groups) and to apply logical tools to them, and nothing more. For example, someone who knows the group of democratic states, or the characteristics of the concept of 'democratic state', and the same applies to 'Jewish state', will be able to know and understand Based on this knowledge alone Also the concept/group 'Jewish and democratic state'. No additional knowledge is required for this.

The basic logical operations are ostensibly analytic operations. The example of the concept 'Jewish and democratic state' demonstrates the analyticity of the operation of conjunction. The same can be done with regard to disjunction. I would now like to examine the nature of the operation of negation, which, as we will see, is exceptional in relation to the other logical operations.[14]

One unusual feature of the negation operation is that it is a one-positional operator, that is, a logical operator that operates on only one data (a set or concept). The other operators listed above are two-positional, that is, they define an operation on, or between, two data.

It seems to me that the negation operation is also exceptional in that it is not an analytic operation, in the sense that we defined above. What I mean by this is to argue that in order to understand the concept/group that results from the operation of the negation operator on another concept/group, it is not enough to know the concept/group that is negated and the nature of the logical negation operation alone. The operation equivalent to logical negation in set theory is finding the complement. It should be noted that this is not a complete parallelism, since the complement is a group with independent content, and not just the negation of the group in question. In order to understand the result of the operation of finding the complement, we must know the entire relevant logical space (the union of the two groups), and then the negation becomes a logical 'subtraction'.

Let us try to illustrate this in the following way: If the group A consists of all individuals with the property P (A is the set of the extent of the property P), then all individuals that do not have the property P are not included in this group. The claim that a particular individual a included in the group A has the property P is a pure analytic claim. This is a deduction that teaches from the general to the particular. In order to discern the correctness of this claim, we only need to look closely 'inside the guts' of the group A, that is, to analyze and refine what we already know. Since we know that all members of A have the property P, it is clear that the individual a, who is one of them, also has this property. In other words: by saying that all members of the group A have the property P, we have actually already said, albeit implicitly, that a is also such. This is a distinct analytic (=analytic) consideration.[15]

Now we would like to discuss group B, which is the scope group of the non-P attribute (=Q). We will ask ourselves whether individual b has attribute P or not. We can conclude this by looking at the entire group A and verifying that he is not a member of it. If this is the case, the obvious conclusion is that he does not have attribute P. However, if we want to say that individual b has attribute Q (which is the non-P attribute), it turns out that we cannot conclude this from looking at group A, since it is possible that this member is outside the relevant logical space, but only from looking at group B, which contains all members of attribute Q. This is actually the complementary group of A (the one that contains all members in the relevant space that are not included in A). In other words, from looking at A, we cannot conclude who the members of group B (its complementary group) are in a purely analytical process. This is not just a focused look at group A and extracting information from what we know about it alone, as in the usual analytical process of learning from the general to the particular.

To know the members of the complementary group, we need knowledge beyond knowing who the members of the group A itself are, and what their properties are. This is not an analysis of the knowledge about the group A that we have only, but a look beyond this group. For example, we need to know who all the relevant individuals are (those to whom the claim that attribute the property P can be asserted or denied), that is, what the entire space (A+B) is. Once we know all of them, we can say analytically that if b is not in the group A, then it is included in its complements, that is, in B. Before we are equipped with more general knowledge of this kind, we can only say that it is not true that it is included in A, but we cannot positively state that it is included in B.

So far we have discussed the possibility of knowing whether a particular item is included in the scope group of the complementary concept. For the same reason, it is also clear that from recognition of the group A and the property of its members (P), it is not possible to positively determine the properties of the members of B (i.e., to positively understand what Q is), beyond understanding that they are members of the complement group of A. The meaning of the concept 'darkness', or the property 'dark' ((Q, cannot be understood only based on understanding the concept 'light', or the property 'illuminated' (P). In other words, the lack of analyticity of negation is its property both when it acts on the meanings of concepts and when it acts on their scope groups.[16]

From this description it seems that negation is not a simple (or pure) analytical operation. Analysis is the analysis of existing knowledge, while synthesis is the use of additional information and its combination with existing knowledge. Here it is a transformation between groups that is reminiscent in some ways of synthetic inference. In this type of consideration, the analysis of the information we have is not enough, and therefore it is similar to synthetic consideration. Our conclusion, so far, is That negation is a synthetic action.

Of course, what we have seen so far is not enough to justify the above argument. My goal is to show that negation is an a priori synthetic operation. So far we have seen, at most, that negation is not an analytic operation, that is, that it is a synthetic operation. The question is whether it is an a priori or a posteriori operation. By the term 'a priori operation' I mean that it is possible to know the properties of the set that is the result of the logical operation, in the case of a negation operation this is the set of complements (B), from the properties of the given set (A), a priori (without additional experience). If this can indeed be done, then the negation operation can rightly be called an 'a priori synthetic operation', in the same terminology defined above.

The very fact that the act of negation is classified as one of the logical acts makes it difficult to understand it as an a posteriori act. The negation of a concept is a logical, not a physical, act, and logic does not deal with the ways of experience but with the ways of thinking of the mind, a priori. In order to further clarify the a priori nature of the act of negation, and as a basis for a better understanding of the 'unity of opposites', we will look a little at the subject of opposites.

Ancient philosophers debated whether light exists and darkness is the absence of light, or whether the opposite is true. The same question can be asked in relation to any pair of opposites, and there are even formulations that discuss the same question in relation to the concepts of 'is' and 'not' themselves (they ask: is 'not' not or is).

A fundamental question in this context is why we assume that only one of these concepts exists, while the other is the absence. Why don't we also consider the possibility that both exist? It seems to me that the explanation for this assumption lies in the fact that light and darkness do not relate to each other in the same way. As the opposite of contrast (1 vs. -1) but like The opposite of island (1 and -0). When we shine light into a dark room, we receive the same amount of light as we put in there, not dim light. There is no offset between light and darkness that lowers the level of light. If we treat such a situation as an 'addition' of light to darkness, it is clear that we interpret the ratio between them as -1 versus 0, since: 1=1+0.

In contrast, the pair cold and heat relate to each other as 1 to (-1), since adding hot water to cold water will give water at an average temperature (0 in the algebraic terms here). We see that cold and heat are a pair of concepts whose addition cancels out their intensity, and therefore they do not relate to each other like being and absence, but rather as opposing qualities.[17]

We have learned that there are two types of inversion relations between properties: a relation of negation, and a relation of opposition. A relation of negation, or 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 operation of negation is interpreted, in different contexts, in two senses: negation by negation or negation by opposition.[18]

The relevant question for our case is how we determine a relation of inversion, or contrast, between two phenomena: is this an a priori or a posteriori determination? Ostensibly, the determination that light is the opposite of darkness is empirical, and therefore a posteriori. And simply put, the same is true for cold and heat.

I think this statement is incorrect. Hume raised the problem 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 advance between them, the relation of causal causation is our conclusion. We can argue similarly about the relation of inversion, as well as about many other relations.[19] The reference to 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 cannot define this pair as a pair of opposites. The determination that they are opposites is made by us not by virtue of observation. The same is true for cold and heat. It seems, then, that the determination that a particular pair of concepts are opposites is not derived from experience: it is a priori and not a posteriori.

Whoever would insist on defining the inversion in a technical way, that is, that whenever there is no darkness there is light, might be able to escape the problem, just as the formal definitions of causality as temporal precedence 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 in 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 he once claimed. Our conclusion now is That negation is an a priori action.

In the last paragraphs we saw that negation is an a priori operation. This seems obvious with regard to the counternegation, but apparently the relation of the animate inversion can be understood as analytic (and not just a priori). In animate negation, the entire content of the inverse concept is the absence of the first concept. It has no independent content, and therefore in order to understand it, it is sufficient to know its inverse (see below in the discussion of the theories of negative adjectives, where it seems that these theories apparently do not accept the analyticity of the animate negation).[20]

I think this is nothing more than an illusion. For example, we saw that light and darkness are opposites of the negative type (1 and -0). Is it really enough to understand what light is in order to know what darkness is? 'The absence of light' is an expression devoid of positive content, and it is clear that it can only be understood based on an understanding of the concept of light. However, 'darkness' is a positive manifestation of this negative concept. I think it cannot be understood only based on an understanding of the concept of light. Regarding the negation relation that exists between them, we have already seen above that it cannot be deduced analytically from their concepts (from David Yom's argument).

If so, only the relationship between the concept of 'light' and the concept of 'non-light' can be considered analytical in the terminology we proposed above.[21] As for 'light' and 'darkness', it is clear that the physical relationship between them is one of negation, but the conceptual-logical relationship between them is one of opposition. The concept of 'darkness' has a positive and independent cognitive content, beyond being the absence of light.

A beautiful example of a two-step process of negating a concept and then pouring positive content into this negation is found in the following quote from Klaus Harms:[22]

Thesis No. 37: I know a term in the religious dictionary that reason only half understands: it is the term 'holiday.' 'To celebrate' means in the language of reason 'not to work,' and so on.[23] However, when the word is translated as "festivity," it immediately escapes the mind as a puzzling word that is beyond comprehension. The same is true of the words "kiddush" and "bracha."

Our conclusion is that negation is essentially an a priori operation. Earlier we saw that negation is a non-analytic, i.e. synthetic, operation. If we combine these two intermediate conclusions, we obtain our final conclusion in this chapter: The operation of negation is a synthetic-a priori operation..

• Logical basis for the 'unity of opposites': two types of contradiction

It follows from our discussion so far that there are two types of logical operations: 'analytical operations' and 'synthetic-a priori operations'. Negation is a synthetic-a priori logical operation, not an analytic one. The basis for this distinction was a distinction between two negation operations: negative negation and counter-negation. Although it seems that negative negation itself is an analytic operation in essence, giving a positive content (or positive interpretation) to the result of the negation is a synthetic operation.

A statement that contains an analytical contradiction, such as: 'A certain object is both illuminated and unilluminated,' is a statement devoid of any meaning, since the entire content of 'unilluminated' is a negation of 'illuminated.' It has no positive content beyond the absence of light. If so, in such a case it is likely that a statement that claims this type of belief cannot be accompanied by cognitive processes. On the other hand, a statement that contains an a priori contradiction that is not analytical, that is, a synthetic-a priori contradiction, such as: 'A certain object is both illuminated and dark,' is certainly meaningful on the logical level (although it is true that such a statement is usually false). A synthetic-a priori contradiction between two concepts describes a situation in which each of them has an independent meaning, independent of the meaning of the dual concept, and therefore their combination together is also meaningful. An object that is simultaneously illuminated and dark is a description that makes sense on a logical level, even though it does not fit 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 analytical one.

It is important to clarify here that for Kant a sentence like 'the illuminated object is not dark' (like the famous 'unmarried bachelor') would be considered an analytic sentence, while here it is classified as synthetic-a priori. Similarly, the contradiction expressed in the sentence 'this illuminated object is dark', which would be considered by Kant to be a meaningless sentence, or an analytic negation, would be classified here as an a priori negation.

The difference derives from the fact that the concept of 'enlightened' and the concept of 'unenlightened' do not have an independent meaning. One cannot be understood without the other, and in order to understand one, no additional assumption is required beyond understanding the other. In contrast, the concept of 'enlightened' and the concept of 'dark' have an independent meaning. Establishing an opposite relation between 'enlightened' and 'dark' is the result of an additional, a priori assumption, but it is not an analytical result of understanding the concepts per se.

Let us now return to the question of knowledge and choice that was given as an example at the beginning of our discussion. We have seen that in order for an argument about the exceptionality of the divine entity to constitute an adequate solution to the paradox, two assumptions must be met: 1. The divine entity tolerates contradictory descriptions. 2. These contradictory descriptions have logical meaning for the person who believes in them.

In light of what we said above, it appears that such solutions may be possible to say about pairs of concepts that are in opposition to each other, since the simultaneous combination of such a pair has meaning on a logical level. God's knowledge of the future is a concept that we have a clear understanding of, independent of the question of free choice, and the concept of choice can be fully understood even without understanding the knowledge of the future. Therefore, each of these concepts can stand on its own. The contradiction between them will be derived from a complex consideration, one of whose links will include a step of counter-negation. A connection between two conceptual worlds, whether a connection of conformity or of contradiction, requires a synthetic 'jump.' A negation from an annihilation cannot create a new conceptual world, but only negate the current one. In order to link the world of concepts of divine knowledge with the world of concepts of free choice, which is different from it, we must 'jump' from one to the other. Such a step will be taken in a link of the argument that will have the nature of counter-negation, or another synthetic nature that is equivalent to such a 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 analytical contradiction between the concepts, that is, when one of them has no meaning beyond being the negative opposite of the other, it is not possible to declare belief in both at the same time.

If we return to the problematic nature of the unity of opposites method, then simultaneous belief in two synthetically-a priori contradictory claims has a clear meaning on the logical level (Proposition 2 holds for them). Although such a belief is usually false, since there is ultimately a contradiction (a priori) between these claims, this is where Proposition 1 comes in, regarding the infinity of God, which allows for the existence of a priori contradictions in his description. The origin of this assumption is religious faith. That is, faith, with its various sources, can form the basis for Proposition 1, but without the existence of Proposition 2, the solution argument has no logical basis even for a believer, and this only holds for synthetically-a priori contradictions.[24]

In effect, we have created here another intermediate sector that divides between physical contradictions, which clearly have cognitive content, and analytical contradictions, which clearly do not. The main claim is that synthetic-a priori contradictions certainly have such content, despite being on an a priori (logical) rather than physical level.

It should be noted that both analytic 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: neither is analytic. This means that the contradiction between the pairs of concepts involved in them is not created by the very definition of the concepts, but by the addition of an additional 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 related to the essence of time, and the inability to skip over it. This is the principle that creates the contradiction between knowledge and choice, and not the meaning of the concepts per se. This principle is a kind of 'law of nature' that must be added beyond the meaning of the concepts themselves in order to conclude 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 these cases is made possible by the ability to waive that additional principle that creates the synthetic contradiction between them (this principle is abandoned when the subject of the discussion is a divine entity. This is assumption 1 in theories of unity of opposites). Such a process is not possible with respect to analytic contradiction, since there is no additional principle involved, and the contradiction is derived from the very meaning of the concepts. In any case, in cases of analytic contradictions, there is nothing to waive, and assumption 2 in Cusanus' argument does not hold, even if it is a divine entity.

One might perhaps look at the argument here from a slightly different angle. There is a feeling that all contradictions constitute a logical problem at the same level, since they are all ultimately productivities in the sense of: P and not P-, and therefore any logical conclusion can be drawn from any such contradiction. In other words, contrary to my argument here, it seems that all types of logical contradictions have no cognitive value.

This feeling is wrong because the generation itself is the result of a synthetic thinking process. Let us consider, for example, a physical contradiction, such as in the situation where I believe that a log is in the fire and yet it does not burn (like the miracle of the burning bush in the book of Exodus). If the logical generation of the sentence 'The log is in the fire' is P, then the generation of the sentence 'The log does not burn' is in no way not-P, since there is no logical connection between the two claims. It is simply a sentence whose generation is Q. In order to arrive at the generation of both P and not-P, we must perform an inference that assumes the physical contradiction between these claims. Therefore, the formal generation of this claim is both P and Q, and only by adding an additional principle that states that this is not possible according to the laws of nature, do we arrive at the contradictory generation: P and not-P. Therefore, this is a synthetic procedure.

Even with respect to synthetic-a priori contradictions, the structure is similar. The generalization of 'I believe in divine knowledge' is not a formal inversion of 'I believe in free choice'. 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, this time an a priori one (the essence of time), which allows me to generalize these two beliefs as both P and non-P.

So, the criterion I propose for contradictions for which the principle of unity of opposites cannot be adopted is that their formal synthesis be both P and not-P, without any further synthetic consideration. The need for some additional principle on the way to contradictory synthesis allows us to claim that for the divine entity this additional principle is not valid (assumption 1, which is based on religious belief), and thus to adopt both contradictory positions simultaneously.

E. Philosophical Precedent: The Doctrine of the Negation of Divine Titles

There is another aspect of religious thought that suggests such logic in the description of God, and therefore it can constitute a philosophical precedent for the argument presented so far. The discussion of it will be brief and not exhaustive, as there is no room here to extend it. Among medieval thinkers (Jews and non-Jews), the attitude prevailed that God cannot be described with human titles, and therefore the use of human descriptions of Him has only a negative meaning. This is the doctrine called the 'doctrine of negative titles.' Maimonides perceived that God's action titles are not negative, but there are thinkers who perceived all of His titles as negative. According to this approach, when I say 'God is a nerd', I mean to say 'it is not true that God is non-nerd'.

Many have addressed the question of whether a description in this way can add to our knowledge of God (the Maimonides himself, in his book The Guide to the Perplexed, addressed this), and other problematic aspects of these teachings. It is clear that the intention is simply to claim that God should not be described in human terms at all, meaning that they are irrelevant to Him, otherwise it would be equally possible to say that God is 'not a nerd', a statement that means, according to this teaching, that it is incorrect to say that God is a nerd. It is clear that there is a higher truth value to the statement that He is a nerd than to the statement that He is not a nerd, meaning that the title 'nerd' does describe God in some sense.

There are two important questions in this context that are relevant to our case, and to the best of my knowledge they are less discussed: a. Is the statement 'It is not true that God is not a nerd' not equivalent to the statement 'God is a nerd'? And if so, what theological gain is gained from the doctrine of negative adjectives. b. What defines an adjective as negative. Why is 'not a nerd' a more negative adjective than the adjective 'nerd', and therefore a sentence containing it constitutes a legitimate description of God? Is it because there is no special word for it in the language, and it is described with a negation operator. I could define the adjective 'not a nerd' as 'nerd', then it would become positive, and the adjective 'not a nerd' (which means 'nerd') would become a negative adjective, and describing God using it would seemingly be legitimate.

It seems that implicitly, in the theory of negative adjectives, there are two assumptions that are relevant to the discussion here: A. The negation of 'not a nerd' is not equivalent to the adjective 'nerd'. This reminds us of the opposite inversion discussed above. There too, we saw that two concepts that negate are not just the analytical absence of each other. There is a meaning to the statement that it is not true that God is not a nerd, which is not equivalent to the statement that God is a nerd. Although 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 the inversion of nothing (X and 'not-X').

B. The additional assumption is that an adjective for which there is a word in the language expresses an independent content. Defining not-X as an independent adjective Y is an arbitrary logical operation. Usually, when there is an independent word for some adjective, and it is used by speakers of the language, this adjective cannot be placed on the negation of another adjective. In such a case, the relationship between them is always one of contrast and not of negation. Therefore, 'nerd' has a positive content, while 'non-nerd' does not have any positive content, it is only the negation of the adjective 'nerd'.[25] The doctrine of negative adjectives states that only adjectives that do not have a positive content can be used to describe God.

This assumption solves the problem only if we add another point to it. The theory of negative adjectives claims that the statement 'God is a nerd' is not analytically equivalent to the statement 'It is not true that He is not a nerd', that is, that the equivalence between them is a priori, but not analytic. We see here a view that even the negative negation is a non-analytic operation.

F. A new meaning for synthetic-a priori

The distinction between logical and physical necessity is accepted in philosophical literature. This distinction, in terms of modal logic, says that physical necessity is a necessity only in our world, while logical necessity is a necessity in every possible world. Similarly, a logical adverb is adverb in every possible world, while a physical adverb is adverb only in our world.

The question is what place the distinction I have made here between analytic necessity and synthetic-a priori necessity occupies. Analytical necessity is certainly a necessity in every possible world, and in this it is parallel to the conventional logical necessity. The question is what is the status of a priori necessity, or more precisely synthetic-a priori. 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 object. This seems to express a weaker strength of contradiction, and not really a different kind of contradiction.

The hierarchy between these sentences is not expressed in terms of possible worlds. To try to define this, one can perhaps think of a different form of observation than the Kantian one regarding the relationship between the analytic and the a priori. In Chapter 3, I presented them, as Kant does, as orthogonal concepts, that is, as two independent axes of reference: the epistemological axis that extends from the a priori to the a posteriori, and the cognitive-linguistic axis that extends from the analytic to the synthetic. Due to the independence of the axes, it seemed to the first rationalist (= a priori?) that four different categories of sentences must be created here, like a table of two squares by two squares. The fact that in reality only two such categories are created (in pre-Kantian philosophy), or three (in Kantian philosophy), is a fact of nature (= non-a priori) that stems from the structure of human cognition and thinking. We as humans have no way to process information except in the form of analytical thinking or in the form of learning from experience. In another possible world, there might be creatures for whom there would be four different categories of sentences.

Kripke[26] He takes this form of observation to an extreme and claims, precisely because of the assumption that there are two perpendicular axes, that in our world too there are indeed all four of these categories. For example, a sentence like 2+2=4 in Kripke is analytic-a posteriori,[27] And not a priori, since an ordinary person learns it from experience, through empirical demonstrations of combining objects. Kripke is right on a technical level, but there is a feeling that this is still only a technical argument, since the child does not know this from experience. It seems that experience only helps him to carry out the conceptual analysis. Therefore, it seems that in terms of the level of certainty there is an analytic theorem here, or at least an a priori one. In fact, Ayer already makes this point,[28] But he is effectively 'throwing the baby out with the bathwater', meaning he refuses to distinguish between the analytic and the a priori at all, and therefore his approach is irrelevant to the current argument.[29]

The hidden point here is that these terms can perhaps be viewed linearly, rather than in the conventional two-dimensional manner. There are not two perpendicular axes here, epistemological and logical, but one axis, and that is the axis of certainty, or necessity. The analytic theorem is a completely necessary theorem, and therefore it is highest on this axis. Its converse has no meaning at all. The converse of an analytic theorem would be an analytic contradiction. An analytic theorem is of course also a priori, but this adds nothing to it.

Second in this linear hierarchy is the synthetic-a priori theorem. It is necessary, we also cannot imagine a situation in which it would not be true, but on a purely semantic level there is a meaning to a statement that contradicts it. Therefore, we can say that there is an a priori necessity here, but not an analytic one. Its negation is a synthetic-a priori contradiction. That is, the combination 'synthetic-a priori' does not mean an intersection between two concepts that belong to perpendicular axes (logical and epistemological), but its real meaning is actually 'a priori', and the addition synthetic only means that we are speaking in a priori that is not analytic. The lowest in the hierarchy is the a posteriori theorem, which is necessary only in our world (due to the laws of nature that prevail in it).[30]

It seems, then, that there is a linear relationship between the level of necessity of three types of sentences, which are the only ones possible at the essential level (as Kant held): the analytic sentence (which is always a priori), the synthetic-a priori, and the a posteriori (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 at things assumes that the distinction between epistemology and thinking (= the two perpendicular axes according to the accepted form of reference) is not as sharp as it is generally accepted. The Kantian explanation, based on transcendence, is not plausible to me, as it is to many others. A more plausible explanation for this, which I can only hint at here, is the adoption of an essentialist approach to concepts. Thinking, according to this approach, is 'observing' concepts, that is, as a kind of epistemic process. The 'objects' that we observe are not tangible objects but concepts. The process of analytical thinking is an analysis of a concept that stands before my cognition, and not an internal, introspective thought process. If this is indeed the case, then thinking and cognition are not two sharply distinguished processes, and therefore it is possible to place all of the above types of sentences in a linear relationship on one common axis.[31]

G. Summary

At the beginning of my remarks, I mentioned that it is common to think that the religious person, in light of his faith, is also willing to accept thinking that seems paradoxical and irrational. At the beginning of my remarks, I briefly alluded to the fact that usually when talking about paradoxes in this context, we mean paradox in a borrowed sense only. Even a believer does not arrive so quickly at paradoxical statements. Often when we find statements about logical paradoxes, we mean historical or physical paradoxes, that is, a posteriori paradoxes, or at most a priori paradoxes, and certainly not analytical paradoxes. This type of thinking, despite its apparent paradoxes, can have a logical basis.

Such thinking has a place in any rational system of thought, not necessarily religious. Ignoring it usually reflects ignoring the principled possibility of synthetic-a priori statements, such as what Ayer suggests in his aforementioned article. This approach precisely prevents the possibility of rational scientific understanding, due to the problems raised by Hume. Religious 'mysticism' and scientific thinking are both based on synthetic-a priori statements, and anyone looking for the differences between them (which certainly exist, although they are not as sharp as is commonly thought) must look for their food in other areas.

[1] I thank the members of the Philosophy Department at Ben-Gurion University for inviting me to lecture on this topic, and especially Dalia Derai. Several of their helpful comments are reflected in the following. I also thank the anonymous reviewer who drew my attention to several important points and necessary clarifications.

[2] Kierkegaard, as well as Rudolf Otto, deal extensively with the irrational and the paradoxical, but often, especially in Otto, the intention is the nominative (=the sublime, the exalted) and not the paradoxical. See, for example, in Otto, Holiness, Carmel, Jerusalem, 1999, translated by Miriam Ron, on page 68, where he deals with the meaning of the irrational, and clearly does not intend paradoxes on a logical level (this is made clear, for example, in note 42 on page 71). For this reason, it is clear that his ideograms, which constitute a form of reference to descriptions of God, do not come to resolve logical paradoxes, but rather to enable reference to that which cannot be referred to in human terms, to the sublime (see, for example, ibid., page 70, and the afterword by Yosef ben Shlomo to Otto's aforementioned book, page 195). In Kierkegaard, on the other hand, there are several places where it is clear that he intends a real paradox, and this is probably the case with Cusanus as well. Many do not distinguish between these two contexts, and see on this below.

[3] I will not go into the details of the problem here. This assumption is also insufficient, since the existence of free choice is an assumption about man and not about God. For the sake of the rest, I will ignore this problematic aspect, and I will assume that this statement, if it does indeed make sense, 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 the article by Yehudit Ronen, in Between religion and morality, Avi Sagi and Daniel Statman (eds.), Bar Ilan University, 1994, page 35. As stated, for the purposes of the discussion that follows, I will assume the position that there is a logical contradiction, not a physical contradiction, between knowledge and choice. I am not sure that I agree with this statement, but that is not important at the moment. I will clarify this point further below.

[5] See the book by Benjamin Ish Shalom, Rabbi Kook between rationalism and mysticism, Am Oved, second printing, 1990. In note 71 to the first chapter and in note 133 to the third chapter, he links Rabbi Kook's positions to the unity of opposites on the logical level, and even brings in Lukashevich's three-valued logic for this purpose.

[6] The most problematic example of this is the principle of complementarity in quantum physics, where it seems that there is a unity of opposites on the logical level, and indeed there is an approach called 'quantum logic', which claims that the quantum way of thinking reflects a different theory of logic. I do not agree with this approach in the scientific field either (see my articles inNoon which is indicated in the following note).

An expression of such logical reductionism is found in the above-mentioned comments of a peacemaker, and in Louis S. Foer's book, Einstein and his contemporaries, Am Oved, 1979, on pages 175-6 where he describes Lukashevich's three-valued logic as the basis for the anti-deterministic mindset of the period, which itself served as fertile ground for the growth of quantum theory.

The reliance on Lukashevich's teachings as an explanation for the phenomena of logical absurdity is very common, and it is important to clarify that this is not an explanation at all. Lukashevich has 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 the understanding of Lukashevich's logical system is itself done (in meta-language) in terms of conventional (two-valued) logic. There is no new 'logic' here, but rather a formal description of a possible formal system. Its scope is a matter of determination that stems from an understanding of ordinary logic. Therefore, as mentioned, Lukashevich's quote does not have any added philosophical value for the explanation of absurdities, the essence of which lies in semantics and not logistics. Of course, the correlation and historical influence can certainly be correct. In this sense, Feuer has what Ish Shalom does not.

[7] See, for example, my articles B.Noon'Volume 2, Tel Aviv, Winter 2009, which provides such an example, and in my reference there to the article by Daniel Weyl, one of the quantum logicians, cited there.

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

[9] See Ben Shlomo's afterword to Otto's book, on page 187.

[10] It is important to note here that I assume an atomistic epistemological view of concepts, and not a network view like that of Quine. The terms 'knowledge' and 'choice' are each examined in their own right. The question of whether there is a contradiction between them will be asked after a separate examination of the meaning of each of them.

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

So, the Kantian assumption of the distinction between a priori and analytic fits well with the assumption of conceptual atomism, so there are no two assumptions here that are completely independent, and it is reasonable to assume one if one assumes the other. I intend to argue that this is not an ad hoc step.

[11] Kripke distinguishes between necessary and a priori, see for example his book Names and Necessity, University Publishing House, 1994 (see also the introduction by the translator and commentator Avishai Raveh). The 'necessary', he claims, belongs to the metaphysical realm, while the 'a priori' belongs to epistemology. Kripke argues that since these two divisions belong to two different realms, it is clear that there are sentences of all four types. I am aiming for a similar argument here, but the argument here deals with the distinction between the a priori and the analytic.

I would like to point out that Adi Tzemach also claims that the three concepts are not identical: the necessary, the a priori, and the analytical. See his article ininspection, Issue 36, 1987, p. 168. Although there is a demonstration there and not a reasoning, therefore the continuation of our discussion here can contribute an additional layer to understanding the picture.

[12] The concepts are of course different. What I mean here is that the group of analytic theorems and the group of a priori theorems are the same.

[13] The characteristics of these concepts must be combined in order to obtain the characteristics of the concept of the result. The more characteristics there are for a concept, the fewer countries there are in its scope group. It is interesting to note that, according to the interpretation of Prof. Aharon Barak, President of the Supreme Court, the result of this logical operation in the example before us, gives the characteristics of the concept of 'democratic state' only.

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

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

[15] I will clarify here that I do not intend to claim that the property P is analytic for every member of a group A, but rather that the fact that each such member has property P is analytically derived from the fact that it is a member of the group that has the property in question.

[16] There is a connection between the two unusual properties of the negation operator. The fact that it operates on one given rather than two necessarily leads to the result of operating the negation operator being outside the given set, and therefore cannot be understood from within it alone.

[17] It should be noted that inversion is a relation between properties of objects and not between the objects themselves. Salty is the inverse of sweet, but salt is not the inverse of sugar. Therefore, it is clear that the discussion here is about the properties of 'light' and 'dark', and not about the objects of light and darkness. In the text, I sometimes refer to properties as entities for the sake of simplicity, although this reference is not precise. See my aforementioned articles inNoon.

[18] For further discussion of these two types of negation, see for example Teacher of the Confused, Maimonides, Part I, Chapter 3, Seventh Introduction, and Part III, Chapter 10. and The doctrine of the rise, by Rabbi Moshe Isserlesh (Rema), Part 3, Chapter 9. For additional places, see Northern decoder, by Rabbi Menachem Mendel Kosher, at the beginning of Chapter 5.

It should be noted that in set theory there are no parallels to these two operations. Contradictory negation is the taking of complements, but negation from negation is related to the (very problematic) concept of an 'empty set'. The emptiness of such a set is universal. There is no particular kind of emptiness in set theory. In contrast, darkness is the absence of light, but not a complete absence.

[19] Relations are not facts, and therefore, as a rule, they cannot be studied empirically. It is true that there is room to discuss certain relations, such as _ is the father of _, and the like, which may be learned from experience alone. In any case, the very concept of the relation, like the concepts of 'causality' or 'contrast', is certainly not studied empirically.

[20] It should be noted that this distinction, even if it is correct, is not so significant in relation to the present discussion. Whoever would like to place logical negation only on the first type, castrates it of all its philosophical meanings. A philosophical argument by way of negation always uses the second type of negation (the contradiction), otherwise nothing constructive can be deduced from such arguments. Any transition from one concept to another (even if it is the opposite of the first) is made only by an operation of contradiction. Arguments that are satisfied with a logic that contains only an operation of negation of negation will be devoid of philosophical value. The principles that will arise from them will, at most, be of the type that P is not true, and not of the type that Q is true. Thus, nothing can be proven, but at most it can be negated, and in fact this is a sharp expression of the emptiness of the analytic. On the other hand, a logic that also allows for a negation of the second type, the negation of a contradiction, 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 (compared to those of light) may even seem a posteriori (they cannot be derived a priori from the properties of light). The fact that darkness is the absence of light is an a priori determination.

[22] See Otto's aforementioned book, page 70.

[23] In Hebrew, this statement seems not to be obvious, although this is indeed the meaning of the word 'holiday' for most users. In English, as well as in other languages, the words are synonymous: HOLLIDAY = (='holy day', literally translated).

[24] I will briefly note here that in fact the argument in the last two chapters begins with the relation of the opposite to the third law of the averted, that is, presents opposites that 'do not satisfy' the third law of the averted (there is a state of 0 between them), and ends with an apparent violation of the law of contradiction. Opposites of this type (=contradictions) are indeed usually contradictory, but a situation can arise, for example with respect to God, who is an infinite entity, in which I would be willing to ignore the contradiction that exists between them. This is not really a violation of the law of contradiction, which, as we say here, deals only with analytic contradictions, but of the 'law of a priori contradiction', which does indeed not hold true for entities like God.

[25] The above proposal to define 'fanon' is artificial, and therefore non-existent. According to the assumption here, every word in the language has a positive meaning. Even if we ostensibly define 'fanon' as 'not a nerd', 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 aforementioned book.

[27] Kant considered these sentences to be synthetic-a priori sentences. 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 arguments that could constitute examples of significant counterweight to Ayer's argument, that is, supporting the existence of a synthetic-a priori sector.

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

[31] For more details, see my book Two carts and a hot air balloon, Bethel, 5762.