Gate Eight: Against Analyticity — Second Move

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

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Against Analyticity: A Second Move

This gate contains four chapters.

1. Chapter 1: Between Deduction, Analogy, and Induction
2. Chapter 2: The Ways the Analytic Thinker Defends the Deductive Mode of Thought
3. Chapter 3: The Meaning and Relativity of Axioms
4. Chapter 4: The Failure of Formal Language to Resolve Paradoxes

Introduction

As we described in the introduction to this part, Gate Seven dealt with the analytic position on the philosophical plane. In the present gate we will examine the analytic position on the logical plane.

In Gate One we saw that the form of argument and inference that represents the thinking of those who hold the analytic position is deductive thought. We will examine this form of inference and try to show that one cannot make meaningful progress from a conception that believes only in knowledge proved deductively.

In many respects, the course of the present gate parallels that of the previous one. The difference is that the previous discussion took place on the general philosophical plane, and chiefly the epistemological one, whereas here the argument will be presented and conducted on the logical and metalogical planes.

Chapter 1: Between Deduction, Analogy, and Induction

Introduction

In Gate One we distinguished between “analytic thought” and the “analytic position.” The form of thought called in this book “analytic thought” is thought of a deductive character, whose conclusions necessarily follow from its premises (although, as we saw there, the premises themselves are not necessarily true). The “analytic position,” by contrast, is the philosophical position that holds that only analytic thought can lead us to valid conclusions, or at least to conclusions with a sufficient degree of certainty.

Let us note again that analytic thought is a tool (or a mode of thought) that is also used by those who hold the synthetic position. They too use deduction where it is relevant and appropriate. The difference between the analytic position and its synthetic counterpart lies only in the legitimacy of non-analytic modes of thought. As stated, the “analytic position” holds that “analytic thought” alone is valid. The “synthetic position,” by contrast, holds that there are additional modes of thought or cognition, and that we must also treat the conclusions that arise from them as claims possessing a sufficient degree of certainty.

In the previous gate we examined the status and validity of the very principle of analyticity itself—that is, the analytic approach which says that no certainty can be derived from synthetic statements. In this gate we will continue and argue that even if we accept the analytic principle itself, it cannot be implemented rationally. The feeling of those who hold the analytic approach, as though they really do operate this way—that is, as though they acknowledge the validity only of proven statements—is an illusion. In the previous gate we showed that the claim regarding the invalidity of synthetic statements or synthetic processes of thought is itself not analytic. Here we will continue and show that the validity of analytic processes of thought themselves is grounded in synthetic principles. We will discuss here not the “analytic position,” but the narrower concept, namely “analytic thought,” the only kind permitted by the analytic thinker’s method (apart from sense observation).

In this gate we will try to show that the validity of analytic thought is no greater than that of synthetic thought, because it too relies on synthetic principles. Therefore, the validity of analyticity is itself dependent on synthetic criteria of validity. Here one can see the parallel between this line of argument and the argument of the previous gate.

Put differently, one may say that the conclusion of the argument in the present gate will be that the analytic thinker, who relies only on analytic thought and rejects synthetic thought as unacceptable, is not consistent. If analytic thought itself is based on synthetic principles (and, as we saw in the previous gate, so too is its rejection), then synthetic thought cannot be rejected on analytic grounds.

From another angle, what emerges from all this is that synthetic thought is no less valid than analytic thought. At the base of both lies a synthetic infrastructure. This is precisely the approach of those who hold the synthetic position, who use both forms of thought. Thus, in the present gate we proceed through an examination of the modes of thought, analytic and synthetic, and from there we arrive at a decision between the philosophical positions, analytic and synthetic.

Is There a Hierarchy Among the Three Modes of Inference?

Let us return to the end of Gate One, where we defined three different modes of inference: deduction (learning from the general to the particular), induction (learning from the particular to the general), and analogy (learning from one particular to another). Deduction is the analytic mode of thought, and according to those who hold the analytic position it alone is valid.

At first glance, the deductive method does indeed appear to be the strongest of the three. Its conclusions possess absolute and necessary validity. Learning from the general to the particular seems safe and secure. Analogy and induction, by contrast, are forms of inference that make claims about the world, and therefore they seem not entirely secure, if secure at all (see Hume’s argument concerning induction, discussed in Gate One).

In Gate One we noted that deduction is a form of inference that in fact does not claim anything at all—the emptiness of the analytic—and it is precisely for that reason that it is completely certain. But even between the two remaining modes of inference, induction and analogy, there would seem at first sight to be a hierarchy of validity and strength. Analogy claims less, and therefore seems safer. Analogy does not purport to assert the truth of a proposition about an entire class, but only about one particular instance. Induction, by contrast, is apparently less secure, for its claim is more far-reaching. It asserts the existence of a property that we have seen in only one instance with respect to an entire group of instances. There is, in effect, a kind of “principle of uncertainty” here: the more far-reaching the claim (the more it asserts), the less certain it is, and vice versa.

Thus, apparently we obtain here a hierarchy of validity and strength among the three modes of inference, as follows:

Deduction > Analogy > Induction

In this gate we will see that the hierarchy among these three forms of thought is nothing more than a collection of claims that are, at most, theoretical; in practice they are mere illusions. We will come to see that, in essence, there are not three modes of inference at all, but only one. All three together make up a single form of inference. We will see that this form of inference is analogy, and in this way we will establish the superiority of the synthetic. Thus we will confirm what we described above: the synthetic stands even at the base of analytic thought.

Analogy and Induction

It is quite clear that analogy and induction belong to the same family. In order to draw an analogy between two particulars, we usually rely on properties that they share, and from this infer concerning an additional property: if it characterizes one of them, it characterizes the other as well. For example, if both Jacob and Socrates are human beings, it seems possible to infer from the fact that Socrates is mortal—that is, destined to die—that Jacob is as well. This follows because in both of them the systems of life seem to us to be built similarly, and therefore they are also candidates to end in a similar way.

One should note that in using this analogy, we covertly used induction. We implicitly assumed that anyone whose life systems are similarly structured will also die, and from this we inferred that Jacob in particular is mortal. In fact, we generalized our knowledge of Socrates’ mortality to all human beings who are similarly constructed physiologically. Only afterward did we infer regarding one individual from within that class—that the same is true of Jacob. Thus, analogy too is based, albeit covertly, on induction.

On the other hand, it seems possible to defend the opposite claim as well: the generalization that every human being is mortal is an inductive inference, but apparently it itself arises from analogy. We in fact assume that Socrates’ being mortal is relevant to the group of beings with similar life systems, and not to some other rule—for example, those whose names begin with the letter Samekh. In performing the induction, we presupposed the relevance of life-system structure to the property of being mortal, and that is a kind of analogy. After all, if I saw three horses whose names all ended with the letter Yod, I would not assume that this is a horse-like characteristic. But their being four-legged is commonly thought to be such a characteristic. That is, horseness is a relevant characteristic with respect to the number of legs, but not with respect to names.

Put differently, the general statement that a being with a human life system is mortal is made up of a collection of particular claims about all the individual beings that possess such systems. We draw an analogy to one similar instance, then to another similar instance, until we reach a generalization about the group that contains all the instances involved in the analogy.

The conclusion that follows is that to the same extent that analogy is based on induction, induction is also based on analogy, or on a collection of analogies. We thus learn that the hierarchy of validity that supposedly exists between analogy and induction is illusory. Induction and analogy possess the same level of validity. Each stands at the foundation of the other, and they are interwoven with one another in a way that makes separation impossible.

Even so, deduction still seems, in any event, to stand beyond all dispute. There, validity is obviously absolute, beyond all doubt, and there seems no question that it is immeasurably stronger than the validity we attribute to induction and analogy.

We will now show that in many respects the absolute validity of deduction is also illusory, for two reasons:

1. Mill’s challenge to deduction.
2. An infinite regress that underlies the justification of deduction’s validity.

The first challenge concerns the question of where we learned the truth of the major premise of the deduction, and the second concerns the deductive method itself. We will now examine these two challenges.

Mill’s Challenge to Deduction

The English philosopher John Stuart Mill raised an argument that undermines the validity of deduction. We will illustrate it by means of the classic structure of deduction as presented in Gate One:

Premise A (the major or general premise): All human beings are mortal.
Premise B (the minor or particular premise): Socrates is a human being.
Conclusion: Socrates is mortal.

The question Mill raised was: from where did we derive our certainty regarding the truth of the major premise? Or, in other words, from where can general truths be derived at all? Apparently this can be done only by means of a process of induction (or analogy, which, as stated, underlies inductive inference). In order to know that all human beings are mortal, we must observe a number of members of the human population, see that they are mortal, and then generalize concerning all human beings.

Now we must note that a deduction of this kind is based on at least one general premise. If so, at the base of every deduction lies an induction (by whose power we reach the major premise of the inference). The validity of the conclusion of a deductive argument cannot exceed the validity of the premises on which it is based. And if the major premise has the validity of the conclusion of an inductive inference, then the deductive conclusion built upon it cannot be more valid.

One may ask the question differently: if we already know that all human beings are mortal, then we should already have known that Socrates is mortal. If so, the statement that appears at first sight to be the conclusion of the deductive argument is, in the final analysis, an assumption that lies at the basis of the major premise. Therefore it is clear that the certainty of the conclusion cannot be greater than the certainty of the major premise, and that premise was, as stated, obtained by induction.

In fact, this challenge is nothing more than the claim that deduction does not assert the truth of the conclusion of the deductive argument, but only its necessary derivation from the premises. We saw this in Gate One when we discussed the emptiness of the analytic, and of deductive inference. In the observation that follows we will discuss a halakhic (Jewish legal) aspect of Mill’s challenge.

Observation 22: A Fortiori as a Syllogism¹

At the beginning of the Sifra—a tannaitic halakhic midrash (rabbinic exposition of law) on the book of Leviticus—there appears a baraita of Rabbi Ishmael that lists thirteen principles (forms)² by which the Torah is interpreted. These are thirteen rules by means of which one can derive laws from verses in Scripture through interpretive exposition. Most of these principles appear to be rules of interpretation that are not easily intelligible to the modern person.³

There is, apparently, one striking exception among these principles: the principle of a fortiori reasoning. In order to clarify what an a fortiori argument is, I will bring an example of its use from the Babylonian Talmud, Bava Kamma 24b. The Mishnah there presents three well-known laws:

1. In a case where an animal causes damage with its foot (“foot,” in standard halakhic terminology) in the public domain, the owner of the damaging animal is exempt from payment.
2. Damage caused by a foot in the courtyard of the injured party obligates the owner to pay the full amount of the damage.
3. Damage caused by an animal’s horn (“horn,” in standard halakhic terminology) in the public domain obligates the owner of the animal to pay half the amount of the damage caused to the injured party.

The question with which the Mishnah deals is: what is the law if a person’s animal caused damage with its horn in the courtyard of the injured party?

There is an opinion in the Mishnah that in such a case the animal’s owner must pay full damages. This conclusion is learned through the well-known halakhic form known as a fortiori inference. The argument works as follows: from the laws of damage in the public domain we see that damage by horn (which requires payment of half the damage) is more severe than damage by foot (which does not require payment at all). From this one infers that it is reasonable that the same hierarchy is preserved in the courtyard of the injured party—that the laws of damage by horn are more severe than the laws of damage by foot. Therefore, if damage by foot in the injured party’s courtyard requires full payment, the same law should apply to damage by horn in the injured party’s courtyard, obligating the damager to pay the full amount.

This inference seems understandable even to contemporary eyes. Some have held that a fortiori reasoning is merely a different garb for the Aristotelian syllogism (the deductive argument we presented in the main text).⁴ But this is clearly not so, for the three premises in the a fortiori argument (see above) do not include any general major premise. The description of a fortiori reasoning as a syllogism ignores the fact that from the first and third premises the Mishnah generalizes and derives—by induction—a conclusion that says: in all contexts, damage by horn is more severe than damage by foot.

That conclusion of the generalization in the first part of the a fortiori argument is a general claim. It serves as the general premise for the argument in the second, deductive part of the a fortiori argument in our example.

The second part of the a fortiori argument infers, from the conclusion of the previous part—which is the general premise—and from the third halakhic datum, which is the particular premise, the particular conclusion: that horn damage in the injured party’s courtyard requires full damages. But as we have seen, this deduction is only the second part of the argument, whose validity is based on the fact that in the previous stage we derived the conclusion—or major premise—by means of induction. Thus the sting of the a fortiori inference (and also its weak point) lies specifically in its first, inductive part: in the generalization from two particular laws, which creates the general premise for the deductive inference. In other words, the interpretive rule—that is, the hermeneutical principle—of a fortiori reasoning is essentially an instruction to perform an induction that will yield the major premise of the deductive argument. This is an exact reflection of Mill’s argument against deduction. At the basis of deduction stands a major premise reached by way of induction, and therefore the validity of the deductive conclusion cannot be greater than that of the induction.

One may say even more than this. If we treat a fortiori reasoning as a deductive inference, then all thirteen interpretive principles can be presented similarly. Every rule can be formulated as a general premise, so that every specific use of it will be a particular case of that rule.

As an example, let us look at another principle, one less comprehensible to ordinary logic: any two subjects that appear in adjacent verses may have their laws compared (this is one version of the talmudic principle of comparison by juxtaposition). A use of this principle can be presented as follows:

Premise A (general/major): The construction of the Tabernacle and the Sabbath are two subjects that appear in adjacent verses, and therefore they are equivalent with respect to all their laws.
Premise B (particular/minor): Plowing was done in the Tabernacle.
Conclusion: Plowing is forbidden on the Sabbath.

And so one can continue with parallel arguments regarding all thirty-nine primary categories of labor. This too is a fully deductive argument. Thus, not only a fortiori reasoning but also comparison by juxtaposition can be presented as a deductive argument. The same, of course, can be done for any interpretive principle or any other rule of inference whatsoever.

But there is a similar mistake in describing this kind of reasoning as deduction. At the basis of the deductive argument lies a non-deductive consideration that creates the major premise of the deduction. It is precisely this consideration that constitutes the sting of the inference, since it is its point of vulnerability. Deduction possesses absolute and necessary validity. The refutation or challenge to this sort of inference is always found in the transformation of the halakhic assumption into the major premise of the deductive argument.

The mistake made in identifying a fortiori reasoning with Aristotelian deduction lies in a failure to understand that the main novelty conveyed by the interpretive rule is the very existence of the major premise, not the deductive possibility of drawing conclusions from it. In the case of comparison by juxtaposition, the novelty lies in the fact that two subjects appearing adjacently in Scripture may be compared. In the case of a fortiori reasoning, it lies in the fact that one may generalize from the relation between two particular laws to a more general relation between the two subjects under discussion. Once these rules are accepted, every use of them can be presented as a deduction, and at that point the principle of a fortiori reasoning is no different from the other principles.

The reason a fortiori reasoning seems more intelligible to us than the other principles is not that it is a deductive argument. The real reason is that the inferential rule that creates the major premise—in the first part of the use of a fortiori reasoning—is indeed an intuitive rule. But it is not deductive; it is inductive. The Torah allows us to perform an inductive generalization, something that sounds reasonable even to a modern person. In the case of the other principles, the interpretive rule that leads to the major premise appears arbitrary to modern eyes—adjacent verses, similarity between words, and so forth. It usually cannot be understood as an ordinary process of thought, such as induction or analogy.

There can be no doubt that using the general premise obtained from induction in relation to a specific case—that is, applying a deductive rule of inference—does not require any special authorization from the Torah, or any special interpretive rule. Reasoning of this kind is used by students of Torah just as it is used by all human beings, all the time. The Torah’s special innovation in the principle of a fortiori reasoning, as in all the other principles, lies in the way we reach the major premise (the first stage), and not in the use of that premise (the second stage).⁵

What we have seen in this example is a talmudic-halakhic consequence of Mill’s challenge to deduction. The challenge is made by pointing out that the sting of the argument lies in the way we reached the general premise, not in the use of the deductive process itself. This parallels the misunderstanding to which Mill points with regard to the syllogistic inference. There too, the main novelty is already found in the major premise itself. At this point one might say that a fortiori reasoning is a logical syllogism—but not as praise of a fortiori reasoning; rather, as criticism of logical syllogism.

In conclusion, it is worth noting that there is a different type of a fortiori reasoning which really is nothing but an Aristotelian syllogism. This is the a fortiori form known as “if it contains two hundred, it certainly contains one hundred.” This subject was discussed above in Observation 2. The a fortiori reasoning discussed here is the ordinary kind (see there for an explanation of the difference between the two types).

Returning to the Distinction Between the Three Modes of Inference

We saw above that every analogy presupposes an implicit inductive process at its base, and vice versa. Here we have seen that the same is true of deduction as well—that is, within every deductive consideration there is also an implicit induction.

The conclusion that emerges from this is that the distinction between the three forms of inference that we presented is not at all sharp. Each uses the others, and therefore it is hard even to separate them. The correct description of the relation among these three forms of inference is as one long inferential process whose basis is analogy. This process begins with a transition from pieces of knowledge about individual particulars, proceeds to a generalization of those particulars by induction into a general principle, and ends with a deductive inference from that principle to other particulars.

For example, we know that Socrates is mortal. We generalize inductively and determine that all human beings are mortal. From here we specify, by means of deduction, regarding another particular—Jacob—and infer from the general premise that Jacob too is mortal.

This entire process is an analogy from one particular—Socrates—to another particular—Jacob. Induction and deduction are two different stages that together compose the analogical inference. According to the picture presented here, these three modes of inference are not three alternative kinds of reasoning whose status can be compared and ranked. They are different stages of thought, all three of which are present in almost every intellectual process. This may be described by the following formula:

Analogy = Induction + Deduction

From this it is clear that if someone tries to grant legitimacy only to deductive inference, while undermining the legitimacy of the other processes of thought, he thereby loses the possibility of making deductive inferences as well. As we have seen, deductive reasoning usually relies implicitly on induction and analogy.

The essence of Mill’s challenge to deduction is that at the base of every deduction lies a premise—the major premise—whose validity is the validity of induction. If so, we have two possibilities. The first is to accept that the major premise was indeed obtained by generalizing observations of several members of the human species. In that case, applying the conclusion of the deductive argument to Socrates, who was not part of the group originally observed, has only the validity of induction. Deduction is only an intermediate inference, or an intermediate stage in the inferential process. We conclude that there is no pure deduction in the world, and every deduction is disguised induction. If so, we find ourselves compelled to acknowledge the legitimacy of the other modes of inference as well, and thereby to adopt the synthetic position.

There is, however, also a second possibility. One may refuse to accept the validity of this premise, and therefore also of the conclusion that follows from it. One commits only to the conclusion’s following from the premises, not to the validity of the conclusion as such. This is precisely the analyticity that places trust in no claim whatsoever.

We have already seen that the analytic position cannot advance us toward any new knowledge that we did not already possess beforehand. Such knowledge can be reached only by synthetic means. The conclusion of Mill’s challenge is that no analytic process of thought can truly say anything new about the world (in Gate One we called this “the emptiness of the analytic”).⁶ We now move to the second challenge to deduction.

The Second Challenge to Deduction: Infinite Regress

Another claim that can be raised against deduction is that there is an infinite regress in the justification of the deductive form of argument itself. This is an argument parallel to the claim in the previous gate that the statement expressing the analytic position itself is itself a synthetic (a priori) statement. Here that argument refers to a particular deductive argument, not to analytic thought as a whole.⁷

In Gate One we saw that a deductive argument of the previous type is based on accepting the following formal schema as valid in every case:

Premise A: Every Q is P.
Premise B: a is Q.
Conclusion: a is P.

This rule is a variation on the law of inference called in logic Modus Ponens, and we will therefore call it MP.⁸

The question we will now address is: what is the logical justification for the validity of this rule—MP—itself? How can we assume that this schema is valid? What justifies that assumption?

One may formulate this differently and say that at the base of this argument lies an additional premise that we did not emphasize, namely the rule MP itself. According to this, a more precise presentation of the above deductive argument would be as follows:

Premise A: Every human being is mortal.
Premise B: Socrates is a human being.
Premise C: The rule MP is correct.
Conclusion: Socrates is mortal.

Now the argument is valid, since we are also explicitly assuming the validity of the MP schema itself. But at this point one can return and say that this argument too is based on another general schema—a meta-schema—which may be called MP1. This schema says that every argument of the following type is valid:

Premise A: Every Q is P.
Premise B: a is Q.
Premise C: The rule MP is correct.
Conclusion: a is P.

Now we must once again present the argument more precisely:

Premise A: Every human being is mortal.
Premise B: Socrates is a human being.
Premise C: The rule MP is correct.
Premise D: The rule MP1 is correct.
Conclusion: Socrates is mortal.

Of course, one can continue in this way endlessly. This is a chain of argument and justification known in philosophy as “infinite regress.” The question is whether such a chain can be regarded as an adequate justification of the deductive argument. Philosophers usually reject an infinite regress as an inadequate form of justification.⁹

The argument presented here is a formal dress for the challenge to the validity of deduction itself. Here we have only seen that in order to justify the use of deductive logical inference one must go on presenting an infinite chain of justifications. One should note that we have not even asked the simpler question: why exactly is MP1 true? Is it because of MP2, which we have not yet justified? If so, how do we know that at all? All these are unnecessary stages of dialectical hair-splitting, and therefore we will not deal with them here. The problem is perfectly clear.

Let us add that for the proponent of the synthetic position this challenge is irrelevant, for exactly the same reason that the previous challenge did not attack his position. The synthetic thinker will say that the rule MP arises from contemplation of the concept “valid.” This rule does not require justification. By means of it one can justify other claims that are not self-evident to us.

If we formulate this differently, it appears that the validity of logical schemata derives from the certain feeling—“evident,” in Descartes’ terminology—of their correctness. According to this, the supreme criterion of certainty is evidence, the feeling of clarity, and it is this that grants force even to the certainty we have in the logical laws. Thus, the logical laws are not the supreme criterion of certainty; rather, sound intuition, or common sense, is. The validity of the logical laws too depends on intuition, and not the reverse.

We continue to see consistently that the distinction between epistemology and logic becomes increasingly blurred. Below, when we discuss receptive logic, we will continue along this line of argument.

Chapter 2: The Ways the Analytic Thinker Defends the Deductive Mode of Thought

Introduction

In the previous chapter two challenges were presented to deduction, which is the only mode of thought the analytic thinker regards as legitimate. In what ways can the analytic thinker defend deduction against these two arguments? (We have seen that the synthetic thinker is exempt from this defense, even though he too uses deductive tools of thought. These challenges do not attack synthetic use of analytic tools.) It seems that with respect to each challenge separately, the analytic thinker can in principle return to one of the two basic options we presented in the previous gate: the analytic-conventionalist path, and the path of transcendental arguments (which, as we have already seen there, is itself also a type of analyticity). We will now examine these two types of analytic defensive arguments with respect to each of the two challenges to deduction that were presented in the previous chapter.

In the previous gate these two directions of argument were examined in general, and the conclusions were applied there in order to refute the analytic strategies of defense against the challenges presented there. In the present gate we can make use of the conclusions of that earlier examination. We will see that these two strategies of defense do not work here either, for exactly the same reasons we encountered there. The arguments in this gate are merely specific examples of the reasons for which we rejected the analytic strategies of defense in general in the previous gate.

Defending Against Mill’s Challenge

In the analytic mode of defense against Mill’s challenge, one may say that the word “human being” serves in our language to describe entities that are also mortal. In other words, this is itself an analytic statement, and therefore valid. It is clear that in such a case no new knowledge about the world has been added. That is, the deductive mode of inference cannot add to our knowledge beyond what is already in our possession. It is also clear that if we know that Socrates is a human being, then included in that very determination is already the knowledge that he is also mortal. For this reason, the deductive argument loses much of its significance.

The proponent of the synthetic position will agree with this as well. He too agrees that deductive inference does not add to our knowledge of the world, or, in other words, that the analytic is empty. All the knowledge is already found in the major premise, which one reaches by induction, as Mill argued. The difference is that for the proponent of the synthetic position the paths of induction or analogy remain open as ways of accumulating valid knowledge. Even the major premise in the deductive argument is obtained by him through these methods. For this reason Mill’s challenge is not relevant to the proponent of the synthetic approach. For him, the use of analogy and induction is not a defect, and grounding oneself in these modes of inference does not undermine his position.

As mentioned earlier, for the analytic thinker these paths have no importance in the context of objective justification. He will have to say that only the conditional statement saying that acceptance of the premises requires acceptance of the conclusion is correct, and not the conclusion as such. Let us note here that this restrictive claim will of course be subject to the second challenge we presented above—the infinite regress—which will be discussed further below.

The analytic thinker can try to defend himself against Mill’s attack also by way of the transcendental path, and say that there can be general premises that are synthetic a priori statements. That is, they can be known a priori, without any inductive generalization from prior particular observations. Inferring a deductive conclusion from such a statement seemingly adds to our knowledge something we did not previously possess. Here the major premise is not the product of inductive generalization, but of transcendental reflection. Deduction is not merely the second part of an analogical inference.

At this point we can use the arguments from the previous gate, according to which transcendental reflection is nothing but a disguised analytic argument. As we saw there, there is no principled difference between using logical tools to analyze given knowledge and using the results of an analysis of our cognitive tools. If so, this defense too on the part of the analytic thinker is only an illusion. As we explained in the previous gate, a synthetic a priori statement is a slight variation on an analytic statement. In other words, knowledge accumulated in this way too was already implicitly in our possession beforehand. One who knows the structure of his cognition, and in addition the factual premises, in effect already knew the “synthetic a priori” conclusion long ago. Thus the transcendental argument too does not help the proponent of the analytic position add valid information beyond what is already in his possession.

Beyond this claim, it seems that in practical terms it will be difficult to find examples of synthetic a priori statements that can serve us in building useful scientific knowledge. Scientific generalizations that lead to general laws of nature are usually—and perhaps always—not transcendental arguments but rather inductions.¹⁰

It follows, then, that the analytic thinker has no way to add knowledge about the world, at least not knowledge that includes general laws about reality—that is, scientific knowledge. The only thing he can learn is simple particular facts, such as that the table before him is green. It is clear that if he has no way to accumulate knowledge beyond what is already in his possession, then he also has no way to be equipped with knowledge at all. Even initial knowledge has to be acquired somehow.¹¹ We will now move on to discuss the application of these two modes of defense against the second challenge.

Defending Against the Challenge of Infinite Regress

The situation regarding the second challenge to deduction—the infinite regress—appears similar. There too the analytic thinker can say that “valid” in our language is used in the following way: “conforming to the rule MP and to the other rules of formal logic,” and therefore there is no need whatsoever to justify the rule MP itself. Since the term “justification” is used in the language as an abbreviation for “subsumption under the rule MP and other logical rules,” it would seem rather absurd to demand that someone justify this rule by subsuming it under itself.

As in the previous gate, here too this form of explanation is insufficient. Once again we arrive at the conventionalist approach that the analytic thinker must adopt if he wishes to claim that the terms “true” and “valid” are in fact abbreviations for a collection of rules that have no connection with one another. That is, it is an abbreviation for the expressions “subsumable under MP,” “subsumable under MT,”¹² and the other logical rules of inference. Already in the first part, as well as in the previous gate, we pointed out the absurdity of this view: why would one choose a single abbreviation to denote so many expressions that have no relation to one another? Beyond that, we must ask ourselves where these rules themselves came from. Is each logical rule itself also nothing more than an arbitrary conventionalist definition? Is trust in the result of logical inference also a convention? This is a patently unreasonable explanation. Its meaning is that we adopt conventions because of convention.

Our conclusion, then, is once again essentialist. The expressions “true” or “valid,” like every other meaningful term, describe a content. This content is the concept “true” itself—the matter of the concept—as we feel it, while its various expressions—the form of the concept—are the specific logical rules of inference. These rules are not “truth” itself, but expressions of the way to verify its presence. If so, the question arises again why all these apparently different laws are expressions of a single concept—“true.” The synthetic thinker, who is also an essentialist, as we saw in Gate Two, will answer that this is a simple result of observing, with the eye of the intellect, the substance—the essence—of the concept “true.” This observation is what yields the rules of logic.¹³

Observation 23: Definition by Extension and by Intension¹⁴

The problem presented in the main text regarding the definition of the concept “true” by means of the totality of its appearances in the world is a particular case of a more general problem in logic. In logic one distinguishes between two ways of defining a concept: by its extension, and by its intension.

The extension of a concept is all the subjects of which it can serve as a predicate. For example, the extension of the concept “democratic state” includes all democratic states, that is, the set: Israel, England, the United States, and so on. All these states can serve as the subject in the sentence “___ is a democratic state.” In this description, the extension appears as determined by the intension of the concept. One who knows what a democratic state is can list all the states that belong to the extension of this concept.

The view that the intension determines the extension—that is, the priority of intension over extension—raises certain difficulties. When we have a concept whose use is not fixed and defined sharply and unambiguously, we tend to clarify its meaning by examining the collection of subjects within its extension. That is, apparently the extension here determines the intension. This is indeed a methodological consideration that is meant to help only in the definition of vague concepts, but in modern logic there were those who argued that in principle the intension of a concept should be grounded in its extension, because of the very lack of clarity concerning the intension of some concepts. There is no doubt that this way of handling concepts is useful at the technical level for analyzing the relation between different concepts—questions such as whether their ranges overlap partially or completely, and so forth—without becoming entangled in the difficulties raised by a concept’s intension.

There is no real need to explain why this may be an efficient formal approach yet one that is mistaken in principle. In order to determine the extension of a concept, one must understand its intension, for otherwise how will we determine who belongs to its extension and who does not? If we want to compile a list of democratic states, we will have to go through the list of countries and decide regarding each one whether it is democratic or not. But if we have no way of making such a decision prior to assembling the list, then it is unclear how the list itself will be assembled. In addition, how will we decide from this list what classification should apply to a new country? This approach, which attempts to give extension priority over intension, is astonishing. It is unclear how such a distorted form of thinking, so far removed from sound logic, came into being.

The root of this strange phenomenon lies in an analytic tendency. We have already pointed out several times that according to the analytic conception, concepts have only use, not meaning or sense. We saw that those who hold the analytic approach are conventionalists in their relation to concepts. Every concept is nothing but an abbreviated and arbitrary definition in language. According to this, the concept “democratic state,” for example, is merely an abbreviation for the collection of states in its extension. If so, we need not be surprised to find an approach that gives priority to extension over intension. The tendency toward analyticity leads, among all its other distortions, to such an approach as well. We continue to see that the desire to flee from common sense—which is manifestly synthetic—can lead one very far afield.

Here too, in order to sharpen the absurdity of this approach, we can use an argumentative technique we have used before: we may ask why a single linguistic abbreviation—“democratic state”—was chosen for a collection of states that have no shared substantive and essential connection. To explain the nature of the connection, we would have to descend to the intension of the concept “democratic state,” which is what characterizes specifically these states and not others. But in doing so we are assuming that the concept “democratic state” has content, or meaning, and not only use.

Let us note that this strange phenomenon is not necessary. Even those who hold an analytic position need not use extensional logic specifically. They can certainly define the concept “democratic state” as an abbreviation for a collection of characteristics—not a collection of entities or objects—such as elected government, protection of civil rights, separation of powers, and so forth. But here too the analytic thinker will have to explain what all these properties have in common, and why their combination was given a common collective name.¹⁵ This argument resembles the one we used more than once in the main text in order to reject the analytic-conventionalist approach.

What I wish to claim here is precisely the opposite: although conventionalism does not necessarily lead to extensional logic, extensional logic is necessarily based on conventionalism, and therefore on analyticity.

Let us now turn to the transcendental direction of defense against the second challenge to deduction. A possible argument in this direction would say that the rule MP has meaning and not merely use, but its meaning is a result of the structure of our cognition. We are compelled to use it, and therefore there is no room to demand that someone explain or justify this rule. The meaning of “valid” or “true” is “conforming to the collection of logical rules of inference.” Here, unlike previous contexts, this direction at first seems plausible, because common sense too accepts the fact that we are compelled to think in accordance with the basic laws of logic. More than that, the determination that a certain statement is true or valid does indeed derive from a feeling or judgment of our thought or cognition.

And yet, in light of what was said above regarding the correspondence between the intellect—and cognition—and the world, even here there is a deeper layer in which a person “observes” the concept “truth” and from this recognizes the logical rules of inference. It seems that this is how a reasonable person experiences the laws of logic. Beyond that, as noted earlier, such an approach uproots Kant’s distinction between the synthetic a priori and the analytic. Even in determining an analytic claim we use an analysis of the concept by means of tools imposed upon us because of the structure of our cognition—that is, because of the laws of logic.

Here we encounter an important point. We already noted in Gate One, and afterward as well, that the proponent of the synthetic position also uses analytic tools of thought. He simply opposes the conception that sees them as exclusive. Transcendental arguments too are, of course, legitimate tools in the world of the synthetic thinker. The claims presented here are directed only against the exclusivity of the transcendental argument—that is, against seeing it as the sole tool for reaching valid generalizations about the world. Beyond that, such an argument is not a sufficient reason for accepting the synthetic a priori conclusion as true. In order to grant validity to synthetic a priori statements, one must acknowledge the correspondence between the human being and the world, or that the intellect functions also as a kind of sense (see Chapter 3 of Gate Two), and not merely as a detached faculty of thought nourished only by the data of sense cognition.

Kant proposed the transcendental form of argument as the only tool that could answer the problem Hume posed to the scientific world: how can general statements be accepted as valid, given that induction cannot be proved? The synthetic approach holds that a transcendental argument is one among several possible ways of making valid generalizations. It is clear that even according to the synthetic position there may be rules whose root lies in human cognition and thought. In Gate Eleven we will see that induction, as distinct from causality, is in fact such a rule (although we will see there that even the use of induction requires an assumption of correspondence between intellect and cognition and the world). On the other hand, scientific laws, for example, are generally not transcendental arguments, but inductive generalizations that speak about the world itself.

We have seen, then, that the analytic approach is inconsistent, because it is itself based on a synthetic determination regarding the exclusivity of the deductive-analytic mode of thought. We have also seen that even those who hold this approach cannot operate honestly and consistently within it, because every deduction they make contains within it induction and other synthetic processes. In conclusion, let us note that Gate Eleven will contain a continuation of this discussion.

Chapter 3: The Meaning and Relativity of Axioms

Axiomatic Systems

Up to this point we have dealt with examining the deductive argument itself. Now we wish to show that our conclusions can be applied more broadly to axiomatic systems. In Gate One the concept of an “axiomatic system” and its meanings were presented briefly. Here we want to point to a difference between the attitude of those who hold an analytic position and those who hold a synthetic position toward the validity of statements and arguments in an axiomatic system.

The case of a deductive argument is a particular case of organizing knowledge and presenting it in the form of an axiomatic system. In a deductive argument we start from premises and arrive at a conclusion by deriving it from them. An axiomatic system is the logical and conceptual framework within which deductive arguments are conducted. Such a system, as we defined it in Gate One, is composed of foundational premises (axioms), rules of derivation that may be applied to any statement within the system, and derived statements (theorems).

Plane geometry, which is taught in high school, is a clear example of the axiomatic mode of thought. The process of learning in geometry proceeds from the axioms to the theorems by analytic derivation. As we noted in Gate One, in the various sciences as well there are tendencies to organize knowledge and present it in the form of an axiomatic system, even though the gathering of the knowledge and its learning did not occur through a process proceeding from premises to conclusions, but rather the reverse—from the conclusions, which are the empirical facts, to the premises, which are the general laws.

Many people feel that this is the ultimate form of learning and of gathering valid knowledge. In addition, many people think that in every axiomatic system the premises are arbitrary, since they have no justification—or more precisely, no proof—whereas the conclusions, the theorems, are proven.¹⁶

The Relation Between Premises and Theorems

This view is a reflection, though admittedly a rather blunt one, of the analytic position. First, one should point out the absurdity of such claims. The theorems cannot be more valid than the premises on the basis of which they are proven. The entire concept of proof is the grounding of theorems in axioms. It is impossible for the premises to be regarded as arbitrary and unjustified, while the conclusions derived from them receive a superior status, as though they were justified. The analytic thinker can rightly argue that what is valid is the derivation of the statements from the premises; but if he wishes to remain consistent, he must admit that the statements themselves are not necessarily true—neither the axioms nor the theorems.

Let us now move from mathematics to life. It is quite clear that in ordinary life we do not treat our fundamental assumptions as something arbitrary. Only in mathematics, whose subject is the analytic aspect of thought, can premises be arbitrary. This is the emptiness of the analytic, described in Gate One. In Chapter 3 of Gate Four we called it “analytic pluralism.” This phenomenon is expressed in formal logic by a basic rule—sometimes called “Rule P”—which says that one may assume any premise whatsoever. In mathematics, Euclidean geometries—that is, geometries corresponding to Euclid’s assumptions—and non-Euclidean geometries—those with opposite assumptions—can develop at the same time, even by the same people. This fact stems from the fact that mathematics examines only the analytic process by which conclusions follow from premises, and not the truth of the conclusions, or of the premises themselves.

One must not extrapolate from mathematics to the other domains of thought, which deal with the truth of the contents themselves. In those other domains the premises are the clearest part of the system of thought. The axioms are not the collection of arbitrary statements in the system, nor its weakest link. On the contrary, axioms are specifically the statements that are so self-evident that they do not require proof at all.¹⁷ Those who hold the analytic approach understand all fields of thought as branches of mathematics, and therefore their attitude toward the status of premises is identical in every field. This is precisely why they hold that it is impossible to accumulate valid knowledge, since such knowledge cannot be accumulated on the basis of “arbitrary” premises. This is the essence of the logical basis of postmodern philosophy, as described in the previous part.

Geometry and the Theory of Relativity as Clear Examples Against Pluralism

Geometry is often brought as an example of intellectual pluralism, which advocates the idea that each person has his own truth. For this purpose, analytic thinkers enlist the fact that Euclidean geometry is indeed the common geometric theory, but at the same time there are also non-Euclidean geometries, whose premises differ from Euclidean theory and even contradict it, and yet all of them are consistent.

This is an argument involving a considerable measure of misunderstanding. Every mathematician, and certainly every physicist, knows that in straight space only Euclidean geometry is correct, and no other geometry. On an ordinary sheet of paper it is impossible in any way to construct a triangle whose angles sum to anything other than 180 degrees. More generally, in any given and specific reality there is only one correct geometry, and no other. A non-Euclidean geometry is based on different assumptions because it describes the properties of a different kind of space—“curved space,” in scientific terminology.

In modern physics, geometries—including non-Euclidean geometries—are used in order to try to discover the geometric properties of our space. The assumption of the physicist who does this is that our real space has objective geometric properties, which he is trying to discover, and that these do not depend on the observer’s views.

Thus, the example of the various geometries is a beautiful proof of precisely the opposite thesis from pluralism: for a given space there is only one correct geometric description. That is, when one leaves abstract mathematics and tries to understand what it is saying, one sees that its premises are not arbitrary but necessarily true. The mathematical working assumption that the premises are arbitrary is only for the sake of convenience. We have seen, then, that even mathematical premises, and certainly our premises in life, are not arbitrary at all.

The study of the geometric properties of the world is carried out in physics within the framework of the theory of relativity. This theory too is often used as an example by advocates of pluralism to support their claim that each person has his own truth.

As in the example of geometry, the theory of relativity too can actually be seen as providing clear support for the opposite position. Different views of the world are derived from different objective data. A different measurement of time and place indicates motion of the measuring system at a speed or acceleration different from that of its counterpart. All the relativity in the theory of relativity is intended solely to coordinate the pictures of the world observed from all the different frames of reference. That is, the basis of relativity is in fact absolute objectivism. In order to ensure that we are all observing exactly the same physical laws, one must alter and adjust the concepts of time and space to the condition of one’s frame of reference.

The conclusion that follows, similar to the description above regarding geometry, is this: from the given state of a particular frame of reference, there is only one correct way of viewing things, and it does not depend at all on the philosophical or other opinions of those inhabiting that frame. There is no room here for pluralism.

Those who use the above arguments from geometry and the theory of relativity do not really understand the significance of those two fields. Both are clear examples specifically of the synthetic thesis, which holds that there is only one truth.

Another example, which appears somewhat different, is the Copernican revolution—this time the original one. As is well known, until Copernicus the solar system was described geocentrically, with the earth at the center and the other heavenly bodies, including the sun, revolving around it. Since Copernicus, the description has been heliocentric: the sun at the center, with the other heavenly bodies, including the earth, revolving around it. Apparently, then, we have here a decision between two descriptions, with the claim that only one of them is correct.

Precisely in this example, many use it to attack ancient religious outlooks that employed a different description. Copernicus, they claim, proved that the ancient conception was mistaken. Here, for some reason, everyone agrees that pluralism is irrelevant.

But surprisingly, here specifically pluralism is very relevant. The notion that Copernicus proved something is simply mistaken. In any situation in which two objects revolve around one another, their relative motion can be described in both ways: if the origin—the fixed point, or the observer’s location—of the coordinate system is defined to lie on object A, then the picture is that object B revolves around object A. If the origin is located on object B, then the picture is that object A revolves around object B. In such a case there is no way to decide who revolves around whom, for both descriptions are entirely equivalent. What Copernicus innovated was the possibility of looking at the solar system also in a heliocentric way, and it turns out that this is mathematically more convenient. The convenience of the description has nothing to do with its being a truer description. Both descriptions are equally true; the heliocentric description is simply mathematically more convenient.

It should be noted that even here this seemingly pluralistic picture does not express any genuine pluralism. Here too, both pictures describe two ways of observing the same reality. When someone stands on object A, he will always see object B revolving around him. When he stands on object B, he will always see object A revolving around him. The two pictures are different because they describe the picture from two different vantage points. There is no room for any pluralism whatsoever in this description. Everyone agrees that if one observes from object A, then object B is the one revolving, and vice versa.

Summary

An axiom is not merely a statement without a proof. It is a statement that does not need a proof, because it is self-evident. In Kantian terminology, this may be a synthetic a priori statement. Of course, the meaning of this claim here is not Kantian. Axioms do not arise only from the structure of our cognition; they can also arise from “intellectual observation” of the world.

This discussion is really a general expansion of the picture described above in the first challenge to deduction. That challenge dealt with a single argument; here we expanded it to entire logical systems. In the next gate we will discuss in greater detail an interesting and surprising conclusion regarding axiomatic systems, one that has implications for the concept of truth in general.

Chapter 4: The Failure of Formal Language to Resolve Paradoxes

Introduction

The second challenge, and the discussion that followed it, is also an archetype of analytic thought and of its limitations. I will now try to generalize the conclusions of that discussion as well, and to examine what happens when a problem arises that constitutes a paradox, or a logical loop. In the next gate we will discuss the assumptions underlying this method. A reader who feels that the coming discussion interrupts the continuity of his reading may skip to the end of the chapter.

What Is a Paradox, and What Is the Goal of This Chapter?

A paradox is an argument that on its face appears valid, yet whose conclusion contradicts some clear assumptions. Alternatively, one may say that it is an argument whose premises and whose steps all appear reasonable, yet one still cannot live with its conclusion. A logical loop is an extreme case in which one cannot even point to the argument’s conclusion, or to the truth value of the statement in question. These matters will become clearer from the examples below.

The accepted way of solving paradoxes, or of stopping logical loops, is by means of an analytic argument. The transcendental direction is generally not useful with paradoxes. In order to illustrate this claim, we will examine several characteristic examples of paradoxes. Our goal here is to show that analytic solutions cannot solve real problems. That is, one who adopts an analytic solution to a philosophical problem is assuming from the outset that the problem is not essential, but rather a result of defective use of language, and therefore its solution will come by means of manipulations of language and of basic definitions. Through this perspective we wish to show once again that analytic methods are usually not merely a technique for solving problems, or a form of philosophizing. They are a different conception of the problems themselves, and of the world in general.¹⁸

Self-Reference Paradoxes — Russell’s Solution

Let us begin with the barber of Seville paradox, one of the most famous: in Seville there is a barber who shaves all those who do not shave themselves. The question is whether that barber shaves himself. It is easy to see that this is a logical loop. If he shaves himself, then he belongs to the group of people whom he does not shave, and vice versa.

There is an entire class of paradoxes of similar form, called paradoxes of self-reference,¹⁹ and the most famous of them all is the liar paradox.²⁰ One formulation of it is:

Statement (A): Statement (A) is false.

This is a statement that assigns its own truth value as false, and therefore it is nothing but a logical loop: if it is true, then it is false; and if it is false, then it is true. There are many more such paradoxes to be found in any book that deals with the subject.²¹

We saw in Observation 19 that Bertrand Russell proposed solving all these logical loops by using a hierarchical logical language that would prevent them from arising. He proposed a hierarchy among statements in the language, so that each statement could refer only to statements below it in the hierarchy, and certainly not to itself or to other statements that could in turn come back to refer to it.²²

This proposal, discussed above in Observation 19, is a specific implementation of the approach proposed by Leibniz. He too tried—needless to say, unsuccessfully—to find a formal language free of logical problems. The price paid for accepting such a language, even if one could find one, is giving up a very large set of statements that have perfectly clear meaning.

In fact, one can go even further and argue that even the paradoxical statements themselves have meaning; what they lack is only a clear truth value. The solution of finding a formal language would only prevent us from formulating the problems, but it would not solve them. As was already argued above in Gate Five, Chapter 1, a formal language is an intellectual trick that cannot solve real philosophical problems. Russell, as an adherent of the analytic position, thought that statements and concepts have only use and not meaning, and therefore he proposed solving philosophical problems by changing the rules of use. In the next gate we will see from a different angle that Russell was, apparently, mistaken. Not only is it impossible to solve all philosophical problems through a formal language, but there is a statement—Gödel’s theorem, which will be described there—that states, under certain conditions, that it is impossible to create a language that is completely formal.

As another example, let us now take the formulation of a “real-life problem”:

A law lecturer signed a contract with his student according to which, if that student were to win the first case he conducted after finishing his studies, he would be obligated to pay his tuition. After the student finished his studies and did not pay the tuition, the lecturer sued him to compel payment. Clearly, if the judge rules in favor of the lecturer—meaning that he determines that the student is indeed obligated to pay—then the result will be that the student lost the first case he conducted, and therefore he is exempt from payment; and vice versa.

In such a case Russell can only lament the foolishness of those who signed such a contract, but he cannot offer them a fair analytic solution. He might claim that the contract was invalid from the outset because such a situation could arise, and therefore one should behave as if no contract had been signed at all. Alternatively, Russell might say that “the first case” mentioned in the contract does not include a case of this kind—not because of the intention of the signatories, as any reasonable person would say, but because of the hierarchical meaning of the language. This is a formal solution to the problem, identical to those discussed above in Observation 19. Such a formal solution does not satisfy one who feels that there is a genuine problem here. A synthetic judge will of course determine his position in light of the intention of those who signed the contract, and not according to this or that set of formal linguistic rules. In his view such a contract has meaning, and he will strive as far as he can to find it.

Observation 24: Legal Systems — An Analytic Solution to Problems of Life

I would like here to present a very current problem, one that is illustrated indirectly in the main text but also has direct implications for it. The cumbersome system of law and jurisprudence, as we know it today, is intended to solve the problem created by the ambiguity of expression in everyday language. To this end, standards were established for binding legal drafting, so that—at least on the theoretical legal plane—every claim is supposed to have a binding formal meaning according to the law, and every case is supposed to fit a clear legal ruling.

One should note that here too this is in fact an analytic solution to a problem of life. Legal language is a formal language, whose principal purpose is to prevent ambiguity, paradoxes, and multivalent meaning.

This is the true root of the familiar problem that law does not necessarily coincide with justice. In a very large majority of cases, it is entirely clear what natural justice requires in a given legal case. Because of cases—few in number—in which this is not clear, we have accepted upon ourselves the burden of a cumbersome legal system that is a kind of formal language. The price we pay for using it is a renunciation of justice in many cases, including cases in which justice is entirely attainable. We are forced to acquit the guilty and convict the innocent because of formal legal rules or procedural regulations. This is exactly the situation created by proposals to solve philosophical problems by means of formal languages that restrict and “clean up” ordinary language. There too, as described above, the price is that we will be unable to express quite a few legitimate and reasonable things because of these draconian restrictions.

The result of this situation is that in a modern state today almost nothing can be done without a lawyer or legal expert. These people take over the lives of citizens in a kind of conspiracy that seems impossible to stop. Today judges examine what the cumbersome legal language in which the contract was drafted requires. Whoever equips himself with the better lawyer—which of course also costs more money—is the “just” one.

In the present situation, any connection between law and justice is entirely accidental. Judges often do whatever they please, since they are committed not to justice but to law, and they are not chosen according to standards of justice, uprightness, and morality, but according to standards of law. On the other hand, judges who nevertheless try to achieve justice, and often find themselves limited by the law, are sometimes forced to circumvent or distort it.

A widespread modern phenomenon is that law and the rule of law have become the modern substitute for values and sanctity. Everything defined as law thereby becomes binding as though it were sacred. Any value may be violated, except the law and the rule of law.

As we saw in the second part—especially in Gate Five—in an analytic society there is no agreement on any value whatsoever, and in fact there is no trust, and no faith, in values at all. As a result, society must define binding standards that will permit orderly common life. These standards—namely, the law—through the Copernican revolution described in Gate Five, Bokononism, become absolute values. Today a person may do anything, provided it is within the law. Every wrong is justified, provided it does not contradict the law. This religious faith in a human creation, often produced merely on the basis of interests and strange coalitions, can be very amusing and grotesque, especially when it appears against a background of analytic skepticism with respect to every other principle.

At a meeting of the leadership of a certain institution in which I participated, a problem arose concerning a person who was disrupting the institution’s functioning. The lawyers among the participants proposed restricting that person’s authority and defining in writing his powers, rights, and duties. The solution was patently ridiculous, and it was clear that it could not truly solve the problem. I remarked there that there is a well-known tendency among jurists to solve problems of life by means of formulations, laws, contracts, and similar afflictions. Problems of life are often not amenable to solutions of this kind. The helplessness the analytic thinker feels when confronted with such problems causes him to try to flee into formulations, contracts, or other formal devices, instead of addressing the problem itself.

This is also the meaning of the growing tendency in our society to solve every problem through legislation and through the High Court of Justice. As we saw in Gates Four and Five, the path of dialogue and understanding is closed before the analytic thinker, and therefore he is forced to use coercive formal means. In Gate Five we pointed out that violence too is a result of the impossibility of dialogue in an analytic world.

It is interesting to note that Gadi Taub, in his book The Crouching Rebellion, comments on the widespread postmodern tendency to deal with problems of formulation while ignoring the solution of problems in life themselves. There is a concentration on terminology, such as “African-American” instead of “black,” or “Mizrahi” instead of “Sephardi,” “golden age” instead of “old age,” and other examples, some of them quite ridiculous. Often, in the practical sphere, these “new enlightened” people will take no action at all to solve the actual situation, and changing the wording to one that is “politically correct” will satisfy them. Taub argues there that the politically correct live off deprivation and therefore try to perpetuate it. This is somewhat exaggerated, although it is quite clear that they do indeed live off and feed on the existence of the deprived. See Gates Five and Six, where it was clarified that Taub’s own approach is, to a considerable degree, of that kind as well.

Here we propose an interpretation according to which this phenomenon—the resort to formulations and the fixation of law as a supreme value—stems from conventionalism. Because of the inability to solve real problems of life, since each person has his own truth, people try to cling to formulations. This is true, as stated, also of liberals who advocate the “old Enlightenment”—whom Taub sees as the antithesis of the “new enlightened,” see the detailed discussion in Gate Six—and not only of the “new enlightened.”

A return to synthetic thought and the abandonment of formal-analytic solutions has implications in this sphere as well. Just as one must return to thinking in accordance with common sense—“mature dogmatism,” to use the terminology of Gate Three—so too one must return to judging on the basis of natural justice. The laws in the law books can be reduced to the minimum required, though of course that minimum must be examined carefully. Afterward, a number of judges should be chosen in each district who need not have any legal education at all. These judges should be people of integrity, whose honesty and morality are accepted by most of the public. They would decide every problem solely according to the dictates of their conscience, and not according to any formal system. They would examine what the true intention was of the two parties who signed a contract drafted without lawyers.

Of course, even in the structure proposed here there would be distortions and disagreements in not a few matters, by the very fact that society contains different components with different worldviews. Yet it is highly likely that their number would be significantly lower than it is in the present system.

The Vagueness of Basic Everyday Concepts

An example of another kind of paradox, not connected with self-reference, can be seen in the following paradox, sometimes called the “heap paradox”:

Premise A: A collection of two pebbles is not a heap.
Premise B: If there is a collection of pebbles that is not a heap, the addition of one pebble cannot change its status so that it becomes a heap.
Premise C: A collection of a million pebbles is called a heap.

It is quite obvious that these three premises cannot all coexist; that is, one cannot accept all of them together. Yet, on the other hand, none of them seems dubious, and certainly none appears clearly rejectable. The required conclusion is that the concept “heap” is not one that can be given a mathematical definition. The analytic thinker will of course propose an artificial definition of this concept, so that coherent use of the concept “heap” becomes possible, even if it is remote from common sense—for example: every number of pebbles over 1,000 will be called a “heap.” This indeed solves all the linguistic problems, but not the real one.

The vagueness demonstrated here exists fundamentally in all ordinary concepts. One can think, for example, of the concept “afternoon.” When my children ask me, “From when does ‘afternoon’ begin?” I of course have no clear answer for them. The same is true of questions such as: from when is something “late”? Or: how much is “bright”? Or “heavy”? Or “a lot”? And so on and so on. There is no basic concept in everyday language that is not bound up with vagueness. This is a welcome phenomenon, which allows us to express in language something beyond a finite combination of well-defined meanings. Human language is a complex thing, perhaps even an infinite one, and no formal basis can be created for it. Perhaps Henry Kissinger’s expression—though of course said in a somewhat different context, though not entirely—is relevant here: “constructive ambiguity.”

Basic concepts are “basic” precisely in the sense that they cannot be reduced to other concepts. See Rabbi Zini’s discussion below in Gate Ten. Descartes states in his Principles of Philosophy, Principle 10:[^51]

10. There are concepts so clear in themselves that they are obscured by the very attempt to define them in the scholastic manner. They are not acquired through study; they are born with us.

…When I said that this proposition—“I think, therefore I exist”—is the first and most certain to appear before one who directs his thoughts in an orderly way, I did not thereby deny the need first to know what thought, certainty, and existence are, and likewise the need to exist in order to think, and similar things. But since these are concepts so simple that in themselves they do not bring us to the cognition of any existing thing, I did not see the need to take them into account here.²³

Conclusions

There are certain analytic philosophers who try—absurdly enough, by analytic means—to demonstrate the truth of various synthetic claims, in order to escape the tangle created by the emptiness of the analytic. We saw an example of this in Leibniz at the beginning of Gate Two, where he tried to refute the individuation of objects analytically. In the note there we hinted at the attempts of Shteinits, which will be discussed in the appendix, and which suffer from a similar problem: they try to arrive at synthetic determinations by analytic means.

It may perhaps be possible to show theoretically the very existence of synthetic claims by such means, but in all these cases one can see clearly that there is no chance of arriving at the specific synthetic content itself. In fact, most of the arguments we raised in this book against analyticity themselves employ methods that are somewhat analytic in character. This serves only to point to the problems in the analytic approach, and to the need for additional paths in order to reach certainties. But these methods provide no way to arrive at specific synthetic contents. This is another aspect of the emptiness of the analytic. The conclusion of an analytic argument is generally a framework without content, or the absence of such a framework.

A classic example of this type of argument is Kant’s ethics. By a very convoluted route, full of philosophical obstacles, Kant arrives at the principle he calls the “categorical imperative.” This is the supreme moral principle that obligates every person, and which, in Kant’s view, is derived solely from pure reason. According to him, the derivation is a purely a priori analysis of reason without empirical observation, something like a synthetic a priori statement. This principle states that a person should do only what he would want to be established as a general mode of conduct in human society.

Beyond the questionable claim that this principle is indeed derived solely from pure reason, one should point out that this principle contains no foothold whatsoever for determining what specific conduct is required of the moral person. A person who wants murder to become a general mode of conduct in society is morally permitted, on this basis, to act that way himself. Kant himself, when he begins to discuss the specific contents of the moral imperative, slides quickly from the heights of the Olympus of pure reason and fails—heaven forbid—into arbitrary synthetic determinations that he cannot justify a priori, not even in the form of a transcendental argument.

This is a striking example of the fact that in order to escape the emptiness of analytic thought, transcendental arguments are not enough, and certainly not analytic proofs of the existence of valid synthetic statements. In order to reach those valid synthetic statements, we must use a different method. This subject will be discussed and illustrated at length in the appendix. For this reason, after we complete the critique of analyticity, we will try in the fourth part to propose a real synthetic alternative that is not based on analytic or transcendental arguments against analyticity.

Summary of the Discussion in This Gate

In this gate we described the argument against analyticity on the logical plane. We saw that the hierarchy among deduction, induction, and analogy is, on the practical plane, merely illusory. In fact there is only one process of thought—analogy—of which the other two modes of inference are partial stages. We then illustrated this in a brief discussion of the validity of the various parts of axiomatic systems—axioms and theorems. Afterward, we once again examined the two modes of defense available to the analytic thinker: the analytic one, based on the definition of concepts, or convention; and the transcendental one. We saw that in the present context too, as in previous situations, these two paths cannot supply an acceptable solution to essential problems.

As a continuation of this, we tried to demonstrate specifically in the area that more than any other seems to call for analytic treatment—the area of paradoxes and logical loops—the powerlessness of analyticity. We came to see that even in this area our salvation will not come from analytic tools as such.

In the next gate we will see that analyticity suffers a blow specifically by mathematical means, which are supposedly the hard core and great success of this mode of thought. The next gate is an expansion and deepening of the discussion of paradoxes conducted in the present gate, but it deals with the heart of the issue, and therefore it is recommended not to skip it, even for one who skipped the present discussion.²⁴

Footnotes