Logical Hermeneutical Principles as the Building Blocks of Non-Deductive Inferences – Part II

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Originally published: July 30, 2017

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Contents of the Article

With God’s help

The Logical Hermeneutical Principles as the Foundation Stones of Non-Deductive Inferences^[1]

A Logical Model for A Fortiori Inference, Paradigm Construction, and the Common Denominator

Part Two

Michael Abraham, Dov Gabbai, and Uri Schild

Introduction to the Second Part of the Article

In the first part of the article, we dealt with the opening of the huppah discussion in Kiddushin 5a-b, and through it developed a logical model for analyzing midrashic inferences that use the logical hermeneutical principles (a fortiori inference and the two forms of paradigm construction). We saw that this is a model relevant to all fields of non-deductive thought, since they are composed of the same logical building blocks.

In this part we will examine more complex inferences through the continuation of that Kiddushin discussion. After that we will analyze objections that relate directly to the microscopic parameters, and finally we will see several implications of the model for explaining various phenomena in the world of legal exegesis. We will conclude with a brief discussion of the distinction between deductive thought and thought that is not deductive.

For the sake of brevity, we will forgo a detailed summary of the first part, and any reader who wishes to follow the discussion is hereby referred to the previous issue of Badad.

E. More Complex Inferences

Introduction

In this chapter we will deal with more complex inference tables and diagrams, and we will apply the model we developed to those cases. For that purpose we will continue to follow the next stages in the Kiddushin discussion.

The complex common denominator

There is no point in describing stage 7 of the discussion in detail, since it is an ordinary a fortiori inference from document, and it is exactly identical to what we did in stage 1. The objection to it from divorce as well (that a document effects divorce whereas huppah does not), which is made in stage 8 of the discussion, is equivalent to what was done in stage 2. Therefore it is clear that both these inferences can be described and will come out valid in our model as well. We therefore move immediately to stage 9, where the discussion constructs a complex common-denominator inference.

This is a common-denominator inference, based on the two sub-inferences that compose it: a common denominator (an a fortiori inference from money and a paradigm construction from intercourse) combined with the paradigm construction from document. In effect, there is here a fourth kind of common denominator, in which one of the basic teaching cases is itself a pair that teaches by means of a common denominator.

After presenting two columns for the objections (benefit against the simple common denominator from money and intercourse, and divorce against the paradigm construction from document) and presenting rows for all the teaching cases (=the acts), the data table obtained in this case is:

Y

H

G

Table 7 (the complex common denominator)

We will now find the optimal models for the two ways of filling the lacuna cell:

__Optimal model for Diagram 7a – complex common denominator with filling 1 __

The solution for the acts is:

Money: (3,0,0)

Huppah: (1,1,0)

Intercourse: (2,1,0)

Document: (1,0,1)

__Optimal model for Diagram 7b – complex common denominator with filling 0 __

The solution for the acts is:

Money: (3,0,0)

Huppah: (0,1,0)

Intercourse: (2,1,0)

Document: (1,0,1)

To see whether the inference is valid, we now compare the two fillings. In both of them the dimension is 3, and the valence is also 3. In both, the connectivity is 1, and the number of points in the diagram is 6. The only difference is in the number of direction changes: in filling 1 there is only one direction change, whereas in filling 0 there are two again, from P to N. Therefore this inference is valid because filling 1 is preferable with respect to the direction-change index.

We note that this situation is exactly identical to the character of the preference we found for filling 1 in the simple common-denominator inference when the two teaching cases are a paradigm construction and an a fortiori inference (the common denominator from stage 5). It turns out that the same disputes we mentioned among the commentators on the Talmud regarding the status of such a common denominator, whether it is strong like an a fortiori inference or weak like a paradigm construction, will also exist regarding this inference.

Objection to a complex common denominator

In stage 10 of the discussion an objection is presented to the complex common denominator. The objection is that all the teaching cases operate, under certain circumstances, against the will of the woman, unlike huppah. This is an advantage that expresses a strength they possess, and therefore it rebuts the inference from them to huppah, where that stringency or strength does not exist. The intuition here is very similar to that involved in the objection to the simple common denominator; see our comments there.

The data table for this case contains one more column, containing the objection:

Y

H

G

K

Table 8 (objection to the complex common denominator)

We will now present the two optimal models, for the two fillings:

__Optimal model for Diagram 8a – objection to the complex common denominator with filling 1 __

The solution for the acts is:

Money: (4,0,0)

Huppah: (1,1,0)

Intercourse: (3,1,0)

Document: (2,0,1)

__Optimal model for Diagram 8b – objection to the complex common denominator with filling 0 __

The solution for the acts is:

Money: (3,0,0)

Huppah: (0,1,0)

Intercourse: (2,1,0)

Document: (1,0,1)

To confirm the existence of the objection, we must compare the two fillings. In both cases the dimension is 3, and the connectivity in both is 1. The valence is 4 in filling 1 and 3 in filling 0. The total number of points in the graph is 7 in filling 1 and 6 in filling 0. And the number of direction changes is 1 in filling 1 and 2 in filling 0.

Thus filling 0 is preferable with respect to the valence and total-number-of-points indices, while filling 1 is preferable with respect to the direction-change index. There are preferences in both directions here, and therefore according to Rule 10 this is an objection. Thus this objection too is confirmed in our model.

The final inference

In stage 11 of the discussion, the Talmud reestablishes Rav Huna’s position, claiming that in his opinion money does not acquire coercively. To be sure, everyone agrees that it does acquire in the case of a Hebrew maidservant, but one must distinguish between a maidservant and an ordinary woman. The objector thought that no such distinction should be made, and on this explanation Rav Huna disagrees with him. In principle we ought to construct a new data table, in which there are two columns of coercion, Kama for a maidservant and Kisha for an ordinary woman. The column for the maidservant is like the previous table, money acquires coercively, while the column for an ordinary woman differs only in the money cell, which does not acquire coercively. But with respect to intercourse, document, and huppah as well, the situation changes, since their laws for a maidservant differ from their laws for an ordinary woman.

But in the formal structure of the discussion, the analysis seems slightly different. If they had raised an objection, ‘what about money, since it does not acquire against her will in the case of a woman,’ we would have had to build a table with two K variables. But the discussion rejects the objector’s assumption, and it seems to see the situation as though there were only one variable of coercion here. It seems that, in its view, the variable of the maidservant is not relevant to our discussion, which deals only with marriage.

Under this assumption, for this situation we get a data table identical to the one we presented for stage 10, except for one cell, since the assumption now is that money does not operate coercively:

Y

H

G

K

Table 9 (revalidation of the complex common denominator)

We will now examine the two models for the two fillings:

__Optimal model for Diagram 9a – revalidation of the complex common denominator with filling 1 __

The solution for the acts is:

Money: (3,0,0,0)

Huppah: (1,0,1,0)

Intercourse: (2,1,1,0)

Document: (1,1,0,1)

__Optimal model for Diagram 9b – revalidation of the complex common denominator with filling 0 __

The solution for the acts is:

Money: (3,0,0,0)

Huppah: (0,0,1,0)

Intercourse: (2,1,1,0)

Document: (1,1,0,1)

And again, to confirm the validity of the inference, we must compare the fillings. In both fillings the dimension is 4, and the valence is 3. In both, the connectivity is 1, and the total number of points is 7. The difference is only with respect to direction changes: in filling 1 there is only 1, whereas in filling 0 there are 2, between P and N or between G and N.

The inference is valid, again because of preference in the direction-change index, as in the two common-denominator inferences, the simple and the complex. Thus the meaning of the change in the datum regarding money under coercion is that we have revalidated the complex common denominator, and as we saw, if it is valid, then it returns and validates the simple common denominator as well. We note that its preference is absolute, that is, there is not even an offset by valence here. Below we will see that this has legal implications, at least in the opinion of the mishnaic sage Rabbi Yehuda.

Summary

To summarize our discussion in this chapter, we present here in table form the list of inferences discussed here, and the conclusions that emerge from our model regarding the preferences in the different indices for them:

Complex common denominator

Objection to a complex common denominator

Revalidation of a complex common denominator

Diagram

Filling

Dimension

Direction change

Connectivity

No. of points in the graph

Valence

Result

1 preferred

Equivalent

1 preferred

Summary Table 3

F. Objections on the microscopic plane

Introduction

Up to this point we have dealt with a discussion conducted entirely on the phenomenal plane, that is, in the legal domain. Examination of different laws gave us indications regarding the microscopic composition of the acts and the legal outcomes, that is, regarding the parameters that characterize each of them and the relation between those parameters. As we have already noted, this is a process parallel to what is done in science, where too we observe scientific phenomena and infer from them theoretical conclusions regarding theoretical entities and the relations among them.

However, as we already noted at the outset, there is also another kind of discussion, which deals directly with the microscopic-theoretical domain, not with the phenomenal domain, what we called an ‘a priori objection,’ as opposed to an ’empirical objection.’ In this chapter we will deal with two such phenomena: the objection from a stringent aspect, and constraint objections.

The objection from a stringent aspect: to the heart of the problem

There are several discussions, for example Makkot 4a, Ketubot 33a, and parallels, in which the Sages disagree about a common-denominator inference. Some of them, Rabbi Yehuda, rebut it with an objection from a stringent aspect, while others do not accept the existence of such an objection.

An ‘objection from a stringent aspect’ challenges a common-denominator inference by arguing that the two teaching cases each contain a stringent aspect, whereas the target case does not. In the physical example we brought: one cannot learn from a table and a ball to a book, since in each of the two teaching cases there is a stringent aspect that the target case does not have: the table has legs, and the ball is round. Therefore one cannot learn from both of them that a book too will fall to the earth.

We emphasize that this objection is essentially different from the objection we encountered in stage 6 of the Kiddushin discussion, since the objection there presented a stringent aspect that was identical in the two teaching cases, in money and intercourse there is benefit, unlike huppah. Such an objection does indeed rebut the common-denominator generalization, since it suggests tying the outcome, the effecting of betrothal, to that stringent aspect, that is, to the microscopic parameter that is absent from huppah. As we saw, both intuitively and formally, this is an alternative equivalent to the common denominator, and therefore there is an objection. By contrast, the objection from a stringent aspect with which we are dealing here is different, since it relies on two different stringent aspects that exist in the two teaching cases. The claim is that the two teaching cases each have a stringent aspect, though in each case a different one from that of the other, and therefore one cannot learn from them to the target case. If we accept such an objection, this undermines the very possibility of generalizing inference, since we would never be able to generalize from two particular examples to a general law. As we saw above in the discussion of the universality of Table 5, every generalizing inference is based on two examples, each of which has a unique characteristic, and if the existence of such characteristics is a legitimate objection, then we have entirely eliminated the possibility of inferences of this type.

We further note that in the overwhelming majority of Talmudic discussions, when a common-denominator inference is brought, such an objection does not arise, and it is agreed that one can learn from the common denominator of the two teaching cases rather than from their differing sides. General reasoning too is constantly generalizing from different examples to a general law; in other words, we do not make use of objections from a stringent aspect. This is a cornerstone of our thought, and it is hard to believe that anyone really contests it on the practical level, beyond various philosophical challenges such as David Hume’s attacks; see Two Wagons.

It appears that there is something special about those discussions in which the objection from a stringent aspect does arise, and this is why such an objection is generally not presented, except in those places. What distinguishes those places? First we will explain this on the intuitive plane, and afterward we will analyze it in our formal model.

The objection from a stringent aspect: an intuitive explanation^[2]

As we explained, on the intuitive plane the generalization of a common denominator is based on Occam’s razor. The preferred possibility is that there is some common aspect in the teaching cases and in the target case that causes the legal outcome, rather than the possibility that each of the two unique properties of the teaching cases can generate the legal outcome. The preference is based on the fact that the absence of the unique property of one teaching case in the other teaches us that it is not that property that is relevant for generating the legal outcome, and likewise with respect to the unique property of the second teaching case.

For example, the fact that money redeems second tithe and intercourse does not, although both effect betrothal, teaches that the property that causes the redemption of second tithe is not what effects betrothal. And similarly with regard to the microscopic parameter responsible for acquisition in the case of a levirate widow, since it does not exist in money, which does not acquire in the case of a levirate widow. From here we prove that there is a third microscopic parameter, common to all of them, that effects betrothal.

The objection from a stringent aspect claims that there are indeed two different legal phenomena in money and intercourse, but it is possible that both are caused by one microscopic parameter, a common property that money and intercourse have and huppah lacks, and therefore a proposal of a common denominator arises here too. This alternative is equivalent to the alternative of a common denominator for all three acts, since in both cases there is only one factor for the legal outcome. This is how Rabbi Yehuda understands it.

But what happens when those unique properties are themselves microscopic parameters? For example, intercourse has the property of being a physical act between the pair, and money has the property of involving the transfer of value from his hand to hers. Huppah, by contrast, has neither this nor that. These are not legal properties, like the fact that money redeems second tithe and huppah does not, because they do not belong to the phenomenal plane. They are properties connected to the microscopic parameters that characterize the legal acts under discussion themselves.

One can immediately see that in such a case one certainly cannot raise an objection from a stringent aspect, since we are speaking here about the microscopic parameters themselves, and here we plainly see that these are two different parameters. That is, there is no basis here for claiming that there might be a parameter common to money and intercourse that effects betrothal, since in the objection itself we are pointing to two different microscopic properties of theirs as effecting betrothal. For this reason, in such cases one cannot raise an objection from a stringent aspect.

And indeed, in all the Talmudic discussions in which an objection of this type arises, it always concerns legal characteristics of the teaching cases, and not their microscopic parameters. In discussions where the unique characteristics are microscopic, an objection from a stringent aspect does not arise at all. In discussions like the Kiddushin discussion with which we are dealing, there are situations where the unique characteristics are legal, and in our case: redemption of second tithe in money, and acquisition of a levirate widow in intercourse, and therefore in such discussions there would seemingly be room to raise an objection from a stringent aspect. Why, then, in most of those discussions is it not raised? To understand this, one must remember that in Jewish law we rule in accordance with the Rabbis against Rabbi Yehuda, and therefore the anonymous voice of the Talmud does not raise such an objection even in these cases. According to the Rabbis there is no room for an objection from a stringent aspect in any case. The distinction we made between microscopic, a priori, objections and legal, empirical, objections exists only in the view of Rabbi Yehuda.

We note that the same distinction can be made in scientific contexts. If we bring unique physical-empirical phenomena that characterize the ball and the table, such as their different reactions to different physical forces, we can attribute those properties to a single theoretical parameter, and propose that it is what causes the fall toward the earth; in that case there will be an objection to the generalization. But if we point to different microscopic-theoretical characteristics that they possess, for example that one is made of leather and the other of wood, then we cannot claim that there is a single microscopic parameter that characterizes both, since those two characteristics are themselves microscopic characteristics, and it is not plausible to attribute them to a single microscopic characteristic different from both. Therefore, in such a case the generalization is agreed upon and is much stronger.

This description points very clearly to the vital need to address the microscopic model that lies in the background of the inference. Looking at an inference while ignoring the microscopic background underlying it does not allow us to distinguish between factual characteristics, namely the microscopic parameters of which we are speaking, these are the parameters that are properties of the legal acts, and they are what cause the imposition of the outcomes, and legal outcomes, that is, the legal characteristics of the acts. The inability to distinguish between these two causes an inability to distinguish between the two cases of generalizing inferences, and consequently an inability to understand the concept of an ‘objection from a stringent aspect’ and where it should be applied. The contradictions between the discussions regarding the objection from a stringent aspect are based on ignoring the microscopic plane. The distinction we proposed in Rabbi Yehuda’s opinion is another indication of the importance of the model that takes into account the existence of microscopic parameters in the infrastructure of these inferences.

The objection from a stringent aspect: a formal explanation

When we deal with the objection from a stringent aspect, no new datum is added to the picture. Rabbi Yehuda’s claim is that, in the data of Table 5, the conclusion is not that filling 1 is preferable; rather, this is an equivalent situation, that is, the inference is not valid. Rabbi Yehuda claims that in the inference at stage 5 there is no proof that the correct filling is 1. As we saw there, the preference for filling 1 lies in the fact that it has fewer direction changes, but on the other hand it is inferior in valence.

We may therefore now propose an explanation of Rabbi Yehuda’s view, namely that in his opinion valence is on a par with the other indices. In other words, he does not accept our Rules 6 and 9, concerning the weakness of the valence index. The clash between the preferences of the valence index and the direction-change index leads him to the conclusion that the common denominator is not a valid inference, and therefore the two sides are equivalent. The problem remains open.

If our proposal is correct, then a common-denominator inference based on two a fortiori inferences, Inference 5.2, is exceptional, for there there is an unequivocal preference for filling 1, without an offset by the valence index. If so, we would expect Rabbi Yehuda to disagree with the Rabbis only in inferences 5 and 5.1, but not in Inference 5.2. Examination of the discussions in which Rabbi Yehuda disagrees shows that the generalizing inferences that appear there are indeed of these two types, that is, there is at least one paradigm construction among the two basic inferences that compose the common denominator; see Makkot 4b and Ketubot 33a and others. With regard to the more complex inferences as well, one may infer that Rabbi Yehuda agrees with them if the weight of the preferable filling is not offset by an inferiority in valence. Thus, for example, in Rav Huna’s final inference, Table 9. This must be checked for each inference on its own.

Result 5: According to Rabbi Yehuda, an objection from a stringent aspect is admissible, but this is true only with respect to common-denominator inferences in which at least one of the two basic inferences is a paradigm construction. Even Rabbi Yehuda agrees that a common-denominator inference based on two a fortiori inferences is valid. This matter requires clarification in light of Result 4 above.

It should further be noted that in the Talmud it is accepted that common-denominator inferences can be rebutted by ‘any slight objection,’ that is, by a weak objection that does not necessarily point to relations of leniency and stringency.^[3] It appears that such objections rebut only a common-denominator inference whose status is like that of a paradigm construction, since this is a weaker inference, see Results 2 and 3. A common-denominator inference from two a fortiori inferences is an inference whose status is stronger, and we expect that with respect to it the rule that one can rebut with any slight objection will not hold. In this sense it would resemble an a fortiori inference.^[4]

Result 6: Any slight objection to the common denominator is said only with respect to the two weaker types, those based on basic inferences in which there is at least one paradigm construction. Again, this must be examined in light of Result 4.

In light of the intuitive explanation, what remains for us to examine is whether, when the objections to the two teaching cases, that is, their unique properties, relate directly to the microscopic parameters, Rabbi Yehuda also agrees to common-denominator considerations that include a paradigm construction. To do so, we must present the application of our model to objections of this type, and then examine whether the decision in them can likewise be challenged on the basis of the valence index. We expect that it cannot, that is, that Rabbi Yehuda will not disagree with the Rabbis regarding the validity of these inferences.

As we will immediately see, we have a similar example in the Kiddushin discussion itself. A second look shows that the objection from benefit to the simple common denominator, stage 6, is not really a column objection as we presented it, but rather a microscopic objection. We will therefore now have to test our model in relation to it as well.^[5]

Benefit as a microscopic objection: solving tables with a constraint

As stated, the claim that there is benefit in money and intercourse but not in huppah is not a legal claim but a factual one. As such, this is a microscopic objection and not an ordinary column objection. An ordinary column objection should contain an additional legal outcome relevant to money and intercourse and not to huppah, and usually the discussions do bring such an objection. For this reason, in the previous chapters we continued the analysis as though there were a column objection here, because we wanted to develop our formal model for ordinary objections to the common denominator, which appear as an additional column, as we presented the objection from benefit. We will now continue the discussion as it really is, and we will treat benefit as a microscopic objection.

Such an objection does not add a column to the table, but rather imposes a constraint on the microscopic model. When the Gemara says that there is benefit in money and intercourse but not in huppah, the meaning is that there is a microscopic parameter, benefit, that exists only in money and intercourse and not in huppah, and perhaps the law depends precisely on it. Therefore one cannot learn from the two teaching cases to the target case. This is an a priori objection, since the fact that those two acts have benefit and huppah does not is not derived from examining any legal datum, but from examining them themselves, a priori, that is, prior to empirical legal inquiry.

The way to find the model that solves this case is to look at the table without benefit, Table 5, the common denominator without the objection, and to search for a model for the diagram generated from it under the constraint that one of the microscopic parameters exists only in money and intercourse and not in huppah. From here on we will proceed along the next stages of the discussion and ignore the benefit column when drawing the diagram. The fact that there is benefit in money and intercourse will be expressed in the fact that we seek a solution for every model, for every table, at every stage, under the above constraint.^[6] Thus, such a microscopic objection enters not as a column but as an a priori constraint on the solution, and this constraint accompanies us throughout all the stages until the end of the discussion.

A microscopic objection to the common denominator

We now return to stage 6 of the discussion, the objection to the common denominator. As stated, the table we are dealing with is Table 5 above:

Y

Table 6.1 (a microscopic objection to the common denominator)

Naturally, the diagrams for the two fillings can also be taken from Diagrams 5:

Diagram 6.1a – objection to the common denominator with filling 1

We seek the solution under the constraint that money and intercourse have a parameter that is absent from huppah. Inspection of the table shows that for this purpose we must impose a constraint that P and Y have a parameter that N does not have, because huppah effects N. If we begin by attaching the parameter to A, we obtain the following solution:

Constraints and structure of an optimal model for Diagram 6.1a – objection to the common denominator with filling 1

where and are variables that we must determine under our constraints.

The solution for the acts is:

Money: , ,

Huppah: ,

Intercourse: , ,

From inspection of the solution for the acts, it is clear that the following must hold here: ; 0 , ; .

Under these constraints, and on the assumption that there are only two parameters, the following solution necessarily arises: = ; =

However, in this case we obtain valence increasing in two different parameters, and this is contrary to Principle 2. This situation forces us to move to a 3-dimensional model, that is, to add another microscopic parameter, and to choose the solution:

= ; =

Thus, the optimal model obtained for this stage is:

Optimal model for Diagram 6.1a – objection to the common denominator with filling 1

The result of the model for the acts is obtained from the table, and it is:

Money: (1,1,1)

Huppah: (2,0,0)

Intercourse: (2,0,1)

This solution satisfies the constraints, and it is optimal for this diagram. It should be noted that benefit () indeed appears in money and intercourse and not in huppah, and it is not what effects marriage, nor betrothal either, because in filling 1 what effects them is the common denominator, .

We now move to the diagram of filling 0:

Optimal model for Diagram 6.1b – microscopic objection to the common denominator with filling 0

The solution for the acts is:

Money: (0,2)

Huppah: (1,0)

Intercourse: (1,1)

Here we find that benefit () indeed appears in money and intercourse, in different measures: there is more benefit in money than in intercourse, interesting, and not in huppah. But in the alternative of filling 0, as expected, it is what is responsible for effecting betrothal, and therefore huppah does not succeed in effecting betrothal.

These are exactly the two alternatives that the objection proposes as equivalent to one another, see the intuitive explanation given above. To check whether there really is an objection here, we must compare the two models in terms of the five preference indices. Filling 1 is preferable with respect to direction changes, exactly as in Diagram 5, and the preference of filling 0 in valence has diminished, in our case the valence of the two fillings is equivalent. But now it turns out that, because of the objection, filling 0 becomes preferable with respect to dimension.

The conclusion is that the common-denominator inference, Table 5, is preferable with respect to the direction-change index, with valence offset, and therefore Rabbi Yehuda disputes it. He will claim here an objection from a stringent aspect, exactly as we saw in the previous paragraph when we analyzed the results we obtained. But the microscopic constraint balances that preference by forcing the addition of a dimension for the diagram in filling 1.

The complex common denominator

We now move on to discuss stage 9 of the discussion, and we will discuss it without the benefit column. Benefit is now a constraint on the solutions. The table in this case is as follows:

Y

G

Table 7.1 (the complex common denominator with a microscopic constraint)

The diagrams obtained for the two fillings are:

__Optimal model for Diagram 7.1a – complex common denominator with a microscopic constraint in filling 1 __

The solution to the model here assumed the above constraints, that there is a parameter that exists in money and intercourse and not in huppah, but here one must also assume that there is no benefit in document, and in addition it does not appear in N. We solve in the same way as above, and obtain what is written on the diagram.

The solution for the acts is:

Money: (2,0,1)

Huppah: (1,1,0)

Intercourse: (1,1,1)

Document: (3,0,0)

As expected, benefit appears in money and intercourse and not in huppah, and it is not what effects betrothal. Exactly as in the previous stage.

__Optimal model for Diagram 7.1b – complex common denominator with a microscopic constraint in filling 0 __

The solution for the acts is:

Money: (2,0,1)

Huppah: (0,1,0)

Intercourse: (1,1,1)

Document: (3,1,0)

Benefit does indeed appear in money and intercourse and not in huppah and document. However, here it does not effect betrothal, because document too effects betrothal, and there is no benefit in it. In this respect it differs from the previous stages.

To examine whether there is a valid inference here, we must compare the two fillings. In both diagrams the dimension is 3 and the connectivity is 1, and the total number of points is 5, and the valence is 3. But the direction-change index favors filling 1, in filling 0 there are two direction changes, from G to N, when crossing A and Y. Thus the inference is valid, and filling 1 is preferable.

Objection to the complex common denominator

We now move to stage 10 of the discussion, again with microscopic constraints. The table obtained here is as follows:

Y

G

K

Table 8.1 (objection to the complex common denominator with a microscopic constraint)

We now present the two optimal models, for the two fillings:

__Optimal model for Diagram 8.1a – objection to the complex common denominator with a microscopic constraint in filling 1 __

The solution for the acts is:

Money: (3,0,1)

Huppah: (1,1,0)

Intercourse: (2,1,1)

Document: (4,0,0)

Optimal model for Diagram 8.1b – objection to the complex common denominator with a microscopic constraint in filling 0

This graph is identical to Graph 7.1b, and therefore the solutions can be taken from there.

The solution for the acts is:

Money: (2,0,1)

Huppah: (0,1,0)

Intercourse: (1,1,1)

Document: (3,0,0)

To examine whether there really is an objection here, we compare the two fillings. In both cases the dimension is 3, and the connectivity is 1. There is a difference in valence in favor of filling 0, and in direction changes in favor of filling 1, there are two direction changes in filling 0, from P to N, when crossing Y and K. The number of points in filling 0 is smaller, 5 as against 6 in filling 1. Thus we have direction change against number of points and valence, and therefore this is indeed an objection.

Revalidation of the complex common denominator with a microscopic constraint

We have reached the end of the process, stage 11 of the discussion. We will now test the revalidation, only this time with a microscopic constraint. The table for this case is:

Y

G

K

Table 9.1 (revalidation of the complex common denominator with a microscopic constraint)

We will now examine the two models for the two fillings, and this time we will begin with filling 0:^[7]

__Constraints for Diagram 9.1b – revalidation of the complex common denominator with a microscopic constraint in filling 0 __

The parameters written on the diagram are the result of the above constraints. We must now consider whether it is possible to fill the entire model using three parameters, or not. We will now prove that it is not.

It is a variable, and it cannot be or multiples thereof, because in that case there would be an order relation between K and N. It also cannot be or multiples thereof, because then there would be an order relation between K and P. It also cannot be because then the order relation between K and Y would be reversed. And if we increase the intensity of in Y, then an order relation is created between it and P. And if we prevent that by raising the intensity of in P, then an order relation will be created between P and K.

Therefore cannot be any of these three parameters, , , and we are compelled to add a fourth parameter to the model. The solution can now be the following:

__Optimal model for Diagram 9.1b – revalidation of the complex common denominator with a microscopic constraint in filling 0 __

The solution for the acts is:

Money: (1,0,1,1)

Huppah: (0,1,0,0)

Intercourse: (2,1,1,0)

Document: (3,1,0,0)

Although at first glance one could have deleted here, and parameter would then be the parameter found in money and intercourse but not in huppah, and likewise with respect to the outcomes, it is found in P and Y but not in N. If so, filling 0 would be three-dimensional, and it would have an advantage. But on closer inspection it is clear that this is not a possible solution, because there is an additional constraint that the parameter common to money and intercourse, which we identify as benefit, not also be present in document. But parameter is also present in document, and therefore it cannot be identified with benefit, and we are necessarily forced to add the parameter . We note that this also accords with our general assumption, according to which parameter defines the strength relations in the model, and other parameters are required to define the different qualities, see the discussion above regarding the requirement that there be a change in valence in only one parameter.

__Optimal model for Diagram 9.1a – revalidation of the complex common denominator with a microscopic constraint in filling 1 __

The solution for the acts is:

Money: (1,0,1,1)

Huppah: (1,1,0,0)

Intercourse: (2,1,1,0)

Document: (3,1,0,0)

To examine whether there is a valid inference here, we must compare the models for the two fillings. In both of them the dimension is 4, the connectivity is 1, the valence is 3, and the number of points in the graph is 6. The decision is in favor of filling 1 because of direction changes. In filling 1 there is one direction change, and in filling 0 there are two direction changes, between G and N, when crossing K and Y.

Thus this inference too is confirmed in our model.

Summary

To summarize our discussion in this chapter, we present here in table form the list of the inferences discussed here, all under a microscopic constraint, and the conclusions that emerge from our model regarding the preferences in the different indices for them:

Objection to a simple common denominator

Complex common denominator

Objection to a complex common denominator

Revalidation

Diagram

6.1a

6.1b

7.1a

7.1b

8.1a

8.1b

9.1a

9.1b

Filling

Dimension

Direction change

Connectivity

No. of points in the graph

Valence

Result

Equivalent

1 preferred

Equivalent

1 preferred

Summary Table 4

With God’s help

Section Two: Special Cases of A Fortiori Inference and Objections

Introduction

After developing the general model, we will now turn to deal with several special cases that appear in Talmudic literature. We will treat several cases here:

1. Offset claims against objections, the discussion in Bava Metzia 41b: ‘The principal without an oath is preferable to double payment with an oath.’ 2. Absorbing objections into an a fortiori inference, Tosafot on Bava Kamma. 3. Non-binary parameters: a. rotation of an a fortiori inference, Mishnah Bava Kamma 24. b. The laws of the ‘it is enough’ principle and the dispute about them. 4. Parameters that operate cumulatively, the end of our discussion: huppah after money. 5. Problems with multiple lacuna cells, ‘derived from what is itself derived,’ and a double objection.

A. An offset claim against objections^[8]

The basic a fortiori inference

The discussion in Bava Metzia 41b deals with the law of misappropriation by bailees:

Do not state the law of misappropriation in the case of a paid bailee, and let it be derived from an unpaid bailee: if an unpaid bailee, who is exempt in cases of theft and loss, is nevertheless liable if he misappropriated it, then a paid bailee, who is liable in cases of theft and loss, is that not all the more so? For what legal purpose did the Merciful One write them? To tell you that misappropriation does not require diminution. And I say: it is not superfluous; it follows Rabbi Elazar, who said that this and that are one. What does ‘this and that are one’ mean? Because one can object: what is special about an unpaid bailee? That he pays double payment when he raises a thief-claim. And the one who does not raise the objection holds: the principal without an oath is preferable to double payment with an oath.

At first, the Gemara proposes learning the law of misappropriation in the case of a paid bailee, that is, a paid bailee, by an a fortiori inference from an unpaid bailee: the unpaid bailee is exempt in theft and nevertheless liable in misappropriation, so the paid bailee, who is liable even in theft, should all the more so be liable in misappropriation. The structure of the data table for this a fortiori inference is exactly like Table 1 above. In our example, we know the following data:

Datum A: the unpaid bailee is exempt in theft.
Datum B: the unpaid bailee is liable in misappropriation.
Datum C: the paid bailee is liable in theft.
Unknown law: is the paid bailee liable in misappropriation?

For purposes of the discussion below, the legal acts are paid guardianship and unpaid guardianship, and the outcomes are liability for misappropriation and liability in relation to theft. We present the picture in table form:

Theft

Misappropriation

Unpaid bailee

Paid bailee

Table 10.1 (a fortiori inference)

In this case we fill the lacuna cell in exactly the same way as we did in Diagram 1.

The objection: first presentation – a column objection

Immediately afterward an objection is raised to this a fortiori inference. On its face it appears to be a column objection: the unpaid bailee is liable in the case of a thief-claim. Therefore, in one approach, we will assume that this is indeed a column objection that adds another legal outcome, and now the table is as follows:

Theft

Misappropriation

Thief-claim

Unpaid bailee

Paid bailee

Table 10.2 (a column objection to an a fortiori inference of acts)

This objection too has already been discussed by us above, and we explained why in such a situation the two filling possibilities are equivalent. The resulting diagrams are:

Model for Diagram 10.2a – objection to an a fortiori inference with filling 1

Model for Diagram 10.2b – objection to an a fortiori inference with filling 0

As we saw, filling 1 is preferable with respect to connectivity, but filling 0 is preferable with respect to the number of points in the graph, valence, and direction changes. Therefore they are equivalent, and there is an objection here.

The objection: second presentation – an alternative objection

But one can view the dispute here in another way as well. The initial a fortiori inference is as we saw above, Table 1. But against it there arises another way to look at the liability for theft. That is, the claim that an unpaid bailee who makes a thief-claim is liable for double payment indicates that Table 10.1 is not correct, since even in ordinary theft the unpaid bailee is more stringent than the paid bailee, for when he makes a thief-claim he becomes liable for double payment, whereas a paid bailee who makes that claim is not liable for double payment. Thus the correct table is the following:

Double-payment liability for a thief-claim

Misappropriation

Unpaid bailee

Paid bailee

Table 10.3 (an alternative objection to an a fortiori inference)

The objection here proposes seeing the data of the table differently from the way proposed by the a fortiori inference, and the result is a paradigm-construction table, though different from the one we saw in Table 2. There the comparison can operate in both directions, whereas here the comparison is only between outcomes and not between acts.^[9] In any event, it appears that here too the result will be 0. The diagrams are as follows:

Diagram 10.3a – an alternative objection to an a fortiori inference with filling 1

Diagram 10.3b – an alternative objection to an a fortiori inference with filling 0

The preference here clearly tilts toward filling 0. That is, here this is not merely an objection, but a proof in the opposite direction, namely in favor of filling 0. The objection arises when we ask which of the two tables we ought to use. This is what we called here an ‘alternative objection.’ Since there are considerations in favor of using each of the two tables, and each yields a different filling, the situation remains open, and therefore there is an objection.

The offset claim

And now the Gemara rejects the objection, and raises the claim, at least according to one Talmudic opinion, that there is an offset in the stringency of the objection. The unpaid bailee’s liability in the case of a thief-claim does not indicate the greater stringency of the unpaid bailee’s liabilities, because that liability is bound up with the fact that he also swears falsely. This reduces the force of the objection.

In the second presentation of the objection, the alternative objection, one can understand this claim as telling us to use the first formulation, Table 10.1, and not the second, Table 10.3. The reason is that the law of the thief-claim points to a lesser stringency in the unpaid bailee, and therefore it is less significant, so it is preferable to use the first table, 10.1.

In the first presentation of the objection, the column objection, see Table 10.2, it seems that the claim that the principal without an oath is preferable to double payment with an oath means that the table should be updated and a different weight assigned to the unpaid bailee who raises a thief-claim. This will express the fact that his liability is for a more severe act, one that includes a false oath, since the punishment here is more severe, double payment. From this it follows that his liability for the theft itself is actually lighter, since it is only with the aid of the oath that he becomes liable for double payment.

If we place the numbers in the data table, they will reflect this consideration. The resulting table is as follows:

Unpaid bailee

Paid bailee

Table 10.4 (an offset column objection to an a fortiori inference)

We now draw the diagrams and find the model:

__Optimal model for Diagram 10.4a– an offset column objection to an a fortiori inference with filling 1 __

Optimal model for Diagram 10.4b – an offset column objection to an a fortiori inference with filling 0

Comparison of the two models yields that filling 1 is preferable in direction change, there are no direction changes in it, and in dimension, there is one parameter in the model, whereas filling 0 is two-dimensional, while filling 0 is preferable only in valence. We have already seen in Rule 9 that valence does not overturn a preference that exists in the other indices. Thus, after the offset claim, it indeed emerges that filling 1 is preferable, and the a fortiori inference again becomes valid.

Summary: an offset claim against objections

We saw two presentations of the objection in the discussion: a. a column objection. b. an alternative objection. In each of the two formulations we saw how the offset must be introduced, and that it indeed revalidates the original a fortiori inference. The difference between the formulations lies in the question whether one can raise an offset claim against an objection that is not an alternative objection, that is, against an ordinary column objection. For example, in the case of huppah and money, with respect to betrothal, marriage, and redemption, can one argue that the fact that money is effective for redemption does not indicate greater importance, and is less significant than the fact that it is not effective for marriage, and therefore on balance it is less strong than huppah. According to the second presentation here, such claims cannot be raised against column objections, and indeed we do not find in the Talmud offset claims against column objections. An offset claim arises specifically with respect to the objection to the a fortiori inference in our case, which is an alternative objection.

Result 7: In principle, one cannot raise an offset claim against an objection, except in a place where the objection lies on the same axis of stringency as one of the data points, and as a result the data in the table can be treated as three-valued. In such a situation, the offset claim is expressed through three-valued data, and the filling 1 that is learned from the basic a fortiori inference again becomes preferable.

B. Absorbing objections into an a fortiori inference

Introduction

Absorbing objections is a process that is not carried out in the Talmud itself, but the early authorities, especially the Tosafists, do it in a number of places. Here we will discuss one example, in order to see how this mechanism too is integrated into our model. For this purpose we will follow the course of the discussion in Tosafot on Bava Kamma, and we will see there several new and interesting phenomena.

The basic a fortiori inference

The Mishnah in Bava Kamma 25a presents an a fortiori inference:

And just as in a place where it was lenient regarding tooth and foot in the public domain, it was stringent regarding horn, then in a place where it was stringent regarding tooth and foot in the injured party’s domain, is it not all the more so that we should be stringent regarding horn!

There is here an a fortiori inference with three data points:^[10]

An animal that caused damage through tooth, that is, it ate something, and foot, that is, it stepped on something in the course of walking, in the public domain: the owner is exempt from payment.
An animal that caused damage through horn, that is, it gored with intent to cause damage, in the public domain: the owner is liable for payment.
An animal that caused damage through tooth and foot in the injured party’s domain: the owner is liable to pay.
Unknown law, the lacuna cell: what is the law of horn in the injured party’s domain?

The data table here is the following:

Public domain

Injured party’s domain

Tooth and foot

Horn

Table 11.1 (an a fortiori inference)

The diagrams for this case are as follows:

Optimal model for Diagram 11.1a – an a fortiori inference with filling 1

The solution for the legal acts is:

Tooth and foot:

Horn:

Model for Diagram 11.1b – an a fortiori inference with filling 0

And with respect to the acts, we obtain:

Tooth and foot: (1,0)

Horn: (0,1)

The opening of Tosafot: a microscopic objection to an a fortiori inference

In Tosafot, s.v. ‘I will not adjudicate,’ there, he asks why this a fortiori inference is not rebutted as follows:

And if you say: what is special about tooth and foot, whose damage is common? Will you say the same about horn, whose damage is not so common, since animals are presumed guarded according to the view that half-damages are a fine?

Tosafot proposes a refutation: tooth and foot have a unique property, namely that their damage is common (as opposed to horn, whose damage is exceptional). We should note that although we have already encountered microscopic refutations, we have not yet encountered them with respect to an a fortiori inference (such refutations certainly do appear in several places in the Talmud). There is no additional column here, that is, no different legal result; rather, there is a reference here to the microscopic characteristics of tooth and foot that create greater stringency than horn.

As we have already seen, a microscopic refutation is handled differently from a column refutation. The data table remains as it was without the refutation, and the refutation merely identifies a relation among the parameters, thereby imposing a constraint on the model and on the solutions for the given diagram. If so, the data table and the diagrams for this case are exactly as in the previous section, except that this time there is a constraint on the models. The constraint is that there is one parameter with respect to which tooth and foot are more stringent than horn. The remaining parameters must be determined by the diagrams, under that constraint.

The solutions obtained are exactly like those we obtained above. Except that in filling 1 we must add another parameter that exists in tooth and foot but not in horn. Looking at the tables, it emerges that this parameter does not affect the results at all, since the diagram continues to show that the public domain is harder for liability than the injured party’s domain. The conclusion is that in filling 1 a two-parameter model is created, but the second parameter does not affect the results and appears only in the actions. And in filling 0 this parameter is simply one of the parameters that appear in the solution. That is, the constraint merely identifies one of the parameters obtained in the previous solution, and nothing more.

If so, in filling 1 the solution for the results does not change, and the solution for the legal actions is:

Tooth and foot: (1,1)

Horn: (2,0)

We see that there is one parameter () with respect to which tooth and foot are indeed more stringent, but it does not affect liability in the public domain and in the injured party’s domain. There is, however, another parameter () in which horn remains more stringent. This is apparently the intention to damage, which the Sages identify as the special stringency that exists in horn. This is the stringency that affects liability in the public domain and in the injured party’s domain.

For filling 0, nothing changes as a result of the constraint. The solution is the same solution, both for the results and for the actions. The constraint merely identifies the stringency that exists in tooth and foot but not in horn () as common damage.

This is the place to note that the case of monetary damages is unique, because the Talmud itself explicitly identifies the microscopic parameters (there are unique characteristics for each primary category of damage that is written in the Torah). We arrive at the existence of such parameters through an empirical examination of the phenomenal aspect (= the legal phenomena). In the case of betrothal and marriage, we truly arrived a priori at a structure of four microscopic parameters, and the learner must identify them independently through reasoning. Here the Talmud does this for us, and therefore our method can guide learners to identify and characterize those parameters. This is a demonstration of the great usefulness of the microscopic model, whose results can guide the learner to identify the basic components underlying the Talmudic discussions.

If so, in our case the competition between filling 1 and filling 0 concerns the question of which parameters affect liability in monetary damages. The question is whether the additional parameter (common damage) affects the results, or whether only the parameter of intent to damage affects them. Under the alternative of filling 0, the commonness of the damage turns out to be the parameter that affects liability in the injured party’s domain, whereas under the alternative of filling 1 it turns out not to affect them at all. Intent to damage affects both fillings; the question is whether it affects only liability in the public domain or also liability in the injured party’s domain.

We must now examine how there is a refutation here. To do so, we must compare the indices of the two fillings. The diagrams remain as in the a fortiori inference, and therefore the advantage of filling 1 is in dimension and connectivity, while it is inferior in valence. All that remains is to add that because of the constraint the models for the two fillings perhaps have the same dimension (if any at all)^[11]. We are left with an advantage for filling 1 because of connectivity (and perhaps also because of dimension). Admittedly, that advantage runs against valence, but according to Rule 9 valence does not impair a preference determined by the other indices. If so, filling 1 remains preferable even after the refutation.

At first glance we seem to have a problem. Tosafot raises a refutation, and our model does not manage to reflect it. But we must not forget that Tosafot is challenging the Gemara, and indeed the Gemara itself does not take this refutation into account. What we obtained here is that this disregard is in fact justified, and Tosafot’s question is incorrect. Below we will see that this itself is probably what Tosafot answers to the question. In any event, this ‘failure’ is actually a great success of our model, since it immediately shows us that such a refutation does not overturn the inference. In our model, Tosafot’s question does not arise at all.

Rejecting rotation of the a fortiori inference

Later in his comments, Tosafot raises the possibility of rotating the a fortiori inference, and thereby escaping the refutation:

And one cannot say that we learn from one domain to the other: just as the public domain, in which the Torah was lenient with regard to tooth and foot, it was stringent with regard to horn, so too the injured party’s domain all the more so. For in any event it is still possible to raise a proper refutation, as is implied below, where it seeks to derive full ransom payment for a harmless ox in the injured party’s courtyard from foot-damages, and it refutes: what distinguishes foot-damages? That they also exist in fire.

He proves from the continuation of the discussion that rotation does not neutralize refutations, which fits our conclusions very well so far. If some inference is valid, then it is valid in both of its aspects (from the side of the actions and from the side of the results), and if an inference is not valid then it is not valid in both. Everything depends on comparing the fillings, and on the existence of a preference of one of them over the other.

Below we will discuss rotation of the a fortiori inference, and we will see in which cases and how it can nevertheless be relevant.

The mechanism of absorbing a refutation

Tosafot resolves the difficulty as follows:

And one may say that this is not a refutation, for this stringency does not help to obligate it in the public domain; and thus we reason the a fortiori inference: just as tooth and foot, whose stringencies do not help to obligate them in the public domain, nevertheless pay full damages, etc.

Tosafot explains that this refutation does not overturn the inference, but is absorbed into it. His claim is that tooth and foot are more stringent than horn because their damage is common, and nevertheless they are exempt in the public domain. If so, horn is more stringent than tooth and foot, despite the fact that tooth and foot possess that stringency. Apparently horn has another stringency that overrides it (as indeed we saw above, that the stringency is the intention to damage, and it has a different parameter from the stringency of common damage that characterizes only tooth and foot). If so, in the injured party’s domain as well one may infer that if tooth and foot are liable, then horn, which is more stringent than they are (despite the stringency they possess), will certainly be liable there too. This is the mechanism of ‘absorbing a refutation.’

How is this expressed in our model? It seems that this is precisely the meaning of the result we obtained above: that a ‘refutation’ of this type does not overturn the inference at all. It leaves filling 1 preferable. The meaning of the matter on the intuitive plane is that there is another stringency parameter in horn, which overrides the stringency found in tooth and foot and leaves the inference valid. Such a refutation is not a refutation at all; it is only an illusion of intuitive thinking. In our model this emerges naturally (from filling 1 above): horn has a special stringency () that does not exist in tooth and foot, and it overrides the stringency in tooth and foot. Therefore this refutation is merely a mistake. No wonder that within our model it was impossible at all to explain Tosafot’s question (who thought that such a refutation overturns the a fortiori inference). When one thinks about these inferences intuitively, such a question can arise, but within our model it does not arise at all. This is an excellent demonstration of the power of the formalization we are proposing.

Absorbing a legal refutation

The authors of the rules write a principle that one does not absorb a refutation that is written in the Torah. The source of this is the continuation of Tosafot’s comments here. They challenge the mechanism of absorbing a refutation with the following difficulty:

And in the first chapter of Zevahim (10a), regarding one who slaughters for its proper sake in order to sprinkle its blood not for its proper sake, where it is disqualified by an a fortiori inference from one who slaughters outside its proper time, which is valid, it refutes: what distinguishes outside its proper time? That it incurs excision. Even though this stringency does not help outside its proper time to disqualify, a stringency that the Torah itself imposed is different, for since the Torah imposed this stringency, it imposed another stringency as well.

Tosafot bring the discussion in Zevahim, where they do not absorb a refutation in an a fortiori inference. To resolve the difficulty, Tosafot formulate another rule: one does not absorb refutations that are written in the Torah. It seems that their intention was not to distinguish between the status of refutations derived from reason and the status of refutations written in the Torah, but rather to say that one absorbs only microscopic refutations (which concern the characteristics of the different actions) and not legal refutations (which deal with legal characteristics).

Refutations that deal with legal characteristics are in fact ordinary column refutations, and therefore clearly there is no room to absorb them, for we saw that such a refutation really does overturn the a fortiori inference.

As a side remark, let us say that it is not entirely clear why Tosafot cites specifically the discussion in Zevahim, and not any other column refutation in the Talmud. It seems that he thought that perhaps there is a different legal refutation there, one that resembles the microscopic refutation discussed here. Perhaps this is because the punishment of excision is not an additional legal characteristic (another column, like redemption of the tithe and consecrated property), but a general feature (admittedly legal, and not factual) of the law of slaughtering outside its proper time in sacrificial slaughter. When one slaughters in order to sprinkle outside its proper time, there are two legal ramifications: the punishment of excision and the disqualification of the sacrifice. If so, this is not another independent data column, but rather a property of the actions discussed in the basic data table. In this sense there is here a similarity to our a fortiori inference, which deals with factual properties of the basic data

But, as stated, Tosafot’s conclusion is that even such a refutation is in fact an ordinary column refutation, and therefore it does overturn the a fortiori inference and cannot be absorbed.

Result 8: Absorbing a refutation.

Absorbing a refutation in an a fortiori inference occurs only when the refutation presents a microscopic characteristic that is relevant to an entire row or column. In such a situation, the intuitive understanding of the a fortiori inference may see a difficulty in such a refutation, but within our model the possibility of presenting such a refutation does not arise at all.
We also saw from our model that when the characteristic is legal there is no absorption of the refutation, as the early authorities and the rule-writers write.

C. Non-binary parameters: rotation of the a fortiori inference and the debate over the ‘it is enough’ principle

Introduction

Until now we assumed that the values filled into the data table are binary: 0 or 1. We saw only one exceptional case, in the discussion of offsetting claims, where there was also a value 2 in one of the cells. We will now discuss such cases in greater detail. The main importance of these cases, which are very rare, is that for some reason דווקא in these cases we find that the Talmud itself tries to ‘rotate’ the a fortiori inference. In these very cases the question of the ‘it is enough’ principle also arises, in both of its forms, and this chapter deals with that.

The main discussion in which such a case appears is the Mishnah in Bava Kamma, where an a fortiori inference appears with a multivalent data table, and there the Mishnah itself performs a ‘rotation’ of the a fortiori inference. There too the question of the ‘it is enough’ principle from the beginning of the derivation and from the end of the derivation is discussed in detail.

The Mishnah’s line of argument

The Mishnah in Bava Kamma 24b deals with monetary damages. Above, in the discussion of absorbing a refutation, we saw an inference that derives liability for horn-damage in the injured party’s domain. Already there we noted that we were presenting a simplified picture of the course of the argument, which was sufficient for our purposes there. We will now move to the full picture. This is the language of the Mishnah:

Mishnah: An ox that causes damage in the injured party’s domain—how so? If it gored, butted, bit, crouched, or kicked, in the public domain it pays half damages; in the injured party’s domain, Rabbi Tarfon says: full damages, and the Sages say: half damages. Rabbi Tarfon said to them: If in a place where the Torah was lenient regarding tooth and foot in the public domain, where it is exempt, it was stringent regarding them in the injured party’s domain, requiring full damages, then in a place where it was stringent regarding horn in the public domain, requiring half damages, is it not all the more so that we should be stringent regarding it in the injured party’s domain, requiring full damages! They said to him: It is enough for what is derived by inference to be like that from which it is derived; just as in the public domain it pays half damages, so too in the injured party’s domain it pays half damages. He said to them: I too do not derive horn from horn; I derive horn from foot. And if in a place where the Torah was lenient regarding tooth and foot in the public domain, it was stringent regarding horn, then in a place where it was stringent regarding tooth and foot in the injured party’s domain, is it not all the more so that we should be stringent regarding horn! They said to him: It is enough for what is derived by inference to be like that from which it is derived; just as in the public domain it pays half damages, so too in the injured party’s domain it pays half damages.

The Mishnah deals with the law of horn in the injured party’s domain. The data are as follows:

Horn (k) in the public domain pays half damages.
Tooth and foot (sr) are exempt in the public domain.
Tooth and foot in the injured party’s domain incur full damages.
The law of horn in the injured party’s domain is unknown.

The data table for this case is the following:

Table 12.1 (a fortiori inference)

At first glance this table can be analyzed exactly like an ordinary a fortiori inference. But here there is a feeling that there should be a difference between the directions in which the a fortiori inference is applied. The a fortiori inference of actions infers from the laws in the public domain (the two cells in the right-hand column) that horn is more stringent than tooth and foot, and therefore in the injured party’s domain too horn should obligate more than tooth and foot, that is, at least 1. But in the a fortiori inference of results we infer from the laws of tooth and foot (the two cells in the top row) that the injured party’s domain is more stringent than the public domain. Therefore in horn too the injured party’s domain should obligate more than the public domain, that is, at least ½. If so, in this case the two directions of the a fortiori inference yield different results for the liability of horn in the injured party’s domain.

When one looks at the Mishnah, one sees that Rabbi Tarfon rules that horn is liable for full damages, that is, on the face of it he seems to derive the a fortiori inference in the direction of the actions, whereas the Sages derive the a fortiori inference in the direction of the results. To be sure, in the course of the Mishnah it appears that both sides are prepared to stand by their conclusion according to both directions of inference of the a fortiori. For that purpose the Sages need to arrive at the ‘it is enough’ principle from the end of the derivation, which is a principle that seems puzzling. At first glance it would seem that Rabbi Tarfon is right, since one formulation of the a fortiori inference suffices to prove that here full damages are required, and in the a fortiori inference of actions that is apparently what emerges. The ‘it is enough’ principle cannot stop the reasoning from the side of the actions, and therefore from the inference of the actions one should conclude that one who causes horn-damage in the injured party’s domain is liable for full damages.

However, according to the conclusion of the Gemara it seems that in fact neither of the tannaitic authorities rotates the a fortiori inference, and both derive the inference from both sides together and are unwilling to make their conclusion depend on learning from the side of the results or the actions. In this sense, it seems that our conclusion still stands: there is indeed only one inference here.

Still, we must understand what the dispute between the Sages and Rabbi Tarfon is. As stated, the dispute does not revolve around the question of rotation, or around which of the two inferences to make, but rather around the question of what to fill into the lacuna cell given the table under discussion, independently of the direction of inference. To understand this, we will now examine what our model yields for such a table. We must check the diagrams for the three possible types of filling. We must check the principles of preference with respect to the three possible fillings of the lacuna cell: 0, ½, and 1.

‘It is enough’ from the beginning of the derivation and ‘it is enough’ from the end of the derivation

The whole question of the ‘it is enough’ principle that arises in the discussion here is also unclear. One can certainly understand the consideration of ‘it is enough’ when we make an a fortiori inference of results (= domains), because in that case the final inference derives the law of horn in the injured party’s domain from the law of horn in the public domain. If the injured party’s domain is more stringent than the public domain, then horn payments in the injured party’s domain should be greater than or equal to horn payments in the public domain, that is, to 1/2. And since we have no indication of how much to rise

above 1/2, it is reasonable to determine that the liability is exactly ½. To be sure, in this case Rabbi Tarfon’s position is unclear, for even with respect to this consideration he obligates full damages.

But the other side of the a fortiori inference, which compares actions (= primary categories of damage), derives the law of horn in the injured party’s domain from the law of tooth and foot in the injured party’s domain. By the same logic, here the result should be full damages, and there is no room for the consideration of ‘it is enough.’ Yet the Sages in the Mishnah determine that even with respect to this inference one applies ‘it is enough,’ and the liability remains at ½. For this purpose they define an additional ‘it is enough,’ and distinguish between ‘it is enough’ from the beginning of the derivation and ‘it is enough’ from the end of the derivation. The early commentators explain these two types of ‘it is enough,’ but at bottom, on the intuitive level, this remains entirely unclear.

In our model the whole matter requires explanation. As we have already seen, our model does not distinguish between the directions of inference (actions or results), and consequently it is not expected to explain to us the mechanisms of ‘it is enough’ from the beginning of the derivation and ‘it is enough’ from the end of the derivation. On the other hand, if we succeed in showing the position of the Sages and the position of Rabbi Tarfon regarding the table as it stands, then the principle of the ‘it is enough’ rule for both directions will automatically emerge for us. The conclusion will be that in such a table the result is 1 or ½, and that is what is called on the intuitive plane ‘it is enough.’ This will be an implicit explanation of the two types of ‘it is enough.’

Testing the inference in our method: the ‘it is enough’ principle

As we saw, the table for a multivalent a fortiori inference of this type is the following:

Table 12.1 (a fortiori inference in a multivalent table)

The diagrams for this case are the following:

Optimal model for Diagram 12.1a – multivalent a fortiori inference with filling 1

The solution for the legal actions is:

Tooth and foot:

Horn:

Optimal model for Diagram 12.1b – a fortiori inference with filling 0

And with regard to the actions, we get:

Tooth and foot: (1,0)

Horn: (0,1)

__Optimal model for Diagram 12.1c – multivalent a fortiori inference with filling ½ __

The solution for the legal actions is:

Tooth and foot:

Horn:

The solution is exactly like the one we obtained for filling 1. The dispute between the Sages and Rabbi Tarfon revolves precisely around the question of how we are to decide which type of filling is preferable in this case.

According to the Sages, the solution seems simple. Since the two graphs for filling 1 and filling ½ give the same result (both are preferable to filling 0, and they are equivalent to one another), we should choose the lesser of them. This is exactly the meaning of the principle of ‘it is enough.’ As long as we have no proof that the filling must be more than 1/2, we set it at ½. Whoever wants to increase it further, the burden of proof is on him.

In other words, the general method according to the Sages for a multivalent table is as follows: one must begin with the lowest possible filling and raise its value slowly upward. When we reach the preferred filling, we must stop. All the fillings at higher values that give the same level of preference are equivalent to this filling, and we must choose the lowest among them. Thus, according to the Sages, we arrive here at the determination that the correct filling is ½.

In light of what we said above, it should be noted that in this way we have explained both types of the ‘it is enough’ principle. The reason the Sages apply ‘it is enough’ against both types of inference is that in their opinion, in this given table the result for the filling in the lacuna cell is 1/2, and it does not matter from which direction

of inference we approach it. Therefore, on the intuitive plane one must innovate two types of ‘it is enough,’ but מבחינתנו this follows by itself. In such a table the result is 1/2, however we may call it.

Result 9: The two types of the ‘it is enough’ principle emerge naturally from our model. In fact, there are no two types of ‘it is enough’ at all, and the arguments of ‘it is enough’ from the beginning of the derivation and ‘it is enough’ from the end of the derivation are nothing but a reflection of the fact that the optimal model for the data table is the filling corresponding to ‘it is enough.’

Rabbi Tarfon’s position: ‘rotating’ the a fortiori inference

Rabbi Tarfon does not accept the principle of ‘it is enough,’ and as we noted he does not accept it with respect to the two paths of inference. It seems that his position can be understood as follows: without the principle of ‘it is enough,’ the level of optimality of the two fillings (1 or 1/2) is equivalent, and therefore there is no way to decide which of them should be chosen. In this case it seems that the preferred value can be chosen by looking at the actions (= rows) instead of the results (= columns).

If we look at the table above, we see that the order relation between the actions is the following:

__Optimal model for Diagram 12.2c – multivalent a fortiori inference by actions with filling ½ __

We described the solution according to the method in which we worked on the results. But if we now consider what the solution should be for the domains, so that it will explain the data of the table, we must define a model in which a result of 1/2 can be explained systematically. Here we propose the following model: if a certain action has half of the strength required for liability in the domain under discussion, then the result is 1/2. In that case, the solution to the diagram is more complicated (the solution that assigns one parameter to each point in the diagram will not succeed in offering a model for the results that explains all the data of the table). Therefore we must choose a more complex model (though still two-dimensional), and what we get is the following solution:

Here too there are two dimensions and independence between the actions (apart from valence).

To obtain the results in the table, we will define the solution for the domains as follows:

N: (2,0)

R: (2,1)

It should be noted that this solution also maintains the relation between the columns (described in the previous description of the a fortiori inference).

__Optimal model for Diagram 12.2b – multivalent a fortiori inference by actions with filling 0 __

With regard to the results, we get:

Injured party’s domain: (1,0)

Public domain: (0,2)

Here too the result 1/2 is explained in exactly the same way.

__Optimal model for Diagram 12.2a – multivalent a fortiori inference by actions with filling 1 __

In this case the arrow connects actions and not results, and therefore it is clear that the solution is reversed, that is, the action k __receives higher values than the action __sr. The one-dimensional solution does not work, and an additional microscopic parameter has to be introduced.

Therefore we will write the solution as follows:

And with regard to the domains, we get:

Injured party’s domain: (1,0)

Public domain: (2,1)

In this case all three models are two-dimensional, but there is a topological difference in favor of filling 1 (connectivity). The two models for filling 0 and 1/2 are equivalent, and both are inferior to filling 1. Therefore, according to Rabbi Tarfon, filling 1 is preferable, and that is indeed how he rules.

A note on the direction of inference: the dispute between Rabbi Tarfon and the Sages

It is important to understand that the model we found would also have been obtained if we had drawn the diagrams for the domains and not for the actions, because ultimately the model is supposed to explain the data table. So what is the difference between an analysis by actions and one by domains? The difference is only in terms of the topological indices. Connectivity and disconnectedness relate to the relation between the rows (the actions) and not to the relation between the columns (the domains). Therefore, despite the identity with regard to the model obtained (dimension and valence), the result of the preference may be different (as indeed we saw happens).

If so, there is here a kind of rotation of the a fortiori inference. To be sure, it is not rotation in the intuitive sense, that is, there is no different inference here of rows or columns. But there are two ways to present a logical analysis of the inference, one in the orientation of rows and the other in the orientation of columns. If so, the expectation that in an asymmetric table there will be significance to the direction of inference is justified. Two different analyses may lead to two different results.

In precisely such a situation the dispute between the Sages and Rabbi Tarfon arises. Rabbi Tarfon’s opinion is now clear. מבחינת the model, all the fillings are in dimension 2, and therefore they are equivalent. The decision is made on a topological basis. The question is on which topological basis to perform the analysis, whether on that of the actions or on that of the results. According to Rabbi Tarfon one analyzes by the actions, and according to the Sages one analyzes by the results.

It may be that the correct formulation is the following: according to the Sages, for the result to be 1, it must emerge from both directions (and not specifically from the results). If there is a difference in the result, we choose the lower result (= this is exactly their principle of ‘it is enough’). Rabbi Tarfon, who does not accept the ‘it is enough’ principle in this case, holds that one proof that the result is 1 suffices for that to be the result. The fact that a second analysis leads to a result of 1/2 is unimportant. In this sense, the dispute does indeed resemble its intuitive appearance (that according to Rabbi Tarfon one proof that the result is 1 is sufficient, whereas for the Sages a double proof is required).

Result 10: Rotation of an a fortiori inference does not exist within the model. It can, however, be expressed on the level of choosing the criterion of preference (columns or rows). Rabbi Tarfon uses such a ‘rotation’ in his criterion of preference, as is implied by the language of the Mishnah. This does not contradict the fact that throughout the Talmud the possibility does not arise of ‘rotating’ an a fortiori inference in order to escape a refutation. This is done only when the data of the table are ternary.

Summary

The conclusion is that these results explain both the question of the ‘it is enough’ principle in our model and the question of ‘rotation,’ which apparently could not arise within it and yet enters by the back door. The ‘it is enough’ principle is interpreted exactly as in intuitive thinking. According to the Sages, one takes the lowest value among the acceptable values, since any higher value requires proof. ‘Rotation’ of the a fortiori inference in our model appears only in Rabbi Tarfon’s reasoning, and its meaning is not a different inference, but an analysis based on an order relation between rows instead of the order relation between columns (precisely because he rejects the ‘it is enough’ principle). Our consistent conclusion remains in place: there is no separating the directions of inference in any a fortiori inference, and even in an asymmetric case the distinction between the two types of the ‘it is enough’ principle does not arise at all.

D. Parameters that operate cumulatively

Rejecting Rav Huna’s argument: an intuitive explanation

After the Gemara in the Kiddushin discussion reaches, at stage 11, Rav Huna’s conclusion that a canopy effects betrothal (on the basis of the consideration validating the complex common denominator, which we discussed above), Rava disagrees with him and maintains that a canopy does not effect betrothal. He raises against him the following argument:

Furthermore, does a canopy complete anything except by means of betrothal? And shall we derive a canopy without betrothal from a canopy with betrothal?

That is, one cannot derive the law of canopy regarding betrothal from canopy regarding marriage, since canopy is effective in marriage only because it comes after betrothal has already been performed.

First, we must note that now the discussion in the passage returns to the first stage: Rav Huna’s simple a fortiori inference. The Gemara assumes that after all the complicated course we have gone through, in the end we returned and validated the original a fortiori inference. Rav Huna’s source for saying that canopy effects betrothal is the original a fortiori inference from money, and all the refutations were rejected during the logical course that followed, which we traced in the first chapters of the first gate.

Rava now raises a refutation against Rav Huna’s a fortiori inference. His claim is that creating the marital bond is a process composed of two stages: betrothal and marriage. Each part of this process requires something that will effect it, and the effecting of the second part is done with the aid of the fact that the first part has already been effected.

Explanation in terms of our model

After we applied betrothal through money, the canopy now completes the process and effects marriage as well. But the action performed by the canopy is done with the aid of the money that was already given at the previous stage. The reason the canopy succeeds in effecting marriage does not mean that the microscopic component found in it is stronger than the component found in money. The component found in money is already operating, since it was created in the stage of betrothal when the money was given to the woman, and it continues to operate after betrothal and assists the component found in the canopy in order to complete the process and effect marriage. The canopy operates with the aid of the money and not alone, and therefore it should not be measured by itself against the strength of the money. If the money had not been given before the canopy, the canopy alone would not have succeeded in effecting marriage. From here Rava argues that if the canopy succeeds in effecting marriage after betrothal, this does not mean that it must necessarily succeed in effecting betrothal itself.

In our language we would say it as follows: from Rav Huna’s a fortiori inference (in which filling 1 is the preferable one, and therefore we assume it is the correct one) it emerges that to effect betrothal the parameter is required, and it is present in money. To effect marriage, a higher strength of the parameter is required: , and that is what is present in the canopy. Therefore, clearly, if canopy effects marriage then it certainly can also effect betrothal. And to this Rava argues that there is indeed still the possibility that the canopy has a lower strength, such as ½ , and yet it nevertheless succeeds in effecting marriage with the aid of the strength that the money has already introduced into the matter. Together with the strength given by the money there is here 1½, and therefore marriage takes effect through the canopy. But from this one cannot infer that the strength of the – found in the canopy alone is higher than that found in money. Therefore the question of whether canopy can effect betrothal (which requires ) remains open.

This still does not explain the matter in terms of our model, but only formulates Rava’s refutation in terms of relative strengths of a microscopic parameter.

‘Rotation’ of the a fortiori inference

Let us now examine whether Rava’s argument refutes both a fortiori inferences. The a fortiori inference of the actions, which assumes that canopy is stronger than money (from examining the marriage column), is indeed refuted. Canopy succeeds in effecting marriage only with the aid of the money that had previously been given for the purpose of betrothal. By contrast, the a fortiori inference of the results, which assumes that marriage is harder to effect than betrothal (from examining the money row), is not refuted at all. The fact that marriage is effected only after betrothal does not attack that assumption. On the contrary, this fact leads to the conclusion that marriage is easier to effect, since the validity of betrothal assists in effecting it. And nevertheless, money does not effect marriage, while it does effect betrothal.

Abaye answers him at the end of the discussion:

Abaye said to him… And as for what you said: Does a canopy complete anything except by means of betrothal? Rav Huna too meant this: If money, which does not complete marriage after money, nevertheless effects betrothal, then canopy, which does complete marriage after money, is it not all the more so that it should effect betrothal.

It should be noted that he does not rotate the a fortiori inference, as we might have expected. In his answer he rescues the validity of the a fortiori inference of the actions as well, and does not need rotation. He argues that one can compare canopy to money and infer that canopy in total, with the addition of the money of betrothal, is stronger than money in total (with the addition of the money of betrothal).

We should note that this approach fits very well with our conclusion here, that rotation of an a fortiori inference does not save it from any refutation. To validate an a fortiori inference, one must validate both inferences together. If one of them is refuted, then neither of them is valid.

Abaye’s proposal uses the mechanism we saw above, of absorbing Rava’s refutation into the a fortiori inference. The refutation that money assists canopy is absorbed into the a fortiori inference and leaves the relations of leniency and stringency in place after the refutation is absorbed: canopy by itself (after neutralizing the added strength that comes from the money of betrothal) is stronger than money, for after money is given for betrothal, canopy succeeds in bringing about marriage and money does not. That is, the money of betrothal does not help money of marriage to operate, but it does help canopy.

A ternary table

Up to this point we have stated the matter in words, but in the formal model itself it appears that it cannot be incorporated, because we do not have a mathematical tool that expresses a chain of consecutive legal actions that operate one after the other. Our entire calculation is conducted at a single stage of the process. Still, it may be that the verbal explanation is sufficient, because the model can be defined as consecutive slices, in each of which we perform the calculation defined up to this point. Sometimes the process is not Markovian (= it has ‘memory’), and therefore there can be a residue from the previous stage that takes part in the current stage (as the money of betrothal assists canopy in effecting marriage).

On the other hand, the fact that rotation of the a fortiori inference could have helped Rav Huna and Abaye against Rava’s attack hints to us that there is probably a ternary a fortiori inference here, as we saw above regarding the Mishnah in Bava Kamma. And indeed, on further inspection it seems that here too there are three levels of values for data in the table: effects betrothal, effects marriage after betrothal, effects both together (or effects marriage directly, without prior betrothal), exactly as in the Bava Kamma discussion. Moreover, the data table looks very similar:

Table 13.1 (a fortiori inference of cumulative parameters)

The meaning of the value ½ is that canopy does not really succeed in effecting marriage by itself, but only part of the marriage process (after betrothal has already taken effect).

We can now apply everything we saw above in the dispute between Rabbi Tarfon and the Sages. According to the Sages, in such a table the conclusion is ½. That is, one cannot infer that canopy indeed effects betrothal (for there is nothing that assists canopy in effecting betrothal. The question is whether it succeeds in doing so by itself). And according to Rabbi Tarfon the conclusion is 1, that is, that one can infer that canopy effects betrothal.

We saw that in the Bava Kamma discussion the halakhah is ruled in accordance with the Sages, against Rabbi Tarfon. If so, it is clear that in our case too the halakhah will be ruled like Rava, that indeed canopy cannot effect betrothal. And so it is ruled in halakhah.^[12]

Result 11: We again confirmed that rotation of an a fortiori inference arises only when the data of the table are ternary. A refutation about a cumulative situation means that the data of the table are de facto ternary.

General method

To be sure, in the most general case it is difficult to see how this method will operate. What will happen if we know that a certain action (m) succeeds in effecting a certain result (X) only after another action (n) has been performed, or after another result (Y) has already been achieved. Or perhaps in a more general case, it is known that the action also succeeds in effecting another result (Y), only after the action n has been performed, and so on.

In such cases, it seems that we must construct the model for the diagrams under constraints. We will search for a model for the diagram in each of the two fillings (which are still binary, 0 or 1), while to the microscopic components present in the action m there are added the components found in the action n. In our example, canopy effects marriage because in addition to the component present in it, the component present in money also operates.

Let us take our a fortiori inference as an example. The table is the following:

__h __

Table 13.2 (a fortiori inference – general method)

We now assume that canopy at the stage of marriage operates with the assistance of the money. If so, the model for filling 1 is exactly as in the regular a fortiori inference:

Betrothal:

Marriage:

And if we move to the actions we can suggest here (and this already differs from the regular a fortiori inference):

Money:

Canopy:

As stated, canopy effects marriage only because it is assisted by the money, and therefore overall it has a de facto strength of . Already from the model for filling 1 one can see that the conclusion does not follow from this that canopy is stronger than money.

We now examine filling 0. If we look at the diagram for this case, it deals with two separate points. In the regular case the solution is two different parameters. But here that may be misleading. We will construct the model from the table and not from the diagram. The results obtained are:

Betrothal:

Marriage:

And for the actions:

Money:

Canopy:

If we now draw the relevant diagrams, we will see that even in filling 0, where the table yields a picture of two separate points, a different diagram is obtained, completely similar to filling 1. This solution shows that even in filling 0, betrothal comes out stronger than marriage, and there is a single simple arrow here.

When the diagram is identical, all the topological indices are equal, and therefore we are left only with comparing dimension and valence. The dimension is 1 in both cases, and regarding valence the situation is somewhat problematic. Looking at the results shows that in both cases there are two values of the parameter, but looking at the actions shows a different valence for filling 0. That is, filling 0 is marked by some inferiority, but we already saw that valence alone is not sufficient to determine a filling’s preference. Therefore the conclusion is that these two fillings are equivalent, and one cannot decide which of them is preferable. בכך we have proved that this is indeed a refutation.

There are cases in which the dependence will be not on the action but on the result. For example, it may be that Rava, who argues against Abaye that one should compare canopy after money (which helps to effect marriage) to money after money (which does not help to effect marriage), is really saying that it was not the money that helped canopy effect marriage but the very fact that there is betrothal. Even if the betrothal was effected by deed or intercourse, canopy could effect marriage after them, whereas money could not do so. Here it is already clear that the entire discussion concerns the application of canopy after betrothal, and the comparison between canopy and money remains intact, and therefore such a proof rescues Rav Huna’s a fortiori inference.

In such cases, we will have to find a solution for the given table under the constraint that when the action is applied, all the microscopic components accumulated when the previous result (the betrothal) was effected are available to us, and not necessarily the components of another action (such as money).

In the most general case, this is a constrained solution that adds parameters to those found in the action under discussion and helps it effect the result. Sometimes this will be an entire column, and it will be a fairly complex solution. In any event, in all these cases a solution can be found from the table in various ways. In a case where there are no such constraints, the diagram is the way to find the solution from the table. In cases with constraints, we must find a solution from the table (without the help of a diagram, as we saw in the example of the a fortiori inference above), and only afterwards create a diagram from the solution obtained (in order to define the topological indices needed for determining the preferred filling). In any event, everything still remains within our model.^[13]

E. ‘Deriving from the derived’

Problems of multiple lacunae

After outlining the entire model and its different applications, one important and fundamental problem still remains. Until now we assumed that in every data table all the legal data can be extracted from Scripture except for one. The problem was how to fill the lacuna cell in light of the other data. What happens if we have a table with more than one lacuna cell?

‘Deriving from the derived’

At first glance, when there are two lacuna cells, we can fill one by the method developed here, and from it fill the second. This proposal touches on what in Talmudic literature is called the problem of ‘deriving from the derived,’ that is, making a midrashic inference on the basis of another midrashic inference. There are chains of midrashic inferences here: an a fortiori inference on the basis of an analogy, or a common-denominator argument on the basis of an a fortiori inference, and so on.

The Sages’ assumption is that there is no impediment to doing this, and every such chain is valid. This itself requires treatment in our model. At first glance, in our model, if there are two lacuna cells, we will have to examine four possible fillings (two for each cell). We will then have to weigh the different fillings against each other and choose the preferable one. We need to check whether one can prove a theorem that if each stage separately is valid, then their combination will also necessarily be valid.

This is essentially parallel to the relation between conditional probability and absolute probability. Assuming that the filling for the first cell is 1, we know that the preferable filling for the second cell is also 1. The question is whether from this it is also correct to infer the unconditional conclusion that the filling 1,1 is the preferable one for the overall table.

In the discussion in Zevahim (around page 50), the Gemara deals extensively with these combinations. The conclusion is that the realm of sacred matters is exceptional, and there there is a limitation regarding ‘deriving from the derived.’ In the realm of sacred matters there are possible combinations and combinations that are not possible, whereas in the realm of ordinary matters all combinations are possible.

Our hypothesis is that the combinations possible in sacred matters are combinations whose unequivocal preference can be proven. By contrast, the combinations that are not possible in sacred matters are apparently combinations whose validity is problematic, or special in some sense. And that sense is legitimate with respect to ordinary matters but not with respect to sacred matters. For example, perhaps valence is considered in the realm of sacred matters to be an index of equal weight to the others (as we saw in the opinion of Rabbi Judah regarding ‘the refutation from the stricter side’), and therefore all combinations of inferences based on the superiority of one index against valence, which was acceptable to us up to now, are not acceptable in the realm of sacred matters. By contrast, the combinations that are possible even in sacred matters are combinations whose relation of preference is unequivocal. Such combinations are of course valid in all areas of law.

Another possibility is that an index such as changes of direction is not significant in the realm of sacred matters, and all relations of preference based on them are not valid there. To be sure, this hypothesis is less plausible, because if there is indeed no preference based on changes of direction, then we must also give up a considerable part of the basic inferences (which are not ‘deriving from the derived’) that we have discussed up to now, with respect to the realm of sacred matters. But there is no indication of this at all in Talmudic literature. Therefore it is more reasonable that the previous hypothesis (regarding valence), or something like it, underlies the distinction between the realms.

In order to test those hypotheses, the possible and impossible combinations in sacred matters must be classified on the basis of the discussion in Zevahim, and the relations of preference in all those cases must be examined. As stated, we are dealing only with combinations of the two types of analogy and the a fortiori inference layered one on top of the other. The attitude toward the other hermeneutical rules does not concern us now.

The data from the discussion

A review of the discussion in Zevahim 49b–51a shows that it deals only with the following four hermeneutical rules: juxtaposition (which does not appear at all in Rabbi Ishmael), verbal analogy, a fortiori inference, and analogy. Moreover, the discussion does not distinguish between the two types of analogy (except for one passage on 50a, where the possibility of learning from one verse is rejected and the possibility of learning from two source texts is raised), and it seems that it refers only to the simple analogy. From this it follows that there are four relevant combinations that we must discuss:

Analogy from analogy. This remains an open question (51a).
Analogy from an a fortiori inference. The Gemara concludes that this is valid (51a).
An a fortiori inference from an analogy. An open question (51a).
An a fortiori inference from an a fortiori inference. Valid (50b).

We must now examine these matters in light of our model.

An a fortiori inference from an a fortiori inference

As stated, the Gemara concludes that an a fortiori inference from an a fortiori inference is effective even in sacred matters. The question is how a situation can בכלל arise in which the question is asked whether one may derive an a fortiori inference from an a fortiori inference.

A table of an a fortiori inference is as follows:

Table 14.1

The first a fortiori inference fills the lacuna cell with 1. We now want to derive from it another law by another a fortiori inference. What could the situation בכלל be? Adding a column on the left or a row below does not create a situation of an a fortiori inference, because that adds two adjacent cells, and however we fill them they will not create a sub-table of an a fortiori inference when they stand adjacent to a row/column of 1’s. For example, a table in which we try to derive an a fortiori inference from this a fortiori inference is obtained if we add another action to the problem, as follows:

Table 14.2

However we fill the cells in the third row (one of them is another lacuna cell), this will not create a situation that allows it to be filled by the consideration of an a fortiori inference based on the previous filling. In no way is a structure created here like that of Table 14.1 around the lacuna cell. In exactly the same way, it is clear that this cannot happen if we add a column with another result (instead of the row).

To examine how a situation of deriving an a fortiori inference from an a fortiori inference is nevertheless created, we must return to the discussion itself and see what example is brought there. The discussion does not bring examples of an a fortiori inference from an a fortiori inference, and therefore it is hard to find an example to analyze. But it turns out that the discussion clarifies the questions of deriving from the derived by means of inferences that are all on the methodological, meta-legal plane. That is, the considerations of an a fortiori inference from an a fortiori inference brought here concern the very discussion of whether to derive an a fortiori inference from an a fortiori inference, and so on. If so, from the course of the Gemara there emerges an argument of an a fortiori inference from an a fortiori inference that we can analyze.

This is the language of the Gemara there, 50b:

May something learned by an a fortiori inference itself teach by an a fortiori inference? An a fortiori argument: if verbal analogy, which is not itself learned by juxtaposition according to Rabbi Yohanan, nevertheless teaches by an a fortiori inference, as we said, then an a fortiori inference, which is learned from juxtaposition according to the school of Rabbi Ishmael, should it not all the more so teach by an a fortiori inference? And this is an a fortiori inference from an a fortiori inference. It is the grandson of an a fortiori inference! Rather, an a fortiori argument: if juxtaposition, which is not learned by juxtaposition, whether according to Rava or according to Ravina, nevertheless teaches by an a fortiori inference according to the school of Rabbi Ishmael, then an a fortiori inference, which is learned from juxtaposition according to the school of Rabbi Ishmael, should it not all the more so teach by an a fortiori inference? And this is an a fortiori inference from an a fortiori inference.

That is, the consideration of whether an inference of an a fortiori inference from an a fortiori inference is valid is itself learned by an a fortiori inference. Two possibilities arise here for learning it:

Verbal analogy is not learned after juxtaposition, and it teaches by an a fortiori inference. Then an a fortiori inference, which is learned after juxtaposition, all the more so teaches by an a fortiori inference.
Juxtaposition is not learned after juxtaposition, and nevertheless it can again teach by an a fortiori inference. Then an a fortiori inference, which is learned after juxtaposition, all the more so teaches by an a fortiori inference.

The relevant tables are the following:

Inference 1 (the grandson of an a fortiori inference)

Learned from juxtaposition

Teaches by an a fortiori inference

Verbal analogy

A fortiori inference

Table 14.3

__Inference 2 __

Learned from juxtaposition

Teaches by an a fortiori inference

Juxtaposition

A fortiori inference

Table 14.4

The Gemara rejects the first table because it is circular. This table itself is based on an a fortiori inference from an a fortiori inference (which is precisely the problem we came to examine). In conclusion, the Gemara learns that an a fortiori inference from an a fortiori inference is valid from the inference in Table 14.4.

But we are interested in an example in which an inference of an a fortiori inference from an a fortiori inference appears. Here, at the beginning of the discussion, we happen to have such an example, since the Gemara derives the law of an a fortiori inference from an a fortiori inference by means of an inference that is itself an a fortiori inference from an a fortiori inference. Therefore, for our purposes, we will examine דווקא Table 14.3.

For this purpose, we must ask ourselves from where the law that verbal analogy teaches by an a fortiori inference is learned. Earlier in that same discussion (50b) it is said that this law too is itself learned by an a fortiori inference:

May something learned by verbal analogy teach by an a fortiori inference? An a fortiori argument: if juxtaposition, which does not teach by juxtaposition, whether according to Rava or according to Ravina, teaches by an a fortiori inference according to the school of Rabbi Ishmael, then verbal analogy, which teaches by juxtaposition according to Rav Pappa, should it not all the more so teach by an a fortiori inference? That works according to the one who accepts Rav Pappa, but according to the one who does not accept Rav Pappa, what can be said? Rather, an a fortiori argument: if juxtaposition, which does not teach by juxtaposition, whether according to Rava or according to Ravina, teaches by an a fortiori inference according to the school of Rabbi Ishmael, then verbal analogy, which teaches its fellow verbal analogy according to Rami bar Hama, should it not all the more so teach by an a fortiori inference.

The tables for these two inferences are the following:

Inference 3 (according to the one who accepts Rav Pappa, that verbal analogy teaches by juxtaposition)

Teaches by juxtaposition

Teaches by an a fortiori inference

Juxtaposition

Verbal analogy

Table 14.5

Inference 4 (according to the one who does not accept Rav Pappa)

Doubled (= teaches twice)

Teaches by an a fortiori inference

Juxtaposition

Verbal analogy

Table 14.6

The upper-left cell in Table 14.3 is filled by force of the a fortiori inference in Tables 14.5 and 14.6. The mathematical structure is exactly the same, and therefore we will make one analysis for both cases. Looking at the combination of the tables shows that in order to present the full picture we must present a 3X3 data table, with three rows of actions (juxtaposition, verbal analogy, and a fortiori inference), and three columns of results (teaches by juxtaposition [or is doubled], learned by juxtaposition, and teaches by an a fortiori inference). The data table for this case has two lacuna cells, and it looks like this:

An a fortiori inference from an a fortiori inference (according to Rav Pappa) (filling 1 in the lower-right cell)

Teaches by juxtaposition

Teaches by an a fortiori inference

Learned by juxtaposition

Juxtaposition

Verbal analogy

A fortiori inference

Table 14.7

We see that in this table there are two lacuna cells, and after filling x by force of the a fortiori inference in the upper-right sub-table, one can return and fill y by an a fortiori inference on the basis of the lower-left sub-table.

How did we arrive at the other data in the table? The fact that juxtaposition is not learned by juxtaposition (upper-left cell) instructs us that juxtaposition is not learned from juxtaposition (upper-right cell). This is the very same law. Regarding the question whether an a fortiori inference teaches by juxtaposition, there is a doubt in the discussion (50b). The Gemara brings evidence for this (from Rav Pappa), and afterwards they reject it (according to the one who does not accept Rav Pappa), and it remains unresolved. If so, in order to show that one may derive an a fortiori inference from an a fortiori inference, this has to be done under two different assumptions:

According to Rav Pappa, we must fill this cell with 1 (and derive from Table 3, which fits his position). The resulting table is the one above.
And according to the one who does not accept Rav Pappa, the filling of this cell is 0 or 1 (because according to him an a fortiori inference does not necessarily teach by juxtaposition—it remains unresolved), but the table we use here is not Table 3 but Table 4 (because Table 3 follows Rav Pappa), and therefore in the relevant table created for him there is no reference at all to the question whether an a fortiori inference teaches by juxtaposition (but rather whether it is doubled). In that case the datum in this cell is a doubled a fortiori inference, that is, an a fortiori inference from an a fortiori inference. But this is precisely the question we are clarifying now. Therefore we must leave this cell as a lacuna, and impose a constraint that it be identical to the lacuna cell to its left (for that cell too asks whether one derives an a fortiori inference from an a fortiori inference). The resulting table is the following:

An a fortiori inference from an a fortiori inference (according to the one who does not accept Rav Pappa) (constrained lacuna in the lower-right cell)

Doubled

Teaches by an a fortiori inference

Learned by juxtaposition

Juxtaposition

Verbal analogy

A fortiori inference

Table 14.8

We will now proceed to see that in both of the tables of an a fortiori inference from an a fortiori inference brought above, our model indeed yields the correct decision. The examination of a table with two lacuna cells must be carried out by comparing the four diagrams obtained from the four possible fillings (the left datum in the vector is the filling examined for x, and in the right place is the filling for y): (1,1) (1,0) (0,1) (0,0)

Diagrams 14.7

Filling: 1,0

Filling: 0,0

Filling: 1,1

Filling: 0,1

A glance makes it clear that the filling (1,1) is the preferred one, and the meaning is that in this case an inference of an a fortiori inference from an a fortiori inference is valid even from a direct look at the two lacuna cells.

We now move to the examination of Table 14.8 (with the constraint on the lacuna):

Diagrams 14.8

Filling: 1,0

Filling: 0,0

Filling: 1,1

Filling: 0,1

Here too one immediately sees that the filling (1,1) is the preferred one even from a direct look at the two lacuna cells.

Result 12: An a fortiori inference from an a fortiori inference is a valid argument for both opinions, in complete accordance with the conclusion of the Gemara in Zevahim.

A note on the direction

To complete the picture, let us look at a table in which the two cells to be filled for an a fortiori inference from an a fortiori inference are horizontal and not vertical:

An a fortiori inference from an a fortiori inference in the horizontal direction

Juxtaposition

Verbal analogy

A fortiori inference

Table 14.9

At first glance, here too there is an a fortiori inference from an a fortiori inference: first, one fills the cell x with 1 by force of an a fortiori inference from the upper right. Afterwards one fills y with 1 by force of an a fortiori inference from the lower left. Here the two considerations each work separately. However, when we examine the corresponding diagrams, we see that the direct consideration is not valid here:

Diagrams 14.9

Filling: 1,0

Filling: 0,0

Filling: 1,1

Filling: 0,1

Here all the diagrams are split (connectivity = 2), but the fillings 1,0 and 1,1 are preferable because of the identity of points in the diagram. That is, it can be proved that the left filling (y) is 1, but the right filling (x) remains open.

This too can be seen from looking at the table, since a filling of 1 in the cell x is refuted by the two rightmost cells in the third row. Alternatively, a filling of 1 in the cell y is refuted by the two leftmost cells in the top row (unless it is not an a fortiori inference but some other inference, as indeed we saw, for the cell x is not necessarily filled with 1).

A note regarding filling 0: there is no general theorem about an a fortiori inference from an a fortiori inference

We should note that a filling of 0 in the cell of an a fortiori inference that teaches by juxtaposition leads to a situation in which there is no preference for the filling (1,1). This is so even though each of the two basic a fortiori inferences remains valid. This is another example of the fact that there is no general theorem that one can always derive an a fortiori inference from an a fortiori inference, exactly as we saw in case 14.9.

The resulting table is the following:

An a fortiori inference from an a fortiori inference (according to the one who does not accept Rav Pappa) (filling 0 in the lower-right cell)

Teaches by juxtaposition A

Teaches by an a fortiori inference B

Learned by juxtaposition C

Juxtaposition

Verbal analogy

A fortiori inference

Table 14.10

The diagrams obtained in this case are (the filling is indicated by a vector in which the first number from the left is the filling for y, and the second for x):

Diagrams 14.10

Filling: 0,0

Filling: 1,0

Filling: 0,1

Filling: 1,1

One can see that the filling (1,1) is not preferable (there is a change of direction there, as against the inferiority of the splits in the other diagrams).

To be sure, this is also very easy to see from looking at the table. If indeed we fill (1,1), then the first a fortiori inference fills the cell of verbal analogy in an a fortiori inference with 1, but an additional filling of 1 in the next cell will create a row refutation (like a column refutation, except that here one adds a row with an axis of stringency opposite to the desired one) against the first a fortiori inference.

Result 13: At least in the cases we examined, when a filling of the table refutes an a fortiori inference found within it, our analysis also yields that same result.

Result 14: There is no general theorem regarding an a fortiori inference from an a fortiori inference. The matter depends on the other data of the 3X3 table within which the inferences are made.

Analogy from an a fortiori inference

We now turn to examine the other three inferences of deriving from the derived. In a table of an a fortiori inference, the outer row/column is (1,1), and that is what created the problematic nature of attaching another a fortiori inference to it. But attaching an analogy to such a structure is simple. The result is a table like this:

Analogy from an a fortiori inference

Table 14.11

We fill the cell x with 1 by the a fortiori inference from the right, and from it we continue to fill y with 1 by analogy. Another possibility is to add a similar row below (instead of column C). This is clearly a valid inference (the filling (1,1) is manifestly preferable), and this fits well with the words of the Gemara in Zevahim 50b.

Analogy from analogy

And the table of analogy from analogy also looks almost identical (except for Aa):

Analogy from analogy

Table 14.12

Here the filling of x is made by analogy and so is the continuation. In both cases it is also clear that the filling (1,1) comes out preferable. To be sure, in the Gemara this is not explicitly resolved (but only as one possibility among several—see 51a).

An a fortiori inference from an analogy

The second complex inference that can raise a problem is an a fortiori inference from an analogy. The table in this case must be 3X3, because it is impossible to attach an a fortiori inference to an analogy from this side (exactly as with an a fortiori inference from an a fortiori inference). The resulting table is:

An a fortiori inference from an analogy

Table 14.13

Here we copied the table of an a fortiori inference from an a fortiori inference (14.7-8). The cell A__a__ changes to 1, because here this is an a fortiori inference from an analogy. The cell A__c__ can be 0 or 1, and so too C__a__ (which we took from the tables above). When A__c__ is filled with 1, it seems clear that the filling (1,1) is the preferable one (because it identifies the two rightmost columns), and therefore we wrote here a filling of 0. We now obtain the following diagrams:

Diagrams 14.13

Filling: 0,0

Filling: 1,0

Filling: 0,1

Filling: 1,1

(0,1) is a preferable result compared to (1,1) because of a change of direction and because of one fewer point, but in terms of connectivity דווקא (1,1) is the preferable filling. The conclusion is that there is here an invalid inference, and the question remains open. And indeed, examination of the discussion shows that an a fortiori inference from an analogy is not necessarily valid (it too is resolved as one of the possibilities on 51a).

Result 15: The inferences of deriving from the derived as they emerge from our model are these: an a fortiori inference from an a fortiori inference comes out as a valid inference, as in the conclusion of the Gemara. An a fortiori inference from an analogy comes out as invalid (and in the Gemara there is no unequivocal resolution of this question). An analogy from an analogy is valid (and regarding this too the Gemara gives no unequivocal resolution). And an analogy from an a fortiori inference comes out valid, as the Gemara proves. In the final analysis, one may say that there is no contradiction from the Gemara to our results, although we arrive at conclusions that in the Gemara remained open.

Double refutation

Above, when we described the intuition behind the qal va-homer, we noted that two different inferences lie behind it (a qal va-homer of actions and a qal va-homer of results). On the intuitive plane, it seemed that a different type of refutation operates on each of these inferences: the qal va-homer of actions is challenged by a column refutation, in which an additional column is added to the left of the table (see Table 2). A different refutation operates on the qal va-homer of results, obtained by adding a row at the bottom (without the column refutation on the left; see Table 2.1). In our model we found an explanation for why the Sages do not distinguish between these two types of refutation, and in fact also not between these two types of inference. The qal va-homer is a single inference, and each of these refutations refutes it.

We can now ask what happens if the qal va-homer is attacked by these two refutations simultaneously. Such a situation arises when we find another relevant result and another relevant action, and in both of them the hierarchy among the data is the reverse of what is sought in the qal va-homer. The table that results in this case is the following (see above in Table 2.2, and the discussion around it):

Double refutation (see Table 2.2)

Table 14.14

The datum in cell C__c__ is not fixed. Although in some cases it may perhaps be possible to extract its value by examining Scripture or halakhah, in principle either of the two fillings may appear there in different cases. Therefore we would not expect the result to depend on the value of the datum in that cell.

When one analyzes such a table, a very interesting result is obtained. Here too this is a case of a double lacuna, since there are two lacuna cells. Here as well, let us define the filling as a vector in which the left number is the filling of cell x, and the second number is the filling of cell y. The resulting diagrams are the following:

Diagram 14.11

Filling: 0,0

Filling: 1,0

Filling: 0,1

Filling: 1,1

What we obtain is that the filling (0,0) is the most preferred, and the filling (1,1) is the least preferred. As noted, the cell that interests us is x, and the conclusion is that its filling is 0, regardless of the filling of cell y. In other words: regardless of the missing datum (which in any case must be filled separately by examining Scripture or halakhah), there is always a preference for filling 0 in the lacuna cell.

The meaning of this result is that a double refutation does something stronger than each of the refutations separately: each of the refutations by itself leaves the question open (this is how we defined the term ‘refutation’ throughout our discussion). After an ordinary refutation is presented, the filling of the lacuna cell can be 0 or 1 with equal force. By contrast, a double refutation proves that the filling is specifically 0. If so, this is not merely a refutation, but a positive proof in the opposite direction, or a counterproof.

The conclusion is that indeed it makes no difference which refutation we raise against the qal va-homer, a row refutation or a column refutation. But if we raise two such refutations simultaneously, the result is definitely different, and stronger.

Result 16: A double refutation against a qal va-homer is not a refutation but a counterproof.

VI. A note and a question about the universality of the model

The universality of the model

As we saw in the introduction, the modes of thought with which we have been dealing are not unique דווקא to talmudic literature or halakhic midrash. These three rules are logical forms used by human beings in every field of thought (in science, economics, psychometric testing, law, and so forth). If so, our model is a general model of non-deductive reasoning, and not a model of one particular field.

The criteria of preference defined here did not assume anything special drawn from talmudic thinking. These are general characteristics of analogy and induction, which are entirely general tools of thought. The criteria for the preference of diagrams also appear general and universal. Transitivity, together with Occam’s razor, underlies the indices we defined. These are requirements that apply in every field of thought, except for pathological cases.

The problem itself

If so, apparently we are offering here a universal mechanism that describes analogical and inductive reasoning in general. Moreover, it apparently seems to be an entirely mechanical model, that is, a model that mechanizes inductive and analogical thinking, which seems patently unreasonable. It is commonly assumed that only deductive reasoning can be fully mechanized, and now it apparently seems that inductive and analogical reasoning can be mechanized as well. Have we not thereby turned all human thought into something mechanical? Is it not true that there is room for creativity and subjectivity in drawing inductive or analogical conclusions? Moreover, the conclusions of a non-deductive inference are not necessary, and apparently the existence of a rigid model shows that there is one necessary correct answer.

It is worth noting that all the scholars mentioned in note 15 at the beginning of our discussion pointed to their open-ended character, in contrast to the Greek syllogism. For example, Maccoby writes there:

We see from this that in a qal va-homer argument there may sometimes be an uncertainty arising from the choice of appropriate terms. This choice of terms may be a matter of intuition, rather than strict logic, and thus one person’s valid qal va-homer may be another’s fallacy. This does not mean that this method of argument should be condemned as subjective, but only that it belongs to the area of rationality rather than strict logic.

By contrast, here we see that this inference is not open at all, and can be placed on a rigid logical mechanism. Admittedly, he brings there the words of Heinrich Guggenheimer:

Heinrich Guggenheimer ( pp. 181-85) gives a cogent account of the dayyo rule in terms of pure logic, saying that, in virtue of this rule, the qal va-homer argument is ‘an admirable solution (the only one known to me) of the problem of making an analogy an exact reasoning’.

But an examination of his remarks shows that he too does not mean that it is exact in the same sense as classical logic. For example, both of them see the dispute of the tannaim in the mishnah in Bava Kamma concerning ‘dayyo’ as an indication of the open-ended character of the midrashic inference. But according to our account, these are different logical starting assumptions. Each side is compelled to reach its conclusion on the basis of its logical method. Moreover, even in classical logic the premises are the product of different intuitions, and only the inference is mechanical. According to our model, such a description is equally true of synthetic thinking (analogy and induction).

They all also point to the possibility of presenting refutations as an indication of the open-ended character of these inferences. We did the same at the beginning of our discussion, במסגרת the critique of Schwartz’s approach, which identified the qal va-homer with the Greek syllogism. But in light of the conclusions we have reached, it seems that this does not imply that the inference is open-ended. As we have seen, once all the relevant data are taken into account there is only one correct answer. The refutations are merely a way of presenting, one after another, the data relevant to the problem. The error in the inference before the refutation was presented stemmed from the fact that we had not taken all the data into account. Theoretically, one can present the table of relevant data immediately, and the conclusion will arise from it unambiguously. A complete table will never be subject to refutation.

Let us emphasize again that if we were really speaking only about a certain field, in which very specific non-deductive modes of inference apply, and which just happen here to be open to mechanization, then the matter would not be so troubling. However, the universality of our model, which apparently assumes nothing from the talmudic data and modes of thought, seems very troubling indeed. How can one mechanize generalizations from empirical data and arrive at general laws in a way that leaves no degree of freedom at all? Are our generalizations forced upon us? If they are, this raises a question about the accepted distinction between analytic thought (logic, or mathematics) and synthetic thought (science, law, and so forth).

A counterexample: the example of mutual attacks

What, then, is not forced here after all? It turns out that in the special cases in which we would prefer graphs (=diagrams) with different characteristics, the considerations of preference will be different. For example, if there is a theory in which every point attacks the other, the criteria of preference there will be different. For example, when the diagram describes claims, each of which attacks/contradicts the other, and the arrow connecting the claims indicates a contradiction between them, there we will strive for maximal separation between the parts of the diagram.^[14] There we will also not demand transitivity, nor identification between different points in the graph. Such identification, as well as a change of direction (=lack of transitivity), will be a disadvantage rather than an advantage.

For example, if we have a model in which each claim attacks the other, and the diagram looks like this:

B attacks A, A attacks C, and C attacks B. Each of the claims A and B does not attack itself. We now ask whether claim C attacks itself.

For the sake of simplicity, let us choose a data table for a simpler case, dealing with two claims, each of which attacks the other. Claim A does not attack itself, and the question is whether claim B attacks itself. The data table here is the following:

Model of mutual attacks

At first glance, this is a qal va-homer table, and if our model were universal we ought to fill in 1 here. But it is clear that in this case the situation is different, since when claims attack one another, the requirement of the diagrams is that they be split and non-transitive, and therefore the preference goes in the opposite direction. In this case we must define an opposite preference consideration, that is, the diagram that is less preferred by our criterion is the correct filling in the lacuna cell in this case. The resulting conclusion is the expected one: claim B also does not attack itself.

The conclusion is that there is a degree of freedom in our model, and it is not wholly mechanical. The degree of freedom lies in the choice of the criteria of preference, and they can vary from field to field and from problem to problem.

And yet, a problem

But this does not really solve the problem. First, we see that even in the pathological case presented here, the logic of the problem itself lets us understand in advance that our criteria are not relevant to it. Second, it is clear that this is a pathological case. But in all the standard fields of thought, when we want to infer a conclusion by analogy or induction, the consideration of preference apparently really should be the one we defined here.^[15] It is still troubling that analogy and induction, which appear to be non-rigid modes and to depend on the degree of creativity of the person employing them, should be subject to a rigid structure that can be mechanically modeled. In other words: it seems that a computer will be able to perform analogies and inductions in all fields of thought, and in effect also to issue halakhic rulings and interpret and expound Scripture for us (at least to fill logical lacunae). The suggestions we raised, to make the degree of freedom and creativity depend on the choice of model, do not seem sufficient, since as we noted this is an almost universal model, and the exceptions for which the model is not suitable are rather esoteric.

Choosing the columns

There is another degree of freedom in the model, and it is which columns and rows are to be chosen for the inference. There are very many possibilities for choosing actions and halakhic results as the basis for our inference. If so, it may be that the choice of actions and results is what conceals the non-mechanical parts of the inference.

However, it seems that this too is not a real solution. Let us think what would happen if, theoretically, we were to consider the full body of halakhic data and take everything that has any connection at all to our discussion. For example, if we want to discuss whether huppah effects betrothal, we would take everything that huppah does across all of halakhah, and add everything that effects betrothal. From the set of actions and results we thus obtain, we would examine all the actions that relate to any result and all the results that relate to any action, and add those too to the pool. After that we would continue in the same way, expanding and taking everything connected to the actions and results we added, and so continue until we accumulate all the halakhic data that are related, on some level, to the problem.

Up to this point, apparently, the process was mechanical, and if our model indeed works, then from here onward the decision once again becomes completely mechanical. If so, filling the lacunae can still be done by a completely mechanical algorithm, even if it is fairly complex.

The fact that it is very difficult to collect all the relevant halakhic data is only a technical question. On the essential level, there is still an entirely mechanical inference here, and now even the degree of freedom involved in choosing the relevant actions and results no longer exists. It is not reasonable to reduce the problem of mechanizing non-analytic thought solely to the question of complexity (which is only a matter of difficulty, and not an essential obstacle to a process of mechanization). Quite simply, there is an additional component here, one that is essentially different, because of which these forms of thought and inference are not mechanical in their essence. This does not seem to be only a problem of complexity. Therefore the philosophical problem remains.^[16]

So what, then, is the significance of our model?

It seems that the answer to these difficulties lies in a distinction usually made in the field of artificial intelligence. There are two principal approaches there regarding the goal of the artificial-intelligence algorithm: 1. The goal of the algorithm is to arrive at the correct answer (that is, the answer that matches the facts). 2. The goal of the algorithm is to arrive mechanically at the answer that an ordinary person would have reached by intuition, or by informal reasoning (that is, the answer that corresponds to what a person would have reached).

Our model is meant to reflect the way human thought works. It was built מתוך following reasoning processes as they are carried out by us in everyday life, in law, and in science. It does this mechanically, but its goal is to reach the results we would have reached in our ordinary (non-deductive) thinking.

The results of applying the model are not necessarily correct in the factual sense, just as ordinary human inferences are not necessarily correct. Unlike deduction, in analogy and induction the conclusions are not contained in the premises, and they constitute an expansion of the data found in the premises. Therefore the derivation of the conclusions from the premises is not necessary. What the rigid model does is represent the mode of human inference, and the conclusion the model reaches is the conclusion that a person should reach if he correctly applies his non-deductive thinking.

At the root of the matter, we must know that the risks of error in our non-deductive thinking can stem from two reasons: 1. An error in applying the inference. 2. The inference itself (analogy or induction) is not the right tool for dealing with the problem.

For example, I see a green frog, and from this I infer that the other frog is green as well. It may be that I have made an error in the inference, that is, that I did not apply it correctly (I should also have paid attention to the shape of their ears). But equally, it may be that I applied it correctly (according to the rules of human thought), yet it did not lead to the correct conclusion, because it is simply not right to infer the property of color from one frog to the other. For example, if I infer the length of frog A from frog B, I will err not because there is no place for analogy, but because analogy is not the right tool to use with respect to the property of frogs’ length. Put differently, even if we use all the data that can be gathered regarding the two frogs, and make a perfect analogy, there is still no guarantee that the conclusion of the inference will be correct. In this respect deductive reasoning differs from its non-deductive counterpart.

The second type of problem expresses a problem in human thought, not a problem in applying the inference. Our model, if it is indeed correct, constitutes at most a precise implementation of the inferences, but of course it cannot solve the problems built into our thinking, which this model merely describes, but certainly does not replace.

In other words: if our intuitive thinking does not match the conclusion that emerges from applying the model we proposed, then there is a problem in the operation of human thinking (assuming the model is correct), or else the model must be corrected. But if human thinking does indeed fit what emerges from the model, yet both of them failed to reach the factually correct conclusion, then this is not a problem of the model but of our form of thinking (the non-deductive form), which is not completely precise and reliable.

Footnotes

The reader interested in precise mathematical definitions, more detailed logical justifications, and examples of applications of the method proposed here to additional fields is referred to our article in English:

‘Matrix Abduction with applications to argumentation theory and the argumentum A Fortiori inference rule (Kal-Vachomer)’, M. Abraham, D. Gabbay, U. Schild, Bar-Ilan University, Israel, and King’s College London.

Hereinafter: ‘the English article’
See Middah Tovah, parashat Shemot, 5766.
See Encyclopaedia Talmudit, entry ‘Binyan Av,’ around notes 67-70. See also in __Middah Tovah__, parashat Shoftim, 5766.
In the sugya in Hullin 115b, some refutation is raised against the common denominator whose one source case is a qal va-homer (orlah) and whose other source case is not (mixed planting in a vineyard).
True, the unique properties of money and intercourse are halakhic (redemption of second tithe and acquisition of a levirate widow), and therefore in our sugya Rabbi Judah will certainly disagree with the Rabbis, and will raise a ‘stricter aspect’ refutation against the common-denominator inference, as we explained above.
We shall number the models and the tables that are solved under this constraint with numbers parallel to those given them when we treated benefit as a separate column. Table 6, which represented the refutation of the common denominator, will now be marked Table 6.1, and likewise 7.1 instead of 7, and so on. This is a move entirely parallel to what we have done so far, and our aim is to test the consistency of the model with a priori refutations.
We begin with the filling 0, because if we prove that four parameters are required there, then clearly 1 is preferable, whether the filling 1 is three-dimensional or four-dimensional.
See Middah Tovah, parashat Devarim 5765, where an example from aggadic midrash is discussed and compared to the halakhic phenomenon. For another phenomenon of offsetting, see Appendix C to the English article, where the paradox of judgment aggregation is discussed.
Such a table presents a comparison of the sort that appeared in the introduction at the beginning of the first section, when we compared qal va-homer and binyan av. The example we used was the inference that if Reuven succeeded in law more than in physics, one may infer from this that Shimon too will succeed in law more than in physics. Already there we noted that such an argument lies midway between qal va-homer and binyan av.
Below we shall see that the data are more intricate than we presented here, since there are several levels of payment. For the time being we are ignoring this aspect for the sake of simplicity.
At present our assumption is that if there is an additional parameter that appears in the actions, even if it does not affect the results, it participates in determining the dimension. If we decide that the dimension, too, does not change because of the constraint, the advantage of the filling 1 is even greater, and our remarks above remain intact.
And further study is required as to whether Abaye rules like Rabbi Tarfon, or whether he did not understand the Sages in that way. In any event, for practical halakhah this presentation of the problem remains consistent.
For the general mathematical formulation of the problem, see the English article.
See the discussion of this in the English article, in two places: when we presented the criteria of preference, and at the end of the article when we commented on Dung’s diagrams.
See examples brought in the English article: regarding tornadoes, regarding a data table for purchasing screens and electronic devices, regarding medical diagnosis, and many more.
These matters take us to questions of Gödel’s theorem, and to the halting problem for Turing machines, since the basic question is whether the halakhic system can indeed be mechanized automatically in this way. We leave this logical-philosophical discussion for another place.

Contents of the Article

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Y

H

G

K

Y

Y

G

Y

G

K

Y

G

K

Footnotes

שתף

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