Clearly, something is off in this argument. To be sure, the problem is not its formal structure, since it has exactly the same structure as the ordinary a fortiori inference. If so, it seems that the problem stems from the contents and not from the logical structure of the inference. The problem appears to be that the two axes involved in the discussion are not relevant to one another. That is, there is no direct relation between love/hate of jazz (= hierarchy on axis A) and love/hate of fine literature (= hierarchy on axis B). The conclusion is that an a fortiori argument should assume the relevance of the axes of stringency, or some sort of parallelism between them.^[26]

With respect to the question of relevance, it is worth examining other inferences as well, for example an a fortiori inference based on only two data points. There are cases in which we infer conclusions by a fortiori reasoning on the basis of two data points, and not three as in the ordinary a fortiori inference.^[27] For example, the sugya in Berakhot 21a deals with blessings over food and over Torah study. The scriptural data are as follows:

1. One must recite a blessing before studying Torah (= the blessing over Torah study).
2. One must recite a blessing after eating (= Grace after Meals).

The question now arises whether one must also recite a blessing after studying Torah, and whether the same is true before eating. The Gemara proposes answering this by means of an a fortiori consideration: if Torah, which does not require a blessing afterward, does require a blessing beforehand, then food, which does require a blessing afterward, certainly should also require a blessing beforehand. Of course, one can also suggest the opposite consideration (which contradicts the first): if food, which does not require a blessing beforehand, does require a blessing afterward, then Torah, which requires a blessing beforehand, is certainly required to have a blessing afterward as well. This argument contradicts the previous one, because the previous argument assumed that Torah does not require a blessing afterward, whereas this argument proves that it does require a blessing afterward. And the same applies on the opposite side.

In this example too, one could say that we do not recite a blessing before eating because a blessing is not relevant before eating at all, but only afterward. Likewise, the reason we do not recite a blessing over Torah afterward is not that there is an exemption from blessing, but that a blessing is not relevant after Torah study, only before it. We see that a lacuna does not necessarily indicate leniency, but sometimes rather irrelevance.^[28]

The fact that only two data points are involved indicates that there is an additional missing datum. Our hypothesis is that the absence of the additional datum is due to its irrelevance. However, as the jazz example shows, it is not only a fortiori reasoning based on two data points that suffers from irrelevance. Sometimes even an a fortiori argument based on three data points does. Moreover, even if we had three data points regarding the blessings, in principle we could still raise the possibility that there is no relevant connection between the axes of stringency. Still, the usual assumption of halakhic exegesis is that if there are three data points, then there is a relation of relevance between the two axes. By contrast, when there are only two, this is often rooted in a lack of relevance. This is also how some writers on the rules explain (see, for example, Rabbi M. Ostrovski, The Hermeneutic Principles by Which the Torah Is Interpreted, Jerusalem, 1924, in the section dealing with a fortiori reasoning) the flaw in the following argument, which seeks to obligate every doorpost in fringes: if a four-cornered garment, which is exempt from mezuzah, is obligated in fringes, then a doorpost, which is obligated in mezuzah, surely should be obligated in fringes as well. Here it seems fairly clear that the fact that there are only two data points (a doorpost is obligated in mezuzah, and a four-cornered garment is obligated in fringes. There are no explicit data about the opposite exemptions, and they follow only from the absence of a binding source) indicates a lack of relevance, or an absence of connection between the two axes of stringency (the stringency regarding obligation in fringes and the stringency regarding obligation in mezuzah). A doorpost is not exempt from fringes; rather, the obligation of fringes is irrelevant to it, and similarly for a four-cornered garment with respect to mezuzah.

What is this relevance of which we are speaking here? In terms of microscopic parameters, we would say that the a fortiori inference is valid only if the components that cause love of jazz and love of reading fine literature are the same components (perhaps at different intensities). Hence the relations between the phenomena are purely quantitative, that is, dependent only on the dosages of the microscopic parameters in each datum, and therefore one can infer from one to the other by a fortiori reasoning. The claim of irrelevance means that these components differ qualitatively, and not only quantitatively, and therefore one cannot infer from one to the other. These are different components, not different intensities of one component. And once two different components are involved, the actions or results can exhibit a different hierarchy on parameter A than on parameter B. In the example above, with respect to the blessing beforehand, Torah is more stringent than food, since that is controlled by one parameter, whereas with respect to the blessing afterward, food is more stringent, since it is controlled by another parameter. We may infer that food contains the first component at intensity ½ and the second component at intensity 1, whereas Torah contains the first component at intensity 1 and the second component at intensity ½. Therefore there is no unambiguous hierarchy between these axes, and so one cannot make an a fortiori inference between them in either direction. All this, even though we did not identify these two components at all (whether they involve some kind of benefit, or the manifestation of the Holy One, blessed be He, and the like). It is enough for us to point to the very existence of the microscopic parameters, and to the relation, or absence of relation, between them.

In summary, the question of relevance is a clear indication of the existence of microscopic components. The split among the microscopic components means that one cannot treat an a fortiori inference on a purely quantitative plane, and we must also take into account the qualities measured by those intensities.

Two different inferences at the basis of the a fortiori

Let us now look at another implication of the existence of microscopic parameters in the background of a fortiori inferences. First, we should note that every quantitative a fortiori argument can be presented in two different ways:

1. The inference from the actions: from the two data points in the right-hand column one can infer that huppah is stronger than money (because it effects marriage, and money does not do so). From there one moves to the third datum and infers that if money, the weaker one, effects betrothal, then huppah, the stronger one, certainly can effect it.
2. The inference from the results: from the two data points in the top row one can infer that betrothal is easier to effect than marriage (since money effects it but does not effect marriage). From there we move to the third datum and infer that if huppah can effect the harder result to effect, namely marriage, then it certainly can effect betrothal.

The first inference makes an assumption about the actions: that huppah is stronger than money, and it does not depend at all on the question of the relation between the results (marriage and betrothal). By contrast, the second inference makes an assumption about the results: that betrothal is easier to effect than marriage, but it assumes nothing at all about the relation between the actions (huppah and money).

At first glance, then, these appear to be two entirely independent inferences. Each is based on a different assumption, and so it is clear that each constitutes a different inference. On the other hand, it is enough that one of them be valid in order to prove that the correct filling of the empty cell (= the lacuna) is ‘yes.’ We will now illustrate this by raising a refutation against these inferences.

Let us think of a possible refutation of this a fortiori inference (in the example above, we saw such a refutation from the study of physics). We can imagine something like this: there is a third halakhic action, neither marriage nor betrothal, which דווקא money can perform and huppah cannot. This is exactly what the sugya in Kiddushin does there, at stage 3. There it brings a refutation from the redemption of consecrated property and the second tithe, regarding which Scripture tells us that money can redeem them whereas huppah cannot.

To present the data now, an additional column is required in the table. The resulting table is as follows:

N

A

P

M

0

1

1

H

1

?

0

Table 2 (A column refutation of an a fortiori inference from actions)

Why does such an argument refute the a fortiori inference? In light of the conventional explanation, it would seem at first glance that it cancels the assumption that huppah is stronger than money, since the data regarding redemption show that the situation is the reverse. If so, the assumption of inference A (from the actions) has been refuted, and thereby canceled. But what about the assumption of inference B (from the results)? It would seem that the additional column in the table does not affect the assumption of inference B, which states that betrothal is easier to effect than marriage. The data regarding redemption do not touch the question of the relation between betrothal and marriage. If so, adding such a third column does indeed refute the inference from the actions, but the inference from the results still remains valid. Hence the conclusion of the a fortiori inference remains valid: the correct filling of the empty cell is 1 (as noted above, it is enough that one of the inferences be valid in order to prove this).

If so, once a refutation of this type is raised against the a fortiori inference from the actions, one can ‘rotate’ the a fortiori and raise the ‘orthogonal’ inference, the a fortiori inference from the results, thereby leaving the conclusion valid despite the refutation. It seems that in order to refute the conclusion completely, we must raise another refutation that will operate against the second inference (that is, find another action, in addition to money and huppah, that effects marriage and not betrothal). Such a refutation would appear as a third row in the table, and the filling of its first two cells (regarding betrothal and marriage) would be the reverse of that in the first row.

In theoretical terms, such a refutation would contain an additional action X that succeeds in effecting marriage and does not succeed in effecting betrothal. The resulting table is now as follows:

N

A

M

0

1

H

1

?

X

1

0

Table 2.1 (A row refutation of an a fortiori inference from results)

This refutation too, of course, refutes only one of the a fortiori arguments, and not both of them. The conclusion is that in order to refute an a fortiori inference, one must raise both refutations and enlarge the table by one row and one column (we will not now enter into the question whether the action x succeeds in effecting redemption or not):^[29]

N

A

P

M

0

1

1

H

1

?

0

X

1

0

Table 2.2 (A comprehensive refutation of both inferences)

As stated, apparently each of the two refutations by itself would not succeed in undermining the conclusion we inferred from the a fortiori argument, that is, in changing the filling of the table. Only the combination of both can do that.

The major problem that arises here is empirical. In the overwhelming majority of the dozens and hundreds of a fortiori inferences that appear in talmudic literature, once a single refutation is raised against the a fortiori argument, it is regarded as nullified. That is, a column refutation cancels the conclusion regarding the filling of the empty cell, as though placing the third column also refuted the inference from the results. There are extremely rare cases (see, for example, the Mishnah in Bava Kamma 24b, to be discussed below) in which the talmudic sages themselves raise the possibility of ‘rotating’ the a fortiori inference.^[30] In the huppah sugya as well, once the refutation is presented, the inference is treated as though it had been refuted.

On the other hand, precisely the fact that the talmudic sages were aware of this possibility makes the question all the sharper: why, in the overwhelming majority of cases, do they nonetheless ignore it? Our microscopic model answers this difficulty as well.

Microscopic model: basic definitions

The problem of rotating the a fortiori inference leads us to the insight that at the basis of the a fortiori there lies a different consideration, one that unifies the two inferences above. We claim that what created the problem is the fact that, in presenting the a fortiori inference up to now, we ignored the microscopic parameters that stand behind the halakhic phenomena. So far, we have treated the a fortiori inference only at the phenomenal level, that is, we have seen it as based solely on a quantitative relation between the degrees of stringency of the factors involved, and nothing more. From this came our conclusion that the assumption that huppah is stronger than money (the assumption of a fortiori inference A) is different from the assumption that marriage is harder to effect than betrothal (the assumption of a fortiori inference B). However, it seems that in order to understand the logic and mode of operation of this inference, we must descend to the microscopic-theoretical level and examine the factors that underlie these relations of stringency. At that level, it appears that this is indeed one inference and not two different inferences, and a column refutation refutes it.

For that purpose, we must define some basic definitions regarding microscopic models. Afterward, we shall return to the table that presents the a fortiori inference and analyze it.

Definition 4: A ‘microscopic model’ for a data table – a collection of microscopic parameters, each of which is found at some fixed intensity in every action (= the intensity of the action) and in every result (= the intensity required in order to effect the result) in the table. This collection of parameters provides an explanation, or realization, of the data table.^[31] There are, of course, several models for every specific data table. We shall choose among them the optimal model, that is, the simplest one (in a manner to be defined below).

A picture of these hidden parameters will give us the correct filling of the empty cell in the table. An incorrect filling ought to be contradicted by the microscopic model; that is, the model that leads to the incorrect filling of the table will be rejected for some reason. In other words: the model required to explain the correct filling should be more correct, or simpler, than the model that explains the incorrect filling, and that will be proof that the correct filling is the one yielded by the simpler model.

To find the correct filling for the lacuna cell, we must examine the two data tables, one with filling 1 and the other with filling 0, seek an optimal microscopic model for each, and determine which of the two optimal explanations is the more elegant and simple (superior).

Definition 5: The conclusion of the inference is the finding of the correct filling for the lacuna cell. This is done by comparing the models for the table with the two possible fillings. The preferred model (in the sense to be defined below) gives the correct filling. This process is the expression, in our model, of the exegetical inference.

To complete the picture, let us add the definition of a refutation of an inference. After we have made an inference and filled the lacuna cell, it is possible to challenge it by adding another action (= another data row), another result (= another data column), another datum (= another cell), or an external constraint on the microscopic parameters in the model. A refutation creates a new table or a constraint on the existing table, and in the new situation the two models for the two filling possibilities are equivalent (that is, there is no preferred explanation). This is the expression of refutation in our model.

Definition 6: ‘Refutation’ – the addition of data, actions, or results, or a constraint on the microscopic parameters, in a way that creates a new table in which the two models for the two ways of filling the lacuna cell are equivalent.

It should be noted that a refutation does not prove that the conclusion is wrong, but only refutes the proof that one of the possible conclusions is ‘correct.’ In other words, it leaves the discussion open. This point will recur again and again below.

The goals from here on:

What remains for us to examine are three things:

1. What is the way to create possible explanations (= how do we get from a data table to an ‘explanation’ that is a model for it)?
2. How should we define and choose the optimal model for a specific data table?
3. We must find a criterion for establishing preference between two optimal models for the two competing fillings of the lacuna cell.

We shall now return to building the microscopic models for the three basic inferences and for the refutations of them. In the course of this move, and by examining the course of the entire Kiddushin sugya, we will be able to extract some insights regarding these three goals.

Microscopic model for an a fortiori inference

As stated, the a fortiori inference from stage 1 of the sugya is represented by Table 1. Assuming that the correct filling of the lacuna cell is 1, we get the following table:

N

A

M

0

1

H

1

1

Table 1A (a fortiori with filling 1)

Our claim now is that Table 1 represents the result of the activity of microscopic parameters that are present in money and huppah, and that these are what cause the effectuation of marriage and/or betrothal. For the filling in the lacuna cell to be 1, we must propose a model in which the intensity of the parameters required to effect the result in that column (= betrothal) is lower than or equal to the intensity of the parameters present in the action in that row (= huppah). In other words, the intensity present in the action under discussion (= huppah) suffices to effect the result under discussion (= betrothal).

Recall that we are looking for an optimal model, that is, a model with a minimum number of parameters, and with parameters that are as simple as possible (in a sense that will be clarified below). Therefore, at the first stage we shall try to build a model for the a fortiori inference that assumes a single parameter. The present assumption is that both marriage and betrothal occur by virtue of a single parameter (unidentified at this stage), and the data in the a fortiori table are based on differences in intensity of that parameter. A first question: could this parameter be Boolean (this is the parameter of the simplest type), that is, could it take only the values 1 or 0? To test this, we will try to find a model that explains the table under the assumption of a single Boolean parameter.

Looking at the money row, it appears that in order to effect marriage one needs an action in which the intensity of the parameter is 1, whereas for betrothal an action in which the intensity of the parameter is 0 is sufficient. Likewise, it follows from the table that money has parameter intensity = 0, and therefore it effects betrothal but not marriage. From the marriage column it follows that huppah has the value of the parameter at intensity 1, and therefore it is clear that it also effects betrothal (for intensity 1 is sufficient to effect it).

At first glance, we seem to have succeeded in finding a one-parameter Boolean model that offers an adequate explanation of the table and proves that the correct filling is ‘yes’ (= 1). However, this picture is problematic, because this analysis yields that we have no information about the factor relevant to effecting betrothal. We are explaining why money effects betrothal even though for it the parameter = 0. Apparently, this means that any action whatsoever could effect betrothal, since nothing at all is needed for that. This is, of course, impossible, and therefore it follows that there must be another parameter that causes betrothal to take effect. Thus, once again, we find that the one-parameter Boolean model is not sufficient to provide an explanation of this table, and we must construct here a two-parameter model.

For this purpose, let us return now to Table 1A. A two-parameter model that explains this table is the following: the parameter that effects marriage is one parameter, and what effects betrothal is another. Money is represented by the vector (0,1) and huppah is represented by the vector (1,1). The left number in these vectors is the value of the first parameter in the action represented by the vector, and the right number is the value of the second parameter in it. The meaning of these vectors is that money has only the second parameter, whereas huppah has both parameters.

But if we are indeed moving to a two-parameter model, then the a fortiori inference is not valid, because now the possibility arises of writing different values of the two parameters for money and huppah, and explaining by means of them the opposite table 1B as well (a fortiori with filling 0):

N

A

m

0

1

H

1

0

Table 1B (a fortiori with filling 0)

The results for the actions are:

Money (0,1)

Huppah (1,0)

where the first parameter is the parameter responsible for effecting marriage, and the second is responsible for effecting betrothal. Clearly, this is a consistent realization of filling 0, since it explains all the data. Again, we are assuming a model of microscopic parameters in the actions and the results, without identifying what they are. This is a purely logical-formal treatment.

We have thus found two optimal explanations for the tables under the two fillings, and found them equivalent (both are two-parameter and Boolean). If so, there is no inference here that shows superiority, and therefore we have no proof for filling 1. Apparently, there is no inference here that proves the superiority of filling 1 over filling 0, and this cannot be a description of an a fortiori inference.

However, this conclusion is based on the assumption that the model contains two microscopic parameters. If an a fortiori inference is a valid inference within the framework of halakhic exegesis, we may infer that a model of an a fortiori inference must assume a single microscopic parameter, and in order for it to explain the superiority of filling 1 over filling 0 (without adding another parameter) we must arrive at two positive values of that parameter, which we shall denote 1 and 2. That is, the parameter cannot be Boolean. We treat it as ternary, because in principle it can also appear at intensity 0 (when it is absent from some result, or from some halakhic action).

Under these assumptions, the picture that explains Table 1A is the following: money has intensity 1 and huppah has intensity 2. Marriage requires an action whose intensity is 2, whereas for betrothal intensity 1 is sufficient. It is now clear that money effects betrothal but not marriage, and huppah effects marriage. We shall denote this model as follows:

Marriage:

Betrothal:

Money:

Huppah:

The meaning of this notation is that money has a characteristic marked by the parameter at intensity 1, and huppah has the same characteristic at double intensity. With respect to the results, the notation expresses how much intensity is required in order to effect them (unlike the actions, where the notation signifies how much intensity they have).

These are the conclusions from the three data points recorded in the table. We now ask whether huppah also effects betrothal. From the data it is clear that it does, for to effect betrothal an intensity of 1 is sufficient, and huppah has a greater intensity. So this is a model that explains the entire table with filling 1.

What about the second filling (0)? Is there a one-parameter explanation for the table of the alternative hypothesis? The answer is no. It is easy to see (we will show this below by means of diagrams) that in order to offer an adequate explanation for Table 1B, we must use a model that is at least two-parameter. Therefore the model of Table 1A is preferable to it, and this is the proof that the correct filling is 1. This is how the a fortiori inference is described in the microscopic model.

Rule 1: The optimality of a model for a table with a given filling is determined, among other things, by the number of parameters presented in it and by their valency (= the number of values they take: Boolean or ternary).

Rule 2: A model with fewer parameters is preferable to a model with more parameters, even if the more numerous parameters are Boolean and the fewer ones are, or some of them are, ternary.

Rule 3: The superiority of a model for one filling of a lacuna cell over a model for another filling of the same cell is determined by the same criteria that determine the optimality of the model (for a table with a given filling), namely by the number of parameters and their valency.

In summary, the two fillings of the a fortiori table are not equivalent. Filling 1 has a simple microscopic model (a single ternary parameter), whereas filling 0 has no model at all in terms of a single parameter. Therefore it is an inferior filling. This is the proof, in terms of our model, that the halakhic result in this case is 1, and this is the explanation that our model offers for the a fortiori inference.

Order relations between rows/columns and their implications

What is it that causes filling 0 to force us to add a microscopic parameter? The possibility of explaining the entire table by means of a single parameter stems from the fact that in filling 1 there is an order relation between the columns or the rows of the table. The result of marriage is harder to effect than betrothal, and this is true of all the actions. That is, in each of the rows the value of marriage is greater than or equal to the value of betrothal. Alternatively, in each of the columns the value of money is lower than or equal to that of huppah. In such a situation, one may infer that the same parameter governs the two rows/columns, and the difference is only in the intensity of the parameter. By contrast, in filling 0 the order relation is broken, and this forces us to introduce an additional parameter into the microscopic model. We shall use this result below, and therefore we define here another important definition:

Definition 7: An ‘order relation’ between columns/rows in a data table – a relation in which all the values of one column/row are greater than or equal to the corresponding values of another column/row. If there are identical columns or rows, we shall leave one column or row of each type and remove the others. After computing the model for the remaining table, the microscopic values assigned to the retained column/row should also be assigned to the rows/columns that were deleted.

One should note that an order relation between columns expresses the opposite situation to an order relation between rows. An order relation between columns means that the higher column is effected more easily (more actions succeed in effecting it), that is, it is the more lenient one. By contrast, an order relation between rows means that the higher row is the stronger (more stringent) one, since it succeeds in effecting more results than the other row.

Diagrams of the order relations between the results

To generalize this method, let us define a scheme in which each result (= column) is represented by a point (SITE) in the diagram,^[32] and the points are connected by arrows. The direction of the arrow marks the order relation between the columns, that is, it shows which result (= column) is higher, meaning which halakhic result is effected more easily. For example: if the order relation between two results is 2S 1S, then the resulting scheme contains two points, with an arrow between them pointing toward * 1S.*

For example, let us take the a fortiori table with filling 1 (Table 1A), where there are two columns that represent two results (marriage and betrothal), and there is a clear order relation between them: 1S 2S (that is, all the values in the marriage column are greater than or equal to those in the betrothal column). Therefore the picture obtained for this filling is as follows:

Diagram 1A – a fortiori with filling 1

The left circle (below: ellipse) represents betrothal, and its values in the table are lower than those of marriage (which are represented by the right circle).

We now present the diagram for the a fortiori inference with filling 0. Examination of Table 1B shows that there is no order relation between the columns. Therefore there are two separate points here, and there is no arrow between them:

Diagram 1B – a fortiori with filling 0

How do we get from the diagram to the model that explains it? For that, we must note the following rule.

Principle 1: When, in the scheme of some table, there is an arrow leading from result A to result B, there are two possible explanations of this intensity relation:

1. The entire diagram is explained by one non-Boolean parameter that takes two different values (the higher value for the lower result).
2. There are two Boolean parameters, where the higher receives one of them and the lower is the conjunction of both.

Of course, in a situation where both possibilities exist, we will choose A, because it is optimal (it requires fewer parameters). Below we shall see that one cannot always choose between them.

Practically speaking, we must begin with the point toward which the arrow points (its values are higher, and therefore it is effected more easily). We attach to it one microscopic parameter. The second point, the one that imposes more stringent demands, can then receive in our model either the same parameter at intensity 2, or a conjunction of two parameters, each at intensity 1. When both possibilities are admissible, we naturally choose the first, because it is optimal (according to Rules 2-3).

When there are two points between which there is no order relation, they should receive values that do not stand in directional intensity relations to one another (neither is stronger than the other). In the simplest case, we attach one microscopic parameter to one ellipse and another microscopic parameter to the second.

Applying Rule 1 to Diagram 1A drawn above immediately yields the optimal model:

Model for Diagram 1A – a fortiori with filling 1

This is the optimal model for the halakhic results (the columns). The results of this optimal model for the halakhic actions can be derived by applying the table to the results in the graph. Looking at Table 1A, money effects betrothal but not marriage, and therefore it clearly has the parameter at intensity 1 (which, as we see from the diagram, is sufficient to effect betrothal but not marriage). By contrast, huppah (under the assumption of filling 1) succeeds in effecting both results, and therefore the intensity of the parameter in it is 2. Hence the result for the actions is:

Money:

Huppah:

In the same way, with regard to Diagram 1B, we obtain:

Model for Diagram 1B – a fortiori with filling 0

And for the actions, we obtain:

Money: (1,0)

Huppah: (0,1)

As we have already seen, the consideration of superiority is clear: filling 1 is preferable because it yields a one-parameter model, as opposed to filling 0, which yields a two-parameter model.

Microscopic examination of the two inferences in the a fortiori table

We shall now see that, in this formulation, an a fortiori inference is a single one, and there are not two different alternative formulations here. This will explain why, in talmudic literature, a refutation of one of the formulations rules out the other as well. For that purpose, we shall formulate the two inferences presented above (the inference from the actions and the inference from the results) in terms of the relevant parameter. The translation of the traditional account into microscopic terms is natural and clear.

For example, the inference from the actions was: from the right-hand column we infer that huppah is stronger than money. Then, from the datum that money effects betrothal, we infer that huppah certainly effects betrothal. The translation into microscopic terms is as follows: the assumption of the a fortiori inference is an assumption about the intensity of the parameter present in the actions (huppah and money). What about the intensity of the parameter that is required in order to effect the results (marriage and betrothal)? As stated, in this inference we have no assumption at all regarding the relation between the results. Whatever the value required in order to effect the results (betrothal or marriage), if money does it then huppah certainly does it as well. All that we assume with respect to the second meaning of the parameter’s intensity (with respect to the results) is only that this parameter is the only relevant parameter for effecting these two results.

In other words: in principle, it is possible that the value required in order to effect marriage is higher than the value required in order to effect betrothal, and the a fortiori inference would still be valid. Admittedly, the other data (see in the second inference) indicate that this is not the relation between the values, but those are data that we did not use in this inference.

The translation of the inference from the results is very similar. But this is still only a translation of the traditional account, and from it it follows that a refutation of the inference from the results leaves the inference from the actions valid, and vice versa. In order to see why this is not correct, we must go one step further and use the definitions of our microscopic model for inference and refutation. In the general formulation we proposed, the a fortiori inference that concludes that the correct filling is 1 is based on the fact that the optimal model for filling 1 is preferable to the optimal model for filling 0.

This formulation unifies both inferences together, that is, it does not distinguish between them. In both cases we use the same data table and compare the optimal explanations for the two possible fillings of the table. Now it is no longer possible to separate them, and when one of them is refuted, this means that the model no longer provides a preferable explanation for filling 1, and thereby the second inference is refuted as well. We shall now see this by presenting the model for a column refutation of the a fortiori inference.

Our model proposes that the a fortiori inference is based on providing a full explanation for the data table, and therefore it does not allow us to ignore part of the data. The separation between the two inferences, which seems natural from the intuitive perspective, simply ignores data from the data table. Our assumption is that every explanation of the data must explain all of them, and therefore every proposal must be backed by a model that explains the entire table. For that reason, the inference from the actions must also offer an explanation for the relation between the results, and vice versa. For this reason, if a column refutation does indeed refute the inference from the actions, it thereby refutes the inference from the results as well. We shall now see this explicitly.

A column refutation of an a fortiori inference

Let us now examine the data table obtained after presenting the refutation (stage 2 in the sugya above). As stated, the refutation presents an additional result (= redemption of the second tithe), which is achieved only by money and not by huppah. The table obtained for this case is as follows:

N

A

P

M

0

1

1

H

1

?

0

Table 2 (A column refutation of an a fortiori inference)

We must now examine the two models for the two possible fillings of the lacuna cell and compare them. In order to show that this is a refutation, we must show that the two models are equivalent (that neither is preferable to the other).

The data table for filling 1 is as follows:

N

A

P

M

0

1

1

H

1

1

0

Table 2A (A column refutation of an a fortiori inference with filling 1)

Examining the table shows that here the following order relations exist (the numbering of the columns is from right to left):

3S 2S

1S 2S

1S and 3S are independent.

The resulting scheme is as follows:

Diagram 2A – a refutation of an a fortiori inference with filling 1

We fill in the solutions for the halakhic results according to the principle in Rule 1. Here we cannot choose the option of increasing the valency of the variables in both directions of the arrows, since that would create an order relation between P and N, which is not correct according to the table. Moreover, one cannot apply the two-parameter option twice, since that would force us to move to three parameters (or to depart from Booleanity in more than one single parameter. See below). The conclusion is that when there is a junction of this type, we must use the two possible tools for the two arrows in the diagram. The model obtained here is the following:

Model for Diagram 2A – a refutation of an a fortiori inference with filling 1

It should be noted that the model must satisfy all three conditions of the order relations. That is, there must be order relations between A and N and between A and P, and in addition there must not be an order relation between N and P.^[33]

For filling 0, Table 1B shows us that columns 2S and 3S are identical. In such a situation, we treat the table without column 3S (according to Definition 7), find the model for it, and then assign the result obtained for 2S also to 3S (again, according to Definition 7). In our case this gives the diagram of an a fortiori inference with filling 0 (see above in Diagram 1B), which we have already solved. Therefore we do this in one step, and draw the graph together with the model obtained for it:

Model for Diagram 2B – a refutation of an a fortiori inference with filling 0

From examining Table 2B, the vectors obtained for the halakhic actions are:

Money: (1,0)

Huppah: (0,1)

We must now ask ourselves which of the two diagrams is preferable. It seems that they are equivalent, since in both of them there is a two-parameter model.

By way of a side remark, let us note that although in the first diagram (2A) the valency rises more (because one parameter is not Boolean), one may say that this is a second-order phenomenon, and from our point of view the two diagrams are equivalent, which leaves the refutation intact. As we saw above, the two fillings of the a fortiori table receive models that differ from one another in the number of parameters and in valency. Filling 1 is one-parameter and ternary (= non-Boolean), whereas filling 0 is two-parameter and two-valued (= Boolean). There too we saw that valency does not outweigh an advantage in the number of parameters. Here we see that it also does not affect equality when the number of parameters is the same.

Rule 4: As we have seen in the previous rules, superiority (between two different fillings) and optimality (of a model for a given filling) are affected by considerations of the number of parameters and of valency. For the time being, we see that the number of parameters in the model is the decisive criterion. Up to now we have seen that valency does not affect considerations of superiority and optimality (it only makes it possible to reduce the number of parameters, as we saw regarding the a fortiori inference with filling 1, in Diagram 1A).

Below we shall add further criteria to the considerations of superiority, and we shall then see the matter more precisely (and then we shall see that there is no need for the assumption we proposed here regarding valency). For the time being we continue with the course of the sugya, and move on to analyze the consideration of a paradigm derived from one verse (= analogy).

In any event, the first result we obtain from our model is that the two a fortiori inferences are refuted together. The picture that we presented for the a fortiori consideration unifies them, and therefore any refutation of the a fortiori inference (column or row) refutes them both together. This fits well with what we find in rabbinic literature, as was explained above.

Result 1: The two directions of the a fortiori inference are not two different inferences but a single inference. Therefore, a refutation that refutes one of them necessarily refutes the other as well. A row refutation alone, or a column refutation alone, is sufficient to refute completely any a fortiori inference.

Microscopic model for analogy: ‘a paradigm derived from one verse’

As we have seen, the sugya in tractate Kiddushin tries to derive the conclusion that huppah is effective in effecting betrothal from money by an a fortiori inference (stage 1). It then rejects this derivation by means of the refutation from the redemption of the second tithe and consecrated property (stage 2). Immediately afterward, at stage 3, it proposes a different route for deriving this law (for filling the lacuna cell), and this time it brings intercourse as the source.

The data learned from Scripture regarding intercourse are different, since intercourse effects both betrothal and marriage, and therefore the data table for this analogy is the following:

N

A

B

1

1

H

1

?

Table 3 (analogy)

One immediately sees that there is no a fortiori inference here, since no part of the table indicates that huppah is stronger than intercourse, or that betrothal is easier to effect than marriage. What is called for here is an analogy between intercourse and huppah, or between marriage and betrothal (see the examples and illustrations brought in the previous chapter).

The conclusion of the inference is that the correct filling is 1. The explanation of this analogy in microscopic terms is simple. First, we must build the diagrams for the two fillings. With filling 1, Table 3A is obtained, and the diagram that realizes it is as follows:

Model for Diagram 3A – analogy with filling 1

From examination of the table we obtain:

Intercourse:

Huppah:

This is the simplest possible model, a one-parameter Boolean model. No diagram other than this trivial diagram implements such a model.

For comparison, let us present the diagram for filling 0. Examination of the table above shows that this is a table and diagram of an a fortiori inference with filling 1 (see above Table and Diagram 1A), except that the roles of the columns and rows have been reversed:

Model for Diagram 3B – analogy with filling 0

The results for the actions are derived from the table (as in the case of 1A):

Huppah:

Intercourse:

We must now ask: the model of which filling is preferable? The answer is that filling 1 is preferable, since the parameter is Boolean (two-valued), whereas filling 0 requires a ternary parameter.

But now a new conclusion arises: valency does play a role in determining superiority of the fillings, contrary to what we saw above. In this case, valency is what determines the superiority of filling 1.

Rule 5: Given two models for the two fillings, in both of which there is the same number of parameters, it is valency that decides superiority.^[34]

Between a fortiori and analogy

As we noted, an a fortiori inference, although it too constitutes a kind of comparison, is considered a stronger argument than ordinary analogy (to refute an analogy, it is enough to raise any sort of refutation)^[35]. We may now perhaps understand the reason. In the a fortiori inference, the superiority of filling 1 over filling 0 is stronger than the superiority in a paradigm derived from one verse. In the a fortiori inference, superiority was decided by the number of parameters (and the valency, whose significance there pointed in the opposite direction, did not interfere), whereas in the paradigm derived from one verse it was valency that decided the matter. It turns out that superiority in valency is weaker than superiority in the number of parameters.

From this we can also confirm our claim above (regarding a refutation of an a fortiori inference), according to which when there is a clash between superiority in the number of parameters and inferiority in valency, the number of parameters is decisive.

Result 2: A paradigm derived from one verse is a weaker inference than an a fortiori inference.^[36]

In any event, we must now soften Rule 4 and say that valency does play a role, but that its relative weight is lower than that of a difference in the number of parameters.

Rule 6: A difference in valency is a relevant criterion in determining optimality and superiority, but its relative weight is lower than that of a difference in the number of parameters.^[37]

Filling 1 was explained by a one-parameter ternary model, and filling 0 was explained by a two-parameter two-valued model. That is, we were required to introduce an additional parameter into the picture, and this is a very significant inferiority. By contrast, in ordinary analogy the superiority of 1 over 0 is weaker, since in both cases we are dealing with a one-parameter model, and the difference is only in the number of values the parameter can take (two-valued as opposed to ternary).

Expanding the considerations of superiority

We have seen that valency is a relevant criterion in determining superiority between fillings, that is, that given equality in the number of parameters, the determination of superiority takes differences in valency into account.

However, this rule contradicts what we saw regarding a refutation of an a fortiori inference (Table and Diagram 2). There we saw that, given that both fillings are explained by two-parameter models, it is irrelevant that one of them is ternary and the other Boolean. If such a difference mattered, then there would be no refutation there, because the two fillings would not be equivalent. On the assumption that valency is important, the column refutation would become a counterproof: it would prove that 0 is the preferable filling, and not merely refute the proof that 1 is preferable.^[38]

From this it follows that there must be additional criteria in the considerations of superiority, beyond the number of parameters and valency. For that purpose, we must turn to graph theory and ask ourselves why Diagram 2A, which describes filling 1, is not inferior relative to Diagram 2B, which represents filling 0. Looking at the two diagrams from this angle reveals that there are three additional important topological differences between them:

1. Connectedness. In Diagram 2B, the graph splits into two parts that are disconnected from one another. Such a phenomenon obviously points to greater complexity (or less simplicity) of the model, and therefore it should be seen as a criterion for the inferiority of filling 0.
2. Changes of direction. In Diagram 2A there are points whose logical connection is too complex. There is no simple relation among the three vertices, since there is no clear relation between P and N. Clearly, a graph in which there is a hierarchical relation among all three points is simpler. Such a picture runs contrary to the natural direction of a fortiori inferences, since an a fortiori inference assumes that the directions of preference are preserved. Therefore a change of direction is a disadvantage of a model. From this standpoint, the diagram explaining filling 1 is inferior.
3. The number of distinct points. Another consideration regarding the simplicity of a model is the number of points in the graph. If two halakhic results (= columns) can be identified, the graph that results is simpler (it contains fewer points). Therefore we shall regard this criterion as well as a consideration in the superiority or inferiority of a graph. For example, the model for Diagram 3A has an advantage over that of 3B, and likewise graph 2B over 2A.

In the English article we discuss the topological significance of these criteria from graph theory, and the domains in which they are expected to work properly. In the next chapter of this article, we shall prove the relevance and logical significance of each of these three topological indices separately, when we examine the relation among the three types of generalization inferences (= the common denominator). See the discussion of Tables and Diagrams 5, 5.1, and 5.2.

The general consideration of superiority

As stated, in light of what we have said so far, we must expand the considerations of superiority beyond the question of the number of parameters or their valency. The consideration that emerges is based on five different parameters that characterize every diagram, as we have defined them thus far:

Rule 7: The superiority of a model for one filling over a competing filling (as well as the optimality of a model versus another model for a given filling) is defined by the following five indices:

1. Dimension. The number of parameters required for the model to explain the diagram. This is the dimension of the vector that describes the actions. A small dimension is an advantage and a large dimension is a disadvantage.
2. Changes of direction. In the diagram, we take the two points between which the route involves the maximal number of changes of direction, and that is the characteristic of the number of changes of direction. A small number of changes of direction is an advantage and a large number is a disadvantage.
3. Connectedness. How many disconnected parts there are in the diagram. A small number of parts is an advantage and a large number is a disadvantage.
4. The number of points in the graph.^[39] A small number of points is an advantage and a large number is a disadvantage.
5. Valency. As we have seen, high valency is a disadvantage and low valency is an advantage.

After defining all the considerations of superiority, we define our superiority algorithm:

Rule 8: The superiority of one filling over another is defined as a state in which it has superiority or equality over the other in all the indices above (except valency, which will be discussed below). In such a state, we say that there is a valid halakhic inference here. If the considerations of superiority tilt in both directions, that is, if there is at least one index in favor of each of the two fillings, then this is a state of refutation (or an invalid inference).

We now add an important remark regarding valency, and then return to building the basic elements of the model.

A note on valency

In the basic picture, all of our parameters are Boolean, since that is the simplest situation. In principle, however, increases in valency are possible in one parameter or in several microscopic parameters (mainly in more complicated diagrams, as will appear below). In many cases, there are several different possibilities for increasing valency and still keeping the model optimal. Therefore we must define a rule that will determine how to measure the valency of a diagram.

For reasons that will be better understood later (see below, in the discussion of Diagram 6.1, the implications of this rule), we shall stipulate that increasing valency is possible only in one of the parameters in the model. We can now define the valency of the diagram (the fifth index in the criterion of superiority above) as the valency of that parameter whose valency we increased.

Principle 2: A constraint on building the model for a given diagram: increasing valency is possible in at most one microscopic parameter. If it is impossible to build a model for the diagram without increasing the valency of two parameters, we instead add further Boolean parameters (increase the dimension).

Let us note that in most of the cases we shall deal with, it is possible to reach the same results even without this restriction (except for inference 6.1), but this requirement can be supported by the fact that it makes sense in itself. It indicates that the relations between the halakhic results and actions have a defined axis along which they are all measured (see more on this in the English article). This is an expression of the question of relevance discussed above. When the two axes of stringency are relevant, that is, when there is some kind of parallel relation between them, this means that there is only one parameter that determines the intensity and its direction. This point will be clarified immediately.

We must now examine a question that might arise in light of this principle. At first glance, when we encounter a diagram in which there is more than one long chain, this seems to require us to add a large number of microscopic parameters instead of increasing valency. At first glance, this seems an unreasonable complication of the model. However, when we examine diagrams for two extreme cases, we will immediately see how the problem is solved, and from this we will also better understand the meaning of this principle.

Case A:

Case B:

The meaning of the models we proposed for these two diagrams is that there is one parameter responsible for the intensities in both branches of the graph, and it is the ‘engine’ that determines the intensification along the route in the diagram, and in addition there are two other parameters, each of which determines the special quality that characterizes one of the branches. This follows from the fact that we assume there is a shared axis for all the branches (= the parameters) that determines the direction of the increase in intensity. For example, when we translate a problem from some field into the logical terminology proposed here, we must decide what is 1 and what is 0. Is the taking effect of betrothal higher or lower than its non-occurrence? At the same time, the same must be decided with regard to marriage. Is there room to reverse the order, and to establish a priority order regarding betrothal in which occurrence is higher than non-occurrence, while regarding marriage non-occurrence would be the higher one? This is not reasonable, because in both marriage and betrothal a marital bond is created, and the creation of that bond defines the common direction. Both betrothal and marriage are governed by parameters, which may perhaps be different, but the directions of their intensities must be parallel (this is the requirement of relevance discussed above), that is, both are higher/lower in the direction of creating the marital bond. For this reason, there must be a shared axis that determines the relative intensities in each of the parameters. This is the meaning of one parameter that changes intensities along both branches in the two diagrams that we have brought here.^[40]

From this one can readily understand the meaning of the requirement in Principle 2 that only one parameter should change its valency in a given diagram. The reason is that in every problem there is only one parameter that determines intensity, and it serves as an ‘engine’ for the progress along the diagram. The other parameters determine the different qualities involved in the problem (the qualities that ‘ride’ on the engine. It is the intensity parameter that determines the intensity with which they appear at each point). If the intensity parameter were not shared, there would be no basis for learning from one result/action to another. There is no logic in learning that if money effects betrothal, then Torah study requires a blessing afterward. Here the two axes of intensity do not touch one another at all (see above the discussion of the question of relevance).

A refutation of analogy

At stage 4, the sugya raises a refutation against the derivation from intercourse (stage 3), and says that intercourse acquires a levirate widow whereas huppah does not. The data table now obtained is the following:

N

A

Y

H

1

?

0

B

1

1

1

Table 4 (A column refutation of analogy)

In order to understand why there is a refutation here, we must propose a microscopic model for this table under both fillings, and show that the models for the two fillings are equivalent.

This case too is very simple, since it can be mapped onto the previous cases. If we fill the lacuna cell with 1, we get that columns 1S and 2S are equivalent. If we fill the cell with 0, we get identity of columns 2S and 3S. In both cases, this is a table of an a fortiori inference with filling 1 (Table 1A).

Model for Diagram 4A – a refutation of analogy with filling 1

Huppah:

Intercourse:

Model for Diagram 4B – a refutation of analogy with filling 0

Huppah:

Intercourse:

One immediately sees that the two fillings are completely equivalent, and therefore there is a refutation here.

We now turn to examine the application of the general consideration of superiority to the four inferences we have described so far (that is, whether it changes the results obtained from dimension and valency alone, before the expansion of the criterion). We shall see that the picture remains consistent, and even fits better.

Testing the application of the general criterion of superiority to the four basic inferences

The test will be carried out in the following way. In each of the diagrams we have already written the corresponding model. Therefore the dimension and valency have already been calculated above. We shall now calculate the three additional indices, and see whether all the inferences fit what we find in the Talmud. We present the results in a table:

A fortiori

Refutation of an a fortiori inference

Analogy

Refutation of analogy

Diagram

1A

1B

2A

2B

3A

3B

4A

4B

Filling

1

0

1

0

1

0

1

0

Dimension

1

2

2

2

1

1

1

1

Change of direction

0

0

1

0

0

0

0

0

Connectedness

1

2

1

2

1

1

1

1

No. of points in graph

2

2

3

2

1

2

2

2

Valency

2

1

2

1

1

2

2

2

Result

1 preferred

Equivalent

1 preferred

Equivalent

Summary Table 1

By examining the table, we can derive several rules regarding the considerations of superiority:

From here one can see Rule 8 that we discussed above, according to which a model will be considered preferable if all the parameters that are not equivalent (except for valency) tilt in its favor. When there are indices that tilt in both directions, we regard the two fillings as equivalent, that is, this is a refutation. This follows from the fact that a refutation is supposed to rest on the existence of doubt regarding the superiority of filling 1, and in order to create doubt it is enough that one index tilt the other way. Valency is an exception, and by itself it cannot contend with the tilting of other indices in the opposite direction, whether for creating a refutation or for creating superiority.

In fact, after we added the other three indices to the consideration of superiority, it turns out that valency plays no role at all (previously it was significant in the analysis of analogy, but now the work is done by the superiority in the number of points). The same is true of the index of changes of direction. In principle, one could have dispensed with them as relevant indices for determining superiority, and then the reservation we added would not have been necessary. However, below we shall need both of them, and therefore we retain them for future purposes.^[41]

Rule 9: Valency can determine superiority or optimality only if there is no superiority against it in any other index. Equality (= refutation) is never based on valency against another index. In such a case, superiority belongs to the direction indicated by the other index.

To conclude this chapter, let us define what a refutation is:

Rule 10: A refutation is a situation in which there is no superiority, that is, there is no valid halakhic inference here for filling the lacuna cell. Such a situation occurs in one of two cases:

1. When all the indices are identical for the two fillings.
2. When there are contradictory superiorities in different indices in favor of the two fillings. This is an equivalent state, and from our point of view this is a refutation of the inference. There is no reason to assume that the number of superiorities on each side matters, since the Talmud does not offset refutations. Once there is superiority in favor of one side and superiority from another standpoint in favor of the other, there is again no way to compel the inference, and therefore, in the intuitive reasoning of the Talmud as well, this is a refutation.^[42]

At the end of the article we shall see that in certain cases a third type of logical conclusion also appears: a counterproof. As we have seen, a proof is a situation in which there is a consideration of superiority in favor of filling 1. A refutation is a situation of equivalence between the models for the two fillings. A counterproof is a situation in which there is a consideration of superiority in favor of filling 0.

D. Complex inferences: the three generalization inferences and the refutations of them

Introduction

In this chapter we shall apply the model that we built from the four simple inferences to more complex inferences. For that purpose, we shall continue following the next stages in the Kiddushin sugya. The main type of inference with which this chapter will deal is ‘the common denominator’ (= a paradigm constructed from two verses), that is, generalization. This inference appears at stage 5 of the sugya, after we refuted the two attempts (the a fortiori inference discussed at stage 1 and the paradigm-from-one-verse discussed at stage 3). It combines them, and from the combination it proves by a more complex consideration that huppah does indeed effect betrothal. As we have already noted, this is a canonical structure that appears often in talmudic literature, but in fact it is not unique to it. We saw in the introduction that this inference represents generalization, scientific or otherwise, which is itself a universal form of thought.

Three types of generalization inferences: ‘the common denominator’

In the example we brought in the introduction regarding generalization, it seems that in such an inference there are two teaching cases (a ball and a chair) and one learned case (another object, such as a book). Each of the two teaching cases has a unique feature (stringency) that does not exist in the other teaching case, and not in the learned case either (the chair has legs but is not round, and the ball is round but has no legs. Books have neither legs nor a round shape). As we have already noted, each of the two teaching cases can relate to the learned case by way of ‘a paradigm derived from one verse’ or by way of an a fortiori inference. This will have implications for the shape of the table, but as we shall see it does not affect the validity of the inference.

Let us now return to the sugya in Kiddushin. After the two attempts to derive the filling of the lacuna (huppah in betrothal), one by an a fortiori inference from money, which was rejected by a refutation from the redemption of consecrated property and the second tithe, and the other by a paradigm from intercourse, which was rejected by the levirate widow, the Gemara again tries to do so from both teaching cases together.

The data table obtained in this case is the following:

N

A

P

Y

M

0

1

1

0

H

1

?

0

0

B

1

1

0

1

Table 5 (the common denominator)

It is important to understand that this is not a table created here from accidental data. This is a universal table, which appears in all common-denominator derivations in talmudic literature, and in fact in every generalization we make in any field. The two left columns are the unique properties of the teaching cases. As we have already explained, the property special to the first teaching case (money, which redeems the second tithe and consecrated property) is never present in the second teaching case (intercourse) and in the learned case (huppah). The same applies to the property special to the second teaching case (intercourse, which acquires a levirate widow), which is never present in the first teaching case (money) and in the learned case (huppah). On the other hand, both money and intercourse (= the teaching cases) always effect betrothal, for otherwise it would not be possible to derive anything from them regarding huppah. As for huppah (= the learned case), the situation with regard to betrothal is unknown. This is the lacuna cell in the table, and filling it requires the inference.

So far we have explained why the three leftmost columns in Table 5 are universal. The only degree of freedom in this table lies in its rightmost column. There the filling of the target (huppah) is always 1, since this is the anchor for the two basic inferences (the a fortiori inference from money, and the analogy from intercourse). But the filling of the two cells that belong to the two source cases can vary: when the basic source is an a fortiori inference, then the filling of the cell corresponding to it is 0 (as with money). And when the inference from the basic source is by analogy – then the filling in the cell corresponding to it is 1 (as with intercourse).

From this it follows that there are three types of common-denominator inferences, and their data differ from one another solely in the rightmost column of the table: in the case where the two basic inferences are both analogies, the entire rightmost column is 1. In the case where the two basic inferences are both a fortiori inferences, the two outer cells in the rightmost column are 0. And if one basic inference is an a fortiori inference and the other is an analogy (as in our case), then the rightmost column contains two cells of 1 and one cell of 0.^[43]

The fact that emerges from the halakhic literature is that each of these three types of tables represents an inference that proves that filling 1 is the correct filling. We must test this in the method we have developed here, but before that we will describe these inferences in the traditional, intuitive way.

The traditional explanation of the generalization inference

In fact, we have two source cases, each of which has a unique property (a stringency) that prevents us from learning from it alone to the target. But the fact that there are two source cases, and that the property of one does not exist in the other, shows us that these properties are not what cause the derived rule. Rather, some other property, common to both source cases, is what is responsible for the derived rule, and because of it the derived rule exists in both of them. But this property also exists in the target, hence the conclusion that the derived rule should exist in the target as well.

The question that arises here is why we should not assume that there can in fact be two different causes for the rule under discussion. For example, money has a unique property (= a microscopic parameter) that causes the redemption of the second tithe. This property does not exist in intercourse, and therefore intercourse does not effect the redemption of the second tithe. Conversely, intercourse has another unique property (= a microscopic parameter) that causes acquisition in the case of a levirate widow. This property does not exist in money. These two unique properties do not exist in huppah, since huppah neither acquires a levirate widow nor redeems the second tithe.

In addition, there is an assumption that there is something common to the three halakhic acts (like mass, in the case of a book, chair, and table), otherwise there would be no basis for trying to learn something from one of them about the other. We denote it by the letter .

Up to this point we obtain the following picture:

Money: (1,0,1)

Huppah: (0,0,1)

Intercourse: (0,1,1)

The case of the common denominator from two analogies is simpler in the intuitive picture, since in this case the entire rightmost column is 1. From this it can be inferred that marriage is effected by the microscopic parameter . This is the ‘common denominator’ that exists in all of them, and it is what effects betrothal.

We now have two possibilities before us: A. that the parameter is also responsible for betrothal, and from this it follows that huppah indeed effects betrothal. B. betrothal is effected by each of the two parameters that exist in money and intercourse but not in huppah, or , and therefore huppah cannot effect betrothal.

The intuitive preference is for possibility A, since according to Occam’s razor it is preferable to assume that there is only one cause for a given result, rather than that either of two different causes alone can produce it. This is the accepted explanation of the ‘common denominator’ inference.^[44]

When at least one of the basic source cases is an a fortiori inference, the cell corresponding to it in the rightmost column is 0, and then the parameter that is common to all the halakhic acts does not by itself produce the marriage, and the picture becomes a bit more complicated. We will discuss this case below.

We note that the description we proposed offers a three-parameter model for the common denominator, and the preference is not formulated in terms of the model we developed. We must now see whether our model is indeed appropriate also for common-denominator inferences. To do this, we must apply the method defined above to the data tables of the common-denominator inferences.

A microscopic model for the generalization inference

We now turn to modeling the generalization inference (the common denominator). This inference appears in the sugya at stage 5, and we have already seen that the following data table is obtained for it:

N

A

P

Y

M

0

1

1

0

H

1

?

0

0

B

1

1

0

1

Table 5 (the common denominator)

We will now find an optimal model in the two fillings:

Optimal model for Diagram 5a – the common denominator with filling 1

We built the model by assuming that betrothal is effected by the microscopic parameter , and afterward increasing the valency in one of the two directions.

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

Money: (1,1)

Huppah: (2,0)

Intercourse: (3,0)

One should note that in our solution as well, what effects betrothal () exists in all three halakhic acts (albeit in different amounts). This is a more efficient (=optimal) expression of the common-denominator inference. The intuitive inference above gives a non-optimal solution, since it assigns the common denominator an independent parameter, thereby increasing the dimension of the model. Here this is a more optimal model, since the common denominator lies on the same axis as the parameters that generate some of the refutations, and the differences are only in intensities. The intuitive solution creates a model with more parameters, and is therefore less optimal. Even so, our solution also satisfies the requirement that there be a ‘common denominator’ that effects both betrothal and marriage. It is fairly clear that there will be differences here between a common denominator built on two analogies or on two a fortiori inferences and a common denominator built on an a fortiori inference and an analogy (and above we already saw that the intuitive explanation is not entirely clear for the other two cases). We will analyze this below.

Just to illustrate the possibilities, we will add here another possible solution for this diagram:

Alternative optimal model for Diagram 5a

In this case we chose to increase the valency in the direction of P, and then we had to increase it to the value 3 in order to create a situation in which P bears no order relation to Y. Increasing the valency of in addition to is not possible under the restriction we imposed above (Principle 2).

The solution for the acts is:

Money: (3,0)

Huppah: (1,1)

Intercourse: (2,1)

In this solution too there is a parameter () that effects betrothal and exists in all the halakhic acts, that is, it fits the intuition of a ‘common denominator’ (here money does not effect marriage, since this is an a fortiori inference, and therefore this parameter governs not marriage but only betrothal, unlike the case of two analogies, which we will see immediately). And again, there is no additional isolated parameter shared by the three acts () as there is in the intuitive explanation, since here this is a more optimal model for this filling.

In both solutions for filling 1 the valency is 3, and the model is two-parameter. The graph is of course the same graph (the shape of the graph follows from the data table, and it does not depend on the different solutions). Therefore, for the purpose of determining the preference of filling 1, there is clearly no difference between these two models.

We must now draw the diagram for filling 0:

Optimal model for Table 5b – the common denominator with filling 0

And the solution for the acts:

Money: (2,0)

Huppah: (0,1)

Intercourse: (1,1)

Here the inferior alternative presented also in the intuitive explanation exists, since money and intercourse have something that huppah does not have (), and that is what effects betrothal. Therefore, according to this proposal, huppah does not effect betrothal.

We must now examine whether the inference is indeed valid. To do so, we must compare the five indices of the models for the two fillings and see whether filling 1 is באמת preferable. Both models are two-parameter. The valency in Diagram 5a is 3, and in Diagram 5b it is 2. But in Diagram 5b there are two direction changes (in the transition from P to N, the direction changes both when crossing Y and when crossing A), whereas in Diagram 5a there is only one direction change (in the transition from P to N, there is a direction change only when crossing A).

Thus Model 5a is preferable in terms of direction changes, even though it is inferior in terms of valency. As Rule 9 states, such a situation is treated as a preference for filling 1. In this way we have also confirmed the validity of the generalization inference (= the common denominator) in our model. This is the formal expression of Occam’s razor in the terminology of this model. This inference also shows for the first time the logical meaning of the direction-change index. It is what creates the preference for filling 1 in this type of inference.

Checking the two additional types of common-denominator inferences

We must now examine what is obtained in the two remaining types of the common denominator (when the basic derivations that combine into the common denominator are two analogies, and when they are two a fortiori inferences).

For the common denominator based on two analogies, the resulting table is the following:

N

A

P

Y

M

1

1

1

0

H

1

?

0

0

B

1

1

0

1

Table 5.1 (the common denominator from two analogies)

(We note that the rule for money with respect to marriage is not correct according to the halakhah, but we assume it for the sake of a full presentation of the model.)

The optimal models for the diagrams for this table are:

Optimal model for Diagram 5.1a – the common denominator based on two analogies with filling 1

This is a picture like Diagram 2a (of a refutation of an a fortiori inference with filling 1). The solution for the acts is:

Money: (2,0)

Huppah: (1,0)

Intercourse: (1,1)

We see here the intuition that there is a parameter shared by all three acts (), and as expected, when the basic inferences are two analogies, it is this parameter that effects both betrothal and marriage.

Optimal model for Diagram 5.1b – the common denominator based on two analogies with filling 0

(We note that if we had put in betrothal, we would have obtained in Y and P valency 2 in two different parameters, and this would have contradicted Principle 2.)

The solution for the acts is:

Money: (3,0)

Huppah: (1,0)

Intercourse: (2,1)

There is here something common to money and intercourse and not to huppah (in that both of them have at least ), and the assumption is that betrothal is effected by it (and indeed this is what one sees in the diagram, that betrothal is effected by ). This is exactly the inferior alternative of the generalization, as we explained intuitively.

We must now compare the two fillings and decide whether the inference is valid. In terms of valency, filling 1 is preferable (valency 2) to filling 0 (valency 3). In terms of dimension and connectivity they are identical. In terms of the total number of points, filling 1 is preferable (it has 3 distinct points in the diagram, whereas filling 0 has 4 distinct points). And in terms of direction changes they are equivalent (one direction change from Y to P, when crossing A).

Thus filling 1 is preferable to filling 0 because of the number of points in the graph. This generalization inference too is confirmed in our model. This consideration also shows the meaning of the index of the number of points, since it is what determines the preference of filling 1 in this type of inference.

We still have to check the common denominator based on two a fortiori inferences. The data table in this case is:

N

A

P

Y

M

0

1

1

0

H

1

?

0

0

B

0

1

0

1

Table 5.2 (the common denominator from two a fortiori inferences)

The diagrams are:

Optimal model for Diagram 5.2a – the common denominator based on two a fortiori inferences with filling 1

The solutions for the acts are:

Money: (2,0,0)

Huppah: (1,1,0)

Intercourse: (1,0,1)

Here too there is a parameter shared by all the acts () that effects betrothal.

Optimal model for Diagram 5.2b – the common denominator based on two a fortiori inferences with filling 0

The solutions for the acts are:

Money: (2,0,0)

Huppah: (0,0,1)

Intercourse: (1,1,0)

Here too there is a parameter that exists in money and not in the other two (), and there is a parameter that exists in intercourse and not in the other two (); but here what effects betrothal is a parameter shared by both of them that is not found in huppah ().

Is this inference also valid? To answer this, we must compare the models for the two fillings. In both cases the dimension is 3, the valency is 2, the number of independent points is 4, and the number of direction changes is 1. The only difference is in connectivity, which is preferable in filling 1. The diagram of filling 0 is split into two independent parts, and that places it in an inferior position.

Thus this inference too is valid in our model. From this we see the logical meaning of the third index, connectivity, since it is what causes the preference for filling 1 in an inference of this type.

Brief summary: a logical-talmudic confirmation of the results obtained from graph theory

In this chapter we examined the three possible types of generalization inferences. We saw that all of them are valid, but each for a different reason. In the first case, the preference for filling 1 was based on direction changes; in the second, on the number of points in the graph; and in the third, on connectivity.

As we noted, one can see here a further confirmation of the importance and the logical significance of the three topological indices we defined above in the general preference consideration. As we noted above (see in greater detail in the English article), these indices have a solid basis in graph theory. Here we see that each one affects a different type of inference, and therefore each has a clear justification as an index that determines inferential preference. The two additional indices (dimension and valency) have an obvious meaning, as we have seen from the outset.

We further note that, from the standpoint of valency, these three inferences again divide among the three possibilities: in a common denominator based on two a fortiori inferences, the two fillings are equivalent in terms of valency. In a common denominator based on an a fortiori inference and an analogy, filling 0 is preferable in terms of valency. And in a common denominator based on two analogies, filling 1 is preferable in terms of valency. That is, the three generalization inferences are fundamental building blocks that sharply distinguish among all the indices we defined. Each has unique properties of its own, complementary to those of the other two. Below we shall see that the a fortiori inference is the inference that completes all three (for in its case the preference is not topological, but in terms of dimension).

Result 3: The generalization inferences are fundamental inferences that are distinguished from one another precisely in terms of the topological measures we defined. Each has its own unique property, linked to a different one of our three topological measures. As we have seen, they are also distinguished from one another in terms of the valency index.

A note on the status of generalization inferences

We mentioned that it is customary to assume that an a fortiori inference is a stronger inference than analogy. Within our model, our conclusion is that it is hard to see why, since in an a fortiori inference the preference for filling 1 is based on dimension and connectivity against valency, whereas in analogy the preference is based on the number of points and valency. Still, it seems that this can be explained in two ways:

1. It may be that preference in two strong parameters (dimension and connectivity) is stronger than preference based on one strong parameter (number of independent points. We have already seen that valency is a weak parameter).
2. Another possibility is that preference in dimension is stronger than the topological preferences (that is, those that come from the indices of direction change, connectivity, and number of points).

One implication concerns the question of any slight refutation against analogy from one verse, regarding which the early authorities disagreed (see Hullin 116, and in __Encyclopedia Talmudit__ under the entry ‘Binyan Av,’ notes 68-70).

But what about the three generalization inferences? The commentators on the Talmud (see Encyclopedia Talmudit there, note 60) disagree on the question whether a common-denominator inference that begins with an a fortiori inference (and is then refuted and restored from an additional verse) retains the status of an a fortiori inference, or whether it drops to the status of an analogy from two texts (a generalization inference based on two analogies). To examine this we must analyze each type of generalization separately. We do see that each of the three bases its preference on a different topological index, but in each of them the preference is based on one index, and therefore it seems that their logical strengths ought to be similar to one another. This of course confirms the claim that even if one begins with an a fortiori inference, the structure of a common denominator is always of the strength of analogy, which is weaker than an a fortiori inference.

We saw that in an a fortiori inference the preference for filling 1 (that is, the strength of the inference) is based on the connectivity and dimension indices. But in these three inferences the preference comes by virtue of one of the following three indices: direction change, connectivity, number of points. Therefore the strength of the a fortiori inference over the three generalization inferences can be explained here as well in the same two ways:

1. It may be that the parameter of dimension is the most important for determining the strength of the inference (for it is indeed the most intuitive one, and it is where we began: how many microscopic parameters are needed to explain the graph).
2. The strength of the a fortiori inference is based on an advantage for filling 1 in two indices, whereas all the generalization inferences are based on an advantage in a single index. In this sense they are similar to analogy.

We again see the full parallel that exists between the three generalization inferences and analogy from one verse. The inferiority of all of these relative to the a fortiori inference is exactly of the same type, and therefore it is no surprise that the commentators write that common-denominator inferences belong to the category of analogy and not to the category of an a fortiori inference. In particular, one sees here that even a generalization inference based on two a fortiori inferences loses its force. It is worse than a single a fortiori inference, and in fact it is similar to a common denominator based on analogy/analogies, exactly as some of the commentators wrote. This follows from the same two explanations we gave above.^[45]

Thus all these results are well confirmed by our model.

Result 4: The a fortiori inference is a stronger inference than analogy, as well as than the three common-denominator inferences. These are identical in strength to analogy from one verse. We proposed two explanations for the strength of the a fortiori inference:

1. The superiority of dimension over topological indices.
2. An advantage in two indices is stronger than an advantage in a single index. It seems that any slight refutation will be effective against these three inferences, as well as against analogy from one verse.

We note that in this context there may perhaps also be room to examine valency considerations. Although the superiority of an a fortiori inference over analogy is despite a reversal in the valency relations (see Summary Table 1), when we compare it with the three common-denominator inferences we will see that the inference that stems from analogies has an advantage in valency. It is not clear whether this has any significance (for example, that any slight refutation would not help refute it, but only the other two generalization inferences).

Refutation of the common denominator: an intuitive explanation

At stage 6 the sugya raises a refutation of the common-denominator inference (from stage 5). Intuitively, when one wishes to refute an inference of generalization (a common denominator), one must find a unique characteristic that exists in both source cases but not in the target. In such a case we attribute the result (halakhic or scientific) to that characteristic, and this prevents us from applying the conclusion of the generalization to the target. For example, if we find something common to the table and the ball that does not exist in the book, this raises the possibility that precisely that component is what causes the fall to the earth, and not the side common to all of them.

As we saw above, from the intuitive standpoint the alternatives that compete in a generalization inference are:

1. There is a factor that exists in all three objects, and it is what causes the fall to the earth.
2. The fall to the earth is caused by a factor that exists in the ball or by a different factor that exists in the table, and neither of them exists in the book. As stated, here Occam’s razor gives preference to possibility 1.

But what if we find a third alternative:

1. There is one factor common to the two source cases (= table and ball) that does not exist in the target (= book).

In such a case it is clear that this alternative will be no less good than alternative 1. Occam’s razor does not distinguish between 1 and 3. This is exactly the meaning of a refutation, since it leaves two alternatives on equal footing, and thereby undermines the inference that claims the superiority of one of them.

For the sake of simplicity, from here on we will deal only with the common denominator that was brought in the Kiddushin sugya, and we will not need to return each time to the two additional types discussed above.

A microscopic model for a refutation of the common denominator

In the case we brought from the Kiddushin sugya, the refutation found by the Gemara is that in money and intercourse there is substantial benefit, which does not exist in huppah. Therefore their ability to effect betrothal may depend precisely on the benefit, and the conclusion regarding huppah is no longer necessary; that is, the inference was refuted.

In such a case the table is as follows:

N

A

P

Y

H

M

0

1

1

0

1

H

1

?

0

0

0

B

1

1

0

1

1

Table 6 (a column refutation of the common denominator)

We will now find the optimal models for the two fillings of this table:

__Optimal model for Diagram 6a – a refutation of the common denominator with filling 1 __

The solution for the acts is:

Money: (3,0)

Huppah: (1,1)

Intercourse: (2,1)

__Optimal model for Diagram 6b – a refutation of the common denominator with filling 0 __

In fact, this is a table identical to Table 5b. The solution for the acts is:

Money: (2,0)

Huppah: (0,1)

Intercourse: (1,1)

We see that a value of 1 is removed from parameter in all the acts. The meaning of this is that there is a common parameter that exists in money and intercourse but not in huppah, and it is what effects betrothal. This is exactly the intuition of the refutation as we explained it above.

To check whether the inference is valid, we must compare the two fillings. The dimension in both cases is 2, the valency in filling 1 is 3 and in filling 0 it is 2. The connectivity in both cases is 1. The total number of points is 5 in filling 1, and 4 in filling 0. By contrast, the number of direction changes in Diagram 6a is 1 and in Diagram 6b it is 2 (from P to N, when crossing H. Exactly as in Diagram 5b).

Thus filling 1 is preferable in terms of direction changes, and inferior in terms of the number of points and valency. Therefore there is no way to decide which filling is preferable, and we remain in a state of refutation. Thus, a refutation of the common denominator is also confirmed in our model.

Summary

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

Common denominator from analogy and a fortiori

Common denominator from two analogies

Common denominator from two a fortiori inferences

Refutation of a common denominator from analogy and a fortiori

Diagram

5a

5b

5.1a

5.1b

5.2a

5.2b

6a

6b

Filling

1

0

1

0

1

0

1

0

Dimension

2

2

2

2

3

3

2

2

Direction change

1

2

1

1

1

1

1

2

Connectivity

1

1

1

1

1

2

1

1

No. of independent points

4

4

3

4

4

4

5

4

Valency

3

2

2

3

2

2

3

2

Result

1 is preferable

1 is preferable

1 is preferable

Equivalent

Summary Table 2

Interim summary at the end of the first part of the article

Up to this point we have dealt mainly with developing the basic tools of our model. We have seen the meaning of the microscopic parameters, the topological and other indices, and their implications for the relative strengths of the various inferences. In the second part of the article there will appear the continuation of the present gate, where we will continue with the Kiddushin sugya, and through it with an analysis of more complex inferences (a common denominator with analogy, a refutation of it, another generalization, and a further refutation and argument). After that we will discuss refutations of a different type from the one we encountered here (those that bear directly on the microscopic parameters), and we will see how to handle them within our model. In the second gate, which will appear later in the second part of the article, we will examine several implications of our model, and we will use it to explain several more talmudic phenomena in the area of midrashic inferences and the refutations raised against them. Finally, as stated in the introduction, we will return to discuss the question of the relation between deductive thinking and non-deductive modes of thought in light of our results.

1.

Footnotes

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

   ‘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’

2. We would like to thank Dr. Avi Lipshitz of Efrat for his help at the beginning of the הדרך (the formulation of the sugya in tables was his suggestion, and this picture ultimately led us to our model).
3. Both parts together are an adaptation of part of a book that will be published on this topic.
4. Voice of Prophecy, Rabbi David Cohen, Mossad Harav Kook, Jerusalem 5730.
5. M. Abraham, in his book Two Wagons and a Balloon (__hereinafter: __Two Wagons), Beit-El, second revised edition, Jerusalem 5767, especially in gate eleven, chapter 3. See also the two articles by the same author, ‘The logical status of the methods of exegesis,’ Tzohar 12, Tishrei 5763 (and see also the continuation article in Tzohar 15).
6. We did not take into account the correspondences of all the basic and complex inferences and refutations brought in rabbinic literature, since that is the subject of the article itself. The results are only the additional correspondences, beyond the inferences and refutations themselves.
7. From here on we will deal only with halakhic exegesis. The relation to aggadic midrash is sometimes self-evident, but it will not be discussed here explicitly.
8. For a survey of this, see the excerpts from the manuscripts of Rabbi HaNazir that appear in Dov Schwartz, ‘Voice of Prophecy Part II,’ in Book of Logic, Bar-Ilan University and Yeshiva University, Tzomet Institute, Alon Shevut (no year of publication indicated), pp. 34-63, in chapter 2. For a comprehensive scholarly survey, see the sources and discussion in volume 1 of Sifra de-Vei Rav, edition of A. A. Finkelstein, New York 5749, and also in the article by Menahem Kahana, ‘Lines for the history of the development of the rule of general and particular in the Tannaitic period,’ in Studies in Talmud and Midrash (Memorial volume for Tirzah Lifshitz), Moshe Bar Asher, Yehoshua Levinson, and Berekhyahu Lifshitz (editors), Mossad Bialik, Jerusalem 5765, 173-216, and in the sources cited there.
9. For the definition of the various hermeneutical rules themselves, and an initial understanding of how they operate, beyond what is found in the rule books, see my article A Good Measure, for the years 5765-5766. The articles are collected in four books (two volumes for each year), called A Good Measure 1-4. The books were published by Tam, Kfar Hasidim, and are arranged according to midrashic rules on the weekly Torah portions (one can search for each rule according to the index at the end of the books).
10. To the best of our understanding, universal meanings can be found there as well, and we will address this in articles to be published in the future.
11. We note that the distinction is not always clear. For example, see Kahana’s above-mentioned article, note 151, and the discussion around it.
12. For example, some have investigated the rules of legal inference, and of course included these three rules among them. See Tarello, in Chaim Perelman, Legal Rhetoric, p. 38.
13. For a survey, see: Handbook of Defeasible Reasoning and Uncertainty Management Systems – Volume 4: Abductive Reasoning and Learning by Philippe Smets and Dov M. Gabbay, Kluwer Academic Publishers Sep 2000.
14. An extremely rare exception is the hermeneutical rule ‘two verses that contradict one another,’ where there are two contradictory scriptural data. In such a case the Sages sometimes perform abduction, that is, they infer that different assumptions underlie the two data (or that they deal with different situations). It can be shown that this is abduction, but it is done only under the compulsion of a contradiction between data. In halakhic exegesis there is no abduction that is done just like that, without the background of contradiction. In any case, the application of this hermeneutical rule is extremely rare (some three or four cases, and even they are in aggadic exegesis and not in halakhic exegesis), and therefore it is less important for understanding the world of midrashic inference. Regarding the application of this hermeneutical rule, the Sages of the Talmud disagree (see A Good Measure, Michael Abraham and Gabriel Hazut, Tam Press, Kfar Hasidim 5765, parashat Bo): according to Rabbi Akiva there is a third verse that decides for one of the sides, and according to Rabbi Ishmael they find a resolution to the contradiction, so that each verse is placed in a different context. According to Rabbi Ishmael this is a process of abduction.
15. See: Techniques and Assumptions in Jewish Exegesis Before 70 CE David Brewer,
16. M. Abraham, ‘The a fortiori inference as a syllogism – an arithmetic model’, Logic 2, Aluma, Jerusalem 5753, pp. 9-22 (hereinafter: the article in __Logic__). For criticism from an unexpected direction, see the article (which deals with considerations akin to an a fortiori inference in Hindu literature):

    An overlooked type of inference, Arnold Kunst, Bulletin of the school of oriental anf African studies, Uni. Of London, Vol. 10, No. 4 (1942), pp. 976-991; Studies in Talmudic Logic and methodology, Louis Jacobs, London 1961; Susan A. Handelman__, The Slayers of Moses: The emergence of rabbinic interpretation in modern literary theory__, State Unuversity of New York Press’ Albany 1982; See also ‘Some Problems in the Rabbinic Use of the Qal Va-Homer Argument’, Hyam Maccoby, Center for JEWISH Studies, Uni. of Leeds. At the Cjs Homepage in the internet.

17. We note that in these works there are quite a few errors and inaccuracies, and this is not the place.

    See: Heinrich Guggenheimer, ‘Logical Problems in Jewish Tradition’, in Confrontation with Judaism, ed. Philip Longworth (Blond, 1967). See also Avi Sion, Judaic Logic, Editions Slatkine, Geneva 1995.

18. For example, Saul Lieberman, in his book Greeks and Greekness in the Land of Israel, Jerusalem 5723, pp. 190-196, argues that the gezera shava, which is known to us today as a hermeneutical rule built on textual comparison, began as a substantive comparison. In any case, it is a comparison, and as such, on the plane of logical description, it too is included in the rule of analogy as we will present it below.
19. Regarding the biblical a fortiori inference, which is always based on one datum and a consideration of stringency based on reason, see my article A Good Measure, parashat Noah and parashat Shemini, 5765, and also in:

    Louis Jacobs, ‘The “Qal Va-homer” argument in the old Testament’, Bulletin of the school of oriental anf African studies, Uni. Of London, Vol. 35 No. 2 (1972), pp. 221-227.

20. There are a fortiori considerations based on one datum (the two types we saw above here) and on two data (which will be discussed below, when we deal with the relevance of the axes of stringency. See in chapter 3, in the section ‘Relevance of data – a reflection of the need to assume microscopic parameters’), but these are not the common a fortiori inferences in rabbinic literature. See on this in several of my A Good Measure articles, for example the two mentioned above, and also in M. Abraham’s above-mentioned article in __Logic__. We should note that the inferences called in Muslim law ‘qiyas’ (which are mentioned and described in the English article) are of a similar character. Halakhic (‘middot-based’) inference is of a more complex character, and we will deal here mainly with it.
21. On this matter, see the article by M. Abraham, ‘Induction and analogy in halakhah’, Tzohar 15, summer 5763, pp. 23-34. Also see Two Wagons, gate eight, chapter 1, and A Good Measure, parashat Genesis 5766, at the end of the article in the discussion of the relation between an a fortiori inference and analogy.
22. In Encyclopedia Talmudit, under the entry ‘Binyan Av,’ a dispute among the early authorities is brought regarding the rule of analogy. It is customary to identify it with ordinary analogy, from one source. But some related to such an analogy as simple interpretation, and also saw the rule of analogy from one verse as a mechanism of ‘the common denominator’ (= analogy from two sources). See a discussion of this also in Kahana’s above-mentioned article, pp. 176-181. Below we will refer to the rule of analogy as ordinary analogy, according to the accepted meaning, since this changes nothing with respect to our actual discussion.
23. The structure is more complex, since at the beginning of the discussion there is an attempt to derive the law of huppah from another source, but it is rejected out of hand and does not join the body of the logical discussion conducted later in the sugya. Likewise, at the end of the sugya a dispute between Abaye and Rava is presented, which returns and concerns Rav Huna’s original a fortiori inference. We will discuss this dispute below separately, since it contains an unusual refutation of an a fortiori inference, and in order to explain it we will need the tools that will be developed in our model. For the purposes of the discussion at the current stage we present only the logical core of the sugya.
24. The course of the sugya can be read in two ways:

    The sugya continually returns to Rav Huna’s original a fortiori inference. After it is presented, it refutes it, then reaffirms it, and then refutes it again, and so on. According to this proposal, in the end the derivation is from Rav Huna’s original a fortiori inference, which in the conclusion is found to be valid.

    Rav Huna’s a fortiori inference was indeed refuted and failed, but the sugya succeeds, through a combination of additional inferential means, in validating Rav Huna’s conclusion. At the end of the sugya a dispute between Abaye and Rava is brought regarding Rav Huna’s original a fortiori inference, and this implies that the first way of reading the sugya is the correct one.

    Our model ostensibly reads the sugya in the second way, since we will present a chain of inferences that become more and more complex, and only the last one succeeds in proving Rav Huna’s conclusion regarding huppah in betrothal. However, from the logical standpoint there is no obstacle to claiming that after we have proved the validity of the complex inference, this means that Rav Huna’s original a fortiori inference is valid.

25. Not every a fortiori inference involves acts and results, but for the sake of simplicity we will use these terms as general ones. These are the two kinds of entries of the problem.
26. For a more detailed discussion of this point, see the English article. Also see our remarks below when we explain the requirement that valency can be high only in one parameter in the microscopic model.
27. See on this the article A Good Measure, parashat Shemini, 5765.
28. In the case of the blessings, the Sages assume that there is relevance, and therefore they are willing to use an a fortiori inference even though there are only two data. See on this the article A Good Measure, on parashat Shemini, 5765.
29. At the end of the article, after we develop our method, we will return to discuss this table, and we will see that it reflects an unusual midrashic inference.
30. The two examples in which the Sages themselves reverse the a fortiori inference in order to escape a refutation are exceptional cases in which the data are not symmetrical, that is, the values of the data are not only ‘yes’ or ‘no,’ but are themselves three-valued (the issue is not the microscopic parameters, but the values of the data themselves. For example, if intercourse effected only ½ marriage in contrast to huppah, which effects it completely, 1). We will deal with such cases below.
31. In the English article this implementation is defined in a more mathematically precise way (through multy-sets). Here we will make do with an illustration that clarifies the idea.
32. The same can be done with respect to the acts, except that there the higher row will receive a higher parameter value. For our convenience later on, we choose to present the schemes through the results.
33. Therefore it would have been impossible to propose for P the model, or , because that would have satisfied the requirement of an order relation with A, but it would have created a situation in which there is also an order relation toward N, which does not fit the data of the table. For this reason there is no escape from introducing an additional parameter into the model. The order is of course arbitrary, and it would have been possible to exchange the functions of P and N.
34. Below we will define a broader and more sophisticated preference criterion, and there valency will lose most of its role and significance. This conclusion is only temporary.
35. However, see the dispute regarding this matter in __Encyclopedia Talmudit__, under the entry ‘Binyan Av,’ notes 69-70. Also below we will see that our model דווקא supports the position that analogy is not weaker. With regard to the common denominator the situation is different, and see on this in the next chapter.
36. This result too is only temporary. See below after we define the full preference criterion.
37. As stated, this is a temporary conclusion.
38. Toward the end of the second part of the article we will see that a double refutation constitutes a counter-proof and not a refutation. But a column refutation or a row refutation are merely refutations, and not counter-proofs.
39. As we have already seen (see Diagram 3a), and will also see below, sometimes there are several results at the same point (when there are identical columns).
40. In fact, the solutions we proposed here for the two diagrams are not optimal. We will note two things:

    A. It is possible to propose for both of them (as long as they are of finite length) a two-dimensional model. The optimal solution is that along one branch the parameter progresses, and along the second branch the parameter progresses in increasing intensities, each of which is multiplied by the parameter . And in order that there be no dependence between the branches, the progression on the branch where appears alone must begin at an intensity one beyond that at which the progression in the other branch ends. But such a model too expresses the same phenomenon we described in the text above. The progression is in the quantity parameter, and it is the same parameter that progresses in both branches. Only the qualities of the two branches differ.

    B. It is possible to define the product of two parameters as a new parameter, so that, for example, 4=4 As long as we have not explicitly identified the meaning of the parameters (the semantics), there is no obstacle to combining them with one another consistently, and defining the combination as a new parameter. In this way we can obtain another solution for the graph from the solutions displayed above. It turns out that this is the optimal solution for this graph.

41. We note that Result 2, according to which analogy is a weaker inference than an a fortiori inference, now does not arise naturally from the model. However, we already noted above that this is disputed among the commentators (whether one may raise any slight refutation against analogy). Our conclusion here is that apparently analogy is not weaker. It is important to emphasize that what is explicit in the Talmud is that one may raise any slight refutation against an inference of the common denominator (= analogy from two texts). Three types of common-denominator inferences will be discussed below, and there we will return to this point.
42. An exceptional case appears in the sugya of Bava Metzia 41b (and see there also 94b), ‘principal without an oath is preferable to double payment with an oath.’ There the Gemara apparently offsets refutations. This is a very exceptional case in the Talmud, and we will deal with it in the second part of the article.
43. There is no need to distinguish between the case in which the upper one is an a fortiori inference and the lower one is an analogy, and the opposite case. In such a case we will simply interchange the rows and columns of the two acts and their two refutations, and we will again obtain the same table.
44. See Two Wagons, pp. 479-482.
45. We note that our model supports the view of the Ran in the above-mentioned sugya in Hullin, and this is somewhat contrary to what appears to follow from Tosafot there (see the above-mentioned Encyclopedia Talmudit).