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The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

Michael Abraham

A. Introduction

At the beginning of the Sifra, the famous baraita of Rabbi Ishmael is cited:

‘Rabbi Ishmael says: The Torah is expounded by thirteen principles: by a fortiori inference, by verbal analogy, by binyan av, etc.’

According to most views among the decisors, these principles by which the Torah is expounded are a law given to Moses at Sinai 1.

All of these principles appear to be a kind of scriptural logic intended solely for expounding verses, and to have nothing whatsoever to do with the logic that describes deductive thought as used in other areas of thinking, including Jewish law itself in talmudic-legal dialectic.

Many tend to think that the principle of a fortiori inference is an exception, since it apparently represents the syllogisms familiar from Aristotelian logic 2.

In this article we shall try to analyze the logical structure of the a fortiori inference and the various refutations directed against it; in the course of doing so, our view will become clear: this principle is like all the other principles, grounded at Sinai, and its deductive force is no different from theirs 3.

B. The Sugya of an A Fortiori Inference Derived from an A Fortiori Inference (Zevahim 50a):

The Gemara in the chapter ‘Eizehu Mekoman’ discusses whether the laws of sacrificial matters can be learned from what has itself been learned; that is, whether a law derived through one of the thirteen principles can itself teach another law by means of one of the thirteen principles.

The Gemara there states that one may derive an a fortiori inference from an a fortiori inference 4.

In the book Birkat HaZevah 5, a difficulty is raised against this Gemara: there cannot be a case in which we would need to arrive at learning an a fortiori inference from an a fortiori inference, for if the hierarchy is of the form C←B←A, one can always formulate a direct a fortiori inference: C←A.

And if there is a refutation that rules out the possibility of a direct a fortiori inference, for example some leniency that exists in C, then if that leniency also exists in B, the first a fortiori inference B←A collapses; and if it does not exist in B, the second a fortiori inference C←B collapses. If so, we will not be able to learn from the double a fortiori inference either 6.

Birkat HaZevah understands the form of the double a fortiori inference as follows:

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30 Logic

– a

+ b

– c

A

AF I

+ a

+ b

– c

B

AF II

+ a

+ b

+ c

C

Figure 1

AF I: if A, which is lenient with respect to a (a⁻), is stringent with respect to b (b⁺), then B, which is stringent with respect to a (a⁺), is certainly stringent with respect to b (b⁺).

AF II: if B, which is lenient with respect to c (c⁻), is stringent with respect to b (b⁺), then C, which is stringent with respect to c (c⁺), is certainly stringent with respect to b (b⁺).

And he proposes, instead, a direct a fortiori inference: if A, which is lenient with respect to a (or c), is stringent with respect to b, then C, which is stringent with respect to a (or c), is certainly stringent with respect to b.

– a

+ b

– c

A

+ a

+ b

+ c

C

Figure 2

He adds that if there is some stringency (d⁺) in A (or d⁻ in C), then if in B the law is d⁻, AF I falls; and if in B the law is d⁺, AF II falls.

And he concludes there as follows: ‘In the final analysis, one whose heart is broad and whose mind is clear should apply himself to resolving this difficulty; as of now it requires much study from me.’

In fact, one could have answered his question by positing a case in which there are additional data missing from the table of the a fortiori inference:

– a

+ b

?

A

AF I

+ a

+ b

– c

B

AF II

?

+ b

+ c

C

Figure 3

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31 The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

At first glance: it is clear that A-c must be negative, for otherwise AF I is refuted. And similarly C-a, otherwise AF II is refuted. But filling these boxes is itself a process of a fortiori inference. Hence the alternative proposed by Birkat HaZevah, namely to learn C←A directly, rests on data that were themselves learned by a fortiori inference. It is therefore still a double a fortiori inference, and without these data there is no way to derive a direct a fortiori inference 7.

At first glance, this form of a fortiori inference is not transitive and thus constitutes an answer to Birkat HaZevah. But he sensed this and argued that even in such a case one can refute AF I by means of c, even though the law of A-c is unknown (and indeed there are several examples in the Talmud of such refutations); and likewise one can refute by means of C-a.

In the diagram we proposed, the impossibility of a direct a fortiori inference stems from the fact that there is no relation of leniency and stringency between A and C, and not from a refutation. Therefore the initial argument of Birkat HaZevah did not, apparently, take this sort of a fortiori inference into account. In fact, in this a fortiori inference one may skip the derivation of the law B-b in the following way: B is more stringent than A with respect to law a, C is more stringent than B with respect to law c, therefore C is more stringent than A, and therefore if A has b, clearly C also has b.

This move uses twice only the deductive part of the a fortiori inference and not the principled component within it (as will be explained below), for we did not infer from AF I the law B-b. Perhaps Birkat HaZevah would call this a single rather than a double a fortiori inference, since the Gemara was clearly not asking whether one may use a logical consideration twice, but whether one may use the hermeneutic principle twice. This is perhaps the reason he did not see such an a fortiori inference as a solution to his difficulty.

In the book __Tzon Kodashim__ 8, Birkat HaZevah’s question is answered by giving an example of a double a fortiori inference that cannot be converted into a direct a fortiori inference.

If we denote the a fortiori inference of Tzon Kodashim in tabular form, it will take the following shape (the significance of the directions of the arrows will be clarified below):

a – b + c + A

AF I

a + b + c – B

AF II

a – b + c + C

Figure 4

All the laws except for B-b and C-b are given in Tzon Kodashim’s case, including the two boxes that were empty in Figure 3; here they are filled in—contrary to intuition—by force of various derivations.

The a fortiori inference is used to learn the law that the receiving of the blood in the Passover offering must be for the sake of one fit to eat, that is, that the person for whom the offering is brought must be able to eat an olive’s bulk of the sacrifice.

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32 Logic

of the sacrifice).

A – slaughtering

B – sprinkling the blood

C – receiving the blood

The given laws:

a – for the sake of the owners (and not for the sake of one who is not the owner of the offering).

b – for the sake of one fit to eat.

c – requires the north side (it must be performed in the northern part of the Temple courtyard).

Slaughtering: intent for the owners is not indispensable, intent for one fit to eat is indispensable, and it requires the north.

Sprinkling: intent for the owners is indispensable, it does not require the north, and by AF I intent for one fit to eat is indispensable.

Receiving: intent for the owners is not indispensable, it requires the north, and by AF II intent for one fit to eat is indispensable.

At first glance, one can refute AF II by arguing: what is special about C, where (ā) applies? But then one would say that A proves the point, since there too (ā) applies and nevertheless b⁺. This is the derivation called ‘the common element’.

Likewise, one can refute AF I by arguing: what is special about B, where (c̄) applies? Tzon Kodashim answers that we are not learning the sprinkling of the blood from slaughtering, but rather ‘for the sake of one fit to eat’ from ‘for the sake of the owners’.

That is, instead of formulating AF I as follows: if in slaughtering, where intent for the owners is not indispensable, intent for one fit to eat is indispensable, then in sprinkling, where intent for the owners is indispensable, intent for one fit to eat is certainly indispensable. Against this formulation one can refute: what is special about sprinkling, in that it does not require the north and therefore is not more stringent than slaughtering as AF I assumes.

We formulate AF I instead as follows: if intent for the owners, which is not indispensable in slaughtering, is indispensable in sprinkling, then intent for one fit to eat, which is indispensable in slaughtering, is certainly indispensable in sprinkling.

That is, we are not learning one action from another but one intention from another; and against this formulation the claim that sprinkling is not more stringent than slaughtering is not a refutation, for AF I did not assume that at all.

This explains the vertical lines in AF I in Figure 4.

It is clear that the direct a fortiori inference is entirely impossible in Figure 4, since A and C have identical laws (and again, this is not because of a refutation, and therefore was apparently not taken into account by Birkat HaZevah).

C. A Fortiori Inferences of Places

Tzon Kodashim’s answer to Birkat HaZevah’s difficulty assumes that one may rotate the direction of derivation in an a fortiori inference in order to escape a refutation of the original form.

At first glance, it would seem that every a fortiori inference can be presented in two formulations ‘perpendicular’ to one another, and when one is refuted one may ‘rotate’ the direction of the argument and evade the refutation. If so, to refute any given a fortiori inference we would have to refute both directions every time.

If we present the schematic a fortiori inference in the form of a table composed of three given data and one derived box:

a

– +

b

+ ⊕

A B

Figure 5

we immediately see that one can, apparently, formulate a ‘horizontal’ and a ‘vertical’ a fortiori inference in every case. Such an a fortiori inference is called by Sefer HaKeritut, Halikhot Olam, and others, ‘an a fortiori inference of places’. But not in every case do both forms have meaning.

It is interesting, for example, to look at the a fortiori inference in tractate Bava Kamma: ‘If one is liable for uncovering, is he not all the more so for digging?’ That is: in the case of a pit in the public domain, one who removes its cover is liable even though he did not dig the pit, and all the more so the one who digs a pit in the public domain should be liable to pay for the damage done to one who falls into it.

Such an a fortiori inference cannot, apparently, be presented in a table like Figure 5, for there is only one given law here, not three as in the regular a fortiori inference.

Whoever nevertheless wishes to draw a formal comparison to Figure 5 may write the following key:

a – stringency A – one who opens the cover of an existing pit

b – liability for payment B – one who digs a pit

In words: A is not stringent and nevertheless is liable for payment, so all the more so B, which is more stringent, should be liable for payment.

But in such a case a very strange formulation emerges if we ‘rotate’ the a fortiori inference:

The stringency that does not exist in the opener exists in the digger; all the more so payment, which exists in the opener (and is therefore more stringent than stringency (!?)), should also exist in the digger. That is, payment liability is learned from stringency on the basis of the assumption that payment liability is more stringent than stringency, as would seem from position A.

What characterizes such an a fortiori inference is that the stringency of the derived case (B) relative to the source case (A) arises from reasoning and not from law; and such a thing cannot really be ‘rotated’ (unless we become extreme formalists).

But apart from a fortiori inferences of the above type, it seems to us that every a fortiori inference can be presented in a table and therefore can also be rotated in order to escape a refutation.

The clearest example of such a rotation is found in the Mishnah in Bava Kamma 24a. There Rabbi Tarfon and the Sages dispute the damages paid by one who causes damage through horn in the injured party’s courtyard.

The given laws: damage by foot in the public domain is exempt from payment.

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34 Logic

Damage by horn in the public domain is liable for half damages.

Damage by foot in the injured party’s courtyard is liable for full damages.

We shall explain the course of the dispute according to the straightforward reading of the Mishnah (though see the commentators there, some of whom explain it differently).

Rabbi Tarfon derives by a fortiori inference: from the laws of foot we see that the injured party’s courtyard is more stringent than the public domain; therefore one should say the same regarding the laws of horn, and thus horn, which in the public domain is liable for half damages, should in the injured party’s courtyard be liable for full damages.

The Sages answer him with the rule that ‘it is enough for what comes from the derivation to be like that from which it is derived’; that is, from this derivation one can prove liability only for half damages in the injured party’s courtyard, for although it is indeed more stringent than the public domain, since it is not known by how much, one imposes only the minimum payment.

Rabbi Tarfon answers them: let us learn by the reverse a fortiori inference: from the laws of the public domain we see that horn is more stringent than foot; if so, the same must be true in the injured party’s courtyard, and therefore if foot in the injured party’s courtyard is liable for full damages, horn, which is more stringent than it, is certainly liable for full damages.

The Sages answer this as well by invoking the rule of dayyo, but for our purposes we see in Rabbi Tarfon a rotation of the a fortiori inference in order to escape dayyo 8.

Tosafot s.v. ‘I’ on 25a, and likewise Nimukei Yosef there, ask why Rabbi Tarfon’s a fortiori inference is not refuted by the claim: what is special about foot, whose damage is common? In this respect foot is more stringent than horn.

They go on to say that this claim would refute the rotated a fortiori inference as well, and they bring proof from 27a; see there.

These early authorities answer that this is not a refutation, since it can be absorbed into the a fortiori inference. One can say: horn is more stringent than foot despite the fact that the damage caused by foot is common; and therefore in the injured party’s courtyard as well that hierarchy will remain in force. It follows that a refutation of this kind does not refute the a fortiori inference 9.

From the words of Tosafot and Nimukei Yosef just cited, it appears that one does not rotate an a fortiori inference in order to escape a refutation, and this seems to be a kind of general rule, for which they bring proof from 27a in Bava Kamma. If so, the words of Tzon Kodashim would, apparently, be contradicted.

Penei Yehoshua in Bava Kamma 26a, on Tosafot s.v. ‘Is it not a law’, asks against Tosafot from several places in the Talmud where Tosafot himself proposes rotating an a fortiori inference in order to escape a refutation 10.

Penei Yehoshua answers that Tosafot also agrees that one can rotate an a fortiori inference, except that in our case (and likewise in the case on 27a), if Rabbi Tarfon rotates the a fortiori inference in order to escape the refutation, he runs into the problem of dayyo that the Sages raise against him. Therefore, in this specific a fortiori inference, the perpendicular direction also needs the horizontal one, and so it is enough to refute one of them. But in all other a fortiori inferences in the Talmud, everyone agrees that any a fortiori inference can be ‘rotated’. If so, it is clear that Tzon Kodashim does not contradict those early authorities according to Penei Yehoshua’s explanation.

At first glance, however, the plain meaning of Tosafot and Nimukei Yosef is that this is a general rule throughout the Talmud: rotation will not help, and not only in an a fortiori inference burdened with problems of dayyo, as Penei Yehoshua says. Moreover, his words are difficult to reconcile with the plain sense of the Mishnah there 11. We shall therefore propose a new resolution of Tzon Kodashim’s view, and through an analysis of the structure of the a fortiori inference and the refutations directed against it, it will become clear that this is in fact the simple explanation of his words.

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35 The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

D. The A Fortiori Inference as a Syllogism

One of the early sources that discuss the nature and force of the a fortiori inference is the book Korban Aharon on the Sifra, in the section on the baraita of Rabbi Ishmael called Middot Aharon. Schwartz, and following him Rabbi Ostrovski, rejected the attempts of the author of Korban Aharon because they lead to the conclusion that the a fortiori inference is merely possible and not necessary, whereas in their opinion it is a fully deductive inference. The essence of their position is that the a fortiori inference is in fact only a different garment for the Aristotelian syllogism, whose form is:

1. (∀x) P(x) → G(x) premise
2. P(a) premise
3. |- G(a) conclusion

where G and P are predicates, and a is a particular subject. Premise 1, called the major premise, states that every subject x possessing the property P must also possess the property G. Premise 2, called the minor premise, states that the subject a has the property P. Conclusion: the subject a also has the property G.

When we wish to compare the course of the a fortiori inference with the argument described above, we say that the existence of a stringency in the derived case that is not found in the source case indicates the major premise:

1. Every stringency that exists in the source case certainly exists in the derived case.

In addition, there is a third given law that teaches us:

1. A certain stringency exists in the source case

and from this follows the conclusion of the a fortiori inference:

1. This stringency also exists in the derived case.

And in tabular form:

H G

R – +

T + ⊕

Figure 6

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36 Logic

The horizontal argument: (∀x) H(x) → G(x)

H(t)

|- G(t)

And the vertical argument: (∀x) R(x) → T(x)

R(g)

|- T(g)

Here we have used lowercase letters to denote subjects and uppercase letters to denote predicates. G(T) and g(t) refer to the same concept: G describes it as a predicate, while g describes it as a particular subject. The major premise in the horizontal argument is derived by looking at row R, and in the vertical argument by looking at column H. It is clear that if one rotates the table, the horizontal becomes vertical and vice versa, that is, the rotation is around the diagonal [- +]. An a fortiori inference whose source is reasoning rather than law would be described in the same way, except that the major premise would then derive from reasoning and not from a law, that is, from a row or column in the table. From this argument Schwartz and Ostrovski conclude that the a fortiori inference is natural deduction.

There is an important point that these authors apparently failed to notice: the very use of a column or row in the table to infer a rule that applies to all subjects is plainly an inductive process. The line of thought of Schwartz and Ostrovski would lead to the conclusion that verbal analogy, too (and the rest of the principles as well), is deduction, since the major premise would derive from the rule of finding the same word in two different subjects, the minor premise would be a law found in one of the subjects, and the conclusion would be that the law is found in the second subject as well. The difference among the various principles lies only in the way one arrives at the major premise. It is therefore also clear that a refutation can attack only the major premise and not the minor one, which is given, and certainly not the deductive course of the argument itself, for that lies beyond all doubt, at least in the talmudic context. In our opinion, the claim that deduction was given at Sinai sounds utterly absurd; it is clear that this is not what the early authorities cited in note [1] meant. We should also note that even an a fortiori inference that is not refuted at all is still not deduction, for the very possibility of refuting it indicates that there are non-necessary elements in the inferential process.

E. The Refutations

At the end of the previous section we explained that refutations attack the major premise of the a fortiori inference, and there are several different ways to do so, each with its own characteristics.

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37 The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

If we look at the table in Figure 6, we can distinguish three main kinds of refutations:

1. Box refutations.

1. Row/column refutations } refutations based on reasoning

1. Refutations from an external law.

It is clear that every property of a row corresponds to a column as well, when we exchange the horizontal a fortiori inference for the vertical one and vice versa; in the same way, the GR box and the HT box are dual.

1. Box refutations are based on a rationale relevant only in one box, and explain why it is more stringent than its two nearby neighbors, or more lenient in the case of the HR box.

For example: H – liability in the public domain R – tooth and foot

G – liability in the injured party’s courtyard T – horn

If we explain a rationale relevant only in the GR box, for example: foot, whose damage is common, must be prevented only in the injured party’s courtyard, since in the public domain imposing payment liability would prevent the public from using the road for fear of damage by their animals. This rationale explains why this box indeed imposes payment, and such a rationale__ refutes both directions of the a fortiori inference.__

Explanation: in the horizontal a fortiori inference, we cannot infer from the fact that, with regard to foot, the injured party’s courtyard is more stringent than the public domain, that the same is true of horn; for in the case of horn there is no rationale unique to the injured party’s courtyard such as exists in the case of foot.

In the vertical a fortiori inference: it is true that horn is more stringent than foot, as seen in the public-domain column, but this is only in relation to foot where the rationale of the refutation is absent. In the injured party’s courtyard, where foot has the rationale of the refutation, perhaps foot is more stringent than horn, and therefore one cannot obligate horn in the injured party’s courtyard by a fortiori inference.

It is clear on grounds of symmetry that a rationale in HT will do the same thing; and a rationale in HR plainly refutes both directions of the a fortiori inference, as simple intuition indicates, and likewise by symmetry, for there is no preference for the direction of a column over that of a row. Of course, a rationale in HR would explain why this box is more lenient than its companions.

2. Row/column refutations refute neither direction13, for they can be absorbed, as Tosafot and Nimukei Yosef say in the passage cited above. For example, if we say that the damage caused by tooth and foot is common, we can formulate the horizontal a fortiori inference thus: from the laws of foot we see that the injured party’s courtyard is more stringent than the public domain, and likewise with respect to the laws of horn. The fact that the damage caused by foot is common poses no problem, since this is true in both domains. In the vertical a fortiori inference: from the laws of the public domain we see that horn is more stringent than foot, even though the damage caused by foot is common; and if so, in the injured party’s courtyard as well horn will be more stringent than foot, and it does not matter that there too the damage caused by foot is common.

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38 Logic

1. Horizontal/vertical refutations from an external law mean, in effect, adding a row/column to the table in Figure 6, filled in with logic opposite to the other data.

For example, a refutation of the form: ~G(b) •H(b)

would refute the horizontal a fortiori inference represented by Figure 6, for it plainly contradicts the major premise of that argument, namely:

(∀x)[H(x) → G(x)]

This is a horizontal external law that refutes the horizontal a fortiori inference.

And similarly, a vertical external law would refute the vertical a fortiori inference.

If we draw the case of a vertical external-law refutation, we obtain:

R + – +

T – + ⊕

Figure 7

I H G

That is, there is another domain in which foot is liable and horn is exempt.

The vertical a fortiori inference, at stage 1, infers from column H that horn is more stringent than foot, and this is directly refuted by column I. By contrast, the horizontal a fortiori inference is apparently not refuted at any stage, for from row R we really do see that G is more stringent than H, that is, that the injured party’s courtyard is more stringent than the public domain, and therefore one infers that the same holds for horn.

Of course, one can argue that a parallel process, claiming that I, which in R is more stringent than H, should also be more stringent in T, would lead to error; hence one should not judge the a fortiori inference from H to G either. But this would only force us to admit that in domain I there is a refutation interfering with the a fortiori inference in that direction; and this is already clear from the data even without column G. Yet this refutation is not necessarily connected to column G. If it is a leniency of I, it is certainly unrelated; and if it is a stringency of H, then perhaps it is related, for perhaps H is also more stringent than G in that same respect.

Therefore, on this matter there can certainly be a dispute whether a vertical/horizontal external-law refutation also refutes the horizontal/vertical a fortiori inference, that is, the a fortiori inference that is not parallel to the direction of the external law.

In the case of Tzon Kodashim, we are dealing with a refutation of this type, namely that slaughtering requires the north. Hence Tzon Kodashim argues that one can rotate the a fortiori inference and escape it.

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39 The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

Birkat HaZevah, who remained with the difficulty unresolved, apparently did not accept this; perhaps he holds that a refutation of this sort refutes both directions, which, as we explained, can indeed be argued.

Tosafot and Nimukei Yosef in Bava Kamma, who said that one cannot rotate an a fortiori inference, were speaking of a refutation based on reasoning; and there, as we explained, the two directions are indeed equivalent. In their answer they actually pointed out that this is a refutation based on reasoning, in order to absorb it into the a fortiori inference 14.

It follows that these early authorities do not contradict the words of Tzon Kodashim at all, even without the answer of Penei Yehoshua. More than that: Penei Yehoshua is, apparently, very puzzling, for he wishes to say that in every refutation, in principle, ‘rotating’ the a fortiori inference will help, including a refutation based on reasoning, which is precisely what is discussed in Bava Kamma, and this is problematic for the reasons we explained.

In the logical model we have presented for the a fortiori inference, one can see how a refutation of the type based on external law operates, as we saw above, just before Figure 7; the other types of refutation cannot be described in that model.

Likewise, it is not easy to see that a legal refutation as presented there does not refute the perpendicular a fortiori inference, because the notations are different.

In order to justify these arguments clearly, it is preferable to define an algebraic model that allows greater penetration into the details of the argument and clear discrimination regarding the way each refutation operates, and as a result regarding the dispute between Birkat HaZevah and Tzon Kodashim.

F. The Algebraic Model

Let us look at the a fortiori inference from tractate Bava Kamma described in the following table:

Public domain – +

Injured party’s courtyard + ⊕

Foot Horn Figure 8

Let us define two functions from the space of damaging agents to the space of real numbers, namely the degrees of stringency:

g1(z) – the function estimating the degree of stringency for liability in the public domain.

g2(z) – the function estimating the degree of stringency for liability in the injured party’s courtyard.

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40 Logic

The relevant elements in the space of damaging agents are: c – foot, d – horn. And let us define two threshold levels:

v – the minimal level of stringency for liability in the public domain. w – the minimal level of stringency for liability in the injured party’s courtyard.

For example: a damager z, for which g₁(z) > v, will be liable for payment in the public domain.

Let us now formulate the horizontal a fortiori inference:

1. Given g₁(c) < v 2. Given g₁(d) > v 3. 1,2 |- g₁(d) > g₁(c)

At this stage, several possibilities stand before us for describing the inductive component that underlies the a fortiori inference, that is, the transition from g₁ to g₂: I g₁(z) = g₂(z) II (∀ᵢⱼ)[gᵢ(d) > gᵢ(c)] → [gⱼ(d) > gⱼ(c)]

The first requirement is, of course, the stronger one, for it identifies the two functions. Hence, for example, any conclusion in which this equality does not hold would refute the a fortiori inference.

Requirement II is the more moderate choice, and in effect means requiring increasing monotonicity of the functions gᵢ(z) [15]. From either of these two assumptions the following conclusion can be derived:

1. From either I or II, g₂(d) > g₂(c) 5. Given g₂(c) > w 6. 4,5 |- g₂(d) > w

At line 6 we have reached the conclusion of the a fortiori inference, namely that horn is liable in the injured party’s courtyard.

Let us turn to the formulation of the vertical a fortiori inference:

If we were to accept the stronger assumption I, and perhaps one that is too strong, as will be seen below, it would be possible to describe the vertical a fortiori inference as follows:

1. Given g₂(c) > w 2. Given g₁(c) < v 3. 1,2,I |- v > w 4. Given g₁(d) > v 5. 3,4 |- g₁(d) > w 6. 5,I |- g₂(d) > w

But if we adopt assumption II, it may already be impossible to write any order relation between v and w, since perhaps they are not measured at all in the same units. Likewise, one cannot write a relation between g₁(z) and g₂(z’), since they too are not necessarily measured in the same units of stringency. It is therefore clear that lines 3 and 6 as well, which were based on full equality between the two functions in accordance with assumption I, will no longer necessarily be correct.

Under assumption II, let us define the functions relevant to the vertical a fortiori inference and formulate it in a manner parallel to the horizontal one:

P₁(z) – the function estimating stringency for liability as foot.

P₂(z) – the function estimating stringency for liability as horn.

a – public domain; b – injured party’s courtyard

x – the threshold of liability for horn; y – the threshold of liability for foot.

The vertical argument will be formulated in the same form as the horizontal one, except that we substitute

{ gᵢ → Pᵢ

c → a

d → b }

1. Given P₁(a) < y

1. Given P₁(b) > y

1. 1,2 |- P₁(b) > P₁(a)

1. 3,II |- P₂(b) > P₂(a)

1. Given P₂(a) > x

1. 4,5 |- P₂(b) > x

Assumption II for this a fortiori inference will be: (∀i,j)[Pᵢ(b) > Pᵢ(a)] → [Pⱼ(b) > Pⱼ(a)]. We note that throughout the horizontal and vertical arguments in this last formulation, we did not compare values of two different functions, nor did we compare the values of the liability thresholds to one another, thus leaving open the possibility that the units are different and incomparable.

The claim that one may rotate an a fortiori inference in order to escape a refutation sounds, at first glance, very strange. After all, the horizontal and vertical arguments seem to be merely two different formulations of the same argument, and a refutation, being a logical process, should not depend on formulation. Both kinds of argument start from the same three data, use all three of them, and infer the same conclusion by the same logic. How, in principle, can a refutation that attacks one of them fail to touch the other?

To understand this, we should note that, as explained at the end of the previous section, a refutation attacks the assumption of the a fortiori inference that was reached inductively, and not the deductive part of the argument. If so, one readily sees that the assumptions of the horizontal and vertical arguments are completely different from one another. Hence the impression that this is the same argument presented in two ways is only an illusion.

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The following:

Foot in the public domain: P₁(a)→g₁(c)

Foot in the injured party’s courtyard: P₁(b)→g₂(c)

Horn in the public domain: P₂(a)→g₁(d)

Horn in the injured party’s courtyard: P₂(b)→g₂(d)

What we would obtain would indeed be a valid argument, but it would be based on assumption II translated as follows:

II vertical (∀z₁,z₂)[g₂(z₁) > g₁(z₁)] → [g₂(z₂) > g₁(z₂)]

From comparison with the assumption of the horizontal argument as presented above

II horizontal (∀i,j)[gᵢ(d) > gᵢ(c)] → [gⱼ(d) > gⱼ(c)]

one sees immediately that this is a completely different assumption. Therefore it is clear that these are two entirely different arguments, and not two formulations of the same argument as a superficial glance might suggest. Consequently, it is entirely possible that a refutation contradicting II horizontal will not interfere with II vertical, and vice versa.

Below we shall see that a refutation from an external law attacks assumption I or II, and therefore rotating the a fortiori inference can help against it. By contrast, a refutation based on reasoning, whether of row, column, or box, attacks the very manner in which the data are recorded, and therefore it refutes both directions, or else neither of them, and does not distinguish between them. In the algebraic model presented here, there is a way to describe all the types of refutations set out in the previous section. A vertical legal refutation will be the addition of a damager e with different values of g₁(e) and g₂(e), values that refute monotonicity.

A horizontal legal refutation will be the addition of a function g₃, with values g₃(d), g₃(c) that do not preserve the monotonicity of g₃. A horizontal refutation will not refute the vertical argument, and vice versa, as we explained in presenting the approach of Tzon Kodashim.

Birkat HaZevah will assume a somewhat different model, as we shall explain below.

Let us look, for example, at a vertical refutation, namely the addition of a damager e, say fire, which is supposedly liable in the public domain and exempt in the injured party’s courtyard.

In the terms of the horizontal a fortiori inference:

1. New datum g₁(e) > v

1. New datum g₂(e) < w

1. Because g₁(c) < v in the argument, |- g₁(e) > g₁(c)

1. Because g₂(c) > w in the argument, |- g₂(e) < g₂(c)

And this refutes nothing in the horizontal a fortiori inference, for its assumption II concerned only c and d.

By contrast, in the vertical a fortiori inference, if we translate the refutation into ‘vertical’ terms:

1. Given P₃(a) > P₃(b)

1. I,II |- P₁(a) > P₁(b)

and this contradicts the conclusion (P₁(b) > P₁(a)) in line 3 of the vertical a fortiori inference.

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43 The ‘A Fortiori’ Inference as a Syllogism – An Arithmetic Model

We see that a vertical refutation contradicts only the vertical a fortiori inference.

We should also note that if we had assumed I, then by force of lines 3 and 4 in the description of the refutation that did not succeed against the horizontal a fortiori inference, it would follow that the horizontal a fortiori inference is also refuted, since the functions would then necessarily not be equal.

Likewise, if in assumption II of the horizontal a fortiori inference we had added a universal quantifier over the damagers, namely (∀z₁z₂), then the refutation would also refute both directions.

A third way of making the refutation apply to both directions would arise if the data were written using the signs {≤,≥} instead of {<,>}, and the assumption were I.

Accordingly, there are three possibilities for explaining the dispute between Tzon Kodashim and Birkat HaZevah:

1. How many quantifiers appear in the monotonicity assumption II.16

1. Whether one assumes monotonicity or equality, that is, I or II

1. Whether, when one assumes equality I, one writes the data using ≥ or using <.

To complete the picture, let us turn to describing refutations based on reasoning.

The general form of refutations based on reasoning will be: gᵢ(z) → gᵢ(z) + hᵢ(z), and similarly in vertical terms:

Pᵢ(z) → Pᵢ(z) + hᵢ(z)

That is, the function describing the stringency must change as a result of the rationale.

For example, saying that tooth and foot cause common damage means that an additional stringency is added to them beyond what we initially took into account. Therefore, in the vertical a fortiori inference we will have to change P₁(z); or, translating this into horizontal terms, we will have to change the values of g₂(c), g₁(c), while gᵢ(d) remain unchanged.

It is obvious that a refutation that finds a leniency in horn is equivalent to one that finds a stringency in foot, and so we need not discuss it separately.

Let us consider, for example, the refutation of Tosafot and Nimukei Yosef in Bava Kamma: what is special about tooth and foot, whose damage is common?

1. Refutation P₁(z) → P₁(z) + h₁(z)

1. Substitution into datum 1 P₁(a) + h₁(a) < y

1. Substitution into datum 2 P₁(b) + h₁(b) > y

1. 2,3 |- P₁(b) + h₁(b) > P₁(a) + h₁(a)

1. Transposing sides: |- P₁(b) > P₁(a) + h₁(a) – h₁(b)

Let us discuss three possible cases:

1. If h₁(z) = Cons., that is, h₁ is a constant function, then it is clear that the a fortiori inference remains valid

for: P₁(b) > P₁(a).

And one can continue the argument as usual.

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And one can continue the argument as usual.
This is in fact a row or column refutation, since $h_i$ is a fixed addition of stringency to the entire row/
column. And as we explained the words of Tosafot and Nimukei Yosef, this is not a refutation at all; it is absorbed into the a fortiori inference.

1. $h_1(a) > h_1(b)$: the a fortiori inference is only strengthened, for there certainly holds: $P_1(b) > P_1(a)$,
   and the gap between them even grows. That is, this is not a refutation but a strengthening, since finding an additional stringency by reasoning in the injured party’s courtyard as compared with the public domain supports the a fortiori inference.

1. $h_1(a) < h_1(b)$: this is a box refutation, saying that one can no longer write with certainty
   $P_1(b) > P_1(a)$, and therefore one can no longer prove the conclusion of the a fortiori inference.
   As one can readily verify, this refutation also refutes the horizontal a fortiori inference.

In the same way one can describe the dual refutations in the horizontal a fortiori inference, and thus we have proved mathematically all the intuitive claims explained in the previous section.
If we pay attention, we will see that in examining the box refutation we did not use assumption II. That means that this refutation does not attack the monotonicity assumption, but rather says that we did not correctly understand the legal data. From this it is clear
that this refutation will work in the same way even if we assume I and not II.
That is, the difference between assumption I and assumption II, which according to one of the explanations above is the dispute between Tzon Kodashim and Birkat HaZevah, does not
affect the results regarding refutations based on reasoning, but only regarding legal refutations.

G. Summary

In this article we attempted to illuminate the logical structure of the talmudic a fortiori argument and the various refutations directed against it,
through an explanation of the dispute between Birkat HaZevah and Tzon Kodashim as to whether one can escape a refutation by ‘rotating’ the a fortiori inference.
We presented an algebraic model that illustrated and explained, in three different ways, the reasoning behind this dispute. This model
also proved the results that had previously been explained intuitively:
a. A legal refutation refutes only the argument parallel to it, unless one assumes Birkat HaZevah’s model.
b. A box refutation based on reasoning refutes both directions of the a fortiori inference.
c. A row/column refutation based on reasoning refutes neither direction.
We also explained and proved by means of this model that the two arguments, the vertical and the horizontal, are not equivalent.
Possible directions for the further development of this model include describing a somewhat different type of a fortiori inference, such as the a fortiori inference in Temurah 28 concerning an animal that copulated with a human or was copulated with by a human, where the admission of one witness disqualifies them from being offered.
Likewise, one can examine ways of describing more complex arguments and refutations, such as ‘the common element’ and ‘dayyo’,

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Notes:

1. Maimonides in his introduction to Seder Zera’im, Rashi on Pesahim 24a, R. Shimshon of Kinon in Sefer HaKeritut, and others.

1. This view was chiefly adopted by R. Schwartz in his book ‘Kal VaHomer’, who went so far as to disqualify opinions of Tannaim and Amoraim as incompatible with natural deduction, in the sugya of dayyo in tractate Bava Kamma 24a-25a. See also Rabbi Ostrovski’s book ‘The Principles by Which the Torah Is Expounded’.

Those who tend to see the a fortiori inference as a syllogism explain in this way the fact that this is the first principle listed in the baraita cited above, and also the fact that it is written explicitly in the Torah. See Genesis Rabbah, parashah 92, which brings ten a fortiori inferences written in the Torah; see also Netivot Olam, p. 67 in the novellae, and see note 3.

1. See the book Berurei HaMiddot by R. H. Hirschensohn, part 1, entry A (a fortiori inference), chapter 1, where he argues, as we do, that the a fortiori inference is a principle like all the others, and proves this from the Gemara and the decisors. On this understanding it is also clear why this principle was given at Sinai like the other twelve principles, and is not assigned to the ordinary rules of thought grounded in simple reasoning, notwithstanding the fact that a person may formulate an a fortiori inference on his own.

1. The Gemara there has a very interesting structure, for it learns this rule by means of a derivation that is itself an a fortiori inference from an a fortiori inference, and in the conclusion by means of a single a fortiori inference that itself raises a certain difficulty, since the law ultimately learned will, even in that case, rest on two a fortiori inferences.

1. It appears in the book Asefat Zekenim on Seder Kodashim, published by the Hafez Hayyim of blessed memory. He was among the earliest of the later authorities to explain the tractates of Seder Kodashim in sequence.

1. It should be noted that this difficulty also applies to combinations of binyan av and a fortiori inference, for binyan av too is refuted in the same way as the a fortiori inference. We further note that the double a fortiori inference by means of which the rule of deriving an a fortiori inference from an a fortiori inference is learned cannot be represented by a direct a fortiori inference unless it undergoes a slight modification, as the Gemara itself does there.

1. There are additional answers to Birkat HaZevah’s difficulty, but they do not arise from an a fortiori inference of a special structure; rather, they come from local examples, and therefore we shall not discuss them here. See Ginat Veradim and Hak Natan, among others.

1. R. Schwartz in his book ‘Kal VaHomer’ sharply attacks Rabbi Tarfon’s reasoning, and it seems, at first glance, that in his eagerness to criticize he failed to notice that it is precisely the Sages who are arguing in accordance with the path we have just taken in explaining the Mishnah, whereas Rabbi Tarfon’s reasoning is perfectly clear that dayyo does not apply to the ‘rotated’ a fortiori inference.

1. And see their words there: this is only in the case of a refutation based on reasoning, but a refutation based on a law written in the Torah cannot be absorbed. See our explanation below. We note only that according to this rule one can answer Birkat HaZevah’s question, as was done in the book Hak Natan, by means of AF 1, whose refutation is based on reasoning and can therefore be absorbed. See there.

1. Penei Yehoshua does not specify what these places are, apart from Tosafot in Bava Kamma on which he relies. Apparently he means Tosafot Hullin 23b s.v. ‘And what’, and 23a s.v. ‘And let it be’, and others.

1. Tosafot s.v. ‘I’ explains that according to Rabbi Tarfon there is no rule of dayyo at all, and the discussion in the Mishnah is only for the sake of the view of the Sages. If so, Rabbi Tarfon himself does not need the vertical direction, from foot to horn, even when he learns from the public domain to the injured party’s courtyard, as Penei Yehoshua says. Hence for Rabbi Tarfon this is an ordinary a fortiori inference like those found throughout the Talmud, and yet in Tosafot itself the rule is established that rotation does not help against a refutation. It therefore seems difficult for Penei Yehoshua.

1. We explained above Figure 4 that Birkat HaZevah too agrees that the deduction within the a fortiori inference may be applied twice, and that this will not be called an a fortiori inference derived from an a fortiori inference, for it is clear that the Gemara did not come to clarify whether we are allowed to perform an inference such as this:

1. premise (∀x) P(x) → G(x)
2. premise (∀x) G(x) → H(x)
   1,2 .3 |- (∀x) P(x) → H(x)

The Gemara, of course, wishes to clarify whether, when the conclusion of the second inference follows from the results of the first inference, the process of derivation is legitimate.

1. One still needs to examine Tosafot and Nimukei Yosef in Bava Kamma, who explain that a row/column refutation written in the Torah is a refutation. This is obviously a rule, and it does not follow from simple logic. See Penei Yehoshua, Kiddushin 3a, s.v. ‘Gemara’, who remains in doubt regarding Tosafot. According to our words, perhaps it works out that when they say ‘a refutation written in the Torah’ they mean an external-law refutation; therefore there they absorb it even though it is written in the Torah, because it is a row refutation. Penei Yehoshua is indeed consistent with his own view. See also Tosafot

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s.v. ‘for’ in Kiddushin 5a, which sounds like Penei Yehoshua.

1. The Tosafot that rotate an a fortiori inference generally do so in cases of refutations from an external law, except perhaps Tosafot Hullin 23a; see note 10.

Otzar HaHokhmah

1. The usual expression in mathematics for increasing monotonicity is

(∀z₁z₂)[z₁ > z₂] → [g(z₁) > g(z₂)], except that in our case one cannot write an order relation between the various damagers without some ‘translation’

of their value into numbers, and this is done by placing one of the functions g(z) in the antecedent of the implication above.

1. Even two quantifiers [(ij)(z₁,z₂)] still do not make the two types of argument identical, as can be seen by comparing the horizontal and vertical versions of assumption II presented above.

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