A FORTIORI LOGIC
CHAPTER 29 – Yisrael Ury
Yisrael Ury’s Charting the Sea of Talmud: A Visual Method for Understanding the Talmud, published in 2012 is a work with a very interesting logical innovation. But, though it has a chapter on qal vachomer (a fortiori) argument, it is not very innovative in that particular domain. We shall begin our exposition of it with an analysis of his contribution to logic in general and then deal more specifically with his comments on a fortiori.
Ury describes his work as “a revolutionary visual method for understanding and summarizing Talmudic discussions, conclusions, and laws.” His method is, as far as I know, indeed original; and moreover, it really does greatly clarify Talmudic discourse. Of course, he applies it only to a few sugyas (Talmudic discussion units), by way of example. Ideally, he should have applied it to all the Talmud’s discussions; or he or others should do that in the future in a separate work. If a sea is chartable, it should be charted.
But what is more amazing, to my mind, is that he has developed a truly practical way to represent all ‘if–then’ statements diagrammatically, a way hitherto unknown (so far as I know) to logical science. Ury deserves heaps of praise for this brilliant idea of general value. Everyone has heard of Euler diagrams. They are commonly used by logicians and students of logic. They allow us to express relationships of inclusion and exclusion visually, by means of circles within circles, partly intersecting circles and non-intersecting circles. Now, we shall have what we may henceforth call Ury diagrams. These are able to do things that Euler diagrams cannot do, though they cannot replace them.
The simplest Ury diagram is a small square, or “box.” This unit represents a subject, say A. If the subject is shaded a certain way, it means that A has a certain predicate, say X; i.e. it means that ‘A is X’. If the subject is not so shaded, the meaning is accordingly that ‘A is not X’. If the square is shaded some other way, e.g. with vertical stripes instead of diagonal ones, then another predicate, say Y, is intended. Thus, the square might be devoid of shading (in which case, A is neither X nor Y), or shaded one way but not the other (in which case, A is X but not Y, or Y but not X, as appropriate), or shaded both ways (in which case, A is X and Y).
Note that, because he is dealing with Talmudic discourse, Ury himself thinks of the subject (e.g. A) primarily as a legal context, a situation subject to some law, and of the predicate (e.g. X) as the law applying (or not applying, as the case may be) to that context. More precisely, “A is X” would here mean “case A is subject to law x.” However, he is clearly aware that the concept has broader application than legal discourse, since he gives an example dealing with fuel efficiency of cars.
Also note, even if Ury does not himself do so, that although the square (A) and its shading (X) are here used to refer to the two components of a copulative proposition, i.e. a subject and a predicate (terms), they could equally well be used for an implicational proposition, i.e. with reference to an antecedent and a consequent (theses). In such cases, instead of ‘A is X’ we would have ‘A implies X’.
Now, the next development is the most significant one. An ‘if–then’ relationship can be visually represented by means of the following convention. The two subjects concerned are represented by two contiguous squares. The square for the antecedent subject is placed to the left of (or below, or more forward than) the square for the consequent subject (which is therefore on the right, or above, or behind). If we label the two squares as A and B, and a predicate concerning them as X, the implication intended is “If A is X, then B is X.” This logically signifies the following three possibilities:
- A and B are both shaded – this means we first discovered that ‘A is X’ and thence inferred that ‘B is X’. We pass the shading of A on to B.
- A and B are both blank, – this means we first discovered that ‘B is not X’ and thence inferred that ‘A is not X.’ We pass the non-shading of B on to A.
- A is blank and B is shaded – this means that ‘A is not X’ and ‘B is X’ (neither of these implies the other). We cannot pass the non-shading of A on to B, or the shading of B on to A.
Note the dynamic intent these diagrams. As regards the fourth alternative, viz. A is shaded and B is blank, this is by definition impossible – i.e. the very meaning of “If A is X, then B is X” is that ‘A is X’ and ‘B is not X’ are incompatible.
Notice that an Ury representation of implication goes only one way – from ‘A is X’ to ‘B is X’, i.e. from left square to right one (or equally well, from lower to upper, or from closer to further), or vice versa (from ‘B is not X’ to ‘A is not X’). A two-way (mutual) implication would mean that the third combination above (i.e. ‘A is not X’ and ‘B is X’) is also impossible. Of course, too, we can get fancier, and consider implications from X to not-X or vice versa, but there is no need here for us to get into such complications. The important thing is to realize the ingenuity and value of the diagram as a tool of representation and communication.
An Euler diagram cannot perform quite the same feat. We can put a smaller circle A within a larger circle B, and represent the above three alternatives by shading both circles or leaving both blank or leaving A blank and shading B – so that the impossible fourth case is that with A shaded and B blank. But here the meaning of the implication is significantly different – it is: if all (or even some) A are X, then some B are X; and by contraposition, if no B are X, then no A are X. Notice the reference to some of B, instead of to B as a whole. What is new in the Ury diagram is our ability to address B as such. This is very useful in certain forms of discourse.
Note that Ury does not explicitly identify the relationship between the squares as one of implication, as done here. Rather, he describes the square to the right as “more likely” than the one to the left, meaning that there is “more reason” for the law (i.e. predicate X) found in A to be found in B. But this is effectively implication, since we definitely infer (deductively, or eventually inductively) one proposition (B is X) from the other (A is X) on this basis. The defining feature of such implication, to repeat, is that the combination “A is X and B is not X” is being excluded.
Note that the preceding diagrams refer to copulative implications, i.e. those between categorical propositions (‘A is X’ implies that ‘B is X’). Although Ury does not mention this, they could also refer to implicational implications, i.e. those between hypothetical propositions (‘A implies X’ implies that ‘B implies X’). This is said by me in passing, so as to be exhaustive.
Now, the above presentation of Ury’s idea is only the beginning of it for him, though we shall essentially leave it at that. He goes on, and explains how such diagrams can be used to describe the theoretical positions of opponents in a dispute (e.g. rabbis debating some point of law) or the changing theoretical position of an individual (e.g. the Gemara, as it considers different options and possible objections). Usually, the theoretical positions are constituted by not one but two implications, so that the best way to represent them is by means of what Ury calls “two by two Diagrams.” The following is an example of such a more complex diagram:
As Ury explains, there are sixteen possible ways to shade such a diagram, but only six of them are ‘allowed’ (i.e. logically consistent) and ten of them are not, given that the boxes to the left imply those to the right and the lower boxes imply those above them. He formulates the implicit restrictions in what he calls “The Shading Rule.” This rule states: “In a Diagram, if a box is shaded, all boxes above it and to its right are also shaded. If a box is blank, all boxes below it and to its left are also blank.” Later, he introduces three dimensional diagrams, though he also shows how these can be reduced to a set of two dimensional ones.
By means of such two by two diagrams, Ury manages to greatly clarify theoretical debates or developments in the Talmud. Moreover, he shows how the diagrams constitute a new “language,” insofar as, once we know the meaning of particular diagram (i.e. what its boxes and their shadings signify), we can abstract it from its labeling, and use the graphic form to quickly identify a theoretical position and compare it to others with comparable labeling. We can thus, by placing different diagrams side by side see in one glance how various positions differ; or we can show the inconsistency of a position or predict additional positions.
This is obviously a very valuable tool, not only for Talmud study, but for thought in general. However, as we shall see, its value is not unlimited.
The limits of Ury’s diagrammatic tool become evident when we consider its application to a fortiori argument, even though to be sure this tool remains valuable in many contexts.
To begin with, it should be said that nowhere does Ury actually delve deeply into the nature of a fortiori argument. He defines the kal vachomer as “a logical argument that proves a proposition to be true under one set of circumstances based on its being true under a less compelling set of circumstances.” As he sees it, “if a certain Law applies to a certain situation, we are allowed to apply that Law to a different situation where it is even more likely to apply.” For him, “The kal vachomer is valid because it is one of the Thirteen Hermeneutic Principles by which Torah Law is derived, not simply because of ‘logic’.” Notice his putting the term logic in inverted commas. Let us examine these ideas.
First, Ury does not say how we know that one set of circumstances is more “compelling” or more “likely” than another. In some cases, this is obviously known ‘intuitively’; e.g. as one knows that four apples weight more than two of them. In other cases, presumably, we have to assume that the rabbis were handed the information down by oral tradition since the Sinai revelation. But the use of words like “compelling” or “likely” can only in this context be taken as figures of speech. For, according to formal logic, the conclusion of an argument can never literally be more forceful than the premise(s) which give rise to it. By way of a given argument, the conclusion can only be as or less probable than the premise(s), even if it may turn out to be more probable through additional reasons external to the argument. Thus, Ury’s language here is misleading, and shows that he has not reflected on this issue.
Second, he does not tell us by what means the law is passed on from one context to another. In his view, it suffices that the words “kal vachomer” are listed as one of the thirteen rabbinic “principles” of hermeneutics for the question to be answered. The reason why he puts ‘logic’ in inverted commas, is that though he sees intuitively that the argument is somehow logical, he cannot say exactly what it is that makes it seem so. For this reason, he falls back on the traditional argumentum. While Ury defines the kal vachomer as “a logical argument that proves etc.,” the only proof he provides for it is the say-so of the rabbis – it is thanks to them (or ultimately the revelation at Sinai, which they claim to merely pass on) that “we are allowed to apply, etc.” But as I have shown in my Judaic Logic, this “Sinai connection” argumentum does not stand up to closer scrutiny; it constitutes a circular argument.
To say that a certain rule of inference is, albeit not proven by formal logic, valid by virtue of a Divine decree, one would have to first show that there was a decree and that it was indeed of Divine origin. But the Tanakh (the Jewish Bible) contains no such explicit decree for a fortiori argument or any other rabbinic hermeneutic principles. There are many examples of a fortiori discourse in the Tanakh, which can be taken as teachings of logic; but such examples are not express decrees, they are subject to interpretation. Moreover, even if the Tanakh did explicitly decree such principles, since belief in the Tanakh’s Divine origin is a matter of faith and not scientific proof, such content would not constitute proof. Faith is certainly necessary and valuable in religious matters, but it is not as reliable as proof. Likewise, there is no credible proof for the claim that the hermeneutic principles were Divinely-decreed orally, and then handed down by tradition, orally and then in writing, unscathed by temporal events. Our acceptance of this historical claim can only be based on faith, since there is no other way to justify it. Thus, the Sinai connection thesis is reasonably open to doubt.
Reason should never be made subservient to apologetic purposes; its independence, objectivity and integrity should always be defended if we really want to pursue truth. To be absolutely clear: if a certain form of argument, such as a fortiori argument, cannot be validated by logical means, then it is invalid; i.e. it is either a non sequitur (i.e. its conclusion does not follow from its premises) or antinomic (i.e. its conclusion contradicts its premises). There is no way around this principle – it is a law of nature. If an argument form cannot be validated by logic, then there is no way to validate it by other means; and conversely, if an argument form can be validated by logic, then there is no need to validate it by other means, such as written Divine decree or subsequent oral tradition. Reasoning is not an arbitrary matter, but a matter of logic.
Following rabbinic precedent, Ury identifies two forms of kal vachomer (lenient and stringent) argument: the first type argues that a lenient law (kula) that applies to a more stringent case (the chomer) is bound to apply to any more lenient case (the kal); and the second type argues that a stringent law (chumra) that applies to a more lenient case (the kal) is bound to apply to any more stringent case (the chomer). These two arguments may be graphically represented as follows:
Notice the different order of the kal and chomer items, and the different shadings, in the two diagrams. In the first diagram, a certain leniency is passed from the chomer item to the kal item, so both boxes remain blank. In the second diagram, a certain stringency is passed from the kal item to the chomer item, so both boxes are shaded. If we take these diagrams as Ury does as simply statements that the application of the “law” concerned to one item implies its application to the other, they are quite okay. But if we ask how come the law is passed on from one item to the other, we are left completely in the dark. Or if we ask why a leniency (rather than a stringency) is passed from chomer to kal, and a stringency (rather than a leniency) is passed from kal to chomer, again no explanation is forthcoming.
Clearly, while this graphical approach is capable of recording the fact of implication in a useful manner, it is apparently not able to explain or justify the implication. Thus, Ury diagrams are heuristic instruments, rather than hermeneutic ones. They can be used to superficially ‘describe’ a fortiori arguments for us, but they do not tell us just how they function or why they work. They are, as he claims them to be, useful as learning or teaching tools, facilitating conceptualization and memorization; but they are not useful for theoretical exploration of a fortiori argument, at least not in the way they have thus far been presented to us.
It is evident that Ury conceives a fortiori argument as some sort of direct implication. The shading of the box to the left is passed on to the box to the right without an intermediary stage, i.e. without apparent reason. Even more, Ury seems to think that all implication is really a fortiori argument. As he puts it:
“The principle of kal vachomer is built into the fabric of Diagrams. Every time we completed a Diagram, we did so using the principle of kal vachomer, and the Shading Rule itself can be viewed as nothing more than a restatement of the principle of kal vachomer.”
In this conception of all implication as essentially a fortiori argument he does not diverge from the traditional view – that is precisely why the rabbis of the Talmud regard a fortiori argument as the very essence of logic, and why such argument is so frequently used in Jewish discourse. But this is untenable from the perspective of formal logic.
If we examine actual a fortiori discourse, even arguments formulated in the Tanakh, in the Talmud and other rabbinic literature, it is evident that it does not consist merely of a minor premise and conclusion. There is also, of necessity, a major premise to take into consideration, which contains a middle term. Furthermore, the minor premise is necessarily more complex than it appears at first sight: it informs us of a threshold value of the middle term as of which the predication is (or is not) feasible. It is precisely this extra information, which is absent in Ury diagrams, that makes possible the putative conclusion. Even if the information is not always explicitly given, it is always implicitly assumed; otherwise, the argument would simply not be convincing.
The reason why Ury cannot validate a fortiori argument is that he does not take the trouble to look for all its implicit components, i.e. to first formalize it. We could say that his diagrammatic approach imprisons him in a simple conception that prevents him from seeing the logical nature of a fortiori argument, and moreover from seeing the other types of arguments that may underlie an implication. If he had pondered the obvious question, regarding the two types of kal vachomer argument, why a leniency is passed from the chomer to the kal, whereas a stringency is passed from the kal to the chomer, he might have begun to go deeper into the logic involved.
Clearly, these drawings are intended to represent a fortiori argument from minor to major and from major to minor, respectively. Stringencies and leniencies are not “laws” in the same sense. By “stringency” is meant some definite imperative or prohibition, whereas by “leniency” is meant a corresponding exemption (i.e. non-imperative) or permission (i.e. non-prohibition). Thus, strictly speaking, the two terms do not refer to different kinds of law, but respectively to the presence or absence of a law. That is, a kula is a non-chumra. If we put the rabbinic formulations in standard a fortiori form it becomes much clearer how such arguments proceed.
“Type 2”: The chomer (P) is more serious (R) than the kal (Q);
and Q is serious enough to be subject to a certain stringency (S);
therefore, P is serious enough to be subject to S.
“Type 1”: The chomer (P) is more serious (R) than the kal (Q);
and P is serious not enough to be subject to a certain stringency (S);
therefore, Q is serious not enough to be subject to S.
Note that I have placed Ury’s type 2 before his type 1, because they are respectively positive subjectal and negative subjectal in form. Ury placed them in the reverse order because he did not realize that the type 2 argument is really negative in form, and therefore a derivative of the positive type 1 argument by means of reductio ad absurdum validation. So, to him, either order was okay, and he happened to choose the said order.
Note also my labeling of the four terms involved as P. Q, R and S, in accord with my standard practice. The chomer (P) is the major term and the kal (Q) is the minor term. The implicit more-less relationship (quantitative comparison) between these two terms has to be expressed in a separate proposition, called the major premise. Furthermore, that proposition must contain a middle term (R), which tells us in respect of what precisely P is more than Q. Although the middle term is not explicitly given here, it must still be provided – I use the vague term “serious” here, because no more specific term is suggested to us.
The subsidiary term (S) is explicitly given in type 2 as “a certain stringency;” this intends a certain legal imperative or prohibition. Although the subsidiary term (S) explicitly given in type 1 is “a certain leniency,” meaning a certain exemption or permission, the real subsidiary is still “a certain stringency” because this argument cannot be positive, since it is subjectal and a positive subjectal cannot validly proceed from major to minor as this argument does. This issue of polarity presents no great problem, since we can obvert the conclusion of type 1 to “Q is not-S,” and not-imperative means exempted and not-prohibited means permitted. Nevertheless, to repeat, acknowledging the underlying negative form of the argument is essential to its validation.
The middle term (R) must also appear in the given proposition, called the minor premise, that “Q is subject to stringency S” (in type 2) or “P is not subject to stringency S” (in type 1), as the case may be. For a certain value of the middle term (what is “R enough to be” S) serves as the sine qua non condition for a subject’s access to the subsidiary term (S). It is only through our knowing this condition from the minor premise that we are able to conclude that the remaining term (P or Q, as the case may be) is also subject or not-subject (as the case may be) to the subsidiary term (S). Thus, even if the middle term is not mentioned or even hinted at anywhere in the argument, it has to be assumed present for any conclusion to be credibly drawn.
Needless to say, an Ury diagram does not comprise these various details and conditions concerning a fortiori argument. It is not a very subtle tool. All it tells us is: in type 2: “if the kal is subject to a certain stringency, then the chomer is so too;” and in type 1: “if the kal is not subject to a certain stringency, then the chomer is also not subject to it.” There is no inherent explanation or justification of these processes. Furthermore, we are not told why we could not equally well say: “if the kal is subject to a certain leniency, then the chomer is so too;” and “if the kal is not subject to a certain leniency, then the chomer is also not.”
The only way to understand why the former two moods are valid and the latter two are not is to refer to the middle term (R), which (as already stated) underlies both the major and minor premises and the conclusion. Rationally, the more “serious” (R) a context is, the more stringent the law concerning it is likely to be; and conversely, the less “serious” (R) a context is, the more lenient the law concerning it is likely to be. This is not a formal matter, but an expression of the principle of justice or of ‘measure for measure’. In general, there might be inverse proportionality between the middle term and the subsidiary term; but in contexts of morality and law, such inversion would be contrary to reason. Since Ury’s diagrams do not formally make room for the middle term, they cannot explain why two moods are valid and two are invalid.
In fact, we could integrate the middle term into his diagrams, as follows. Instead of saying, for instance (in the case of type 2): “if the kal is subject to a certain stringency, then the chomer is so too,” we would say: “if the less serious context is serious enough to be subject to a certain stringency, then the more serious context is so too.” That is, we would label the boxes as “the less serious context” or “the more serious context,” i.e. in such a way that the intended middle term is clearly designated in them; and moreover we would label their shading as “serious enough to be subject to a certain stringency,” i.e. in such a way that the necessity of a sufficient quantity of the middle term for access to the subsidiary term is clearly acknowledged. Thus, we can by such semantic artifice improve the accuracy of an Ury diagram, even though the diagram per se is technically simplistic.
Another weakness of Ury diagrams that we must consider is that they apparently refer only to subjectal argument, and make no mention of predicatal argument. Each box represents a subject and its shading (or lack of it) represents a predicate, and the inference consists of transfer of shading (i.e. predicate) from one box (subject) to another. Yet in real discourse – including Biblical, Talmudic and other rabbinic discourse – argument often proceeds in the opposite direction, by change of predicate for one and the same subject. In pictorial terms, the latter would mean that the same single box, say A, lacks and then has a certain shading (signifying X), or vice versa; or has a certain shading (X) at one time and another shading (Y) at another time:
To try and resolve this problem, i.e. how to express predicatal arguments through Ury diagrams, let us look more closely at the arguments we wish to put in diagrammatic form. They are, mirroring those proposed above by Ury, the following:
‘Type 4’: More seriousness (R) is required to be subject to the chumra (P) than to the kula (Q);
and a certain context (S) is serious enough to be subject to P;
therefore, S is serious enough to be subject to Q.
‘Type 3’: More seriousness (R) is required to be subject to the chumra (P) than to the kula (Q);
and a certain context (S) is not serious enough to be subject to Q;
therefore, S is not serious enough to be subject to P.
These two arguments are respectively positive and negative predicatal. Notice that here the subsidiary term (S) is the subject of the minor premises and conclusions and refers to “a certain context,” whereas the major and minor terms (P and Q) are now predicates and refer respectively to a chumra (stringency) and a kula (leniency); as we have seen, a kula is a non-chumra. The middle term (R) is the same as before (“seriousness”), but it now serves as the precondition for S to be (or not be) P or Q, as the case may be. Notice that the positive form goes from major to minor and the negative form goes from minor to major. I have numbered the arguments as 4 and 3 in accord with Ury’s preference (even though the opposite order would be logically more accurate).
If we tried to illustrate these two arguments by means of Ury diagrams in the usual way, we would have to call both boxes S, and make the shading change from box one to the other. This would be from chumra shade to kula non-shade in type 4 argument, and from kula non-shade to chumra shade in type 3 argument. But clearly, this diagram would not be in accord with Ury’s diagrammatic scheme and his Shading Rule. It follows that Ury diagrams are not able to represent predicatal a fortiori argument as such. Therefore, their power of representation is essentially limited to subjectal a fortiori argument.
We could however try and rephrase the minor premises and conclusions of the latter two a fortiori arguments as follows. For type 4: “If the chumra (P) is applicable to a certain context (S) – because S is serious (R) enough to be subject to it – then the kula (Q) is applicable to that context (S).” And for type 3: “If the kula (Q) is not applicable to a certain context (S) – because S is not serious (R) enough to be subject to it – then the chumra (P) is not applicable to that context (S).” This is leaving the major premise unchanged. In this way P and Q are made to appear as subjects and S as (part of) the predicate. The explanations – that S is or is not serious enough to be P or Q, as the case may be – are then effectively left out of the arguments, note well. In that event, we could propose the following Ury diagrams to illustrate them:
In these diagrams, note well, the shading refers to the positive predicate “applicable to context S” and the lack of shading to the negative predicate “not applicable to context S.” Although the arguments are really predicatal, they are made to seem subjectal in order to be fitted willy-nilly into Ury diagrams. Notice how these two arguments (types 3 and 4) differ from the previous two (types 1 and 2). Here, each argument is concerned with legal rulings of different severities relative to one and the same case, whereas previously each argument was concerned with legal rulings of one and the same severity relative to different cases.
As I have demonstrated formally and in detail in an earlier chapter (3.5) such ‘traductions’ (as I called processes of this sort) are artificial and somewhat misleading. In fact, a predicatal argument cannot be recast in subjectal form (nor, incidentally, can a subjectal argument be recast in predicatal form). The reason for that is that each argument form lacks some of the information needed to construct the other argument form. Although we can superficially, by verbal manipulation, make a predicatal argument look subjectal (or vice versa), in fact its underlying rationale, which determines its structure, remains unaffected.
Nevertheless, Ury diagrams can, as just explained, be used in practice to visually represent predicatal arguments, because Ury diagrams are anyway rough tools, which cannot display the fine distinctions needed to distinguish differently structured arguments. In conclusion, although Ury has not addressed the question of diagrammatic representation of predicatal a fortiori argument in his book, we have here done it for him (albeit with appropriate warnings), and thus extended the utility of Ury diagrams.
All the above concerns copulative a fortiori arguments, but the same could obviously (possibly with appropriate adjustments) be repeated with reference to implicational a fortiori arguments.
We saw in the previous section that there are definite limits to the power of representation of Ury diagrams when it comes a fortiori argument, because such argument is in fact not as simple and straightforward as it is commonly depicted to be. In the present section, we shall reinforce this observation with reference to a crescendo argument and the dayo principle. In his book, Ury gives no hint of being at all aware of the issue of ‘proportionality’, i.e. of the difference between purely a fortiori argument and a crescendo argument. Consequently, it is no surprise to see that he does not deal with the dayo principle, either.
Ury warns that “The kal vachomer also has certain special rules and limitations”, but adds that he “will not deal with all the aspects” of the argument (“in this chapter” he says, but in fact he means in the whole book). He views it as “compelling… unless refuted,” and gives as example of a refutable argument: “If it is healthy to consume a certain amount of a nutrient then is it certainly healthy to take a larger amount?” However, though this reflects the rabbinic view, it is not an accurate statement. When, as in this example, the argument does not work (in this particular case due to an erroneous major premise), it is not due to a weakness in a fortiori argument as such, but due to the proposed particular a fortiori argument being not well formed. It is not the argument form that makes the example less compelling, but the fact that the example fails to conform to the argument form.
Ury mentions the “concept of dayo” only once, in passing, in a footnote, even though he there acknowledges it to be an “important concept.” The reason for Ury’s silence on proportionality and the related dayo (sufficiency) principle is, I suggest, that they are not easy to fit into his diagrammatic scheme. The reason is not that these matters are unknown to rabbinic reasoning – indeed, they are very present in it. We find in the Mishna (Baba Qama 2:5) R. Tarfon reasoning a crescendo from half payment for certain damages on public grounds to full payment for similar damages on private property, and we find there his colleagues, the Sages, rejecting his conclusion by saying that “It is enough (dayo)” to conclude with the same penalty (i.e. half payment).
Then in the Gemara commentary on this issue (Baba Qama 25a), a baraita is cited, according to which the dayo principle is based on the example of Numbers 12:14-15, in which the prophetess Miriam is sentenced to only seven days punishment for a sin which, the baraita claims, deserves fourteen days. The Torah passage cited makes no mention of fourteen days, and could easily be read as purely a fortiori argument. Nevertheless, the Gemara insists on reading it in accord with the said baraita, and thus seemingly adopts the viewpoint that all a fortiori argument is essentially a crescendo argument, and therefore that the dayo principle is always needed to limit the conclusion to a non-proportional magnitude.
Never mind that many a fortiori arguments within the Tanakh, the Mishna, and even the Gemara, not to mention other rabbinical discourse, are clearly non-proportional – and indeed, some cannot be interpreted in any other way than as non-proportional – the Gemara’s clear position in this sugya of the Babylonian Talmud, through which it explains the dispute between R. Tarfon and the Sages in the Mishna, is that a fortiori argument is essentially proportional. The Gemara’s explanation is in fact deficient, since it only addresses one of the two arguments reported in the Mishna, as I show in an earlier chapter (7.5).
Be that as it may, it is evident from all this that it is impossible to claim to be able to visually represent Judaic (Biblical, Mishnaic, Talmudic or later rabbinic) a fortiori argument if one has not duly considered and assimilated the complications of ‘proportionality’ and dayo. Yet Ury does just that, blithely glossing over these crucial issues. He must have studied the Mishna and Gemara in Baba Qama 25a-b regarding these topics. Possibly he considered them, but found them intractable; but if so, he should have said that. Let us therefore look and see if we can apply his diagrammatic scheme to these issues.
The two a fortiori argument forms Ury deals with pictorially in his book are purely a fortiori positive and negative subjectal arguments. That is to say, the minor premise and conclusion in them have the exact same subsidiary term. The argument, as he sees it, consists in passing a given law from one context to another. If the ‘law’ is a leniency (i.e. it grants an exemption or permission) applicable to a chomer (more serious) case, it can be passed on to a kal (less serious) case; and if the ‘law’ is a stringency (i.e. it makes something imperative or prohibited) applicable to a kal case, it can be passed on to a chomer case.
In both these arguments, note well, the ‘law’ remains unchanged in the transfer from one context to the next. Pictorially, in the corresponding Ury diagrams, this means that the shading or non-shading of one box is carried over unchanged to the next box, as appropriate (i.e. as specified in the Shading Rule). This is purely a fortiori argument – it involves no ‘proportionality,’ and therefore no dayo restriction is applicable to it, since dayo can only be applied where a ‘proportional’ interpretation is attempted.
A crescendo argument differs from purely a fortiori argument in having a ‘proportional’ conclusion. That is to say, if the minor premise tells us that predicate S is applicable to the major or minor term, then the conclusion’s predicate will have a magnitude smaller or greater than S (i.e. S– or S+), as the case may be, for the remaining term (the minor or major, as the case may be), in proportion to the different magnitudes of the major and minor terms (or more precisely, to the amounts of the middle term that they respectively have). This information, which differentiates a crescendo argument from purely a fortiori argument, is usually left tacit, but is logically essential to draw a ‘proportional’ conclusion, note well.
The dayo principle is applied in some though not all cases of a crescendo argument (specifically, in cases where a penalty for a greater crime is being derived from the penalty for a lesser crime given in the Torah). Its application consists in saying ‘no’ to the putative proportional conclusion (i.e. it demands that the concluding penalty remains quantitatively the same as the textually given penalty from which it is derived). This is not a rule of logic, but a rabbinic moral rule (or ultimately a Divinely-decreed one, according to the rabbis). Our sense of justice, or the moral principle of ‘measure for measure’ (midah keneged midah) we derive from it, would have us infer a proportional penalty; but the rabbis (rightly, I think) prefer to take no chances and avoid all possible errors of human judgment (which could occasionally result in unjust punishments) by interdicting such inference where applicable.
While, as we have seen, purely a fortiori argument can be expressed (briefly put, leaving aside important details) as: “If A is X, then B is X,” the a crescendo form of such argument has the form: “If A is X, then B is Y, where Y is greater than X” (again, this is putting it briefly, leaving aside the middle term and the premise about proportionality). This means that the visual representation of a crescendo argument using Ury’s diagrams would require that the shading within a box to the right of (or above or behind) another box be different. This would be contrary to his Shading Rule.
Note the arrow and the difference in shading in the above diagram. Of course, in cases where the dayo principle is applied, the shading would again be made the same, and the Shading Rule would be obeyed. Nevertheless, there are cases where a crescendo argument is used, and the dayo principle is not appropriate or merely not applied, which therefore continue to break the Shading Rule. Thus, Ury’s scheme, as it appears in his book, is deficient in not having taken into account such a crescendo arguments, which are far from uncommon in Judaism or elsewhere.
This deficiency can possibly be remedied by changing the conventions involved in Ury diagrams. We could reformulate the Shading Rule, saying that in certain cases the shading might vary and be different on the left and right (or below and above, or in front and behind, depending on the orientation of the diagram), being perhaps made more or less intense in proportion to the magnitudes of the predicates in the minor premise and conclusion. Of course, this greatly diminishes the visual impact of Ury diagrams, since whether its conventions have been fully obeyed or not cannot be known merely by glancing at the diagrams.
Indeed, this reminds me that there are examples of a fortiori argument where we argue from a negative to a positive or a positive to a negative. I have discussed such arguments in detail in an earlier chapter, under the heading of a crescendo argument with ‘antithetical subsidiary terms’ (3.4); and I there found them to be usually invalid but in certain cases valid. In valid such instances, the Ury diagram might thus have a shading on the left which is not passed on to the right, or a blank on the right which is not passed on to the left – in direct contravention of the Shading Rule. This is explicable with reference to ‘proportionality’ (i.e. the blank and the shading are in such cases effectively two degrees of shading).
Thus, to conclude our critique of Ury diagrams, it appears that, though they function well in simpler cases of a fortiori argument, there are not infrequently cases of a fortiori argument that are considerably more complex and which therefore cannot be adequately represented by means of such diagrams and in accord with the Shading Rule regulating their use. Furthermore, remember, while Ury diagrams do visually represent some of the information contained in ordinary a fortiori arguments, they do not visually represent all the information in them – so even with respect to simpler arguments (which can easily be fit into Ury diagrams and do obey the Shading rule) there are relevant aspects that remain invisible and therefore susceptible to misunderstanding or error.
Thus, in the final analysis these diagrams may not be as generally useful as initially thought. They are doubtless of use in many contexts, but are better avoided in certain contexts.
Actually, Yisrael Ury does mention and partly discuss Baba Qama 25a, but he does so only incidentally while dealing with another sugya. He does not deal with the issues it involves relating to ‘proportionality’ and dayo, but focuses principally on an argument proposed by Tosafot through the expression: “kol zeh achnis bakal vachomer,” which Ury translates as: “I will fold all this into the kal vachomer.”
I have formally analyzed the passage of Tosafot concerned, thanks in part to the explanation provided by Ury of the Ri’s objection an a fortiori argument by R. Tarfon, and placed my analysis in an earlier chapter of the present volume (9.7), where it rightly belongs. I refer readers of the present chapter to that preliminary analysis, because its study greatly facilitates the present discussion. Here, what interests us primarily is to see how Ury diagrammatically expresses this Tosafot commentary, which he cites in full in both Hebrew and English.
The Tosafot commentary consists of three distinct stages. In the first stage (i), an argument by R. Tarfon is paraphrased (the argument is attributed to the Gemara, but actually it comes from the Mishna); this is his first argument: his second is not mentioned. In the second stage (ii), an objection by the Ri to the preceding is introduced, and then the Ri’s own reply to such objection. In a third stage (iii), we are taught the kol zeh achnis argument proposed by the unnamed Tosafist, the author of the commentary, which argument is supposed to conflate the components of the previous two stages.
(i) Ury’s diagram for the argument by R. Tarfon (fig. 6.15a) is a two by two square similar to the following:
As can be seen, the horizontal distinction is that between tooth & foot damage and horn damage, and these are placed with the kal (more lenient) item on the left and the chomer (more stringent) item on the right. The vertical distinction is that between damage occurring on public grounds and that on private property, and these are placed with the kal (more lenient) item below and the chomer (more stringent) item above. Two shadings are used here, single hash for half payment and double hash for full payment. No shading (i.e. blank) signifies that no payment is required. We are given three boxes (A, B and C), and asked to infer the fourth, namely the box in the top right hand corner (D). The conclusion proposed by R. Tarfon is that box D should have double hash shading, i.e. that horn damage on private property entails full payment. Thus, the shading in box C is passed on to box D, in accord with Ury’s Shading Rule.
However, as far as I can see, the Shading Rule would equally well allow us to pass on the single hash shading in box B to box D, and thus conclude that horn damage on private property entails half payment. Ury does not make clear why we prefer to inferring D from C (as R. Tarfon indeed does) to inferring D from B (as the Sages do when they say “dayo—it is enough,” although Ury does not mention them, and indeed the Tosafot passage at hand does not mention them). Is there some unstated additional clause to the Shading Rule that justifies such preference? He does not say. Clearly, he is content here to illustrate; he makes no effort to explain the a fortiori reasoning involved.
Actually, the argument by R. Tarfon that Ury is here illustrating is not his first, but his second. This is evident, since the arrow of inference in Ury’s diagram goes horizontally across from box C to box D, i.e. from the full payment required in the case of damage by tooth & foot on private property (the minor premise) to full payment for damage by horn on private property (the conclusion). See for yourself:
Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].
Tooth & foot damage (Q) implies legal liability (R) enough to necessitate full payment for damage on private property (S).
Therefore, horn damage (P) implies legal liability (R) enough to necessitate full payment for damage on private property (S).
The major premise of this argument is based on comparison of the legal liabilities for damage in the public domain by tooth & foot and horn, respectively. It is this premise, obtained by generalization, which makes the conclusion follow from the minor premise. The major premise is represented in Ury’s diagram by the lower two boxes, A and B. The argument as a whole is purely a fortiori. Thus, Ury does not illustrate the argument actually cited by Tosafot, i.e. R. Tarfon’s first, but R. Tarfon’s second argument. R. Tarfon’s first argument may be stated formally as follows:
Private property damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].
Public domain damage (Q) implies legal liability (Rq) enough to necessitate half payment for damage by horn (Sq).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, private property damage (P) implies legal liability (Rp) enough to necessitate full payment for damage by horn (Sp = more than Sq).
The Ury diagram for this argument could be the same as the one above, but here the arrow of inference would go vertically up from box B to box D, i.e. we would be inferring full payment for horn damage on private property (double hash shading) from half payment for horn damage on public grounds (single hash shading), by analogy to the transition from no payment for damage by tooth & foot on public grounds to full payment for damage by tooth & foot on private property. And this would, of course, constitute an a crescendo argument, i.e. a proportional a fortiori argument.
However, such representation would clearly be contrary to the Shading Rule formulated by Ury for his diagrams! Thus, the Shading Rule would have to be modified to assimilate a crescendo argument in Ury diagrams, as already pointed out. As things stand, Ury diagrams cannot handle a crescendo argument. So, it is no surprise that Ury opted unconsciously for R. Tarfon’s second argument which is purely a fortiori, i.e. non-proportional. But it should be said that the more relevant argument in the present discussion, as we shall presently see when we deal with the next two stages of it, is in fact R. Tarfon’s second. So Ury was thinking straight when he opted for the latter. It is Tosafot, not Ury, who is to blame for evoking R. Tarfon’s first argument.
(ii) The objection of the Ri (another Tosafist, called R. Isaac, whence Ri) is, briefly put, that the major premise of R. Tarfon’s second argument could well be reversed, i.e. it may argued be that Horn damage implies less, instead of more, legal liability than tooth & foot damage. This alternative hierarchy of liability would be based on the idea that tooth & foot damage being rather common, the ox’s owner should be especially vigilant to prevent it; in comparison, horn damage being rather uncommon, the ox’s owner is not expected to be so careful. Given this new major premise, the objection goes, it is not logically possible to infer (as R. Tarfon proposed) full payment for damage by horn on private property from the full payment required in the case of damage by tooth & foot on private property. This objection (pircha, in Aramaic) is illustrated by Ury (Fig. 6.15b) roughly as follows:
Here, horn is placed to the left of tooth & foot, so that (by the Shading Rule) we cannot transfer the double hash shading (i.e. full payment) in the latter to the former (i.e. from box C to box D).
Moreover, the private domain is placed by Ury below the public domain, so as to prevent transfer of the single hash shading for horn on public grounds to horn on private property (i.e. from B to D). The latter move is surely an overreaction by Ury, since it is not mentioned in the Ri’s objection. Perhaps that is why Ury explains it in a footnote, rather than in his main text. His idea seems to be that if the columns were switched but the rows were left in their original order, we would be tempted (given the Shading Rule) to infer half payment for horn damage on private property from the half payment for horn damage on public grounds. But I do not see why he would want to preempt such inference, since it would be quite compatible with the Ri’s objection and not legally problematic (since, in fact, the Sages end up with that conclusion through their dayo objection). Certainly, the payment for horn damage on private property cannot be zero (less than on public grounds), but must be at least half.
Furthermore, in the configuration thus proposed by Ury, we might be tempted (given the Shading Rule) to infer from the half payment for horn damage on public grounds, a like half payment for tooth & foot damage on public grounds (i.e. to pass single hash shading horizontally from B to A); or alternatively, to infer from the full payment for tooth & foot damage on private property, a like full payment for tooth & foot damage on public grounds (i.e. to pass double hash shading vertically from C to A) – whereas the Torah specifies exemption from payment for this case (i.e. no shading in box A is allowed)! Ury does not address this issue, but simply leaves box A blank.
In any event, Ury surprisingly does not at all mention the Ri’s own reply to the objection. The Ri’s retort is that if the reversed major premise that tooth & foot damage implies more legal liability than horn damage were adopted as the objection advocates, then we could infer from the half payment for horn damage on public grounds a like half payment for tooth & foot damage on public grounds (i.e. we could pass single hash shading from B to A) – which, as already said, would be contrary to the Torah specification of zero payment in such case. This retort by the Ri must, of course, be taken into consideration in our diagram. But since it refers to a horizontal inference, it does not matter to it whether we place the ‘public’ row above the ‘private’ row, as Ury has it, or vice versa.
Thus, in conclusion, I would say that Ury erred in this matter, out of inattention (i.e. he did not think his diagram through far enough). He should have left the row B-A below the row D-C, and only switched the two columns around, as in our diagram above. By switching the two rows around also, unnecessarily, Ury created an additional problem (namely, the possibility of passing double hash shading vertically from C to A). Our drawing is preferable, because it avoids this added problem. Moreover, it excludes the possibility of zero payment for damage by horn on private property, which Ury thoughtlessly allows. Ury also made a mistake in not mentioning the Ri’s own reply to the objection, which is a crucial part of the whole commentary. It looks like he did not realize its exact significance.
The Ri’s retort to the objection is in fact tacitly included in Ury’s diagram for the objection, in that in the row labeled ‘public’ he has left the box on the right (for tooth & foot, here called A) blank, even though the box on the left (for horn, here called B) is shaded – whereas, according to the Shading Rule, given this configuration, the boxes should be either both shaded or both blank. But Ury does not point this out in so many words. He does, admittedly, write:
“Even though shein and regel damages are commonplace, and therefore the owner should be liable, nevertheless the owner is exempt in the public domain, proving that it is very difficult to make the owner liable for shein and regel damages.”
However, this statement is not made as a presentation of the Ri’s explicit retort to the objection. Moreover, Ury here shows that he misunderstands the Ri’s objection. It is not that it is “very difficult” to blame the ox’s owner for tooth & foot damage – it is that it is impossible to do so, since doing so would lead through a fortiori argument to a conclusion of half payment for such damage in the public domain, which would contradict the Torah given of zero compensation. Thus, I would say, Ury does not duly take stock of and acknowledge the Ri’s full intervention; he read the objection, but skimmed over the retort to it.
(iii) Instead, he jumps fast forward to the kol zeh achnis argument proposed by the unnamed Tosafot commentator. He presents this pictorially (Fig. 6.15c) roughly as follows:
The column placed most to the left represents the kol zeh achnis argument, while the two columns to the right of it are the original two by two diagram. Ury refers to the new column as “reality,” and to the middle column as the “original conception.” Notice that the arrow in the top row goes across all three boxes in it. The message of this diagram is that the kol zeh achnis argument reinforces the initial argument, by rendering it immune to the objection raised by the Ri. However, this diagram far from clarifies just what Tosafot’s new argument is! This is clear from the fact that the two columns in it labeled “tooth & foot” are visually identical, even though the argument in fact changes. We see from this the weakness of Ury’s diagrammatic method, at least with regard to a fortiori argument. As already pointed out, we cannot by this method make fine comparisons and contrasts between arguments, and discern precisely where they agree and differ.
In fact, as I show in my own analysis, the kol zeh achnis argument proposed by Tosafot differs from the original argument by R. Tarfon in an important respect, namely in the middle term involved. The argument by the Tosafist can be stated formally as follows:
Tooth & foot damage (P) is more common (R) than horn damage (Q) [since the former is common and the latter not so].
Yet, tooth & foot damage (P) is common (R) not enough make the ox’s owner exempt from full payment for damages on private property (S) [since tooth & foot damage on private property necessitates full payment].
Therefore, horn damage (Q) is common (R) not enough to make the ox’s owner exempt from full payment for damages on private property (S) [whence, horn damage on private property does necessitate full payment].
From this formulation, we can see clearly what purposes the argument serves and what novelties it contains. By using the middle term “common,” Tosafot integrates both insights of the Ri, the possible objection regarding frequency of occurrence and the retort to it that this would lead to absurdity, since he avoids inferring the middle term “legal liability” from such frequency. Moreover, so doing, Tosafot is able to arrive at the same conclusion as R. Tarfon, namely that damage by horn on private property calls for full compensation. In this sense, Tosafot manages to allude to the whole discussion in one argument.
However, though this kol zeh achnis argument is consistent with the earlier argument by R. Tarfon, it is no substitute for it. It cannot either be said that the Tosafot argument is needed, as Ury seems to think by placing it far to the left, to buttress R. Tarfon’s case, since in fact the doubt sown by the Ri’s objection is dissolved by the Ri’s own reply to the objection. Once this dissolution occurs, any further mention of ‘frequency of occurrence’ (i.e. of what is common or uncommon) becomes redundant, because it in fact has no relevant consequence on the issue of legal liability. This means that the Tosafot argument is an embellishment, but not a very useful one. To fully understand its utility, we must still evoke the Ri’s objection and his own retort to it. If anything is important in this context it is the Ri’s objection and his own retort to it.
It is evident from Ury’s characterization of the Tosafot argument as “reality” and as able to “prove forcefully,” in comparison with R. Tarfon’s argument, that Ury takes the words “all this I will fold into the kal vachomer” quite literally, i.e. as signifying that the Tosafot argument contains “all” the thoughts preceding it. But these words are an exaggeration – the kol zeh achnis argument is in fact qualitatively inferior to R. Tarfon’s original argument because it does not have (not even implicitly) the latter’s reference to “legal liability.” Clearly, Ury allows the prestige of Tosafot to affect his judgment here. He allows himself no critical thoughts.
Moreover, according to Ury’s explanation, Tosafot considered that “when R’ Tarfon stated his kal vachomer, he was well aware of the presence of a potential pircha, but already had it included into his kal vachomer using the principle of kol zeh achnis!” But this seems to be just a projection of the thoughts of later commentators onto earlier protagonists. Such retroactive rationalizations are very common in traditional readings, giving the impression that there are ultimately no innovations in Torah discourse. But to my mind, they are dishonest (if their historicity is not established) and unnecessary. Certainly, R. Tarfon need not have thought of the said complications to formulate his own arguments; they stand quite well on their own two feet.
Note that Ury gives no evidence from the Mishna that R. Tarfon was ex ante facto “well aware” of the said problem and the said solution to it. He does not specify just what formal logical relations he is referring to when he says that R. Tarfon’s Mishna argument “included” the pircha “using” the kol zeh achnis; these are overly vague expressions. He gives no evidence that the kol zeh achnis form of argument was ever used before the Tosafist’s time (about 12th century CE, well over a millennium after R. Tarfon’s time). Note also that he seems to think that R. Tarfon formulated only one argument (“his kal vachomer”), whereas in fact the latter formulated two quite distinct arguments (though they came to the same conclusion).
Ury’s approach reflects a common fault in traditional hermeneutics – namely, anachronism. Reading his presentation, one does not see a clear distinction between the contributions of R. Tarfon, the Ri and the more recent Tosafist. They are treated synchronically instead of diachronically. Ury seems to think that R. Tarfon already anticipated and took into consideration both the Ri’s and the later Tosafist’s contributions. But, to repeat, there is no textual evidence to that effect, and no logical need to assume such foreknowledge on R. Tarfon’s part. There is only the need to perpetuate the ‘orthodox’ fantasy that earlier rabbis were well-nigh omniscient, so that later rabbis are only involved in making explicit information that is timelessly already there implicitly.
Furthermore, as I explain in my earlier detailed analysis of Tosafot’s argument, this argument is much more radically in conflict with the Sages’ dayo principle than R. Tarfon’s two arguments were, because unlike them it does not resort to information concerning the ox owner’s half legal liability following horn damage on public grounds to draw a conclusion on his full liability following such damage on private property. Tosafot’s argument is thus unaffected by dayo objection, and could be used to challenge the dayo principle. Ury – like Tosafot and subsequent rabbis (as far as I know) – does not point this out, nor show how Tosafot’s argument might be scrapped and the dayo principle might be saved.
In conclusion, although Ury’s diagrams can no doubt be very useful in many contexts, they can only roughly depict the state of affairs in certain contexts, and not at all do so in still other contexts. It is noteworthy that he does not attempt to analyze in detail how he would pictorially handle other hermeneutic rules, besides qal vachomer (R. Ishmael’s #1). What of gezerah shavah (#2) and the other contextual rules (#12), binyan av (#3), the inclusion and exclusion rules (#s 4-7), and the rules of harmonization (#s 8-11, 13)? How would he distinguish these various discursive techniques, and others still, from each other in his pictorial approach? He can only deal with ‘implication’ in a general way, without distinguishing its apparent varieties. Clearly, Ury’s diagrams constitute too rough an instrument to reflect the complexities of more intricate forms of reasoning.
Logical precision is only possible through formal verbal discourse. Every component of every argument must be brought out into the open and credibly shown to be relevant, truthful and valid. Diagrams of various types may be helpful, but they can never suffice.
 Jerusalem: Mosaica, 2012.
 On pp. 107-111.
 See chapter 12.1 there.
 Simply put, although the sentence “X is applicable to A” has X in the position of subject and A in that of predicate, if we consider the meaning of the relationship “applicable to” it is clear that X is still in fact the predicate and A the subject.
 A certain quantity x of this nutrient may have beneficial effects on the organism, while larger quantity y (an overdose) of it may be detrimental, and for that matter a lesser quantity z (too small a dose) of it may be useless. We cannot infer from the fact that x is an amount sufficiently beneficial to cause health that y is also so, because we lack the needed major premise that y is more beneficial (as well as a larger quantity) than x. This example is mentioned by Ury on p. 95; but he does not attempt to understand why it does not work.
 Ury also, in the same footnote, mentions in passing the concept of tzad hashaveh. This rabbinic term refers to the common feature between two items, which makes possible an analogical inference from one to the other. This concept is in fact more relevant to binyan av argument than to kal vachomer. See the section on analogical argument in an earlier chapter of the present work (5.1).
 A possible illustration given there is 2 Samuel 12:18. “Behold: while the child was yet alive, we spoke unto him, and he hearkened not unto our voice; then how shall we tell him that the child is dead, so that he do himself some harm?” The relevant antithesis here is between David ‘being distracted’ and ‘harming himself’.
 See pp. 113-118 in his book.
 Of course, religious debate can – just like philosophical debate – be considered as essentially transcending time. I am not denying that. Nevertheless, it cannot be taken for granted without evidence that earlier protagonists actually anticipated, took into consideration and then dismissed objections raised by later protagonists.