A FORTIORI LOGIC
CHAPTER 27 – Andrew Schumann
Andrew Schumann has edited two collections of essays by various authors, Logic in Religious Discourse and Judaic Logic (both published in 2010), to which I was invited to contribute articles. Though I am grateful to him for this friendly gesture, it does not exempt him from honest criticism on my part for the serious faults in his approach to logic in general and his alleged theory of a fortiori argument in particular. I regret having to do it, but I must. If an engineer discovers dangerous deficiencies in a bridge designed by a fellow engineer, or alleged engineer, he is duty bound to blow the whistle. The same is true regarding logicians. It is not a personal matter, but a professional responsibility.
Let us examine and evaluate how Andrew Schumann understands the a fortiori argument. In the Introduction to the volume called Judaic Logic that he has edited (not to be confused with my much earlier work of the same name), he presents various ideas regarding Judaic logic and its relation to general logic to which we shall return further on. With regard to the a fortiori argument specifically, he begins (p. 7) by analyzing an example found in the Talmudic tractate Baba Qama.
The rabbis there (on p. 2b) distinguish three types of damages/nezeqin that an ox might cause, namely: by horn/qeren (i.e. goring), by tooth/shen (i.e. eating) or by foot/regel (i.e. trampling). They derive from the Torah (Ex. 22:4) that damages caused by foot or by tooth on public property result in no financial liability for the ox’s owner, while for such damages on private property there is full liability. With regard to damages caused by horn, the Torah (Ex. 21:35) suggests that there is half liability on public property; while nothing is said about such damages on private property. The question the rabbis then ask (on pp. 24b-25a) is: what of the latter? A debate takes place in the Mishna between R. Tarfon and some unnamed Sages, with the former advocating full liability while the latter advocate half liability. The latters’ answer, which appears to be the authoritative one, is considered as obtained by them through a qal vachomer (i.e. a fortiori) argument regulated by a principle referred to as dayo (sufficiency).
Now, Schumann represents this argument as follows:
“In order to draw up a conclusion by qal wa-homer, we should define a two-dimensional ordering relation on the set of data: (i) on the one hand, according to the dayo principle, we know that payment for horn action in a private area [which amount is to be determined] cannot be greater than the same in a public area [which amount is given as 50%], (ii) on the other hand, payment for horn action at a private place [which amount is to be determined] cannot be greater than foot/tooth action at the same place [which amount is given as 100%]. Hence, we infer that payment of compensation for horn action at a private place is equal to 50% cost of damage.” (My italics and my bracketed comments.)
Schumann is here saying that, based on the dayo principle (as he takes it): (i) horn in private places (unknown) is equal to or less than horn in public places (50%), and (ii) horn in private places (unknown) is equal to or less than foot/tooth in private places (100%); therefore, horn in private places = 50%. Though we shall later challenge his premises, let us accept them for a moment so as to show first that his conclusion does not follow from them. According to his premise (i), the amount for horn in private places must be between 0% and 50%; and according to his premise (ii), the amount for horn in private places must be between 0% and 100%. From these two premises, all we can conclude is the common ground that the amount for horn in private places must be between 0% and 50% – we cannot conclude, as Schumann does, that the conclusion is precisely 50% as the rabbis have taught.
Another flaw in Schumann’s treatment is that neither of his premises (i) and (ii), which he announces grandiloquently to be “a two-dimensional ordering relation on the set of data,” mention the given that foot/tooth damages in public places cost nothing (i.e. 0%); yet this fact is not generally considered irrelevant to the deduction. Perhaps Schumann has made an error of inattention while typing his two premises? If we look at the description of his argument in what he calls “a formal notation,” we do find mention of the given 0% for foot/tooth damages in public places. Here, he writes:
x = 50% ”
This schema implies four premises, not two: the two referred to earlier in Schumann’s text and two more which we shall label as (iii) and (iv). Let us look at each in turn:
(i) Horn in private places (unknown, x%) has some here unstated relation to horn in public places (50%), although that relation was earlier specified as “cannot be greater than.” It is not stated how this proposition is established.
(ii) Foot/tooth in private places (100%) is here symbolically specified as more than or equal to (≥) horn in private places (unknown, x%); this is equivalent to the earlier specification that the latter “cannot be greater than” the former. It is not stated how this proposition is established.
(iii) Foot/tooth in private places (100%) and foot/tooth in public places (0%) have some here unstated relation; but since the numbers are given we can readily say that the former is more than the latter; note that the clause ‘or equal to’ is not an option here, since the exact quantities are known.
(iv) Foot/tooth in public places (0%) is here symbolically specified as less than or equal to (≤) horn in public places (50%); however, this is an error, since the clause ‘or equal to’ is not an option here, since the exact quantities are known.
However, since the two latter premises, labeled (iii) and (iv), do not mention the unknown quantity x%, they cannot directly affect the conclusion we can draw, which (granting the previously specified relation for (i), even though it is here not mentioned) therefore remains as before, viz. that the amount for horn in private places must be between 0% and 50% – and not as Schumann claims (so as to appear in agreement with the rabbis’ conclusion) precisely 50%. Note well that Schumann does not tell us precisely how his operative premises, viz. (i) and (ii), were established, other than by vague appeal to “the dayo principle.”
Now, at first sight, this schema might be indicative of an argument by analogy. However, upon scrutiny, it is obviously not so. Looking at Schumann’s tabulation, one might think that (a) just as (iii) foot/tooth in private places is greater than same in public places (100% > 0%), so (i) horn in private places ought to be greater than same in public places (x% > 50%), from which we would conclude that x = 100% or more. Similarly, (b) just as (iv) horn in public places is greater than foot/tooth in public places (50% > 0%), so (ii) horn in private places ought to be greater than foot/tooth in private places (x% > 100%), from which we would again conclude that x = 100% or more. Yet, Schumann tells us that, in both cases (a) and (b), x cannot be greater than 50%.
In fact, looking at the Mishna debate, the analogies here proposed coincide with R. Tarfon’s position, while the limitation given by Schumann attempts to reflect the Sages’ position, which is based on a statement which has come to be called the dayo principle. According to the rabbis, only three options are conceivable, namely zero, half or full compensation for damages. For this reason, though the two analogies, (a) and (b), conclude with “x = 100% or more” – the ‘or more’ clause can be dropped without further ado. R. Tarfon and the Sages both agree that x = 0% is not the solution. So the issue is whether to conclude with “100%” like R. Tarfon, or with “50%” like the Sages. Schumann’s conclusion of “50% or less” does not match either traditional position.
Schumann’s approach does not tell us what the dayo principle is all about. He does not mention R. Tarfon’s apparent analogies at all, or explain why the Sages preferred the lesser conclusions on the basis of that principle. What he does instead is to take what he imagines to be the Sages’ two intermediate conclusions (in reply to R. Tarfon’s arguments (a) and (b)) as his own premises. Moreover, he does that erroneously, thinking that they concluded “x = 50% or more,” whereas in fact they concluded “x = 50% exactly.” Maybe he did this, consciously or unconsciously, in order to artificially imbed R. Tarfon’s opinion into his representation. In any case, he does not at all perceive or analyze the two a fortiori arguments underlying the Sages’ discourse, nor see where exactly in that discourse their dayo principle applies. All he does, to repeat, is calculate the net result of their two conclusions (as he sees them); and even that he does erroneously. This is not a fortiori argument, as we have seen.
This explains why Schumann has only one argument, namely (a), which corresponds to R. Tarfon’s first argument. This also explains why he does not mention propositions (iii) and (iv) in his initial representation of the argument. Although they are pictorially implicit in his “formal notation” schema, he does not explicitly reflect on them and they in fact play no role in his pursuit of the final conclusion. I suggest that their presence here is just eyewash, to give the impression he is taking all known factors into consideration. In any case, note well, he in fact has not found a way to logically integrate them into his reading; i.e. he does not use all the given information traditionally considered relevant. Schumann has, in other words, like many other commentators before him, not taken due note of argument (b), i.e. R. Tarfon’s second argument, let alone realized its crucial significance in the debate.
Actually, the derivation occurs in two different ways relative to the two qal vachomer arguments dealt with here. In relation to the first argument, the derivation has ‘horn in public places’ in the minor premise and ‘horn in private places’ in the conclusion. But in the second argument, though the conclusion is the same, ‘horn in public places’ is not present in the minor premise – but only in an inductive preliminary to the major premise. Thus, in fact, the dayo principle is not merely related to a fortiori argument functioning, but also to a fortiori argument preparation. It is therefore not to be confused with or equated to, as often done, the principle of deduction, according to which “the conclusion of a deductive argument cannot contain more information than what is already given in the premises,” although they do happen to intersect occasionally.
It is evident from the formulation of his premises that Schumann misconstrues the dayo principle. The Sages put it like this: “It is quite sufficient if the law in respect of the thing inferred is equivalent to that from which it is derived.” It is not as he claims that the amount in the conclusion merely “cannot be greater than” the amount in the premise it is derived from. The dayo principle is, rather, that as far as the logical inference is concerned the two amounts must be identical, no more and no less. If any premise predicates some amount, then so must the conclusion have it. In this case, since ‘horn in public places’ entails 50% penalty, then ‘horn in private places’, being derived from that information, must also entail 50%. There is no ‘or more’ clause attached to the Sages’ conclusions.
Schumann also evidently misconceives a fortiori argument, as inference based on the intersection of two ranges of value. This is why he defines it as “parallel concurrent deduction.” We have seen that he does not by this method succeed in obtaining the expected conclusion (though he fakes the result to make it look as if he does). He claims the said intersection to be a single point (as his conclusion x = 50% implies), whereas in fact the two ranges overlap with one (0-50%) being entirely within the other (0-100%), so that his conclusion should have been the lesser range, i.e. x = 0-50%. The lesser, more precise range is preferred to the greater, vaguer one, because the former requirement excludes part of the latter (viz. the 100% part). If we correctly formalize his schema for him we obtain the following argument:
x is equal to or less than A (50%).
x is equal to or less than B (100%).
A is less than B.
Therefore, the range ‘≤ A’ is wholly included in the range ‘≤ B’.
Therefore, x is equal to or less than A (the common ground).
Schumann, to repeat, wrongly concludes that x = A. But in any case, we must ask, is this a fortiori argument? It may appear to be so, because he is verbally referring (albeit erroneously) to the dayo principle and because he is using comparative propositions (i.e. propositions involving quantitative relations like ‘is equal to or less than’). But in fact it is not a fortiori argument – it does not have any acknowledged form of such argument. Schumann just labeled as ‘a fortiori argument’ something that vaguely seemed to him to resemble one.
It is amazing to me that this writer, who I have been assuming has read my work on the subject, completely ignores it and presents a manifestly useless and false theory of a fortiori argument. In my book Judaic Logic, in a footnote to chapter 4.3, I explain clearly enough how the (first) a fortiori argument in Baba Qama, p. 25a is to be construed:
“…we have 3 givens: (1) relatively unintentional damages by animals (shen and regel) in the public domain imply owner’s liability to pay none of the damages; whereas, (2) the same on private property imply his liability to pay all of the damages; and (3) relatively intentional damages by animals (keren) in the public domain imply liability to pay half of the damages. The first two givens serve to induce the major premise of our actual a-fortiori: make the comparison ‘for unintentional acts, acts committed on private property imply more liability than those committed in the public domain’, then generalize to ‘for all acts, the same’, then educe the new particular ‘for intentional acts, the same’. This result, combined with the third given, which serves as minor premise, form the a-fortiori argument proper, whose conclusion is ‘intentional acts on private property imply liability to pay half the damages’.”
Notice that all available data is used. First, we induce the required major premise from the given data on foot and tooth damages; then, we combine it with the given minor premise on goring and draw the desired conclusion from a perfectly standard a fortiori argument, namely the following positive subjectal (from minor to major), where P is horn damage on private property, Q is horn damage on public property, the damages here being intentional damages because the goring ox intends (insofar as an animal can intend) to damage, R is the scale of liability for damages, S is the given or inferred amount to pay for the damages concerned:
Major premise: [intentional damages committed on private property] (P) imply more [liability] (R) than [intentional damages committed in the public domain] (Q).
Minor premise: [intentional damages in the public domain] (Q) imply enough [liability] (R) to [pay half of the damages] (S).
Conclusion: therefore, all the more, [intentional damages on private property] (P) imply enough [liability] (R) to [pay half the damages] (S).
In that earlier account of the first argument in BQ 25a, I identify the fact that we can only infer in the conclusion the same amount (S) as given in the minor premise with the dayo principle. Upon reflection, this explanation is best attributed to what I have lately called the principle of deduction. This can be stated as: in a fortiori argument, as in syllogism or any other form of deductive inference, the conclusion cannot contain more information than what is already verbally or tacitly given in the premises. Given only the above shown two premises, and no additional information, if the term S in the conclusion was more or less than the term S in the minor premise, there would be illicit process – i.e. fallacious reasoning. Clearly, the principle of deduction is not an artificial, arbitrary or conventional limitation, but a natural, rational one.
Today, having reexamined the whole issue much more deeply, I would rather explain this debate as follows. R. Tarfon may be construed as having initially formulated an a crescendo argument – that is an a fortiori argument with an additional premise justifying proportionality – which concludes with full damages; while the Sages may be construed as having formulated the above described purely a fortiori argument in opposition – thus, effectively, by virtue of their dayo objection, rejecting the additional premise about proportionality. It just so happens in this first exchange that the dayo principle corresponds to the principle of deduction. But in the second exchange, this equation falls apart, because R. Tarfon’s argument has the same conclusion whether construed as a crescendo or as purely a fortiori, which means that it is effectively the latter. In that case, the Sages’ new dayo objection, though formulated in exactly the same words, must mean something different; namely, as already mentioned, it must concern the generalization of the major premise preceding the (second) a fortiori argument. The latter explanation is suggested by a Tosafot commentary, by the way.
Whence it follows that the dayo principle has to be understood as a principle specific to Judaic law, either revealed by the Torah or established by rabbinic authority, and not as a purely logical and therefore universal principle. I did not realize this at the time I wrote Judaic Logic. I only came to this realization in the present work, when I wrote the chapters called ‘In the Talmud’ (7-8). There, after careful examination of the Mishna and Gemara in question, I got to realize that the rabbinic viewpoint is not as monolithic and the dayo principle is not as uniform as depicted by me before. The views of R. Tarfon and the Sages in the Mishna, and of the later Gemara, cannot credibly be merged into one “rabbinic” opinion, as I earlier naïvely assumed. The divergences between them are rich in lessons for us about a fortiori argumentation and other interesting topics.
This is, briefly put, my revised viewpoint today. Because Schumann does not inquire into all these issues, his treatment of this important Talmudic passage is simplistic and prone to error.
Although the above examination already proves that Schumann has not grasped the nature of a fortiori argument, to be thorough in my critique of his views I would like us to look at the rest of his discussion of this subject. Without mention or rebuttal of objections to the idea by Louis Jacobs and others, he writes: “As A. Schwarz and M. Mielziner showed, an Aristotelian syllogism may be presented as the simplest case of qal wa-homer” (notice the word “showed” suggesting the matter is settled). He then proceeds to justify this claim with reference to the syllogism “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” – whose “qal wa-homer analogue” he presents as follows:
“general notion (predicate)
particular notion (subject)
x = Socrates’ mortality ”
He explains this schema as follows:
“Continuing our reasoning in the same way, we should define a two-dimensional ordering relation: (i) by the dayo principle, we know that the notion ‘Socrates’ is more general than the notion x, (ii) the notion ‘mortal’ is more general than this x as well. Hence, x is ‘Socrates’ mortality’.”
Now, I (Avi Sion) ask you: Where did x come from, since it was not mentioned in the given syllogism? And how does “the dayo principle” inform us that Socrates and mortal are more general notions than the mysterious x? And where does concluding the equation of x to Socrates’ mortality spring from? I would have thought by consideration of symmetry that x is like ‘mortal’ a predicate; indeed, according to the syllogism, x should be the predicate ‘mortal’.
Clearly, what we have here is the sight of Schumann working hard to load the dice in support of his utterly artificial schema. In what way does this schema of his render the reasoning involved more comprehensible or valid to us? It sounds impressive, but just what is a “two-dimensional ordering relation”? What has this schema of his to do with a fortiori reasoning, other than the fatuous appeal to a sort of omniscient “dayo principle” and the use of mathematical comparatives like ‘≥’? How do we know what notion is “more general” than which, other than by reference to Aristotelian syllogisms? Of what additional use is Schumann’s schema in this regard? Exactly how does the conclusion follow from the premises? The whole thing is contrived and useless – it proves neither the putative conclusion nor the claim that syllogism is “the simplest case” or “analogue” of a fortiori argument. I am sorry to say it, but it is just an empty show.
He goes on to propose to us, as “the [!] Biblical example of Aristotelian syllogism” (exclamation mark mine), and by implication as an example of a fortiori, the argument in Ex. 6:12: Behold, the children of Israel have not hearkened unto me; how then shall Pharaoh hear me, who am of uncircumcised lips? Schumann takes Moses’ speech impediment as the syllogistic middle term and recasts the argument into his unnatural schema, as follows:
“(Jews do not hear Moses)
(Moses is of uncircumcised lips)
(does Pharaoh hear Moses?)
(Moses is of uncircumcised lips)
No, he does not ”
Observe, now. There is no mention this time of a “dayo principle” setting relationships in order for us; and no explanation is given for this omission. Here, the symbol of comparative size ‘≥’ is suddenly, without explanation, joined by or interpreted as the causative conjunction ‘because’. There is no unknown quantity ‘x’ this time. Instead, the two items on the right are one and the same proposition “Moses is of uncircumcised lips.” One of the items on the left is an assertion “Jews do not hear Moses” and the other is a question “does Pharaoh hear Moses?” The putative conclusion confidently appears out of nowhere: “No, he does not.” We are not told how it was derived from the apparent premises, or what role each element played in obtaining it. We are supposed to think that this schema is somehow a magical formula that “proves” whatever is forced into it. All this I submit is not logic, let alone syllogistic or a fortiori logic.
Also, Schumann has evidently read this passage of the Torah very superficially, as saying: “Because Moses has a speech impediment, the Jews do not understand him. For the same reason, Pharaoh won’t understand him either.” Even if this may at first sight seem what Moses was arguing, we should ask (here as always) how does the conclusion proceed from the premises? The simplest answer would be to say: by analogy; “just as the Jews are befuddled by Moses’ speech impediment, so will Pharaoh be.” But this is neither syllogism nor a fortiori argument. It is not even a deduction, since it is conceivable that Pharaoh could have been more tolerant or skillful in relation to people with speech impediments than the Jews allegedly were.
If we want to formally generate a syllogism from this material, we would have to begin with an induction. We would have to generalize the given proposition “Jews cannot understand Moses’ speech” to “No one can understand Moses’ speech.” Using the latter as our major premise, we could then engage in the following syllogistic inference:
No one can understand Moses’ speech (by generalization).
Pharaoh is someone (obviously).
Therefore, Pharaoh cannot understand Moses’ speech.
This would be a valid syllogism, subsuming Pharaoh under the previously induced general statement. But have we thereby engaged in a fortiori reasoning? Clearly, not. What is missing in this syllogism is the sense that Pharaoh would for some reason be less prone to listen to Moses than the children of Israel, who admittedly were recalcitrant enough (though, obviously, not entirely so, since they do listen to him quite a bit). Schumann could of course reasonably deny that such relative proclivity is intended in Moses’ statement, since it is not explicitly mentioned in it. In that case, the argument would involve a syllogism (as just explained), but it would not be a fortiori (as he admits it to be). What is sure in any case, is that the rabbis interpreted Moses’ statement as an a fortiori argument, and Schumann has tried but in no way succeeded in representing it as such.
I analyzed Ex. 6:12 in my Judaic Logic, explaining that the clause “who am of uncircumcised lips” could not be viewed as part of the a fortiori argument and that the argument required a fourth term more variable than that clause to serve as a fortiori middle term (R). I there proposed to use “fear of God” as middle term, but any other appropriate term would do as well (e.g. “spiritual closeness to Moses”). The resulting a fortiori argument was a negative subjectal, as follows:
The children of Israel (P) fear God (R) more than Pharaoh (Q) does.
Yet, they (P) did not fear Him (R) enough to hearken unto Moses (S).
all the more,
We have here a perfectly natural and comprehensible a fortiori argument, capable of formal validation. It explains why Pharaoh will be less inclined to listen to Moses than the children of Israel have been. It interprets, as the overall context suggests, the phrase “hearken unto Moses” in a moral sense, as “taking Moses’ instructions to heart and following them,” and not merely as a physical issue of difficulty to understand Moses’ speech due to his difficulties with speech. Moses’ mention of his speech impediment is easily pushed aside as a lame excuse on his part, an expression of his humble hesitation to take on the gigantic political and spiritual task he is being given by God.
If we take the subsidiary term “hearken unto Moses” as an either-or predicate, then just as the children of Israel do not listen to Moses as much as they ought to (if at all), so Pharaoh can be expected not to listen to him as much as he ought to (if at all). This would be an acceptable purely a fortiori reading, suggesting that Pharaoh can be expected to disobey Moses at least to the same degree as the children of Israel do. We could however go further and read the argument as a crescendo. If we understand the subsidiary term “hearken unto Moses” as allowing of degrees, we can say that the suggestion that the children of Israel “do not” listen to Moses is intended hyperbolically rather than literally, and grant that since Pharaoh fears God less than the Jews, he is more prone to dismiss Moses’ injunctions. In that case, we are tacitly admitting an additional premise that the degree of disobedience to Moses is inversely proportional to the person’s fear of God.
This is the correct explanation, which Schumann has totally ignored. How can anyone, having read the above and other examples, and copious explanations and proofs, manage to come up instead with such a confused concoction as his? Baffles me. What is clear that he has thoroughly misunderstood the nature of a fortiori argument. He misconceives it as some sort of table-filling mechanism, whereby given three cells, the fourth cell automatically follows; and he entirely ignores the key variable (viz. the middle term, R), the quantitative aspect underlying the other terms which makes the inference in fact possible. Moreover, he imagines the dayo principle to be some sort of premise-formation tool, whereas its function is to limit the conclusion to information given in the core premises.
And the main reason Schumann has misunderstood a fortiori argument is that he is trying to artificially impose on Judaic logic modes of thought or techniques (of anyway doubtful value) that he has supposedly gleaned in modern symbolic logic. Instead of going bottom-up from empirical study to fitting theory, he is functioning top-down from some rationalistic prejudice, forcing things into imaginary forms without regard to their actual content.
We have thus far demonstrated that Schumann has grossly distorted the nature of a fortiori argument, not to mention his falsifying its applications in Judaic contexts. Allow me to continue with the present analysis, so as to better understand the methodological failures that have led that author into such serious error. For this purpose, I shall have to indulge in some literary and psychological analysis, in conjunction with purely logical analysis. Consider to start with the following paragraph:
“All examples of qal wa-homer regarded above show that this inference rule cannot be presented in a linear form and assumes a multi-dimensional ordering relation (the simplest case of two-dimensional order was considered in instances above). As opposed to this, usual inference rules in modern logic suppose linearity, therefore by combining these rules we obtain conventional proof trees. Qal wa-homer provides us with an algorithm for massively-parallel proofs. Hence, a deductive system of Judaic logic may be presented as a hybrid cellular automaton.”
Here, Schumann communicates to us that he is not modestly content with his above ad hoc treatment, but has greater ambitions. He aims first to generalize his nonsensical findings and find ways to mass produce equally pseudo-logical “proofs.” Observe the flashy verbiage he uses here and throughout. Its intent is of course firstly to excite the reader’s awe and admiration. But moreover, its purpose is to suggest that he is fully plugged in to the latest trends in logic research and that his work fits right in there on the cutting edge; this serves to buttress his earlier, more specific claims, and to conceal their weaknesses. Psychologically, what is happening in Schumann’s mind at this stage is that he is working himself up into a sort of intellectual frenzy. He is delighting his ego with the thought of the mind-boggling possibilities implied by his discoveries. Not only has he, in his mind’s eye, solved the till now intractable problem of a fortiori argument, but he sees that he can take it all much farther and expand the horizon of logic into hitherto unexplored fields.
With this goal in mind, Schumann now proposes a complex-looking symbolic formula for “the inference rule of qal wa-homer,” again using impressively abstract and modernistic terminology, which I will spare you here. I do not bother to reproduce it, for four reasons. First, if it is at all meaningful and comprehensible (which, I assure you, is open to debate), it is not at all clear that it has any relationship whatsoever (other than his say-so) with a fortiori argument. Second, assuming that it is as Schumann claims a generalization of a fortiori argument as he sees it, we can predict without further ado that it is bunkum. Why? Simply because, if what is generalized is known to be devoid of substance and logically erroneous, it follows that its generalization is equally (if not even more) empty and mistaken.
He, of course, is convinced that the mere putting of a thought into symbolic language, however superficial in content and logically fragile that thought may be, gives it profundity and legitimacy. This is very important to realize – this belief that symbolization is somehow, by some mysterious magic, able to confer on abstract ideas a truth and power that they lack in ordinary language. But this faith, though widespread in today’s academia, is without basis. Third, Schumann makes no attempt to prove his proposed formula, i.e. to validate it from already established principles. He just asserts it, as if its truth is self-evident; and that just won’t do, if logic is to be a scientific enterprise.
Fourth, he makes no effort to demonstrate on paper the practical unfolding of the “multi-dimensional ordering relation” that he claims performs “massively-parallel proofs.” He just vaguely imagines this fantastic animal in fancy words and symbols, but does not actually show it in action or give us an example of its application and utility. Just what does a “multi-dimensional” a fortiori argument look like? Exactly what are its premises and what is its conclusion? Those “massively-parallel proofs” that we are promised – they are proofs of what, exactly? All we are offered is an alluring chimera. Nevertheless, impressed with his linguistic tour de force, Schumann concludes:
“As we see, Judaic reasoning may be formalized only by using the non-well-founded mathematics and process semantics, [which] both for the first time began to be used in computer science. In particular, we can assume that the (sic) Judaic semantics and Judaic formal logic can be developed within the framework of interactive-computing / concurrency paradigm. One of its means is the coinduction principle. For more details see [various references]. However, the concepts of coalgebra and coinduction have not yet had much impact in the pure logical investigations as well as in the logical-historical studies.” (Italics his.)
Note here again the high density of glitzy words and the implication that all this is part of some larger avant-garde effort. Schumann is here revealing one aspect of his programme for Judaic logic, namely to extract from it hidden jewels of logic (like the a fortiori argument), and then inject them into general logic studies – of course in a more developed form capable of mechanically processing any number of elements, indeed even an “infinite” set of them. The a fortiori argument is important to him, but only as a stepping stone to bigger and better things. He is doubtless sincerely interested in Judaic texts and practices, but he also perceives them as potentially rich sources of new ideas for modern logicians to investigate. This goal is of course acceptable, and has long been pursued by many logicians. Nevertheless, the fact remains that, as we have above shown with regard to Schumann’s work on a fortiori argument, he is unequal to the task.
What I find galling is Schumann’s pretentiousness. After putting forward various “themes and objectives” for Judaic logic studies, he declares (my italics): “These aims were not fulfilled in other books devoted to Judaic reasoning and methodology.” Earlier on in the same essay he dismisses offhand as “informal traditional logic” the work of “Ronen Reichman, Louis Jacobs, Avi Sion et all (sic).” I am not acquainted so far with the work of Ronen Reichman (who writes in German). Regarding Louis Jacobs, though it is true that the large majority of his work is informal (though very perspicacious), he has nevertheless made an effort at formalization of qal vachomer and binyan av Talmudic argumentation – an effort, I should add, much more interesting than Schumann’s paltry attempt. As for my own work on Judaic logic, it is almost entirely formal, not to mention its wide scope and originality. I can only suppose that Schumann misrepresents it (“informal”) and disparages it (“traditional,” “aims… not fulfilled”) in order to avoid having to compare his work to it and in order to claim originality.
As regards originality, I would like to point out that the phrase “Judaic logic” was, I think, coined by me; certainly, it was the title of my book on the subject published some 15 years before Schumann used it for his collection. Long before Schumann conceived it, my book explicitly set as its goal the study of the interaction of Judaic and general logic, as e.g. in its abstract:
“Judaic Logic attempts to honestly estimate the extent to which the logic employed within Judaism fits into the general norms, and whether it has any contributions to make to them.”
Even so, I did not and do not claim that such research was original with me. I was well aware then, and have become even more aware since, that many competent people have been making similar efforts at comparison and correlation between Talmudic reasoning and Aristotelian logic, since the Middle Ages if not earlier. My point here is only that I resent the impression given by Schumann that this is some kind of new vision that he has personally come up with. A science cannot advance if those who wish to contribute to it do not duly acknowledge the significant work of others – or, if they disagree with it (as of course they may well), do not explain why they do so. It is silly for someone to pretend to be an innovator or a leader when he is not – the truth will come to light sooner or later.
Attempting to dig deeper into why Schumann failed in his attempt to formalize a fortiori argument, I would suggest that his methodological tool is fundamentally flawed. His reliance on the ways and means of “modern symbolic logic” is bound to lead him astray, for reasons I explain elsewhere (in Appendix 7.1). The fault is not his personally; it is to be blamed on the educational system that taught him; he is a victim of intellectual fashion, or more precisely perhaps of a new and infectious cognitive disease. My kindly advice to him (and others like him) is to quit the make-believe world of modern symbolic logic and learn some real logic – classical formal logic – e.g. by studying my books, starting with Future Logic. Ask yourself whether you want to be a logician, or just appear to be one; whether you want to merely talk the talk, or really walk the walk. Steer clear of posturing and imposture, and seek to master the fine and demanding art of true logical analysis.
Of course, Andrew Schumann deserves respect for his work as an editor, bringing together essays by many writers (of varying competence) on logic relative to religion in general and Judaism in particular. Such initiatives stimulate often valuable research and writing, and acquaint writers with each other. I am sorry to have found it necessary to expose the shortcomings in his logic, here; it gives me no pleasure. I do believe he is well-meaning, and sincerely devoted to both logic and Judaism, values which I share. So, this issue was a hot debate in my conscience for weeks. But, I decided, intellectual integrity should be our paramount value, above all personal or ideological considerations.
My view is that logic, like philosophy, cannot be made a handmaiden to religion – any religion, my own included. It is not servile to any domain, be it secular or religious; it is our common precious tool for the objective and impartial rational judgment of all claims of human knowledge. Logicians cannot adapt logic to fit preconceived notions, however desirable they seem; that would be dishonest, unscientific and unethical. Logic is something universal; it is not the particular possession of any people, culture, class, place or time. Different people may well and do display different degrees of use, mastery and understanding of logic; it is possible to say this person is not as logical as that one. But it is not permissible to say that all ‘logics’ are on equal footing.
I am not sure that Schumann is of the same opinion on these issues, judging by his statements and performance in his contributions to the Judaic Logic book he edited. Already, in his Preface to it, he strikes me as excessively apologetic and relativistic, writing of Judaic logic as “a methodology for deducing religious laws [from the Pentateuch]” and as an “original logic that is not less deductive than Aristotle’s logic.” Admittedly, he is not here speaking in the first person, but his immediate emphasis on the deductive claim is clearly indicative of his own perspective. Indeed, this is confirmed further on when he proposes to “develop general approaches to formalizing Judaic logic” and adds: “This consideration of Judaic logic has relevance for modern logic and analytic philosophy and may be compared with the contribution made by the formalization of Ancient Greek logical systems to 20th-century logic and language philosophy.”
My complaint here is firstly that he takes for granted that the explicit and implicit Talmudic midot (the rabbinic hermeneutic principles and practices) are all deductive and all valid, whereas I have shown systematically in my book Judaic Logic (see its chapter 12 for a summary of its conclusions) that, while some midot can truly be said to be deductive and valid (notably the a fortiori argument), other midot can only at best be conceived as inductive (and therefore compatible with alternative conclusions to those imposed by rabbinical fiat) and still others must be admitted to be invalid (i.e. phony logic or even contrary to logic). Many researchers agree with these overall conclusions from their own perspectives. It is therefore unacceptable that Schumann here abstains from critical judgment and seems to give a blanket sanction to all of Judaic logic. This perhaps reflects an apologetic stance, or alternatively an ideological relativism.
Secondly, Schumann tries to put over that Judaic logic is “original,” i.e. has made contributions unknown to Aristotelian logic. There is some truth to that claim, to be sure. The a fortiori argument, for instance, though not unknown to Aristotle and his predecessors and successors, was not extensively used or systematically studied by them, whereas it plays an important role in rabbinic discourse and the rabbis have over time made considerable efforts to understand its workings. Nevertheless, as I showed in my Judaic Logic (chapter 4.2), to formalize and validate a fortiori argument, we must refer to the methods developed by Greek logic. Some other midot can similarly be favorably compared, and found to contain some valuable novelty. One of the interesting findings in my study of the harmonization midot (see chapter 11 of my work) was that these arguments are built distinctively around four terms instead of three. Nevertheless, here again, it was necessary to refer to syllogistic logic to formalize and evaluate them.
Come to think of it, if I were to propose one midah as the most original in comparison to Aristotelian logic it would perhaps be the thirteenth rule of R. Ishmael, which can be defined in general terms as dialectical reasoning, i.e. thesis-antithesis-synthesis. The reason why this midah is worthy of attention is that it is distinctively inductive, whereas the a fortiori argument and the other harmonization rules are, or at least are claimed to be, deductive. This 13th rule teaches us (by implication) that if we come across a contradiction in our thinking, we should find a way to reconcile the theses concerned, either by modifying them both, or by preferring one over the other, or by rejecting them both in favor of some third thesis. Of course, Aristotle practiced such adaptation; but somehow he does not seem (as far as I can recall) to have enshrined it as a crucial principle in his theory of knowledge. Reconciliation of conflicting theses has remained a little noticed aspect of induction till modern times, and yet in rabbinic debate it played a very prominent and conscious role (not only through the thirteenth midah, but more frequently in the drama of kushya and terutz – difficulty and resolution – in rabbinical discussions).
If we leave aside this important inductive process, and here consider only deductive processes as Schumann does, we can surmise from what was said before that his claim to the originality of Judaic logic is exaggerated. I do not of course claim that the deductive aspects of Judaic logic are historically derived from Greek logic. Though there may have been some trickles of influence, there is no doubt that they emerged independently. What I am contending is that the distinctive deductive elements were not necessarily true logic – and that those that were true logic can only be so assessed by referring to Aristotelian logic. The rabbis originally never engaged in formalization and validation of their logical discourse; it is only much later in time under the influence of Aristotelian logic that some efforts were made to do so. Their justification of logical arguments was either revelation (see e.g. Baba Qama p. 25a: “Is not dayo of Biblical origin?”), or rabbinic decision through rational insight (svara) or convention (rov, majority vote). Therefore their discourse, though it may well be characterized by us as logical, cannot strictly speaking be classified as an exercise in formal logic.
Schumann seems aware of that defect, since he proposes to formalize Judaic logic. However, what he means by “formalize” is “symbolize and axiomatize.” As far he is concerned, even “Ancient Greek logical systems” were not formal and had to be formalized by “20th-century logic and language philosophy” – and he proposes to do the same job of updating for Judaic logic. He can thus ignore my attempts at formalization and validation of Judaic logic, and the attempts by others, since that is not what he means by formalization. What he has in mind, in fact, is symbolization and axiomatization. We shall presently see what this means in practice. We have already in fact seen it with regard to the a fortiori argument. We saw how Schumann misconstrues the form of such argument and its justification. We saw how, when things do not fit into his conception of them, he forces them in and fakes the conclusion.
Let us now look at another case in point – his treatment of the klalim uphratim (the hermeneutic rules referring to combinations of general and particular terms). In his Introduction, he writes:
“In this volume we are going to present a modern logical analysis of basic middot used by Talmudists…. These rules have nothing in common with Ancient Greek logics. For instance, truth and falsity cannot be regarded as meaning of (sic) Scriptural passages. Evidently, the latter are viewed as absolutely true. By Judaic inference rules [here listed], other meanings of Biblical statements are introduced, namely the phrases may be either ‘general’ or ‘particular.’ As a result, logical connectives are defined in a unique way.”
Needless to say, I do not agree that “These rules have nothing in common with Ancient Greek logics” – Schumann is only saying that to enhance the value of his forthcoming “formalization” of some midot. As for his suggestion that since the rabbis viewed the Scriptures as “absolutely true,” truth and falsity must have to them referred to “other meanings,” viz. the distinction between “general” and “particular” – he is here obviously trying to give a modernistic spin to the rabbinic interest in the klal-prat distinction, again for the purpose of enhancement. In fact, the klal-prat distinction simply refers to the scope of terms. The interest of the rabbis is in interpreting how far a Biblical statement is applicable in practice. This is not, as Schumann hints, a theory of truth or of meaning.
Anyway, he then presents tables for the logical connectives “…and…,” “…or…” and “if… , then…” (which he labels respectively as “Judaic conjunction,” “Judaic disjunction” and “Judaic implication,” because they seemingly have different properties to the corresponding ordinary, Western connectives) with reference to different combinations of “general” and “particular” contents. The first table, based on Hillel’s fifth rule and R. Ishmael’s rules 4, 5 and 6, looks like this:
A and B
Now what I notice here is that this table is both more extensive and less extensive than what is given in actual Judaic logic. It is correct that klal uphrat yields prat (rule 4 of R. Ishmael), prat ukhlal yields klal (rule 5 of R. Ishmael). But the table is more extensive, in that it mentions a conjunction of two particulars with a concluding particular (the first row) and a conjunction of two generals with a concluding general (the last row); these alleged conjunctions are, to my knowledge, nowhere treated in the listings of rules or anywhere else. Also, the table is less extensive in that it does not mention double conjunctions, i.e. those with three conjuncts. Rule 6 of R. Ishmael refers to klal uphrat ukhlal, and later additions to this rule (not listed) include prat ukhlal uphrat, klal ukhlal uphrat and prat uphrat ukhlal. The conclusions from these various conjunctions more or less follow the last conjunct – more or less, I say, because in practice this is somewhat subjective, depending on the rabbis’ preferred interpretation in a given context. Schumann possibly intended his two interpolations (the first and last rows) to cover these various cases of double conjunction; but he does not say so. Moreover, he makes no mention of R. Akiva’s ribui umiut and miut uribui alternatives to the R. Ishmael rules, although they play a prominent role in Talmudic discourse (initially as a rival system, but later as a complementary one).
Therefore, what appears to have happened here is that Schumann derived his first table from the single conjunctions of rules 4 and 5 of R. Ishmael, ignoring the double conjunction of rule 6 and its later variants. He added in the first and last rows of his table to fill in the blanks symmetrically, perhaps thinking to himself that rule 6 could be assumed to roughly fit in there somewhere. He skipped R. Akiva altogether, no doubt because distinguishing his approach from R. Ishmael’s requires some rather subtle discussion. What Schumann wanted for his “formalization” demonstration was something manageable, like his first table, which would somehow reflect Judaic logic and also differ sufficiently from the conjunction rules of Western logic. Having more or less forced the given data into his simplistic scheme, he can now claim to have discovered an alternative “logic” in Jewish sources.
After that, Schumann proposes two more tables of similar construction, one for disjunction (“A or B”) and another for implication (“If A, then B”), which I will spare you. Well, if you must know, the disjunction table is identical to the conjunction one, except for the first row, which tells us that “particular or particular” results in “general” (which makes sense only if we know that the two particulars are an exhaustive listing of alternatives). The implication table is identical to the conjunction one, except for the third row, which tells us that “if general then particular” results in “general” (which makes sense only if the terms of antecedent and consequent are the same).
Now, whereas the first table is in part based on explicit midot, Schumann gives no hint as to what Judaic texts or even just rabbinic practices these two additional tables are based on. He does give various examples as to how these tables might be interpreted, but these examples seem to be his own inventions, without basis (to repeat) in Jewish tradition. Furthermore, the examples he gives seem to me very arbitrary, with the sentence clauses he claims to be “particular” or “general” very doubtfully so. For my part, I have no recollection in my studies of any doctrine of disjunction or implication such as that which Schumann here proposes – correct me if I am wrong (I certainly do not claim great erudition). So what we have here, again, are figments of Schumann’s imagination presented as Judaic logic.
In short, in his effort to subject Judaic logic to “modern logical analysis” and to “formalize” it, Schumann reinvents “Judaic logic.” He confuses fiction with fact. He lacks scholarship. He does not have a scientific, empirical approach, of patient sifting through a mass of given data and then tentatively summarizing it and then drawing conclusions from it. No: he decides the way it is, the way it ought to be, and that’s that. He evidently does not have a clear idea in his mind as to the difference between description and prescription. Is he saying that these tables truly reflect the way the Talmudic rabbis actually thought – or are these tables intended to teach them how they should have thought? Is he accepting Talmudic logic at face value, or criticizing it? Perhaps he is not interested in history or religion, but is only intent in partly extracting from Talmudic sources a new logic for modern secular consumption. In that case, is he telling us that this new logic is better or truer than the one we have already – or is he just a relativist who considers all logics equally valuable or valid?
From what he tells us, with an impressive tone of authority, his view is that: “Judaic logic for inferring legislative statements from the Pentateuch is closed under connectives defined above.” So it looks that Schumann takes it all very seriously and considers that he has correctly represented the logical connectives of Judaic logic. He then goes on to deal with the a fortiori argument as we saw earlier, first stating with characteristic bombast: “This inference rule differs from all conventional inference rules involved in a deduction procedure of classical and non-classical logics. It is connected with parallelism and non-recursiveness assumed in its using.” So it also looks like he considers that he is teaching modern logic new tricks.
What is evident, in any case, is that Schumann is greedy for territory; he wants to stake his claim on Judaic logic, or the modern symbolic logic derivative of it, and take for himself the tacit title of its conqueror. If no single logician can do it, how can a pseudo-logician do it? His efforts remind me of a recent funny news item about some people laying claim to ownership of the moon or sun. With this, I am tempted to say, having already deconstructed his various claims: my case rests. But I notice he has another article, called “Judaic Syllogistics” (15p.), in the same volume, in which he proposes with his usual gusto “a formal syllogistics that verifies reasoning of the Baba Qama, one of the books belonging to the Talmud. The formal logical system of such kind is built for the first time. This system is a version of non-Aristotelian syllogistics.”
Now, what can “non-Aristotelian syllogistics” possibly refer to? Presumably, it refers to syllogism of some sort – but to syllogism not valid by Aristotelian standards, i.e. under all three of his laws of thought (identity, non-contradiction and the excluded middle). What else could it mean, if appropriately named? Categorical syllogism (for example) is composed of two premises and a conclusion comprising three terms. There are four conceivable ‘figures’ of it, i.e. ways in which the terms might be disposed. Considering (for example) the four plural non-modal forms A, E, I, O, there are 64 (4x4x4) mathematically possible combinations of these propositions in each figure. As I show in my Future Logic (chapter 8-10), there are in all only 28 valid moods (13 primary and 15 secondary) in this sort of syllogism (out of 256 = 11%). All other moods are invalid, i.e. yield a non-sequitur or an antinomy. To claim any of the latter arguments valid, as Schumann apparently does when he boasts of “non-Aristotelian syllogistics,” is not logic but sheer sophistry. Anyone who thinks by such means is not a logician but a sophist. Need anything more be said?
Or maybe by this term Schumann intends forms of mediate inference that were unknown to Aristotle (or at least not formally treated by him), perhaps having in mind hypothetical syllogism (as against Aristotle’s categorical syllogism), hypothetical and disjunctive apodoses (i.e. modus ponens and modus tollens), and other sorts of inference through a middle term or thesis, including a fortiori argument. If so, why does he not say so; why does he opt for such a misleading term? Maybe he thinks that this makes him seem superior to Aristotle.
Some months after writing the above analysis of Schumann’s approach to a fortiori argument and Judaic logic, and logic in general, I happened to come across another essay of his posted on the Internet, which I think worth examining briefly because it is very revealing of the author’s mental processes and motivations. The essay serves as Preface to a recent collective work called History and Philosophy of Logic. Many of the things Schumann says in this essay are historically, philosophically and logically delusive nonsense, as we shall see. He thinks he is being clever, but is only making a fool of himself in public. The assertion of his that most caught my attention and incited my indignation was the following:
“In Aristotle’s syllogistics we are moving from general knowledge to particular conclusions. For example, in the following syllogism
All men are mortal,
Socrates is a man;
therefore Socrates is mortal.
we are concluding the facts, which can be verified by experience, from the less obvious general knowledge. Indeed, ‘Socrates is a man’ and ‘Socrates is mortal’ are singular facts which we can know from the direct observation or from testimonies borne by reliable witnesses. However, the statement that all people are mortal cannot be verified on experience. What is its truth then grounded on? Is that all people we knew did not live longer than 120 years? But we know not all people and on a broader scale it is a popular induction of type ‘all swans are white’ and ‘all ravens are black’.”
What we have here is a profound (though common enough nowadays) failure to understand the logical and epistemological role of the syllogism in human thinking. In Schumann’s view, the singular minor premise and conclusion of the above syllogism are empirical facts, whereas the general major premise “cannot be verified on experience.” According to him, then, all general propositions are groundless; claims of induction such as ‘all swans are white’ are unreliable (as is suggested by his “but we know not all people”) and ultimately fallacious (as is suggested by his mention of ‘all ravens are black’, which is the statement used in Hempel’s paradox of confirmation).
There are many crass errors here. For a start, he presents the syllogism as a purely deductive tool (“moving from general knowledge to particular conclusions”), whereas in fact it plays a major role in induction. I have already explained this in some detail in the past and will not repeat myself here. But briefly put, in the above example of syllogism, if the major premise were based on complete enumeration (as some claim), the argument would be circular, since the conclusion would be a necessary support for the major premise. The conclusion is indeed an inference from the major premise, because the latter is based on generalization. We believe that all men are mortal from past observation of all men dying; until Socrates dies, he cannot logically be used as a basis of such generalization, so his death is a prediction from it. Once, however, Socrates dies, he passes over from an inferred case to one of the many cases that buttress the generality. Notice the involvement of temporal factors in this understanding.
Schumann, on the other hand, presents the minor premise and conclusion as both known “from the direct observation or from testimonies borne by reliable witnesses.” This means, for a start, that the syllogism he presents involves no deduction at all, since its “conclusion” that Socrates is mortal, being singular, is necessarily known by direct or indirect observation (for which, of course, we must wait till Socrates actually dies). The general major premise then becomes, in his way of looking at syllogism, totally inexplicable, playing no role in the emergence of our belief in the “conclusion.” Moreover, he tries to put in doubt the validity of all generalization by pointing out (rightly or wrongly) that some people live for more than 120 years (as if this was the question, as if such people do not eventually die too), and giving as another example the claim that all swans are white (when of course we all know that some swans are black). As regards the justification of generalization, I have already written a lot about it, and will not repeat myself here.
Moreover, Schumann seemingly alludes to Hempel’s “raven’s paradox,” trying to appear slick and knowledgeable. But as I have shown in the past this so-called paradox is based on foolish misconceptions and easily resolved. Thus, Schumann’s attack on “Aristotle’s syllogistics” turns out to be just so much posturing, and a demonstration of his ignorance and stupidity. If we examine his own discourse, we observe that even he unconsciously uses syllogism. For example, when presenting the ‘Socrates is mortal’ syllogism as an example of syllogism in general, he obviously expects readers to apply his critique concerning it to all other syllogisms they may encounter; and this is nothing other than an implicit act of syllogistic application!
The truth is that Aristotle’s syllogism is essential to many aspects of human knowledge. It is obviously used (among other things) to include (first figure positive moods) or to exclude (first figure negative moods) observed percepts in/from concepts, or more narrow concepts in/from larger concepts. This function, as just briefly explained, may be deductive or inductive, according as we are using the syllogism to define the scope of a concept through sample percepts, or to make new predictions on the basis of past generalizations. Without this intelligent tool, we could not organize our knowledge or ensure its consistency, and would be faced with a mass of disorderly and potentially contradictory data. Syllogism is not something arbitrary, as Schumann attempts to imply. Aristotle did not invent this instrument of thought; he merely discovered it by observing his own thinking processes and the thoughts of others.
Schumann does further on say a few pious words about induction, but he does not identify this as the source of our major premises:
“For the first time Karl Popper (1934) paid attention that there is an asymmetry between verification (modus ponens) and falsification (modus tollens): no general (empirical) statement can be proved or even definitively confirmed, but only it can be falsified [sic]. Thus, according to Popper giving priority to confirmation before falsification is not grounded logically. On the contrary, in scientific theories we should prefer modus tollens, instead of modus ponens.”
Schumann, who poses (confusing name-dropping with scholarship) as a historian of logic and philosophy, should have known that it was, long before Popper, Francis Bacon who “first” explicitly realized that a negative instance is sufficient to eliminate a scientific hypothesis, whereas an immense number of positive instances can only confirm it but never definitely prove it. We could also in this context mention the earlier Robert Grosseteste and the later John Stuart Mill, and indeed many more. Even Immanuel Kant, whose main focus was (like David Hume’s) the positive side of induction, shows some awareness of the negative side. In a Jewish context we might mention Saadia Gaon, who wrote in Emunot veDeot: “Whatever leads to the rejection of the perception of the senses or rational faculty is false.”
But in truth, we can probably find earlier roots than those. The scientific method is a broad field of study, with a long and rich history; it certainly did not start from scratch with Popper as Schumann claims. Moreover, it is practitioners of science who have developed it the most even as they pursued their more specific scientific goals. Logicians and philosophers have developed their theories ex post facto, mostly by observation of scientists in action. Though sometimes, of course, the same person had both roles. Science developed for centuries, and indeed millennia, without need for Popper’s guidance. And long before science, common man very often thought in a scientific manner, as he had to in order to at all survive on this planet. The scientific method is only a distillation of ordinary human cognitive means, a selection of the most reliable aspects of them.
In any case, from Schumann’s presentation here it is clear that, though he mentions falsification, he has not grasped the significance of the absence of falsification, i.e. of the power of confirmation inherent not only in positive instances, but in the fact of having sought and not found a negative instance. It is this absence of any negative instance that justifies generalization from the presence of positive instances. That is, it is not the positive instances alone, but their conjunction with an absence of negative instance, that logically allows us to generalize. This is something known from way back, as evidenced by the statement by Jean Buridan that the principles of natural science “are accepted because they have been observed to be true in many instances and to be false in none” (my italics). There are many people, still today, who do not understand this.
Schumann has obviously not understood this crucial point, since he presents the issue quite superficially, as a contest between the force of conviction of a negative instance and that of any number of positive instances. In truth, the modus ponens and modus tollens only occur simultaneously in the event of actual occurrence of a negative instance; as long as there is no such occurrence in a given case, only the modus ponens is relevant to it and the modus tollens plays no role. Thus, Schumann’s claim that the priority of confirmation “is not grounded logically” merely reveals his own misunderstanding of inductive logic.
Positive instances are essential to induction, and the more of them the better; and most of the work of science relates to finding them, even if the scientist must always be on the lookout for any negative instance that might belie, or even just put in doubt, the generalities they suggest. The elimination of hypotheses that are inconsistent with empirical findings is just one aspect of induction, and certainly not all of it. Schumann’s minimal interest in positive instances is indicative of his lack of acquaintance with the history of science. Positive data has always played a crucial role in the formulation and testing of theories. For instance, Johannes Kepler could not have traced the orbit of Mars in 1600 without the data previously collected by Tycho Brahe, and his more accurate prediction of the transit of Mercury in 1631 did much to gain support for his astronomical model as against Ptolemy’s.
As regards Schumann’s motivation in mentioning induction here, he apparently imagines that the Aristotelian method is antithetical to Popper’s. He thinks that (as above exposed) Aristotle’s syllogism is entirely deductive, and therefore irreconcilable with Popper’s induction. But the truth is, they are complementary, Popper’s induction providing the major premises needed for Aristotle’s deduction; and indeed, in the opposite direction, syllogism is often used to make predictions from general hypotheses, in order to test them empirically. The reason why Schumann is unable to integrate the two is, ironically, because his own method is (as we have shown elsewhere) entirely “deductive” – being a figment of his imagination with no authentic relation to actual human reasoning processes. That is, while posing for the cameras as virtuously opposed Aristotle’s deductive logic and favorable to Popper’s inductive logic, his own method is even more blindly “deductive.”
But aside from his questionable grasp of theory, let us take a look at Schumann’s practice of induction. This is very telling. What we see him do right here, in the present essay (see further down), is hastily and erroneously generalize from a single example of syllogism with a totalitarian theme to “all totalitarian discourse is syllogistic,” and moreover conversely to “all syllogistic discourse is totalitarian.” There is no visible effort of falsification on his part. He does not ask whether any totalitarian discourse is non-syllogistic (whereas most of it is), nor whether any syllogistic discourse is non-totalitarian (whereas most of it is). On the basis of this one content of syllogism, and maybe a few more examples like it in his mind, he draws a general negative conclusion about the validity or value (the distinction is not mentioned by him) of the form of syllogism. Now, tell me – is this the practice of someone who has understood and assimilated inductive logic, or of someone who hasn’t a clue? This is manifestly a tyro pretending to be an expert.
What becomes evident, watching the theatrical rhetoric of Schumann (and many others like him, to be sure), is that he is basically against logic. It is not just Aristotelian syllogism that he seeks to obliterate from our culture, but human reason as a whole. He poses as a hip and with-it proponent of non-Aristotelian logic – but Aristotle is not his real target; his real target is logic itself. Even though he labels what he does as logic, it is not only non-logic but anti-logic. He has obviously made no effort to study and understand true logic, but has all his life fed himself (or allowed himself to be fed) a diet of delusional nonsense dishonestly labeled as logic. Genuine logical theory, starting with Aristotelian logic, is an account of the means available to mankind to get in contact with reality. Schumann’s pseudo-logical discourse is a formula for cognitive disaster; it does not get him or anyone in closer contact with reality, but sentences all who use it to the nuthouse. It is discourse emanating from and leading to acute mental derangement. It is not logic, but lunacy.
Evidently, he imagines it is all a power game that one can cynically play with impunity. He thinks that a theory of logic is something arbitrary established by authoritarian means. His failure to see the objective justification for the major premises of syllogisms is a cause of this attitude. He thinks that Aristotle’s logic, comprising mainly the three laws of thought and the syllogism, was widely accepted merely because he spoke with authority. Schumann imagines that if he advocates another sort of logic, he may one day likewise be regarded as an authority. If all logic is arbitrary, why not? Maybe he can get away with it. Look around: how many people who should know better are publicly objecting to what he says? But what Schumann does not realize is that he is cognitively incapacitating and psychologically hurting himself as well as others. His writings are irresponsible.
We can see some of this in his ridiculous ideas about syllogistic argument being typical of Christianity and Communism, while excluded from Judaism:
“While the Christian thinking is totalitarian and holistic, the Judaic thinking is massive-parallel and concurrent, it prefers a singular differentiation. For example, in Judaism there is differentiation between those animals which may be eaten and those which are unclean, between things devoted to the temple and not devoted, etc. This feature of the Judaic thinking has entailed the impossibility of using Aristotelian deductive syllogistics and in fact the latter has been never used in Judaism.”
It should be mentioned that Schumann is from Belarus and of Jewish origin, so no doubt he and his forebears have suffered considerably at the hands of the totalitarian regimes that have plagued that country and region in the recent and more distant past. But that does not justify making empirically inaccurate statements. Firstly, Christianity is not intrinsically totalitarian. Even if it has in the past been pretty nasty (to put it mildly) in various times and places, today in the various countries of Western Europe and North America that I have lived in, and in particular in Switzerland where I happily currently live, no such characterization is remotely appropriate. Most Christians (whether Protestant or Catholic), or people of Christian origin, in the West can safely be said to be essentially liberal and even libertarian; and let us not forget it is their recent forebears who conceived of and established our modern political freedoms, including the freedom to be Jewish without persecution or even discrimination.
I do not know about Eastern Europe nowadays, maybe things are different there; but in any case, Orthodox Christianity is not the whole of Christianity. I do not of course mean to imply that anti-Semitism has disappeared in the West. It is present enough, inexcusably so, mostly in the guise of rabid anti-Israeli propaganda and activism, in many mainstream media, among many (mostly leftwing) politicians and academics, and in many grassroots organizations, including some with Christian orientation. But this is not totalitarianism. If any religion in the world today is totalitarian, it is surely Islam. Most of its preachers, it seems, promote the blind hatred, oppression and murder of Jews, Christians, and any other non-Moslems, and even recalcitrant Moslems, that their followers (who are relabeled as ‘Islamists’ once they concretely commit themselves to this course) can get their bloody hands on. Only God knows how long more this terrorist plague on our poor planet will last, for it will not stop till the Moslem peoples themselves get wise and energetically suppress all those in their midst who advocate or engage in such extreme intolerance.
Both Christianity and Islam – and indeed Judaism too – have given the world many famous scientists and philosophers of science. I have already mentioned Grosseteste, an important contributor to the scientific method, who was a Christian bishop. We could also here mention Albertus Magnus, a monk and later a bishop, who (it is interesting to note in the present context) wrote: “Syllogisms cannot be made about particular natures, of which experience alone gives certainty,” by which he surely meant that knowledge of nature cannot be obtained through reasoning alone but mainly relies on experience. And there are countless more, such as Gregor Mendel, the founder of the modern science of genetics, who was a Christian friar. To depict Christianity as inherently syllogistic, deductive, unscientific and totalitarian is surely a distortion. Islam, too, in the first few centuries of its existence produced important contributors to logic, philosophy and science, even if today it is more intellectually paralyzed than it has ever been, with some of its preachers even still claiming the earth is flat and the sun revolves around it!
With regard to the question as to why, despite the insights of medieval Christian philosophers of science like Grosseteste and Roger Bacon, science did not progress much at the time, Kneale and Kneale (who were genuine scholars and historians of logic) proposed the following answer: “The chief obstacle to steady scientific progress was not the influence of Aristotelian logic or anything else derived from Greece, but a lack of sustained curiosity about things which were not mentioned by ancient authors and did not appear to contribute in any way to salvation” (p. 241).
Secondly, use of the syllogism is not limited to totalitarian régimes as Schumann alleges. He gives the following inane illustration of the syllogistic thinking he considers typical of such régimes:
“Enemies of Soviet people must be taken into camp,
Non-content with Stalin’s policy are enemies of Soviet people;
Therefore, non-content with Stalin’s policy must be taken into camp.”
But this is nothing special; it is just subsumption of a particular case under a general rule. Similar rules and reasoning occur in democratic societies. For instance: ‘people who ride on a bus without a ticket must pay a fine; you did so, so you must pay up!’ Similar arguments can be formulated in any field of human endeavor, even in sports. Only the terms are different; the form of reasoning is the same, and there is nothing intrinsically dreadful about it.
I would have rather thought that the difference between totalitarian and democratic régimes lies in people being forced to obey arbitrary decrees emanating from a powerful minority. Such obedience involves the following reasoning: ‘if I do not do what I am told to do, I will be severely hurt; therefore, although I don’t want to, I will do it’. This is apodosis, not syllogism. But even if we call it syllogism in a broad sense, it is certainly not the formal essence of syllogism, but merely a particular case of it, one with specific terms. Moreover, such reasoning concerns the victims; as regards the dictators and their henchmen, they rarely think rationally. The Nazis were furiously anti-rational, in the name of pre-Christian obscurantism. The Communists may have occasionally paid lip service to science and progress, but in actual practice were just as moved by unbridled passion, imprisoning or killing anyone they imagined to be a menace to their power.
Thirdly, Schumann’s claim that such thinking is an “impossibility” in Judaism and has “never been used” in it is quite untenable. Judaism is certainly filled with rules (mitzvoth) that have to be obeyed under threat of punishment (and with promises of reward). Surely more than in the Christian religion, though perhaps less than in the Moslem one. Moreover, even if the various lists of hermeneutic principles developed during and after Mishnaic and Talmudic times do not explicitly mention syllogism, syllogistic thought processes underlie many of the rabbinic rules. As I explained in my earlier work, Judaic Logic, any inclusion or exclusion of a particular in/from a generality is a mental act of syllogism. Moreover, several of the rules of Rabbi Ishmael having to do with harmonization are clearly syllogistic. Many people fail to see that, because each such rule involves two syllogisms in series, with a total of four terms; but the fact is undeniable. Additionally, in earlier chapters of the present work, in a section on analogical argument (5.1) and in one on the syllogistic middot (8.8), I show how the rabbinic rules of gezerah shavah, binyan av, and others, rely on syllogism. Thus, syllogism is very present, even if often only implicitly, in rabbinic thinking.
Fourthly, contrary to Schumann’s claim, the thought process of differentiation is not specific to Judaism, but found equally in Christianity, Islam, Communism, Nazism, Democracy, astrology, science, sports and what have you. To take an extreme example, Nazi “selections” at the gates of death involved differentiation. Just as no human thought is possible without syllogism, so none is possible without differentiation. It is not an ideological matter, but a cognitive one. To classify things, we must make distinctions between them – observe what features each thing has and what each lacks. To make decisions, and proceed to act on them, we must be aware of differences as well as commonalities. Syllogism can be used equally well for integration and for differentiation. Schumann’s attempt to reserve syllogistic reasoning to Christianity and Communism and differentiation to Judaism is totally unfounded.
Therefore, Schumann’s claims concerning Judaism are also just so much balderdash. Evidently, he is settling personal scores not only with Christianity and Communism, but also with Aristotelian logic, when he tries to associate them. As regards Judaism, he maybe imagines he is doing it a favor by associating it with his brand of non-Aristotelian “logic,” which in his estimate is a more advanced level of logic; but he is in fact just giving Judaism a bad name, making it seem irrational and even anti-rational. He is rather in fact ‘using’ Judaism’s good reputation, as at least somewhat logical, to try to plug his pseudo-logic as a logic with valuable practical applications. The truth is that nowhere does he in a scientific manner substantiate any of his offensive assertions; all his fancy claims are arbitrary and moreover nonsensical. Like the contemporary avant-garde “artists” whose works are exhibited in costly museums, he relies on bluff.
 Respectively: Heusenstamm: Ontos Verlag, 2010; and Piscataway, N.J.: Gorgias, 2010.
 Actually, according to Louis Jacobs in Structure and form in the Babylonian Talmud (p. 56), although “not obliged to pay the full value of the food consumed,” the owner is “obliged to pay for the benefit he has received in that he has been spared the cost of feeding the animal” (Baba Qama 20a).
 It should be said that Schumann is not the first to represent a fortiori argument by means of a table. He is preceded in this by Michael Avraham in 1992 (chapter 20), by Gabriel Abitbol in 1993 (chapter 21), and by Abraham, Gabbay and Schild in 2005 (chapter 25). Also note that, as we have explained in our analysis of Abitbol’s work, while such representation may be (roughly) applicable to some a fortiori arguments (notably those by R. Tarfon here discussed, where the major premises need to be constructed by generalizations), it is not universally applicable (it does not apply to an argument whose major premise is already given or immediately obvious).
 Notice that these two premises are oddly placed in this tabular schema: one is represented by the column on the right, while the other is represented by the bottom row. This asymmetry is significant in that the reasoning involved is made more awkward by this table. Yet such devices ought to facilitate our thinking, not impede it. Furthermore, notice that while we can express comparative relations (like ≥) between elements horizontally (as in premise (ii)), we have no way of doing so vertically (as in premise (i)). Note also the asymmetry in comparisons, with the lower row having “≥” while the upper one has “≤” – note the reversal of direction. All this shows how artificial a construct Schumann’s schema is.
 Note that this is effectively the posture adopted by Abraham, Gabbay and Schild. However, they too, as we saw earlier, did not arrive at the logical conclusion (100%), but instead drew an erroneous foregone conclusion (≥ 50%) so as to appear to reflect the Mishna!
 I did not know all this (and more) at the time I wrote my Judaic Logic, so Schumann can be excused for not having found it there.
 However, this should not be taken to mean that the dayo principle excludes a priori the possibility that greater amounts might be discovered in some other Scriptural text. It just means that no inference is capable of guaranteeing greater amounts.
 To repeat, at the time I wrote my Judaic Logic I had not taken due note of R. Tarfon’s second argument, let alone realized its crucial significance in the debate. But I had already correctly perceived the form of a fortiori argument in general, and thence the form of R. Tarfon’s first argument.
 Schumann does not specify where such statements were supposedly made. As regards Mielziner, all I found him saying is that a fortiori is “a kind of syllogism” (p. 130); but it is clear from the rest of his treatment that he did not say or mean that “an Aristotelian syllogism may be presented as the simplest case of qal wa-homer” as Schumann suggests here, but was simply using the term etymologically (as ‘syn’ + ‘logism’, implying a combination of propositions bound by a common term).
 Note by the way how the symbol of quantitative comparison has changed direction without forewarning or explanation. In the first schema, the top row had ≤ and the bottom row had ≥; whereas here and the Biblical example below both rows have ≥. Furthermore, in the first schema, the second premise referred to the bottom row; whereas here, it refers to the top row; and in the Biblical example below it refers to the bottom row again, though in the opposite direction to previously! Schumann manifestly manipulates his information as convenient to his ideas, hoping to fool everyone. Or alternatively, his thinking is very confused.
 Chapter 4.2. See also its mention in the present volume in the chapter on Louis Jacobs (16.1).
 See Ex. 4:10, where Moses says that he is “not a man of words” and he is “slow of speech and of a slow tongue,” and then v. 13, where he begs God to send someone else on this mission. See also Ex. 6:9, where the word “to hearken” (lishmoa’) is clearly used in the mental sense of internalizing (which the children of Israel fail to do due to “impatience of spirit” etc.) – and not in a physical sense.
 As for the dayo principle, its application in this example would be as follows. If the argument is read as purely a fortiori, invoking the dayo principle in relation to it would mean no more than reminding us of the universal principle of deduction. If, on the other hand, the argument is read as a crescendo, invoking the dayo principle in relation to it would signify that the assumed proportionality is to be rejected. However, in truth, the dayo principle would not be invoked in such a case, because the a fortiori argument is not being used to justify application by a human court of a penalty by inference from a penalty found in Scripture.
 Note too, he makes no mention of the distinction between subjectal and predicatal a fortiori or that between positive and negative a fortiori. All a fortiori look formally alike to him, it seems.
 Another idea expressed by Schumann that shows his misunderstanding of a fortiori argument is found in his explanation of Hillel’s first hermeneutic rule. Here, he writes (p. 14): “…one deduction of the set of concurrent deductions is much more certain. As a result, a certainty of that deduction is expanded to cases of other concurrent deductions.” By “deductions” he here apparently means the propositions concerned (i.e. the minor premise and conclusion); this may be explained by remembering that in his mind they are inferred from the dayo principle. Anyway, what this citation shows is that Schumann thinks that a fortiori argument transmits the greater certainty of the (minor) premise to the heretofore uncertain concluding proposition. This is a common error people make – confusing an ‘ontical’ quantity with an epistemic one (see my treatment of this issue earlier on).
 It is interesting to note that Schumann did not defend his thesis when I posted (in mid-December 2010) the first two sections of the present chapter in my website (I of course wrote to him telling him I did so). If he believed his theory of a fortiori argument was correct, one would have expected him to indignantly protest; but he did not. Alternatively, if my exposition of its faults was convincing to him, one would have expected him to publicly admit them, and indeed thank me for exposing them; he did not do that, either. From his silence we can infer a non-scientific attitude, and maybe that he consciously or subconsciously tried to fool people in the first place.
 Here, we are referred to Schumann’s article: Non-Archimedean Valued and p-Adic Valued Fuzzy Cellular Automata, in the Journal of Cellular Automata, 3(4) (2008), pp. 337-354.
 Though I might mention some of its terms: “a hybrid cellular automaton for Judaic deduction,” “finite or infinite set of elements, called the states of an automaton… collected from statements of the Pentateuch,” etc. All this is, to my mind, bluff intended to conceal the underlying sophistry.
 Perhaps his idea is that a fortiori arguments might be strung together, so that long chains of quantitative comparisons occur, like A > B > C > D > E…. Wow, revolutionary!
 It is formal, though not symbolic – more on this distinction later.
 For more detail, see chapter 15.1, Epilogue: Motives of the Present Research.
 Worth mentioning here as recent examples are the articles by numerous authors in the journal Higayon: Studies in rabbinic logic, edited by Moshe Koppel and Ely Merzbach of Bar Ilan University, which was published in 1989-2001 by Alumna, Jerusalem. It is a pity that this publication has been discontinued. Unfortunately, too, its articles (most of which are in Hebrew, and have not been translated into English) have not been posted on the Internet. See: u.cs.biu.ac.il/~koppel/higayon.html.
 Most of those in Judaic Logic were interesting, but I particularly disliked the essay: “In Search of the Logic of Judaism: From Talmudic Chaos to Halakhic Linearity” by Tzvee Zahavy.
 That example is from the Talmud. Of course, long after the Talmud, Saadia Gaon (882-942 CE), in an effort to resist Karaite skepticism, claimed that all the midot were Divinely revealed to Moses at Mount Sinai.
 Although Schumann does not mention Guggenheimer, his ambition to impose on Talmudic logic the terms and principles of modern symbolic logic is reminiscent of the latter’s.
 He even goes so far in manipulation as to define a fortiori argument as “parallel concurrent deduction” (in inverted commas, yet) in his listings (in an Appendix) of the hermeneutic rules of Hillel and R. Ishmael, as if this idea of his is obvious and already generally accepted.
 See my detailed treatment of this topic in my Judaic Logic, chapter 10.3.
 Note, by the way, that since he makes no attempt to “formalize” all the midot, he can hardly claim to know for a fact that they “have nothing in common with Ancient Greek logics.”
 I can here, offhand, point to one instance where the outcome of a disjunction is an issue. At one point in his Studies (chapter 8, p. 150), Jacobs discusses an expression found in a Gemara (B.Q. 2a-3b) that pertains to interpretation of a disjunction: “they are equally balanced and both of them can be included, for which will you exclude?” Just with reference to this case, one can see that the issue is more complex than Schumann pretends (i.e. it is not a mere issue of particularity or generality of the disjuncts). Moreover, looking at this example, it occurs to me that Schumann makes no distinctions between inclusive disjunction and exclusive disjunction, or between logical (de dicto) disjunction and factual (de re) disjunction.
 London: Taylor & Francis, 2011. The preface here examined (written: May 2010) is posted at: www.tandfonline.com/doi/pdf/10.1080/01445340.2010.506079. The full list of authors and articles in the book can be seen at: www.tandfonline.com/toc/thpl20/32/1.
 I chided him about it by e-mail on 20-3-2012, but received no reply from him.
 See the essay called ‘Syllogism Adds to Knowledge’ in my Phenomenology, chapter 7.4, posted online here: www.thelogician.net/PHENOMENOLOGY/Logic-has-Active-Role-7.htm.
 See for instance the essay called ‘Generalization is Justifiable’ in my Phenomenology, chapter 7.2, posted online here: www.thelogician.net/PHENOMENOLOGY/Logic-has-Active-Role-7.htm.
 See for this my Logical and Spiritual Reflections, book 1, chapter 8, posted online here: www.thelogician.net/LOGICAL-and-SPIRITUAL-REFLECTIONS/Hume/Hempel-Confirmation-Paradox-A8.htm.
 Roughly speaking, the second figure serves to take note of differences between things and ideas or between ideas, while the third figure serves to notice common grounds between them.
 England, 1561-1626; author of the Novum Organon (1620).
 England, ca. 1168-1253, who (according to Freely, pp. 139-141) was “the first medieval scholar to deal with the methodology of science, which for him involved two distinct steps. The first of these was a combination of deduction and induction… The second step was what Grosseteste called verification and falsification… it was one of the basic tenets of his scientific method that if a theory was contradicted by observation it must be abandoned.”
 England, 1806-1873; author of A System of Logic, Ratiocinative and Inductive (1843). Note in his “method of difference” reference to “an instance in which a phenomenon does not occur” (see my The Logic of Causation for an analysis of his methods).
 Prussia, 1724-1804. In Lectures on Logic (p. 176), he says: “By means of hypotheses one… assumes something, and investigates whether from it one can explain the known consequences or not; if the first occurs, then one accepts the hypothesis; if the latter occurs, one rejects it.” (Tr. Michael Young. Cambridge: UP, 1992.)
 Egypt-Irak, ca. 882-942. See my essay on this author earlier in the present volume.
 We could even refer to the Torah, since the laws given Deut. 13:2-4 and 18:21-22 clearly exemplify respectively the positive and negative aspects of adduction, as I have explained in my Judaic Logic, chapter 2.2. See also Appendix 6 of the present volume.
 We should not, by the way, get overly fixated on the fashionable word “falsification,” or for that matter on the word “verification.” These words, signifying the negative and positive aspects of research, are relatively recent (dating from about the 14th-16th centuries). Before their emergence, people no doubt used other words or phrases to signify that they were, by reference to experience and reason, checking out whether a certain idea was true or false, with the implicit intent to opt for it if found true and to drop it if found false. Also, the word “verification” is often intended in the sense of “falsification,” or in a double (positive and negative) sense. Popper increased the separation between the two words so as to emphasize the difference between the positive and negative sides of research; but the underlying concepts they refer to were already there.
 France, ca. 1295-1358. He is quoted by Freely, p. 153.
 I refer the reader to my essay “The Principles of Adduction” for a more detailed analysis of the nuances involved in the process of theory selection. It is not a mere matter of confirmation or rejection, but there are many situations in between, which strengthen or weaken a hypothesis in comparison with other hypotheses. See my Phenomenology, chapter 7.1, at: www.thelogician.net/PHENOMENOLOGY/Logic-has-Active-Role-7.htm.
 I marvel at the incompetence of publishers who publish such drivel. It goes to show the low level knowledge and understanding of their editorial staff.
 I will not bother here to discuss Schumann’s silly claim that Christianity is distinctively “holistic.” For a start, he misuses that word, when by it he means monistic, or pantheistic if not monotheistic (since he quotes Colossians 3:11, “the Messiah is all, and in all,” and the messiah in Christian belief is the same as God). Secondly, these ideas are not peculiar to Christianity; many similar ideas can be found in earlier religions (Judaism, Hinduism) and philosophies (Xenophanes, Parmenides). He obviously does not know what he is talking about. He should seriously study the history of ideas before having the pretention to pontificate on the subject. He should also improve his English.
 For instance, the founding fathers of the U.S.A. were Christians.
 Sharia (Islamic law) is nothing other than an instrument of total control, penetrating every nook and cranny of the lives of people living under its power.
 Judging by videos and texts seen on the Web over the years. See the many examples in jihadwatch.org, palwatch.org, and memri.org, among others. Note that these remarks of mine are written only a few days after the wanton murder of defenseless Jewish children by an Arab jihadist in Toulouse, France.
 Roger Bacon (c. 1214–1294), who is also credited with advances in scientific method, and who seems to have been a student of Grosseteste, was a Franciscan friar.
 Germany, ca. 1200-1280. He was canonized in the 20th century and made “patron saint of all those who cultivate the natural sciences.”
 Austria-Hungary, 1822-1884.
 The paralysis of Islamic thought since the middle ages can be attributed mainly to the limitations on freedom of thought and speech, under the guise of anti-blasphemy laws. At first, Islam showed considerable intellectual energy; but in time the most reactionary elements in Islamic societies imposed their will, stifling any velleity of independent thought. This has been the situation in the countries concerned for hundreds of years.
 Note that, though he refers to it as “often used,” this example is made up by Schumann: he is not quoting any historical document or speech.
 See discussion there in chapter 9.1 (on the elucidation of terms) and chapter 10.3 (on the scope of terms). These rules are not exclusively syllogistic, but certainly include much syllogistic thinking.
 See chapter 11 there. That such rules are not always valid, i.e. in accord with the formal canons of syllogism, is another issue. An attempted syllogism is an object of study under the heading of syllogism (in a broad sense) even if it is found invalid.
 In Schumann’s own earlier example, concerning Soviet practice, there is differentiation between people “non-content with Stalin’s policy” and the content, between “enemies of Soviet people” and non-enemies, between people “taken into camp” and those not taken. Or again, Schuman’s distinction here between, as he has it, Christian syllogistic thinking and Judaic thinking by differentiation, is itself an act of differentiation, though not one given in Judaism.
 As for his statement that “the Judaic thinking is massive-parallel and concurrent” – we have already seen in a previous section the meaninglessness and invalidity of this “massive-parallel and concurrent” thingamajig. The words refer to an imaginary multiplication of a logic formula he alleges is used in Judaism, which we however found to be a quite inaccurate representation of actual practice. It is a hasty generalization from a false observation. In Schumann’s mind, it is something established, as real to him as a perpetual motion machine is no doubt to the dreamer who, having seen it move repeatedly for a while in his mind’s eye, thinks this implies that it can move perpetually in the real world. If anything is massive here, it is his incompetence.
 Just as he ‘uses’ contributors to the collections of essays he edits and prefaces. His introductory remarks are given some appearance of legitimacy, by being published side-by-side (and indeed in a cadre position) with other people’s more thoughtful work. Those thus ‘used’ don’t mind too much, so long as they are getting published. And the publishers don’t care, as long as they look set to make money. ‘To use’ here means ‘to exploit’, as against ‘to serve’.