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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

Appendix 6 – Logic in the Torah

There is evidently quite a bit of logic – inductive as well as deductive – to be found in the Tanakh (the Jewish Bible). Although this document aims, of course, primarily to convey narratives (they did this, they said that) and legislation (do this, don’t do that), it also incidentally – whether intentionally or not – contains quite a few lessons in logic. Perhaps for this reason, Jews have traditionally been rather logical-minded people. However, to date no one (so far as I know) has made a systematic study of this topic, looking for all information of logical interest in the Tanakh. The term ‘logic’ should here be taken in its most comprehensive sense, including not only inductive and deductive processes capable of formal representation, but also logical intuitions, methodological guidelines and epistemological theories, and indeed anything that may conceivably improve cognitive efficacy, for that after all is the main purpose of logic.

As we have seen, researchers have over time found 46 instances of a fortiori argument, including 5 in the Torah proper (i.e. the Pentateuch). In my earlier work, Judaic Logic, I focused on passages in the Torah which contain a couple of very important principles of inductive logic. But there is bound to be much more material of logical significance than that. If a fortiori argument, a relatively subtle and complex argument, is so frequent, we can reasonably expect a great many simpler and more common arguments to be found. Noticing them and correctly identifying them requires some logical skills, of course; not everyone can do this job.

With this thought in mind, I have read once through the Torah looking for new material of logical significance and commenting on it. My findings are given below. The result is a grab bag of miscellaneous logical techniques or principles; it is clearly not an exhaustive toolbox, let alone a systematic teaching. Still, these findings are worth noting, as they may well have influenced and continue to influence Jews and others, consciously or otherwise, into more logical thinking. The findings listed below are, to repeat the findings from merely a first reading of the Torah; more readings will very likely yield additional findings, some perhaps more subtle and profound than those initially identified.

Moreover, I have not yet looked into the rest of the Jewish Bible (the Nakh) for similar material; no doubt quite a bit more should be found there. Nevertheless, I think the work below should pave the way for further research of the same sort in the future, by myself and others.

The first words of Genesis 2:16, “And the Lord God commanded the man, saying,” suggest the possibility of communication between God and mankind, i.e. of prophecy. From a secular point of view, this possibility is not very significant, since most of us do not nowadays lay claim to prophetic powers. But from a religious point of view, it is of course of major significance, and it is implied not only here but throughout the Torah and Tanakh.

In Genesis 2:16-17, God gives Adam the first of all commandments, viz.: “Of every tree of the garden [of Eden] thou mayest freely eat, but of the tree of the knowledge of good and evil, thou shalt not eat of it; for in the day that thou eatest thereof thou shalt surely die.” This passage is of interest to logic, because of the form of its discourse. The second part of it may be stated as: If you eat of this tree, the consequence will be death; therefore, do not eat of it; and the first part as: If you eat of any other tree, no such consequence will follow; therefore, you may eat of it. More symbolically put: if you do X, Y will then occur, and Y is undesirable, therefore avoid X; and if you do something else X’, Y will not then occur, therefore, no need to avoid that other thing X’. Thus, this passage makes use of an if–then proposition, and implicitly of an inverse if–not-then proposition. Moreover, it teaches that if something undesirable would follow a certain action, one should not do it; whereas if nothing undesirable would follow a certain action, one may do it. The latter lesson concerns ethical logic.

Genesis 2:19 tells us that after God formed the beasts of the field and the fowl of the air, he paraded them before Adam “to see what he would call them; and whatsoever the man would call every living creature, that was to be the name thereof.” Now this obviously a statement with logical significance; but what does it mean, exactly? Well, the simplest reading would be that words are arbitrary labels (initially composed of sounds, and later of written symbols) that man, for his own purposes, mentally attaches to the things he perceives or conceives. And this corresponds to the commonsense view of words; so we can say that the author of this statement was saying the obvious. The implication of this simple view, note well, is that words in themselves tell us nothing about the objects they refer to: they are conventional.

Very different is the mystical interpretation of this verse that has developed among Jewish commentators[1]. According to them, language is originally made up of Hebrew letters, which were used by God during Creation to fashion the things created. When Adam named things, he was actually seeing into their essence and intuiting the verbal elements constituting them. This is clearly a much more complex theory of words, with metaphysical as well as epistemological implications. And, note well, it is in direct opposition to the simpler, commonsense view, since it claims that words (or at least, the Hebrew words used by God and Adam) are significant in themselves, as against conventional.

My point here is not to judge the matter, but merely to draw attention to the philosophical issues raised by this statement. Nevertheless, I would personally opt for the non-mystical theory, in view of the practical difficulties inherent in trying to prove the mystical one. The latter can only be classed as speculative, since (as far as I can see) we have no way to ever inductively prove it. Of course, we can conceive the things of this world as consisting of vibrations (of force fields, presumably) – but how would we scientifically establish that these vibrations correspond to those of Hebrew letters? Anyway, the commentators who make this claim have not demonstrated the correspondences. Therefore, by elimination, the commonsense view is more convincing.[2]

Genesis 2:21-24 tells us that God made the first woman (Eve) by taking one of the ribs of the first man (Adam), then declares: “Therefore (al ken) shall a man leave his father and his mother, and shall cleave unto his wife, and they shall be one flesh.” This passage is logically interesting because it says “therefore,” implying that an inference is taking place. But what inference is intended here is far from clear! Why should the fact that the first man and woman were “of one flesh” imply that men should, or even merely would, thenceforth leave their parents and cleave to their wives? For a start, Adam and Eve had no parents, so they provide no example to follow. Secondly, women after the first were not formed from a rib of their husband, but by ordinary reproduction; so the unity of later couples is not as literal.

Thirdly, leaving aside those first two objections, why would couples who are “of one flesh” not stay with the husband’s parents? In fact, they often do stay with his parents (or at least, did so till modern times), and even sometimes with hers (though this is not excluded here). No explanation is given, in the text or in subsequent commentary. Sure, one can readily see the purpose of the statement – to teach that the new couple is, or is to be, a new psychosocial unit, distinct from the parental units. But this is a supplementary insight, rather than a deduction; so why say “therefore”? The translation of al ken into “therefore” seems sound; the literal meaning of the phrase is ‘on yes’ – i.e. on this basis, because of this, this implies.

So we may have here an early instance of ‘dud’ inference; that is to say, of something presented as inference, which is in fact not at all inference. The use of a marker like “therefore” makes it look like inference, but it is not really so. In short, this is rhetoric, rather than logic. It is interesting to find this mode of communication being used in the Torah. Such use is perhaps innocent – not intended to fool anyone, but merely to make the narrative seem more continuous.

Another tack. Needless to say, all the above is said within the framework of the Torah world view. But of course, modern science has a very different world view. The Torah view is one claimed to be based on revelation, and deduction and to a lesser extent induction therefrom; whereas the scientific view is one based on publicly accessible empirical data, and induction and to a lesser extent deduction therefrom. According to modern science, there was no first man and woman, and the world was not created less than 6000 years ago. The Big Bang, which started the expansion of the material universe, occurred some 13.8 billion years ago (based on complex astronomical measurements). Our planet, Earth, was born some 4.6 billion years ago, soon after the Sun (long after most stars visible in the sky, note) and before the Moon. Life on Earth began about 3.9 billion years ago (we do not know exactly how). Mankind is the product of a long evolution of life forms since that time. The age of our species is difficult to gauge, simply because it gradually emerged in an evolutionary continuum; but we can say, even if somewhat conventionally, that it is about 200,000 years old (which means, very, very recently).

As regards sexual differentiation, it has a long and rich natural history. Certainly, males did not precede females in our species, but both existed in earlier species well before our species arose. Therefore, the Torah here, as in many other contexts, is a world apart – a factually inaccurate report. This is, of course, indicative of logical errors; especially, not enough inductive logic was used. The Torah account may seem convincing to some people, due to their limited scientific knowledge; but the conclusion of inaccuracy and error is inevitable to anyone who studies the matter objectively. I take no pleasure is saying this, and I am certainly not the first to say it. Therefore, we must view many of the Torah claims – especially concerning prehistoric times – as myths, legends and opinions. We can still learn many valuable lessons from this venerable document, which is after all a message from our ancestors; but we should remain coolheaded. It may conceivably have been Divinely inspired, but some admixture of human ideas must have occurred in the process, since it contains factual errors.

In Genesis 3:1-5, the serpent tries to tempt Eve by means of the following argument (here paraphrased): “since God did not say ‘you shall not eat of any tree of the garden’[3], then you may eat of this tree;” to which Eve rightly retorts, briefly put: “He said we may eat of all trees except this one.” We can discern in this a teaching of logic, namely that the serpent’s inference from ‘not all X are Y’ (i.e. ‘some X are not Y’) to ‘this one X is not Y’ is fallacious, and learn from Eve’s reply that a proposition may be general and exceptive, i.e. have the form ‘all X except this one are Y’. It is easy to see how readers of such discourse absorb, over time, if only incidentally, lessons in logic.

Genesis 4:15 provides the first Biblical example of syllogism and apodosis in the Torah. In response to Cain’s complaint (in the preceding verse) that “whosoever findeth me will slay me,” God declares: “whosoever slayeth Cain, vengeance shall be taken on him sevenfold.” The form of this statement is: anyone who does X, will be subject to consequence Y. Understanding such a statement requires the ability to subsume a particular case under a generality Although this is not explicitly exemplified here, it is understood that if ever any individual does indeed do X, the consequence Y will befall him. That is to say:

Any man who does X will receive Y;

and this man did X,

therefore, this man will receive Y.

This is syllogism (1/ARR) if the major premise is viewed as categorical, or apodosis (modus ponens) if it is viewed as conditional. Clearly, the argument may be described as application of a general rule to a particular case. The same arguments are implicit in all the Biblical statements that detail a negative consequence of disobeying a commandment. For example, in Exodus 12:15: “whosoever eateth leavened bread from the first day until the seventh day, that soul shall be cut off from Israel.” There is no need to list all cases.

This is admittedly not a fully explicit demonstration of syllogism and apodosis in the Torah, since the minor premise and conclusion are not stated, but obviously the Torah takes for granted that we understand the word ‘whosoever’ and can grasp the practical significance of an if–then statement. We do have an example of concrete application of an abstract rule, in the case of breach of the Sabbath, described in Numbers 15:32-36, where a man disobeys the interdiction to work on the day of rest and is consequently put to death[4]. Further on, e.g. in Leviticus 21:21, we do find more explicit apodoses.

In Genesis 4:23-24 we find the Torah’s first example of a fortiori argument, which is (more precisely) an a crescendo argument. Lamekh, who was the father of Noah (5:28-29), argues: “If Cain shall be avenged sevenfold, truly Lamekh seventy and seven-fold.” The statement about Cain being “avenged sevenfold” is a reference to 4:15, which reads: “whosoever slayeth Cain, vengeance shall be taken on him sevenfold.” This interpretation and translation[5] is not very clear. The statement is by God, who intends thereby to prevent people from slaying Cain, who murdered his brother Abel. Fair enough; but how would the person who slays Cain be punished “sevenfold” – he can only be executed once, unless we admit of reincarnation! For this reason, I gather, Rashi (and other commentators follow suit) reads the statement as meaning that the punishment of Cain must be put off for seven generations.

Given this interpretation, we can better understand Lamekh’s thinking. Rashi[6] reads his statement as meaning: “If Cain killed intentionally, [and yet] his punishment was delayed for seven generations, [then] I, who killed unintentionally[7], surely will have my punishment deferred for many periods of seven generations.” This is, as already mentioned, an a fortiori argument; or more precisely, it is an a crescendo argument, since the conclusion has a larger quantity (77) than the premise (7). And the argument is logically credible: unintentional killing is far less culpable than intentional killing, and so deserves proportionately less punishment, and/or longer deferral of punishment. All this may be viewed as, incidentally at least, a teaching of logic.

There are four more examples of a fortiori argument in the Torah, namely: Genesis 44:8, Exodus 6:12, Numbers 12:14, Deuteronomy 31:27. Each of these contains some interesting lesson of logic, since their forms vary. The first is positive predicatal in form, the second negative subjectal, and the last two are positive subjectal. All four are (in my opinion) purely a fortiori, unlike the above argument by Lamekh. The argument in Numbers plays a big role in Talmudic discussion, and is there considered by the Gemara author(s) as a crescendo. I need not here say more about them, since I deal with them in plentiful detail elsewhere.

Genesis 8:6-12 may be viewed as a lesson in inductive logic, since it describes a deliberate experiment. This passage is part of the deluge story: Noah, while still in his ark, together with his family and samples of all animals, at some point “wanted to see if the waters were abated from off the face of the ground.” To test the matter, he first “sent forth a raven;” this bird “went forth to and fro, until the waters were dried up from off the face of the earth” – meaning that the raven kept flying around, either close to the ark or elsewhere, without returning into the ark[8]. Noah then “sent forth a dove,” but she “found no rest for the sole of her foot, and she returned unto him to the ark, for the waters were on the face of the whole earth.” A few days later, “he again sent forth the dove out of the ark;” and this time she “came to him at eventide, and lo in her mouth an olive-leaf freshly plucked.” In this way Noah “knew that the waters were abated from off the earth.” A few days later, he “sent forth the dove” again, “and she returned not again unto him any more.”

This story gives its readers a clear example of experimental method. Noah sends birds out to scout the surrounding countryside, and judges the level of water there by observing their subsequent behavior. The experiment seems to be a flop with a raven: its behavior is not conclusive. He instead tries a dove, three times. When she first returns, he assumes this means that she found no rest elsewhere (this conclusion is a bit doubtful, if doves behave like homing pigeons or if they are attached to their mates; moreover, the world is a big place, which it would take a long time for a dove to cover and a dove’s energy is not unlimited). When she next returns with an olive leaf in her beak, he assumes this means she found some dry land or at least an olive tree branch above the waters (though strictly it might have been floating on the water, even if it seemed fresh to him). The last time, she does not return, and he presumably concludes that it went off to get on with its life (though strictly it might have died – it is after all surprising that it did not return for its mate). The conclusions Noah draws are reasonable, even though only probable.[9]

Of course, in view of its divergences from scientific fact, we can hardly doubt that this story is legendary[10]. For a start, where would the quantity of water required to cover the earth’s surface to the peaks of the highest mountains come from? Gen. 1:6 mentions “waters above the firmament” and “waters below the firmament,” and 7:11 states that “the fountains of the great deep [were] broken up, and the windows of heaven were opened.” As regards the “fountains of the great deep,” people in antiquity[11] believed the waters of rivers and seas were overflows from subterranean reservoirs of water. As regards the “windows of heaven,” our forefathers thought that rain came from a heavenly reservoir of water[12]. But these beliefs are now known to be empirically inaccurate. This is not said to deny God and his kindly providence – it only means that the Bible’s particular theory as to how that providence proceeds is fanciful. The text cannot be taken literally; it is at best a poetic statement.

Secondly, there are no geological or other traces of a worldwide flood about 4000 years ago. Of course, God may have erased all traces of the event – but why would He do that? Note that four millennia are not so long ago; yet, there is no evidence that history was suddenly interrupted. Peoples across the globe did not disappear, to be replaced by descendants of Noah, as is obvious from genetic and cultural evidence[13]. Thirdly, I have not made the calculation, but I very much doubt that two specimens of each and every animal species existing in the world at the time concerned would have fitted into a space the size of Noah’s ark. People at the time this story was told were evidently not aware of the enormous number of animal species in the world. What of animals in the Americas, in Australia and other far off places – how did they get to the ark? Of course, God might have performed miracles, teleporting the beasts, and shrinking them or expanding the ark (and of course, undoing all that after the flood) – but to affirm this would be sheer conjecture and interpolation, for it is nowhere hinted at in the proof-text.

Genesis 18:16-32 can be viewed as a lesson in logic, if we understand the term logic in a broad sense including methodology. What is taught here, by means of an example, is the need to ask questions, and not passively accept things that seem unreasonable, if we want to get to know reality. Inversely, we should know when to stop asking questions, for we cannot expect answers to an infinite number of questions. When an answer satisfies one’s rational faculty, there is no need to press further. One may of course press a little bit further, just to make sure; but at some point one must stop. It may, however, be that some time later, one does feel the need to ask more questions, because one’s context of knowledge has changed somewhat and so one’s reason requires revised answers. Different people may, of course, due to having different contexts of knowledge, ask different questions and different numbers of questions.

All this can be seen in the story told in this passage. God tells Abraham that He may destroy Sodom and Gomorrah if their sin is as grievous as the cry of it suggests. There is a moment of silence, during which Abraham reflects on this and decides to ask a question: what if there are some righteous men among the wicked people there – would God refrain from destroying the cities for their sake? Abraham first asks the question with reference to 50 righteous men, and God answers He would forgive for their sake. Abraham then tries 45, then 40, then 30, then 20, then 10 righteous men – God answers every time that in such case He would not destroy[14]. Below 10, Abraham asks no further question – he is satisfied with that number as a minimum; there has to be some justice, after all.

It is noteworthy that God does not preempt Abraham’s question-asking outburst by specifying at the outset that He would refrain from destruction even if there are only 10 righteous men. Nor does God answer Abraham’s first to fifth questions that way. Nor does He state that the answer to Abraham’s sixth question is the last. This suggests that God’s relationship to Man is interactive. God leaves blank spaces in His discourse, inviting Man to ask questions about them and eventually to try filling them in. This preference of God for dialogue is also evident in the passages relating to the daughters of Zelophehad (later on, in Numbers). God did not create human beings as automatons, never questioning anything, born only to shut up and obey. He endowed each one of us with a rational faculty and some intelligence, giving us means to think for ourselves, to ask questions and look for answers.

Genesis 23:18-27 provides another lesson in inductive logic. The patriarch Isaac asks his son Jacob to draw near so that he may feel him and tell whether he is indeed Esau (Jacob’s brother, whom he is impersonating). This is also experiment – using one’s senses to answer some material question. This story could be viewed as a forewarning that while one can test ideas empirically, one must be careful to construct the trial in a sufficiently conclusive manner. For here, Isaac was indeed (luckily) fooled by Jacob, being led to believe he was Esau by means of fake hair. And this, even though Isaac suspected something was wrong, since he reflected that the voice of the person before him sounded more like Jacob’s than Esau’s. From a logical perspective, he should have investigated the matter further before drawing a conclusion, even if it was in this case a good thing that he did not.

Exodus 2:11-14 may be construed as illustrating legal reasoning by analogy, and more specifically a possible illustration of the hermeneutic principle of gezerah shavah. Here, we are told that when Moses “saw an Egyptian smiting (makeh) a Hebrew, one of his brethren” he “smote (vayakh) the Egyptian, and hid him in the sand.” Later on, seeing two Hebrews “striving together,” he says to the bad guy: “Wherefore smitest thou (takeh) thy fellow?” To which, the man retorts: “Who made thee a ruler and a judge over us? Thinkest thou to kill me, as thou didst kill the Egyptian?” This causes Moses to worry that his earlier summary execution of the Egyptian might be common knowledge; and eventually, when Pharaoh hears of the matter and orders Moses be killed, he flees the country.

Now, the analogy here is proposed by the bad Hebrew man, but he is suggesting that Moses might be thinking along those lines. The argument conceived is: just as Moses smote the Egyptian who smote a Hebrew slave, so he might smite the bad Hebrew who smote another Hebrew. The situations might superficially be construed as analogous, in that in both cases there is one man striking another and a third comes along and kills the striker. The analogy is perhaps not perfect, since the first case seems to have to do with political injustice (an Egyptian oppressor smiting an oppressed Hebrew), while the second case seems to concern a more private dispute (a Hebrew, supposedly unjustly, smiting another); but both events seem to involve a bully severely beating up and perhaps killing an innocent person.

The repeated use of the same verb to smite (lehakot) suggests the hermeneutic process of gezerah shavah. To begin with, since the violence by the bad Hebrew (harasha) is described with the same verb (to smite) as the violence by the Egyptian, we may suppose that the two crimes were the same in their severity – this is the main gezerah shavah. But what was the crime involved – was it merely a severe beating, or was it blows resulting in death? We are not told whether the two victims died or not. Yet Moses evidently considered the Egyptian sufficiently criminal to kill him. Moreover, the bad Hebrew asks whether Moses intends to kill him too. What legal principle was Moses appealing to?

Two explanations of Moses’ response are possible. One is to suppose both victims to have been actually killed; the other is to suppose that Moses intervened to prevent their likely murder. In the former case, Moses was punishing or about to punish a murderer with the death penalty[15]. In the latter case, Moses was acting preemptively, on the principle that a ‘pursuer’ (rodef) can be killed if that is the only way to save the life of his victim[16]. The Torah’s use of the verb to smite to characterize Moses’ killing of the Egyptian may suggest that this word is intended to refer to killing; and this may be viewed as confirmation that the two aggressions (the Egyptian smiting his victim and the bad Hebrew smiting his) were also killings. Alternatively, smiting may here mean, more loosely, striking – possibly, but not necessarily, to the extent of killing – in which case, Moses’ smiting of the Egyptian may be known to have been a killing by inference from the fact that he thereafter buried the corpse.

That Moses’ approach is legalistic seems implied by the bad Hebrew’s question: “Who made thee a ruler and a judge over us?” The bad Hebrew, of course, is not aware of the Torah’s use of the verb to smite. When he asks Moses: “Thinkest thou to kill me, as thou didst kill the Egyptian?” he characterizes Moses’ killing of the Egyptian – which he presumably witnessed, or maybe just heard about – as killing, using the less ambiguous verb leharog, and he uses that same verb for what he anticipates to be Moses’ reaction to his own deeds. That this argument by analogy offers a template for the hermeneutic technique of gezerah shavah is evident from the important role that verbal equation plays in it. This is quite an interesting finding, suggesting that gezerah shavah has indeed, like qal vachomer, a Biblical origin.

Exodus 3:2-3 contains an important teaching of inductive logic; namely that if you have some theoretical belief in mind, and it turns out to be belied by empirical observation, you should look for a way to harmonize the two (either by particularizing the theory or by revising the initial observation by means of additional observations).

Here, Moses comes across a bush that apparently burns with fire and yet is not consumed; and naturally he wonders “why the bush is not burnt.” All of us would, on the basis of past experience, firmly expect the burning of a bush (cause) to be invariably followed by its consummation (effect). If this expectation seemed belied by the facts, we would wonder what optical illusion was involved: maybe the fire we glimpsed was not really fire, but a dance of colorful butterflies; or maybe the bush was not really a wooden bush, but a metal replica of one; or maybe both the fire and the bush were real, but their conjunction was illusory, made possible by some arrangement of mirrors. How would we find out? Normally, we would do so by more careful observation of the facts on the ground.

And this is what Moses was evidently about to do, moving towards the burning bush to observe it more closely. But in this particular case, a miracle was involved, and he was advised not to come closer. A miracle, by the way, would not annul the categorical statement that ‘all burning bushes are consumed’ or the hypothetical statement that ‘if a bush is burning, it must be consumed’ – for these statements are generalizations from experience in a natural context, and miracle is by definition non-natural or supernatural. In miracle, God willfully prevents nature from taking its normal course, so this is outside the range of ordinary human knowledge. But we can anyway, with reference to Moses’ initial reaction to the surprising phenomenon, learn from this episode a valuable lesson about inductive reasoning – viz. to research the matter further, when something unexpected occurs.

Exodus 18:13-26 exemplifies the logical process of formulating causal propositions relating means to ends. Such propositions are composed of if–then clauses. Although there are, of course, many if–then propositions in the Torah, before and after this passage, here we witness the reasoning through which one was constructed. Jethro, Moses’ father-in-law, observes that Moses spends much of his time, and the time of his people, sitting in judgment in particular cases. This is, in Jethro’s eyes, “not good,” because it will unnecessarily fatigue both Moses and the people. He recommends that Moses, instead of this, concentrate on obtaining from God the general laws, and appoint and train qualified men to deal with their application to particular cases. These men are placed in a hierarchy, with Moses at the top. In difficult cases, where all lower-level judges experience doubt, Moses might still be called upon to decide. Moses follows his advice.

From this story we learn that, seeing a faulty state-of-affairs (say, not-Y), one must look for ways (say, X) to correct it (i.e. produce Y). That is, one must find the causes (X) that give rise to the desired effect (Y). This is causal logic: ‘if not-X, then not-Y, but if X, then Y’; that is, ‘if the corrective measures (X) are not done, the negative situation (not-Y) will endure, whereas if the corrective measures are indeed done, the negative situation will be eliminated’. Of course, the appropriate measures cannot be chosen out of context. This is made clear by Jethro when he wisely adds: “If thou shalt do this thing, and God command thee so, then…” (my italics), which may be taken to mean: but remain aware of the wider context, i.e. your ultimate values (in this case, God’s will). To avoid collateral damage from the pursuit of particular goals, one must not proceed ‘with tunnel vision’, but must act in harmony with the totality of one’s value-system. In the event of conflict between values, obviously the higher values ought to be preferred to the lesser ones. Such reasoning is central to ethical logic.

Exodus 20:12 and Exodus 23:1-3, 6-8, and similar passages in Leviticus and Deuteronomy, proffer guidelines or rules addressed primarily to judges functioning within a formal justice system. But they can also be read more broadly as applicable to anyone at all times, since we are all throughout our lives called upon to judge between truth and falsehood. There are two aspects to this cognitive imperative: accuracy in collection and reporting of facts and wisdom in judgment. Regarding the first aspect, viz. ensuring the factual accuracy of our knowledge, we can be inspired by the following instructions of the Torah:

Ex. 20:12 – “Thou shalt not bear false witness against thy neighbour.” Ex. 23:1 – “Thou shalt not utter a false report; put not thy hand with the wicked to be an unrighteous witness.” Ex. 23:2 – “Thou shalt not follow a multitude to do evil; neither shalt thou bear witness in a cause to turn aside after a multitude to pervert justice.” Ex. 23:7 – “Keep thee far from a false matter; and the innocent and righteous slay thou not; for I will not justify the wicked.”

We may be factually inaccurate either through negligence in our cognitive ways or through more or less deliberate prevarication. This may seem without consequence to some people, but that is because they do not realize how much they harm themselves as well as others through such policies. In the short term, it may seem innocuous; but over time, the harmful consequences come round. Logic is founded on the idea of utter respect for reality; on the idea that knowing reality is certainly a major component of the highest good, if not the summum bonum itself. To divorce one’s mind (and eventually the minds of others) from reality is to sentence oneself (and others) to a sort of prison, where the soul is out of contact with the world at large – that is, to a delusional state[17]. The business of cognition is a delicate matter, requiring constant attention and conscientious observation and memorization. Moreover, one should have respect for and confidence in one’s own faculties, and not let one’s perceptions and conceptions be influenced unduly by the say-so of other people, be they a few authorities or a big crowd. One may of course learn from others, but one should do so in a careful manner and not irrationally. By training oneself, one can develop a mind that is both independent and balanced. Regarding the second aspect, viz. judging perspicaciously, we can be inspired by the following instructions of the Torah:

Ex. 23:3 – “Neither shalt thou favour a poor man in his cause.” Ex. 23:6 – “Thou shalt not wrest the judgment of thy poor in his cause.” Ex. 23:8 – “And thou shalt take no gift; for a gift blindeth them that have sight, and perverteth the words of the righteous.”[18]

This can be taken to mean: Don’t let your judgment be swayed by extraneous considerations, but judge honestly in accord with the facts of the case at hand. Good judgment is an exercise in common sense. The poverty, or the richness, of a man has nothing to do with his innocence, or with his guilt – for a man, whether outwardly rich or poor, is subject to the same inner constraints and liberties, and may do good or bad equally. Similarly, ideological preference for the underprivileged or for the elite, or greedy desire for social or material rewards, should not be allowed to impinge on the impartiality of one’s judgments. Emotion is all very nice, but when it comes to judgment one should try to be cool-headed. Objectivity is necessary to good judgment, and every effort should be made to ensure one has it.

Leviticus 10:9-11 teaches us a principle which has to do with epistemology, even though not with formal logic. This reads: “Drink no wine nor strong drink… that ye may put the difference between the holy and the common, and between the unclean and the clean, and that ye teach the children of Israel all the statutes, etc.” This passage is followed by a great many laws relating to allowed and forbidden foods, to ritual purity and impurity, and so forth – so it may be viewed as an introduction to that field of religious study. Nevertheless, we can also regard it more broadly as a recommendation relating to the pursuit and transmission of knowledge in general.

If we want to distinguish things accurately, we must have a clear head. Alcohol can, at least when consumed in excess, negatively affect our cognitive abilities. Moreover, although the text does not explicitly say so, we can generalize from it and say the same for all psychotropic drugs – marijuana and the like. We can reasonably do that, since the Torah explicitly tells us why alcohol is undesirable – because (that, ve) it affects our ability to discriminate and to teach. Thus, this passage can be viewed as a general warning that, if we want to discover and transmit truth (in particular religious truth, but more generally any truth), it is wise to avoid alcohol (and more broadly, any substance abuse that affects the mind adversely).

The reasoning through a common cause proposed here suggests that the rabbinic hermeneutic principle of binyan av (lit. ‘father construct’) has some root in the Torah. When the reason for a Torah commandment is explicitly given, as here, or (presumably, by extension) implied with sufficient clarity, the same reason might be used to generate other commandments. This would constitute inductive reasoning, since we are in effect generalizing from the given Torah commandment (whether positive or negative) to a new ruling; i.e. we are assuming that the reason given (or implied[19]) is applicable not only to the specific Torah commandment, but to other, similar situations. Such assumption is inductively reasonable, provided no counterevidence makes it unreasonable; i.e. unless or until we find the apparently general rule belied elsewhere in the proof-text.

Leviticus 10:16-20 provides us with another epistemological guideline. This passage recounts how Moses “diligently inquired” regarding the correct performance of some ritual, and “he was angry” when he discovered that an error had been made. He asks Aaron and his sons why they did things as they did, and Aaron gives him a reply he finds satisfactory. Finally, “when Moses heard that, it was well-pleasing in his sight.” We learn two things, here, of value to all pursuit of knowledge: (a) it is virtuous to diligently inquire into every significant issue, and not just let apparent problems pass without trying to solve them; and (b) when one discovers one was wrong in some contention, one should be happy to have learnt the truth, and be grateful rather than resentful to one’s opponent.

Leviticus 11:3-4 – and a similar passage in Deuteronomy 14:6-8 – is of some logical interest. This passage allows Jews to eat beasts that “chew the cud” and “part the hoof,” but forbids them those that “only chew the cud” or “only part the hoof.” There is, however, no explicit mention here of beasts that neither chew the cud nor part the hoof: can they be eaten or not? Presumably, the implicit intent is that subsequent verses will clarify the matter – i.e. some may be eaten, and some not.

In any case, this passage can be used to exemplify the various ways two items X and Y and their negations can be combined: in “X and Y,” in “X but not Y” (i.e. “only X”), and in “Y but not X” (i.e. “only Y”). This list is missing a fourth possibility, viz. “neither X nor Y” (that is, “both not X and not Y”). Thus, the list of three combinations really refers to the inclusive disjunction “X and/or Y.” Although the Torah contains no explicitly formal logic, by providing this example it teaches people immersed in it how to think of the different possibilities of combination between any two items, X and Y.[20]

Leviticus 13 is of logical interest in that it describes how the plague of ‘leprosy’ in people or objects may be recognized through certain indices by a qualified priest. Needless to say, I am not here granting credibility to the description of this so-called leprosy and its symptoms. All that I wish to draw attention to is the example of looking at alleged symptoms of an alleged disease, and deciding whether they are indeed indicative of the disease or not. It is this logical relation between a sign and what it signals that are of interest from a logical perspective. Under what conditions do symptoms point to a disease?

The answer, according to causative logic, is as follows. The disease X may be said to be the cause of its symptoms Y1, Y2, Y3, etc. That means that the relation: “if X, then Y1 and Y2 and Y3, etc.” is true. It does not follow from this that the symptoms Y1, Y2, Y3, etc., individually (i.e. separately) imply their underlying cause, or even that they collectively (i.e. in conjunction) do so. For it may be that these very phenomena, Y1, Y2, Y3, etc., individually or collectively, are also implied by a cause other than X. However, in some cases we might know by observation (or, in a Torah context, by revelation) that these symptoms, either individually or collectively, are exclusively implied by the cause X. In such event, the presence of any one of them, or certain combinations of them, or all of them, as the case may be, may be taken to imply the presence of the cause X.

Although the Torah does not say all that in the referenced chapter, it does sufficiently hint at it in its precise descriptions of what signs are or are not conclusive for the diagnosis of ‘leprosy’. Note that in some cases the passage of a certain amount of time stands as one of the indicators. The reader who pays attention will take note of the intricacies involved, and absorb a lesson about causative logic in more abstract terms.

Leviticus 18:24, which reads: “Defile not ye yourselves in any of these things; for in all these the nations are defiled, which I cast out from before you,” contains a teaching of logic – namely, that the quantity ‘all’ does not necessarily mean ‘all together’ (all collectively), but may in some situations mean ‘every one of’ (all distributively). Although in the Hebrew original the same word (kol) is used in both instances, the context makes clear that there is no need to do literally all the deviant sexual acts referred to in the preceding verses (viz. 6-23) to be defiled; doing any one or more of them is enough to be defiled. This is evident from the repetition of the word ‘defiled’ in a couple of the listed cases (namely, in v. 20 and 23). Note that this understanding of the passage is explicitly upheld by the rabbis in the Talmud (Makkoth 24a).

Leviticus 19:11 has obvious logical implications. “Ye shall not steal; neither shall ye deal falsely, nor lie one to another.” When we lie to other people, whether outright or by intentionally misleading them in our discourse, we deal falsely with them and steal their credulity. Lying implies dishonesty in the speaker and disrespect for the one spoken to. A person with respect for reality and for other people avoids such behavior, knowing that in the long run it is sure to cause him confusion and loss.

Leviticus 19:15 and 19:35 are passages of the Torah primarily addressed to judges, but which can be interpreted more generally. The first verse reads: “Ye shall do no unrighteousness in judgment; thou shalt not respect the person of the poor, nor favour the person of the mighty; but in righteousness shalt thou judge thy neighbour.” This is a teaching that we should not allow our judgment to be distorted by favoritism for this or that prejudice, but judge matters in a lucid and honest manner.

The second verse reads: “Ye shall do no unrighteousness in judgment, in meteyard, in weight, or in measure.” This gives us two very important teachings in inductive logic. The lesson of the second verse is that empirical studies must be accurate: if we want to discover reality and not get lost in illusion, we must make thorough and precise observations or experiments. The lesson of the first verse is that rational interpretation of the data thus collected must be objective, impartial and intelligent; we have no right to ignore or twist facts and fake conclusions. Truth is a paramount value, not to be cynically compromised for whatever motive. (Similar lessons can be found in Exodus and Deuteronomy.)

In Leviticus 21:21, we are given an explicit positive apodosis (modus ponens), i.e. an argument consisting of a hypothetical major premise, a categorical minor premise that affirms its antecedent, and a categorical conclusion that affirms its consequent. This passage first tells us that no priest who has a blemish “shall come nigh to offer the offerings… made by fire” – this is the major premise; then it adds: “he hath a blemish; he shall not come nigh to offer the bread…” – these are the minor premise and conclusion. Although in truth the major premise concerns offering “the offerings… made by fire” and the conclusion concerns offering “the bread,” so there is a change of term that is not strictly deduced, nevertheless I would say that a partial deduction (as regards a priest with a blemish not coming nigh) is indeed intended, because of the categorical form of the statement “he hath a blemish.” This statement clearly serves as an explanation for the inference from the previous sentence to the next.[21]

Numbers 13:17-20 may be looked at as example of planned research – i.e. as a lesson in methodology. Here, Moses tells the twelve men chosen to spy out the promised land what information to look for: they are “to see the land, what it is; and the people that dwelleth therein, whether they are strong or weak, whether they are few or many; and what the land is that they dwell in, whether it is good or bad; and what cities they are that they dwell in, whether in camps or in strongholds; and what the land is, whether it is fat or lean, whether there is wood therein, or not.” Thus, when we have a research project, we should start by thinking about the kind of material information we shall need to collect in order to arrive at an answer to the question asked.

Numbers 14:11-16 has significance for inductive logic, in that it shows that an event might have different explanations. Here, God is angry at the children of Israel for doubting in His capacity to get them into the promised land; and He proposes to wipe them out and replace them with another people, descended from Moses. Moses argues with Him that if He carries out this threat, other peoples will think that He killed the Israelites because He was unable to do with them as He had sworn, i.e. unable to give them the promised land. Moses thus points out that the destruction of the Jewish people by God (the event referred to) would be wrongly interpreted by the nations: instead of understanding God’s true motive (exasperation with the Israelites’ weak faith), they would assume that God was motivated by lack of power (inability to achieve what He set out to do).[22]

Symbolically put, an event E may have alternative causes C1, C2, etc. Each cause implies the same event: i.e. ‘if C1, then E’; ‘if C2, then E’; etc. are all true. Thus, observing the occurrence of E, one cannot logically tell whether its cause is C1 or C2 or what. That is to say, positive apodosis from the presence of the consequent to that of the antecedent is logically invalid. We can also learn from this that the absence of any one antecedent (say, C1) to the absence of the consequent (E) is invalid, since the consequent might occur through the agency of another antecedent (say, C2). In other words, if C1 implies E, it does not follow that E implies C1, or that not-C1 implies not-E; and similarly for C2, etc. Thus, to correctly identify the causes of events, one must remain aware that there may alternative explanations and not rush to judgment. We should not assume the first hypothesis that comes to mind to be correct, but consider what other hypotheses might be equally (or more or less) credible.

Numbers 16:28-33 can be viewed as another lesson in inductive logic. Moses tells the children of Israel to observe what will befall the rebels Korah, Dathan and Abiram: “If these men die the common death of all men… then the Lord has not sent me. But if the Lord make a new thing, and the ground open her mouth, and swallow them up, with all that appertain unto them, and they go down alive into the pit, then ye shall understand that these men have despised the Lord,” by doubting that He chose Moses. And indeed the latter prediction is realized. This argument can be clarified by putting it in more symbolic terms. There are two conflicting theories: that God chose Moses (T) and that He did not (not-T); and they make two conflicting predictions: that the rebels will perish in ordinary circumstances (not-P) and that they will perish in an unusual manner (P). The argument is, then, as follows: If not-P, then not-T; but if P, then T. Since P came true, rather than not-P, it follows that T is true, and not-T is false.

The argument is formally valid and its content is reasonable (assuming that natural means like high explosives were not available to Moses and that these events indeed took place). However, it should be noted that this is a special case, in that the relation between the theories and their predictions is here exclusive and exhaustive; this is what makes possible the inference from the predictions to the theories. In most cases, the problem is more complicated in that the theories may number more than two and they may have some predictions in common, and their other predictions may differ but not be known to be in conflict. In such cases, inference from a prediction to a theory is not logically possible – and we decide between the theories on the basis, rather, of comparatively how many successful predictions they respectively make, and especially with reference to unsuccessful predictions if any. In other words, the truth of a prediction is less decisive that the falsehood of a prediction, for whereas the former cannot prove the theory implying it (unless only that one theory is compatible with it), the latter makes possible immediate elimination of the theory implying it (or at least calls for modification of the theory).

The latter, which are principles of adduction, are implied more explicitly in the Torah in Deuteronomy 13:2-4 and 18:21-22 (see further on).

Numbers 17:16-24 provides us with yet another example of inductive logic. In order to preempt further doubt and rebellion, Moses, under instructions from God, had the princes of the tribes of Israel prepare rods, each man’s rod with his name on it, including Aaron the high priest, of the tribe of Levi; the twelve rods were then placed overnight in the tent of meeting, and the next morning Moses found that the rod of Aaron “was budded, and put forth buds, and bloomed blossoms, and bore ripe almonds,” which miracle demonstrated that God’s choice for the post of high priest was definitely Aaron.

Even though this story relates to a miracle (and of course it contains a weakness in that the miracle occurred behind the scenes, so that the onlookers could not be sure Aaron’s rod was not switched by human hand), it can be viewed as a lesson in controlled experiment. The conflicting predictions made by different theories are tested empirically: the winning theory is the one with the most successful prediction. Here, there were twelve theories regarding which prince merited the high priest’s position, with twelve predictions regarding which rod would stand out from the rest. The miraculous budding, blooming and bearing fruit of just one rod, namely Aaron’s, indicated that the latter was selected for the job.

Numbers 27:1-11 and the follow-up in Numbers 36:1-12 demonstrate again the methodological importance of asking questions. A group of women, the daughters of Zelophehad, of the tribe of Manasseh, approach Moses demanding that they be given the portion of the land of Israel that would have been given to their father, who died earlier on without sons. Moses brings their cause before God, who agrees with the women and decrees that women may inherit from their father if he has no son. Further on, the leaders of the family grouping to which the daughters of Zelophehad belong come to Moses and complain that if any of these women married outside her tribe, the land she inherited would go to her husband’s tribe and so be lost to her father’s tribe. Once again, Moses consults with God on the issue, and the latter agrees and decrees that the women may marry who they wish but only within their own tribe.[23]

In these passages we see, as we did in relation to Genesis 18:16-32, that God often waits for a question before giving us its answer. If the daughters of Zelophehad and the leaders of their family grouping had not had the curiosity and courage to ask their respective questions, they (and we) would not have known the answer. People, women as well as men, are graced with thinking minds; and they have to use them to get at the truth. The truth of a matter will not always be placed in their laps ready-made. Research is usually necessary. If this is true with regard to matters subject to revelation, it is all the more true with regard to natural issues. Notice also Moses’ open-mindedness. He does not reject the appeals offhand, but gives them serious consideration, even going so far as to bring them before God.

Numbers 32:24 – “Do that which has proceeded out of your mouth!” – may be viewed as a lesson in methodology, insofar as harmony between word and deed, or more broadly between thought and action, or consciousness and will, is essential to proper functioning of the human mind and to human relations. Someone who often indulges in lies, to self and/or others, is ultimately bound to suffer psychologically, losing contact with reality and self-confidence, and socially, losing credibility and respect. Of course, that does not mean one should be rigid-minded at all costs – sometimes it is wise and good to change one’s mind. But as a general rule of behavior, integrity is a key to mental health and social well-being. One’s word, to oneself or to others, should be one’s bond.

In Numbers 35:30, and further on in Deuteronomy 17:6 and 19:15, we are told that a matter cannot be established by the testimony of only one witness, there has to be more than one witness for credibility. This rule is of course stated in the Torah primarily with regard to issues that come before a court of law; but we can also assume it to have a broader intent. What does ‘witnessing’ mean? Many events in the world occur without having been observed by any man or woman. Some events are only observed by the person or persons participating in them. Some events are observed by non-participant third parties. A ‘witness’ is someone who has observed an event, and thus can and eventually does describe it to other people. The observation may occur through any combination of the five senses; and it may be direct or made through the intermediary of some instrument.

The alleged witness may or may not be reliable. Reliability depends, firstly, on the witness’ skills in observation, memory and verbal expression. If someone lacks these abilities, then no matter how earnest their testimony, it is unreliable. Secondly, reliability depends on the witness’ personal interest in the matter, i.e. on whether he or she has something to gain (e.g. money) from the testimony or is proffering it purely as a public service. It is of course theoretically possible for someone to be able to speak truthfully in spite of interests to the contrary; and indeed this is ethically ideal. But few people are able to do this; and it is not always possible for third parties (who must judge the matter) to tell whether they are trustworthy or not. For these reasons, the testimony of one witness is rarely enough evidence to establish a matter; and in serious cases, there has to be more than one.

In science, such reflections give rise to the principle of reproducibility of experiments. In most matters of internal experience, scientists have to rely on the say-so of people. If someone claims to believe or disbelieve something, or to love or hate something, scientists can hardly contradict him or her (although they might check the claim by observing the person’s actual behavior over time). However, in matters of external experience, scientists generally do not believe the say-so of people, but endeavor to recreate the experience for themselves (and thus become first-hand witnesses too). For instance, if a scientist claims he observed so and so in a certain experiment, he is in principle not believed until at least one other scientist has made the same experiment with the same results. Thus, we can say that testimony just provides confirmation for a hypothesis – never proof. The more such confirmations, the more confidence we have in the hypothesis; but it is never definitely proved.

In Deuteronomy 1:17 and 16:18-20, Moses reminds the appointed judges some of the ways to attain just judgment, saying: “Ye shall not respect persons in judgment; ye shall hear the small and the great alike; ye shall not be afraid of the face of any man;” and again: “Judge the people with righteous judgment. Thou shalt not wrest judgment; thou shalt not respect persons; neither shalt thou take a gift; for a gift doth blind the eyes of the wise, and pervert the words of the righteous. Justice, justice shalt thou follow.” As we remarked in relation to similar passages in Exodus and Leviticus, such injunctions may be taken as general cognitive principles, for everyone to follow in all circumstances. If we want to get in contact with reality, and remain there, we must all always make every endeavor necessary to that end, and not allow any consideration to lower our determination and deflect us from the truth. Knowledge of the truth is a great value, not to be abandoned out of fear of some loss or out of hope for some gain. Integrity is a virtue not only for judges, but for all people.

Deuteronomy 5:17, which forbids us to “bear vain witness” against our neighbors, is a repetition of the same commandment in Exodus 20:12 against bearing “false” witness, i.e. of the ninth of the ‘ten commandments’. The intent is primarily to refrain from lying in a court of law. Commentators explain the difference in wording here by saying that even if the lie seems harmless, it should be avoided. But here again, we may generalize the imperative to all cognitive contexts. The importance of scrupulous accuracy, honesty and integrity in the pursuit and transmission of knowledge cannot be overemphasized.

In Deuteronomy 5:21-23, referring to Moses statement: “we have seen this day that God doth speak with man and he [i.e. the man spoken to] liveth [i.e. survives the experience],” the children of Israel retort: “if we hear the voice of our Lord our God any more, then we shall die. For who is there of all flesh, that hath heard the voice of the living God speaking out of the midst of the fire, as we have, and lived?”[24] Now, these statements are somewhat in conflict, in that Moses points out that in fact God spoke to the Israelites and they survived, yet the latter fear that if God addresses them directly any longer they risk dying nevertheless, since in their view this is ultimately impossible[25]. The resolution of that conflict is of course simply that the Israelites were momentarily graced with an exception to the rule, but they are not confident that they merit further direct revelation after that brief preview.

What is interesting in this passage, for the purposes of logic theory, is that it contains an effective definition of implication, albeit in material terms. The first statement of the Israelites: ‘if we hear the voice (etc.), then we shall die’ signifies an implication of the form ‘if X, then Y’. Their next statement: ‘For no one who hears the voice (etc.) survives’ constitutes an explication of the first statement, since it says ‘for’ (ki, in Hebrew), and it has the form of a negation of conjunction ‘not-(X and not-Y)’. Thus, we have here a clear hint that ‘if X, then Y’ means the same (to the ancient Israelites, or at least to the author of that Biblical passage) as ‘not-(X and not-Y)’, even though that equivalence is stated in material terms. Moses’ earlier empirical statement may be symbolized as ‘X and not-Y’, and so apparently belies the Israelites’ statements. That being the case, the implication advocated by the Israelites must be viewed as more limited in scope than it seems at first – i.e. as applicable thenceforth (i.e. after Moses’ statement) – to avoid inconsistency.

All this may be taken as a teaching of logic in a book of the Torah, a document traditionally dated as ca. 1300 BCE (though some modern commentators consider it as having been composed some 500 years later). This is of course (in any event) quite a while before the time of Megarians Diodorus Cronus (d. ca. 284 BCE, some say later) and his disciple Philo (fl. ca. 300 BCE), as well as their successor Chrysippus of Soli (279-206 BCE). I am not, of course, saying that the Torah teaching was as thorough as that of these Greeks[26]; but I am saying that it was potentially as effective. I am not saying, either, that the said finding in the Torah excludes the possibility of similar findings in more ancient Greek or other literature; obviously, the latter remains possible (though of course, if we accept the traditional dating of the Torah, we would have to look into very early literature). It is just an interesting finding. So far as I know, the rabbis never consciously discussed the logical aspects of this or similar passages of the Torah. In any case, they do not include the definition of implication among their hermeneutic principles. But one can well suppose that they were at least subconsciously affected by this passage and others like it.

Deuteronomy 6:7, “thou shalt teach them diligently unto thy children,” refers primarily to teaching the commandments of the Torah to one’s children. But we can take this verse as exemplifying the importance of education more generally. Human knowledge depends largely on education. Human knowledge is of course primitively an individual enterprise, meaning that we can and must to some extent engage in self-education. But human knowledge would be very limited if it were limited to that one source or process. In fact, human knowledge is a social enterprise, stretching in time as well as space. Much of what we know we have learned from others, people from other countries as well as people in the past we descend from. The riches of our individual knowledge were made possible by all the information and skills we were fortunate to receive from others.

A well-rounded education would consist of what we might call ‘The Six Rs’ – Reading, wRiting, aRithmetic, Reasoning (logic), Righteousness (ethics), and Religion (spirituality). Through literacy, we can receive knowledge from others (reading) and transmit knowledge to others (writing). Though it is true that receipt and transmission of knowledge of various sorts can be done orally, oral communication is relatively limited in scope and also not entirely reliable (since information might more easily be forgotten or modified or added on). Writing and reading are manifestly more powerful means overall, opening the way to a vast pool of world literature that everyone can contribute to and draw from – more than ever nowadays due to technological developments. Arithmetic, and more broadly mathematics (including geometry), is needed to understand quantitative relations. Logic trains the mind to think correctly and avoid error, thus improving one’s own production of knowledge and also learning to be critical and selective in one’s absorption of knowledge from others.

Ethics is needed not only to develop worthwhile values, but even for the pursuit of purely factual knowledge. To obtain true knowledge of any kind, we have to observe accurately, report honestly, make an intellectual effort, be realistic, be open-minded, be courageous, and so forth. Methodology is impossible without appeal to many ethical imperatives of this sort. And of course, ethics in a broader sense also serves to build within us the mental health necessary to knowledge. In this context, the value of meditation should be stressed: it calms the mind and ensures its most efficient working. Also, the social and political offshoot of ethics, jurisprudence, serves to ensure optimum conditions around us for knowledge. Without freedom of thought and speech, for instances, we obviously cannot hope to attain and pass on true knowledge. Again, a society where no one cares for his fellows, or worse still where violence is widespread, is obviously not conducive to education.

Last but not least is spirituality, which consists in awareness that we humans are not merely bodies with minds, but moreover spirits or souls – i.e. that we have a dimension of being that transcends the material living organism with some mental capabilities that we seem to be or inhabit at first sight. Such awareness is needed to take into consideration facts relating to consciousness, volition and valuation. Many scientists doggedly refuse to take these facts into consideration, out of fear of spirituality. This is understandable in view of the fact that religions have often misled mankind, by dogmatically denying material facts. But there is no wisdom in throwing out the baby with the bath-water. Human spirituality is also a fact – to ignore it and opt for a narrow materialistic viewpoint is to condemn ourselves to another sort of stupidity. Open-mindedness on both sides is essential to true cognition.

Deuteronomy 13:2-4 and 18:21-22 contain two extremely important principles of inductive logic. I drew attention to these two passages of the Torah and explicated them in detail, years ago in my Judaic Logic (chapter 2.2). The first passage clarifies the positive aspect of adduction, viz. that a thesis which makes a correct prediction regarding certain empirical events is thereby confirmed – although this does not mean that it is proven: it is only made more probable than it previously was. The second passage clarifies the negative aspect of adduction, viz. that a thesis which makes a wrong prediction regarding certain empirical events is thereby disproved – and not merely undermined, i.e. not just made less probable than it previously was. These are, to repeat, two crucial logical principles.

I’d like to add here a few remarks concerning the form of reasoning called, in Latin, post hoc ergo propter hoc (meaning, in English, ‘after this, therefore because of this’)[27]. Aristotle drew attention to such reasoning as fallacious, in his Rhetoric (2:24.7): it “consists in representing as causes things which are not causes[28], on the ground that they happened along with or before the event in question. They assume that, because B happens after A, it happens because of A.” With regard to Deut. 13:2-4, this means that reasoning from “a sign or wonder” that “comes to pass” to the authenticity of an alleged prophet, would be fallacious[29]. However, it should be stressed that this severe judgment by Aristotle is made specifically with reference to deductive logic: we cannot deduce from B following A that B is caused by A.

However, from an inductive perspective, we might well and often do induce from B following A that B is caused by A. On this basis, popular confidence in a prophet is naturally encouraged by the accuracy of his predictions. Such reasoning can be justified by generalization, i.e. by assuming that since A is followed by B in this case, and maybe a few more similar individual instances, we may expect A to be followed by B in all cases, or all cases of a certain sort, which would indeed imply that A causes B, at least for cases of that sort. Such generalization may, of course, turn out to be wrong – and often enough does. But then, in those cases where it fails, we would simply particularize; i.e. we would retract the generalization. This is indeed the teaching of Deut. 18:21-22, according to which if what an alleged prophet’s prediction “follow not, nor come to pass,” then he may confidently be considered not to be a true prophet.

It should also be pointed out that there are two distinct areas of causality where reasoning post hoc ergo propter hoc might be used. One is causation and the other is volition. When by ‘cause’ we mean causation, then the argument from ‘after this’ to ‘because of this’ always relies on generalization, as above detailed. However, when by ‘cause’ we mean volition (i.e. acts of will), we rely on such generalization only in some situations; very often, we rather rely on introspection (and then, of course, generalize from there). The reason we have to do so is that volition, unlike causation, is inherently a form of causality that is not necessarily repetitive. To will something means to cause it, but we could equally have willed the opposite thing; this is the meaning of ‘freedom of the will’. Every act of will is individual, rather than the product of a law of nature as in causation. Therefore, in volition, the notion that “like causes have like effects” does not always hold[30].

In volition, there is room for real variety emerging from one and the same cause (i.e. the person doing the willing). Will is usually motivated, but it may occasionally be quite devoid of purpose. Motives are variously influential, but never determining; which is why the person willing may logically (and morally and legally) be held responsible for the act. Therefore, we can often only guess at the motive of an action, by resorting to introspection (or more broadly, by asking a sample of people to introspect and answer our questions) and assuming that the action occurred in such and such a context. This is also inductive reasoning, note well, for we may well later be proven wrong by discovering that the person concerned could not have had that particular motive. In the latter case, we might then doubt that the person did the action, or we might suspect that he or she did it but with some other motive in mind. Clearly, causal reasoning becomes more speculative and complex when dealing with volition. Needless to say, the issues involved are very significant in legal contexts when, for instance, judges attempt to determine who committed a crime and why; but they are also relevant in everyday life.[31]

Deuteronomy 13:15, “then shalt thou inquire, and make search, and ask diligently; and, behold, if it be truth, and the thing certain…,” is of course stated in a specific context, with intent to prevent idolatry in Israel. But we can also learn from it the attitude of respect for truth: we should not base our beliefs on mere imagination or hearsay, but make inquiries, do all necessary research, and also pay attention to what others have said or say on the subject, and then only, if and only if the empirical evidence and rational considerations in favor of the belief are sufficiently convincing and comparatively strong may we opt for the belief in question. Our beliefs should be products of pondered judgment, not fanciful prejudices.

Deuteronomy 18:10-11 states “There shall not be found among you… one that useth divination, a soothsayer, or an enchanter, or a sorcerer, or a charmer, or one that consulteth a ghost or a familiar spirit, or a necromancer.” This may be read as an epistemological guideline: truth is not to be found through such superstitious means. Here, the Torah is way ahead of other documents in its civilizing influence. Of course, it recognizes prophesy as a means of knowledge; but that is a much grander route, to which these various superstitious means cannot be compared.

Deuteronomy 25:13-15, which commands us to have perfect and just, and not diverse, weights and measures, is a passage that has been cited by some commentators in support of the application of logic to legal matters. This makes sense, in that logic is an essential tool to get at the truth and avoid falsehood in any and every context. The command regarding weights and measures is, of course, primarily aimed at accuracy in perception and in factual reporting. A dishonest merchant attempts, by means of adulterated weights and measures, to buy more quantity of a good for less money or to sell less quantity for more money. In the same way, by analogy or by generalization, we can say that a dishonest speaker attempts, by means of fallacious arguments, to defend a false thesis or to condemn or put in doubt a true one. The value of logic is its ability to spot and unmask such dishonest attempts to fool people. And it is important not only as a protective shield against being fooled by others, but also against being fooled by oneself. For we do often out of various motives, fool ourselves. For this reason, training in logic is essential. Even though it does not guarantee perfection, it safeguards us against much dishonesty and foolishness.

Deuteronomy 27:18 says: “Cursed be he that maketh the blind to go astray in the way.” There are in this world people who, when asked by a blind person for directions, think it is very funny to misguide him or her; and this is of course sick behavior[32]. We can generalize this moral judgment to all people who knowingly misinform their neighbor – and of those there are a lot more. Think of all the politicians and journalists who daily, for whatever motives, grossly lie to the people: hiding relevant information, inventing patent falsehoods, and distorting what they know to be the truth; they are indeed despicable. The Torah here again, then, stresses the importance of commitment to truth – for this is one of the foundations of social cohesion and peace. It is one of the prime expressions of benevolence towards one’s fellows.

Everyone, including philosophers and sundry academics, and indeed religious teachers, are required by the ethics of logic to pursue and propagate truth, because we are all responsible for the collective knowledge of mankind as well as for our individual state of knowledge. This means that we should always admit our ignorance or uncertainty when applicable, and not give other people the false impression that we have knowledge or certainty when we in fact lack it. Most if not all of us are unaware of very many things, and thus deserving to be qualified as “blind” to varying degrees. When stupid or malicious people propagate falsehood, it is like “the blind leading the blind.” If they do not know for sure something to be true, they should not pretend they do; all the more, if they do know for sure something to be untrue, they should not dish it out as true.

The Sayings of the Fathers counts these virtues among the seven characteristics of the true scholar: “Where he has heard no information he says: ‘I have not heard’” and “He admits that which is true;” to which it adds: “The contrary of these attributes marks the boor” (5:10). Another saying worth quoting here is: “Who is wise? He who learns from all men” (4:1).

Conclusion. We can summarize the above information, if only roughly, concerning the Torah, as in the following table. As can be seen, we found altogether some 50 passages of broadly logical interest (respectively 11, 6, 9, 9 and 15, in each of the books of Moses); and these we roughly classified as: deduction (15), induction (10), causative logic (2), ethical logic (2), methodology (16) and, more vaguely, epistemology (6). This may not seem like a very rich harvest, but it is considerable anyway in view of the great antiquity of the document concerned.

Reference

Content (roughly)

Type (roughly)

Gen. 2:16

Possibility of prophecy

epistemology

Gen. 2:16-17

If-then discourse

ethical logic

Gen. 2:19

Theory of words

epistemology

Gen. 2:21-24

Dud ‘therefore’ marker

deductive logic

Gen. 3:1-5

‘Not all are’ does not imply ‘this one is not’

deductive logic

Gen. 4:15

Syllogism (fig. 1) or apodosis (ponens)

deductive logic

Gen. 4:23-24

A crescendo argument (+s)

deductive logic

Gen. 8:6-12

Experiment

inductive logic

Gen. 18:16-32

Asking questions

methodology

Gen. 23:18-27

Experiment

inductive logic

Gen. 44:8

A fortiori argument (+s)

deductive logic

Ex. 2:11-14

Analogy (gezerah shavah)

deductive logic

Ex. 3:2-3

Harmonization of theory with observation

inductive logic

Ex. 6:12

A fortiori argument (-s)

deductive logic

Ex. 18:13-26

Means and ends

ethical logic

Ex. 20:12

Factual accuracy

methodology

Ex. 23:1-3, 6-8

Factual accuracy, perspicacious judgment

methodology

Lev. 10:9-11

Stay sober and analogy (binyan av)

method, deduction

Lev. 10:16-20

Diligent inquiry and admitting one’s errors

methodology

Lev. 11:3-4

Inclusive disjunction

deductive logic

Lev. 13

Sign and signaled

causative logic

Lev. 18:24

‘All’ as collective or distributive

deductive logic

Lev. 19:11

Truthfulness

methodology

Lev. 19:15

Unprejudiced judgment

methodology

Lev. 19:35

Factual accuracy

methodology

Lev. 21:21

Apodosis (modus ponens)

deductive logic

Num. 12:14

A fortiori argument (+s)

deductive logic

Num. 13:17-20

Planning research

methodology

Num. 14:11-16

Alternative explanations

causative logic

Num. 16:28-33

Theory and predictions

inductive logic

Num. 17:16-24

Controlled experiment

inductive logic

Num. 27:1-11

Asking questions and open-mindedness

methodology

Num. 36:1-12

Asking questions and open-mindedness

methodology

Num. 32:24

Word and deed

epistemology

Num. 35:30

Testimony

inductive logic

Deut. 1:17

Unprejudiced judgment

methodology

Deut. 5:17

Factual accuracy

methodology

Deut. 5:21-23

Definition of implication

deductive logic

Deut. 6:7

Education

epistemology

Deut. 13:2-4

Positive adduction

inductive logic

Deut. 13:15

Respect for truth

epistemology

Deut. 14:6-8

Inclusive disjunction

deductive logic

Deut. 16:18-20

Unprejudiced judgment

methodology

Deut. 17:6

Testimony

inductive logic

Deut. 18:10-11

No superstition

epistemology

Deut. 18:21-22

Negative adduction

inductive logic

Deut. 19:15

Testimony

inductive logic

Deut. 25:13-15

Factual accuracy and use of logic

methodology

Deut. 27:18

Commitment to truth

methodology

Deut. 31:27

A fortiori argument (+s)

deductive logic

Table A6.1

So much, for now, as regards logic in the Torah[33]. I will not here propose a list of cases in the rest of the Tanakh, but I predict that many more cases are there to be found. As regards a fortiori argument alone, 41 instances have been identified.

The following is, offhand, one example which provides evidence of syllogistic reasoning in the Tanakh. In Malachi 1:6, God argues: “A son honors his father, and a slave his master. If I am a father, where is the honor due me? If I am a master, where is the respect due me?” This may be rendered as: “Fathers/masters are generally [to be] honored by their sons/slaves” (major premise, ‘All Y should receive Z from their counterparts’) and “I am their Father/Master” (minor premise, ‘X is Y’), therefore “I ought to be honored by them” (tacit conclusion of the syllogism, ‘X should receive Z from his counterparts’) – “yet they do not honor Me” (i.e. ‘X does not receive Z from his counterparts’, contrary to the tacit conclusion of the syllogism, i.e. to rational expectation). There is no other way to understand this discourse than through syllogism.

I believe I have over the years noticed many other Biblical passages that may be likewise viewed as logic teachings of various kinds. The Tanakh did not wait for Aristotle or other Greeks to engage in logical thought. Such thought is an integral part of the human condition – something that existed long before logicians brought it into full light by distinguishing the form of logical argument from its content. Certainly, once logicians did highlight argument forms, ordinary people thereafter made use of them more easily, correctly and frequently. Similarly, though the examples of reasoning offered in the Tanakh were not new to mankind, nevertheless they helped Jews and other readers of that document to fortify their reasoning powers.



[1] Notably Radak (David Kimchi – Southern France, 1160–1235), according to a comment ad loc. in The Stone Chumash (ArtScroll, NY).

[2] It should be added that the Biblical view that all languages are derived from Hebrew (since the incident described in Gen. 11:6-7) is open to doubt. In this regard, I would cite for instance John McWhorter’s The Power of Babel: A Natural History of Language (London, Heinemann, 2002). This explains the technical difficulties inherent in such reductionism, no matter what the initial language is assumed to be.

[3] He puts it as a rhetorical question: “Hath God said ‘you shall not eat of any tree of the garden’?” – suggesting that God did not say that.

[4] The rule is given earlier, in Ex. 31:14: “everyone that profaneth it [the Sabbath] shall surely be put to death.” Nevertheless, in the episode of Num. 15:32-36, the question is put to God before the rule is applied.

[5] It is the American-Jewish (A.J.) translation, given in The Soncino Chumash.

[6] Based on Tanchuma Bereshit 11.

[7] The story traditionally told in this context is that Lamekh, who was blind, shot an arrow at what he thought (because his son Tubal-Cain suggested it) was a deer – but his arrow shot and killed Cain (and then, when he discovered his mistake, he killed his own son).

[8] Rashi, referring to the haggadah of [chapter] Chelek [from Sanh. 108b], reads “to and fro” as meaning that “the raven kept returning to the ark, because it [the raven] suspected him [Noah] concerning its mate [i.e. as having sexual interest in the raven’s mate].” This interpretation strikes me as so fanciful as to be ridiculous.

[9] Notice that Noah uses inductive means; he does not try to deduce reality from previous Divine utterances, using gematria (numerology) or any other mystical means.

[10] The only way this story might conceivably be sustained is by claiming the apparent world to be illusory, and the reality behind it to be as described in the Bible. But that is really a farfetched defense!

[11] At least until Aristotle objected to the idea (see his Meteorology 2:2, given below in Appendix 4). Note that Ex. 20:4 and Deut. 4:18 mention “the water under the earth.” There is no mention anywhere of ice at the earth’s poles melting, and indeed they did not.

[12] They apparently did not realize that rainclouds are formed by evaporation of water from the seas, lakes and rivers. See also Deut. 11:17 and 28:12.

[13] For instance, there are Amerindian tribes in North, Central and South America which are genetically proven descendants of a human whose bones were unearthed on the west coast of the U.S.A. and found to be 13000 years old. At that time, the Bering Strait was frozen, and people could cross over it (on a coastal route) from Asia. Today’s tribes could not have descended from Noah, since the Bering Strait was no longer frozen 4000 years ago. Think likewise of Australia’s Aborigines, and many other remote peoples. As for cultures, they continued without a break; for instance, in nearby Egypt (where, by the way, most pyramids, including the three big ones at Giza, were already in existence). The flood story obviously doesn’t hold water.

[14] Note in passing that forgiveness is only promised for 50 righteous men, whereas for 45-10 such men only non-destruction is promised.

[15] See Ex. 21:13: “He that smiteth a man, so that he dieth, shall surely be put to death.”

[16] This is based on Deut. 22:25-27, according to BT Sanhedrin, 73a.

[17] This is not to be confused with meditative states, where one temporarily distances oneself from sensory and mental phenomena in order to better get in touch with aspects of reality that are accessible more intuitively.

[18] A similar message is given in Deut. 1:17 – “Ye shall not respect persons in judgment; ye shall hear the small and the great alike; ye shall not be afraid of the face of any man; for the judgment is God’s.” This is commented on further on.

[19] Obviously, reliance on a relatively implicit reason is even more inductive than an explicit reason.

[20] The Torah does make use of disjunctions with more than two alternatives. To give an example at random, Leviticus 22:19 says: “Ye shall offer a male without blemish, of the beeves, of the sheep, or of the goats.”

[21] There may be better examples of explicit positive apodosis, and maybe also of negative apodosis. But at least this one is a start.

[22] Moses therefore asks God for mercy for the Jews; and God pardons them to some extent, agreeing not to wipe them out immediately, but at the same time resolving to wait for the next generation before taking them into the promised land (v. 17-24).

[23] Women are evidently not treated entirely as equal to men in these contexts. They do inherit from their fathers, but only if they have no brothers. They can freely marry within their tribe, but not outside it. Or maybe (I do not know) the law is that they may marry into other tribes, but in that case they lose their share of their father’s estate.

[24] Similarly, in Ex. 20:16, the people ask Moses to speak with them instead God – “lest we die.”

[25] The people believed this perhaps based on tradition; or maybe it seemed obvious to them for some reason. For although in Ex. 33:20 God says: “man shall not see Me and live,” this is well after the ten commandments episode (which occurs in Ex. 20 and is retold in Dt. 5). Ex. 33-34 details God’s revelation to Moses of the thirteen attributes of mercy, after the episode of the golden calf, when he went back up Mount Sinai with two new stone tables on which to write the ten commandments again.

[26] For Diodorus Cronus a conditional proposition was true only if the consequent (Y) always followed the antecedent (X), whereas for Philo, it sufficed for such a proposition to be true if the consequent followed the antecedent at a given time. These two kinds of implication are today labeled respectively ‘strict’ and ‘material’, or necessary and actual. Given that actual implication means ‘It is at this time false that X and not-Y’, we might ask how it can be known – for knowledge of such ‘negation of a conjunction’ cannot be arrived at independently. Either (a) we know this because we know (either by generalization from past experiences or as the resolution of some paradox) that ‘it is impossible for X to be true and Y to be false together’, or (b) we know it through specific knowledge that ‘X is true and Y is true’ or that ‘X is false and Y is true’ or that ‘X is false and Y is false’ is true at this time, or (c) someone else knows one of the preceding facts and tells us its implication that ‘X is true and Y is false’ is false at this time. Thus, though so-called Philonian implication is conceivable, it is a rather artificial form. Chrysippus went further than his predecessors by clarifying the arguments (which I call ‘apodoses’) that could be formulated with conditional propositions, such as the modus ponens (affirming the antecedent, and concluding with the consequent) and the modus tollens (denying the consequent, and concluding with denial of the antecedent).

[27] When the intent is ‘with this, therefore because of this’, the Latin phrase would strictly speaking be cum hoc ergo propter hoc; but we may take post hoc as including this special case. Note that a distinction might also be drawn between events that are successive or simultaneous, and events that are noticed successively or simultaneously, and the discussion further deepened.

[28] Note that Aristotle’s wording here suggests that post hoc ergo propter hoc is a specific fallacy within the larger category of non causa pro causa (taking a non-cause for a cause). We might include under the latter heading not only unjustified assumptions of causality, but also mistaken causal identities – by which I mean, for instances, taking a partial cause for a complete (sufficient) cause or taking a contingent cause for a necessary (sine qua non, lit. ‘without which not’) cause.

[29] Clearly, if a predictor is a genuine prophet, then his predictions should “come to pass;” but if his predictions “come to pass” it does not logically imply that the predictor is a genuine prophet. A scientist may repeatedly make correct predictions without thereby being considered prophetic, for he remains naturally capable of human error like anyone else. An alleged prophet, on the other hand, has to be invariably right to be accepted as genuine.

[30] The statement “like causes have like effects” is sometimes referred to as the law of causation. I call this a mere notion, because it is too vague: it does not clarify how much alike “like” means. Furthermore, it refers to things as causes or effects without having first proven them to be causes or effects. The statement is nevertheless useful as a practical guide.

[31] For a more detailed study of this topic see my book Volition and Allied Causal Concepts.

[32] Lev. 19:14 – “You shall not curse the deaf nor place a stumbling block before the blind.” – is a broader warning not to take malevolent advantage of people’s weaknesses. The reference to the blind in both passages (and to the deaf in the one from Lev.) is obviously metaphorical – and has indeed been so taken by Jewish commentators. “Plac[ing] a stumbling block before the blind,” being more calculating and active, seems generally a bit more criminal than “Mak[ing] the blind to go astray in the way,” although of course in some cases losing one’s way might be more harmful than being tripped over.

[33] Note that some passages cited are only intended as representative, without attempting to find and list all similar instances.

2016-06-14T05:09:03+00:00