1 Introduction

Logic in Judaism is mainly used for the determination and application of Jewish law, though also for the interpretation of the stories in holy texts. Before we begin our reflections on Jewish logic, therefore, let us very briefly look into the development of Jewish law. To begin with, we must of course consider how this development is perceived and conceived within Judaism itself.

The founding document and proof-text of the Jewish faith and religion is, as is well known, the Torah (translated as the Law, or Doctrine). This refers to the Five Books of Moses or Pentateuch (Chumash, in Hebrew), which Jews^[1] believe was handed down by Gd^[2] to the Jewish people, through Moses, at Mount Sinai^[3], some 3,300 years ago. The five books are Bereshith (Genesis), Shemot (Exodus), Vayikra (Leviticus), Bemidbar (Numbers), Devarim (Deuteronomy).

The Jewish Bible, or Tanakh, consists of this 5-volume Torah, together with the 8 other prophetic books (of which one includes twelve minor prophets) and 11 other holy scriptures (counting the books of Ezra and Nehemiah as one), written under Divine inspiration over the next 800 years or so, mostly in the land of Israel and in a few cases in the first Babylonian exile. TaNaKh is an acronym, including the initials T of Torah, N of Neviim (Prophets) and K of Ketuvim (Scriptures); the books of the Bible other than those written by Moses are therefore simply known as the Nakh^[4]. The latter play a relatively secondary role in the development of Jewish law, being referred to occasionally to resolve certain questions of detail^[5] or to provide illustrations.

The Talmud (which means, teaching) is an enormous compilation of legal discussions between Rabbis, stretching over several centuries, starting about 2,100 years ago (at least). It includes two main components: the Mishnah (meaning, learning by repetition - pl. Mishnaiot), which was edited by R. Yehudah HaNassi in the 1st century CE, followed by the Gemara (meaning, completion - pl. Gemarot), which was redacted by R. Ashi in the 5th century. Actually, there are two Talmuds: the Bavli (or Babylonian), which is the one we just mentioned, and the parallel Yerushalmi (or Jerusalem^[6]), which was closed in Israel some 130 years earlier, in the 4th century, and carries relatively less authority.^[7]

The Mishnah is divided into six so-called Sedarim (Orders - sing. Seder)^[8], to which there corresponds sixty-three Gemara commentaries called Masekhtot (Tractates - sing. Masekhet), found in one or both of the Talmuds. The names of the Orders and corresponding numbers of Tractates are as follows: Zeraim (Seeds), 11; Moed (Appointed Time), 12; Nashim (Women), 7; Nezikin (Damages), 10; Kodashim (Consecrated Objects), 11; Taharot (Purities), 12.

Jewish law, or the Halakhah (meaning, the Path, or the 'Way to go'), as it stands today, is (as we shall see) the outcome of a long historical process of debate and practise, in which the above mentioned documents, mainly the Torah and the Talmud, have played the leading roles. Jewish law, note, concerns not only interactions between individuals (be they civil, commercial or criminal) and societal issues (communal or national structures and processes), but also the personal behavior of individuals (privately or in relation to Gd) and collective religious obligations (which may be carried out by selected individuals, such as the priests or Levites).

Many people, not well-acquainted with normative Judaism^[9], believe that Jewish law was derived purely and exclusively from the Torah (or, more broadly, perhaps, the Tanakh). In this view, the Torah (or Tanakh) was the totality of Gd's message to the Jewish people in particular, and Humanity in general; so that only what was explicitly written in it, or strictly deductively inferable from that, qualifies as Divine Will. However, in fact, it would be technically impossible to derive in that way all of existing Jewish law from the Torah (or Tanakh) alone; more data would be required - and more was actually used....

Orthodox Jews believe that, at the same time as the Written Torah (Torah Shebekhtav) was given, an Oral Torah (Torah Shebealpeh) was inaugurated, by Moses, which served to clarify and amplify the written law, by consideration of more specific cases. The existence already in Sinai of an unwritten component to the Law is suggested within the written Torah itself (see, for instance, Exodus, ch. 18). The Tradition (Hamasoret, in Hebrew) was, orthodox Jews believe, faithfully transmitted across the centuries, through popular practise and verbal repetition, until it was largely committed to writing in the Nakh, the Talmud and other Rabbinical texts. Existing Jewish law, then, claims logical descent from, not only the Written Torah, but also the Oral Torah^[10].

It should be noted that, although written laws would seem more reliable than oral laws, nevertheless, some oral laws (for instance, the laws defining Sabbath observance) are considered as equal in force to written laws. Such 'as-if written' oral laws are called deoraita, in distinction from oral laws which are regarded as based on Rabbinic authority, called derabbanan. This distinction plays an important role in Halakhic decision-making, in the event of doubts concerning the tenor of a law or the facts of a case^[11]. A similar distinction is made with reference to inferences from Scripture, those with mere Rabbinic force being classed as asmakhta^[12].

What concerns us, in the present study, are the thought-processes which have been used to construct the Halakhah. This issue has several levels. The simplest is an uncritical description of the ways Jewish law is derived from the first principles claimed by Jewish tradition as having been given in the Sinai Revelation, in writing or orally. At a more advanced stage, we will want to determine to what extent these thought-processes, or methods of 'derivation', have been truly logical. And ultimately, we will have to scrutinize more carefully the bases of the 'first principles' themselves - which implies an investigation of hidden or unexplicited thought-processes, which in turn must be assessed from a purely logical point of view.

Let us now look more closely at the course of events, as taught within Judaism. The Torah, written and oral, is supreme, not open to doubt or review. Some aspects of the oral Torah make their appearance in the Nakh, if only incidentally within stories. Next in importance comes the Mishnah, which is the condensed essence of Jewish oral tradition, as it stood at a specific point in time. The Mishnah faithfully reports, not only legal positions generally agreed on by the Rabbis of the time and earlier, but also where they disagree, their points of controversy. The authority of the Rabbis stemmed from the Torah itself; for instance, Deuteronomy 17:8-13 (emphasis added):

If there arise a matter too hard for thee in judgment (...); then shalt thou arise and get thee up, unto the place which the Lrd thy Gd shall choose (...), unto the Levitical priests, and unto the judge that shall be in those days; and thou shalt inquire; and they shall declare unto thee the sentence of judgment (...); and thou shalt observe to do according to all that they shall teach thee.

The religious authorities were, first of all, the trustees of the oral transmission (many of these people, in the long line since Moses, are identified by name^[13]). And secondly, it was foreseen that there would be gaps in knowledge, or changing circumstances, which would require wise and considered judgment by competent and recognized spiritual leaders^[14].

The decisions set down in the Mishnah, once it was closed, became binding for all future generations, and thus acquired the status of first principles, like the Scriptures, not open to challenge, and serving as top premises in the inference of further Halakhah. Although the Mishnah provided more practical detail than the Torah, it was written very telegraphically, and therefore could itself give rise to misunderstandings or disagreements. Furthermore, historical events - namely, Roman wars and persecutions, which caused the death of many major Rabbis of the time (known as Tanaim - sing. Tana) - created serious gaps in the collective memory, as to the Halakhah concerning many issues; and a fear that the still-oral portions of the tradition might be lost.

Such considerations motivated the next generations of Rabbis (known as Amoraim - sing. Amora), some still in Palestine, but many in Babylonian exile - to compare notes and memories, and debate outstanding issues, and report their collective findings and decisions in writing, in what became known as the Gemara. This was based, then, on an interplay of Torah and Mishnah - as well as, to some extent, on the living memories of eyewitnesses and the unwritten pronouncements of earlier teachers remembered by their later disciples, known as the Tosefta (additional material) and Baraitot (sing. Baraita - material left out of the Mishnah). The latter included lists of hermeneutic principles. Using these three sources, the Rabbis developed the Gemara.

Again, the clear decisions in the Gemara, once made, became binding on all future generations. They in turn became unassailable first principles in the system of inference of Jewish law. The reason for this privilege of earlier authorities is that they were closer to the source (the Revelation at Sinai), in touch with a relatively unbroken chain of tradition, compared to succeeding Rabbis. The latter were still left with work to do, however; some questions had been left unanswered, some answers were open to conflicting interpretations, and also new situations arose which required Halakhic decision.

Thus it is that the law developed, layer upon layer; there were the Savoraim (6th-7th centuries CE), the Gaonim (to the mid-11th century), the Rishonim (to the mid-15th century), the Acharonim (since the mid-15th century). Each era's Rabbis basing themselves on the decisions and suggestions of their predecessors, as deductively as they could, refined and developed the Halakhah. And almost always, the work of previous authorities acquired the status of well-nigh incontrovertible major premises for those that came after them. The latter could only comment or codify, or at best fill in gaps left by the former.

However, it should be noted parenthetically that when we today encounter an apparent contradiction between an earlier authority and a later one, we as a rule take the more recent as the more authoritative. It is taken for granted that the latter made his ruling with full awareness of the former's positions. Thus, while in principle the earlier personality has more prestige, in practise the later personality once established as an authority is more to be relied on. The status of authority is not of course acquired arbitrarily, but is a function of proven scholarship.

There have been attempts by some Jewish thinkers, at various times, to challenge many of the principles presented above, and try to liberalize the law. Especially of interest to us are the efforts made in this respect over the past couple of centuries, under the influence no doubt of the surrounding European culture of Enlightenment. The authors were generally free-thinking laymen, and there is no denying that most of them eventually gave up many religious observances, if they did not end up totally indifferent to religion. Such philosophers of religion were behind non-Orthodox Jewish movements, including Conservative Judaism, Reform Judaism, and the non-religious Haskalah.^[15]

Usually beginning with a critical review of traditional claims, pointing out logical weaknesses or factual inaccuracies or uncertainties, such attempts would often include proposals for legal change, generally with a view to making life easier for Jews, allowing them to adapt more readily to the modern world. However, the authors were in all evidence rather frequently unacquainted with the traditional answers to their questions; furthermore, even when their critique might be convincing, they often allowed themselves to draw conclusions more radical than their premises made possible.

To give an example^[16]: one might argue that even acknowledging that Biblical passages like the one cited above (Deut. 17:8-13) effectively grant legislative power to the Tanaim, Amoraim, and subsequent Rabbis, it is not manifestly evident why such past judgments should be irreversible. All one might affirm, logically, is that so long as the judges in each generation, appointed by those in the previous generation, continue to confirm these judgments, they hold; otherwise, they would cease to be binding. Claims that the Talmudic generations were necessarily wiser, because closer in time to the Sinai revelation, are rather circular arguments, based on a prejudicially positive evaluation, rather than on a logical connection; one could equally well claim (even if just as prejudicially) that most of these people were rather ignorant and superstitious by modern standards.

However, it must be noted that such objections do not really make possible a breach in the continuity of Rabbinical authority as such. Even if changes in the law, through reassessments of the logic or consideration of new data or new conditions, were in principle permissible, they would have to come specifically from within the line of succession of Jewish authority, to be in fact permissible. Anything else would effectively be an illicit attempt to takeover an institution, a misappropriation of the name "Judaism" by a new religion. There is no license to invent (as happened historically) a new line of spiritual guides called "Rabbis", not linked by education and appointment to the original line, and unable to claim direct descent from Moses. So long as the legitimate authorities consensually reaffirm the same judgments, they would seem to remain binding.

The reader of the present volume does not need to have previously studied logic in depth to be able to follow the discussion fully, but will still need to grasp certain concepts and terminologies. We will try to fulfill this specific task here, while reminding the reader that the subject is much, much wider than that.

Broadly speaking, we refer to any thought process which tends to convince people as 'logical'. If such process continues to be convincing under perspicacious scrutiny, it is regarded as good logic; otherwise, as bad. More specifically, we consider only 'good' logic as at all logic; 'bad' logic is then simply illogical. The loose definition of logic allows us to speak of stupid forms of thought as 'logics' (e.g. 'racist logic'), debasing the term; the stricter definition is more demanding.^[17]

Logic, properly speaking, is both an art and a science. As an art, its purpose is the acquisition of knowledge; as a science, it is the validation of knowledge. Many people are quite strong in the art of logic, without being at all acquainted with the science of logic. Some people are rather weak in practise, though well-informed theoretically. In any case, study of the subject is bound to improve one's skills.

Logic is traditionally divided into two - induction and deduction. Induction is taken to refer to inference from particular data to general principles (often through the medium of prior generalities); whereas deduction is taken to refer to inference from general principles to special applications (or to other generalities). The processes 'from the particular to general' and 'from the general to the particular' are rarely if ever purely one way or the other. Knowledge does not grow linearly, up from raw data, down from generalities, but in a complex interplay of the two; the result at any given time being a thick web of mutual dependencies between the various items of one's knowledge.

Logic theory has succeeded in capturing and expressing in formal terms many of the specific logical processes we use in practise. Once properly validated, these processes, whether inductive or deductive in description, become formally certain. But it must always be kept in mind that, however impeccably these formalities have been adhered to - the result obtained is only as reliable as the data on which it is ultimately based. In a sense, the role of logic is to ponder information and assign it some probability rating between zero and one hundred.

Advanced logic theory has shown that what ultimately distinguishes induction from deduction is simply the number of alternative results offered as possible by given information: if there is a choice, the result is inductive; if there is no choice, the result is deductive. Deductive logic may seem to give more certain results, but only because it conceals its assumptions more; in truth, it is merely passing on probability, its outputs being no more probable than the least probable of its inputs. When inductive logic suggests some idea as the most likely to be true, compared to any other idea, it is not really leaving us with much choice; it is telling us that in the present context of knowledge, we decisively have to follow its suggestion. These are the reasons why the word "proof" is often ambiguous; do we mean deductive proof or inductive proof, and does it matter which we mean?

The first task of logicians is to observe actual thought and speech, and take note of recurring linguistic formulas. At first, the variety may seem bewildering; but, starting with the most common and simple items, and gradually considering more detailed issues and more complex cases, Logic has grown and matured. A great breakthrough, which we owe to Aristotle (4th century BCE, Greece), was the discovery of an ingenious artifice, which clarified all subsequent discussion. In everyday discourse, we make statements with specific contents, like "swans are white"; Aristotle developed logical science by focusing on forms, substituting variables like "X" and "Y" for specific values like "swans" and "white". Such a formal approach signifies that certain aspects of reasoning can be justified without reference to content; they are abstract truths for all propositions of a certain kind.

We shall here first consider some of the simplest of the forms called categorical propositions. (It is worth memorizing the symbols, traditionally used since the Middle Ages to abbreviate theoretical discussions. A and I come from the word affirmo; E and O, from nego - these are Latin words, whose meanings are obvious. Note that IO refers to the sum of I and O.)

"X" and "Y" (or any specific equivalents) are referred to as the terms, the former being called the subject and the latter the predicate. The relational expressions "is (are)" and "is (are) not" are known as copulae, the former having positive polarity and the latter negative (note that the "not" is used here to negate the "is", even though placed after it). Expressions like "all", "some" are called quantifiers: they serve to tell us the extension (i.e. the number or proportion) of the subject which the predication (i.e. copula and predicate) refers to. So much for the various features of individual propositions.

A and E are characterized as general (or universal) propositions, because they each concern the whole of the subject, each and every instance of it which ever has appeared or may ever appear. A may also be expressed in the form "Every X is Y". It should be clear that "No X is Y" means "Every X is-not Y", the only difference between A and E being the polarity of their copulae. I and O are called particular propositions; they each concern at least part of the subject, and again differ only in their polarity; note well that such propositions are ambiguous with regard to just how much of the subject they address. Often, in practise, we fail to explicitly specify the quantity involved, taking for granted that it is well understood (as in "swans are white"); in case of doubt, such a statement may be dealt with as, minimally, a particular.

IO represents the conjunction of I and O, and may be classed as (extensionally) contingent. Though here presented as a compound, IO is also a proposition in its own right; it could equally be expressed in exclusive form, as "Only some X are Y" or "Only some X are not Y" (different emphasis, same logical significance). What distinguishes IO from its elements I and O, is that it is more definite about quantity than they are. It follows from the various definitions, and it is important to note, that I can be interpreted to mean "either A or IO" (that is, "either All X are Y or Only some X are Y"), and likewise O can be read as "either E or IO" (that is, "either No X is Y or Only some X are not Y").

The foregoing definitions and correlations, together with certain self-evident principles, enable us to infer the following oppositions, as they are called. (Note that the expression "opposition", in the specialized sense used in logic, does not necessarily signify conflict, but is intended in the sense of 'face-off'.)

* A implies I; that is, the first cannot be true without the second being also true. Remember that "all" is one of the possible outcomes of "at least some", and therefore conceptually presupposes it. Logic demands that we acknowledge the meaning and implications of what we say^[18] (this principle is known as the Law of Identity). Likewise, E implies O; and of course both I and O are implicit in IO. But note that these relations are not reversed: I does not imply A, nor IO; O does not imply E, nor IO.

* A and O are contradictory; that is, they cannot be both true and they cannot be both false, one must be true and the other false. The general statement "All X are Y" tells us that every single X is Y, and is therefore incompatible with any claim that "Some X are not Y" which would mean that one or more X is not Y; for we must admit that nothing can at once and in the same respect both have and not-have a given characteristic (this principle is known as the Law of Non-contradiction). Also, since there is no alternative to either being or not-being (this principle is known as the Law of the Excluded Middle^[19]), we are forced to assert one or the other of our two sentences in any given case. Similarly, E and I are contradictory.

* A, E and IO are all contrary to each other, mutually exclusive; that is, only one of them may be true, and the other two must then be false. Furthermore, they are taken together exhaustive; that is, one of them must be true, since there are no available forms besides them. It follows that the contradictory of the conjunction of I and O is simply a disjunctive statement of the form "either A or E"; for IO signifies a denial of all universality, whether that of A or that of E.

Lastly, what is the relation between I and O as such? They are obviously compatible, since they combine together within IO; that is, they may both be true at once. But they cannot both be false at once, for then their contradictories A and E would both be true, which is impossible. Their special opposition is therefore given a distinct name; they are said to be subcontrary. Note that the concept of subcontrariety applies to a pair of propositions, while the larger concept of exhaustiveness (above defined) applies to any number of propositions.

These concepts of opposition are applicable to other forms, besides those above, note. Also, there are other, related such concepts worth mentioning. Two propositions are mutual implicants, if the truth of either implies the truth of the other and the falsehood of either implies the falsehood of the other. If, however, the implication is only one-way, they are said to be subalternatives, and the one which implies but is not implied is called the subalternant, while the one which is implied but does not imply is called the subaltern. Two forms are said to be incompatible, if they are contrary or contradictory; in all other cases, they are said to be compatible. The latter class includes forms which are unconnected, or neutrally related, meaning that they are related neither by mutual implication or subalternation, nor by contrariety or contradiction, nor by subcontrariety.

It should be noted, too, that the same concepts of opposition can be applied to terms, as well as to propositions. Two terms, say X and Y, are mutual implicants, if all X are Y and all Y are X (in such case, X and Y are equivalent classes or coextensive). X subalternates Y, if all X are Y but not all Y are Y, in which case, Y is called a genus or overclass of X, and X is called a species or subclass of Y; mutadis mutandis for the reverse case, of X subalternated by Y^[20]. X and Y are contradictory, if no X is Y, and no nonX is nonY. They are contrary, if no X is Y, but some nonX are nonY. They are subcontrary, if some X are Y, though no nonX is nonY. And, finally, they are unconnected, if all the categorical propositions relating them or their negations are contingent (note, however, that they may still have conditional connections in such case).

There are of course many other forms, besides those listed above. Propositions may also be singular (these involve an indicated instance of the subject "This X" or a proper name; symbols R and G are used for the positive and negative variants, respectively). All propositions other than singular are called plural; this class includes not only A, E, I, O, but also majoritive or minoritive forms (those are introduced by the quantifiers "most" and "few", respectively) and with indeed any number or proportion we please ("lots of", "a few", "17", "two thirds of", etc.)^[21]. Propositions may involve relations other than the copula "is" or its negation - for instances, "becomes" or "is-greater-than". Also, propositions are not all categorical in form, as above, but may be more complex constructs, such as the forms of conditioning. As well, all propositions are implicitly or explicitly qualified by modality.

By modality is meant the attributes of relations we signify by using words like: necessarily, possibly, actually, actually-not, possibly-not, impossibly. These categories of modality, as they are called, are collectively of many types (or modes); and ordinary language reflects this variety in meaning somewhat. They may have a logical sense (referring to the various contexts of our knowledge), a natural sense (referring to causal relations within/among things themselves), a temporal sense (referring to the times of the existence of a thing), an extensional sense (referring to the cases of classes of things), or even an ethical sense (referring to the available standards of value).

Thus, for instances, the logically necessary is what is true in all knowledge contexts, and the logically possible (or conceivable) is something true in some knowledge contexts; in contrast, the naturally necessary is what occurs in all circumstances, and the naturally possible (or potential) is something occurring in some circumstances; and so forth. Modal considerations inevitably emerge in all human knowledge, as expressions of its limitations (logical mode), and in the external world itself, as expressions of its diversity (extensional mode) and change (natural and temporal modes). The study of modality is a vast and fundamental domain, which has important repercussions in every issue of concern to logic and epistemology.

Note that though all propositions have underlying modal attributes, these modalities are not always explicitly stated, nor are they automatically known. The wording is in practise pretty mixed up, but to develop the theory of modality, some expressions may be reserved for one or the other mode - e.g.: must, can, cannot, for natural relations; always, sometimes, never, for temporals; all, some, none, for extensionals; should, may, mustn't, for ethicals.

The building block of conditional propositions is the relation of conjunction (signaled by use of the word "and") and its negation; from the latter we derive the various types of implication (usually signaled by "if-then-") and disjunction (signaled by the relation "-or-"), and their respective negations. The formal study of this field is reported to date from at least the 3rd century BCE in ancient Greece (notably, with Philo the Megarian). We cannot here cover this wide field; only a few remarks concerning it will be made.

With regard to the relation of implication. The expression "P implies Q", where P and Q refer to any two propositions, signifies that "P" and "not-Q" cannot be true both together (in the same body of knowledge, or with reference to the same instances of some concept, or in the same natural circumstances or times). This means that P is incompatible with the negation of Q; and it can be stated in the hypothetical form "if P is true, then Q is true", or more briefly as "if P, then Q". Here, P and Q are called theses, P being the antecedent and Q being the consequent.

The contradictory of such a proposition has the form "P does not imply Q" or "if P, not-then Q", which is defined by the statement that "P" and "not-Q" can be true both together, i.e. they are compatible. Note that both these propositions can be freely contraposed; that is, "if P, then Q" implies "if not-Q, then not-P", and likewise "if P, not-then Q" implies "if not-Q, not-then not-P"; this is easy to prove, merely by comparing the definitions of the original propositions and their contraposites (which state the incompatibility or compatibility, respectively, of "not-Q" and "not-not-P").

With regard to disjunctive propositions, they have the form "P or Q or R or...", in which P, Q, R, etc. are two or more alternatives (or disjuncts). Considering the simplest case, with two theses; we should note the distinction between inclusive disjunction ("P and/or Q", equivalent to "if not-P, then Q") and exclusive disjunction ("P or else Q", equivalent to "if P, then not-Q"). In the former case, the two theses cannot be both false, but may eventually be both true; in the latter case, they cannot be both true, but may eventually be both false; if the disjunction is both inclusive and exclusive, the theses are in contradiction (and we tend to use the form "either P or Q"). Generally speaking, disjunctions can be defined precisely by stating explicitly how many of the available alternatives may or must be true and how many of them may or must be false.

The study of propositional forms is merely a preparatory to the study of the logical processes involving them, which we shall now consider.

All such processes, taken together, are widely referred to as the scientific method, but the word "science" in this expression must of course be understood as referring to knowledge as distinct from pseudo-knowledge or ignorance; and not to any professional body with privileged claims to truth. It should be clear that all inductive and deductive processes are commonly used by everyone, not just people involved in scientific enterprises. The scientist is, if at all distinguishable from others, distinguishable by his attitudes, as someone who (ideally) makes just a little more effort to be careful with his methodology - to be open-minded and objective, clear and precise in language, and strict and perceptive in logic.

How do propositions, such as those described above, come to be known? This is the question inductive logic tries to answer. The way we commonly acquire knowledge of nature, as ordinary individuals or as scientists, is by a gradual progression, involving both experience or perception, whether of external phenomena (through the sense organs somehow) or of mental phenomena (with what we often call the "mind's eye", whatever that is), and reason or conceptual insight (which determines our evaluation and ordering of experience).

At the simplest level, we observe phenomena, and take note, say, that: "there are Xs which are Y" (which means, "some X are Y" = I), leaving open at first the issue of whether these X are representative of all X (so that A is true), or just special cases (so that IO is true). The particular form I is needed by us as a temporary station, to allow us to express where we stand empirically thus far, without having to be more definite than we can truthfully be, without being forced to rush to judgment.

If after thorough examination of the phenomena at hand, a continued scanning of our environment or the performance of appropriate experiments, we do not find "Xs which are not Y", we take a leap and presume that "all X are Y" (A). This is a generalization, an inductive act which upgrades an indefinite particular I to a universal of the same polarity A, until if ever evidence is found to the contrary. The justification of such a leap is that A is more uniform with I than O, and therefore involves less assumption: given I, a move to A requires no change of polarity, unlike a move to O, whereas with regard to quantity, the degree of assumption is the same either way.

If, however, we do find "Xs which are not Y" (i.e. that "some X are not Y" = O), we simply conclude with a definite contingent IO. If the discovery of O preceded any assumption of A, so well and good, the induction of IO proceeded in an orderly fashion. If on the other hand, we had assumed A, and then discover O, an inconsistency has effectively occurred in our belief system, and we are forced to reverse a previously adopted position and effect a particularization of A back to I, to inductively conclude IO. Needless to say - and we need not keep pointing out such parallels between positive and negative polarities - the sequence of such harmonization might equally have been O followed by E, and then I followed by IO.

Note that the particulars involved, I or O, may be arrived at directly, by observation, as suggested above, or, in some cases, indirectly, by deduction from previously induced data. The inductive processes we have so far described, of observation followed by generalization and particularization, are only a beginning. Once a number of propositions have been developed in this way, they serve as premises in deductive operations, whose conclusions may in turn be subjected to deductive scrutiny and additional inductive advances and retreats.

But we are not limited to the pursuit of such "laws" of nature; we have a broader inductive method, known as the process of adduction^[23].

This consists in postulating propositions which are not arrived at by mere generalization and particularization, but involve novel terms. These novel terms are put forward by the creative faculty, as tentative constructs (built out of more easily accessible concepts^[24]) which might conceivably serve to explain the generalities and particularities (the "laws") developed more directly out of empirical evidence, and hopefully to make logical predictions and point the way to yet other empirical phenomena. The imagination, here, is not however given free rein; it is disciplined by the logical connections its postulates must have with already available data and with data which might eventually arise.

Scientific theories (complexes of postulates and predictions) differ from wild speculations in that (or to the extent that) they are grounded in experience through rational processes. They must deductively encompass accepted laws, and they stand only so long as they retain such a dominant position in relation to newly discovered phenomena. If logical predictions are made which turn out to be empirically true, the postulates are regarded as further confirmed - that is, their own probability of being true is increased. If however any logical predictions are found to be clearly belied by observation, the postulates lose all credibility and must be rejected, or at least somehow modified. Theories always remain subject to such empirical testing, however often confirmed.

Thus, knowledge of nature proceeds by examining existing data, making intelligent hypotheses as to what might underlie the given phenomena, showing that the phenomena at hand are indeed deductively implied by the suggested postulates, and testing our assumptions with reference to further empirical investigations. However, there is one more component to the scientific method, which is often ignored. It is not enough to adduce evidence in support of our pet theory; and the fact that we have not yet found any grounds for rejecting it does not suffice to maintain it....

We must also consider all conceivable alternative theories, and if we cannot find grounds for their rejection, we should at least show that our preferred theory has the most credibility. This comparative and critical process is as important as the constructive aspect of adduction. To the extent that there are possible challenges to our chosen theory, it is undermined - that is, its probability of being true is decreased. Evidence adduced in favor of one set of postulates may thus constitute counter-evidence adduced against other hypotheses. We may regard a thesis as inductively "proved", only if we have managed to eliminate all its conceivable competitors one by one. Very rarely - though it happens - does a theory at the outset appear unchallenged, the exclusive explanation of available information, and so immediately "proved". Also note, at the opposite extreme, we are sometimes stumped, unable to suggest any explanation whatsoever.

Now, two kinds of deduction are possible from the categorical propositions we considered earlier: eduction and syllogism. Eduction (or immediate inference) consists in drawing out from a single given proposition, some implicit information concerning the same terms. Syllogism (or mediate inference) consists in drawing out from two given propositions which have a term in common, some implicit information concerning the other two terms involved. We call a given proposition, a premise, and an inferred one, a conclusion; and all these propositions considered together are said to constitute an argument. We need not here go into a systematic and exhaustive listing and analysis of these processes.

An eductive conclusion merely changes the polarity of one or both terms in the premise and/or their positions; the polarity of the copula may change or remain the same, as appropriate. In all, there are seven processes; the primary two are obversion and conversion, all other kinds being reducible to combinations of them. Though the various processes are always applicable to the general forms; they are often inapplicable to one or both of the particular forms. Note well that not all the processes are reversible; in some cases, though the premise implies the conclusion, the conclusion does not imply the premise. An example of eduction is: "All Kohens are Levites; therefore, all non-Levites are non-Kohens" (this is contraposition of an A form, as defined below). In the following definitions, "S" and "P" symbolize the initial subject and predicate, and "-" a positive or negative copula:

1. Obversion merely changes the polarity of the predicate, moving from S-P to S-nonP. A, E, I, O become E, A, O, I, respectively (all reversibly). The validity of obversion proceeds from the laws of thought: "S is P" and "S is not nonP" are equivalent, because P and nonP are mutually exclusive and together exhaustive alternatives; the quantity remains unaffected by it, because plural propositions are just sets of singulars.

2. Conversion merely transposes the subject and predicate, moving from S-P to P-S. A, I both become I (only the latter reversibly); E becomes E (reversibly); but O is not convertible. The validity of conversion, in the case of I, proceeds from the equivalence of the conjunctives "Some things are S and P" and "Some things are P and S", which are respectively identical with "Some S are P" and "Some P are S". A is convertible by virtue of its implication of I; it is not, however, fully convertible, note well. E is convertible, because it contradicts I.

3. Obverted conversion is achieved, where possible, by converting then obverting, moving from S-P to P-nonS. A, I both become O (only the latter reversibly); E becomes A (reversibly); but the process is inapplicable to O.

4. Conversion by negation is achieved, where possible, by obverting, then converting, from S-P to nonP-S. A becomes E (reversibly); E, O both become I (only the latter reversibly); but the process is inapplicable to I.

5. Contraposition is achieved, where possible, by obverting, then converting, then obverting, moving from S-P to nonP-nonS. A becomes A (reversibly); E, O, both become O (only the latter reversibly); whereas I is not contraposable. (Note well that E is not contraposable to E.)

6. Inversion is movement from S-P to nonS-nonP. This is achieved (irreversibly, note): for A, by contraposing, then converting, to obtain I; for E, by converting, then contraposing, to obtain O; whereas the particulars I, O are not invertible.

7. Obverted inversion is achieved (irreversibly, note), where possible, by inverting, then obverting, moving from S-P to nonS-P. A becomes O; E becomes I; while the process is inapplicable to the particulars, I and O.

Syllogistic arguments are distinguished by the interplay of their figures and moods. The figure of a syllogism is the way its three terms are arranged in the two premises and the conclusion. The mood of a syllogism is an expression of the quantity and polarity of its three propositions. A specific figure/mood of syllogism is said to be valid if its premises, whether they are materially true or false, together formally imply its conclusion; otherwise it is invalid, even if the premises and conclusion happen to be true. The issue of validity is primarily an issue of dependence, not of truth, note well.

The premises are named the minor premise and the major premise. The term found in both premises is known as the middle term (we will symbolize it by a Y) - this term is absent in the conclusion; the remaining terms are known as the minor term (symbol, X) and major term (symbol, Z), like their corresponding premises - these terms reappear in the conclusion, as its subject and predicate respectively. Traditionally, a mood is identified by explicitly listing its major premise, its minor premise, and its conclusion, in that specific order; although in practical discourse they may appear in any order, and sometimes one or two of them may be left tacit. As shown below, there are four conceivable figures of syllogism:

Now, if we take as our propositional arsenal the forms A, E, I, O, which we defined earlier, and we examine all their combinations closely (a simple calculation shows that there are 43=64 conceivable moods in each figure), we find that only 23 syllogisms are logically valid (that is, a mere 9% of the total)! Of these, only 13 need concern us (the other 10 are of derivative importance); they are:

More explicitly, we have the following significant valid moods in the various figures. Notice the symmetries and asymmetries.

The mood 1/AAA (nicknamed Barbara), is in fact the prototype of all syllogism, and may be regarded as intuitively obvious; all others can be reduced to it, directly or indirectly. We may use eductive processes and transpositions of premises for purposes of validation; and we may also use a method called reduction ad absurdum. For instance, to validate the argument "if no Z is Y and some X are Y, then some X are not Z" (2/EIO), we would say "for if all X were Z, then no X would be Y".

Here is an example of syllogism (mood 1/AAA, in which the middle term is, as it happens, a compound predicate): "All fishes with fins and scales are kosher; sardines are fishes with fins and scales; therefore, sardines are kosher".

A few words on the logic of change are necessary, here, as we shall have occasion to refer to this field in a later chapter. Change has two forms 'getting to be' and 'becoming'; propositions involving these relations are known as transitive categoricals, in contrast to attributive categoricals, which involve the copula 'is'. While 'X is Y' refers to a static relation between the terms, 'X changes to Y' refers to a dynamic relation such that something X was previously not Y, and later Y. In 'X gets to be Y', what is 'X and not Y' initially, is 'Y and still or again X' finally; whereas in 'X becomes Y', what is 'X and not Y' initially, is 'Y and no longer X' finally. Thus, the former concerns superficial change (the thing remains X when it appears as Y), the latter fundamental change (or metamorphosis, the thing ceases to be X before it reappears as Y). The distinction is most evident in the limiting case: while X cannot be or get to be nonX, it can become nonX; note this well.

The concept of change and the distinction between superficial and fundamental change are of great significance to syllogistic logic. For instance, with reference to natural modality, the premises 'All Y must be Z, and X can be Y' yield a valid conclusion 'X can be Z' (in the first figure, with necessary major premise and potential minor premise, the conclusion is potential). However, the premises 'All Y can be Z, and X can/must be Y' do not similarly yield the conclusion 'X can be Z', as we might at first sight imagine, but 'X can get to be or become Z'; in the case of a potential major premise (first figure), the valid conclusion is still potential, but it is not a single categorical, it is a choice of two categoricals! For here, the given premises, though they affirm the minor and major terms separately, do not guarantee them capable of coexistence: it is conceivable, and it happens, that they are incompatible.

We see from this pivotal case that the relations of being and becoming are formally interlaced in the theory of syllogism (the role played by modality in this is secondary: it merely abstracts the temporal element). Many other syllogisms of the same kind are valid, notably 'All Y can become Z, and X can/must be Y; therefore, X can get to be or become Z', or 'All Y must become Z, and X can be Y; therefore, X can get to be or become Z', or 'All Y must become Z, and X must be Y; therefore, X must get to be or become Z'. (Note that 'get to be' may replace 'be' in such arguments, since the former logically implies the latter.) We need not go further into this field for our purposes here.

With regard to conditional propositions, various deductive processes have also been identified. We shall here briefly focus only on logical conditioning, and mainly on propositions of specifically hypothetical form, like "if P, then Q" or "if P, not-then Q".

We have, to begin with, eductions, the most notable of which are conversion (shown below) and contraposition (shown earlier, when the forms were first defined):

"If P, then Q" (=P and not-Q impossible) is normally convertible to "If Q, not-then not-P" (=Q and not-not-P possible), for if Q implied not-P, then P would be self-contradictory.

"If P, not-then not-Q" (=P and not-not-Q possible) is convertible to "If Q, not-then not-P" (=Q and not-not-P possible), for if Q implied not-P, then P would imply not-Q, contradicting the premise.

We also have syllogistic arguments, very similar to those encountered previously; some important examples:

All such arguments can normally be validated with reference to the first of them here listed, by direct or indirect reduction. Note that the first two examples here given, involving only positive hypotheticals, may be referred to as uppercase syllogisms, whereas the others, involving negative hypotheticals, may be referred to as lowercase moods; because, effectively, positive hypotheticals are analogous to general categoricals, while negative hypotheticals have properties similar to particulars.

Another important kind of deduction is apodosis, which has two essential moods, one positive, one negative^[25]:

Hypothetical propositions can often be derived from non-hypothetical propositions by a process called production. All the results of formal logic can be viewed as productive in this sense. For example, the premises "All Y are Z and some X are Y," within the primary, Aristotelian perspective, yield the categorical conclusion "some X are Z". However, one can also draw a secondary conclusion of extensional conditional form, "if any X is Y, it is Z"; or again tertiary conclusions of logical conditional form, like "if all Y are Z and some X are Y, then some X are Z".

In dealing with logical conditioning, it is important to distinguish normal and paradoxical forms. Some arguments involving hypothetical propositions are valid independently of this issue, while others are only valid specifically in cases where neither thesis is internally inconsistent. We cannot go into this matter in detail here, only briefly touch upon it....

In its broadest sense, the form "if P, then Q" allows for paradox: if we substitute "not-P" for "Q" we obtain the proposition "if P, then not-P". This proposition is formally possible, since it merely tells us, according to our original definition, that "P and not-not-P cannot be true" - and from it we can infer that P cannot be true. Note well that the hypothetical "if P then not-P" is not itself faulty, but clearly reveals to us the logical impossibility of the antecedent P, and therefore incidentally the logical necessity of the consequent not-P. We say, therefore, that "if P, then not-P" is a paradoxical proposition or argument, yielding that P is self-contradictory and not-P is self-evident. Needless to say, the form "if not-P, then P" would have the opposite result (namely, P).

Paradoxical argument is seldom encountered in practise; but when it is feasible, it usually justifies some fundamental pillar of human knowledge. An example: If we assert that "humans cannot know anything for sure", we are effectively claiming certain knowledge for ourselves in that case at least; our explicit assertion is therefore implicitly self-contradictory and false; it follows that "humans can know some things for sure" is true and self-evident. Note that 'self-evident' means immediately evident or obvious, independently of any knowledge context, and therefore evident in all knowledge contexts; this may be contrasted to 'contextually evident', which refers to what seems evident in some contexts, but might eventually not be so in others.

Such inference may also, more commonly, occur indirectly, in the process known as simple dilemma. This kind of argument was historically one of the first noticed by the Greeks^[26]. Two moods have been identified:

Here again, a relatively categorical conclusion is drawn from conditional premises^[27]. A similar kind of inference, also with two moods, which however yield only disjunctive conclusions, is complex dilemma:

Much more can of course be written about all the forms and processes mentioned above^[28], and others still. The science of logic, as above briefly presented, dissects our thinking processes into small units, studying each one carefully. In practise, these formal units are combined together and permuted in every possible way, usually with many intermediate steps left unuttered (whether out of dishonesty, ignorance, laziness or merely to avoid saying the obvious). The person who studies logic becomes adept at analyzing any reasoning he/she encounters with reference to its formalities, and is thus able to evaluate it accurately.

I refer readers open to further study of formal logic to text-books found in the market, and especially to my own work Future Logic. In the present volume, however, we must move on to more specific concerns.

[1] Or believing Jews, if you prefer; and many non-Jews, of course.

[2] The names of our Divinity are commonly written incompletely, even in their non-Hebrew forms, so as to avoid their destruction (which is prohibited on the basis of Deut. 12:4) should a copy of the book be damaged. I am not sure that merely leaving out the vowel, as in G-d or the L-rd in English, suffices, but it at least shows respect.

[3] Apparently, not only at Mt. Sinai, but also earlier at Marah and later on the plains of Moab. We shall just say 'Sinai', in the way of a collective term. See Lewittes, p. 38.

[4] However, sometimes the word Torah is broadened to refer to the whole Tanakh; indeed, sometimes it is used even more broadly to include all Jewish law.

[5] The fact that some laws were Prophetic rather than Mosaic in origin is of course a problem, in that Judaism is supposed to be essentially unchanged since Sinai. The Sages explained this by claiming them oral traditions dating from Sinai, which were written down by the prophets, or else forgotten and again revealed to the prophets. See Lewittes, pp. 32-33.

[6] Also known as the Palestinian Talmud. 'Palestine' refers to the land of Israel, which at the time was under Roman occupation. Although Jews were then the large majority of the inhabitants, the country was named after the Philistines, a non-Arab people who had by then disappeared.

[7] Nowadays, most editions of the Talmud include a mass of later commentaries and supercommentaries.

[8] The expression haShas is an acronym for the 'shishah sedarim' (six orders), and thence a name for the oral law.

[9] I am not here alluding to the Zadokim (Sadducees), or Karaim (Karaites) or even to certain like-minded modern Conservatives and Reformists, who more or less believed that Jewish Law should have been derived exclusively from the Torah.

[10] Oral transmission may be thought at first less reliable than the written word, but if one thinks about it, there is no real reason to regard documents as any more reliable. Sooner or later, an act of faith is necessary, that the document or the spoken report was indeed of Divine origin. This is the faith of Judaism, as we have said, and we shall take it as our starting point. (Of course, it is to some extent easier to date documents than oral traditions, and thus to some extent verify claims concerning their authorship; but there are often difficulties and disagreements, which leave us with doubts, anyway.)

[11] See Lewittes, p. 91. But note well that it is the Rabbis themselves who tell us which oral laws are deoraita and which are derabbanan; there is no way to independently audit their pronouncements in this respect, since by definition they refer to oral and not to written laws.

[12] See Lewittes pp. 33, 57. This presumably refers to non-deductive inferences, since purely deductive inferences are logically bound to have Biblical force. But it does not follow that only deductive inferences have been granted Biblical force.

[13] See, for instance, the Pirkei Avot, ch. 1.

[14] Philosophers will note that such innovation implies, to some extent, a delegation of creative powers by Gd to the human authorities; for the power to make a legal ruling is nothing less than the power to create an ethical fact which was previously non-existent.

[15] Some of those who ventured to look at Judaism critically ended up converting to other religions.

[16] The illustration here given is rough, and should not be taken as a thorough analysis of the issues touched upon, pro or con.

[17] We may also speak of 'a logic' in a non-pejorative way, when referring to intelligent forms of thought which are found especially in certain areas of knowledge or scientific fields; e.g. logistics is the logic of willed deployment of (material or mental) objects in space and time, mathematics is the logic of numbers and spatio-temporal relations. Similarly, historians of logic may objectively refer to the logic of (used by or known to) different geographical or cultural groups or periods of history. All specific logics, good or bad, may be subjected to objective study, of course.

[18] And indeed, 'call a spade a spade'.

[19] The three laws: Identity, Non-contradiction, and Exclusion of a Middle, are known as the Laws of Thought. They were first formulated by Aristotle, who identified them as the foundations of all logic.

[20] Note well here that the subclass implies the overclass, and not vice-versa; so that the subclass is the subalternant and the overclass is the subaltern. The concept of subalternation is not to be confused, as is easily done, with the concept of subordination. While the species implies the genus (because whatever falls under the species is subsumed under the genus), the genus is said to include the species (because species is narrower than genus). To indicate the inferiority, in the latter sense, of the species to the genus, one may say that it is subordinate. Thus, to repeat, a subclass is not subaltern to an overclass, but subordinate to it.

[21] It should be clear that plural propositions are here understood as statistical summaries of independent singular propositions. That is, "All/Some S are/aren't P" is equivalent to "This S is/isn't P" and "That S is/isn't P" and..., etc. (which, however, need not all be true at once). Such quantifiers are characterized as dispensive (or distributive), and are distinguished from collective reference (e.g. "All S, taken together, are P") and collectional reference (e.g. "All S, separately but simultaneously, are P").

[22] Inductive logic is also (though rarely) known as epagogic; in which case the term 'logic' is limited (as is often the case in common discourse) to deductive logic.

[23] This is also called the hypothetico-deductive method or the scientific method.

[24] A good example of this, is the Newtonian concept of 'force'. At the root of this scientific concept are the notions obtained through our intimate experience of push and pull, speeding and slowing. These intuitions give meaning to the idea of invisible attractions and repulsions between physical bodies, which cause them to accelerate or decelerate as they visibly do. The invisible factor of force is then quantified with reference to measurable changes of velocity. (Positivistic philosophy regards the invisible factor as superfluous; but it is convenient and we do use it, and furthermore, positivism itself makes use of such abstracts.) The 'novel terms' used in adduction are always based on notions recycled from experience, through the imagination, by analogy, into a new context. What gives the process scientific legitimacy is the check-and-balance provided by adduction.

[25] Note that the valid moods of apodosis consist in 'affirming the antecedent' or 'denying the consequent'. To 'affirm the consequent' or 'deny the antecedent' is invalid from a deductive point of view; but note that such reasoning has inductive value, respectively 'confirming the antecedent' or 'weakening the consequent' (see comments on adduction in the next chapter).

[26] Sophists, if I remember rightly; long before Aristotle, anyway. A Biblical example of such arguments is 2 Kings 7:4, which goes roughly like this: if we enter the city, we die; if we stay at the gate, we also die; but if we go to the enemy’s camp, we might be spared. This may be viewed as partly a dilemmatic argument (whether we enter city or stay at gate, we die), but more broadly as a disjunctive argument (listing three alternatives, eliminating two, leaving one).

[27] With appropriate substitutions, it is easy to show that paradox is a special case of simple dilemma. Put "not-P" instead of "Q" and "P" instead of "R"; the constructive mood involving "if not-P, then P" yields P, and the destructive mood involving "if P, then not-P" yields not-P.

[28] We may, for instance, note the formal continuity between apodosis and dilemma. Simple dilemma uses a vaguer premise than apodosis (disjunctive, instead of categorical) to yield an as definite conclusion (categorical), while in complex dilemma the conclusion is vaguer (disjunctive).

JUDAIC LOGIC