CHAPTER 33 – Conclusions and Prospects

1. My past errors and present improvements

2. Historical research into logic

3. Assessing contemporaries

4. Perspectives

We shall here briefly recount the main theoretical innovations, historical findings, and critical assessments put forward in the present work (including the Appendices), and reflect on possible future directions of research.

1.My past errors and present improvements

In my bookJudaic Logic, first published in 1995, I devoted four chapters (out of fifteen), or some 65 A4 pages (out some 300), to the a fortiori argument because of the important role such discourse plays in Jewish thinking. The first of these chapters is devoted to ‘the formalities of a fortiori logic’; the second, to ‘qal vachomer’; the third, to a ‘revised list of Biblical a-fortiori’; and the fourth, to ‘the language of Biblical a-fortiori’. The present work devotes about ten times as much space to the same subject.

As regards the formalities, I distinguished inJudaic Logiceight primary moods of the argument. These were of two forms, the copulative and the implicational. The structure of the former could be subjectal or predicatal, and that of the latter could be antecedental or consequental. Furthermore, for each of these possibilities, the polarity of the argument might be positive or negative. There were thus four moods with the orientation ‘from minor to major’ and four with that ‘from major to minor’. The distinctions and terminology there introduced have been retained in the present work and proven very useful. Moreover, the formal analysis that I made in the past – of each of the said eight moods, into a ‘commensurative’ major premise, together with a ‘suffective’ minor premise, yielding a ‘suffective’ conclusion – has endured and proven perfectly accurate in the present work. However, I have here gained a better understanding of implicational moods, after realizing that the ‘middle thesis’ in them, being a proposition, cannot strictly speaking have degrees (i.e. be implied more or less); so there was some inaccuracy in this specific area.

But the most important improvements have emerged in the area of validation. The idea of reducing a fortiori argument to a complex of implications and quantitative comparisons has been carried over. But, whereas my analysis of ‘commensurative’ propositions has stayed essentially the same, that of ‘suffective’ propositions has evolved considerably. Whereas previously I assumed the latter forms to be composed of three simpler propositions, I have here come to realize that a fourth constituent must be acknowledged to fully capture the crucial notion of ‘sufficiency in relation to a threshold’ involved. For instance, it is not enough to say that ‘something X is R enough to be Y’ implies,inter alia, the positive constituent: ‘whatever is at least to a certain measure or degree R (say, Ry) is Y’ – we must also acknowledge the corresponding negative constituent: ‘whatever isnotat least to that measure or degree R (i.e. is not Ry) isnotY’.

Another important finding in the present work is the idea that the middle item R may range in value from minus infinity through zero to plus infinity, i.e. that R need not necessarily be positive, but may well be zero or negative. This is something that I had some difficulty with before, but which I realized the need for when I tried to explain Louis Jacobs’ distinctive construction of a fortiori argument. Once the idea of a full range of values emerged, it became very useful – in the development of secondary moods of the argument, in dealing with relative and antithetical middle items, and in efforts of ‘traduction’ (i.e. correlation of moods). Similarly, my theory of quantification of a fortiori argument was drawn up in response to the report by Wiseman of a hot dispute on this issue between Mercier, Schiller and others, in early 20thcentury (1915-1918) issues ofMind. As regards my correlations and comparisons between a fortiori argument and syllogism, they were of course proposed in opposition to Schwarz’s reported claim that these forms of argument are identical or interchangeable. Similarly, my comparison and contrast between a fortiori argument and argument by analogy was made in reaction to the apparent equation of the two by several commentators, notably by al-Ghazali. My distinction between ontical, logical-epistemic and ethical-legal a fortiori arguments was likewise motivated by the wish to correct erroneous thinking found in various commentaries, such as that of Lenartowicz and Koszteyn.

Another important new breakthrough in the present work is the formal description and validation of a crescendo argument. I had in the past mentioned the possibility of ‘proportional’ a fortiori argument, but I had not tried to present and analyze such reasoning in formal terms. The formal study of this topic in the present work greatly clarifies the issues involved. Also new in the present work is a thorough examination of what constitutes valid inductive a fortiori reasoning. Wiseman tried to give credence to proportional a fortiori argument, but in fact failed. He also tried to give voice to inductive a fortiori argument, but again in fact did not succeed. With all these new formal tools in hand, it becomes easier from now on to interpret a fortiori arguments one comes across in material discourse – i.e. to estimate the probable intent of the speaker or writer in each case – and of course to judge their validity. This emphasis on hermeneutics – in the widest sense – is a valuable aspect of the present work.

Turning now to my studies more specifically aimed at Judaic a fortiori argument, there are many significant improvements in the present work. For a start, thanks in part to Louis Jacobs’ contribution, I now have an even longer list of Biblicalqal vachomer. Moreover, thanks largely to Samely’s hard work, I now have a long list of Mishnaicqal vachomer. But the most important novelty here is the greater understanding, and more precise pinpointing, of thedayoprinciple. It is not, as many have thought, including myself in my previous work, a general logical principle; it is a much more specific, essentially moral principle that rabbis should not infer, measure for measure, from the lower penalties given in the Torah for certain crimes, higher penalties for greater crimes. This understanding was made possible through detailed study of the Mishnaic origin of this principle, and is amply confirmed with reference to other uses of it in the Talmud.

Also to be found in the present work is a first attempt by me at listing the a fortiori arguments in later, Talmudic literature. For the time being, our study proceeds only in relation to Rodkinson’s English edition of the Babylonian Talmud, which contains only part of that major cultural asset. But by this means we provide future researchers with valuable ways and means that they can apply in a more exhaustive study of both Talmuds and of other literature of the same period. Also new in the present work, as contributions to historical research: detailed listings of a fortiori discourse in the works of Plato and Aristotle, and in the Christian Bible and in the Koran, as well as evidence of such discourse in Chinese and Indian spiritual literature.

Additionally, in the final appendix, I present novel analyses of some topics in general logic, including symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox.

2.Historical research into logic

If we introspect, and watch our thought processes carefully, we can well see that rational insight can be non-verbal; indeed, logically, the insight must precede its verbalization, even if verbalization of an earlier insight greatly facilitates the next insight. The proof is that sometimes we have a thought in mind, but have difficulty putting into words (and in the opposite direction, we may hear or read words and yet not immediately understand them). This psychological sequence of events tells us something about the history of logic and of language. When did some pre-human species of animal evolve into the “rational animal” we call man? Two million years ago (when Australopithecus made and used simple tools)? Four hundred thousand years ago (when Homo Erectus used fire)? Or two hundred thousand years ago (when Homo Sapiens emerged)? It is hard to say. But we can say that even before linguistic abilities distinguished mankind from other animals, we must have had some simple logical thoughts. Since the understanding and use of words is itself a complex logical act, speech must have evolved later. It emerged as an additional species skill, greatly enhancing our reasoning powers. Much later still writing appeared, further distinguishing us from other species.

Logic is not only implicit in language, but in all human behavior that calls upon reason. All tool-making and much tool usage involves rational activity. To understand, invent, design and perfect, and even use, a hunting instrument (e.g. a trap, a spear, a bow and arrow) requires rational reflection, which need not be verbal. Similarly, fire-making, cooking, preparing pigments, agriculture and other such activities present in very early human societies, are all evidence of reasoning of sorts. Some animals can use natural tools, e.g. a stone to break a nut; but only man can make artificial tools. Perhaps if we analyzed precisely, through careful introspection today, the reasoning necessary for these creative activities, we could draw up an inventory of the earliest expressions of logic in mankind. By dating the earliest tools or their products, we might then be able to date the logic they imply. For instance, there are paintings in cave dwellings that are estimated to be some 40,000 years old. Obviously, man was already, at least that far back, a “rational animal.”

In this way, by referring to our present mental and physical behavior, we can somewhat trace the historical development of logic. However, to say that logical abilities are inherited from our biological forebears does not mean that such potential is automatically actualized in us. For instance, the hypothetical (if–then) form of proposition is not somepre-existinggenetically transmitted mental tool, but emerges initially, consciously or almost unconsciously, from reflection (which may be preverbal) on the laws of thought. The negations of conjunction found in the laws of non-contradiction and of the excluded middle move us toinvent and definehypotheticals – in the ways I explained in my past work,Future Logic. If you look at the order of the chapters in that book, you will see that the chapter on conjunction precedes the chapter on hypotheticals – it has to, because we cannot mentally truly understand hypothetical forms without first grasping the idea of conjunction.

I do not of course mean that each of us individually invents hypotheticals. Some individuals presumably did, probably in prehistory, in the hazy dawn of mankind. Eventually, these logical tools were passed on as cultural heritage, through language. But such discovery and transmission was not all done at once. There has been a development through historical times, which can indeed be traced to some extent. You can see it, for instance, by comparing the wording for if–then in the Tanakh and that in Talmudic literature; or again the latter to modern wording. Language changes over time in response to changing awareness of the nature of things. Verbal changes reflect changes in understanding. You can see also the growing awareness of the meaning and importance of hypotheticals in modern logical science, compared to earlier phases of logic. Aristotle hardly reflected on hypotheticals, whereas later logicians made a big thing of them.

The same can of course be said concerning a fortiori discourse, which is more complex than if–then and syllogistic forms of discourse, since it appeals to them among others. It has a history, no doubt a very long and geographically scattered one. I have tried in the present work to trace some of that history. There is no doubt much more work to be done in this respect. The present treatise does not pretend to be a sweeping history of the a fortiori argument, of its use and discussion of it through all ages and in all cultures. Such a detailed, definitive history still needs to be researched and written – for a fortiori, and indeed for all forms of argument. Nevertheless, I believe the present work shows the way to carry out this mission, with some novel methods and applications. In particular, note well that it is not enough to just describe past ideas – they must also be judged[1].

We have seen in the present volume that Aristotle (4thcentury BCE) was probably the first to discuss a fortiori argument, though clearly not the first to use it. However, as far as we know he did not study such argument in as much detail as he did the syllogism. A fortiori argument appears in Jewish literature probably in the 13thcentury BCE; in Greek literature probably in the 7thor 8thcentury BCE; and in Indian and Chinese literature probably in the 5thcentury BCE. But these dates of course refer towrittenlanguage; they do not exclude the existence of a fortiorispeechcenturies or even millennia before them. My own belief is that a fortiori discourse antedates the splitting of humanity into different linguistic groups. This is suggested by its wide geographical distribution. It should be sought in proto-languages, and in languages with little or no written component, though of course all languages change over time. Did aboriginal African, Asian, European, American and Australasian languages in the distant past include a fortiori phrases?

In any study of a fortiori logic, a large place must perforce be accorded to Jewish logicians. Because of the prime position of such argument throughout Talmudic and other rabbinic discourse, and in all lists of hermeneutic principles used in Judaism, it is natural for Jews to have had an especial interest in this thought form. These ancient practical and theoretical expressions of human thought, however imperfect some of them might have been found upon further scrutiny, were without doubt a strong stimulus to Jewish interest in logic in general and a fortiori logic in particular. A considerable expertise emerged over time, in the way of a ‘family business’, or more broadly put a ‘tribal specialty’! (This is just a statistical observation, nothing more, I hasten to add.[2]) Doubtless, Christian and Islamic thought were strongly affected by Judaic logical practices in religious and legal contexts, even if other cultures also had their influences.

The present investigations show that whereas a fortiori expressions (like ‘a fortiori’, ‘all the more’, etc.) are found in all sorts of ancient Greek and other non-Jewish literature (philosophical and otherwise) – they are in such literature mostly used as ‘turns of speech’, i.e. as elegant rhetorical devices that could conceivably have been dispensed with, rather than as logical arguments crucial to proving some significant assertion. In this sense, it may be more appropriate to refer to such use more vaguely as a fortioridiscourse, rather than argument. On the other hand, in Talmudic and rabbinic literature such discourse is usually indicative of anargument, in the true sense of the term. It is used to discover or prove something, something important to the speaker or writer (such as a legal ruling).

As regards religion, the following perhaps needs be said here. Much of human discourse over the centuries has occurred in religious contexts, so it is necessary for historians of logic to study religious documents in search of logic. This is, however, a delicate and touchy subject. My own approach to religion is, I think, objective. I do not hesitate to criticize where criticism is due. To my mind, spirituality is a natural dimension of man; man needs to earnestly reflect on the meaning and direction of his existence. Such reflection has historically produced a great variety of man-made religions and sects. All religions (including the religion called atheism) claim omniscience of sorts, to establish their authority. All indeed contain some truth in some degree; but in fact none contains all truth or even only truth. This is evident, if only at the material level of logical and factual blunders; in many cases, there are also (in my personal opinion) serious spiritual blunders.

The logician or philosopher being a human being cannot be entirely objective on such crucial issues; but he or she has to try to. A good start is to notice and admit factual errors. There are many errors of that sort in all religious documents, though to varying degrees. The next thing to notice and admit is logical inconsistencies or gaps. Again, these occur in all religions, to various degrees. As regards spiritual blunders, their discovery is perhaps best ensured by comparative studies. If one studies only one religion (or no religion), one remains locked in a single conception of things; but when one broadens one’s perspective, one is better able to see things differently, and eventually judge by oneself what is best point by point. That is ‘natural religion’ – one remains open to influence by the spirituality of our forefathers, but in a more lucid and rational manner, without fanaticism.

Nothing is further from true spirituality than blind faith, especially when this results in attempts to impose one’s views on others by force. In this perspective, the religion called Islam has to be singled out, and regarded by all honest and moral men and women as a criminal organization, and treated as such. For Islam is exceptional in this day and age in its political ambition of world dominion. Whereas Judaism in the distant past (in its war against the Canaanites) and Christianity in the more recent past (in its persecution of Jews, for instance) have been guilty of religiously-motivated violence, Islam still today advocates and works towards mass conversion by coercion, and the enslavement or murder of those who refuse to submit to it. That Islam may contain some spiritually positive elements, which make it attractive to many people, does not change the fact that its political intentions and ways are evil. This fact must never be glossed over.

3.Assessing contemporaries

If philosophy – and its various branches, including epistemology and logic – is to be a scientific enterprise, it has to be subjected to strict standards of judgment, comparable to those of more materialistic disciplines. The way things stand, still today, is that anyone may say anything, however patently silly and contrary to fact, and hardly have to face any reasoned criticism from his or her peers. The reason is that many of those peers, even those with advanced university degrees and high academic positions, are themselves ignorant and/or confused, and so unable to tell the difference between wheat and chaff. As we have seen in the present volume, reputable journals, book publishers and encyclopedias publish papers and treatises that are full of glaring factual and logical errors. One wonders who sits on their editorial committees, till one realizes that the publishers themselves cannot tell the difference between those who have and those who haven’t a clue, and so are content to rely on the qualifications ‘on paper’ of the staff they hire.

Too many of those who write articles or books simply do not know what they are talking about. Their motive is not pursuit of truth, but mere posturing. Rather than wanting to know, they want to appear as if they know. They seek glory, or at least position, not true knowledge. They are too proud to ask questions of those who know, because that would imply admission of ignorance. They would rather say nonsense than humbly face the fact of their nonsense and learn how to think straight. Sure, they do read – skim through – other people’s work, but only enough to drop names. They especially like to sow famous names all over their discourses, so as to effortlessly bask in reflected glory. They think that they need only mention Kant, Frege or Wittgenstein to seem like cognoscenti. This is the way they were educated, how they gained their credits and diplomas. Their teachers do not know any better, having themselves been brought up that way.

I was constantly struggling with my conscience while writing the assessments in the present work, especially the assessments of the work of my mostly younger contemporaries, including Alexander Samely, Dov Gabbay, Stefan Goltzberg, Andrew Schumann, Allen Wiseman, Yisrael Ury, and Hubert Marraud, many of whom I have corresponded with and look upon with friendliness. Jewish ethics forbids the practice oflashon hara(the evil tongue). Telling people their faults to their face; reproving them in public and with harsh words – this is not moral behavior, even if no lies are told. Even just hurting a person’s feelings is bad; but such speech is especially reprehensible if it may cause the person targeted loss of a job or of an economic opportunity. Yet, I finally gave myself free rein and allowed myself total frankness. My argument was that if these high moral standards were applied to scientific debate, it would spell the end of science, since science depends on open criticism. If I did not draw attention to the mistakes and foolishness I came across, I would be allowing them to persist indefinitely, causing many people harm. A distinction must be made betweenad hominemargument and argument against a thesis. The former attacks the thesisindirectly, by defaming the person defending it; the latter attacks the thesisitself, even if doing so may cause vexation to its defender.

My whole writing career in logic and philosophy has been motivated by the desire to increase the use of logic in the world. I have spent, hours, days, weeks, months and years, unhealthily stuck at my desk, with this one goal in mind. Logic and philosophy today are still sorely lacking in intelligence and consistency. To become sciences in the best sense of the term, they need to be tamed by logic and empiricism. These fields should not remain playgrounds for dilettantes and poseurs. High standards of discourse must be imposed and maintained, as is done in mathematics or the physical sciences. I sincerely apologize to the people, alive or dead, that I have criticized, for hurting them personally in whatever ways. But in this conflict of values, the value of truth must take precedence over that of solidarity. The moment someone emits an opinion concerning matters of logic or philosophy, he must be ready to receive criticism. If the criticism is undeserved, he can and should defend his thesis; but if it is deserved, he should admit his intellectual transgressions without resentment, indeed with gratitude[3]. Criticism looks destructive; but it is necessary, because one cannot build the new without tearing down the old.

Here is the consolation: even as I criticize other writers for their errors, I must say that I am grateful to them for their errors, because were it not for their errors, I would not have thought of the corrections to their errors. It was by detecting and examining their mistakes that I found out how such mistakes could be avoided. These people were therefore to some extent my teachers or collaborators. Thus, people can contribute to a subject-matter even through errors. They may have made errors, but their work was not in vain. One thing they can be sure of, and that is that I have given their work a fair hearing. Very often, my analysis is longer that the passage or essay or treatise that I analyze. My criticism is never unjustified, always carefully reasoned. Remains to be seen how many of these people will have the grace to publicly admit their errors, and maybe even thank me for pointing them out! Those of them who were really curious about a fortiori logic, and wished to know the answers to questions at all costs, will study the present work and be grateful to me for it. As regards those whose foray into this field was entirely narcissistic, motivated by the desire to appear – to themselves if not to others – like great thinkers and discoverers: they will lose interest in the subject of a fortiori logic, and will not read the present work but instead be angry with me for having written it.

In any case, I take no ultimate credit for my work, considering that all intelligence, all insight and understanding, comes from God, since it says in the Torah: “In the heart of all that are wise-hearted, I have put wisdom” (Ex. 31:6). I naturally had to sacrifice time and generate effort, to discipline myself and persevere; but my abilities were given to me, as to everyone else, as a gift. Moreover, to repeat, the innovations made in the present work were not produced out of the blue. Many or even most of them emerged in reaction to the work of others, duly mentioned in it. Therefore, to repeat, it can be said of even the authors here greatly criticized that their work was far from useless and they participated indirectly in the formation of these innovations. Without their mistakes, I would have taken for granted many things that seemed obvious to me. Their errors or lacunae stimulated my responses, and turned them from potential to actual doctrines.

The theory of a fortiori presented in myJudaic Logicsome 18 years ago was, I thought then, pretty thorough. However, in the past few years I came across some old and some new theories on the subject that I did not at that time know of – and these now needed to be commented on by me. When I first presented my own wholly original theory, I considered and claimed, implicitly if not explicitly, that other moods than those I validated were not valid and that other interpretations of the argument as a whole could no longer be upheld. Yet, still today, I find some writers trying to revive some old notions or to propose new ones inferior to or in conflict with mine. I had to respond to their implicit challenge, because after having read their theories I still viewed mine as the definitive and leading statement on this topic. I had no desire to ‘defend my turf’ egotistically – but I did feel responsible as a logician who has contributed considerably to this particular field.

It shocks me that some barely competent, if not quite incompetent, people try to make a name for themselves by formulating some new theory of a fortiori argument, thinking the field still virgin territory. They pursue innovation ‘for its own sake,’ even when there is no call for it. This is not a syndrome peculiar to a fortiori logic, but is unfortunately found in all fields of logic. I have often remarked in my works on the ridiculous tendency of some alleged logicians to look for some completely new approach to logic that would constitute the Copernican revolution for that field. Everybody seems to want to be the Einstein of logic theory, so as to impress the whole world with his incomparable genius. Such motivation is a sure formula for failure. A logician must be moved by awareness of his basic ignorance and stupidity, and a strong desire to improve his situation somewhat through hard work. He must regard himself as a humble servant, and seek to honestly clarify and understand the subject-matter at hand.


I will finally briefly state what directions I think research in a fortiori logic should take – in general and in specific areas.

The first thing I would recommend to future researchers is to fully study and absorb the present work. Then they should continue collecting past ideas and commentaries on the subject of a fortiori argument, carefully assessing the contribution of each, looking to see if it offers any valuable new insights and cataloguing the errors and lacunae in it. In this way, future researchers can develop full understanding and skill in this field. Further empirical research into the history of use and discussion of a fortiori argument is also very important work. Finding evidence of this argument (and indeed others) in the phraseology of different languages – particularly exotic ones – would greatly help us determine how far back in human history it appeared. All of world literature should be scanned, with the aid of computers at least, for words or phrases indicative of a fortiori intent. Then each concrete argument found should be analyzed and duly classified.

Each cultural context should be studied separately, by knowledgeable researchers. Thus, for instance, Jewish literature, including the two Talmuds and all related works, requires specialized research. Similarly, Christian literature, Islamic literature, Buddhist literature, Hindu literature, etc. In order for results to be made readily available to all interested parties, they should ideally be made freely accessible in the Internet. The latter is a tall order, seeing as many people like the prestige and the income that ideally comes from publishing hardcopy articles or books. But if your motive is to accelerate the spread and pursuit of knowledge, nothing beats posting your findings for all to see on the Internet. In this context, translation into an international language – meaning English, today’slingua francaof science – is essential. Of course, the research work should preferably be done in the original language – but when it is done, make the results available to all through translation into English. Brief English abstracts are not enough. Without a common language progress is bound to be slow.

Many modern authors dealt with in the present volume attempt to define or describe a fortiori argument in symbolic terms – and they all, as we have seen, woefully fail to do so convincingly. This should not be viewed as a mere coincidence, but as a failure almost inevitable if a subject is approached through the medium of modern logic. People generally wrongly assume that what is stated in symbolic terms must have been “scientifically” pondered. The opposite is closer to the truth. Symbols more often than not are designed to hide the ignorance or triviality or confusion or errors of those who indulge in them. They are a smokescreen behind which those who have not understood a subject hide, hoping no one will find them out. One purpose of this book has been to demonstrate the superiority of ordinary-language logic over symbolic logic. It is best to use ordinary language to discover and deeply analyze logical forms; only after this main job has been completed is it permissible, although not necessary, to express the results by means of symbols.

Unfortunately, symbolism is all too often a refuge for pretense by not very bright people. But let the child in us blurt out that their emperor’s new clothes are non-existent. The main sin of modern, symbolic logicians is to try to circumscribe and finalize something which is far too big for human control. The domain of logic is something that must always remainopenfor new insights and further evolution; it cannot be locked down with simplistic formulas akin to those of mathematics. Symbolic logic based on paltry human analysis can only produce paltry results. What counts is the logician’s depth of understandingbeforehe starts concocting symbolic formulas. Otherwise such formulas are merely the frozen evidence of the superficiality of his thought. There is room for symbolic logic in logic – but it is something to be done, if at all, at the tail end of the logician’s work. Once a fortiori argument is fully understood, as in the present study, it is easy to reword the results in symbolic terms; but understanding must come first.

If universities continue to produce students who can restate and reshuffle worthless symbolic formulas ad nauseam, but are unable to intelligently analyze actual human discourse, there is little hope for the survival of logic in our culture. I would propose we replace the current fashion of ‘modern, symbolic’ logic with what might be called ‘neo-classical’ logic. The latter is a logic whose main goal is deep understanding, rather than superficial posturing.

As regards the development of intelligence, I would like to here say just how big a role sitting meditation has played in my writing of the present book. Most of the significant insights to be found in it were obtained in the course of my daily meditations. It is possible that many of the problems here solved would have eventually been solved anyway; but they were certainly solved more readily and efficiently thanks to meditation. When the mind is relatively quiet, it is able to concentrate with much greater force on any issue put before it. So, another methodological recommendation I would offer to anyone wishing to get into logic research is: learn and regularly practice meditation so as to sharpen your intellectual capabilities and develop patience.

My regular practice of tai chi and other sporting activities should also be mentioned as a source of health and equilibrium. Having mentioned meditation, I should also mention the Sabbath: this weekly day of obligatory rest has always been for me a source of renewed energy and inspiration. Finally, I wish to thank God for kindly allowing me to complete this arduous task. I pray that this work, together with my other works, improves many people’s minds and help them to live their lives more wisely.

[1]The Kneales’The Development of Logicprovides a good example of the critical approach to logic history.

[2]A similar statistical observation I’ve made in the past (in a footnote to chapter 4 of myJudaic Logic): “I have an impression, for instance, that modern French discourse involves more use of a-fortiori than modern English discourse. To what extent that is true, and why it should be so, I cannot venture to say.”

[3]I certainly practice what I preach here. Indeed, I beg all readers to draw my attention to any error they may happen to find in my work, be it typographical, grammatical, logical, or whatever. And when they do, I hasten to correct it (as soon as time allows), and to thank the readers for their solicitude.