1. Innovations

2. History

3. Assessments

When I started writing the present work, in late 2010, I thought it would take a dozen pages and a couple of weeks at most to say what I felt the need to say. I had, I believed, said most of what needed to be said in my previous foray in the field of a fortiori logic, in my 1995 study of Judaic logic. But having noticed that some people were still writing on the subject without reference to my work, and to boot were making serious mistakes, I felt the need to show them the errors of their ways. However, as I proceeded in this set task, I found myself more and more involved in its intricacies.

For a start, to be fair the critiques had to be detailed, and show exactly what had been said and where lay the errors and lacunae. Secondly, I kept discovering more and more commentaries which needed to be similarly reviewed and evaluated. Thirdly, it became obvious that I needed to expand my theoretical investigations, to be able to answer various questions these commentaries brought up, consciously or unconsciously. Eventually, I realized that I had to aim for a history of the subject and a survey of more recent contributions to it, to be able to demonstrate precisely who said what first.

Thus, the work ended up taking me three years to complete. Three parts emerged. The first presented my new, much more detailed theory of a fortiori argument. The second part traced the early history of use and discussion of such argument, so far as I could make it out with the resources available to me. The third focused on modern commentaries on the subject. However,these parts did not emerge separately, but repeatedly impinged on each other, so that many chapters or sections had to be written more than once to be adapted to new findings. For this reason, it was impossible to publish any part of the work before it was all done.

It should be stressed that the work did not proceed in the order that the chapters are now set out. Whereas now all commentators are ordered chronologically, I did not comment on their work in their order of appearance in history. It was all a matter of chance encounter and personal mood. Moreover, my theoretical baggage at each stage was different. For this reason, some earlier chapters may appear more perspicacious or analytically cutting than some later ones. I tried, of course, to harmonize things as much as I could; but as the book grew in size, it became more and more unwieldy. No doubt my memory in these later years is not what it was once; so I may have missed some things.


The present work is replete with valuable innovations in the field of a fortiori logic, and in other, related subjects. The present, wider ranging research confirms that my past work in this field, in my 1995 bookJudaic Logic, was novel and important. But moreover, the present work corrects some inaccuracies in that past work, and greatly enlarges and sharpens our theory of a fortiori argument, so that it may be said to address almost every nook and cranny of the subject. There is not a single topic that I worked on here that did not yield some new insight or new theoretical development in a fortiori logic. This means that the research was certainly worthwhile and interesting; it is not a mere collection and rehashing of old material.

‘Formalities’, part one of the present volume, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged.

The present work also contains, in a final appendix, valuable innovations relating to certain topics in general logic; namely, symbolization and axiomatization,existential import, the tetralemma, the Liar paradox and the Russell paradox.


Logic science, properly conceived, is not just a theoretical enterprise, but also an investigation into the historical roots of the forms of human discourse. The present work on a fortiori logic constitutes an excellent case study of how a particular form of thought is rooted deep in antiquity (in history), and probably much earlier, in language itself (in prehistory), and then gradually develops as awareness of it dawns, expands and intensifies. There is ample evidence that a fortiori discourse existed in very ancient times and in very diverse cultures. A fortiori reasoning was present in early Greek literature (Homer, Aesop), long before Aristotle first discussed it (in hisRhetoricandTopics); and it was present before that in Jewish literature (the Torah and other Biblical books). Aristotle did not invent the a fortiori argument, any more than he invented the syllogism; he ‘merely’ observed, described and explained them, as a botanist might notice and catalogue interesting plants.

‘Ancient and Medieval History’, part two of the present volume, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaicqal vachomerargument, and a reassessment of thedayoprinciple relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah.


When I started to study a fortiori logic, I was little aware of the number of people who have since the late 19thcentury attempted to describe and explain this common form of reasoning. The field seemed nearly empty of contributors, a desert yet to be explored. Only little by little did I realize that many people have indeed tried their hand at solving the enigma of a fortiori argument – some, to be sure, more competently than others. It gradually became clear that a survey of existing contributors needed to be made, and their work had to be carefully studied and assessed. Such assessment depended, of course, on the theoretical and historiographical work undertaken earlier. It was interesting to see how many of the contributors studied past work very little before proposing their own ideas. Each apparently thought he was one of the first explorers.

‘Modern and Contemporary Authors’, part three of the present work, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is astonishing, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us hownotto proceed.