1.Summary of Findings.

2.Gaps to Fill.

3.Concluding Words.

1.Summary of Findings.

Let us now, finally, try and summarize the information presented in this book on logic, part by part and chapter by chapter.

I.Actual Categorical Logic.This is classical, Aristotelean logic, embellished somewhat over the centuries.

1.We distinguished between the art of logic, and the science of it. The former is commonly practised, the latter is intended to guide practise, as well as serve to provide theoretical grounds for human knowledge.

2.We discussed the three Laws of Thought instituted by Aristotle, the founder of logical science as we know it. They are our principal equipment in sorting out the phenomena appearing before us.

3.The basic tools of logic were introduced: the concepts of truth and falsehood, and logical relations like implication, incompatibility and exhaustiveness.

4.We discussed how words are related to the things they refer to, the concepts of sameness and difference, and the role of definition.

5.The features of the propositional forms called actual categorical were described, distinguishing the terms and copula, and the polarity and quantity. The traditional notation for these various propositions was introduced.

6.The various oppositional relations of propositions were defined, and these concepts were applied to actual categoricals, by means of diagrams and tables, and the findings were validated.

7.The various eductive processes which propositions may be subjected to were defined, and these concepts were applied to actual categoricals, and the findings were validated.

8.Syllogistic deduction was defined, its figures and moods, and the discrimination between valid and invalid such arguments.

9.The main valid moods, plural and singular, of actual categorical syllogism were listed; and the less significant moods of the fourth figure and the subaltern moods, as well as imperfect syllogism were also mentioned. The common attributes of the valid moods were noted.

10.Finally, why and how syllogism are validated was considered. Also, derivative arguments, like sorites, were mentioned.

II.Modal Categorical Logic.This is a broadening of Aristotelean logic, with the addition of modality. Though to some extent known since antiquity, this field has never been properly and fully developed as here done.

11.We distinguished the categories of modality — necessity, presence, possibility, and their negations, as well as other degrees of probability. We distinguished various types of modality, concentrating to begin with on three — the extensional, the temporal and the natural. Tense and duration were discussed incidentally.

12.We discussed certain phenomena underlying these concepts of modality, namely diversity, time and change, and causality.

13.A full list of propositions involving the various categories and types of modality under discussion was presented, and a new notation facilitating our reference to them was introduced.

14.We devised a general theory for predicting the oppositions between plural and modal forms, from the known oppositions of singular and actual forms. We applied these findings to the forms previously listed, and determined all their interrelations methodically, in enlarged diagrams and tables. We also investigated the eductions feasible from modal propositions.

15.The main valid moods of modal syllogism were listed for each type of modality, and their validations were described.

16.Valid moods of lesser or derivative significance were also listed, including moods of mixed modal types. The statistics of validity were looked into, and general principles formulated.

17.The concepts of being and becoming were analyzed, and new propositional forms concerning change (transitives) were introduced. Some important syllogistic arguments involving them were pointed out.

18.Permutation was discussed in more detail, and other copulae than those thus far considered were mentioned.

19.We looked into substitutive processes, comparative propositions, and the differences between dispensive, collective and collectional quantities; also, the doctrine of quantification of the predicate was discussed.

III.The Logic of Logical Conditioning.This is a closer inspection of the logical relations used in practise, a field which may also be described as the self-analysis of logic. Its beginnings date from Aristotle and Philo in Ancient Greece, but it has been especially developed formally in modern times. However, our own treatment of the subject is considerably novel, on many counts.

20.We discussed the genesis and role of the three Laws of Thought in logic, and the distinctive functions of each of them. The notion of phenomenal credibility was further highlighted.

21.We made original formal definitions of the categories of logical modality, and justified the knowability of these concepts.

22.We discussed the various aspects of contextuality, which affect logical modality.

23.The formal treatment of logical relations begins with the concepts of conjunction. The factual forms of conjunction were distinguished with reference to polar considerations, and their oppositions to each other were identified. The modal forms followed accordingly, and so did their oppositions, by means of the general theory of opposition earlier presented.

24.Conjunction gives rise to various kinds of conditioning. We began by analyzing logical conditionals, known as hypothetical propositions or ‘strict’ implications. Novel negative forms were also considered, which in the next chapter led us to a hierarchy of forms.

25.We discerned that hypotheticals are defined not only by the connection they signify, but often also with reference to certain logical bases. Thus, we distinguished between hypotheticals with unspecified bases, and those with normal or abnormal specified bases. The oppositions between hypotheticals, and the eductions from hypotheticals, were described and validated.

26.Disjunctive, as distinct from subjunctive, conditioning was considered. Various manners of disjunction were described and interrelated.

27.Then we looked into various intricacies of logic, expanding on what had so far been presented. Conjunctive, hypothetical and disjunctive propositions form a broad continuum of relations, affected by the number of theses involved, their respective polarities, the polarities and modalities of their relations. Nesting and mixed-form relations were looked into, and we evolved the unifying method of matrixual logic.

28.Next, using some modern symbolic techniques, we investigated the principal interactions between logical relations. We did so with reference to matrices, and thus demonstrated the precise intellectual goal of all such manipulations, and the limits of their practical utility.

29.We developed a full list of hypothetical syllogisms, including those with negative forms, showing how they are derived from the most obvious case. We also introduced the novel process of production, drawing conditional conclusions from unconditional premises.

30.Logical apodosis was dealt with, including its modal forms. Dilemmas and their rebuttals were analyzed.

31.Paradoxical propositions were considered. They allowed us to formally define self-contradiction and self-evidence, and some important philosophical applications were pointed out. We also evolved a more thorough theory of hypotheticals, listing all the normal and abnormal forms which may arise, and investigating their distinct properties in opposition, eduction and deduction.

32.We considered apparent double paradoxes, which are not as legitimate as single paradoxes, showing their logical function and how they are to be dissolved. The examples of the Liar Paradox and the Barber Paradox were dealt with.

IV.The Logic of De-re Conditioning.This field may be viewed as an expansion of the modal categorical logic presented in part II (by considering propositions with more than two terms), or as a broadening of the logic of conditioning presented in part III (by considering non-logical types of modality); in either case, it seems to be entirely original. The value of this field to all future science and ontology is inestimable — it sets new, very high standards of precision for any discussion of causal relations.

33.Just as logical modality gives rise to logical conditioning, so the natural, temporal and extensional modalities give rise to their own quite distinct types of (de-re) conditioning — and thence in turn to various types of causal relation.

34.We began our analysis with reference to natural conditioning, proceeding much as we had done for logical conditioning. The features of natural conditionals were distinguished, including their bases and connections and their quantities. We also noted the issues of sequence of the theses, modalities of actualization, and acquisition and loss of powers. Natural disjunction was also mentioned.

35.We looked into the relations between natural conditionals and categoricals; and we investigated the oppositions among, and eductions from, these new forms.

36.We developed the main valid moods of natural conditional syllogism, and also listed the subaltern and invalid moods. We described the productive arguments, which enable us to infer, and thus form, natural conditionals from modal categorical premises.

37.Natural apodosis, both actual and modal, was analyzed; and so was natural dilemma.

38.We similarly investigated the structure and properties of temporal conditionals, showing their exact relation to naturals (especially noteworthy, was their discontinuity), and various arguments of mixed modal-type were presented.

39.Then we considered extensional conditionals — their (analogous yet distinct) features, oppositions, and eductions.

40.And we analyzed, for extensional conditionals, syllogism, production, apodosis and dilemma in some detail.

41.Development of the logics of various types of conditionals allowed us to review certain issues concerning categorical propositions, with more powerful formal tools. Until then, the modalities of subsumption by the terms of categoricals had been skimmed over; now, a more nuanced approach became possible. Included here was discussion of imaginary terms.

42.We were also now able to analyze certain condensed forms, which are thought of as categorical but involve some kind of conditioning, including forms with complex terms, and those having to do with aetiology, teleology and ethical modality.

V.The Logics of Classification and of Adduction.These two topics were lumped together, without intent to imply a close relation between them. They are fields of logic which derive from the previous, though important in themselves. There are many significant innovations in our treatments of class-logic. The novelty in our treatment of adduction lies in its modal orientation.

43.We saw that class-logic takes terms ‘nominally’, in a way distinct from the subsumptive approach of Aristotelean forms. However, classes and classes of classes are easily defined with reference to Aristotelean forms; and the features, immediate inferences, and deductive arguments of forms with such terms are readily derivable from these definitions.

44.We distinguished classes and classes of classes as two separate ‘orders’ of classes, each with its own though parallel ‘hierarchy’ of classes. The relational aspect of these concepts was stressed, when we sought to clarify their extreme manifestations.

45.We analyzed the concept of self-membership both conceptually and with reference to examples, and found it wanting. We then considered the famous Russell Paradox, and demonstrated that the solution of the problem lay in the concept of permutation (rather than in issues of membership), whose ontological significance was also clarified.

46.Adduction is the general method by which we induce the logical probability of any information. We discussed its well-known form of argument, which is similar to apodosis, only with less established premises and/or conclusions. We showed how it provides and weights evidence, and thus validated it. We also discussedde-readduction.

47.We looked into the psychology of theorizing, described the structures of theories and various criteria we use in making them, and we suggested ways theories may be more purposefully formed and tested.

48.We described in formal terms the scientific method of judging between theories, but also indicated the pragmatic compromises that are often called for, and how theories may gradually be changed. Theories with exclusive empirically-tested predictions were granted formal certainty.

49.Under the heading of Synthetic Logic, we advocated a healthy skepticism and flexibility, which transcends rigidly formal standards of theory-evaluation — an open-mindedness to more far-fetched hypotheses which are not definitely disproved.

VI.The Logic of Factorial Induction.This is a completely new field of logic — the first genuinely formal theory of generalization and particularization in history. Again, this sets entirely new, extremely precise standards for all future science, and answers some of the most fundamental questions of epistemology (if I may be forgiven for sounding such a loud trumpet).

50.The problem of induction was to begin with posed with regard to actual categorical propositions — how are they known, whether particular or general? The solution is so simple, with relation to actuals, that it seems puerile; but as we later see, when modal propositions are considered, the solution appears much more interesting.

51.In order to deal with modal induction, we first had to develop a precise theory of all the logically possible combinations of modal propositions. The forms considered thus far were mere elements, that may be compounded in a certain number of ways, according to their oppositions. We noted that compounds give rise to special arguments; also, that directional issues may be raised.

52.Next, we introduced the concepts of fractions and integers, which describe states of being more definitely than elements or compounds can do. The former are parts of the latter. These concepts and the resulting formulas depend on the logics ofde-reconditionals, and different systems evolve according to the mixtures ofde-remodality we choose to consider, and whether we ignore or include directional issues.

53.These preliminaries led us to a formal theory of factorial analysis of elements and compounds, and indeed of all states of knowledge concerning anything. A factorial formula consists of all the alternative integers which logically may come out of any given item of knowledge. In some cases, only one alternative is formally acceptable, so that an unexpected deductive situation occurs.

54.Thereafter, we proceeded to formally demonstrate the knowability of all types of necessity, whether extensional (generality), natural or temporal, or any combination of these. We described the stages of induction, and defined the central issue of induction as a pursuit of solitary integers.

55.After discussing the philosophical aspects of induction, we proposed an exact ‘Law of Generalization’ in formal, factorial terms — one which precisely determines the factor selection from any given datum whatsoever. As later shown, this same Law also controls the formula revisions we call particularization.

56.We applied the Law of Generalization to all possible elementary and compound forms, and showed the predicted valid inductive conclusion in each and every case to be rationally credible, thus also demonstrating the correctness, value and validity of the Law as a whole.

57.We then considered context changes, which require us to amplify previous conclusions or harmonize them with new data. This was called formula revision, and the difficulties it involves were clearly described, in order to show the power of their formal resolution.

58.We applied the Law of Generalization to all inconsistent conjunctions of elements or compounds, and obtained precise inductive conclusions from all of them.

59.Lastly, we applied the Law of Generalization to other situations requiring formula revision, like adding fractions to integers, reconciliation of integers, indefinite denial of integers, among others.

In this way, we demonstrated that we have evolved a single, uniform, consistent tool for dealing with all knowledge contexts. We used the specific example of categorical propositions involving different mixtures ofde-remodalities; but we also indicated how expanded application to still more complex situations is to be effected.

VII.Perspectives.In this final part of the book, we dealt with wider, more philosophical or historic issues. Chapters 60-62 together (with reference of course to all previous chapters) sketch my theory of cognition. Chapters 63-67 could be viewed as a separate volume, calledFor Future Logicians; it is a small-print commentary mostly intended for academics rather than general readers.

60.We looked into various ontological issues. What do we mean by phenomena? How do we distinguish the empirical from the hypothetical, the physical from the mental, the concrete from the abstract? Representation and analogy were briefly discussed.

61.Next, we pointed out the primarily relational nature of consciousness, and on this basis evolved a novel systematic classification of the kinds of consciousness, defining many of the epistemological terms used in the course of our logical treatise. We thereby proposed a more logically consistent theory of the mind, than that suggested by popular psychology and many philosophers and biologists.

62.We also considered some important logical issues surrounding sense-perception, and recognition and memory. These insights, together with those made previously, about the various kinds of phenomena and consciousness, and about imagination, allowed us to arrive at some understanding of ‘universals’.

63.We reviewed a history of logic, from Ancient Greece to the present, making positive or negative evaluations as we proceeded.

64.We analyzed concepts like formalization, symbolization, systematization, and axiomatization; and thus we began our critique of modern logic, mentioning also the more constructive contributions.

65.Continuing, with reference to literature on the subject, we tried to estimate the level of knowledge in current modal logic, its achievements and its weaknesses and blank areas.

66.We looked more closely into current views on metalogic (and incidentally class-logic), countering them with our own views of language and meaning, definition and proof, and indeed of the foundations and role of logic as a whole.

67.We then deepened our understanding of all modality, as signifying different kinds and degrees of being; and we indicated how our theory of factorial induction can be expanded to include logical modality — to precisely solve the problem of induction from logical possibility, and thus explain the essential continuity between this mode and thede-retypes. Then we looked into the current state of knowledge in inductive logic, endorsing or disagreeing as appropriate, and pointed out a methodological standard.

68.The whole was finally summarized here, and in the next section I mention some possible areas of research for future logicians.

2.Gaps to Fill.

This treatise, though evidently somewhat encyclopedic in scope, makes no claim of exhaustiveness. We have pointed out, as we proceeded, various directions of possible further research, often spelling out the major parameters for it. The general reader was invited to personally try and do the job; one does not have to be a certified expert, one learns by doing. The value of such research is not merely that it takes the science of logic forward, but especially the exercise in the art of logic that it provides the researcher.

Let us now, therefore, mention some of the major opportunities for further theoretical inquiry. The field of logic is still wide open, the task is still enormous; no one person can do it all.The goal is nothing less than clarifying the formalities of all human knowledge, starting on the nonmodal, categorical level, then fanning out into modal and conditional considerations; and moving from deductive to inductive issues.

a.Categorical Logic.

The main focus of logicians through the ages has been the ‘is’ copula, which was gradually understood to be intended in a purely subsumptive sense. The sense of ‘X is Y’ before permutation may however contain formal properties which have yet to be thoroughly investigated. We have to go beyond the simplistic, merely quantitative aspects of logic, into its more conceptual, qualitative aspects.

The possessive ‘X has Y’, the active ‘X does Y’, and other similar verbs, are all copulae of broad applicability which are open to further analysis, including their oppositions, their eductions, their syllogisms, and so forth. Substitutive arguments, the logic of comparatives, the logic of collectives and collectionals (as distinct from dispensives), all still require work.

The class-membership copula has received a lot of attention in modern times, but as I have shown some serious conceptual errors were made. The whole field needs to be reconstructed, in the framework of my new definitions and initial analyses, avoiding the flights of fantasy which have characterized previous incursions. It is also important to keep a humble perspective on things; this field is not as crucial as has been construed, but a very limited and derivative one.

Issues of tense and duration have to be more systematically clarified (though much work has admittedly been done by some moderns in this field). My work in the formal logic of change is just a beginning, a sketch of some of the main components; much is still to be done, to cover all eventualities. For all categoricals, once the actual logic is dealt with, the corresponding modal logics must also be developed.

b.The Logic of Conditioning.

Once the categorical manifestations of any family of forms has been thoroughly formalized, its conditional versions must in turn be investigated. Each type of modality requires separate treatment, first on a categorical level, and then in the distinct type of conditioning it gives rise to. Mixtures of modes must also be dealt with. With regard to the more generic conditional logics considered in the present work, many details have been left out.

Even in logical conditioning, which moderns have investigated extensively using symbolic techniques, and which I have handled in some detail in ordinary language, there are still opportunities. Context comparisons could be elucidated in more formal terms. The dynamics of changes in logical modality can be further investigated. Also, although the various forms of modal conjunctives, hypotheticals, and disjunctives, may strictly be equated to each other, so that analyzing their interactions is somewhat repetitive — they nevertheless inform thought in different ways, and therefore it would be valuable to treat them as if distinct; interesting lessons might transpire.

I have developed the logics ofde-reconditioning in some detail, but not fully. In view of the novelty of these doctrines, my prime concern was to put the concepts across clearly, avoiding fatiguing minutiae; but logical science is eventually obligated to consider every little thing, however outwardly repetitive it seems. Thus, for instance, temporal conditioning might be looked into more thoroughly.

De-reconnections devoid of basis might be profitably investigated, or again unusualde-reconnections of one modal-type combined with bases of another modal-type. But especially important, in any case, is a thorough investigation of the oppositions and syllogistic mutual-impacts of forms of entirely different modal-types. Productive argument is a new field, which may be open to further expansion.

More work needs to be done on the modalities of subsumption in categorical propositions.De-reconditioning involving more than three terms, or other special constructions, is worth pursuing further. The temporal sequences ofde-reconditionals which we considered were very simple; more complex ones should be looked into. The issue of modalities of actualization is still wide open to formal treatment, as is that of acquisition and loss of powers.

Condensed propositions involving complex terms may be subjected to more systematic treatment (working out their separate and combined oppositions, eductions, deductions). Other forms deriving from the various types of conditional logic, like ‘making possible’ or ‘causing’ or ‘requiring’, are important in themselves, and should be exhaustively dealt with. A complete taxonomy of causal relations has to be developed.

The field of causal logic cannot of course be completed without reference to volition — but, though I have indicated how this type of modality is to be formally defined with reference to natural modality (by denial of all deterministic antecedents), I have here avoided the big job of developing this field. The auxiliary field of influences on volition (including habit), which is definable with reference to temporal modality, has also been purposely ignored. I have many preparatory notes on aetiology which I may one day use, but the reader is welcome to try and do these jobs independently.

Ethical modality has been all but ignored here, and I find the treatments of it by modern logicians to be very simplistic. Yet this field of formal logic follows very naturally and systematically from that of aetiology (via teleology), and the job is not overly difficult. My old notes on the subject show very interesting ways that absolute standards can be formally defined and logically induced, and I may one day publicize them. Meanwhile, go ahead, try and do the job.

In any case, it is my conviction that we can formally demonstrate the harmony of Ethics and Science. That is, I consider illogical, the view that scientists may pursue any research, however harmful to human welfare; the view that the pursuit of scientific truth sometimes necessitates ethical compromises. Ethical modality is ultimately just as ‘factual’ as logical orde-remodality, and the findings in all fields are bound by logic to be harmonious. Thus, if ethical logic concludes ‘do not research this matter further’, science might be slowed by obedience, but in the long run can discover as much if not more than it would by disobedience.

c.Inductive Logic.

My work in this area, though extensive, merely opens the door to a host of new possibilities. The science is now founded, the model is sharply drawn and part of the edifice is built; but it is far from complete. Especially here will the enterprising future logician find rich rewards.

Logic until now has focused on elementary forms, but compound forms are also propositions in their own right, though describing more specific relations. I have indicated some of the oppositions between compounds, but a more thorough treatment is required. Compounds involving singulars should be researched further. I have dealt with one-directional (‘flat’) compounds, but two-directional (‘stereo’) compounds may also be looked into.

Fractions and integers are of course still more definite compositions than gross compounds, and their formal interrelations may be looked into further. But in any case, those I have presented and dealt with here are flat; the field of stereo fractionating and integration is a large and important one, which has yet to be developed; patience, imaginativeness, and a big-CPU computer and powerful software are required for this work. Once stereo factors have been developed, all gross compounds should be factorized in their terms.

Another avenue of expansion for factorial analysis, as we mentioned, is to take logical modality into consideration, as well as thede-retypes. The value of this is more theoretical than practical, since logical modalities are inherently implicit in the concept of ‘strength’ of factors; nevertheless, this work would serve to demonstrate the formal character of induction from logical possibility tode-remodalities. Here again, the volumes of data involved require appropriate computing equipment, as well as intellectual capacities.

In any case, all such expansions of factorial analysis will engender enormous growth in the fields of factor selection and formula revision. However, the Law of Generalization we introduced remains the sole operative principle, so the job is reasonably straightforward. Even without going into such expansions, I have left quite a bit work for future researchers to do in the field of formula revision; for instances, solving indefinite denial of gross compounds, or interactions between integers and gross formulas.

But all this is only the beginning: it concerns categorical propositions (with the standard copula). A greater challenge still is the inductive logic of conditional propositions of all types and forms. These can, as we have shown, be effectively deduced from categoricals, through the process of production. However, that does not mean that they cannot be and are not also induced.

Thus, a factorial logic for all conditionals is also required, which will be much more complicated than that for categoricals, yet may obviously be developed along the same lines. First consider compounds, then fractions, then integers, then factorization, then inductive decisions, as before. Again, better have a big machine at your disposal!

As for changes in copula, they are only significant at the initial, experiential stage of induction; subsequent processes of generalization and particularization are as far as I can see exactly identical.

d.Logical Philosophy.

With regard to ontology and epistemology, we have seen some of the issues and conditions logic implies for them. Above all, logic demands of any philosophical suggestion concerning the presuppositions or implications of human knowledge that it be self-consistent, that it include itself in its considerations, that it explain itself. Beyond what has been said, many questions of course arise, which are relevant to logical science, but were outside the chosen scope of our treatise.

In the process of clarifying the history of logic, in order to determine what was old and what was new in our own findings, we incidentally came across issues relating to the methodology of historical judgment. This is an interesting field in my view, which yet requires more pondered and systematic consideration, at least in my case. I learned from this discovery that indeed, as modern thinkers maintain, each science has its own methodological issues, and logicians may be called upon to set specialized standards.

In any case, logicians, philosophers, historians, and special scientists, indeed all of us, should of course aim for as wide a perspective as possible in theories. By this I mean, that it is not enough to be very knowledgeable in one field, like Western Philosophy or Modern Physics say, but to keep in mind (as far as possible) the beliefs and insights of all peoples and all periods of history and all disciplines (including religion). There is no profit in moving from one closed-minded dogma to another; open-mindedness is of the essence of knowledge.

I do not of course advocate that we pay attention to obviously deranged or perverse ideas, or clearly discredited ones; but just that unfashionable ideas, which are less probable in our eyes, be kept in mind. Our criteria in theory selection are not all binding; many theories are effectively discarded, not because they are intrinsically inconsistent or discordant with empirical data, but because they are more far-fetched (they require more confirmation than their alternatives) or because their practical value is relatively minimal. Such theories are not strictly-speaking out of the running, though of course we need not be committed to them or endorse them.

With regard to the philosophy of logic, I have argued vehemently — and I hope very convincingly — against modern extreme symbolism and axiomatism. These trends are confused and futile. Logicians must be able to build their case essentially in ordinary language terms, which do not obscure what they are saying if anything; they must consider whether the epistemologies and ontologies they imply are internally consistent and in accord with common experience; and they must strive for a more genetic, conceptual understanding of the development of logic.

Historians of logic, for their part, should keep in mind the distinction between the art of logic and the science of logic, between implicit logical skills and overt (formal or informal) pronouncements about logic. These are separate issues, which do not always travel in tandem. There is still much work to do in this field, as indeed in that of philology. The self-centeredness of many Western evaluations have to be guarded against, without of course going to the opposite extreme of belittling evident differences and imposing interpretations which were never intended.

3.Concluding Words.

No human can claim omniscience, and no human achievement can be claimed to be perfect. I am of course human, more ignorant than a great many people, and of limited intelligence and virtue. Still, I believe this work on logic has considerably revived a virtually moribund enterprise. Important advances have been made in traditional fields, major new fields have been pioneered and developed, and the science as a whole has been set on much stronger foundations. Much work remains to be done, and it is my fond hope that many will take up the challenge, for I consider logic to be of high importance to human knowledge, personal improvement, and social development.

Logic is of course not everything, but it helps. It will not make a person or society virtuous, but it will facilitate such pursuits. Many people avoid logic precisely because they are afraid of the imperatives it may impose on them; they would rather not know, than find themselves forced to accept what they refuse to believe, or to behave in demanding ways. Also, logic requires an effort of thought, just what the mentally lazy wish to avoid. But the important thing is to realize its benefits; knowledge is enlightening, it is pleasant, it saves us from wasteful or irrevocable mistakes, it improves one’s life.

I ask myself now, what is the specific message of this book? To the general reader, it is:think more clearly. Consider the alternatives, the possibilities, the strictures, the necessities, the conditions under which things take place or are true, and those in which other things come to the fore. Consider the sources or bases of your beliefs, opinions or knowledge; and evaluate them.

Educators are to be encouraged toteach children, youths, and adults formal logic— the classical, and the new modal, conditional, and inductive doctrines. Using ordinary language, not esoteric symbols. Each age group should of course be addressed at its own level; children can learn simple syllogisms, Ph.D. students can research factorial formulas.

The goal is not to produce thinking machines, people enslaved to verbal mental processes, but merely to strengthen the natural faculty of logical intuition we all share. If we teach mathematics, the ability to reason with numbers, should we not also teach logic, the ability to reason with concepts? The latter would surely seem more important, since we are more than merely economic beings.

Even economic thinking is of course improved by logic. A house is better planned, a business better run, by a person better trained in the art of thought. But furthermore, in this era of democracy and ecology, people are called on to make all sorts of complex judgments, for which it is to everyone’s advantage that they be well-equipped. The habit of reasoning, thinking things through, is also of moral value — teaching people to resolve their differences in rational ways, to judge each other more objectively and dispassionately (which does not of course mean cruelly or without regard for feelings).

With regard to special scientists, I am here calling fora more modality-conscious level of Science. It is time for the sciences to specify the precise types and categories ofde-remodality of their statements. I believe this is the next step in the evolution of the scientific enterprise, one which will greatly enhance its capabilities, generating new ideas and solving many outstanding and future problems. Thus far, science has proceeded with the limited tools of thought put at its disposal by Aristotle and Philo, and their successors to this day. It seems obvious that with more sensitive cognitive equipment it successes will be still more impressive.

To give an example. Everyone will agree that the more rigorously we reason, the more likely are our conclusions to be correct. It should therefore be obvious that if we distinguish, say, between a natural causal relation and an extensional one, we save ourselves from error. For if these two types of conditioning are lumped together in a generic if-then statement we may assume them to be in disagreement while they are in fact compatible, or harmonious while they conflict. Only by specifying the types involved (mentally if not explicitly), and knowing the precise formal oppositions between forms of these types, can a proper judgment be made.

Where our thinking is not modality-conscious, we may thus assume a theory to be consistent when it is not, and lose the opportunity to improve it or reject it. Or we may assume a theory to be inconsistent when it is not, and adhere to an unproved alternative in its stead. Clearly, the more perspicacious, the less naive, our logical tools are, the more intelligent and interesting are our theories. Knowing faults in one’s theories stimulates the imagination; and knowing that alternatives are still viable also expands one’s consciousness.

Furthermore, I challenge scientists to make the effort to specify the exact logical modalities of their statements, with reference to inductive logic. I believe I have made available to them unequivocal tools for this purpose, namely the processes of factorial analysis, factor selection and formula revision. My vision is of a science which knows precisely where it stands logically, which can precisely trace and honestly displays its inductive as well as deductive justifications. Again, such efforts can only be rewarding, because they promote open-mindedness, imagination, awareness of extra problems, awareness of alternative solutions.

Scientists no longer have any philosophical reason to doubt causality, or to feel embarrassed in referring to it. We have here shown in clear and indubitable formal terms its definability, its varieties, and just how it is to be established case by case. Likewise, the Cartesian ideal of deductive science, which was so far from the empirical standards of practising scientists, has been demonstrated to be absurd; instead formal logic now demands the organization of knowledge in accord with the model set by factorial induction. It is possible, and it is necessary.

All this concerns not only natural scientists, like physicists, biologists, psychologists, but also social scientists, like historians or political scientists (though in their case considerations of ethical logic impinge more directly). Needless to say, it also applies to less officially theoretical professionals, like doctors, business leaders, politicians or journalists, who are to some extent — like all people, though perhaps more than most — making daily theoretical judgments of their own.

Philosophers too are subject to the same confines of inductive and deductive logic. Speculations are of course often valuable, and unavoidable. Sometimes even, to be sure, paradoxical insights, like some of Kant’s or some mystical philosophies, have enormous creative potential, generating ideas in diverse fields. We do not advocate naive and impatient over-simplifications. But still, ultimately, the goal is to make genuine sciences out of epistemology and ontology, and in order to achieve this philosophers must make an effortto keep track of their methodological grounds.

My concluding message to future logicians, the theoreticians of logical science, is to keep in mind above alltheir role and responsibility as teachers. It is useless to write something most or all other people will never understand. Our goal is not to impress others with our exclusive knowledge. The logician is not supposed to be a solipsistic manipulator of virtually meaningless symbols, divorced from reality and from society.

The logician is a scientist and communicator. As a scientist, he or she must refer to the widest possible perspective on things, and find non-naive and neutral methodological information. As a communicator, he or she must find ways to pass this information on to anyone open to it. The goal of logic is to clarify and improve knowledge, not to obscure it and effectively leave people without methodological ways and means. Thus, symbolism must be eschewed so far as possible (it is admittedly not always easy); today, symbolization has become almost synonymous with academic respectability and proof — but our attitude should be the very opposite.

Furthermore, the logician must be philosophically aware, and never range out too far from common-sense ideas in epistemology and ontology. There must after all be reasons why the human psyche has cumulatively come to these conclusions, these views; the job is to understand what these reasons might be. The logician must in any case learn to apply logic to his or her own thinking, and thus avoid arriving at ridiculous conclusions like those of certain ‘linguistic’ schools. Only ideas which somehow or other vindicate the human faculties of knowledge, admitting their essential effectiveness and objectivism, have any logical credibility.

It is my hope that the present manifesto for future logic will change things for the good, G-d willing.

Completed[1]on Denman Island, B.C., Canada,

on 26 June, 1990 (the 3rd of Tammuz, 5750),

with G-d’s help.Baruch HaShem.

Here, in gratitude for the knowledge received, is a beautiful psalm of praise by David, lovely king of Israel:

The heavens declare the glory of G-d,

And the firmament tells of His handiwork.

Day unto day utters the tale,

Night unto night unfolds knowledge.

There is no word, no speech,

Their voice is not heard,

Yet their course extends through all the earth,

And their theme to the end of the world.

The symbols of the L-rd are faithful,

Teaching the simple man wisdom.

[1]Writing started in late 1988, but was based to some extent on notes made in 1968-74.