The next few chapters, 63-67, contain material which is perhaps more of interest to academics (teachers and students of philosophy and logic, mainly), than to the ordinary reader. I would prefer the latter to skip this segment of the book and just go to the final chapter. The ‘flavor’ of Logic as it appears thus far, is I believe very clear and pleasant; one is also left with a practical tool. Those who read on, will experience a change of taste, as we enter into concerns and disputes, which fatigue the mind rather unnecessarily (at least, that was my experience). There can be too much of a good thing, as the saying goes.

Nevertheless, it was of course a scholarly duty for me to write this segment, and of course theoreticians are well-advised to study it closely. It may be viewed as a philosophical commentary on all the preceding chapters. As well as re-evaluating the work of past logicians, it deals with broader issues, like metalogic, and induction from logical possibility. The smallness and the errors of many doctrines of modern logic are demonstrated. Historical logic is also touched upon here and there.

1.Historical Judgment.

2.Aristotle, and Hellenic Logic.

3.Roman, Arab, Medieval European Logic.

4.Oriental Logic.

5.Modern Tendencies.

6.In The 20th Century.

1.Historical Judgment.

As stated from the beginning, a detailed history of logic is beyond the scope of this work. My concern here has been substantive — to develop a wide-ranging system of logic with an emphasis on the role played by modality in its main senses. My approach has been very independent, a research in the sense offresh thought, rather than one based on scrutiny of past achievements.

It was inevitable that there would be overlap between my own discoveries, and those of past logicians. However, the prospect did not bother me, as I felt that enough of my findings would be completely original, so that I could afford not to lay claim to all of them. One can never, in any case, be sure that one was not influenced indirectly — or directly, in some forgotten reading — by the work of anterior researchers.

Thus, to be fair, one must acknowledge the authorship of any idea which even remotely resembles one’s own. I will here try to minimally satisfy this standard of judgment, even knowing that without very extensive scholarly research into all the original sources, I can never hope to properly do so. I have only so far attempted to be a logic theorist; the label of logic historian is still well beyond my grasp.

I therefore hereby make a general disclaimer for the benefit of historians of logic: any idea which they find to have been put forward by previous logicians, they may attribute to them without any argument from me! The true scientist, after all, is not so much interested in personal aggrandizement (though one naturally wants one’s own contributions recognized), as in getting the job done and the science of one’s choice moved forward.

However, even without having read all that has been written on a subject, one can estimate what is generally known and accepted by its experts, by observing not only what they say about it, but also what theyomitto say, directly or indirectly. This is an important point, to keep in mind. For example, if no mention is made in current presentations or histories of logic of the various modal-types of conditioning, or of something resembling factorial analysis, one can fairly assume that these insights have not been previously arrived at.

For it is inconceivable that certain doctrines of such high importance should remain without mention, even in elementary presentations or histories intended for the general public. It is of course conceivable that the discoveries in question may have been made independently by different researchers too recently to be widely known and accepted.

Note that the issue is never one of terminology, but of essences. Different theorists may give different names to the same insights, and one of the tasks of the historian is to recognize the similarities in essence irrespective of terminology used. Of course, one should not dilute any significant differences. But naming or renaming an already known concept can hardly be called a notable advance.

Also, the historian has to distinguish seminal, pivotal, or comprehensive contributions, from derivative or trivial ones. I mean, there has to be some perspective in evaluating material, a sense of proportion. One may of course be ungenerously critical, and ignore valuable points. But to linger overmuch on artificial or superficial schemes is a disservice to logical science, making it appear quaint and redundant, and diminishing its respectability.

The reader is now referred toThe New Encyclopaedia Britannica: Macropaedia(henceforth occasionally referred to asNEB), to the article ‘History of Logic’ byCzeslaw Lejewskiof Manchester University (23:235-250). I chose that historical overview of logic, because it is very recent (1989) and readily available to a reader wishing to consult it, in most public libraries. It will be used as our first, and to begin with virtually our sole, source of information. That is, we will assume that:

a.It presents a reasonably complete picture of the ‘state of the science’, in its main lines, to date;

b.It accurately reflects the opinions of current historians of logic, with regard to authorship of ideas;

c.It accurately reflects the attitudes of current theoreticians in the field, with regard to substance.

These assumptions are no doubt debatable. We may well suppose that many significant doctrines have already been published, which are not taken into consideration in that article, and that there are divergent historical viewpoints and theoretical perspectives. But, for now, let us look on this one prestigious source as the academic mainstream; we can later broaden the picture somewhat.

My purpose here is not only to recount the salient facts and features of logic as it is currently perceived, butto reevaluate these current perceptions in the light of the insights arrived at within my own work. Thus, I am to some extent reviewing the article critically, and suggesting how the history of logic might be rewritten.

Unless otherwise stipulated, all quotations are from the said article. References to previous chapters and sections of my own work, will be distinguished by using the abbreviation ‘ch’. — for instance, this is ch. 63.1.

2.Aristotle, and Hellenic Logic.

The ancient Greeks gave birth to logical science as we know it, discussing both categorical and conditional logic in considerable detail (NEB23:235-238).

a.The logic ofcategoricalpropositions is generally attributed toAristotle(c. 350 BCE), the great Greek philosopher, the pioneer of Logic. He is credited with founding the science, through several major breakthroughs, among which: his description of the laws of thought, and introduction of symbols for terms, as well as the ensuing formalities for this class of proposition. Aristotle also discussed common fallacies of argumentation.

Aristotle’s model of formal logic included: analysis of the structure of categorical propositions: terms and copula; polarity, quantity (and, to some extent, modality); and systematic treatment of opposition, eduction, and most importantly syllogistic deduction. He discovered the figures and moods of syllogism, and methods of validation like exposition, andreductio ad absurdum. These doctrines were developed in several treatises, collectively known as theOrganon.

Aristotle’s main concern was withactualcategoricals, though he ‘initiated the development of modal logic’, through philosophical exploration of concepts like potentiality, change and causality in hisMetaphysics(which is not counted as part of theOrganon), as well as in more formal discussion of the categories of modality and the inferences which may drawn from them in other works.

In everyday discourse, we commonly use the indefinite form ‘X is Y’, which tacitly intends but does not specify a quantity. It can be taken to mean minimally that ‘at least one X is Y’ (a particular proposition); but often it is meant universally as ‘all X are Y’. By formalizing these alternative interpretations through quantification of the subject, Aristotle made possible the systematic development of categorical logic.

However, with regard to modality, Aristotle does not seem to have decisively opted for a similarly hard and fast distinction. As earlier mentioned (ch. 11.1), he seems to have on the whole intended, as we all often do, the copula ‘is’ in a timeless sense, more akin to the ‘must be’ of natural laws, than to the mere ‘here and now’ sense of temporal events. But this ambiguity and hesitation retarded formal development of modal logic.

There is no doubt in my mind that Aristotle did have a rich philosophical understanding of the various types of modality. For instance, he pointed out that, in a certain sense, things are ‘possible prior to the event, actual then, and necessary thereafter, so that their modal status is not omnitemporal… but changes in time’; also, that futuristic statements, like ‘there will be a sea-battle tomorrow’, often cannot in advance be judged true or false (thus implying admission of some indeterminism).

I am only saying that he did not fully exploit his knowledge in formal logic. His modal syllogism seems to refer specifically to logical, rather thande-re, modality; or perhaps his intent was to develop general principles applicable equally to all types of modality — but then, he was presuming that there are no significant technical differences between them.

In any case, he seems to permit drawing a possible conclusion from a possible major premise in the first figure(see for instance hisPrior Anylytics, book 1, chapter 14)— a serious error (see ch. 15.3, 17, 36.3), which more precise definition of the modalities would have allowed him to avoid, since he had already discovered the invalidity of such argument with a particular (extensional possibility).

Likewise, his profound conceptual analyses of change and causality were never translated into formal logical doctrines, concerning transitive categoricals and conditioning of various types. I guess Aristotle had so much to say about so many things, that he could not find time, in the few years of his writing career, to say everything within his grasp.

It is safe to say, anyway, that the contribution to logic by Aristotle is incomparable, in its breadth and depth. He is the grand master of Western logic. All anterior and subsequent work in the field seems like mere footnote or embellishment in comparison, because after all the initial impetus, theideaof aformallogic, is an unrepeatable feat.

I say this, because it seems to me that many modern logicians (best not named) try to put down and belittle Aristotelean logic, seemingly in an attempt to raise the value of their own contributions. Logicians also should acknowledge their teachers, and behave without ego. It is just a matter of respect for the effort and work of others. Criticism and improvement are of course not excluded; only, the boast of revolution is unnecessary.

There has of course been other great logicians and philosophers, before and since Aristotle. Earlier, the Sophists made various distinctions between sentences.Socratessearched heuristically for definitions.Platoencouraged axiomatization as ‘the best method to use in presenting and codifying knowledge’, and addressed the philosophical problems of universals. There was a broad cultural heritage to draw from; a practise, at least among intellectuals, and especially lawyers, of argumentation.

Later, among the ancients,Theophrastusof Eresus, a Peripatetic, indeed a direct pupil of Aristotle, and his successor as head of the Lyceum, is reported to have emphasized doctrines like the fourth figure (see ch. 9.2); also the (not entirely accurate) principle that the conclusion follows the modality of the weakest premise; and ‘prosleptic syllogism’, or the process of substitution (see ch. 19.1).

DiodorusCronusof Megara (4th century BCE) explored modality with reference to time, under ‘the influence of Socrates and the Eleatics’. Tense, the before-after aspect of chronology, was discussed. Temporal modalities were defined; ‘the actual is that which is realized now, the possible… at some time or other, the necessary… at all times’ (see ch. 11).

Porphyry, a Neoplatonist, clarified classification by the use of ‘trees’ dividing genera into species.Aristonof Alexandria is said to have introduced the subaltern moods.Galencontributed ‘compound syllogism’ (ch. 10.4).

Although many of these contributions were no doubt original, many of them were also or already known to Aristotle, but considered by him to deserve relatively little attention. Work of the latter sort may be viewed as a process of digesting the received information, making it more explicit.

b.The logic ofhypotheticalpropositions is rooted in Aristotle to a much lesser extent. As a logician he reasoned clearly in hypothetical form, giving later theoreticians an example of and opportunity for their theories. He effectively was the first to formallyproducehypotheticals, since all his inferences had the form ‘if the premise(s), then the conclusion’; and skillfully used thereductio ad absurdummethod of validation. But apparently, apart from mentioning ‘syllogism from hypothesis’, to show his self-awareness as a thinker and perhaps hint at a possible line of further inquiry, he did not go into this area of logic in detail.

The dilemma, and rebuttal, were forms of hypothetical argument used long before Aristotle, as evidenced by the case of Protagoras vs. Euathlus in the 5th century BCE (Copi, 258-259).

Paradoxical hypotheticals made their appearance early in the history of logic (5th-4th century BCE), though they were not formally understood. The paradoxes ofZeno, an Eleatic, gave rise to analytic conclusions seemingly contrary to experience and common-sense (but these were single paradoxes).Eulibedes, a Megarian, developed paradoxes like that of The Liar (a double paradox), which revealed purely conceptual contradictions (ch. 32). But the significance and logical acceptability in principle of single paradoxes, and the dialectic of self-inconsistency, is a modern realization (ch. 31).

Theophrastus is credited with the development of hypothetical syllogism, in figures similar to those of categorical syllogism (ch. 29.1). The hypothetical propositions were positive logical relations (if/then), with positive or negative theses; negative hypotheticals (if/not-then) were apparently ignored, then and since (ch. 24.2).

Diodorus defined implication by pointing out that the antecedent is always followed by the consequent (note the modal definition, albeit with reference only to what seems like temporal modality). His pupil,Philo(called the Megarian), considered it enough to just deny conjunction of antecedent and negation of consequent. He thus defined what is today called ‘material’ implication and is credited with the ‘truth-functional’ analysis of positive hypothetical propositions (see ch. 24.3). Some hundred years later,Chrysippusof Soli, a Stoic philosopher, developed logical apodosis (ch. 30.1), and considered hypotheticals with conjunctive theses and nesting (see ch. 27.3).

All the above seems to have concerned only logical conditionals. Perhaps, however, the conditional relation was unconsciously intended (as we commonly do) as not only logical, but generic, with the (incorrect) presumption that all types behave identically. But on the whole, natural, temporal or extensional conditionals were not clearly distinguished or treated in any detail. Diodorus did, it is true, analyze temporal conditioning, through the form ‘all times after {the sun has risen} are times when {it is daytime}’ (see ch. 33-40).

Let us continue now, and consider logic in later periods, or in some other cultures.

3.Roman, Arab, and Medieval European Logic.

The article now deals with the ensuing centuries, between the end of antiquity and the end of the Middle Ages (NEB23:238-240).

After Greek logic, there was a period of some five centuries during which ‘little or nothing happened in the field of logic’. There was ‘little creative work’. Some compendiums and scholarly commentaries on Aristotle’s logic were prepared, many of them later to be lost. There were, ‘on occasion, improvements in the minutiae’, but no major contributions. However, we ‘are indebted to[these authors]for salvaging numerous fragments from the lost writings of earlier logicians’.

Some works were translated into Latin. Roman authors, likeCicero(106-43 BCE) andBoethius(d. 524/5 CE), ‘transmitted the achievements of the Greek logicians to logicians of the middle ages’. But by the 4th century of the common era, ‘logic was treated as a subsidiary subject, providing useful training for students of law and theology’.

In the ensuing centuries, interest in logic seems to have been more lively, with the translation into Latin of some more Aristotelean works. Textbooks were written. Logical studies spread deeper into Europe, as far as England. The Scholastic period, as it is called, was more active, in that theoretical innovations were attempted. Ancient logic was brought into focus, clarified, systematized, extended and improved upon in various ways.[1]

Among the new inputs were: the distinction between subsumptive and nominal use of terms (see ch. 43.1); the concept of distribution of terms (ch. 5.3); recognition of the fourth figure; and, I might add, the use of symbolsA,E,I,Ofor the four actual, plural categorical propositions, which in my view was an important novelty.

There was a heightened understanding of logical relations: premises and conclusions were seen to be theses of a hypothetical form; hypotheticals could have conjunctive or disjunctive theses; a distinction was apparently made between formal and material implication. The latter concept was, in my view, an unfortunate development, which may have helped them to grasp some properties of implication simplistically, but also produced some at best trivial if not downright misleading results.

More impressive, however, was the interest in modal logic, by logicians like ‘Pseudo-Scotus‘ andWilliam of Ockham(14th century). They seem to have concentrated their efforts on logical modality, since their concern was with concepts like truth, falsehood, knowledge, opinion.

They are credited with defining the oppositional relations between the categories of modality, including actuality; for instances, that necessity implies actuality and actuality implies possibility, or that possibility of negation contradicts necessity (see ch. 11.2, 13.1, 14.2). They also discovered that (in logical modality) an impossible proposition implies all other propositions and a necessary proposition is implied by all others (ch. 21.2).

Scholastics also studied fallacies, and ancient paradoxes were confronted. The Liar paradox was seen as a meaningless proposition, in an anonymous manuscript of the 14th century. It is clear that logicians of this period were quite creative.

There was also a transmission of data by way of Byzantium, and the Arab lands. Arab logicians ‘played an important role in reviving the interest of Western scholars’ in Aristotle and other Greek logicians. Included areAvicenna(of Persia, 11th century) andAverroes(of Muslim Spain, 12th century), both of whose work on ‘temporal’ modality (by which is meant a mixture of natural and temporal modalities, to be precise) was of some value. But overall, according to the article, there does not seem to be work of any great originality or importance emerging from these logicians.

Incidentally, Jewish residents of those countries also played a part in this process, some of them as translators bridging the Moslem and Christian cultures. One may also mention logicianIsaac Albalag(13th century).[2]

4.Oriental Logic.

With regard to the Indian and Chinese logics, here are some of the findings of research mentioned in the article under review (NEB23:240-242):

Indian logic dates from as early as the 5th century BCE, with grammatical investigations. Later on, it also evolved in the framework of religious studies. Thinkers were ‘interested in methods of philosophical discussion’, although ‘logical topics were not always separated from metaphysical and epistemological topics’.

In a Hindu text of the 1st century CE, we encounter sophisticated philosophical concepts, among which some of a very logical character, like ‘separateness, conjunction and disjunction, priority and posteriority, …motion, …genus, ultimate difference, …inherence, …absence’. In the 2nd century, examples of arguments akin to syllogism appear, which enjoin that generalities be applied to specific cases. In the 7th-8th century, the various ways statements can be negated are explored.

A Buddhist text of the 5th century teaches that a mark found exclusively in a certain kind of subject may be used to infer that subject. Another, appears to describe some properties of implication and logical apodosis: an if-then statement is presented, and it is pointed out that admission of the antecedent coupled with rejection of the consequent is wrong; although the if-then statement has a specific content, its elucidation uses logical terminology.

But variables were not consciously used. The article concludes that, though Indian logic ‘developed independently of Greek thought’, its achievements were comparatively ‘not very impressive’. I may add that I find it curious that Greek-Indian contacts, at least after Alexander the Great, did not result in transmission of logical science.

With regard to China, in the 5th to 3rd century BCE, during ‘the controversies between the major philosophies of Confucianism, Taoism, and Moism’, there was some logical activity, especially by the latter school. However, once Neo-Confucianism became well-established, in the 11th century CE, the subject was virtually abandoned; the authority of that philosophy was so overriding, that there was nothing to argue about[3]. Though Taoism survived to some extent, it was not a philosophy of a kind inclined to intellectual argumentation.

The Moists (followers ofMotzu) made distinctions, like those between personal and common names, the various senses of philosophical terms, or absolute identity and sameness in a specified respect; they even explored inferences by added determinants or complex conception. But, ‘in developing logic, the Chinese thinkers did not advance beyond the stage of preliminaries, a stage that was reached in Greece by the Sophists in the 5th century BCE’.

I would like to add that this assessment may be somewhat harsh. My own minimal acquaintance with Asian philosophy (I read many books and received some practical training, years ago), including some aspects of Yoga, Tai Chi Ch’uan, and various meditations of Hindu, Buddhist and Taoist origin, incite me to greater respect for its achievements[4]. I am not prepared to go into this issue in detail here, but here are two examples which come to mind.

TheTao Teh ChingofLaotzu(China, 7th century BCE; some regard the work as a 2nd century BCE compilation), may be viewed as a treatise on holistic logic. One need only consider the opening sentence to see the truth of this claim:

Existence is beyond the power of words to define:

terms may be used, but none of them are absolute.

…and there are many more such profound insights in it. The lesson of the limits of verbalization, in particular, should be taken to heart by modern axiomatic logicians.

If we view one of the practical functions of logic to be the efficient completion of one’s everyday tasks, by the most direct and least entangling route, then we may well regard the art ofTai Chias a teaching of logic in action. The ideas of Yin and Yang may be considered as logical tools, akin to polarity and modality. Yin is the potential but not quite actual, hence the receptive; Yang is the necessarily actualizing, the willed to be; actuality is a certain balance between these two components of being.

Ultimately, no civilization can take shape without logic. One cannot plant a field, build a house, develop a language and laws, or produce the marvels of contemporary Japanese technology, without some sort of logical culture. We are sure to find some kind of logical knowledge, in the native cultures of Africa, America, and Australasia, as well as in Asia.

All the more, any philosophy or religion is bound to involve logical presuppositions or implications, at least within its epistemological and ontological pronouncements, or in its practical guidelines. The use of formal variables, or explicitly logical principles, are just possible ways for logic to find expression; the Orientals used other, more abstract or practical ways to achieve the same educational ends.

For instance, Zen Buddhism’s belief in the efficacy of meditation or in spontaneity, in the pursuit of mystical ‘Illumination’, is very obviously a logical doctrine, since it prescribesa way of knowledge. Ontologically, it posits ultimate reality to be a unity and of a spiritual nature; epistemologically, it to varying degrees opposes structured knowledge, enjoining wordlessness and unselfconsciousness.

This logic is in contrast to the Occidental, which posits a more step-by-step and intellectual procedure, but whose ultimate goal or result may yet be the same. It may well be that both ways, and still others, are equally efficacious (that is the premise of multiculturalism).

Buddhism has it of course, as a basic epistemological and ontological premise, that the world is ultimately created by the (solipsistic) individual living being. This idea is contrary to Judaism, which acknowledges that we are humble creatures of a single universal Creator. But it may be that Buddhism by this thesis refers to an ultimate return of all souls to their Maker….

In any case, even by Western standards, Oriental philosophies, particularly Buddhism, are clearly internally consistent world-views. They are large-scale systems of epistemology and ontology, which if not explicitly, at least implicitly, demonstrate the logical-mindedness of those who constructed them.[5]

5.Modern Tendencies.

I continue passing on data from the ‘History of Logic’ article previously mentioned, interspersing comments emerging from my own perspective (NEB23:242-247). It tells us: ‘the logical tradition of the Middle Ages survived for about three centuries after it had reached its maturity in the 14th century’. Thereafter, ‘the advent of the Renaissance and Humanism did not enhance logical studies’.

Petrus Ramusdiscussed concepts and judgments, and introduced singular syllogism (see ch. 9.4). The Port-Royal logicians formulated in 1662 ‘rules of the syllogism’, summarizing the common attributes of valid arguments in each of the figures; this was an important development, in my view, because it showed that logic could be expressed conceptually instead of formally: using ordinary language without need of symbolic terms (ch. 9.6). This approach ‘continued to be popular to the mid-19th century’; it did not yield new moods of the syllogism, but offered a system of explanation.

The great French philosopherRené Descartes(early 17th century), whose epistemology is summarized by the statement ‘cogito, ergo sum‘, (‘I think, therefore I am’) and who founded coordinate geometry, insisted on the definition and ordering of scientific knowledge, in accordance with the model of Euclidean geometry: clear and precise terminology, (‘self-evident’) axioms, and (not so ‘self-evident’) corollaries.

The German philosopherGottfried Leibniz(late 17th century), who discovered mathematical calculus independently of Isaac Newton, conceived of logic as ‘a general calculus of reasoning’; it would be an algebra for all thoughts, with ‘unanalyzable notions’ expressed as numerals and signs, from which more complex notions would be derived.

While both the Cartesian method and the ‘universal mathematics’ of Leibniz were valuable contributions, of course, they were from our point of view overly rationalistic. They seemed to regard knowledge as essentially given, needing only to be manipulated; they did not yet quite grasp the gradual and empirical apprehension of most data. A limited number of words in a limited number of combinations, of obscure origin, would somehow suffice to define and prove all others.

Although Leibniz’ ideas of ‘an artificial language’ and of ‘reducing reasoning to computing’ led to mathematical logic and computing science, I must say that his specific logical insights (those known to me) seem trivial to me; as far as I can see, he was just repeating the known in other words — at best, it was work of clarification.

Leibniz also did research on the whole-part relation. The use of diagrams to represent categorical propositions, though ‘already in evidence in the 16th century’ and after, ‘has come to be associated with the name ofLeonhard Euler, an 18th century Swiss mathematician’. I find these to be useful learning instruments, though they can be misleading at times (see ch. 5.3).

In the 19th century,Joseph Gergonne(French) analyzed these diagrams through concepts of co-extension, inclusion, intersection and mutual exclusion. In Britain, quantification of the predicate was proposed byGeorge Benthamand SirWilliam Hamilton(ch. 19.4);Augustus de Morganfocused on complementary propositions using antithetical terms, like ‘all X are Y’ and ‘all nonX are nonY’ (ch. 51.5), and the interactions of eduction and opposition, as in ‘the contradictory of the converse is the converse of the contradictory’ (this only applies toEandI).

These efforts were in my view far from remarkable. They were accompanied by elaborate symbolic languages, but added little to Aristotle’s findings. I see them as assimilation of received information, but in a manner whichmore and more divorced logical science from logical practise. Only logicians have occasion to wonder about the quantity of the predicate or to compare the outcomes of immediate inference; these issues are one step removed from ordinary thought processes.

The movement toward symbolic logic, and the preoccupation with extensional issues, was further accentuated byGeorge Boole, who constructed logical formulas using symbols like those of mathematics. ‘Boolean algebra was subsequently improved by various researchers’, includingWilliam Jevons(who ‘constructed a “logical piano”, a forerunner of the modern computer’) and American philosopherCharles Pierce. I will not go into the details (see ch. 28 for one version of such algebras); much of this was essentially just rewording known things, as far as I am concerned.

These logicians began to see the common grounds between categorical and hypothetical logic, and Pierce also became aware of ‘the notion of a proposition that implies any other’. Within the system of the present treatise, the explanation is modal: quantity expresses the portion of the extension of the subject addressed by a categorical proposition (see ch. 11.4), and implication has a similarly quantitative aspect with reference to logical modality, namely the contexts applicable to the antecedent (ch. 21).

In the late 19th century, in Germany,Ernst Schroederdeveloped an algebra of logic, with reference to the notion of inclusion; this was a more systematic system, using axioms like ‘a non-empty class contains at least one individual’. The notion of inclusion, by the way, can be rather ambiguous: a proposition which implies another is said to include it, and a class is said to include its subclasses or its members; yet, these senses are antiparallel, since we may also say that the presence of a member or a subclass implies that of the classes above it.

At the end of the 19th century, we findGuiseppe Peano(Italian) sought ‘to base arithmetic on axiomatic foundations’. This enterprise may be viewed as an application of logic to mathematics, rather than a work on logic as such.

6.In The 20th Century.

TheNew Encyclopaedia Britannicaarticle goes on to describe developments in the 20th century (NEB23:247-250). I am sorry if my evaluation of modern logic seems at times overly critical; it is not my intention to put anyone down. More will be said about modern logic in the succeeding chapters; for now, I will only make brief comments.

I acknowledge the advances made, the refinements in definition and the more rigorous systematization, but I must take the long view for the centuries to come. What counts for me, the bottom-line ortachlis(as they say in Yiddish) is: is there anything really new in it from the point of view of logic (like new moods of syllogism, say)? Philosophical or mathematical findings are all very well, but our concern here is with logic as such.

Georg Cantor(1845-1918) introduced a theory of ‘sets’ or classes; this was ‘just another version of’ the logic dealt with by Aristotle, except for an emphasis on the denotative rather than connotative aspect of terms (ch. 18.1). These systems were later found to contain double paradoxes; but their goals were not abandoned.Gottlob Frege(1848-1925) worked to systematically reduce arithmetic, which concerns natural numbers, to logic; as evidenced by axioms like ‘for all p and q, if p then if q then p’, the hypothetical relation was taken in the petty sense of ‘material implication’ (see ch. 24.3).

The British philosopherBertrand Russell(1872-1970) set out ‘to show that arithmetic is an extension of logic’, with the publication in 1903 of hisThe Principles of Mathematics.Later, he together withAlfred Whitehead, ‘produced the monumentalPrincipia Mathematica, 3 vol. (1910-13), which has become a classic of logic’. A ‘work of impressive scope’, including ‘topics such as the logic of propositions, and the theories of quantification, of classes, and of relations’. It ‘marked a climax of the researches in logic and the foundations of mathematics’, and ‘provided a starting point for… development… in the 20th century’.

Several double paradoxes had been discovered in the 15 years prior, including those in the systems of Frege and Cantor, and those ofBurali-Forti,Berry,Richard, andGrelling, and the ancient Liar paradox was still of interest. These antinomies ‘cast doubt upon men’s logical intuitions’, and Russell wanted to resolve them. I quote at length, to leave as-is the language used in the article, with an emphasis on ‘the paradox of the class of all classes not members of themselves’:

Russell argued that they result from a “vicious circle” that consists in assuming illegitimate totalities. A totality is illegitimate when it is supposed to involveallof a collection but is itselfoneof the same collection… such totalities cannot be generated because of… the theory of logical types,[which]demands that… a class belongs to a higher logical type than that to which its elements belong. (Similarly, a predicate belongs to a higher logical type than the object of which it is predicated.) Consequently, to say that a class is an element of itself is neither true nor false but simply meaningless…. Although the theory of types obviated the paradox…, it raised certain problems of its own. It was by no means clear whether the theory was a kind of ontology that classified extra-linguistic entities or a kind of grammar that classified expressions of a logical language. Moreover, some critics charged that it was an ad hoc palliative… and that the ramifications… were unduly complicated.

Although I found myself in agreement with some of Russell’s viewpoints and findings with regard to class logic, my sense was that he confused many of its concepts. Rather than try to define our differences point by point, it seemed more effective to develop a consistent system of my own from scratch. This work took me about a month, and the results are to be found in ch. 43-45. Even though this is intended as a reply to Russell, including a definitive solution of his paradox, I recognize he is to be credited with initiating such research. My conclusion is that, although the logic of classes qualifies as an important field in its own right, it is merely a derivative of Aristotle’s logic.

The view that mathematics is an offshoot or segment of logic (known as ‘logicism’), was countered byL. BrouwerandArend Heyting, of Holland, who regarded it as an independent field. These ‘intuitionists’ (as they are called) felt that Aristotle’s Law of the Excluded Middle tended to be overemphasized, and doubted that every problem is soluble; they rejected the ‘elimination of double negation’, which ‘allows its proponents erroneously to infer the provability of a proposition from the unprovability of its negation’.

Though I agree to some extent with these views, I can also see that there is a confusion, here, between deductive and inductive logic. The third law of thought is an ultimate goal, set by deductive logic, which is not always easy to apply in the interim, during induction. According to strict logic, a double negative is equivalent to a positive; whereas the extrapolation from unprovable to absent is just a generalization.

Next,David Hilbertcame up with the claim that ‘freedom from contradiction in an arbitrarily posited axiom system is the guarantor of the truth of the axioms’, less concerned than his predecessors ‘with the meaning of the axioms’ (a position known as ‘formalism’). In 1931,Kurt Godelreportedly argued that no theory can be both complete and consistent, at least in mathematics.

As I see it, these logicians had become so concerned with the problems of systematization of mathematics, that their view of logic was very narrow, intent on deduction, and ignoring the inductive aspect of concept formation on empirical grounds. Consistency is only one of the tests of truth; and no theory is ‘complete’, anyway. Needless to say, principles proved with mathematical terms do not necessarily apply to other, more conceptual, terms.

Deeper down, their ‘philosophy of logic’ was faulty. There was a misapprehension of the ‘laws of thought’ as axioms, the ordering of logic on the model of geometry; as I have argued, that model is inapplicable to logic, whose grounding has to be much more subtle, since one of its roles is to justify that very model (see ch. 2, 20).

Continuing, efforts were made to investigate all the varieties of logical relations, ‘the logic of propositions’ (see ch. 23-27). The method of ‘truth tables’ was developed by Pierce,Jan Lukasiewicz(Polish, d. 1956),Emil Post(U.S.),Ludwig Wittgenstein(Austrian-British, d. 1951), and others, with the aims of defining logical relations and evaluating intricate formulas. Since relations like implication and disjunction were interdefinable, the question arose as to which came first. Pierce, and laterHenry Sheffer(of Harvard), andJ.-G.-P. Nicod(French), considered disjunction to be the most fundamental.

In my view, the T-F values of truth tables do not define, but are merely implied by, logical relations; they are effects, not causes. The primary relations in logic are plain conjunction and the negation of such conjunction; both implication and disjunction are thereafter derived with reference to logical modalities, so that their order is irrelevant. Intuitively, they are different angles: implication suggests forced appearance of a thesis, and disjunction suggests mutual replacement of theses without regard to underlying forces.

Logicians like Lukasiewicz, of the Warsaw school, and others in Dublin, traced the order of derivation of logic from a few well defined axioms. Others with an interest in systematization included Hilbert,Ingebright Johansson(Oslo), and Heyting. Conscious of the weakness of the current concept of implication, so-called Philonian implication,WilhelmAckermanntalked of ‘rigorous implication’ andAlan Anderson(Pittsburgh) of ‘entailment’, to bind the theses together more firmly.

Logicians like the PolesStanislaw Lesniewski(d. 1939), and later Lukasiewicz andAlfred Tarski, attempted to express logical relations in terms of quantifiers; but they do not seem to have had a clear definition of logical modality, which would explain why quantifiers work (ch. 21).

Much more impressive, was the work ofClarence I. Lewisin 1918, and later as ‘author (withC.H. Langford) ofSymbolic Logic(1932), a classic of modal logic’. He revived the definition of implication in its ‘strict’ sense [by Diodorus], in contrast to the pretensions of ‘material’ implication [by Philo], with reference to the modal category of possibility; this seemingly refers to the logical type of modality, though the intention may have been generic.

However, since he apparently had no clear definition of the concept of possibility, but used it as his undefined starting point, he developed ‘several different systems of strict implication (S1-S5), and many more have been constructed since’. The word ‘system’ is here used in a limited sense, not in the grand sense of an overall, methodical presentation of the science of logic.

Godel built an alternative system, ‘with necessity as the primitive notion’, andE.Lemmon(of Oxford) constructed one ‘with the notion of contingency’. The same criticism can be repeated: these categories of modality are interrelated, so that which one takes as one’s starting point is arbitrary; these different systems just reshuffle the data. What matters, is to define modality first, with reference to nonmodal notions; the rest follows.

Lukasiewicz analyzed Aristotle’s notions of modality, and suggested a ‘many-valued logic’, which in addition to truth and falsehood, would recognize ‘intermediate’ states of knowledge. His intent may have been to formalize ‘indeterminism’, but he seems to have been unaware of the distinction between logical contingency and problemacy. I have not taken a similar approach, because I regard it as redundant. There is nothing it can do which cannot be done through two-valued logic, so why complicate things? And in any case, intermediate concepts are only definable with reference to the extremes (see ch. 20, 21, 46).

However, we are told, ‘many-valued systems that have no such defects have been developed byBoleslaw Sobocinski… and…Jercy Slupecki‘ (both of Poland). Post constructed ‘purely formal systems of many-valued logics… independently’. But not all logicians have approved of these tendencies. Note, my own position is not that this threatens two-valued logic, but merely that we do not need it; I see no harm in building multi-valued logics in the way of exercise or for cybernetic purposes.

Other notable developments were, an interest in ‘deontic’ logic (see ethical modality, ch. 11.8), and the work ofA. Priorin tense logic (ch. 11.6).

The work done after Russell with regard to categorical propositions, seems mostly paltry to me. Lukasiewicz rehashed the syllogism.‘Edmund Husserl, the founder of Phenomenology, …was concerned with the grammar of logical language and not with any extra-linguistic entities’. Lesniewski attempted to define the copula ‘is’ (while, I notice, using the word ‘is’ in his definition), claiming to use a language which ‘implies no existential assertions’.

He also worked on ‘the theory of part-whole relations’, (which he labeled ‘mereology’). Work in this field was also done, independently, byJ. Woodger,Henry Leonard, andNelson Goodman.

In the U.S.Ernst Zermelo, of Germany, ‘axiomatized the theory[of sets]and succeeded in avoiding the paradoxes without resort to the theory of types’. This work was continued byAbraham Fraenkel, of Israel, andJohn Von Neuman, a Hungarian-American ‘pioneer in computer mathematics’. The latter ‘distinguished between sets, which can be members of other sets, and classes which have no such property’.

More recently,Paul Bernays, of Switzerland, ‘made important contributions’ to this field. Harvard’sW. Quinereplaced the theory of types ‘by the much simpler theory of stratification’. However, the article points out, these ‘axioms have scarcely made the notion of a set any clearer. If[they]describe some aspect of reality, it is not known what these aspects are. Thus, the theory is becoming detached from any interpretation’.

These ideas are cast in obscure and esoteric language, which make them hard to judge by common mortals. Presumably, if they were used in computer work, they have been found to have some practical value. They seem to have concepts in common with my own theory, but my sense is that we differ, and the said hermeneutic complaint confirms it. The question of interpretation is answered by my theory at the outset, and a comprehensible and comprehensive system is easily derived. I leave the comparison to others. In any case, more detailed critiques are presented in later chapters.

‘Once systems of logic were axiomatized, …logicians began to examine the systems themselves’, in search of ‘problems of consistency, completeness, and decidability, …as well as… the independence of the axioms… the definability of primitive terms’. These studies were called ‘metalogic’, and among the people engaging in them wereAlessandro Padoa(d. 1938), Post, Lukasiewicz,Alonzo Church(U.S.), Godel,Paul Cohen(Stanford).

Among the disciplines of metalogic were syntax and semantics, and work in those areas was done byRudolph Carnap(German-American), Lesniewski,KazimierzAjdukiewicz(Polish), Godel, Tarski,Karl Popper(Austrian-British). Also mentioned areGerhard GentzenandStanislaw Jaskowski, for their ‘systems of natural deduction’.

These issues are dealt with in more detail in later chapters. However, for now, my overall impression is that these logicians had a very a priori view of logic. My view is that the role of intuition and empirical data is much stronger, at all levels of the enterprise. It is vain to pursue consistency and order in logical science, in a purely rationalistic manner, without a broader epistemological and ontological outlook.

The copula ‘is’, the early notions of ‘conditioning’, and many other concepts, are irreducible primaries, which however do not exist in some detached plane, but are continuously reinforced by intuitive and empirical input. Likewise, identity, incompatibility, and exhaustiveness, are not strictly abstract ‘laws of thought’, but repeatedly ‘experienced’ concepts, given within phenomena. There is no such thing as deduction without induction; the human mind does not function like a computer, merely processing bits of information, but involves an ongoingconsciousnesswhich gives meaning to the data.

[1]An example is the traditional doctrine ofthe Fallacies. The ‘fallacies’ have been of interest since the time of Aristotle and before, but their study as a body rather dates from the Scholastic period. The whole approach is somewhat antiquated; and, though it still has didactic value, it has become less relevant, as strictly-formal logic has evolved. Common errors of reasoning and rhetoric, were named, listed, described and classified — traditionally as follows (according toNEB, 23:280-281). I do not consider this analysis perfect, but proposing an alternative has not been high on my list of priorities.

·Material‘ fallacies: making erroneous analogies (secundum quid, accidensand its converse), errors in causal judgment (non causa pro causa, post hoc ergo propter hoc, non sequitur), circularity in definition or argument, (petitio principii,begging the question, a vicious circle), as well as various kinds of intimidation or appeal (ad hominem, ad populum, ad misericordiam, ad verecundiam, ad ignorantiam, ad baculum,and also bribery).

·Verbal‘ fallacies: equivocation, figures of speech taken literally, ambiguity, distortive accentuation, confusing collective and dispensive senses of terms.

·Formal‘ fallacies: like errors of syllogism (‘the four terms’, ‘illicit processes’), or apodosis (‘denying the antecedent’ or ‘affirming the consequent’).

[2]TheNEBtreatment of logic history is, in my view, defective, in that it makes virtually no mention of Judaic logic. Yet there is evidence of logic use in the Jewish Bible and the Talmud, and later Rabbinical writings are replete with logical discourse, including theoretical statements; and these manifestations of logic are bound to have had someinfluence on Western logic. But the subject is too vast for me to try and deal with it here. See my workJudaic Logic.

[3]This explanation was suggested to me by a Japanese acquaintance, Matski Masutani.

[4]Note that this is not intended as a blanket endorsement; there are doubtless invalid forms of reasoning in these philosophies.

[5]I unfortunately no longer remember the names of the books this judgment was based on.