Appendix 7 – SOME LOGIC TOPICS OF GENERAL INTEREST

1. About modern symbolic logic

2. The triviality of the existential import doctrine

3. The vanity of the tetralemma

4. The Liar paradox (redux)

5. The Russell paradox (redux)

Although the present work is not a general study of logic, but essentially restricted to a fortiori logic, I have in the course of writing it reflected on more general issues, and wish to share these reflections here rather than hold on to them till a future work, all the more so since they significantly either correct or amplify my comments on these issues in my past works.

 

1. About modern symbolic logic

Since the later decades the 19th century, and more and more so throughout the 20th century, “modern symbolic logic” has gradually discarded and displaced “classical formal logic.” What is the essential difference between them? Classical formal logic, which was discovered or invented by Aristotle (4th century BCE) and further developed and improved on over time by many successors, is based on the idea of studying the logical properties of propositions by replacing material propositions with formal ones. A categorical proposition is formal, if its terms are variables instead of constants – e.g. “All X are Y” is formal, because the symbols X, Y represent in theory any terms that might arise in practice. A hypothetical proposition is formal, if its theses are variables instead of constants – e.g. “If X, then Y” is formal, because the symbols X, Y represent in theory any theses that might arise in practice.

Now, whereas classical logic symbolized terms and propositions, it did not similarly symbolize the other components of propositions, such as their quantities, their modalities or their relational operators. In “All X are Y,” the words “all” and “are” remained in ordinary language (in our case, plain English). Similarly, in “If X, then Y” the words “if” and “then” were not symbolized. In modern symbolic logic, on the other hand, the trend developed to symbolize every aspect of every proposition[1]. This was, to be sure, a new school of logic, which considered that only in this way could utter precision of language be achieved, and all ambiguity or equivocation be removed from human discourse. Modern symbolic logic, then, advocated the adoption of an altogether artificial language comparable to the language of mathematics.

Some of the pros and cons of this approach are immediately obvious[2]. One advantage of symbolization, already mentioned, is the sense of precision sometimes lacking in natural languages. However, this impression is surely illusory – for if one’s understanding of the matter at hand is vague and uncertain to start with, how can symbols improve on it? A patent disadvantage of symbolization is the esoteric nature of artificial language. Logic was originally intended as a teaching for the masses, or at least the intellectuals, to improve their daily thinking. Nowadays, logic has become the exclusive domain of a few specialists, and has little to do with human cognitive practice. Moreover, communication is not always easy even among symbolic logicians, because each of them quite naturally prefers a different set of symbols, so that there is in fact not one artificial language, but many of them. Another disadvantage is the slow adaptability of any artificial language to forms of discourse newly discovered in everyday usage. An example of this is the a fortiori argument, which is still without convincing symbolic expression.

The main activity of ‘modern logicians’ nowadays seems to be to translate ordinary language into their favorite symbolic language. Most of the time, they seem content to rewrite a perfectly comprehensible plain English sentence into a purely symbolic one, as if this is some great achievement that will earn them their place in history, or at least in the profession. Just that act of translation or rewriting in symbolic terms seems to satisfy and thrill them tremendously, as if it confers scientific status onto the sentence. Additionally, they resort to pompous terminology for window-dressing and intimidation purposes. One gets the impression that symbols play for them the role of magic incantations in ancient times – ‘abracadabra!’ they would chant in pursuit of mystical insights and powers.

But, think about it a moment. In truth, when modern logicians rewrite a sentence in symbolic terms they have achieved exactly nothing other than to use shorter ‘words’ (i.e. the symbols they invent) in place of ordinary words, and (ideally) drawn up a table telling us what symbols correspond to what ordinary words. All they have done is abbreviate the given sentence. Apparently, they are too lazy to write long sentences and prefer concise ones. They have produced no new information or insight. They cannot credibly argue that the ordinary language statement was essentially deficient, since it must have been understandable enough for them to have translated it into symbols. If it was understandable enough for them, why not for everyone else? What absolute need have we of the artificial language(s) they so insistently try to sell us?[3]

Moreover, note this well, whenever we (and they) read the symbolic statement they have concocted, our minds have to translate it back into ordinary language in order to understand it. We have to mentally repeatedly refer back to the ordinary language definitions of the symbols. We have to remember: “Oh! This funny symbol means so and so, and that weird doodle means this, and the zigzag means that,” and so forth. This means that our mental process of understanding is made more difficult and slowed down considerably. We are further from the object of study than we were to start with – more removed from the reality we are trying to think about. This increased distance and waste of time is not accidental or incidental, however – it serves to cloud the issues and prevent critical judgment. Errors are hidden from sight, and if we spot them we hardly dare point to them for fear of admitting we may have missed something. In this way, foolishness is perpetuated and spreads on.

But all that is not the worst of it. The worst of it is twofold. First, modern logicians usually symbolize much too soon, when their level of insight and analysis is still in its early stages. They typically do not give the subject-matter studied time to develop and mature in their own minds, but impatiently rush into their more orderly looking and comfortable world of symbols. The result is that their symbols are usually representative of a very naïve, elementary, immature level of understanding of the object at hand. Secondly, once their symbolic representation is done, it freezes all subsequent work at that childish level. Since the symbolization is already settled, all they can do is play around with it in different ways. All they can do is manipulate and reorder and recombine their symbols this way and that way, and this is what they pass the rest of their time doing. They cannot feed on new experiences and insights from the world out there or actual human discourse, since they have already confidently and reassuringly separated themselves from all that. Their symbols thus blind them and paralyze them. The paucity of their results testifies to it.

Another aspect of modern symbolic logic important to note is its pretensions of ‘axiomatization’. In classical formal logic, the Laws of Thought (Identity, Non-contradiction and the Excluded Middle) were sovereign; these were axioms in the original sense of irreducible primaries of rational knowledge, together constituting the very essence of logic. In syllogistic validation by Aristotle, all syllogisms could be reduced directly or ad absurdum to a minimum number of primary moods – mainly the first figure positive singular syllogism: “This S is M, and all M are P, therefore, this S is P.” The latter argument was not perceived as an axiom in its own right or even as an arbitrary convention, but as a logical insight in accord with the laws of thought that what is claimed applicable to a concept must be acknowledged to apply to the things it subsumes. The relation of human reasoning, and more deeply of formal logic, to the laws of thought was progressively ‘systematized’, but it was allowed to remain essentially open and flexible. Though integrated, it was not rigidly fixed, so as to allow for its constant evolution and adaptation as knowledge developed.

Modern logicians, on the other hand, focusing on the more geometrico, the method of proof used in Euclidean geometry, sought a more predictable and definitive arrangement of knowledge. Their simplistic minds demanded rigid rules and perfect orderliness. A hierarchy was established between thoughts – with those at the top of the hierarchy (the laws of thought) being viewed as ‘axioms’ and those lower down (syllogisms, and eventually similarly other arguments) as ‘theorems’. This may work well for mathematics, which is a relatively special science, but it caused havoc in general conceptual logic, which is the science of science. The question naturally arose as to where those apparent axioms came from and whether they could be replaced by contrary ones as was done in non-Euclidean geometry. It did not take long for these simpleminded people to decide that logic was a conventional mental game, with no apparent connection to the empirical world. This philosophy (known as Logical Positivism) was largely justified by Immanuel Kant’s analytic-synthetic dichotomy[4], so it could hardly be doubted.

What is lacking in this model of knowledge is the understanding that formal logic is not deduced from the laws of thought. The laws of thought are not premises of formal logic; they are not contents from which other contents are deduced. The laws of thought refer us to the autonomous logical insights through which we naturally judge what constitutes appropriate inference. They are what justifies the processes of deduction (and more broadly, of induction) from premises to conclusions. The forms of syllogism and other arguments are not deduced from the laws of thought. The forms are induced from actual thought contents. The thought contents can be judged correct or not without reference to the forms, using ad hoc logical insights. What formal logic does is simply collect under a number of headings recurring types of thoughts, so that again using ad hoc logical insights we can once and for all predict for each type of thought (e.g. syllogism 1/AAA) whether it is correct or not. There is in fact no appeal to general ‘laws of thought’ in this validation (or invalidation) process; honest ad hoc logical insights are sufficient. The ‘laws of thought’ are merely ex post facto typologies of ad hoc particular acts of logical insight. For that reason, they are not top premises is a geometrical model of knowledge.

The question these pseudo-logicians did not ask themselves, of course, is why their allegedly logical insights in the course of ‘axiomatization’ (including their skepticism towards the objectivity of the laws of thought) should be preferred to the logical insights of the ‘non-axiomatization’ logicians. Is any discussion of logic possible without use of logic? Can logicians ever rightly claim to transcend logic? Can they logically deride and nullify logic? For instance, some have argued that appeal to the laws of thought is either circular argument or infinite regression. They stopped their reflection there, and never asked themselves why the rejection of circular argument or infinite regression should be considered primary logical acts not needing justification, while the laws of thought are to be rejected precisely on the ground that (according to them) there are no logical acts not needing justification. Is that not a double standard (another primary logical insight)?[5]

The radical blunder of the Kantian legacy is the belief that there is such a thing a ‘purely analytic’ or ‘a priori’ knowledge. Logicians influenced by this inane idea remain blind to the empirical aspects of all knowledge development. Even apparently purely symbolic systems of logic rely on perceptions. Some symbols refer to concrete objects (e.g. individuals a, b) and some to abstract ones (e.g. classes x, y); but every symbol is, as well as a sign for something else, in itself a concrete object (whether as a bit of ink on paper, or of light on a computer screen, or as a shape conjured in our mind’s eye). If follows that symbolic formulas, whether inductively or deductively developed, always depend on some empirical observation. The observation of symbols is not a transcending of experience; it is an empirical process just like the observation of cows; it requires physical or mental perception. Thus, if I count symbolic entities or I imagine them collected together, that is not purely analytic work – it is quite synthetic work. Moreover, such logicians tend to ignore the countless memories, imaginations and rational insights that form the wordless background of all discourse concerning logic.

Clearly, axiomatization was a con-game on a grand scale, through which shallow but cunning pseudo-logicians wanted to take power in the domain of logic. And they have indeed managed to do that almost completely. But the fact remains that primary logical insights like the laws of thought, or the rejection of circularity, infinite regression and double standards, or again the acceptance of subsumption (syllogistic reasoning) and the many, many other foundations of human thought, are not open to doubt. No amount of ‘axiomatization’ can either prove or disprove them, because all proof and disproof depends on such insights to at all convince us. The conclusion to draw from that is certainly not relativism or conventionalism, for that too would be a claim to the validity of some logical insight – i.e. the insight that all insights are arbitrary must itself be arbitrary. Logic cannot be refuted by logic. Logic can however be justified, by honest acknowledgment that some thoughts are primary logical insights. And these insights, which together constitute what we call ‘human reason’, cannot all be listed in advance, but emerge over time as knowledge develops.

Aristotle said all that long ago, but many have preferred to ignore him or misrepresent him because they dearly want to belittle him and supplant him, being envious of his achievements. Consider for instance the following statement about the law of non-contradiction drawn from his Metaphysics (Book 4, part 3. Translated by W. D. Ross.):

“For a principle which every one must have who understands anything that is, is not a hypothesis; and that which every one must know who knows anything, he must already have when he comes to a special study. Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect.” (Italics mine.)

As a result of symbolization and axiomatization, modern logic is essentially a deductive logic enterprise. However complicated or complex it may look, it is inevitably superficial and simplistic. Even when modern logicians pretend to discuss induction, they are stuck in deductive activities. Their ‘logic’ is thus more and more divorced from reality. For this reason their thinking on issues of metalogic is thoroughly relativistic. Things have to be the way they think they are, since symbols are somehow omniscient and omnipotent. They see no idiocy or harm in ‘paraconsistent logic’ (i.e. in breach of one or more of the laws of thought), since to them it is all a game with conventional symbols with no connection to any reality. When things do not fit into their preconceived schemes, they blithely force them in and use florid terminology to keep critical judgment at bay. They do not look upon practical deviations from their arbitrary theoretical constructs as problems, as signals that they have made mistakes somewhere on their way; they just add more symbols and make their theories more abstruse. Please don’t think I am exaggerating – that’s the way it is.

Why are so many people drawn to and impressed by modern symbolic logic? Part of the problem is of course that this is what the universities demand from their teaching staff and teach their students; papers have to be written in symbolic terms to be even considered. But why this preference? Perhaps because pages filled with esoteric symbols seem more ‘scientific’, reminding readers of mathematical formulae in the physical sciences. It matters little that in logical science the subjacent subject-matter becomes less transparent and comprehensible when translated into symbols. Indeed, part of the aim is to befuddle and intimidate the reader, so as to conceal weaknesses and faults in the treatment. The grandiloquent language is similarly useful as eyewash. Modern symbolic logic boasts of superiority to classical formal logic, to give itself authority; but the truth is that most good ideas the former has it has stolen from the latter, reworking them a little and renaming them to seem original and independent. The whole enterprise is a massive ongoing fraud; or, alternatively, a collective delusion of epidemic proportions.

I am not, of course, saying (as, no doubt, some will rush to accuse me of saying) that everything modern symbolic logic tells us is false and irrelevant, or stolen. What I am saying is that whatever is true and significant in it is certainly not due to symbolization and axiomatization, and can equally well be (could be and probably was) developed by classical formal logic. Moreover, to repeat, excessive symbolism tends to simplistically lump things together and gloss over important nuances, and condemns its users to rigid and abstract thinking processes out of touch with the empirical domain.

We have seen, in the course of the present treatise on a fortiori logic, how some budding or experienced logicians strayed or failed due to their attempt to solve problems by means of modern symbolic logic. In the following pages, I present some more examples of the relative ineffectiveness of modern symbolic logic compared to classical formal logic. I show how the issue of ‘existential import’ is far less significant that it is touted to be; how attempts to bypass the laws of thought are futile; how the liar paradox is not only due to self-reference; and how the Russell paradox is due to the acceptance of self-membership.

 

2. The triviality of the existential import doctrine

A term is, nowadays, said to have ‘existential import’ if it is considered to have existing referents; otherwise, it is said to be ‘empty’ or a ‘null class’. For examples, ‘men’ has existential import, whereas ‘dragons’ does not. This concept is considered original and important, if not revolutionary, in modern symbolic logic; and it is often touted as proof of the superiority of that school over that of classical formal logic. We shall here examine and assess this claim. As we shall see, although the concept has some formal basis, it is in the last analysis logically trivial and cognitively not innocuous.

The founder of formal logic, Aristotle, apparently did not reflect on the issue of existential import and therefore built a logical system which did not address it. The issue began to be raised in the middle ages, but it was not till the latter half of the nineteenth century that it acquired the importance attached to it today by modern logicians.

a. Based on Aristotle’s teaching, classical formal logic recognizes six basic categorical forms of proposition: the general affirmative, “All S are P” (A), which means that each and every S is P; the general negative, “No S is P” (E), which means that each and every S is not-P; the particular affirmative, “Some S are P” (I), which means that each of an indefinite number (one or more) of S is P; the particular negative, “Some S are not P” (O), which means that each of an indefinite number (one or more) of S is not-P; and the singular affirmative, “This S is P” (R), and the singular negative, “This S is not P” (G), which refer to a specifically pointed-to or at least thought-of individual instance. Note that general (also called universal) propositions and particular propositions are called plural, in contradistinction to singular ones[6]. The labels A, E, I, O, R and G come from the Latin words AffIRmo and nEGO; the first four are traditional, the last two (R and G) were introduced by me years ago[7].

The symbols S and P stand for the subject and predicate. The verb relating them is called the copula, and may have positive (is or are) or negative (is not or are not) polarity[8]. In the present context, the copula should be understood very broadly, in a timeless sense[9]. When we say ‘is’ (or ‘is not’) we do not mean merely “is (or is not) now, at this precise time,” but more broadly “is (or is not) at some time, in the past and/or present and/or future.” The expressions ‘all’, ‘some’ and ‘this’ are called quantities. Obviously, the general ‘all’ covers every single instance, including necessarily ‘this’ specific instance; and ‘all’ and ‘this’ both imply the particular ‘some’, since it indefinitely includes ‘at least one’ instance. The ‘oppositions’ between the six forms, i.e. their logical interrelationships, are traditionally illustrated by means of the following ‘rectangle of oppositions’:

Diagram 7.1 – Aristotelian oppositions

Although Aristotle did not, to our knowledge, represent the oppositions by means of such a diagram, we can refer to it as a summary his views. It is taken for granted that, on the positive side A implies R and R implies I (so, A implies I), and on the negative side E implies G and G implies O (so, E implies O), although these implications cannot be reversed, i.e. I does not imply R or A, and R does not imply A, and so forth. This is called subalternation[10]. The core opposition in this diagram is the contradiction between R and G; from this assumption, and the said subalternations, all else logically follows[11]. A and O are contradictory, and so are E and I; A and E, A and G, E and R, are pairs of contraries; I and O, I and G, O and R, are pairs of subcontraries. Two propositions are contradictory if they cannot be both true and cannot be both false; they are contrary if they cannot be both true but may be both false; they are subcontrary if they may be both true but cannot be both false.

b. Shockingly, the above traditional interpretation of the basic categorical forms (Diagram 7.1) has in modern times been found to be problematic. The above listed propositions are not as simple as they appear. The form “Some S are P” (I) means “Something is both S and P,” while the form “All S are P” (A) means “Something is both S and P, and nothing is both S and not-P;” similarly, the form “Some S are not P” (O) means “Something is both S and not-P,” while the form “No S is P” (E) means “Something is both S and not-P, and nothing is both S and P.” Seeing the forms I, A, O, E, in this more detailed manner, we can understand that A implies I since I is part of A (and likewise for E and O), but then we realize that A and O are not truly contradictories (and likewise for E and I).

The exact contradictory of “Something is both S and not-P” (O) is “Nothing is both S and not-P” (i.e. only part of A, with no mention of its I component) and the exact contradictory of “Something is both S and P, and nothing is both S and not-P” (A) is “Nothing is both S and P, and/or something is both S and not-P” (i.e. a disjunction including O, but also E). Note this well[12].

It should be pointed out that “All S are P” (A) can be defined more briefly as: “Something is S, and nothing is both S and not-P;” for given this information, it follows logically that the things that are S are P (for if this was denied, it would follow that some things are both S and not-P), Similarly, “No S is P” (E) can be defined more briefly as: “Something is S, and nothing is both S and P,” without need to specify explicitly that “Some things are both S and not-P.” Thus, all four forms A, E, I, O, imply, or presuppose (which is logically the same), that “some S exist(s).” Also, the positive forms, A and I, imply that “some P exist(s).” On the other hand, the negative forms, E and O, do not imply that “some P exist(s),” since the negation of a term is not informative regarding its affirmation[13].

Thus, in the above diagram, the diagonal links between the corners A and O, and between E and I, should not be contradiction but contrariety. For, while to affirm one proposition implies denial of its opposite, to deny one proposition does not imply affirmation of the other. To remedy this real problem of consistency, modern logicians have proposed to redefine the general propositions A and E as the exact contradictories of O and I, respectively. That is to say, the new meaning of A is only “Nothing is both S and not-P” and the new meaning of E is only “Nothing is both S and P.” It follows from this measure that A (in its new, slimmer sense) no longer implies I, and likewise E (in its new, slimmer sense) no longer implies O. This redefinition of symbols A and E can, to my mind, lead to much confusion. In my view, it would be better to re-label the forms involved as follows:

  • Keep the traditional (old) labels A and E without change of meaning; i.e. old A = A, old E = E.
  • Label the modern (new) senses of A and E as respectively not-O and not-I.
  • That is, new ‘A’ = not-O ≠ old A. Whereas, old A = new ‘A’ plus I = I and not-O.
  • Likewise, new ‘E’ = not-I ≠ old E. Whereas, old E = new ‘E’ plus I = O and not-I.

Thus, when we say A or E in the present paper, we mean exclusively the traditional A and E; and when we wish to speak of the modern ‘A’ and ‘E’ we simply say not-O and not-I, respectively. Note this convention well[14]. Actually, such propositional symbols are effectively abandoned in modern logic and the propositions are expressed by means of a symbolic notation, including the existential and universal quantifiers, (there exists) and (for all), respectively; but we do not need to get into the intricate details of that approach here, because we can readily discuss the issues of interest to us in plain English. Now, let us consider the formal consequences of the above findings in pictorial terms.

One way for us to solve the stated problem is to merely modify the traditional rectangle of oppositions, by showing the diagonal relationships between A and O and between E and I to be contrariety instead of contradiction; this restores the traditional diagram’s consistency, even if it somewhat dilutes its force (Diagram 7.2). Another possibility, which is the usual modern reaction, is to change the top two corners of the rectangle to not-O and not-I, instead of A and E respectively; this allows us to retain the contradiction between diagonally opposed corners, although now the lateral relation between the top corners is unconnectedness instead of contrariety, and the vertical relations in the upper square are unconnectedness instead of subalternation (Diagram 7.3).[15]


7.2 – modified traditional


7.3 – modern version

Notice that the lower square of the modern version is unchanged. This is due to the judgment that the forms I and O, i.e. “Something is both S and P” and “Something is both S and not-P,” both imply that “some S exist” (or “some things are S” or “there are things which are S”) meaning that if they are true, their subject ‘some S’ has existential import. Moreover, in the case of I, the predicate P is also implied to have existential import, since it is affirmed; but in the case of O, the predicate P is not implied to have existential import, since it is merely denied. Until now, note well, we have not mentioned the issue of existential import in our formal treatment. Now, it comes into play, with this interpretation of particular propositions.

The same applies to R and G – their subject ‘this S’ has existential import, whereas the predicate P has it if affirmed but lacks it if denied. On the other hand, since not-O (as distinct from A) is a negative statement, i.e. means “Nothing is both S and not-P,” it has no implication of existential import. Similarly, since not-I (as distinct from E) is a negative statement, i.e. means “Nothing is both S and P,” it has no implication of existential import. Clearly, if not-O was thought to be contrary to not-I, then if not-O were true, it would imply the negation of not-I, i.e. it would imply I; but this being erroneous, not-O and not-I cannot be contrary, i.e. they must be unconnected. Similarly, if not-O was assumed to imply R, it would then imply I, since R still implies I; therefore, not-O must also be unconnected to R; and similarly for not-I and G. On the other hand, not-O remains contrary to G, since if not-O is true, then O is false, in which case G must be false; similarly as regards not-I and R.

It is now easier to see why the traditional rectangle of oppositions (7.1) seemed right for centuries although it was strictly-speaking wrong. It was tacitly assumed when drawing it that the subjects of general propositions always have existential import, i.e. imply that “some S exist (s).” When this condition is granted, then in combination with it not-O becomes A and not-I becomes E, and A implies I and E implies O, and A exactly contradicts O and E exactly contradicts I – in other words we happily return to the original rectangle of oppositions (7.1). The problem is that this condition is not always satisfied in practice. That is, not-O or not-I can occur without their subject S having existential import.

Effectively, the forms “Nothing is both S and not-P” (not-O) and “nothing is both S and P” (not-I) signify conditional propositions (“Whatever is S, is P” and “Whatever is S, is not P”) which, without the minor premise “this is S,” cannot be made to conclude “this is P” or “this is not P” (respectively). In other words, they record a ‘connection’ between an antecedent and a consequent, but they have no ‘basis’, i.e. they contain no information affirming the antecedent, and thence the consequent. Obviously, if that information is provided, the condition is fulfilled and the result follows. Once we realize that the traditional rectangle remains true in the framework of a certain simple condition (viz. that some S exist), we see that its hidden ‘inconsistency’ is not such a big problem for formal logic.

It is interesting to also consider the significance of the above revisions in the field of eduction (i.e. immediate inference). Whereas A, which implies I (“Some S are P”), is convertible to “Some P are S” – not-O, which does not imply I, is not so convertible. Also, whereas not-I is convertible to “No P is S,” since “Nothing is S and P” and “Nothing is P and S” are equivalent and have no implication of existential import for S or P – E is not likewise unconditionally convertible, since in its case even if we are given that “some S exist” we cannot be sure that “some P exist” (but only that “some not-P exist”). Note well, just as O does not imply predicate P to have existential import, since it merely negates it, so is it true for E; therefore, the traditional conversion of E is really only valid conditionally. We can also look into the consequences of the above revisions in the field of syllogistic reasoning; the main ones are pointed out further on.

c. Let us now go a step further in the possible critique of Aristotelian oppositions, and suggest that all terms may be denied to have existential import, whatever the forms they occur in, and whatever their positions therein. That is to say, not only the subjects of general propositions, but even the subjects of singular or particular propositions might conceivably lack existential import. Although R and G, and I and O, do formally imply that some S exist(s), it is still possible to deny them in pairs without self-contradiction. That is, R and G cannot be claimed strictly-speaking contradictory, because if “this S exists” is false then they are both false; this means that their traditional relation of contradiction is valid only conditionally (i.e. provided “this S exists” is true) and their absolute relation is in truth only contrariety. Similarly, I and O are only relatively subcontrary and their unconditional relation is really unconnectedness.

Indeed, it happens in practice that we reject a singular subject altogether, when we find that some predicate can be both affirmed and denied of it. This is dilemmatic argument: finding both that ‘this S’ is P and that it is not P, we must conclude that either one of these predications is wrong, or both are wrong because ‘this S’ does not exist. Particulars, of course, do not necessarily overlap; but if we can show by other means that “no S exists,” we can be sure that neither the set of S referred to by I nor that referred to by O exist, and thus deny both propositions at once. Granting all this, the above diagrams (7.2 and 7.3) can be further modified as follows:


7.4 – re-modified traditional


7.5 – modified modern version

In both these diagrams (7.4 and 7.5), all relations are the same as before, except the one between R and G (contrariety), and those between R and O, G and I, and I and O (which are now unconnected pairs). Notice that in the second diagram (7.5), although R and G are no longer contradictory, the pairs not-O and O, and not-I and I, remain contradictory, since if we deny that “Something is both S and P” (I) on the basis that “No S exists,” we can all the more be sure that “Nothing is both S and P” (not-I), and likewise regarding O and not-O.

d. We have thus proposed two successive dilutions (weakening revisions) of the traditional rectangle of oppositions. In the first, we followed modern logic in no longer assuming with Aristotle that the subjects of universal propositions have existential import. In the second, we went further and additionally denied that singular and particular propositions may well lack existential import. Clearly, if our goal is to formulate an absolute logic, one applicable equally to propositions with existential import and those without, the successive dilutions of the Aristotelian diagram are justified and important. But are such logics of anything more than academic interest – are they of practical interest? The answer must clearly be no, as I will now explain.

A difficulty with the ideas of existential import and emptiness is immediately apparent: these are characterizations that may be true or false. Different people at the same time, or the same person at different times, may have different opinions as to the existential import or emptiness of a certain term. Some people used to think that dragons exist, and maybe some people still do, yet most people today think dragons never existed. So, these characterizations are not obvious or fixed. Yet modern logicians present the question of existence or non-existence as one which has a ready answer, which can be formally enshrined. They fail to see that the issue is not formal but contentual, and thus in every given material case subject to ordinary processes of testing and eventual confirmation or disconfirmation.

It follows that the issue of existential import is not as binary as it is made out to be. The issue is not simply existence or non-existence, as modern logicians present it. The issue is whether at a given time we know or not that existence or non-existence is applicable to the case at hand. A term with existential import may be said to be ‘realistic’, in that it refers (or is believed to refer) to some existing thing(s). An empty term, i.e. one without existential import, may be said to be ‘unrealistic’, in that it refers (or is believed to refer) to a non-existent thing. In between these two possibilities lies a third, namely that of ‘hypothetical’ terms, for which we have not yet settled the issue as to whether they are (in our opinion) realistic or unrealistic. Moreover, this third possibility is not monolithic like the other two, but comprises a host of different degrees.

Our knowledge is mostly based on experience of physical and mental phenomena, though also on logical insights relating to such experience. Roughly put, we would regard a term as realistic, if we have plentiful empirical evidence as to the existence of what it refers to, and little reason to doubt it. We would regard a term as unrealistic, if we have little empirical evidence as to the existence of what it refers to, and much reason to doubt it. And we would regard a term as hypothetical if we are thus far unable to decide whether it should be characterized this way or that. In any case, the decision is usually and mostly inductive rather than purely deductive as modern logicians effectively imagine it.

How are terms formed? Very often, a term is formed by giving a name to a circumscribed phenomenon or set of phenomena that we wish to think about. Here, the definition is fixed. More often, a term is applied tentatively to a phenomenon or set of phenomena, which we are not yet able to precisely and definitively circumscribe. In such case, we may tentatively define it and affirm it, but such a term is still vague as well as uncertain. Over time we may succeed in clarifying it and making it more credible. Here, the definition is variable. Thus, the formation of terms is usually not a simple matter, but an inductive process that takes time and whose success depends on the logical skills of the thinker(s) concerned.

Of course, as individuals we mostly, since our childhood, learn words from the people around us. This is effectively fixed-definition terminology for the individual, even if the term may have been developed originally as a variable-definition one. In this context, if we come across an obscure ready-made term, we cannot understand it till we find some dictionary definition of it or someone somehow points out for us the referent(s) intended by it. But even then, inductive acts are needed to understand the definition or the intent of the pointing. When you point at something, I cannot immediately be sure exactly what it is you are pointing at; I may have to ask you: ‘do you mean including this, excluding that?’ and thus gradually zero in on your true intent.

Each of us, at all times, retains the responsibility to judge the status of the terms he or she uses. The judgment as to whether a term is realistic, or unrealistic is not always easy. In practice, therefore, most terms are effectively hypothetical, whether classed as more probably realistic or more probably unrealistic. Even so, some terms are certainly realistic or unrealistic. All terms that are truly based exclusively on empirical evidence or whose denial is self-contradictory are certainly realistic, and all manifestly counterfactual or self-contradictory terms are certainly unrealistic. So, all three of these characterizations are needed and effective.

Let us suppose the formation of realistic terms is obvious enough, and ask how imaginary ones are formed. Imaginary terms are not formed ex nihilo; they are formed by combining old terms together in new ways. A new term T is imagined by means of two or more existing terms T1, T2…. We would call term T realistic, if all the terms (T1, T2…) constituting it are realistic and their combination is credible. But if all the terms on which T is based are realistic, but their combination is not credible (e.g. we know that no T1 is T2, so the conjunction T1 + T2 is contrary to fact), we would call T unrealistic; and of course, if one or more of the terms constituting T is/are unrealistic, we would call T unrealistic. If T is made up of hypothetical elements or if its elements are realistic but their combination is of uncertain status, we would call T hypothetical.

Now, our thinking in practice is aimed at knowledge of reality. That is to say, when we come across a term without existential import, i.e. when we decide that a term is unrealistic, whether because it goes against our empirical observations or because it is in some way illogical—we normally lose interest in it and drop it. We certainly do not waste our time wondering whether such a subject has or lacks some predicate, since obviously if the subject is non-existent it has no predicates anyway. If we regard a term as empty, the oppositions of its various quantities and polarities in relation to whatever predicate are henceforth totally irrelevant. An empty term, once established as such, or at least considered to be such, plays no further role in the pursuit of knowledge. This attitude is plain common sense, except perhaps for lunatics of various sorts. For this reason, the oppositions between propositions involving empty terms are trivial. That is, the above detailed non-Aristotelian models of opposition are insignificant.

The net effect of the successive ‘dilutions’ is to make the strong, Aristotelian rectangle of oppositions (concerning propositions with existential import) seem like a special case of little importance, and to give the weaker, non-Aristotelian rectangles (concerning variously empty propositions) a disproportional appearance of importance. The reason why this occurs is that the weaker oppositions represent the lowest common denominator between the Aristotelian and non-Aristotelian oppositions, which we need if we want to simultaneously discuss propositions with and without existential import. But the result is silly, for the Aristotelian diagram (7.1) is the important one, teaching us to think straight, whereas the non-Aristotelian ones are really of very minimal and tangential academic interest.

Practical logic is focused on terms that are believed to be realistic or at least hypothetical – it is not essentially concerned with empty terms. Contrary to the accusations made by modern logicians, Aristotelian logic is not only concerned with realistic terms. It is in fact mainly used with hypothetical terms, since (as already pointed out) most of the terms which furnish our thoughts are hypothetical – tentative constructs in an ongoing inductive enterprise. We do not think hypothetical thoughts by means of some special logic – we use the same old Aristotelian logic for them. That is to say, in accord with the principle of induction, we treat a hypothetical term as a realistic term until and unless we have reason to believe otherwise.

The reason we do so is that a hypothetical term, i.e. one not yet proved to be realistic or unrealistic, is a candidate for the status of realism. This being the case, we treat it as we would any realistic term, subjecting it to the strong, Aristotelian model of oppositions, rather than to any watered-down model with wider aspirations, in the way of an inductive test. If the hypothetical term is indeed deserving of realistic status, it will survive the trial; if, on the other hand, it does not deserve such status, it will hopefully eventually be found to lead to contradiction of some sort. In that event, we would decide that the hypothetical term should rather be classed as an unrealistic term, and we would naturally soon lose interest in it. Thus, there is only one significant and useful model for oppositions between propositions, namely the Aristotelian one.

Indeed, we sometimes use Aristotelian logic even for unrealistic terms. Very often, we remove the stigma of unrealism by rephrasing our statement more precisely[16]. Alternatively, we might just keep the imaginary intent in mind: say a novelist wishes to write about fictional people, or even science-fiction creatures, he would not logically treat his subjects as empty terms – but rather subject them to the logic applicable to realistic terms, so as to enhance the illusion of realism in his novel. Thus, the logic applicable to empty terms which we have above investigated is in practice never used.

Whatever the alleged existential import of the terms involved, our thoughts remain guided by the demanding model of Aristotelian oppositions. The rational pursuit of knowledge still indubitably requires the clear-cut logic of Aristotle enshrined in the traditional rectangle of oppositions (diagram 7.1). The reason why Aristotle took the existential import of the subjects of categorical propositions for granted is, I suggest, because naturally, if there is nothing (i.e. no subject) to talk about (i.e. to predicate something of) we will not talk about it; and if we are talking, then that presumably means we do have something to talk about, i.e. a subject as well as a (positive or negative) predicate. This is manifest common sense.

If Aristotle – as far as we know, or at least as far as readers of his extant works have so far managed to discern, or so we are told by historians of logic – did not ask the question regarding the existence of the subject, it is probably simply because he quite intelligently had no interest in empty subjects. He was rightly focused on the pursuit of knowledge of the world facing him, not some non-existent domain. Modern logicians are rather, I suggest, more intent on impressing the yokels with their intellectual brilliance. With that overriding purpose in mind, they fashion systems of no practical significance whatever. They make mountains out of molehills, presenting trivia as crucial discoveries, so as to draw attention to their own persons.

e. Modern logic is a complex web of static relationships, most of them irrelevant. It ignores the dynamics of human thinking, the fact that our knowledge is constantly in flux. It is, we might say, a science of space irrespective of time. In an effort, on the surface praiseworthy, to formally acknowledge the issue of existential import, it gives undue attention to empty terms, elevating them from a very marginal problem to a central consideration. Instead of dealing with existential import parenthetically, as a side issue, it erects a logical system that effectively shunts aside some of the most important logical processes in the human cognitive arsenal.

The traditional universal propositions are cognitively of great importance. They cannot just be discarded, as modern logic has tried doing under the pretext that formal logic had to be expanded to include consideration of counterfactual terms. There are logical processes involving these propositional forms that are of great practical importance, and which logic must focus on and emphasize. It is absurd to henceforth effectively ignore these venerable and indispensable forms while making a big thing of a theoretical consideration of no practical significance whatever. The universals A and E cannot be retired under any pretext; they are not mere conventional conjunctions of more primitive forms.

For a start, universal propositions are essential to the crucial logical processes of subsumption and non-subsumption, which are enshrined in Aristotle’s syllogistic. First figure syllogisms serve to include an instance in a class or a subclass in a wider class; they teach us the notion that ‘all X’ includes every individual ‘this X’ and any possible set of ‘some X’. If, instead of an argument such as “All X are P and this S is X, therefore this S is P” (1/ARR) we propose the modern major premise “Nothing is X and not-P,” with the same minor premise, we obviously (even though the minor premise implies the existential import of an X) can no longer directly draw the desired conclusion! We are forced to stop and think about it, and infer that “this S is not not-P” before concluding that “this S is P.” Similarly, second figure syllogisms serve to exclude an instance from a class or a subclass from a wider class, and third figure syllogisms to identify overlaps between classes; and the moods of these figures become inhibited or greatly distorted if universal propositions are reinterpreted as modern logicians suggest.

Again, universal propositions are essential to the crucial logical processes of generalization and particularization. If ‘this X’ and ‘some X’ are not implied by ‘all X’, then we cannot generalize from the former to the latter. Of course, given ‘this X’ or ‘some X’, we do have existential import, and thus can anyway generalize to ‘all X’. But the fact remains that if, in accord with modern logic, we conceive our generalization as a movement of thought from “This/Some X is/are Y” to “Nothing is X and not-Y,” we miss the point entirely, even if admittedly the existential import of X is implied by the premise. For in such case, the formal continuity between premise and conclusion is lost, there being two inexplicable changes of polarity (from something to nothing and from Y to not-Y)! Similarly, particularization requires formal continuity. To move freely from I to A, and then possibly to IO, we need the traditional opposition (contradiction) between A and O.

Another issue that is ignored by modern logicians is modality. Although modern logic has developed modal logic to some extent, it has done so by means of symbolic notations based on very simplistic analyses of modality. Although it has conventionally identified the different categories of modality (necessity, impossibility, actuality, inactuality, possibility, unnecessity), it has not thoroughly understood them. It has not clearly identified and assimilated the different types of modality (the logical, extensional, natural, temporal, and spatial modes), even if human discourse has included them all since time immemorial. Notably lacking in its treatment is the awareness that modality is an expression of conditioning and that the different types of modality give rise to different types of conditioning.

Consideration of modality is manifestly absent in the doctrine of existential import. The latter (as we saw) is built around the timeless (or ‘omnitemporal’) forms of categorical proposition, which are non-modal. It does not apply to modal categorical propositions, for these do not formally imply (or presuppose) the actuality of their subject but only its possibility. Thus, a universal proposition with natural-modality, “All S can (or must) be P,” does not formally imply that “Some things are S” but only that “Some things can be S;” likewise, one with temporal modality, “All S are sometimes (or always) P” does not imply that “Some things are S” but only that “Some things are sometimes S;” and so forth.

This may be called ‘existential import’ in a broadened sense, acknowledging that being has degrees; but it is certainly not the actual sense intended by modern logicians: they apparently imagine that use of such modal propositions implies belief that “Some things are S.” And of course, the modality of subsumption, as I have called this phenomenon in my book Future Logic (chapter 41), is very relevant to the processes of opposition, eduction (immediate inferences), syllogistic deduction (mediate inference) and induction. Regarding the latter, see my detailed theory of factorial induction in the said work. Thus, we may well say that the proponents of the doctrine of existential import constructed an expanded system of logic based on a rather narrow vision of the scope of logic. Even if their expansion (for all it is worth—not much, I’d say) is applicable to non-modal propositions, it is not appropriate for modal ones.

f. The critique of the Aristotelian rectangle of oppositions began apparently in the middle ages, with Peter Abelard (France, 1079-1142). According to the Kneales, further input on this issue was made over time by William of Shyreswood, by Peter of Spain and St. Vincent Ferrer, and by Leibniz. They also mention Boole’s interest in it, and many people attribute the modern view of the issue to this 19th century logician. However, E. D. Buckner suggests that the modern idea stems rather from Franz Brentano (Austria, 1838-1917), in a paper published in 1874[17]. And of course, many big name logicians such as Frege and Russell have weighed in since then.

Even though the new logic that ensued, based on the concept of existential import, is today strongly entrenched in academia, the switchover to it was epistemologically clearly not only unnecessary but ill-advised. The doctrine of existential import has been woefully misnamed: it is in fact not about existential import, but rather about non-existential import. It gives to empty terms undue importance, and thus greatly diminishes the real importance of non-empty terms. To be sure, this innovation fitted the anti-rational ‘spirit of the times’, and it kept many people happily busy for over a century, and thus feeling they existed and were important – but it was in truth emptiness and vanity.

Apparently, none of these people reflected on the obvious fact that once a term is identified as empty, it is simply dumped – it does not continue affecting our reasoning in any significant manner. This being so, there is no need to abandon the universal forms A and E because they imply (presuppose) the existential import of their subject. Even if the Aristotelian framework, which is built around non-empty terms, occasionally ‘fails’ due to the appearance of an empty term in discourse, such event is taken in stride and dealt with by summarily eliminating the discredited term thenceforth, and certainly not by switching to a non-Aristotelian framework as modern logicians recommend to do. In any case, the issue of existential import does not apply to modal logic, and so lacks generality.

Moreover, these people failed to realize that Aristotelian logical processing relates not only to realistic terms, but more significantly to hypothetical terms, i.e. terms in process. They viewed logic as a deductive activity; they did not realize its essentially inductive character. If, due to an immoderate interest in empty terms, the science of logic abandons the universal forms A and E, it deprives people of a language with which to accurately express the movements of thought inherent in the processes of syllogistic inference and of generalization and particularization. The science of logic must acknowledge the forms of actual human thinking, and not seek to impose artificial contraptions of no practical value. Otherwise, natural processes essential to human cognition cannot be credibly expressed and logic will seem obscure and arbitrary.

Modern logic has sown confusion in many people’s minds, turning the West from a culture of confident reason to one of neurotic unreason. The purpose of logic studies ought to be to cognitively empower people, not incapacitate them. If logicians err in the forms of thought they describe and prescribe, they betray their mission, which is to intelligently and benevolently guide and improve human thinking. If they err, whether out of stupidity or malice, they turn logic from a responsible science and a fine art to a vain and dangerous game. They do not merely cease benefitting mankind; they positively harm people’s minds.

 

3. The vanity of the tetralemma

The most radical assault on reason consists in trying to put in doubt the laws of thought, for these are indeed the foundations of all rational discourse. First, the law of identity is denied by saying that things are never quite what they seem to be, or that what they are is closer to grey than black and white. This is, of course, an absurd remark, in that for itself it lays claim to utter certainty and clarity. Then, the laws of non-contradiction are denied by saying that things may both be and not-be, or neither be nor not-be. This is the ‘tetralemma’, the fourfold logic which is favored in Indian and Chinese philosophies, in religious mysticism, and which is increasingly referred to among some ‘scientists’. To grasp the vanity of the tetralemma, it is necessary to understand the nature of negation and the role of negation as one of the foundations of human logic.

The first thing to understand is that everything we experience is positive phenomenon. Everything we perceive through our senses, or remember or imagine in our minds, or even cognize through ‘intuition’ – all that has to have some sort of content to be at all perceived. Each sense organ is a window to a distinct type of positive phenomenon. We see the blue sky above, we hear birds sing, we smell the fresh air, we taste a fruit, we feel the earth’s texture and warmth, etc. Similarly, the images and sounds in our heads, whether they come from memory or are produced by imagination, are positive phenomena; and even the objects of intuition must have some content that we can cognize. Secondly, we must realize that many positive phenomena may appear together in space at a given moment. This is true for each phenomenal type. Thus, the blue sky may fill only part of our field of vision, being bounded by green trees and grey buildings; we may at once hear the sounds of birds and cars; and so on. Thirdly, many positive phenomena may at any given time share the space perceived by us. Thus, superimposed on visual phenomena like the sky may be other types of phenomena: the sound of birds in the trees, the smell of traffic in the streets, the feelings in our own body, and so on. We may even hallucinate, seeming to project objects of mental perception onto physical space. For example, the image of one’s eyeglasses may persist for a while after their removal. Fourthly, each positive phenomenon, whatever its type, varies in time, more or less quickly. Thus, the blue sky may turn red or dark, the sounds of birds or traffic may increase or decrease or even stop for a while, and so forth.

In order to express all these perceptual possibilities – differences in space and in time and in other respects, we need a concept of negation, or more precisely an act of negating. Without ‘negation’, we cannot make sense of the world in a rational manner – it is the very beginning of logical ordering of our experience. Thus, in a given visual field, where (say) blue sky and trees appear, to be able to say ‘the sky ends here, where the trees begin’ we need the idea of ‘negation’ – i.e. that on one side of some boundary sky is apparent and on the other side it is not, whereas on the first side of it trees are not apparent and on the other they are. Likewise, with regard to time, to be able to describe change, e.g. from blue sky to pink sky, we need the idea of ‘negation’ – i.e. that earlier on this part of the sky was blue and not pink, and later on it was pink and not blue. Again, we need the idea of ‘negation’ to express differences in other respects – e.g. to say that ‘the sounds of birds singing seem to emanate from the trees, rather than from buildings’. Thus, negation is one of the very first tools of logic, coming into play already at the level of sorting of experiences.

Moreover, negation continues to have a central role when we begin to deal with abstractions. Conceptual knowledge, which consists of terms and propositions based directly or indirectly on perceptual phenomena, relies for a start on our ability to cognize similarities between objects of perception: ‘this seems to resemble that somewhat’ – so we mentally project the idea of this and that ‘having something in common’, an abstract (i.e. non-phenomenal, not perceived by any means) common property, which we might choose to assign a name to. However, to take this conceptual process further, we must be able to negate – i.e. to say that ‘certain things other than this and that do not have the abstract common property which this and that seem to have’, or to say that ‘this and that do not have everything in common’. That is, we must be able to say not only that one thing resembles another in some way, but also that these or other things do not resemble each other in that way or in another way. Thus, negation is essential for making sense of information also at the conceptual level of consciousness.

Now, what is negation? To answer this question we first need to realize that there are no negative phenomena in the realm of experience. Everything we perceive is positive phenomenon – because if it was not we obviously would have nothing to perceive. We can only ‘perceive’ a negative state of affairs by first mentally defining some positive state of affairs that we should look for, and then look for it; if having looked for it assiduously we fail to find it, we then conclude inductively that it is ‘absent’, i.e. ‘not present’. Thus, positive phenomena come before negative ones, and not after. Existence logically precedes non-existence. Negative phenomena are ‘phenomena’ only metaphorically, by analogy to positive phenomena – in truth, negative phenomena are not: they do not exist. ‘Negation’ is not a concept in the sense of an abstraction from many particular experiences having a certain property in common. Negation is a tool of the thinking observer, as above described. It is an act, an intention of his.

To illustrate how confused some people – even some scientists – are with regard to negation, I offer you the following example drawn from Richard Dawkins’ The Greatest Show on Earth: The Evidence for Evolution[18]. He describes an experiment by Daniel J. Simons, in which some people are asked to watch a brief video and observe how many times a certain event takes place in it; but at the end they are asked another question entirely, viz. whether they noticed the presence of a man dressed up as a gorilla in the course of the movie, and most of them admit they did not[19]. According to Dawkins, we may infer from this experiment how “eye witness testimony, ‘actual observation’, ‘a datum of experience’ – all are, or at least can be, hopelessly unreliable.”

But this is a wrong inference from the data at hand, because he confuses positive and negative experience. The people who watched the video were too busy looking for what they had been asked to observe to notice the gorilla. Later, when the video was shown them a second time, they did indeed spot the gorilla. There is no reason to expect us to actually experience everything which is presented to our senses. Our sensory experiences are always, necessarily, selective. The validity of sense-perception as such is not put in doubt by the limited scope of particular sense-perceptions. The proof is that it is through further sense-perception that we discover what we missed before. Non-perception of something does not constitute misperception, but merely incomplete perception. ‘I did not see X’ does not deductively imply ‘I saw the absence of X’, even though repetition of the former tends to inductively imply the latter.

A negative ‘phenomenon’ is not like a positive phenomenon, something that can directly be perceived or intuited. A negation is of necessity the product of indirect cognition, i.e. of an inductive (specifically, adductive) process. We mentally hypothesize that such and such a positive phenomenon is absent, and then test and confirm this hypothesis by repeatedly searching-for and not-finding the positive phenomenon[20]. If we were to at any time indeed find the positive phenomenon, the hypothesis of negation would immediately be rejected; for the reliability of a negation is far below that of a positive experience. We would not even formulate the negation, if we already had in the past or present perceived the positive phenomenon. And if we did formulate the negation, we would naturally retract our claim if we later came across the positive phenomenon. Therefore, the content of negative phenomena is necessarily always hypothetical, i.e. tentative to some degree; it is never firm and sure as with (experienced) positive phenomena.

Negative assertions, like positive assertions, can be right or wrong. If one looked diligently for a positive phenomenon and did not find it, then one can logically claim its negation. Such claim is necessarily inductive – it is valid only so long as the positive phenomenon is actively sought and not found. The moment the positive phenomenon is observed, the negation ceases to be justified. If one did not look for the positive phenomenon, or did not look with all due diligence, perhaps because of some distraction (as in the example cited above), then of course the claim of negation is open to doubt; certainly, it is inductively weak, and one is very likely to be proved wrong through some later observation.

How, then, is negation to be defined? We could well say that negation is defined by the laws of non-contradiction and of the excluded middle. That is, with regard to any term ‘X’ and its negation ‘not-X’, the relation between them is by definition the disjunction “Either X or not-X” – which is here taken to mean that these terms (X and not-X) cannot be both true and cannot be both false, i.e. they are exclusive and exhaustive. What do I mean here by ‘definition’? – is that an arbitrary act? No – it is ‘pointing to’ something evident; it is ‘intentional’. Here, it points to the instrument of rational discourse which we need, so as to order experience and produce consistent conceptual derivatives from it. The needed instrument has to be thus and thus constructed; another construct than this one would not do the job we need it to do for us. That is, the only conceivable way for us to logically order our knowledge is by means of negation defined by means of the laws of non-contradiction and of the excluded middle. Without this tool, analysis of experience is impossible.

Suppose now that someone comes along and nevertheless objects to the preceding assertion. Well, he says, how do you know that the dilemma “either X or not-X” is true? You just arbitrarily defined things that way, but it does not mean it is a fact! Could we not equally well claim the tetralemma “Either X or not-X or both or neither” to be true? The reply to that objection is very simple. Suppose I accept this criticism and agree to the tetralemma. Now, let me divide this fourfold disjunction, putting on the one side the single alternative ‘X’ and on the other side the triple alternative ‘not-X or both or neither’. I now again have a dilemma, viz. “either ‘X’ or ‘not-X or both or neither’.” Let me next define a new concept of negation on this basis, such that we get a disjunction of two alternatives instead of four. Let us call the complex second alternative ‘not-X or both or neither’ of this disjunction ‘NOT-X’ and call it ‘the super-negation of X’.

Thus, now, the objector and I agree that the disjunction “either X or NOT-X” is exclusive and exhaustive. We agree, presumably, that this new dilemma cannot in turn be opposed by a tetralemma of the form “Either X or NOT-X or both or neither” – for if such opposition was tried again it could surely be countered by another division and redefinition. We cannot reasonably repeat that process ad infinitum; to do so would be tantamount to blocking all rational thought forever. Having thus blocked all avenues to thought, the objector could not claim to have a better thought, or any thought at all. There is thus no profit in further objection. Thus, the tetralemma is merely a tease, for we were quite able to parry the blow. Having come to an agreement that the new disjunction “Either X or NOT-X” is logically unassailable, we must admit that the original disjunction “Either X or not-X” was logically sound from the first. For I can tell you that what I meant by not-X, or the ‘negation of X’, was from the beginning what is now intended by NOT-X, or the ‘super-negation of X’!

I was never interested in a relative, weak negation, but from the start sought an absolute, strong negation. For such utter negation, and nothing less radical, is the tool we all need to order experience and develop conceptual knowledge in a consistent and effective manner. In other words, whatever weaker version of negation someone tries to invent[21], we can still propose a strong version such that both the laws of non-contradiction and of the excluded middle are applicable without doubt to it. If such negation did not exist, it would have to be invented. No one can destroy it by denying it or diluting it. Those who try to are merely sophists who do not understand the source, nature and function of negation in human discourse. They think it is a matter of symbolic manipulation, and fail to realize that its role in human discourse is far more fundamental and complex than that. Negation is the indispensable instrument for any attempt at knowledge beyond pure perception.

 

4. The Liar paradox (redux)

I dealt with the Liar paradox previously, in my Future Logic[22], but now realize that more needs to be said about it. This paradox is especially difficult to deal with because it resorts to several different discursive ‘tricks’ simultaneously.

a. The statement “This proposition is false” looks conceivable offhand, until we realize that if we assume it to be true, then we must admit it to be indeed false, while if we assume it to be indeed false, then we must admit it to be true – all of which seems unconscionable. Obviously, there is a contradiction in such discourse, since nothing can be both true and false. But the question is: just what is causing it and how can it be resolved? We are not ‘deducing’ the fact of contradiction from a ‘law of thought’ – we are ‘observing’ the fact through our rational faculty. We cannot, either, ‘deduce’ the resolution of the contradiction from a ‘law of thought’ – we have to analyze the problem at hand very closely and creatively propose a satisfying solution to it, i.e. one which indeed puts our intellectual anxiety to rest. As we shall see, this is by no means a simple and straightforward matter.

The proposition “This proposition is false” is a double paradox, because: if it is true, then it is false; and if it is false, then it is true. Notice the circularity from true to false and from false to true. The implications we draw from the given proposition seems unavoidable at first sight. But we must to begin with wonder how we know these implications (the two if–then statements) to be true. How do we know that “it is true” implies “it is false,” and that “it is false” implies “it is true”? Apparently, we are not ‘deducing’ these implications from some unstated proposition. We are, rather, using ad hoc rational insight of some sort – i.e. in a sense directly ‘perceiving’ (intellectually cognizing) the implications of the given proposition. But such rational insight, though in principle reliable, is clearly inductive, rather than deductive, in epistemological status. That is to say, it is trustworthy until and unless it is found for some reason to be incorrect. This means, there may be one or more errors in our thinking, here; it is not cast in stone. And indeed there must be some error(s), since it has led to double paradox. Therefore, we must look for it.

Perhaps use of the pronoun “it” is a problem, for it is a rather vague term. Let us therefore ask the question: more precisely what does the pronoun “it” refer us to, here?

At first sight, the “it” in “if it is true, then it is false; and if it is false, then it is true” refers to the whole given statement, “This proposition is false.” In that event, we must reword the double paradox as follows: if ‘this proposition is false’ is true, then ‘this proposition is false’ is false; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true. Here, the subject of the two if–then statements is more clearly marked out as “this proposition is false,” and so remains constant throughout. But this clarification reveals an abnormal changes of predicate, from “true” to “false” and from “false” to “true,” which cannot be readily be explained. Normally, we would say: if ‘this proposition is false’ is true, then ‘this proposition is false’ is true; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is false. The reason we here reverse the predicates is that we consider the original proposition, “this proposition is false,” as instructing such reversal.

However, whereas a proposition of the form “‘this proposition is false’ is true” is readily interpretable in the simpler form “this proposition is false,” a proposition of the form “‘this proposition is false’ is false” cannot likewise be simplified. How would we express the double negation involved? As “this proposition is true”? Clearly, the meaning of the latter is not identical to that of the former, since the subject “this proposition” refers to different propositions in each case. So the formulation of the liar paradox in full form, i.e. as “if ‘this proposition is false’ is true, then ‘this proposition is false’ is false; and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true,” does not make possible the reproduction of the initial formula expressed in terms of the pronoun “it.”

b. Let us therefore try something else. If the pronoun “it” refers to the term “this proposition”, then the double paradox should be reformulated as follows: if ‘this proposition’ is true, then ‘this proposition’ is false; and if ‘this proposition’ is false, then ‘this proposition’ is true. But doing that, we see that in each of these two if–then statements, though the subject (“this proposition”) remains constant throughout, the predicate (“true” or “false,” as the case may be) is not the same in the consequent as it was in the antecedent. There is no logical explanation for these inversions of the predicate. Normally, the truth of a proposition P does not imply its falsehood or vice versa.

We might be tempted to use the given “This proposition is false” as a premise to justify the inference from the said antecedents to the said consequents. We might try to formulate two apodoses, as follows:

If this proposition is true, then it is false (hypothesis),

If this proposition is false, then it is true (hypothesis),

and this proposition is false (given);

and this proposition is false (given);

therefore, this proposition is true (putative conclusion).

therefore, this proposition is true (putative conclusion).



Obviously, in the first case we have invalid inference, in that we try to deny the antecedent to deny the consequent, or to affirm the consequent to affirm the antecedent. In the second case, the putative conclusion does follow from the premises; but we can still wonder where the major premise (the hypothetical proposition) came from, so we are none the wiser. So, this approach too is useless – i.e. it proves nothing.

Alternatively, we might try formulating the following two syllogisms:

This proposition is false (given),

This proposition is false (given).

and this proposition is true (supposition);

and this proposition is false (supposition);

therefore, this proposition is false (putative conclusion).

therefore, this proposition is true (putative conclusion).



Clearly, these arguments are not quite syllogistic in form; but they can be reworded a bit to produce syllogisms. The first two premises would then yield the conclusion “there is a proposition that is true and false” (3/RRI), which is self-contradictory (whence, one of the premises must be false); the second two premises, however, being one and the same proposition, would yield no syllogistic conclusion other than “there is a proposition that is false and false” (3/RRI), which is self-evident (and trivial). But these are not the conclusions we seek, which must concern “this proposition” and not merely “some proposition.”

A better approach is to look upon the latter two arguments as follows. In the first case, the premises “this proposition is false” (given) and “this proposition is true” (supposition) seem to together imply “this proposition is both true and false;” and the latter paradoxical conclusion in turn indeed suggests that “this proposition is false,” since contradiction is impossible. And in the second case, the premises “this proposition is false” (given) and “this proposition is false” (supposition) agree with each other that “this proposition is false,” and so this is their logical conclusion. Since both arguments conclude with “this proposition is false,” the latter must be the overall conclusion.

However, the latter result is not as conclusive as it seems, because upon closer scrutiny it is obvious that “this proposition is false” and “this proposition is true” do not refer to the same subject, since the predicate changes. The first “this proposition” refers to the proposition “this proposition is false” and the second “this proposition” refers to the proposition “this proposition is true.” So, these two propositions in fact have different subjects as well as different predicates (viz. false and true, respectively). The subjects superficially look the same, because they are verbally expressed in identical words; but their underlying intent is not the same, since they refer to significantly different propositions (propositions with manifestly different, indeed contradictory, predicates). This means that when the predicate changes, the subject effectively changes too. When the predicate is “true,” the subject means one thing; and when the predicate is “false,” the subject means something else. Although the words “this proposition” are constant, their underlying intent varies. That is to say, the term “this proposition” does not have a uniform meaning throughout, and therefore cannot be used as a basis for the inferences above proposed.

c. Let us now try another angle. If we examine our initial reasoning in terms of the pronoun “it” more carefully, we can see what is really happening in it. Given that ‘this proposition is false’ is true, we can more briefly say: ‘this proposition is false.’ Also, given ‘this proposition is false’ is false, we can by negation educe that ‘this proposition is not false’ is true, which means that ‘this proposition is true’ is true, or more briefly put: ‘this proposition is true’[23]. In this way, we seem to argue, regarding the subject “this proposition is false,” from ‘it is true’ to ‘it is false’, and from ‘it is false’ to ‘it is true’. But in fact the use of the pronoun “it” or the term “this proposition” as abbreviated subject is a sleight of hand, for the underlying subject changes in the course of the second transition (that ending in “this proposition is true”). When abbreviation is used throughout, we seem to be talking about one and the same proposition throughout as being both true and false. But seeing that this is based on hidden equivocation, the paradoxes disappear.

It is interesting to note that when the reasoning is viewed more explicitly like that, the proposition “this proposition is true” also becomes paradoxical! We can argue: if ‘this proposition is true’ is true, then obviously ‘this proposition is true’. And: if ‘this proposition is true’ is false, then its contradictory ‘this proposition is not true’ must be true, which means that ‘this proposition is false’ is true, i.e. more succinctly: ‘this proposition is false’. Here, superficially, there seems to be no paradox, because we seem to argue, regarding the subject “this proposition is true,” from ‘it is true’ to ‘it is true’, and from ‘it is false’ to ‘it is false’. But if we look at the final conclusion, viz. “this proposition is false,” we see that it corresponds to the liar paradox![24] And here again, the explanation of the double paradox is that the apparent subject “it” or “this proposition” changes significance in the course of drawing the implications.

Notice that, in both these lines of reasoning, the first leg is ordinary self-implication, mere tautology, while the second leg is the operative self-contradiction, the paradox. If the given proposition (whether “this proposition is false” or “this proposition is true”) is true, we merely repeat the proposition as is (without need to add the predication “is true”). But if the given proposition is false, we cannot drop the additional predication (i.e. “is false”) without changing the original proposition. Thus, we could say that the two propositions, “this proposition is false” or “this proposition is true,” present no problem when taken as true; and it is only when they are hypothetically taken as false that the problem is created. So we could say that the way out of the liar paradox (and its positive analogue) is simply to accept the two claims as true, and not imagine them to be false!

We could furthermore, if we really want to, argue that “this proposition is false” and “this proposition is true” differ in that the former explicitly appears to put itself in doubt whereas the latter does not do so. On this basis, we could immediately reject the former and somewhat accept the latter, even while admitting that the latter is equally devoid of any useful information. That is to say, since the former appears ‘more paradoxical’ than the latter, the latter is to be preferred in extremis. But this, note well, ignores the equally insurmountable difficulties in it. It is better to resolutely reject both forms as vicious constructs.

d. To grasp the illusoriness of the liar paradox, it is important to realize that the two forms, “this proposition is false” and “this proposition is true,” are not each other’s contradictory; and that, in fact, neither of them has a contradictory! This is a logical anomaly, a fatal flaw in the discourse of the liar paradox; for in principle, every well-formed and meaningful proposition is logically required to have a contradictory. If a propositional form lacks a contradictory form, it cannot be judged true or false, for such judgment depends on there being a choice. We do not even have to limit our propositions to the predicates “true” or “false” – any predicate X and its negation not-X would display the same property given the same said subject. That is, “this proposition is X” and “this proposition is not-X” are not each other’s contradictory, and are therefore both equally deprived of contradictory.

We could, of course, remark that “this proposition is X” can be denied by “that proposition (i.e. the preceding one) is not X,” or even introduce a symbol for the original proposition in the new proposition. In such case, although the subjects would be verbally different, their intents would surely be the same. But the form “that proposition is not X” is more akin to the form “‘this proposition is X’ is not X,” in which the whole original proposition is given the role of subject and its predicate is given the role of predicate. However, though these two forms are somewhat equivalent in meaning to each other and to the original proposition, their logical behavior patterns are not identical with that of the original proposition, as we have already seen. The fact remains that “this proposition is not X” is not the contradictory of “this proposition is X.”

Clearly, any proposition involving the special subject “this proposition” exhibits a very unusual property, and may be dismissed on that basis alone. The reason why such a proposition lacks a contradictory is that its subject refers to the proposition it happens to be in, and that proposition is evidently different when the predicate in it is the term “false” and when it is the term “true” (or more generally, any pair of predicates ‘X’ and ‘not-X’). When the predicate changes, so does the subject; so the subject cannot be pinned-down, it is variable, it is not constant as it should be. The term “this proposition” has a different reference in each case, which depends on the predicate; consequently, each subject can only be associated with one predicate and never with the other (i.e. its negation).

From this we see that when at the beginning we thought, looking upon the statement “This proposition is false,” that if we take it at its word, then it is must be regarded as false, and so we have to prefer to it “This proposition is not false,” i.e. “This proposition is true,” and so forth, we did not realize that we were in fact, due to the ambiguity inherent in the term “This proposition” or “it,” changing its meaning at every turn. This change of meaning passes by unnoticed, because the term used is by its very nature not fixed. The pronouns “this” and “it” can be applied to anything and its opposite without such change of meaning being verbally signaled in them. They are not permanently attached to any object, but are merely contextual designations. In the technical terminology of linguistics, they are characterized as ‘deictic’ or ‘indexical’.

Thus, it appears that the liar paradox arises, however we understand its terms, as a result of some sort of equivocation in the subject. Although we seem superficially to refer to one and the same subject in the antecedent and consequent of our if–then reasoning, there is in fact a covert change of meaning which once we become aware of it belies the initial appearance of contradiction. The suggested impossible implications are thus put in doubt, made incredible. The contradictions apparently produced are thus defused or dissolved, by virtue of our inability to make them stick.

e. Another, and complementary, way to deal with the liar paradox is to point out the logical difficulty of self-reference. This is a tack many logicians have adopted, including me in my first foray into this topic in Future Logic. The argument proposed here is that the term “this proposition” refers to an object (viz. “This proposition is false” or “This proposition is true”) which includes the term itself. A finger cannot point at itself, and “this” is the conceptual equivalent of a finger. Effectively, the expression “this” has no content when it is directed at itself or at a sentence including it. It is empty, without substance. It is as if nothing is said when we indulge in such self-reference.

Thus, “This proposition is X” (where X stands for false, or true, or indeed anything) is in fact meaningless; and a meaningless sentence cannot be true or false. Such a sentence can reasonably be described as neither true nor false, without breach of the law of the excluded middle, because neither of these logical evaluations is applicable to meaningless sentences. “This proposition is false” looks meaningful because its four constituents (i.e. “this,” “proposition,” “is” and “false”) are separately normally meaningful. But in this particular combination, where one of the elements (viz. “this”) does not refer to anything already existent, the sentence is found to be meaningless.

The apparent contradictions that self-reference produces help us to realize its meaninglessness. And it is through the intellectual realization of the meaninglessness of self-reference that we explain away and annul the apparent contradictions. On this basis, we can say that even though the sentence “This proposition is true” does not at first sight give rise to any paradox (as people think: “if it is true, it is true; and if it is false, it is false”), nevertheless, since it involves self-reference as much as “This proposition is false,” it is equally meaningless and cannot be characterized as true or false. In fact, as I have shown above, “This proposition is true” does also give rise to double paradox.

Someone might object: What about the propositions: “this statement is self-referential” and “this statement is not self-referential”? Surely, we can say that these are meaningful and that the former is true while the latter is false! The retort to that objection is that the two propositions “this proposition refers to itself” and “this proposition does not refer to itself” are not mutual contradictories, because (just like in the liar paradox) their subjects differ radically, each referring to the proposition it is in and not to the other. Thus, while the positive version may seem more self-consistent than the negative one, and therefore to be preferred in extremis, they are in fact both fundamentally flawed, because (just like in the liar paradox) neither of them has a contradictory, and without the logical possibility of negating a discourse it is impossible to judge whether it is right or wrong.[25]

f. Not long after the preceding reflections, I happened to come across another interesting example of paradoxical self-reference, namely “Disobey me!”[26] This involves the ‘double bind’ – if I obey it, I disobey it and if I disobey it, I obey it. To resolve this paradox, we need to first put the statement in more precise form, say: “you must disobey this command!” We can then disentangle the knot by realizing that the order being given has outwardly imperative form but inwardly lacks content. It does not define a specific, concrete action that is to be done or not-done. If we wished to obey it, or to disobey it, we would not know just what we are supposed to do or not-do! It is therefore an order that can neither be obeyed nor be disobeyed. Ruminating on this case led me to what I now believe is the trump card, which convincingly finalizes the resolution of the liar paradox, even as the preceding reflections all continue to be relevant.

It occurred to me then that this is precisely the problem with the liar paradox. It says “this proposition is false” – but it does not tell us anything about the world that can be judged as true or false. A ‘proposition’ is a statement that makes some claim about the world. If the statement makes no such claim, if it ‘proposes’ nothing, it cannot be logically assessed as true or false. If it refers to nothing – whether physical, mental or spiritual, perceptual, intuitive or conceptual – it has no meaning. A meaningless statement does not qualify as a ‘proposition’. The attributes of ‘true’ or ‘false’ are not ordinary predicates, like ‘white’ or ‘black’, which can be attached to any subject and then judged to be truly or falsely attached. The attributes of ‘true’ or ‘false’ require a precise claim to be made before they can at all be used.

The truth of this explication can be seen with reference to the ‘propositional forms’ used in logic theory. Take, for example, “All X are Y.” Such a propositional form cannot be judged true or false because it manifestly has no content. Only when such an abstraction is given some specific content, such as “All men are mortal,” can we begin to ask whether it is true or false. A propositional form is too vague to count as a proposition. It does not tell us anything about the world, other than implying that there are (or even just that there may be) concrete propositions which have this form. Just as we cannot disobey or even obey an imperative without content, so we cannot judge a purely formal expression true or false.

The same applies to the liar paradox: like a formal proposition, it has no concrete content, and therefore cannot be judged true or false. The liar paradox has no content partly due to its having a self-referential subject (“this proposition”). But the truth is, even if its subject was not self-referential, it would still have insufficient content. This is so, because its predicate “false” (and likewise its opposite, “true”) is not an ordinary predicate; it is more like a formal predicate. It can only be used if another, more concrete predicate has already been proposed for the subject at hand. For example, “this proposition is interesting” could be judged true or false (if it was not self-referential) because it already has a predicate (viz. “interesting”). Thus, the problem with the liar paradox is not only the self-reference it involves but also its lack of a predicate more concrete than the logical predicate “false” (or “true”).

All this illustrates how the ‘laws of thought’ are not axioms in the sense of top premises in the knowledge enterprise from which we mechanically derive other premises. Rather the expression ‘laws of thought’ refers to recurring insights which provide us with some intellectual guidance but cannot by themselves determine the outcome. The individual in pursuit of knowledge, and in particular the logician, is driven by the obviousness or by the absurdity of a situation to look for creative solutions to problems. He or she must still think of possible solutions and test them.

 

5. The Russell paradox (redux)

Logic is what helps us transmute scattered concrete perceptions into well-ordered abstract concepts. Human knowledge, or opinion, is based on experience, imagination and rational insight. The latter is a kind of ‘experience’ in the larger sense, a non-phenomenal sort of experience, call it logical ‘intuition’. Reason was for this reason called by the ancients, in both West and East, the ‘sixth sense’ or ‘common sense’, i.e. the sense-organ which ties together the other five senses, those that bring us in empirical contact with phenomenal experience: colors, shapes, sounds, smells, tastes, touch-sensations, etc., whether they are physically perceived or mentally imagined. The five senses without the sixth yield chaotic nonsense (they are non-sense, one cannot ‘make sense’ of them); and conversely, the sixth sense is useless without the other five, because it has nothing about which to have rational insights. Imagination reshuffles past experiential data and reasoning, making possible the formation of new ideas and theories which are later tested with reference to further experience and reasoning.

Elements of class logic. Logic initially developed as a science primarily with reference to natural discourse, resulting in what we today refer to as predicate logic. In natural human discourse, we (you and me, and everyone else) routinely think of and discuss things we have perceived, or eventually conceived, by means of categorical propositions involving a subject (say, S) and a predicate (say, P) which are related to each other by means of the copula ‘is’. Such propositions have the form “S is P,” which may be singular or plural, and in the latter case general (or universal) or particular, and positive or negative, and moreover may involve various modes and categories of modality[27].

A proposition of the form ‘S is P’ is really a double predication – it tells us that a thing which is S is also P; thus, S and P are really both predicates, though one (the subject S) is given precedence in thought so as to ‘predicate’ the other (the predicate P) of it[28]. Primarily, S refers to some concrete phenomenon or phenomena (be it/they physical, mental or spiritual), i.e. an individual entity or a set of entities, and P to a property of it or of theirs. For examples, “John is a man” and “All men are human beings” are respectively a singular predication (about one man, John) and a plural one (about all men).

Additionally, still in natural discourse, the subject of our thoughts may be predicates as such, i.e. predicates in their capacity as predicates; an example is: “‘men’ may be the subject or predicate of a proposition.” The latter occurs in specifically philosophical (or logical or linguistic) discourse; for example, in the present essay.

Now, logicians through the ages, and especially in modern times, have effectively found natural discourse somewhat inadequate for their needs and gradually developed a more artificial language, that of ‘classes’[29]. This type of discourse exactly parallels natural discourse, but is a bit more abstract and descriptive so as to facilitate philosophical (or logical or linguistic) discourse and make it more precise. In this language, instead of saying “this S is P,” we say “this S is a member (or instance) of P” (note well the lengthening of the copula from ‘is’ to ‘is a member (or instance) of’. If ‘this S’ symbolizes a concrete individual, then ‘P’ here is called a ‘class’; but if ‘this S’ symbolizes an abstract class, then ‘P’ here is called a ‘class of classes’.

A class, then, is an abstraction, a mental constructs in which we figuratively group some concrete things (be they physical, mental or spiritual). Although we can and do temporarily mentally classify things without naming the class for them, we normally name classes (i.e. assign them a distinctive word or phrase) because this facilitates memory and communication. Naming is not the essence of classification, but it is a great facilitator of large-scale classification. The name of a class of things does not ‘stand for them’ in the way of a token, but rather ‘points the mind to them’ or ‘draws our attention to them’; that is to say, it is an instrument of intention.

A class in the primary sense is a class of things in general; a class in the secondary sense is more specifically a class of classes. Membership is thus of two kinds: membership of non-classes in a class, or membership of classes in a class of classes. Alternatively, we may speak of first-order classes and second-order classes to distinguish these two types. There are no other orders of classes. When we think about or discuss more concrete things, we are talking in first-order class-logic; when we think about or discuss first-order classes, we are talking in second-order class-logic, and the latter also applies to second-order classes since after all they are classes too. The two orders of classes should not be confused with the hierarchy of classes within each order.

The relation between classes of classes and classes is analogous to the relation between classes and concretes; it is a relation of subsumption. When a lower (i.e. first-order) class is a member of a higher (i.e. second-order) class, it does not follow that the members of the lower class are also members of the higher class; in fact, if they are members of the one they are certainly not members of the other. Thus, for example, you and me, although we are members of the class ‘men’ because we are men, we are not members of the class ‘classes of men’ because we are not ‘men’. Also, the class ‘men’ is not a man, but is a member of the class ‘classes of men’. The members of the class ‘classes of men’ (or more briefly put, ‘men-classes’), which is a class of classes, are, in addition to the broad class ‘men’, the narrower classes ‘gardeners’, ‘engineers’, ‘sages’, ‘neurotics’, and so on.[30]

Hierarchization, on the other hand, refers to classes within a given order that share instances, not merely by partly overlapping, but in such a way that all the members of one class are members of the other (and in some but not all cases, vice versa). For example, since all men are animals, though not all animals are men, the class ‘men’ is a subclass (or species) of the class ‘animals’, and the class ‘animals’ is an overclass (or genus) of the class ‘men’. If two classes have the same instances, no more and no less, they may be said to be co-extensive classes (a class that serves as both species and genus in some context is said to be sui generis). If two classes merely share some instances, they may be said to be intersecting (or overlapping) classes, but they are not hierarchically arranged (e.g. ‘gardeners’ and ‘engineers’). If two classes of the same order have no instances in common, they may be said to be mutually exclusive classes.

It is important to grasp and keep in mind the distinction between hierarchy and order. Since you and I are men, each of us is a member of the class ‘men’; this is subsumption by a first-order class of its concrete instances. Since all men are animals, the class ‘men’ is a subclass of the class ‘animals’; this is hierarchy between two classes of the first order. Since ‘men’ is a class of animals, it is a member of the class ‘classes of animals’ (or ‘animal-classes’); this is subsumption by a second-order class (i.e. a class of classes) of its first-order-class instances (i.e. mere classes). Since all ‘classes of men’ are ‘classes of animals’, the class ‘men-classes’ is a subclass of the class ‘animals-classes’; this is hierarchy between two classes of the second order, i.e. between two classes of classes. The relation between classes of the first order and classes of the second order is never one of hierarchy, but always one of subsumption; i.e. the former are always members (instances) of the latter, never subclasses. Hierarchies only occur between classes of the same order.

Thus, in class logic, we have two planes of existence to consider. At the ground level is the relatively objective plane of empirical phenomena (whether these are physical, mental or spiritual in substance); above that, residing in our minds, is the relatively subjective plane of ideas (which are conceived as insubstantial, but do have phenomenal aspects – namely mental or physical images, spoken or written words, and the intentions of such signs), comprising ideas about empirical phenomena and ideas about such ideas. Classes are developed to facilitate our study of empirical phenomena and classes of classes are developed in turn to facilitate our study of classes – for classes (including classes of classes) are of course themselves empirical phenomena of sorts. Classification is a human invention helpful for cognitive ordering of the things observed through our senses or our imaginations or our introspective intuitions. Although classes are products of mind, this does not mean that they are arbitrary – they are formed, organized and controlled by means of our rational faculty, i.e. with the aid of logic.

Clearly, to qualify as a class, a class must have at least one member (in which case the sole member is “one of a kind”). Usually, a class has two or more members, indeed innumerable members. A class is finite if it includes a specified number of instances; if the number of instances it includes is difficult to enumerate, the class is said to be open-ended (meaning infinite or at least indefinite). What brings the instances of a class together in it is their possession of some distinctive property in common; the class is defined by this property (which may of course be a complicated conjunction of many properties). A class without instances is called a null (or empty) class; this signifies that its defining property is known to be fanciful, so that it is strictly speaking a non-class.

Thus, note well, the term ‘class’ is a bit ambiguous, as it may refer to a first-order class (a class of non-classes, i.e. of things other than classes) or a second-order class (a class of classes, i.e. a mental construct grouping two or more such mental constructs). A class (of the first order) is not, indeed cannot be, a class of classes (i.e. a class of the second order). There is, of course, a class called ‘non-classes’; its instances are principally all concrete things, which are not themselves classes; for example, you and I are non-classes. ‘Non-classes’ is merely a class, not a class of classes, since it does not include any classes. Thus, ‘non-classes’ may be said to be a first-order class, but does not qualify as a second-order class.[31]

The realm of classes of classes is very limited as an object of study in comparison to the realm of mere classes. For what distinctions can we draw between classes? Not many. We can distinguish between classes and classes of classes, between finite and open-ended classes, between positive and negative classes[32], and maybe a few more things, but not much more.

An apparent double paradox. Bertrand Russell (Britain, 1872-1970) proposed a distinction between ‘a class that is a member of itself’ and ‘a class that is not a member of itself’. Although every class is necessarily co-extensive with itself (and in this sense is included in itself), it does not follow that every class is a member of itself (evidently, some are and some are not). Such a distinction can be shown to be legitimate by citing convincing examples. Thus, the class ‘positive classes’ is a member of itself, since it is defined by a positive property; whereas the class ‘negative classes’ is not a member of itself, since it is also positively defined (albeit with general reference to negation). Again, the class ‘finite classes’ is not a member of itself, since it has innumerable members; while the class ‘open-ended classes’ is a member of itself, since it too has innumerable members.

What about the class ‘classes’ – is it a member of itself or not? Since ‘classes’ is a class, it must be a member of ‘classes’ – i.e. of itself. This is said without paying attention to the distinction between classes of the first and second orders. If we ask the question more specifically, the answer has to be nuanced. The class ‘first-order classes’ being a class of classes and not a mere class, cannot be a member of itself, but only a member of ‘second-order classes’; the members of the ‘first-order classes’ are all mere classes. On the other hand, since the class ‘second-order classes’ is a class of classes, it is a member of itself, i.e. a member of ‘second-order classes’. Thus, the class ‘second-order classes’ includes both itself and the class ‘first-order classes’, so that when we say that the wider class ‘classes’ is a member of itself, we mean that it is more specifically a member of the narrower class ‘classes of classes’. As regards the class ‘non-classes’, since it is a class and not a non-class, it is not a member of itself. Note however that Russell’s paradox does not make a distinction between classes of the first and second orders, but focuses on ‘classes’ indiscriminately.

Russell asked whether “the class of all classes which are not members of themselves” is or is not a member of itself. It seemed logically impossible to answer the question, because either way a contradiction ensued. For if the class ‘classes not members of themselves’ is not a member of the class ‘classes not members of themselves,’ then it is indeed a member of ‘classes not members of themselves’ (i.e. of itself); and if the class ‘classes not members of themselves’ is a member of ‘classes not members of themselves,’ then it is also a member of ‘classes which are members of themselves’ (i.e. of its contradictory). This looked like a mind-blowing double paradox.

The solution of the problem. The pursuit of knowledge is a human enterprise, and therefore one which proceeds by trial and error. Knowledge is inductive much more than deductive; deduction is just one of the tools of induction. There are absolutes in human knowledge, but they are few and far between. When we formulate a theory, it is always essentially a hypothesis, which might later need to be revised or ruled out. So long as it looks useful and sound, and does so more than any competing theory, we adopt it; but if it ever turns out to be belied by some facts or productive of antinomy, we are obliged to either reformulate it or drop it. This is the principle of induction. When we come upon a contradiction, we have to ‘check our premises’ and modify them as necessary. In the case at hand, since our conception of class logic is shown by the Russell paradox to be faulty somehow, we must go back and find out just where we went wrong. So, let us carefully retrace our steps. We defined a class and membership in a class by turning predication into classification, saying effectively:

If something is X, then it is a member of the class ‘X’, and not a member of the class ‘nonX’.

If something is not X, then it is not a member of the class ‘X’, but a member of the class ‘nonX’.

Where did we get this definition? It is not an absolute that was somehow cognitively imposed on us. We invented it – it was a convention by means of which we devised the idea of classes and membership in them. Knowledge can very well proceed without recourse to this idea, and has done so for millennia and continues to do so in many people’s mind. It is an idea with a history, which was added to the arsenal of reasoning techniques by logicians of relatively recent times. These logicians noticed themselves and others reasoning by means of classification, and they realized that this is a useful artifice, distinct from predication and yet based on it somehow. They therefore formally proposed the above definition, and proceeded to study the matter in more detail so as to maximize its utility. The ‘logic of classes’, or ‘class logic’, was born.

However, at some stage, one logician, Bertrand Russell, realized that there was an inherent inconsistency in our conception of classification, which put the whole edifice of class logic in serious doubt. That was the discovery of the paradox bearing his name. That was a great finding, for there is nothing more important to knowledge development, and especially to development of the branch of knowledge called formal logic, than the maintenance of consistency. Every discovery of inconsistency is a stimulation to refine and perfect our knowledge. Russell deserves much credit for this finding, even if he had a lot of difficulty resolving the paradox in a fully convincing manner. Let us here try to do better, by digging deeper into the thought processes involved in classification than he did. What is classification, more precisely?

If we look more closely at our above definition of a class ‘X’ and membership of things in it by virtue of being X, we must ask the question: what does this definition achieve, concretely? Are we merely substituting the phrase ‘is a member of’ for the copula ‘is’, and the class ‘X’ for the predicate X? If this is what we are doing, there is no point in it – for it is obvious that changing the name of a relation or a term in no way affects it. Words are incidental to knowledge; what matters is their underlying intent, their meaning. If the words change, but not the meaning, nothing of great significance has changed. No, we are not here merely changing the words used – we are proposing a mental image.

Our idea of classification is that of mental entities called classes in which things other than classes (or lesser classes, in the case of classes of classes) are figuratively collected and contained. When we say of things that they are members of class ‘X’, we mean that class ‘X’ is a sort of box into which these things are, by means of imagination, stored (at a given time, whether temporarily or permanently). That is to say, our ‘definition’ of classification is really a formal convention used to institute this image. What it really means is the following:

If something is X, then it is in the class ‘X’, and out of (i.e. not in) the class ‘nonX’.

If something is not X, then it is out of (i.e. not in) the class ‘X’, but in the class ‘nonX’.

Clearly, to ‘be’ something and to ‘be in’ (within, inside) something else are not the same thing. Our definition conventionally (i.e. by common agreement) decrees that if X is predicated of something, then we may think of that thing as being as if contained by the mental entity called class ‘X’. But this decree is not an absolute; it is not a proposition that being subject to predication of X naturally and necessarily implies being a member of class ‘X’. For the whole idea of classification, and therefore this definition of what constitutes a class and membership therein, is a human invention. This invention may well be, and indeed is, very useful – but it remains bound by the laws of nature. If we find that the way we have conceived it, i.e. our definition of it, inevitably leads to contradiction, we must adjust our definition of it in such a way that such contradiction can no longer arise. This is our way of reasoning and acting in all similar situations.

As we shall presently show, since the contradiction is a consequence of the just mentioned defining implication, we must modify that implication. That is to say, we must decree it to have limits. Of course, we cannot just vaguely say that it has limits; we must precisely define these limits so that the practical value of our concept of classification is restored. We can do that by realizing that our definition of classification with reference to something ‘being in’ something else means that class logic is conceived of as related to geometrical logic. This is obvious, when we reflect on the fact that we often ‘represent’ classes as geometrical figures (notably, circles) and their members as points within those figures. This practice is not accidental, but of the very essence of our idea of classification. Classification is imagining that we put certain items, identified by their possession of some common and distinctive property, in a labeled container[33].

Let us now examine the concept of self-membership in the light of these reflections. What is the idea of self-membership? It is the presupposition that a class may be a member of itself. But is that notion truly conceivable? If we for a moment put aside the class logic issue, and reformulate the question in terms of geometrical logic, we see that it is absurd. Can a container contain itself? Of course not. There is no known example of a container containing itself in the physical world; and indeed we cannot even visually imagine a container containing itself. So the idea of self-containment has no empirical basis, not even in the mental sphere. It is only a fanciful conjunction of two words, without experiential basis. For this reason, the idea strikes us as illogical and we can safely posit as a universal and eternal ‘axiom’ that self-containment is impossible. A nonsensical term like ‘the collection of all collections’ is of necessity an empty term; we are not forced to accept it, indeed we are logically not allowed to do so; we can only consistently speak of ‘the collection of all other collections’[34].

A container is of course always co-extensive with itself, i.e. it occupies exactly the space it occupies. But such ‘co-extension’ is not containment, let alone self-containment, for it does not really (other than verbally) concern two things but only one; there is no ‘co-’ about it, it is just extended, just once. We refer to containment when a smaller object fits inside a larger object (or in the limit when another object of equal size neatly fits inside a certain object). The concept of containment refers to two objects, not one. There has to be two distinct objects; it does not suffice to label the same object in two ways. To imagine ‘self-containment’ is to imagine that a whole object can somehow fit into itself as a smaller object (or that it can somehow become two, with one of the two inside the other). This is unconscionable. A whole thing cannot be a part (whether a full or partial part) of itself; nothing can be both whole and part at once. A single thing cannot be two things (whether of the same or different size) at once; nothing can simultaneously exist as two things.

You cannot decide by convention that something is both whole and part or that one thing is two. You cannot convene something naturally impossible. You can only convene something naturally possible, even though it is unnecessary. Thus, the concept of self-containment is meaningless; it is an inevitably empty concept, because it assumes something impossible to be possible. There is no such thing as self-containment; a container can never contain itself. If this is true, then it is of course equally true that no class includes itself, for (as we have seen) classification is essentially a geometrical idea. Given that a container cannot contain itself, it follows that the answer to the question as to whether a class can be a member of itself is indubitably and definitely: No. Because to say of any class that it is a member of itself is to imply that a container can be a content of itself. Just as no container which is a content of itself exists, so no class which is a member of itself exists!

Now, this is a revolutionary idea for class logic. It applies to any and every class, not just to the class ‘classes not members of themselves’ which gave rise to the Russell paradox. Moreover, note well that we are here denying the possibility of membership of a class in itself, but not the possibility of non-membership of a class in itself. When we say that no container contains itself, we imply that it is true of each and every container that it does not contain itself. Similarly, when we say that no class is a member of itself, we imply that it is true of each and every class that it is not a member of itself. What this means is that while we acknowledge the subject of the Russell paradox, namely the class ‘classes that are not members of themselves’, we reject the notion that such a class might ever, even hypothetically for a moment, be a member of itself (and therefore paradoxical) – for, we claim, no class whatever is ever a member of itself.

How can this be, you may well ask? Have we not already shown by example that some classes are members of themselves? Have we not agreed, for example, that the class ‘classes’ being a class has to be a member of the class ‘classes’, i.e. of itself? How can we deny something so obvious? Surely, you may well object further, if the class ‘classes that are not members of themselves’ is not a member of itself, then it is undeniably a member of itself; and if it is a member of itself, then it is undeniably not a member of itself? To answer these legitimate questions, let us go back to our definition of classification, and the things we said about that definition. As I pointed earlier, our definition of classes and membership in them has the form of a conventional implication. It says:

If and only if something is X, then it is a member of the class ‘X’.

Now, since this conventional implication leads us inexorably to paradox, we must revise it, i.e. make it more limited in scope, i.e. specify the exact conditions when it ‘works’ and when it ceases to ‘work’. What is essentially wrong with it, as we have seen, is that it suggests that a class can be a member of itself. For example, since the class ‘classes’ is a class, then it is a member of ‘classes’; in this example, the variable X has value class and the variable ‘X’ has value ‘classes’. But, as we have shown, the claim that a class can be a member of itself logically implies something geometrically impossible; namely, that a container can be a content of itself. So, to prevent the Russell paradox from arising, we need to prevent the unwanted consequences of our definition from occurring. Given that our concept of classification is problematic as it stands, what are the conditions we have to specify to delimit it so that the problem is dissolved?

The answer to this question is that when the subject and predicate of the antecedent clause are one and the same, then the consequent clause should cease to be implied. That is to say, if the antecedent clause has the form “if the class ‘X’ is X” then the consequent clause “then the class ‘X’ is a member of ‘X’ (and thus of itself)” does not follow. This ‘does not follow’ is a convention, just as the general ‘it follows’ was a convention. What we have done here is merely to draw a line, saying that the consequent generally follows the antecedent, except in the special case where the subject and predicate in the antecedent are ‘the same’ (in the sense that predicate X is applicable to class ‘X’ which is itself based on predicate X). This is logically a quite acceptable measure, clearly. If an induced general proposition is found to have exceptions, then it is quite legitimate and indeed obligatory to make it less general, retreating only just enough to allow for these exceptions.

Since the initial definition of classification was a general convention, it is quite permissible, upon discovering that this convention leads us into contradiction, to agree on a slightly narrower convention. Thus, whereas, in the large majority of cases, it remains true that if something is X, then it is a member of the class ‘X’, and more specifically, if a class (say, ‘Y’) is X, then it (i.e. ‘Y) is a member of the class of classes ‘X’ – nevertheless, exceptionally, in the special case where the class that is X is the class ‘X’ (i.e. where ‘Y’ = ‘X’), we cannot go on to say of it that it is a member of ‘X’, for this would be to claim it to be a member of itself, which is impossible since this implies that a container can be a content of itself. Note well that we are not denying that, for example, the class ‘classes’ is a class; we are only denying the implication this is normally taken to have that the class ‘classes’ is a member of the class ‘classes’. We can cheerfully continue saying ‘is’ (for that is mere predication), but we are not here allowed to turn that ‘is’ into ‘is a member of’ (for that would constitute illicit classification).

In this way, the Russell paradox is inhibited from arising. That is to say, with reference to the class ‘classes not members of themselves’: firstly, it is quite legitimate to suppose that the class ‘classes not members of themselves’ is not a member of itself, since we know for sure (from geometrical logic) that no class is a member of itself; but it is not legitimate to say that this fact (i.e. that it is not a member of itself) implies that it is a member of itself, since such implication has been conventionally excluded. Secondly, it is not legitimate to suppose, even for the sake of argument, that the class ‘classes not members of themselves’ is a member of itself, since we already know (from geometrical logic) that no class is a member of itself, and therefore we cannot establish through such supposition that it is not a member of itself, even though it is anyway true that it is not a member of itself.

As can be seen, our correction of the definition of classification, making it less general than it originally was, by specifying the specific situation in which the implication involved is not to be applied, succeeds in eliminating the Russell paradox. We can say that the class ‘classes not members of themselves’ is not a member of itself, but we cannot say that it is a member of itself; therefore, both legs of the double paradox are blocked. In the first leg, we have blocked the inference from not-being ‘a member of itself’ to being one; in the second leg, we have interdicted the supposition of being ‘a member of itself’ even though inference from it of not-being one would be harmless. Accordingly, the answer to the question posed by Russell – viz. “Is the class of all classes which are not members of themselves a member of itself or not?” – is that this class is not a member of itself, and that this class not-being a member of itself does not, contrary to appearances, make it a member of itself, because no class is a member of itself anyway.

Thus, to be sure, though it is true that the class ‘classes’ is a class, it does not follow that it is a member of itself; though it is true that the class ‘classes of classes’ is a class of classes, it does not follow that it is a member of itself; though it is true that the class ‘positive classes’ is a positive class, it does not follow that it is a member of itself; though it is true that the class ‘open-ended classes’ is an open-ended class, it does not follow that it is a member of itself; though it is true that the class ‘classes that are not members of themselves’ is a class that is not a member of itself, it does not follow that it is a member of itself. As for the class ‘classes members of themselves’, it has no members at all. It should be emphasized that the restriction on classification that we have here introduced is of very limited scope; it hardly affects class logic at all, concerning as it does a few very borderline cases.

The above is, I believe, the correct and definitive resolution of the Russell paradox. We acknowledged the existence of a problem, the Russell paradox. We diagnosed the cause of the problem, the assumption that self-membership is possible. We showed that self-membership is unconscionable, since it implies that a container can contain itself; this was not arbitrary tinkering, note well, but appealed to reason. We proposed a solution to the problem, one that precisely targets it and surgically removes it. Our remedy consisted in uncoupling predication from classification in all cases where self-membership is assumed, and only in such cases. This solution of the problem is plain common sense and not a flight of speculation; it is simple and elegant; it is convincing and uncomplicated; it does not essentially modify the concept of class membership, but only limits its application a little; it introduces a restriction, but one that is clearly circumscribed and quite small; it does not result in collateral damage on areas of class logic, or logic in general, that are not problematic, and therefore does not call for further adaptations of logic doctrine. Note moreover that our solution does not resort to any obscure ‘system’ of modern symbolic logic, but is entirely developed using ordinary language and widely known and accepted concepts and processes.

A bit of the history. Let us now look briefly at some of the history of the Russell paradox, and see how he and some other modern symbolic logicians dealt with it[35].

Georg Cantor had already in 1895 found an antinomy in his own theory of sets. In 1902, when Gottlob Frege (Germany, 1848-1925) was about to publish the second volume of his Grundgesetze, he was advised by Russell of the said paradox. Frege was totally taken in and could not see how to get out of the self-contradictions inherent in “the class of classes that do not belong to themselves.” He perceived this as very serious, saying: “What is in question is … whether arithmetic can possibly be given a logical foundation at all.” Frege first tried to fix things by suggesting that there might be “concepts with no corresponding classes,” or alternatively by adjusting one of his “axioms” in such a way that:

“Two concepts should be said to have the same extension if, and only if, every object which fell under the first but was not itself the extension of the first fell likewise under the second and vice versa”[36].

Clearly, Frege’s initial suggestion that there might be “concepts with no corresponding classes” can be viewed as an anticipation of my uncoupling of predication and classification in specific cases. However, Frege did not identify precisely in what cases such uncoupling has to occur. This is evident in his next suggestion, which, though it points tantalizingly to the difficulty in the notion of self-membership, does not reject this notion outright but instead attempts to mitigate it. He speaks of two concepts instead of one, and tries to conventionally exclude the extension as a whole of each from the other, while of course continuing to include the objects falling under the extension; this shows he has not realized that self-inclusion by an extension is not even thinkable.

It should be stressed that Russell’s paradox pertains to a certain class (viz. that of all classes not members of themselves) being or not-being a member of itself – not of some other class. Frege tries to resolve this paradox with reference, not to a single class, but to a pair of equal classes, even though (to my knowledge) he has not demonstrated that co-extensive classes result in a paradox comparable to the Russell paradox. It follows that his attempted solution to the problem is not germane to it. Moreover, Frege seems to have thought that if all items that fall under one class (say, ‘Y’) fall under another class (say, ‘X’), then the class ‘Y’ may be assumed to fall under the class ‘X’; and vice versa in the event of co-extension. This is suggested by his attempt to prevent such assumption, so as to avoid (in his estimate) the resulting Russell paradox. But in truth, it does not follow from the given that all Y are X that the class ‘Y’ is a member of the class ‘X’ – it only follows that the class ‘Y’ is a subclass of the class ‘X’, or an equal class if the relation is reversible. Thus, it appears that Frege confused the relations of class-membership and hierarchization of classes, using a vague term like ‘falling’ to characterize them both.

We may well ask the question whether an equal class, or a subclass, or even an overclass, might be a member of its hierarchically related class. Offhand, it would seem to be possible. For example, all positive classes are classes and therefore members of the class ‘classes’, and the class ‘positive classes’ is a subclass of the class ‘classes’; however, although not all classes are positive classes (some are negative classes), nevertheless the class ‘classes’ is a positive class (being defined by a positive statement), and so is a member of the class ‘positive classes’. But although this example suggests that an overclass might be a member of its subclass (and therefore, all the more, an equal class or a subclass might be a member of its hierarchical relative), we might still express a doubt by means of analogy, as Frege perhaps intended to do. We could argue inductively, by generalization, that if a class cannot be a member of itself, then maybe it cannot be a member of any coextensive class (as Frege suggests), and perhaps even of a subclass or an overclass. For the issue here is whether the instances referred to by the first class can be thought to occur twice in the second class (as members of it in their own right, and as constituents of a member). So, Frege may have raised a valid issue, which could lead to further restrictions in class logic. However, this need not concern us further in the present context, since (as already explained) it is not directly relevant to resolution of the Russell paradox.

Russell described his paradox in his book Principles of Mathematics, published soon after. Although at first inclined to Frege’s second approach, he later preferred Frege’s first one, proposing that there might be “some propositional functions which did not determine genuine classes.” Note here again the failure to pinpoint the precise source and remedy of the problem. Subsequently, Russell thought that “the problem could never be solved completely until all classes were eliminated from logical theory.” This, in my view, would be throwing out the baby with the bath water – an overreaction. But then he found out (or rather, he thought he did, or he convinced himself that he did) that the same paradox could be generated without “talk of classes,” i.e. with reference to mere predicates – that is, in terms of predicate logic instead of in terms of class logic. As Kneale and Kneale put it (p. 655):

“Instead of the class which is supposed to contain all classes that are not members of themselves let us consider the property of being a property which does not exemplify itself. If this property exemplifies itself, then it cannot exemplify itself; and if it does not exemplify itself, then it must exemplify itself. Clearly, the nature of the trouble is the same here as in the original paradox, and yet there is no talk of classes.”

But even if classes are not explicitly mentioned here, it is clear that they are tacitly intended. How would a property ‘exemplify’ itself? Presumably, property X would be ‘a property which exemplifies itself’ if property X happens to be one of the things that have property X. That is to say, X exemplifies X if X is a member of the class of things that are X. We cannot talk about properties without resorting to predication; and once we predicate we can (given the initial definition of classification) surely classify. So, this attempt is just verbal chicanery; the same thought is intended, but it is dressed up in other words. It is dishonest. Moreover, the way the paradox is allegedly evoked here does not in fact result in paradox.

We cannot say, even hypothetically, “if this property [i.e. the property of being a property which does not exemplify itself] exemplifies itself” for that is already self-contradictory. To reconstruct a Russell paradox in ‘property’ terms, we would have to speak of ‘the property of all properties which do not exemplify themselves’; for then we would have a new term to chew on, as we did in class logic. But clearly, this new term is quite contrived and meaningless. Here again, we must mean ‘the class of all properties which do not exemplify themselves’ – and in that event, we are back in class logic. Thus, note well, while Russell was right in looking to see whether his paradox was a problem specific to class logic, or one also occurring in predicate logic, and he claimed to have established that it occurred in both fields, in truth (as we have just demonstrated) he did not succeed in doing that. In truth, the paradox was specific to class logic; and he would have been better off admitting the fact than trying to ignore it.

In response to certain criticisms by his peers, Russell eventually “agreed that the paradoxes were all due to vicious circles, and laid it down as a principle for the avoidance of such circles that ‘whatever involves all of a collection must not be one of the collection’.” Thus, Russell may be said to have conceded the principle I have also used, namely that a collection cannot include itself as one of the items collected, although in truth the way he put it suggests he conceived it as a convention designed to block incomprehensible vicious circles rather than a logical absolute (notice that he says ‘must not’ rather than ‘cannot’). He viewed the paradoxes of set theory as “essentially of the same kind as the old paradox of Epimenides (or the Liar).” This suggests that, at this stage, he saw his own paradox as due to self-reference, somehow. It does look at first sight as if there is some sort of self-reference in the proposition ‘the class of all classes that are not members of themselves is (or is not) a member of itself’, because the clause ‘member of itself’ is repeated (positively or negatively, in the singular or plural) in subject and predicate[37]. But it cannot be said that self-reference is exactly the problem.

A few years later, in a paper published in 1908, Russell came up with a more elaborate explanation of the Russell paradox based on his ‘theory of types’. Russell now argued that “no function can have among its values anything which presupposes the function, for if it had, we could not regard the objects ambiguously denoted by the function as definite until the function was definite, while conversely … the function cannot be definite until its values are definite”[38]. In other words, the question “the class of all classes that are not members of themselves, is it or is it not a member of itself” is inherently flawed, because the subject remains forever out of reach. We cannot take hold of it till we resolve whether or not it is a member of itself, and we cannot do the latter till we do the former; so, the conundrum is unresolvable, i.e. the question is unanswerable. Effectively, the subject is a term cognitively impossible to formulate, due to the double bind the issue of its definition involves.

Here, we should note that the purpose of Russell’s said explanation was effectively to invalidate the negative class ‘classes not members of themselves’, since this is the class giving rise to the double paradox he was trying to cure. The positive class ‘classes members of themselves’ clearly does not result in a double paradox: if we suppose it is not a member of itself, self-contradiction does ensue, but we can still say without self-contradiction that it is a member of itself. In fact, if Russell’s explanation were correct, the positive class ought to be as illicit as the negative one. For if we claim the impossibility of a class referring to something that is not yet settled, as Russell did with reference to the negative class, then we must admit this characteristic is also found in the positive class, and we must reject it too. Russell does not seem to have realized that, i.e. that his remedy did not technically differentiate the two classes and so could be applied to both. For this reason, his attempt to solve the Russell paradox with reference to circularity or infinity must be judged as a failure. In my own theory, on the other hand, it is the positive class (that of self-membership) which is invalid (and empty), since it is geometrically unthinkable, while the negative class (that of non-self-membership) remains quite legitimate (and instantiated), as indeed we would expect on the principle that all claims (including that of self-membership) ought to be deniable.

Anyway, Russell concluded, briefly put, that a function could not be a value of itself; and proposed that function and value be differentiated as two ‘types’ that could not be mixed together indiscriminately. But this theory is, I would say, too general, and it complicates matters considerably. As we have seen, we cannot refuse to admit that, for instance, ‘classes’ is a class; the most we can do is to deny that this implies that ‘classes’ is a member of itself. This is a denial of self-membership, not of self-predication or of self-reference. As regards the notion of ‘types’ and later that of ‘orders within types’, these should not be confused with the more traditional ideas of hierarchies and orders of classes, which we laid out earlier in the present essay. In truth, the resemblance between the Russell’s concepts and the latter concepts gives Russell’s theory a semblance of credibility; but this appearance is quite illusory – these are very different sets of concepts. Russell’s notion of ‘types’ is highly speculative and far from commonsense; while it might appear to solve the Russell paradox, it has ramifications that range far beyond it and incidentally invalidate traditional ideas that do not seem problematic[39]. In short, it is a rough-and-ready, makeshift measure, and not a very convincing one.

Every paradox we come across is, of course, a signal to us that we are going astray somehow. Accordingly, the Russell paradox may be said to have been a signal to Frege, Russell, and other modern logicians, that something was wrong in their outlook. They struggled hard to find the source of the problem, but apparently could not exactly pinpoint its location. All the intricacy and complexity of their symbolic and axiomatic approach to logic could not help them, but rather obscured the solution of the problem for them. This shows that before any attempt at symbolization and axiomatization it is essential for logicians to fully understand the subject at hand in ordinary language terms and by means of commonsense. To my knowledge, the solution of the problem proposed in the present essay is original, i.e. not to be found elsewhere. If that is true, then the theory of class logic developed by modern symbolic logicians, which is still the core of what is being taught in universities today, needs to be thoroughly reviewed and revised.

A bit of self-criticism. As regards the resolution of the Russell paradox that I proposed over two decades ago in my Future Logic, the following needs to be said here. While I stand, in the main, by my theory of the logic of classes there (in chapters 43-44), I must now distance myself somewhat from my attempted resolution of the Russell paradox there (in chapter 45).

I did, to my credit, in that past work express great skepticism with regard to the notion of self-membership; but I did not manage to totally rule it out. I did declare: “Intuitively, to me at least, the suggestion that something can be both container and contained is hard to swallow,” and I even postulated, in the way of a generalization from a number of cases examined, that “no class of anything, or class of classes of anything, is ever a member of itself,” with the possible exception of “things” or “things-classes” (although it might be said of these classes that they are not members of any classes, let alone themselves[40]); but still, I did not reject self-membership on principle, and use that rejection to explain and resolve the Russell paradox, as I do in the present essay.

This is evident, for instance, in my accepting the idea that “‘self-member classes’ is a member of itself.” The reason I did so was the thought that “whether self-membership is possible or not, is not the issue.” Superficially, this is of course true – the Russell paradox concerns the ‘class of all classes that are not members of themselves’, and not ‘the class of all classes that are members of themselves’. But in fact, as I have shown today, this is not true; acceptance of self-membership is the true cause of the Russell paradox, and non-self-membership is not in itself problematic.

Anyway, not having duly ruled out self-membership, I resorted to the only solution of the problem that looked promising to me at the time – namely, rejection of ‘permutation’ from “is (or is not) a member of itself” to “is (or is not) {a member of itself}” (notice the addition of curly brackets). That is to say, I proposed the logical interdiction of changing the relation of self-membership or non-self-membership into a predicable term. Now I see that this was wrong – it was an action taken too late in the process of thought leading up to the Russell paradox. It was a superficial attempt, treating a symptom instead of the disease. I did that, of course, because I thought this was “of all the processes used in developing these arguments, [the] only one of uncertain (unestablished) validity.” But in truth, it was not the only possible cause of the effect – there was a process before that, one of deeper significance, namely the transition from ‘is’ to ‘is a member of’. I did not at the time notice this earlier process, let alone realize its vulnerability; and for that reason, I did not attack it.

Clearly, I was on the right track, in that I sought for a place along the thought process at which to block development of the Russell paradox. But my error was to pick a place too late along that process. In fact, the right place is earlier on, as advocated in the present essay. The Russell paradox does not arise due to an illicit permutation, but due to the illicit transformation of a predicate into a class in cases where a claim of self-membership would ensue. And while the remedy proposed is even now in a sense ‘conventional’, the flaw it is designed to fix is quite real – it is that self-membership is in fact impossible and therefore can never be assumed true. My previous proposed solution to the problem only prevented the Russell paradox; it did not prevent self-membership, which is the real cause of the paradox. Thus, the solution I propose in the present essay is more profound and more accurate.



[1] For examples: the word “all” became an upside down capital A, the word “some” became a laterally inversed capital E (for existence, as in ‘there are’), the words “if–then” (implication) became an arrow pointing from antecedent to consequent, and so forth.

[2] For more on this topic, see my Future Logic, chapter 64: Critique of Modern Logic.

[3] Ideally, of course, symbols are useful to summarize large amounts of information. I would dearly love to develop terse symbolic formulas that summarize my findings in the logic of causation. So I am not entirely rejecting symbolism. What I am saying here is that it is not necessary (i.e. we can well do without it) and indeed can be a serious hindrance to logical thinking and logic theory.

[4] See my Logical and Spiritual Reflections, book 2, chapter 2, posted online at: www.thelogician.net/LOGICAL-and-SPIRITUAL-REFLECTIONS/Kant/Analytic-Synthetic-Dichotomy-B2.htm.

[5] We should of course in this context mention Kurt Gödel, who showed the incompleteness of axiomatic systems like that of David Hilbert.

[6] Singular propositions are often called particular, but this usage is inaccurate, since they refer to an indicated individual.

[7] One can remember these six labels by means of the phrase ARIEGO.

[8] What I have called ‘polarity’ is traditionally called ‘quality’, but the latter term is inaccurate and confusing and should be avoided.

[9] This approach allows us to momentarily ignore the issue of modality, and reflects common usage in many contexts. A fuller treatment of categorical propositions must of course deal with modality; I do that in my earlier work, Future Logic.

[10] The implying proposition being called the subalternant and the implied one the subaltern, and the two being called subalternatives.

[11] If A is true, then R is true, then G is false, then E is false; whence, the contrarieties shown on the diagram. If I is false, then R is false, then G is true, then O is true; whence, thus the subcontrarieties shown. Since R and G are incompatible (cannot both be true) and exhaustive (cannot both be false), it follows that A and O, and likewise E and I, whose instances overlap somewhat, must be contradictory, since, if they were both true or both false, R and G would in at least one case be accordingly both true or both false (this is proof by exposition).

[12] The Kneales propose a similar analysis of the problem in The Development of Logic (Oxford, London: Clarendon, 1962), chapter II, section 5. Further on (on p. 211), they say that Peter Abelard “should have the credit of being the first to worry about the traditional square of opposition, though he did not work out all the consequences of the change he advocated.”

[13] We could say that nothing in the world is conceivably P, without affecting the truth of “Some S are not P” or “No S is P.” Clearly, in the special case where “nothing is P,” the latter propositions are true for any and every value of S.

[14] Of course, we could introduce modified symbols for the new A and E, such as A’ and E’, but I prefer to stress their underlying meanings, viz. not-O and not-I. In my view, it is dishonest and misleading to redefine the symbols A and E themselves as meaning only not-O and not-I. This is like a hostile takeover, permanently blocking further reflection and debate.

[15] A third possible approach is, of course, to draw a rectangle with A and E in the top two corners, and not-E and not-A (instead of I and O) in the bottom two corners. In that case, it is the lower square that would suffer changes, with not-E and not-A as unconnected to each other and to R and G respectively. This possibility is however not very interesting, as the forms not-E and not-A are disjunctive. That is, not-E = not-(O and not-I) = not-O and/or I; and not-A = not-(I and not-O) = not-I and/or O. Note that this position is historically found in Peter Abelard, who insisted on distinguishing between “Not all S are P” (not-A) and “Some S are not P” (O), and who apparently denied that “No S is P” (E) implies anything to be S let alone P (even while regarding “All S are P” (A) as implying that something is S); see Kneales, p. 210.

[16] For example, we might say (instead of “unicorns are horses with a horn”) “the imaginary entities called unicorns look like horses with a horn on their forehead” or (instead of “some unicorns are white, some black”) “some of the unicorn illustrations I have seen involve a white horse, but some involve a black one”. Note that both the initial propositions (given in brackets) have empty terms, even though one is general and the other is particular. Clearly, after such corrective rephrasing the two propositions do have existential import, although they do so with reference to imaginary (mental) entities rather than to real (physical) ones. Consequently, while the initial propositions cannot be said to be true, the more precise ones replacing them can be said to be true, and we can apply Aristotelian logic to them without qualms. Note also in passing that even a seemingly eternally imaginary entity may one day become real – for example, we might by artificial selection or by some genetic manipulation one day produce real unicorns.

[17] For Buckner’s account of the history, see: www.logicmuseum.com/cantor/Eximport.htm. Notice his pretentious characterization of “the traditional ‘syllogistic’” as “a historical curiosity.” Brentano’s position is to be found in his Psychologie vom empirischen Standpunkt, II, ch. 7. The Kneales do mention the latter reference in passing, in a footnote on p. 411.

[18] New York: Free Press, 2009. Pp. 13-14.

[20] Not-finding is the non-occurrence of the positive act of finding. Objectively, note well, not-finding is itself a negative phenomenon, and not a positive one. But subjectively, something positive may occur within us – perhaps a sense of disappointment or continued relief. See more on this topic in my Ruminations, chapter 9.

[21] There are people who say that the law of non-contradiction is logically necessary, but the law of the excluded middle is not. Clearly, this claim can be refuted in the same way. If they claim the three alternatives “Either X or not-X or ‘neither X nor not-X’” – we can again split the disjunction into two, with on one side “X” and on the other side “not-X or ‘neither X nor not-X’” – and then proceed as we did for the tetralemma. The same can be done if anyone accepts the law of the excluded middle but rejects the law of non-contradiction. All such attempts are fallacious nonsense.

[22] See there chapter 32.2. (See also Ruminations 5.1.)

[23] Some logicians have tried to deal with the liar paradox by denying that true and false are contradictory terms, i.e. that not-true = false and not-false = true. Such a claim is utter nonsense; the attempt to shunt aside the laws of non-contradiction of the excluded middle so as to resolve a paradox is self-contradiction in action.

[24] That ‘this proposition is true’ is implicitly (if only potentially) as paradoxical as ‘this proposition is false’ is, so far as I know, a new discovery. Note well how both paradoxes occur through quite ordinary eductions: viz. if ‘P is Q’ is affirmed, then P is Q; and if ‘P is Q’ is denied, then ‘P is not Q’ is affirmed, then P is not Q (where P stands for ‘this proposition’, and Q for ‘false’ or ‘true’ as the case may be).

[25] Another objection (which was actually put to me by a reader) would be propositions like “this statement has five words” and “this statement has six words” – even though they contain the demonstrative “this,” the former looks true and the latter false! Here, we might in reply point out that though the propositions “this statement has five words” and “this statement does not have five words,” seem to mean opposite things, they cannot be contradictories, since both appear true. Also compare: “this statement has five words” and “this statement does have five words” – the former is true while the latter is false, though both mean essentially the same. Clearly, the behavior of these propositions is far from normal, due to their unusual dependence on the wording used in them. On one level, we get the message of the proposition and count the number of words in it, and then check whether this number corresponds to the given number: if yes, the proposition is judged ‘true’, and if no, it is judged ‘false’. But at the same time, we have to be keep track of the changing reference of the demonstrative “this,” which complicates matters as already explained, and additionally in this particular context we must beware of the impact of wording. The Kneales give “What I am now saying is a sentence in English” as an example of “harmless self-reference” (p. 228).

[26] I found this example in Robert Maggiori’s La philosophie au jour le jour (Paris: Flammarion, 1994); the author does not say whether it is his own invention or someone else’s (p. 438).

[27] We need not go into the details of these distinctions here, for they are well known. There are also many fine distinctions between different sorts of terms that may appear in propositions as subjects or predicates; but let us keep the matter simple.

[28] ‘Predication’ refers to the copula and the predicate together as if they were an action of the speaker (or the statement made) on the subject.

[29] The following account of class logic is based on my presentation in Future Logic, chapters 43-45. The word ‘class’ comes from the Latin classis, which refers to a “group called to military service” (Merriam-Webster). I do not know whether the Ancients used that word in its logical sense, or some such word, in their discourse, but they certainly thought in class logic mode. Examples of class thinking are Aristotle’s distinction between species and genera and Porphyry’s tree.

[30] Note that saying or writing the word men without inverted commas refers to a predicate. When we wish to refer to the corresponding class, we say the class of men, or the class men; if we are writing, we may write the same with or without inverted commas, or simply ‘men’ in inverted commas. When dealing with classes of classes, we say the class of classes of men, or the class of men-classes, or the class men-classes, and we may write the same with or without inverted commas, or simply ‘classes of men’ or ‘men-classes’ in inverted commas.

[31] Note that, whereas positive terms are easy enough to translate into class logic language, negative terms present a real difficulty. For example, whereas the term men refers only to non-classes, its strict antithesis, the term non-men in its broadest sense, includes both non-classes (i.e. concrete things other than men) and classes (i.e. more abstract things). Again, whereas the term finite classes refers only to classes, its strict antithesis, the term non-finite-classes in its broadest sense, includes both open-ended classes (abstracts) and non-classes (concretes). Thus, we must, for purposes of consistency, admit that some terms do cover both non-classes and classes (including classes of classes). Practically, this means we have to make use of disjunctives which reveal the implicit alternatives. This of course complicates class logic considerably.

[32] Positive classes are defined by some positive property and negative classes are defined by a negative one. For examples, ‘men’ is defined with reference to rational animals (positive), whereas ‘bachelors’ is defined with reference to not yet married men (negative).

[33] This is a pictorial ‘representation’, an analogical image not to be taken literally.

[34] To give a concrete image: a bag of marbles (whether alone or, even worse, with the marbles in it) cannot be put inside itself, even if the bag as a whole, together with all its contents, can be rolled around like a marble and so be called a marble.

[35] I am here referring principally to the account by William and Martha Kneale in The Development of Logic (Oxford, London: Clarendon, 1962), ch. XI.1-2.

[36] Kneale and Kneale, p. 654. Italics theirs.

[37] Note that if self-reference were the crux of the problem, then the proposition ‘the class of all classes that are members of themselves is (or is not) a member of itself’ would be equally problematic, even though it apparently does not result in a similar paradox.

[38] Quoted by the Kneales, p. 658.

[39] See for a start the Kneales’ critique of the ‘theory of types’ in ch. XI.2.

[40] Note that in this context I come up with the idea that the definition of membership might occasionally fail. But I did not at the time pursue that idea further, because I did not then analyze what such failure would formally imply.