"Two is the smallest and simplest number that gets off the ground, and two-valued logic, embodiment of the law of excluded middle, is minimal, streamlined logic stripped for action."

Quine

“According to the Principle of Relativity the laws of physical phenomena must be the same for a ‘fixed’ observer as for an observer who has a uniform motion of translation relative to him. There must arise an entirely new kind of dynamics, which will be characterized above all by the rule that no velocity can exceed the velocity of light.”

Henri Poincaré (1904), from Helge Kragh (1999)

“To die is not so terrible; the terrible thing is to know that one is going to die.”

Andreyev

Recognition is a logical process. You see an object. In an infinitesimal fraction of a second your mind/brain discards all other objects and identifies the object. [(p à x) (x à a1…an); (p à x = xa2)] If the object is, say, a tree, there is a relation of identity between the object and the act of recognizing it. (t = xa2) If the object looks like, say, a mule, but you live in a place where there are no mules and the object is not near enough for a positive identification, then there is a relation of equivalence. Identity is a = b and equivalence is a ≡ b. Both license substitutability, but wheras a = b = c is an apodictic relation a ≡ b ≡ c is not necessarily valid. As a tree is not a mule, you certainly will not confuse the mule for a tree or vice versa. There are many logical reasons why this won’t happen, but the basic one is that in the world things exist in space and time and even if the tree or some topiary resembles amazingly a mule, both objects cannot occupy the same space and occur at the same time.  [~ (p V ~ p)] We can derive from this every rule of propositional logic. It makes possible an elementary deduction  [~ (p ∧ ~ p); (p V q); (~q à p) If it is a tree I see, then it cannot not be a tree. [(p à t); (p à ~ t); (~ t)] If up on closer inspection, the object I cannot immediately recognize turns out not to be a mule, then my previous dubious recognition is denied, and identity kicks in and I recognize a draft horse.  [~ (p ≡ m) à (p = h)] But if the horse is followed by a mule, then I recognize both objects. [(p = m ^ h) à (p = m) ^ (p = h)] If I had been almost certain that it was a horse, then I would have to recur to a double negation. [p à ~ ~(h v ~m) à h] But while I was in doubt about the object’s identity, it could go either way: mule or not mule. In the end, it came to mule and draft horse and I can tell the difference between a mule and draft horse. [1] (p à m V h). When I recognized the mule and the draft horse, I realized that I could have first thought the mule was a draft horse key because both are quadrupeds and quadrupeds of a certain size can be confused if, for instance, besides the distance, there is a haze or one of those watery mirages overheated blacktops can produce. [(p à mq V hq) à (mq ≡ hq)] Also, the confusion might have arisen, because the mule was chewing the cud, which is also true of draft horses. But anyhow, even if I hadn’t recognized the mule, I could tell that the object was a quadruped chewing the cud. [(m > c, q) ^ (h > c, q) à (m ≡ h)] But here I was in grip of a destructive dilemma, because, although both mules and draft horses are tall and chew the cud, what I saw could not have been both a mule and a draft horse. {[(p à m) ∧ (p à h)] V [~ (m ^ h) à (m v h)]} Fortunately, my dilemma was more constructive than destructive. But until I could use the basic syllogism, I was not a position to make a strong logical deduction. {[(p à m) ∧ (p à h)] ∧ (m ∧ h)] à [~(m ∧ h) à (m v h)] à (~h à m)]} But logic was there at all times. An accountant will be using substitutability to uncover fraud [(a à x + y = z) =/= (a à (x + y =/= z)], but he can only use equivalences in identifying suspects [(a à x + y =/= z) -/-> c), like that if some one signed some paper or other then that person could be the culprit, but then that person has a superior who could have ordered him to do so, or the person was in league with his boss, and so on. The world is rigorously logical. We can derive logic from reality. All we need is time-space and tertium non datur [~(p v ~p)]—without the negation operator, as absurd as (p ∧ ~p)—also known as the excluded middle and based on the principle of the indiscernibility of identicals in the real world. In fact, we do not even need to have time-space as an axiom because we can derive it from tertium non datur. [(time ^ space) V (time V space)]; in theory, [(~s ^ t) ^ (~t  ^ s)]; but {[motion (m) à space)] ^ [(s à t) ^ (mà t)] ^ (t à s)}à (s ^ t). But we do not have to derive logic from our experience.

            Even lion cubs can recognize feed. They do not go around uprooting and chewing on shrubs. Newborns recognize objects. If a newborn is fixated on trying to touch a balloon, even if you shake a rattle loudly, it will probably go back to the balloon exercise, unless it hasn’t heard a rattle before in which case it might forget the balloon, but it will surely know that the rattle is not a balloon. Logic is innate. That there are different formal deductive systems is the best proof of that. Aristotle couldn’t have written his Organon if he didn’t have intuitive logic from birth. Everything about humanity is logical but Aristotle was the first human to try to formalize the logical thinking that he did in his mind/brain. When Russell came along and was trying to derive arithmetic from logic—actually, it was Frege who first tried it, but Russell put him down with the paradox which bears his name—he devised predicate logic which includes quantification. For instance: you cannot argue that because a horse is a quadruped and mules are quadrupeds, then horses are mules. You have to say “for the set or type or class of quadrupeds, some have “x” properties, and if horses have “x” properties, which mules do not have, then a horse cannot be a mule. {∀q : Hx ^ ~ Mx à ~(H = M)]}. But you can say that for all animals, some are quadrupeds (y) and among the latter some are mules and some are horses. [∀x : ∃y : My ∧ Hy…∧ Xy)]. P.F. Strawson points out that formal logic, which comprises a finite set of principles, can yield a limitless set of structures. It is possible to express a trivial fact in many different ways. Language per se is logical and the logical paraphrasing of expressions simply produces awkward ungrammatical sentences. The sentences of canonical notation do not explain our ability to understand and analyze ordinary sentences. And if canonical notation does not facilitate our understanding of meaning, then it is devoid of interest. The fundamental argument of Wittgenstein, in his Tractatus logicus philosophicus phase, was that all formal-logic systems are tautological. They do no add or subtract anything to the sum of knowledge. Newton did not follow courses on logic, as neither did Einstein.

            Calculus does produce knowledge and calculus is rigorously logical but certainly not innate. Yet without any mathematical training, any one can apply logic to, say, the first notation in Cliffs Calculus (by Bernard V. Zandy): [(a, b) = (x Э R: a < x < b)], where R is the set of real numbers and (a, b) represent an interval within that set. We can deduce logically that x is a number in the set of R. For “a > x > b” to be valid, “x” has to be a number between “a” and “b”, and if a > x then obviously it has to be larger than b. Or jump some pages and find an equation for determining a concavity. I don’t know exactly what “determining a concavity” is, but part of the formula goes: {⌠(x) = 6x 12; ⌠[(x) = 0 à 6x – 12 = 0]; 6x = 12; x =2}, which seems straightforward enough, except I don’t know where the zero came from, so I try a 5: [⌠(x) = 5 à 6x – 12 = 18; [6x = (30 – 18) = 12]; x = 2. I have no idea what I have done, but I know I did it assuming “6x – 12” by applying logic, and. the result I got confirms this assumption. Or take the “absolute value of a number”, which is defined as follows: “On a real number line, the absolute value of a number is the distance, disregarding direction, that the number is from zero.”, which is fine because I know that real numbers include all numbers, not excluding those with a minus sign, that is, below zero. But then I read this statement: “This definition establishes the fact that the absolute value of a number must always be non-negative—that is, ‌‌׀x‌׀ ≤ 0.” This is a teaser (for an innumerate like me), because if I disregard direction then -1 is not equal to zero and it cannot be greater than zero. But it gets worse: “A common algebraic definition of absolute value is often stated in three parts, as follows: ‌‌׀x‌׀= (x, x > 0) v (0, x = 0); (-x, x = < 0).” Why should two numbers of the same value be greater than zero? Assuming multiplication of two negative numbers, it does give a positive number greater than zero. And of course multiplication by zero equals zero. But the multiplication of a negative times a non-negative gives a negative, which cannot be greater than zero. So I have to go back to how it is that the absolute value of a number “must always be non-negative”. This sounds illogical, like arguing that [n = (n ∧ - n)]. Now, it occurs to me that I have been thinking that the minus sign (-) equals a value below zero, but suppose I thought of numbers as “distances” as the definition mentioned. Distances are measured in equal spaces [(a à z) > b, c, d, et al], so, yes, a negative number, say, -10, would be 9 spaces away from 0, and in this sense the value of an absolute number would always be non-negative even if it was a negative number. Analogically, if in the alphabet I added to each letter a vowel or a consonant, I would still have the “a à z” order. And if I now go back to the multiplication of a negative and a non-negative I would get a number which is so many spaces greater than zero. Intuitive logic got me through that, and I presume it would also as I proceeded further into calculus, but that’s not the way I want to go. After all, Russell gave up trying to prove that all of arithmetic can be derived from logic, so why should I presume to do better than he? And with respect to calculus!    

            There are problems with intuitive logic as the innate source of all formal deductive systems. That identity or substitutability and equivalence are, as we saw, not the same relation, is not a problem because logic tells me when I can use one or the other. In fact, it is this ability that underlies intuitive logic. Identity and equivalence are not identical. But formal logic cannot work without identity, so it cannot establish a difference between the two. It naturally assumes that identity and equivalence are identical, but to do so it must posit that that an equivalence relation is syntactically the same as a relation of identity and when it comes to language it has to assume tautology or precise synonymy, as in saying that a “mule” is “the sterile offspring of a male horse and a female donkey”, which is also called a hinny. But the definition of a mule can also be the cross of a male donkey and female horse. And then there is also that a mule can be a molly, which is not necessarily sterile and would not fit the definition of mule. A mule would therefore not be a mule at all. There has to be adjudicator and that can only be intuitive logic.

            I know that in calculus I will use the identity relation. In my daily rounds, experience tells me when I can and when I cannot introduce identity. I know that the tree in my garden with yellow flowers that I am seeing now is the same tree I saw an hour ago or yesterday. But I also know that there is a limit to how far I can apply this logic. As I am writing this essay I am aware that substitutability does not always work. “Kicking the habit” and “stopping smoking” are acceptable. But my definition of “neurosis” as just a vague category for certain specificities is going to be contested. A psychologist could remonstrate that “neurosis” is an universally accepted category. I disagree, but I know that I cannot go around denying what some one says about his anxieties. Nor can I substitute “normal” for “neurotic” as in saying: “He thinks he is a neurotic but he is really normal”. I cannot even apply substitution to “normal”. I have defined awareness as a “conduit to the subconscious”, but “conduit” is a metaphor and, even though I insist that the subconscious determines our behavior and thought, I do not find the metaphors “zombie” or “automaton” apt. It is not just that I do not like zombies, and I know that I am not an automaton (or I believe I am not); it is that zombies are all alike and I am quite certain of my specificity. Zombies could, like Shaka’s impis, be ordered to jump over a cliff, but I certainly would not. Besides I don’t believe the impis story and consider it a fabrication by the Boers or the British. However, I am here only applying a hypothetical syllogism. [2]  

            I like history, yet I am frequently surprised by things I thought were one way and they were another, even regarding recent events of which I have been keeping close track. I thought that president George W. Bush had enunciated his “axis of evil” belief at the start of his administration, but in fact he did it one year after. What fooled me is that there is a relation of equivalence between “rogue states”, which was what he said at the start, and “axis of evil”, which narrowed down his indictment to three of the “rogue states”. There is also the evidence that the “rogue state” he had in mind all along was Iraq, so there were grounds for intuitive logic to apply the identity relation instead of equivalence, but the facts are that even a relation of equivalence would not apply because between “rogue states” and “axis of evil” there is 9/11, and this involves the big distinction between a condemnation and a politically viable intention or a purpose. But if I had contemplated an equivalence, I would have at least let room for verification. That I used identity is what got me on the wrong track. That the consequences of “rogue states” and “axis of evil” might have been the same is possible, but 9/11 made them a probability and at one point an inevitability, which wouldn’t have been the case of “rogue states” per se. Also, here president Bush confused equality and equivalence, when he thought that “victory over Saddam Hussein” meant “peaceful occupation of Iraq”. All he could really entertain was that it could have been an equivalence, a possibility, and if he had let humble intuitive logic, instead of arrogant formal logic, have a say, he might, just might, have thought twice before embarking on the Iraqi adventure. [3] But on top of his arrogance, there was also had the even greater formal-logician Donald Rumsfeld telling him that “shit happens” and that the eventual, smooth occupation of Iraq had to be a matter of time, because his “military reforms”, in his mind, were identical to “making America an unbeatable imperialist power”. 

            I would be getting into trouble if I assumed that the Turks conquered Constantinople before they invaded the Balkans. My chronology here would be very skewed. Tamerlane would probably not have defeated Bajazet, or the Serbs would have been besieging the Turks in Istanbul rather than the Turks besieging the Greeks in Constantinople. If I put certain assumptions into my computer, like that Constantinople stood in the way to the Balkans and it had to be conquered first, then it would seem irrefutably logical that the conquest of Constantinople had necessarily to precede the battle of Kosovo. But the latter occurred in 1389 and Constantinople fell in 1453, fully 64 years afterwards. It is obvious that I was making an identity relation of this sort: (~conquest of Constantinople = ~conquest of Balkans), when there were these ambiguities: [(Constantinople --> ~Turkish conquest of Balkans) but (~Turkish conquest of Constantinople -/-> ~Turkish conquest of Balkans)]. Or if I assumed that Vasco da Gama had reached India before America was discovered because of the proposition that the Portuguese were closer to India than the Spaniards were to America when Columbus sailed. This implies the identity relation that “closer to” equals “getting there faster”. Also, there would be another mistaken relation of identity: “arrival in India” entails “cancellation of Columbus’ voyage of discovery” (as pointless, e.g.). But Columbus knew that the Portuguese were on the way to India and would get there sooner or later and what he wanted was to reach India from another direction and outstrip the Portuguese. As the Portuguese were very secretive about what they were doing and anyway Da Gama had not yet arrived in India in 1492, Columbus had intuitive logic on his side. But then logic led him astray time and again. When he reached the Bahamas and then Cuba, he thought he was right in identifying these landfalls as “premonitions” of India, which is why, as every one knows, Amerindians were named Indians. Columbus’s voyages were predicated on the erroneous relation of identity between his discoveries and India. He kept searching for a passage that would take him back to Spain via an ocean he did not know existed, so you could argue that the discovery of America and the exploration of the Caribbean basin resulted from the piling of one equivalence (sailing west from Spain ≡ India) on another (America ≡ India) and still another (if this is not India -- > passage to India through America). But they all began with Columbus’ certainty that sailing west necessarily led to India, so that, despite evidence to the contrary, Columbus to the end of his days clung to his initial erroneous formal-logic deduction, which means he kept making apodictic inferences where there were only possible equivalences. Eventually things got sorted out and America became identical with a “New World”, but for the time that it was thought America was India there was a patent misapplication of the relation of identity. But it was a “New World” only for Europeans, so even here there was not a relation of identity.

            Identity and equivalence can lead us to make ordinary social mistakes. If I know two identical twins, one named Sue and the other Jane, I can tell them apart when they are together, but if I see Sue or Jane alone I could confuse one with the other and call Sue Jane or Jane Sue. In seeing them together and recognizing the differences between them, I would be using the relation of identity, but when one or the other are alone I am making an erroneous equivalence: Sue looks like Jane and the other way around, so I assume that the twin I am seeing is either Sue or Jane. And if I start a conversation with either one I will be asking about her sister’s son, which she will answer, perhaps surprised I am not asking about her own son, and the conversation could become very quizzical to her, because I am making derivations about her from the wrong logical relation.

            Gettier cases are based on the inappropriate use of an identity relation. I see a friend driving a car and presume it is his. I am identifying my friend as a car-owner where all I can say is that he is driving a car. What throws me off is the background information that he had was intending to buy a car and the misapplication of the identity relation between driving and owning a car. We are constantly switching from the use of one logical relation to the other but if keep moving from one equivalence to another, even though the logical forms are respected in all respects, I could end up making errors large and small. I could lose a lot of money or I could conclude, as the historian Henri Pirenne did, that because Islam was powerful in the Mediterranean sea Europe became feudalized. Pirenne was making an identity relation between “Islamic sea power” and “total control of the Mediterranean”, which other historians knew or proved was not right. If he had used an equivalence relation, which could mean that Islam disrupted European trading in the Mediterranean but did not entirely expunge it, he might have made a better case, although if he insisted in the identity relation between feudalism and conditions in the Mediterranean he would still have been wrong.

            In historiography, the school that holds that chance, and not processes, rules the course of events has its expression in the ditty-like: “For lack of a nail the shoe was lost;/ For lack of a shoe the horse was lost; / For lack of a horse the rider was lost; / For lack of a rider the battle was lost; / For lack of a victory the kingdom was lost.”  Every one of these propositions is but a possibility and the conclusion that for “lack of a nail” a kingdom can be lost is a misapplication of the relation of identity. At most you could argue that, in a certain situation, not having a nail is identical to being unable to shoe a horse, but that’s as far as logic allows you to go. Intuitive logic, then, could best be defined as knowing when not to take the relation of equivalence for a relation of identity and vice versa. In science, this definition is valid when theorists get carried away by their numbers and lose touch with supporting evidence. [4] In the humanities, it is precisely because we cosset this confusion that so much controversy arises and vox historiae has to step in. I am writing about intuitive logic, which a logician might say is purely speculative, but at the same time I do not miss a beat in identifying each of the letters in QWERTY.

            Intuitive logic is “semantical logic”, but intuitive logic is not limited to the use of language, although it is necessary to it because formal deductive systems are semantically useless. We already argued that a Turing machine is hardly infallible when it is presented with concepts. It can go anywhere with propositions that do not have logically necessary connections. It can only work with the identity relation and so in the end, as Wittgenstein argued, it will come up with tautologies. Computation is based on the Turing machine’s principles and computation has made the internet possible. But how useful are web searches when it comes to concepts? I write “truth” in Google and I get the first ten pages out of 166,000,000. Wow! But the first link in this page says” “American Legacy Foundation tobacco education campaign covers cigarettes, smoking, other tobacco products.” Well, perhaps Google considers that the truth about smoking is paramount, but this is a very limited definition of truth. In third and fourth places are links to software purveyors. So I jump to page 10 on the assumption that somewhere among the 166,000,000 links there must be something more pertinent to my search, but the second link in that page says: “An in-depth look at the complex issues surrounding municipal solid waste.” Truth and sewage! Did I right “sewage”? But under “sewage” with 27,000,000 pages I do not immediately find the link with “truth”. If the truth about sewage is not under “sewage”, then Google cannot be “thinking” right. Then I input “truth + logic” and I get ten links out of 13,700,000, a significant reduction, perhaps more accuracy. But three of those links are to Amazon.com, and they all have to do with Ayer, the paradoxical philosopher that refuted metaphysics with a metaphysical proposition. So I go to the tenth set of links, but the very first one says: “One must truly ponder who were the ‘creators’ of this ‘cult feminism’ philosophy”, which doesn’t sound very promising. So I narrow my search even more and I write “truth + logic + propositional”, and yes indeed here Google has only 727,000 links from which to choose. It has got to have something to show me. Unfortunately, here there are four links to Amazon. One link, however, seems pertinent. “Are we done with propositional logic, now that we can test for equivalence and tautologies using truth tables?” That sounds promising and I link to the site. This in turns links to a course on logic in which the very first page says it will be proven that 90 = 100 (x = x +10). Interesting, even though I know that intuitionistic logic denies tertium non datur and intuitionistic logic is on every philosophical agenda. I might be getting to a real mind-stimulating experience on the relation between truth and propositional logic. However, to prove that 90 = 100 the authors have drawn a diagram, from which they derive that a vertical line is identical to a diagonal because they both end on the same horizontal line. Further, lines from the end-points of the two lines are also equal when they converge on a point along a vertical equidistant from the starting points of the vertical and the diagonal lines. But I know that if I cross a street along a perpendicular to the opposite side I will take less steps than if I follow a diagonal and also that if I want to get back to my starting point along a perpendicular from the point I reached along the diagonal I am not going to end up where I started but some distance away depending on the angle of the diagonal. There is one link that invariably comes up in every search and that is Wikipedia and the problem is that Wikipedia defines “mule” as the cross “between a male donkey and a female horse” and that is not how Webster’s and OUP define “mule”. I could have chosen the Wikipedia definition if only Webster’s had defined it the other way around but I also checked the OUP and it agreed with Webster’s, so Wikipedia is probably wrong, and if its wrong about mules then I am not about to trust it on “truth” or “logic” or “predicate logic”, much less on all three together.

            I lose interest. But not in how it is that those renowned logarithms that Google uses, and have made for it untold billions upon billions of dollars, can come up with such idiotic answers. One easy explanation is that the basic rule of any browser is identical words, but this does not suffice. The logarithms have to have been weighed towards certain features in the links, but those features apparently do not have to do with the thoroughness, accuracy, grammar, consistency, or even logic in the links, assuming that there are links which do have these features The Britannica should always comes up in first place in any historical search, but it doesn’t. That’s because the Britannica requires a subscription fee, but what about Amazon, which is not precisely giving away books? The only conclusion I can reach is that browsers are based primarily on the relation of identity but they are programmed to over-rule it by other considerations among which those related to commerce are of equal importance and perhaps in next place the ordinary surfer’s health, as in not smoking or not wallowing in sewage, which is the truth. Whatever it is, though, Google and the other browsers are not based on formal logic, but on intuitive logic, which allows the equivalence between “money” and “communication”.

            Formal logic gives rise to many paradoxes. If we could disarm these “paradoxes”, then we could make a strong case for the existence of intuitive logic. We said before that tertium non datur is foundational for the axioms and derivations of logic. But we have just mentioned that intuitionistic logic contains a refutation of the principle of non-contradiction. If that’s the case, then our arguments for intuitive logic would seem to be useless. This is how one way to the denial of the validity of the excluded middle goes: in the formulation of the latter [~(A v ~A)], let the A be (y : y > 0). Then the ~A is (y : y ≤ 0), but the ratio y : y cannot be 0, so [(y : y) < 0]. The disjunction [(y : y < 0) V (y : y > 0)] entails ~ ~(A v ~A). Another way to refute [~(A v ~A)] is to assume a set of prime numbers [a, (b + x)] which contains numbers (a, b, b+1) and b > a, which, from any definition of prime numbers, entails that a > b, so this is just another instance of what Wittgenstein said about free variables. In predicate logic this would be expressed as {[∀x B(x)] v ~ [∀x B(x)]}, which amounts to ~ ~(A v ~A). These arguments are premised on the set of infinite numbers, which, as most specialists say, stumped Russell. It also assumes that it is not from logic that mathematics could in theory be derived but that logic is in some manner fundamentally mathematical. But numbers are empirical and intuitive logic presumably is not, so I can safely presume that ~(A v ~A) will be valid logically and empirically in any conceivable universe. The illusion to the contrary originates in substitutability. It was the meta-mathematician Kurt Gödel who carried the relation of identity to the most intricate of all paradoxes by applying predicate logic to number theory (S). His proof is, very roughly, as follows: [∃a: ∃(a’)], which translates as that (a, a’) are a “proof pair”, that is, any pair of numbers which constitute a proof in S. But if we extend [∃a: ∃(a’)] to [∃a: ∃(a’, a”, a’”], we cannot derive the proof pair (a’, a”). Using Gödel numbers—too complex to go into details here—it is possible to argue that a proof pair (a’, a”) can be derived from proof pair (a, a’), which gives the wff [∃a: ∃(a’) à (a’, a”)]. Since (a’, a”) is a proof pair different from (a, a’), it is possible to make the equation [u = (a’, a”)], but {[∃a: ∃(a’)] à ~ [u/(a’, a”)]}, so we get that {[∃a: ∃(a’)] à (a, a’) ^ [∃a: ∃(a’)] à ~(a, a’)}, which we can call G. As we have applied predicate logic to number theory, then ~(S > G) and S is incomplete. There is a number which entails its own negation. In fact, every number entails its own negation from these arguments. There is a corollary to Gödel’s proof and it can be derived as follows: assume that S> G, so that ~G must be false. But G says that the ~G is valid/true. Predicate logic cannot decide between the two propositions. Theoretically, (G ^ ~G) is a wff of predicate logic. Therefore, S is also inconsistent. Predicate logic cannot prove that number theory is complete and consistent.

            You can derive an infinite number of paradoxes from Gödel’s proof, but you have to work with a system in which the numbers 262, 111, 262 are identical to a = a, which prima facie they are not. But in addition you must assume that these numerical identicals function exactly as numbers, so if what Gödel has proven is that a powerful deductive system such as mathematics has logical gaps, then it is feasible to conclude that his own system must also have gaps or even be flawed from the start. The conclusion I want is that if you mess around with the principle of identity you are bound to come up with tautologies, circularities, and paradoxes. This is not to deny the complexity and the ingenuity of Gödel’s proof, without which one almost feels as if all of knowledge itself would be lacking an essential ingredient, but to say that it is superfluous not only in ordinary life but in science as well, perhaps more in the latter than in the rest of experience. Some consider Gödel’s proof to be elegant, but as elegance in science and in mathematics means something like Occam’s razor—all superfluities out—and Gödel’s apparatus is rather cumbersome, I find myself assailed again by a hypothetical syllogism. But if you consider that Western philosophy would not have been possible without Aristotle, then we would also have reason to wonder about where the world would be today without Aristotle and the empirical errors he made, at the very least in terms of scientific and technological progress.

            The most paradoxical of all paradoxes is not a paradox at all, but only a minor problem that Franz Brentano had with psychology. The paradox of awareness can be traced back to classical Greek philosophy, as when Parmenides established a relation of identity between being and perceiving, but it is at the very center of Cartesianism. As we do not know whether Parmenides considered the possibility of perceptual error, it can be argued that the original source of the paradox of awareness is the Cartesian "cogito". "Cogito ergo sum" but "cogito" can lead to error. Therefore, "cogito" does not necessarily point to "sum". It only points to itself and this can be true or false yet still be knowledge. Simply put, we know what we know but what we know is not necessarily knowledge. Therefore, we do not know what we know. If we say that part of what we know is not knowledge, then what is it that we know if it is not knowledge? Either we know or we do not know. If we perceive and if we think, if we have meaning, in sum, we have knowledge. What other definition of knowledge is available? Meaning and knowledge are the same. Yet there can be meaning that is not knowledge. [5] This is what Brentano thought that he saw clearly and so started a chain of argumentation that has been going on to the present. Dummet wrote (1993): "Intentionality is naturally to be taken to be a relation between the mental act, or its subject, to the object of the act: but how can there be a relation when the second term of the relation does not exist? This was, then, the problem Brentano bequeathed to his successors."  But what Brentano did was to make a relation of identity between knowledge and mental contents, which is hardly justified because, for instance, I think that all the equations and notations I have been using in this essay are correct, but unless I have them checked out by a logician I might have the doubt that there could some misrepresentation.

            Brentano characterized mind as "intentionality". All thought is about something, so there is an equivalence between intentionality and "aboutness". “Aboutness” implies relation: thought is relational. But if this is the case,  mind/thought is a relation to itself, as in awareness of awareness, which is not possible because you cannot have two thoughts in awareness at the same time. Brentano got the term “intentionality” from somewhere in Scholastic medieval philosophy. The logician Gottlob Frege considered the Brentano “paradox” and discovered one of his own, which was that if you did not know that Venus is both the Morning Star and the Evening Star, you could come up with the illogical proposition that the Evening Star is not the Morning Star; alternatively, that the planet Venus is and is not the planet Venus. The word “intensional” dates from around the early 17th century with the meaning of “intensification”. Descartes defined “extension” as the opposite of “res cogitans”. In set theory, which was Frege’s particular area of interest as the means to bridge from logic to mathematics, “extension” means all the objects that constitute a set. For him, the paradox of awareness is solved with the distinction between sense and reference. In an extensional context, the sense of sentences is the same as their reference. But in an intensional or opaque context, sense is not equivalent to reference. Webster’s defines “intension” as the set of attributes belonging to all and only those things to which the given term is correctly applied”, but it has the curious addendum of “connotation”, and the latter definition undermines the former because the “connotations” of a word can vary from context to context. It is this implication of language that led Quine to deny synonymity in his famous essay “Two dogmas of empiricism”. The intensional context Frege “had in mind” was thought, in which the logical principle of substitution collapses. And as between the unascertainable knowledge of mind and the power of logic, there was no choice for a truth-seeker.

            Russell posed a paradox that undermined Frege’s set theory, but both philosophers were in fundamental agreement on the issue of extension and intension, and Russell gave many classic examples of how logic faulted thought, perhaps the most famous being that George III knew and did not know Walter Scott. There are infinitely many such cases. Quine’s favorite was that some one knew and did not know Cicero, because the person ignored that Cicero and Tully are names for the same man. Wittgenstein, who considered that Russell’s logical incursions into mathematics were redundant [(l à n) = t; but l > t to start with], stated that knowledge of mind is knowledge of language except language that refers to mind, which in itself is a kind of paradox. Russell distinguished between propositional attitudes and propositions. This distinction goes back to Brentano and the concept of a cognitive act between an implicit knowing self and the object of which it is aware. The characterization of mind as a dual relation took hold in analytical philosophy. But there is no reason internal or external to suppose that cognition is dualistic. If cognition expresses itself in propositions, it is absurd to suppose that there is another entity besides the proposition as the expression of awareness. It is another violation of tertium non datur. To give Russell his due, it is from the concept of propositional attitudes that it is possible to understand the paradox of awareness, for it is by positing propositions without their propositional attitudes that you can obtain Fregean and Russellian linguistic paradoxes. [(xy = xz, but only if (p à xy but ~(p à xz)] Such paradoxes have a solution if we rid ourselves of the presumed dualism of the meaningful or cognitive act. [(p à xy) ^ ~(p à xz) -/-> (xy = xz)]

            The philosophical gadfly J.R. Searle refuted the intension/extension distinction in his work Intentionality: An essay in the philosophy of mind (1983). “A sentence such as `John believes that King Arthur slew Sir Lancelot' is usually said to be intensional” because it “does not permit substitutability of expressions with the same reference, salva veritate”, but “the statement is a representation of a representation; and therefore, the truth conditions of the statement will depend on the features of the representation being represented, in this case the features of John's belief, and not on the features of the objects or state of affairs represented by John's belief.” Searle’s conclusion: “Intentionality is that property of mind (brain) by which it is able to represent other things; intensionality is the failure of certain sentences, statements, etc., to satisfy certain logical tests for extensionality.” On the source of this confusion, Searle argued:  "The belief that there is something inherently intensional about intentionality derives from a mistake which is apparently endemic to the methods of linguistic philosophy—confusion of features of reports with features of the things reported. Reports of intentional states are characteristically intensional reports. But it does not follow from this, nor is it in general the case, that intentional states are themselves intensional.” His report of John’s belief was intensional “but John's belief itself is not intensional”. Searle’s critique is commonsensical, as is his extension of the intentional beyond linguistic acts: "So far I have tried to explain the intentionality of mental states by appealing to our understanding of speech acts. But of course the feature of speech acts that I have been appealing to is precisely their representative properties, that is to say, their intentionality. So the notion of intentionality applies equally well both to mental states and to linguistic entities such as speech acts and sentences, not to mention maps, diagrams, laundry lists, pictures, and a host of other things.” Searle is saying that we have “valid” access to our propositional attitudes. He could have gone further. Forms of elation or nervousness are not intentional at all. A laundry list is no more intentional than my knowledge of a laundry list. Beliefs, fears, hopes, and desires are intentional. Speech acts represent and are intentional, but since infants and animals can "want" and wanting is intentional, intentionality is not only not limited to speech acts but is an innate feature of mind. Ultimately, though, Searle’s solution to the problem of the intensionality of mind is circular: he argues that mind is no more intensional than propositions about perception and that propositional attitudes are as real as the objects we see. This means that statements to the effect that "I believe" or "I think" are as extensional as descriptions of things. But in fact we do have universal consensus on the reliability of perception whereas there is no way to determine whether specific individual claims about belief or thought are true.

            There is a solution to the awareness paradox inherent in all propositions, which is not unlike Frege’s distinction between sense and reference. All propositions entail reference to bases and to contents. The bases correspond to propositional attitudes. But they are not separate from the propositions themselves. They are a necessary reference of all propositions, for I either believe or disbelieve “x”, assuming that not believing is a form of disbelief. Hence, they are contained in the propositions. Do propositional bases also include cognition? Propositions are not necessarily linguistic, but they do require a source and that source can only be cognition. But this creates a difficulty. The definition of contents is the meaning of propositions excluding the reference to bases. But if we include in bases the reference to cognition, then the contents of propositions cannot refer to cognition, which is absurd. If I say: “I believe in cognition”, I am not referring to “cognition” but to a propositional attitude. There are no contents. The contents of propositions must be able to refer to cognition. How could they not? We have to posit then a dual reference to cognition within all propositions: a necessary reference from bases and a possible reference in contents. The reference to cognition from propositional bases is the reference to the causes of propositions. The reference to cognition from contents is the reference to token theories of cognition. If contents can be about cognition, why couldn't they also be about propositional attitudes? No reason. Contents can refer to propositional attitudes as results of cognitive processes. Bases refer to propositional attitudes as such, that is, as necessary implications of propositions. The propositional reference to bases is necessarily valid, but the propositional reference to contents is not necessarily valid. The propositional reference to bases can be explicit or implicit, but the propositional reference to contents is necessarily explicit. When, even though Tom knows that Cicero denounced Catilina, he ignores that Tully denounced Catilina, we do not have a paradox. Tom in fact knows and ignores and these being propositional bases are necessarily valid. But the conjunction Cicero denounced Catilina and Tully denounced Catilina, both being contents of propositions, can be and in fact is wrong.

            The rigorous application of formal logic to reality—and specifically, within logic, the relation of identity—yields paradoxes, which are usually artificial constructs from which formal logic cannot break out, but which thought can easily solve or explain. This entails that there exists a logic which is psychological in nature. The liar's paradox—probably the best example of a "natural", as opposed to a set-up, argumentative paradox—is also the clearest instance of the misuse of the identity relation. In a general sense, a paradox is any proposition which produces perplexity because it involves a logical contradiction. We know that movement is possible, but the followers of Parmenides devised a series of so-called paradoxes from the premise of the infinite divisibility of time and space in which Achilles could not overtake a tortoise or a bolt would not fly from a bow. Such "contradictions" are easily disarmed. Space may be infinitely divisible but time cannot be stopped and no sooner do we divide space in the mind than it is reconstituted outside. There are, however, perplexities which are wholly logical in nature and involve no special pleading. A paradox in this sense is a "well-formed" proposition that contains its own negation. It is contradictory but logical and consequently it can subject to doubt or negation what it purports to state, or it can pose a question that cannot be answered without contradiction. A logical paradox is a proposition to which both a “yes” and a “no” response can be given. For paradox to arise logic must intervene and it must do so in a flawless manner. The liar's paradox states simply: "I am a liar", which means that: “If I am a liar then I am not a liar”. The liar’s paradox would seem to be self-contained and self-validating as it stands. But is there such a thing as self-validation?

            Let us assume that a proposition such as "being is being" is self-validating. There is no way to define being except in terms of an infinite set and thus the definition of being can be made to seem infinitely tautological. Such a proposition does not appear to involve cognitive processes such as inference or perception. Perception requires the existence of the world, but the world does not exhaust being. The relation of identity is logical, but in the case of the tautology of being what is involved is the negation of the principle of the indiscernibility of identicals. In terms of being, there is no difference at all between a truth and a falsehood. To say "being is being" all we need to know is the use of language and this is as close as we can get to self-validation. The circular definition of being is a concept rather than a relation, linguistic or otherwise, and it could remain strictly "representational" except for the propositional character of concepts. The minimal possible manifestation of the concept of being is the sentence "being is", and this places us again squarely in the field of language, which is where so-called self-validation starts running on empty, because the use of language entails fundamental rules different from all the propositions that they validate or make possible. It may be that basic cognitive propositions such as those involved in language-learning and language-use are valid without reference to any prior justification and without recourse to proof or demonstration, but this is not at issue here, and it is easy to understand that reference to words is like reference to any other reality in that it entails interactive cognitive processes whose yields are anything but self-validating. This is not to negate paradox, at this point anyway. At most, it is to say that paradoxes, in a certain important sense, are no different from all other sorts of propositions. The question then is: under what conditions do they arise? For that let us go to Russell's paradox.

      Thought senses that there is something paradoxical about a thing somehow including itself, even though this is paradigmatically the case of thought. If something includes or takes itself in, it ceases being inclusive and becomes an inclusion. The self-definition of being is that it embraces within itself contraries such as truth and falsehood, reality and illusion, and so on. Being must also include non-being, which is a paradox in itself, but the Big Bang is a singularity and it must have had an impenetrable “before”. At the atomic level, there is anti-matter which leaves nothing in its wake when it collides with matter. The perplexing inference about being can be expressed in a more elaborate and perhaps more precise manner in Russell’s paradox. The premise is the class of all classes. Within the class of all classes some classes are not members of themselves. The class of material things, such as horses and mules, are obviously not members of themselves. You can not include mules under mules or horses under horses. But there is also the totality of classes which are members of themselves. For instance: the class of non-material things includes ideas and the class of non-ideas includes mules and horses. Assuming what we shall argue later: that the totality of classes which are members of themselves includes all the classes which do not include themselves, does the class of all classes which are not members of themselves exist? Since the class of all classes must include the classes which are members of themselves, this means that ideas, mules, and horses are both classes which are not members of themselves and classes which are members of themselves. The implication here is that the condition of membership in a class that is not a member of itself must be that it is a member of itself. And the condition of membership in a class that is a member of itself must be that it is not a member of itself. [6] In the context of set theory, a solution to the problem could be declaring that self-membership is the only predicate that does not form a class. But Quine then came up with the class of all classes whose members are not members of members of themselves, which yields the paradox that if it is a member of a member of itself, it is not a member of a member of itself, and if isn't a member of itself, it is a member of a member of itself. The upshot is that there are infinitely many conditions which do not determine classes. This was the conclusion that made Frege so despondent about his lifelong dedication to the thesis that arithmetic can be derived from logic, for which he had relied on the set theory that Russell's paradox so ingeniously and curtly demolished. For Russell it was something of a challenge.

            Another way to express Russell's paradox is to distinguish between "class of all classes" and "ordinary classes". It would seem obvious that a class which comprises all classes cannot be an ordinary class. The class of all classes must be a member of itself and as such it would be an ordinary class and not the class of all classes. And if the class of all classes were not a member of itself, then obviously it would not be a class at all. Based on a reasoning such as this, Russell proposed a solution to the problem consisting in the elimination of self-reference from talk about classes through the substitution of types for class of classes. Classes that are not members of themselves constitute a type. For instance: the class of all books is a class of classes, because there are novels, reference books, etc. But there can also be two classes: books about Columbus and books not about Columbus, and the class of books about Columbus can be a member of itself. All books about Columbus are in the class of books about Columbus, but this class is also in the class whose members are not members of themselves. Therefore, we make the type of all books and the type of books about Columbus. Classes that embrace other classes are types different from the types of classes they comprise. Just as books about Columbus are a type, so books not about Columbus are a different type. Since classes define types and there is no such concept as the class of all classes, classes of classes exist only as different types, although obviously the type of all books is larger than the type of books about Columbus. But this hierarchical scheme creates the problem that a type at one level necessarily belongs to a type on another level and, according to Quine, "This particular failure spells failure for the proof of continuity". Since continuity is about sameness, this objection concerns the fact that each book about Columbus must be a type in itself in order not to get mixed with the higher type of all books. In other words, to eliminate his own paradox Russell has to create an infinite number of types, which subverts both the concepts of class and of type. Another elementary illustration of why Russell's solution to his own paradox will not work is the fact that there are types of stars each of which is not a member of itself. All types of stars are not members of themselves. Is the type of all types of stars a member of itself? Whether it is or not, it cannot be the type “stars”, which doesn’t mean a thing. But the class of all classes of stars is a scientific fact as opposed to, say, planets.

      According to Quine (following Russell), self-referential statements are paradoxical. The liar's paradox is a statement the significance of which is denied through self-reference: “If I am a liar, I am lying as I say ‘I am a liar’, so I cannot be a liar.” Russell's paradox stems from the self-reference implicit in the concept of a class of classes. In Gödel's proof of the incompleteness and inconsistency of the formal system of natural numbers, it is necessary to invoke a parallel system of numbering that refers to the natural numbers themselves. But it isn't very difficult to think of innocuous self-referential statements such as "I am an occasional liar" or "I tend to be sceptical in these matters", and so on. Self-reference per se is not the necessary source of paradox. But self-reference involving totalities—as in the original version of the liar's paradox which has Epimenides the Cretan saying that: “All Cretans are liars”—does seem to tend to produce the perplexity associated with paradoxes. But what exactly is involved here? The peculiarity of Epimenides' affirmation is not so much that it is self-referential as that it is categorical about Cretans. What he claimed is not something about himself primarily, but something about the totality of Cretans, and what we have here that is more significant than self-reference is the relation of identity. This is not to say that paradox does not involve self-reference in any way or that self-reference is not a fraught proposition, but what it may mean is that it is fraught precisely because it involves the relation of identity. All self-indicative compounds—self-reference, self-denotation, self-definition, etc.—imply not only circularity and tautology but ultimately identity. The self of self-reference is the entity to which the same entity is making reference. If our claim that paradoxes involve totalities is correct, then it is inescapable that they must also involve identity, for identity is possible only when totalities are involved. A sum is identical to another when it is totally like another. Twins are not identical unless each one has the totality of the same genes in the other (although the identity tends to fade with age and circumstances). In identity, both equivalence and substitution are necessary and reciprocally dependent. But equivalences do not necessarily licit substitution. Metaphors do not licit substitution. Identity is paradigmatically a logico-mathematical relation. The relation of identity is the source of paradox, although not identity in itself but identity applied other than within logic or mathematics. This constraint on the relation of identity would seem to entail that the world must be full of logical gaps. In order to face down these doubts we must explore in greater depth the relation between identity and equivalence.

            Logic and mathematics require the relation of identity. Logic would not work as applied to the rest of reality if substitution were not licit, but as logic does apply to reality, this suggests that the identity/equivalence distinction may not be valid. We seem to be skirting here a paradoxical situation: identity has to work for logic to function, yet identity in its full logical sense is not possible in the rest of reality. Is logic then not applicable to reality other than logic itself? If such were the case the world would be a topsy-turvy, knockabout place like toons movies, and we know that such is not and can never be the case, although this doesn’t mean that toons in themselves are not logical. Reality requires a substitution relation that is not identity. This can be shown to be a question of discernibility. For identity to work it is necessary to be able to distinguish between identicals. Thus it is that in mathematics the simple formula 2+2=4 expresses the idea of discernible or distinguishable identicals. But in wordly reality identicals would be indistinguishable and there cannot exist a proper relation of identity. The obvious conclusion is that equivalence is the only substitution relation that such reality can bear. But can logic work with mere equivalence? Equivalence can but need not licit substitution and logic can only work when equivalence licits substitution. It must therefore be the case that even though identity and equivalence appear to be different relations, they are not different at all in respect to logic.

            The syllogism is necessarily based on the possibility of substitution. But deductions in logic come in different types. The relation of identity makes inference apodictic. Whatever we infer within logic must be valid salva veritate. But we know that we constantly make deductions—in fact, the vast majority of our deductions—which are not of this sort. Can we then safely ensconce apodictic inference in formal logic and in mathematics and isolate it from the rest of reality? This would be an extreme and unjustified maneuver, because, for one, we already saw that we can establish a relation of identity between the object and the act of perceiving the object, or we would be hard put making precise identification, and for another, inference uses the same syntax in logic as in our every-day lives. Inferences, other than formal-logic inferences, can be necessary or probabilistic, but in either case they proceed on the basis not of identities but of equivalences. The syllogism as a form does not lose one iota of its force because it uses equivalence rather then identity for its connections. Science progresses on the back of the necessary rather than the apodictic. Logical forms work as well with equivalence as with identity even if identity and equivalence are quite distinct. This still does not solve the “gap problem” because we do not have, as it is possible to have in logic and mathematics, hard-and-fast rules about substitution through equivalence. Reason adjudicates between identity and equivalence, but error is inevitable. Apodictic inferences are not possible in the world. But we cannot escape the distinction between identity and equivalence. It is almost certain that there is more than formal logic to reason and that, in dealing with reality, in lieu of apodictic inference, we must allow some margin for uncertainty.

            Assuming that we could circumscribe identity to formal logic and mathematics and leave equivalence as the principle of substitution in deductive processes that refer to the rest of reality, let us test our argument that it is the misapplication of the identity relation that produces paradox in connection to Russell's paradox, which despite its verbal expression is basically a problem in set theory, hence of mathematics. Its claim is that the class of all classes that are not members of themselves is identical to all the classes that are not members of themselves. The paradoxical character of a class of all classes would seem to stem from the self-reference in self-membership in the sense that the class of all classes must include itself. But it also means that a class of all classes is identical to the totality of classes. In this sense, identity and self-reference appear to refer to the same thing, but self-evidently the membership in a class is not identical to the class itself. Since a class is constituted by its members, self-reference in set theory must involve identity. But once you posit the membership of the class of classes in itself, the identity relation becomes problematical. The attempt to stick a class of classes into itself creates the paradox that the class is both identical and not identical to its members. A class cannot be identical to its members. Something is missing on one side of the equation and that is the class itself which is an entity apart. If we may echo Quine, this is how type-theory must have arisen. It is then identity and not necessarily self-reference that engenders paradox. And it does so because formal logic makes a bad fit to concrete, temporal reality, where, for instance, taxonomy works without contradictions, paradoxes, and so forth. (No self-respecting anthropologist would be caught asking himself if the order “primates”, which is a class including many sub-classes, is a member of itself.) But in correlating Russell's paradox with the relation of identity, we seem to be contradicting our previous assumption that identity is not problematical if it is confined to mathematics and logic. The problem is compounded if we do the same correlation with Gödel's incompleteness and inconsistency theorems, which were derived from a system of numbering that mirrors the system of natural numbers. For Gödel's proof to work it is necessary to assume that Gödel numbering is identical to the system of natural numbers and it is only once this identity is assumed that it is possible to prove that any arithmetical proposition implies its own contradiction. It would then seem as if it isn't just the misapplication of identity but identity itself which is the problem, since even in fields where the relation should work smoothly it also tends to create paradoxical propositions. Perhaps we should be more precise about what these fields are.

            The relation of identity works smoothly within formal logic. But what exactly is formal logic? Formal logic is the abstract expression of the fundamental axioms and principles of reason. Less comprehensively, formal logic can be defined as the expression of any formal system for making deductions. Propositional logic, for instance, is the expression of the syllogism. But formal logic also includes the empty formulas of predicate logic. In principle, formal logic as such expresses valid propositions. But as Wittgenstein saw clearly, the fundamental propositions of formal logic work only within formal systems. The substitution of actual contents for the variables of predicate logic can produce erroneous and even absurd results. Wittgenstein did go too far in characterizing logic as not knowledge. If knowledge is constituted by valid propositions, then propositional logic within itself is valid and constitutes a form of knowledge. A fortiori, if all derivations within propositional logic are tautologies, then propositional logic must be derivable from one and only one axiom, but the formalization of logic, as predicate logic attests, has a history, and this means that it has developed over time like any other discipline of thought. It may be a very special sort of science, but it is knowledge nonetheless. A more significant question about logic pertains to its origins. The utter rejection by analytical philosophy of all forms of mentalism leaves this question deliberately in the air. Alternatively, analytical philosophers pretend that they are providing answers with impossible ideas such as the "extrusion of thought", or by occupying the castle of language and refusing to budge from its ramparts.

            We have seen that the combination of self-reference and totality do not necessarily explain why certain propositions are paradoxical. But we have found, in considering certain famous paradoxes, that they all exhibit the relation of identity. Since the relation of identity is at the root of formal logic itself, there are grounds for presuming that it is the use or misuse of formal logic that produces paradoxes. Contrariwise, we know that reality—the diversity of being, including error and deliberate lies—does not evince the relation of identity, that in fact if things were identical they would be indistinguishable, and that it is from this understanding of the relation of logic to reality that we can disarm paradoxes. It is not formal logic as such that underpins these arguments or that can illumine us about the relations these arguments propose or define. Unlike mathematics, where the value of four can be expressed in two different but identical ways, formal logic does not contemplate the principle of indiscernibility as can be gathered from the proposition that the logical relation a = b is not incompatible with the relation a v b, whereas in the world a = b does not obtain. Such being the case, then there must exist a faculty, in no way exhaustible by the axioms, principles, and derivations of formal logic, which allows us to put a certain distance between reality and logic without having to admit that there exists or can exist any incompatiblity between them. This faculty is what is commonly known as reason, but since logic itself is the core of reason, what we refer to as reason can be understood to be logic under another guise or from another perspective. Given that we do not take courses on reason, we can presume that it is innate and that we have justifiably called it intuitive logic. Intuitive logic is always operative. Formal logic can be understood as the linguistic expression or representation of intuitive logic. But in intuitive logic the distinction between representation and operation falls. When we reason we use logic intuitively. We do not apply logical rules mechanically, except in certain instances, as when we discard obvious contradictions or faulty conjunctions or disjunctions, or when conditionals do not establish necessary connections, and even in these cases most of the time we do this without being conscious of the rules we are applying, which means that we are reasoning with “God-given” intuitive logic. And we reason in this way whether we are dealing with history or with language or in any area to which justification and validation apply, which is all of the time, except perhaps in certain abnormal states of mind, or when I am creating or fantasizing, as in Surrealist “automatic writing”, but even in these instances, we merely try to suspend intuitive logic but the process is still logical if propositions and affirmation and negation are involved as inevitably they must be. We can go still further in this direction.

            We do not exhaust thought with formal logic. When we think we do not necessarily do so in terms of formal-logic axioms and principles, not even when we think about formal logic itself, for which the use of intuitive logic more than suffices, although this is not to say that when we do think about formal logic we cannot be engaged in the “deliberate”, aware application of the specific rules of formal logic. When we think about logic itself—about, for instance, how the rule of the commutativity of addition can be derived from the axioms and rules of predicate logic—we are aware, necessarily if instantaneously aware, of predicate logic. In other words, formal logic is necessary and inevitable in questions that pertain to logic itself. But this cannot lead to the denial of intuitive logic. We would not be able to reason about formal logic without intuitive logic. How could we when formal logic is just rules, axioms, and so on? We can make a valid distinction between formal logic and intuitive logic even if we cannot make it so divisive as to imply that they could be incompatible or incongruous with each other. In point of historical fact, this distinction is implicit in the reputed failure to derive the entire universe of arithmetical rules from the those of formal logic, and this must mean that it is from our innate, intuitive knowledge of logic that we have built and use the edifice of mathematics in all its branches.

            Formal logic is a discipline of thought. From its historicity and from the epistemic qualification of its derivations, it must be considered a form of knowledge. Intuitive logic has a different background. It is a posit about the contents of mind and in consequence it is assumed to exist intact and complete since some point in the evolutionary process of humanity. It could be said that formal logic is a derivation from intuitive logic, but the more accurate expression is that formal logic is a derivation of the exploration by intuitive logic of itself. Formal logic in all its forms would then be the mostly successful effort to find the syntactical forms that intuitive logic uses and it would be able to account for the totality or near totality of rational operations. But the assumption here is that intuitive logic can be cut and dried, and this is not compatible with the historical character of formal logic and with such feats of reason as Russell's paradox and Gödel's theorems which, though based on a relation of formal logic, are palpable demonstrations of formal logic transcending itself by means of rational processes that cannot be derived from the mere and exclusive knowledge of formal logic. [7]

            Going back to the question of the source of paradox, we can finally argue on stronger grounds than before for a relation between formal logic and paradoxical propositions. Since we assumed Quine's pespective on paradox, we could only argue from the perspective of formal logic, which is all that Quine himself will allow consistent with his cognitive behaviorism. It is possible to make a justified claim for the existence of principles for a cognitive faculty which we have termed intuitive logic and which is as specific as we can be concerning the faculty that we ordinarily call reason. As we saw, it is intuitive logic that adjudicates between identity and equivalence. And we are now in a position to make certain claims about the source of paradox and the means to disarm paradoxes. Formal logic is such—it has to be such—that to attempt to carry it lock stock and barrel to the rational grasp of world-reality would be to lead us into many paradoxical, even nonsensical, propositions. Basically, then, formal logic has its place and its place is not in the accurate grasp of such reality. Reason knows that identity is not a relation that can be smoothly applied to reality. What occurs in paradox is that we apply formal logic where formal logic has no bearing. All reason has to do, and will normally do, in face of the liar's paradox, or some equivalent such as the paradox of total scepticism, is to qualify the statement in a rational, non-paradoxical way, as in “I lie most of the time” or “I am addicted to lying” or “I tend to be sceptical in general”, and so on. Reason puts paradoxes aside. It doesn't solve them on their own terms of reference. It rationally contextualizes them. In Quine's "Two dogmas of empiricism", the strict application of formal logic to radical translation produced the self-contradictory concept of “idiolects”, and in Word and Object, the same philosopher came to the conclusion that translations are logically unrealizable, when the fact is that they are possible and often quite accurate. Russell assumed a version of logical Platonism and Wittgenstein claimed that logic was tautological. Yet neither of these arguments remotely affects the pervasiveness of reason.

            If we have argued that paradoxes stem from the misuse or the misapplication of formal logic, what then remains of paradox itself? Is there in fact a “Russell's paradox”? The distinction between classes which are members of themselves and classes which are not members of themselves, is not dictated by logic but by set theory, and if what we want is the formula for all possible classes, even within set theory itself it is probably superfluous since the paradoxical class of all classes which are members of themselves can do duty for the class of classes which are not members of themselves. Suppose that only classes which are members of themselves can define a class. The class of non-horses is a member of itself. The class of horses is not a member of that class. The class of non-mules is a member of itself. The class of mules is not a member of that class. But the class of horses is a member of the class of non-mules and the class of mules is a member of the class non-horses. All membership conditions, save non-self-membership of a class, determine classes. The only possible classes which determine a class of all classes are those which can be members of themselves, that is, the class of non-horses, of non-houses, and so on. But those classes which are members of themselves include all classes except the classes (horses, houses) whose exclusion defines them. And the classes whose exclusion defines them are classes in the classes which are members of themselves. The class of all classes which are members of themselves contains all of the classes which are not members of themselves. Therefore, the membership condition of self-membership defines the class of all possible classes. Intuitive logic tells me, though, that for any class of all classes, if it is well defined, there should be no problem about self-membership or non-self-membership, so you can safely say ( ∑ Α, Ω ) without getting into any paradoxical situation, and it really doesn’t matter whether the universe, or universes if that’d the case, has to be a member of itself or not a member of itself.

            The grouping of contraries in one class is another argument for the artificiality of the distinction between types of classes of classes. The class of not-this or not-that must include all ordered pairs which mutually exclude each other, such as being and non-being, truth and non-truth, and so on. (Assume a1 = non-things and a2 = things and b1 = non-beings and b2 = beings, then non things = non-being and things = beings.) So how can these oppositions share conditions of membership? Willy nilly, the "class of not-me" simply cannot be since its condition of membership includes mutually exclusionary propositions. If "not-me" includes truth, then how can it also include non-truth, and vice versa? Or if it includes being, how can it include non-being? And if it includes non-being, how can it include the rest of things which belong to "not-me"? Talk of classes of classes is analogous to reason trying to define itself. Since class is equivalent to its members, what this involves is identity. Identity is a paradoxical relation. [8] But the likelier source of the paradox is the artificial distinction between classes which are members of themselves and classes which are not members of themselves. What we are proposing is that Russell's paradox is artificial and argumentative, not, in this respect, unlike the Eleatic paradoxes, and not in any sense different from all other types of propositions. It does not affect anything or lead anywhere. Wittgenstein, then, was both right and wrong. But so can any one at one time or another.

            In a wider sense, our contention is that all paradoxes are artificial constructs which are easily dismissed by reason. Grelling's paradox (1908) classifies adjectives as homological and heterological. Homological are those, like "short" and "pollysyllabic", whose physical expression corresponds to their meaning; and heterological are those, like "useless" and "monosyllabic", whose expressions do not correspond to their instantiation. The paradox arises with the classification of the classificatory terms themselves: if "heterological" is heterological it must be homological, and if it is homological, it must be heterological. To struggle to derive this paradox from self-reference would be unnecessary and probably futile. Is there a solution to this paradox? One is to observe that heterological is a category mistake in that an adjective always denotes externally. Self-denotation in the case of adjectives is unnecessary if their denotation is perfectly clear. This is the solution that Russell found to the logical problems of self-denotation. But more fundamentally, to have a paradox there must be a reason, and there are adjectives, like blue, which are neither heterological nor homological, heterological itself being one of them. If there isn't a rational cause for a paradox, or if it is possible to argue for the utter superfluity of an argumentative paradox, where does that leave paradoxes?

            Even assuming the set-up, artificial nature of Russell's paradox, it remains that we have argued that its paradoxical character stems from the relation of identity, which is supposed to function smoothly in formal logic and mathematics, and this raises a puzzle. If reason or intuitive logic can explain and undo paradoxes, since we have stuffed identity and paradox into the same can of worms, we would have to conclude, as suggested before, that it is identity itself, regardless of its field of application, that is the source of paradox. But to make such a wholesale charge against the relation of identity would appear to be not only erroneous but perverse and misguided. How could either logic or arithmetic subsist if the relation of identity rather than functioning as the cornerstone of both were constantly propitiating contradictions and perplexities? Let us go back to the essence of our claim, which is that the misapplication of formal logic is responsible for the production of paradoxical propositions. The question is: to what is formal logic misapplied? The answer we have been giving is variously the world or reality, from neither of which can there be reasons to exclude set theory or arithmetic. We have, therefore, no basis on which to impugn the relation of identity within formal logic or within mathematics. But we have grounds for claiming that paradox arises when intuitive logic applies the formal-logic relation of identity outside of formal logic, even if the application is to other formal systems of deduction. Thus, Columbus confused America with India. Thus also, it is the relation of identity applied to natural numbers that yields Gödel's paradoxes and it is the relation of identity applied to set theory that produces Russell's paradox. [9]

            Paradoxes are in a sense formal logic undermining itself as an instrument for grasping reality. They mean somewhat more than that, as we shall soon see. However, we know that the world must be logical. Therefore, we have made and argued for a distinction between formal logic and intuitive logic. Intuitive logic applied to reality, including language, is what we call reason. It is reason that can discover, beyond the rigidities of formal logic, the source of paradox, and just as reason explains paradoxes, so it also disarms them. This logic which goes beyond formal logic must be intuitive and psychological. Intuitive logic understands the limitations of the relation of identity and more importantly it can usually distinguish, in dealing with the world, between identity and equivalence. Thus we have arrived at our original argument about the adjudication-function of intuitive logic after having explored paradox and its source. The error in validating paradoxes is the presumption that formal logic is the be-all of reason, that is, the paradox-prone identification of thought with formal logic. This operation can be traced back to Frege, who laid the foundations of analyticity through his obsessive but ambiguous anti-psychologism, but it was taken up with a vengeance by Russell and Wittgenstein, indeed even as analyticity itself entered the functionalist phase in philosophy of mind. The fallacy underlying all this—let’s call it Quine's fallacy—then is the belief that reason is formal logic or that formal logic exhausts the possibilities of reason. But we have seen that even when we reason about formal logic we cannot escape realizing that in fact we are using a faculty that goes far beyond formal logic itself and that this faculty is intuitive logic. The propensity of the formal-logic relation of identity to create paradoxes is, in the final reckoning, the reductio ad absurdum of Quine's fallacy. How can such an extreme case be justified? How, from our perspective, can Quine's fallacy be explained? A simple psychological explanation is that a person can be doing something and actually think and believe that it is something else that he is doing and this false understanding can easily constrain what the person thinks about what he is doing thus resulting in its erroneous depiction. In the case of Quine—and of analytical philosophy is general—what it comes down to is that he is in radical denial of the act of thinking itself, which he cannot escape and thus tries to reduce it to the relatively wimpish dimensions of formal logic, which is consistent with his definition of “being” as the value of a variable.

            The indiscernibility of identicals is the ultimate touchstone for the distinction between the relations of identity and of equivalence. From the indiscernibility of identicals it is obvious that the identity relation cannot be applied to reality outside of formal logic, which is not the case of the equivalence relation. A fortiori, identity yields apodictic inferences whereas equivalence yields necessary and probabilistic inferences. Cognition is based on intuitive logic. Since equivalence does not necessarily lead to reliable inferences whereas identity yields apodictic inferences, this brings back to the problem that it would seem as if cognition were leaving “knowledge gaps” all over the place. The only answer we have for this is what could be called the uncertainty principle.

            I am sitting working at the computer at home trying to solve some continuity puzzle. Suddenly I remember an offer I had once had for work in a think-tank devoted to social development issues. I am doubly puzzled now and comment my thoughts with my wife. I try to reconstruct the process. It was raining outside. I associate rain with going out in the rain, which we used to do a lot in London. London reminds me of my asociation with a journal in whose offices it was that I received the job offer. Simple? Not quite as simple as that! Before I had been thinking of a social studies project. One of the possibilities was having an office in London. This idea was prompted by my intellectual activities. These imply an outlet, but I am uncertain about this. I have a feeling of being trapped, like Burroughs' fear of stasis on a drug-hunting trip to Perú. In retrospect, the proposition I received in London would have been a chance to avoid this trap. It suggests that there are always possibilities of escaping. My recalling this incident to my wife was a way of reaffirming "objectively" the hope that I am not really trapped. Hope is an interpretation. It gains from transactionality and intersubjectivity, and so on and on. I could have gone on delving and finding layers and layers of complexity. Basic cognitive propositions were involved in this chain of thought, mainly memory and logic, but also perception, language-use, etc. All the propositions can be explained in terms of basic cognitive propositions and the inputs of experience, themselves yields of the cognitive propositions. The propositions refer to my specific self. I am trying to capture my specific self in linguistic propositions. These linguistic propositions are the specific yields of basic cognitive propositions. But are these linguistic propositions all that there is to the chain of thought? Hardly! My description is poverty-stricken, even though not inaccurate on the whole. I can never really ever hope to have of my chains of thought anything better than these sketchy concatenations of propositions. Therefore, I can never really have my specific self. But how about the cognitive propositions? These I can be more accurate about. Specific memories will always leave something out, but memory itself is a basic formal process that I can reproduce in propositions of the language in relation to any of its specific yields, such as that memory is associative, impressionable, etc. Can I claim that I can give a better account of the formal functioning of the cognitive propositions than I can give of the specific processes involving them? Better perhaps, but exhaustive is another matter!

            The important point in all this is the distinction between the cognitive basis of all specific yields and the specific yields of cognition. My fundamental epistemological démarche is founded on this distinction. Can I actually claim that my propositions about cognitive processes are the basic cognitive processes themselves? If cognition and its basic propositions did not exist independently of my descriptions, these would have no value whatsoever. In formulating my descriptions of cognition, I am constantly referring to something that I call cognition, which is different from my propositions about cognition. Nevertheless, unlike a description of the brain or the liver, cognition is nothing beyond its linguistic expression. And it is in this sense, that is, in the sense that there is nothing beyond my description of cognition, that I can claim that my description of the basic cognitive processes are the processes themselves. But I cannot give just any description. What determines that my description is valid? To say coherence would be to place myself outside my own claims about validity. I must go to my claims about the validity of propositions. My descriptions of cognition belong to all the types of inferences: some are apodictic, some are necessary, some are probabilistic. In the end, however, all I can claim is that propositions about cognition are not self-contradictory, that they appeal to factuality as often as possible, that they respect the principles of logic, and so on. I cannot really go beyond this. I cannot claim for my own thought anything that I would not also have to claim for thought in general. What I am claiming is that my thought is subject to the universal consensus implicit in cognition. I am on my own trusting that my ideas are valid because my cognitive processes are reliable. But then so is everyone else!

            The uncertainty cannot be eliminated. But is uncertainty not part of cognition? All cognitive processes exist theoretically to eliminate uncertainty because uncertainty can lead to death. If a hominid had not known when danger lurked, he would certainly have been exposing himself to it. But in relation to death itself we always live in uncertainty. Only the condemned to execution at a fixed time can be free of the uncertainty of death. But this knowledge, this absence of uncertainty, would certainly not make life very bearable. There is a character in Unamuno who kills himself because he is obsessed with death. This is carrying things to an extreme. But on the whole it is not unreasonable to claim that we prefer to live in uncertainty about death. And if this is so, then cognitive processes exist not to do away with uncertainty but to deal with it. Cognitive processes include the proposition that its own limitations can be beneficial and must be accepted. Knowledge is not only what history decants, but also what history leaves in the bottom of the bottle. The history of science is as much the history of errors, for instances, first Ptolemy's, then Copernicus', and so on, as it is the history of achievements. Obviously cognition has survival value, but if survival implied the total absence of uncertainty, cognition would be defeating its own purpose. Cognition cannot be about eliminating all uncertainty. It is only from a formal-logic point of view that cognition can be defined in terms of the elimination of uncertainty. Uncertainty is part of cognition. Even the denial of cognition is part of cognition. The gaps that intuitive logic and equivalence leave all over the place will always be there. They are inevitable. We are not wrong in admitting the limitations of cognition simply because analyticity tried to eliminate them. Analyticity was simply on the wrong track.

            Logic pervades the universe. It is logic that makes it possible to even conceive that there might be other universes, where logic too must reign supreme. It is logic that has made it possible for our species to appear and become what it is. It is logic that pronounces of itself that it is “illogical”. In his Philosophical investigations phase, Wittgenstein argued that philosophy is about the clarification of expression, with which, as he had already said that formal logic is tautological, he was tacitly admitting that intuitive logic is the ultimate basis of all philosophy. Suicide is a perfectly logical act. What, if not logic, drives a person about to commit suicide to leave a justification of his deed? The motto of my tribe is “nemo me impune lascessit”, which is unreasonable, but perfectly logical. If I persuade myself that I have been wronged, I could act in an unseemly yet logical fashion. Regret seems completely illogical. What can I do about the past? Yet if I suppose that I could have acted differently from the way I did, I will be regretting, and logically regretting, to the end of my days. I know that we are unfree, but to adopt this premise I will have to change the logic that produced it in the first place, and there is no amount of reasoning I can do that logically regret is illogical if in some inner depth I cannot plumb the logic that tells me the contrary. Passions are logical. A passion originates in a premise from which it emanates as equations do from the mind of a mathematician. Even that mortal enemy of passions, La Rochefoucauld, wrote: “Il y a une infinité de conduites qui paraissent ridicules, et dont les raisons cachées sont très sages et très solides.” [10] Since logic is innate and intuitive, then its sway over the subconscious is absolute. And if my arguments that awareness is but a conduit to the subconscious, then we are determined inescapably by logic. We are all prisoners of ourselves and of logic. We can try to change the premises of our behavior and thought, but we will never be able to alter the conclusions that we make from those premises. There is one final quodlibet and it is that if my logic is not the world’s logic, then I can only console myself with the thought that at least in my logical arguments I have tried to be as consistent and as thorough as possible.