If there is a constant thread through the work of the American logician W.V.O.Quine, it is that logic is the light of superior intelligent life. More accurately, logic is the tool of science and science is our guide to reality. Just to cite one example, but a telling and witty one, in Quiddities he writes: "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." His very definition of being--the value of a variable--is imbued by logic. Since Quine has always been more or less behavioristic about mind, the only logic he can be referring to is expressible formal logic. The psychological representation and function of logic is obviously of no concern to him.

      The object of this paper is to argue that 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 and is in rank contradiction to Quine's fundamental tenets. The liar's paradox--probably as near as we can get to a "natural", as opposed to a set-up, argumentative paradox--is also the clearest instance of the misuse of the identity relation. In showing that paradoxes are artificial constructs that result from the misapplication of formal logic we strike at Quine's logicism and in arguing that reason can disarm paradoxes we strike at Quine's cognitive behaviorism.

WHAT IS A PARADOX?

      In a general sense, a paradox is any proposition which produces perplexity because it involves a logical contradiction. We know, e.g., 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. In sum, 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 of course that if I am a liar then I am not a liar. The liars 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. What this implies is that there is no way to define being except in terms of different cognates of the word being itself. 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 of course logical, but in the case of the tautology of being what is involved is 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. It is not exactly either conventionalism or analyticity, because there is no question here of lexicology or of synonymy. In fact, 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 least 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. Nevertheless, this is not to negate paradox. 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 or taking itself in or defining 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. We already saw in the case of the self-definition of being that it embraces within itself such contraries as truth and falsehood, reality and illusion, etc. This perplexing intuition can actually be expressed in a precise manner in Russell's paradox. It is based on the distinction between classes which are not members of themselves, such as things, ideas, and so forth, and classes which are members of themselves, such as non-things, non-ideas, and so forth. Does the class of all classes which are not members of themselves exist? The answer to this question is logically self-contradictory. If it is a member of itself, its condition of membership in a class is that it not be a member of itself. If it is not a member of itself, then its condition of membership in a class is that it be a member of itself. In the context of set theory, a solution to the problem could be declaring that non-self-membeship is the only predicate that does not form a class. However, Quine then comes 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 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 of entities. 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 obviousy it would not be a class at all. Based on a reasoning of such as this, Russell himself proposed a solution to the problem consisting in the elimination of self-reference from talk about classes through the substitution of types for classes of classes. Classes that are not members of themselves are a type of class. Classes that embrace other classes are also types different from the types of classes they comprise. Since classes define types and there is no such concept as the class of all classes, classes of classes exist only as different types. But this hierarchical scheme creates the problem that a class formed of classes within one level necesarily belongs to another level and, according to Quine, "This particular failure spells failure for the proof of continuity". An elementary illustration of why Russell's solution to his own paradox will not work is the fact that there are classes of stars each of which is not a member of itself. All classes of stars are not members of themselves. Is the class of all classes of stars a member of itself? Whether it is or not, it cannot be a class of stars. 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. However, 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 is claiming is not something about himself primarily, but something about the totality of Cretans, and not only about the totality of Cretans, but also about what defines the totality of Cretans, i.e., that they are all without exception liars and nothing but liars, 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, e.g., is the entity to which the same entity is making reference. Finally, 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. And twins are not identical unless each one has the totality of the same genes in the other. Identity is equivalence plus substitution. In identity both equivalence and substitution are necessary and reciprocally dependent, i.e., for identity to exist between two entities they must be equivalent in such a way that either one can stand in for the other. However, equivalences as such do not necessarily licit substitution. Metaphors, e.g., the mainstream of history, do not licit substitution. Identity, in sum, is a logico-mathematical relation. Our claim is that 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.

IDENTITY AND EQUIVALENCE

      Logic and mathematics require the relation of identity. But logic would not work as applied to the rest of reality if substitution were not licit, and 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. 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, and 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. We can therefore safely ensconce apodictic inference in formal logic and in mathematics and isolate them from the rest of reality. Other inferences can be necesary 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 and logical axioms 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,  but then one may well ask whether that problem is solvable at all? 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.

      We have argued that the misapplication of the identity relation produces paradox and we have assumed that we can safely circumscribe it to formal logic and mathematics leaving equivalence to be the principle of substitution in deductive processes that refer to the rest of reality. Let us test our argument 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. It is this claim that makes the paradox for a class of classes cannot be identical to the totality of its members. The paradoxical character of a class of all classes would seem to stem from the self-reference in self-membership, i.e., 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. The problem lies in the misapplication of identity. 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 engenders paradox because identity is a formal-logic relation and formal logic does not apply adequately to concrete, temporal reality. However, in correlating Russell's paradox with the relation of identity, we seem to be contradicting our previous claim 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 refers to 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. Formal logic is, e.g., the expression of the syllogism. It is also the empty formulas of predicate logic. In principle, formal logic as such expresses valid propositions. However, as Wittgenstein saw clearly, the fundamental propositions of formal logic work only within formal systems. The substitution of actual contents for the symbols of predicate logic can produce erroneous and even absurd results. However, Wittgenstein went too far in characterizing formal logic as tautological. If knowledge is constituted by valid propositions, then formal logic within itself is valid and constitutes a form of knowledge. A fortiori, if all derivations within formal logic are tautologies, then formal logic must be derivable from one and only one axiom, but formal logic, as propositional 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 formal 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 going to the castle of language and refusing to budge from its ramparts.

INTUITIVE LOGIC

      We have seen that the combination of self-reference and totality do not necessarily explain why certain propositions are paradoxical. However, we have found, in considering certain famous paradoxes, that they all evince or exhibit the logico-mathematical 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 propose or define. 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 th world a=b simply 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 can justifiably call 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 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 and 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.

      As a general rule, we cannot identify 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, conscious application of the specific rules of formal-logic. When we think about logic itself--about, e.g., how the rule of the commutativity of addition can be derived from the axioms and rules of predicate logic--we are aware, necessarily 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? The point is that 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 would then be the mostly succesful effort to find the inferential forms that intuitive logic uses and it would be able to account for the totality or near totality of rational operations. 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.

      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. 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 worldly 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, e.g., I lie most of the time or I am addicted to lying or whatever 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.

PARADOX DISCARDED, IDENTITY VINDICATED

      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, e.g., 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 class of all classes which are members of themselves can do duty for the paradoxical 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-houses is a member of itself. The class of houses is not a member of that class. But the class of horses is a member of the class of non-houses and the class of houses 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, e.g. 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.

      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 pairs which mutually exclude each other, such as being and non-being, truth and non-truth, and so on, 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 probably identity. Identity is a paradoxical relation. 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.

      In a wider sense, our contention is that paradoxes are artificial constructs which are easily dismissed by reason. Grelling's paradox (1908), e.g., classifies adjectives as homological and heterological. Homological are those, like "short" and "pollysyllabic", whose physical representation corresponds to their meaning; and heterological are those, like "useless" and "monosyllabic", whose meanings 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 autological, 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. This is presumably borne out by Gödel's mathematical arguments concerning the system of natural numbers. 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 itself, even if the application is to other formal systems of deduction including itself. Thus, 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.

QUINE'S FALLACY

      Whether the product of a deliberate philosophical argument or whether the result of spontaneous language-use, paradoxes involve logic, but how can it be that logic creates paradoxes, which is precisely the opposite of what it is supposed to do? We have argued that it is the misapplication of the relation of identity that gives rise to paradoxes. Since identity is a relation within formal logic, our argument amounts to saying that formal logic is not adequate in dealing with reality outside of formal logic itself. 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, 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 distinguish, in dealing with the world, between identity and equivalence. The error in validating paradoxes is the presumption that formal logic is the be-all of reason. The source itself of paradox is ultimately the identification of thought with formal logic. This operation can be traced back to Gottlob Frege, who laid the foundations of analyticity through his obsessive but ambiguous anti-psychologism, but it was taken up with a vengeance, after Russell and Wittgenstein, indeed even as analyticity itself entered the functionalist phase in philosophy of mind, by Quine. 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 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 to the relatively wimpish dimensions of formal logic.

COGNITION AND THE PRINCIPLE OF UNCERTAINTY

      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 where as identity yields apodictic inferences, it would seem as if equivalence, and consequently cognition, were leaving cognitive gaps all over the place. The only answer we have for this is 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 Peru. 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 propostions. 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, e.g., 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! However, 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, i.e., 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. However, 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. Therefore, 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. However, in relation to death itself we always live in uncertainty. Only the condemned to execution at dawn 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. What this means is that 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, e.g., 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. 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.

Texts omitted or modified

QUINE'S FALLACY: PARADOX AND THE RELATION OF IDENTITY

by C. Ramirez-Faria

(1) Russell's paradox surges when you ask a certain question. The idea for that question came to Russell from Cantor's propositions on infinity, which posit that there are more classes than members of classes. If the classes in point are integers there are more classes of integers than integers--because of the class of all integers, which is not an integer--and there are more classes of classes of integers than integers, and so on. Quine explains: "The argument hinged...on the class of nonmembers of own correlated classes; Russell's paradox hinged on the class of nonmembers of selves. Take the correlation in particular as self-correlation, and Cantor's argument yields Russell's paradox. This is in fact how Russell discovered his paradox".

(2) As A. Flew puts it: "A statement referring to other statements must, Russell says, be of a different type from, a higher order than, the statement it is about. So we must say that the class of all first order classes which are not members of themselves is a second order class, and hence it will be `obvious nonsense' to say of a class either that it is or that it isn't a member of itself. Thus the paradox disappears."

(3) Quine claims to have come up with the general expression of self-referential paradoxes: "Nothing of a given kind can bear a relation to all and only the things of the given kind that do not bear it to themselves". Assuming this formula, which means nothing as it stands, does it work when paraphrased? Some substitutions do appear to be paradoxical: "if a human being helps other human beings and belongs to the class of human beings that help other human beings but do not help themselves, then he does not help other human beings"; or "no member of the class of vehicles can be a member of the class of vehicles whose members are not members of themselves", which licits the following contradictory reasoning: "assuming all classes of members of the class of vehicles cannot be members of themselves and that automobiles are a class that cannot be a member of itself, then an automobile is a member of the class of vehicles but cannot be a member of the class of vehicles". But using the same verbal formula it is also possible to come up with opinions or truisms or even nonsense. To say that "no human being can help all and only other human beings if they do not help themselves" is to express a very common opinion; just as the claim that "no mathematical entity can have a relation to other mathematical entities that they cannot have to themselves" is a truism; and the statement "no person can be conscious of a person that is not conscious of him or herself" is patent nonsense, for I can be conscious of a sleeping person. It would seem then that self-reference even involving totalities such as all or nothing, does not necessarily lead to paradox.

(4) However, we must be clear as to which identity we are referring. Quine's version is as follows: "Evidently to say of anything that it is identical with itself is trivial, and to say that it is identical with anything else is absurd. What then is the use of identity?...When I say that the hiding place is known to Ralph and only to him, nobody else, I mean to say two things: that Ralph knows the hiding place and that whoever knows the hiding place is identical with Ralph." Is this definition good? It is self-evidently wrong, for Ralph--if we are talking about some one and not just about a name--is much more than the only person that knows a certain hiding place. Since there cannot exist identicals in reality and since Ralph cannot simply be a person who knows a hiding place, then the identity that Quine is defining or describing or illustrating is more in the nature of the practical equivalence between someone who alone knows a certain hiding place and the conventional designation "Ralph", which could as well be "Peter", "Paul", or "Mary".

(5) Newton's laws are principles of nature. But each one of its applications is different from the other. The specifications of dropping balls of lead from the Tower of Pisa are not those that would allow calculations about the movements of the planets. No rocket launch is exactly identical to another. Yet we know that in every case the force of gravity will be exerting its influence in varying degrees. Similarly, assuming the existence of laws of perception applicable to sense-data, it is these laws that yield specific acts of perception. Hence, sense-data yield perception, even though sense-data and perception are equivalent but distinct. In these cases it is the syllogism which is at work. Logical forms and logical axioms work as well with equivalence as with identity even if identity and equivalence are quite distinct.

(6) We shall be returning shortly to these question, but what we want to demonstrate at this stage is that it is the misapplication of the relation of identity that yields paradoxes. The liar's paradox works only because it assumes that it is not possible to distinguish at any time between myself and a liar. The same reasoning applies to the sceptic's paradox, in which total scepticism precludes scepticism itself; and to the statement "nothing exists", which would not allow even for its own existence. Another instance of identity creating paradox is the barbers paradox. Does the village barber who only shaves people who do not shave themselves shave himself? If the barber shaves himself, then he does not shave himself because he only shaves people who do not shave themselves. And if he does not shave himself, then he shaves himself because he shaves all those and only those who do not shave themselves. What is involved here is the assumption that no one who goes to barbers ever shaves himself. Since an implicit total negation, as in the liar's paradox or in the total scepticism contradiction, totally denies a totalizing affirmation leaving nothing to doubt, it is evident that the logical concept of identity must be involved, i.e., total negation of total meaning. Statements implying total negation if taken literally engender the paradox that their meaning is not their meaning.

(6) I am sitting working at the computer at home. I get up and ask my wife if she remembers when Echeverría offered me a job in his think-tank in Mexico. She says she does and asks me why I am telling her that. I try to reconstruct the process. It was raining outside. I associated rain with going out in the rain, which we used to do a lot in London. London reminds me of Kofi and his Third World review, which was where I spoke to Echeverría. Simple? Not quite as simple as that! Before I had been thinkling of a journal of Venezuelan studies. One of the possibilities was having an office in London. This idea was prompted by all 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 Peru. In retrospect, the Echeverría offer would have been a chance in the past to avoid this trap. At all events, it suggests that there are always possibilities of escaping the trap. 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 go on delving and finding layers and layers of complexity.

 

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