Reijication Fallacies and Inappropriate Totalities NICHOLAS RESCHER University of Pittsburgh Abstract: As Russell's paradox of "the set of all sets that do not contain themselves" indicated long ago, matters go seriously amiss if one operates an ontology of unrestricted totalization. Some sort of restriction must be placed on such items as "the set of all sets that have the feature F' or "the con- junction of all truths that have the feature G." But generally, logicians here introduce such formalized and complex devices as the theory of types or the doctrine of impredictivity. The present paper argues for the informal and elementary idea that the items invoked in a proper identification have themselves already been identified. Even as an explanation is not satisfactory that proceeds in terms of items that them- selves require prior explanation, so the same holds with identification. And heed of this elementary idea suffices to sideline those oth- erwise paradoxical perplexities. Resume: Le paradoxe de Russell, < to abbreviate "the totality that embraces (that is, C-relevantly 'embraces') all of the C-type items." Thus, for example, if C(x) "x is a letter of the (Roman) alphabet," then is the totality of letters of the (Roman) alphabet, which is to say it is that alphabet itself. If C(x) "x is a cat" then is the totality of cats, the entire genus Felix that includes all the felines there are. Or again, if C(x) = "x is a color" then is the totality of colors, which is to say it is the whole color spectrum. On the basis of the preceding deliberations it is clear that whenever the totality that is presumably being defined is "itself" seen as something that meets condition C-which is emphatically not the case with any of the preceding examples-then the totalization process in question goes awry since it is now unable to realize a well-defined result. A purported item that is not subject to a discussion-introducing identification in terms of reference inde- pendent of itself has simply not been introduced meaningfully into the discussion at all. Examples of putatively totalized wholes that violate the proscription of self- presupposition include the following pseudo-totalities: The supertruth understood as the truth that conjoins all truths. s The cosmic fact about the world understood as a fact that encompasses all facts about the world. The megaset understood as the set that includes all sets. Eternity understood as the timespan that contains all timespans. The pan explanation understood as the explanation that encompasses all explanations (that explains everything). The protocause understood as the cause that causes all causes. The mega-story understood as the story that encompasses all stories. The superadjective understood as the adjective that applies to all adjec- tives. The world-all (Weltall) understood as the physical region that includes all physical regions. 48 Nicholas Rescher The omni-cause understood as the cause that causes literally everything that is real. In each case we have a putative item that is identified by means of a common format: "The C-type item that 'embraces' all C-type items." And this clearly pre- supposes not only that there indeed is (a unique) something that "embraces" all C- type items, but that this something is itself of type C. Throughout, the proscription of self-involving totalization is thus violated. The boundaries ofthat purported total at issue are not something that can be settled unproblematically until we have determined the viability of that total itself as an item of its purported kind. There yet remains the prospect of objecting: "But if it is defined as an X, if its . X-hood is something that is explicitly specified in its definition, then surely it's got to be an Xl" This plausible protest does not, however, hold water. As the longstanding critique of the Ontological Argument shows, substantive questions cannot be settled by definitional fiat. (This is a cardinal principle of rational in- quiry.) Calling something an X does not mean that this is an actual item that is actually an X. Thus "the integer that is larger than any other" is a formula that purports to specify an integer but actually fails to do so, seeing that there is no integer larger than all others. And similarly to call a supertruth a truth or a megaset a set does not mean that this is actually so-that there really is a set or a truth that answers to the specification at issue. For that item may simply not exist as such. The assumption that something exists under a certain description-for example, as the set of all sets that do not include themselves-may well be false. Anything that answers to this description-whatever it might be--cannot be a set. That supposed specification simply fails to identify. To be sure, hypostatization (item-introduction) is one thing and mere descrip- tion (of a pre-identified item) something else again. Once we have an item at hand (say 2 as "the smallest prime") one can describe it in self-inclusive terms (as "the prime that is not undersized by any prime"). But we cannot properly introduce it in this way until the issue of existence has been (independently) resolved. There is thus a crucial disanalogy between identifying something on the one hand and de- scribing it on the other. We can only consider "the thing identified" once some "identification ofthe thing" has been given: any possible entertainment of the former (the thing identified) is contingent upon the latter (the identification of the thing). By contrast, nothing whatever about the describability of something rests upon the description of this thing. The hypothetical removal of its identification creates problems for discourse about something in view in a way that the hypothetical removal of its description would not. Accordingly, there is no problem about (say) characterizing a pre-identified being (i.e., God) as that which is "the ultimate reason of all being" or again as that which "self-caused-causa sui-the cause of itself." Describing an already individuated God by such formulas could in theory qualify as perfectly meaning- ful. But we nevertheless cannot identifY God by a decision-introducing characteri- zation as "the reason-for-being of all reasons-for-being" or as "that being which is Reification Fallacies and Inappropriate Totalities 49 the cause of all being itself included." For when seen as an identifYing specifica- tion self-reference must be rejected as counter-productively vitiating. We can cer- tainly say-truly and meaningfully- of some pre-identified item that "it is the result of the cause that produced it." But we cannot use this sort of formula when the initial identification of the item is at issue ("the item that is to be at issue is the product of its cause"). For here nothing is as yet available to serve as referential for that ultimate anaphoric back-reference. Unlike inappropriately described items, inappropriately introduced items are no more than illusions. A significant lesson emerges. Not only are there pseudo-questions like "Have you stopped cheating on your taxes?" that should not be asked (because they rest on inappropriate suppositions) but there are pseudo-identifications that should not be considered (investigated, taken seriously) because their very conception is flawed in that it rests on the erroneous presupposition of a particular answer to an inap- propriate question. For successful identification requires and presupposes an af- firmative answer to the question: "Have all the items being referred to already been identified effectively?" Let us be somewhat more explicit about the ramifications of this state of af- fairs. 2. The Route to Paradox All contexts of rational assertion and deliberation are governed by an at least tacit supposition that its terms of reference are meaningful. Should this presupposition prove to be unwarranted, the chasm of paradox yawns wide-open before us. This is illustrated by a conundrum that has received much attention is The Barber Paradox. 6 It is based on the following riddle: A certain village has a barber whose practice it is to shave all the adult male villagers who do not shave themselves. He himself, of course, is an adult male who lives in the village. Does he or does he not shave himself? The evident paradox here is that if he is a self-shaver, then (by hypothesis) our barber does not shave himself, while ifhe is not a self-shaver, then (by hypothesis) he does shave himself. Either way we are in difficulty. Putting it more explicitly, our barber B is (by specificatory hypothesis) such that: ('dx)[S(B, x) iff -Sex, x)]. And when the x at issue is instantiated as B himself, this yields: S(B, 11) iff -S(B, B). This result plunges us into paradox, seeing that we now can maintain neither S(B, B) nor -S(B, B). It seems that consistency is beyond our reach here. The following theses constitute the aporetic cluster that defines paradox: 50 Nicholas Rescher (l) There is-or can be-a barber who answers to the specifications of the narrative. (2) This barber at issue in the narrative (like any other) either does or does not shave himself, but not both. (3) If the barber shaves himself then, by the narrative's stipulation he is not someone who shaves himself. (4) If the barber does not shave himself then, by the narrative's stipula tion he does shave himself. (5) Thus either way a contradiction ensues. The optimal point at which to break the chain of inconsistency here is clearly at its very start with that hypothetical barber himself. For there is not and cannot be a barber who answers to the specified conditions. The barber paradox thus is vitiated from the very outset, by being predicated on the supposition of a barber who cannot possibly be. That purported introduction of a barber upon the stage of discussiory misfires-it fails to introduce. And what holds for inappropriate identi- fications also holds for inappropriate totalizations. With such self-encompassment in view we can now project. Specifically, con- sider what might be characterized as the X-totalization of self-nonencompassing X"s the X of all those X" s that do not X-wise encompass themselves (the set of all those sets that do not set-include themselves, the shaver of all these shavers who do not shave themselves, or the like). It is clear that any such totalization is always paradoxical. Thus consider the con- tainer (set) of all containers (sets) that do not contain (set-include) themselves, the list of all lists that do not list themselves, the picture of all pictures that do not depict themselves, etc. In each case we have a totality Z of such a sort that: Z enc X iff -(X enc A). But now whenever Z itself belongs to the range of the X"s at issue we shall arrive at the upshot: Z enc Z iff -(Z enc Z). And now we confront the question: Does this totalitarian X-item Z encompass itself X-wise or not: does that set include itself, that list list itself, that picture picture itself, etc. And paradox occurs immediately because in virtue of its very definition that totality of self-exclusive X" s is itself anX that neither can nor cannot encompass itself. The problem, of course, is that the presupposition that such a negatively totalitarian Z does or can exist is simply false. The item purported to be an instance of the kind at issue (a set, a list, etc.) just does not exist at all: to call it such does not make it such. The preceding difficulties relate to the mis-identification of items purported to be of a type to which they could not possibly belong in view of the way in which that identification is presented. Throughout this range of cases, we have paradox Reijication Fallacies and Inappropriate Totalities 51 of inappropriate presupposition that incorrectly assumes that something is appro- priately identijied by a specification which actually does not and cannot succeed. Any body of reasoning or argumentation presumes, explicitly or tacitly, that its propositions are meaningful and thus that its relevant terms are well-defined. And when terms become unraveled on the basis of an inappropriate totalitarian self- involvement, this crucial presupposition is falsified and the paradox dissolved. When the charge of illicit totalization can be made to stick, it is a highly effective paradox-buster. 3. The Root of the Problem Recall that the identification operator (tx)Px so functions that (l) it is undefined unless there is just exactly one specifiable item that has the property P, and (2) when there is just exactly one such item, then (tx)Px is (identical with) that item. And here saying that (tx)Px is properly defined-symbolically E!(tx)Px-is to say that it represents a unique item belonging to the range to our universal quantifier "i. Thus understood, (tx)Px will be defined only when P plainly and unequivocally applies to a single object. And it remains undefined whenever: (i) P is uninstantiated: it applies to nothing at all (Px is always false). (ii) P is multiply instantiated: it applies to several distinct objects (Px is true for several values of x). (iii) P is problematic: it is unclear where it applies (Px is not clearly determinable for some potential values of x). Now let Il represent membership in a collectivity (of some, i.e., any sort). Then we may define: (T) (u)Px = df (ty)Vx(xIlY == Px). Accordingly, (u)Px-or

as symbolized above-is the (Il-correlative) collectivity of all the items that have the property P. Specifically when Il is E the set membership of the mathematical theory of sets, then (u)Px = df (ty)Vx(x E y == Px). And it should be emphasized that (tx)Px is undefined whenever-as in this E- oriented case, thanks to Georg Cantor's Power Theorem-there fails to be just exactly one single unique x such that Px obtains. This of course means that (u)Px will also be undefined in analogous circumstances. In particular, when P is an "unsuitable" predicate-say because it is equivocal or ill-defined-then (u)Px will remain undefined. On this basis, a properly-defined total exists whenever-but only whenever-the individuals that have the property at issue constitute a unique, well-defined set. The unfeasibility of totalization in unsuitable conditions is now not a distinctive and characteristic phenomenon but simply a natural consequence of the definitional specification of't in terms of t. Accordingly, the principle at issue with the present approach is a limitation on the introduction (or specification or definition) of totalities. It reads: 52 Nicholas Rescher Illicit Totalization Principle (lTP): The attempted specification of a total of items of a certain type is not meaningful unless-when itself is purported of the type at issue-the question of its self-membership can be resolved unproblematically in its favor. To be meaningfully and viably introduced into the discussion (that is, adequately specified, identified, defined) a totality (collection or whole) must not be purported in its introdu.cing definition to include itself. If this principle is violated, then we certainly run the risk of paradox. For note that this principle so functions as to preclude all of the following item specifica- tions: • the set of all sets [including that set itself] (Cantor's Paradox); • the set of all sets satisfying a certain condition whose satisfaction by the set itself is unresolved (Curry's Paradox). Given the inappropriateness ofthe item-specification at issue, no separate mecha- nism need be adopted to avert those associated paradoxes. The issue of identifica- tion-legitimacy is a powerful tool that suffices our needs here. No separate more elaborate mechanism is required to avert paradox in these indicated cases. And so, given definition (T) it follows theorematically that totalization hinges on the nature of the factor at issue: unsuitable predicates demonstrably do not totalize. Accordingly, no special machinery along the lines of a theory of set-types need be adopted to resolve the paradoxes posed by "illicit totalities": those para- doxes simply do not arise because these problematic totalities are not defined. 4. Illicit Totalities There is no problem about the idea of a list of lists or a bibliography of bibliogra- phies; such things exist in plenty and are altogether unproblematic. In principle we can even contemplate a list of all lists or a bibliography of all bibliographies: a complete list of lists or a complete bibliography of bibliographies. But what of the idea of: • a complete list of all incomplete lists; • a complete bibliography of all incomplete bibliographies. If such a work includes itself, then it is ipso facto ineligible for self-inclusion. But if it does not include itself, then it is eligible for self-inclusion and for that very reason fails to answer to its own specification. What this clearly means is that the very idea of such works is self-contradictory, and the work supposedly at issue cannot possibly exist. A list of incomplete lists or a bibliography of incomplete bibliographies represent practical ideas, but totalization is inherently impracticable here. For what we have here is a fallacy of erroneous presupposition analogous with the erroneous presuppositional "loading" at issue in the question "Have you stopped cheating on your taxes?" F or the fact of the matter is that not every otherwise plausible item-specifica- tion can be totalized. Consider such specifications as: Reification Fallacies and Inappropriate Totalities 53 • a list of self-omitting lists • a bibliography of self-omitting bibliographies ~ an inventory of self-omitting inventories • a set of self-omitting sets • a discussion of self-ignoring discussions So far, so good. However, otherwise meaningful item-specifications cannot be totalized. The very idea of a complete list of all self-omitting lists (or bibliogra- phies, etc.) is absurd. What we have here throughout is a problem of one uniform phenomenon: the illicit totalization of the item at issue. But just why should it be that we cannot totalize on those otherwise practicable and unproblematic characterizations? This question needs and deserves a detailed scrutiny. Observe that the Illicit Totalization Principle is not one that governs the mem- bership of totalities-let alone their existence as such-but one that merely ad- dresses the proprieties of how totalities can meaningfully be introduced upon the stage of discussion and consideration. It deals with issues of communicative pro- cedure, not issues of existence as such, and is accordingly a principle of logical grammar rather than one of ontology.7 The principal lesson of this discussion, however, is that recourse to elaborate formal devices for paradox evasion such as the theory of types can in various cases be averted by the employment of a very straightforward and far less elabo- rate informal device, namely the eminently plausible proscription ofidentificatory self-reference. 5. A Russellian Digression What is at issue here is closely allied to-but yet not identical with-a principle that Bertrand Russell took over from the French mathematician Henri Poincare and which, following him, Russell characterized as the Vicious Circle Principle (VCP): No collection (whole or totality) can contain members that are defined in terms of itself: specifically, no col- lection can ever be a constitutive part of itself.8 As it stands, this is clearly a limitation upon the constitution of totalities-and a very strong limitation at that. To quote Russell: "Whatever involves all of a collec- tion must not itself be one of the collection."9 Saying that such a "collection" "has no total" is to say that it does not exist as a collection. What we have here is a restriction on the sorts of collections that can exist-that is, upon how authentic collections can validly be constituted. And Russell's approach has a serious drawback. As he saw it, we must not stake claims about "all propositions" or about "all properties" because such locu- tions involve violations of his Vicious Circle Principle. lo However, he was so intent on barring illicit self-involvement that he insisted on barring self-involvement in 54 Nicholas Rescher general. In seeking to eliminate paradox, Russell also similarly dismisses a great deal of innocuous stuff as well-including such tautologically harmless universalizations as "All meaningful propositions make an assertion of some sort" or "All properties can be attributes to some sort of object," or "All objects can be members of collections." Someone prepared to subject logic to the requirements of common sense might well see this consequence alone as vitiating his version of the Vicious Circle Principle on grounds of throwing out the baby with the bath water. To recapitulate: while the Russellian Vicious Circle Principle is ontological in its nature, the presently contemplated Illicit Totalization Principle is merely com- municative or semanttcal-it only deals with the expository proprieties of how items of discussion can meaningfully be placed upon the agenda of consideration. The one principle deals with matters of actual existence, the other merely with matters of appropriate specijication. Against this background it should be ob- served that the bearing of our present analysis of the implications of "vicious circularity" is purely terminological. It pivots upon showing that various puta- tively identifactory specifications will not succeed in placing a certain putative item upon the stage of consideration. And the rationale at issue is simply that a particular individuative presupposition is violated by the use of such an expression, thereby failing to enable the expression in question successfully to establish its intended reference. An instructive lesson thus emerges. Russell sought an existence-negatory ra- tionale for rejecting certain totalities. And he thought to find this in a conception of inappropriate existence claims based on a complex theory of types. But in fact a much simpler rationale is available. It lies in construing totalization (,;) in terms of definite description (t) as per definition (T) above. For then the fact that definite descriptions fail to identify in certain conditions automatically provides a rationale for seeing totalization in the same light. What is now at issue with improper totals is a problem of identification rather than one of being or existence--an epistemic rather than ontological principle. And it is fortunate that the present approach is far less drastic. What is wrong with the "set of all sets that do not contain themselves" is-on the present ac- count-not its excessive inclusiveness as a presumptive set but rather the anaphorically self-invoking back-reference-via the expression "themselves"- that is at work in its formulation. In short, what the present approach proscribes is not self-referential impredicativity in general, but only its presence in identijicatory contexts. (There is nothing wrong with a descriptive report along the lines that some pre-identified set "is a set that does (or does not) contain itself' as such.) The proscription of impredicative self-involvement in item-specification is thus to all appearances a sensible policy. But now in taking this line in the present context we place a restriction not on existing totalities as such, but on the sorts of predications that can meaningfully be employed in the course of identifying totali- ties so as to introduce them into the discussion. Reification Fallacies and Inappropriate Totalities 55 To avoid paradoxes, then, we need not reject the existence of certain kinds of thing; it suffices to reject the appropriateness of certain particular ways of talking about things. The inappropriateness at issue turns on answering questions based on untenable presuppositions. For the principle we are violating in matters of that totalization is not that of faithfulness to the facts of existence, but that of keeping to conformity with the conditions of discursive meaningfulness. It is-to reemphasize-the linguistic proprieties that are being violated. And this perspec- tive makes it possible to take a more narrowly targeted approach that is not com- mitted to wholesale object dismissal in the manner of Russell's approach. We need not proscribe all talk about "all sets" or "all propositions" but rather must merely be careful about what we endeavor to say with these locutions. II We need to embargo impredicative characterizations only in identificatory, definitional, and similarly item-specificatory contexts where ontological applications are at issue. Russell's wholesale rejection of self-inc lusion--