Notes on Ontology

Notes on Ontology Gustav Bergmann Noûs, Vol. 15, No. 2. (May, 1981), pp. 131-154. Stable URL: http://links.jstor.org/sici?sici=0029-4624%28198105%2915%3A2%3C131%3ANOO%3E2.0.CO%3B2-W Noûs is currently published by Blackwell Publishing.

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http://www.jstor.org Fri May 18 08:18:23 2007

Notes o n Ontology

UNIVERSITY OF IOWA

In a recent book ([9]) Professor Hochberg attempts to derive a contradiction from my assay of acts, first articulated in an essay published in 1955 ([I]) and then developed through three books ([2], [3], and [4]) during the Fifties and Sixties.' One purpose of this essay, officially the major one if I may so put it, is to show that the attempt fails. Both Hochberg's argument and my counterargument will be presented in the second of the three sections into which the essay is divided. Yet this middle section is the shortest of the three. The first consists of two long notes without which the second could not be understood. The last section consists of a series of seven numbered notes of varying length. Both the first and the last, particularly the last, also contain some indications about the major changes that have taken place in my world (ontology) since, more than a decade ago, I stopped reporting them in print. For without these indications the deep-lying ground and the far-ranging ramifications of the present disagreement between Hochberg and myself would almost certainly be missed. T o provide some such indications, however ~ u c c i n c tis, ~the other purpose of this essay. But I shall, as I should, also show that the hard core of my counterargument does not depend on what happened to my world during the Seventies.

Natural, improved, and ideal languages. The English we use, also when thinking and speaking about philosophy, is a natural language. So are French, German, and so on. The ideal language (IL) so-called, or the ideal schema, as I would rather call it, is neither a natural language nor even an artificial one that we could learn to use as if it were a natural language, except about universes of discourse so limited that learning it would not be worth our while. The IL is merely a class of diagrams (the elements of) which some philosophers, including myself, design and,

according to certain rules they also lay down, employ when doing philosophy. The main features of this device are by now familiar; the more obvious issues have been thrashed out. So I call attention to only four such features. ( F l ) Withfour exceptions, of which more i n due course, every primitive mark of the ZL stands for an existent. Phenomenologically. on the side of the act, these existents are allsimples. Ontologically, on the side of the intention, they are either things. i.e.,. either particulars or universals, which are among the determinates; or they are subdeterminates, such as, e.g., the connectives and the quantifiers. The latter I call subdeterminates rather than, as is more usual, subsistents, in order to emphasize that there are no kinds or degrees of "existence" but only categories of existents, distinguished by how they combine and by their different degrees of "separability" and "independence." I call them subdeterminates because every member of this category, so characteristic of my world, is much less separable and independent of determinates than any two or more members of the latter category among them~elves.~ Adeterminate, finally, is either a thing or a complex or a class; and a complex is either a fact or a circumstance.

( F 2 ) Every well-formed "string" of the IL standsfor a determinate, i.e., for either a thing or a complex or a class. But note that I use 'string' merely provisionally and, since all the schemata ontologists have heretofore proposed are written linearly, also for the sake of intelligibility. Yet I have been convinced for quite some time that no linear notation can accomodate an ontologically adequate assay of linear order. For this reason alone, unless I am mistaken, all the candidates so far proposed for IL are merely improved languages. Of this more at the very end. In the meantime, lest all sense of proportion be lost, keep in mind how much good ontology has been done with the help of improved languages. Such a "language", with an image that needs no unpacking, is something in between a natural language and the ideal schema, such as, most conspicuously, some variant of PM with constants added. (F3) Every primitive standing for a thing and every well-formed string of the ZL stands for one and only one existent; and, with a single exception, conuersely. The use of 'same' corresponding to this

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use of 'one and only one' which is the strictest possible, is at the heart of the present disagreement between Hochberg and myself; so I shall soon restate it where it fits best. The single exception, not surprisingly, accomodates the extensionality of classes. Of it briefly much later. (F4) (a) Since in my world conjunctions exist, so do molecular facts such as, say, this'-being-green-and-this'being-square. Yet there are in it no complex characters such as, say, green-and-square. In the IL, therefore, there are, by (Fl), (F2), no derived predicates (as I called them), whether introduced by an abstraction operator or (as one says, metalinguistically) by definitions. (b) Since. in natural as well as in improved languages 'Calpurnia' and 'Ceasar's wife' stand for one and only one existent, there are by (F3) in the IL none of the expressions Russell called incomplete symbols. T o epitomize both (a) and (b) in a way that throws some light on the reasons for them, the ILcontains no abbreviations. The unwieldiness produced by this restriction is truly staggering, so staggering indeed that even an improved language as serviceable for many ontological purposes as the first-generation descendants of PM (one of whidh until the end of the Sixties I mistook for the IL) are much, much closer to a natural language than what I now take to be the IL. (c) In the ZL there are no variables: The reason is that, since by itself a variable stands for nothing, any notation using them prevents one from arriving at an ontologically adequate assay of quantification. The "new" notation for the latter, that I have been using during the Seventies, is fortunately only a bit clumsier than the usual. In this essay, though, it will be easy to get along without the new notation and without variables, by resorting to schemata, written in Greek letters, rather then, as the syntacticists call them, to axioms. So I shall virtually ignore (c). What is the purpose, if any, and what is the possible use of a tool so monstrously unwieldy? The only purpose of the IL, or, if you insist, of the rules by which its designs are constructed and interpreted, is to reflect the world in which we live "perspicuously," not of course in a psychological sense but, rather, structurally. Words such as "picture" or "isomorphism," from which I have long ago learnt to keep away,

have been, and sometimes still are, used to unpack this use of 'perspicuous'. The only use of the IL is to serve as a standard. The IL-philosopher will, and practically must, whenever the difference makes no difference, use mere notations of convenience, i.e., those of some merely improved language. The IL serves merely asthe standard by which he decides whether or not the notation he adopts for what he is about does make a difference for it. Unless he does that, he may get caught in the snare that nowadays catches so many. He will not, as an ontologist talk (in Husserl's phrase) about the things but, rather, as a linguist, about the words people use in thinking and talking about them. Needless to say, I shall follow my own prescription. The assay. A simple paradigm will recall the gist of the assay proposed for the act. Let g,(a) be an atomic fact; then an act that (as one says) intends this fact will be another fact, viz., a second particular exemplifying (at least) a second and a third simple character, g, and A,, g, being the thought, or as I first called it. the proposition that (as I technically use 'intend') intends g,(a), while A, is perceiving, or believing, and so on, depending on what I called and still call an act's species, or less circumstantially, depending on whether the act is a perceiving, a believing, and so on. g,Jtlg,(a) is in the paradigm the connection (not: relation!) between the thought and "its" intention. g2JtlSl(a)isanalytic, while, for every g3# g2, g3Jtlg1(a)is contradictory; just as-I resort for the first time to a semi-schema-every instance of g,Jtl6 except g,Jtlg,(a), is contradictory. As for the paradigm, so mutatis mutandis for every thought and intention. The notation I just used and with one exception have always used is the one Hochberg now also uses. The exception is the notation for thoughts, deplorably counterstructural and therefore now rightly criticized by Hochberg, which I have banned from the IL only in the early Seventies. This improvement, though, makes no difference for either his argument or my counterargument. So I leave for the second note what I think should be said about that hapless notation. Hochberg and I both appeal to a notion of analyticity. The rethinking of this notion I have done during the Seventies produces an explication of it more complex than any one so far proposed. Yet again, this development in my world makes no difference for any context here relevant in which he or I speak of something's being analytic or contradictory. So we can ignore the later e~plication.~ Hochberg and I both use instances of the schemata d p and a! = p; we both take them to stand for something's-intending (meaning)something and something's-being-the-same-as-something, respectively. Yet I had better alert the reader at this first occasion to two interconnected questions we must face. How are the existents, ifany,

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which the instances of the two schemata stand for, to be assayed? What, anything, do 'A' and '=' stand for? When first proposing the assay, I took 'A' to stand for a subdeterminate of the sort to which, in my world, the connectives and the quantifiers belong. Hochberg still attributes to me this view which I abandoned during the early Seventies. For argument and counterargument this change, once more, fortunately makes no difference. Yet, with a phrase from the opening paragraph, if I ignored this particular change, the import of our present disagreement would only too easily be missed. So there will be a good deal about the view I now hold about 'A' (not: A)in the second and the third note. The case of sameness is even more sensitive. The st.rictest possible notion of sameness, already implicit in my 1955 essay.,first fully articulated in a joint paper with Hochberg published in 19576 and here first mentioned in connection with (F3), is crucial for our present disagreement. Upon this notion, "two" determinates that are either things or complexes7 are the "same" if and only if they are one and only one. Or, equivalently on the side of the IL, an instance of a = /3 is, by (F2) and (F3), analytic or contradictory depending on whether or not the expressions in it to the left and the right of '=' are, as one says, two tokens of the same type; i.e., more explicitly for once, if and only if these expressions, whether primitive marks or strings, are mark for mark, including parentheses, if any, different tokens of the same types in the same order. (As you see I still suppose, as I said I would until the very end, that the IL is linear and that parentheses occur in it.) This notion and nothing else I mark by the instances of a = p. Clearly, it differs radically from the familiar defined notion I have called Russell-Leibniz identity, marking it by '=R<. Equally clearly, this older notion, on which we have all cut our teeth, is useful in a world whose IL is (unramified) PM, with extensionality, supplemented by primitive marks to stand for some things. Its usefulness in such a world--call it for brevity's sake a PM-world-rests essentially on the actuality, or if you please, analyticity, of all instances of the schema, 4(a).(a=~~P)>4(P) The argument, letting the two primitive marks 'a' and 'b' stand for one particular, takes for granted, or, as I would rather say, it supposes, first, that a = b is analytic; and it supposes, second, that all the instances of the (half-) schema 4(a).(a = b) >&b) are analytic. ([9]) Then it considers the instance

and concludes-if both suppositions are' granted, correctly-that (1) is a contradiction. With respect to thefirst supposition, I put the matter as I did because I cannot think of any but a misleading senses in which, with the two primitives so employed, a = b would be analytic, and, also, because its being actual, which may indeed be taken for granted, will do as well. Or is it not sufficiently damaging to an assay that one can derive from it what is potential? I turn to the counterargument. If in the (half-) schema 4 (a) . (a = b) >+(b), which the argument uses, '=' stands for strict sameness, then its instances are of course all actual, or, if you please, analytic. The trouble is that, by (F3), there are in the IL no two primitive marks that stand for one particular unless they are two tokens of one type. Thus, if, as I do, one insists that IL satisfy (F3), the argument fails on this ground alone. Yet two further considerations in support of so summary a dismissal are in order. For one, with respect to the second supposition, I cannot but believe that the argument does not use the above (half-) schema but instead +(a).(a=RLb)>+(b). NOW,in a PM-world this schema is indeed analytic, even if the first supposition is granted; in my world, however, with (I speak succinctly) M-contexts being regulated as they are, if the second supposition is again granted, at least one instance, that happens to be the crucial one, is of course potential; hence the second supposition cannot be supported. For another, those prepared to jettison (F3) may wish to inquire whether in the improved language (as I would call it) that may be obtained by adding to the IL two primitive marks--call them again 'a' and 'b'-standing for one particular, the job done in a PM-world by the traditionally defined a= RLbcould not perhaps in my world be done by a stand-in, i.e., by a complex modeled upon the traditional definiens, (9) [B(a)=B(b)],by somehow so broadening, or supplementing, or modifying it that if the complex it stands for is actual, so is not only, for every gn,gn(a)=gn(b)but also the equivalence between any two complexes that falls (still speaking concisely) under 4(a)=+(b). As far as I know there is not and there cannot be such a complex. But suppose a contrario that there is one and try it as a stand-in for a=RLb.Since 'g2.&,(a) =g2&,(b)' falls under 4(a)=+(b), the stand-in will be potential. Hence there is no contradiction where the argument claims that there is one. So again it fails. That much for the hard core of the argument. Hochberg, I should also mention. does not use (1) but adds a further expression as a third conjunction term on each side. I do not see what difference that makes.

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In view of the present disagreement it will help if we look at the joint paper from 1957, where the new strict notion of sameness is fully articulated. The only difference is that we then called it identity. Yet we distinguished it clearly from the older notion I call Russell-Leibniz identity; and we even emphasized the distinction by writing for the new notion and the familiar older one '-' and '=', respectively. (I, a bit later, started writing, as I still do, '=' and '=RL) for the former and the latter, respectively. Nor does the paper leave any doubt that if 'a' and 'b' are two marks for one particular, the thoughts, g, and g3, that intend g,(a) and gl(b), respectively, are two and not one and that, g2Agl(a) and g3Jtlgl(b)therefore will be both analytic. The only change I would now make is to replace "are" and "will" by "were" and "would". The need for the change shows that we had then not yet grasped the importance of (F3). This, though, is not by any means the only flaw of the paper. There is also a good deal of fuss, vacuous in view of (F3) and (F4), about complex characters, definite descriptions, and the Russell-Leibniz notion. Objectively, I submit, these flaws are all part of the price one may have to pay if, at a place where it matters, one fails to appreciate how huge the difference is between the IL on the one hand and, on the other, a merely improved language, even if the latter is at many other places ontologically as serviceable as the one Hochberg and I then employed. Subjectively and speaking of course only for myself, the common cause of all these flaws was the reluctance or even fear to get too far away from PM and, therefore, the inclination to make the Russell-notion of identity appear more useful than it really is.1° This, though, after more than twenty years, can be only a conjecture. As to how Hochberg came to use now in his argument the schema >+P) as (I believe) he did, it is not my place to offer any +(a). conjecture.

The objectionable notation writes ,rpl for the thought, if any, whose intention is p; it makes well-formed all instances of cpJtlp such that the instances of cp and p stand for a nonrelational character of the first Russell type and a determinate that is a fact, respectively." The notation is very convenient, so convenient indeed that I still use it at my desk whenever a fragment of an improved language will do. Yet it is objectionable for at least three reasons. First, it is syntactically counterstructural. Second, it reveals a severe structural weakness of what was then my ontology. Third, and worst, it encourages by omission the belief that in that earlier world of mine there are certain existents I

never meant to be there and which, therefore, I should have taken care to exclude most explicitly. I shall next by my present standards identify these defects. But do not mistake anything I shall say for a defense of that notation. All I may show, incidentally, is that with some luck one may use a confusing notation without being snared by any of the confusions it invites. First.In keeping with the paradigm, since 'g,(a)' is a string, so is ' rg,(a)ll". Yet the formation rules of 1955 and 1957independently of, although of course with a view toward interpretation-endow ' rg,(a) l i v with all the "syntactical" features of the marks intended to stand for the members, all simple, of a single category. In effect, therefore, the marks standing for the members of a fundamental category are, with a view toward interpretation, divided into two kinds; those of one are quite properly primitive, those of the other, most improperly, strings. That is, to say the least, a case of very bad syntactical manners. There is still another, although much lesser flaw of the same kind. Syntactically, the corners are introduced as a new member of the category some call logical constants. In the interpretation, however, there is no explicit statement as to what, if anything, they stand for. This omission, though, is easily accounted for and, I submit, excusable. on the ground that at that time no member of that category had in my world achieved ontological status. Implicitly, therefore, the new logical constant is specified as, were it otherwise defensible, I would still specify it, namely, as standing for nothing. Second. Both the 1955 and the 1957 paper distinguish between the formation rules of the calculus and their interpretation. By virtue of the former ' rgl(a)l Ap1(a)' is well-formed. By virtue of the latter and because of what the world is like, rgl(a)l Agl(a) is analytic. That is of course as it ought to be. A cue as to what is wrong with the way the analyticity is secured is provided by how in this context the author of the 1955 paper used 'define'. Having stated the formation rules of the new calculus, to serve as the IL of his new world, he proceeds to "define" a new notion of analyticity by adding, ad hoc and (as one says) mechanically, to the collection of determinates picked out in his old world by his old notion of analyticity, the collection of all determinates instances of rp7 A@ stand for (and, of course, what becomes deducible from their "union"). Such ad hoc modification, even if as in this case for a good reason, diminishes the credibility of an ontology.12 The thing to do instead is what I have tried to do since, namely, search for a new explication satisfying more than just a single systematic need and such that what (in this case) had to be "added" to the

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extension of the old notion "by definition" falls under the new one, if I may so put it, a principio et ex principiis. Third. In 1957 we said that if '. . . . . 'is a well-formed sentence (of the IL) then ' r. . . . ' will be a well-formed predicate, without adding, either immediately or at least somewhere in the paper, the proviso that ' r. . . . .l ' will be admitted into the IL only if there is in the world at least one act intending (as one says). . . . . . , in which case ' r. . . . .T ' will stand for this thought. Hochberg now correctly observes that omitting the proviso may encourage the belief that in my world there are thoughts unthought. I deplore this unfortunate byproduct of bad syntactical manners that do not either acknowledge or keep away from everything not strictly syntactical. For the rest, I can only say that I have always taken the proviso for granted and, fortunately, can adduce two items of evidence for my good faith in saying it. For one, I have always held and still hold that by theprinciple of Exemplification there is no universal not exemplified by a particular.13 For another, I have always rejected the view, attributed to Frege by many students including myself,14that what he calls thoughts (and concepts) are existents "outside of minds." And I have, with or without reference to Frege, emphasized this rejection by maintaining that, like all "mental" universals, the thoughts of my world are exemplified only by the particulars in acts and therefore "in minds." Thus I had no structural motive against and two very potent ones for adopting the proviso.

..

How are the existents the instances of a = pstand for to be assayed; and what, if anything does '=' stand for? Stay for the moment with the simples I call things. Nonphilosophers will rarely, if ever, have occasion to intend some particular's being thesame as itself. Not so for diversity. We all are quite commonly, at the very phenomenological rock bottom and even with immediate evidence, presented with one thing's being diverse from another, universal o r particular. Because of this phenomenological advantage of diversity ( a # P) over sameness ( a = P) I attend to the former, using 'a # p' merely as a notation of convenience (abbreviation) for '-(a # P)'. So we had better restate the questions. How is what (an instance of) a # p stands for to be assayed; and what, if anything does '#' stand for? Turn once more to our paradigm, the fact g,(a). Its constituents are not only the two things, g, and a, but also a subdeterminate, of the sort I call exemplification, which (as I used to say) "ties" g, and a into the complex that is the fact. This feature, so characteristic of my world, I

have already defended so often that I shall not here defend it again. But consider against this background a # g,. Given the two things, a and g,, there obviously is no need for an existent, subdeterminate or whatever, to tie them into the complex a # g,.Or, to say the same thing positively, given "two" things, there is eo ips0 a third, I a l l it their diad, that is a complex whose only constituents they are, and that is actual or potential depending on whether the "two" are diverse or the same.,'#' thus stands literally for nothing. l 5 Such marks, of which there are in the IL all together four, I call diacritical. They are of course the four exceptions mentioned in (Fl). As for things, so for all determinates. There is for any "two" of them eo ipso a third. viz., the diad whose immediate16 constituents they are. A diad is one of the three sorts of complexes I call circumstances. The other two we shall encounter in the next two notes. A few further comments on sameness will fit better after these notes. First of all, though, for two features of it. First. Remember the condition, imposed earlier, requiring the two terms of a sameness to be determinates. On the side of language, that makes, say, ' v # 3'and '3# a' ill-formed. The two grounds of the requirement are, first, on the side of the intentions, that only determinates are sufficiently "independent" for any two of them to sustain a diad,17 and, second, on the side of the act, that only a determinate is sufficiently "separable" to be the whole of an intention. This is of course merely a part of the unpacking of the two metaphors (see fn. 3); yet it should give the flavor. Second. PM assays sameness and diversity as what the instances of a = R L P and stand for, even though upon this assay there will be an instance if and only if the two determinates are things of the same Russellian type. Hence, if one accepts Russell's assay, he cannot literally say18 that two things of different types, or a thing and a fact, or a thing and a class, and so on, are diverse. Yet are we not sometimes with equal evidence presented with actual diversities of these sorts? Some may even insist that in these cases the evidence is still stronger than in those to which Russell limits us. T o me, who in this respect uses 'evidence' like Brentano, the comparative makes no sense. Be that as it may, why did so many philosophers accept Russell's assay? One cause I can think of is the tremendous influence which, quite justifiably in many respects, he has had on us; another is the pervasiveness of a more o r less inarticulate nominalism.

-

How are instances of COUPto be assayed and what, if anything, does 'JU' stand for? Stay with the paradigm; suppose that there is a thought, gz,

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intending9, (a).ThengdB,(a)will be analytic; all other instances of the two (half-) schemata g&p and awUBl(a)will stand for contradictions. Hence, always provided there is a thought intending a determinate, the two single each other out, uniquely and without any appeal to what is not either analytic or contradictory. In view of this feature, no "tie" is needed to connect a thought and its intention with each other. Nor, considering also that the thought is a simple, is there anything else the ontologist needs to do about it, except, of course to recognize that in this world 'JU' stands for literally nothing and that the complexes the instances of awUP stand for, i.e., as I call them, the meaning nexus, are a second sort of circumstances. But once again the argument is rather summary; so three comments are in order. First. I repeat for emphasis, the only further existents, in addition to the thoughts themselves, the ontologist needs to admit into his world in order to account for the meaning nexus is a new sort of circumstances whose members are actual (and analytic) if and only if their immediate constituents are a thought and its intention. Some may think that just a round;about way of reluctantly recognizing what Descartes called the indivisibility of mind and some more recent thinkers the unity of consciousness. As far as I am concerned, there is nothing round-about and no reluctance. I am pleased to have made a place in my world for what corresponds to one of the fundamental gambits of a much earlier ontology I greatly admire. Second. As long as 'A' was classified as a "logical constant" without any of the marks traditionally so called being given ontological status, I was at ease. As soon as the latter were made to stand for something, the need for a clean-cut and perspicuous taxonomy put me under pressure to ontologize 'A. Yet I was reluctant. The reason I was is that to grant 'A' the same ontological status as exemplification, the connectives, and the quantifiers overextends man's place in the world by spreading all over it, if only after a fashion, a feature "anthropological" in the wrong sense. If you ask me what the right sense is, I shall'answer that the phenomenological rock bottom of our conscious states is the only natural as well as indispensable taking-off place of all ontology. Third. The particulars of this world, being the only determinates that exemplify temporal universals, have long been the only ones literally "in time." Yet, by the Principle of Exemplification, some universals, including surely all mental ones, may, in a sense, be said to have a "beginning in time." For universals, mental or otherwise, that does not bother me at all; for something presumably of the same sort as exemplification, however, it then did and still would.lg

The ontology of classes is as elaborate as their phenomenology is delicate. The latter we must ignore; for the former a small finite class will serve the purpose. Consider, then, the class such that every element of it is either a or b or c. Attend first to a feature classes share with thoughts and with nothing else. Just as there is in our natural language no way of speaking about a thought except by such phrases as 'the thought that.. . . ', 'the belief that. . . ', and so on, so there is no way of speaking about a (finite) class except by means of a phrase 'the class such that. . . ', where the dots (in the finite case) stand for an enumeration clause corresponding to the one just italized. Yet there is also a difference. Thoughts are simples, classes, having elements although no constituents, clearly are not.20That brings us up against a new question. Which are the strings that, either in the IL or in an improved language that will serve, stand for classes? For a little while I shall dodge it by writing for our paradigm '{a. b, c)', a notation mathematicians sometimes use in elementary textbooks. Then, if we agree to mark being-an-element-of by the customary 'E', we can tackle the old question raised in the first sentences of the two preceding notes about '#' and 'A'. How shall we assay elementhood, i.e., in the paradigm the instances of the (half-) schema ae{a, b, c), and what, if anything does 'E' stand for? It is (as one says) the very nature of a class, or, more strongly still, it exhausts its nature that it gathers into a single determinate, viz., itself, all those and only those determinates that satisfy "its" enumeration clause, or, should there be several, of which presently, all of them. Thus there is no need for a subdeterminate 'E' would stand for, by virtue of which (an instance of) a ~ { a b, , c) will be actual or potential depending on whether or not a satisfies the enumeration clause. Also, as we shall presently have occasion to notice in the finite case, a-determinate's-satisfying-an-enumeration-clause is not just actual (or potential) but analytic (or contradictory). That is why in this world elementhood is assayed as the third sort of circumstance and why e standsfor literally nothing. Or, equivalently , a circumstance that is eo ips0 there. Which string or strings stand in the IL for {a, b, c)? One cannot answer unless one first familiarizes oneself with the notion of a "canon." Again a paradigm will do. "Given two complexes, there is, given that the subdeterminate marked by ' v ' is also there, the third complex which we call their disjunction." That is one of the "canons" that regulate how from certain determinates, either simples or already built ones, certain others are build. Each canon corresponds to a formation rule of the IL and conversely. Such a rule, we know, is one of those that specify how the strings of the IL are stepwise build from its primitive~.~~

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The search for the string or strings standing for {a, b, c) falls naturally into two parts. We must (1) find an expression of the IL standing for a determinate that does the job which in our natural language is done by the enumeration clause. Call such a determinate a selector. We must ( 2 ) state the formation rule corresponding to the canon by which a class is built from a selector and whatever else, if anything, is required. (1) Pick any determinate diverse from a, b, c, say 9,(d) and consider the diad22b,(d), b,(d) = a) v (8, (d) = b) V b5(d)= (c).Anticipate that it can do the job and save words by calling it a selector of {a, b, c); call 9, (d) the dummy of this selector; the disjunction, its enumerator.

(2) Consider what has just been said about the very nature of classes and you will see that, given a selector, there is eo ips0 the class it selects. Yet a selector and "its" class are two and not one. One of the two things to be done, therefore, is to choose a design, say 'A', make 'A' our fourth diacritical mark and write for the paradigm the string

The other thing still to be done is to make explicit, within the IL and without appealing to anything not either analytic or contradictory, that the class picked out by (2') is in fact {a, b, c). The way to do that is to rethink (as I did) the notion of analyticity so that all instances of

(3') ffeA(95(d),b5(d) = a) Vb5(d) = b) Vb5(d) = c)) = ( a = a) .v ( a = b) V(a = C) are analytic. As for the paradigm, so mutatis mutandis for each finite class. Nor is that all. The selectors of all finite classes are of a certain sort. T o characterize syntactically the expressions standing for determinates of this single sort, the IL must, as of course it does, accomodate at least one infinite class of type w. Making sure that it does goes beyond the limits I set to this essay. But if for a moment you take it for granted, you will be able to state concisely the canon for all finite classes: I f S is a determinate of a certain specifiable sort, then there is eo ipso the class,XS, of which S is a elector.^^'^^ I turn next to three comments.

First. What precedes has been so worded that it allows for several selectors of one class. Let us now look into the matter. (1) "Two" classes are one if and only if every element

of the one is also an element of the other, and conversely. That is of course the familiar extensionality of classes. I know of no notions of class and sameness upon which it is not actual. In this world it is even analytic. (2) Consider two selectors that differ only in their dummies. Upon our notion of sameness they are thus two and not one. Yet they select the same class. That follows by elementary logic from (1) in conjunction with the appropriate variant of the complex marked (3'). (3) Take two instances of (cpv +) v x and cp v (+.\I x), x stand in both for the same such that cp, complexes. The disjunctions are of course analytically equivalent. Yet upon our notion of sameness they are two and not So therefore are the selectors; and one will, by reasoning as in (2), for any finite class with at least three elements arrive at a second multiplicity of selectors such that, while any two of them are diverse from each other, they all select the same class. That is of course the exception mentioned in (F3),

+,

Second. In choosing g5(d) as the dummy of the pardigm I planted a cue. One who accepts (I speak concisely) the PM-way of assaying classes, also accepts the restriction which already Zermelo threw off, that all elements of a class must be either things or classes of the same Russellian type. The restriction is unnatural because, first, classes have no type, and, second and even more fundamentally, it is part and parcel of the very idea of a class that no two elements of one need have anything in common, except of course that they must all be determinates. Notice. then, that by using the strict notion of sameness in the enumerators we have without any further effort got rid of that unnatural restriction. Third. Some think it outright absurd to ontologize exemplification, some others find it hard to understand how an ontologist could fail to do so. Now I do not expect in a brief comment to settle this ongoing debate to which I am a party. Yet one such comment may throw some light on it and reduce the heat. Write, then, as we presently shall, 'r)(g,, a)' instead of the customary 'g,(a)'

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145

and consider it together with 'ae{a, b, c)'. One side ontologizes exemplification by making '7'stand for a subdeterminate. Both sides agree that 'Z stands for nothing. Yet there are nowadays many ontologists in whose worlds there is only one sort of existents, viz., classes, or at most two, viz., individuals and classes. Hence they must, and do, assay universals as classes; thus from where I stand they mistake '7'for 'E'. That, I submit, identifies the real burden of the debate, namely, whether or not there are radical differences between universals and classes.

We are ready for what still ought to be said about the issue of diversity and sameness. (1) When designing an artificial notation one may decide to introduce two marks, say again, 'a' and 'b', for one thing. In this case 'a = b' does not stand for anything in the world but merely reflects a "linguistic rule." (2) 'a = a' is ill-formed. Jointly (1) and (2) paraphrase what I take to be the position the author of the Tractatus (4.241, 4.243) has taken on the issue. Diversity and sameness, unless I misread him, are literally nothing. If the formula is read to assert that '=' and '.#' do not stand for anything in the world, 1 agree. Notice, though, that when it comes to 'transcribing" into the IL a certain intention, one must look at the world and not just at "linguistic rules"26in order to decide whether to write 'g,(a)' o r 'gl(a) . g,(b)'. That alone suffices to convince me that the circumstances called diversitites and therefore also their negations, called samenesses, have ontological status. With (2), therefore, I disagree. Yet one may fairly ask of what use, if any, are the expressions that stand in the IL for either a strict diversity or a strict sameness. I divide my answer into three parts; the first is general, the second and the third are specific. First. Every ontologist using an IL cannot but be impressed with the distinction between what can be said literally, i.e., in his IL, on the one hand, and, on the other, although he may not put it this way, what can be said only nonliterally in ontological discourse (see fns. 3 and 18).Nor, considering the style in which he has chosen to philosophize, can he be insensitive to the challenge of establishing the exact limits of what is literally sayable.

Second. We have already twice used the new assay of sameness and diversity in order to do what (I claim) cannot be done without them. We have, for one, got rid of a most unnatural restriction on all classes; and we have, for another, ontologically grounded all finite ones, which is indispensable for grounding elementary a r i t h m e t i ~ .But ~ ~ let me hurry to add, in order to dispel the suspicion of exaggerated claims, that the mathematicians, who quite properly for their purposes proceed axiomatically, do not and need not bother with that restriction. Nor is it their business to ground ontologically either classes, or order, or anything else. This privilege, however, works both ways. Axiomatics as such cuts no ontological ice. The third and last part of my answer will take some time. So I had better anticipate the outcome: If there were in my world no diads, I would not know how to ground order ontologically. The assay which does that is the last major change my world has undergone in the early Seventies. Let r, be a binary relation either asymmetrical or nonsymmetrical, a and b two particulars such that the two intentions, r, (a, b) and r, (b, a), differ in a familiar way. When presented with either we are also presented with which of the two it is. Hence there must be in each intention an ingredient such that the difference between the latter two "ontologically" grounds the difference between the former. From where I stand, rejecting this inference amounts to rejecting the very idea of ontology. Positively, therefore, the task in this paradigm case is to specify these ingredients and, in the general case. to uncover the ontological ground of all finite linear order. But let me first, negatively, block the entrance to a blind alley. The order in a paradigm, one may at first blush surmise, is in our natural as well as in all artificial languages, either spoken or written, "represented" by the linear order, either temporal or spatial-'a, b' versus 'b, a ' - o f the marks for the particulars. The trouble is that, if one stops there, order is "represented" by order and thus remains ungrounded. Or, equivalently, it remains unassayed because it appears, say, on paper not just "represented" but i n ipsissima persona. This is not to say that when an ontologist speaks literally about the geometrical patterns called calculi, he may not speak about the order or orders he finds in them. The point is, rather, that when in ontological discourse he presents his candidate for the IL, he cannot without circularity make an order in it stand for the ingredient which he wants to claim grounds order.28As for order, so for sameness and diversity, the integers, and so on. In the Tractatus, for instance, we saw that sameness and diversity appear on paper i n ipsissima persona.

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Turning to the positive, let us first recall what the axiomatic set theorists do about order. Starting from the class {a, b), they form the two classes {a, {a, b)) and {b, {a, b)), call them ordered pairs, writing for them the abbreviations '(a, b)' and '(b,a)'; proceed to define ordered triples such as (a, (b, c)), writing for it '(a, b, c)'; and so on. Then they prove that the classes (a, b), (a, b, c), and so on, satisfy the axioms of (finite) linear order. For the purposes that are their exclusive concern this will do very well; ontologically, however, this "axiomatic reconstruction" of order does not amount to an assay of it.29Its idea we owe to Kuratowski; Bernays first fully developed it.30 This machinery of the mathematicians I take over without any formal change. I merely shift its "interpretation," or, as I would rather say, using a serviceable notation of convenience with Greek-letter variables, I let the latter stand for any determinate whatsoever; '(a, P)', as once before, for the diversity or, as I also called it, the diad of a and P; '( a. p, 7)' as an abbreviation for (a, (P, y)) and so on. As you see, I even borrow the classical notation. But since, as I use it, (a, p), (a, P. y), and so on, are all diads, I call them all n-tuples, in order to set them apart from the mathematicians' ordered pairs, triples, and so on, which are all classes. Formally, the shift seems minor, ontologically it is decisive. So I shall next try to convince you of that and only then insert the 2-tuples (a, b) and (b, a) in the assays of rl(a, b) and r1 (b, a), respectively. The foundation or basis of this world are its things. From this basis every existent is built, or generated, by a series of steps, each step being regulated by one canon. As a result of this mode of generation, this world is,roughlyspeaking, a layered structure. Yet, rough as the image is, it suggests another, equally rough, such that the two illuminate each other. Moving upward from the basis, some categories come "before" some others; some are "closer" to the top or to the basis as some others. Two examples will help and at the same time serve the immediate purpose. Since, given two things. there is eo ips0 their diad, what could be closer to the bases than certain diads? At the other end, it is very easy to produce from the IL of our world that of one otherwise like ours but without classes. All one has to do is to delete 'A'and 'E' and, of course. all the expressions in which they occur. That surely puts classes at or close to the top. Suppose now that you have the choice between assaying the ground of order in, say r,(a, b) by either the diad (a, (a, b)) or by the class{{a), {a, b)).31 Upon the first alternative, which this ontology permits, you find the building stone needed in a layer still closer to the basis. Upon the second, you would have to reach up for it into a layer at or close to the top. That makes the second alternative counterstructural in a very bad sense; for it cannot but undermine the confidence in an ontology that so "scrambles" the taxonomy of its existents. I, for one,

had I to make a class a constituent of an atomic fact, would rather give up. That is one reason why I think the shift from order classes to order diads is so important. The other reason, though perhaps less striking, is also structural and equally important. Since any two determinates, of whatever sorts, are the immediate constituents of a diad, the latter are scattered throughout the structure. Thus, if for instance in the two schemata aJtlj3 and aep. a and p range over all expressions standing for determinates, the corresponding diads will always be available; similarly, with the ranges limited to complexes, for the schema a>P; and so on. Of that a bit more in the last note. Suppose I am presented with this'-being-green. In the familiar artificial notation 'g,(a)' stands for this fact. At the rockbottom phenomenological level I am in this case presented with two things and a characteristic togetherness of theirs. Both the phrase of the natural Ianguage and the artificial notation reflect that very neatly and very nicely. In the latter, 'g,' and 'a' stand for the two things; the two halves of the parenthesis, for the togetherness I ontologize as a subdeterminate. The way the four marks are arranged on the line even suggests the metaphor of a "tie." Yet, if one wants to arrive at the IL, he must once more transcribe, or, as I shall say, he must standardize the current transcription, even though the result-call it briefly the standardization-does not fit the phenomenological rock bottom as neatly and as nicely as, in the paradigm, 'g,(a)'. Such a standardization will be acceptable if and only if it satisfies two conditions. (i) The rearrangement of the assay it represents contains all those and only those existents that either are in what it is supposed to stand for or are eo ips0 there if some of the former are there. (ii) With the exceptions mentioned in (Fl) and (F3) the one-one correspondence between the standardized expressions and what they are made to stand for remains one-one. (ii) is, if anything, even more obvious than (i). Keeping all this in mind, I now standardize 'g,(a)' as follows: (1) I start from the diad (s,, a).

(2) I ontologize the characteristic togetherness of exemplification not as a "tie" but, rather, as a subdeterminate ('7') that in its dependence "clings" to a single determinate and for which by

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the standardized canon, (g,, a) is an appropriate determinate to cling to.32 (3) I write in the IL not 'gl(a)' but, instead, 'r)(g,, a). If you check you will see that this standardization satisfies (i) and (ii). What for nonrelational atomic facts is a possibility, becomes for relational ones a necessity; hence in the next to last paragraph the italicized 'must.' Instead of 'r,(a, b)' I write in the IL 'q(r,, (a,b))';instead of 'r,(b,a)', 'q(r,, @,a))',and so on. Again, conditions (i) and (ii) are satisfied.

By ( F l ) there are four diacritical marks, '#', 'A', ' E ' , 'A'. The last three will do as they stand. On = we can improve.33It has in fact already been displaced by the diad's being written '(a, P)'. But the latter, too, you remember, was introduced as a provisional notation. The reason I called it that is that, the two ultimate constituents of a diad not being ordered, instances of (a, P)and (P, a)stand for one and only one diad. That violates, as I have so far deliberately ignored, our standards for representation in the IL. Why then not replace the parenthesis by a circle and inscribe a and /3 into it in any arrangement whatsoever, e.g., a in the lower right, 0 in the upper left of the circle; or both in the right half; or one above the other around the center; and so on? Clearly, as far as diads are concerned, linear order has no representative role left in this notation. Should the idea carry through, I am thus on the verge of making good my promise to construct a nonlinear IL. How shall we "delinearize" 'q(gl, a)'? Here the standardization comes in handy. Draw first a circle around the two primitive marks '9; and 'a', then enclose both this inner circle and the primitive mark '7' by an outer circle. T o see that this does the job, remember first that q is unary, and notice, second, that if one tried to read the outer circle as standing for the "diad" of q and the diad e l , a), the expression would be ill-formed, since by the corresponding canon no subdeterminate is sufficiently independent to become an immediate constituent of a diad. or, for that matter, of any circumstance. Recall now how freely parentheses are used in strictly formal syntax. Even the simple disjunction ordinarily written '9, (a).Vg2(b),' becomes, when standing by itself, '(gl(a)) v (g2(b))'; while when entering into a larger complex it becomes '(& (a)) V b2(b)))'. My new notation uses with the same cautionary profusion, not ofcourse the two halves of a parenthesis, but a circle. Circles are thus used in two ways, once like a parenthesis, once as a diacritical mark for being-diversefrom. So I must next show that the double use does not produce any

'

ambiguity. The structural reason it doesn't is, broadly speaking. that a parenthesis, merely directing us what to read together in an inscription, must not and cannot be o n t ~ l o g i z e d But . ~ ~ we must look more closely. Negation (-) and the diacritical 'A' are unary to begin with; The schemata a Vb and a>&I replace first, as already mentioned, by V (a, p)and >(a, p), respectively,and then transcribe the latter info the circle notation. As for "or" o r "if-then," so for (all binary connective^^^ and) the two subdeterminates that take in the new assay of quantification without variables the place of "all" and "some." That takes care of all the subdeterminates represented by primitive marks on their own. There is nowhere any ambiguity due to the double use of the circles. As for these, so a fortiori, since they stand for literally nothing. for 'e' and'&. 'aep', for instance, first becomes '€(a, P)'; the latter in turn is transcribed by an outer circle surrounding 'E' and an inner circle for '(a@)'; the latter in turn surrounds two circles, one for 'a', one for '(a, p)'; and so on. Stop to reflect and you will see that I have already made good the promise to delinearize the IL. Yet one question remains. Granted (a critic may say) that you have produced such a pattern, is there not in virtually all your designs an "order," in the mathematicians' broad sense of the word? Are there not, in particular, as parts of such an "order," in the design for every complex of some "complexity" several longish threads, or fibers, of linear order; one circle's being inside another, the latter inside a third, and so on? I answer in two steps. The first is by now obvious, the second perhaps less so. In the first I point out that whatever the syntacticist, whom we know to be just a geometrician with a special inierest, wants to say about the "order" and the several linear orders in my diagrams, can be said literally in the IL, in which order has been assayed as above. The critic grants that, too. Yet, he insists, there remains the feature that this order "reflects" the order in which the appropriate canons are involved in building the complex. This challenge I meet, in the second step, by making a claim. A canon is not an existent; nor, therefore, is any order of any number of such. Hence. there is nothing for the geometrical order to "reflect," or, as it was put, to make its appearance in ipsissima persona. The claim is very plausible indeed. If you accept it, this essay concludes with a satisfying kind of closure. But that makes it only more appropriate for me to conclude by pointing out that to embed the claim accurately into its full context and then support it, phenomenologically and dialectically, requires another essay. largely but by no means exclusively phenomonological, of the same length.

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[ l ] Gustav Bergmann, "Intentionality" Archivio di Filosofi, 1955. Meaning and Existence (Madison: University of Wisconsin Press, 1960). [2] , Logic and Reality (Madison: University of Wisconsin Press, 1964). [3] , [4] , R e a l i s m (Madison: University of Wisconsin Press, 1967). "Sketch of an Ontological Inventory," Journal of the British Society for [5] , Phenomenology, 1979, vol. 10, 3-8. [6] , The Metaphysics ofLogica1 Positivism (Madison: University of Wisconsin Press, 2nd ed., 1967). [7] P. Bernays, "A System of Axiomatic Set Theory-Part I," Journal of Symbolic Logic, 2, 1937. [8] Hector-Neri Castaneda, ed., Action, Knowledge, and Reality (Indianapolis: BobbsMerrill. 1975). [9] Herbert Hochberg, Truth, Facts, and Reference (St. Paul: Minnesota University Press, 1978).

'[I] is reprinted as the opening essay of [2]. .2Asit happens, though, there is another, [5], that supplements this one; or, rather, the fit is so close that the two unpack each other, thus remedying to some extent the concision of either. And there are reasons, if not excuses, for the concision of either. The present essay is written around the reply to a specific criticism. The other is the English text of a lecture delivered in February 1978 at the University of Valencia, hereafter referred to as the Valencia paper, of which a Spanish translation is scheduled to appear in Teorema . 3Classes, you will gather from the way matters are stated above, occupy a position intermediate between complexes and things. Among subdeterminates only the subcategory comprehending the connectives and the quantifiers is sufficiently "independent" and "separable" to allow and require representation by primitives of the IL. Among the rest, the least "separable" and most "dependent" are the two modes, actuality and potentiality, one and only one of which "pervades" every complex. The words between double quotes are of course merely metaphors. To unpack them is among the tasks of ontological discourse. Such discourse, by the way, is not a fourth kind of language but, rather, a very special kind of discourse in a natural language. For a concise explication of its nature as well as some indications about the taxonomic schema as a whole, see the Valencia paper. 4Isay virtually because of a few glancing comments later. Russell's representation of classes by incomplete symbols I ignore here completely. 5Noticethat, as I prefer to speak, what isanalytic or ontradictory is not an expression of the IL but the complex it stands for. Again it suits me better to call a complex either actual or potential rather than to say that the expression standing for it is either true or false. About the two modes, actuality and potentiality, one of which pervades every complex virtually nothing will be said in this essay. Yet I cannot resist mentioning a most strategic development within my world during the early Seventies. In [4] (p. 281) I still lamented vociferously that no complex is ever presented to us with its mode and justified, most apologetically, the modes nevertheless being given ontological status as a "dialectical necessity." In the meantime I got rid of this most irksome feature. 6"Concepts,"Phil. Studies, 8. 1957, reprinted in [2]. The first adumbration of the assay, including the notion of strict sameness, occurs in an essay first published in 1950, reprinted in [6]. See ibid. p. 28. There, though, the thought is still a particular and the place of 'A' is taken by 'Des', from 'designates'; which shows that in 1950 I was still a reluctant nominalist and logical positivist. 'Convince yourself that the limitation to things and complexes also controls the working of (F2) and (F3). That the exclusion of classes has something to do with their extensionality has already been mentioned and will be dealt with later. The exclusion of even those subdeterminates that are in the IL represented by primitive marks of their

own, is, on the side of the IL, accounted for by such expressions as 'I=v ' 's being ill-formed. The features of the world, or equivalently, on the side of the intentions, that in turn account for their being ill-formed, will be more profitably discussed after we shall have become acquainted with the four exceptions mentioned in (Fl). SButdo not forget the familiar limitation. If the calculus contains variables, say 'x', 'y', and so on, an instance of the (half-) schema ~+(x).(x=,~y)>I+(y) may not be analytic if a free occurrence of 'x' in 4 lies within the scope of a quantification over 'y'. gFor support of this judgment see later (fn. 26). 1°For clinging to the ontological status of "derived characters," I did have specific and intellectually respectable motives. Yet, having finally discovered, again at the beginningof the Seventies, how to do without them thejobs I had until then thought only they could do, I promptly got rid of them, some time before, independently, Reinhardt Grossmann dispatched them in his contribution to the Sellars Festschrift. ([8]): pp. 129-46). "But notice that if in this world of 1955, say ' ra7 ' and ' rg17 'had been well-formed, they too would have been strings, and let me use the opportunity to report still another change of the early Seventies. In that change, every instance of aA/3became well-formed provided only (the instances of) both a and /3 are determinates; which of course makes all the members of several subcategories transparently potential. Yet the inconvenience, if any, of such "ontological wasteland" is, I believe, many times outweighted by the structural, or, if you please, systematic reasons for the change. I21n the case at hand a probable cause of this shortcoming was, I submit, that the explication of the notion of analyticityto which I clung until the early Seventies did not, as its successor does, contain as an essential part a painstaking phenomenological description of the conscious state to which a complex i s presented with its mode, i.e., as either analytic or contradictoly. Nor should it come as a surprise that, on the side of the act, it involves a notion of evidence structurally descended from Brentano's. See also the Valencia paper. 130r,in case the universal is relational or of a higher type, by either several particulars or either one or several universals of the next lower type. In this respect I am still as good an anti-Platonist as any. But I would rather avoid the term, saying instead that, by the Principle of Acquaintance, we cannot intend an universal unless it either comes exemplified or has been so presented to us at an earlier occasion. For the rest, everything not merely "speculative" (and in which I am therefore not interested) is taken care of in the IL, by the unabbreviated equivalents, however cumbersome, of Russellian description. 14Seemy papers on Frege. ([2]: pp. 205-34 and [3]: pp. 124-57). 15Using for once a traditional phrase in order to call attention to a structural similarity, this feature makes diversity and the other two sorts of circumstances the only internul rehtiow of this world. Ordinarily, though, I avoid the phrase, lest it blur the distinction between what is literally nothing, on the one hand, and, on the other, either relational universals, which are things among things, or certain subdeterminates such as, say, exemplification and disjunction, which, once given ontological status, may because of their (apparent) binary feature be mistaken for relational universals. (The reason for the bracketed 'apparent' will become clear only in the sixth note. IeI add 'immediate' because a determinate that is a complex has itself constituents and because in ontological discourse the use of 'constituent' is most conveniently so regulated that constituency is transitive (and irreflexive). "Or any other circumstance; so we shall not need to return to this feature; but see fns. 7 and 11. I8As I use "literally sayable" in ontological discourse, what can be literally said can be expressed in the IL, and conversely. See also the Valencia paper. lgThisthird comment is really but acomment on the second. But notice, finally, that the corners, as in ' rg,(a)i ', having been introduced as "logical constants," were never meant to stand for anything. Nor, if for a moment we resurrect that notation, do they. That, though, does not make the notation lesscounterstructural. Rather, it shows that by introducing two diacritical marks, 'At and ' r. . . 1', where one will do, it is redundant. 20Constituency,we remember (fn. 16),is transitive; elementhood, every one knows, is not. Also, classes, like things but unlike complexes, have no modes. All that goes to show

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that, as already mentioned, classes occupy an intermediate position between complexes and things. 21'Building'is in this paragraph used in two ways, once syntactically and therefore literally, once ontologically in ontological discourse. If you are aware of the ambiguity, both metaphors are harmless. 'Canon', although convenient, is anything but harmless. The double quotes, hereafter dispensed with, are to warn you of the troubles one gets into when assuming that there is in the world anything, whether or not literally sayable, that a canon stands for. But I cannot, and we fortunately need not. here pursue this intriguing point. See also the last paragraph of this essay and, for some indications, again the Valencia paper. 22Theonly purpose of this move, which has no other significant consequence, is to provide a single determinate. If it were not for this move and for the dummy, which as you probably sensed stands in for a variable, the notation I am about to introduce for finite classes is the one which, to the best of my knowledge, Alonzo Church was the first to propose for all classes. 23Noticethat, unlike diversity and the meaning nexus, elementhood requires two diacritical marks, 'r' and 'A'; and, also, how intimately connected the two are by the analyticity of (all instances of) the schema aeA(6,v(6)) = v(a) ,of which (3') is an instance. 24Turningto infinite classes in general, once comes, not surprisingly, upon a hornet's next of difficult questions. Are there "selectors" that secure, more or less in the same style, the ontological status of at least some sorts of them? Suppose there are, how is the analytic connection, mentioned in the preceding footnote, to be secured? What, if anything, beyond making them the counters of purely syntactical games, can be done about infinite classes that cannot be so grounded? And so on. Fortunately once more, for our purpose we need not stay for the answers. Nor of course do I think I have them all. Or, with a twist, not surprising in view of the intuitionist-constructivist position which largely on phenomenological grounds I have come to take, many would disagree with the few answers I believe I have. 25Sincethe one-one correspondence between thought and intention is one of the most fundamental gambits of this ontology, that is also required by the two thoughts intending the two complexes being indeed two and not one. Nor is, in a strict syntactical formulation, writing a "running" disjunction or conjunction of more than two terms without parentheses more than an abbreviation, justified by the analytical equivalence of what any two of the unabbreviated expressions stand for. But, then, in this world of mine analytical equivalence is not sameness! 26Thosewho not only in case ( 1 ) called 'a = b' a "linguistic truth" but also held that all"analytic truths" are "linguistic truths," courted confusion. That is not to say, though, that every one who in this century ever used the confusing phrase entertained so trivial a notion of analyticity; but see fn. 9. 271fyou infer from the last clause that I disagree with the author of the Tractatus (6.031), when he calls classes all together superfluous in the ontology of arithmetic, you will of course be right. 28Noticethat this is but a way of unpacking in a particular case the familiar warning against a familiar vicious circle, or, if you would rather look at it this way, infinite regress. 291n order to see that accurately, one must consider what else they do. They "formalize" their axiomatic system by embedding it into the lower functional calculus. I.e., they add to the latter '=' and 'r' as the only two "primitive predicates" together with the axioms in which they occur, and do, as one says, interpret the result by letting the individual variables run over what they call classes (or sets). Yet they do not '~ormully reconstruct" the order in such fundamental contexts as lacy', 'ya', 'p 3 q'. 'q 3 p', which are outside of their exclusive concern. Ontologically, therefore, order remains unassayed. 30Kuratowski'sclassical paper dates from 192 1. Bernays' ([7]) is the first in a famous series of four papers. 311avoid here for once and for obviousreasons the wedges, '(. . . . . )', I have borrowed from the mathematicians. 32Thedifference between the tying and the clinging metaphor corresponds in our natural language to that between "and" and "not." A mathematician may think of the difference between unary functions (i.e., of one variable) and binary ones (i.e., of two

variables). But I would rather avoid 'function' in this context. Just think of the ways some eminent ontologists have used 'function'. Yet I shall borrow "unary" and "binary." 33'=', since it occurs only in abbreviations, stands of course for nothing; yet, since it does not occur in the IL, it is not counted as a diacritical mark. 34Fora more challenging formulation of this idea see the concluding paragraph. 3 5 Y owill ~ gather that, as already implied, for reasons of my own I ontologize not only the syntacticists' minimum of one but all thirteen binary connectives and negation. But, just as at the very outset we agreed that we may here take quantification without variables for granted, so we need not pursue this piste either.