Synthetic A Priori¹

class would be false. Call one truth more fundamental than another if the subclass corresponding to the sentence representing it is in some specifiable sense larger than the subclass corresponding to the sentence which represents the other. To say that the truths some philosophers call a priori are "merely" more fundamental than those they call a posteriori is to say that the difference in question is "merely" one of degree. Those who disagree must propose a criterion. That is the problem of the synthetic a priori.
   What sort of criterion is acceptable? Let F₁ be a fact. A mind's intending F₁, i.e., thinking of it, either believing or disbelieving it, being more or less certain of it, and so on, is another fact, F₂. While F₁ is a constituent of F₂, the latter also has other constituents, i.e., constituents which are neither F₁ nor constituents of F₁. Call F₂ a knowledge situation. A criterion will be unacceptable if, applied to F₁, it mentions any constituent of any knowledge situation into which F₁ may or may not enter as a constituent, which is not also a constituent of F₁. Positively, one may state this requirement by saying that the problem is ontological; negatively, by saying that to violate it is to fall into the error of psychologism.
   "A truth belongs to the core if and only if the evidence for it is not increased by the number of its instances." That is one of the criteria proposed. Since it applies only to generalities, it applies only to (2) and (4) not to (1) and (3). At least, it does not obviously apply to (1) and (3). But let that pass. The point is that what is meant by 'increasing evidence' could as well be expressed by 'enhancing certainty' or by 'strengthening belief'. The criterion is psychologistic. So are most criteria that have been proposed. I say most rather than all only because I do not want to dismiss the idealistic systems, or, for that matter, Kant's, as mere variants of psychologism. But there is a connection. My now believing that all dogs are carnivorous ( F₂) is a mental fact; their being carnivorous (F₁) is not. Commonsensically, the distinction between the two kinds of facts mental and nonmental, is very sharp. Idealists by blurring this distinction, also blunt that between F₁ and F₂. Or, at least, they are in danger of blunting it. That is the connection.
   What one calls the solution of the problem is always or almost always only the last of several steps. Yet there are differences. In some cases the last step is the most difficult. In some others the preparatory steps require the great effort; the last is easy. As for effort, so for importance. In some cases some of the earlier steps are of much greater intrinsic interest that the last. The problem of the synthetic a priori has never bothered me. Nor I judge it to be of great intrinsic interest.

   Yet I shall at the end propose a criterion which (if I am right) solves the problem. What happened was that, when I finally saw the connection between two other questions or problems, which I do judge to be very important, I found not only the answers to both of them but also, as an easy last step, that criterion. One of the two questions concerns a crucial point in the ontology of space; the other, bare particulars. That accounts for two major preparatory steps in the argument which is officially the burden of this essay. Perfect particulars and elementarism account for two others. In the first step we shall attend to the distinction between internal and external relations.
   While Peter loves Mary, they jointly exemplify the relation of loving. Four and two jointly exemplify the relation of being larger (between numbers). This is the broad use of 'relation'. As I now use it, loving is a relation, being larger (between numbers) is not. Loving and larger are both entities. But larger (between numbers) is a subsistent. Things and subsistents are in several respects profoundly different from each other. From these differences the broad use of 'relation' diverts the attention. One may preserve it, of course, and, as is often done, distinguish between descriptive (loving) and logical (being larger) relations. As it happens, not I think by chance, all those who do that either explicitly or implicitly deny that what they call logical relation has any ontological status whatsoever. That alone strongly recommends the narrow use I now follow. I shall call a relation only what some others call a descriptive relation.
   Many philosophers have tried to establish a dichotomy among relations. One kind they call internal; the other, external. Quite a few divisions have been proposed. The subclasses they yield overlap. Being higher (in pitch), for instance, as far as I know, is an internal relation in all these divisions. Thus, not surprisingly, there is again a common core. But I am not prepared to argue that the criterion I am about to state explicates the idea. Perhaps it does, perhaps it doesn't. Perhaps there is no common idea. However that may be, my criterion explicates one idea of internal relation which (I believe) is important although (or should I say because?) we shall see that by this criterion there are no internal relations.
    The criterion is ontological. Assume that a and b stand in the relation p. Relations obtain between things. If a and b are facts, p is a pseudorelation. Nothing will be lost if we ignore pseudorelations, assume that a and b are things. What are the constituents of the fact p(a,b)? Or, rather, since a,b and their constituents obviously are among them: What, if any, are the additional constituents of this fact?

   Should there be none, I shall say that the relation needs no (additional) ontological ground. If there are some, they may either (a) all be subsistents or (b) at least one is a thing. Accordingly, p is either (a) a subsistent or (b) a thing. If p were a subsistent, it would be a fundamental tie. Such ties can occur between entities which are mere possibilities. A relation cannot be exemplified by such entities. That is just one of the several profound differences between things and subsistents. Hence, p cannot be a subsistent. (b) The thing p, which is of course a (relational) character, tied to a and b by relational exemplification, is the ontological ground of the relation obtaining between them. Relational exemplification, which is a subsistent, is a further additional constituent of the fact. No harm will be done if, together with all other such constituents, we ignore it. Thus, a relation either needs no (additional) ontological ground or its ground is a "third" thing. A relation is internal if and only if it needs no (additional) ontological ground. That is the criterion.
   Sameness and diversity need no ontological ground. Everything else does. Hence, the ontological ground of an internal relation must lie "in" one or both of the two entities related, in the sense in which an entity's constituents and only they lie "in" it. As philosophers have used 'nature', an entity's nature is "in" it. "An internal relation between two entities is grounded in their natures." That is the classical formula. But, then, what is a nature? The use undoubtedly is philosophical; thus it needs explication. Moreover, there are several philosophical uses of 'nature', and their explications are anything but simple. But there is also a minimal use whose explication is not at all difficult, i.e., there is a necessary condition satisfied by any entity that has ever been said to have a nature. The difference between two entities (of the same ontological kind) is either qualitative or numerical; if it is qualitative, the entity can be recognized as such; if it is qualitative, the entity can be recognized as such; if it is "merely" numerical, the entity cannot be so recognized. That is the commonsensical idea. An entity has a nature if and only if it is qualitatively different from all others (of the same ontological kind). That is the condition. I shall not use 'nature' in any other sense.
    Some philosophers, including the great Leibniz, said that there are no relations. What they meant, I believe, was that all relations are internal. This latter view has been held by quite a few, very explicitly. Some others, including myself, hold that all relations are external. So I shall make it my next business to show, by means of example, that there are, upon my criterion, no internal relations. The example, to be fair, ought to be simple; it ought to belong to the common core; the entities related ought to be recognizable as such, i.e., in the minimal

sense, have natures. (1), e being higher than c, satisfies all these conditions. Nor could anyone reasonably object if I modify (1), assuming 'higher' to be an abbreviation for 'higher-by-a-third' (and neglecting the difference between the two kinds of musical thirds). I shall also assume that pitches are things. In my world (ontology) they are. At this point some might object. Let them for a short while hold their peace.
   Consider (c,e), (d,f,), (e,g), and so on. Each of these ordered pairs exemplifies the relation. Take (c,e). If the relation is internal, there are three possibilities. Its ground is either (a) one entity which is a constituent of one of the two pitches; or (b) one entity which is a constituent of both pitches; or (c) two entities, one a constituent of one of the two pitches, the other a constituent of the other. (a) Assume the entity to be a constituent of c. Then the pair (c,f) should also exemplify the relation. Yet it doesn't. f is the fourth of c. (b) In view of the first pair, the entity is a constituent of c and of e; in view of the second pair, a constituent of d and f; and so on. Hence any pair of pitches should exemplify the relations. (One could also argue that in this case the relation would be symmetrical, which it isn't.) That leaves (c). A thing is simple if and only if no other thing is a constituent of it. Assuming that the two entities which are the ground of the relation are things, that leaves two possibilities. (c1) Pitches are simple things. (In my world they are.) Then the two things which are the ground of the relation as exemplified by the first pair must be c and e themselves. But c and e are not constituents of d and f, respectively. (c2) If pitches are complex , then one of two things which are the ground, call it the lower, is a constituent of the lower pitch; the other, call it the higher, a constituent of the higher pitch in each pair. Thus, in view of the pair (c, e), the lower thing is a constituent of c; in view of the pair (d,f), it is also a constituent of d. As for the lower thing, so for the higher. It is a constituent of both e and f. Hence c and f should also exemplify the relation, which they do not. I conclude that there are no internal relations.
   This type of argument is of course familiar. Even so, two comments should help to appraise it. One. Remember the objector I asked to hold his peace when assuming that pitches are things? If you examine the argument, you will see that no use is made of the assumption - except that in my world the dichotomy simple-complex does not even make sense for substitents while, on the other hand, even simple things have constituents which are subsistents; e.g., in the case of individuals, individuality, and existence. Two. Could not the two constituents mentioned in (c) be two subsistents jointly "exemplifying" a "subsistent (logical!) relation"? That possibility has been excluded by another

assumption, namely, that the two constituents are things. To anyone who objects to this assumption I make a gift of that possibility. For he must surely be a "nominalist" who, even though he cannot stomach universals and exemplification among things does countenance them among subsistents. Everyone else, if faced with this sort of "nominalistic" world, will reach for Occam's razor.
   D₁. All relations are internal. D₂. All relations are external. Call D₁ and D₂ the strong doctrines of internal and external relations, respectively. D₁, we saw, is dialectically untenable. One refutes it by showing that no relation can be internal. D₂ follows. It does not follow that one may comfortably settle down with D₂ without mastering its dialectic. The critics of D ₂ press many questions. One among these is (I believe) the structural root of all the others. The critics hold it to be unanswerable. This is the one question I shall try to answer.
   Pitches have natures. Otherwise the example I used to refute D₁ would not have been fair. Being equally fair to D₂ requires that in the example used to defend D ₂ the two things claimed to be externally related have no natures. Are there such things? The question leads rather quickly to another, which is the one, supposedly unanswerable, which I undertook to answer.
   In some ontologies, call them Scotist, all simple things are characters. All characters have natures. In a Scotist world, therefore, there are no things which have no natures. In a nonscotist world some things are individuals. All individuals are simple. But the individuals of most nonscotist ontologies, including all classical ones, have natures. In such a world, therefore, there is again no thing without a nature. The classical entity without a nature is, of course, Aristotelian matter. Structurally, the problem we must eventually face and which no serious philosophy can dodge is indeed the one Aristotle first tried to solve when he invented his notion of matter. A bare particular is an individual without a nature. It is, as one says, a mere individuator and cannot be recognized as such. Some recent ontologies share two features. Their individuals are more or less explicitly bare particulars; their characters all enjoy the same ontological status; there being no difference between properties and relations except the obvious one, namely, the number of entities required to exemplify either the one or the other. I say more or less explicitly because none of these ontologies has fully faced the dialectic of the bare particular. Nor is any of them free of the taint of nominalism. That is why I did not say that all their characters are things but merely that they enjoy the same ontological status. My ontology is very explicit in both respects. It's characters are things; its individuals are particulars.
    Every individual exemplifies at least one character; there is no chara-

actor which is not at least once exemplified. That is the Principle of Exemplification. It states a categorial feature of my world. Since only very few philosophers rejected it. I shall take it for granted, even though Plato was one of those few. From this principle in conjunction with the classical formula from which I took my cue, namely, that an internal relation has its ground in the natures of the things related, it follows immediately that all relations (and peroperties!) exemplified by bare particulars are external. For, bare particulars having no natures, what would an internal relation between any two of them be grounded in? It does not follow that in a world with bare particulars all relations are external. (D₂. Relations among characters, e.g., higher among pitches, might still be internal. That is one reason I chose my first example as I did. Even so, if there are bare particulars then D₁, the strong doctrine of internal relations, is false. All one needs to do, therefore, in order to show that it is false is to present a relation between two bare particulars.
   Suppose someone asks me what c is . I strike the right key, strike some others, strike the first again, tell him that c is what has been presented to him on the first and last occasion but on none of the others. There are some complications; e.g., not only a pitch but also a loudness may have been presented twice. None of these complications is very serious; they are all familiar; and they are beside the point. So I shall not worry them once more, assume that the question is satisfied. In my ontology, what is presented on each of these two occasions is a fact, namely, a particular exemplifying a ptich, a loudness, and so on. The pitch is one; the particulars are two. Suppose now that the questioner asks me to direct his attention to the particular in the way I just directed it to a pitch. Particulars, or, at least, this sort of particular being momentary, they cannot be presented twice. The questioner appreciates the point but insists that what he was in fact presented with on each of the two occasions is a pitch, a loudness, perhaps some other qualities and nothing else. (That shows the appeal of scotism!) Thus he keeps asking me what a bare particular is, demanding that his attention be directed to one. This is the question the critics of (D₂ hold to be unanswerable. So far the defenders have not known how to answer it. Eventually I shall propose an answer.
   (Let it be said once and for all that even if the question were unanswerable, it would not follow that there are no bare particulars. Should they turn out to be dialectically indispensible, an argument could be made for "postulating" their existence. The proper place for such postulation, though, is in science and in science only. Thus it is much, much better not to have to make the argument.)

   Suppose I had answered that the particular is the tone, i.e., the thing that "has" this pitch, that loudness, and so on. The questioner recalls that my particulars are bare and is amazed. You hold, then, he asks, that a tone is a tone only because it happens to exemplify a pitch but wouldn't be one if it happened to exemplify a color or smell. I agree, and, delighted that he understands so well, add that what in such discourse one ought to mean by "tone" is indeed a complex characer exemplified by those and only those bare particulars which exemplify a pitch, a loudness, and so on. He, though, listens no longer. For by now he is convinced that the notion of a bare particular is not commonsensical. Nothing is gained by glossing things over; better for once to heed the advice of the French; reculer pour mieux sauter. The way we ordinarily speak, e.g., the way I just used 'tone', is stacked against the bare particular, makes it doubtful whether the notion is commonsensical. Does the doubt spread to the notion of external relations? To show that it doesn't , I introduce the second example.
   Take two small pieces of paper, call them spots. Each has a color (hue) and a shape. Ignore whatever other properties it has. Let us agree to use 'shape' and the shape words, 'round (circular)', 'square', 'triangular, and so on, not of contours but of areas and so that, say, two circles of different size have different shapes. One of the spots is red and round; the other, green and square. Put them on the table before you so that the red spot is to the left of the green one, then shift them around so that the red spot is to the left of the green one, then shift them around so that the first is is above the second, to the right of it, and so on. There is nothing "in" either spot in which to ground any of these (successive) relations between them. Common sense agrees, even insists. The way we ordinarily speak supports this bit of common sense. We say, or may say, that the spots are (successively) at different places, some of which are to the left of some others, and so on. Spatial relations, we see, do for (D₂) what relations among simple characters such as pitches do for (D₁). The former provide prima facie plausible evidence that at least some relations are external; the latter, that at least some are internal.
   Two comments will show where we stand; two more, in structural history, should add perpsective.
      1.   Using 'spot' as I earlier used 'tone', I spoke, as we ordinarily do, as if the spatial relation obtained between the two spots. If, as I hold, there is a bare particular "in" each spot, then the relation obtains between them. Or, at least, that is in this case the structurally obvious way to assay the situation. But, then, had I now so assayed it, I would have prejudged the question which is presumably unanswerable. There is no need for doing that at this point. The prima facie plausible evi-

dence provided by the spatial relation between the two spots is not at all affected by what the ontological assay of a spot may or may not yield. At worst, the connection might turn out to be a pseudorelation. But it would still be an external pseudorelation.
       2.   A spot's being at a place must have an ontological ground. The connection between the crucial question about particulars and some others in the ontology of space is being impressed upon us. I shall indeed soon turn to the latter.
       3.  In some ontologies such "ordinary" characters as colors and shapes are not the only characters "in" the spot. There is still another constituent, or class of such, of the kind Scotus calls hacceitates and which I like to call coordinate qualities. Two spots agreeing in all nonrelational "ordinary" characters are two and not one because the coordinate qualities in them, one in each , are two and not one. As far as I know, that is in a Scotist world the only way of solving the problem of individuation. As to relations, there are two possibilities. They may be either internal or external relations between coordinate qualities. Making them internal, you obtain Leibniz's assay of space. If you make them internal, then there is something "in" the spot which, together with something else "in" the other spot, is the ground for the one spot being to the left of the other. But even if you make them external, the ground for the spot's being at this rather than at another place is still "in" the spot. That is why these assays of space are repugnant to common sense. As to 'place', some philosophers have used it so that the spot's "place" is not "in" it. That is indeed one of the two differences between coordinate qualities and "places." The other is that while coordinate qualities are characters, "places" have been thought of as individuals. I, of course, used 'place' without any ontological commitment whatsoever, merely taking advantage of a locution to make the point that if the spatial relations were not commonsensical examples of external relations, ordinary language would not contain the locution.
       4.   To one not familiar with the refutation of D₁ the two examples may suggest that there are internal as well as external relations, the former all between simple characters, the latter all spatial. (As for space, so for time. I shall, however, up to the very end ignore time.) Turn now to Kant. 'Concept' is one of the most mischievously slippery words in the philosophical vocabulary. My concept of something is of course in my mind. Horse and green are not. That is the only clear use of the word. Yet we have been subtly conditioned to use the word so that horse and green are also concepts. That is why in philosophy I do not use it at all. Kant uses it, of course. Green and horse, he tells us, are discursive concepts( Begriffe). These are the only real concepts. The

spatial ones, being merely limitations of (Einschraenkungen) of the single representation (Vorstellung) of space, are not really concepts, just as space itself is not a concept but an intuition (Anschauung). Accordingly, while a discusive concept subsumes under itself many representations, the "manifold in space," e.g., this being to the left of that, is merely a limitation of, or, rather, in a single representation. As it is put in this jargon, we cannot "think" a point without thinking it in its relations to all others. After a fashion, that makes all spatial relations internal. We see what went on. In the tradition to which Kant belongs all relations are internal. Commonsensically, the spatial ones are external. The jargon drowns that bit of common sense. Structurally, I believe, this is the reason why transcendental aesthetics is split off from transcendental analytics. The reason we are given is that while in intuition the mind is passive, in conception it is active. I am not impressed. In the system space as well as the discursive concepts are contributions of the mind; the only thing not contributed by it is sensation (Empfindung). And whenever the mind contributes anything it is active in the sense which is relevant in the system.
   Presently I shall turn to space. First, though, I shall use what has been said so far to give a twist to the dialectic of the perfect particular. Perhaps that is a digression. But it can be done so easily and fits so well that it seems justified. It also yields a point that will come in handy.
   Consider two spots of the same color (hue), say, red. To assay them so that a single entity, red, is a constituent of both spots is to make red a character (universal). This character is the ontological ground of the respect (color) in which the two spots agree. If you make characters things you are a realist; otherwise, a nominalist. Characters are not localized; the example shows what that means. "What exists is localized." If one is dominated by this pattern he will reject characters. Thus he must find another ontological ground for the respect in which the two spots agree. Some hold that this ground is provided by two entities, red₁ and red₂, one in each spot. That is the doctrine of perfect particulars. Obviously, it conforms to the localization pattern. Its proponents also hold, either explicitly or implicitly, that perfect particulars are simple things. The perfect particulars in the example are, of course, red₁ and red₂.
   Even though a perfect particular be simple, any expression purporting to represent it must have two parts, one indexical, one adjectival. The indexical part is needed because (presumably) the entity does the job which in other ontologies is done by bare particulars; the adjectival part, because (presumably) it does the job which in other ontologies is

done by characters. The crucial difference between a perfect and a bare particular is indeed that the former does and the latter does not have a nature; the crucial difference between perfect particulars and characters is that the former are and the latter are not localized. The purpose the entity is meant to serve is transparent. Things bare as well as things not localized are to become expendable. I shall next show that the perfect particular cannot do either job.
   Take three perfect particulars, red₁, red₂, blue₇. The first two share something which neither of them shares with the third. The dead-end nominalism which does not even see that this something must be ontologically grounded we may safely dismiss. The sophisticated defenders of the perfect particular place that ground in a single relation of (exact) similarity jointly exemplified by red₁, red₂ but not by either red₁ and blue₇ or red₂ and blue₇. Thus they do not really get rid of universals. They merely replace the class of all nonrelational characters by a single relational one, (exact) similarity. All this is familiar. Nor is it difficult to assign a structural motive to the preference for a single relational character over a class of nonrelational ones. That, though, is not the line I want to pursue.
   The relation of (exact) similarity is either (a) external or (b) internal. (a) A relation, we know, is external if and only if, negatively, the ground for two things exemplifying it lies wholly outside these two things, and, positively, if it lies wholly in a third thing, namely, the relation itself and its being tied by (relational) exemplification to the two others. It follows that there is nothing in either red₁, red₂ which the adjectival part of the expressions purporting to represent them could represent. That part is, therefore, merely an arbitrary name of one of the subclasses of the exhaustive and nonoverlapping division which the relation called (exact) similarity produces in the class of all perfect particulars. I say arbitrary, and not of course just in the sense in which every name is arbitrary, because, although each member of a subclass is "similar" to all other members and to no member of any other subclass, there is, in spite of the relation's misleading name, nothing "in" red₁, red₂ themselves. But the relation is symmetrical, which it couldn't be if red₁, and red₂ were two things and not just one. That shows that if the relation is internal the perfect particular is in all but name a character and the problem of individuation is left unsolved. I conclude that there are no perfect particulars; or, if you insist, if there were any, they couldn't do the job they are supposed to do. That is the twist I

wanted to give to the dialectic. Now for the point that will come in handy.
   In some ontologies there are perfect particulars as well as substances; the former inhere in the latter; the latter are individuals though of course not bare. Upon the most important philosophical use of 'individual', an individual cannot either ihnere in or be exemplified by anything else. In this respect the perfect particulars of those worlds differ from individuals. But the difference makes no difference for the point. So we may safely ignore it.
   Let L₁ and L₂ be two schemata, of the kind called artificial languages, constructed and interpreted by the same rules. A primitive predicate represents a simple character. That is one such rule. 'x' marks the places at which the primitive signs representing individuals stand. That is another. L₁ contains the primitive predicate 'sim' and 'a', purporting to represent (exact) similarity and red₁, respectively; L₂, the primitive predicate 'red representing red; L₂ does not contain either 'sim' or 'a'; L₁ doesn't contain ' red'. Suppose, finally, that both L ₁ and L₂ are extensional. No schema representing mental entities can be; but the point does not depend on that; thus, if anything, the supposition strengthens the argument. Introduce now into L₁ the definitional schema (D₁)      'RED(X)' for ' sim(a,x) V (x=a)' ; 1 be a sentence of L₁ containing 'RED'; P₂, the sentence one could obtain from P₁, if L₁ contained 'red', by substituting 'red' for 'RED'. Everyone knows that 'P₁ <---> P₂' could be used as if it were analytic. I shall express this by saying that "formally" there is no difference between an ontology with perfect particulars and single relational character of (exact) similarity on the one hand, and, on the other, an ontology with bare particulars and a class of simple nonrelational characters.
   'Formal' has been put to several philosophical uses. The one just marked by double quotes stems from mathematics. There it causes no trouble. In philosophy it may and has become a source of error. "Since there is no formal difference between the two ontologies, there is none; the question which of the two is right is a pseudoquestion (meaningless)." The sentence between the double quotes exhibits both source and nature of the error. Its name is formalism. Formalism is an error as fatal as psychologism. That is the point. To reject formalism is one thing; to deny that artificial languages (formalisms) can be put to good philosophical use is quite another thing. Two comments will show that I mean that.
      I. L₁ reflects the attempt to avoid nonrelational characters. 'RED',

   being a nonrelational predicate expression, represents such a character. This is just another of the rules by which both schemata are built. Does that mean that on this ground alone the attempt has failed? Far from so. (D₁) shows that, being defined, the character represented by 'RED' is complex. "What exists is simple." Either consciously or unconsciously some philosophers are dominated by this pattern. Such a one, while rejecting 'red', may yet countenance 'RED'.
    2. The definition in L₂ which corresponds to (D₁) is
   
(D₂)        'sim(x,y)'    for     '(Ef)[f(x) & f(y)]'.

   The definition contains no sign that represents a thing. That makes the entity represented by 'sim' a subsistent of the kind some call a "logical relation." Some of the predicate expressions in the range of the bound variable represent subsistents of the kind some call "logical properties," including that of being identical with itself, which makes any two individuals "similar." That trivializes 'sim', as defined by (D₂). If, on the other hand, one says that two hues are similar, he does not use 'similar' trivially. (Remember that we agreed to let 'red' stand for a single hue rather than, as usually, for a range of such.) In its nontrivial use, 'similar' no doubt represents, however succinctly, a complex relational structure. The formalism not only reflects the distinction between these two uses of 'similar' but also very effectively draws attention to it. If one misses it, he is in danger of doing what the more sophisticated champions of the prefect particular did. He may mistake a trival subsistent for a relation which is not there.².
       Some of the world's features are spatial. Some facts, call them spatial, have some of these features. Rome being south of Milan is such a fact; being south is its spatial feature. Everyone knows what these features are. In this sense everyone knows what space is. The ontological assay of spatial fact must yield at least one entity which is the ground of its spatial feature. There is thus a class of entities, call them spatial, such that a fact is spatial if and only if its assay yields at least one member of the class. An inventory of all spatial entities, or, rather, as usual, of all kinds of such entities answers the ontological question: What is space? It could not be answered, or, for that matter, it could not even asked unless everyone knew what space is. (That is the point of Augustine's aphorism about time.)

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    ² A bare particular exemplies exemplifies more than one property. Hence, even if you called two of them similar if and only if they exemplify the same simple property, similarity would not yield a class division. But one could replace the several primitive predicates by several primitive relations of "similarity (similar in a certain respect)." The two languages will again be formally equivalent.

   We cannot and need not begin at the beginning. I take it for granted that the spatial entities do not merely subsist but are things. Things are either individuals or characters. Characters are either properties or relations; and they are either simple or complex. That makes for quite a few possibilities. Are all simple spatial things individuals? Are they all relations? Either properties or relations? Either individuals or properties? And so on. Each of the so-called theories of space propounds one of these alternatives. Since the one clear and important use of 'theory' occurs in science, I would rather avoid the word, speak instead of the several alternatives. Since the one clear and important use of 'theory' occurs in science, I would rather avoid the word, speak instead of the several ontological assays of space that have been proposed. But it may be best to compromise. So I shall speak of the several views of space. These views have been classified in several ways. Each classification consists of, or, at least, starts from a dichotomy. Some of these classifications are helpful. But it is dangerous, to say the least, if in different classifications the same labels are attached to different dichotomies. Yet this is what happened. Everyone tells us that a view is either absolute or relative (nonabsolute). Distinguishing the several ways in which the two words may and (I believe) have been used in this context is as good an approach as any, if not perhaps the best of all, to the issues themselves.
   If all "things" now ceased to exist, would space be left? The view of those who answer negative is relative₁. Those answering affirmatively hold the absolute₁, or, as it is also called, the container view. The use of 'thing' in this traditional question clearly is not the one I make of the word in ontology. For, if it were, since all spatial entities are things, the answer would be trivially negative. Yet the container view is anything but trivial. There is thus an ambiguity that must be eliminated. Remember the small pieces of colored paper, we called them spots, which were shifted around on a table. Suppose that the "things" mentioned in the traditional question are all spots (and, if you please, that space is two-dimensional). Making this supposition, we shall lose nothing that matters for our purpose while, on the other hand, we can now so restate the question that the ambiguity disappears. If all spots now ceased to exist, would space be left? (In other words, we retain the ontological use of 'thing', have found a way of getting along without the one marked by double quotes.) Now we can see what is and what is not at stake.
   What is not at stake is the assay of the spots themselves. All the relativist₁ is committed to is the view that all spatial entities are either things "in" the spots or relations among such things. The absolutist₁ is merely committed to holding that at least some spatial entities are neither things "in" the spots nor relations among such things. Or, as

   I shall say for the moment, an absolutist₁ must hold that there are space-things. the most interesting kind of absolutism₁ is the view that all spatial things are space-things. Since there are three kinds of things, there are still some alternatives left. The most interesting among them is the view that at least some space-things are individuals. As some philosophers have used the word, these space-individuals are "places." If there are such entities then a spot's "occupying" a "place" must have an ontological ground. That is one obvious weakness of this view. But we need not dwell on it. For, once more, I take something for granted, dismiss all kinds of absolutism₁ out of hand. The alternatives to which I turn are all relativistic₁.
   In a nonscotist world there is an individual "in" each spot. Is this individual a spatial thing? Depending on whether your answer is affirmative or negative, you are an absolutist₂ or a relativist₂. Unfortunately the question that produces the dichotomy is not a good question. In a world of bare particulars it does not even make sense. That is indeed one of the major points of the essay. Of this more presently. For the time being, we shall be well advised not to scrutinize the question too closely. For it provides us with an access not only to the dialectics of the space issue but also to one of the fundamental gambits on which all issues hang.
   Why should one care whether or not the individual "in" the spot is spatial? "Only what is independent exists. (The more independent an entity is, the higher is its ontological status.)" The pattern is familiar. There are quite a few philosophical uses of 'dependent' and 'independent'. The pattern shows their ontological import. In a world of bare particulars there is no structural reason not to accept the Principle of Exemplification as a categorial truth. That makes individuals and characters equally dependent₁³ on each other. But the individuals of the classical ontologies are not bare. In these ontologies only individuals of the classical ontologies are not bare. In these ontologies only individuals exist "independently." Their characters exist only "dependently." The idea is that while characters need individuals to inhere in, the latter can or could somehow manage without the former. I say can or could and I say somehow because the idea is irremediably vague. Its root is anthropomorphic. The individual "creates" its characters. Yet, vague as this idea is, it helps us to understand why one may care. Space, or at least some of it, exists "independently" if and only if the indviduals "in" the spots are spatial; it exists only "dependently" if and only if they are not spatial. That is why the issue of absolutism₂ versus rela-

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   ³ In this essay only two of the philosophical uses of 'independent' play a role. That makes it convenient to represent them by 'independent₁' and 'independent₂', disregarding other notations occurring elsewhere in this book.

tivism₂ has been so sensitive for so long, and also, why absolutism₁ and absolutism₂ have not always been clearly distinguished. (Leibniz was a relativist₂. The view he so vigorously opposed was, I believe, absolutism₂. His opponent, Newton, was an absolutist₁.)
   Relativism₂ is the view that simple spatial things are either properties or relations. Are they all properties or are some of them relations? If all relations are internal then no spatial thing is a relation. If the spatial relations are external then it is plausible, to say the least, that some of them are simple. The issue, we see, is part and parcel of the issue of relations. So we need not pursue it.
   Remember those peculiar spatial properties I called coordinate qualities, hereness, thereness, and so on. The Scotists need them to solve the problem of individuation. That is why I also called them haecceitates. If there are such properties then they are simple; or at least some of them are simple. Are there such properties? Depending on whether your answer is affirmative or negative you are an absolutist₃ or relativist₃. There are no coordinate qualities. We are neither presented with them nor dialectically forced to "postulate" them. Absolutism₃ is wrong. (Leibniz, though not a Scotist, was an absolutist₃.)
   Round is a shape; triangular is another. If the wording startles you, think of color. We say that green is a color. Shape is a kind of spatial property. Some shapes are simple. When presented with a triangular spot, I am presented with the simple property triangular. That is not to say that 'triangular' always represents a simple. As carpenters and geometricians often use the word, the property it represents is complex. I merely insist that some shape words have two uses. The best labels are probably phenomenological and geometrical. Used phenomenologically, the word represents a simple. (Remember that we agreed to use the space words so that they represents contours, not shapes, and that, say, two circular areas of different size also differ in shape. Thus there is a phenomenological use of 'triangular' in which it represents neither a shape nor, since there are many sorts of triangles, a kind of shape but, rather, a kind of kind of shape. But it is merely pedantic to speak more accurately than the context requires.)
   Geometry, i.e., the Euclidean calculus is an axiomatic system. Call it the schema. With the appropriate idealizations it can be interpreted into space. The interpretation is very successful. Because of its success a certain way of thinking, call it the schematic way, has become very deeply ingrained in all of us. That makes it difficult to see space as it is phenomenologically (ontologically). Our vision is blocked by the schema. Yet, space is one thing, the schema is another. Space is in the

world; the schema is nowhere. In the schema there are points, lines, areas, and so on. In the several axiomatizations 'point', 'line', 'area', and so on, may or may not be undefined. In the schematic way of thinking the single element or building stone of space (!) is the extensionless point of the schema (!). A line becomes a class of points; an area another such class; and a class is a line rather than an area because its members stand in certain relations to each other. I speak, vaguely, of elements or building stones rather than simples. The vagueness is deliberate. The schematic way itself is irremediably vague, if only because in space there are no extensionless points. The extensionless points of the schema are interpreted into certain series of spots. (That is the gist of extensive abstraction.) That there are such series is an idealizing assumption.
   Some shapes are simple and all simple spatial properties (of the first type) are shapes. To me that has become obvious. Those who have not freed themselves of the schematic way will probably object. Or, at least, they will be puzzled. The cause of their trouble is that they see the schema, not space. If the schema were space and if the schematic way were not as vague as it is, there would indeed be only one simple spatial property (of the first type), namely that of being an extesionless point. All other simple spatial characters would be relations. (Historically, I surmise, the unpalatability of coordinate qualities (absolutism₃) has cast a shadow on the very idea that some spatial properties are simple. Thus the quest for these entities has been neglected. Leibniz did slay absolutism₁. But that is not to say that the acceptance of absolutism₃, which was the view he proposed, has ever been both wide and articulate. That is just one of the many ironies of history.)
   In my world there are neither space-things nor coordinate qualities. That makes my view relative₁ as well as relative₃. Whether or not I am also a relativist₂ depends on whether or not the individuals which exemplify shapes are themselves spatial. That takes us back to the bare particular and the question which supposedly is unanswerable.
   Remember the questioner who, when presented with middle c, insisted that all entities presented to him were properties. Suppose he gives me another chance, asks me to direct his attention to the bare particular "in" the spot. I first acquaint him with my use of 'shape', then tell him that the bare particular is the spot's area. He grants that this time I direct his attention to an entity which is presented to him and that, therefore, if this entity were a bare particular I would, in the case of spots, have answered the "unanswerable" question. Then he

offers two arguments. The first purports to show that the entity is not bare; the second, that it is not an individual.

"Being a certain area is a character. The entity to which you directed my attention is this character. Thus it is not bare. If it were, how would I know that it is an area and not, say, a tone." This is the first argument. I answer as follows. I know that the entity is an area because it is round. Or, rather, to be an area is to be an entity that has a shape. Otherwise I would not know what 'area' meant. (When in our first exchange I made this point about 'tone' he broke off. This time he lets me go on.) Assume that you are presented with two spots. If they agree in all nonrelational characters, including shape, they will also agree in the character you claim the entity is. How then would you know that they are two and not one? The questioner has no answer.
   "Call the areas bounded by the inner and the outer of two concentric circles a and b, respectively. a is part of b. (P(a,b)). Individuals are simple. What is simple has no parts. Hence b is not an individual." That is the second argument. I answer as follows. 'Part' has many uses; some are literal; some, metaphorical. A thing is simple if and only if it has no constituent which is a thing. As I use 'constituent' in ontology, a constituent of an entity is not a part of it in the sense in which a is part of b nor in any other literal sense of the word. Since all individuals are simple, the only complex things are complex characters. Stone is such a character; hard, a constituent of it. yet, clearly, hard is not part of stone in the sense in which a is a part of b. P, on the other hand, is a simple spatial relation between areas, just as higher is a simple relation between pitches (or tones). In this respect there is no difference between 'P(a,b)' and 'e is higher than c'. So I ask the questioner whether that makes either pitch a constituent of the other. He agrees, of course, that it doesn't but points out that there is a difference. If I am right, P is of the first type, while 'higher' is of the second. Now it is my turn to agree. I add, though, that I cannot see what difference this difference makes. If he remains unconvinced, I can do no more than try to release his block. What may block him is the way of thinking which the success of the schema has so deeply ingrained in us.
   The schema produces not just one block or likely block, but two. In the schematic way of thinking an area is a class of points which stand in certain relations to each other. If this were so, then areas would indeed be complex. More precisely (whatever that means once one has taken the schematic way) area would be a complex character. That may make it hard to see that areas are simple. This is one likely block. In the schematic way, a and b "are" two classes of points, the former

being included in the latter. Class inclusion is one of the literal meanings of 'part'. It is also a subsistent, of the kind some call "logical relations." That may make it hard to see that P is a relation, i.e., a thing of the kind some call "descriptive relations." This is the other likely block.
   The spot's area is not only round, it also is red. I take it, then, that the bare particular "in" the spot is its area. At this point for reasons which are to transpire, I discard 'area'. The two reasons I used it up to now are that it is suggestive and that either 'individual' or 'bare particular' would have verbally prejudged the case. In geometry 'area' has many uses with which of course I have no quarrel. But we have learned to be wary of geometry in the ontology (phenomenology) of space.
   Absolutism₁ and absolutism₃ are rightly discredited. Nor are the three absolutisms always distinguished. Thus one may try to discredit my answer to the "unanswerable" question by pointing out that it commits me to "absolutism." If particulars are spatial things, then I am indeed an absolutist₂. Are they spatial things? (Since I consider as yet only those particulars which are constituents of spots, I suppress the qualifying clause: "in" the spot.
   There are two kinds of reasons, one psychological, one verbal, why particulars may seem to be spatial things. In the first and the second of the four comments which follow I shall show that two such reasons, one of each kind, which are at least as good as any other I can think of, are really bad reasons. In the third and the fourth comment I shall give two reasons why, even if particulars were spatial things, I would not be an "absolutist" in any sense in which "absolutism" is rightly discredited.
   First. Let a be a particular; b, c and so on, parts of it. When presented with a fact of which a is a constituent, one more often than not becomes very quickly and more or less clearly aware of (is being presented with) quite a number of other facts such as P(a,b), b being contiguous to c, and so on. P and contiguous are are spatial relations. These facts, therefore, are all spatial facts. Clearly, this is not a good reason for a itself being a spatial thing. (I call this bad reason psychological rather than phenomenological because the laws by which presentations succeed each other are psychological.
   Second. Call a word a space word if and only if, when used phenomenologically, it represents a simple spatial thing. 'Area' has geometrical uses. Clearly that is not a good reason for its being a space word. The illusion that it is one has two sources; one is purely verbal; the other is

the schema. 'Area' obviously is a predicate. Hence, if its were a space word, area would be a character. Thus it would be either of the first or of a higher type. Round is one of the first type; shape, of the second. Round, which is a shape, is "in" the spot, shape is not. Nor can I think of any other property of a higher type which is "in" it. Yet, the spot's area undoubtedly is "in" it. I conclude that area is not a higher type character. If one holds that it is of the first type then I turn the tables on him, ask him to direct my attention to the individual that exemplifies the spot's area. This question is unanswerable. Your only way out is to become a Scotist. Then you will need coordinate qualities.⁴
   Third. The individuals of my world are bare; the Principle of Exemplification is a categorial truth; individuals and characters are equally dependent₁ on each other; neither can "manage" without the other. In other words, one of the traditional philosophical uses of 'dependent-independent' (as in "Only individuals exist independently, characters only dependently") does not even make sense. Nor does therefore the formula, which as we saw discredited absolutism₂, that space exists only dependently if and only if individuals are not spatial things. True, if the entities "in" the spots which I claim are individuals were spatial things, then my view of space would be absolute₂. But, since the formula makes no sense, why should that be discreditable?
   Fourth. Each individual exemplifies at least one character. That is one half of the Principle of Exemplification. At least one of the char-

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    ⁴ Notice that 'area' does not behave like 'shape'. One can say that round is a shape but not that it is an area. One could of course construe 'area' as the (impredicatively) complex property (A) of having a property which is a shape in analogy to 'colored' and, if 'extension' is used instead of 'area', to 'extended'. In a world otherwise like ours but without time and minds, every individual has the property A. That is not to say, though, that even in this truncated world to be an individual would mean to be an A. In an improved language, we know, 'individual' cannot even occur without futility. 'A' can be defined in such a language. That is not to say that in speaking about the truncated world it would have any ordinary use. But the possibility of defining it, jointly with the truth of '(x)A(x)', explicates the aphorism that in that truncated world space is the principium individuationis, just as space and time are in ours.
   Instead of 'area' one could use 'spatial extension' or 'piece of space'. From the former we better keep away until time is brought in. Use of the latter would have strengthened the suspicion of "absolutism." To dispel the suspicion, one merely has to point out that the phrase is metaphorical. There is, as one says, no Space with capital S. In our world not all individuals are areas (extensions). That makes 'extended' a very convenient synonym for 'having a shape'.

acters an individual exemplifies is a property. That is a most reasonable refinement of this half.⁵ Since among (simple) spatial things there are properties as well as relations, the Principle, even if so refined, does not exclude the possiblity that the characters some individuals exemplify are all spatial. The view that there are some individuals of this strange sort is a kind of "absolutism." If these strange individuals were themselves spatial things, it would be a strange variant of absolutism₁. There are of course no such individuals. Yet one may feel that unless a view excludes this possibility it smacks of "absolutism." So we may be asked to exclude it. When one philosopher is asked by another to "exclude a possibility," he is not only asked to acknowledge a certain truth but also expected to show that in his way of thinking this truth lies rather "deep."⁶. If one counts black, white, and greys as colors, then it is of course true that what is extended is colored (and conversely). It is also, we shall see, an a priori truth. Such truths lie "deep." This particular one excludes the possibility which smacks of "absolutism."
   Let us take stock. Formally, the purpose of this essay is to solve the problem of the a priori. Once the preparatory steps required have been taken, one sees that neither the solution nor the problem itself is either difficult or very important. Some of the steps are both. What has been said about internal relations, bare particulars, and space covers three of the four steps required. Some comments about elementarism will cover the last.
   Are all simple characters of the first type? The affirmative answer is elementarism. If one does not make other mistakes then his answer to other philosophical questions will not depend on whether or not he is an elementarist. Thus the issue is not important. Probably that is the most important thing to be said about elementarism.⁷ Yet, since philosophers do make mistakes, it will help if we connect it with the issues at hand.
   Remember the first example. (1) e is higher than c. Phenomenally higher is simple. Prima facie, therefore, in view of (1) elementarism seems wrong. I say prima facie and I say seems because, if in an improved language 'higher' (and all other such predicates) can be represented by a predicative expression (i.e., as one says, by a defined

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  ⁵ H. Hochberg. "Exemplification, Independence, and Ontology," Philosophical Studies, 12, 1961, 36-43.
  ⁶ To say that a truth lies deep is to speak metaphorically, of course. The metaphor will be unpacked in the first concluding remark.
  ⁷ See "Elementarism," Journal of Philosophical and Phenomenological Research, 18, 1957, 107-14 (reprinted in Meaning and Existence).

predicate) containing no primitive of a higher type, then elementarism nevertheless is right.
   This is the place where an earlier digression comes in handy. Remember the two artificial languages L₁ and L₂. Let L₁ contain a class of primitive predicates of the first type as well as one of the second type, 'hg^2', which have been made to represent the pitches and higher, as used in (1), respectively. L₂ is like L₁ except that instead of 'hg²' it contains a first-order predicate, 'hg¹', which is so interpreted that 'hg¹(a,b)' is true if and only if a and b both exemplify pitches such that the one exemplified by a is higher than the one exemplified by b. Add to L₁ the definition

    (D₃ )         'HG¹(x,y)'     for     '(Ef)(Eg)[hg²(f,g) & f(x) & g(y)]';
   
add to L₂ the definition

    (D₄)    'HG²(f,g)'    for       '(x)(y)[f(x) & g(y) --->
                                                        hg¹(x,y)] & (Ex)f(x) & (Ex)g(x)' .
   L₁ and L₂ are formally equivalent.⁸ A formalist, therefore, will dismiss as meaningless the question whether elementarism is right or wrong.⁹ I merely said that if one makes no other mistakes it is not important. Since the mistakes philosophers make are often interesting, the following question may be and in fact is of interest. Which motives may lead one to embrace either elementarism or nonelementarism?
   The dead-end nominalists tell us that characters do not exist. The more sophisticated nominalists try to depress their ontological status. In these attempts, wittingly or unwittingly, they are often guided by two patterns. By the localization pattern, what exists is localized in

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   ⁸ Some may wish to argue that this time the equivalence is merely "factual," or, even, that L₁ and L₂ are not equivalent at all because (1) the proof that two individuals exemplify hg¹ if and only if they also exemplify two pitches requires the premiss that two properties exemplify higher if and only if they are pitches; and (2) this premiss introduces the simple second-type character pitch.

Ad(1). The premiss, we shall see, is an a priori truth. That is why one may not notice that it must also be stated in L₁, where it becomes the closure of

    hg²(f,g) V hg²(g,f) <---> pitch ²(f) & pitch²(g) & not(f=g)

    Ad(2). If in L₂ one defines

'PITCH²(f)' for '(x)(y)[f(x) & e(y) ---> (hg¹(x,y) V x=y] & (Ex)f(x)' ,

where e of course represents a pitch, then the closure of

    hg¹(x,y) <---> (Ef)(Eg)[f(x) & g(y) & PITCH²(f) & PITCH²(g) & not(f=g)]

will do as premiss. Thus the argument is fallacious.

    ⁹ Twenty years ago, since the logical positivists were my first teachers, I myself dismissed it on this ground. See "Remarks concerning the Epistemology of Scientific Empiricism," Philosophy of Science, 9, 1942, 283-293.

   space and time. No character is so localized, of course. Yet there is a difference. If I may use a metaphor which needs no unpacking, higher-type characters are even less localized than those of the first type. By the simplicity pattern what exists is simple. That makes nominalism a potent motive to incline one toward elementarism.
   We often say that this tone is higher than that. We also say that e is higher than c. Tones are individuals. On the other hand, a tone is what has a pitch. There is also the a priori truth that if one individual is higher than another then they are both tones. Some may wish to argue that the (phenomenally) simple relation sometimes presented to us is hg². I do not engage in this sort of argument. But I can see that it may incline one toward nonelementarism.
   In L₁ (1) becomes (1') hg²(e,c): in L₂ it becomes (1'') (x)(y)[e(x) & c(y) ---> hg ¹(x,y)]. (1') is atomic; (1'') is synthetic generality. (1) belongs to what I called the common core of truths that were claimed to be a priori. Such truths supposedly are more "certain" than "mere" synthetic generalities. This provides a very potent motive for rejecting elementarism. Those who believe that it also provides a good reason commit the error of psychologism. To say the same thing differently, the use of 'certain' which is marked by double quotes is philosophical. In one such use, to be certain is to be an atomic fact; in another, to be certain is to be a priori. The argument on which the supposedly good reason rests thus suffers from a quaternio terminorum. Notice, finally, that just as (1) belongs to the common core of a priori truths, being higher in pitch belongs to the common core of supposedly internal relations.
   Speaking nonelementaristically not only makes it easier to state the solution of the a priori problem, it also makes the connections of this solution with the various strands of the traditional dialectic more perspicuous. These are two appreciable advantages, if only, characteristically if I am right, with respect to an issue which like the elementarism issue is not very important. In view of these two advantages I shall henceforth speak nonelementaristically. But, then, advantages of speaking nonelementaristically are one thing, reasons for choosing between elementarism and nonelementarism another thing. I see no need for a choice. (This strengthens my conviction that elusive phenomenological details such as those on which the choice depends¹⁰ never make a dialectical difference.) I once presented in a lecture the idea of the solution which I am about to propose. In the discussion a distinguished colleague asked me whether "my" a priori was a "real" a priori such as, say, Kant's. I

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   ¹⁰ Is the simple relation with which we are sometimes presented hg¹ or hg²?

readily assured him that it was nothing of the sort. The difference has been explained in the introduction of this essay. Yet it may help if with a twist I explain it once more. Some synthetic truths (facts) are felt to lie "deeper" than some others. They are those called a priori. The proponents of the "real" a priori account for that felt difference by so assaying these privileged facts that they alone contain an ingredient (I avoid 'constituent') which is a contribution of the minds that know them. Or, since anyone who puts "in" the act (mind) what belongs "in" the intention is either an idealist or on the way to idealism, perhaps I had better say that this spurious ingredient is a contribution which minds make to these facts alone. This is the idea of the "real" a priori. I ask myself whether there is in those privileged facts themselves something which distinguishes them structurally and not merely as a matter of degree from all others. The solution I propose is therefore quite different. The problem obviously is the same.
   Each truth of the common core belongs to one of six kinds. The following list contains examples of each:

    (A) Round is a shape. Green is a color. e is a pitch. Of two pitches one is higher than the other; only a pitch is higher than anything else.
    (B) e is higher than c. (This shade of) brown is darker than (that shade of) yellow.
    (C) If the first of three pitches is higher than the second and the second is higher than the third, then the first is higher than the third. (Higher is transitive.)
    (D₁) What has pitch has loudness and conversely. (Pitch and loudness depends_2¹¹ on each other.) What has shape has color and conversely. (What is extended is colored and conversely. Color and shape depend₂ on each other.)
    (D₂) Nothing (no tone) has two pitches. Nothing (no area) has two shapes. Nothing (no area) has all over two colors.
    (E) If the first of three things (areas) is a part of the second and the second is a part of the third, then the first is a part of the third. P is transitive.)
   By the idea that produces this classification there is still another class and there is something to be learned from the fact that (as far as I know) the truths of this seventh class have never been called a priori. First, though, I must state the idea.
   The simple properties of the first type fall into several classes such that all the members and only the members of a certain class exemplify

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   ¹¹ This explicates one philosophical use of 'dependent'. For its importance, see E. B. Allaire, "Berkeley's Idealism," Theoria,29, 1963.

a certain simple property of the second type, e.g., shape. In this sense the second-type property constitutes the class. In some cases at least the class is also constituted by a simple relation of the second type, e.g., higher. The psychologists call these classes modalities. Since in philosophy that word is preempted, let us call them dimensions.
    The truths of (A) are those and only those constituting the several dimensions. Some of them are atomic; some are general. A simple relation obtaining between two (or more) members of a dimension is an atomic fact. (B) is the class of all such facts. The truths of (C) are all general. They are all those and only those which connect the properties of a single dimension with the simple relations between them. The truths of (D₁) are the general truths by which the members of two dimensions depend₂ on each other. The truths of (D₂) are those and only those by which the members of one or two dimensions exclude each other. The last two classes are labeled with the same letter because they have long provided the most popular examples of a priori truths. The class (E) corresponds to (C). The reason for setting it apart is that the spatial relations "in" the facts of (E) are the only relations of the first type that are mentioned in the list.
   Taken together, these six characterizations are the idea of the classification. They are all structural. Nor is the sum of the six classes an arbitrary collection. There is a sense of 'simple' in which the facts in the sum are the simplest we know. I also understand what one means when he says that we are more certain of these facts than we are of any other. One may reasonably doubt that what he says is true. That, though, is unimportant as long as psychologism is kept out by the two little words which are italicized.
   Let once more a and b be two areas such that P(a,b). The seventh class, the one which has been overlooked, contains such spatial (and temporal) atomic facts as P(a,b). The only difference between this class and (B), which contains hg^2(e,c), is one of type. Our confidence in the idea of the classification will be strengthened if we can find a likely reason why this seventh class has been overlooked. When seeing a and b I am (sometimes) presented with P(a,b). When hearing two tones, one an e, one a c, I am of course not presented with all such pairs. Express this difference by saying that while one may be "wholly presented" with P(a,b), one is never so presented with hg²(e,c). One may easily take for granted that what is "wholly presented" is as "certain" as an a priori truth. And what one takes for granted he easily overlooks.¹²

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    ¹²The double quotes around 'certain' mark again the connection with psychologism. For the connections between elementarism, nominalism, and the philosophical uses of 'wholly presented' see the essay on "Elementarism" cited in fn. 7.

   Some of the truths in the list are atomic; some are general. What is the difference between one of the latter and a general truth which is a posteriori? Dogs being carnivorous is general and does not belong to the common core. That makes it an example. But the usual examples are too complicated. It will be better to invent one that is as simple as one can make it. Suppose that what is round is also red. Round is a shape; red is a color. The general fact that color and shape depend₂ on each other is a priori. The (fictitious) fact that whatever has a certain shape (round) also has a certain color (red) is general as well as a posteriori. As for dependence₂ (D₁) so for exclusion (D₂). Just consider the (fictitious) fact that nothing is both yellow and triangular. That shows the difference. Nor is it chance that there are no truths which are general, a posteriori, and as simple as these examples. Otherwise those which are a priori wouldn't be the simplest truths we know. ¹³
   To say that an entity is in spaceis to use a metaphor as inconspicuous as it is convenient. An entity is in space if and only if it stands in spatial relations to others. It follows that only individuals are in space. All areas are in space. Areas also have shapes. Tones do not. Yet they are spatially localized. Are they in space? If you can make that a good question, you will discover that the answer depends on some of those phenomenological details and subtleties which dialectically make no difference. That makes it safe to suppose that in the truncated world without time and minds which we have so far considered all particulars are in space without depriving ourselves of the musical examples which because of their simplicity are so attractive. Or, to use the traditional phrase, in the truncated world the principium individuationis is space and nothing else.
   A spot is a bare particular exemplifying two simple properties. In this respect spots and acts are alike. In the case of the spot the properties are a shape and a color. In the case of the act they are a species

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   ¹³A little reflection will show that if the three claims I am about to make are justified and the seventh class is included in the classification then all atomic facts are a priori.
   ¹⁴ That supposes that the spot is one color all over. No harm is done if in this context we "schematize" by ignoring this complication, just as we ignore all other properties which the individuals may or may not exemplify. Philosophy, being dialectical, always schematizes. That is why those who get lost in phenomenological details and subtleties so often do so badly in philosophy.
    This is as good a place as any for mentioning another detail or complication lest some be puzzled. The transitivity of pitch appears in the list, under (C), as an a priori truth even though, because of the so-called threshold phenomena, it is literally a falsehood! The details which at this point the schema neglects are in their different aspects the proper concern of scientists and mathematical logicians.

(perceiving, believing, and so on) and thought. When the questioner asked me to direct his attention to the bare particular "in" the spot, I directed it to the spot's area, i.e., to the entity which has a shape. Had he asked me to do the same thing for an act, I would have directed him to the entity "in" it which has a duration. A particular is in time if and only if it has a duration. That unpacks another inconspicuous and convenient metaphor. The bare particulars "in" acts are in time but not in space. In the truncated world, the bare particulars in the spots are in space but not in time. In our world they are in space and time.
   'Duration' corresponds to 'shape'. The bare particulars "in" spots I also called areas. This use of 'area', we know, is phenomenological, not geometrical. Yet there is nothing startling about it. It is, as one now says, an ordinary use. The bare particulars "in" acts I call awarenesses. This use of 'awareness' is technical. As things now stand, I would improve my case if I could produce a word that corresponds 'area' as 'duration' corresponds to 'shape'. As it happens, there is no such world. Ordinary language seems to be against me. Yet I can do even better. Instead of 'area' I could have used '(spatial) extension'. To have a shape is to be a spatial extension. to have a duration is to be a (temporal) extension. (There is, of course, a chronometric use of 'duration' which mutatis mutandis requires the same clarifications as the geometrical use of 'area'.) We do use a single word, 'extension', to speak of particulars irrespective of whether they are in space and time or in time only! Or we use the word with one of the two qualifying adjectives, 'spatial' and 'temporal', depending on the characters the particular we talk about exemplifies. As it happens, ordinary language really is on my side. (Some may be impressed. Or they should be impressed. I am not.)
   Considering time and minds makes new examples available. We can enlarge the list. Shape and color constitute two dimensions. Thought and species constitute two others. Remembering being a species goes into (A); the transitivity of later goes into (E). The dependence₂ of thought and species, or, as one says, the intentionality of mind becomes an a priori truth in (D₁). Nothing being both a thought and spatially extended goes into (D₂).
   I now make three claims. Each truth of the common core belongs to one of the six classes. Or, equivalently, each such truth fits one of the six characterizations which are the idea of the classification. This is the first claim. (I have made it once before. But it bears repeating.) If this claim is justified then the problem of the a priori is solved. The relations "in" the facts of the first five classes are all those and only those

which belong to the common core of relations which were held to be internal. This is the second claim. The particulars we have so far considered all either in space and time or in time only. Briefly, they are all extensions. The third claim is that all particulars are extensions.
   Is the first claim justified? The enlarged list supports it impressively. I, for one, cannot think of a truth in the common core that does not fit into one of the six pigeonholes. That, though, doesn't prove the claim. The only way of literally proving it is to start a huge historical investigation, compile a complete list of all truths in the common core and show that one of the six characterizations applies to each of them. That is hardly practical. Nor, if it were, is it the sort of thing I would do. As for the first claim so for the second. One would have to compile another complete list and check it against the first.
   Is the third claim justified? I, for one, cannot even think of an individual (of our world) that is not an extension. Again, that does not prove the claim. What, though, does it mean to "prove" a claim of this sort? How would one go about "proving" it? But one can, and I shall, do something else. I shall show that the three claims dialectically support each other.
   Unless the first claim is justified the second cannot even be made. If they are both justified, then the common core of relations which were held to be internal contains those and only those of (at least) second type which are mentioned in the list. Hence, even though there are no internal relations, those in the common core do have something in common. We not only understand why there was a problem of internal relations; we have completely unraveled its connection with that of the a priori. That shows how, even though the second depends on the first, the two claims dialectically support each other. In our world the principium individuationis is space and time and nothing else. All individuals are extensions. This is the third claim. In our world space and time are uniquely pervasive. That is indeed one of its most striking features. Without it, the container view of space and time would never have seemed plausible to anyone. Because of it, in spite of some fundamental differences¹⁵ between space and time, philosophers have always put the two together. By the Principle of Exemplification, every simple character is exemplified. Hence, of whatever type the character may be, there is a "descent" from it, which leads to individuals. That shows how in view of that striking feature the third claim and the Principle of Exemplification dialectically support each other. The Principle, though, is more fundamental than

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   ¹⁵See the essays on Leibniz and on time in Meaning and Existence.

either claim or feature. That makes the claim the root of the feature and a structural reason for putting time and space together.
   If the first and the second claim are justified, then the only simple relations of the first type are the spatial and temporal ones between bare particulars. All other simple relations are of (at least) second type. This is, or would be, another structural reason for putting space and time together. Nor is that all. The spatial and temporal relations, we saw, are not only the simplest examples but quite probably also the commonsensical source of the philosophical idea of external relations, just as the simple relations of the second type, higher, darker, and so on, provide not only the most plausible examples but quite probably also the commonsensical source of the very idea of internal relations. Thus, if all three claims are justified, then we have completely unraveled the connections between the ontology of space and time on the one hand and the three problems of internal relations, of bare particulars, and the a priori on the other. That shows how all three claims dialectically support each other.
   The two structural reasons for putting space and time together are also reasons for setting them apart from everything else. The second, all simple first-type relations being either spatial or temporal, also explains why in the grammar of philosophical uses the two pairs 'extensive-intensive' and 'external-internal' are sometimes interchangeable. Kant surely set time and space apart. Remember his distinction between Einschraenkungen and discursive Begriffe. He also tells us that while the spatial and temporal "magnitudes (Groessen)" are "extensive," all others are "intensive." What he meant by these two words remain obscure. We see what one could reasonably make them mean.
   Now for some concluding remarks.
   Some truths were said to lie deeper than some others. The metaphor must be unpacked. The Principle of Exemplification was called a categorial truth. The opening paragraph contains a classification of all truths. A truth is either analytic or synthetic. A synthetic truth is either a priori or a posteriori (contingent). To which of these three classes do categorial truths belong? As 'deep' has been used and as (I believe) most philosophers have used it, analytic truths lie deeper than those which are a priori; the latter in turn lie deeper than those which are a posteriori. That unpacks the metaphor. A categorial truth clearly is not contingent. Since "analytic" and "synthetic" are essentially mathematical (combinatorial) notions, one must in explicating them make use of an improved language. In such a language a priori and analytic truths can be stated. Categorial truths cannot (without

futility) even be stated in it. Nor is that surprising. The distinction between what is analytic and what is synthetic and the very possiblity of the distinction itself depends on what the categorial truths are. That puts all the latter on a fourth level, which is the deepest of all. ¹⁶
   Is what is said about time in this essay compatible with what is said elsewhere ¹⁷ in this book? I believe that it is. But I shall not pursue the matter.
   We may have stumbled upon a fundamental difference between space and time. If "two" particulars (areas) coincide in space, as one says, then they are one and not two. If two awarenesses, e.g., one of mine and one of yours, are exactly simultaneous, i.e., as one says, if they coincide in time, they are still two particulars and not one.
   Philosophers have long speculated that time and mind depend₂ on each other. The later Husserl even tells us that they "constitute" each other. Awarenesses are in time but not in space. Being particulars, they are also bare. What is bare is neither mental nor nonmental. Yet, awarenesses are the only particulars which exemplify the simple characters which are mental. If one accepts the Principle of Exemplification then he may express this by saying that if there were no time there would be no minds. That explicates one half of the speculative proposition. The other half is at the root of idealism.

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   ¹⁶ These are bare hints, of course. Several aspects of this issue are in considerable detail discussed at several places in this book. The cliche in the opening paragraph will (I hope) be excused as an expository device.

¹⁷ I refer to the essay on "Duration and the Specious Present."