1 1 Introduction Michael Friedman (1987, pp.89f.) famously defended the Aufbau 2 against Quine's criticism (1950, p.40), saying that Carnap's strategy had more affinity with Kantianism (Friedman 1987, p.98) 3. Indeed, Carnap showed a strong empathy to Kant's philosophy: (1)4 By categories, we mean the forms of synthesis of the manifold of intuition into the unity of an object. [... The] manifold of intuition is called "the given" in [our] constitution theory [...]. [...And t]he synthesis of the manifold into the unity of an object is[, in our theory,] regarded as "the constitution of an object from the given" (Carnap 1928, §83). "Categories," "synthesis," and "the manifold"- these terms of Kant's were inherited by Carnap as well. Meanwhile, he attempted even modernization of Kant's philosophy, according to Friedman: 科学哲学 47-1(2014) The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective 金子裕介 Abstract The foremost aim of this paper is to realize the fourth part of the Aufbau. This part, which provides an actual phenomenalistic constitution system, is interpretable from a Kantian perspective (§§1-4). But Carnap plotted to overcome Kant's old style of philosophy as well. We review this aspect of his constitution, focusing on space (§§7-13) and time (§§5-6), especially. 20 (2) [Carnap's] project is not strictly Kantian [...]. For Kant himself, merely formal logic is quite inadequate for the constitution of objectivity, and we need to supplement it with a "transcendental logic" that makes essential reference to intuition: the "pure intuitions" of space and time. Now, in the context of the much more powerful conception of formal logic bequeathed to him by Frege and Russel, Carnap finds such an independent appeal to the "forms of intuition" quite unnecessary [...] (Friedman 1987, pp.98-99). The bequest of Frege and Russel made Carnap think Kant's framework of space and time unnecessary. Did Carnap succeed in this strategy? Hereafter, we review Carnap's course of thought exclusively from this perspective 5. This inquiry provides, on the one hand, a systematic study 6 of the Aufbau Friedman never attempted. On the other hand, it leads to a new perspective filling the gap between a traditional philosophy once completed by Kant and a modern analytic philosophy opened up by Carnap. I strongly hope this study is read by researchers of Kant as well, since herein could be a modernization or formalization of Kant's epistemology. The central figure carrying it out henceforward is nobody but Carnap. Did he fail or succeed? I want readers to make sure of it by themselves. Carnap's strategy was, as far as this article is concerned, to abstract the Kantian notion of space and time from our personal, primitive experiences, using his original concept of similarity. By doing so, he thought, the Kantian "pure intuitions" of space and time could be removed. But as we shall see, Carnap never succeeded in it: Kant's notion of space and time must be presupposed even in his formalization, because without the notion, the elementary experiences are never provided (§6, §13). Was, then, Carnap's attempt in the Aufbau completely frustrated? No. It still remains a great precursor and role model of a modernization of traditional epistemology. And I myself think it could be applied to the explication of analyticity as well if we improve the disputable concept of similarity (§14). 2 The Manifold Now, let us embark on the modernization of Kant's epistemology. What we take as such was adequately stated in the preceding citation (=(1)). Kant dealt with 21 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective it, notoriously, in naïve psychological terms, according to which the manifold is integrated into a unity of an object, and then recognition is formed from the unity (cf. Kant 1787, B137). Carnap's formalization of Kant's theory began with the clarification of this naïve terminology, so to speak. First, he worked on that of a typically Kantian term, the manifold 7. Köhler & Wertheimer's Gestalt theory and Schlick's analysis seem to have influenced him at that time. As a result, he reached the following view: (3) [The given] are the personal experiences 8 themselves in their totality and closed unity (Carnap 1928, §67). That is, the manifold is personal (private) and never decomposed into atoms. Carnap put this notion of the manifold at the bottom of his system, calling it an autopsychological basis 9. 3 Egocentricity By taking the manifold as a basis, however, Carnap once broke with Kant (Carnap 1928, §§64-65). It is because, in so doing, his picture became fully subjective. Then, how on earth can our recognition be objective? This question eventually brought him back to a Kantian picture, which provided him the distinctive notion of form/content (Carnap 1928, §66). Hereby, the subjectivity of the manifold was attributed to its content alone, to which the form was applied successively. This is how the manifold turned objective. Additionally Carnap introduced the core of Kant's theory, the transcendental apperception 10, as the bearer of the form. It was stated by Kant with a Cartesian twist in the following way: (4) "I think" must be capable of accompanying all representations of mine (Kant 1787, B131-132). Carnap favored this picture, calling it egocentricity 11. 4 The Recollection of Similarity The universal ego gives a form to private experiences. The form is nothing but 22 what Carnap called "category" above (cf. (1)). But what exactly is the form? Carnap adopted only one thing as such: the recollection of similarity 12. With the manifold (the content) newly named elementary experiences 13, now the basis of his system was decided: (5) Phenomenalistic constitution system 14 (i) Basic relation 15 : the recollection of similarity, i.e., Er(x,y) (or xEry)16. (ii) Basic element 17 : elementary experiences, i.e., erl={x1, x2,...}. The more fundamental of the two was the recollection of similarity; elementary experiences were no more than the elements of its field 18, i.e., erl=fldEr. 5 The Constitution of Time Now we came to the starting point of the Aufbau. From this basis, other objects are constituted 19. In the constitution, we deal with that of space and of time above all. As for time, however, Carnap constituted it with the basic relation, Er, alone. The ground for it was that he thought Er to parallel the order of time. He says: (6) When [we recall x and find it similar to y], the memory image of the earlier, i.e., x, must be compared with y. Therefore, this recognition process is not symmetric; the way x appears when we compare it with y is different from that when we compare y with x conversely (Carnap 1928, §78). The recognition of similarity is not symmetric, not reversible. In this respect, it resembles the course of time. It should be noted that the time in question is our inner state of time. Precisely here, Kant's theory steps into our picture: (7) [T]ime decides the relationship of representations in our inner states (Kant 1787, B50). But did this parallelism work so well? It seems difficult at first sight, because temporal order is fundamentally linear 20, i.e., transitive 21 and trichotomous 22. Similarity lacks, in the first place, transitivity. Suppose Tom and Mike are similar, and so are Mike and John. Yet it is not always the case that Tom and 23 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective John are similar 23. But regarding this defect, Carnap provided a relief measure, the power relation 24 symbolized as Erpo. Its definition is as follows. (8) 25 Er1(x,y)←→ def.Er(x,y) Er2(x,y)←→ def. z1[Er 1(x,z1) Er(z1,y)]←→ z1[Er(x,z1) Er(z1,y)] Er3(x,y)←→ def. z2[Er 2(x,z2) Er(z2,y)]←→ z2[ z1{Er 1(x,z1) Er(z1,z2)} Er(z2,y)] ...Ern+1(x,y)←→ def. zn[Er n(x,zn) Er(zn,y)] Here Erk+1 is defined so that it complements the lack of transitivity (of Erk). Carnap added extensions further to these; that is, Er1=def.{<x,y>|Er(x,y)} 26, Er2=def.{<x,y>|Er 2(x,y)} etc. Lastly, these extensions are connected into a union: Er0 Er1 ... Ern= ni=0 Er i. This union is called a chain 27 with the symbol Erpo. Erpo is the attribute of this class: Erpo={<x,y>|Erpo(x,y)}. It is certainly t r a n s i t i v e i n d i s t i n c t i o n f ro m E r ; f o r e x a m p l e , i f To m E r p oM i ke (<Tom,Mike> Er1 Erpo) and MikeErpoJohn (<Mike, John> Er 1 Erpo), then TomErpoJohn (<Tom, John> Er 2 Erpo). This is how Carnap followed up the lack of transitivity. But another defect remained. Trichotomy did not hold even of Erpo. There could be a pair, xi and xj, of which none of xiErpoxj, xi=xj, and xjErpoxi holds; simply put, they could be not similar at all. Nevertheless, Carnap optimistically expected that this defect would be overcome as his constitution develops (1928, §120)28. 6 Goodman's criticism This is the constitution of time by Carnap. Probably, the author can make any excuse for the technical defect of this kind. Was the constitution then successfully made? Nelson Goodman, who wrote the best commentary (1951, V), objected: (9) [I]t is questionable whether [Carnap's arguments on the recollection of similarity] make possible a satisfactory constitution of temporal order [...]. Carnap's argument [...] would seem to assume [against his will] that memory images and afterimages [which are temporally specified in advance] are epistemologically as fundamental as [the recollection of similarity] (Goodman 1951, p.132). 24 "[M]emory images and afterimages" 29 mean the same thing: past experiences. Regarding them, Goodman insists: to recognize the past experiences as such, i.e., to specify them in time, we need more than the recollection of similarity. This makes sense, practically. Consider the case where we recognize a similarity between two past experiences. How exactly could we know one is temporally precedent to the other? It is impossible unless we specify them in time in advance. For this very reason, we cannot but say Carnap's argument is quite unsatisfactory. 7 The Other Point at Issue: Space This is how Carnap's theory of time is criticized. It did not supersede Kant's theory, either 30. Then, what about space? Let us continue our inquiry. Space is concerned with our visual field in particular. Hence we need more detailed information of the manifold. But unfortunately, it is not found in the Aufbau (cf. Kleinknecht 1980, p.23 note1). Nevertheless, by reference to other researches (cf. Leitgeb 2007, p.190, Goodman 1951, p.141), we can provide it: (10) Now I see a red spot in the upper left place of my visual field and a blue spot in the lower right place. This is an example of the manifold. How is space constituted from this coarse, raw experience? This is our concern below. 8 Color Spot For the discussion, let us first segment the preceding example: (11) <now, <red, the upper left>, <blue, the lower left>> Although these expressions are merely for simplicity, it is quite interesting that these ordered pairs 31 show a similar structure to protocol sentences 32. In this formulation, we realize that an elementary experience consists of two parts: specification of time like "now," and a color spot like "<red, the upper left>." The color spot is so important in the following discussion, which was introduced by Goodman 33 and defined as a pair of a color like "red" and a place like "the upper left." Clearly, this latter factor is concerned with our present 25 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective interest: space. Our experience could have different color spots at one time like "<red, the upper left>" and "<blue, the lower left>" in (11)34. So the elementary experience is fundamentally a colorfully spotted plane (two-dimensional visual field). We are largely indebted to Goodman for this interpretation (1951, p.141). Following him, we can arrange elementary experiences in the following way: (12)35 Here, to each color-spot, e.g., "<red, the upper left>," one alphabet, e.g., "a" is assigned. And "xi" stands for one elementary experience, on the right side of which its content is shown like "ah." 9 Similarity Circle However, the most outstanding above is surely the balloons. They stand for the similarity circles based on Ae 36. This is an abstraction concept in Carnap's system 37 and defined as follows: (13)38 ähnli is a similarity circle 39 ←→ def. x y[((x ähnli y ähnli)→xAey) ((x ähnli y ähnli)→¬(xAey))] This formula says, "In a similarity circle, every element is similar to any of the other members." Take another look at (12). Certainly, in one balloon, every element is similar to any of the other members. But what exactly does it mean? What is "similarity" in the first place? In (13) just stated, Ae stands for part similarity 40, which is reduced to Er: x1) ah x4) bl x7) chp x1 0) dlt x1 3) ep x1 6) f tw x2) ai x5) bm x8) ciq x11) dmu x1 4) eq x1 7) fu x3) aij x6) bmn x9) ciqr x1 2) dmuw x1 5) eqr x1 8) fuw x1 9) afim ähnl1 ähnl2 ähnl4 ähnl3 26 (14) For any x and y, xAey ←→ def. xEry yErx x=y (Carnap 1928, §110)41 But there is room to discuss the notion of Er further. Actually, Er is definable here in a stricter manner than before (§5): (15) For any x and y, xEry ←→ def. at least one alphabet of x is adjacent to at least one alphabet of y (cf. Goodman 1951, p.127)42. See, e.g., ähnl1 (cf. (13)). Certainly, x1Erx3 holds since a in x1 is the same as (thus, adjacent to) a in x3 ; again, x1Erx9 holds since h in x1 is adjacent to i in x9. This is how we realize every element is similar to any of the other members in a similarity circle. 10 Quality Class The constitution of space is made by abstracting places from the similarity circles. But places are still inside color spots, which consist of elementary experiences in similarity circles. So next, we must take color spots out of similarity circles, and then, abstract the places. As we see in (13), each color spot is already arranged neatly. For example, {x4, x5, x6} brings b into relief. Interestingly enough, these arrangements occupy vertical spaces alone, where similarity circles overlap each other. Carnap called them essential overlaps 43 (cf. Goodman 1951, pp.134f.). Thus, if we can take out essential overlaps, it soon leads to the abstraction of color spots. On this procedure, Carnap had two obstacles in mind 44. One is the case where a class not fitting into the vertical space is wrongly chosen. For example, {x19, x5, x6, x11, x12} is possibly chosen to abstract m. However, in the present situation, it is not favorable. The other is the case where a subclass of an essential overlap is wrongly chosen. For example, {x17, x18} is possibly chosen to abstract u. But in the present situation, it is not favorable. To avoid the first obstacle, Carnap laid down the following regulation. (16)45 [{( is a similarity circle) x(x x )}→ ( )] " " stands for "a quality class for one color spot," which we call a color-spot 27 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective class hereafter (cf. Goodman 1951, p.140). Now, (16) says, "If some members of belong to a similarity circle, as a whole must be included in the circle." By this regulation, {x19, x5, x6, x11, x12} is excluded. Take this class as ; x19 ähnl1, but {x19, x5, x6, x11, x12} ähnl1. To avoid the second obstacle, Carnap laid down the following regulation: (17)46 x[(x )→ {( is a similarity circle) ( ) (x )}] Roughly speaking, this means: "For any x outside , there must be a bigger circle (= ) to which x does not belong, either." By this regulation, {x17, x18} is excluded. Take this class as ; x16 {x17, x18}, but there is no similarity circle which {x17, x18} is wholly included in, and x16 does not belong to. By these regulations, the abstraction of color spots seems to be made smoothly. But practically, the first regulation (=(16)) was too strong. Have a look at x19 in (12). Although this is not illustrated, ähnl1 and ähnl4 overlap at x19. In other words, a member (=x19) of {x1, x2, x3, x19}, which is a subset ofähnl1 and promising for the abstraction of a, belongs to ähnl4 as well. But {x1, x2, x3, x19} is not wholly included in ähnl4. So it violates regulation (16). Carnap called this kind of accident an accidental overlap 47 (cf. Goodman 1951, pp.134f.). To keep {x1, x2, x3, x19} promising for a, he then turned his eyes to the number of the members belonging to a similarity circle. As for {x1, x2, x3, x19}, the number of its members belonging to ähnl4 is only one (=x19), while that belonging to ähnl1 is all of the four. Distinguishing these two cases, Carnap laid down the bar of half, through which the accidental overlap is avoided. With this modification, the color-spot class is defined as follows: (18) ( is a color-spot class) ←→ def. [{( is a similarity circle) ( | | ―| | > 1―2 )}→ ( )] x[(x )→ {( is a similarity circle) ( ) (x )}] 48 (Carnap 1928, §112). The first conjunct on the right side (=" ...") is the modified version of (16), and the second (=" x...") is the same as (17). Hereafter, we symbolize the class of all color-spot classes, i.e., { 1, ..., n}, as qual 49. 28 11 Similarity between color-spot classes Now then, suppose that we obtain some color-spot classes, 1, 2, ..., from (18). The next step is to partition them into "similar" groups. For this purpose, we must define the "similarity" concept between color-spot classes in advance: (19) For any i and j, iAq j ←→ def. x y[(x i y j)→xAey] 50 Definiendem Aq is the similarity between color-spot classes 51. Taking Def. (14) of Ae into account, simply this formula reduces Aq to Er, which was defined earlier (=(15)). But oddly enough, Carnap introduced a new similarity concept here again: (20)52 Let x be <t1, <c1, p1>>, and y be <t2, <c2, p2>> 53. Then, xAey ←→ def. [(c1 is similar to c2) (p1=p2)] [(c1=c2) (p1 is near (similar to) p2)]. Although incompatible with previous arguments, this new concept of similarity actually work very well, which properly partitions qual, i.e. { 1, ..., n}, into equivalence classes: (21)54 Make a power relation of Aq, i.e., Aqpo, which becomes an equivalent relation as well 55. Thereby, we can partition qual into {{ 1, ...}, { 2, ...},...}, that is, {{ i| 1Aqpo i}, { j| 2Aqpo j},...} 56. This latter class is called a partition of qual modulo Aqpo (a quotient set of qual modulo Aqpo), symbolized as qual/Aqpo or {[ i]Aqpo| i qual} 57. 12 The Visual Field Place Each member of this class, [ i]Aqpo ( qual/Aqpo), is called an equivalence class 58. Let us then take one, [ 1]Aqpo) (={ 1, 3, ...}={ i| 1Aqpo i}) 59, numbering its members all over again: { 11, 12,...} (={ 1, 3, ...}). Its content is, taking (19) and (20) into account, supposed to be as follows (Given that the class is composed of only four elemenets): (22) 11=the color-spot class for <red, the upper left> 12=the color-spot class for <pink, the upper left> 29 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective 13=the color-spot class for <orange, the upper left> 14=the color-spot class for <red, the left> Here, 11Aq 12 holds, because any member of 11, which has the form <t, <red, the upper left>, ...>, is similar (from Def. (20)) to any member of 12, which has the form <t, <pink, the upper left>, ...>, so that 11 and 12 are taken to belong to the same equivalence class, [ 1]Aqpo, in accordance with (21). How can we then take the spatial place "the upper left" out of this class? This is the final stage of our abstraction. Carnap focused on a certain feature here: (23) Two color-spot classes indicating the same spatial place cannot have any elements in common 60. For example, 11, 12, and 13 above cannot have any elements (elementary experiences) in common, because they refer to the same place, "the upper left." This could be realized if we admit one place, even if referred to by different experiences, cannot have different colors. Surely it can if we admit a certain length of time in the experiences. But Carnap excluded such cases by adding a proviso "at the same time" 61 to (23)62. Thus, to abstract the spatial place, we should partition { 11, 12, 13, 14} into subclasses the members of which do not have any elements in common. We introduce the following relationship to carry it out: (24)63 For any i, j qual/Aqpo, iFre j ←→ def. i= j ¬ x[x i x j] Using this Fre, we can partition qual/Aqpo into the following k's: (25) k is a certain subclass of an equivalence class in qual/Aqpo the members of which do not have any elements in common ←→ def. For any i and j qual/ Aqpo, [(( i k j k)→ iFre j) (( i k j k)→¬( iFre j))]. Compare this with (13) above. As is soon realized, k is taken a similarity circle based on Fre. And we symbolize the class of all k's as Sim'Fre (Carnap 1928, 30 §117). This k is much the same as the spatial places sought for. The following definition becomes a finish: (26)64 P is a place ←→ def. i( i P) k(k Sim'Fre) P=(k- (Sim'Fre-{k})) A concrete example may help to understand this definition: (27) x1=<t 65, <red, the upper left>> x2=<t, <red, the upper left>, <green, the lower right>> x3=<t, <red, the upper left>, <blue, the center>> x4=<t, <pink, the upper left>> x5=<t, <pink, the upper left>, <black, the just lower part>> x6=<t, <orange, the upper left>> x7=<t, <orange, the upper left>, <green, the just upper part>> x8=<t, <red, the left>> x9=<t, <red, the left>, <pink, the upper left>> x10=<t, <red, the left>, <red the upper left>> Following the preceding notation (§8), x1 can be symbolized as "b," x2 as "bi," x3 as "bm," x4 as "c," x5 as "cu," x6 as "d," x7 as "dw," x8 as "a," x9 as "ac," x10 as "ab." These are, in accordance with (13) and (15), put into one similarity circle, while the color spot <red, the upper left> is common among x1 to x3 and x10, <pink, the upper left> among x4, x5 and x9, <orange, the upper left> between x6 and x7, <red, the left> among x8 to x10, respectively. Suppose that these groups are located in the essential overlaps (cf. §8). Then, color-spot classes corresponding to (22) above are constituted. (28) 11={x1, x2, x3, x10}, 12={x4, x5, x9}, 13={x6, x7}, 14={x8, x9, x10} The preceding definitions, (25) and (26), are understandable from this instance. First, recall that { 11, 12, 13, 14} forms an equivalence class. That is, { 11, 12, 13, 14}={ i| 1Aqpo i} qual/Aqpo. Then, we can constitute "the similarity circles based on Fre" in accordance with (25): (29) k1={ 11, 12, 13}, k2={ 13, 14} 31 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective For a finish, the spatial places are constituted in accordance with (26): (30) Sim'Fre={k1, k2} (i) Let k1 be k in (25). Then, (Sim'Fre-{k1}= {k2}=k2 66. Therefore, k1- (Sim'Fre-{k1})=k1-k2={ 11, 12, 13}-{ 13, 14}={ 11, 12} 67=P. (ii) Likewise, when k2 is k in (25) instead, { 14}=P. In the case of (i), the place "the upper left" is abstracted. In the case of (ii), the place "the left" is abstracted. Each of them is called a visual field place 68, which is nothing but the spatial place we have sought for. 13 Evaluation This is how space was constituted in the Aufbau. Let us then ask: Could it supersede Kant's picture? Kant's picture here means the following: (31) Through external senses, we represent objects as outside of us and in space. It is in the space that we recognize the form of the objects [as far as they appear to us], their size [as far as they appear to us], and their mutual relations [as far as they appear to us] (Kant 1787, B37). In this passage, Kant defines space not as something like a coordinate, but as a fundamental framework for our recognition of external objects. It implies that even initial recognition of spatial locations, such as "right," "left," "upper," "lower," "in front of," and "behind," is unfeasible without that framework 69. It is true that Carnap succeeded in the abstraction of space. But it never follows that Kant's framework of space is no longer necessary. The fact is the opposite. Kant's framework is indispensable even for the Aufbau. For see the original example of the elementary experience (=(10)); therein, the spatial locations, "the upper left" and "the lower right," are inscribed in an inerasable manner, which means: the spatial location is indispensable even for the elementary experience, since otherwise we would be lost in regard to where each color is. For this reason, we should say, the Aufbau never superseded Kant's philosophy; it still needs the latter framework. 32 14 Conclusion We have seen Carnap's course of argument in the Aufbau from a Kantian perspective. After all, it never superseded Kant's picture. This does not mean, however, Carnap's theory was useless. Its significance remains. At the cutting edge, Hans Leitgeb (2007, 2011) has worked on its revival. As for me, I think the Aufbau is more suitable to explicate analyticity. Analyticity cohesively concerns the concept of the objects, which is also the specialty of the constitution theory, as we have seen so far. While details are left to another paper, there remains a few parts to be corrected in Carnap's theory. In particular, its central notion, similarity, is still unclear. As much as three characterizations of it are presented heretofore; that is, (7), (15), and (20). We will not be able to apply the theory of the Aufbau to the explication of analyticity until this ambiguity is removed. Notes 1. Each section is referred to with the symbol "§." But "§" also stands for "section" of the Aufbau, for example. 2. As for the abbreviation of titles, see REFERENCES. 3. Pincock's survey (2009) is informative for the overview. 4. The translation of German texts (of Carnap's and of Kant's) is arbitrarily made by the author. 5. However, we do not deal with the relationship of our argument to modern physics like Einstein's relativity theory. 6. Hans Leitgeb (2007, 2011) is one of the researchers who develop the technical aspect of the Aufbau. But his study is not concerned with Kant's philosophy. 7. "Das Mannigfaltige" (Carnap 1928, §83). 8. "Die Erlebnisse" (Carnap 1928, §64). 9. "Eine eigenpsychische Basis" (Carnap 1928, §§63-64). 10. "Transzendentales Subjekt" (Carnap 1928, §66). 11. "Ich-Bezogenheit" (Carnap 1928, §65). 12. "Die Ähnlichkeitserinnerung" (Carnap 1928, §66). 13. "Elementarerlebnis" (Carnap 1928, §66). 14. A constitution system is a system according to which almost all the objects are "constituted" (cf. note19) from more basic ones. We could imagine many systems of that kind, but they are theoretically unified into the constitution theory, which 33 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective Carnap elaborated on in the first half of the Aufbau (Carnap 1928, §§25-26). 15. Grundbeziehung" (Carnap 1928, §61, §78). 16. Relations are symbolized as xRy or as R(x,y) depending on contexts. 17. "Grundelement" (Carnap 1928, §61, §67). 18. "Das Feld" (Carnap 1928, §34). Carnap symbolized it as "C'Er" (1928, §109, §34) in accordance with Principia (Whitehead&Russel 1910, p.35). In any case, C'Er=fl dEr=domEr ranEr={x| y(xEry) z(zErx)} (cf. Enderton 1960, p.40). 19. Constitution is to define a less basic (so abstract) object from more basic ones by the constitutional definition (Carnap 1928, §35). The constitutional definition has two kinds. One is the explicit definition. The other is the definition in use or the contextual definition (Carnap 1928, §§38-40, Whitehead & Russel 1910, p.25, p.69). All the definitions below are the definitions in use, because they define the objects in the context of a biconditional sentence as a whole. 20. See Sugihara's analysis, for example (1974, pp.38f.). It is naturally true of Kant's theory (Sakai 1978, p.64, Kant 1787, B46). As for the notion of linear ordering, see Enderton's explanation (1977, p.170). 21. For any t1, t2, and t3, if t1 t2 and t2 t3, then t1 t3. ("ti tj" stands for "ti is temporally prior to tj.") 22. For any t1 and t2, exactly one of the following three holds: t1 t2, t1=t2, or t2 t1. 23. This feature (intransitivity) was stated by Carnap in his logical definition of similarity (Carnap 1928, §11). But there, in contrast with Er, symmetricity and reflexivity were admitted. 24. "Potenzrelation" (Carnap 1928, §34). This concept is attributable to Principia (Whitehead & Russel 1910, pp.35-36). 25. We use logistic notation a bit sloppily. For example, " " is sometimes replaced with "for any...," and often omitted in the case of definition especially. Again, we do not observe the distinction between the object and the meta-language, the application of Quine's quasi-quotes, and so on. 26. Carnap distinguished Beziehung from Relation (1928, §28, §34). The latter is the extension of the former. He symbolized the former as Er, for example, and the latter as Er (Carnap 1928, §109). Then, Er={<x,y>|Er(x,y)}. 27. "Eine Kette" (Carnap, 1928, §34). 28. Carnap called trichotomy "Zusammenhang" (1928, §11). 29. Goodman sometimes says "a memory image or afterimage" (1951, p.132). 30. Not supporting the Kantian view, actually Leitgeb (2011, p.280) adopted the temporal order as a primitive term in his "Aufbau-like" system. 31. Carnap defined the sensation (die Empfindung) in a similar manner (1928, §93, §116), but it is a pair of an elementary experience and a quality class or color-spot (cf. Goodman 1951, p.145). 34 32. "Protokollsätze" (Carnap 1932, p.438). 33. Goodman's notation is "color-spot" (1951, p.134). Carnap generally called it a quality class (1928, §81). 34. Although there are naturally far more color spots in reality, we ignore this point in the present discussion. 35. This figure is made from Goodman's rough arrangement (1951, p.135). 36. "Die Ähnlichkeitskreise in bezug auf Ae" (Carnap 1928, §80, §111). 37. Russel's principle of abstraction (1937, §210) is one of its predecessors (Carnap 1928, §97), which is originally an application of the partition into equivalence classes in set theory (cf. Enderton 1977, pp.55f., Leitgeb 2007, p.181). On the other hand, Carnap (1928, §§69-73) called this method quasianalysis (Quasianalyse), because abstracting a quality from an elementary experience is contradictory to his doctrine of totality (cf. (3)). Whether this self-criticism is taken seriously or not, most researchers later challenged this part exclusively. Among them were Goodman's famous companionship difficulty (1951, p.123) and difficulty of imperfect community (1951, p.125). Since this criticism, most researchers customarily have dealt with Goodman's argument (Kleinknecht 1980, Leitgeb 2007; 2011). 38. Carnap did not articulate this definition except in an informal style (1928, §71). We are indebted to later researchers for this definition (Leitgeb 2007, p.214, Kleinknecht 1980, p.24, Goodman 1951, p.121). 39. Instead of "ähnli," we can use "Sim'Aei" (Carnap 1928, §111). 40. "Teilähnlichkeit" (Carnap 1928, §77). 41. As Goodman suggested (1951, pp.132-133), in this definition, Carnap is said to have withdrawn his doctrine on the temporal order of Er (cf. §5). But now it does not matter since we have already discarded that doctrine (§6). 42. But this definition is exclusively concerned with the preceding figure (=(12)). So Carnap did not state it. 43. "Wesentliche Überdeckungen" (Carnap 1928, §80). 44. We owe the following argument to Goodman (1951, pp.135-136). 45. See the first proviso of Goodman's (1951, p.135). 46. See the second proviso of Goodman's (1951, p.135). 47. "Eine zufällige Überdeckung" (Carnap 1928, §80). 48. "| |" stands for 's cardinality. 49. This is originally used for the class of all quality classes (Carnap 1928, §112). 50. We can obtain our formulation of (19) from Carnap's original (1928, §114) by reference to Principia (Whitehead&Russel 1910, p.278). 51. In Carnap's terminology, "die Ähnlichkeit zwischen Qualitäten" (1928, §114). 52. This was stated only in an informal style (Carnap 1928, §88, Goodman 1951, p.140). 35 The Constitution of Space and Time in the Aufbau Viewed from a Kantian Perspective 53. See (12). "t" stands for time, "c" for a color, and "p" for a place. Strictly speaking, x must be <t1, <c1, p1>,...> to make it general. But in the present discussion, we omit "..." for simplicity. 54. Carnap provided this definition as the constitution of the sense class (die Sinnesklasse) symbolized as "Aeq'Aqpo" (1928, §115). 55. The relation which is reflexive, symmetric and transitive (cf. Enderton 1962, p.56). As for the power relation, see (8) above. 56. Here, we step up from a class (qual) to a class of classes (qual/Aqpo). This is nothing but constitution (Carnap 1928, §40). 57. In detail, see Enderton's explanation (1962, p.57), for example. 58. In detail, see Enderton's explanation (1962, p.57), for example. 59. " 1" is replaceable with other members, e.g., " 3." 60. This is not stated by Carnap directly (1928, §88). See also Goodman's commentary (1951, p.140). 61. "zugleich" (Carnap 1928, §88). 62. Here, we can realize what kind of experience Carnap had in mind. 63. In Carnap's notation, iFre j ←→def. iI j iFr j (1928, §97, §117). 64. In Carnap's notation, " i( i P)" was " !P." But this does not stand for the unique existence (cf. Canrap 1928, §97, Whitehead&Russel 1910, p.229). 65. These are supposed to be experienced within a short length of time (cf. note62). 66. {a}=a (Enderton 1962, p.25). 67. Recall the calculation of the relative complement (Enderton 1962, p.27). 68. "Sehfeldstelle" (Carnap 1928, §88, §117). 69. Recall the famous argument in Prolegomena as well (Kant 1783, §13). REFERENCES (Letters in square brackets mean abbreviations) Carnap, R. 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