A New Analysis of Quasianalysis - Semantic Scholar

A New Analysis of Quasianalysis Hannes Leitgeb

Abstract We investigate the conditions under which quasianalysis, i.e., Carnap’s method of abstraction in his Aufbau, yields adequate results. In particular, we state both necessary and sufficient conditions for the socalled faithfulness and fullness of quasianalysis, and analyze adequacy as the conjunction of faithfulness and fullness. It is shown that there is no method of (re-)constructing properties from similarity that delivers adequate results in all possible cases, if the same set of individuals is presupposed for properties and for similarity, and if similarity is a relation of finite arity. The theory is applied to various examples, including Russell’s construction of temporal instants and Carnap’s constitution of the phenomenal counterparts to quality spheres.

Keywords: Quasianalysis, Aufbau, Carnap, abstraction, similarity.

1

Introduction

In his Der Logische Aufbau der Welt[7] – from now on briefly: Aufbau – and in unpublished manuscripts before the Aufbau (see [5], [6]), Rudolf Carnap introduced the so-called method of quasianalysis as an extension of Frege’s and Russell’s method of abstracting mathematical entities from equivalence relations.1 Since the empirical domain seemed to demand descriptions in terms of similarity relations rather than in terms of the more restrictive equivalence relations, the standard method of abstraction had to be adapted in order to enable also the logical (re-)construction of empirical entities. Several years later Nelson Goodman critized Carnap’s quasianalysis for not delivering the right distribution of qualities from a similarity relation under certain conditions (see section V.3–V.5 in [14], p.557 of [15], and [16]): sometimes qualities are not introduced by quasianalysis, since they cannot be separated with respect to the similarities that they induce (“companionship difficulty”), or they are introduced unjustifiedly because several individuals are mutually similar without sharing a single quality (“difficulty of imperfect 1

community”). Subsequent papers on this topic strengthened and elaborated Goodman’s criticism in terms of further examples and observations (see e.g. Eberle[11], Kleinknecht[22]). In fact, Carnap already anticipated and dealt with these problems in his unpublished “Die Quasizerlegung”[6] and also in the Aufbau itself, i.e., long before Goodman. However, the conditions under which quasianalysis fails are not characterized by Carnap in apt detail and have not been characterized in such form since then.2 The topic of this paper is to fill this gap and thus also to describe in a precise manner the conditions under which quasianalysis indeed succeeds. As we are going to see, several necessary, sufficient, and even both necessary and sufficient conditions of this kind can be found. Moreover, the outlines of these conditions will in some sense explain why Carnap’s application of quasianalysis in the context of the Aufbau is bound to fail, whereas Russell’s application of the same method in Our Knowledge of the External World [51] and of a related method in his The Analysis of Matter [52] succeeds. Quasianalysis is not to be abandoned generally as a method of logical construction, but the class of properties or qualities that a given similarity relation is intended to reflect has to be of a particular kind in order to make quasianalysis yield faithful and full results. We will also show that there is no method of constituting properties from similarity that succeeds under all conditions as long as the underlying set of individuals for properties is the same as for similarity and the similarity relation is of fixed finite arity. Our results vindicate the recent rise of interest in quasianalysis (cf. Rodriguez-Pereyra[49], Hazen&Humberstone[18], Danut[9]) and support the positive assessment of the method in Moulines[42] and Mormann[35], [36], [37], [38], [39], [40]. However, contrary to Moulines and Mormann, we think our analysis also shows clearly that quasianalysis indeed cannot be applied as intended by Carnap in section §111 of his phenomenalistic constitutional system in the Aufbau, i.e., as a method of constituting the phenomenal counterparts to quality spheres on the basis of similarity. Thus our findings ultimately support Goodman’s criticism of the Aufbau, although it takes an elaborate theoretical account to justify this criticism and not just a set of examples as given by Goodman himself.

2

Similarity and Properties

In the subsequent section we will present quasianalysis – actually, the Aufbau version of quasianalysis of the first kind : see section 4 below – as a formal procedure by which so-called property structures may be generated from similarity structures. In this section we introduce the necessary terminology

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that is presupposed later. As we will see, Carnap’s typical examples of quasianalysis are instances of the more general theory that we are going to develop. According to the Aufbau, p.35, a similarity relation is characterized formally as a binary reflexive and symmetric relation. The paradigm case example that Carnap thinks of is metrical similarity: if x is similar to y and y is similar to z in the sense that their respective distances d(x, y) and d(y, z) as being given by a metric d are less than or equal to some fixed boundary ǫ > 0, it is not necessarily the case that also x and z are similar according to this metrical notion of similarity since distances might add up in a way such that d(x, z) > ǫ. Therefore, similarity need not be transitive, although reflexivity (since d(x, x) = 0 6 ǫ) and symmetry (if d(x, y) 6 ǫ, then also d(y, x) 6 ǫ) hold unrestrictedly. We follow the terminology of Mormann[35] when we define: Definition 1 (Similarity structure) A pair hS, ∼i is a similarity structure on S if 1. S is a non-empty set, 2. ∼ ⊆ S × S is a reflexive and symmetric relation on S. If hS, ∼i is a similarity structure, the members of S will be called ‘individuals’. In case that x ∼ y we say that x and y are similar (according to hS, ∼i). Since reflexivity and symmetry are presupposed, we can always “depict” similarity structures in a unique way by their underlying (undirected) graphs G∼ = hS, {{x, y}|x ∼ y, x 6= y}i. In most cases S will be assumed finite.3 Let us now contrast similarity structures with structures of a different kind, i.e., property structures: Definition 2 (Property structure) A pair hS, P i is a property structure set on S if 1. S is a non-empty set, 2. P is a set of subsets of S, ∅ ∈ / P , and for every x ∈ S there is an X ∈ P , such that x ∈ X. If hS, P i is a property structure, we call the members of S again ‘individuals’, while the members of P are called ‘properties’ (according to hS, P i). Note that the latter are properties in the extensional sense, i.e., sets.4 For 3

reasons of convenience, we assume that there is no “empty” property which does not apply to any individual, and we take for granted that every individual in S has at least one of the properties in P . As we are going to see later, the empty set could not be constructed by means of quasianalysis, so if the empty set were included as a property, the notion of adequacy that we are interested in would have to be restricted to adequacy with respect to non-empty properties only. Against a background such as Carnap’s application of quasianalysis in the so-called “autopsychological” section of the Aufbau, it might be more appropriate to call the members of P ‘qualities’ rather than ‘properties’, but since we aim at a more general viewpoint on the method, we prefer the latter expression.

3

Quasianalysis

Once the classes of similarity structures and of property structures have been introduced, the next natural question is whether some sort of correspondence can be established between them. In particular, we are going to consider how similarity structures can be determined by property structures and how property structures can in turn be determined by similarity structures. We are not interested at this point in arguing that any of these two directions of determination is actually to be regarded “the preferred one”, for whatever empirical or philosophical reasons. It is clear that Goodman regarded qualities as being prior to similarity, therefore he always takes the determination of similarity structures by quality structures for granted while the determination in the other direction is just a matter of reconstruction. Proust[44], [45], and Mormann[35], [36], [37], [38], [39], [40] have a viewpoint that is closer to Carnap’s earlier presentations of quasianalysis in [5] and [6] where similarity relations – or binary relations in general – are considered to be prior to properties or qualities. According to this latter account, the determination of property structures by similarity structures is not about reconstruction at all but rather about construction or representation: construction of properties on the basis of similarity or representation of a given similarity relation in terms of properties. There is no external standpoint that would allow for a comparison of the properties that were constructed from similarity with anything beyond the given similarity relation itself. As we are going to argue in section 9, Carnap’s view in the Aufbau is sophisticated and does not coincide with either of these two interpretations: while similarity is prior to properties or qualities on the level of the phenomenalistic constitution system, properties or qualities are prior to similarity on the

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level of Carnap’s extrasystematic description of the phenomenalistic constitution system. When Carnap discusses the intended results of quasianalysis (cf. §81, 72) and distinguishes them from unintended ones that might occur under unfavourable circumstances, he takes the extrasystematic standpoint. Accordingly, we are going to present quasianalysis as a method of reconstructing properties from a similarity relation that is itself determined by a given property structure. The notions of faithfulness, fullness, and adequacy that we will introduce in section 5 are motivated by this latter point of view. However, our results will be of relevance for discussions of quasianalysis in general, independent of how the method is interpreted or what its aims are understood to be. In section 5 we will also consider cases in which a property structure is determined by a given initial similarity structure. It is easy to see in what sense a given property structure may be said to determine a similarity structure. The idea can already be found in some of Leibniz’ handwritings (see [24], A64 107/P 13): as Leibniz remarks (this is our translation), “Peter is similar to Paul” reduces to “Peter is A now and Paul is A now”. If we disregard the indexical ‘now’ and if we make the hidden existential quantifier explicit, we have in fact set up our intended correspondence principle: two individuals are similar iff they share a common property. The same idea may of course also be found in much older literature (e.g. Bonaventura[3], p.35) and in much more recent literature (cf. Goodman[16], pp.441–442). Similarity in this sense is dual to identity which satisfies Leibniz’ principium identitatis indiscernibilium: two individuals are identical iff they share every property. Put formally, this account of determining similarity from properties amounts to:5 Definition 3 (Determined similarity structure) hS, ∼P i is determined by hS, P i if for all x, y ∈ S: x ∼P y iff there is an X ∈ P , such that x, y ∈ X. Instead of ‘hS, ∼P i is determined by hS, P i’, we also say ‘ hS, P i determines hS, ∼P i’. Obviously, every property structure on S determines a unique similarity structure on S. Thus, we are entitled to refer to the similarity structure determined by hS, P i. Let us take a look at a simple example: Example 1 We consider a property structure hS1 , P1 i with just four individuals and two properties: S1 = {1, 2, 3, 4}, P1 = {{1, 2, 3}, {3, 4}}. Graphically: 5

✬ r2 r1 ✫

✩

✗ r3 ✖

✪

r4

✔ ✕

Fig. 1: Property Structure hS1 , P1 i The similarity structure hS1 , ∼1 i = hS1 , ∼P1 i that is determined by hS1 , P1 i is in this case given by: S1 = {1, 2, 3, 4}, ∼1 = {h1, 1i, h2, 2i, h3, 3i, h4, 4i, h1, 2i, h2, 1i, h1, 3i, h3, 1i, h2, 3i, h3, 2i, h3, 4i, h4, 3i}, i.e., depicted as a graph: r2 ◗ r ✑

◗

✑

◗ ◗ ◗r3 ✑ ✑ ✑

r4

1

Fig. 2: Similarity Structure hS1 , ∼1 i (From now on we will always specify similarity structures simply by their underlying graphs.) Note that every similarity structure hS, ∼i on S can be determined by at least one property structure on S in the manner of definition 3: e.g. consider the (trivial) property structure in which unordered pairs {x, y} are used as properties if and only if x ∼ y (and x 6= y) (cf. Mormann[38], p.79). Under certain conditions there is a particularly nice isomorphic copy of hS, ∼P i as follows: replace each individual x in S by the class of properties in P that include x as a member; define these “new” individuals to be similar if and only if they have non-empty intersection. In this way one ends up with a similarity structure of the form hS ′ , ∼′ i where S ′ = {A|∃x ∈ S with A = {X ∈ P |x ∈ X}}, A ∼′ B iff A ∩ B 6= ∅. The members of S ′ might be regarded as the individual concepts of the original individuals in S (relative to the properties of the given property structure hS, P i). It is easy to see 6

that if for every two members of S there is a property in P which applies to the one but not to the other, hS, ∼P i is isomorphic to hS ′ , ∼′ i (i.e., there is a bijective mapping f : S → S ′ , such that for all x, y ∈ S: x ∼P y iff f (x) ∼′ f (y)). The graphs that are underlying similarity structures such as hS ′ , ∼′ i are called ‘intersection graphs’ in the graph-theoretic literature (see McKee&McMorris[34] for an introduction); the nodes of intersection graphs are sets and their edges are given by non-empty intersection. It is well known that every graph is, up to isomorphism, an intersection graph. There are also interesting results concerning parsimonious set representations: define the intersection number i(G) of a graph G to be the minimum cardinality of a set S such that G is (isomorphic to) an intersection graph of a family of subsets of S. Let θ(G) be the minimum cardinality of a family {G1 , G2 , . . . , Gk } of complete subgraphs of G such that every edge in G is included in at least one of G1 , G2 , . . . , Gk . As P. Erd¨os, A.W. Goodman, L. P´ osa have observed, for every graph G it holds that i(G) = θ(G) (see [34], p.10). Such results are particularly relevant for quasianalysis in the version of [6] (see section 4). Mormann has studied intersection graphs intensively, though not by that name (see also Brockhaus’ uniqueness result of [4] which is further analyzed by Mormann; cf. [35]. See p.977 of Rodriguez-Pereyra[50] for another instance of an intersection graph). Let us now turn to the other direction: in what sense may a similarity structure be said to determine a property structure? That is where quasianalysis enters the picture. We need some further formal terminology in order to describe the method: • X ⊆ S is called a clique of hS, ∼i iff for all x, y ∈ X: x ∼ y. • X ⊆ S is a maximal clique of hS, ∼i iff X is a clique of hS, ∼i and there is no Y ⊆ S, such that X & Y and Y is a clique of hS, ∼i as well.6 The main idea of quasianalysis can be explained easily in terms of an informal example: think of a room with coloured objects, where sharing a colour is used as a similarity relation. Each colour may be supposed to embrace a certain range of hue, brightness, and intensity, and colours are permitted to “overlap”. A set X of individuals which are brown (partially or completely) will then certainly be a clique with respect to similarity, since every two members of X share a colour. In order to turn from a set such as X to the set of brown individuals in this room, and accordingly for the other colours, one might take maximal cliques rather than just cliques simpliciter in order to constitute the colour properties. That is essentially 7

the core of the method of quasianalysis, a procedure by which Carnap’s socalled ‘similarity circles’ (see Aufbau, sections §70–73, 80–81, 97, 104), i.e., our maximal cliques, are constituted. In our context, instead of ‘similarity circles’, we might call these set-theoretic constructs ‘quasiproperties’ (cf. Carnap[6]): properties as being given by quasianalysis. As indicated above, we take a point of view according to which quasianalysis in the Aufbau (though not so much in [6]) aims at determining those properties that have originally determined the very similarity structure from which they are now to be (re-)constructed. The formal definition is as follows: Definition 4 (Determined property structure; Quasianalysis) hS, P ∼ i is determined by hS, ∼i if P ∼ = {X ⊆ S|X is a maximal clique of hS, ∼i}.7 In this way every similarity structure hS, ∼i determines a unique property structure hS, P ∼ i (which is again to be read as being synonymous to: hS, P ∼ i is determined by hS, ∼i). Accordingly, we will sometimes speak of the property structure determined by hS, ∼i. As we are going to show later, not every property structure on S can be determined by a similarity structure on S in the manner of definition of 4. Definition 4 stipulates what it means to say that a property is determined by a similarity structure. But from time to time we will refer to definition 4 also as explaining the method or procedure of quasianalysis, although taken strictly no particular method has been outlined. In such a case consider any constructive way of generating all maximal cliques in a given similarity relation as “the“ method in question.8 In Aufbau terminology: definition 4 allows several interpretations or manners of being read, one of which is as a method of “fictive construction” (cf. Aufbau, p.152). In order to give an example of quasianalysis we may again turn to hS1 , P1 i and hS1 , ∼1 i of example 1: Example 2 If we consider hS1 , ∼1 i now as a given similarity structure, it is easy to see that hS1 , P1 i = hS1 , P ∼1 i is its determined property structure, so quasianalysis leads us back from fig. 2 to fig. 1. We can also make use of maximal cliques in order to derive a further representation result for similarity structures in terms of intersection graphs: let hS, ∼i be given; this time replace each individual x in S by the class of those maximal cliques (not properties as we did above) of hS, ∼i that include x as a member; define these “new” individuals to be similar again if and 8

only if they have non-empty intersection. This gives us another similarity structure hS ′′ , ∼′′ i of intersection graph form and if no two members of S are similar to precisely the same individuals in S, hS, ∼i may as well be proved isomorphic to hS ′′ , ∼′′ i. Therefore, if hS, ∼i = hS, ∼P i is at the same time determined by hS, P i, the intersection similarity structure hS ′ , ∼′ i from above must be isomorphic to the intersection similarity structure hS ′′ , ∼′′ i that we have just introduced. We will see in section 5 that these two isomorphic structures can be – but by no means need to be – equal, because P need not be identical to the set of maximal cliques of hS, ∼i.

4

Other Versions of Quasianalysis

We have already pointed out that there is not just one version of quasianalysis in the Aufbau but actually two: quasianalysis of the first kind (I) and quasianalysis of the second kind (II). But historically, Carnap’s first outline of quasianalysis in his unpublished [6] was neither quasianalysis I according to definition 4 nor quasianalysis II: rather he suggests a more elaborate version of quasianalysis I in which not every maximal clique necessarily counts as a quasiproperty but where additional maxims of “parsimony” with respect to the number and structure of quasiproperties are adopted. Mormann gives an excellent survey and analysis of the formal properties and the philosophical interpretation of this original method of quasianalysis (see his papers in our references; [37] and Mormann’s unpublished manuscript “Carnap’s Quasi-analysis Revisited” contain extensive material on the formal applications of the method). We are nevertheless going to concentrate on quasianalysis I as developed in the Aufbau since it is simpler to analyze, historically more influential, and it always yields unique results (which the version in [6] does not do necessarily). In spite of these differences between the two versions of quasianalysis of the first kind, we can see that a version of quasianalysis I had already been considered by Carnap before he later added quasianalysis II as a further procedure of logical construction. Let us turn now to the differences between quasianalysis I and quasianalysis II (in the Aufbau). The typical Aufbau set S of individuals would be a set of so-called elementary experiences, i.e., total momentary slices through a subject’s stream of experience in a specified interval of time. Carnap’s primary application instance of quasianalysis I is the case of a similarity relation which holds between two elementary experiences if and only if they realize a common quality point in a common quality space, like e.g. one spot in the visual field in the one elementary experience having the same color

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and location as a spot in the visual field in the other experience. In such a case, Carnap would call the similarity relation a relation of part identity (see Aufbau, §76). Quasianalysis of the second kind is to be applied if the given similarity relation is one of part similarity (see Aufbau, §77): two elementary experiences are (part-)similar if and only if they realize some quality points in a common quality space such that the points are “close” according to the metric that is associated with the space, i.e., where the distance of the points in the quality space is less than or equal to some fixed ǫ. Assume e.g. a light red spot to lie in the upper-left corner of the visual field in the one elementary experience and a dark red spot to be in the upper-middle part of the visual field in the other experience: the two realized quality points in the visual quality space would be close with respect to location and colour and the elementary experiences would therefore count as similar (we return to this in more formal detail in section 9). Carnap suggests that quasianalysis of the first kind, as being stated by our definition 4, is appropriate for relations of part identity, whereas a more complex procedure, i.e., quasianalysis of the second kind, is to be used in the case of part similarity. For the rest of this section we are going to outline why we will still focus in this paper just on quasianalysis I while only sketching briefly the characteristics of quasianalysis of the second kind. First of all, despite of Carnap’s considerations on the distinctive features of part identity and part similarity, both part identity and part similarity relations may be subsumed under the scheme of definition 3 if only the determining properties are selected accordingly: Let S be the set of elementary experiences of a particular subject at a particular interval of time: if P is “induced” by quality points in the sense that for every property X in P there is a point p in one of the quality spaces (visual, auditory,. . .), such that X is the set of elementary experiences which realize p, then the determined similarity relation is one of part identity; call the corresponding properties “pointlike”. But if P is not “induced” by quality points but rather by closed quality spheres of fixed radius 2ǫ in the sense that for every property X in P there is a sphere Sph of radius 2ǫ in one of the metric quality spaces such that X is the set of elementary experiences which realize at least one point in Sph, then the determined similarity relation is one of part similarity: for let x and y be elementary experiences in S; it follows that

10

x ∼P y

iff iff

iff

∃X ∈ P , s.t. x, y ∈ X ∃ quality points p, q, s.t. x realizes p, y realizes q, and ∃ sphere Sph of radius 2ǫ , s.t. p, q ∈ Sph ∃ quality points p, q, s.t. x realizes p, y realizes q, and d(p, q) 6 ǫ

(def. 3)

(choice P)

of

(def. of spheres of radius 2ǫ )

In this latter case where properties correspond to spheres let us call the respective properties “spherical”: so if we employ “spherical” properties instead of “pointlike” ones, we automatically end up with what Carnap calls a relation of “part similarity”. We find that both the relation of part identity and of part similarity can be subsumed under the similarity structures of definition 1 which may in either case be determined by property structures in the manner of definition 3. The study of part similarity relations does therefore not by itself exceed the formal framework that was introduced above. Accordingly, Carnap’s method of quasianalysis of the second kind starts with applying quasianalysis of the first kind as its first step, even though the similarity relation in question is one of part similarity. Hence quasianalysis II presupposes quasianalysis I and our aim of finding necessary and sufficient conditions under which quasianalysis of the first kind delivers the right set of (e.g. “spherical”) properties is thus important for the study of quasianalysis of both kinds. This warrants giving quasianalysis I priority treatment. What distinguishes quasianalysis II from quasianalysis I is in the end just that its final output ought not to be the “spherical” properties that determine the underlying relation of part similarity on the basis of definition 3, but rather the additional “pointlike” properties: therefore, quasianalysis of the second kind combines quasianalysis according to our definition 4 with an additional mechanism that tries to recover the “pointlike” properties from the similarity circles that have already been constituted. In very restricted but still important cases, this second step of quasianalysis II can in fact be regarded as another application of quasianalysis I but now applied to a similarity relation that differs from the given one (see section 8). However, in the general case, the second phase of quasianalysis II is somewhat more complicated. Before we turn to this further complication, let us assume for simplicity that quasianalysis of the first kind has indeed delivered all and only the 11

“spherical” properties from the given relation of part similarity: whether or not the “pointlike” properties can now be reconstructed from the “spherical” ones is therefore only dependent on the original underlying system of “pointlike” and “spherical” properties. The part similarity relation itself does not play a role anymore since it has already enabled us to determine the spherical properties correctly and that is everything which is necessary to let the second phase of quasianalysis II begin. Since we are interested in the general question of when properties may be reconstructed from similarity adequately, and since similarity does not really play a role in the second step of quasianalysis II if quasianalysis I has delivered the intended results, this gives us another reason why we rather focus on quasianalysis of the first kind. E.g.: if the “pointlike” properties simply coincide with maximal non-empty intersections of “spherical” properties, then the second step in quasianalysis II will simply consist in generating maximal non-empty intersections of similarity circles. Unfortunately, as far as the intended application of quasianalysis II in the Aufbau is concerned, this latter method of generating quasi-“pointlike“properties will not do, because there may be maximal non-empty intersections of similarity circles that do not coincide with any of the original “pointlike” properties: Carnap calls this the problem of “accidental intersection” (see Aufbau, §80–81). In order to overcome this difficulty he includes a quantitative condition which amounts essentially to this: look for maximal intersections of similarity circles by taking intersections in a step-by-step manner, but do only take an intersection step if the settheoretic overlapping of a similarity circle with the previously generated intersection is not “too small” compared with the number of elements of the previous intersection (see Aufbau, §81). This is the core idea of the procedure (cf. Moulines[42], pp.278f); it can be made precise and expressed formally in a way analogous to definition 4, but the details are not relevant for our present purposes. In any case, as Goodman and others have shown (see Goodman[14], Eberle[11], Kleinknecht[22]), even this elaborate method of quasianalysis II in the Aufbau still does not avoid accidental intersections and thus does not always give the intended results (for a further improvement of the method see Moulines[41], [42]). Moreover, the quantitative intersection check that is included in quasianalysis of the second kind makes a transparent analysis in terms of necessary and/or sufficient conditions of success rather unlikely. That is the main reason why we are not going to deal with this method in the following. The further analysis of quasianalysis II becomes even more complicated if quasianalysis I is not assumed to give us the right set of “spherical” properties as its output: it might still be the case that the addi12

tional second step of quasianalysis II somehow “compensates” the failures of the first step and determines the “pointlike” properties correctly, although that would not be the manner of operating intended by Carnap. In such a case, whether or not the second part of quasianalysis II succeeds does not only depend on what the underlying “spherical” and “pointlike“ properties look like but also on the given similarity relation. However, we do not know of any criteria, necessary or sufficient, which would characterize the circumstances in which quasianalysis of the second kind generates the right output while the intermediate step of applying quasianalysis I does not.

5

A Notion of Adequacy for Similarity Structures

So let us concentrate just on quasianalysis as being given by definition 4. In particular, we want to examine the cases where hS, ∼P i is determined by a P given property structure hS, P i and where in turn a structure hS, P ∼ i of quasiproperties is determined by hS, ∼P i. Our aim is to find out how quasianalysis performs in reconstructing the properties in P from the similarity relation ∼P . Kleinknecht[22], p.27, who regards properties or qualities as intensional entities while only considering the constructed quasiproperties as sets, calls a property represented by a quasiproperty if and only if the latter is the extension of the former. Furthermore, he defines (what we call) a determined similarity structure to be adequate with respect to the property structure by which it is determined iff all and only the determining properties are represented as quasiproperties within the similarity structure. The adequacy of a similarity structure with respect to its determining property structure is thus equivalent to quasianalysis’ yielding precisely the intended result. In order to get a more fine-grained picture of the possible merits or deficits of quasianalysis, we suggest to split up this notion of adequacy for determined similarity structures into two further notions: faithfulness and fullness (and of course we presuppose extensional properties again). A similarity structure is faithful if and only if all quasiproperties that it determines are among the original properties by which it was determined. Thus in the case of faithfulness, quasianalysis does not add any “improper” properties. A similarity structure is full if and only if all properties by which it was determined are among the quasiproperties that it determines. So given fullness, quasianalysis does not omit any of the “actual” properties. Adequacy is just the conjunction of faithfulness and fullness. Here is the precise definition: Definition 5 (Faithfulness; fullness; adequacy) 13

Let hS, P i be a property structure on S, let hS, ∼P i be determined by hS, P i, P and let hS, P ∼ i be determined by hS, ∼P i: P

1. hS, ∼P i is faithful with respect to hS, P i if P ∼ ⊆ P .9 P

2. hS, ∼P i is full with respect to hS, P i if P ⊆ P ∼ . P

3. hS, ∼P i is adequate with respect to hS, P i if P ∼ = P .10 Kleinknecht[22] (following Eberle[11]) considers a third possible deficiency of a determined similarity structure: cases where a quasiproperty represents more than just one property. This kind of failure can occur only if properties are intensional entities and therefore does not affect quasianalysis within our framework. Using Goodmanian terminology, we could e.g. consider two qualities a and b, such that every individual in S has a if and only if it has b. But then a and b would simply be identical to each other, given our presumption that properties or qualities are subsets of S. If there is such a “problem” of manifold representation at all, it ought to be regarded as a consequence of Carnap’s extensionalist account in the Aufbau (which we subscribe to) rather than as a difficulty of Carnap’s method of quasianalysis itself (cf. Rodriguez-Pereyra[49], chapter 8, who is very clear on this point). All possible combinations of faithfulness/unfaithfulness vs. fullness/nonfullness can actually be realized, and lots of such examples are to be found in the relevant literature, though perhaps not systematized as such. Our subsequent three examples are obvious choices and have been used in several of the papers that we cite in our bibliography (we refrain from quoting them separately in this context). hS1 , ∼1 i in example 1 proved to be adequate with respect to hS1 , P1 i. Here are three examples of the other combinations: Example 3 (Faithful, but not full) Let S2 = {1, 2, 3, 4}, P2 = {{1, 2}, {1, 2, 3}, {3, 4}}:

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✬ ✛✘ ✩ r2 ✗ r3 ✖

r1 ✚✙ ✪ ✫

r4

✔ ✕

Fig. 3: Property Structure hS2 , P2 i hS2 , P2 i is simply hS1 , P1 i from example 1 plus the additional property {1,2}. On the other hand, the similarity structure hS2 , ∼2 i = hS2 , ∼P2 i which is determined by hS2 , P2 i is indeed identical to hS1 , ∼1 i. But since, as we have seen, hS1 , P1 i = hS2 , P ∼2 i is determined by hS2 , ∼2 i and P1 $ P2 , it follows that hS2 , ∼2 i is faithful but not full with respect to hS2 , P2 i. Example 4 (Not faithful, but full) Let S3 = {1, 2, 3, 4, 5, 6}, P3 = {{1, 2, 4}, {2, 3, 5}, {4, 5, 6}}: ✬ 6r

✩

✫

✪

✬ r4 r1

✫

✬ ✩ 5r

2r

✫ ✪

✩

3r

✪

Fig. 4: Property Structure hS3 , P3 i

The graph that corresponds to the similarity structure hS3 , ∼3 i = hS3 , ∼P3 i as determined by hS3 , P3 i is in this case G∼3 = hS3 , {{1, 2}, {1, 4}, {2, 3}, {2, 4}, {2, 5}, {3, 5}, {4, 5}, {4, 6}, {5, 6}}i, i.e.:

15

r ✡❏ ✡ ❏ ❏ ✡ ❏ 5 4 r✡ ❏r ✡ ✡✡ ❏ ✡ ❏❏ ✡ ❏ ❏ ✡ ❏2✡ ❏ ✡ 1✡ ❏✡ ❏r3 r r

6

Fig. 5: Similarity Structure hS3 , ∼3 i Hence hS3 , P ∼3 i with P ∼3 = {{1, 2, 4}, {2, 3, 5}, {4, 5, 6}, {2, 4, 5}} is determined by hS3 , ∼3 i. Because of the “new” triangle {2, 4, 5} which is a maximal clique, P3 $ P ∼3 , and therefore hS3 , ∼3 i is not faithful but full with respect to hS3 , P3 i. Example 5 (Neither faithful nor full) Let S4 = {1, 2, 3}, P4 = {{1, 2}, {1, 3}, {2, 3}}:

✩✩ ✬✬ r

3

✬

✩

2r r1 ✫✫ ✪✪ ✫ ✪

Fig. 6: Property Structure hS4 , P4 i

The similarity structure hS4 , ∼4 i = hS4 , ∼P4 i that is determined by hS4 , P4 i is of course given by the graph G∼4 = hS4 , {{1, 2}, {1, 3}, {2, 3}}i, i.e.:

16

r ✡✡❏❏ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ 1✡ ❏r2 r

3

Fig. 7: Similarity Structure hS4 , ∼4 i hS4 , ∼4 i is neither faithful nor full with respect to hS4 , P4 i, because hS4 , P ∼4 i consists of a single property (P ∼4 = {{1, 2, 3}}) which is not a member of P4 . In example 3, property {1,2} is covered or accompanied by {1,2,3}; Goodman called this the “companionship difficulty”. Example 4 exemplifies Goodman’s “difficulty of imperfect community”: individuals 2,4,5 are mutually similar but for three different “reasons”. Example 5 is the simplest example of this latter kind. Let us consider some further instances of inadequacy. In some sense the most extreme case of inadequacy is the one where P is identical to the largest set of properties on a given set S of individuals, i.e., where P is equal to the powerset ℘(S) minus the empty set. Since in this case every two members of S share a property, the determined similarity structure is simply given by the complete graph on S which in turn determines a single quasiproperty: S. The similarity structure is faithful but lacks fullness to the greatest possible extent. Beside his well-known attack on quasianalysis, Goodman[16] used this simple fact as an additional argument against the respectability of the notion of similarity in general. This is certainly unjustified: the point only shows that in order to determine a non-trivial similarity relation the determining property structure has to be non-trivial, too (cf. Leitgeb[25], [26]). Speaking metaphysically: an account of similarity as the one developed above only makes sense if the properties in question are sparse or natural (cf. Quine[46], Lewis[30], Dunn[10], Hirsch[19], G¨ardenfors[13]) rather than abundant; not every set of individuals must count as an (extensional) property. So while perhaps ‘red’ expresses a property in this sense, ‘not red’ and ‘red or green’ presumably do not. This might even be another aspect in which the analysis of similarity is close to the analysis of identity: as Mates[32] argues, according to Leibniz’ own intentions, the principle of the identity of indiscernibles ought to be understood as referring to natural properties as well, 17

since Leibniz did not regard it as merely trivially true but as having proper content. In view of Carnap’s discussion of similarity circles and quality spheres in the Aufbau, the following example of inadequacy is particularly interesting: Example 6 (Carnap’s spherical properties) Let V = Rn be the n-dimensional Euclidean space with Euclidean metric d, such that n > 2. If ǫ > 0 is a real number, Sph(x, 2ǫ ) = {y ∈ V |d(x, y) 6 ǫ ǫ 2 } is the closed 2 -sphere around x ∈ V (we say ‘sphere’ independently of whether V is two-dimensional or of any higher dimension). Now let ǫ > 0 be fixed, S5 = V : the similarity structure hS5 , ∼5 i = hS5 , ∼P5 i that is determined by hS5 , P5 i, with P5 = {Sph(x, 2ǫ )|x ∈ S5 }, is not faithful, but full. 1. Consider three points x, y, z ∈ S5 which are the vertices of an equilateral triangle T , such that d(x, y) = d(x, z) = d(y, z) = ǫ: obviously, every two points contained in the triangle are members of an 2ǫ -sphere in P5 and thus are similar according to ∼5 ; in particular, it follows that x ∼5 y, y ∼5 z, x ∼5 z. However, there is no single member of P5 which includes x, y, z as members: the only eligible candidate is the circumcircle of the triangle, but the latter has radius √ǫ3 > 2ǫ . So we find that T is not a subset of any property in P5 . By the lemma of Zorn, T can be extended to a maximal subset T ′ of S5 , such that for all a, b ∈ T ′ : d(a, b) 6 ǫ.11 For that reason, T is a maximal clique of hS5 , ∼5 i, but because of what we have shown before, it is not a member of P5 . Hence, hS5 , ∼5 i is not faithful with respect to hS5 , P5 i. 2. Let Sph(x, 2ǫ ) be an arbitrary 2ǫ -sphere in P5 : every two members of Sph(x, 2ǫ ) have a distance 6 ǫ, hence every two members of Sph(x, 2ǫ ) are ∼5 -related. For every point y ∈ S5 that is not contained in Sph(x, 2ǫ ) there are two points a, b ∈ Sph(x, 2ǫ ), such that a and b are the intersection points of the straight line l that leads through y and x with the boundary of Sph(x, 2ǫ ), such that a is the point of Sph(x, 2ǫ ) which is closer to y, and b is the antipode of a with respect to Sph(x, 2ǫ ). But that entails that d(y, b) > ǫ, thus y is not ∼5 -similar to every member of Sph(x, 2ǫ ). So we see that Sph(x, 2ǫ ) is indeed a maximal clique of hS5 , ∼5 i. hS5 , ∼5 i is therefore full with respect to hS5 , P5 i. 1 of example 6 proves Carnap incorrect when he says on p.72 of the Aufbau: 18

The largest possible parts of the color solid, which contain nothing but colors that are similar to one another, are spheres which partially overlap each other, and whose diameter is the arbitrarily fixed maximal distance of similarity. T ′ in 1 of example 6 is a counterexample to this claim. Summing up, we have found that if we start with a given property structure, if we let the property structure determine a similarity structure, and if this similarity structure determines a property structure by means of quasianalysis, the initial property structure does not necessarily coincide with the final one; there are more or less illustrative counterexamples. But what if we begin such a determination chain with a similarity structure? The subsequent observation shows that the result is quite a different one: Observation 6 (From similarity to adequacy) Let hS, ∼i be a similarity structure on S, let hS, P ∼ i be determined by hS, ∼i, ∼ and let hS, ∼P i be determined by hS, P ∼ i: ∼

1. hS, ∼i is identical to hS, ∼P i. 2. hS, ∼i is adequate with respect to hS, P ∼ i. Proof. 1. For all x, y ∈ S, we have: x ∼P

∼

y

iff iff iff

∃X ∈ P ∼ , s.t. x, y ∈ X ∃ maximal clique X of hS, ∼i, s.t. x, y ∈ X x∼y

(def. 3) (def. 4)

The last equivalence holds because: (⇒) If x and y are members of a common clique of hS, ∼i, they are similar according to ∼. (⇐) If x ∼ y, then x and y are members of at least one common clique of hS, ∼i, i.e., {x, y}. But every clique can be extended to a maximal one. 2. Obvious. In a nutshell: if we restricted ourselves to property structures which are determined by similarity structures, quasianalysis would always yield adequate results. The philosophical importance of this fact is that if similarity were in some sense prior to properties, such that the determination chain that is presupposed in observation 6 reflected this priority order, quasianalysis would necessarily deliver adequate results.12 19

Note that observation 6 implies that no two distinct similarity structures can determine the same property structure. On the other hand, two distinct property structures might determine the same similarity structure (just reconsider example 3). The final example of this section shows that quasianalysis is indeed a generalization of the well-known method of abstracting equivalence classes from equivalence relations (cf. §73 of the Aufbau): Example 7 (Equivalence relations) Let S be a non-empty set and let P be a partition on S, i.e: (i) for all X, Y ∈ P : X ∩ Y = ∅; (ii) for all x ∈ S there is an X ∈ P , such that x ∈ X: Then hS, P i is a property structure on S, the similarity structure hS, ∼P i that is determined by hS, P i is clearly adequate with respect to hS, P i, and ∼P is an equivalence relation on S. Given that a method of abstraction for similarity structures is aimed to be a generalization of the method of generating equivalence classes from equivalence relations, we see that several possible alternatives of quasianalysis are ruled out: e.g., if such a method did not only determine maximal cliques as “quasiproperties” but rather all sets that can be generated from maximal cliques according to some fixed scheme of determination (like open sets can generated from unions of open neighbourhoods), this method would still have to determine just the maximal cliques in the case of a given equivalence relation. Furthermore, if we took e.g. all cliques – not only the maximal ones – as the output of our intended abstraction method, the application of this method to an equivalence relation would yield more than just its equivalence classes and would thus fail to be a generalization of the standard method of abstraction for equivalence relations.

6

Why Inadequacy Abounds

We have seen that many determined similarity structures are either not faithful, or not full, or neither with respect to some property structure by which they are determined. Furthermore, every similarity structure is adequate with respect to the property structure it determines. It might seem now that quasianalysis is simply not as good a method as Carnap and others have thought. Why not have a method of construction that works on every property structure? Indeed, the variety of subtle suggestions of how to improve quasianalysis of the first or of the second kind 20

(see e.g. Goodman[14], Lewis[27], Eberle[11], Kleinknecht[22], Moulines[42]) indicates that this is the usual response to exemplifications of inadequacy such as examples 3–5 in our last section. However, there are good reasons to believe that this response is itself inadequate. It is not so much quasianalysis that is deficient, but rather the hope that every property structure on a set S of individuals could be represented adequately by a binary relation of similarity on the same set of individuals at all. The point is that for a given set S the number of property structures on S is much larger – in fact, exponentially larger – than the number of similarity structures on S (we think primarily of finite sets S but the same holds for infinite S). Therefore, since every similarity structure determines a unique property structure, not every property structure on S can be determined by a similarity structure on S (whereas, as we have seen, the converse does hold). Consider one of those property structures hS, P i which are not determined by any similarity structure: hS, P i determines a similarity structure hS, ∼P i which thus does not determine hS, P i. In other words: hS, ∼P i, which is determined by hS, P i, is not adequate with respect to hS, P i. This general fact does not depend on the manner of determination – of property structures by similarity structures or vice versa - but only presupposes (i) our definitions of ‘property structure’ and ‘similarity structure‘ (on the same set of individuals), and (ii) that ‘determines’ expresses the graph of a function, in particular, that every similarity structure determines a unique property structure (as it is indeed the case if quasianalysis is the method in question). Put differently: while, as we have concluded from observation 6, determination by a similarity structure is a one-to-one mapping from similarity structures to property structures, determination of a similarity structure is not a one-to-one function from property structures to similarity structures and cannot be so plainly for cardinality reasons. Let us strengthen this point by some further quantitative considerations. We might e.g. ask: what is the proportion of property structures hS, P i that determine a similarity structure which is adequate with respect to hS, P i, amongst the totality of property structures (for a fixed finite set S of individuals)? The following observation gives the answer: Observation 7 (Proportion of Success) Let S = {1, . . . , n} be a finite set of individuals (so the cardinality card(S) of S is n): 1. Let cardadeq be the number of property structures hS, P i on S which n determine a similarity structure that is adequate with respect to hS, P i. 21

n It follows that cardadeq = 2( 2 ) . n

2. Let cardPn rop be the number of property structures on S. It follows that n−1 cardPn rop > 22 −1 . 3. The proportion

cardadeq n rop cardP n

converges to 0 for n → ∞.

Proof. 1. Let P adeq be the set of property structures hS, P i on S which determine a similarity structure that is adequate with respect to hS, P i. We already know that no similarity structure determines two distinct property structures, let alone two different members of P adeq , and no two distinct similarity structures determine one and the same property structure, let alone one and the same member of P adeq . We have also seen that every similarity structure determines a property structure by which it is itself determined and with respect to which it is adequate, i.e.: every similarity structure determines a member of P adeq . Finally, every member of P adeq is determined by some similarity structure, by definition of ‘P adeq ’. It follows that f , where f : {∼ | ∼ is a reflexive and symmetric relation on S} → P adeq with f (∼) = hS, P i iff hS, P i is determined by hS, ∼i, is a one-to-one and onto mapping. cardadeq , the n adeq cardinality of P , is therefore identical to the cardinality of the class of similarity structures on S. But the number of similarity structures on S of course equals the cardinality of the class of undirected graphs n having S as their set of vertices. This latter cardinality is just 2( 2 ) , as
n(n−1) n n! is the number of “potential” edges which might 2 = (n−2)!2! = 2 be contained in the set of edges of such a graph. 2. By standard combinatorial arguments, the number of sets P − of subsets n−1 of {1, . . . , n − 1}, such that ∅ ∈ / P − , is just 22 −1 ; every such set P − can be extended to the property set P − ∪ {S} on S (and if P1− 6= P2− n−1 then P1− ∪ {S} = 6 P2− ∪ {S}). Therefore, cardPn rop > 22 −1 (this inequality is even strict for n > 1). n

3. As we have seen, 2( 2 ) grows polynomially in the exponent whereas adeq n−1 22 −1 grows exponentially in the exponent, which entails that cardPnrop cardn is zero in the limit. If we identified limit proportions such as the one of 3 in observation 7 with probabilities – which is highly problematic – we might say: the probability 22

that quasianalysis is adequate on an unspecified but finite set of individuals is 0. “Almost” no property structure hS, P i determines a similarity structure which is adequate to hS, P i. So the main reason for quasianalysis’ inadequacies is not a flaw in the method but rather that the content of information that is coded by a property structure simply cannot be coded by a binary relation on the same set of individuals in each and every case. The same holds if we used instead a ternary, quaterny or any similarity relation of fixed arity, since analogous cardinality estimations can be proved. This is not without consequences: e.g., in order to overcome Goodman’s difficulties, Eberle[11] introduced a primitive ternary similarity relation that might be expressed in the form ‘x is similar to y in a respect in which z is neither similar to x nor to y’. Kleinknecht[22] has shown that there are also Goodman-type examples which prove Eberle’s version of quasianalysis as adopted for his similarity relation deficient. The considerations from above explain why this result is simply unavoidable as far as Eberle’s basic relation is concerned. The same applies to any comparative concept of similarity of the form ‘x is more similar to y than to z’ (as used by Lewis[28], p.48, Quine[47], p.18, Clark[8], p.91, and various others) and to any other kind of ternary or quaterny similarity relation (cf. Williamson[59]). Of course, this is not say that the constitution of properties from similarity is impossible per se. If e.g. similarity is assumed to be a relation which is both “contrastive” and has variable – finite or infinite – arity, or if the domains of similarity structures are somehow extended beyond the original domain of individuals and at the same time a numerical concept of similarity is used, a substitute of quasianalysis can be found which is always adequate. The first option was suggested by Lewis[30], the second by Rodriguez-Pereyra[49]. There are further ways of avoiding this problem, but they always presuppose a major change of setting. In this paper, however, we are going to stick to binary and qualitative similarity in the lines of Carnap’s Aufbau. As discussed after observation 6, a further possible response to the problem of inadequacy is not to require that all possible property structures be determined by quasianalysis but only those that can themselves be determined by similarity.

7

Adequacy Criteria

We have seen that quasianalysis – as any other possible method of reconstructing similarity structures from property structures on the same set of

23

individuals – is bound to fail in the vast majority of cases. But there are still infinitely many property structures that in fact do determine similarity structures which are adequate relative to them. What do they look like? Obviously, if one property is a proper superset of another one, the latter cannot be recovered anymore from the determined similarity structure. This is just Goodman’s companionship problem again; it gives us a trivial necessary condition for fullness and thus for adequacy: for all X, Y ∈ P , X 6⊆ Y . Hazen&Humberstone[18] (see pp.29–32) have joined this condition with a more interesting one and prove that the two together are both necessary and sufficient for adequacy (this is of course stated in a different terminology): for all Y ⊆ S, if for all x, y ∈ Y there is an X ∈ P such that x, y ∈ X, then there is an X ∈ P such that Y ⊆ X. As Hazen&Humberstone note, Schreider[55] has given a similar characterization of adequacy. While this latter condition is already nice and useful, it is still relatively close to the original definition of adequacy. But there is a theorem in the literature on hypergraph theory which states a simpler, and thus more surprising, necessary and sufficient condition for adequacy. Correspondingly, the proof of the theorem needs induction over the cardinality of S rather than just direct argumentation. According to Berge[2], section 1.7, 1 and 3 of theorem 8 below were first proved by Gilmore (but Berge’s reference to the original paper remains unclear). A proof of Gilmore’s theorem – actually a generalization thereof – can be found in Berge[2], pp.22–31, and in much more condensed form in Berge[1], pp.396f. We have added the straightforward (and less interesting) item 2 for fullness. Here is the theorem: Theorem 8 (Criteria; Gilmore) Let hS, ∼P i be determined by hS, P i, and let S be finite: 1. hS, ∼P i is faithful with respect to hS, P i iff for all A, B, C ∈ P there is an X ∈ P , such that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) ⊆ X 2. hS, ∼P i is full with respect to hS, P i iff there are no X ∈ P, x ∈ S with x 6∈ X, such that for every y ∈ X there is a Y ∈ P with x, y ∈ Y 3. hS, ∼P i is adequate with respect to hS, P i iff each of the following two conditions is satisfied:

24

(a) for all A, B, C ∈ P there is an X ∈ P , such that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) ⊆ X, (b) there are no X, Y ∈ P , such that X & Y . ✬✩ A✬✩ B ✬ ✩ ✬✩ ✫✪ ✫✪ ✫❅✪ ❅ ✫✪ X

C

Fig. 8: Faithfulness Criterion Note that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) in 3a is identical to (A ∪ B) ∩ (A ∪ C) ∩ (B ∪ C) by distributivity. 3 of theorem 8 proves Goodman correct in focusing on the difficulties of imperfect community and of companionship; 3a deals with the former, while 3b obviously deals with the latter. Fig. 8 is a graphical sketch of what the faithfulness criterion amounts to: X “glues” A, B, C together and thereby guarantees that the difficulty of imperfect community does not arise. If the right sides of the equivalences in 1 and 2 were joined by conjunction, they would of course yield a necessary and sufficient condition for adequacy as well. But the weaker necessary condition 3b for fullness suffices if joined with the condition in 1. This is the case essentially because faithfulness implies that every maximal clique of hS, ∼P i is not only a property in P but even a maximal property with respect to settheoretic inclusion. By 1 of theorem 8, every set P of properties (on S) can be extended to a set P ′ , such that the similarity structure as being given by P ′ is faithful with respect to hS, P ′ i: just close P under the operation m : hA, B, Ci 7→ (A ∩ B)∪(A∩C)∪(B ∩C). However, the members of P ′ are no longer guaranteed to be “natural properties” in the sense of being members of P , and the members of P are not necessarily represented within the similarity structure determined by hS, P ′ i even if they were represented within the similarity structure determined by hS, P i. In the literature on convex structures, the operation m is called a “median operation” and every set closed under such an m a “median algebra” (see e.g. Van de Vel[58]). The adequacy or indequacy of many of the determined similarity structures that we referred to in our examples above could easily be shown by applying theorem 8, but, as we have seen, such simple instances may also be 25

checked “directly” without much effort. The subsequent example demonstrates the merits of theorem 8 more clearly: Example 8 (Carnap’s spherical properties revisited) Let S6 be a finite subset of the two-dimensional Euclidean space R2 with S k Euclidean metric, ǫ > 0, and let P6 = ( k∈Z P6 ) \ {∅} where: for every even integer k, ǫ ǫ ǫ P6k = {S6 ∩Sph(x, ) | x = hx1 , x2 i with x1 = m , x2 = k , for even m ∈ Z} 2 2 2 for every odd integer k, ǫ ǫ ǫ P6k = {S6 ∩Sph(x, ) | x = hx1 , x2 i with x1 = m , x2 = k , for odd m ∈ Z} 2 2 2 (for Sph(x, 2ǫ ) as defined in example 6); see fig. 9 for an illustration of a fragment of this tesselation of spheres: ✬✩ ✬✩ ✬✩ ✬✩

✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✫✪ ✫✪ ✫✪ ✫✪

✫✪ ✫✪ ✫✪ ✫✪ ✫✪ ✫✪ ✫✪ ✫✪ ✫✪

Fig. 9: Property Structure hS6 , P6 i

For every A, B, C ∈ P6 it holds that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) is a (whether empty or non-empty) subset of either A or B or C: in each case, there is an X ∈ P6 , such that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) ⊆ X and by theorem 8 the similarity structure hS6 , ∼6 i = hS6 , ∼P6 i that is determined by hS6 , P6 i must be faithful with respect to hS6 , P6 i. Since furthermore no member of P6 is a proper superset of any other member of P6 , hS6 , ∼6 i is even adequate with respect to hS6 , P6 i, by theorem 8 again. It follows that if Carnap had only omitted some properties of his structure of spherical properties, the determined similarity structure would have been adequate with respect to it. But of course in that case the similarity relation would not have been one of metrical similarity in the strict sense: if x were similar to y, then x and y would be “close” to each other, but the other direction would not necessarily hold. 26

There are several variations of example 8 which yield adequate similarity structures, too: e.g. the spheres in P might be redistributed in an arbitrary manner as long as it is still guaranteed that for every A, B, C ∈ P the set (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) is a subset of A or B or C. Moreover, according to Johnson’s theorem (see [21]), if three planar circles of the same radius intersect in one common point and each two of the circles intersect in some additional point, the three latter points of pairwise intersection always lie on a circle with the same radius: this may be used in order to construct alternative sets of spheres which satisfy the desired condition. Moreover, the Euclidean spheres in P could be replaced by ǫ-spheres according to a different metric on R2 . The subsequent example does not really present an application of theorem 8 to a particular property- and similarity structure, but rather contemplates the possibility of a much more fundamental application of theorem 8 in metaphysics where S is regarded to be a finite set of, say, space-time individuals: Example 9 (An Application in Metaphysics?) Let us assume that we formulate an axiomatic system for natural properties or classes; we use • (Ax1) For all properties A, B, C there is a property X, such that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) ⊆ X • (Ax2) There are no properties X, Y , such that X & Y . • (Ax3) All properties are sets of individuals, ∅ is not a property, and for every individual x there is a property X, such that x ∈ X. as our basic ontological axioms. If we employed • (Def ) For all individuals x, y: x ∼ y iff there is a property X, such that x, y ∈ X. as a definition of similarity, we would be able to derive the adequacy of quasianalysis as a corollary along the lines of theorem 8 (given some additional formal background theory that is needed to derive theorem 8). Remark 9 Ought we adopt Ax1 and Ax2 (plus Ax3) as plausible axioms for natural properties? No. There does not seem to be any compelling argument for Ax1; indeed, why should it always be possible to combine similarities given 27

by two out of three “natural” respects into a joint “natural” similarity respect X? Assuming Ax2 would be even be disastrous: no natural property could be a subproperty of another13 ; no law-like sentence of the form ∀x(A(x) → B(x)) where A and B express natural properties would be a law.14 Another characterization of faithfulness can be given in terms of the dual property structure of a given property structure hS, P i (actually we just translate notions of hypergraph theory such as ‘dual hypergraph’ into our context and we apply some results which are well-known there; see Berge[2], p.2 and p.30). While this criterion is technically more complicated, it will be particularly useful in section 8 where we are going to apply it in order to warrant Russell’s logical construction of time instants. But first some definitions: Definition 10 (Dual property structure) A pair hS ∗ , P ∗ i is the dual property structure of hS, P i if 1. S ∗ = P , 2. P ∗ = {Φ ⊆ P |∃x ∈ S : Φ = {X ∈ P |x ∈ X}}. So the dual property structure of a property structure hS, P i takes the members of P as its new “individuals”, while its new “properties” are just maximal collections of “old properties” that share a common member of S (this type of construction also plays a role in Mormann’s account of quasianalysis; see e.g. [35]). Every such dual property structure is again a property structure according to definition 2. It is easy to see that if for every x, y ∈ S there is an X ∈ P , such that x ∈ X but y 6∈ X, the dual property structure hS ∗∗ , P ∗∗ i of the dual property structure hS ∗ , P ∗ i of hS, P i is isomorphic to hS, P i again (i.e., there is a bijective mapping f : S → S ∗∗ , such that for all X ⊆ S: X ∈ P iff f (X) ∈ P ∗∗ ), where f (X) = {y|∃x ∈ X, s.t. f (x) = y}). This last point is particularly interesting since it implies that property structures hS, P i, with S being a set of concrete spatially or temporally bounded particulars, have as their duals property structures hS ∗ , P ∗ i where S ∗ is now a set of nonconcrete qualitative elements, and – though only up to isomorphism – vice versa. In Goodman’s terminology: the duality operator turns “particularistic systems” into “realistic systems” and the other way round (cf. Goodman[14], p.142). The notion of k-Helly applies to a property structure if its properties show a particular pattern of intersection. We are going to apply the notion just to dual property structures of some given property structures, but the 28

definition holds for property structures in general (see Berge[2], p.22 for the hypergraph counterpart): Definition 11 (k-Helly) hS, P i is k-Helly (for k ∈ N) if for every Φ ⊆ P the following two conditions are equivalent: T 1. for all Ψ ⊆ Φ with card(Ψ) 6 k: Ψ 6= ∅, T 2. Φ 6= ∅. T Here we presuppose the convention that the intersection Φ of a set Φ is just the intersection of all members of Φ (accordingly for ‘Ψ’). 2 of definition 11 obviously entails 1, so the crucial direction is the other one. Note that the property of being k-Helly is preserved under isomorphism of property structures. The faithfulness of a similarity structure with respect to a determining property structure hS, P i can be proved to be equivalent to the 2-Hellyness of the dual property structure hS ∗ , P ∗ i of hS, P i. The proof is by Berge and is to be found in [2], p.30: Theorem 12 (Faithfulness and duality; Berge) Let hS, ∼P i be determined by hS, P i: hS, ∼P i is faithful with respect to hS, P i iff the dual property structure hS ∗ , P ∗ i of hS, P i is 2-Helly, i.e., for every Φ∗ ⊆ P ∗ the following two conditions are equivalent: • for all X, Y ∈ Φ∗ : X ∩ Y 6= ∅, T • Φ∗ 6= ∅. Our final criterion states a necessary condition for fullness. We already know that the representation capacity of similarity structures is restricted compared to that of property structures. How many properties at most can be represented fully by a similarity structure S with n members? The following observation which is a consequence of Sperner’s classic combinatorial theorem gives the answer: Theorem 13 (Representation capacity; Sperner) Let S be a set of n individuals:
n If P is a set of properties on S with card(P ) > ⌊n/2⌋ , then the similarity 15 structure determined by hS, P i is not full. 29

Proof. By 2 of theorem 8, every set P of properties of a property structure that determines a similarity structure which is full with respect to it satisfies the constraint that there are no X, Y ∈ P , such that X & Y . Put differently, every such set P is a so-called antichain, i.e., the members of P are pairwise set-theoretically incomparable. According to the classic theorem of Sperner[56] (see also Berge[2], p.6, where a different proof than Sperner’s is to be found), the maximal cardinality of an antichain P on a set S with
n n members is just ⌊n/2⌋ .
4 E.g., assume that S has four members: if P contains more than ⌊4/2⌋ = 4 6 properties (out of 2 − 1 = 15 possible), the similarity structure that is determined by hS, P i is not full.

8

A Note on Russell’s Construction of Temporal Instants in Our Knowledge of the External World

Carnap counts B. Russell’s Our Knowledge of the External World ([51]) among the exemplary predecessors of his Aufbau (see Aufbau, §3). In this section we are going to show that Russell’s[51] method of constituting temporal points from extended events on the basis of a relation of overlapping or compresence is an instance of adequate quasianalysis; in order to do so we will make use of the framework and the results of the previous sections. Kleinknecht[23], p.63, already noted that Russell’s method is formally analogous to quasianalysis. Russell’s construction of time instants is only a minor part of his constitution of temporal entities, properties and relationships – see e.g. Thomason[57] Kleinknecht[23], L¨ uck[31], for detailed reconstructions and improvements of Russell’s efforts on this topic (the original sources being [51], but also [52] and [53]). Now and in the sequel we refer to Russell’s first method of constructing points of time on p.124 of Our Knowledge of the External World, not to the second one on p.127 which is an instance of Whitehead’s method of extensive abstraction.16 Russell’s idea is to reconstruct time instants logically as maximal sets of mutually overlapping events. However, Russell does not prove that this logical construction is adequate. In filling this gap we make use of the following abbreviations: Let St be the set of time instants in the usual sense of physicists: so we may simply identify St with R, i.e., with the set of points of the real “time” axis. Let Pt be the set of formal counterparts to what Russell calls ‘events’: for convenience, we identify Pt with the set 30

of (i) bounded, (ii) open or left-open or right-open or closed intervals of positive length (“duration”) in R. This is presumably not precisely what Russell meant by ‘event’, but it is a plausible mathematical model of what he could have had in mind when he set up his constructional definition of time instants. We might exclude events the duration of which is “too short”, in fact, if we wanted events to be epistemically accessible without further theoretical means, we even should do so. Furthermore, Russell would surely regard the question of which events there are as an empirical question, not a mathematical one as it seems to be the case when we identify events with real intervals; but let us keep issues reasonably simple. Our aim is simply to outline one plausible possible model of Russell’s construction and to show that his method of constituting points of time is at least adequate in this model. Let us now consider the dual property structure hSt∗ , Pt∗ i of hSt , Pt i and ∗ the similarity structure hSt∗ , ∼t i = hSt∗ , ∼Pt i that is determined by hSt∗ , Pt∗ i. We find: • St∗ = Pt is the set of events again, ∼t is precisely Russell’s relation of overlapping or compresence of events, i.e., two intervals overlap if and only if they have non-empty intersection. ∼t is the basic relation that Russell presupposes in his theory. • hSt∗ , ∼t i is adequate with respect to hSt∗ , Pt∗ i: the dual property structure of hSt∗ , Pt∗ i is isomorphic to hSt , Pt i, but hSt , Pt i is 2-Helly by a famous theorem in Convex Geometry (namely, Helly’s theorem: see Matousek[33], p.10);17 we conclude that also hSt∗ , Pt∗ i is 2-Helly and by theorem 12 hSt∗ , ∼t i is faithful with respect to hSt∗ , Pt∗ i. Since no member of Pt∗ is a proper superset of another, theorem 8 implies that hSt∗ , ∼t i is adequate with respect to hSt∗ , Pt∗ i. • Pt∗ is Russell’s set of “constructed” temporal instants; by the last item, Pt∗ is the class of maximal cliques of hSt∗ , ∼t i, i.e., Russell uses quasianalysis in order to determine his “constructed” time point structure hSt∗ , Pt∗ i from the given event-overlap structure hSt∗ , ∼t i. • Since the dual property structure of hSt∗ , Pt∗ i is isomorphic to hSt , Pt i, hSt∗ , Pt∗ i is, up to isomorphism, the dual “copy” of hSt , Pt i. While the members of St∗ are not strictly identical to the members of St , they can play the same scientific roles given that all of the usual temporal predicates are translated to their dual counterparts.

31

Thus, Russell’s application of quasianalysis is vindicated. Although the intended interpretation of ∼t is not similarity but overlapping, Russell’s construction of time instants in Our Knowledge of the External World can thus be viewed as an adequate example of quasianalysis. Moreover, hSt∗ , ∼t i is an intersection graph (recall section 5): in this case, taking maximal cliques of hSt∗ , ∼t i corresponds one-to-one to taking maximal non-empty intersections of events. Therefore, this particular application of the quasianalytic procedure is not just an instance of quasianalysis I but also of a version of quasianalysis II. We could just as well have used theorem 8 in order to prove the above adequacy result. But in this particular context, the faithfulness criterion in terms of 2-Hellyness is more illuminating since it enables us to understand the contrast between Russell’s construction of points of time in Our Knowledge of the External World and his constitution of points of four-dimensional space-time in his Analysis of Matter (see [52], chapter XXVIII): curiously, he does not use the same method as above in order to construct points of space-time and he also presupposes a new quintary basic relation of compresence or “copunctuality” of events. Why this difference in method and framework? Russell simply notes that taking groups of events, such that any two members of a group are compresent and no event outside the group is compresent with every member of the group, does not work in four dimensions: “When we pass beyond one dimension, this method is no longer applicable.” ([52], p.295). Russell justifies this claim by stating some examples. But we can also explain these findings by means of the theory developed above: Let Sst now be the set of points of R4 , let Pst be the set of bounded convex space-time regions which are spatially spherical (“space-time events”; see [52], p.295); then it follows that a construction analogous to the one ∗ , P ∗ i of above does not go through because the dual property structure hSst st hSst , Pst i is not 2-Helly anymore but rather 5-Helly, as can be proved by another application of Helly’s theorem. That is why Russell has to presuppose a quintary relation of copunctuality between space-time events in The Analysis of Matter ; the method he uses may in fact be proved adequate by the theorem of Helly, but since it is not really quasianalysis that is at work here we refrain from pointing out the details.18

32

9

Discussion: Reconsidering Quasianalysis in the Aufbau

Now that we have a theory of quasianalysis at hand, it is time to reconsider the application of quasianalysis (of the first kind) in the phenomenalistic constitutional system of the Aufbau. In order to get a clear picture of what is going on there, we have to find a way of dealing with quasianalysis that explains how Carnap can on the one hand regard the so-called recollection of similarity relation as basic and primitive in the Aufbau, but on the other hand explicates the relation informally in terms of sensory fields, distances, quality spheres and the like. How can he say that quality spheres are to be constituted from similarity by quasianalysis while at the same time referring to quality spheres in order to explain the meaning of his basic predicate? Accordingly: Carnap claims that elementary experiences have no parts, such that there is no analyzing procedure that would detect components of them but only a quasianalyzing procedure that constructs quasicomponents from a given binary relation between them. But then he speaks of part-identical or part-similar experiences that have a common qualitative component, or which have qualitative components that are proximate to each other with respect to an associated quality space (see Kleinknecht[22], pp.33f, for similar worries). In fact, there is no real circularity, let alone contradiction here. The problem dissolves once one has introduced a distinction that Carnap could not have used at the time the Aufbau was written: the distinction between object- and metalanguage. In §52 and §75–81, Carnap himself comes rather close to this distinction when he separates the expressions of the “language of the constitutional system” and those of the “psychological language” by marking them with a ‘K’ and a ‘P’ sign, respectively. Although Carnap himself did not fix a particular object language in the Aufbau – let alone a metalanguage – by determining its vocabulary and its formation rules in a now standard manner, we regard it as helpful to do so in order to avoid apparent perplexities as the ones sketched above. Furthermore, it is surely in the spirit of the later Carnap when we take this “Tarskian” move towards his Aufbau. Transitions between object level and metalevel may indeed sometimes be subject to curious twists: it is possible that, say, a predicate A is defined in terms of predicates B, C, D on the metalevel, although B, C, D are defined on the basis of A on the object level. These different manners of defining terms might correspond to different aims of the two theories on the object and on the metalanguage level: e.g., the object theory might be

33

phenomenalistic and A is regarded “epistemically prior” to B, C, D, while the metatheory is realistic and B, C, D are regarded “ontically prior” to A. A procedure that is described to be “analyzing” by the metatheory might have an inverse procedure that is described to be “quasianalyzing” by the object theory; and so forth. Our suggestion is that something like this is going on in the corresponding sections of the Aufbau and that any confusion about these object-/metalevel distinctions ought to be avoided or the proper assessment of quasianalysis will be hampered. So let us reconsider quasianalysis now on both language levels. Although the core of the Aufbau metatheory is open to various kinds of constitutional object languages – phenomenalistic, physicalistic, and so forth (cf. chapters III.B and III.C of the Aufbau) – we are going to concentrate just on the phenomenalistic object language that is also used by Carnap as his main example. Our reconstruction will of course be close to the relevant passages of the Aufbau, but not too close – we take the freedom to put them into a contemporary context: e.g., we do not use simple type theory but rather modern set theory (some version of ZFC extended by urelements). Finally, we try to avoid getting too much into details that would distract us from the topic of quasianalysis itself. The counterparts to our subsection 9.1 in the Aufbau are essentially the formal parts of §106–§111 of chapter IV; the parts which correspond to our subsection 9.2 are §67–§80 of chapter III and the informal parts of §106–§111 of chapter IV. We use more or less the original Aufbau symbols at the object level and partly also on the metalevel. Our metatheory can be regarded as a kind of model theory of the constitutional system on the object level. The final aim of the following two subsections is to specify – and thus to distinguish – clearly (i) which sets of elementary experiences are quasianalytically defined as similarity circles, i.e., as the members of the extension of the predicate aehnl, and (ii) which sets of elementary experiences are the intended members of the extension of the predicate aehnl. We are going to denote the former set by ‘I(aehnl)’, the latter set by ‘P’; the set of basic elements, i.e., the set of elementary experiences for a given agent, will be denoted by ‘erl’, the relation of qualitative similarity of elementary experiences by ‘∼’. Since we will finally concentrate just on the visual experiences of the agent in question, the similarity structure herl, ∼i will actually be the phenomenalistic approximation of its visual space counterpart hQualvis , ∼vis i, where ‘Qualvis ’ denotes the set of visual quality points and where ‘∼vis ’ denotes the metrical similarity relation on the agent’s visual space; hQualvis , ∼vis i = hQualvis , ∼Pvis i is determined by the property structure hQualvis , Pvis i where ‘Pvis ’ denotes the set of visual quality spheres with 34

a fixed small diameter. As we are going to show, I(aehnl) is identical to Pvis according to the intended model of Carnap’s constitutional system if and only if the similarity structure herl, ∼i is adequate with respect to herl, Pvis i, which in turn depends on the structural similarity between herl, ∼i and hQualvis , ∼vis i. Each of the terms that we have just referred to will be dealt with thoroughly in sections 9.1 and 9.2. Those readers for whom these terms are already sufficiently clear might consider turning directly to subsection 9.3.

9.1

Object Language and Object Theory

Object language OL: first-order language with two primitive binary predicates: Er (“. . . is recollected to be similar to . . . ”) and ∈ (“. . . is a member of . . . ”).19 There are many further symbols which are introduced by definition on basis of Er and ∈ (see below).We do not consider these definitions as metalinguistic abbreviations, therefore the defined signs are members of the vocabulary of OL as well. Each of them may be taken to be an individual constant denoting a particular set. Object theory OT : first-order theory, including: set theory with urelements, where the field of Er functions as the set of urelements (actually we do not need set theory in its full strength, but rather just a weak fragment of it); moreover, various definitions involving both Er and ∈, starting with definitions of erl (“the set of elementary experiences”), Ae (“the partsimilarity relation between elementary experiences”), and aehnl (“the set of similarity circles”). We focus solely on these three definitions, the last of which explicates quasianalysis: erl = {x|∃y(x Er y ∨ y Er x)}

(1)

Ae = {hx, yi|x Er y ∨ y Er x ∨ x = y}

(2)

 (i) ∀x, y ∈ X : x Ae y    (ii) ∀z : aehnl = X if ∀x ∈ X x Ae z,    then z ∈ X

9.2

   

(3)

  

Metalanguage and Metatheory

Metalanguage ML: first-order language with a primitive binary predicate ∈ (“. . . is a member of . . . ”) and a number of further descriptive predicates 35

for “. . . is a quality space”, “. . . is the visual quality space of . . . ”, “. . . is a quality point of . . . ”, “the distance of . . . from . . . in . . . is . . . ”, “. . . is a quality sphere in . . . ”, “. . . is the set of elementary experiences of . . . in time interval . . . ”, “. . . is before . . . ”, “. . . realizes . . . ”, and so forth. Not all of these syntactic items need to be primitive. Indeed, there are many further metalinguistic symbols which are introduced by definition, including “. . . is recollected to be similar to . . . ” and “. . . is similar to . . . ”. Additionally, the metalanguage has various syntactic resources by which the items of the object language can be referred to, i.e., names for object language expressions, predicates expressing the syntactic categories of OL as well as the way its members are syntactically composed of others, etc. Finally, the metalanguage includes semantic predicates for the object language, in particular the (defined) relative-truth predicate “. . . is true in . . . ” and related expressions. These semantic terms are needed in order to determine the intended model of the constitutional object theory precisely; Carnap himself describes this intended model only informally when he justifies the choice of his basic elements and his basic relation and when he explains how the method of quasianalysis is to proceed. Note that we are going to employ sans serif notation if we want to make clear that we are using metalinguistic terminology. Metatheory MT : first-order theory, including: some version of set theory that extends the set theory on the object language level; additionally, a number of eigenaxioms and definitions, where the axioms may be of general conceptual or empirical nature or may just be singular descriptions of a particular cognitive agent. The Aufbau metatheory includes the somewhat idealized description of a concrete cognitive agent, say, Ag, and its corresponding (finite) set erl of elementary experiences in a fixed interval of time; furthermore the description of a finite family C = (Qi )i∈I of quality spaces Qi = hQuali , Stri , Pi i (i.e., a conceptual space C in the sense of G¨ardenfors[13]) given by (i) a set Quali of quality points, (ii) a formal structure Stri (order, topology, metric, norm,. . .) on Quali , (iii) a set Nati ⊆ ℘(Quali ) of “natural regions” of the space (which might satisfy certain formal constraints, like being bounded, closed, convex, extended, spherical, and the like). Axiomatically, each quality space Qi is associated with a particular manner of experiencing of the agent Ag (visual, auditory, tactile,. . .). In particular, one of the quality spaces is a 5-dimensional visual space, where three dimensions are reserved for colour parameters (hue, brightness, saturation) and two dimensions for place parameters (x, y-coordinates in Ag’s twodimensional visual field). Let Qvis = hQualvis , Strvis , Pvis i be this visual space: Qualvis is a specific subset of R5 , Strvis is the metric space structure on Qualvis 36

as being given by the Euclidean metric d, and Pvis = {Sph(q, ǫ)|q ∈ Qualvis } for fixed ǫ>0 where Sph(q, ǫ) = {p ∈ Qualvis |d(p, q) 6 ǫ}. The axioms on the metalevel tell us that each of Ag’s elementary experiences in erl realizes certain quality points in the quality spaces of C. We concentrate on the realization of visual quality points: Let Real ⊆ Erl × Qualvis be the realization relation between experiences in Erl and quality points in Qualvis . E.g., if x is a member of Erl, and if p ∈ Qualvis corresponds to a light-red colour point in the left-upper corner in the visual field of Ag, then Real(x, p) if and only if Ag experiences in x a light-red colour point in the left-upper corner of his visual field. Moreover, the set erl of Ag’s elementary experiences is assumed to be ordered by a strict total order < according to temporal predecessorship. The quality points in p ∈ Qualvis are endowed axiomatically with a proximity notion ∼vis of similarity as explained in example 6 of section 5: two quality points are similar according to ∼vis iff they are close to each other (with respect to a given threshold value). As we have seen, we can consider this similarity structure hQualvis , ∼vis i as being determined by the property structure hQualvis , Pvis i in which quality spheres figure as the corresponding “spherical” properties. However, the similarity structure that we are actually interested in is a phenomenalistic one that is defined on elementary experiences: two elementary experiences are (part-)similar iff at least one of the quality points that are realized by one experience is close to at least one of the quality points that are realized by the other experience. Let us simplify matters and concentrate again just on the visual case: we may thus consider the similarity structure herl, ∼i = herl, ∼P i as being determined by the property structure herl, Pi the properties of which are our spherical properties of section 4, i.e., for every property X in P there is a sphere Sph in Pvis , such that X is the set of elementary experiences which stand in relation Real to at least one point in Sph. herl, ∼i and herl, Pi are the phenomenalistic counterparts to hQualvis , ∼vis i and hQualvis , Pvis i; the former pairs conform to the latter pairs via the realization relation Real and existential quantification. Now we are finally in the position to describe the intended model of our object theory OT . Of course, this model has to be a first-order structure for the language OL: Let the domain D of this structure be the cumulative hierarchy based on the set erl of urelements up to some ordinal; we suppose that this is sufficient in order to satisfy the weak set theory at the object level.20 Let the interpretation I(∈) of the object language membership predicate ∈ be defined as expected. Finally, let the interpretation I(Er) 37

of the object language predicate Er for recollection of similarity be defined by: I(Er) = {hx, yi ∈ erl × erl|x ∼ y, x < y}. Since this is to be understood in the way that the agent Ag remembers an elementary experience as being similar to another one if and only if the two are qualitatively similar and the former occurs earlier than the latter, we are obviously assuming Ag to have perfect memory capabilities; let us accept this idealization just for the sake of simplicity (Carnap calls such idealizing assumptions ‘fictions’; cf. §101, 102). So the universe and the interpretation of the basic predicates of OL have been chosen as intended by Carnap. In order to satisfy the definitions that are included in OT , we simply need to determine the interpretation of the defined terms accordingly. This yields for the first three defined terms: I(erl) = {x|∃y(x I(Er) y ∨ y I(Er) x)}

(4)

I(Ae) = {hx, yi|x I(Er) y ∨ y I(Er) x ∨ x = y}

(5)

   

(6)

 (i) ∀x, y ∈ X : x I(Ae) y    (ii) ∀z : I(aehnl) = X if ∀x ∈ X x I(Ae) z,    then z ∈ X

  

The first-order structure hD, Ii is now the intended model of our object theory OT . It is intended in so far as its domain is chosen just as described in the Aufbau, the interpretations of the primitive terms for membership and recollection of similarity are selected accordingly, and the structure as a whole satisfies OT , in particular, the definitions of OT . However, it is not so clear that the interpretations of the defined terms are thereby assigned intended interpretations as well. After all, these terms are interpreted in a – in fact uniquely determined – manner that ensures that the definitions of OT are satisfied, but that does not entail that their extensions in hD, Ii conform to what their definitions are intended to be definitions of. E.g.: I(erl) was set to {x|∃y(x I(Er) y ∨ y I(Er) x)} in order to satisfy erl = {x|∃y(x Er y ∨ y Er x)}. But the intended interpretation of the object language term erl is, according to what Carnap says in the Aufbau, erl. Thus, I(erl) is intended if and only if {x|∃y(x I(Er) y ∨ y I(Er) x)} = erl. Is that the case? Not necessarily. Assume that there is an elementary experience of the cognitive agent Ag, i.e., a member of erl, such that this elementary experience is dissimilar to all other members of erl (with respect to ∼). This is certainly not impossible: Ag might at one time have experienced one and the same colour at all places of his visual field, but never again have 38

experienced that colour or a similar colour at any place of his visual field. In such a case the elementary experience in question is not included in the field of I(Er) and therefore it is also not a member of I(erl). Thus, under certain conditions, I(erl) is not as intended. Whether these conditions do arise, is an empirical matter. Since we are only interested in quasianalysis, let us nevertheless assume now that the interpretations of erl and especially of Ae are indeed equal to their intended interpretations, i.e., I(erl) = erl and I(Ae) = ∼: what can we say then about the interpretation of aehnl? Carnap tells us that the intended interpretation of aehnl is just the set P of spherical properties from above (cf. §80, 72); indeed, many OT -definitions on higher levels of the constitutional system presuppose that the interpretation of aehnl is this intended one. Since hI(erl), I(aehnl)i = herl, I(aehnl)i is simply the (quasi)property structure that is determined by herl, ∼i, and since herl, ∼i is determined by herl, Pi as explained above, we conclude that I(aehnl) = P if and only if herl, ∼i is adequate with respect to herl, Pi. So whether I(aehnl) is intended or not is finally a matter of whether a particular similarity structure is adequate with respect to a particular determining property structure.

9.3

Quasianalysis and the Intended Model

Is herl, ∼i adequate with respect to herl, Pi? This is again not to be settled a priori – it is an empirical question the answer to which depends on what experiences the cognitive agent in question has. However, if we turn to our theory of sections 2–7 as being formulated on the metalevel, we can at least derive some metalinguistic qualifications for adequacy. Let us focus on the visual quality space again. There are two possible cases: either herl, ∼i is some coarse-grained, finite, phenomenalistic approximation of its conceptual space counterpart hQualvis , ∼vis i or it is not. In the latter case, it is more or less unpredicable what ∼ looks like and therefore one cannot seriously judge its representational adequacy. In the former case – which is the one that is expected by Carnap – we may suppose that herl, ∼i inherits the merits or defects of hQualvis , ∼vis i, whatever they may be. In fact, we know what they are: as we have shown in example 6, hQualvis , ∼vis i is not faithful though full with respect to hQualvis , Pvis i. Thus we may expect herl, ∼i to be in this case not faithful but full with respect to herl, Pi. We may also argue more directly: theorem 8 characterizes faithfulness in the way that for all triples of properties in P there would have to be a further property which covered the union of each pairwise intersection of the three, therefore whenever two of three properties were realized together, 39

this further property would have to be realized accordingly. This implies that (i) the faithfulness of herl, ∼i would presuppose an overly high degree of variation of quality points realized within each member of erl, while simultaneously (ii) peculiar law-like relationships between the members of P would have to hold. We illustrate this by an example: let A be a property in P and let SphA be a sphere in Pvis , such that A is the set of elementary experiences which stand in relation Real to at least one point in SphA ; accordingly for B and SphB , C and SphC . E.g., SphA might be a red colour sphere in the left-upper quarter of the visual field, SphB an orange colour sphere in the right-upper quarter, and SphC a violet colour sphere in the left-lower quarter. If there is no X ∈ P, such that (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) ⊆ X, then (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) can be extended to a maximal clique of herl, simi that is not identical to any member of P and therefore herl, simi is not faithful. If there is such an X, there must be some colour sphere, say, blue, at some part of the agent’s visual field, say, its right-lower part, such that whenever the agent experiences some quality points in SphA and SphB , or in SphA and SphC , or in SphB and SphC , he also experiences a quality point that is blue and that is located in the right-lower section of his visual field. Every elementary experience which realizes this particular pattern of quality spheres SphA , SphB , SphC also has to realize at least one point of SphX ; these experiences must therefore realize at least three different quality spheres, and the same holds of course for all other triples of properties in P if faithfulness is to prevail. This seems to be a severe constraint that is not likely to be satisfied. Why should Ag experience a blue quality point in the right-lower section of his visual field whenever he experiences either something red in the left-upper quarter and something orange in the right-upper quarter, or something red in the left-upper quarter and something violet in the left-lower quarter, or something orange in the right-upper quarter and something violet in the left-lower quarter? Why should corresponding statements be true of all further triples of colour spheres? In particular: why should cases be excluded in which Ag experiences something red in the left-upper quarter and something orange in the right-upper quarter but has no experience whatsoever of something blue in the right-lower quarter of his visual field (accordingly for all other colour-and-place combinations)? Neither the structure of our human cognitive system nor the structure of the physical world seems to satisfy the strange law-like relationships between the members of P which would be entailed if quasianalysis in the Aufbau were faithful. Fullness, on the other hand, would only presuppose (i) a high degree of discriminability of realized quality spheres in terms of members of erl 40

(2 of theorem 8) together with (ii) the absence of phenomenal “laws” of a particular form. E.g., for every pair of properties in P there would have to be an elementary experience in erl which realizes the one but not the other, i.e.: at least at one point of time, Ag should experience something red in the leftupper quarter of the visual field but not anything orange in the right-upper quarter, while at another time he should experience something orange in the right-upper quarter of the visual field though not anything red in the leftupper quarter. Since neither the structure of our human cognitive system nor the structure of the physical worlds seems to satisfy law-like entailment relations between the members of P, fullness is relatively easy to satisfy and it is certainly much more likely to be satisfied than faithfulness. While the lack of simple phenomenal “laws” concerning the occurrence of colour qualities speaks, on the one hand, in favour of non-faithfulness, it speaks on the other hand in favour of fullness. In both cases our conclusions seem to be robust if we consider “approximate” faithfulness and “approximate” fullness rather than faithfulness and fullness simpliciter, i.e., if we focus on the overlapping patterns of “most” triples of properties and “most” pairs of properties, respectively. Note that observation 7 certainly does not raise the chances of approximate adequacy, although it has to be kept in mind that this result is particularly problematic with respect to drawing any real-world conclusions from it because it concerns a proportion “in the limit”. Moulines[42], pp.281–284, argues that if certain “extrasystematic assumptions” are satisfied, at least approximate adequacy may be expected to hold. In our reconstruction such assumptions would correspond to the antecedents of metalinguistic implications of the form ‘if . . . , then herl, ∼i is (approximately) adequate with respect to herl, Pi’; these implications would have to be derived or at least shown to be likely on the basis of the axioms of our metatheory MT (of course including some empirical hypotheses). One assumption which Moulines considers to be of special importance is that the cardinality of erl is high (say, ten million), whereas the cardinality of P is considerably smaller (say, several thousand; cf. Moulines[42], p.283). While theorem 13 of section 7 adds some support to the necessity of this assumption as far as the satisfaction of fullness is concerned, it is certainly not sufficient for faithfulness, let alone adequacy: suppose that just a few thousand quality points in Qualvis are realized by a huge number of elementary experiences; so our agent’s experiences would include a vast number of possible combinations of these few quality points. Why would that exclude – at least to a high extent – the occurrence of examples like the one stated above for the quality spheres SphA , SphB , SphC ? By assumption, it would certainly be highly probable that each pair out of SphA , SphB , SphC 41

is realized together with many of the other few thousand quality spheres, but it would not likely be the case that there is a particular quality sphere SphX , such that every experience of some quality points in SphA and SphB , or SphA and SphC , or SphB and SphC , is at the same time an experience of a quality point in SphX . Whether herl, ∼i is approximately adequate with respect to herl, Pi, and whether the extension of aehnl in Carnap’s intended model of his phenomenalistic constitutional system is therefore approximately the intended one or not, is in the end an empirical matter. Our theoretical results indicate that there are good chances for herl, ∼i being full with respect to herl, Pi; however, herl, ∼i is likely to be not even approximately faithful. As we have already pointed out earlier, Mormann (just as Proust) defends quasianalysis as a general structural method of constitution rather than as a particular means of constructing the phenomenalistic counterparts to quality spheres. Indeed, as Friedman[12], Richardson[48], Moulines[42], and others have shown, there is ample historical evidence that Carnap’s Aufbau is not primarily the upshot of the traditional programme of phenomenalism or empiricism but has much more general aims. Furthermore, or so Mormann argues, Goodman’s so-called “failures of quasianalysis” might be interpreted as cases of empirical underdetermination: similarity is epistemically prior to properties but at the same time our theoretical knowledge of properties is underdetermined by our empirical data, so it is no wonder that a given similarity structure does not uniquely determine a corresponding property structure. Our criticism of quasianalysis in this section was solely devoted to Carnap’s application of the method in section §111 of his constitutional system. As far as this particular instance of quasianalysis is concerned, it does not lead to the results that were intended by Carnap.

10

Conclusions

We are left with a mixed assessment of the method of quasianalysis. First of all, it is definitely wrong to say that quasianalysis does not work as such: indeed it yields adequate results under certain conditions while it does not do so if these particular conditions are not satisfied. The necessary and sufficient conditions for adequacy have been described in theorem 8 and other necessary or sufficient conditions have been stated in section 7. The corresponding notion of adequacy can be defined and analyzed thoroughly (cf. section 5). Secondly, it is wrong to say that the inadequacy of quasianalysis under 42

certain conditions is due to a flaw that affects the details of the method and that could and should be “repaired”. As we have seen in section 6, quasianalysis is bound to fail in the majority of cases for simple cardinality reasons and that the same is true of any other method of constructing properties from a similarity relation on the same set of individuals as long as the similarity relation has a fixed finite arity. Given this unavoidable constraint, quasianalysis actually does rather fine. It is a natural extension of the abstraction of equivalence classes (example 7), it is adequate if similarity is in some sense prior to properties (observation 6) and it is also adequate in the case of abstracting points of time from the overlapping of events (section 8). Where quasianalysis is inadequate (example 6) there is sometimes a reasonably close setting on which it is adequate again (example 8). But quasianalysis fails – at least partially – in the field that it was to be applied to within Carnap’s phenomenalistic constitutional system of the Aufbau (our section 9). Quasianalysis is definitely to remain in the philosopher’s – and perhaps also the scientist’s – toolbox; if so, future applications in semantics, epistemology, or metaphysics may be expected not to be a long time in coming. As every other tool it is not good or bad by itself, but its qualities depend on what it is used for. Acknowledgements: This paper was supported by the Erwin-Schr¨odinger Fellowship J2344-G03, Austrian Research Fund FWF. Drafts were discussed in seminars in Salzburg, Munich, Leuven, Berkeley, and Stanford; we want to thank the participants for several helpful suggestions. We are particularly grateful to Reinhard Kleinknecht, Leon Horsten, Michael Friedman, Ed Zalta, Thomas Mormann, Uwe L¨ uck, Simon Huttegger, Ronald Ortner, Hans Rott, and an anonymous referee.

Notes 1

It is an interesting historical question how this development of methods of abstraction actually took place (see e.g. Moulines[43] for some historical references). In particular, the additional impact of Whitehead’s method of extensive abstraction must not be neglected (see Gr¨ unbaum[17] for the collected references to Whitehead’s work). Russell was obviously influenced by Whitehead and the same is likely to hold – whether directly or indirectly – for Carnap. Whitehead intended extensive abstraction to apply to nonsymmetrical part-of relations. However, it is easy to see that if the method of extensive abstraction is applied to a reflexive and symmetric relation, its results correspond to the outputs of Carnap’s so-called quasianalysis of the first kind.

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2 Hazen&Humberstone[18] are exceptional in stating some results in that direction; Mormann, who proves various important results on quasianalysis, concentrates on a version of quasianalysis that is different from the one in the Aufbau and does not subscribe to Goodman’s way of presenting quasianalysis; compare our discussions at the beginnings of section 4 and section 3, respectively. 3 The set S of elementary experiences, i.e., the set of basic elements according to the phenomenalistic constitution system in the Aufbau, is considered to be finite by Carnap; cf. §180 of the Aufbau. 4 Formally, a property structure hS, P i is a so-called hypergraph (see Berge[2] for the standard reference on hypergraphs). Hypergraph theory can be employed successfully as a means of investigating Carnap’s method of quasianalysis. Unfortunately, quasianalysis seems to have passed unnoticed by hypergraph theorists and the same holds vice versa. As far as the usefulness of graph-theoretical results for the study of quasianalysis is concerned, we regard this paper as an example of the surprising benefit that one may get from applying mathematical methods and results in a strictly philosophical context. 5 In what follows, expressions such as ‘hS, ∼i’ and ‘hS, ∼P i’ denote similarity structures on the set of individuals denoted by ‘S’, and expressions such as ‘hS, P i’ and ‘hS, P ∼ i’ denote property structures on the same set of individuals. 6 What we call a ‘maximal clique’ is simply called a ‘clique’ in part of the literature on graph theory. We follow the terminology that is used e.g. by Berge[2]. 7 Actually we should distinguish ‘determined’ according to definition 3 and ‘determined’ according to definition 4 syntactically, e.g. by defining ‘determinedP rop ’ and ‘determinedSim ’ instead. However, it will always be clear from the context whether a similarity structure is to be determined by a property structure or the other way round. 8 Note that there is no efficient construction procedure for maximal cliques: in fact, the problem of finding out whether there is a clique of cardinality k in a given graph is well-known to be NP-complete. 9 What we call ‘faithful’ is termed ‘conformal’ in hypergraph theory, in particular in Berge[1], [2]. Conformal hypergraphs have recently been studied in other fields, too, including so-called guarded logics (see e.g. Hodkinson&Otto[20]). 10 Our definitions of ‘faithful’, ‘full’, and ‘adequate’ are conditional definitions. Since every property structure determines a unique similarity structure that in turn determines a unique (quasi-)property structure, we could have used unconditional definitions instead which would define ‘faithful’, ‘full’, and ‘adequate’ as unary predicates that apply simply to property structures. However, the motivation for these definitions seems to be clearer if we say that a determined similarity structure is faithful or full or adequate with respect to the determining property structure rather than saying that the determining property structure is itself faithful or full or adequate. 11 Every clique X in a graph can be extended to a maximal clique X ′ such that X ⊆ X ′ : this is trivial in the case where S is finite; in the infinite case it follows from a straightforward application of Zorn’s lemma. We omit any details; see e.g. Schoch[54], pp.124f, or Hazen&Humberstone[18], p.29, for the proof). 12 We might even give this a Kantian twist: if properties are derived by the structure of our understanding and if the mechanism of our understanding is quasianalysis, then we can come to know a priori that the world can never have a property structure that determines a similarity structure that is inadequate with respect to the former. We owe this way of looking at observation 6 to Leon Horsten. 13 Quine[46], p.118, notes this problematic consequence. 14 We thank Gerhard Schurz for bringing our attention to this consequence.

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15

For real x, ⌊x⌋ is the largest integer less than or equal to x. We thank Thomas Ryckman for pointing this out to us. 17 The theorem applies since events in our sense are convex subsets of R. A subset X of n R is called convex if and only if for every x, y ∈ X, the straight line segment between x and y is included in X, i.e., for all λ ∈ [0, 1] : λx + (1 − λ)y ∈ X. In our case, n = 1, and obviously every real interval is convex. 18 Russell does not prove his method of constructing points of space-time adequate. He only derives from the well-ordering theorem that every copunctual group of events can be extended to a maximal such group. Nowadays it is surely Zorn’s lemma that would be put to use here. 19 We are going to employ ‘∈’ both when we use and when we mention the membership symbol. For all other symbols we will distinguish occurrences in the object language from those in the metalanguage syntactically. 20 For our current purposes we might simply consider the cumulative hierarchy up to the first infinite ordinal ω. Whence all urelements would be members of D, but also all sets of finite rank that can be built up from urelements cumulatively would be members. 16

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