The Atlas of Reality
Page 35
Similar problems face both Trope Connectionists and Reductive Separatists. Trope Connectionists have to suppose that whenever a relational truth of the form Rab is true, then there are two relational tropes involved, one for R and one for R's converse. Similarly, Natural Class Separatists have to suppose that there are two distinct natural classes involved in grounding the truth that Rab: the natural class corresponding to R (to which the ordered pair
The only account that is immediately immune to these problems is Ostrich Separatism. Ostrich Separatists believe that all of these relational truths are metaphysically fundamental. Hence, they don't have to offer any explanation of what grounds their truth, and thus they don't have to introduce ordered pairs at all. Ostrich Separatists can suppose that the truth that Cyrano loves Roxanne and the truth that Roxanne is loved by Cyrano are the very same truth, merely expressed in different ways.
Resemblance Separatists and certain Connectionists could avoid this problem by denying that there are any natural non-symmetric relations. For Resemblance Separatists, this would mean that non-symmetric relational truths are never grounded directly in truths about the resemblances among ordered pairs, and, for Connectionists, it would be that there are no non-symmetrical relational universals or tropes. All non-symmetric relations would have to be unnatural and definable in terms of the more fundamental symmetric relations. For example, we could use the symmetric relations of resemblance and co-bundling to define the non-symmetric relation definable as: x resembles something that is co-bundled with y.
However, there are many non-symmetric relations that seem to be fully natural and not definable in terms of more fundamental symmetric relations. Consider, for example, the relations of temporal priority (x is earlier than y) or causal priority (x causes y). These seem to be natural and real non-symmetric relations, not artificial relations definable in terms of more fundamental symmetric relations. Similarly, the three-way relation of betweenness (x is between y and z) is non-symmetric and, in many popular developments of the foundations of geometry, more fundamental than such symmetric relations as distance.
Connectionists and Reductive Separatists have three options in trying to even the score with Ostrich Separatists by avoiding the use of ordered pairs. First, they can postulate the existence of ordered pluralities that are ordered by various relations (the taking things in order solution). Second, they can introduce a family of distinct instantiation relations (or modification relations), making the instantiation relation between a non-symmetric relation and its relata incorporate the non-symmetry within itself. Third, they can use a process of composition to build up complex states of affairs representing non-symmetric relational facts.
10.2.2 Taking things in order
The first solution is to take non-symmetric relations to be properties, not of ordered pairs, but of pairs of things taken in a certain order. Thus, it is Oklahoma and Texas that instantiate the north of relation, when taken in a certain order.3
This seems to be an attractive way to develop Relational Connectionism, Modifying Trope Connectionism, and Reductive Separatism.
In cases of non-symmetric relations like north of or loves, it would involve an undesirable duplication if Classical Connectionists had to posit both the universal NORTH-OF and the universal SOUTH-OF, as well as if we had to posit both the universal LOVES and the universal IS-LOVED-BY. It would be much better to suppose that there is a single universal in both cases. If so, we cannot say that it is Oklahoma and Texas taken in that numerical order (i.e., Oklahoma first and Texas second) that instantiate the relational universal NORTH-OF, since this would mean that the two states taken in that order would also instantiate the identical relation SOUTH-OF. Instead, we have to build an appropriate order into the pair: it is Oklahoma and Texas, taken in the order of Oklahoma to the north of Texas, that instantiate the NORTH-OF and the SOUTH-OF relation. However, this would seem to be an unsuccessful reduction of the primitive kind being to the north of to the kind instantiating, since we need the being to the north of relation itself to specify the ordered pluralities that are supposed to instantiate the supposed relational universal.
There are other cases, however, in which the reduction might be made to work. Consider the relation loving (and being-loved-by). We could say that Cyrano and Roxanne, taken in the order of Cyrano the agent to Roxanne the patient, stand together in the loving relation. We have to treat the agent/patient relation as a primitive, irreducible kind, in order to specify the ordered pluralities that then instantiate the various action relations: loving, moving, creating, destroying, healing, harming, and so on. The Classical Connectionists can now achieve a significant gain in qualitative ontological economy by positing action-universals in place of the Ostrich Separatists' primitive relational kinds of loving, moving, and so on. This would be true even if the Classical Connectionists had to posit two or three primitive kinds of instantiation: the instantiation of monadic universals by single things, the joint instantiation of symmetrical relational universals by pairs of things, and the joint instantiation of non-symmetric action-universals by pairs of things taken in an agent/patient order.
Modifying Trope Connectionists, assuming that they adopt Substrate Theory (9.1T.1A) rather than Bundle Theory (9.1T.1T), can claim a similar economy for their account. A single love modifying trope could stand in the agent modifying relation to Cyrano and the patient modifying relation to Roxanne.
10.2.3 A family of distinct, complex instantiation relations
We can accomplish much the same effect by replacing ordered pluralities with a family of complex instantiation relations. So, instead of supposing that there is a single instantiation relation that connects snow with WHITENESS and the ordered plurality of Cyrano and Roxanne (with Cyrano as agent and Roxanne as patient) with LOVE, we could instead suppose that there are at least two distinct instantation relations: a binary relation that connects individuals with monadic properties or qualities (like WHITENESS) and a three-place relation that combines two individuals with an action-universal (like LOVES). We will also need a distinct instantiation relation for each class of thematically structured non-symmetric relations: a patient-agent-action instantiation relation (for action relations like LOVES), a greater-less-comparison instantiation relation (for comparative relations like IS GREATER THAN), an extreme-extreme-intermediate-placement instantiation relation (for 3-place relations like BETWEEN), and so on. It is difficult to say which of the two theories, the ordered-plurality theory or the multiple-instantiation-relation theory, is most economical in terms of qualitative parsimony. We think it's plausible that the ordered-plurality view has a slight edge here, especially if the number of thematically structured relational families is relatively large. Each new instantiation relation introduces two or three new categories of things (e.g., agents and patients, things greater and lesser, things intermediate and extreme), while the corresponding ordered pluralities introduce just one new thing (agent-patient ordered pluralities, greater-lesser ordered pluralities, extreme-intermediate ordered pluralities).
In his 1913 manuscript on the theory of knowledge, Bertrand Russell (1984) offered what seems, at first glance, to be an alternative account of non-symmetric relations. This account is usually called ‘positionalism’. On Russell's account, each binary relational universal has two positions or poles, with one relatum attached in some way to each of the two positions on the universal. So, the fact that Romeo loves Juliet involves Romeo's being attached to the lover position of LOVES and Juliet's being attached to the beloved position. However, Russell's account still leaves the fundament
al problem unsolved: what is it for Romeo to be “attached” to a position within a particular fact? It's obviously not sufficient to say that the fact that Romeo loves Juliet consists simply in Romeo's instantiating the lover position and Juliet's instantiating the beloved position, since this leaves undecided whom Romeo loves or who loves Juliet. It provides no connection between Romeo as lover with Juliet as beloved. Whether the universal LOVES has two positions or not, we still need some account of what it is for the pair to occupy those two positions jointly, as part of a single fact.
Accounting for this joint occupation of the two positions requires something like associating each of the pair with one of the two roles (which is what the ordered-plurality theory proposes) or providing a single relation that simultaneously associates Romeo with one position and Juliet with the other (which is what the multiple-instantiation-relation theory proposes). In either case, the positing of different positions within the universal seems unnecessary. There is, however, a third option, one that may come closer to Russell's intention: we could think of particular facts as constructed from particulars and parts of universals. We will take this possibility up as option 2 in the next sub-section.
10.2.4 Constituent Ontologies and non-symmetric relations
With a little imagination, Constituent Ontologists can solve these problems, whether they take the constituents of states of affairs to be universals or modular tropes. For simplicity's sake, we will consider two options for Classical Connectionism. Trope Connectionism can also employ counterparts of these two options.
Both options make use of the non-symmetric part-whole relation, and thus are faced with an obvious objection: Doesn't appealing to part-whole make both accounts viciously circular?
In short, No. Constituent Ontologists can make use of the familiar distinction between real and merely logical relations. The part-whole relation is not a real relation, in the sense that there is no PART-WHOLE universal (nor are there any part-whole relational tropes). In a sense, the part-whole relation is so metaphysically fundamental that it cannot be analyzed in terms of instantiating or including any universal or trope.
Constituent Ontologists ask us to accept one non-symmetric relation, the part-whole relation, as a primitive, with no analysis of what makes it true. In effect, Constituent Ontologists treat the part-whole relation the way Ostrich Separatists treat all relations. By reducing all other relational facts to facts involving only the part-whole relation, Constituent Ontologists offer a much simpler theory of the world, in terms of the qualitative diversity of things.
In addition, a version of the Extrinsicality Objection (Section 9.3.1.1) can be wielded by Constituent Connectionists against Relational Connectionism. A natural relation is intrinsic to its relata, taken together. A binary relation is intrinsic to the pair, taken as a pair. We shouldn't have to introduce a separate, third thing to act as the truthmaker for a relational truth. However, this is just what Relational Connectionists have to do. Any R-related pair is related by R in virtue of the relata being extrinsically related (by the instantiation relation) to R.
In contrast, Constituent Connectionists can distinguish between thin and thick pairs. Thin pairs consist of two relata and their constituents. Thick pairs include two relata and any relational universals that belong to the pair as a pair. Relational facts are intrinsic on this view to the thick pair, which is as it should be: they are intrinsic to the pair taken together, as a pair.
Having dispatched the circularity worry, we turn to a consideration of the two options for solving the problems to with non-symmetric relations. The first option involves two-step constructions; the second involves divided universals.
OPTION 1: TWO-STEP CONSTRUCTIONS The first option borrows an idea from set theory and the representation of ordered pairs. Standard set theory represents the order of an ordered pair by building up the ordered pair in two steps. First, the first member of the pair is put into a unit set, and both members are put into a pair set. Second, the unit set (containing the first member) and the pair set (containing both members) are combined into a single set. Constituent Connectionists can do something similar, using the part-whole relation. If Cyrano loves Roxanne, they could suppose that there is a complex state of affairs, comprising exactly two parts. The first part combines Cyrano with the relation of LOVE, and the second part consists of Roxanne alone. The order of the loving relation corresponds to the fact that it is Cyrano, and not Roxanne, who forms a whole with LOVE. The fact that it is Roxanne who is the object of Cyrano's love corresponds to the fact that the Cyrano-LOVE complex combines with Roxanne to form a single state of affairs. (This solution presupposes the denial of Mereological Universalism (22.3T.1), the thesis that there is a whole composed of any set of members whatsoever.) Here is a tree-diagram (Figure 10.1) representation of two-step constructions (with parts arranged below the wholes that contain them):
Figure 10.1 A Two-Step Construction of a Binary Relational Fact
The same relationship can also be represented by means of circles, with parts inside their containing wholes (Figure 10.2):
Figure 10.2 A Pictorial Representation of the Two-Step Construction
The main drawback with this account is one that it shares with the various standard, ordered-pair accounts of non-symmetric relations: it requires that we deny the Metaphysical Parity of Converse Relations. We have to assume that, in reality, there is a metaphysical order that corresponds more closely to one of a pair of converse relations than to the other. If we think of the inner relatum as prior to the outer relatum, then we have to decide whether the truthmaker of the truth that Cyrano loves Roxanne puts Cyrano first or Roxanne. It would seem to be impossible for us to tell which is in fact the case.
However, this constituent account of order has one advantage over the standard account: it doesn't entail that any sets (such as ordered pairs) have non-mathematical natural properties. So it doesn't contradict Membership the Only Fundamental Set Relation (PMeta 3).
OPTION 2: DIVIDED UNIVERSALS In option 2, Constituent Connectionists suppose that each relational universal consists of two parts, one representing one pole of the relation and the other the second pole. This is a version of what Kit Fine (2000) calls “positionalism”. For example, in the case of the universal LOVE, it would contain two parts: a LOVER part and a BELOVED part. When Cyrano loves Roxanne, there will be a single state of affairs with two parts: the first part containing Cyrano and LOVER, and the second part containing Roxanne and BELOVED. Here is a diagram of divided universals (Figure 10.3):
Figure 10.3 A Divided-Universal Construction of a Binary Relational Fact
Figure 10.4 A Pictorial Representation of the Divided-Universal Construction
The top node of the diagram represents the total relational state of affairs, the state of a's being R-related to b. The left node just below the top combines the R+ universal with a (or some essential part of a), and the corresponding right node combines the other part of R, R−, with b (or some essential part of b). Here is the same relationship, in a circle (Venn-like) diagram (Figure 10.3):
PROS AND CONS OF THEORIES OF NON-SYMMETRIC RELATIONS We've seen that there are three solutions to the problem of non-symmetric relations: Ostrich Separatism, ordered pluralities (with either Reductive Separatism or Relational Connectionism), and Constituent Connectionism with divisible universals. Which of the three is best? There are two considerations. First, the Extrinsicality Objection tells against Relational Connectionism. Second, we can consult Ockham's Razor, in its qualitative economy version (PMeth 1.4 and 1.4.1). That is, we can try to minimize the number of metaphysically fundamental relations. To answer this question, we have to look at two sub-questions. First, how many natural relations are there? Second, how many types of basic ordering (e.g., agent/patient, greater/lesser, medium/extreme) are there?
If there are very few natural relations, then Ostrich Separatism is best. If there are very many, it is worst.
If there are very few basic ord
erings, then both ordered pluralities and Constituent Connectionism are viable theories. If there are very many, then Constituent Connectionism is the best theory.
10.3 Structural Universals and Constituent Ontology
David Armstrong (1983, 1989b) has argued for the existence of structural universals, universals that correspond to the realization of complex structures. For example, consider the property of being a methane molecule. This property can be defined in terms of having four hydrogen atoms and one carbon atom as parts, with each of the hydrogen atoms bound chemically to the central carbon atom. There is certainly such a property, but is there in addition a universal, BEING METHANE, shared by all individual methane molecules? A very simple case of structural universals is that of conjunctive universals, like the universal RED AND ROUND. If we have universals REDNESS and ROUNDNESS, do we need a third universal present in cases of things that are both? Ockham's Razor suggests not, so we need to consider whether there are any countervailing considerations.
David Lewis (1986b) collected six arguments for structural universals.
Providing meanings of complex predicates. A complex universal could serve as the meaning of compound predicates, like RED AND ROUND. However, this reason is incompatible with adopting a sparse theory of universals.
Accounting for resemblance. One might think that accounting for resemblance is not something that could justify structural universals. For example, if two things are both red and round, we can use the universals REDNESS and ROUNDNESS to explain their similarity. We seem to need a third universal to ground their similarity in having both properties. The sharing of the two simple universals seems to be sufficient. However, to push back against this, one might appeal to what the nineteenth-century German psychologist Christian von Ehrenfels (a student of Franz Brentano) called ‘Gestalt similarities’. A Gestalt quality is a feature of a structured whole that emerges from and supervenes on the instantiation of simpler universals by the parts. Gestalt properties are real and objective and are not in principle reducible to simple facts about the parts. Some examples of Gestalt qualities are artistic styles of individual artists or schools. Paintings can have a holistic quality that marks them as products of the Renaissance, or of Jan Brueghel the Younger. The similarity of two Brueghel paintings is not reducible to pixel-by-pixel comparisons of paint pigments, even though the quality does supervene (Def D2.6) on the details of both paintings, in that once you fix the distribution of colors and strokes on the painting, you have also determined whether it has that ineffable Brueghelish quality. Ethical qualities of actions might also be examples of Gestalt properties.