Def. D10.1.1 Weakly internal. R is a weakly internal relation if and only if, necessarily, for every x and y, whether R holds between x and y weakly supervenes on the set of facts about the intrinsic properties of x and y.
Def. D10.1.2 Strongly internal. R is a strongly internal relation if and only if, necessarily, for every x and y, whenever R holds between x and y, the fact that R holds between x and y is wholly grounded by the facts about the intrinsic properties of x and y.
Monadists must not only deny that there are any external relations: they must also deny that there are any weakly internal relations. If Monadism is true, all relations must be strongly internal.
What about the relation between parts and wholes? Is this inclusion relation strongly or weakly internal? That is, if x is a part of y, is this part-of relation external or internal? If internal, is it weakly or strongly internal? This question will turn on whether the relations of identity and distinctness are internal or external. If x is part of y, and if being identical to x is an intrinsic property of x, then x's being part of y will count as in intrinsic property of y, since, for any F that is an intrinsic property of a part of x, there will be a corresponding intrinsic property of y, namely, the property of having a part that is F. So, if being identical to x is an intrinsic property of x, and x is a part of y, then having a part that is identical to x will be an intrinsic property of y. But is being identical to x an intrinsic property of x? This is precisely the question we examined in some detail in Chapter 8. As we saw there, all Bundle Theorists, including those who believe in fundamental haeacceities, hold that identity is an intrinsic property. All others must treat being identical to x as an extrinsic property of x, and so they must count the part-whole relation as an external relation.
One thing to notice immediately is that only Ostrich Nominalists and Bundle Theorists can be Monadists, since all Substrate Theorists and all Reductive Nominalists accept the existence of some fundamental relational truths: either truths about instantiation (Classical Relational Ontologists 9.1A.1T) or composition (Constituent Ontologists 9.1A who are also Substrate Theorists) or resemblance or class membership (Class or Resemblance Nominalists 8.1T.3/8.1T.4). All of these relations will be either external or only weakly internal. For example, instantiation (for Classical Relational Ontologists) and resemblance (for Resemblance Nominalists) must be only weakly internal relations, since whether a particular instantiates a universal (for Classical Relational Ontologists) or resembles another particular (for Resemblance Nominalists) cannot be grounded in the intrinsic character of the particular, since it is the instantiation or resemblance relation that grounds that intrinsic character (not vice versa).
In addition, spatial relations pose a problem for Monadists. Being spatially contiguous or being a certain distance apart appear to be fundamental relational truths about material objects. Leibniz believed that this was not so, that each monad carried within it a spatial representation of the entire universe, with some element in that representation standing for that monad's own unique perspective. However, Leibniz's account seems to require some fundamental spatial or quasi-spatial relations between the parts of these spatial representations. In addition, his account seems to require representation itself as a fundamental relation.
An initially more promising direction for Monadists to take is to adopt the Theory of Spatial Qualities (see Chapter 17.1T.1T). On this view, to be located somewhere in space is simply to instantiate some intrinsic quality that is uniquely associated with that region. Distance between regions corresponds to the degree of dissimilarity between two such spatial qualities. On this view, spatial contiguity and spatial distance are internal relations, fully determined by the intrinsic character of the two material objects.
However, this merely pushes the problem of accounting for spatial relations back a step, since we still have to consider what makes one spatial location close to another one. We will need to posit spatial relations between spatial qualities, but the existence of such fundamental relational facts, even facts between universals, is inconsistent with Monadism.
Leibniz proposed a version of Monadism in which everything was conscious. The world consisted of an infinite number of souls, each of which represented a spatially-ordered world centered upon itself by a simple and non-relational act of representation. The public space of the physical world was, for Leibniz, a kind of projection pieced together from the internal representations of each of the conscious monads.
The most difficult problem for Leibniz's version of Monadism is giving a full and adequate theory of monadic representation. A monad's representing the world as being a certain way must consist of a set of intrinsic mental acts. These acts cannot be related to each other in the way that pixels in a picture relate to each other, or in the way words in a sentence relate to each other, since these would sneak fundamental external relations back into the picture. Since Leibniz was a Bundle Theorist, he can help himself to the part-whole relation, and to the relations of identity and distinctness (since these will all be strongly internal relations, given his commitments). Nonetheless, it's going to be a very difficult task to construct an adequate theory of mental representations of space without anything like spatial relations to work with.
Given these problems, few philosophers have embraced Monadism. So we turn to varieties of Anti-Monadism.
When we turn to relational properties, we find that some Realists about monadic universals and some Moderate (or Trope) Nominalists embrace Separatism (an Extreme Nominalistic attitude toward relational universals), in contrast to Connectionism:
10.1A.1T Connectionism. Some relational universals or relational tropes are fundamental entities.
10.1A.1A Separatism. There are no fundamental non-symmetric relational universals or relational tropes.1
Connectionism comes in several varieties. First, there is the distinction between Universal Connectionists, who believe in non-symmetric relational universals, and Trope Connectionists, who believe in non-symmetric relational tropes (either modular or modifying).
10.1A.1T.1T Classical Connectionism. There are relational universals.
10.1A.1T.1A Trope Connectionism. There are relational tropes that are fundamental, but no relational universals.
Second, among Classical Connectionists we can distinguish between those who embrace relational nexuses and those who instead rely on relational states of affairs as truthmakers. (We'll ignore for the sake of simplicity Classical Connectionists who reject Truthmaker Theory entirely.) A Nexus Connectionist adopts what is essentially a version of a Relational Ontology (9.1T) for non-symmetric relations, treating instantiation itself as a weakly internal relation, while State of Affairs Connectionists adopt a version of Constituent Ontology, with the pair (corresponding to the state of affairs) actually containing the relation as a proper part.
10.1A.1T.1T.1T Relational (Nexus) Connectionism. When some particulars instantiate some relational universal, the truthmaker for the corresponding proposition is a relational nexus that ties those things to the universal. The nexus does not contain either the relation or the relata as parts.
10.1A.1T.1T.1T.1A Constituent (State of Affairs) Connectionism. When some particulars instantiate some relational universal, the truthmaker for the corresponding proposition is a state of affairs that contains those things and that universal as proper parts.
We could draw a similar distinction between two kinds of Trope Connectionists: Relational Trope Connectionists, who posit modifying relational tropes that modify each of the two relata, and Constituent Trope Connectionists, who posit states of affairs that contain a relational trope (either modular or modifying).
Connectionists have typically made a distinction between real relations and merely logical relations. The real relations are universals that are instantiated by pairs of things, while logical relations (including instantiation itself) are irreducible relational kinds, with no universal involved. This enables Realists to claim a further advantage in econom
y over Ostrich Nominalists, reducing the number of primitive relational kinds to one. We must avoid making the mistake of assuming that the “real” relations are somehow more real than the “merely logical” ones. In fact, the opposite is the case: it is the merely logical relations (like instantiation or resemblance) that are metaphysically more fundamental than the real relations. The use of the term ‘real’ here reflects an older, Latin tradition. It might be better to translate think of real relations as things-ish relations. A real relation is a relation that can be reified, that is, thought of as a thing (the Latin word for ‘thing’ is ‘res’, the root of ‘real’).
A similar move is available to Trope Nominalists. They can also distinguish between real relations, corresponding to relational tropes (either modifying or modular), and logical relations (such as resemblance and being bundled).
Separatists also come in two varieties. Just as there are Ostrich Nominalists and Reductive Nominalists, so are there Ostrich Separatists and Reductive Separatists. Ostrich Separatists take relational truths that involve external relations, like ‘Oklahoma is north of Texas’, to be metaphysically fundamental, with no need for explanation in terms of the instantiation of universals, resemblance of tropes or any other metaphysical device. (Internal relations, like resemblance, can be explained in terms of the monadic properties of the two relata.)
10.1A.1A.1T Ostrich Separatism. Separatism is true, and there is no general explanation of why some things taken in a certain order resemble other things taken in a certain order.
10.1A.1A.1A Reductive Separatism. Separatism is true, and there is some general explanation of why some things taken in a certain order resemble other things taken in a certain order.
As in the case of Reductive Nominalism, the most plausible version of Reductive Separatism would be Resemblance Separatism, positing some fundamental relation of resemblance between the things involved in a certain kind of relational fact.
There is an obvious parallel between the accounts of monadic properties that we examined in Chapters 7–9 and the various theories of relations. In general, those who adopt a more nominalistic account of monadic properties should adopt an equally or more nominalistic account of relations. It would be very weird, for example, to combine Ostrich Nominalism for monadic properties with a Classical Connectionist account of relations. Among Realists and Trope Theorists, it makes sense to take a position on relations that exactly mirrors their views about monadic properties. Here is a table indicating which theory combinations are plausible.
Table 10.1 Combining Theories about Monadic Properties and Relations
Ostrich Separatism Resemblance Separatism Relational Connect- ionism Trope Constituent Connect-ionism Classical Constituent Connect-ionism
Ostrich Nominalism Yes No No No No
Resemblance Nominalism Yes Yes No No No
Classical Relational Ontology Yes Yes Yes No No
Constituent Trope Theory Yes Yes Yes Yes No
Classical Constituent Ontology Yes Yes Yes No Yes
Now that we have surveyed the various metaphysical options for Anti-Monadism, we shall next turn to the most difficult test case: that of non-symmetric relations (or orderings).
10.2 Non-Symmetrical Relations and the Problem of Order
A symmetric relation R is one of such a kind that Rxy always entails Ryx.
Def D10.2 Symmetric Relation. A relation R is symmetric if and only if, necessarily, for all x and y, if x stands in R to y, then y stands in R to x.2
A non-symmetric relation involves no such entailment. Being next to is symmetric, while being to the north of is non-symmetric. Belonging to the same club is symmetric, but loving and hating are non-symmetric.
A symmetric relation can always be thought of as a simple property of two things taken collectively. Universal Connectionists could suppose that Texas and Oklahoma jointly instantiate the relational universal BEING NEXT TO. However, non-symmetric relations are different. Connectionists cannot simply say that Texas and Oklahoma jointly instantiate BEING TO THE NORTH OF. Instead, it is Oklahoma and Texas in that order that instantiate the relation. The same two states, taken in the reverse order, do not instantiate the BEING TO THE NORTH OF relation, since Texas is not to the north of Oklahoma. In a same way, Modifying Trope Connectionists cannot suppose that Oklahoma and Texas are jointly modified by a relational modifying trope of being to the north of.
A similar problem besets Modular Trope Connectionists who take particulars to be bundles of modular tropes. They cannot simply say that there is a being to the north of modular trope that belongs to both the Oklahoma-bundle and the Texas-bundle, since this would fail to distinguish the case where Oklahoma is to the north of Texas from the case where Texas is to the north of Oklahoma.
Similar problems beset Reductive Separatists' account of non-symmetric relations. For example, the Resemblance Separatist cannot simply suppose that Cyrano's loving Roxanne consists simply in the fact that the pair of Cyrano and Roxanne resemble other lover-beloved pairs, since this would fail to distinguish Cyrano's loving Roxanne from Roxanne's loving Cyrano, which are obviously two distinct and independent truths.
There is a standard solution to all of these difficulties, one which involves introducing a new kind of set, the ordered pair. An ordered pair is a certain kind of set that has sets as its members (a second-order set). In the most popular version (the Kuratowski construction), the ordered pair
10.2.1 The twin problems of converse relations
As Kit Fine (2000, 2007) has argued, the standard or ordered-pair solution, in both its Reductive Separatist and Connectionist versions, encounters two problems involving converse relations. If R is a non-symmetric relation, then a relation S is the converse of R if and only if whenever Rxy then Syx, and vice versa.
Def D10.3 Converse Relation. R and S are converse relations if and only if, necessarily and for all x and y, Rxy if and only if Syx.
Here are some examples of relations and their converses:
North of South of
Parent of Child of
Loves Is loved by
Greater than Less than
Above Below
Suppose that at least one of each of these pairs is a real and natural relation (i.e., a relation corresponding to a universal). If so, which is it? Is north of natural and south of unnatural or vice versa? Is loves natural and is loved by unnatural or the other way around? In each case, two converse relations seem so much alike that it is hard to believe that they could differ in their degree of metaphysical fundamentality. If so, then the Classical Connectionists are forced to say that both relations are fundamental.
The Metaphysical Parity of Converse Relations. If R and S are converse relations, then either both are natural or neither is (that is, either both correspond to relational universals or neither do).
Moreover, if Connectionists and Reductive Separatists were to deny the Metaphysical Parity of Converse Relations, they would have to suppose that some ordered pairs have natural properties other than mathematical ones. We'd have to assume that the ordered pairs
ed pairs
PMeta 3 Membership the Only Fundamental Set Relation. If S is a set, then the only fundamental relation involving S is the membership relation between S's members and itself. (Principle of Metaphysics 3)
If Classical Connectionists accept the Metaphysical Parity of Converse Relations (as it seems they should), then they immediately face two further problems. First, they must suppose that there is some sort of brute metaphysical necessity that ensures that whenever a non-symmetric relational universal exists, there must also exist a second relational universal, corresponding to its converse. As we have seen, Ockham's Razor directs us to minimize such brute necessities.
Second, the Metaphysical Parity of Converse Relations forces Classical Connectionists to posit a massive duplication of truthmakers. The truth that Cyrano loves Roxanne will now be made true by both the fact that the ordered pair of Cyrano and Roxanne stand (in that order) in the loves relation and by the separate fact that the ordered pair of Roxanne and Cyrano stand (in that order) in the distinct is loved by relation. Such duplication of truthmakers seems indefensible.
The Atlas of Reality Page 34
