Abelard’s loving Eloise is obviously not the same as Eloise’s loving Abelard. A distinguishing feature of non-symmetric relations, like the love relation, is that they admit of differential application, i.e., they may apply to the same things in multiple ways. A crucial question is, what makes differential application possible? How can things be arranged one way rather than another?

The answers given depend on the view on relations one adheres to. There are three basic accounts of relations: the standard view, the positionalist view, and the antipositionalist view.

In brief, the standard view says that the arguments of a relation come in a linear order, e.g., Abelard comes first and Eloise comes second in Abelard’s loving Eloise. The positionalist view says that a relation comes with positions to which arguments may be assigned, e.g., for the love relation we have the positions Lover and Beloved. The antipositionalist view says that a relation is a network of complexes interrelated by substitutions, e.g., substituting Anthony for Abelard and Cleopatra for Eloise in Abelard’s loving Eloise gives the complex of Anthony’s loving Cleopatra.

In his seminal paper “Neutral Relations,” Kit Fine made clear that the standard view and the positionalist view give rise to problems (2000). His answer was a new view on relations, the antipositionalist view. However, the antipositionalist view has also been heavily criticized (Donnelly 2016; Gaskin and Hill 2012; MacBride 2007, 2014; Orilia 2011). In my opinion, however, the criticisms arise from a fundamental misunderstanding of the position. In this paper I want to clarify some of the misconceptions. In particular I will show that positionalism and antipositionalism are not really opposite views.

For simplicity I will assume throughout the paper that all relations are of finite degree.

1 Views on Relations

The views presented here contain some aspects that have not been described before. For the positionalist view we make a distinction between thick and thin positionalism, where only in thick positionalism objects may occupy positions.

A note in advance: in Leo (2013), I made a sharp distinction between relational states and relational complexes, and conceived of relational complexes as a structured perspective on relational states. I argued that a state may have more than one corresponding complex. For example, the state of Abelard’s loving Eloise corresponds not only with a complex from the binary love relation with two relata, but (among others) also with a complex from the unary relation of loving Eloise with one relatum. For the argumentation in this paper relational states do not play an essential role. However, occasionally I will not only talk about relational complexes but about relational states as well.

1.1 Standard View

The standard view assumes that the arguments of a relation always come in a given linear order. For example, in each instance of the love relation one of the arguments comes first and the other comes second. One might also say that relations have a direction. In the instance \(aRb\) of a relation \(R\) the relation runs from \(a\) to \(b\), and in \(bRa\) the relation runs in the opposite direction. Different directions make differential application possible.

A nice feature of the standard view is that it corresponds straightforwardly with natural and most formal languages. For example, for the relation loves, we have a direct match with linguistic expressions of the form “___ loves ___.”

Unfortunately, there are also problems with the standard view. In the states “out there” there is no linear order or direction between the arguments. The linear order is just a representational artifact. Already in 1913 Russell rejected the idea that all relations have a “natural” direction. For example, this is not the case for right and left, up and down, and greater and less (Russell 1984, 87).

This problem may also be formulated in different terms. The standard view makes it plausible that for each binary relation \(R\) there is a converse relation \(R'\), where \(aRb\) holds iff \(bR'a\) holds. For example, for the relation on top of, we have the converse relation beneath, where the state of \(a\)’s being on top of \(b\) is the same as the state of \(b\)’s being beneath of \(a\). We would like to regard this state as a relational complex consisting of a single relation in combination with the two relata. However, this relation can neither be on top of nor beneath, because there is no good reason to choose one over the other (Fine 2000, 3–4).

1.2 Positionalism

According to positionalism, each relation comes with a collection of positions to which objects may be assigned and with no intrinsic order between the positions. Such an assignment results in a relational complex. We distinguish two forms of positionalism: thick positionalism, which is the “normal” positionalist view, and thin positionalism, a new variant introduced in this paper.

1.2.1 Thick Positionalism

In thick positionalism, a relation comes with positions to which objects may be assigned. Such an assignment may result in a relational complex with objects occupying positions.

As pointed out by Fine, a problem with this view is that symmetric relations like the adjacency relation have distinct complexes that intuitively should be the same (2000, 17). We would, for example, like to regard \(a\)’s being next to \(b\) as the same complex as \(b\)’s being next to \(a\). But suppose that the adjacency relation has two positions Next and Nixt. Then assigning \(a\) to Next and \(b\) to Nixt gives a complex which is distinct from the complex obtained by assigning \(b\) to Next and \(a\) to Nixt if in the complexes objects occupy positions. In one complex, \(a\) occupies Next and \(b\) occupies Nixt, and in the other complex it is the other way around.¹

1.2.2 Thin Positionalism

In thin positionalism, a relation comes with positions for which objects may be substituted. Such a substitution may result in a relational complex with occurrences of the objects involved.

Positions are not boxes in which you can put an object; rather they are substitutable places in a structure or form. The relevance of this distinction can be illustrated with an example.

For the adjacency relation with positions Next and Nixt substituting \(a\) for Next and \(b\) for Nixt results in a complex with an occurrence of \(a\) and an occurrence of \(b\). The complex is the same as the one that we get when we substitute \(b\) for Next and \(a\) for Nixt. It is as if the positions disappear once we assign objects to them.² So we don’t get too many complexes as in thick positionalism. This makes thin positionalism preferable over thick positionalism.

A relation itself is viewed as an entity and its positions as occurrences of some kind of entity. Though it is not essential, positions might perhaps best be seen as occurrences of arbitrary objects. What is essential is that we may substitute objects for positions. The result of a substitution (if any) is a complex with occurrences of the objects substituted for positions.

The notions of substitution and occurrence are taken as primitive.

In appendix A a general composition principle for substitutions is given. In the principle substitution is conceived of as an operation on occurrences of entities within an entity.

We will assume that thin positionalism endorses the Composition Principle in appendix A.

The Composition Principle does not speak about complexes and positions for which objects may be substituted, but about entities and occurrences of entities for which entities may be substituted. However, because positions are conceived of as occurrences of some kind of entity, and because objects can be substituted for positions, the principle applies in a straightforward way to thin positionalism.

If \(s\) is taken as a substitution in a complex, then a similar statement holds.

We call \(\mu\) a co-map of substitution \(s\).

The Composition Principle of Thin Positionalism has the interesting consequence that substitutions in complexes can be derived from the substitutions for the positions and their co-maps.

A single substitution of objects for positions may have more than one co-map. For example, if for a symmetric relation like the resemblance relation substituting an object \(a\) for both positions \(p, p'\) results in a complex with two occurrences of \(a\), then this substitution has two co-maps; one that maps \(p\) to an occurrence \(\alpha\) and \(p'\) to an occurrence \(\alpha'\), and another that maps \(p\) to \(\alpha'\) and \(p'\) to \(\alpha\).

One could in principle allow that a co-map \(\mu\) is not injective. For example, one could argue that for the love relation with positions Lover and Beloved, substituting Narcissus for both positions results in a complex with just one occurrence of Narcissus.

If for a given substitution \(s\) of objects for positions a co-map \(\mu\) is not injective, then we say that the substitution results in a coalescence of occurrences.

We call a relation coalescence-free if it has no coalescence of occurrences. So each complex of an \(n\)-ary coalescence-free relation will have \(n\) occurrences of objects. If the love relation is coalescence-free, then the complex of Narcissus’ loving Narcissus would have one occurrence of Narcissus in the role of lover and another one in the role of beloved.

As we have seen, the adjacency relation is symmetric in a strict sense. Switching the arguments does not change the complex. More generally, we say that \(R\) is strictly symmetric if there is a non-identity permutation \(\pi\) of its positions such that for every substitution \(s\) for the positions resulting in a complex \(\xi\), substitution \(\pi \cdot s\) results in \(\xi\) as well.³

Thin positionalism may appear to be more complicated than thick positionalism. Nevertheless, I think it is a much more natural view than thick positionalism. Having relational complexes in the world as a result of substituting objects for positions seems to make more sense than having complexes “out there” containing objects in a kind of boxes, called positions.

1.3 Antipositionalism

Relational complexes have constituents. But this does not necessarily mean that we can directly speak about how these constituents occur in a given complex. According to antipositionalism, the structure of a relation can be fully expressed in terms of structure preserving connections between its complexes. There is no need to say anything about the internal structure of the complexes. This may sound a bit vague, so let us look at an example.

For the love relation, one of the complexes could be Paris’ loving Helen. In this complex we have one occurrence of Paris and one of Helen. By substituting Venus for the occurrence of Paris and Adonis for the occurrence of Helen we get the complex of Venus’ loving Adonis. With this substitution corresponds a structure preserving map between the occurrences of Paris and Helen in Paris’ loving Helen and the occurrences of Venus and Adonis in Venus’ loving Adonis. By taking all possible substitutions into account, we get a network of interrelated complexes.⁴

Networks like this are conceived of as relations. Isomorphic relations are not necessarily identical, as the monadic relations of having a heart and having a kidney make clear.

As in thin positionalism, the notions of substitution and occurrence are taken as primitive. Likewise, we assume that antipositionalism endorses the Composition Principle in appendix A.

To make the Composition Principle appropriate for antipositionalism, we only have to make a slight change in terminology. Instead of using a phrase like “a substitution of entities for the occurrences of entities in an entity \(\xi\)” we say “a substitution of objects for the occurrences of objects in a complex \(\xi\).”⁵

We call a map \(\mu\) with this property a co-map of substitution \(s\).

We call a complex an initial complex if any complex of the relation can be obtained from it by a substitution. If a relation has an initial complex, then it follows from the Composition Principle of Antipositionalism that for any complex \(\xi\) of the relation the substitution in \(\xi\) that maps each occurrence \(\alpha\) to the object of \(\alpha\) results in \(\xi\) itself.⁶

More principles could be given. An interesting, but controversial one says that all complexes of a relation are connected via a substitution. This may not hold for certain relations of variable degree, like the relation of forming a circle. It is not obvious how to characterize for such relations the unity of its complexes.

Like thin positionalism, antipositionalism does in principle not exclude a coalescence of occurrences, i.e., two or more occurrences of objects in a complex may be mapped to the same occurrence of an object in another complex. For example, substituting Narcissus for the occurrence of Paris as well as for the occurrence of Helen in the complex of Paris’ loving Helen could result in a complex with one occurrence of Narcissus.

A coalescence of occurrences is very natural for set-like relations. For the relation of forming a group we may want the complex for the group consisting of Athos, Porthos, and Aramis to have three occurrences and the group of Batman and Robin to have two occurrences. If this is the case, then the second complex may be obtained from the first by a substitution, but there is no substitution the other way around.

Also for the ternary relation \(R\) where \(R abc\) is the complex of \(a\)’s loving \(b\) and \(b\)’s loving \(c\) it may seem natural to assume that a coalescence of occurrences can take place. For substituting in \(R abc\) the object \(a\) for \(c\) gives the complex \(R aba\), and substituting in \(R abc\) the objects \(b, a, b\), for the occurrences of \(a, b, c\) gives the complex \(R bab\). These complexes are obviously empirically indistinguishable, but if a coalescence of occurrences is allowed they can be identical (cf. Leo 2010, 147–148).

It should be noted that not always all complexes in a relation are empirically distinguishable. This is obvious for mathematical relations, but it is also the case for some other relations, like the conjunction of the binary love relation with the unary relation of loving \(d\), where \(d\) is a fixed object.⁷ For this relation, the conjunction of \(a\)’s loving \(d\) with \(d\) substitutable and \(b\)’s loving \(d\) with \(d\) fixed is a complex that is distinct from the conjunction of \(b\)’s loving \(d\) with \(d\) substitutable and \(a\)’s loving \(d\) with \(d\) fixed, but the two complexes are empirically indistinguishable (cf. Leo 2013, 364).

Under antipositionalism, different substitutions in a complex may result in the same complex, which is a defining characteristic of strictly symmetric relations. For the adjacency relation, for example, we have the complex of \(a\)’s being adjacent to \(b\). Substituting in this complex \(b\) for (the occurrence of) \(a\) and \(a\) for (the occurrence of) \(b\) gives the same complex. This means that in the network of the relation we have a map from each complex to itself that switches the two objects involved.

One may worry that antipositionalism is less able to identify complexes than positionalism because in antipositionalism we don’t have positions with meaningful names like lover and beloved. However, in antipositionalism we could give occurrences equally meaningful names like lover in complex \(\xi\) and beloved in complex \(\xi\). Besides, names can be freely chosen; in both views on relations the meaning of names do not play a constitutive role.

There are alternative antipositionalist accounts possible. One could, for example, assume that any complex has for each object at most one occurrence. Then there is not really a need to talk about occurrences and one can simply substitute objects for objects in complexes.

2 Intertranslating the Views

In this section the translatability from positionalism to antipositionalism and vice versa will be examined. Particular attention will be given to the question whether the translations respect the Composition Principle in appendix A. By examining the translations back and forth, we get a clear picture of the relative expressive power of positionalism and antipositionalism.

2.1 From Positionalism to Antipositionalism

Can a positionalist express himself in antipositional terms? We will describe what kind of networks of interrelated complexes a thick and a thin positionalist can construct, and discuss whether these networks are all acceptable for an antipositionalist as networks of relations.

2.1.1 From Thick Positionalism to Antipositionalism

Let us first assume you are a thick positionalist. Let \(R\) be a relation with positions \(p_1, \ldots, p_n\). Then you can simply create a network of interrelated complexes as follows. Let \(\xi\) be the complex obtained by assigning \(a_1, \ldots, a_n\) to \(p_1, \ldots, p_n\). Identify the pairs \(\alpha_i = \langle\xi, p_i\rangle\) with occurrences of objects in \(\xi\). If \(\xi'\) is the complex obtained by assigning \(b_1, \ldots, b_n\) to \(p_1, \ldots, p_n\), then define the assignment of \(b_1, \ldots, b_n\) to \(\alpha_1, \ldots, \alpha_n\) as a substitution in \(\xi\) resulting in \(\xi'\).

By repeating the construction for every assignment of objects to the positions of \(R\), you get a network of complexes interrelated by substitutions.

It is easy to verify that the resulting network of complexes satisfies the Composition Principle of Antipositionalism.

The construction is adequate for non-symmetric relations, but not for symmetric relations since in thick positionalism different assignments of objects to positions always result in different complexes.

A way out could be the use of equivalence classes of complexes to express strict symmetry of relations. The equivalence classes could be identified with what the antipositionalist regards as complexes. There is, however, a complication; not for every relation, occurrences of objects can be defined non-arbitrarily in set theory in terms of positions, complexes, and objects. This will be discussed in the last part of this section.

2.1.2 From Thin Positionalism to Antipositionalism

Now assume you are a thin positionalist. Let again \(R\) be a relation that comes with a set of positions. Without any adjustment, the complexes of the relation already form a network of complexes interrelated by substitutions—at least, if there are complexes. So, for the translation, we just retain the network of complexes.

The network of complexes satisfies the Composition Principle of Antipositionalism. But is it always acceptable as a relation for the antipositionalist?

If the relation \(R\) is not coalescence-free, then it might happen that not all the complexes are interrelated by substitutions. For example, let \(R\) be a ternary relation with only two assignments to its positions \(p_1, p_2, p_3\) resulting in a complex, namely \(a, a, b\) and \(a, b, b\), respectively. If the resulting complexes both have only two occurrences, then the complexes cannot be connected via a substitution.

It may be questionable whether an antipositionalist would regard such a network of complexes with unconnected parts as a relation. If not, then a thin positionalist who allows coalescence of occurrences could have relations for which an antipositionalist has no counterpart.

It is also possible that the thin positional relation has no complexes. So also in this case a thin positionalist has relations for which there is no antipositional counterpart.

In all other cases, the relations of the thin positionalist do have an antipositional counterpart.

2.1.3 Identifying Occurrences

As I said in section 1.2, a thick positional relation may have distinct complexes that intuitively should be the same. In translating such a relation to thin positionalism or antipositionalism, we may want to translate such similar complexes to the same complex. If so, then the question is how to define the occurrences of objects for the reconstructed complexes. In particular, we may ask whether the occurrences can be defined in a non-arbitrary way in terms of the positions, complexes, and objects of the original or the reconstructed relation.

If in the reconstructed complexes each object occurs at most once, then occurrences may simply be defined as ordered pairs \(\langle\xi, a\rangle\), with \(\xi\) a reconstructed complex and \(a\) an object. But if we want the reconstructed relation to be coalescence-free, we have to distinguish different cases.

For coalescence-free relations without strict symmetry, we can define occurrences in a complex \(\xi\) as ordered pairs \(\langle\xi, p_1\rangle, \ldots, \langle\xi, p_n\rangle\), with \(p_1, \ldots, p_n\) the positions of the relation. This is the translation depicted above in figure 6.

For coalescence-free relations with complete strict symmetry, we can define the occurrences of an object \(a\) in a complex \(\xi\) as triples \(\langle\xi, a, 1\rangle, \ldots, \langle\xi, a, k\rangle\), where \(k\) is the number of positions to which \(a\) is assigned to obtain \(\xi\).

However, for some other strictly symmetric coalescence-free relations, we cannot define occurrences for certain complexes in a non-arbitrary way in terms of positions, complexes and objects within the context of set theory. This is, for example, the case for a quaternary cyclic relation for which the complexes may be depicted as four objects equally spaced on a circle and such that rotating them over \(90^\circ\) gives the same complex.

The proof is given in appendix B.2.

2.2 From Antipositionalism to Positionalism

The name “antipositionalism” suggests that the view is against positions, but it is certainly not against a reconstruction of this notion within the confines of its theory.

2.2.1 Reconstructing Positions

According to Fine (2000, 29) the antipositionalist can reconstruct positions as abstracts with respect to the equivalence relation co-positionality, where object \(a\) in state \(s\) is co-positional to object \(b\) in state \(t\) if \(s\) results from \(t\) by a substitution in which \(b\) goes into \(a\) (and vice versa). But this reconstruction is not satisfactory for cyclic relations, where the objects are arranged clockwise in a circle, because for such relations all objects in a state are co-positional with each other, and therefore we would get just one position (Leo 2008a, 357).

Here we will follow a different approach. Let \(R\) be an antipositional relation with an initial complex \(\xi_0\) (i.e., a complex from which any complex of the relation can be obtained by a substitution). Then we could treat the occurrences of objects in \(\xi_0\) as positions, but there are more elegant approaches; one makes use of abstraction and the other of subtraction.

Suppose that we may abstract from the nature of the objects of the occurrences. Then, by simultaneously abstracting in \(\xi_0\) from the nature of the objects of all occurrences, we get a kind of skeleton complex.⁸ What remains of the occurrences an antipositionalist may call the positions of the relation.

Instead of abstracting from the nature of the objects of the occurrences, we may perhaps also simultaneously subtract the objects from the occurrences. If so, then the result is again a skeleton complex with “empty” occurrences that can be taken as positions.

In my view the operation of abstraction and the operation of subtraction are both quite natural. It’s hard to say what is the best choice. An advantage of abstraction is that it does not necessarily commit you to the existence of additional entities. It may be seen as just a way of speaking about a class of complexes (cf. Russell 2009, 33–34).⁹ In favor of subtraction it may be argued that substitution is in fact a two-step operation, where in step one objects are subtracted and in step two objects are added. If so, then subtraction is an operation we implicitly already had.

2.2.2 From Antipositionalism to Thick Positionalism

We start with an antipositional relation \(R\) with an initial complex \(\xi_0\), and assume that the operation of abstraction or subtraction yields a skeleton complex \(\zeta\) with reconstructed positions, each corresponding with exactly one occurrence of an object in \(\xi_0\). Then there is a bijection \(\pi\) from the occurrences in \(\xi_0\) to the positions in \(\zeta\).

For an assignment \(f\) of objects to the positions, we define as resulting complex (if it exists) the complex obtained by the substitution \(\pi \cdot f\) in \(\xi_0\) together with the positions being occupied by the assigned objects.

The translation may give more complexes than in the original relation. For example, if \(R\) is the adjacency relation, then the corresponding positional relation has two positions \(p_1, p_2\), and for \(a\)’s being adjacent to \(b\) it has two complexes, one with \(p_1, p_2\) being occupied by \(a, b\), and another with \(p_1, p_2\) being occupied by \(b, a\).

2.2.3 From Antipositionalism to Thin Positionalism

For translating antipositionalism to thin positionalism, we follow the same route, except that we simply use the original complexes as the complexes for the positional relation. So we start again with an initial complex \(\xi_0\), and we assume that by abstraction or subtraction we obtain reconstructed positions and a corresponding bijection \(\pi\) from the occurrences in \(\xi_0\) to the positions. Then, for any assignment \(f\) of objects to the positions, define as resulting complex (if it exists) the complex obtained by substitution \(\pi \cdot f\) in \(\xi_0\).¹⁰

This completes the translation. To be acceptable for a thin positionalist, the reconstructed relation must satisfy the Composition Principle of Thin Positionalism.

This can be proved as follows. Let \(\xi_0, \pi\) be as in the translation, and let \(f\) be a substitution of objects for the reconstructed positions resulting in a complex \(\xi\). Then substitution \(\pi \cdot f\) in \(\xi_0\) results in \(\xi\) as well. Let \(\mu\) be a co-map of \(\pi \cdot f\). Then, by the Composition Principle of Antipositionalism, for every substitution \(s'\) in \(\xi\),

By the reconstruction of the positional relation, \(\mu \cdot s'\) is a substitution in \(\xi_0\) resulting in \(\xi'\) iff \(\pi^{-1} \cdot (\mu \cdot s')\) is a substitution for the positions resulting in \(\xi'\). So, because \(\pi^{-1} \cdot (\mu \cdot s') = (\pi^{-1} \cdot \mu) \cdot s'\),

From this fact and the observation that \(\pi^{-1} \cdot \mu\) is a surjective map from the positions to the occurrences of objects in \(\xi\) mapping each position \(p\) to an occurrence of the object substituted by \(f\) for \(p\), it follows that \(\pi^{-1} \cdot \mu\) is a co-map of \(f\). This completes the proof.

If a relation has more complexes from which all of its complexes can be obtained by substitution, then any of them could be chosen for abstracting from the nature of the objects of the occurrences. As you might expect, the reconstruction of a positional relation is essentially independent of the choice of \(\xi_0\). More specifically, the reconstructed sets of positions may perhaps be different for different choices of \(\xi_0\), but it is not difficult to show that the reconstructed relations are all the same up to isomorphism.

Nevertheless, there is a subtle complication; in set theory the positions cannot always be reconstructed “neutrally,” i.e., without an arbitrary choice in terms of the basic ingredients of antipositionalism. This will be shortly discussed at the end of this section.

A serious restriction of the given reconstructions is that it only works for relations with an initial complex. But there might be more sophisticated reconstructions that also work for certain relations without initial complexes. However, for relations with a variable number of objects in different instantiations, like the relation of forming a circle, there may not be equivalent positional relations. This might mean that antipositionalism is a richer theory that offers more possibilities than positionalism.

2.2.4 Identifying Positions

An interesting question is whether for any antipositional relation with an initial complex a reconstruction of positions can be made with no arbitrary choices.

For relations without strict symmetry a non-arbitrary reconstruction of positions is possible. We can, for example, identify a position for such a relation with the equivalence class of occurrences of objects in initial complexes that can be mapped to each other by co-maps.

For strictly symmetric relations this reconstruction does not work. For some strictly symmetric relations there is simply no reconstruction of positions possible in set theory without an arbitrary choice. This is, for example, the case for a quaternary cyclic relation for which the complexes may be depicted as four objects equally spaced on a circle and such that rotating them over \(180^\circ\) gives the same complex, but rotating them over \(90^\circ\) gives a different complex when the objects are not all the same.

The proof that for this relation no non-arbitrary reconstruction of positions is possible is given in appendix B.3.

2.3 Translations Back and Forth

That positionalism and antipositionalism are translatable into each other is nice, but it doesn’t say that much. With translations relevant information can in principle get lost. Therefore it is very interesting to investigate if translations back and forth yield a structure that is isomorphic to the original relation. If this is the case, then the translation is really good.

First we translate back and forth starting from a positional relation, and then we translate back and forth starting from an antipositional relation.

2.3.1 From Positionalism to Antipositionalism and Back Again

We have the following results:

This is easy to see. The translation to antipositionalism gives a coalescence-free network of reconstructed complexes without any strict symmetry, where the reconstructed complexes correspond one-to-one with the original complexes. By translating it back to thick positionalism we get a structure of reconstructed complexes and positions that matches the original relation, up to isomorphism.

We may prove this claim as follows. The translation to antipositionalism retains all complexes and the substitutions between them. Because the original relation has at least one coalescence-free substitution for the positions, it has an initial complex \(\xi_0\). We use \(\xi_0\) for the reconstruction of the positions. Then, for some bijection \(\tau\) from the reconstructed positions to the original positions, any \(s\) with co-map \(\mu\) is a substitution for the reconstructed positions iff \(\tau \cdot s\) with co-map \(\tau \cdot \mu\) is a substitution for the original positions. This proves the claim.

2.3.2 From Antipositionalism to Positionalism and Back Again

A minimal requirement for a translation of an antipositional relation \(R\) to positionalism and back again to result in essentially the same relation as the original one is that \(R\) has an initial complex, i.e., a complex from which any complex of the relation can be obtained by a substitution.

We prove this as follows. Assume that \(R\) is an antipositional relation with an initial complex \(\xi_0\). Furthermore assume that \(R\) is coalescence-free and without any strict symmetry. Translating \(R\) to thick positionalism gives complexes being a combination of the original complexes and positions being occupied by the assigned objects. Because \(R\) is without any strict symmetry, these reconstructed complexes correspond one-to-one with the original complexes. Because \(R\) is coalescence-free, translating back to antipositionalism gives a network of complexes in which the complexes have occurrences that correspond one-to-one to the occurrences in the complexes of \(R\). From this it follows that the reconstructed relation is the same as the original relation, up to isomorphism.

The “only if” part of the claim follows because the translation of a thick positional relation to antipositionalism always gives a coalescence-free relation without any strict symmetry. This completes the proof.

The proof is straightforward. By translating from antipositionalism to thin positionalism, the original complexes and the substitutions between them are fully retained. Translating back to antipositionalism gives as a result again the original relation.

3 Conclusion

In this paper we compared positionalism and antipositionalism. The main conclusion is that, contrary to what the names suggest, the views are not really opposites of each other. In fact, a specific form of positionalism, which I called thin positionalism, is very similar to antipositionalism.

In thin positionalism as well as in antipositionalism substitution is taken as a primitive operation. In thin positionalism we have substitution of objects for positions of a relation, and in antipositionalism we have substitution of objects for occurrences of objects in relational complexes.¹¹ Substitution is in both cases used to characterize the structure of a relation.

As we have seen, the translations back and forth show that there is a very close relationship between thin positionalism and antipositionalism. The class of thin positional relations with at least one coalescence-free assignment of objects to its positions matches perfectly with the class of antipositional relations with at least one initial complex; they are translatable into each other without any loss of information.

In summary, the relationship between thin positionalism and antipositionalism may be expressed as follows:

1. both views rely upon the notion of substitution, which I regard as a fundamental operation for expressing relatedness between complexes;

2. the main difference between the views is that in positionalism the relatedness between complexes is expressed via positions and in antipositionalism it is expressed directly between complexes;

3. the views are for a significant range of relations translatable into each other in a natural way with complete preservation of structure.

What about the standard view? Relations of the same significant range could also be translated from the standard view back and forth to positionalism and antipositionalism. However, in this case the end result is not necessarily isomorphic with the original relation. The reason is that in translating from the standard view to positionalism or antipositionalism some constitutive information—namely the order of the arguments—is lost. This puts the standard view apart from positionalism and antipositionalism.

It may go too far to say that thin positionalism and antipositionalism are essentially the same. In thin positionalism, a relation is seen as a universal and positions belong to the fundamental furniture of the world, whereas in antipositionalism no ontological commitment to relations as universals and to positions is needed.¹²

Because antipositionalism is apparently less demanding with respect to ontological commitments, I am inclined to regard it as the preferable view. Furthermore, a strong feature of antipositionalism is that it may accept relations with a variable number of objects involved in the complexes, as in the relation of forming a group and forming a circle, for which the positionalist may have no equivalent counterpart.

But there may perhaps be reason for not jumping to the conclusion that antipositionalism is in every way superior, because a positionalist may accept relations with no complexes and relations for which the translation to antipositionalism yields an unconnected network of complexes. Such relations may be unacceptable for an antipositionalist.

Despite the differences, I consider the agreement between positionalism and antipositionalism as fundamental. The analysis given in this paper shows that the views are essentially two sides of the same coin. Therefore I regard the name “antipositionalism” as misleading. A better name might be “apositionalism.”

4 Substitution Principles

In (1989, 235–238), Fine made a start for developing a general theory of constituent structure. The key notion of the theory is the operation of substitution. A substitution takes an entity \(\xi\) and a map from the occurrences of entities in \(\xi\) to entities as input, and gives an entity as a result (if any).

Fine gave the following example of a basic principle for the theory:

If \(F'\) is the result of substituting \(E'\) for the occurrence \(e\) of \(E\) within \(F\), then there is an occurrence \(e'\) of \(E'\) within \(F'\) such that the result of substituting any expression \(E''\) for \(e'\) within \(F'\) is identical to the result of substituting \(E''\) directly for \(e\) in \(F\). (1989, 236)

The notions of substitution and occurrence are taken as primitive.

Because we may simultaneously substitute entities for occurrences, I propose the following more general principle.

We call a map \(\mu\) with this property a co-map of substitution \(s\).

5 Neutral Reconstructions

In this appendix we show two things: (1) for some relations, occurrences of objects cannot be reconstructed set theoretically in a non-arbitrary way in terms of basic notions of thick positionalism, and (2) for some relations, positions cannot be reconstructed set theoretically in a non-arbitrary way in terms of basic notions of antipositionalism.

We will not give a precise definition of non-arbitrariness, but we will give a formal definition of neutrality that obviously any non-arbitrary construction in set theory should fulfill. This notion of neutrality, which was introduced in Leo (2008b), is interesting in its own right since it may be more generally applicable for showing that certain things cannot be modeled in set theory in a non-arbitrary way.

All reconstructions in this appendix are understood to be within the context of standard set theory with urelements. Other modeling media may provide more possibilities.

5.1 The Notion of Neutrality

I will define neutrality in the context of set theory with urelements \(A\). The idea is as follows. Let \(X\) and \(Y\) be sets. Suppose that \(Y\) is constructed in a non-arbitrary way on the basis of \(X\). Let \(\pi\) be a permutation of the urelements for which replacing in \(X\) each occurrence of each urelement \(a\) by \(\pi(a)\) doesn’t change the set. Then—since all urelements are set-theoretically indiscernible—replacing in \(Y\) each occurrence of each urelement \(a\) by \(\pi(a)\) doesn’t change this set either.

If \(Y\) has the property that each permutation of the urelements that keeps \(X\) unchanged also keeps \(Y\) unchanged, then we say that \(Y\) is neutral with respect to \(X\).

The notion of neutrality may in principle be used to show that certain things cannot be constructed in a neutral way with respect to other things, and we will do that in the next sections, but first we give a formal definition of neutrality.

Let \(\mathsf{V}[A]\) be the cumulative hierarchy with urelements \(A\). Any function \(u: A \rightarrow A\) can be lifted to a function \(\widetilde{u}: \mathsf{V}[A] \rightarrow \mathsf{V}[A]\) in an obvious way:

We may regard \(\widetilde{u}(X)\) as the result of replacing in \(X\) each occurrence of each urelement \(a\) by \(u(a)\).

So if \(A = \{a, b\}\), then any set in \(\mathsf{V}[A]\) is neutral with respect to \(\{ a \}\), but \(\{ a \}\) is not neutral with respect to \(\{a, b\}\).

I do not claim that the definition of neutrality completely characterises non-arbitrariness of a set-theoretic construction, but it should be clear that any non-arbitrary construction of \(Y\) on the basis of \(X\) will be neutral with respect to \(X\).

5.2 Reconstructing Occurrences

We will show that not for every positional relation the occurrences of objects in the complexes can be neutrally reconstructed in terms of positions, complexes, states, and objects.

Let \(R\) be a positional relation for which the states may be depicted as four not necessarily distinct objects equally spaced on a circle and such that rotating them over \(90^\circ\) gives the same state.

A set-theoretical positional model for \(R\) is a tuple \(\mathcal{M} = \langle C, S, O, P, \Gamma, \Omega\rangle\), with complexes \(C\), states \(S\), objects \(O\), positions \(P\), a map \(\Gamma\) from \(O^P\) to \(C\), and a map \(\Omega\) from \(C\) to \(S\), where \(\Gamma\) maps assignments of objects to positions to complexes, and \(\Omega\) maps complexes to their corresponding states.¹³

We assume that \(C\), \(S\), \(O\), and \(P\) are mutually disjoint sets of urelements, and that \(O\) has at least four objects.

The symmetry of \(R\) can be expressed in terms of the model \(\mathcal{M}\) as follows.

The set \(P\) can be written as \(\{p_1, p_2, p_3, p_4\}\) such that for the permutation group \(G\) generated by the map taking \(p_1, p_2, p_3, p_4\) to \(p_4, p_1, p_2, p_3\), we have for every \(f, g \in O^P\), \(\Omega(\Gamma(f)) = \Omega(\Gamma(g))\) iff \(g = f \circ \pi\) for some \(\pi \in G\).

Let us now try to reconstruct a coalescence-free thin positional or antipositional model for \(R\) with the same states as in \(R\) and for each state just one corresponding reconstructed complex.

For every reconstructed complex \(\xi\) with four distinct objects \(a, b, c, d\) we may define its occurrences non-arbitrarily as pairs \(\langle\xi, a\rangle, \langle\xi, b\rangle,\langle\xi, c\rangle,\langle\xi, d\rangle\), but, if each complex has four occurrences, then no neutral reconstruction of all occurrences is possible with respect to \(\mathcal{M}\). This can be shown as follows.

Select two objects \(a\) and \(b\). Let \(\delta: O \rightarrow O\) switch the objects \(a\) and \(b\) and leave all other objects unchanged. Define \(u: C \cup S \cup O \cup P \rightarrow C \cup S \cup O \cup P\) by:

\[u(x) = \begin{cases} \delta(x) & \text{if } x \in O \text{,}\\ \Gamma (\delta \circ f) & \text{if } x = \Gamma(f) \text{ for some } f \in O^P \text{,}\\ \Omega( \Gamma (\delta \circ f)) & \text{if } x = \Omega(\Gamma(f)) \text{ for some } f \in O^P \text{,}\\ x & \text{otherwise.} \end{cases}\]

It is not difficult to see that \(u\) is a bijection, \(u \circ u = \mathsf{id}_{C \cup S \cup O \cup P}\), and \(\widetilde{u}(\mathcal{M}) = \mathcal{M}\).

Let \(Rabab\) be the state with objects \(a, b, a, b\) arranged in a circle (in that very order) and let \(\alpha_1, \beta_1, \alpha_2, \beta_2\) be the occurrences of \(a, b, a, b\) in the corresponding reconstructed complex \(\xi_{abab}\).

Now suppose that the occurrences are neutrally reconstructed with respect to \(\mathcal{M}\). More specifically, suppose \(R\) has a coalescence-free thin positional or antipositional reconstruction \(\mathcal{N}\) in \(V[C \cup S \cup O \cup P]\) such that \(\widetilde{u}(\mathcal{N}) = \mathcal{N}\).

Then, because \(u(Rabab) = Rabab\) and \(u\) switches \(a\) and \(b\), \(\widetilde{u}(\alpha_1)\) must be an occurrence of \(b\) in the reconstructed complex \(\xi_{abab}\). So, either \(\widetilde{u}(\alpha_1) = \beta_1\) or \(\widetilde{u}(\alpha_1) = \beta_2\).

Let \(c, d, e, f\) be distinct objects in \(O\) and let \(\xi_\mathit{cdef}\) be the complex obtained by substituting \(c, d, e, f\) for \(\alpha_1, \beta_1, \alpha_2, \beta_2\) in \(\xi_{abab}\). From \(u(Rabab) = Rabab\) and \(\widetilde{u}(\mathcal{N}) = \mathcal{N}\) it follows that substituting \(c, d, e, f\) for \(\widetilde{u}(\alpha_1) , \widetilde{u}(\beta_1), \widetilde{u}(\alpha_2), \widetilde{u}(\beta_2)\) in \(\xi_{abab}\) results in \(\xi_\mathit{cdef}\) as well.

From this it follows that \(\widetilde{u}\) must preserve the relative order of the occurrences in \(\xi_{abab}\). This means that either \[\widetilde{u} \text{ \;maps\; } \alpha_1, \beta_1, \alpha_2, \beta_2 \text{ \;to\; } \beta_1, \alpha_2, \beta_2, \alpha_1\] or \[\widetilde{u} \text{ \;maps\; } \alpha_1, \beta_1, \alpha_2, \beta_2 \text{ \;to\; } \beta_2, \alpha_1, \beta_1, \alpha_2.\]

In both cases \(\widetilde{u}(\widetilde{u}(\alpha_1)) \ne \alpha_1\). But, since \(\widetilde{u} \circ \widetilde{u} = \widetilde{u \circ u}\),¹⁴ this contradicts that \(u \circ u = \mathsf{id}_{C \cup S \cup O \cup P}\).

So we conclude that if each state has just one reconstructed complex and each complex has four occurrences, then the occurrences cannot be neutrally reconstructed with respect to \(\mathcal{M}\).

5.3 Reconstructing Positions

In a similar way as we did for occurrences, we can prove that not for every relation positions can be neutrally reconstructed in terms of the notions of antipositionalism.

I will show this again for a cyclic relation, but not for the same one. For an antipositional relation that holds of objects \(a, b, c, d\) when \(a, b, c, d\) are arranged in a circle (in that very order) it is possible to reconstruct the positions in a non-arbitrary way. I leave this as an exercise.¹⁵

Let \(R\) be an antipositional relation for which the states may be depicted as four distinct objects equally spaced on a circle and such that rotating them over \(180^\circ\) gives the same state, but rotating them over \(90^\circ\) does not give the same state.

We assume that each state of \(R\) has just one corresponding complex.

The symmetry of \(R\) can be expressed as follows.

For every complex \(\xi\) the occurrences of objects can be written as \(\{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}\) such that for the permutation group \(G\) generated by the map taking \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\) to \(\alpha_3, \alpha_4, \alpha_1, \alpha_2\), we have for every substitution \(s, t\) in \(\xi\), if \(s\) results in a complex, then \(\xi \cdot s = \xi\cdot t\) iff \(s = \pi \cdot t\) for some \(\pi \in G\).

In Corollary 7.8 of Leo (2008b) it is shown that for this relation positions cannot be neutrally reconstructed in terms of the notions of antipositionalism if substitution is directly done on objects. Here we show that it is also impossible when substitution is done on occurrences.

A set-theoretical antipositional model for \(R\) is a tuple \(\mathcal{M} = \langle C, S, O, \mathsf{Oc}, \Pi, \Theta, \Omega\rangle\), with complexes \(C\), states \(S\), objects \(O\), occurrences \(\mathsf{Oc}\), a map \(\Pi\) from \(\mathsf{Oc}\) to \(O\), a partial map \(\Theta\) from \(C \times O^\mathsf{Oc}\) to \(C\), and a map \(\Omega\) from \(C\) to \(S\), where \(\Pi\) maps occurrences to their objects, \(\Theta\) represents the substitutions in complexes of objects for occurrences, and \(\Omega\) maps complexes to their corresponding states.

We assume that \(C\), \(S\), \(O\), and \(\mathsf{Oc}\) are mutually disjoint sets of urelements, and that each occurrence occurs in only one complex. Furthermore, we assume that \(O\) has at least four objects, and that \(R\) holds for any selection of four distinct objects in \(O\) in any order, but not for any other selection.

We call two states siblings if each can be obtained from the other by rotating the objects over \(90^\circ\). Furthermore, we call two complexes siblings if their corresponding states are siblings, and we call two occurrences siblings if they are occurrences of the same object in complexes that are siblings. Note that by our assumptions each state, each complex, and each occurrence has exactly one sibling.

Define \(u : C \cup S \cup O \cup \mathsf{Oc} \rightarrow C \cup S \cup O \cup \mathsf{Oc}\) by:

\[u(x) = \begin{cases} \text{sibling of } x & \text{if } x \in S \text{,}\\ \text{sibling of } x & \text{if } x \in C \text{,}\\ \text{sibling of } x & \text{if } x \in \mathsf{Oc} \text{,}\\ x & \text{otherwise.} \end{cases}\]

It is not difficult to see that \(u\) is a bijection, \(u \circ u = \mathsf{id}_{C \cup S \cup O \cup P}\), and \(\widetilde{u}(\mathcal{M}) = \mathcal{M}\).

Let \(P = \{ p_1, p_2, p_3, p_4 \}\) be reconstructed positions for \(R\) and let \(\Psi\) be a partial map from \(O^P\) to \(S\) that maps assignments of objects to positions to corresponding states. We may assume that the assignment of distinct objects \(a, b, c, d\) to \(p_1, p_2, p_3, p_4\) results in the same state as the assignment of \(c, d, a, b\) to \(p_1, p_2, p_3, p_4\). We denote this state as \(Rabcd\).

Now suppose that \(P\) and \(\Psi\) are neutral with respect to \(\mathcal{M}\).

Let \(f\) be an assignment of \(a, b, c, d\) to \(p_1, p_2, p_3, p_4\). Then \(\widetilde{u}(f)\) is the assignment of \(a, b, c, d\) to \(\widetilde{u}(p_1), \widetilde{u}(p_2), \widetilde{u}(p_3), \widetilde{u}(p_4)\), and \[\begin{aligned} \Psi(\widetilde{u}(f)) & = (\widetilde{u}(\Psi))(\widetilde{u}(f)) && \text{because } \widetilde{u}(\Psi) = \Psi\\ & = \widetilde{u}(\Psi(f)) && \text{because if } g \colon x \mapsto y \text{, then } \widetilde{u}(g) \colon \widetilde{u}(x) \mapsto \widetilde{u}(y)\\ & = \text{sibling of } \Psi(f) && \text{by the definition of } u\\ & = Rdabc.\end{aligned}\]

From this it follows that \(\widetilde{u}\) preserves the relative order of the positions. This means that either \[\widetilde{u} \text{ \;maps\; } p_1, p_2, p_3, p_4 \text{ \;to\; } p_2, p_3, p_4, p_1\] or \[\widetilde{u} \text{ \;maps\; } p_1, p_2, p_3, p_4 \text{ \;to\; } p_4, p_1, p_2, p_3.\]

In both cases \(\widetilde{u}(\widetilde{u}(p_1)) \ne p_1\). But, since \(\widetilde{u} \circ \widetilde{u} = \widetilde{u \circ u}\), this contradicts that \(u \circ u = \mathsf{id}_{C \cup S \cup O \cup \mathsf{Oc}}\).

So we conclude that positions for \(R\) cannot be neutrally reconstructed with respect to \(\mathcal{M}\).

References