CHAPTER VII

The Myth of Relations

§1. Necessity and Relations. Belief in relations has had the authority of so many distinguished minds of the twentieth century behind it that it has by now become dogma. If an argument is needed to support belief in relations, one feels secure in merely repeating old points.

Great weight, for example, is still attached to the fact that Aristotle did not develop a logic of relations. His ontological emphasis on things and their (monadic) properties is thought to have rendered his logic incomplete. Even the validity of so simple an argument as “All doves are birds; so all wings of doves are wings of birds” cannot be proved in his logic. Yet it is far from evident that this logical deficiency resulted from an insufficient ontology. In fact, the validity of the argument can be proved in a logic appropriate to an ontology without relations. One gets such a logic by strengthening monadic predicate logic with a scheme for eliminating relational predicates. Part of my task, then, will be to develop such a scheme.

Moreover, great weight is still given to the seeming impossibility of accounting for facts of serial order solely on the basis of the properties of the entities ordered. For, without relations, a series seems to reduce to a mere collection. But perhaps it is possible to have relatedness─that is, a condition an entity has of being related to another─without relations, and, in particular, to have seriality without relations. I shall attempt to show that this is indeed the case. Relations, thought to be so secure, are supported only by a handful of arguments that falter under scrutiny.

The case against relations is as strong as the one for them is weak. So far, our reason for excluding relations from any ontology that includes the required ontology has been the following. Natures require an ontology of components rather than an ontology of simples. Yet a relation cannot be a component of its relata, where its relata are distinct entities. For a relation is unitary, and if it were a component, it would be a component of each of its relata. If it were a component of only some of them, then not all of them would be brought into the relationship. So, as a component of each, it has to be multiple in contradiction to its being only one.¹ But relations and properties should, it would seem, be given the same ontological status. If properties depend on individuals for their sameness and are thus components of individuals, then relations would seem to be similarly dependent and thus have the status of components. Thus there are no relations, since they cannot be components, and since components are the only things they could be.

It is the analogy between relations and properties that is the weak step in this argument. Can we not recognize natures and properties as components and make an exception for relations? After all, the argument of Chapter III for an ontology of components establishes only that natures must be components. Other entities are given the status of components, not out of any necessity but for the general coherence of the view. So a stronger argument is now needed against treating relations as distinct entities.

The image suggested by the doctrine that relations are distifict from their relata is that of a network of wires. Individuals are like rings to which segments of wire have been anchored. Since individuals stand in many relations to other things, rings anchor numerous wires linking them to other rings. Even nominalistic thinkers who do not admit that properties are different or distinct from what has them succumb to the network model of relations. Hume denies that a cube’s blackness is different from the cube itself, but the idea of time is for him derived from a “succession of objects.” This succession must be different from, and thus for Hume distinct from, the distinct objects in the succession, for otherwise there would be as many successions as there are objects succeeding one another, which is absurd.²

The idea of the universe as a collection of entities kept from being worlds apart by a constraining network has, then, become an unquestioned part of many different strands of philosophical thought. But inevitably the network model discourages reflection on the question of why entities have the relationships they do have. As distinct entities, relations are simply there between their relata; the relata provide no basis for selecting which relations will be there. You can link any two rings by any wire, and any wire between two rings can be replaced with another. That this is inevitable on the network model must now be shown.

This calls for showing that relations that are distinct from their relata are external. A relation is “external” if possession of it is not essential for any one of its relata. Traditionally, a relation was called external, in a second sense, when it was not a component of any individual.³ Correspondingly, a relation is internal in this second sense when it either is a component of its relata or is a component of an encompassing individual of which its relata are also components.

Now it was thought that these two senses of externality had the same implications. The reason for this was the mistaken Humean belief that what is not distinct is inseparable and what is inseparable is not distinct. Thus, when relations are components of their relata and hence not distinct from them, their relata are inseparable from these relations, and hence have them essentially. Con־ versely, when relations are essential to their relata, they are not distinct from their relata and are hence components of them. Moreover, when a relation is a component of a whole encompassing its relata, neither the relation nor the relata are distinct from the whole; therefore, they are not distinct from one another, and are thus inseparable. Conversely, if they are inseparable, the relation is not distinct from the relata, which are then not distinct from one another, and thus the relation and the relata are components of an encompassing whole.

Since I reject the Humean premisses here, I must insist on keeping the two senses of externality apart. In fact, I shall henceforth take ‘externality’ only in the first sense, the one implying contingency. If it can be shown that relations are external when they are entities distinct from their relata, then it turns out that certain necessary connections needed for the practice of action on prior experience are denied. But, first, to show they are external.

If relations are distinct from their relata, what is the nature of their relation to their relata? There seem to be only two answers. Either there is a distinct entity that must be inserted between the relation and each of its relata, or there is, instead, either a component entity or no entity at all. Each alternative leads to the conelusion that if relations are distinct entities, then nothing has a relation to anything else.

The first alternative raises a difficulty made familiar by Bradley.⁴ To posit, in addition to the relata a and b and the relation R, a third entity to relate R to a is like requiring that the wires in a network of wires and rings be tied to rings with additional wires. The third entity between a and R will be an entity signified by the word ‘has’ when we say that a has R to b. Let us call it the having entity, or simply having. There is now the question as to how a and having are related. How is the wire that ties the main wire to the ring itself tied to the ring?

Since a must have having to R, the above logic calls for a second having entity. There is then a regress of havings of having, since a further having entity is called for at each stage. However far one traces the regress, one gets no closer to satisfying the conditions for the truth of Rab. As is customary, this regress is taken here as a reductio ad absurdum of the view that relations need a distinct entity, having, in order to be related to their relata. But is there a viable alternative?

As for the second alternative, the truth condition for Rab ineludes no more than the three distinct entities, R,a,b. Here it is as though the main wires were simply twisted around the rings they connect in the network. There are no wires between the main wires and the rings. One way of describing this is to say that relations relate simply of themselves. One might then suppose that the ontological gulf between relations and relata has been successfully bridged without any entity at all. But has it been bridged? If a relation relates, then something needs to be said about how this comes about. Otherwise, one is not sure that is a viable alternative to the view Bradley expressed. There are several possibilities for describing how relations relate. The relating that a relation does might be a further distinct entity. It might, however, be a component of the relation. Or, finally, it might be that the relation is just the relating it does and nothing more.

If the relating is a distinct entity, then we are back to the case Bradley dealt with. There will be a relating of relating to the relation, and this pattern will go on endlessly. Suppose, however, the relating that the relation does is a component of the relation. Can a regress then be avoided? Relating, the component, would insure that the entity with it, the relation, did relate. But relating itself would not relate, as the relation does. A regress would not then get started by there being a need for a second relating as a component of the first relating. However, a regress does get started in a quite different manner. Relating, as something the relation does, is like an action of the relation. The relationship of a to b is then a result of the relating R does. The relation result of must in turn relate the relationship to the relating. And so the regress starts. The relationship of the relationship of a to b to the relating R does is a result of the relating result of does. So the quasi-agent view of relations, which attributes relating to them as a component, leads to a regress of relating result-of relations.

We are down, then, to the alternative of collapsing the difference between the relation and relating. This involves a change in perspective whereby a relation is no longer viewed as a quasi-agent that relates individuals to one another but is viewed as a quasiprocess in which they come together. Now it is odd to suppose that the quasi-process of relating in which a and b are involved is an entity distinct from them. Yet on the supposition that relations are distinct entities, we are forced to say this. Our original problem was to find out how three distinct entities could form a unitary condition of relatedness. And replacing the relation with relating does not begin to solve this problem. For how can a relating that is distinct from both a and b become a relating of a to b? This is the same as the problem of how a relation that is distinct from a and b becomes a relation of a to b. To solve this problem, there is no choice but to revert to our already rejected alternatives. That is, relating, to which the relation is now reduced, will itself have to have relating as a component or as an entity distinct from it in order to bridge the gap. But this just leads back to the preceding difficulties.

In short, if relations are entities distinct from their relata, then nothing has a relation to anything else. But if nothing has a relation and if relatedness requires relations, then nothing is related to anything else. But clearly entities are related. So either relations are not distinct entities or relatedness does not require relations. If relations are not distinct, they are components. Yet Leibniz's argument makes it impossible to view them as components. So relatedness must exist without relations. However, the reason for positing relations is to account for entities being related. So since relatedness is incompatible with relations, there are no relations.

It is worth noting that the argument of the last few pages can be varied to show that if properties are distinct from the entities they qualify, nothing can have a property. Suppose the assumption of a distinct entity relating an individual to a property leads to absurdity. One is then led to ask how a property can by itself qualify an individual. It can if it has qualifying as a component. But qualifying is an action, just as in the above argument relating was an action. We are thus faced with a regress of result-of relations. Likewise, to say that the property just is qualifying is to raise the original question again, for one still wants to know how qualifying and the individual are joined, despite their being two distinct entities, into a unitary condition. So if properties and relations are distinct from individuals, nothing is qualified or related. To avoid this conclusion, I say that properties are components. But since relations cannot be components, we are still left with the result that, if relatedness is based on relations, nothing is related.

It is now time to observe two facts about necessity. First, the network model, which treats relations as distinct entities and yet as the only possible basis of relatedness, has the consequence that nothing is related to anything else. It certainly follows then that, if relations are still allowed to exist at all, they are external to individuals. For there is no relation an individual must have to something else, since there is no relation at all that it has to anything. And since relations are assumed to be needed for relatedness, yet are not had by any individuals, there is no way an individual must be related to some other individual, since there is no way it is related.

Second, the practice of acting on the basis of prior experience requires that there be necessities. Certain individuals will have certain properties of necessity, and they will thus be of necessity in the conditions of having these properties. But if there are relations, condition of is a relation. So at least this relation could not be external. For if no condition was a condition of any individual of necessity, all properties would be contingent. (Recall that conditions are distinct from their individuals.)

But we have just shown that, since relations must be distinct entities, entities are not related by relations and hence are not necessarily related by relations. If relations are the only bases for relatedness, nothing can be necessarily related to anything, and hence, in view of condition of, there are no necessities of any kind. An ontology of relations and the practice of acting on prior experience are incoherent.

This conclusion was clearly stated by Whitehead when he argued that unless “the relationships of an event are . . . constitutive of what the event is in itself” induction fails “to find any justification within nature itself.”⁵ By a constitutive relationship he meant not a relation, but what in the next section I shall call a relational property. Since if there were relations, there would be no basis for relatedness other than relations, inductive practice is reasonable relative only to an ontology without relations. Strange as it sounds, a condition for there being entities that are necessarily related is that there be no relations. To make sense of this, we must now provide an interpretation of being related that does not appeal to relations.

§2. Relational Properties and Their Foundations. In my interpretation of how things are related, two kinds of entity will play a crucial role: relational properties and their foundations. If six is greater than five, then six has the relational property greater than fiveג which is to be distinguished from the relation greater than. Relational properties, unlike relations, can be components. But, as components, they are not composites of relations and their relata.⁶ For if relations were parts of relational properties, then relations would be components of individuals, which they cannot be in view of Liebniz’s argument. Moreover, since five is not a part of greater than five, one cannot say there is an x such that six has the relational property greater than x. Positions in relational property predicates are not open to quantification.

To each relational property there correspond several foundations.⁷ To six’s relational property greater than five there correspond the magnitudes of six and five. One foundation alone is insufficient for a relational property. It is six having its magnitude and five having its magnitude that is the basis for six having such a relational property. The same pair of foundations in the relata will also be the basis for a relational property in five, the property smaller than six.

Relational properties are correlated, though foundations need not be. That is, when Smith is taller than Jones, Smith’s having the relational property taller than Jones will coimply Jones’s having the relational property shorter than Smith. But where their respective heights are the foundations, Smith’s having the height he has will not imply, or be implied by, Jones’s having the height he has.

Still, taken together, the foundations coimply the relational properties. If Smith has one height and Jones has a dissimilar height, one of them will be taller than the other. The implication in the opposite direction seems at first sight more controversial. One wants to say that being taller than Jones is associated with many heights, and thus Smith’s having this property cannot imply a particular height for Smith and one for Jones. But this is to overlook the difference between relational properties and relations. Indeed, Smith’s being a relatum of the relation taller than does not imply a specific height for Smith and one for Jones. But here the property of Smith, taller than Jones, is in question. And observe that this property is correlated with shorter than Smith, and is thus not similar to Hayes’ property taller than Jones, since this is correlated with the clearly dissimilar property shorter than Hayes. Of course, it would be appropriate to call them similar if the relation shorter than were a component of both the correlated properties, shorter than Smith and shorter than Hayes. But there are no relations. So Smith’s property taller than Jones is quite specific enough to imply specific heights for both Smith and Jones.

On the one hand, it will not do to interpret relational claims just in terms of correlated properties. The reason for this is that the correlated properties involved must be of a special sort. They must be correlated because they have the same foundations. Correlated properties at the basis of a relationship─that is, properties that by their presence make a relational claim true─rest in their turn on foundational properties. The correlated properties might be correlated simply because the entity with one of them is in causal interaction with the entity with the other. And in this case no foundations would be required for the two properties. Neighboring warm bodies affect each other’s temperature, and for this reason their temperatures are correlated. But it does not at all follow that their temperatures are relational properties.

On the other hand, it will not do to interpret relational claims merely in terms of pairs of properties that happen to be foundations. For it needs to be explicitly mentioned that the properties in these pairs are indeed the basis for correlated properties. Relatedness does not consist in the foundations just by themselves, but in the correlation of the relational properties based on the foundations. To interpret relational claims without making this explicit would be to give the misleading impression that the correlation of relational properties is an accidental by-product of relatedness. Nonetheless, relational properties are ontologically dependent on foundations and thus derivative ontologically from foundations. The asymmetry required for this dependence comes from the fact that, in general, one can have one foundation without the other and hence without either relational property, but never one relational property without one of the foundations.

We can, then, leave out neither the correlation of relational properties nor the basis for relational properties in foundations. Thus where Rab is a singular relational claim, the following equivalence will be true:

(1) Rab ↔ (fa • (fa ↔ gb) • (fa ↔ (Fa • Gb))).

Here ƒ and g are properties correlated between the relata a and b. And F and G are the foundations, in the distinct relata, of these correlated properties. Having foundations, the correlated properties are relational properties. Of course, one need not know which relational properties and which foundations will do the trick for the specific case of R holding from a to b. Nonetheless, when they are found, one knows what it is that makes a relational claim true. (In §3, which follows, I shall make clear that relevant coimplication provides a natural interpreation of at least the ↔’s on the right-hand side of (1).)

Though (1) indicates how singular relational propositions can be true without relations, it is not adequate as a pattern for eliminating relational formulas in quantified propositions. I shall try to indicate why this is so. Consider the quantified proposition that every number is less than some other number. Working on the basis of (1), it might be thought that this is equivalent to the proposition that (x)(Ǝy)(fx • (fx ↔ gy) • (fx ↔ (Fx • Gy))). But this would be a mistake. For ƒ here is some fixed relational property such as less than ten, and as such it is correlated with─that is, it coimplies─another fixed relational property, say greater than eight.

Thus only the number eight has this particular relational property less than ten. The relational property less than ten that the number nine has is not similar to the one eight has since it is correlated with the dissimilar property greater than nine. Moreover, the foundation F here, is some fixed foundation, say the magnitude of eight, and clearly not every number has this magnitude. In short, though it is true that every number is less than some other, our monadic counterpart is plainly false.

This can be taken to indicate that quantified relational claims are incompletely expressed. To say that every number is less than some other number is, in fact, to say that every number is less than some other for some pair of relational properties associated with less than and for some pair of foundations associated with less than. To make this clear, I must clarify what it is for an entity to be “associated with less than.” The four features f,g,F,G are associated with less than when the following two conditions are satisfied. For some pair of individuals, say, a and b, (i) (fa • (fa ↔gb) • (fa ↔(Fa • Gb))) and (ii) (a is less than b ↔(fa • (fa ↔ gb) • (fa ↔ (Fa • Gb)))). I shall symbolize the fact that f,g,F,G are associated with R by ‘f,g,F,G ε R*’.

I am saying, then, that (Qx)(Qy)Rxy incompletely expresses what is completely expressed what is completely exoressed as (Qx) (Qy) (Ǝφ,ψ,θ,ζ є R*) Rxy. Here (Qx) is either the universal or the particular quantifier. Now the quantification over φ,ψ,θ,ζ is intended to capture variables concealed in Rxy. This makes it reasonable to see Rxy in this context as having the structure (φx • (φx↔ψy) •(φx↔(θx • ζy))). Thus there is the equivalence:

(2) (Qx)(Qy)(Ǝφ,ψ,θ,ζ,єR*)Rxy↔(Qx)(Qy)(Ǝφ,ψ,θ,ζ є R*)(φx • (φx↔ψy) • (φx↔(θx • ζy))).

Since R* has been defined on the basis of the equivalence for singular relational claims, its presence does not imply that one must still contend with relations. The view of relational claims embodied in (1) and (2) is easily extended to ternary and other higher cases. Where there are three relata there will be three correlated relational properties and also three foundations that together are sufficient for each of the relational properties.

Philosophers who spoke of foundations for relational properties emphasized the role of quantity and action as foundations.⁸ “Since relation [i.e. relational property] has the weakest existence, because it consists only in being related to another, it is necessary for a relation to be grounded upon some other accident. . . . Now relation is primarily founded upon two things which have an ordination to another, namely quantity and action. For quantity can be a measure of something external, and an agent pours out its action upon another.”⁹ Having already said something about a relational property based on quantity, let us turn to one based on action.

The relational property parent of Otto belongs to Hugo since Hugo has as one of his components a mating action and since one of Otto’s components is his existence, which comes about as a result of the mating action.¹⁰ The foundational components of Hugo’s relational property are his act of mating and Otto’s existence. Now it will be objected that the fact that Hugo mated and Otto exists does not logically imply that Otto was the offspring. Only if we add that Otto’s existence came about as a result of the mating can this logically follow.

But it was not claimed that the implication from foundations to relational properties had to be of this logical character. All I have claimed is that the presence of certain components, by being the ontological base for relational properties, implies the presence of certain relational properties. In the special case of Hugo, his mating and Otto’s existence are physically, though not logically, sufficient for his being a parent of Otto. If we had to rely on the existence of the relation comes to be as a result of between Otto’s existence and Hugo’s mating for it to be the case that Hugo did beget Otto, then Hugo would not only have to mate but he would also have to act to bring about this relation in order for him to acquire the relational property. To avoid the regress implied here, the action of mating and the component of existence (it can be a component without being a property) must be the foundations for the relational property parent of Otto.

It is important to keep in mind that relational properties in respect to distinct individuals─say, parent of Otto as compared with parent of Felix─are not similar. Hugo’s mating and Otto’s existence imply Hugo’s being the parent of Otto. This does not mean that Hugo’s mating and Felix’s existence lead to Hugo’s being the parent of Felix, for clearly Barnabas may be the male parent of Felix. Given the first implication, the second might well hold if the relation parent of were a component signified by a part of each of the consequents, which it is not. Since, however, the relational properties parent of Otto and parent of Felix are dissimilar, the second implication need not hold. Our principles do not lead to the absurdity that Hugo is everyone’s parent.

Conversely, even though Hugo’s mating and Otto’s existence lead to Hugo’s being a parent of Otto, it does not follow that Barnabas’s mating and Otto’s existence lead to Barnabas' being a parent of Otto. For, despite linguistic similarity, parent of Otto as had by Hugo is correlative with child of Hugo and would thus not be similar to parent of Otto as had by Barnabas, if contrary to fact Otto could have two male parents. For as had by Barnabas parent of Otto would be correlative with the clearly dissimilar property child of Barnabas. So our principles do not lead to the absurdity that Otto is the child of everyone who mates.

§3. Asymmetry Without Relations. Apart from objections to the details of my specific program to eliminate relations from relatedness, there are some familiar objections to the general program of doing without relations. Perhaps the best known is Russell’s objection that asymmetry cannot be accounted for if there are properties but no relations, that is, if what he calls monadism is true.¹¹

Russell’s objection contains a misconception about relational properties. His objection can be formulated as follows. Suppose R is asymmetrical, so that if Rab then ~Rba. Let (Rb) and (Ra) be the relational properties R to b and R to a. In view of the asymmetry, if (Rb)a then ~(Ra)b. Since R is irreflexive if it is asymmetrical, it will not be the case that (Ra)a. So (Ra) and (Rb) are not similar properties, for (Ra)a cannot be true though (Rb)a may be true. Since asymmetry involves both relational properties, one cannot account for asymmetry without accounting for their not being similar. But it would seem that only by saying that (Rb) is composed of b and that (Ra) is composed of a can one explain how the two properties are dissimilar. If this is the root of their dissimiliarity, then how can one escape the existence of relations? For if (Rb) is composed of b it is also composed of something that relates a to b. This something must be R itself. Likewise, (Ra) must be composed of R as well as of a.

In the monadistic view of §2, however, there is no need to account for the dissimilarity of the above relational properties by appeal to their composition. Of course, if one supposes they are composite, as Russell never doubts they are, then one will drag relations out of them. If, however, the connection between relational properties and foundations is kept in mind, there is no temptation to resort to the composition of relational properties to explain their dissimilarity. Russell’s mistake¹² was to treat foundations and relational properties as belonging to two different analyses of relations. In fact, any adequate analysis must include both. If Rab is true there will be foundations F and G such that (Rb)a coimplies (Fa • Gb). And if Rba were true, there would be foundations H and K such that (Ra)b would coimply (Hb • Ka). This is the explanation of the dissimilarity between relational properties implied by asymmetry. It is their derivations from different foundations that account for their dissimilarity. Even where F is similar to H and G is similar to K, the derivations are from the two different bases, (Fa • Gb) for the relational property (Rb) and (Fb • Ga) for the relational property (Ra).

One will still want to know how asymmetry comes from dissimilarity, which is symmetrical. Suppose (Rb)a has as its foundational base (Fa • Gb). If this is also the base for (Ra)b, then R is indeed symmetrical. The relatedness is asymmetrical not just when the relational properties are dissimilar. Rather, associated with this dissimilarity must be an incompatibility of foundational bases. If the base for (Ra)b─ say (Hb • Ka)─is incompatible with that for (Rb)a─(Fa • Gb)─then R is asymmetrical.

Let me turn now to some other questions about monadism. Does the monadistic view, as expressed by (1) and (2) of §2, preserve the customary connections between properties of relations? Earlier formulations of the monadistic view were some version of the claim that Rxy is equivalent to (Fx • Gy). But this has the immediate consequence that symmetry implies reflexivity.¹³ For if ((Fx • Gy) כ (Fy • Gx)) then ((Fx • Gy) כ ((Fx • Gx) • (Fy • Gy))). Yet there are symmetrical irreflexive relations such as spouse of. But relatedness involves not just a conjunction of foundations but also the grounding of relational properties in foundations. When treated in this way, the connections among the properties of relatedness are precisely the customary ones. Thus asymmetry implies irreflexivity. For to say that R is asymmetrical is to say that (x)(y)((Ǝφ,ψ,θ,ζ є R*)Ryx ~ כ (Ǝφ,ψ,θ,ζ є R*)Ryx). Replacing ‘Rxy’ here with ‘φx • (φx↔ψy) • (φx↔ (θx • ζy)) and ‘Ryx’ with a corresponding formula allows one to derive the proposition that (x)~(Ǝφ,ψ,θ,ζ є R*)(φx • (θx • (φx↔ψx) • (φx↔(θx • ζx))).

Apparently, one of the most damaging objections to monadism has been that it allows the derivation of quantificational laws that are not valid. Thus it is claimed that, if monadism holds, then the doubly quantified claim that (x)(Ǝy)Rxy will imply that ( Ǝy)(x)Rxy.¹⁴ Since this implication does not in general hold, monadism must be false. However, the objection is based on the assumption that monadism must treat Rxy as a truth function of monadic formulas. On this assumption, the illegitimate switching of quantifiers is indeed sanctioned. But should we grant the assumption?

A consideration of the correlation of relational properties and of the basing of relational properties on foundations reveals that it would be incorrect to treat the conditionals on the right-hand side of (1) and hence of (2), of §2, as truth functions. Relational properties are not correlated simply because of a similarity of truth values of corresponding atomic sentences. They are correlated since a relatum's having one of them depends (in a way that does not involve context) on the other relatum's having the other. To try to express what is involved in a relatedness claim without expressing such a connection between the relational properties would be to give an incomplete account of relatedness. Likewise, basing relational properties on foundations is a matter of (noncontextual) dependency and not of a similarity of truth values. Relational properties are properties that relata have because there are certain foundations in those relata.

Let us consider the exact point at which the kind of conditional involved becomes crucial. To show that (2) allows the illegitimate switching of quantifiers around Rxy, one must show that they can be switched around ((φx↔ ψy) • (φx↔(θx • ζy))). If this is possible, then it is possible to derive ( Ǝy)(x)(φx↔ ψy) from (x)(Ǝy)((φx↔ ψy) • (φx↔(θx • ζy))). But for this to be possible it will have to be the case that ((y)ψy→ (x)φx) implies (Ǝy)(ψy → (x)φx). Though this is notoriously the case for material implication, it is not the case for any implication that expresses the way relational properties are correlated. Intuitively, it does not follow from the premiss, that if everyone is comradely there will be peace, that there is someone such that if he or she is comradely there will be peace. Strict implication satisfies this intuitive requirement, but it suffers the limitations discussed in Chapter II, §2, that make it unsuitable as a device for understanding natural connections generally. This makes relevant implication an obvious candidate for the implication connecting correlative properties among ״ themselves and to their foundations.

A final objection to the general program of monadism concerns the undecidability of first-order quantificational logic. If one eliminates all but monadic predicates by applying (1) and (2) and their counterparts for higher adicity, then what was first-order logic becomes first-order monadic logic, which is decidable.¹⁵ But if (1) and (2) are genuine equivalences, how can they turn an undecidable into a decidable system? So it seems that monadism cannot be correct, whatever form it takes. Though (2) does introduce higherorder quantifiers, they can effectively be ignored in making inferences within first-order logic. So their presence will not introduce the needed undecidability. Moreover, if the → is a relevant implication, as I have taken it to be, then the first-order monadic logic will be undecidable,¹⁶ as the standard first-order monadic logic with truth functions is not. But I do not choose to save the day in this way, since there is a more important mistake lurking behind the objection.

There is simply no reason to expect that the elimination of the polyadic predicates should leave us with an equivalent logical system. For these predicates will, in general, be non-logical expressions. Thus to state a connection between them and an expression for what must be the case for them to apply will not be to state a logical law. (1) is no more a logical law than is the claim that humans are rational animals. However, in order to state the correct connection between necessity and possibility or between disjunction and conjunction one will need to state a logical law. So a propositional calculus from which disjunction has been eliminated might well be equivalent to one with disjunction. The equivalence of the two logical systems will require that it be a law of the system with disjunction that the connection hold between disjunction and conjunction that was used for the elimination.¹⁷

But there is no question of (1) being a logical law. To assume (1) for purposes of eliminating polyadic predicates is not to assume something that can be proved as a logical law in a system with polyadic predicates which purports to be a logical system. To assume (1) and similar equivalences for other polyadic predicates and other relata is to assume a host of non-logical claims. Since the elimination is based on such claims, it is not at all surprising that these claims can be true even though the resulting logical system is not equivalent to first-order logic.

§4. The Interconnection of the World. A philosophy that denies relations as entities distinct from their relata would seem committed to separateness, whereas a philosophy of relations as distinct entities would seem committed to togetherness of entities. But we have seen that a philosophy of relations as distinct entities is committed to the view that there is such a gap between individuals and relations that what the individuals are cannot affect what relations they have. If relations make for togetherness, it is an imposed togetherness, and the relata have nothing to do with its maintainence. In short, it is not a togetherness at all.

On the other hand, the paradox is rounded out when it becomes clear that the denial of relations as distinct entities in the context of our required ontology excludes separateness and introduces togetherness. There are many ways in which individuals are together in the world, and any view that obscures this fact is in some way wanting. The togetherness must be understood in such a fashion that it is internalized by the entities involved. That is, their relatedness is not external to them but is both based in their components─the foundations─and reflected in their components─the correlative properties.

If F and G are foundations, there will be some relational property ƒ such that when a has F and b has G a will have ƒ. This is not just a contingent connection. Suppose Fa means that a is male and mates, and Gb means that b exists. Suppose further that Fa and Gb suffice for fa, where fa means a is the father of b. They suffice for fa not because of any special circumstances. Rather, they suffice simply because it is of the nature of each of the individuals involved─a and b─that when they have these foundations a has the relational property of being b’s father. Of course, if father of b and father of c are mistakenly treated as similar properties, then it will seem that the connection is at best contingent. For where c is not a’s son, (Fa • Gc) does not imply that a is c’s s father.

Now the notion of togetherness, or relatedness, can be interpreted in the light of this necessary connection. Two individuals are “together” when their natures are such that each is determined to have a relational property by a pair of non-relational properties whose members do not belong to the same individual. Since it is the nature of the individuals that is behind such determination, no mere accident accounts for the way they internalize one another. Each “internalizes” the other when it has one of a pair of relational properties as a component. Put another way, it is the natures of the relata that give certain of their components the status of foundations. Those components are foundations when it is the natures of the individuals which have such components that they also have certain relational properties.

This is so far only one side of the story. Not only does (Fa • Gb) necessarily imply fa, but also, in some cases, Fa, under appropriate circumstances, necessarily implies Gb. When this is the case, I say that not only are the individuals together but that they are “necessarily together.” If they have properties from which relational properties are derived, they are together; if, in addition, one of the foundations is necessarily connected with the other, they are necessarily togther. Necessary connections across time between foundations of relational properties are one form of necessary togetherness.

But are relations being reintroduced here in the guise of necessary connections? As always, such a connection is merely the necessity of a conditional. It is the necessity in certain circumstances of a’s having (F_→ Gb). So if relations are abandoned, there will be reliance on properties described by means of logical connectives. But independent reasons were already adduced for the existence of such properties in Chapter VI. These properties are then not an extravagant price to pay for relatedness without relations.

Togetherness does not, in this view, require a “pre-established harmony.” Leibniz was forced to postulate such a harmony, not because of his rejection of relations, but because of the special way he conceived of natures. A Leibnizian nature accounts for every condition the individual has. This conception seems at first to conflict with Leibniz’s view that every individual “mirrors” all the other individuals. There are what appear to be causal reflections, such as heat due to fire. And there are non-causal ones, such as Caesar’s difference in make-up from the Rubicon. But the heat of an individual is not really due to the burning of another, and Caesar’s being of a different stuff from the Rubicon is not really due, even in part, to the watery stuff of the Rubicon. Otherwise, the natures of the hot entity and of Caesar would not fully determine their conditions. So to account for the fact that there is a correspondence, Leibniz must go back to the source of the corresponding natures. God chooses individual natures that do in fact correspond. The correspondence of conditions between individuals cannot result from a mutual dependence of these conditions. Otherwise, not all conditions would flow from the natures of the individuals with them.¹⁸

To avoid making the source of togetherness external, it suffices to modify Leibniz’s notion of a nature. The notion of nature developed here grants that some conditions of an individual are not due to its nature. In particular, an individual might be in the condition of having a relational property contingently. Having that property could then depend on the features of another individual that, in fact, exists only contingently. The source of togetherness is then in the individuals related and not beyond the world of concrete individuals.

It is well to note that if a had the relational property ƒ by its nature then it would be true that □ a(Fa • Gb) and hence that □  a(Gb). There will then be a strict correspondence between a and b. This suggests a way Leibniz might have had of avoiding the preestablished harmony without giving up his conception of natures. For b’s burning might be held to be a consequence of a’s─the heated object’s─nature. And the Rubicon’s wateriness might be held to be a consequence of Caesar’s nature. But such possibilities are too remote to be given serious treatment.

Still, it might seem objectionably mysterious that it is by nature that an individual should have a relational property when it has a certain foundation and another entity with which it need not be interacting has a certain foundation. For how can its nature “know” that the isolated individual has the appropriate foundation? A telepathic power seems to be required. Short of this, and without relations, a pre-established harmony seems needed.

This objection could be sustained if the relevant notion of nature were Locke’s notion of a constitution of small particles. For then, indeed, a nature would not be the source of anything in respect to another entity unless it were directly interacting with that entity. If a’s nature were a constitution of parts, then it might be the basis for a’ s being smaller than b, since a by this nature might act on b to make it grow. But the material composition would not be responsible for a s being smaller than b if there is no causal action between the two and the two possess only a difference of magnitudes.

Clearly, then, the objection derives its bite from just such a scientific conception of natures. However, we were discussing natures of a quite different kind. We were discussing a nature that gave rise to the conditional property ((F_ • Gb) → ƒ_). Having ƒ as a result of having this property is merely a matter of having (F_ • Gb). And this is no more problematic than modus ponens ever is. There is no need for the nature to “know” that b does have G in order for the entity with the nature to have ƒ. In the Lockean view, however, a nature could not give rise to such a conditional property. For one of the conjuncts in the antecedent need have no conceivable causal connection with the entity that has the conditional property. Because we need to account for the fact that such an entity will have the above conditional property by its nature, the Lockean kind of nature is inadequate.

The notion of togetherness developed here differs not only from the Leibnizian notion of togetherness with an external source but also from the view associated with Whitehead of togetherness with no source at all. In the present view, togetherness requires foundations. Suppose though every individual has relational properties in respect to every other, but relational properties are the only kind of property. Whitehead expresses such a view when he says, “The aspects of all things enter into its [the enduring object’s] very nature. It is only itself as drawing together into its own limitation the larger whole in which it finds itself.”¹⁹ There is, in this view, universal togetherness without a source, since there are no foundations for relational properties.

This is unacceptable because of the obvious need to account for the fact that one individual has a property correlated with that of another individual. Individuals are not correlative for no reason at all. What they are of themselves gives them the special kinds of correlativity they have. “. . . The properties of a thing are not the result of its relations to other things, but only manifest themselves in such relations.”²⁰ In addition, there is an important systematic inadequacy in this view that basic individuals have as components only their reflections of the world. For one cannot coherently hold this view and also engage in the practice of action on prior experience.

To see the incoherence, recall the difficulty about relational properties raised in Chapter V,§3. Each individual has so many independent relational properties that, if they are all included, any sort of postulate of limited independent variety, even one for levels, would fail. Thus we excluded the relational properties of any individual and put their foundations in the individual in their place. A finite number of foundations in the individual could sustain an infinite number of its relational properties. This puts no limitation on the kinds of hypotheses to which induction could be applied, since hypotheses about relational properties would be consequent upon hypotheses about individuals with foundations. However, if, as Whitehead's ontology requires, there are no foundations but only relational properties, then one cannot replace relational properties by their foundations in considerations of limited independent variety. Thus there will be no limited independent variety, even of the weak, levels sort. Whitehead’s ontology, which rejects non-relational properties in its eagerness to give importance to relational ones, is then incoherent with the practice of action on prior experience.

§5. Sameness and Unity. It might seem that so long as we still have a component ontology we have not done with relations. For components are components of individuals; components differ from one another; having a component is a condition of an individual; and components of distinct individuals are distinct from one another. Even if relations among distinct relata have been reduced to properties, we have left as a residue the relations of composition, difference, condition, and distinctness. The philosopher of relations has waited thus far to set his trap. For to do away with relations at this point is to do away with all composition, all difference, all conditions, and all distinctness. In short, it is to fall into Parmenidean monism.

But the relation of composition, for example, does not have distinct entities as relata. So our strictures against relations among distinct entities no longer apply. It would, then, be consistent with all the preceding to allow that the relation component of exists. If it exists, it can be a component of the individual that contains its relata. It was only relations among distinct entities that could in no way be components of individuals. But in this case, in which the relata are the same particulars, the relation can be a component without the risk of being multiplied by the number of its relata. However, there is no need to admit relations that are internal in our second sense. I shall now try to show this.

The task can be simplified by noting that composition can be defined in terms of the other relations. Suppose F is a component of a. Then (i) a is an individual or a condition, and (ii) a is the same particular as F but nonetheless is different from F. Thus components are components of individuals or conditions and of nothing else. In particular, properties, actions, and physical parts, though components of individuals, are not entities with components, since they are neither individuals nor conditions. This account of components introduces the relations of difference and sameness. So we are led to concentrate on these.

In Chapter III, distinctness of conditions was said to be a sign of the difference of corresponding entities. But to define the difference of entities by means of the distinctness of conditions would make conditions prior to components, which they are not. Yet, though difference cannot be defined by other relations, such as condition of, it is reducible to relational properties and foundations. Since difference holds, in many cases at least, among components, it seems natural to attempt this reduction by means of relational properties and foundations of the components themselves. I just said, however, that components have no components. Thus in the case of a putative relation among components, one must search for the corresponding relational property and foundations in the individual or condition with those components.

Thus, if the property F of a differs from the property G of a, then (i) a encompasses the difference of F from G in a itself, and (ii) a is both F and G. The encompassing of (i) is a relational property a has in respect to itself. The properties F and G of (ii) are the foundations associated with this self-relational property. These foundations do imply that relational property, for being F and G will imply that a encompasses the difference of F from G in a. The foundations of the difference of components are simply different components.

In case it is an individual that is different from a component, a slightly different approach is needed. The foundation cannot be the individual; it must be, rather, the individual’s sameness with itself. Thus the foundations for F’ s differing from a are F itself and a’s self-sameness. Now sameness also appears when we attempt to make condition of yield to our reductive machinery. If being yellow is a condition of a, then the foundations are the yellow in a and the sameness of the condition with itself. More will be said on the self-sameness of conditions in Chapter XI, §4. For the moment it is clear that all our relations point back to sameness.

But there seem to be no obvious foundations for sameness. Strangely enough, though, if sameness does not exist, we fall back into undifferentiated monism. Yet if sameness exists, it would seem that there are relations. This relation─one that is internal in our second sense─would then be the price to be paid for having composition, difference, and distinctness.

But perhaps sameness is not a genuine relation. Then abandonment of all relations, even the internal ones, leaves sameness intact and does not force monism on us. I think sameness claims are, in fact, attributions of unity and that unity is a non-relational component of individuals and conditions. Thus the foundations of the relation condition of are a component of the individual with the condition and the unity of the condition. To say Cicero is himself is to say Cicero is a unity. To say Cicero is Tully is to say the entity Cicero and Tully is a unity. To say Cicero is not Catiline is to say the entity Cicero and Catiline is not a unity. Moreover, to say Cicero is the same particular as his hunger is to say that the entity Cicero and his hunger is a unity. Without unity as a component, Cicero could not be unified with, could not be the same particular as, his hunger. Not only does the component unity account for the sameness of individuals with their other components, but it also accounts for the sameness of individuals with their unity.

It will be objected that this account of sameness merely introduces the part-whole relation and that we fall into the monist’s trap if the part-whole relation does not exist. When we say the entity Cicero and Catiline is not a unity are we not saying that the whole whose parts are Cicero and Catiline is not a unity? Are we not saying that the whole is a multiplicity rather than a unity? In at least one important respect, a negative unity claim is like a negative existence claim. When one says Pegasus does not exist, one’s claim is significant even though ‘Pegasus’ lacks a referent. Now in the ontology of components, only unities exist. Properties exist only because a property is the same particular as its individual, and is thus the same unity. So when one says the entity Cicero and Catiline is not a unity, what one says is true even though there is no entity, that is, even though there is no entity referred to by the phrase ‘Cicero and Catiline’. If, contrary to fact, there were such an entity, one would call it a whole and say Cicero and Catiline are its parts. Then, indeed, one would be committed to mereological relations. But what about positive unity claims? If there is no entity that is a whole with many particulars as parts, one is not forced in the case of true positive unity claims to say one is referring to wholes with only a single particular as part. One is free to say that in claiming that the entity Cicero and Tully is a unity, or that the entity Cicero and his hunger is a unity, one is referring not to a whole that has one or more parts, but simply to an individual.

Philosophers of different times and different schools have also held that sameness was not a relation. But they have as often as not made something intentional out of it. For Aquinas it was a relation of reason;²¹ for Wittgenstein a sameness claim signified the interchangeability of the terms set out in it.²² Here, by contrast, I have treated sameness claims as ascribing the real component of unity to particulars. Suppose these claims were not true or false on the basis of what is real; suppose, that is, they were true or false on the basis of concepts of relations or on the basis of linguistic interchangeability. Then since distinctness, difference, condition, and composition all rest on sameness, the ontology of components would only describe the structure of particulars as conceived or talked about. It would not be an ontology of the real.