Relational Tropes and Bradley’s Regress Serious Metaphysics Club, University of Cambridge 17/ Markku Keinänen University of Turku

Introduction Relational tropes are introduced by several contemporary metaphysicians to account for the instantiations of external relations. As a particular relation relational trope r has a thin particular nature (e.g., of 1m distance). It is strongly rigidly dependent on two or more particulars. Moreover, the existence of relational trope r is assumed to entail that r connects (or, relates) its relata. In this presentation, I defend relational tropes against Fraser MacBride’s (2010) attack, according to which the advocates of relational tropes fail to avoid Bradley’s relation regress.

Introduction I take up two distinct trope answers to the contemporary formulation of Bradley’s regress problem (the contemporary Bradley’s regress). According to the second standard trope answer (Maurin 2008, Wieland & Betti 2008), it is the nature of relational trope r to connect its relata. The advocate of the second standard answer can claim to deal with the contemporary Bradley’s regress and Bradley’s relation regress (contra MacBride). I argue that the main problem with this answer is to give a satisfactory analysis of relational inherence, not Bradley’s relation regress. In order to bolster the case of relational tropes, I outline the first steps of such analysis.

Strong rigid dependence Let “≤ “ be a relation of improper parthood (cf. Simons (1987: 112) for the definition) and “E!” the predicate of (singular) existence. Following Simons (1987: 303), we can define strong rigid dependence as follows: entity e is strongly rigidly dependent on f iff SRD (e,f): SRD(e, f) = ¬( □ E!f) & □ ((E!e → E!f) & ¬( f ≤ e )). Weak rigid dependence is defined by replacing ¬( f ≤ e ) with ¬( f = e ).

Formal characterisation of relational tropes Let “SRD (x, y)” be an abbreviation of x is strongly rigidly dependent on y. Relational trope t connecting f and g must satisfy the following condition: MRD (t, (f, g)) =  (E!t  (E!f  E!g   (f  t)   (g  t)  ¬(f  g)  ¬(g  f)))  ¬ (  E!f)  ¬ (  E!g)  ¬ (SRD(f, g))  ¬ (SRD (g, f)) vs. Simons RD (t, (f, g)) =  (E!t  (E!f  E!g   (t = f)   (t = g)  ¬(f  g)  ¬(g  f)))  ¬ (  E!f)  ¬ (  E!g)

Formal characterisation of relational tropes I propose two amendments to RD in order to avoid two first of the problems and return to the third problem in the end of the of the presentation. First, the entities related cannot be parts of relational tropes. Secondly, relational tropes connect entities that are not strongly rigidly dependent on each other. Consequently, they cannot connect the trope constituents of the same substance but they typically connect distinct substances.

Contemporary Bradley’s regress vs. Bradley’s relation regress I distinguish between the contemporary Bradley’s regress problem and Bradley’s (1893) original relation regress. The contemporary Bradley’s regress problem is the problem presented and answered in several recent discussions of Bradley’s regress. The contemporary problem concerns external relations and the relata are assumed to be capable existing independent of the relation. By contrast, the original problem is more general because it does not set any constraints on the modal relations between a relation and its relata. Both of these problems appear to show that we must introduce an infinite regress of relations to account for the instantiation of a single relation.

The contemporary Bradley’s regress [CBR1]: Assume that two objects a and b are connected by external relation R. [CBR2]: Since R is an external relation, the existence of a and b does not entail that a and b are connected by R. [CBR3]: The existence R does not entail that a and b are connected by R. [CBR4]: That a and b are connected by R must have an ontological ground. [CBR5]: We need to introduce further relation R’ to connect R, a and b.

The contemporary Bradley’s regress [CBR6]: However, the existence of R’ does not entail that R’ connects R, a and b, and we need to introduce a third external relation to connect R’, R, a and b. An infinite regress ensues. The distinction between the group of entities (relation R, objects a and b) and relational complex (a and b related by R) must have an ontological ground ([CBR4]). The ontological ground is a further relation R’ that connects a, b and R ([CBR5]). This suggestion generates a vicious infinite regress: in order to account for the holding of each relation, we must introduce a further relation that connects the first relation to its relata ([CBR6]).

The factualist and relational trope answers There are two main ways to (try to) deal with the contemporary Bradley’s regress. Both of them try to prevent the conclusion expressed by [CBR5]. Factualists accept [CBR1] – [CBR4] but deny [CBR5]. According to factualists, that a and b are connected by R is a further entity, a relational fact. A relational fact is able to connect its constituents without help of any further entity. The advocates of relational tropes deny [CBR3]. According to them, the existence of (dyadic) relational trope r entails that entities a and b exist and are connected by the corresponding relation. Thus it seems that relational tropes can themselves connect (or “relate”) their terms.

The truhmaker theoretic answer The truthmaker theoretic answer (Simons 2003) [1]: Relational trope r is connected to a and b by the formal relation of multiple strong rigid dependence (MRD (r, (a, b)) holds). [2]: Necessarily, if trope r exists, a and b exist. [3]: Relational trope r has a thin particular nature (it is a trope of kind R). [4]: Relational trope r is a truthmaker of the corresponding proposition that Rab (i.e., that a and b are connected by relation R). As a consequence, we need not introduce any further entities to make proposition Rab true.

The truhmaker theoretic answer The gist of the truthmaker theoretic answer is to replace the talk about relational inherence with the talk about multiple strong rigid dependence Relational trope r inheres in a and b, because it is strongly rigidly dependent on a and b (cf. Simons 2003: sec.3). Since r makes true that Rab, the existence of r entails that a and b are connected by relational trope r of kind R and the relation R. Hence, we need not introduce any further entities to connect a, b and relational trope r.

The second standard answer The second standard answer (Maurin 2002, 2010; Wieland & Betti (2008); Keinänen & Hakkarainen 2010) [1]: Relational trope r is connected to a and b by the formal relation of multiple strong rigid dependence (MRD (r, (a, b)) holds). [2]: Necessarily, if trope r exists, a and b exist. [3]: Moreover, necessarily, if r exists, a and b are connected by r (or r “relates” a and b). [4]: Relational trope r has a thin particular nature (it is a trope of kind R). [5]: Thus, the existence of relational trope r suffices to explain that a and b are related by trope r of kind R and make the corresponding relational proposition Rab true.

The second standard answer The second standard answer shares claims [1] – [2] and [4] with the truthmaker theoretic answer. This answer adds claim [3], according which the existence of relational trope r entails that r connects (or, relates) a and b. Relational trope r has a thin particular nature (it is a trope of kind R) (claim [4]). As a consequence, relational trope r is a truthmaker of the corresponding proposition Rab.

The second standard answer Unlike the truthmaker theoretic answer, the second standard answer tries to block the contemporary Bradley’s regress directly, without recourse to truthmaking. Relational tropes are relata-specific relations (cf. Wieland & Betti 2008: 518): it is the nature of relational trope r to connect its specific relata. Thus, the existence of r entails that a and b are related by r (claim [3]). Hence, [CBR3] is denied and the contemporary Bradley’s regress is blocked.

Truthmaker theoretic vs. the second standard answer Simons (2003) claims to give a generalized account of relational inherence in terms of multiple rigid dependence but there seems to be a explanatory gap between premises [1] – [3] of his account and claim [4]. The existence of 1m distance trope r entails that objects a and b exist. However, prima facie, it need not entail that a and b are in 1m apart from each other. Therefore, trope r is not a sufficient truthmaker of the corresponding proposition that a and b are 1m apart from each other (Rab). In order to obtain this conclusion, we need to assume that relational tropes are relata-specific, i.e., that the existence of r entails that a and b are connected by r.

Truthmaker theoretic vs. the second standard answer As a result, the second standard answer seems to be exactly what we need to close the explanatory gap. The second standard answer seems to be both superior to the truthmaker theoretic answer and is able to avoid the contemporary Bradley’s regress. The second standard answer has some problems of its own. Before we go into them, let us try to find out how the advocates of relational tropes can deal with Bradley’s original relation regress.

Bradley’s relation regress In his original relation regress argument, Bradley (1969 [1893]: 27) does not distinguish between internal and external relations; it is meant as an argument against all kinds of relations (internal and external). Ungrounded internal relations (such as distinctness, mereological relations and relations of existential dependence) are formal relations, which are not further entities to their relata – any two entities are in formal relation simply because they exist. Here, I confine myself to the following issue: whether we can use Bradley’s relation regress argument against relational entities (relation universals or tropes), which are introduced to account for some external relations

Bradley’s relation regress We can present Bradley’s relation regress as follows: Bradley’s relation regress: Assume that a and b are related by R. Either R is nothing to a and b or it is something to them. In the first case, a and b are not related by R. Thus, if a and b are related by R, R is something to them. If R is something to a and b, a and b need to be connected to R by a further relation R’. A vicious infinite regress ensues.

Bradley’s relation regress Bradley’s relation regress argument seems to have the following simple structure: [A]: Assume that a and b are connected by relation R. [B]: If a and b are connected by R, R is something to a and b. [C]: If R is something to a and b, R is connected to a and b by further relation R’. [D]: Consequently, R is connected to a and b by further relation R’. [E]: Again, R’ is connected to R, a and b by a further relation and a vicious infinite regress ensues.

Bradley’s relation regress The phrase “R is something to a and b” is somewhat unclear. One possibility is to leave it out and replace [B] and [C] with a single claim [B*]: [B*]: If a and b are connected by R, R is connected to a and b by further relation R’. However, I suggest a clarification of [B] and [C] by means of the following two premises: [B’]: If a and b are connected by R, this fact must have an ontological ground (cf. [CBR4]). [C’]: The only candidate for the ontological ground is a further relation R’ that connects R to a and b (cf. [CBR5]).

Bradley’s relation regress Prima facie, Bradley’s relation regress argument applies to relational entities (relation universals or tropes) independent of the modal relations them and their relata (independent of whether the existence of a relation entails the existence of its relata or vice versa).

The Second standard answer to Bradley’s relation regress The trope theorists advocating the second standard answer to the contemporary Bradley’s regress can try to answer Bradley’s relation regress as follows (cf. Wieland & Betti (2008); Maurin (2010); Keinänen & Hakkarainen (2010)): The second standard answer to Bradley’s relation regress: Relational trope r is connected to a and b by the formal relation of rigid dependence. Moreover, r is relata- specific: the existence r entails that a and b are connected by r. Therefore, r is a sufficient ontological ground for its connecting a and b (“it is something to its terms”). No further entities need to be introduced.

The Second standard answer to Bradley’s relation regress Thus, if the second standard answer is successful, it is able to deal with both of the regresses: the contemporary Bradley’s regress and Bradley’s relation regress. According to Fraser MacBride (2010: sec. 3.3), the advocates of relational tropes have not succeeded in dealing with Bradley’s relation regress. MacBride’s main target is Simons’s truthmaker theoretic answer but he also addresses the second standard answer.

MacBride’s criticism An advocate of the second standard answer maintains that it is the nature of relational trope r to connect (or relate) its relata. According to MacBride, the main problem is whether relational tropes having such a nature are capable of existing. Bradley’s relation regress argument purports to show that relations (including relational tropes) cannot exist. Therefore, an advocate of relational tropes must address Bradley’s regress directly in order to show that there can be relational tropes.

MacBride’s criticism According to MacBride (2010), the two standard trope answers have not solved Bradley’s regress problem: First, relational trope r is rigidly dependent on its relata a and b. However, this guarantees the necessary co- existence of a and b with r but not that a and b are related by r. Pace Simons, some additional ontological ground need to be introduced to secure that a and b are related by r. Secondly, according to MacBride, the second standard answer just stipulates that it is the very essence of relational trope r to connect its relata a and b. It has not addressed Bradley’s relation regress because we have not explained why r is capable of relating its relata.

Trope theorists vs. MacBride We seem to have arrived at a stalemate: an advocate of the second standard answer claims to deal with Bradley’s regress by means of relational tropes, while the critic (MacBride) counters this claim. Can we decide between these two views? The crucial issue concerns the ontological ground of the (alleged) fact that a relation connects its relata. According to Bradley’s relation regress argument, the only possible ontological ground is a further relation, which leads to an infinite regress. All advocates of relations deny that the ontological ground must be a further relation and there is no specific reason why the ontological ground should be a further relation.

The problem of relational inherence The trope theorists maintain that the required ontological ground is the relational trope. Since multiple rigid dependence does not guarantee relational inherence, the advocates of the second standard answer seem to be obliged to admit that relational trope r stands in two formal relations to its relata a and b: [A]: r is strongly rigidly dependent on a and b (MRD(r, (a,b). [B]: r inheres in a and b. The existence of relational trope r entails that it is multiply rigidly dependent on a and b and that it inheres in a and b.

The problem of relational inherence If both relational inherence and multiple rigid dependence are formal relations (and no further entities), the answer is successful in maintaining that relational trope r is an ontological ground for holding of the relational connection. However, the claim that relational inherence is a further formal relation between relational trope and its relata does not look plausible for two reasons: First, relational inherence entails multiple rigid dependence. Thus, relational inherence is prima facie analysable in terms of multiple rigid dependence and some further component.

The problem of relational inherence Second, as relational trope need not connect its relata at each moment of their existence, the further component has to do with determination of the relative temporal location of the relation to its relata. Compare this to the case of monadic inherence. We can suggest a following kind of analysis of monadic ontic predication (monadic inherence): trope t is property of i, iff t is rigidly dependent on i, necessarily a proper part of i if it exist and necessarily co-located with i when it exists (Keinänen & Hakkarainen 2010: sec.2-3). Thus, we can suggest that monadic inherence of tropes is analyzable in terms of rigid dependence, proper parthood and necessary spatial co-location and temporal sub-location.

The problem of relational inherence Monadic inherence understood in this way is not a (defined) formal relation: monadic inherence is analyzed by means of on spatial co-location and temporal sub- location. We need to introduce further entities to ground the location of tropes and a substance. Therefore, the relation of inherence between tropes and a substance does not obtain solely on the basis of the existence of its relata (it cannot be a formal relation). Similarly, since relational inherence must be analyzed in terms rigid dependence and temporal sub-location, it is not a (defined) formal relation.

The problem of relational inherence We suggest that relational tropes are (as concrete particulars) proper parts of relational complexes that are complex particulars. Let r be a dyadic relational trope. Trope r fulfils the following conditions: [1]: Necessarily, if relational trope r exists, its relata a and b exist. [2]: Dyadic relational trope r is strongly rigidly dependent only on its relata a and b.

The problem of relational inherence [3]: Relational trope r and its relata a and b form a further complex individual, relational complex r – a – b. [4]: Assume that the relata of r are substances (i.e., strongly independent particulars). By claim [2], the strong rigid dependencies of the constituents of relational complex r – a – b are fulfilled by the constituents themselves. Thus the relational complex is itself a strongly independent particular and a complex substance (the Conditioning Principle).

The problem of relational inherence [AR]: Relation trope r is instantiated by objects a and b (at some time T) iff r is proper part of the relational complex formed by r, a and b and the relational complex exists at T. Unsolved Problems: 1) how to ground the location of relational complexes? 2) it seems natural to assume that the temporal location of a relational complex is a proper part of the temporal location of the aggregate of its constituent objects. Does this apply to all cases?

Conclusion Since the ontological ground for a relation connecting its relata need not be a further relation, we can block both Bradley’s relation regress and the contemporary Bradley’s regress. However, the main proposed trope strategies to block Bradley’s regress (i.e., the truthmaker theoretic answer, the second standard answer) have not specified that ontological ground in a satisfactory fashion. The second standard answer, which has been the best trope theoretic strategy, leaves relational ontic predication (i.e., relational inherence) unanalyzed. Hence, it does not offer us any transparent account of the connection between a relational trope and its relata.

Conclusion As a consequence, the second standard answer does not give us satisfactory an explanation of why relational trope r acts as ontological ground of the holding of the relational connection.

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