Relational Tropes and Bradley’s
Regress
Serious Metaphysics Club, University of Cambridge 17/11 2010.
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
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 relataspecific: 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 coexistence 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 sublocation. 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.
References
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References
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