Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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(3) Mutatis mut<strong>and</strong>is, realism has <strong>the</strong> same two<br />
options as trope <strong>the</strong>ory: (3a) Accord<strong>in</strong>g to a bundle <strong>the</strong>ory<br />
based on universals, of which Bertr<strong>and</strong> Russell (1948,<br />
part 4, ch. 8) is a proponent, particulars are understood as<br />
bundles of universals. In this view, a <strong>in</strong>stantiates F iff F is a<br />
member of <strong>the</strong> a-bundle. S<strong>in</strong>ce F is a member of <strong>the</strong> abundle<br />
necessarily, 7 <strong>the</strong> <strong>in</strong>stantiation relation between<br />
a <strong>and</strong> F strictly supervenes on <strong>the</strong> existence of <strong>the</strong> abundle.<br />
(3b) The second type comb<strong>in</strong>es a substance–<br />
attribute view with a <strong>the</strong>ory of nontransferable universals.<br />
Accord<strong>in</strong>g to this position – ma<strong>in</strong>ta<strong>in</strong>ed by, e.g., David<br />
Armstrong (2004a, 2004b <strong>and</strong> 2006) 8 – that<br />
a <strong>in</strong>stantiates F supervenes on <strong>the</strong> existence of a <strong>and</strong> F<br />
alone. 9<br />
Thus, nei<strong>the</strong>r nom<strong>in</strong>alism nor realism is committed<br />
to <strong>the</strong> regress. Three of <strong>the</strong>se positions, namely, (1), (2a)<br />
<strong>and</strong> (3a), agree <strong>in</strong> underst<strong>and</strong><strong>in</strong>g <strong>in</strong>stantiation to be<br />
constituted by class (or bundle) membership. For <strong>the</strong>m<br />
necessity of <strong>in</strong>stantiation – <strong>and</strong> hence <strong>the</strong> possible denial<br />
of (P1) – is built <strong>in</strong>to <strong>the</strong> ontological conception of<br />
<strong>in</strong>stantiation. For <strong>the</strong> substance–attribute views<br />
(2b) <strong>and</strong> (3b), necessity of <strong>in</strong>stantiation is a feature<br />
additional to <strong>the</strong> basic conception of <strong>in</strong>stantiation <strong>and</strong><br />
devised, I presume, specifically to avoid (P1).<br />
All of <strong>the</strong>se five options come with heavy ontological<br />
burdens. Ignor<strong>in</strong>g <strong>the</strong>ir specific difficulties, I shall mention<br />
only <strong>the</strong> problem which <strong>the</strong>y share: necessity of<br />
<strong>in</strong>stantiation makes cont<strong>in</strong>gency impossible. Whe<strong>the</strong>r <strong>the</strong><br />
substitutes on offer 10 are satisfactory is at least doubtful.<br />
So it is worthwhile to <strong>in</strong>vestigate whe<strong>the</strong>r <strong>the</strong>re might not<br />
be ano<strong>the</strong>r way out of <strong>the</strong> regress.<br />
2.2 What is necessary for <strong>the</strong> regress? – A fur<strong>the</strong>r<br />
condition<br />
So far I have acted as if (P1) <strong>and</strong> (P2) were sufficient for<br />
<strong>the</strong> regress, with <strong>the</strong> purpose of show<strong>in</strong>g that realism is no<br />
more committed to (P1) than trope <strong>the</strong>ory is, <strong>and</strong> that <strong>in</strong><br />
fact nei<strong>the</strong>r of <strong>the</strong> two views is committed to (P1). Thus, I<br />
hi<strong>the</strong>rto relied on <strong>the</strong> analysis of Bradley’s regress which<br />
seems commonly accepted. Now it is time to show that this<br />
analysis is flawed. (P1) <strong>and</strong> (P2) by <strong>the</strong>mselves do not yet<br />
yield Bradley’s regress. It is quite obvious but frequently<br />
ignored: <strong>in</strong> order for <strong>the</strong> regress to obta<strong>in</strong>, it must be given<br />
that <strong>the</strong> <strong>in</strong>stantiation relation is itself <strong>in</strong>stantiated (by <strong>the</strong><br />
entities it relates). O<strong>the</strong>rwise, given an <strong>in</strong>stantiation relation,<br />
(P1) does not generate a fur<strong>the</strong>r <strong>in</strong>stantiation relation.<br />
To arrive at a regress, we <strong>the</strong>refore need <strong>the</strong> fur<strong>the</strong>r premise<br />
(P3) The <strong>in</strong>stantiation relation is itself <strong>in</strong>stantiated (by<br />
<strong>the</strong> entities it relates). 11<br />
Conditions (P1), (P2) <strong>and</strong> (P3) are jo<strong>in</strong>tly sufficient for <strong>the</strong><br />
regress. Are <strong>the</strong>y also <strong>in</strong>dividually necessary? I consider<br />
(P2) to be superfluous, s<strong>in</strong>ce any <strong>in</strong>stantiation relation is<br />
7 Aga<strong>in</strong>, I assume that <strong>the</strong> identity of a bundle depends on <strong>the</strong> elements<br />
constitut<strong>in</strong>g it.<br />
8 For Armstrong, not only properties but also particulars are nontransferable;<br />
particulars have <strong>the</strong>ir properties of necessity. Therefore, Armstrong has two<br />
<strong>in</strong>dependent means to secure <strong>the</strong> <strong>in</strong>tended supervenience relation.<br />
9 As <strong>in</strong> <strong>the</strong> case of tropes, <strong>the</strong>re are at least three possible conceptions of <strong>the</strong><br />
nontransferability of universals. The supervenience claim would have to be<br />
restricted to <strong>the</strong> analogues of (i) <strong>and</strong> (ii.a).<br />
10 The best, <strong>and</strong> perhaps only, known way to achieve this is by replac<strong>in</strong>g<br />
transworld identity with a counterpart relation for particulars (as David Lewis<br />
(1968 <strong>and</strong> 1986) <strong>and</strong> Armstrong (2004b) suggest) or for properties, depend<strong>in</strong>g<br />
on <strong>the</strong> dem<strong>and</strong>s of <strong>the</strong> <strong>the</strong>ory. Given a suitable semantics, sentences may turn<br />
out to be cont<strong>in</strong>gent, although <strong>in</strong>stantiations are necessary.<br />
11 One of <strong>the</strong> few to recognize <strong>the</strong> need for this condition is Loux (1998,<br />
p. 38).<br />
Does Bradley’s Regress Support Nom<strong>in</strong>alism? — Wolfgang Freitag<br />
ipso facto also an entity. (P1) <strong>and</strong> (P3) are hence jo<strong>in</strong>tly<br />
sufficient for Bradley’s regress. I consider <strong>the</strong>m also <strong>in</strong>dividually<br />
necessary: (P1) states <strong>the</strong> dem<strong>and</strong> for an <strong>in</strong>stantiation<br />
relation given any <strong>in</strong>stantiation, while (P3) makes<br />
certa<strong>in</strong> that this <strong>in</strong>stantiation relation dem<strong>and</strong>s fur<strong>the</strong>r <strong>in</strong>stantiation.<br />
Thus (P1) <strong>and</strong> (P3) constitute, I th<strong>in</strong>k, <strong>the</strong><br />
proper analysis of <strong>the</strong> basis of Bradley’s regress.<br />
Given this analysis, <strong>the</strong>re is a second way of avoid<strong>in</strong>g<br />
Bradley’s regress: accept (P1) <strong>and</strong> deny (P3); accept<br />
<strong>in</strong>stantiation relations <strong>and</strong> <strong>the</strong>refore take <strong>the</strong> first step of<br />
<strong>the</strong> regress, but block <strong>the</strong> regress by deny<strong>in</strong>g that <strong>the</strong> <strong>in</strong>stantiation<br />
relation is itself <strong>in</strong>stantiated. This option should<br />
be <strong>the</strong> natural path to take for substance–attribute views<br />
operat<strong>in</strong>g with cont<strong>in</strong>gent <strong>in</strong>stantiation, <strong>the</strong>ories of types<br />
(2b) or (3b) albeit without <strong>the</strong> unnatural condition that <strong>in</strong>stantiation<br />
is necessary. There is no space to develop this<br />
option here, 12 yet <strong>the</strong> fact that (P3) is necessary for <strong>the</strong><br />
regress should elim<strong>in</strong>ate any rema<strong>in</strong><strong>in</strong>g doubts: Bradley’s<br />
regress has noth<strong>in</strong>g to do with <strong>the</strong> problem of universals.<br />
Conclusion<br />
To show that Bradley’s regress is nei<strong>the</strong>r specific to nor<br />
<strong>in</strong>surmountable for a realist about universals is one th<strong>in</strong>g.<br />
To expla<strong>in</strong> why <strong>the</strong> opposite view has been so compell<strong>in</strong>g<br />
to many, is ano<strong>the</strong>r. So let me end with a suggestion on<br />
this po<strong>in</strong>t.<br />
The source is <strong>the</strong> confusion of two different <strong>and</strong><br />
logically <strong>in</strong>dependent senses of <strong>the</strong> problem of One over<br />
Many. There is <strong>the</strong> <strong>in</strong>traworld version of <strong>the</strong> problem,<br />
which concerns <strong>the</strong> question whe<strong>the</strong>r different particulars<br />
<strong>in</strong> a s<strong>in</strong>gle world can have <strong>the</strong> very same property F. And<br />
<strong>the</strong>re is <strong>the</strong> transworld version of <strong>the</strong> problem, which<br />
concerns <strong>the</strong> question whe<strong>the</strong>r different particulars <strong>in</strong><br />
different worlds can have <strong>the</strong> very same property F.<br />
The traditional problem of universals is <strong>the</strong><br />
<strong>in</strong>traworld version of <strong>the</strong> problem of One over Many.<br />
Universals can <strong>and</strong> tropes can’t be multiply <strong>in</strong>stantiated<br />
with<strong>in</strong> a s<strong>in</strong>gle world. Bradley’s regress, on <strong>the</strong> o<strong>the</strong>r h<strong>and</strong>,<br />
concerns <strong>the</strong> transworld problem of One over Many.<br />
Transferable entities can <strong>and</strong> nontransferable entities can’t<br />
be multiply <strong>in</strong>stantiated across different worlds. Keep<strong>in</strong>g<br />
<strong>the</strong>se two versions of <strong>the</strong> problem of One over Many apart,<br />
we get a clearer grip on <strong>the</strong> dem<strong>and</strong>s that a satisfy<strong>in</strong>g<br />
metaphysical <strong>the</strong>ory must fulfil.<br />
Acknowledgments<br />
This paper draws on results from an ongo<strong>in</strong>g research<br />
project commissioned by <strong>the</strong> L<strong>and</strong>esstiftung Baden-<br />
Württemberg (Germany). I am very much <strong>in</strong>debted for <strong>the</strong><br />
support provided.<br />
12 In (Freitag 2008) I have fur<strong>the</strong>r explored this possibility.<br />
101