J. Bacon, Supervenience
J. Bacon, Supervenience
J. Bacon, Supervenience
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Studies: An International Journal for Philosophy in the Analytic Tradition
REDUCIBILITY
One feature is supervenient upon others if, while not following deductively from those
others, it nevertheless cannot vary unless they do. (1981, p. 15)
Supervenience has been popularized by Hare in ethics (1952, pp. 131ff)l and
by Davidson in his treatment of the mind-body problem (1970, p. 88). In
the words of Kim, who has led the way in clarifying it, supervenience is to be
a "determinative relationship between properties without requiring correla-
tions between them" (1978, p. 154). It is supposed to provide a happy
medium between dualism and reductionism, for example Yet, as Kim has
pointed out, a variant of supervenience entails necessary coextension: each
supervenient property is necessarily coextensive with some base property
(1983, p. 49 (4); 1985, pp. 23f). To be sure, necessarily coextensive proper-
ties need not be identical or interdeducible. But in the context of would-be
physical theory, nomically necessary coextension does establish a presump-
tion of reducibility. Thus if the 'cannot' in Campbell's definition expresses
nomic impossibility, supervenience threatens to collapse into reducibility.
My purpose here is to work out the logical relationships among several
closely related variants of supervenience. Among other things, I shall prove
a stronger form of Kim's result of necessary coextension. This narrows still
further the options for supervenience theorists.
I. CONCEPTS OF SUPERVENIENCE
Thus the same properties are found (although not necessarily instantiated)
in all words. Finally, I define
The restricted variables s and b will range over S and B respectively. In view
of (1) and diagonal closure, we have the surprising
x- yforYb(bxDby).
(3) If-B[xIB.
(4) BX]BY 4 X-
For y has the conjunction of x's B properties iff it has each of them. Since
B
- is reflexive, we get in particular
strong supervenience2
necessary coextension
propositional coextension
Fig. 1
supervenience
strong suprvenience,
necessary coextension
propositional coextension
Fig. 2
In S5, finally, all these notions collapse into equivalence except propositional
coextension.
1.5. Proofs
Here I sketch proofs of the entailments just claimed. It simplifies things that
nonmodal formulas can be treated set-theoretically, as only the denotations
of predicates matter in that case.
TI. Supervenience entails coextension. From the lemma that - C = entails
S- V{ [XiB: SX} for each s E S. For suppose sy: by (5) [YIBY. Thus [YIB E
{ [X]B: sx}, and [YIB C V{ [XIB: sx} by generalized disjunctive addition.
Since [Y1BY, it follows that y has V{ [XJB: SX}. Conversely, if y has this dis-
junction, then it has one of the disjuncts [XIB such that sx. By (4) we get
B S
X B Y, whence x
establishes the l
ties. By disjunct
tially generalize the conclusion of the lemma to 3b(s b) for each s. Ti
results by conditional necessitation.
T2. Coextension entails determination. Suppose Vs 3b(s b) and sx.
Then x has some b such that b C s. Thus demodalized determination follows
from demodalized coextension. T2 results by conditional necessitation.
T3. Determination entails consequentiality. By (5) x has [x]s, which by
(3) is in S. It follows by demodalized determination that x has some b such
that b C [xis. Since [XlB C b by generalized conjunctive simplification, we
get [XIB C [XiS. Conditional generalization yields Vx([xIB C [x]s), and
conditional necessitation yields T3.
B
T4. Consequentiality entails supervenience. Suppose tnat x = y. By (4)
[xJBy. Demodalized consequentiality then yields [x]sy, or x - y by (4). Thus
demodalized consequentiality entails B C_, so EVx([XiB C [X]S) entails
B S
=8- S.
T5. Supervenience -4 F- coextension - F determination - I- consequen-
tiality. From T1 -T4.
T6. Strong necessary coextension entails strong supervenience2. By a
proof similar to that of T2, necessary coextension entails VsVx{ sx D 3b [bx &
(b 8- s)] }. Conditional necessitation then gives T6.
T7. Strong supervenience2 entails supervenience. By denecessitation the
b 8- s component becomes b C s, yielding determination. Supervenience
follows by T3 and T4.
T8. Supervenience entails necessary coextension. By TI, supervenience
entails LVs 3b(s b), i.e. DVp[Sp D 3 b(up b)] . By Barcan(p) and necessi-
ty distribution, we get Vp[LlS<p D l 3b(p-b)]. (1) Let us erase the left
box. The right box permutes with the existential quantifier by (2), yielding
Vp[S D 3bO(po b)], which is necessary coextension.
T9. Strong supervenience2 entails strong superveniencel. By methods
similar to those of T8. [This is the only place where Barcan(x) is needed.]
TI0. Strong supervenience, entails necessary coextension. From the
lemma that strong supervenience1 entails LO(s V{b: 3 x[bx & (b 8- s)]})
for each s eS. I abbreviate {b: 3x[bx & (b 8- s)] } as B(s). Right-to-left:
by concretion, every b E B(s) is such that b 8- s. Hence, by the Boolean
analogue of constructive dilemma, VB(s) 8- s. Left-to-right: suppose sx.
Then it necessarily follows by strong supervenience1 that there is a b such
that bx & (b 8- s). Existential generalization gives 3 x [bx & (b 8- s)], whence
b E B(s) by abstraction. As x has b, by disjunctive addition x has VB(s). That
establishes s 8- VB(s), the other half of the consequent of the lemma. By
disjunctive closure, VB(s) is a B property. Thus T1O follows from the lemma
by existential and universal generalization of the consequent.
TI 1. Necessary coextension entails propositional coextension. By univer-
sal instantiation, necessary coextension yields a b such that O(sx bx) for
arbitrary x. Existential generalization gives 3 b 3yL(sx by). Finally, we
get VsVx 3b 3yEZl(sx by) by universal generalization.
That establishes the entailment relations in Fig. 1 of ? 14. Next we add the
NN-thesis of S4.
T1 2. In S4, supervenience entails strong necessary coextension. By condi-
tional necessitation, T8 gives us strong necessary coextension from OI(- C
S
_). T12 follows by the NN-thesis.
T13. In S4, supervenience strong supervenience2 - strong neces-
sary coextension. From T1 2, T6, and T7.
Finally, we move to S5 with the principle
Va 3, VYw(aw - )
i.e.,
Haugeland and Post put forward a notion very like (8), except that wand
Y are replaced by "languages". A language is a set of sentences with assigned
truth-values for each possible world in W. In other words, each sentence is
assigned a proposition (subset of W). Now Haugeland defines
Two worlds in W are discernible with language L just in case there is a sentence of L
which is true at one, and not at the other....
I have now exhibited and demonstrated the logical relations among several
different concepts of supervenience. Some of these are fairly obvious, and
some were known before. So far as I know, T8, T10, and T12 are new. By
T8, (weak) supervenience entails necessary coextension, and not just co-
extension as in Kim (1983, p. 49(3)), provided we assume diagonal closure.
Barring that, strong supervenience, still entails necessary coextension (T10)
According to Kim, only strong supervenience2 entails necessary coexten-
sion. And by T12, in S4 (weak) supervenience entails strong supervenience2,
notwithstanding Kim's assertion, "Strong supervenience entails weak super-
II. IMPLICATIONS
Reducibility in principle. S C B
Kim's results (1983, 1985) as extended above might seem to narrow the use-
fulness of supervenience, particularly where potential theories are concerned.
Yet I have found most supervenience theorists unmoved. David Lewis has
suggested to me that they probably get off the boat right at the lemma of TI:
the formation of a possibly infinite disjunction of possibly infinite conjunc-
tions of base properties. (The disjunction will be infinite if the supervenient
property has infinitely many instances. A conjunction will be infinite if the
corresponding instance of the supervenient property has infinitely many
simple base properties.) Teller puts the point forcefully:
Properties which in this way do not show up similarities I call properties in an extended
sense only, an epithet which clearly applies to the disjunctive properties which follow
from strong supervenience as described by Necessary Coextension.
Next, we must keep in mind that these disjunctive properties are going to be awfully
fat....
Now what, I want to ask, makes it appropriate to call a property 'physical' when it
is such a disparate and infinitely long disjunction of disjuncts each of which is prob-
ably already an uncountable [?] conjunction of fimite physical descriptions? Yes, it is
a Boolean combination of physical properties, but I feel that to call it "physical" threat-
ens to be misleading. (1983a, p. 58f)
physical seems a desperate verbal ploy to save the "mental". If that's all
it is for a property to be mental, then ontologically materialism has carried
the day. In any event, the results of part I can all be reinstated by simply
replacing the word 'property' by 'Boolean combination of properties' through-
out.
11.5. Conclusion
ACKNOWLEDGEMENT
NOTES
'Supervenient' in this sense seems to appear first in Hare (1952, p. 80), but Hare
denies originating this use of the term. For further history, see Kim (1982).
2 A single comprehensive domain D is compatible with varying inner domains Eu, EW,...
of "existents" for different words u, w,... "Inner" quantifiers, restricted to existents,
would not satisfy the Barcan equivalence, for existence is a contingent matter. Here I
use "outer" individual quantifiers, ranging over all of D.
3 The symbol for strict inclusion, or property-entailment, is due to Barcan Marcus
(1946), viz. an inverted fishhook, here approximated as '8-'.
4 Sometimes 'base' is used for a more limited set of properties whose closure is B, and
sometimes for the particular property upon which another supervenes.
' In (1982, p. 134) Kim defined strong supervenience as VsVx{sx D 3 b(bx & (b 8- s)] }
(my notation), but this is clearly a slip of the pen for strong supervenience1 or 2
6 Currie himself does not rest with (7), but develops a more complicated notion of
supervenience involving quantification over times and permutations of D. He derives a
corresponding version of coextension (1982, Claim 3).
7 The assertion is refuted by the following countermodel. Let D = {x, y } and let w be
the sole possible world. Let the accessibility relation be universal over worlds. Let s be
true of x and false of y in w. Let b be the universal property. S is the closure of {s} and
B of {b}. [XIB or B* (x's B-maximal property) is then the same as b. In this model,
(7) is trivially true, while b I s.
I am indebted to D. M. Armstrong for this insight. See my (1980).
"Suppose that we say 'St. Francis was a good man'. It is logically impossible to say
this and to maintain at the same time that there might have been another man placed
in precisely the same circumstances as St. Francis, and who behaved in exactly the same
way, but who differed from St. Francis in this respect only, that he was not a good man."
(1952, p. 145, my italics)
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