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White's remarks about the "state of Aristotle's thinking in Metaphysics
V" are meant to show why Aristotle "is neglecting to give a clear account
of the notion of identity":
Rather than thinking about what it is for X and
Y to be identical, he has his mind fixed on what it is for an entity to
be unitary. . . This discussion drags the treatment of sameness along on
its coattails. (ibid., p. 188)
Nevertheless, an account of the notion of identity which specifies what
it is for X and Y to be identical need not dispense, as White
suggests, with an Aristotelian notion of sameness as oneness in substance.
If particular and complex are one and the same entity, not only are the
usual properties of identity engendered by the mutual relations of complexes
and their constituents, but these relations also provide a foundation,
as evidenced by the theorems that follow, for an Aristotelian notion of
sameness as oneness in substance.
T1a,b and T2a,b specify identity-conditions for the constituents of
complexes. T1a spells out when individuals are materially identical; T1b,
when individuals are self-identical; T2a, when haecceities are materially
identical; and T2b, when haecceities are self-identical. Each b theorem
is a corollary of its a counterpart.
-
T1a
 xy(x
= y  wz(w.Z
cont x & w.Z cont y))
-
(Individuals are identical just in case there is some complex of which
they are parts.)
-
T1b x(x
= xwz(w.Z
cont x))
-
(An individual is self-identical just in case there is some complex of
which it is a part.)
-
T2a xy(X
= Ywz(w.Z
emb X & w.Z emb Y))
-
(Haecceities are identical just in case there is some complex of which
they are parts.)
-
T2b x(X
= Xwz(w.Z
emb X))
-
(A haecceity is self-identical just in case there is some complex of which
it is a part.)
The material identity of the individuals x and y is thus sufficient for
the self-identity of x and y, as is the material identity of the haecceities
X
and Y for the self-identity of X and Y.
T3a, T3b, T3c and T3d specify identity conditions for complexes (a complex
being, arguably, an Aristotelian tode ti in modern garb). T3a spells
out when complexes are materially identical; T3b, when a complex is self-identical;
T3c, when C-complexes are materially identical (a C-complex being a complex
whose constituents correspond); and T3d, when a C-complex is self-identical.
Each b theorem is a corollary of the corresponding a theorem, and T3d
is a corollary of T3c.
-
T3a uvwx(u.V
= w.Xyz(y.Z
cont u & y.Z cont w & y.Z
emb V & y.Z emb X))
-
(Complexes are identical just in case there is some complex of which their
constituents are parts.)
-
T3b xy(x.Y
= x.Ywz(w.Z
cont x & w.Z emb Y))
-
(A complex is self-identical just in case there is some complex of which
its constituents are parts.)
-
T3c xy(x.X
= y.Ywz(w.Z
cont x & w.Z cont y & w.Z emb X & w.Z
emb Y))
-
(C-complexes are materially identical just in case there is some complex
of which their constituents are parts.)
-
T3d x(x.X
= x.Xwz(w.Z
cont x & w.Z emb X))
-
(A C-complex is self-identical just in case there is some complex of which
its constituents are parts.)
For complexes as well as their constituents, material identity is thus
a sufficient condition for self-identity. |
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