A-Theory-of-Complexes (Greenberg)

A THEORY OF COMPLEXES*
William J. Greenberg
It is vain to do with more
what can be done with fewer.
Entities must not be multiplied beyond necessity.
--William of Occam
*Epistemologia XIX, 1996, pp. 85-112
It is vain to try to do with
fewer what requires more.
Entities must not be reduced
to the point of inadequacy.
--Karl Menger
Abstract:
In 'A Theory of Complexes', the simple form of the pure theory of identity found in
logic books is subordinated to a complex form of the theory whose background
ontology is one of structured individuals, orcomplexes. In the complex form:
The axioms of the simple form proceed from principles more basic.1
The semantic difference between (true) 'a = b' and 'a = a' supervenes on features
of a and b.2
Identicals such as nine and the number of planets differ in their modal
properties and relations.3
'A Theory of Complexes' puts into question: the privileging of FOL individuals, the
expressive completeness of FOL, and ontology-free logic.
1.1 Introduction1
In his review of Errol Harris's An Interpretation of the
Logic of Hegel,2 John E. Smith highlights what is mainly at issue
between the Hegelian conception of logic and the formalist
conception favored by modern empiricism and many forms of analytic philosophy.
The Hegelian holds logic to be "internally
related at every level to the real content it articulates" (463).
In contrast, the formalist pegs logic as neutral--both with
respect to "the nature of things" and to "what there is" (ibid.).
How fares the logic of identity in this dispute?
Is iden-
tity neutral, both to "the nature of things" and to "what there
is"?
Or is the complexity of this relation a legacy of its terms?
From a formalist standpoint, the identity-relation is indif-
ferent to the nature of its terms:
what a particular is (qua
value of a variable) has no bearing on its relation to itself.
Unrelated to the things it relates, identity thus has its nature
imposed from without--by classical axioms which appear in the
1
I am grateful to Paul Schachter, John Olney, and an anonymous referee for
comments on earlier versions of this paper--and to the College of Humanities of the
University of Puerto Rico at Rio Piedras for reductions in load which helped make
completion of this paper possible.
2
British Journal for the Philosophy of Science, V. 36, No. 4, Dec. 1985:
459-465.
3
guise of empirical observations or arbitrary stipulations, or as
the inexpugnable avatars of a Self-Certifying Logical Truth.
In contrast, from the (broadly) Hegelian standpoint of 'A
Theory of Complexes', the properties of identity have their
source in the kind of entity over which individual variables
range: a particular--the particular-cum-complex--which, unlike
the denatured simple of Russell, Bergmann, Allaire, et al.,3 is
ontologically differentiated and logically complex.
The intrin-
sic features of this particular ground a minimal identity relation4, which ramifies classically and non-classically.5
3
Such a particular is as portrayed by John Baker, in
These
'Particulars: Bare,
Naked, and Nude,' (Nous, V. 1, No. 2, May 1967: 211-212):
Particulars are nude in that they have no natures, that is, they are not
necessarily connected to any specific property or set of properties.
A
nude particular has no nature, and is to be distinguished from the naked
particular which has no properties.
Those who claim that there are bare
particulars, Russell, Bergmann, Allaire, et al., claim that they are nude
of natures... (211)
4
By a minimal identity-relation I understand one whose properties--symme-
try, transitivity, weak reflexivity, indiscernibility for identicals and
identity for self-idenical indiscernibles--are criterial for identity.
5
The classical relation being strongly reflexive, indiscernibles are
identical.
In contrast, in the non-classical relation identity is partially
irreflexive, and so some indiscernibles are not identical.
4
features also ground a difference between formal and material
identity--the identity of a and a and the identity of a and b-which sets 'a = a' and 'a = b' apart in cognitive content.
Such
properties of identity are thus not external to its terms, as in
formalist treatments.
Here they are engendered instead by fea-
tures of the entity which identity self-relates.
A theory of the particular-cum-complex encapsulates a theory
of its constituents: haecceities6 and individuals7.
To the theory
HI of haecceities and individuals I now turn.
1.2.0 The Language of HI
HI is developed within a first-order, multi-sorted language
L.
Among its primitive signs, L counts two predicate constants,
(1.1)"ex" for exemplification
(1.2)"=" for identity
three sorts of variables,
(2.1)variables for individuals: u, v, w, x, y, z, u1, u2, ...,
ui, ...
6
Pick any particular.
Corresponding to your particular is a property
that, as established by T46 (see section 1.4.6.1), only your particular has if
anything does.
This property is a haecceity.
My notion of a haecceity and
Adams's (1979, 1981) notion of a thisness are compared in note 24.
7
A haecceity needs a support, or substratum, if it is to belong to an
actual thing.
An entity which can be such a support, or substratum, is an
individual.
5
(2.2)variables for haecceities: U, V, W, X, Y, Z, U1, U2, ...,
Ui, ...
(2.3) variables for particulars: u, v, w, x, y, z, u1, u2,...,
ui, ...
the usual sentential connectives,
(3.1) ¬,
, &, v,
and the universal and existential quantifiers.
(3.2)
,
In the metalanguage of L, t1, t2, ..., ti range over
individual variables; T1, T2, ..., Ti over haecceity variables;
t1, t2, ..., ti over particular variables; and α, β, γ, ... over
expressions.
A formula of L is any expression provided for by the following conditions:
(4.1)
ti ex Tj, ti = tj and Ti = Tj are atomic formulas.
(5.1)
Every atomic formula is a formula.
(5.2)
If α is a formula, ¬α is a formula.
(5.3)
If α and β are formulas, (α
and (α
(5.4)
8
β), (α v β), (α & β)
β) are formulas.
If β is a formula,
tiβ and
tiβ are formulas.8
In L, the particular variables u, v, w, x, y, z, u1, u2, ..., ui, ..., occur
only in quantifier expressions, binding individual and haecceity variables as
indicated in (1.2.1).
6
1.2.1 Variable-binding in L9
The logic of L departs from that of conventional, singlesorted, first-order languages in its definition of variablebinding and formulation of the Quantifier Rules.
An occurrence
of a variable ti/ti/Ti within a formula φ is a bound occurrence of
ti/ti/Ti in φ if, and only if, it occurs within some part of φ
which is a formula of the form
tiβ or of the form
tiβ.
Other-
wise, that occurrence is a free occurrence of ti/ti/Ti in φ.
formula
tiβ or
If a
tiβ occurs within a formula φ (or is the formula
φ itself), then the scope in φ of that occurrence of the quantifier
ti or
ti is the formula
tiβ or
tiβ itself.
1.2.2 Quantifier Rules in L
The rule of Universal Instantiation sanctions the move from
a formula:
(1)
tiΠ(... ti/ti/Ti ...)10
to a formula:
(2)
Π(... tj/tj/Tj ...)
where (2) results from replacing every occurrence of ti/ti/Ti free
in:
(3)
Π(... ti/ti/Ti ...)
by occurrences of tj/tj/Tj free in (2).
9
UI thus sanctions the
My treatment of variable-binding in L is an adaptation of the treatment
of variable-binding for first-order predicate logic in Rogers (1971).
10
Π is possibly null.
7
move from:
(4)
u v(v ex U
u ex U)
(5)
v(v ex U
u ex U)
(6)
v(v ex W
w ex W)
(7)
v(v ex X
x ex X)
to:
.
.
.
.
.
.
.
.
in each of which generically identical, individual and haecceity
variables u/U free in,
(8)
v(v ex U
u ex U)
have been replaced by other, or perhaps the same, generically
identical, individual and haecceity variables ti/Ti free in:
(9)
v(v ex Ti
ti ex Ti)
UI does not, however, sanction the move from (4) to
(10)
v(v ex W
x ex W),
for (10) results from replacing generically identical, individual
and haecceity variables u and U in (5), by the generically distinct x and W in (10).
The rule of Existential Instantiation sanctions the move
from a formula:
(11)
tiΠ(... ti/ti/Ti ...)
to a formula:
(12)
Π(... tj/tj/Tj ...)
where (12) results from replacing every occurrence of ti/ti/Ti
free in:
(13)
Π(... ti/ti/Ti ...)
8
by occurrences of tj/tj/Tj free in (12), which do not occur free
in any earlier line of a proof.
1.2.3 The Postulates of HI
P1-P3 describe the relations of haecceities and individuals
which determine the properties of identity in the realm of
individuals.
P1
wxyz((w ex Y & x ex Y)
(w ex Z
x ex Z))
(If two individuals co-exemplify any haecceity, they exemplify
the same haecceities.)
P2
xy(x = y
z(x ex Z & y ex Z))
(Two individuals are identical just in case they co-exemplify
some haecceity.)
P3
xy(x ex Y
y ex Y)
(If x exemplifies Y, y exemplifies Y.)
P4-P6 describe relations of individuals and haecceities which
determine the properties of identity in the realm of haecceities.
P4
wxyz((y ex W & y ex X)
(z ex W
z ex X))
(If two haecceities are co-exemplified by any individual, they are exemplified by the same individuals.)
P5
xy(X = Y
z(z ex X & z ex Y))
(Two haecceities are identical just in case they are coexemplified by some individual.)
P6
xy(x ex Y
x ex X)
(If x exemplifies Y, x exemplifies X.)
1.2.4 Identity in HI
P1-P6 found an identity relation which is weakly reflexive,
9
symmetric, and transitive;
T1*
(x = x))11
x( y(x = y)
(Identity is weakly reflexive.)
T2*
xy(x = y
y = x)
(Identity is symmetric.)
T3*
xyz((x = y & y = z)
x = z)
(Identity is transitive.)
identicals that are indiscernible; and indiscernibles that are
identical-if-self-identical.
T4*
xyz(x = y
(x ex Z
y ex Z))
(Identicals are indiscernible.)
T5*
xy((x = x & y = y)
( z(x ex Z
y ex Z)
x = y))
(Self-identical indiscernibles are identical.)
P1-P6 thus found a minimal identity relation.12
From P2 it follows that an individual is self-identical just
in case there is some haecceity it exemplifies; and from P5, that
11
Let α(ti,Tj/Ti,tj) be the result of replacing ti and Tj everywhere in α by Ti
and tj, and then replacing every resultant occurrence of Tk ex tl by tl ex Tk.
The
asterisk ("*") will then be understood to indicate that T1(ti,Tj/Ti,tj), which makes
identity weakly reflexive for haecceities, is also a theorem.
12
By a minimal identity relation, I understand an identity relation whose
properties are criterial for identity.
10
a haecceity is self-identical just in case there is some individual it is exemplified by.
T6*
x(x = x
y(x ex Y))
Hence, every individual is self-identical just in case every
individual exemplifies; and every haecceity is self-identical
just in case every haecceity is exemplified.
T7*
x(x = x)
x y(x ex Y)
From T5*-T7*, it also follows, moreover, that indiscernibles are
identical just in case every individual exemplifies and every
haecceity is exemplified.
T8*
xy( z(x ex Z
y ex Z)
x = y)
x y(x ex Y)
The ontological scaffolding provided by P1-P6 thus lends itself
to further specification.
Is every haecceity exemplified?
every individual exemplify?
Does
In view of T9*13, we have here not
two questions but one.
T9*
x y(y ex X)
x y(x ex Y)
(If every haecceity is exemplified, every individual
exemplifies.)
1.2.5 Extending HI
1.2.5.1 HI+
A world in which every individual exemplifies--and every
haecceity is exemplified--is described by adding P10 to the
13
T9(ti,Tj/Ti,tj) specifies that every individual exemplifies only if every
haecceity is exemplified.
11
postulates of HI, yielding the theory HI+.
P10
x y(y ex X)
(All haecceities are exemplified.)
In HI+, identity is totally reflexive and indiscernibles are
identical, and so HI+ is a classical extension of HI.
T10*
x(x = x)
(Identity is totally reflexive.)
T11*
xy( z(x ex Z
y ex Z)
x = y)
(Indiscernibles are identical.)
1.2.5.2 HIIn contrast, a world in which not every individual exemplifies--and not every haecceity is exemplified--is described by
adding P10'14 to the postulates of HI.
P10' ¬ x y(y ex X)
(Some haecceity is not exemplified.)
In the resulting theory HI-, identity is partially irreflexive
and not all indiscernibles are identical. HI- is thus a nonclassical extension of HI.
T10'* ¬ x(x = x)
(Identity is partially irreflexive.)
T11'* ¬ xy( z(x ex Z
y ex Z)
x = y)
(Not all indiscernibles are identical.)
1.2.6 The ontological significance of HI, HI+, and HIThe reciprocal connections of individuals and haecceities in
14
α' is the contradictory negation of α.
12
HI scaffold properties of identity which, perhaps because they
resist analysis in classical terms, have generally been taken to
be self-evident--and ontology-free.
In HI, however, that identi-
ty is weakly reflexive, identicals indiscernible, and indiscernibles identical if self-identical are propositions which rest
upon yet more basic ontological principles.
Thus, the notorious-
ly "self-evident" dictum that everything is what it is divulges,
upon analysis, ontological conditions whose satisfaction depends
not on a logician's fiat but on the way the world is.
These
conditions hold in HI+, in which every haecceity is exemplified
and every individual exemplifies, so that identity is totally
reflexive and indiscernibles are identical.
HI- posits, in
contrast, a world populated by unexemplified haecceities and nonexemplifying individuals, in which identity is not totally
reflexive and not all indiscernibles are identical.
these worlds is ours?
Which of
Ask not Logic, for Logic will not decide.
But I promised an account of the properties of the identity
relation as this applies to particulars.
nought but individuals and haecceities.
So far we have seen
The road to particulars
is from individuals and haecceities by way of complexes.
theory of complexes I now turn.
13
To a
HI
HI+
HI-
Identity is weakly reflexive: T1*
Yes
Yes
Yes
Identity is symmetric: T2*
Yes
Yes
Yes
Identity is transitive: T3*
Yes
Yes
Yes
Identicals are indiscernible: T4*
Yes
Yes
Yes
Indiscernibles are identical if
self-identical: T5*
Yes
Yes
Yes
There are unexemplified haecceities.
---
No
P10
Yes
P10'
There are 'bare' individuals.
---
No
P10
Yes
P10'
Identity is totally reflexive.
---
Yes
T10*
No
T10'*
Yes
T11*
No
T11'*
Indiscernibles are identical.
-
IMPORTANT THEOREMS
14
--
1.3 A Theory of Complexes
1.3.1 The Language L'
The theory C of complexes is developed within L', a firstorder language which contains L.
In addition to the relational
constants and variables of L, the symbols of L' include:
(1.3) "emb" for embodiment15 and "cont" for containment16.
(1.4) an operation symbol: .
(2.4)variables for complexes, as specified by the rule: if ti is
an individual variable and Tj a haecceity variable, ti.Tj
is a complex variable.
In addition to the atomic formulas of L, L' also includes atomic
formulas, as provided for by:
(4.2) ti.Tj emb Tk, ti.Tj cont tk, and ti.Tj = tk.Tl are atomic
formulas.
The definition of variable-binding and formulation of Quantifier
Rules in L' is as in L.
15
Embodiment relates a particular to an haecceity just in case there is some
complex to which the haecceity and constituents of the particular belong (see T48,
section 1.4.6.2).
16
Containment relates a particular and an individual just in case there is
some complex to which the individual and constituents of the particular belong (see
T49, section 1.4.6.2).
15
1.3.2 The Theory C
To the postulates of HI, C subjoins P7-P9.
P7
wxyz(w.X = y.Z
(w ex X & w ex Z & y ex X))
(w.X and y.Z are identical just in case w exemplifies
X and Z, and y exemplifies X.)
P8
xyz(x.Y emb Z
(x ex Y & x ex Z))
(x.Y embodies Z just in case x exemplifies Y and Z.)
P9
xyz(x.Y cont z
(x ex Y & z ex Y))
(x.Y contains z just in case x and z exemplify Y.)
P7 determines when complexes are identical; P8, when a complex
embodies a haecceity; and P9, when a complex contains an
individual.
In so doing, P7-P9 determine, together with P1-P6,
the properties of identity as this relation applies to complexes.
1.3.3 Identity in C
Identity in C is weakly reflexive, symmetric, and transitive
for individuals and haecceities (T1*-T3*) and complexes (T12T14); identicals are indiscernible (T4*, T15#); and indiscernibles are identical if self-identical (T5*, T16#).
C thus
grounds a minimal identity relation.
T12
wx( yz(w.X = y.Z)
w.X = w.X)
(Identity is weakly reflexive for complexes.)
T13
wxyz(w.X = y.Z
y.Z = w.X)
(Identity is symmetric for complexes.)
T14
uvwxyz((u.V = w.X & w.X = y.Z)
u.V = y.Z)
(Identity is transitive for complexes.)
16
T15#
uvwxy(u.V = w.X
(u.V emb Y
w.X emb Y))17
(Identical complexes are indiscernible.)
T16#
uvwx((u.V = u.V & w.X = w.X)
w.X emb Y)
( y(u.V emb Y
u.V = w.X))
(Self-identical complexes are identical if indiscernible.)18
1.3.4 Reflexivity and Complexes
In HI+, obtained by subjoining P10 to the postulates of HI,
identity was totally reflexive for individuals and haecceities.
However, subjoining P10 to the postulates of C does not suffice
to make identity totally reflexive for complexes.
Thus, from P7
it follows that any complex is self-identical just in case the
individual and haecceity which constitute it are such that the
17
Let α(ti,Tj//Ti,tj) be obtained from α by simultaneously replacing ti and Tj
everywhere in α by Ti and tj, "cont" and "emb" by "emb" and "cont", and then Tk.tl by
tl.Tk.
The symbol "#" will then be understood to indicate that T15(ti,Tj//Ti,tj),
which makes complexes indiscernible in respect of the individuals they contain, is
also a theorem.
18
From T16# it follows that every pair of indiscernible complexes is
identical just in case every complex is self-identical;
T17# uvwx( y(u.V emb Y
w.X emb Y)
u.V = w.X)
uv(u.V = u.V)
a principle which, in view of T5*, holds as well for individuals and haecceities.
17
individual exemplifies the haecceity:
T18
xy(x.Y = x.Y
x ex Y)
Hence every complex is self-identical just in case every individual exemplifies every haecceity:
T19
xy(x.Y = x.Y)
xy(x ex Y)
Moreover, it follows--from P2, P6 and T4*--that every individual
exemplifies every haecceity just in case some haecceity is such
that every individual exemplifies it:
T20
xy(x ex Y)
x y(y ex X)
Therefore, for identity to be totally reflexive for complexes,
not only must every haecceity be exemplified, but some haecceity
must be universally exemplified:
T21
xy(x.Y = x.Y)
x y(y ex X)
Identity in C is thus not totally reflexive for complexes unless
no two complexes are distinct.
T22
wx(w.X = w.X)
wxyz(w.X = y.Z)
P10 thus fails to make identity totally reflexive for complexes
generally (although it does make identity totally reflexive for
individuals and haecceities).
But P10 does make identity totally
reflexive for complexes w.W, x.X, y.Y, z.Z,..., whose constituents correspond.
T23
x( y(y ex X)
x.X = x.X)
T24
x y(y ex X)
x(x.X = x.X)
I will refer to these as C-complexes.
18
1.3.5 Extending C
1.3.5.1 C+
The theory C+ is obtained by subjoining P10 to the deductive
apparatus of C.
In C+, identity is partially reflexive for
complexes,
T25
xy(x.Y = x.Y)
but totally reflexive for individuals and haecceities (T10*), and
for C-complexes:
T26
x(x.X = x.X)
Therefore, in C+ individuals and haecceities (T11*) and C-complexes
are identical if indiscernible:
T27#
xy( z(x.X emb Z
y.Y emb Z)
x.X = y.Y)
1.3.5.2 CIn C-, obtained by subjoining P10' to the deductive apparatus of C, identity is partially irreflexive for individuals and
haecceities (T10'*), and for C-complexes and complexes.
T26' ¬ x(x.X = x.X)
T28' ¬ xy(x.Y = x.Y)19
Consequently, in C- not all individuals and haecceities (T11'*),
19
The following is also a theorem of C:
xy(¬(x.Y = x.Y)
wz¬(x.Y = w.Z))
A complex which is not self-identical is thus diverse from every complex.
19
or C-complexes and complexes, are identical if indiscernible.
T27'# ¬ xy( z(x.X emb Z
y.Y emb Z)
T29'# ¬ wxyz( v(w.X emb V
y.Z emb V)
x.X = y.Y)
w.X = y.Z)
C
C+
C-
Identity is weakly reflexive: T1*/T12
Yes
Yes
Yes
Identity is symmetric:
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
---
Yes
No
T10*
T10'*
Yes
No
T10*
T10'*
Yes
No
T2*/T13
Identity is transitive:
T3*/T14
Identicals are indiscernible:
T4*/T15#
Self-identical indiscernibles are identical: T5*/T16#
Individuals
Identity is totally
reflexive.
Haecceities
C-complexes
---
---
T26
Indiscernibles are
identical.
T26'
Complexes
---
---
No
Individuals
---
Yes
No
T11*
T11'*
Yes
No
T11*
T11'*
Yes
No
T27#
T27'#
---
No
T29'#
Haecceities
C-complexes
Complexes
IMPORTANT THEOREMS
20
---
---
---
T28'
1.4 C-complexes and Particulars
In classical identity theory, identity is totally reflexive,
symmetric and transitive, identicals are indiscernible, and
indiscernibles identical.
Two other theories suggest themselves
here, a restriction of the classical theory, minimal identity
theory, in which it cannot be established that identity is
totally reflexive or that indiscernibles are identical; and a
deviation of the classical theory, which I shall refer to as nonclassical identity theory, in which identity is partially
irreflexive and not all indiscernibles are identical.20
Classical and non-classical identity theory both include and
extend the minimal theory.
They include it because every thesis
of the minimal theory is also a thesis of the classical and nonclassical theories: in the minimal, classical and non-classical
theories, identity is symmetric, transitive, and weakly reflexive; identicals are indiscernible, and self-identical indiscernibles are identical.
They extend it because the classical and
non-classical theories each have theses that are not theses of
20
The class of wffs of the classical theory and the non-classical theory
coincide.
However, the theorems of these theories differ, the Reflexive Law of
Equality and logically equivalent principle of the Identity of Indiscernibles
being provable in the former, while their contradictory negations are provable
in the latter.
Non-classical identity theory is thus a deviation, as Susan
Haack defines "deviation" in her (1974), of classical identity theory.
21
the minimal theory.
In the classical but not the minimal theory,
identity is totally reflexive and indiscernibles are identical;
in the non-classical but not the minimal theory, identity is
partially irreflexive and not all indiscernibles are identical.
C-complexes in C provide a model for minimal identity
theory: a domain in which identity is symmetric (T13), transitive
(T14), and weakly reflexive (T12), identicals are indiscernible
(T15#), and self-identical indiscernibles are identical (T16#).
C-complexes in C+ provide a model for classical identity theory:
a domain in which identity is symmetric, transitive, and totally
reflexive (T26), identicals are indiscernible, and indiscernibles
are identical (T27#).
Finally, C-complexes in C- provide a model
for non-classical identity theory: a domain in which identity is
symmetric, transitive, and partially irreflexive (T26'), identicals are indiscernible, but not all indiscernibles are identical
(T27'#).
C-complexes may thus be taken as the values of particu-
lar variables in each of the aforementioned theories,21 and C, C+
and C- may be enriched, as in the theory P and its extensions, by
treating C-complexes as particulars.22
21
The individual variables of classical identity theory I refer to as
particular variables, a particular being construed here as an entity whose
constituents are individuals and haecceities.
22
C-complexes in C, C+ and C- are intended both as realizations of minimal,
classical and non-classical identity theory, and as schematizations of contrasting
22
1.4.1 The Language of P
P is developed within L*, a first-order language obtained by
adding to the formation rules of L' the clause (4.3).
(4.3) ti emb Tj, ti cont tj, ti = tj, ti = tj.Tk and
tj.Tk = ti are atomic formulas.
The definition of variable-binding and formulation of the Quantifier Rules in L* is as in L' and L.
1.4.2 The Deductive Apparatus of P
P augments the deductive apparatus of C by the inference
rules I1 and I2.
I1: From φ to infer ψ, where ψ results from φ by substituting
for t.T at one or more places of its occurrence.
I2: From φ to infer ψ, where ψ results from φ by substituting
t.T for t at one or more places of its occurrence.
I1 and I2 license the translation of an assertion about t.T/t
into one about t/t.T.
The warrant for I1 and I2 is that the
C-complex t.T and particular t are one and the same entity.
systems of real relations of particulars and their constituents.
C-complexes thus
constitute a model in both senses Jean Ladrière distinguishes for this term in his
(1979).
For Ladrière, a model is:
...une construction idéale, intermédiaire entre une théorie au sens strict
(considerée comme ensemble de propositions munie d'une structure déductive)
et un domaine concret dont il s'agit d'analyser le fonctionnement... (p. 183)
23
t
In HI, the properties of identity are determined by the
reciprocal relations of individuals and haecceities.
In C, the
properties of identity are determined by the reciprocal relations
of individuals, haecceities and complexes.
In P, particulars are
C-complexes, and so the properties of identity (as this applies
to particulars) are determined by the reciprocal relations of
individuals, haecceities, and C-complexes.
1.4.3 Identity in P
For particulars in P, as for individuals and haecceities in
HI and complexes in C, identity is weakly reflexive (T30),
symmetric (T31) and transitive (T32); identicals are indiscernible (T33#); and self-identical indiscernibles identical (T34#).
T30
x( y(x = y)
x = x)
T31
xy(x = y
T32
xyz((x = y & y = z)
y = x)
T33#
xyz(x = y
T34#
xy((x = x & y = y)
x = z)
(x emb Z
y emb Z))
( z(x emb Z
y emb Z)
x = y))
As in HI and C, moreover, in P it cannot be established whether
identity is reflexive or indiscernibles are identical.
For the
self-identity of a particular is bound up in P with the unity of
the individual and haecceity which constitute it.
T35
x(x = x
x ex X)
This unity depends in turn upon a haecceity's being exemplified,
T36
x(x ex X
y(y ex X))
24
so that every particular is self-identical, and identity totally
reflexive if, and only if, every haecceity is exemplified.
T37
x(x = x)
x y(y ex X)
However, whether every, or any, haecceity is exemplified is not
specified in P, for P specifies just those relations of individuals and haecceities which ground properties that are criterial
for identity.
Consequently, whether identity is totally or par-
tially reflexive--or indiscernibles are identical--cannot be
established in P.
For these properties differentiate species
within a genus without being relevant to the genus itself.
1.4.4 Extending P
In discussing the different species of identity, it will be useful to
distinguish classical particulars/individuals/haec- ceities/complexes from
non-classical ones.
Definition 1:
ti/ti/Ti is classical iff
tj(ti/ti/Ti = tj/tj/Tj)
Definition 2:
ti.Tj is classical iff
Definition 3:
ti/ti/Ti is non-classical iff
tkti(ti.Tj = tk.Ti)
¬ tj(ti/ti/Ti = tj/tj/Tj)
Definition 4:
ti.Tj is non-classical iff ¬ tkti(ti.Tj = tk.Ti)
From the foregoing it follows that every particular/individual/haecceity/complex is either classical or non-classical, and that none
is both.
25
1.4.4.1 P+
P+ is obtained by subjoining P10 to the postulates of P.
P10
x y(y ex X)
(Every haecceity is exemplified.)
Theorems of P+ include:
T38
x(x = x)
(Identity is totally reflexive for particulars.)
T39#
xy( z(x emb Z
y emb Z)
x = y)
(Indiscernible particulars are identical.)
T40
x y(x = y)
(Every particular is classical.)
T41
xywz(x.Y = w.Z)
(Some complex is classical.)
1.4.4.2 PP- is obtained by subjoining P10' to the postulates of P.
P10' ¬ x y(y ex X)
(Not every haecceity is exemplified.)
Theorems of P- include:
T38' ¬ x(x = x)
(Identity is partially irreflexive for particulars.)
T39'#
¬ xy( z(x emb Z
y emb Z)
x = y)
(Not all indiscernible particulars are identical.)
T40' ¬ x y(x = y)
(Some particular is non-classical.)
26
T42' ¬ xy wz(x.Y = w.Z)
(Some complex is non-classical.)
1.4.5. Extending P+ and P1.4.5.1 P++
P++ is obtained by subjoining P10* to the postulates of P.
P10*
x y(y ex X)
(Some haecceity is such that every individual exemplifies it.)
Theorems of P++ include:
T28
xy(x.Y = x.Y)
(Identity is totally reflexive for complexes.)
T29#
wxyz( v(w.X emb V
y.Z emb V)
w.X = y.Z)
(Indiscernible complexes are identical.)
T42
xy wz(x.Y = w.Z)
(Every complex is classical.)
1.4.5.2 P-P-- is obtained by subjoining P10'* to the postulates of P.
P10'*
x¬ y(y ex X)
(No haecceity is exemplified.)
Theorems of P-- include:
T43' ¬ x(x = x)
(Identity is totally irreflexive for particulars.)
T25' ¬ xy(x.Y = x.Y)
(Identity is totally irreflexive for complexes.)
T44' ¬ xy(x = y)
(There are no classical particulars.)
27
T41' ¬ xywz(x.Y = w.Z)
(There are no classical complexes.)
1.4.5.3 P-+
P-+ is obtained by subjoining P10'*' to the postulates of P-.
P10'*' ¬ x¬ y(y ex X)
(Some haecceity is exemplified.)
Theorems of P-+ include:
T43
x(x = x)
(Identity is partially reflexive for particulars.)
T25
xy(x.Y = x.Y)
(Identity is partially reflexive for complexes.)
T44
xy(x = y)
(Some particular is classical.)
T41
xywz(x.Y = w.Z)
(Some complex is classical.)23
23
A comparison of P-- and P-+ may here be of interest.
(see P10'*).
In P--, no haecceity is exemplified
As a result, there are no classical individuals or haecceities.
Therefore, in
view of P7 there are no self-identical complexes--and hence, because identity is weakly
reflexive--no classical complexes.
reason:
Moreover, every complex in P-- is non-classical for the same
its constituents are non-classical.
However, this is not the case in P-+, where a
complex can be non-classical for either of two reasons.
one--or both--of its constituents are non-classical.
sufficient to make a complex non-classical.
A complex is non-classical, by P7, if
Non-classical constituents are thus
But non-classical constituents are not a necessary
feature of a non-classical complex.
Consider the particulars Saul Kripke and the author of Waverley.
28
The constituents of Saul
1.4.5.4 P+P+- is obtained by subjoining P10*' to the postulates of P+.
P10*'
¬ x y(y ex X)
(No haecceity is exemplified by every individual.)
Theorems of P+- include:
T28' ¬ xy(x.Y = x.Y)
(Identity is partially irreflexive for complexes.)
T29'#
¬ wxyz( v(w.X emb V
y.Z emb V)
w.X = y.Z)
(Not all indiscernible complexes are identical.)
T42' ¬ xy wz(x.Y = w.Z)
(Some complex is non-classical.)
1.4.6 The Primitive Notions and Relations of P
1.4.6.1 Haecceities and Individuals
P1-P9 specify conditions that every individual, haecceity and complex
Kripke include an individual, saul kripke, and a haecceity, SAUL KRIPKE; and the constituents of
the author of Waverley include the author of waverley and THE AUTHOR OF WAVERLEY.
Now saul
kripke exemplifies SAUL KRIPKE, and the author of waverley exemplifies THE AUTHOR OF WAVERLEY.
Therefore, saul kripke and SAUL KRIPKE, and the author of waverley and THE AUTHOR OF WAVERLEY
are classical.
However, saul kripke does not exemplify THE AUTHOR OF WAVERLEY, for saul kripke
and THE AUTHOR OF WAVERLEY are not one in substance.
Therefore, although saul kripke and THE
AUTHOR OF WAVERLEY are classical--that is, exist; the complex saul kripke.THE AUTHOR OF WAVERLEY
is non-classical--that is, does not exist.
Although non-classical constituents are sufficient
to make a complex non-classical, they are thus not necessary.
29
must satisfy.
These conditions make haecceities identity properties,24 for
from P1-P9 it follows that embodying Y is necessary and sufficient for
identity-with-y:
T46
xy(x emb Y
x = y)
Just as it follows from P1-P9 that embodying Y is necessary and sufficient
for identity with-y, it also follows that containing y is necessary and
sufficient for identity-with-y:
T47
xy(x cont y
x = y)
Haecceities and individuals, and embodiment and containment, are thus dual
with respect to T46 and T47.
24
ences.
is.
Between my haecceity and Robert Adams's (1979, 1981) thisness there are two key differFirst, a thisness depends for its features upon Adams's conception of what a thisness
In contrast, every feature of a haecceity--including its being an identity property--is
engendered by the mutual relations of complexes and their constituents.
While the features of a
thisness are thus the artefacts of arbitrary legislative postulation, the features of a haecceity are conferred upon it by the role it plays in the system of complexes.
Second, Adams makes it clear that a thisness is not "a special sort of metaphysical
component of [a particular]":
I am not proposing to revive this aspect of [Scotus'] conception of a haecceity, because I
am not committed to regarding properties as components of [particulars]. (1979: 7)
But a haecceity retains this feature of its Scotian progenitor, for any haecceity and individual
constitute a complex--the existence of which, but not whose being, depends upon the individual
and haecceity being one in substance:
T45
xy( wz(x.Y = w.Z)
x ex Y)
30
1.4.6.2 Embodiment and Containment
A particular embodies a haecceity just in case there is some complex
to which the haecceity and constituents of the particular belong:
T48
xy(x emb Y
wz(w.Z emb X & w.Z cont x &
w.Z emb Y))
Likewise, a particular contains an individual just in case there is some
complex to which the individual and constituents of the particular belong.
T49
xy(x cont y
wz(w.Z emb X & w.Z cont x &
w.Z cont y))
1.5
Concluding Remarks
The theory of the particular-cum-complex puts in question: (a)
ontology-free thematizations of the identity relation,
(b) the Reflexive Law of Equality (hereafter RLE),25 and
(c) set-theoretic monist restrictions on the interpretation of the
individual variables of classical identity theory.
(a) The truths of classical identity theory are here engendered by an
entity which appropriates for logic the essence of the particularity of
particular things.
Every particular thing is both a that and a what, an
existence and a content.
And every particular thing manifests both the
individual and the universal.
So every particular thing suggests the
particular-cum-complex as a(n) (onto)logically proper simulacrum for its
particularity.
But the unity and interconnection of the logically
pertinent elements of every particular's particularity, as embodied in the
25
And therewith, the logically equivalent principle of the Identity of Indiscernibles.
31
mutual relations of a particular-cum-complex, are irretrievably lost to
ontology-free thematizations of the identity relation. For the point of
departure for such thematizations is the supposition that the nature of a
particular is not to have a nature.26
(b) From individuals and haecceities one can construct
(i) a classical particular-cum-complex, which demonstrably satisfies the
usual axioms of classical identity theory, including RLE;
x(x = x)
and (ii) a non-classical particular-cum-complex, which demonstrably satisfies the other classical axioms27, together with a weak
reflexivity principle,
x( y(x = y)
(x = x))
and the contradictory negation of RLE:
¬ x(x = x)
But the validity of RLE depends on its applicability to classical and non26
The conception of a natured particular clashes with a fundamental trait, according to
André Lichnerowicz (1972), of contemporary mathematical thought: "l'absence de toute métaphysique de l'identité et de la chose en soi" (p. 1502).
A kindred anti-metaphysical strain in
neo-positivist philosophy of logic has perhaps been responsible for the reluctance of
linguistically-oriented analytical philosophers to posit a logical nature for particulars,
these tending to be regarded, as Manuel Sacristán points out in his 1984, as "individuos
puntuales sin intrincación ontológica" (p. 249).
27
Except for the strongly classical principle of the Identity of Indiscernibles, to which
RLE is logically equivalent.
32
classical objects.
For
So RLE is not valid.
bA--where b is an individual variable of classical quantification theory--to be
valid, A must come out true under every interpretation.28
Consequently,
(1) " x(x = x)" is valid iff "x = x" is valid.
For A to be satisfiable, A must come out true under at least one interpretation.
Consequently,
(2) "x = x" is valid iff "¬(x = x)" is not satisfiable.
Therefore,
(3) " x(x = x)" is valid iff "¬(x = x)" is not satisfiable.
But A is satisfiable iff for some domain D, A is satisfiable in D.
Moreover, for any domain
D, A is satisfiable in D iff there is a true interpretation of A wherein D is the Universe of
Discourse.
Consequently,(4) " x(x = x)" is valid iff for no domain D is there a true inter-
pretation of "¬(x = x)" wherein D is the Universe of Discourse.
But the entities which realize P-- constitute just such a domain; and with respect to this
domain--on classical assumptions, the "empty" domain--there is a true interpretation of "¬(x
= x)".
So the vaunted 'soundness' of classical predicate logic with identity goes by the
board, for the contradictory negation of the Reflexive Law of Equality is satisfiable by
means of a construction from individuals and haecceities, of whose mathematical existence
there can be little doubt.29
28
This and the ensuing remarks are based on section 48.0 of Gerald Massey's (1970).
29
The mathematical existence of such entities I take to turn upon the following
stipulations of Hilbert (as stated by O. Becker (1927) and cited by Fernando Gil (1971)):
Déf. 1:
on appelle mathématiquement existants les objectités dont on fait le thème ("Thema")
d'une théorie mathématique et qui peuvent fonctionner sans contradiction dans cette
33
(c) Set-theoretic restrictions on the interpretation of particular
variables exclude objects which do not satisfy RLE, among them nonclassical particulars and other such complexes. But these objects are
consistently thinkable.
Moreover, they
play--as I will show in sequels to this paper30--an indispensable
role in the unification of identity theory.
Complexes thus challenge set-
theoretic restrictions on the interpretation of particular variables which
have held logical semantics in thrall for half a hundred years.31
théorie.
Déf. 2:
on appelle mathématiquement existants les objets qu'avec des moyens déterminés avec
précision peuvent être construits à partir de points de départ fondés.
30
The particular-cum-complex will there be shown to make sense of contingent identity and
modal discernibility, and provide an ontological foundation for existence and non-existence as
modes of being of particulars.
31
The only argument of which I am aware for these restrictions is Georg Kreisel's
argument in his (1969, 1971) that "true in all models" and "true in all set-theoretic models"
are extensionally equivalent.
But Kreisel's argument rests on the "intuitive validity" of
the axioms of predicate logic with identity; and as I suggest here, RLE is not valid.
For
Tory restatements of Kreisel's argument, see John Etchemendy (1990: 144 f) and Daniel Quesada
(1985: 154 f).
For a contrasting view, see 'The Paradox of Identity,' Appendix One
(forthcoming, this journal).
34
Identity is
totally reflexive.
Identity is
partially
reflexive.
Identity is
partially
irreflexive.
Identity is
totally irreflexive.
Indiscernible
are
identical.
Every
is classical.
Every
is nonclassical.
P+
P++
P+-
P-
P--
P-+
Particulars
Yes
T38
(Yes)
(Yes)
No
T38'
(No)
(No)
Complexes
---
Yes
T28
No
T28'
No
T28'
(No)
(No)
Particulars
Yes
T38
(Yes)
(Yes)
---
No
T43'
Yes
T43
Complexes
Yes
T25
(Yes)
(Yes)
---
No
T25'
Yes
T25
Particulars
No
T38
(No)
(No)
Yes
T38'
(Yes)
(Yes)
Complexes
---
No
T28
Yes
T28'
Yes
T28'
(Yes)
(Yes)
Particulars
No
T38
(No)
(No)
---
Yes
T43'
No
T43
Complexes
No
T25
(No)
(No)
---
Yes
T25'
No
T25
Particulars
Yes
T39#
(Yes)
(Yes)
No
T39'#
(No)
(No)
Complexes
---
Yes
T29#
No
T29'#
No
T29'#
(No)
(No)
Particular
Yes
T40
(Yes)
(Yes)
No
T40'
(No)
(No)
Complex
---
Yes
T42
No
T42'
No
T42'
(No)
(No)
Particular
No
T40
(No)
(No)
---
Yes
T44'
No
T44
Complex
No
T41
(No)
(No)
---
Yes
T41'
No
T41
IMPORTANT THEOREMS
35
APPENDIX
Be Glad It's Not Deviant
Mephisto.Thomas.
Thomas. Yes?
Mephisto. Bet you a quarter meaning is reference.
Thomas.Meaning is not reference.
If it were, the cognitive
value of 'a = a' would become essentially that of 'a =b', provided 'a = b' were true.
What sets (true) 'a = b' and
'a = a' apart is not the reference of 'a' and 'b'--that is the
same--but their sense.
Extends hand. Gimme the quarter.
Mephisto, puzzled. What is sense?
Thomas.
Sense...sense is...whatever it is about 'a' and 'b'
that sets 'a = b' and 'a = a' apart in cognitive content.
Mephisto. And what's that?
Thomas, truculent. It's...it's sense!
Mephisto, agitated. I know, I know, I know! I know what sense is!
I know what sets true 'a = b' and 'a = a' apart! Thomas
rolls eyes.
Mephisto. I do, I do, I do!
What sets them apart is they assert
different things!
Thomas, incredulous. Assert different things? Sacré Leibniz! How
can they?
Mephisto. If they do, they can. Right?
Thomas.Right.
Mephisto. Well, they do. So they can.
Thomas, sighs. Look, 'a' refers to a and 'b' refers to b.
36
Right?
Mephisto.Right.32
Thomas.And a is b. Right?
Mephisto.Right.
Thomas, triumphant. So 'a = b' and 'a = a' assert the same thing!
Extends hand.
Mephisto.You didn't say "right".
Thomas, reddening. Right?
Mephisto. Wrong! Approaches Thomas. You wanna know why?
Thomas, trapped. Why?
Mephisto. In Thomas's ear. Because a and b are
complexes!!!33
Thomas, mopping ear. Complexes? Agitated. What's a complex? Did
Frege or Russell say something about complexes? Did Quine or
Kripke? David Lewis didn't. His meanings were functions. Dreamily. Functions from functions from functions to functions from
functions to functions...34
Agitated. Frege Futures are taking
32
Mephisto is no Russellian.
33
What Mephisto is getting at here is that complexes confer mutual consis-
tency upon the (classically) irreconcilable theses (A-C):
(A) 'a = b' and 'a = a' are distinct in semantic value.
(B) The semantic value of 'a' is a, and the semantic value of 'b' is b.
(C) a = b
Of course Thomas, who has imbibed the ontology of classical quantification
theory--apparently, with his mother's milk--is singularly unimpressed.
34
David Lewis, 'General Semantics', in Semantics of Natural Language
37
off. My What You Always Wanted to Know About Sense But Were
Afraid to Ask is hot off the press. And--wouldn't you know it!-along comes some pigeon-toed, circle-squaring anti-Frege, who...(crosses himself)
Mephisto, inspects feet. Pigeon-toed, yes. Eyes on Thomas.
Circle-squaring, no.
Thomas, with feeling.
True 'a = b' says a = a,
So does 'b = b'.35
That may seem fa-arcical,
But dahling, it's cla-ahsical.
Hey! I don't mean maybe,
(Donald Davidson ed., D. Reidel, Dordrecht, 1972): p. 180.
35
"...if a = b predicates the relation of identity between the referent of
'a' and the referent of 'b', then if a = b is true, this sentence predicates the
same relation between the same pair of objects as a = a. The sentences therefore
have the same truth and reference conditions." (Engel, p. 161).
38
It's righteous G. Frege.36
Sound inconvenient?
Be glad it's not deviant.
Bub! Don't make me rough it,
Take your complex and...
Remembers bet, Now can I have my quarter?37
36
But what has happened to Sense?
referentialist?
37
Is Thomas on his way to becoming a direct
Will Mephisto pay for Thomas's Sinn?
quarter?
Or will Mephisto give Thomas no
See 'The Paradox of Identity' (forthcoming, this journal).
39
Infernal Quiz in the Sociology of Knowledge
---To what might one attribute Thomas's attitude towards
Mephisto?
Mephisto is pigeon-toed.
Mephisto spit in Thomas's ear.
Mephisto is known not to referee.
Mephisto is not on Thomas's list of preferred wisdom
providers.
Thomas has an Influential Friend in a High Place.
Thomas is into E & R (exegetics and rehash).
Thomas knows that logic is ontology-free.
Complexes might ruin the transcendental part of
Thomas's free lunch.
Other (please specify)
41
References
Adams, Robert: 1981, 'Actualism and Thisness,' Synthese 49,
3-41.
Adams, Robert: 1979, 'Primitive Thisness and Primitive Identity,' The Journal of Philosophy, Vol. LXXVI, No. 1, 5-26.
Becker, O.: 1927, 'Mathem. Existenz,' in Jhb. f. Phil. u.
phenom. Forschung.
Engel, Pascal: 1991, The Norm of Truth, University of Toronto
Press, Toronto.
Etchemendy, John: 1990, The Concept of Logical Consequence,
Harvard University Press, Cambridge.
Gil, Fernando: 1971, La Logique du Nom, Éditions de l'Herne.
Greenberg, William J: 1982, Aspects of a Theory of Singular
Reference, UCLA Ph.D. Dissertation (published in the Garland
Series Outstanding Dissertations in Linguistics, 1985).
Haack, Susan: 1974, Deviant Logic, Cambridge University Press,
Boston.
Kreisel, Georg: 1969, 'Informal Rigour and Completeness Proofs,'
in The Philosophy of Mathematics, edited by Jaakko Hintikka,
Oxford University Press, London, 78-94.
Kreisel, Georg: 1971, 'Mathematical Logic: what has it done for
the philosophy of mathematics?' in Ralph Schoenman (ed.),
Bertrand Russell: Philosopher of the Century, Allen & Unwin,
London, 1971.
42
Ladrière, Jean: 1979, 'Le Logique et Le Réel,' in Ontology and
Logic, edited by Paul Weingartner and Edgar Morscher, Duncker and
Humbolt, Berlin, 157-184.
Lichnerowicz, André: 1972, 'Mathématique et transdisciplinarité,'
Économie et Société, Tome VI, No. 8, 1497-1509.
Massey, Gerald: 1970, Understanding Symbolic Logic, Harper & Row,
New York.
Quesada, Daniel: 1985, La Lógica y su filosofía, Editorial
Barcanova, Barcelona.
Rogers, Robert: 1971, Mathematical Logic and Formalized Theories, American Elsevier, New York.
Sacristán, Manual: 1984, 'Apuntes de Filosofía de la Lógica,'
in Papeles de Filosofía, Icaria Editorial, Barcelona.
van Heijenoort, Jean: 1985, Collected Papers, Bibliopolis, Napoli.
43