## 1. Formulating the Principle




The Identity of Indiscernibles (hereafter called the Principle) is
usually formulated as follows: if, for every property *F*,
object *x* has *F* if and only if object *y* has
*F*, then *x* is identical to *y*. Or in the
notation of symbolic logic: 



> [?]*F*(*Fx* - *Fy*) -
> *x*=*y*.




This formulation of the Principle is equivalent to the Dissimilarity
of the Diverse as McTaggart called it, namely: if *x* and
*y* are distinct then there is at least one property that
*x* has and *y* does not, or vice versa.




The converse of the Principle, *x*=*y* -
[?]*F*(*Fx* - *Fy*), is called the
Indiscernibility of Identicals. Sometimes the conjunction of both
principles, rather than the Principle by itself, is known as Leibniz's
Law.




Thus formulated, the actual truth of the Principle seems unproblematic
for medium-sized objects, such as rocks and trees, for they are
complex enough to have distinguishing or individuating features, and
hence may always be distinguished by some slight physical
difference. But fundamental principles are widely held to be
non-contingent. We might require, therefore, that the Principle should
hold even for hypothetical cases of qualitatively identical medium
sized objects (e.g., clones which, contrary to fact, really are
molecule for molecule replicas). In that case, we shall need to
distinguish such objects by their spatial relations to other objects
(e.g., where they are on the surface of the planet). In that case the
Principle is consistent with a universe in which there are three
qualitatively identical spheres *A*, *B*, and *C*
where *B* and *C* are 3 units apart, *C*
and *A* are 4 units apart and *A* and *B* are 5
units apart. In such a universe, *A*'s being 5 units from
*B* distinguishes it from *C*, and *A*'s
being 4 units from *C* distinguishes it from *B*. The
Principle often gets called into question, however, when we consider
qualitatively identical objects in a symmetrical universe. Consider,
for instance, a perfectly symmetrical universe consisting solely of
three qualitatively identical spheres, *A*, *B*
and *C*, each of which is the same distance, 2 units, away from
the others. In this case there seems to be no property which
distinguishes any of the spheres from any of the others. Some would
defend the Principle even in this case by claiming that there are
properties such as *being that very object A*. Call such a
property a *thisness* or *haecceity*.




The possibility of resorting to thisnesses might make us query whether
the usual formulation of the Principle is correct. For as initially
stated the Principle told us that no two substances exactly resemble
each other. Yet if *A* and *B* otherwise exactly
resemble each other then, on a common intuition, the fact
that *A* has the property
*being identical to A* while *B* has the distinct
property *being identical to B* cannot result in a respect in
which *A* and *B* fail to resemble each other.




Rather than argue about these intuitions and hence argue as to which
is the correct formulation of the Principle we may distinguish
different formulations, and then discuss which, if any, of these are
correct. To that end a distinction is commonly made between
*intrinsic* and *extrinsic* properties. Here it might
initially seem that extrinsic properties are those analysed in terms
of some relation. But this is not correct. For the property *being
composed of two concentric spheres* is intrinsic. For present
purposes it suffices to have an intuitive grasp of the
intrinsic/extrinsic distinction. (Or see Weatherson, 2008,
SS2.1.)




Another useful distinction is between the *pure* and the
*impure*. A property is said to be *impure* if it is
analysed in terms of a relation with some particular substance (e.g.,
*being within a light year of the Sun*). Otherwise it is
*pure* (e.g., *being within a light year of a star*).
Those two examples are both of extrinsic properties, but some intrinsic
properties are impure, (e.g., *being composed of the Earth and the
Moon*). According to my definitions all non-relational properties
are pure.




Armed with these distinction we may ask which properties are to be
considered when we formulate the Principle. Of the various
possibilities two seem to be of greatest interest. The *Strong*
version of the Principle restricts it to pure intrinsic properties, the
*Weak* to pure properties. If we allow impure properties the
Principle will be even weaker and, I would say, trivialised. For
instance in the three sphere example the impure properties *being 2
units from B* and *being 2 units from C* are possessed
by *A* and only *A*, yet intuitively they do not prevent
exact resemblance between *A*, *B* and *C*. (For
a different classification of principles, see Swinburne (1995.))




Suppose we take identity to be a relation and analyse thisnesses as
relational properties, (So *A*'s thisness is analysed
as *being identical to A*). Then thisnesses will be impure but
intrinsic. In that case the world consisting of the three
qualitatively identical spheres 3, 4 and 5 units distance apart
satisfies the Weak but not the Strong Principle. And the world with
the three spheres each 2 units distance from the others satisfies
neither version.




A further distinction is whether the Principle concerns all items in
the ontology or it is restricted to just the category of
*substances* (ie things which have properties and/or relations
but are not themselves properties and/or relations.) It is usually thus
restricted although Swinburne (1995) does consider, and defend, its
application to such abstract objects as integers, times and places,
without explicitly treating these as substance.


## 2. Ontological Implications




Most formulations of the Principle carry a prima facie commitment to
an ontology of properties, but nominalists of various kinds should
have little difficulty in providing suitable paraphrases to avoid this
commitment. (For instance, by using plural quantification. See Boolos
1984, Linnebo 2009, SS2.1.) Most interesting in this context is
the way the Principle can be stated in terms of resemblance without
any mention of properties at all. Thus the Strong Principle might be
formulated as denying that distinct substances ever exactly resemble,
and the Weak Principle as denying that distinct states of affairs ever
exactly resemble. 




Russell (e.g., 1940, Chapter 6) held that a substance just is a bundle
of universals themselves related by a special relation between
properties, known as *compresence*. If the universals in
question are taken to be intrinsic properties, then Russell's theory
implies the Strong Principle. (At least it *seems* to imply it,
but see O'Leary-Hawthorne 1995, Zimmerman 1997 and Rodriguez 2004.)
And if the status of substances is non-contingent then it implies the
necessity of the Strong Principle. This is important because the most
vulnerable version is clearly the Strong when it is held to be
non-contingent. (See also Armstrong 1989, Chapter 4.)


## 3. Arguments for and against the Principle




(i) The Principle appeals to empiricists. For how could we ever have
empirical evidence for two indiscernible items? If we did, empiricists
might say, then they would have to be differently related to us.
Unless we ourselves have exact replicas, which is implausible, we are
the unique beings with pure
properties *X*, *Y*, *Z* etc. Hence the
empirically distinguishable objects have different pure properties,
namely, being related in different ways to the unique things
with *X*, *Y*, *Z*, etc. From this and the
empiricist premiss that there are no things which are not empirically
distinguishable, we would conclude that the Weak Principle
holds. Presumably the premiss would not be proposed as anything more
than contingently true. For there are possible situations in which
there would be theoretical reasons for believing in indiscernible
items as a consequence of a theory which best explains the empirical
data. Thus we might come to hold a theory of the origins of the
physical universe which had large amounts of empirical support, and
which implied that, in addition to our enormously complicated
universe, various simpler ones had been generated. For some of the
simplest universes this theory might imply that there were exact
replicas. In that case the Weak Principle would fail. 




(ii) If we ignore quantum mechanics, we might well conclude that not
merely the Weak Principle is contingently correct but even the Strong
Principle. For unless we take space to be discrete, the classical
mechanical situation would seem to be summed up by the Poincare
recurrence theorem which tells us that typically we get arbitrarily
close to an exact repetition, but never get to one. (See Earman 1986,
p. 130.)




(iii) Concerning the Weak Principle there has been an interesting
development of a line of argument due to Black (1952) and Ayer (1954)
in which it is proposed that there could be exact symmetry in the
universe. In Black's example it is suggested that there could be a
universe containing nothing but two exactly resembling spheres. In
such a completely symmetrical universe the two spheres would be
indiscernible. Against this has been noted, e.g., Hacking (1975), that
such a completely symmetrical situation of two spheres could be
re-interpreted as one sphere in a non-Euclidean space. So what might
be described as a journey from one sphere to a qualitatively identical
one 2 units apart could be redescribed as a journey around space back
to the very same sphere. Quite generally it might be said that we may
always redescribe apparent counter-examples to the Weak Principle so
that qualitatively identical objects symmetrically situated are
interpreted as the very same object. This Identity Defence, as Hawley
(2009) calls it, is vulnerable to a version Adam's continuity
argument. (1979)




A rejoinder to this is the continuity argument, essentially due to
Adams (1979). It is granted that almost perfect symmetry is possible.
For there could be a space with nothing in it but a sequence of
spheres arranged in a line at equal distance without any intrinsic
difference except that one of them is scratched. The identity defence
is then committed to the counter-intuitive counterfactual "If
there had been no scratch on a sphere the shape of space would have
been different". 




In addition to this rejoinder, it should be noted that in only
slightly more complicated examples the identification strategy is
rather less persuasive than in the two sphere case. Consider the
example of three qualitatively identical spheres arranged in a line,
with the two outside ones the same distance from the middle one. The
identification strategy would first require the two outer ones to be
identified. But in that case there remain two qualitatively identical
spheres, so these must in turn be identified. The upshot is that it is
not merely the two spheres we took to be indistinguishable that are
said to be identical but all three, including the middle one which
seemed clearly distinguished from the other two by means of a pure
relational property.


Adams may be interpreted as providing two arguments, the first
being the continuity argument used above. The second is a modal
argument relying on the Necessity of Identity and a suitably strong
modal logic. Suppose there are two objects that are distinguished by
accidental features, as it might be one of the spheres, *A* has
a scratch, while the other *B* does not. Then it is possible
that *A* has no scratch and hence possible that the spheres be
indiscernible. If the Principle holds of necessity then that entails
that it is possible that *A* = *B*. But by the Necessity
of Identity that in turn entails that it is possibly necessary
that *A* = *B*, so in **S5** modal logic (or the
weaker system **B**), it follows that *A* = *B*,
which is absurd given that one has a scratch and the other does
not. In this argument any accidental difference would suffice in place
of the scratch.


Ignoring quantum mechanics we have, then, arguments that many find
persuasive to show that both the Weak and the Strong Principle are
contingently true but neither are necessarily so. For the relevance of
quantum mechanics, see French 2019.


### 3.1 Recent Developments


O'Leary Hawthorne (1995) redescribes Black's example as
a single sphere with two locations. If we accept either of
Adams' argument it follows that discernible spheres can be
redescribed as a single sphere with two locations but with
incompatible properties in the locations, which is seriously
counter-intuitive if not absurd (Hawley 2009 -- see also her
further criticisms.)


Another ingenious idea, suggested by Hawley, is that the two
spheres be redescribed as a simple extended object, contrary to the
intuition that a simple extended object must have a connected location
(Markosian 1998). Once again, Adam's argument then implies that this
redescription holds even of discernible objects of the same kind,
threatening us with the somewhat counter-intuitive monist thesis that
the universe is just one simple object. (For discussions of this
latter thesis, see Potrc and Horgan 2008 and Schaffer 2008,
SS2.1.)


### 3.2 Identical Collocated Spheres?


Della Rocca invites us to consider the hypothesis that where we
ordinarily think there is a single sphere in fact there are many
identical collocated spheres, made up of precisely the same parts. (If
they were not made up of the same parts then the mass of the twenty
spheres would be twenty times that of one sphere, resulting in an
empirical difference between the twenty sphere hypothesis and the one
sphere hypothesis.) Intuitively this is absurd, and it is contrary to
the Principle, but he challenges those who reject the Principle to
explain why they reject the hypothesis. If they cannot, then this
provides a case for the Principle. He considers the response that the
Principle should be accepted only in the following qualified form:



>  There cannot be two or more indiscernible things with all
> the same parts in precisely the same place at the same time (2005, 488)
> 


He argues that this concedes the need to explain non-identity, in
which case the Principle itself is required in the case of simple
things. Against Della Rocca, it may then be argued that for simples
(things without parts) non-identity is a brute fact. This is in
accord with the plausible weakening of the Principle of Sufficient
Reason that restricts brute facts, even necessary ones, to the basic
things that depend on nothing further. 


### 3.3 The Third Grade Principle


Suppose we grant the possibility of otherwise indiscernible objects
that are asymmetrically related. Then we have not just a
counterexample to the weak Principle, but an interesting further
weakening to the Third Grade Principle, namely that in cases where the
Weak Principle fails the otherwise indiscernible objects stand in a
symmetric but irreflexive relation -- "Third Grade"
because based on Quine's third grade of discrimination (1976). Recently
Saunders has investigated this, noting that fermions but not bosons
are third grade discriminable (2006).


Black's spheres are third grade discriminable because they stand in
the symmetric relation of being at least two miles apart, but this
example illustrates the objection that third grade discriminability
presupposes non-identity (see French 2006). For suppose we identify
the two spheres, treating space as cylindrical, then the geodesic
joining the sphere would still be a geodesic and remain the same
length. So we could quite naturally say the sphere was at least two
miles from itself, unless we analyse that relation negatively as there
being no path joining the spheres of less than two miles. But that
negative relation only holds in the Black case because the spheres are
not identified.


## 4. The History of the Principle




Leibniz prudently restricts the Principle to substances. Moreover,
Leibniz is committed to saying that the extrinsic properties of
substances supervene on the intrinsic ones, which collapses the
distinction between the strong and the Weak Principles. 




Although the details of Leibniz's metaphysics are debatable, the
Principle would seem to follow from Leibniz's thesis of the
priority of possibility. (See Leibniz's remarks on possible
Adams in his 1686 letter to Arnauld, in Loemker 1969, p. 333.) It does
not appear to require the Principle of Sufficient Reason, which
Leibniz sometimes bases it on. (See for example Section 21 of
Leibniz's fifth paper in his correspondence with Clarke (Loemker
1969, p. 699). See also Rodriguez-Pereyra 1999.) For Leibniz takes God
to have created by actualising substances which already exist as
possibilia. Hence there could only be indiscernible actual substances
if there were indiscernible ones which were merely possible. Hence if
the Principle holds for merely possible substances it holds for actual
ones as well. There is, therefore, no point in speculating as to
whether there might not be a sufficient reason to actualise two of a
possible substance, for God cannot do that since both would have to be
identical to the one possible substance. The Principle restricted to
merely possible substances follows from Leibniz's identification
of substances with complete concepts. For two complete concepts must
differ in some conceptual respect and so be discernible.