NAMES AND NOMOLOGICAL OBJECTS
As we noted above, Dalla Chiara and Toraldo di Francia suggest that the practice of physics supports a Descriptivist view of names in general. When it comes to QM they insist there are no proper names to begin with, and hence quantum particles are 'anonymous'. 68 Two important questions then arise: why are there no proper names at the microphysical level? And how can we talk about what happens at this level if there are no names?
With regard to the first question, Dalla Chiara and Toraldo di Francia's answer is that "[t]he lack of proper names is due fundamentally to the fact that the objects of microphysics are nomological". 69 By 'nomological' they mean that not only do particles of the same kind possess the same set of state-independent or intrinsic properties, but that these properties are 'fixed' by physical law. Now, at first sight this seems a little curious, since one might think that classical particles are also nomological in this sense. Before pursuing Dalla Chiara and Toraldo di Francia's account of names in QM, we shall digress a little in order to explicate this notion of 'nomological' objects.
Classical objects, Toraldo di Francia maintains, are not 'nomological', since ". their individual configuration had nothing to do with laws". 70 The idea appears to be that in classical mechanics the value of the mass of a particle, say, is not given by law, so that it can have a contingent range of values. 71 For a nomological object, on the other hand, this value, and also those of other properties such as charge, angular momentum etc., is well determined
end p.221
and 'prescribed' by physical law. 72 The crucial 'discovery' appears to be that of discreteness, in the sense that the value of a particle's charge can be expressed as an integral multiple of the charge of an electron, angular momentum is quantized as multiples of ħ/2 and so on. 73 In this sense, then, the values of a property cannot take any of a continuous range of values. And, of course this has been seen as the re-emergence of a fundamental form of a 23223x239x tomism in physics, in terms of which one might assert that the notion of physical object has been 'recovered'.
However, it is still not clear in what sense these properties are 'well determined' and the objects themselves 'prescribed' by law. After all, the charge of an electron, Planck's constant etc., all represent fundamental constants whose values cannot be obtained by physical law; indeed, attempts to so obtain them (the most famous, perhaps, being Eddington's) are typically dismissed as mere numerology. What Toraldo di Francia seems to have in mind is a broader conception of what counts as a physical law: for instance, one can formulate the law that a mass m = 9,1 × 10−23 g must always be accompanied by an electric charge e = ±4,8 × 10−10 e.s.u., by a spin ħ/2, and so on. 74 But as he acknowledges, with this conception all the objects of physics are 'more or less' nomological, including, for example, planets with rings such as Saturn and Uranus (ibid.). What we can perhaps discern here is the suggestion that the individual is submerged under the kind in modern physics, in the sense that an electron, for example, is defined to be that kind of thing which has a mass of 9,1 × 10−23 g and a charge of 4,8 × 10−10 e.s.u. and so on, so that anything which possesses this set of properties has to be an electron. Furthermore, the discreteness of these values implies that such sets-characterizing muons, quarks and so on-are sharply delineated. Thus, the view of objects as 'nomological' ultimately reduces to a form of bundle theory, as Toraldo di Francia himself acknowledges: "In some way, physical objects are today knots of properties, prescribed by physical laws". 75
The 'recovery' of integral numbers inherent in the above discreteness appears to reduce measurement to counting. 76 But the counting of elementary particles is problematic and here Toraldo di Francia raises concerns about individuality: 77
Can we distinguish this and the other, in a system of two electrons? As is well known, this cannot be done: identical particles are indistinguishable. Here, cardinal numbers
end p.222
seem to take over the role we had previously attributed to ordinal numbers. A system of identical particles has a cardinality; but we cannot tell which is the first, the second, and so on."
Setting aside the conflation of individuality with distinguishability, the conclusion that an assembly of quantum particles may have cardinality but not ordinality has an impact on the foundations of logic: 78
Think, for instance, that in any formalised theory, we must start by defining a universe of discourse, that is a set of objects we want to talk about. For this to make sense, it must be well determined whether an object belongs or not to the set (extension of the set); moreover, the objects must be distinguishable from one another, so we can tell which is which. These requirements can perhaps be met by mathematical objects (although some scholars strongly oppose the idea). But what happens in the case when our universe of discourse is made up of physical objects? How can we tell that it is determined whether a given electron belongs or not to our system, and how can we tell which electron we are talking about?"
The alternatives Toraldo di Francia proposes are to employ fuzzy set theory 79 or develop an intensional semantics appropriate for such objects.
This leads us to quaset theory, but before we return from our digression, we wish to note a number of points concerning the notion of 'nomological objects' and the way it feeds into subsequent considerations of denotation and set theory: first of all, the underlying metaphysics appears to be one of (non-classically) indistinguishable individuals, where the individuality of the particles-as objects-is understood in terms of a form of bundle theory, but without PII, of course. Secondly, there appears to be something of a lacuna between Toraldo di Francia's exploration of the notion of nomological objects and the above conclusion concerning the foundations of logic. Indeed, the latter appears to follow solely from the previous consideration of the nature of indistinguishability in QM.
The situation is muddied somewhat by Dalla Chiara and Toraldo di Francia's subsequent remarks on this issue. As we have already noted, they insist (1993) that the lack of proper names in QM is due to the fact that the particles are nomological objects. There they explicate this notion both in the terms given by Toraldo di Francia-that is, of objects whose characteristics are 'fixed' by physical law-and further, in terms of what they now refer to as the 'identity' of electrons. Indeed, in remarks that recall Weyl's comments
end p.223
about the electrons' lack of an alibi, they ask, "How can you talk of Peter or Paul if you cannot distinguish the one from the other .?". 80 Thus, it appears that it follows from the notion of a nomological object that two or more such objects are 'absolutely indistinguishable'. But if this were the case then Saturn or Uranus could not be nomological. Perhaps we could clarify the situation if we were to characterize their position in the following way: all objects in physics are 'more or less' nomological, in the sense that the metaphysical underpinning of substance has been removed, to be replaced by that of invariants understood in terms of fundamental properties. In this sense, particles are just 'knots' or bundles of such properties. Quantal particles are objects in this sense which are, further, absolutely indistinguishable (and which violate the Identity of Indiscernibles). Thirdly, and in particular, the crucial concern arising from this metaphysics is whether or not the objects in question can be said to belong to a set, in the standard sense. As we shall see, it is on this point that qua-set theory can be viewed as distinct from quasi-set theory.
Let us now consider Dalla Chiara and Toraldo di Francia's second question: if quantal particles are anonymous, how can we talk about them? As we have seen, they consider the notion of a rigid designator to be a dubious one even in the domain of 'macrophysics'. In the microphysical realm it is rejected as 'useless', because it is claimed absolute indistinguishability implies that there is no trans-world identity. 81 Of course, one might object that there are situations in which the overlap of the wave functions of two electrons, say, is sufficiently small that we can treat the particles as if they were distinguishable and assign proper names to them. It is this that Toraldo di Francia refers to as a 'mock individuality' and writes,
This is why an engineer, when discussing a drawing, can temporarily make an exception to the anonymity principle and say: "Electron a, issued from point S, will hit the screen at P, while electron b, issued from T, will land at Q". 82
However, he insists, these names cannot be regarded as rigid designators and cannot denote the same object in all possible worlds. Granted that an electron could be sufficiently isolated that we could-temporarily-distinguish it and regard it as named, if we were to shift to a possible world in which the electron interacts with others, giving rise to a superposition, the distinguishability, and hence the possibility of naming, would be lost. Thus the identity of the electron cannot be maintained across all possible worlds. Note that the basis of this claim-the problematic nature of superpositions-also supports the claim that there is no temporal identity for electrons (since if an electron, again sufficiently isolated as to allow it to be temporarily and approximately distinguished, interacts with another electron in this, the actual world, we cannot tell, even in principle, which electron is which after the interaction). In other words, it is not the re-identification of the electron across possible worlds per se that is the problem, but the re-identification of the electron after entering a superposition.
Mittelstaedt has suggested that we can still speak of a 'weak kind' of trans-world identity in this sort of case if we take names to denote not single objects but classes of indistinguishable systems. 84 With such classes defined in terms of the relevant essential properties of the particles, we might regard them as classes of nomological objects in Toraldo di Francia's sense. If the system in the actual world is prepared in a given state, then those possible worlds which are 'accessible' to the system are defined to be those for which the system can be in states related quantum mechanically to the given state (cf. our previous discussion of 'accessibility', where the symmetry of the Hamiltonian guarantees that for a system in a state of a given symmetry, only those states of the same symmetry will be accessible to the system). Since this relation is typically probabilistic, Mittelstaedt argues that the trans-world identity of such quantum systems can be approximately preserved. 85 Of course, as we have noted, the trans-world identity here is not of a single object, but of a whole class or kind of objects. Thus the semantics being suggested is only Kripke-like, not only with regard to the role of probability in QM 86 but also with regard to the nature of the denotata.
With rigidity deemed inappropriate for describing single quantum mechanical objects, Dalla Chiara also offers a kind of 'liberalized' Kripke semantics, but one which appears to go much further than Mittelstaedt's suggestion. 87 One way of entering this formalism is via a comparison with so-called Kripke
end p.225
models. 88 Consider a language L with
individual names a m , predicates 
,
variables x m , and logical constants ¬, 
and 
.
Following Dalla Chiara, we introduce a set of possible worlds I and an
accessibility relation ± between them which is symmetric and reflexive. Sets of
worlds X are possible propositions containing all and only worlds whose
accessible worlds are not inaccessible to the whole set. We can then construct
sets of possible propositions P which are closed under the operations of
orthocomplement and union, and contain 
and I.
A Kripke model is a system 
=
<I, ±, PU , D, r> with U some
family of subsets of PU closed under infinitary intersection,
where the important features for our purposes are that: (1) For any world i,
the set D represents the domain of i; (2) r gives extensional
meanings to non-logical constants-so r i (a m )
(the denotatum of a m in i) is an individual in i
and 
(the
extension of in
i) is an n-ary relation on D. Names are rigid in that r
i (a m ) = r j (a m
) for all i, j (where individuals in i and j
are of course members of D); (3) An interpretation of the variables of L,
s, is a function assigning a value in the domain to any variable of L.
A pair rs is a valuation and a triplet 
is
a world valuation. It is easy to see that such a model satisfies the Fregean
Compositionality Principle, exemplified by a particular case (which can be
generalized):
The truth-value (extension) of an atomic sentence of the form "a is P" (Pa) is a function of the extensions of its parts: the name a and the predicate P. As a consequence, in order to ascertain whether Pa is true or false, one has first to identify the denotatum of the name a. 89
As we have seen, the reasons for thinking that this sort of condition breaks down within quantum mechanics are not decisive. Dalla Chiara presents a counterexample in the case of a proper description, where the sentence 'there is exactly one F' is true yet the description term 'the x which is F' fails to have a denotatum in the domain. To deal with this sort of case she proposes a "weak theory of descriptions". There are two strategies which she discusses. One involves weakening the constraints on the functions r such that they need only be partial functions. The second involves modifying the requirement that the domain D be a set, treating it instead as a quaset.
The first strategy works by requiring that, although there may be worlds where proper descriptions are denotationless, these worlds can be correlated
end p.226
to accessible worlds in which the description acquires a
precise denotatum in D. This is extended to a treatment of proper names
by the introduction of a liberalized Kripke model, which is a structure
of the sort described above, except that r in this case is a partial
function which associates extensional meanings to some but not all of the
non-logical constants. Thus r i (a m ), if
defined, is an element of D and 
,
if defined, is an n-ary relation on D. Thus there will be worlds
in which a name a m does not pick out a denotatum in D.
However, proper descriptions of the sort 'The a which is F' are well
behaved in the sense that a world in which a proper description is
denotationless can be correlated with a world in which the term does denote
(and denotes uniquely).
This approach retains an essentially classical ontology for which we still have a domain D of individual objects. We can think of this in two ways. It seems a natural way to think about Bohm's version of quantum mechanics. The fact that the functions r i are partial functions expresses epistemic ignorance about the exact behaviour of the particles, but the particles can be individuated and do have properties-their positions-which will serve to distinguish them one from another. 90 The other way of thinking about this is to consider the situation along the lines of Bohr's complementarity. 91 Here the domain D remains a set of classical objects and represents the classical resources we are limited to in our attempts to conceptualize the world. The partial functions then represent the restrictions complementarity places on our use of language if these classical concepts are not to clash with what is given in experience.
However, as mentioned above, Dalla Chiara has a second
strategy for producing a semantics appropriate for quantum logics. Consider
first the classical case of a system of n particles. We can name the
elements and predicate properties of them, such as 
,
which represents the claim that the value of some quantity Q m lies
in E. If the domain of individuals at any time t is D t
then r t (a m ) is some element of D
t and 
is
some subset of D t . What of the situation for a system of n
identical quantum particles in a state f(t)? Dalla Chiara poses
two questions: 92 "(I) Does f(t) determine a
set of n elements in the standard set-theoretical sense? (II) Is the
denotation-function definable in such a way that r t (a n
) univocally determines an element of D t ?"
She defines a liberalized Kripke model as before, where I
is now the set of all unitary vectors in the Hilbert space 
, ±
is the accessibility relation of
end p.227
non-orthogonality, PU is the set of all
sets X of unitary vectors which are in one-one correspondence with
subspaces of and D
f "represents a physical situation consisting of two physical
objects, as is described by the vector f" (we quote Dalla Chiara's
wording here because in view of what follows it seems perhaps
question-begging). Possible worlds are associated with vectors in the Hilbert
space, and, recalling our discussion above, it will be useful to have in mind a
subset of these representing physically possible worlds. 93
A world f verifies a sentence like 
if
the first subsystem certainly has a value for the quantity Q m lying
in E. 94
What sort of admissible worlds do we have? We can have states like
but not states like |u 1 > |d 2
>. Dalla Chiara notes that states like |u 1 > |d
2 > would represent worlds where the labels 1 and 2 had precise
denotata (and proper descriptions such as "the electron with spin up" would
also have precise denotata), but that such states are not admissible states for
real systems. Instead she suggests that we treat these states as what she calls
"Gedanken" states. We can then introduce a sense in which a "real" state (that
is, one representing a physically possible world) verifies a statement of the
type ,
or more generally "a is F". A real state verifies "a is F"
if and only if all Gedanken states compatible with the original state
can be correlated with a compatible state where the name a is defined
and where "a is F" is true.
Consider our spin singlet state. At first sight it might
seem that what we have is the following situation. We correlate the real state 
with
a Gedanken state |u 1 > |d 2 > which
enables us to make sense of the notion that the real state verifies the
sentence "electron 1 has spin up". But this will not do, for the singlet state
is also compatible with a Gedanken state |u 2 > |d 1
> which cannot be correlated to the state |u 1 > |d
2 > in which it makes sense to talk of electron 1 having spin up.
95 Thus it seems that even in this weak sense there will be very
few admissible states (that is, those representing physically
end p.228
possible worlds) in which proper descriptions and singular terms in general will be meaningful, even in this weak sense.
At the conclusion of her account, Dalla Chiara maintains
that the trend in 'traditional' semantics (citing Mill and Kripke) has been to
connect the use of proper names with an ostensive function. 96 However,
she urges, this function is not available in QM and
hence names cannot be understood, in such situations, as mere labels.
What the above considerations demonstrate is that ". one may regain a theory of
individual terms (names and descriptions) which is not necessarily
bounded to a referential commitment". 97 However, as
we have seen, problems arise within the account itself. The source of
these problems lies with the insistence on a descriptivist approach: if names
are just disguised definite descriptions, then it remains opaque how we are to
regard the labels in a superposition such as but
not states like |u 1 > |d 2 >. Dalla
Chiara's own attempt to spell this out by correlating this state with |u
1 > |d 2 > understood as a 'Gendanken' state
runs into the difficulty indicated above. This brings out a tension within the
Dalla Chiara and Toraldo di Francia approach: quantum particles are regarded as
'anonymous' individuals and the issue, as we have noted, is how we can talk
about such individuals. Their answer is two-fold: a liberalized Kripke
semantics tied to a non-standard set theoretical description of the relevant
elements of the domain. We have still to come to the latter, but the former is
hamstrung by the dependence on a descriptivist view of names.
What are the alternatives? One would be to abandon the project altogether and follow Redhead and Teller in their insistence that the appropriate metaphysics for quantum particles is one of non-individuals. If one wishes to cleave to a descriptivist approach, this would appear to be the only option available. Thus we recall Maidens' claim that quantum particles cannot possess qualitative thisnesses, because of the entanglement. Hence, she concludes, they must be regarded as non-individuals. 98
However, we have indicated above that one can still maintain
some form of 'hybrid' approach to naming consistent with the metaphysics of
quantum particles as individuals, particularly if one were to accept Gracia's
distinction between the 'referring' and 'descriptive' functions associated with
names. With states such as |u 1 > |d 2
> these functions coincide, so that we are able to assert that 'electron 1
has spin up'. In these cases, the description may serve to refer to the
individual. However, in cases such as it
may not. In these cases, we simply have to accept that the labels refer, but we
cannot say to which individuals they refer, in the sense of giving an
appropriate description. We can at least explain why we cannot give such a
description (cf. Maidens' remark above regarding the Descriptivist approach):
superpositions can be understood in terms of non-supervenient relations
holding between the individuals, where these relations effectively hide or
'veil' the individuals and thus prevent the ascription of appropriate
descriptions.
This may then enable us to respond to concerns about what the names could mean in the above situations. In particular, if the meaning of names is understood, at least in part, in terms of rigid designation, so that this meaning is cashed out, as it were, in terms of the stipulation of corresponding possible worlds, then it might seem as if we are caught in Kripke's dilemma above: we must then accept states such as |u 1 > |d 2 > as corresponding to legitimate possibilities, but such states are not legitimate from the perspective of QM. The response we have delineated above is to effectively place constraints on this 'cashing out', so that not all such possibilities correspond to 'genuine', physical possibilities. But in that case, it might be insisted, 99 the names must be meaningless. When we are considering just the symmetric or anti-symmetric subspace, having discarded, as it were, the non-symmetric ones as not genuine possibilities, 100 the symbols |u 1 > |d 2 > and |u 2 > |d 1 > are to be understood as perfectly meaningless, as is the expression [|u 1 d 2 > + |u 2 d 1 >], and, hence, also the labels themselves.
Alternatively, however, consider the parallel with well-formed formulae in propositional logic, say, where the string 'P' is taken to be meaningless, but neither the string 'P&Q' nor the symbol '' is. 101 Thus, in the context of the formalism of the symmetric and anti-symmetric subspaces, the strings |u 1 > |d 2 > and |u 2 > |d 1 > are understood as uninterpreted since they denote no states. However, it does not follow either that |u 1 d 2 > + |u 2 d 1 > is meaningless or that the labels themselves are meaningless. In reply, the Compositionality Principle can be appealed to
end p.230
again:
Please explain to me what 'P&Q' means WITHOUT appealing to what the parts mean. Perhaps you will say that you can say what the ultimate parts mean in the QM case: '1' and '2' refer [to] individual particles. [|u 1 >] refers to a state in which particle [1] has spin up and similarly for [|d 2 >] etc. But you have acknowledged that [|u 1 d 2 > and |u 2 d 1 >] are meaningless. So how are you going to explain what [|u 1 d 2 > + |u 2 d 1 >] means? 102
Such an appeal undermines the parallel with propositional logic, since compositionality is obeyed with respect to such logic, of course, but may be viewed as problematic in QM, as we have indicated. If the principle is accepted, then explaining what 'P&Q' means does indeed involve an appeal to what the parts mean: we explicate the truth of 'P&Q' in terms of the truth of 'P' and of 'Q' and the meaning of the symbol ''. On the quantum side, however, we must be careful in working our way back up from the components to the superposition, as we have already indicated. In the case of |u 1 >, we can say that this string of symbols refers to a state in which particle 1 has spin up and in this case, the sense of reference is one in which the label both denotes the individual and 'stands for' the description in terms of which we may pick that individual out. The same can be said for the labels in both |u 1 d 2 > and |u 2 d 1 >, so in this sense, these latter expressions are not strictly meaningless.
Nevertheless, that does not imply that they must be accepted as genuine physical possibilities-this is precisely what the constraints on what can be stipulated as a possible world block. In other words, we can understand what these strings of symbols mean in a modal sense, just as we can understand the meaning of the strings of symbols representing the equations of classical mechanics. The problem lies with |u 1 d 2 > + |u 2 d 1 >, and if one were to understand compositionality in the sense indicated by Dalla Chiara above, namely that in order to ascertain the meaning of 'Pa' (understood in terms of its truth or falsity), one has first to identify the denotatum of the name a, then such an expression would indeed appear to be meaningless, since we precisely have no means of making such an identification. However, if we reject this identification-first approach, then we can understand this expression as meaning that one electron has spin up and one has spin down, but we cannot say
end p.231
which because of the existence of these obscuring non-supervenient relations holding between the individuals concerned. In other words, the labels '1' and '2' have meaning in the sense of denoting individuals, but not in the sense of allowing us to distinguish which individuals have which properties, or satisfy which descriptions. It is only if one takes the sense of meaning to be entirely cashed out in the latter terms that one would be inclined to regard the above expression as meaningless.
Of course, what this suggests is the need for a non-Fregean semantics and, further, a non-standard set theory, since the above considerations raise obvious concerns regarding the principles of compositionality and extensionality. It is just such a theory that Dalla Chiara and Toraldo di Francia have attempted to elaborate in terms of quasets.
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