QUASETS
The central idea of quaset theory is that an assembly of quantum particles-electrons in an atom, say-has a cardinal but not an ordinal. 103 If we consider a helium atom, for example, we can conclude that there are two electrons in the assembly because the relevant state function depends on six coordinates and we can also ionize the atom and strip two electrons off, where we can distinguish them in the 'mock' sense seen above. However, within the assembly, the particles cannot be regarded as 17417q162r so ordered. Nevertheless, one can talk of subassemblies inside the assembly where each subassembly is determined by some property (corresponding to a quantum number); each subassembly possesses its own cardinality but, again, not an ordinality. Such assemblies and subassemblies are quasets and subquasets respectively.
The extension of the natural kind 'electron' cannot, therefore, be given by determining a set with a specified property and assuming the standard comprehension and extensionality principles. Instead, the extension of 'electron' can be defined as a quaset whose subquasets ". are all the electron systems that can-at least in principle-be detected experimentally". 104 The intension of the term 'electron', on the other hand, is given by the conjunction of the bundle of properties defining the nomological object. 105 Such properties are
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essential in that a particle which has all the properties of an electron except a mass of, say, 1.9 × 10−25 g is not regarded as a peculiar electron, but as a different particle entirely (a muon in fact). Corresponding to a given intension, there may be more than one extension in the following sense:
. once the intension of the term 'electron' has been stipulated, we have the possibility of recognizing-by theoretical or experimental means-whether a given physical system is an electron quaset or not; if yes, we can also enumerate all the quantum states available within it. But we can do so in a different number of ways! For example, take the spin. We can choose a z-axis and state how many electrons have s z = +1/2 and how many have s z = −1/2. But we could instead refer to the x-axis, or the y-axis, or any other direction, obtaining different sets of quantum states, all having the same cardinality. 106
Although a particular name might have a well-determined and precise intension, in the case of quantal particles, the extension is 'strongly indetermined' in that the name lacks a precise reference. Thus, the absolute indistinguishability of such particles gives rise to 'uncertain and ambiguous denotation relations', as already noted. We shall now outline the formalism corresponding to this metaphysical package.
5.5.1 Quaset Theory 107
Let us begin by recalling the obvious point that in various situations physicists are able to recognize, by theoretical or experimental means, whether a given physical system is an electron system or not. In these cases, they can also enumerate all the quantum states available within such a system, and they can do so in a number of different ways. For example, by considering the spin, one can choose the x-axis and state how many electrons have spin up and how many have spin down. However, we could instead refer to the y-axis or any other direction, obtaining different collections of quantum states, all having the same cardinality. This motivates the suggestion that micro-object systems present an irreducibly intensional aspect: generally they do not determine precise extensions and are not determined thereby. Accordingly, a basic feature of the quaset theory QST is to incorporate a strong violation of the extensionality principle.
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QST is formalized in classical first-order logic with identity (which is dealt with as a logical constant), and its language contains the following primitive non-logical concepts:
|
(i) |
one monadic predicate: urobject (O); |
|
(ii) |
three binary predicates: the positive membership
relation ( |
|
(iii) |
a unary functional symbol: the quasi-cardinal of (qcard); |
|
(iv) |
a binary functional symbol: the quaset-theoretical intersection ( ). |
Restricted quantifiers are assumed: 
Px
A stands for x(P(x)
→ A), while 
Px
A stands for x(P(x)
A).
Further, the quantifier 'there exists exactly one' (!)
is understood as defined in the usual way.
We will present here only the minimal axiomatic nucleus of the theory. The notion of quaset is defined as follows:
Definition 1 [Quaset] A quaset is something that is not an urobject. In symbols,
Then, the basic idea of a quaset is that of a
collection of objects which has a well-defined cardinal, but in such a way that
there is no way to tell (with certainty) which are the elements that belong to
the quaset. The primitive predicates 
(the
positive membership) and 
(the
negative membership, which is not the negation of the former) help in
expressing this. The postulates, to be stated below, imply that z y
entails ¬z y,
but not the converse. So, it may be the case that it is false that z
certainly does not belong to y, but this does not entail that z
(certainly) belongs to y. The elements z for which it may be said
that 'it is false that they certainly do not belong to y' might act as
members in potentia of y. 108 Since the
cardinal of the quaset is well defined, there is a kind of 'epistemic'
indeterminacy with respect to the elements of a quaset. The postulates below
provide the grounds for the whole theory:
Ax1 All that has elements is a quaset: xy
(y x
→ Q(x)).
Ax2 The inclusion relation 
between
quasets is a partial order (reflexive, anti-symmetric and transitive).
In the intended interpretation, has
an intensional meaning: x y
can be read as 'the concept x involves (or implies) the
concept y'. 109
Ax3 Suppose that something certainly does not belong
to a given quaset. Then it is not the case that it certainly belongs to our
quaset, but generally not the other way around: xy(x
y
→ ¬x y).
As a consequence, a strong tertium non datur
principle (x y
V x y)
fails and indetermined membership relations are possible (in accordance with
the quantum uncertainty relations).
Ax4 Intensional inclusion implies extensional inclusion (but not the other way around):
Ax5 Any quaset has exactly one quasi-extension, where the quasi-extension of a quaset x is the unique quaset that certainly contains all the elements of x and certainly does not contain all the other entities:
Axiom 5 justifies the definition of a unary functional symbol ext (the quasi-extension of x). (Definition 3.2.)
Definition 2 (The quasi-extension of a quaset)
Definition 3 Sets are quasets that are identical with their quasi-extension.
One can easily show that sets satisfy the Extensionality Principle. The extension of an empty quaset (which turns out to be trivially a set) is postulated.
Ax6 [The empty quaset] There exists a quaset that
necessarily does not contain any element: Qy
x
(x y).
QST contains a copy of ZF (Zermelo-Frankel set theory), obtained by restricting the universe of quasets to sets only. For any formula A of ZF, let Az be the corresponding formula of QST relativized to sets. Then,
Ax7 If A is any instance for an axiom of ZF, then Az is an axiom of QST.
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The notion of quasi-cardinality of a quaset is introduced as follows:
Ax8 Any quaset has a unique quasi-cardinal, which is a cardinal number:
A careful remark is in order here. The axiom says that the quasi-cardinal is a cardinal and thus, if the concept of cardinal is defined in the usual way, that is, as a particular ordinal, then a quasi-cardinal is an ordinal and hence it may appear that there is no sense in saying that a quaset may have a cardinal but not an ordinal, as we have suggested above. This should be understood in the sense that the concept of cardinal (or of counting) should be taken differently from standard mathematics (Zermelo-Fraenkel set theory), perhaps in the sense of Frege-Russell. This is of course interesting, but we shall not pursue the point here. It will be recalled in the context of quasi-sets in Chapter 7.
Ax9 Quasi-cardinals and cardinals coincide for sets:
Ax10 The quasi-cardinal of a subquaset is less or equal than the quasi-cardinal of the whole quaset:
Ax11 The quasi-cardinal of a quaset is greater or equal than the quasi-cardinal of its quasi-extension:
Ax12 represents a weak conjunction for quasets. This weak conjunction coincides with the usual set-theoretical intersection in the case of sets:
As a consequence, a separation procedure may be applied. Notice that our axioms do not require that proper quasets (that are not sets) exist. From an intuitive point of view, the quasi-extension of a proper quaset does not represent an adequate semantic counterpart for the usual notion of extension. Think for instance of the fact that the quasi-extension of a quaset, whose quasi-cardinal is greater than 0, might have an empty quasi-extension.
What can be said about the validity of Leibniz's Principle in this theory? Are non-identical objects always distinguished by a property, represented by a quaset? As expected, the answer is negative. Namely, the implication
end p.236
generally fails, and so the classical set-theoretical argument founded on the theorem of ZF:
obviously cannot be repeated here, for nothing guarantees that the singleton of x (which should be the characteristic property of x) exists and that x certainly belongs to it.
5.6 CONCLUSION
What we have tried to do in this chapter is to develop the particles-as-individuals package a little further by considering the treatment given to the particle labels in the quantum context. As we have indicated, there are ways in which such labels can be regarded as names and arguments to the effect that this is problematic are themselves contentious. Gracia's account, in particular, appears to offer an appropriate metaphysical framework in which the particles can be considered as named individuals for which distinguishing descriptions cannot be given. We have concluded by suggesting that Dalla Chiara and Toraldo di Francia's quaset theory provides the appropriate formal counterpart. 110 Understood in this way, it would not be the most appropriate formal framework for capturing the notion of 'non-individuality' as expressed in the Received View. One way of doing that would be to represent this notion by abandoning the condition of self-identity for quantum particles and constructing a formal system on that basis. We shall now consider just such a construction.
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