Abstract
One of the main criticisms of the theory of collections of indiscernible objects is that once we quantify over one of them, we are quantifying over all of them since they cannot be discerned from one another. In this way, we would call the collapse of quantifiers: ‘There exists one x such as P’ would entail ‘All x are P’. In this paper we argue that there are situations (quantum theory is the sample case) where we do refer to a certain quantum entity, saying that it has a certain property, even without committing all other indistinguishable entities with the considered property. Mathematically, within the realm of the theory of quasi-sets \(\mathfrak {Q}\), we can give sense to this claim. We show that the above-mentioned ‘collapse of quantifiers’ depends on the interpretation of the quantifiers and on the mathematical background where they are ranging. In this way, we hope to strengthen the idea that quantification over indiscernibles, in particular in the quantum domain, does not conform with quantification in the standard sense of classical logic.
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Notes
It is disputable whether we really can, as Bohr seems to have proposed, divide the world into ‘micro’ and ‘macro’ parts. Apparently, quantum mechanics would be a ‘totalitarian theory’ (to use Leggett’s expression) which applies not only to electrons and protons but also to macroscopic objects; see (Leggett 2014). But here we shall be concerned, in the examples mentioned, with the microscopic level only. We assume, as usual, that ‘macroscopic’ objects obey classical logic, although the frontier is known to be fuzzy.
We keep the words ‘identical’, ‘identity’, etc. to be used in a logical and mathematical context meaning the same. Today experiments have shown that also ‘big’ molecules such as C\(_{60}\) and C\(_{70}\) present a ‘quantum behaviour’, say in the two-slits experiment (Zeilinger 2005).
I shall leave the discussion about Bohmian mechanics, which encompasses an ontology similar to classical physics, out of this paper and assume the standards in quantum theories. It is disputable whether Bohmian’s positions can be known. As far as I am concerned, they remain hidden.
It should be remarked that Hermann Weyl has called our attention to precisely this point, positing that in quantum physics what imports is an ‘ordered decomposition’ emphasizing precisely the kinds and quantities only; see (Weyl 1949, App.B), French and Krause (2006); Krause (1991). Just to remark, this was also the idea underlying the birth of modern chemistry with Boyle, Hooker, and mainly Dalton, to whom the atoms of a given element are indiscernible – see Dalton (1808).
An alternative definition could be \(\langle a,b \rangle _z {:=}[\llbracket a \rrbracket _z, \llbracket a \rrbracket _z \cup \llbracket b \rrbracket _z]_{\mathcal {P}(z)}\).
In this case, since all elements of q are indiscernible, the n strong singletons are also indiscernible, but the postulates grant that there are n of them. The notation ‘\(\llbracket a \rrbracket _q\)’ stands for a quasi-set with cardinal one and whose only element is indiscernible from a, but the theory is unable to prove that it is really a since there is no identity holding among them.
We shall not enter in this important topic here; the resume is that standard mathematical frameworks, as we shall comment a little bit below, work with individuals, that is, with entities endowed with identity conditions, given by the standard theory of identity, which of course is not the case of quantum systems. Notice that one of the main characteristics of an individual is its re-identifiability, that it, it can be recognized later as being that individual from before; it is clear that this does not happen with quantum objects.
In a ‘pure’ quantificational language, the only terms are the individual variables. We don’t lose generality in assuming this. Of course, we can generalize our simplified example; see Uzquiano (2020).
A Bose-Einstein Condensate is a typical case, obtained when certain atoms or other quantum entities are cooled to quite closer the absolute zero. As far as the temperature decreases, the wavefunctions of the elements become so that at that temperature they behave as a single thing, not presenting any differences at all. Although they are not the same entity, they cannot be discerned by any means. See Ketterle (2007).
Notice that the assignment s is attributing \(d \in D\) to x. Since every variable is indiscernible only from itself -we assume that the elements of the language obey classical logic-, there is just an element being associated with it, namely, d. The interesting fact is that we cannot put our finger over the element designed by d and say ‘this one’; it is simply ‘d’, and we cannot identify it.
Another subtlety: if D comprises indiscernible elements, we cannot characterize S except by indicating that its elements belong to D (or that are of the ‘kind’ of the elements of D), and its cardinality. Whatever ‘other’ subqset of D with the same cardinality than S (that is, whatever subqset indiscernible from S) would act as the extension of P as well. This is an interesting trait of quantum mechanics: it subverts the standard meaning of intensions and extensions, enabling that one intension may have ‘different’ extensions; see Dalla Chiara and Toraldo di Francia (1993).
We could use Hilbert’s \(\epsilon \) symbol for that: \(\epsilon x P(x)\) stands for one electron with such and such quantum numbers. See below footnote 20.
Shall be enough to analyse this case, the most general ones can be got from extending the reasoning to formulas in general in the usual sense.
It could be not necessarily so. We could assume that D comprises sub-collections of indiscernible elements which are discernible from the elements of another collection of indiscernible elements, and even that there could exist ‘classical’ elements in D, that is, elements which obey the rules of classical logic. In more precise terms, D is a quasi-set as described in \(\mathfrak {Q}\).
Since ordered tuples need to be understood adequately within the scope of \(\mathfrak {Q}\); the details are not relevant here and the reader can reason as usual in standard set theories.
Remember again what was said above that the elements of L obey classical logic. So, a variable x is an individual and the q-function s associates to it just one element \(d \in D\), yet ‘hidden’, that is, yet we may be unable to discern it from others.
Given a qset A and an element \(x \in A\), we can form the ‘unitary qset’ of x as being the qset of all elements of A that are indiscernible from x. It is denoted \([x]_A\) and may have more than one element (that is, its cardinal, or ‘quasi-cardinal’, may be greater than one). We also call the ‘strong singleton’ of x the qset \(\llbracket x \rrbracket _D\), which is a subqset of \([x]_A\) and has a quasi-cardinal one (for a proof of its existence, see (French and Krause 2006, chap.7).
These objects are term m-atoms; for them, expressions like \(x = y\) are not well formed formulas when x or y stand for an m-atom.
In our pure logic, we state this theorem as \(P(y) \rightarrow \exists x P(x)\), where y is a variable distinct from x.
Using Hilbert’s epsilon symbol (Avigad and Zach 2020), perhaps we could write it as \(\epsilon x P(x)\), but this needs to be analysed, for by obvious reasons (the lack of identity) the schema of extensionality \(\forall x (A(x) \leftrightarrow B(x)) \rightarrow \epsilon x A(x) = \epsilon x B(x)\) doesn’t hold.
By the way, this shows that the criticisms advanced by da Costa and Bueno to the semantics of ‘non-reflexive logics’ (da Costa and Bueno 2009) should be put within parentheses since they ground their semantic reasoning in a standard set theory, contrary to what is required. The right semantics should be constructed for instance in \(\mathfrak {Q}\) according to their own requirements that a semantics would reflect the aims and presuppositions of the logic (da Costa and Bueno 2009). Non-reflexive logics are non-classical systems that depart from classical logic with respect to the theory of identity. A ‘right’ semantics should be developed within a set theory where identity is limited, as \(\mathfrak {Q}\) does.
Well, you could argue that in ZFA (the Zermelo-Fraenkel theory with atoms, entities that are not set but can be elements of sets) any permutation of atoms conduces to an automorphism of the whole universe. This strategy is useful for the construction of ‘permutation models’, where (inside the model) distinct atoms are made indiscernible by nontrivial automorphisms. Good, but this doesn’t entail that the atoms cannot be discerned; in fact, in the whole universe of ZFA, given any atom a, we can always form the unitary set \(\{a\}\) and distinguish it from any other element of the universe by the property ‘to belong to the singleton of a’, as seen already.
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Acknowledgements
I would like to thank Otávio Bueno, Federico Holik, Silvio Chibeni, and Jonas Arenhart for the helpful discussions and remarks about the topics of this paper, which all of us regard as difficult and subtle one. Thank you, guys.
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Krause, D. Quantifying Over Indiscernibles. Axiomathes 32 (Suppl 3), 931–946 (2022). https://doi.org/10.1007/s10516-022-09646-y
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DOI: https://doi.org/10.1007/s10516-022-09646-y