Décio Krause

One of the main ontological challenges posed by quantum mechanics is the problem of the indistinguishability of so-called “identical” particles, that is, particles that share the same state-independent properties. In the framework of this philosophical problem, a quasi-set theory was formulated to provide a proper metalanguage to deal with quantum indistinguishability; this theory included certain Urelemente called m-atoms, representing essentially indistinguishable objects. In turn, over the last two decades, the Modal Hamiltonian Interpretation proposed an ontology of properties, totally devoid of objects, where quantum systems are bundles of instances of universal properties. Therefore, the original quasi-set theory, with its m-atoms, does not adequately reflect the structure of an ontology devoid of objects. The purpose to the present article is to introduce a new quasi-set theory that does not include atoms at all: elementary items correspond to properties and are also represented by quasi-sets, which can be only numerically different. The final aim is to apply this new quasi-set theory to the MHI ontology.

It is well known that in quantum mechanics we cannot always define consistently properties that are context independent. Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities. In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability. In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen–Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same. We will discuss how the consequences of this move may help us understand quantum contextuality.

Intuitively, an individual is a unity, having identity and being such that it is possible at least in principle to discern it from any other individual. But when we leave the intuitive realm and attempt a logical analysis, we find a cluster of problems that are difficult to overcome within standard logico-mathematical apparatuses. This chapter is concerned with some aspects of this intuitive concept of an individual and some related facts about individuation taken from recent discussions that arise in quantum theory, pushing the discussion to a “logical” view. It characterizes individuals by means of invariance by automorphisms. The final part of the chapter proposes a metaphysics where the notion of identity is substituted, for some objects, by a weaker notion of indiscernibility, and the chapter tries to motivate and justify such a move.

One of the main ontological challenges posed by quantum mechanics is the problem of the indistinguishability of so-called “identical” particles, that is, particles that share the same state-independent properties. In the framework of this philosophical problem, a quasi-set theory was formulated to provide a proper metalanguage to deal with quantum indistinguishability; this theory included certain Urelemente called m-atoms, representing essentially indistinguishable objects. In turn, over the last two decades, the Modal Hamiltonian Interpretation proposed an ontology of properties, totally devoid of objects, where quantum systems are bundles of instances of universal properties. Therefore, the original quasi-set theory, with its m-atoms, does not adequately reflect the structure of an ontology devoid of objects. The purpose to the present article is to introduce a new quasi-set theory that does not include atoms at all: elementary items correspond to properties and are also represented by quasi-sets, which can be only numerically different. The final aim is to apply this new quasi-set theory to the MHI ontology.

At present, there are at least two set theories motivated by quantum ontology: Décio Krause’s quasi-set theory (Q) and Maria Dalla Chiara and Giuliano Toraldo di Francia’s quasi-set theory (QST). Recent work [Jorge-Holik-Krause, 2023] has established certain links between QST and Pawlak’s rough set theory (RST), showing that both are strong candidates for providing a non-deterministic semantics of N matrices that generalizes those based on ZF. In this work, we show that the new atomless quasi-set theory Q^− , recently introduced to account for a quantum property ontology [Krause-Jorge, 2024], has strong structural similarities with QST and RST. We study the level of extensionality that each theory presents, its relation to the Leibniz principle and the rigidity property. We believe that developing common features among these three theories can motivate common fields of research. By revealing shared structures, the developments of each theory can have a positive impact on the others.

This paper introduces the theory QST of quasets as a formal basis for the Nmatrices. The main aim is to construct a system of Nmatrices by substituting standard sets by quasets. Since QST is a conservative extension of ZFA (the Zermelo-Fraenkel set theory with Atoms), it is possible to obtain generalized Nmatrices (Q-Nmatrices). Since the original formulation of QST is not completely adequate for the developments we advance here, some possible amendments to the theory are also considered. One of the most interesting traits of such an extension is the existence of complementary quasets which admit elements with undetermined membership. Such elements can be interpreted as quantum systems in superposed states. We also present a relationship of QST with the theory of Rough Sets RST, which grants the existence of models for QST formed by rough sets. Some consequences of the given formalism for the relation of logical consequence are also analysed.

The so-called Received View (RV) on quantum non-individuality states, basically, that quantum particles are not individuals. It has received an amount of criticism in the recent literature, most of it concerning the relation between the RV and the relation of identity. In this chapter we carefully characterise a family of concepts involved in clarifying the view, indicating how the very idea of failure of identity, commonly used to define the RV, may be understood. By doing so, we hope to dissipate some misunderstandings about the RV, which shall also be seen as evidence of its tenability.