Non-reflexive Logical Foundation for Quantum Mechanics

Abstract

On the one hand, non-reflexive logics are logics in which the principle of identity does not hold in general. On the other hand, quantum mechanics has difficulties regarding the interpretation of 'particles' and their identity, also known in the literature as 'the problem of indistinguishable particles'. In this article, we will argue that non-reflexive logics can be a useful tool to account for such quantum indistinguishability. In particular, we will provide a particular non-reflexive logic that can help us to analyze and discuss this problem. From a more general physical perspective, we will also analyze the limits imposed by the orthodox quantum formalism to consider the existence of indistinguishable particles in the first place, and argue that non-reflexive logics can also help us to think beyond the limits of classical identity.

A Defense of the Paraconsistent Approach to Quantum Superpositions (Answer to Arenhart and Krause)

“…which use the interaction of all terms in the superposition irrespectively of the actual measurement outcome. 13 It is important to remark that some new paraconsistent formalizations of quantum superpositions -which go in line with the proposal presented in [16]-are being developed [17,43].…”

“…12 Although we believe there is plenty of room to use the PAQS in many interpretations of QM, the author of this paper has argued elsewhere in favor of a non-collapse interpretation which considers the potential realm completely independent of actuality. We will discuss the particular relation of this interpretation to PAQS in [17].…”

“…In the same way, the introduction of paraconsistent logics might provide a valuable help to investigate the physical meaning of quantum superpositions. There are some new formal proposals which following this path are starting to seriously explore the possibilities of such logical approach [17,43].…”

“…The problem is that QM does not describe a mere ensemble of individuals. We do not know what is an individual according to QM (see for discussion [17,36]), neither is it clear what is the relation between preexistence and observation. As we know, it makes no sense to assume that the measurement exposes an already preexistent ASA as a known paper of Peres express it, in QM "unperformed experiments have no results."…”

Revisiting the Applicability of Metaphysical Identity in Quantum Mechanics

“…Contrary to this widespread (metaphysical) interpretation, we will provide arguments which show that the equivalent classes of a vector in Hilbert space (related through the congruence relation, '=') cannot, in general, be regarded as making reference to a metaphysical identity. Even though we remark that we are not discussing here the well known problem of identical or indistinguishable particles in QM (see [1,11,25]), our analysis will have deep consequences for that debate which we attempt to discuss in a future work.…”