Algebraic assignments of truth values to experimental quantum propositions

TL;DR

This paper challenges the common view by proposing that quantum propositions are primarily truth-bearers, and demonstrates that algebraic properties of Hilbert spaces prevent dispersion-free truth assignments, leading to a probabilistic semantics.

Contribution

It introduces a novel perspective that quantum propositions are primary truth-bearers and shows algebraic constraints prevent dispersion-free valuations, supporting a probabilistic interpretation.

Findings

Dispersion-free truth assignments are impossible in finite-dimensional Hilbert spaces.

Algebraic properties of Hilbert spaces restrict total truth functions to quantum propositions.

Probabilistic semantics naturally arise from the algebraic structure of quantum logic.

Abstract

Of what are experimental quantum propositions primary bearers? As it is widely accepted in the modern literature, rather than being bearers of truth and falsity, these entities are bearers of probability values. Consequently, their truth values can be regarded as no more than degenerate probabilities (i.e., ones that have only the values 0 and 1). The mathematical motivation for precedence of probabilistic semantics over propositional semantic for the logic of experimental quantum propositions is Gleason's theorem. It proves that the theory of probability measures on closed linear subspaces of a Hilbert space (which represent experimental quantum propositions) does not admit any probability measure having only the values 0 and 1. -- By contrast, in the present paper, it is proclaimed that experimental propositions about quantum systems are primary bearers of truth values. As this paper demonstrates, algebraic properties of separable Hilbert spaces of finite dimension equal or greater than 2 do not allow in valuations (that is, truth assignments) which are dispersion-free, i.e., total functions from the set of atomic propositions to the set of two objects, true and false. Providing a probability function can be interpreted as a measure of the (un)certainty in the assignment of truth values, the fact that valuations cannot be dispersion-free gives rise to probabilistic semantics for the logic of experimental quantum propositions.