# Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions

## TL;DR

This paper explores different algebraic structures for quantum logic, showing that non-bivalent semantics like supervaluationism simplify foundational theorems such as Kochen-Specker.

## Contribution

It demonstrates that supervaluationist algebraic structures allow a trivial deduction of the Kochen-Specker theorem, highlighting the impact of semantics choice on quantum foundations.

## Key findings

- Supervaluationist algebraic structures simplify the Kochen-Specker theorem.
- Different semantics lead to different organizational structures of quantum propositions.
- Non-bivalent semantics provide alternative interpretations in quantum logic.

## Abstract

The failure of distributivity in quantum logic is motivated by the principle of quantum superposition. However, this principle can be encoded differently, i.e., in different logico-algebraic objects. As a result, the logic of experimental quantum propositions might have various semantics. E.g., it might have either a total semantics, or a partial semantics (in which the valuation relation -- i.e., a mapping from the set of atomic propositions to the set of two objects, 1 and 0 -- is not total), or a many-valued semantics (in which the gap between 1 and 0 is completed with truth degrees). Consequently, closed linear subspaces of the Hilbert space representing experimental quantum propositions may be organized differently. For instance, they could be organized in the structure of a Hilbert lattice (or its generalizations) identified with the bivalent semantics of quantum logic or in a structure identified with a non-bivalent semantics. On the other hand, one can only verify -- at the same time -- propositions represented by the closed linear subspaces corresponding to mutually commuting projection operators. This implies that to decide which semantics is proper -- bivalent or non-bivalent -- is not possible experimentally. Nevertheless, the latter allows simplification of certain no-go theorems in the foundation of quantum mechanics. In the present paper, the Kochen-Specker theorem asserting the impossibility to interpret, within the orthodox quantum formalism, projection operators as definite {0,1}-valued (pre-existent) properties, is taken as an example. The paper demonstrates that within the algebraic structure identified with supervaluationism (the form of a partial, non-bivalent semantics), the statement of this theorem gets deduced trivially.