Connectives in Quantum and other Cumulative Logics

TL;DR

This paper explores the structure of cumulative logics, introduces a representation theorem using choice functions, and examines the definability of connectives like conjunction, negation, and disjunction, especially in quantum logic contexts.

Contribution

It provides a new representation theorem for cumulative logics using choice functions and analyzes the definability of connectives within these logics, including quantum logics.

Findings

Choice functions characterize cumulative logics.

Proper conjunction and negation can be defined in these logics.

Quantum logics with orthogonal complement negation lack proper negation.

Abstract

Cumulative logics are studied in an abstract setting, i.e., without connectives, very much in the spirit of Makinson's early work. A powerful representation theorem characterizes those logics by choice functions that satisfy a weakening of Sen's property alpha, in the spirit of the author's "Nonmonotonic Logics and Semantics" (JLC). The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of worlds, not only definable sets and no definability-preservation property is required in the soundness part. For abstract cumulative logics, proper conjunction and negation may be defined. Contrary to the situation studied in "Nonmonotonic Logics and Semantics" no proper disjunction seems to be definable in general. The cumulative relations of KLM that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by Engesser and Gabbay are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation.