Algebraic Approach to Empirical Science and Quantum Logic.

摘要

This paper develops some of the work of Foulis, Randall, Aerts, and Piron in the fields of empirical science and quantum logic from an algebraic point of view. More specifically, it begins with three axioms of what is called a 'subtraction algebra,' and generates various theorems associated with properties which are useful in empirical science. After a foundation is established, it moves on to define the term manual. This term is defined as a dominated, atomic, semi-Boolean algebra which satisfies an additional condition called condition M. Several properties of the manual are discussed, and different types of manuals are given: classical semi-classical and non-classical. The paper defines operational complements, operational perspectively, atoms, events, and tests, before moving on to define a logic, and how it is derived from a manual. Properties of the logic are discussed, including a subtraction operation, a partial order, and an ortho complement. Next, a computer program is presented. Its purpose is to take a finite semi-Boolean algebra and decide if the algebra is a manual. This is followed by a brief non-classical probabilistic discussion, which includes topics such as weights, pure states, and dispersion-free states. Aerts' and Piron's work with properties, states, and questions is briefly discussed before moving on to several examples, some of them arising from navigation problems. The examples include the hook, the square, the Wright Triangle, and the free algebra. Empirical techniques are demonstrated on these examples. The examples comprise the bulk of this paper.