An introduction to lattice based probability theories

详细信息    查看全文

• 作者：Louis Narens ^{lnarens@uci.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Foundations of probability ; Lattice theory ; Intuitionistic logic ; Quantum probability theory
• 刊名：Journal of Mathematical Psychology
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：74
• 期：Complete
• 页码：66-81
• 全文大小：817 K

文摘

The focus of this article is on probability theories based on non-boolean event spaces that have potential for scientific and philosophical applications. Since the early 1930s, such an alternative probability theory has been systematically used to provide a mathematical foundation for quantum physics. This article investigates in a general way feasible alternatives to boolean event algebras as bases for probability theories. Such alternatives permit new kinds of logical relationships among events that are not expressible in boolean event algebras. In psychology, these relationships allow for kinds of modeling methods that help account for various puzzling phenomena found throughout psychology, particularly in cognitive decision theory.

The alternative probability theories presented are investigated through algebraic characterizations of their event spaces and probability functions. This is done using results from a well-developed area of mathematics known as “lattice theory”. This article describes the basics of lattice theory and how the interpretation of a few of its results indicates that generalizations of boolean event spaces are severely limited for useful application in scientific modeling. The overall conclusion is that two types of event spaces appear especially promising for generalization: One that captures key concepts of the logic inherent in quantum mechanics, “quantum logic”, and another that captures key concepts inherent in a generalization of classical logic that is used in the foundations of mathematics, “intuitionistic logic”. Both of these types of logical structures are useful for constructing unorthodox models of troublesome findings in behavioral economics. One theme of this article is the use of intuitionistic logic to model probabilistic judgments and decision making.