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.