Stone algebras, conditional events, and three valued logic

There are many ways to extend the operations on events to operations on conditional events, but it is shown that there is only one way to make such extensions so that the resulting structure is a bounded lattice extension of the space of events.

Abstract

There are several equivalent ways to represent the set of conditional events, and in some the operations proposed by Goodman and Nguyen (1991) become much simpler, making the development of the theory much easier and much more concise. Such a development is carried out here using a representation whose relation to three-valued logic is analogous to that of Boolean algebras to two-valued logic, and in which the operations are simple and intuitive. There are many ways to extend the operations on events to operations on conditional events, but it is shown that there are only nine ways to extend intersection and nine ways to extend union so that the operations are Boolean polynomials of their arguments and are idempotent and commutative. Further, there is only one way to make such extensions so that the resulting structure is a bounded lattice extension of the space of events. These particular extensions turn out to be the operations proposed by Goodman and Nguyen.>