Propositional logic based on the dynamics of belief

Extract

In this article propositions will be identified with a certain kind of changes of belief. The intended interpretation is that a proposition is characterised by the change it would induce if added to a state of belief. Propositions will thus be defined as functions from states of belief to states of belief. A set of postulates concerning the properties and existence of propositions will be formulated. A proposition will be said to be a tautology iff it is the identity function on states of belief. The main result is that the logic determined by the set of postulates is intuitionistic propositional logic.

The basic epistemic concept is that of a belief model, which is defined as a pair 〈, 〉, where is a nonempty set and is a class of functions from to . The elements in will be called states of belief and they will be denoted K, K′,…. A discussion of the epistemological interpretation of the states of belief can be found in Gärdenfors [2]. Here, no assumptions about the structure of the elements in will be made.

The elements in will be called propositions, and A, B, C, … will be used as variables over . Functions from states of belief to states of belief can be characterised as epistemic inputs. The intended interpretation of the functions in is that they correspond to changes of belief where the new evidence is accepted as “certain” or “known” in the resulting state of belief. This means that not all functions defined on can properly be called propositions.