Revising Probabilities and Full Beliefs
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Revising Probabilities and Full Beliefs Sven Ove Hansson1 Received: 15 June 2019 / Accepted: 6 January 2020 / © The Author(s) 2020
Abstract A new formal model of belief dynamics is proposed, in which the epistemic agent has both probabilistic beliefs and full beliefs. The agent has full belief in a proposition if and only if she considers the probability that it is false to be so close to zero that she chooses to disregard that probability. She treats such a proposition as having the probability 1, but, importantly, she is still willing and able to revise that probability assignment if she receives information that gives her sufficient reasons to do so. Such a proposition is (presently) undoubted, but not undoubtable (incorrigible). In the formal model it is assigned a probability 1−δ, where δ is an infinitesimal number. The proposed model employs probabilistic belief states that contain several underlying probability functions representing alternative probabilistic states of the world. Furthermore, a distinction is made between update and revision, in the same way as in the literature on (dichotomous) belief change. The formal properties of the model are investigated, including properties relevant for learning from experience. The set of propositions whose probabilities are infinitesimally close to 1 forms a (logically closed) belief set. Operations that change the probabilistic belief state give rise to changes in this belief set, which have much in common with traditional operations of belief change. Keywords Probability dynamics · Update · Probability revision · Multistate model · Probabilistic learning · Belief change · Infinitesimal probabilities
1 Introduction Formal models of belief states can employ either a dichotomous (all-or-nothing) or a more fine-graded representation of beliefs. The former approach is applied in models belonging to the tradition called belief change or belief revision [1, 5, 9, 15, 18, 35]. In these models, an agent’s current beliefs are represented by a consistent and logically closed set of propositions, called a “belief set”. We will call these models Sven Ove Hansson
[email protected] 1
Division of Philosophy, Royal Institute of Technology (KTH), Teknikringen 76, 100 44 Stockholm, Sweden
S.O. Hansson
dichotomous since they have only two degrees of belief; a proposition is either believed, in which case case it is an element of the belief set, or it is not believed, and not an element of the belief set. The most important fine-graded approach is probability theory, in which a belief state is represented by a probability function with a dense range of infinitely many degrees of belief, between 0 and 1. Both dichotomous and probabilistic representations are usually treated as parts of dynamic frameworks, which also contain operations of change. Such operations specify how the belief state will be modified in response to various inputs. In both approaches, the most important type of input is a proposition to be assimilated as a full belief. If the input is a proposition a, then in
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