Modelling Belief Dynamics

1. Belief dynamics (BD) involves all ways of updating or changing a belief system (BS). This may be expansion, contraction or revision. Partial analyses have been developed by the AGM-approach, Neil Tennant and other authors of the field of 'belief revision', John Pollock and other (AI)-authors dealing with inductive and non-monotonic logic, dynamic epistemic logic and corresponding parts of epistemology. One may see philosophy’s role in the cognitive sciences here again as inter alia providing (formal) models of basic entities and capacities. This formal modelling is continuous to the formal modelling in some cognitive sciences, as (formal) linguistics or 'ontologies' in Artificial Intelligence (there being only a vague boundary between work in Artificial Intelligence and research in pure formal logic). Given the sheer number of researchers working in the fields, however, much more detailed work in formal modelling is done outside of philosophy and more in the cognitive sciences, especially in abstract computer science and the related fields of logic. Some areas like 'knowledge representation' and also belief dynamics fall squarely in the area of traditional epistemology. So, again, the role of philosophy cannot be to supply all these models and results, but to consider there most general aspects and components, and to provide some theoretical general framework wherein to place these approaches.

2. We should distinguish a logical model of a belief system and the accessible belief system. The logical model can contain data structures and information that is not consciously accessible to the agent. Conscious access has to play a part as belief dynamics is part of deliberation and conscious and not only sub-conscious 'online' interaction with reality.

3. A computational approach to issues of belief dynamics, i.e. algorithms and a data formats, proceeds from 'crude' to 'refined'. 'Crude' beginnings try to get a grisp on modelling BD, neglecting issues of storage waste, time complexity and at least some questions of psychological reality. 'Refinement' then tries to proceed to models and algorithms that meet the goal of providing a feasible model.
The web of beliefs can be modelled in a graph. The nodes of the graph are sentences. They are expressed in a First Order Language. The links between sentences are annotated. Each node can have finitely many links to other nodes. Some nodes are basic nodes (e.g. sensory input). Basic nodes depend inferentially only on themselves. Whether they imply e.g. perceptual beliefs, however, may depend on normality assumptions etc. Another type of basic nodes are 'postulates' (like axioms of mathematics). Postulates are not confined to logical truths, but they are treated as those sentences not to be given up, and thus are 'valid' as far as the BS under consideration is concerned.
Some sentence A can be linked to some other sentence B by being part of a premise set that allows to derive B.
The link matrix contains sets of sets of sentences. All these sets are finite as we are dealing with finite BS. The matrix entry indexed by sentences A and B contains all sets of premises such that A together with these premises logically implies B. These sets of sets are the annotation of the links of the graph of beliefs (beliefs expressed as sentences). Sentence A can be linked to B by more than one minimal set because A may be part of quite different arguments, even differing in supposed 'logic' (i.e. one being inductive the other deductive).

4. The links between sentences are inferential steps in a given logic. By Hilbert's Thesis (using the Church-Turing-Thesis and Turing's Theorem that First Order Logic (FOL) is equivalent in computational power to Turing-Machines) one can argue that in as much as the inferential rules of some type of reasoning or some (probabilistic, inductive, non-monotonic, adaptive or otherwise non-classical) logic can be specified algorithmically (be it in the meta-language of some calculus) these rules can be rendered as conditionals that have a representation in FOL. FOL is the upper limit of inference. The rules of FOL are finite. The links between nodes represent steps of reasoning. We should assume that steps are minimal in the sense that all premisses of a step of reasoning are used in that step. Otherwise everything would be connected to everything else by being a irrelevant/vacous premise in some step of reasoning.
Belief systems contain finitely many sentences. They are not deductively closed under FOL. Agents are not logically omniscient. An agent may even follow an inference rule of her own logic that is not sound, the corresponding conditional (expressed in FOL) is then one of her beliefs. Belief systems need not be consistent, but the drive for at least local consistency will be part of belief revision.

5. Belief dynamics consists in cycles of re-computation (CRC). Given a change of belief (one node being activated, deactivated or added) or change of links the repercussions of a local change spread stepwise.
There may be a continous activity of re-computation in the background, and more pressing dynamics in the foreground.
CRCs are connected to algorithms of graph traversal. In each CRC the link matrix is updated.
A consistent link matrix is stable [where 'consistency' as always is understood as maximal in the sense that all remaining inconsistencies cannot be avoided given the basic principles of that belief system]. As long as the links matric is unstable there will be CRCs.

6. The logical model contains the links between the nodes. A psychological model may reckon with the psychological fact that we cannot easily access these links introspectively. Thus the psychological model may rather look like a set of nodes (containing sentences believed). CRCs and BD depend on (sub-doxastic) access to links. Positive undermining may deactivate a belief directly. Loosing justification requires some track keeping of the links towards a belief.
Inquiring into an issue may involve either (i) accessing previously established and still present links in a (logical) graph of beliefs, or (ii) constructing routes to evidence for the issue in questions starting from a semantic parsing of the sentences expressing the issue. In case (ii) the links may either (iia) be more easily be remembered by this prompting, but still preexisting or (iib) constructed on the spot by accessing the repository of other beliefs in a belief set. A repetitive reconstruction of links seems a waste of resources. Online total recall of justifications seems a waste of conscious awareness and may result in a loss of focus. Semantic prompting of analytic and justificatory links may be more economical than a facility of total recall on the spot.

7. The graph structure open to cycles of justification and the existenc of basic nodes together combine elements of coherentism and foundationalism.
BD does not spell out epistemology in terms of either of these structural claims, it can however be considered to be part of an elucidation of coherence. BD models the dynamics of rationality in maintaining a coherent BS. If the ways of connecting beliefs are modelled by the steps (these being explanatory steps or inductive steps or ...) then coherence may come down to two ingredients: (i) from the internal perspective within the BS coherence is maintainance of consistency by appropriate CRCs , (ii) from an external perspective one may ask whether the link structure of a consistent BS µ is more coherent than the links structure of a consistent alternative µ' where µ and µ' share (most of) their nodes. Considerations of type (ii) are the classical desiderata of spelling out 'coherence' by 'explanatory power', 'simplicity' etc.
An account of BD need not consider (ii) in the beginning, maybe (ii) can be part of an account what triggers changes in links or postulates.

MB, 2016.