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.
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.
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
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.
sentence A can be linked to some other sentence B by being part of a premise
set that allows to derive B.
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
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.
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.
may be a continous activity of re-computation in the background, and more
pressing dynamics in the foreground.
are connected to algorithms of graph traversal. In each CRC the link matrix
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
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.
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.
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.