From Degrees of Belief to Defeasible Knowledge
ABSTRACT:
This talk introduces a probabilistic version of the so-called defeasibility theory of knowledge, championed by Lehrer, Klein and others. We take as basic the standard Bayesian framework, given by a probability measure interpreted as giving an agent's ``degrees of belief". In this setting, we define a number of doxastic attitudes: absolute certainty (=probability 1), infallibility or epistemic necessity (=truth in all possible worlds that are consistent with the agent's information), de Finetti's qualitative probability ranking P<=Q (Q is at least as probable as P), Lockean belief B (defined as high confidence, above a fixed threshold), and its conditional version B(P|Q) (high confidence in P given Q). We give a complete axiomatization of the logic of these notions, then move on to defeasibility theory.
A proposition P is undefeated iff its degree of belief stays high (above the Lockean threshold) when any true information is learnt. This is a quantitative version of the ``robustness'' or stability requirement that underlies the defeasibility theory. Undefeated true belief is neither (positively) introspective, nor closed under conjunction: both P and Q can be undefeated without their conjunction being so. However, there is a reflexive version of this concept, definable by a circular definition: let us say that P is known iff both P and the fact that P is known are undefeated. This corresponds to a conscious type of defeasible knowledge: whenever an agent knows something in this sense, he believes (with high confidence) that he knows it.
We study the logic of this concept of knowledge and show that it is very well behaved, and that it is useful for inductive learning. We show that Leitgeb's ``stability theory of belief" can be recovered as a special concept in our setting, namely what we call ``firm belief": the attitude corresponding to ``believing (with high confidence) that you know". We address some possible objections: (1) the context-sensitivity of knowledge (its dependence on the underlying partition), and (2) the apparent failure of qualitative learning to track Bayesian conditioning (problem spotted by K. Kelly). We answer the first problem by arguing that all knowledge depends on the ``relevant questions" (giving the relevant conceptual space).
Time-permitting, we show that only minimal assumptions about probability are in fact needed for developing a version of this theory (namely, de Finetti's axioms of qualitative probability minus the comparability axiom).