Only-Knowing à la Halpern-Moses for Non-omniscient Rational Agents: A Preliminary Report

We investigate the minimal knowledge approach of Halpern-Moses 'only knowing' in the context of two syntactic variants of stable belief sets that aim in avoiding the unreasonably perfect omniscient agent modelled in R. Stalnaker's original definition of a stable epistemic state. The 'only knowing' approach of J. Halpern and Y. Moses provides equivalent characterizations of 'honest' formulas and characterizes the epistemic state of an agent that has been told only a finite number of facts. The formal account of what it means for an agent to 'only know a' is actually based on 'minimal' epistemic states and is closely related to ground modal nonmonotonic logics. We examine here the behaviour of the HM-'only knowing' approach in the realm of the weak variants of stable epistemic states introduced recently by relaxing the positive or negative introspection context rules of Stalnaker's definition, in a way reminiscent of the work done in modal epistemic logic in response to the 'logical omniscience' problem. We define the 'honest' formulas - formulas which can be meaningfully 'only known' - and characterize them in several ways, including model-theoretic characterizations using impossible worlds. As expected, the generalized 'only knowing' approach lacks the simplicity and elegance shared by the approaches based on Stalnaker's stable sets (actually based on S5) but it is more realistic and can be handily fine-tuned.

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