A modal logic of knowledge, belief and estimation

Abstract

We introduce KBE, a modal epistemic logic for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent's decision-making process. In our logic, Knowledge and Belief are captured by S4.2, a modal logic holding a distinguished position among the epistemic logics investigated in AI and Philosophy. The Estimation operator of KBE is a kind of generalized 'many' or 'most' quantifier, whose origins go back to the work of J. Burgess and A. Herzig, but its model-theoretic incarnation ('weak filters') has been introduced by K. Schlechta and V. Jauregui. We work with complete weak filters ('weak ultrafilters') as we are interested in situations where an estimation can be always reached. The axiomatization of KBE comprises 'bridge' axioms which reflect the intuitive relationship of 'estimation' to 'knowledge' and 'belief', several introspective properties are shown to hold and it comes out that believing i¾? can be equivalently defined in KBE as 'estimating that i¾? is known', an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The model theory of KBE comprises a class of frames combining relational Kripke frames with Scott-Montague semantics, in which neighborhoods are collections of 'large' sets of possible worlds. Soundness and completeness is mentioned and a tableaux proof procedure is sketched.