Knowledge Means 'All', Belief Means 'Most'

We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalized ‘majority ’ quantifier, interpreted as truth in many (a ‘majority ’ of the) epistemically alternative states. This doxastic interpretation is of interest in KR applications and it also has an independent philosophical and technical interest. The logic KBM comprises an S4 epistemic modal operator, a doxastic modal operator of consistent and complete belief and ‘bridge’ axioms which relate knowledge to belief. To capture the notion of a ‘majority ’ we use the ‘large sets’ introduced independently by K. Schlechta and V. Jauregui, augmented with a requirement of completeness, which furnishes a ‘weak ultrafilter ’ concept. We provide semantics in the form of possible-worlds frames, properly blending relational semantics with a version of general Scott-Montague (neighborhood) frames and we obtain soundness and completeness results. We examine the validity of certain epistemic principles discussed in the literature, in particular some of the ‘bridge’ axioms discussed by W. Lenzen and R. Stalnaker, as well as the ‘paradox of the perfect believer ’, which is not a theorem of KBM.