A modal logic of knowledge, belief and estimation
Costas D. Koutras, Christos Moyzes, Yorgos Zikos · Journal of Logic and Computation · 2017
The actions of an agent operating in a complex environment are based on her knowledge and beliefs. Epistemic states are typically incomplete, thus the agent has often to estimate whether a fact is true or false before proceeding to actions. We introduce a modal framework for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent’s decision-making process. In this modal account, Knowledge and Belief are captured by |$\mathbf{S4.2}$| which has been advocated as a ‘correct’ logic of knowledge by W. Lenzen and R. Stalnaker. Estimation is a non-normal modal operator interpreted as a ‘majority’ quantifier; the approach employs the ‘weak filters’, a general notion of ‘big’ subsets introduced within KR by K. Schlechta and V. Jauregui. Its axiomatization reveals that this ‘majority’ operator has been introduced by J. Burgess (1969) and used also by A. Herzig (2003). We provide a full account of the logic |$\mathbf{KBE}$| in which estimation is complete: exactly one of |$\varphi$| and |$ eg\varphi$| is estimated to be true. We prove soundness and completeness with respect to a class of frames combining relational Kripke frames with Scott-Montague semantics in which neighbourhoods are ‘weak ultrafilters’ and present a tableaux proof procedure. It comes out that believing |$\varphi$| can be equivalently defined in |$\mathbf{KBE}$| as ‘estimating that |$\varphi$| is known’, an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The assumption on complete estimation is rather strong; yet, it is not hard to define |$\mathbf{KBiE}$|, a relaxed variant in which estimation is still consistent but not complete. We provide the technical details for soundness and completeness with respect to the larger class of frames where neighbourhoods are weak filters. Finally, it is proved that |$\mathbf{KBiE}$| invalidates a rule introduced in W. Lenzen’s probabilistic analysis of weak belief and thus the weak filter semantics is essentially different than its probabilistic counterpart.