On a Simple 3-valued Modal Language and a 3-valued Logic of ‘not-fully-justified’ Belief

In this paper, we advocate the usage of the family of Heyting-valued modal logics, introduced by M. Fitting, by presenting a simple 3-valued modal language and axiomatizing an interesting 3-valued logic of belief. We give two simple bisimulation relations for the modal language, one that respects non-falsity and one that respects the truth value. The doxastic logic axiomatized, apart from being interesting in its own right for KR applications, (i) it comes with an underlying 3-valued propositional logic which is a syntactic variant of the ‘logic of here-and-there’, whose importance in KR and Logic Programming is well-known, (ii) it is endowed from its very inception with a Gentzen-style proof theory from [Fit92] and a completeness theorem from [KNP02], (iii) it can be equivalently seen as a logic describing the epistemic agreement of two interrelated agents: a K45 agent who ‘dominates’ an S5 agent, (iv) as we show here, its satisfiability problem is NP-complete, i.e. at the lower level one can expect for applied logics. This is the first concrete example of an epistemic logic from Fitting's framework, that has been overlooked hitherto, despite its many attractive characteristics.