Tracking truth: Knowledge and conditionals in the context of branching time

Van Benthem has recently remarked that Nozick’s notion of knowledge via ‘tracking the truth’ constitutes ‘[...] an open challenge’ to logicians. It is easy to see that its adoption will block laws such as Distribution or Introspection. But are there any valid inference patterns left? Given some plausible background logic of belief and counterfactuals, what is the complete set of validities of Nozick’s K? [35].’ In a recent paper Arló-Costa and Parikh [1] tackled this problem and offered a parametrical account of the resulting logics of knowledge by considering various such plausible background conditional and doxastic logics. The logics of knowledge that thus arise are all non-normal modal logics, where some aspects of logical omniscience fail [29] and which we modeled by appealing to neighborhood semantics [2]. So the standard relational models inspired by the seminal work of Kripke are inapplicable here. In fact the knowledge operator is not closed under entailment and therefore there is no known technique to formulate a normal logic that can simulate it [19]. Nevertheless other important axioms, like Adjunction, are obeyed. It is important to emphasize here that this is a case where logical omniscience fails for normative rather than descriptive reasons (like lack of computational power). Kyburg and associates [20], [21] also offered normative reasons of different kind for abandoning closure. In this case it is Adjunction that fails, while other forms of logical omniscience, like closure under entailment, are obeyed.