This sort of thing is also studied as part of http://en.wikipedia.org/wiki/Epistemology and it may be of interest that it is studied also as part of computer science for example as http://en.wikipedia.org/wiki/Belief_revision which is in turn closely related to http://en.wikipedia.org/wiki/Reason_maintenance which used to be the basis for general problem solvers. By the way, view maintenance in relational databases could be seen as a subset of reason maintenance.

Note that belief revision and various kinds of paraconsistent logics in a way allow to work around the classical requirements for consistency.

Another really interesting related area is http://en.wikipedia.org/wiki/Modal_logic which allows you to reason about statements such as "I know that you know that I know that the sky is blue." and that can be used to analyze the coordinated attack problem (http://en.wikipedia.org/wiki/Two_Generals'_Problem) which is very much related to the http://en.wikipedia.org/wiki/Byzantine_fault_tolerance of distributed systems.