Explicit and Implicit Knowledge in Neighbourhood Models

Under relational models, epistemic logic agents are logically omniscient. A common strategy to avoid this has been to distinguish between

implicit

and

explicit

knowledge, and approaches based on relational models have used implicit knowledge as a primitive, defining explicit knowledge as implicit knowledge that satisfies some additional requirement. In this work we follow the opposite direction: using neighbourhood models, we take explicit knowledge as a primitive, then defining implicit knowledge as what the agent will know explicitly in an ‘ideal’ state. This approach, though natural, does not satisfy two ‘intuitive’ properties: explicit knowledge does not need to be implicit, and the consequent of an explicitly known implication with explicitly known antecedent does not need to be implicitly known; we discuss why this is the case. Then a modus ponens operation is defined, and it is shown how it satisfies a third ‘intuitive’ property: if the agent knows explicitly an implication and its antecedent, then after a modus ponens step she will know explicitly the consequent.