Some Computationalconstraints In Epistemic Logic

Abstract

Some systems of modal logic, such as S5, which are often used as epistemic logics with the ‘necessity’ operator read as ‘the agent knows that’ are problematic as general epistemic logics for agents whose computational capacity does not exceed that of a Turing machine because they impose unwarranted constraints on the agent's theory of non-epistemic aspects of the world, for example by requiring the theory to be decidable rather than merely recursively axiomatizable. To generalize this idea, two constraints on an epistemic logic are formulated: r.e. conservativeness, that any recursively enumerable theory R in the sublanguage without the epistemic operator is conservatively extended by some recursively enumerable theory in the language with the epistemic operator which is permitted by the logic to be the agent's overall theory; the weaker requirement of r.e. quasi-conservativeness is similar except for applying only when R is consistent. The logic S5 is not even r.e. quasi-conservative; this result is generalized to many other modal logics. However, it is also proved that the modal logics S4, Grz and KDE are r.e. quasi-conservative and that K4, KE and the provability logic GLS are r.e. conservative. Finally, r.e. conservativeness and r.e. quasi-conservativeness are compared with related non-computational constraints.