Using Modal Structures to Represent Extensions to Epistemic Logics

Kripke structures have been proposed as a semantic basis for modal logics of necessity and possibility. They consist of a set of states, informally interpreted as "possible worlds", and a binary accessibility relation between states. The primitive notion of a possible world in this context seems highly intuitive, since necessity can be interpreted as "truth in all possible worlds", and possibility as "truth in some possible world. However, modal logics have also been used to model the epistemic notions of knowledge and belief, where an agent at a particular world is said to "know" or "believe" a proposition if that proposition is true in all possible worlds compatible with its beliefs. In this context, it is not as obvious how to interpret a "possible world". Modal structures have recently been introduced as a formally equivalent alternative to Kripke structures for modeling particular states of knowledge and belief. Modal structures consist of an infinite number of recursively defined levels, where each level contains the possible worlds that model an agent's meta-beliefs of a certain depth. For example, beliefs about the world are modeled at level 1 of a modal structure, and beliefs about beliefs about the world are modeled at level 2. Each modal structure corresponds to a single world of a Kripke structure and contains all the worlds that are accessible from that world in its levels. Modal structures are defined for the classical propositional epistemic logics S4 and S5. Recently, the traditional possible worlds approach has been extended to model "explicit", or limited, belief with partial worlds, called situations, in an appropriately modified Kripke structure. In this thesis, I demonstrate how modal structures can replace Kripke structures to interpret three recent logics of explicit and implicit belief. I also extend modal structures to model a first-order predicate logic which includes quantifiers, equality, and standard names. For each logic, I demonstrate the equivalence of the extended modal structure and the Kripke structure that originally provided the semantics for the logic. I discuss the advantages and disadvantages of using modal structures to model logics of knowledge and belief.