Propositional quantification

Propositional quantifiers are quantifiers binding variables in the position of sentences. For instance, when added to standard propositional logic, they allow us to express the claim that for every proposition p, there is a proposition q such that p is materially equivalent the negation of q.

This course will focus on propositional quantifiers in the context of modal logics, where they are especially useful. For example, in the context of a doxastic interpretation of modal logic, they allow us to make generalizations about what is and is not believed by an agent. With this, we can state that everything the agent believes is the case, that the agent believes that they believe something false, or that everything believed by one agent is believed by a second agent.

Standard possible world models for modal logics can be extended straightforwardly to propositional quantifiers, by letting these quantifiers range over arbitrary sets of worlds. However, in many cases, this straightforward model theory leads to logics which are not recursively axiomatizable. In addition to these simple models, we will therefore consider a range of alternative models, including models based on complete Boolean algebras, and possible worlds models in which propositional quantifiers range over a restricted domain of sets of worlds.

The aim of the course is to show the usefulness of propositional quantifiers in modal logics using examples, to provide a systematic overview of the work that has been done in this field, and to highlight some of the many interesting questions which remain open.