On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

On higher-order logical grounds

Existential claims are widely held to be grounded in their true instances. However, this principle is shown to be problematic by arguments due to Kit Fine. Stephan Kr�mer has given an especially simple form of such an argument using propositional quantifiers. This note shows that even if a schematic principle of existential grounds for propositional quantifiers has to be restricted, this does not immediately apply to a corresponding non-schematic principle in higher-order logic.

The Moral Law and The Good in Temporal Modal Logic with Propositional Quantifiers

The Moral Law is fulfilled (in a possible world w at a time t) iff (if and only if) everything that ought to be the case is the case (in w at t), and The Good (or The Highest Possible Good) is realised in a possible world w' at a time t' iff w' is deontically accessible from w at t. In this paper, I will introduce a set of temporal alethic deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.

COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

Higher-Order Illative Combinatory Logic

We show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.