Yuanzhe Yang

In this paper, we study the $$\Box \exists $$ -bundled fragment of first-order modal logic, in which quantifiers are only allowed to occur within “bundled operators” of the form $$\Box \exists x$$. We characterize the expressivity of the fragment via the notion of $$\Box \exists $$ -bisimulation, and offer complete axiomatizations of the fragment over 15 classes of constant-domain augmented frames in the modal cube, ranging from K to S5. In the axiomatizations, when characterizing certain frame properties, axioms involving the $$\Box \exists $$ -operator, which do not appear in propositional modal logic, are also used. Hence, we also show that these $$\Box \exists $$ -axioms are necessary for the completeness of the axiomatizations in question. Finally, we show under what frame conditions can we partly or completely eliminate the part of the axiomatization which characterizes the constant-domain property.

In epistemic logic, the beliefs of an agent are modeled in a way very similar to knowledge, except that they are fallible. Thus, the pattern of an agent’s true beliefs is an interesting subject to study. In this paper, we conduct a systematic study on a novel modal logic with the bundled operator ⊡ϕ:=□ϕ∧ϕ as the only primitive modality, where ⊡ captures the notion of true belief. With the help of a novel notion of ⊡-bisimulation, we characterize the expressivity of this new language on relational models, and offer various completeness results. We also discuss some interesting connections between our work and previous works on reflexive-insensitive logics and the so-called boxdot conjecture. Finally, we study two generalizations of the ⊡-operator: one is the iterated true belief operator □nϕ:=ϕ∧□ϕ∧⋯∧□nϕ, the other is the neighborhood semantics for the ⊡-operator.