We study the completeness problem for propositionally quantified modal logics on quantifiable general frames, where the admissible sets are the propositions the quantifiers can range over and expressible sets of worlds are admissible, and Kripke frames, where the quantifiers range over all sets of worlds. We show that any normal propositionally quantified modal logic containing all instances of the Barcan scheme is strongly complete with respect to the class of quantifiable general frames validating it. We also provide a suﬀicient condition for the truth of all formulas, possibly with quantifiers, to be preserved under passing from a quantifiable general frame to its underlying Kripke frame. This is reminiscent of both the idea of elementary submodel in model theory and the persistence concepts in propositional modal logic. The key to this condition is the concept of finite generated diversity (Fritz 2023), and with it, we show that if Θ is a set of Sahlqvist formulas whose class of Kripke frames has finite generated diversity, then the smallest normal propositionally quantified modal logic containing Θ, Barcan, a formula stating the existence of world propositions, and a formula stating the definability of successor sets, is Kripke complete. As a special case, we have a simple finite axiomatization of the logic of Euclidean Kripke frames.

**Some General Completeness Results for Propositionally Quantified Modal Logics**[preprint]

Joint with Yipu Li

In most modal logics, atomic propositional symbols are directly representing the meaning of sentences (such as sets of possible worlds). In other words, they use only rigid propositional designators. This means they are not able to handle uncertainty in meaning directly at the sentential level. In this paper, we offer a modal language involving non-rigid propositional designators which can also carefully distinguish \textit{de re} and \textit{de dicto} use of these designators. Then, we axiomatize the logics in this language with respect to all Kripke models with multiple modalities and with respect to S5 Kripke models with a single modality.

**Someone knows that local reasoning on hypergraphs is a weakly aggregative modal logic**[publisher]

Joint with Jixin Liu and Yanjing Wang; Synthese 201, 46 (2023)

Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of WAML. Specifically, we first give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modallogics, we show that each basic WAML system Kn lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of K2 does have Craig interpolation, as an example of amending the interpolation problem of WAML.

**Model Theoretical Aspects of Weakly Aggregative Modal Logic**[publisher]

Joint with Jixin Liu and Yanjing Wang; Journal of Logic, Language and Information 31, 261–286 (2022).

Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of WAML. Specifically, we first give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modallogics, we show that each basic WAML system Kn lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of K2 does have Craig interpolation, as an example of amending the interpolation problem of WAML.

**Hypergraphs, local reasoning, and weakly aggregative modal logic**[publisher]

Joint with Jixin Liu and Yanjing Wang; in LORI 2021.

This paper connects the following three apparently unrelated topics: an epistemic framework fighting logical omniscience, a class of generalized graphs without the arities of relations, and a family of non-normal modal logics rejecting the aggregative axiom. Through neighborhood frames as their meeting point, we show that, among many completeness results obtained in this paper, the limit of a family of weakly aggregative logics is both exactly the modal logic of hypergraphs and also the epistemic logic of local reasoning with veracity and positive introspection.

**Logics of imprecise comparative probability**[preprint] [publisher]

Joint with Wesley Holliday and Thomas Icard, International Journal of Approximate Reasoning, Volume 132, May 2021, 154-180

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

**On the Logic of Belief and Propositional Quantification**[preprint] [publisher] [slides]

Journal of Philosophical Logic (2021)

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 some-thing 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.

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics---the atomicity of the algebra of propositions---but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.

**The Logic of Comparative Cardinality**[preprint] [publisher]

Joint with Matthew Harrison-Trainer and Wesley Holliday; The Journal of Symbolic Logic, vol. 85, no. 3, 972-1005

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

**Weakly Aggregative Modal Logic: Characterization and Interpolation**[preprint]

Joint with Jixin Liu and Yanjing Wang; in LORI 2019.

Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation and show that each basic WAML system K_n lacks Craig Interpolation.

**When Do Introspection Axioms Matter for Multi-Agent Epistemic Reasoning?**[preprint] [slides]

Joint with Wesley Holliday and Cedegao Zhang; in TARK 2019.

The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later literature on epistemic logic in computer science and game theory has focused on multi-agent epistemic reasoning, with the single-agent 4 and 5 axioms largely taken for granted. In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes; but it is rarely important to reason just about what agent A believes about what agent A believes. This raises the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In this paper, we formalize and answer this question. To formalize the question, we first define a set of multi-agent formulas that we call agent-alternating formulas, including formulas like Box_a Box_b Box_a p but not formulas like Box_a Box_a p. We then prove, for the case of belief, that if one starts with multi-agent K or KD, then adding both the 4 and 5 axioms (or adding the B axiom) does not allow the derivation of any new agent-alternating formulas -- in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas.

**On the Logics with Propositional Quantifiers Extending S5Pi**[preprint] [slides]

In Advances in Modal Logic 2018.

Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π.

**Epistemic Logic with Functional Dependency Operator**[preprint]

Studies in Logic, Vol. 9, No. 4 (2016): 55-84.

Epistemic logic with non-standard knowledge operators, especially the "knowing-value'' operator, has recently gathered much attention. With the "knowing-value'' operator, we can express knowledge of individual variables, but not of the relations between them in general. In this paper, we propose a new operator Kf to express knowledge of the functional dependencies between variables. The semantics of this Kf operator uses a function domain which imposes a constraint on what counts as a functional dependency relation. By adjusting this function domain, different interesting logics arise, and in this paper we axiomatize three such logics in a single agent setting. Then we show how these three logics can be unified by allowing the function domain to vary relative to different agents and possible worlds. A multiagent axiomatization is given in this case.

**The axiomatization and complexity of Knowing-What-Logic on model class K**[preprint]

Unpublished note.

Standard epistemic logic studies propositional knowledge, yet many other types of knowledge such as "knowing whether'', "knowing what'', "knowing how'' are frequently and widely used. This paper presents a axiomatization and a tableau for the modal logic of "knowing-what" operator on arbitrary Kripke models. As we are not working on S5 model class, this operator is not technically a "knowing" operator, but the inner structure is clearer in this setting.