An Advanced Course (Logic and Computation) at ESSLLI 2024 (First week 17:00-18:30)

Instructor: Yanjing Wang (http://wangyanjing.com)

Bundled modalities typically combine a quantifier with a modality semantically. In recent years, these constructions have drawn increased attention for capturing non-normal modal logics and have led to the discovery of new decidable fragments of first-order modal logic, as well as modal interpretations of various non-classical logics. This course aims to introduce the concepts, fundamental techniques, and applications of bundled modalities in areas such as epistemic logic, deontic logic, intermediate logic, and first-order modal logic.

## Motivation

Modalities can be more than what they appear to be. They may have intricate semantic structures, especially when their logical behaviors deviate from the standard normal modal logic. One pattern observed again and again in the literature is that a non-normal modality may contain an implicit quantifier. A quantifier and a normal modality bundled together can behave non-normally in a very natural way. We call such packages of a quantifier and a modality bundled modalities.

For example, in the epistemic logic of knowing how [5], a know-how modality can be understood as a bundle of an existential quantifier and a know-that modality together ∃xK, i.e., knowing how to achieve ϕ can be interpreted as “there is a plan such that one knows that it is executable and will guarantee ϕ”. This bundle can intuitively explain why the know-how operator disobeys the conjunction aggregation □ϕ ∧ □ψ → □(ϕ ∧ ψ) that is valid in normal modal logic: the witness plans for ϕ and ψ can be incompatible, e.g., knowing how to get drunk and how to prove a theorem does not mean that knowing how to prove the theorem while drunk. This also explains why (monotonic) neighborhood semantics is considered suitable for giving semantics to such operators there is also a quantifier alternation in the neighborhood semantics. As shown in [1], it turns out that such a bundling idea can give natural semantics to various epistemic logic of know-how/why/who/what etc. Moreover, by using such bundled modalities, we can give other philosophical logics new semantics that can solve interesting puzzles. For example, in [7], by treating the permission as a bundled modality, we not only have a desirable logic consistent with the known linguistic data but can also predict new logical behaviors that have never been discussed in the literature but fit our linguistic intuition.

More generally, as initiated in [4], we can bundle quantifiers and modalities in the setting of first-order modal logic (FOML), which leads to a new way of defining useful fragments of FOML. It is well-known that finding decidable fragments of FOML is extremely hard. The previous state-of-the-art was to allowonly one variable to occur in the scope of a modality, on top of the restrictions on the arity of the predicates and the total number of variables. In contrast, by requiring the quantifiers to occur in combination with the modalities in certain ways, we can obtain expressive decidable fragments of FOML without any restriction on the number of variables nor the arity of predicates. We will survey such bundled fragments and show their (un)decidability as in [3].

Bundled modalities also occur in non-classical logics implicitly. For example, various previous BHK-based semantic analyses of intuitionistic logic and intermediate logics are implicitly based on the notion of uniform solvability, which is essentially a bundle of an existential quantifier and an S5 modality. Inspired by this, we give an intuitive know-how interpretation of various intermediate logics as in [6], and make the hidden modalities more explicit. This echoes Heyting’s epistemic interpretation in the early days of intuitionistic logic. The bundles can thus help us to provide non-classical logics intuitive semantics grounded both philosophically and mathematically. Most of the above developments about bundled modalities have happened in the past ten years. The aim of the course is to give the students a taste of this fascinating and fast-moving new direction and its various applications.

## Tentative outline

• Day 1 To discuss the “prehistory” of bundled modalities in temporal logic and neighborhood semantics and then explain the basic techniques for the later discussions.
• Day 2 To survey various bundles in epistemic logics of know-wh.
• Day 3 To introduce the bundled semantics of deontic modalities and their logics.
• Day 4 To introduce the bundled fragments of first-order modal logic and shows their (un)decidability.
• Day 5 To discuss the bundled interpretations of various non-classical logics.

## Expected level and prerequisites

and first-order logic, in particular, the model theory of modal logic and techniques of completeness proofs.

## References

Survey papers:
1 Yanjing Wang: Beyond knowing that: a new generation of epistemic logics, in Jaakko Hintikka on knowledge and game theoretical semantics, 499-533, (2018) Springer
2 Rendsvig, Rasmus, John Symons, and Yanjing Wang ”Epistemic Logic” (Sect. 4), The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.), 2023
3 Mo Liu, Anantha Padmanabha, R. Ramanujam and Yanjing Wang: Are Bundles Good Deals for First-order Modal Logic. Information and Computation 2023

Key papers:
4 Yanjing Wang: A new modal framework for epistemic logic. in Proceedings of TARK 2017: 493-512 EPTCS
5 Yanjing Wang: A logic of goal-directed knowing how, Synthese 195 (10): 4419-4439. 2018
6 Haoyu Wang, Yanjing Wang, Yunsong Wang: Inquisitive Logic as an Epistemic Logic of Knowing How. Annals of Pure and Applied Logic 173 (10):103145
7 Zilu Wang and Yanjing Wang: Strong Permission Bundled: First Steps. In Proceedings of DEON 2023