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.

Lecture 1: An informal overview

Lecture 2: Bundles in Know-wh (updated)

Ref:

- Hintikka, J., Hintikka, M.: Reasoning about knowledge in philosophy: The paradigm of epistemic logic. In:
*The Logic of Epistemology and the Epistemology of Logic*. Volume 200 of Synthese Library. Springer Netherlands (1989) 17–35 - Hintikka, J. (2003). A Second Generation Epistemic Logic and its General Significance. In V. F. Hendricks, K. F. Jørgensen, & S. A. Pedersen (Eds.),
*Knowledge Contributors*(pp. 33–55). Springer. (IF-like ideas in epistemic logic) - Yanjing Wang: Beyond knowing that: a new generation of epistemic logic in Jaakko Hintikka on knowledge and game theoretical semantics: 499-533. Springer (See the section about Hintikka’s contributions and the references therein)
- Rendsvig, Rasmus, John Symons, and Yanjing Wang, “Epistemic Logic“, The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), .
- Gochet P, Gribomont P (2006) Epistemic logic. In: Gabbay DM, Woods J (eds) Handbook of the History of Logic, vol 7 An early survey about FO-EL
- Jan Plaza: Logics of public communications. In
*Proceedings of the 4th ISMIS*Oak Ridge, TN: Oak Ridge National Laboratory, pp. 201–216. (1989) - Xiwen Ma, Weide Guo: W-JS: A modal logic of knowledge in
*Proceedings of IJCAI83*关于马希文 - H. van Ditmarsch. Comments to ’logics of public communications’. Synthese, 158(2):181– 187, 2007.
- Yanjing Wang, Jie Fan: Knowing that, Knowing what, and Public Communication: Public Announcement Logic with Kv Operators, in
*Proceedings of IJCAI 2013*: 1147-1154. AAAI press. - Yanjing Wang, Jie Fan: Conditionally knowing what, in
*Advances in Modal Logic*Vol. 10: 569-587 (2014), College Publications - Tao Gu, Yanjing Wang: “Knowing value” logic as a normal modal logic. in Advances in Modal Logic Vol. 11:362-381, College Publications
- Jan van Eijck, Malvin Gattinger, Yanjing Wang: Knowing Values and Public Inspection. ICLA 2017: 77-90 Springer
- Alexandru Baltag: To know is to know the value of a variable.in
*Advances in Modal Logic*Vol. 11, College Publications - Yifeng Ding: The axiomatization and complexity of Knowing-What-Logic on model class K, Epistemic Logic with Functional Dependency Operator Studies in Logic
- F. Dechesne, Y. Wang: To know or not to know: epistemic approaches to security protocol verification. Synthese 177(S1): 51-76 (2010), Springer
- Jixin Liu, Yifeng Ding, Yanjing Wang: Model Theoretical Aspects of Weakly Aggregative Modal Logic. J. Log. Lang. Inf. 31(2): 261-286 (2022)
- Yifeng Ding, Jixin Liu & Yanjing Wang: Someone knows that local reasoning on hypergraphs is a weakly aggregative modal logic Synthese 201 (2):1-27 (2023)（WAL over hypergraphs）
- Bo Hong: Knowing the Value of a Predicate. LORI 2023: 149-166

**Lecture 3+4** (Slides) Part of the presentation contained unpublished materials and is omitted. Please check the following papers for technical details on the epistemic interpretation of intuitionistic and intermediate logics:

Haoyu Wang, Yanjing Wang, Yunsong Wang: Inquisitive logic as an epistemic logic of knowing how. Ann. Pure Appl. Log. 173(10): 103145 (2022)

Haoyu Wang, Yanjing Wang, Yunsong Wang : An Epistemic Interpretation of Tensor Disjunction. AiML 2022: 719-739

Key references:

Zilu Wang and Yanjing Wang: Strong Permission Bundled: First Steps. In Proceedings of DEON 2023

Yanjing Wang: A new modal framework for epistemic logic. in Proceedings of TARK 2017: 493-512EPTCS

Anantha Padmanabha, R. Ramanujam, Yanjing Wang: Bundled Fragments of First-Order Modal Logic: (Un)Decidability. FSTTCS 2018: 43:1-43:20

Mo Liu, Anantha Padmanabha, R. Ramanujam and Yanjing Wang: Are Bundles Good Deals for First-order Modal Logic. Information and Computation 2023

Xun Wang: Completeness theorems for ∃ □-bundled fragment of first-order modal logic. * Synthese* 201 (4):1-23 (2023)

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.

- Day 1 To discuss the “prehistory” of bundled modalities in existing work and give a brief overview of what we are going to talk about.
- Day 2 To survey various bundles in epistemic logics of know-wh.
- Day 3 To discuss the bundled interpretations of various non-classical logics.
- Day 4 To introduce the bundled semantics of deontic modalities and their logics.
- Day 5 To introduce the bundled fragments of first-order modal logic and shows their (un)decidability.
- If time permits, we will discuss when and why sometimes we also need to break the bundles.

This advanced graduate-level course presupposes familiarity with modal logic

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