诚邀参加第五届清华逻辑学暑期学校, 6.23 - 27 & 7.14 - 18, 2025, 清华大学

Introduction to Categorical Logic

Lecturer: Ivan Di Liberti (Göteborgs Universitet) and Lingyuan Ye (University of Cambridge)

Time: June 23rd-27th

Abstract: Categorical Logic, and Functorial semantics more specifically, emerged in the 1960s as an alternative framework to capture universal algebra. The framework offers a collection of advantages, including a more flexible and modular approach to semantics, which delivers a perfect correspondence with syntax. These tools offer a more quantitative and conceptual take on completeness results and definability-type theorems.

After a brief introduction to the language of categories, we focus on universal algebra and functorial semantics. We capture the notion of algebraic theory via categories with products (Lawvere theories) and present a syntax-semantics duality between varieties and Lawvere theories. The course ends with some vistas on the theory of sketches which offers a much more general framework, covering the leap from universal algebra to infinitary first order logic.

Prerequisites: The audience is expected to be familiar and have played with the basic definitions of one of following objects: vector space, monoid, group, ring, module, set equipped with operations.

Proof Theory of Modal Logic

Lecturer: Marianna Girlando (ILLC, University of Amsterdam)

Time: July 14th -18th

Abstract: While proof systems for classical and intuitionistic logic can be defined using sequent calculus, the Gentzen-style formalism does not readily extend to the case of modal logic. And indeed, Gentzen-style sequents fail to meet basic requirements in the case of several modal logics: for instance, no cut-free sequent calculus is known for logic S5. In order to overcome these difficulties, various proof systems extending Gentzen’s sequent calculus have been introduced. Namely, labelled calculi extend the language of the calculi with semantic information, while structured calculi, such as hypersequents or nested sequents, employ additional structural connectives.

In this course we will introduce labelled and structured proof systems and illustrate their main properties, including cut-freeness and modularity. Other than normal modal logics, we will discuss logics with neighbourhood semantics, such as conditional logics, and intuitionistic modal logics. We will showcase how labelled and structured calculi are suitable to obtain decision procedures for these logics. To conclude, we will compare the various proof systems by defining formal translations allowing to convert proofs from one formalism to another.

Prerequisites: The course assumes some basic familiarity with modal logic and with sequent calculus for classical propositional logic. The main proof-theoretic notions used in the course will be covered in the first lecture.

Topological Approaches to Epistemic Logic

Lecturer: Aybüke Özgün (ILLC, University of Amsterdam)

Time: July 14th -18th

Abstract: Topological semantics for epistemic logic provides an elegant and natural way to model not only knowledge and belief within a (modal) logical framework, but also the evidence upon which knowledge and belief are based. In this course, we investigate various topological approaches to epistemic logic. We explore topological spaces as information structures naturally suited to modeling evidence and its relationship to knowledge and belief. We focus mainly on the recent developments in the field, elaborate on the conceptual arguments behind using topological spaces as formal models for epistemic logic, and study the technical details of relevant (meta)-logical results. Topics to be covered include:

• basic notions in topology and how they relate to important notions in epistemology.

• the interior-based topological semantics for modal/epistemic logic.

• topological evidence models.

• subset space semantics, topo-logic, and their extensions.

At the end of the course, we will provide an overview of recent developments in the field, highlighting open questions and further avenues for investigation.

Prerequisites: The course will only require background in metalogic and basic modal logic (its syntax, Kripke semantics, some basic results on completeness, and its epistemic/doxastic interpretation). In particular, we will introduce all the necessary topological concepts on the first day. In the course, we will generally focus on technical material but often provide examples, in order to make the material more accessible, and we will also have class discussions about the philosophical issues behind the technical material covered in the course. Students are also expected to have some degree of mathematical maturity and an interest in the themes of the course.