## Events in 2021-2022 Spring Semester

**Abstract:**

The notion of a nonstandard model was introduced by Skolem almost 90 years ago. Since then, nonstandard models have been a subject of study in mathematical logic. In recent years, such structures were successively used to investigate combinatorial problems in reverse mathematics. This talk will give a brief introduction of the key features of a nonstandard model, provide some examples, and discuss the role of nonstandard models in metamathematical investigations, as well as in the philosophy of mathematics.

**Abstract:**The talk is divided into two parts: the first part gives a brief introduction to the topological duality between Boolean algebras with operators and descriptive general frames as well as the discrete duality between complete atomic Boolean algebras and Kripke frames, and we use this framework to discuss canonicity theory. The second part is an introduction of correspondence theory in an algebraic way, which is modular and easy to generalize to other semantic settings.

**Abstract:**This is a theoretical development of epistemic logic to problems concerning the relationship between perception and knowledge. We closely follow the approach of Seligman, Liu and Girard’s “Logic in the Community” which proposes a two-dimensional multi-agent epistemic logic, in which the model operator K (knows) is supplemented with a ‘social’ operators which allow reasoning about relations between agents. The logic also uses operators from hybrid logic, such as nominals n, which name agents, the perspective shifting operator @n, which moves to agent n’s perspective, and the downarrow operator ↓x, which names the current agent a rigid name x. We propose an axiomatisation and completeness proof, using the step-by-step method, first for the basic logic and then for the case of downarrow, which is more involved. While the framework is very general, we are specifically interested in a perceptual agent-oriented operator S (sees). Axioms for the interaction of seeing and knowing are explored. We then consider dynamic extensions of the basic logic with public announcement and “observational” announcement, in which information is given only to agents who can see the announcer.

**Abstract:**

近代来在科学哲学中，因果性已取代定律成为显学。笔者试图论证：James Woodward 的不变性要求太弱，可能有自相矛盾或琐碎无聊的不变性；模态性要求太强，可能出现实际上不太合理的因果关系；定律中所包含的深层次概念，可能无法从因果图中导出；特殊科学实际上很难进行有效操控，数理传统可能比实验传统发挥更大的作用。笔者建议，科学定律与因果性可以有效互补，定律“大处着眼”，因果性“小处着手”。

关键词：定律，因果性，其它情况均同定律

**Abstract:**

Public opinion is a common yet complex phenomenon. We present a formal theory of public opinion for a rigorous platform for the topic, in which opinions are represented by logical formulas. The method of norm forms is used to simplify the problem. We present various aggregation conditions and aggregation functions. We study in detail a specific function. It reflects a common usage that public opinions are the most popular opinions among the public. We prove a characterization theorem for this kind of public opinions, saying that it is the only one that satisfies six fairness conditions.

**Abstract:**In the traditional so-called Tarski’s Truth Definition the semantics of first order logic is defined with respect to an assignment of values to the free variables. A richer family of semantic concepts can be modelled if semantics is defined with respect to a set (a “team”) of such assignments. This is called team semantics. Examples of semantic concepts available in team semantics but not in traditional Tarskian semantics are the concepts of dependence and independence. Dependence logic is an extension of first-order logic based on team semantics. It has emerged that teams appear naturally in several areas of sciences and humanities, which has made it possible to apply dependence logic and its variants to these areas. In my talk I will give a quick introduction to the basic ideas of team semantics and dependence logic as well as an overview of some new developments, such as quantitative analysis of team properties, a framework for a multiverse approach to set theory, and probabilistic independence logic inspired by the foundations of quantum mechanics.

**Abstract:**We study the identification and estimation of long-term treatment effects when both experimental and observational data are available. Since the long-term outcome is observed only after a long delay, it is not measured in the experimental data, but only recorded in the observational data. However, both types of data include observations of some short-term outcomes. In this paper, we uniquely tackle the challenge of persistent unmeasured confounders, i.e., some unmeasured confounders that can simultaneously affect the treatment, short-term outcomes and the long-term outcome, noting that they invalidate identification strategies in previous literature. To address this challenge, we exploit the sequential structure of multiple short-term outcomes, and develop three novel identification strategies for the average long-term treatment effect. We further propose three corresponding estimators and prove their asymptotic consistency and asymptotic normality. We finally apply our methods to estimate the effect of a job training program on long-term employment using semi-synthetic data. We numerically show that our proposals outperform existing methods that fail to handle persistent confounders.

**Abstract:**

Reasoning in social context has many important aspects, one of which is the reasoning about strategic abilities of individuals (agents) and groups (coalitions) of individuals to guarantee the achievement of their desired objectives while acting within the entire society. Several logical systems have been proposed for formalising and capturing such reasoning, starting with the Coalition Logic (CL), the Alternating Time Temporal Logic (ATL) and some extensions of these, introduced the early 2000s.

Coalition Logic provides a natural, but rather restricted perspective: the agents in the proponent coalition are viewed as acting in full cooperation with each other but in complete opposition to all agents outside of the coalition, which are thus treated as adversaries. The Alternating Time Temporal Logic extends Coalition Logic with temporal operators allowing for expressing long-term temporised goals.

The strategic interaction in real life is much more complex, usually involving various patterns combining cooperation and competition. To capture these, more expressive and versatile logical frameworks are needed.

In this talk I will first present briefly Coalition Logic and then will introduce and discuss some more expressive and versatile logical systems, including: (i) the Socially Friendly Coalition Logic (SFCL), enabling formal reasoning about strategic abilities of individuals and groups to ensure achievement of their private goals while allowing for cooperation with the entire society; (ii) the complementary, Group Protecting Coalition Logic (GPCL), capturing reasoning about strategic abilities of the entire society to cooperate in order to ensure achievement of the societal goals, while simultaneously protecting the abilities of individuals and groups within the society to achieve their individual and group goals.

Finally, time permitting, I will discuss briefly conditional strategic reasoning, where agents reason about their strategic abilities conditional on the actions that they expect the other agents to take.

In conclusion, I will take a more general perspective on a unifying logic-based framework for strategic reasoning in social context.

**Abstract:**

I present a topological epistemic logic, motivated by a famous epistemic puzzle: the Surprise Exam Paradox. It is a modal logic, with modalities for knowledge (modelled as the universal modality) and knowability of a proposition (represented by the topological interior operator), and (un)knowability of the actual world. The last notion has a non-self-referential reading (modelled by Cantor derivative: the set of limit points of a given set) and a self-referential one (modelled by Cantor’s perfect core of a given set: its largest subset without isolated points). I completely axiomatize this logic, showing that it is decidable and PSPACE-complete. I point its connections to my older joint work on topological mu-calculus, and finally I apply it to the analysis of the Surprise Exam Paradox (in both its non-self-referential and its self-referential versions). This talk is based on recent joint work with Nick Bezhanishvili and David Fernandez-Duque.

Reference.

Baltag, A.; Bezhanishvili, N.; and Fern´andez-Duque, D. 2022. The topology of surprise. Accepted for presentation at the KR workshop, affiliated with LICS 37. To appear in the KR Proceedings.

Baltag, A.; Bezhanishvili, N.; and Fern´andez-Duque, D. 2021. The topological mu-calculus: completeness and decidability. In Proceedings. of LICS 36, 1–13. IEEE Press.

**Abstract:**Classical philosophical analyses seek to explain knowledge as deriving from more basic notions. The influential “knowledge first” program in epistemology reverses this tradition, taking knowledge as its starting point. From the perspective of epistemic logic, however, this is not so much a reversal as it is the default—the field arguably begins with the specialization of “necessity” to “epistemic necessity”; that is, it begins with knowledge. In this context, putting knowledge second would be the reversal. This work motivates, develops, and explores such a “knowledge second” approach in epistemic logic, founded on distinguishing what a body of evidence actually entails from what it is (merely) believed to entail. We import a logical framework that captures exactly this distinction, use it to define formal notions of (internal and external) justification, and investigate applications to the KK principle, the regress problem, and the definition of knowledge.

**Abstract：**Carnap’s Problem, or Carnap’s Question, as Denis Bonnay and I understand it, is to what extent a consequence relation in some logical language fixes the meaning of the logical constants in that language. This can be seen as relevant to the issue of what ‘logical’ means. Also, it seems that people can have fairly robust intuitions about ‘what follows from what’ without having clear ideas about logicality, so it is of some interest to see if and how the former determines the latter. I will give an overview of what has been achieved in this area so far, concerning classical first-order logic, logics with generalized quantifiers, modal logic, and some partial results for intuitionistic propositional logic (the latter is joint work with Haotian Tong). I end by briefly discussing how this approach fares in comparison to other ideas about logicality.

## Events in 2021-2022 Autumn Semester

**Abstract: **Using a special kind of Birkhoff lattices, we construct a permutation model in which there exists a finite-to-one function from the symmetric group of an infinite set *A* onto *A*, which cannot exist even in the presence of the axiom of countable choice. This is a joint work with Jiachen Yuan.

**Abstract: **新一代人工智能以大数据和机器学习技术为核心，实行的是联结主义的路径。该路径在场景封闭的数据密集型应用中取得了巨大成功，但面临可解释性差、伦理对齐困难、认知推理能力弱等瓶颈问题。为了在一定程度上解决这些问题，不可避免地涉及到对开放、动态、真实环境中信息的刻画，以及对人类推理和解释机制的建模。形式论辩可以提供不一致情境下知识表示与推理的通用机制，与偏好、权重、概率等决策因素的灵活结合机制，局部化和模块化的语义高效计算机制，以及基于论证和对话的可解释机制等。有机结合形式论辩与现有大数据和机器学习技术，有望在一定程度上突破现有技术瓶颈，促进新一代人工智能的健康发展。

**Abstract: **Reasoning with generalized quantifiers in natural language combines logical and arithmetical features, transcending divides between qualitative and quantitative. This practice blends with inference patterns in ‘grassroots mathematics’ such as pigeon-hole principles. Our topic is this cooperation of logic and counting, studied with small systems and gradually moving upward. We start with monadic first-order logic with counting. We provide normal forms that allow for axiomatization, determine which arithmetical notions are definable, and conversely, discuss which logical notions and reasoning principles can be defined out of arithmetical ones. Next we study a series of strengthenings in the same style, including second-order versions, systems with multiple counting, and a new modal logic with counting. As a complement to our fragment approach, we also discuss another way of controlling complexity: changing the semantics of counting to reason about ‘mass’ or other aggregating notions than cardinalities. Finally, we return to natural language, confronting our formal systems with linguistic quantifier vocabulary, monotonicity reasoning, and procedural semantics via semantic automata. We conclude with some pointers to further entanglements of logic and counting in formal systems, in philosophy of logic, and in cognitive psychology.

Paper available at: https://eprints.illc.uva.nl/id/eprint/1813/1/Logic.Counting.pdf

**Abstract: **Much of causality and causal inference can be understood profitably through the lens of modern logic. In this talk we present two applications of this study to artificial intelligence. The first concerns the theoretical and empirical limitations of causal inference from observational and experimental data. The second involves the use of causal-logical tools to derive abstract and human-interpretable analyses of opaque AI systems trained with large, complex data. The broader aim of the talk will be to illustrate the potential for symbiosis between theoretical work in logic and practical work in AI.

**Abstract: **We discuss several questions regarding doxastic logic with propositional quantifiers. First, suppose we take the normal modal logic KD45 as the starting point of the logic of belief, with propositional quantifiers, what logics are available to us, and can we show completeness with respect to some established semantics for belief? To this end, we identify two key principles: the immodesty principle (I believe that everything I believe is true), and the quantificational introspection principle (if no matter what p is, I believe in phi, then I believe that no matter what p is, I believe in phi). We show that, on the one hand, to invalidate the immodesty principle, we need to deviate from the standard possible world semantics, and on the other hand, it is very hard to avoid the quantificational introspection principle since every complete modal algebra validating KD45 also validates it, though it is not derivable in KD45 with the usual axioms for propositional quantifiers. We will also touch on the issue of belief as credence 1. Then, time permitting, we shall consider deviations even from KD45. We will show how rejecting immodesty also puts pressure on introspection (in particular negative introspection) and discuss logical issues specifically with propositional quantifiers coming out of rejecting introspection.

**Abstract: **As concerns the explication of the intuitive notion of truthmaking, Barry Smith has an insight that deserves more attention. Basically, in his view, an object x makes a proposition

true iff (α) x necessitates

and (β)

is representationally closely tied with x. To be more specific, he suggests that (β) is fulfilled only if x is among

’s ontological commitments. I appreciate his basic insight but reject his specific suggestion. I argue that we can make a more attractive proposal from his basic insight if we take into consideration that the close tie can also be realized by

’s being about x.

**Abstract: **Justification logics are closely related to modal logics and can be viewed as a refinement of the latter with machinery for justification manipulation. Justifications are represented directly in the language by terms, which can be interpreted as formal proofs in a deductive system, evidence for knowledge, and so on. This more expressive language proved beneficial in both proof theory and epistemology and helped investigate problems ranging from a classical provability semantics for intuitionistic logic to the logical omniscience problem.

In this talk, we will give an introduction to justification logic and present recent developments in the field such as subset models, conflict tolerant logics, and formalizations of zero-knowledge proofs.

**Abstract: **I will consider different alternatives for giving logic models of ability. First I will explain the difference between possibilities for action and (general) abilities. Then I will focus on the role of knowledge and knowing how in understanding and modelling ability. The importance of modeling these concepts is motivated by discussing their application in symbolic and responsible AI.

**Abstract: **Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in various areas of mathematics such as probability theory, stochastic processes, mathematical physics, functional analysis, and mathematical economics. Nonstandard analysis allows construction of a single object a hyperfinite probability space which satisfies all the first-order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings. We present several applications of this novel approach:

(1) Extending known results for discrete Markov processes to analogues results for general Markov processes (e.g., ergodicity of Markov processes, mixing and hitting times of Markov processes);

(2) Establishing tight connections between frequentist optimality and Bayes optimality for general statistical decision problems;

(3) Existence of Walrasian equilibrium for exchange/production economy models

that are specific to climate change.

**Abstract: **Dynamic Epistemic Logic can be used for Epistemic Planning, as shown in several recent works, ranging from theoretical proposals to actual implementations on robots. A crucial part of automated epistemic planning is to compute perspective shifts that let agents take into account the knowledge of others. So far, these perspective shifts are usually done using explicit Kripke models which may grow exponentially in the number of agents or propositions. I will discuss methods to tackle this state-explosion problem and show how to compute perspective shifts without explicit Kripke models. The two methods I will present are based on symbolic structures and succinct models. Both are compact representations from previous literature showing how to speed up model checking DEL. The new definitions aim to make epistemic planning more efficient in the future. Most of the talk will be based on the article available here: (https://malv.in/2020/EpiP-perspective-shifts.pdf) Time permitting, I will also present related software recently developed by my students and myself.

**Abstract：**传统分析认为“都”表示“总括”（吕叔湘，1980）。我们发现[复数性名词成分+“都”]中“都”是否出现受语境制约。大致说来，当复数性名词成分所在的句子没有独立完整回答语境中的问题时，“都”最好不出现；反之，“都”通常需要出现。从“都”的这一语境适用条件出发，我们提出“都”总括的是语境中的话题/问题（Roberts 1996）；“总括”说的是，“都”表达了与之结合的句子包括了当前话题下的所有内容，因而具有“排除谈话中其他人或事物”(陆庆和，2006)的功能。同时，为了满足“总括”，跟“都”结合的句子必须取分配解读，这造成了“都”的“分配效应”。我们认为，这种解释可以让我们对“都”的“总括义”，“甚至义”和“反预期”效果有统一的认识。更进一步，我们提出“都”的“总括义”是一个预设，这使我们可以通过“强制性预设”现象来解释“都”为什么在某些环境下必须出现（Liu 2021）。最后，本次报告将探讨如何将我们的分析推广至“都”与其他名词成分搭配的情况，以及解释“都，就”为何经常出现在无条件句与条件句的后件中。

**Abstract: **For a long time now, deontic logicians have studied the typology of legal rights that Wesley Hohfeld proposed at the beginning of the 20th century. This has become known as the theory of normative positions. However, one prominent type of legal rights, so-called epistemic rights, has not yet been systematically studied in the hohfeldian typology. In this talk, I will present recent and ongoing work with Réka Markovich (Luxembourg) in which we take the first steps towards filling this gap. I will consider two prominent epistemic rights, the right to know and freedom of thought, and one application of the resulting theory to a recent example in US law.

**Abstract: **集体认定是按一定的规则，综合集体中每个人的意见，对命题的一种断定。法律审判中的陪审员制度，社会政治生活中的选举、决议等都是集体认定的典型例子。

人们很早就发现，集体认定中会出现不一致，合理的认定得到的若干命题放在一起可以是不一致的。人们的研究往往从社会学的角度出发，集中在对于规则合理性的讨论，而不是对不一致现象本身的讨论。

集体认定还有一种类似于不一致的现象：合取原则的失效。而在大多数关于集体认定的研究中，合取原则失效的问题并没有得到充分讨论。

本文从现有的集体认定的规则出发，总结出一些基本的原则，包括认定集体中的个人的原则和认定集体的集体的原则，在这些原则的基础上建立了集体认定逻辑系统，在此逻辑系统中严格定义了不一致和合取原则，给出并证明了不一致现象产生和合取原则失效的条件。

## Events in 2020-2021 Spring Semester

- 2021-5-20 Lian Zhou: Co-reference Without Referent
2021-5-13 Qingbian Ma: Decision Making in the Emergency Room

2021-4-29 Qi Feng: Cantor and Set Theory

2021-4-22 Jialong Zhang: Bertrand Russell and Mathematical Logic

## Events in 2020-2021 Autumn Semester

- 2021-1-10, Lingyuan Ye: Uniformity, Contingency, and Self-reference in Arithmetic & Xiao Li: Towards a Semantic Concept of Aboutness
- 2020-12-25, Duoyi Fei: A Defense for the State of Internal Knowledge
- 2020-12-04, Changsheng Lai: Epistemic Gradualism and the Gradability of Truth
- 2020-12-11, Kang Liu: From Vectors to Symbols.
- 2020-11-12, Thomas Bolander and Lasse Dissing: Implementing Theory of Mind on a Robot Using Dynamic Epistemic Logic.
- 2020-11-06, Zhisheng Huang: Application of Logic in Data Mining

## Events in 2019-2020 Autumn Semester

- 2020-1-08, Fengkui Ju: Towards a Logical Theory of Temporal Conditionals & Xinwen Liu: Jin Yuelin’s Encounter with C.P. Peirce
- 2019-11-28, Shengyang Zhong: On Quantum Logic
- 2019-11-14, Martin Stokhof: Natural Language, Formal Language: a Complex Relationship
- 2019-10-23, Dazhu Li: Dynamic Epistemic Logic of Social Influence; Kaibo Xie (UvA and Tsinghua) : Formal Semantics for Counterfactuals
- 2019-10-10, Frank Veltman: On Imperatives in Natural Language.