Tsinghua Logic Salon

A proof of a theorem can be said to be pure if it draws only on what is “close” or “intrinsic” to that theorem. In this talk, we will introduce the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. After giving an example from number theory and a brief history of purity in mathematics, we will distinguish four types of purity, based on different measures of distance between theorem and proof. We will then discuss reasons for preferring pure proofs, for the varieties of purity constraints that we have presented.

This talk is about a project where we aim to explore the concept of ultimate ignorance of an agent, depending on the underlying logic of knowledge. By “ultimate ignorance” we mean an operator obtained by iterating the ignorance operator \mathcal{L} (possibly transfinitely), where \mathcal{L} _\varphi intuitively saying that “The agent is ignorant whether \varphi is true”, until stabilisation up to logical equivalence, if that stabilisation ever occurs.

Here, we set the stage for the project and explore the logical behaviour of the finite hierarchy of ignorance \mathcal{L}, \mathcal{L}^2, . . . and its limit operator \mathcal{L}^\omega and demonstrate that, contrary to common expectation, that behaviour is generally rather complex, both in terms of the valid consequences between these operators and in terms of the semantic conditions corresponding to the respective implications between them, over relatively weaker underlying logics of knowledge.

This is joint work with Hans van Ditmarsch and Yanjing Wang.

In this talk, I will introduce a rule-based semantics for causal reasoning and a family of modal languages interpreted through this semantics, built around the concept of causal necessity. I will use these languages to formalize causal concepts, including actual causality and counterfactual conditionals. I will then present techniques for automating these languages, based on satisfiability and model checking. Time permitting, I will illustrate how these languages and decision procedures can be applied to causal reasoning in the legal domain. My talk is based on a paper recently published in the Journal of Artificial Intelligence Research (JAIR) and available at https://www.jair.org/index.php/jair/article/view/18960

We examine the question through the lens of modern and future Set Theory.

About the speaker: William Hugh Woodin is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, bear his name. In 2023, he was elected to the National Academy of Sciences.

Much work has been done to develop one system of inductive logic or another, but there has been much less discussion of what inductive logic should be—or should not be—at a high level of generality. Accordingly, this talk formulates, sharpens, and addresses some foundational problems in inductive logic, while being careful about how those problems are interrelated. The problems I have in mind include the following: (1) Should inductive logic be formal (like deductive logic) or material? (2) Should inductive logic concern a binary relation between sets of premises and conclusions (like deductive logic) or a ternary relation? (3) How should inductive logic be related to other branches of logic, such as deductive logic and nonmonotonic logic? (4) Should inductive logic incorporate probability theory and, if so, how—for example, in a Bayesian or Peircean way?

About the speaker: Hanti Lin is a philosopher of science and formal epistemologist, with papers published in philosophy as well as theoretical computer science. Before he joined UC Davis, he was a postdoc at the Australian National University.