Nordic Online Logic Seminar - page 4

In this talk I will discuss a line of work that attempts to understand the expressivity of circular reasoning via forms of proof theoretic strength. Namely, I address predicate logic in the guise of first-order arithmetic, and type systems in the guise of higher-order primitive recursion, and establish a recurring theme: circular reasoning buys precisely one level of ‘abstraction’ over inductive reasoning.

This talk will be based on the following works:

• https://arxiv.org/abs/1807.10248
• https://arxiv.org/abs/2012.14421

In 2005, Ulrich Kohlenbach introduced higher order reverse mathematics, and we give a brief explanation of the what and why? of Kohlenbach’s approach. In an ongoing project with Sam Sanders we have studied the strength of classical theorems of late 19th/early 20th century mathematics, partly within Kohlenbach’s formal typed theory and partly by their, in a generalised sense, constructive content. In the final part of the talk I will give some examples of results from this project, mainly from the perspective of higher order computability theory. No prior knowledge of higher order computability theory is needed.

Six months ago I started to examine this report - I suspect I am the first logician to do that. It contains many surprising things. Besides introducing some new ideas that readers of Avicenna know from his later works, it also identifies some specific points of modal logic where Avicenna was sure that Aristotle had made a mistake. People had criticised Aristotle’s logic before, but not at these points. At first Avicenna had no clear understanding of how to do modal logic, and it took him another thirty years to justify all the criticisms of Aristotle in his report. But meanwhile he discovered for himself how to defend a new logic by building new foundations. I think the logic itself is interesting, but the talk will concentrate on another aspect. These recent discoveries mean that Avicenna is the earliest known logician who creates new logics and tells us what he is doing, and why, at each stage along the way.

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.

An argument is valid when all its inferences are valid, and it then amounts to a proof in case it has no unbound assumptions or variables. The validity of an inference – not to confuse with the conclusion being a logical consequence of the premisses – seems in turn best explained in terms of proofs. This means that the concepts of valid inference and valid argument depend on each other and cannot be defined independently but have to be described by principles that state how they are related. A number of such principles will be proposed. It is conjectured that inferences that can be expressed in the language of first order intuitionistic predicate logic and are implied to be valid by these principles are all provable in that logic.