Nordic Online Logic Seminar  page 4
An online seminar for logicians and logic aficionados worldwide.
The Nordic Online Logic Seminar (NOL Seminar) is a monthly seminar series initiated in 2021 presenting expository talks by logicians on topics of interest for the broader logic community. Initially the series focused on activities of the Nordic logic groups, but has since expanded to offer a variety of talks from logicians around the world. The seminar is open to professional or aspiring logicians and logic aficionados worldwide.
The tentative time slot is Monday, 16.0017.30 (Stockholm/Sweden time). If you wish to receive the Zoom ID and password for it, as well as regular announcements, please subscribe to the NOL Seminar mailing list.
NOL seminar organisers
Valentin Goranko and Graham Leigh

Erich Grädel (RWTH Aachen University)
Semiring semantics for logical statements with applications to the strategy analysis of games
Semiring semantics of logical formulae generalises the classical Boolean semantics by permitting multiple truth values from certain semirings. In the classical Boolean semantics, a model of a formula assigns to each (instantiated) literal a Boolean value. Kinterpretations, for a semiring K, generalize this by assigning to each such literal a value from K. We then interpret 0 as false and all other semiring values as nuances of true, which provide additional information, depending on the semiring: For example, the Boolean semiring over {0,1} corresponds classical semantics, the Viterbisemiring can model confidence scores, the tropical semiring is used for cost analysis, and minmaxsemirings (A, max, min, a, b) for a totally ordered set (A,<) can model different access levels. Most importantly, semirings of polynomials, such as N[X], allow us to track certain literals by mapping them to different indeterminates. The overall value of the formula is then a polynomial that describes precisely what combinations of literals prove the truth of the formula.
This can also be used for strategy analysis in games. Evaluating formulae that define winning regions in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. For this approach, the case of Büchi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the logical definition of the winning region involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorptiondominant strategies – strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables further applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.

Anupam Das (University of Birmingham)
On the proof theoretic strength of cyclic reasoning
Cyclic (or circular) proofs are now a common technique for demonstrating metalogical properties of systems incorporating (co)induction, including modal logics, predicate logics, type systems and algebras. Inspired by automaton theory, cyclic proofs encode a form of selfdependency of which induction/recursion comprise special cases. An overarching question of the area, the socalled ‘BrotherstonSimpson conjecture’, asks to what extent the converse holds.
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 firstorder arithmetic, and type systems in the guise of higherorder 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:

Dag Normann (Oslo)
An alternative perspective on Reverse Mathematics
In his address to the International Congress of Mathematics in Vancouver, 1974, Harvey Friedman launched a program where the aim would be to find the minimal set of axioms needed to prove theorems of ordinary mathematics. More than often, it turned out that the axioms then would be provable from the theorems, and the subject was named Reverse Mathematics. In this talk we will survey some of the philosophy behind, and results of, the early reverse mathematics, based on the formalisation of mathematics within second order number theory.
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.

Wilfrid Hodges (Fellow of the British Academy)
How the teenage Avicenna planned out several new logics
Almost exactly a thousand years ago a teenager known today as Avicenna lived in what is now Uzbekistan. He made a resolution to teach himself Aristotelian logic, armed with an Arabic translation of Aristotle and a centuryold Arabic textbook of logic. A couple of years later, around his eighteenth birthday, he wrote a brief report of what he had learned.
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.

Jouko Väänänen (Helsinki)
Dependence logic: Some recent developments
In the traditional socalled 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 firstorder 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.

Dag Prawitz (Stockholm)
Validity of inference and argument
An account of inferences should take into account not only inferences from established premisses but also inferences made under assumptions. This makes it necessary to consider arguments, chains of inferences in which assumptions and variables may become bound.
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.