Dependence and independence concepts are ubiquitous in natural science, social science, humanities as well as in everyday life. The sentences “I park on this side of the road depending only on the day of the week” and “the time of descent is independent of the weight of the object” are examples of this. The question arises, do these concepts possess enough exactness for us to investigate their logic? There is currently a whole literature on this topic. It is the purpose of this talk to give a non-technical overview of this area of logic.
The 2025 Lindström Lectures will be delivered in November 2025 by Jouko Väänänen, Professor of Mathematics at the University of Helsinki and the ILLC, University of Amsterdam.
The Public Lindström Lecture will occur on Monday, 17 November 2025 at the Faculty of Humanities of Gothenburg University and online. The Research Lecture will be held on Tuesday, 18 November 2025. at the Department of Philosohy, Linguistics and Theory of Science.
See below for information about the talks. Further information about the Lindstrom Lectures are available on the GU page about the lectures.
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The combination of Löwenheim-Skolem Theorem and Compactness Theorem limits the expressive power of a logic to that of first order logic. This maximality principle is the famous Lindström Theorem for first order logic. It reveals that first order logic is at an optimal point of balance: by adding expressive power to it one necessarily loses model-theoretic properties. Soon after Lindström’s result, the question was raised, whether there are other logics at a similar point of equilibrium. More precisely: are there strict strengthenings of first order logic satisfying a Lindström-type characterization? Despite the naturality of this question, it remained unanswered, until recently.
In 2012 Saharon Shelah offered a solution to this problem in the form of a new infinitary logic. It has a Lindström-type characterization in terms of model-theoretic properties combining weak forms of Compactness and a Löwenheim-Skolem type property. In all known cases, a proof of a Lindström-type characterization automatically gives a proof of Craig Interpolation. This is the case for Shelah’s logic, too. There is, however, one aspect where Shelah’s logic seems to be rather weak: the syntax. The logic is derived from a game, in the sense that a sentence is, by definition, a class of structures, closed under a certain Ehrenfeucht-Fraïssé type of a game. This results in the absence of a syntax defined in such a way that the set of all formulas could be obtained by closing the set of atomic formulas under negation, conjunction, quantifiers, and possibly other logical operations. The lack of syntax complicates further study of it and logics in its neighborhood. In the present talk, we address the general question of deriving a syntax from a game, and the more localized question of finding a syntax for Shelah’s logic. Partial answers are provided to both questions. This is joint work with Andres Villaveces and Siiri Kivimäki.