Nordic Online Logic Seminar
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

Wes H Holliday (UC Berkeley)
Nordic Online Logic Seminar

Åsa Hirvonen (University of Helsinki)
Looking at quantum mechanics with model theoretic glasses
The axioms of quantum mechanics present the state of a quantum system as a unit vector in complex Hilbert space. However, when Dirac [1] presented his bra and ketvectors, he had a more general space in mind. Schwartz [4] later gave a rigorous account for Dirac’s “vectors” as distributions, but in elementary physics books one still encounters presentations where ketvectors are presented just as elements of a Hilbert space, and treated with methods from finitedimensional linear algebra.
During the last years Tapani Hyttinen and I [2,3] have been looking at various models justifying the finite dimensional approaches from such textbooks. Our approaches are based on various embeddings of a Hilbert space into metric ultraproducts of finitedimensional Hilbert spaces. In this talk I will present the basic ideas, their benefits and limitations. Time permitting, I will also contrast our approach to Boris Zilber’s work on the same questions, that was the original inspiration for us.
 P.A.M. Dirac. The principles of Quantum Mechanics, 3rd ed, Clarendon Press, Oxford, 1947.
 Å. Hirvonen, T. Hyttinen, On eigenvectors, approximations and the Feynman propagator, Ann. Pure Appl. Logic 170 (2019).
 Å. Hirvonen, T. Hyttinen, On Ultraproducts, the Spectral Theorem and Rigged Hilbert Spaces, to appear in J. Symb. Log.
 L. Schwartz, Théory de Distributions, Hermann, Paris, 1950.

Nina Gierasimczuk (Danish Technical University)
Coordinating quantity terms: a simulation study in monotonicity and convexity
Natural languages vary in their quantity expressions, but the variation seems to be constrained by general properties, socalled universals. Explanations thereof have been sought among constraints of human cognition, communication, complexity, and pragmatics. In this work, we examine whether perceptual constraints and coordination dynamics contribute to the development of two universals: monotonicity and convexity. Using a stateoftheart multiagent language coordination model (originally applied to colour terms) we evolve communicatively usable quantity terminologies. We compare the degrees of convexity and monotonicity of languages evolving in populations of agents with and without approximate number sense (ANS). The results suggest that ANS supports the development of monotonicity and, to a lesser extent, convexity. We relate our results to some classical observations about generalised quantifiers in mathematical logic and to the research on conceptual spaces.
This is joint work with Dariusz Kalociński, Franek Rakowski and Jakub Uszyński.

Eric Pacuit (University of Maryland)
From paradox to principles: Splitting cycles and breaking ties
Voting on two alternatives appears unproblematic compared to voting on three (or more). When faced with only two alternatives, many arguments show that Majority Rule distinguishes itself from all other ways of making a group decision. For three or more alternatives, one faces the socalled “Paradox of Voting”: there may be elections with a majority cycle in which a majority of voters prefer A to B, a majority of voters prefer B to C, and a majority of voters prefer C to A. In this talk, I will explain a series of results that axiomatically characterize rules for resolving majority cycles in elections. These rules avoid the “Strong No Show Paradox” by responding properly to the addition of new voters to an election and mitigate spoiler effects by responding properly to the addition of new candidates to an election.
This talk is based on joint work with Wes Holliday.

Ivano Ciardelli (University of Padua)
Inquisitive modal logic: an overview
Inquisitive modal logic is a generalization of standard modal logic where the language also contains questions, and modal operators that can be applied to them. In this talk, I will provide an introductory overview of inquisitive modal logic. I will review some motivations for the approach, present some prominent examples of inquisitive modal logics, mention some results about them, and outline directions for future work.

Ana María MoraMárquez (University of Gothenburg)
Medieval Aristotelian Logic is Scientific Method
This presentation aims to show that medieval Aristotelian logic can be generally characterized as scientific method. To be sure, this method includes formal logic as one of its parts, but formal logic is by no means the crucial part. In fact, if, as I intend to show, the main aim of medieval Aristotelian logic is to provide methods for knowledge production and distribution, so its crucial parts are the methods for scientific proof provided in commentaries on Aristotle’s Posterior Analytics and Topics.
In the first part of the presentation, I argue for the possibility of talking of medieval ‘science’, ‘scientific knowledge’, and ‘scientific method’ from a modern perspective, and discuss how the modern perspective relates to the Latin ‘scientia’ in its different senses. In the second part, I show the progression from Nicholas of Paris (1240s) and Albert the Great (1250s), who see Aristotelian logic as a systematic scientific method where syllogistic argument is fundamental, but who struggle to coherently organize it around syllogistic argument, to Radulphus Brito (1290s) who, still seeing Aristotelian logic as scientific method, uses the notion of ‘second intention’ in order to coherently structure it around syllogistic argument.

Albert Visser (Utrecht University)
Restricted Sequential Theories
Sequential theories form a fundamental class of theories in logic. They have full coding possibilities and allow us to build partial satisfaction predicates for formulas of bounded depthofquantifieralernations. In many respects, they are the proper domain of Gödelian metamathematics. We explain the notion of sequential theory.
A theory is restricted if it can be axiomatised by axioms of bounded depthofquantifieralernations. All finitely axiomatised theories are restriced, but, for example, also Primitive Recursive Arithmetic. We explain the smallisverysmall principle for restricted sequential theories which says that, whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness, i.e., a witness below a given standard number.
We sketch two proofs that restricted theories are incomplete (however complex the axiom set). One uses the smallisverysmall principle and the other a direct Rosser argument. (The second argument was developed in collaboration with Ali Enayat.)

Lauri Hella (Tampere University)
Game characterizations for the number of quantifiers
A game that characterizes definability of classes of structures by firstorder sentences containing a given number of quantifiers was introduced by Immerman in 1981. In this talk I describe two other games that are equivalent with the Immerman game in the sense that they characterize definability by a given number of quantifiers.
In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the other two games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a oneplayer game, but the other two games are genuine twoplayer games.
The talk is based on joint work with Kerkko Luosto.

Peter Pagin (Stockholm University)
Switcher Semantics and quantification
Switcher Semantics is a semantic framework that is basically characterised by allowing switching: when recursively applying a semantic function \(\mu\) to a complex term \(t\), the semantic function applying to an immediate subterm \(t'\) of \(t\) may be a function \(\mu'\), distinct from \(\mu\). An operatorargumentposition pair is called a switcher if it induces such a switch. Switcher semantic systems do not satisfy the standard form of compositionality, but a generalized form, which allows greater flexibility. In earlier work (mostly published), some together with Kathrin Glüer, some with Dag Westerståhl, it has been applied to natural language constructions like proper names in modal contexts, general terms in modal contexts, indexicals in temporal contexts, quotation, and belief contexts. This talk will focus on quantifiers and quantification. Firstorder quantifiers can be regarded as switchers, switching from truth conditions to satisfaction conditions. The larger topic is quantification into switched contexts. I shall begin by giving an introduction to the framework.

Göran Sundholm (Leiden University)
CurryHoward: a meaning explanation or just another realizability interpretation?
Around 1930 a major paradigm shift occurred in the foundations of mathematics; we may call it the METAMATHEMATICAL TURN. Until then the task of a logician had been to design and explain a fullscale formal language that was adequate for the practice of mathematical analysis in such a way that the axioms and rules of inference of the theory were rendered evident by the explanations.
The metamathematical turn changed the status of the formal languages: now they became (meta)mathematical objects of study. We no longer communicate with the aid of the formal systems – we communicate about them. Kleene’s realizability (JSL 1945) gave a metamathematical (re)interpretation of arithmetic inside arithmetic. Heyting and Kolmogorov (19312), on the other hand, had used “proofs” of propositions, respectively “solutions” to problems, in order to explain the meaning of the mathematical language, rather than reinterpret it internally.
We now have the choice to view the CurryHoward isomorphism, say, as a variant of realizability, when it will be an internal mathematical reinterpretation, or to adopt an atavistic, Fregelike, viewpoint and look at the language as being rendered meaningful. This perspective will be used to discuss another paradigm shift, namely that of distinguishing constructivism and intuitionism. The hesitant attitude of Gödel, Kreisel, and Michael Dummett, will be spelled out, and, at the hand of unpublished source material, a likely reason given.