Nordic Online Logic Seminar  page 2
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

Thomas Bolander (Technical University of Denmark)
Epistemic Planning: Logical formalism, computational complexity, and robotic implementations
Dynamic Epistemic Logic (DEL) can be used as a formalism for agents to represent the mental states of other agents: their beliefs and knowledge, and potentially even their plans and goals. Hence, the logic can be used as a formalism to give agents, e.g. robots, a Theory of Mind, allowing them to take the perspective of other agents. In my research, I have combined DEL with techniques from automated planning in order to describe a theory of what I call Epistemic Planning: planning where agents explicitly reason about the mental states of others. The talk will introduce epistemic planning based on DEL, address issues of computational complexity, and demonstrate applications in cognitive robotics and humanrobot collaboration.

Thomas Ågotnes (University of Bergen and Shanxi University)
Somebody Knows and Weak Conjunctive Closure in Modal Logic
Normal modal logics are closed under conjunctive closure. There are, however, interesting nonnormal logics that are not, but which nevertheless satisfy a weak form of conjunctive closure. One example is a notion of group knowledge in epistemic logic: somebodyknows. While something is general knowledge if it is known by everyone, this notion holds if it is known by someone. Somebodyknows is thus weaker than general knowledge but stronger than distributed knowledge. We introduce a modality for somebodyknows in the style of standard group knowledge modalities, and study its properties. Unlike most other group knowledge modalities, somebodyknows is not a normal modality; in particular it lacks the conjunctive closure property. We provide an equivalent neighbourhood semantics for the language with a single somebodyknows modality, together with a completeness result: the somebodyknows modalities are completely characterised by the modal logic EMN extended with a particular weak conjunctive closure axiom. The neighbourhood semantics and the completeness and complexity results also carry over other logics with weak conjunctive closure, including the logic of socalled local reasoning (Fagin et al., 1995) with bounded “frames of mind”, correcting an existing completeness result in the literature (Allen 2005). The talk is based on joint work with Yi N. Wang.

Magdalena Ortiz (Umeå University)
A Short Introduction to SHACL for Logicians
The SHACL Shapes Constraint Language was recommended in 2017 by the W3C for describing constraints on web data (specifically, on RDF graphs) and validating them. At first glance, it may not seem to be a topic for logicians, but as it turns out, SHACL can be approached as a formal logic, and actually quite an interesting one. In this paper, we give a brief introduction to SHACL tailored towards logicians and frame key uses of SHACL as familiar logic reasoning tasks. We discuss how SHACL relates to description logics, which are the basis of the OWL Web Ontology Languages, a related yet orthogonal standard for web data. Finally, we summarize some of our recent work in the SHACL world, hoping that this may shed light on how ideas, results, and techniques from wellestablished areas of logic can advance the state of the art in this emerging field.

Neil Tennant (Ohio State University)
It's All or Nothing: Explosion v. Implosion
We set out five basic requirements for a logical system to be adequate for the regimentation of deductive reasoning in mathematics and science. We raise the question whether there are any further requirements, not entailed by these five, that ought to be imposed. One possible reply is dismissed: that the logical system should allow one to infer any proposition at all from an inconsistent set—i.e., it should have as primitive, or allow one to derive, the rule Ex Falso Quodlibet. We then propose that the logic should be implosive: it should not allow an inconsistent set to have any consequences other than absurdity. This proposal may appear to be very radical; but we hope to show that it is robust against objections.

Ali Enayat (University of Gothenburg)
Arithmetic and set theory through the lens of interpretability
The notion of (relative) interpretation for first order theories was introduced in a landmark 1953 monograph by Alfred Tarski, Andrzej Mostowski and Raphael Robinson, where it was developed as a powerful tool for establishing undecidability results. By now the domain of interest and applicability of interpretability theory far exceeds undecidability theory owing to its multifaceted interactions with both proof theory and model theory. Special attention will be paid to recent advances in the subject that indicate the distinctive character of Peano Arithmetic, ZermeloFraenkel set theory, and their higher order analogues in the realm of interpretability theory. This talk will present a personal overview of the interpretability analysis of arithmetical and set theoretical theories.

Sven Ove Hansson (KTH)
How to combine probabilities and full beliefs in a formal system
One of the major problems in formal epistemology is the difficulty of combining probabilistic and full (dichotomous, allornothing) beliefs in one and the same formal framework. Major properties of actual human belief systems, including how they are impacted by our cognitive limitations, are used to introduce a series of desiderata for realistic models of such belief systems. This leads to a criticism of previous attempts to combine representations of both probabilistic and full beliefs in one and the same formal model. Furthermore, a formal model is presented in which this is done differently. One of its major features is that contingent propositions can be transferred in both directions between full beliefs and lower degrees of belief, in a way that mirrors reallife acquisitions and losses of full beliefs. The subsystem consisting of full beliefs has a pattern of change that constitutes a credible system of dichotomous belief change.

Vann McGee (MIT)
Boolean Degrees of Truth and Classical Rules of Inference
Compositional semantics that acknowledge vagueness by positing degrees of truth intermediate between truth and falsity can retain classical sentential calculus, provided the degrees form a Boolean algebra. A valid deduction guarantees that the degree of truth of the conclusion be at least as great as every lower bound on the degrees of the premises. If we extend the language to allow infinite disjunctions and conjunctions, the Boolean algebra will need to be complete and atomic. If we extend further by adding quantifiers ranging over a fixed domain, we get the supervaluations proposed by Bas van Fraassen.

Alexandru Baltag (University of Amsterdam)
From Surprise Exams to Topological MuCalculus
I present a topological epistemic logic, motivated by a famous epistemic puzzle: the Surprise Exam Paradox. It is a fixedpoint modal logic, with modalities for knowledge (modelled as the universal modality), knowability of a proposition (represented by the topological interior operator), and (un)knowability of the actual world. The last notion has a nonselfreferential reading (modelled by Cantor derivative: the set of limit points of a given set) and a selfreferential one (modelled by the socalled perfect core of a given set: its largest subset which is a fixed point of relativized derivative). I completely axiomatize this logic, showing that it is decidable and PSPACEcomplete, as well as briefly explain how the same modeltheoretic method can be elaborated to prove the completeness and decidability of the full topological mucalculus. Finally, I apply it to the analysis of the Surprise Exam Paradox and other puzzles.
References:
 A. Baltag, N. Bezhanishvili, D. FernándezDuque. The Topology of Surprise. Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning. Vol. 19 (1), 3342, 2022. Available online in ILLC Prepublication (PP) series PP202206.
 A. Baltag, N. Bezhanishvili, D. FernándezDuque. The Topological MuCalculus: Completeness and Decidability. LICS ‘21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, vol 89: 113, 2021. doi:10.1109/lics52264.2021.9470560. Available online in ILLC Prepublication (PP) series PP202107.

Laura Crosilla (University of Oslo)
On Weyl's predicative concept of set
In the book Das Kontinuum (1918), Hermann Weyl presents a coherent and sophisticated approach to analysis from a predicativist perspective. In the first chapter of (Weyl 1918), Weyl introduces a predicative concept of set, according to which sets are built ‘from the bottom up’ starting from the natural numbers. Weyl clearly contrasts this predicative concept of set with the concept of arbitrary set, which he finds wanting, especially when working with infinite sets. In the second chapter of Das Kontinuum, he goes on to show that large portions of 19th century analysis can be developed on the basis of his predicative concept of set.
Das Kontinuum inspired fundamental ideas in mathematical logic and beyond, such as the logical analysis of predicativity of the 195060’s, Solomon Feferman’s work on predicativity and Errett Bishop’s constructive mathematics. The seeds of Das Kontinuum are already visible in the early (Weyl 1910), where Weyl, among other things, offers a clarification of Zermelo’s axiom schema of Separation.
In this talk, I examine Weyl’s predicative concept of set in (Weyl 1918) and discuss its origins in (Weyl 1910).
Bibliography
 Weyl, H., 1910, Über die Definitionen der mathematischen Grundbegriffe, Mathematischnaturwissenschaftliche Blätter, 7, pp. 9395 and pp. 109113.
 Weyl, H., 1918, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig. Translated in English, Dover Books on Mathematics, 2003.

Melvin Fitting (City University of New York (Graduate Center))
Strict/Tolerant Logic and Strict/Tolerant Logics
Strict/tolerant logic, ST, has been of considerable interest in the last few years, in part because it forces consideration of what it means for two logics to be different, or the same. And thus, of what it means to be a logic. The basic idea behind ST is that it evaluates the premises and the conclusions of its consequence relation differently, with the premises held to stricter standards while conclusions are treated more tolerantly. More specifically, ST is a threevalued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does, though it differs from classical logic at the metaconsequence level. A version of this result has been extended to meta, metameta , etc. consequence levels, creating a very interesting hierarchy of logics. All this is work of others, and I will summarize it.
My contribution to the subject is to show that the original ideas behind ST are, in fact, much more general than it first seemed, and an infinite family of many valued logics have Strict/Tolerant counterparts. Besides classical logic, this family includes both Kleene’s and Priest’s logics themselves, as well as first degree entailment. For instance, for both the Kleene and the Priest logic, the corresponding strict/tolerant logic is sixvalued, but with differing sets of strictly and tolerantly designated truth values. There is a reverse notion, of Tolerant/Strict logics, which exist for the same structures. And the hierarchy going through meta, metameta, \ldots consequence levels exists for the same infinite family of many valued logics. In a similar way all this work extends to modal and quantified many valued logics. In brief, we have here a very general phenomenon.
I will present a sketch of the basic generalizations, of Strict/Tolerant and Tolerant/Strict, but I will not have time to discuss the hierarchies of such logics, nor will I have time to give proofs, beyond a basic sketch of the ideas involved.