See below for a list of all previous and upcoming seminars in the *Gothenburg Research Seminar in Logic* and the *Nordic Online Logic Seminar*. We also run a bi-weekly *Colloquium in Logic* focused in the Masters in Logic; details and schedules are listed on the course page of the Masters Programme.

The research seminar meets on alternate Fridays at 10.15. Talk locations (i.e., Zoom link) are distributed in the GU Logic mailing list. Alternatively, contact Graham Leigh directly.

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.

This talk is part of the Nordic Online Logic Seminar.

Victor Lisinki
(Corpus Christi, Oxford) – Decidability problems in number theory

In its modern formulation, Hilbert’s tenth problem asks to find a general algorithm which decides the solvability of Diophantine equations. While this problem was shown to be unsolvable due to the combined work of Davis, Putnam, Robinson and Matiyasevich, similar question can be posed over domains other than the integers.
Among the most important open questions in this area of research is if a version of Hilbert’s tenth problem for F_{p}((t)), the field of formal Laurent series over the finite field F_{p}, is solvable or not. The fact that this remains open stands in stark contrast to the fact that the first order theory of the much similar object Q_{p}, the field of p-adic numbers, is completely understood thanks to the work by Ax, Kochen and, independently, Ershov. In light of this dichotomy, I will present new decidability results obtained during my doctoral research on extensions of F_{p}((t)). This work is motivated by recent progress on Hilbert’s tenth problem for F_{p}((t)) by Anscombe and Fehm and builds on previous decidability results by Kuhlman.

Juliette Kennedy
(Helsinki) – Logicality and Model Classes

When is a property of a model a logical property?
According to the so-called Tarski-Sher criterion this is the case when the property is preserved by isomorphisms.
We relate this to the model-theoretic characteristics of abstract logics in which the model class is definable, resulting in a *graded* concept of logicality (in the terminology of Sagi’s paper “Logicality and meaning”).
We consider which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi.
We suggest that a logic is the more logical the closer it is to first order logic, and offer a refinement of the result of McGee that logical properties of models can be expressed in *L*_{∞∞} if the expression is allowed to depend on the cardinality of the model, based on replacing *L*_{∞∞} by a “tamer” logic.
This is joint work with Jouko Väänänen.

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 century-old 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.

This talk is part of the Nordic Online Logic Seminar.

Mattias Granberg Olsson
(FLoV) – A proof of conservativity of fixed points over Heyting arithmetic via truth

I will present work in progress (together with Graham Leigh) on a novel proof of the conservativity of the intuitionistic fix-point theory over Heyting arithmetic (HA), originally proved in full generality by Arai [1].
We make use of the work of van den Berg and van Slooten [2] on realizability in Heyting arithmetic over Beeson’s logic of partial terms (HAP).
Let IF be the extension of Heyting arithmetic by fix-points, denoted \hat{ID}^{i}_{1} in the literature.
The proof is divided into four parts:
First we extend the inclusion of HA into HAP to IF into a similar theory IFP in the logic of partial terms.
We then show that every theorem of this theory provably has a realizer in the theory IFP(Λ) of fix-points for almost negative operator forms only.
Constructing a hierarchy stratifying the class of almost negative formulae and partial truth predicates for this hierarchy, we use Gödel’s diagonal lemma to show IFP(Λ) is interpretable in HAP.
Finally we use use the result of [2] that adding the schema of “self-realizability” for arithmetic formulae to HAP is conservative over HA.
The result generalises the work presented at my half-time seminar 2020-08-28.

[1] Toshiyasu Arai. Quick cut-elimination for strictly positive cuts. Annals of Pure and Applied Logic, 162(10):807–815, 2011.

[2] Benno van den Berg and Lotte van Slooten. Arithmetical conservation results. Ind- agationes Mathematicae, 29:260–275, 2018.

Sonia Marin
(UCL) – Focused nested calculi for modal and substructural logics

Focusing is a general technique for syntactically compartmentalising the non-deterministic choices in a proof system, which not only improves proof search but also has the representational benefit of distilling sequent proofs into synthetic normal forms. However, since focusing was traditionally specified as a restriction of the sequent calculus, the technique had not been transferred to logics that lack a (shallow) sequent presentation, as is the case for some modal or substructural logics.

With K. Chaudhuri and L. Straßburger, we extended the focusing technique to nested sequents, a generalisation of ordinary sequents, which allows us to capture all the logics of the classical and intuitionistic S5 cube in a modular fashion. This relied on an adequate polarisation of the syntax and an internal cut-elimination procedure for the focused system which in turn is used to show its completeness.

Recently, with A. Gheorghiu, we applied a similar method to the logic of Bunched Implications (BI), a substructural logic that freely combines intuitionistic logic and multiplicative linear logic. For this we had first to reformulate the traditional bunched calculus for BI using nested sequents, followed again by a polarised and focused variant that we show is sound and complete via a cut-elimination argument.

Jouko Väänänen
(Helsinki) – Dependence logic: Some recent developments

In the traditional so-called 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 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.

This talk is part of the Nordic Online Logic Seminar.

Carlo Nicolai
(King's College) – A New Look at Cut Elimination for Compositional Truth

In the field of axiomatic theories of truth, conservativity properties of theories are much investigated. Conservativity can be used to argue that, despite the well-known undefinability of truth, there is a sense in which a primitive truth predicate can be reduced to the resources of an underlying mathematical theory that provides basic syntactic structure to truth ascriptions. Conservativity is typically proved model-theoretically, or proof-theoretically via the elimination of cuts on formulae containing truth (Tr-cuts). The original Tr-cut-elimination argument for the theory of Tarskian, compositional truth CT[B] by Halbach is not conclusive. This strategy has been corrected by Graham Leigh: Leigh supplemented Halbach’s strategy with the machinery of approximations (introduced by Kotlarski, Krajewski and Lachlan in the context of their M-Logic). In the talk we investigate a different, and arguably simpler way of supplementing Halbach’s original strategy. It is based on an adaptation of the Takeuti/Buss free-cut elimination strategy for first-order logic to the richer truth-theoretic context. If successful, the strategy promises to generalize to the type-free setting in a straightforward way. This is joint work with Luca Castaldo.

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.

This talk is part of the Nordic Online Logic Seminar.

Lance Rips
(Northwestern University) – Experimenting with (Conditional) Perfection

Conditional perfection is the phenomenon in which conditionals are strengthened to biconditionals. In some contexts, “If A, B” is understood as if it meant “A if and only if B.” I’ll present and discuss a series of experiments designed to test one of the most promising pragmatic accounts of conditional perfection. This is the idea that conditional perfection is a form of exhaustification, triggered by a question that the conditional answers. If a speaker is asked how B comes about, then the answer “If A, B” is interpreted exhaustively to meaning that A is the only way to bring about B. Hence, “A if and only if B.” The evidence suggests that conditional perfection is a form of exhaustification, but not that it is triggered by a relationship to a salient question. (This is joint work with Fabrizio Cariani.)

Giacomo Barlucchi and Tjeerd Fokkens
(FLoV) – PhD Project Presentations

Project presentations

Bahareh Afshari
(FLoV) – Cyclic Proof Systems for Modal Logics

A cyclic proof is a, possibly infinite but, regular derivation tree in which every infinite path satisfies a certain soundness criterion, the form of which depends on the logic under study. Circular and, more generally, non-wellfounded derivations are not traditionally regarded as formal proofs but merely as an intermediate machinery in proof-theoretic investigations. They are, however, an important alternative to finitary proofs and in the last decade have helped break some important barriers in the proof theory of logics formalising inductive and co-inductive concepts. In this talk we focus on cyclic proofs for modal logics, ranging from Gödel-Löb logic to more expressive languages such as the modal mu-calculus, and reflect on how they can contribute to the development of the theory of fixed point modal logic.

Rasmus Blanck
(FLoV) – Interpretability and flexible formulas

This talk is about two different generalisations of Gödel’s incompleteness theorems and the (natural?) idea of trying to merge the two into a stronger result. The first generalisation I have in mind is Feferman’s theorem on the “interpretability of inconsistency”. This result strengthens the second incompleteness theorem, by showing that the sentence expressing that PA is inconsistent is not only consistent with PA, but even that PA + “PA is inconsistent” is interpretable in PA. The second generalisation is due to Kripke, who showed that there is a flexible formula: a formula “whose extension as a set is left undetermined” by PA. In particular, he constructed a Σ-1 formula γ(x), such that for every Σ-1 formula σ(x), the theory PA + “γ = σ” is consistent. This result generalises the first incompleteness theorem, since any instance of such a γ must be independent of PA. Unfortunately, the straightforward combination of Feferman’s and Kripke’s theorems — the “interpretability of flexibility” — is a false claim. Indeed, it is easy to show that there can be no Σ-1 formula γ such that for all Σ-1 formulae σ, the theory PA + “γ = σ” is interpretable in PA.

There are, however, several weaker results available, due to Enayat, Hamkins, Shavrukov, Woodin, and me. These results either (1) relax the restriction on γ, (2) further restrict the allowed choices of σ’s, (3) strengthen the interpreting theory beyond PA, or (4) use equality modulo finite differences. I will present these four intermediate results, along with the remaining question that they all fail to answer. Finally, I will discuss why the known proof method does not seem to lend itself to giving a positive answer to this remaining question, as well as explain how any counterexample would have to look.

Rajeev Goré
(Australian National University) – Bi-Intuitionistic Logics: a New Instance of an Old Problem

As anyone who reads the literature on bi-intuitionistic logic will know, the numerous papers by Cecylia Rauszer are foundational but confusing. For example: these papers claim and retract various versions of the deduction theorem for bi-intuitionistic logic; they erroneously claim that the calculus is complete with respect to rooted canonical models; and they erroneously claim the admissibility of cut in her sequent calculus for this logic. Worse, authors such as Crolard, have based some of their own foundational work on these confused and confusing results and proofs. We trace this confusion to the axiomatic formalism of RBiInt in which Rauszer first characterized bi-intuitionistic logic and show that, as in modal logic, RBiInt can be interpreted as two different consequence relations. We remove this ambiguity by using generalized Hilbert calculi, which are tailored to capture consequence relations. We show that RBiInt leads to two logics, wBIL and sBIL, with different extensional and meta-level properties, and that they are, respectively, sound and strongly complete with respect to the Kripkean local and global semantic consequence relations of bi-intuitionistic logic. Finally, we explain where they were conflated by Rauszer.

Stefan Hetzl
(TU Wien) – Formula equations

A formula equation is a formula in first-order logic with unknowns representing predicates. Solving a formula equation consists of finding first-order instances of these unknowns that result in a valid formula. In this talk I will first give an overview of this topic and, in particular, describe its connections to well-known problems in logic and its applications in computer science. I will then concentrate on specific classes of formula equations and present

- joint work with J. Kloibhofer on solutions of Horn formula equations in first-order logic with least fixed points and
- joint work with S. Zivota showing that solvability of affine formula equations is decidable, thus generalising previous results about loop invariant generation.

Benno van den Berg
(ILLC, Amsterdam) – Quadratic type checking for objective type theory

An important fact about type theory, and something which is also heavily exploited in implementations, is that type checking is decidable However, from a complexity-theoretic point of view the worst case upper bounds are quite serious: according to a result by Statman from 1979, the problem is not elementary recursive. Recently, in the context of homotopy type theory people have started to consider a version of type theory in which all computation rules are replaced by propositional equalities. In this talk, I will discuss this version of type theory, which one could call “objective type theory”. In particular, I will show that if one formulates this objective type theory with sufficient care, type checking can be done in quadratic time. (This is joint work with my PhD student Martijn den Besten.)

Mateusz Łełyk
(Warsaw) – Between Disquotation and Compositionality

The talk will be devoted to the formal connections between two canonical types of axiomatizations for the truth predicate: the disquotational scheme and the compositional clauses. We will be interested in measuring the “logical distance” between the two canonical sets of principles and determining whether each of them “determine” the other. Working in the framework of axiomatic truth theories we shall consider various ways of formalizing the above two questions and present the current state of art. The talk initializes the project (under the same name) which started in Warsaw at the beginning of September. At the current time there are still much more questions than answers, but we believe that it is worth uncovering this research perspective. Some questions will be investigated jointly with the Gothenburg logic group.

Paul Kindvall Gorbow
(FLoV) – The Copernican Multiverse of Sets

We develop an untyped semantic framework for the multiverse of set theory and show that its proof-theoretic commitments are mild. ZF is extended with semantical axioms utilizing the new symbols M(U) and M(U,σ), expressing that U is a universe and that σ is true in the universe U, respectively. Here σ ranges over the augmented language, leading to liar-style phenomena that are analysed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles.

Jean-Philippe Bernardy
(FLoV) – A Unified View of Modalities in Type Systems

We propose to unify the treatment of a broad range of modalities in typed lambda calculi. We do so by defining a generic structure of modalities, and show that this structure arises naturally from the structure of intuitionistic logic, and as such finds instances in a wide range of type systems previously described in literature. Despite this generality, this structure has a rich metatheory, which we expose.

Joint work with Andreas Abel.

Fredrik Engström
(FLoV) – Dependence logic and generalized quantifiers

Dependence logic, proposed by Väänänen, is an elegant way of introducing dependencies between variables into the object language. The framework of Dependence logic, so-called team semantics, has turned out to be very flexible and allows for interesting generalizations. Instead of considering satisfaction with respect to a single assignment s, team semantics considers sets of assignments X, called teams. While the semantics of Dependence logic is based on the principle that a formula φ is satisfied by a team X if every assignment s ∈ X satisfies φ, we will replace this principle by the following: a formula φ is satisfied by a team X if for every assignment s: s ∈ X iff s satisfies φ, replacing an implication by an equivalence. When only first-order logic is considered nothing exciting happens, it is only when we introduce new logical operations that things start to get more exciting.

Research seminars prior to 2020 are in a separate archive.