Due to the Coronavirus restrictions in place in Switzerland, all the talks this semester will take place online via Zoom and the access details will be shared by e-mail. If you would like to be added to the mailing list, please contact Thomas Studer or George Metcalfe (contact details can be found in the sidebar).

## Schedule

Date | Time | Speaker | Title |
---|---|---|---|

17.02.2021 | 16.00 - 17.00 | Igor Sedlár - Czech Academy of Sciences | Changing the World, Constructively |

24.02.2021 | 16.00 - 17.00 | Luca Reggio - University of Oxford | Counting homomorphisms between finite structures |

10.03.2021 | 16.00 - 17.00 | Brett McLean - University of Nice Sophia Antipolis | Temporal logic of Minkowski spacetime |

14.04.2021 | 16:00 - 17:00 | Hitoshi Omori - Ruhr University Bochum | Two applications of Herzberger’s semantics |

21.04.2021 | 16.00 - 17.00 | Federico Faroldi - University of Ghent | The Structure of Reasons: Subtraction and Partiality |

28.04.2021 | 16.00 - 17.00 | Iris van der Giessen - Utrecht University | The admissible rules of Lax Logic |

05.05.2021 | 16.00 - 17.00 | Shawn Standefer - Slovak Academy of Sciences | Varieties of necessity in a non-classical setting |

19.05.2021 | 16.00 - 17.00 | Wesley Fussner - University of Nice Sophia Antipolis | Generalized basic logic from a modal point of view |

26.05.2021 | 16.00 - 17.00 | Grigory Olkhovikov - Ruhr University Bochum | A maximality result for bi-intuitionistic propositional logic |

## Abstracts

#### Changing the World, Constructively

Igor Sedlár

The finite tree property of intuitionistic logic entails completeness with respect to posets where each element, seen as a possibly partial situation, is under a maximal element, seen as a possible world containing the situation. This suggests a natural semantics for intuitionistic modal logic based on posets with a binary relation on the set of maximal elements. In this semantics, truth of modal formulas in a situation is determined by looking at worlds containing the situation and worlds accessible from them. In this paper we study modal logics arising from such a semantics. A general completeness-via-canonicty result is provided and various operations on such posets including filtrations are studied. Differences with respect to intuitionistic modal logics known from the literature are discussed. In the final part a completeness result for a version of intuitionistic propositional dynamic logic based on the framework is obtained and the logic is shown to be decidable.

#### Counting homomorphisms between finite structures

Luca Reggio

Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite relational structure C. Categorical generalisations of this result were proved independently in the early 1970s by Lovász and Pultr. I will present another categorical variant of Lovász' theorem and explain how it can be used, in combination with the game comonads recently introduced by Abramsky et al., to obtain homomorphism counting results in finite model theory.

This is joint work with Anuj Dawar and Tomáš Jakl.

#### Temporal logic of Minkowski spacetime

Brett McLean

If we wish to do temporal logic on (flat) spacetime, special relativity suggests we should use an accessibility relation that is independent of the choice of inertial frame, and that there are a limited number of ways to do this. Two possible accessibility relations are 'can reach with a lightspeed-or-slower signal' and 'can reach with a slower-than-lightspeed signal'. We focus on the resulting frames in 2D spacetime (1 space + 1 time dimension). For both frames, validity of formulas in the basic temporal language is a PSPACE-complete problem. I will describe the proofs of these results and also how those proofs can be extended to obtain results on interval logics.

This is joint work with Robin Hirsch. The lightspeed-or-slower case is due to Hirsch and Reynolds.

#### Two applications of Herzberger’s semantics

Hitoshi Omori

In his paper "Dimensions of truth", Hans Herzberger develops a semantic framework that captures both classical logic and weak Kleene logic through one and the same interpretation. The aim of this talk is to apply the simple idea of Herzberger to two kinds of many-valued semantics. This application will be led by the following two questions: (i) Is de Finetti conditional a conditional? (ii) What do CL, K3 and LP disagree about? Note: This is a joint work with Jonas R. B. Arenhart.

#### The Structure of Reasons: Subtraction and Partiality

Federico Faroldi

Practical reasons are central both in everyday normative reasoning and in normative theorizing, but most accounts treat them as atomic and flat. In this talk, I investigate the structure of practical reasons in order to deal with aggregation, double counting, subtraction, and partiality. The ideal aim is to give a unified formal account that is able to serve as a semantic backdrop to construct natural logical systems to reason with reasons, based on a hyperintensional justification logic with a truthmaker semantics.

#### The admissible rules of Lax Logic

Iris van der Giessen

Propositional Lax Logic is a fascinating intuitionistic modal logic with a non-standard modality that combines some properties of a ‘Box’ and some properties of a ‘Diamond’. In this talk I will present recent results about its admissible rules. The admissible rules are those rules under which the set of theorems of a logic is closed. Thereby they give insight in the structure of all possible inferences in a logic. Iemhoff (2001) showed that the so-called Visser rules form a basis for the admissible rules of IPC. Similarly, modal Visser rules are formulated for modal logics such as K4, S4 and GL (Jeřábek 2005). I will characterize a sequent-based proof system for the admissible rules of propositional Lax Logic, containing Visser-like rules. In this work we will see that the structure of the relational semantics will be of great importance.

#### Varieties of necessity in a non-classical setting

Shawn Standefer

In standard modal logics, there are three common conceptions of necessity:, the universal conception, the equivalence relation conception, and the axiomatic conception. theses provide distinct presentations of the modal logic S5, commonly used in metaphysics and epistemology. In standard settings, these presentations coincide, giving three views of a single, unified logic. I will explore these different conceptions in the context of the relevant logic R, explaining when they come apart and why that matters. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be truth in all possible worlds.

#### Generalized basic logic from a modal point of view

Wesley Fussner

Generalized basic logic was introduced through its algebraic semantics (GBL-algebras) in order to provide a natural common generalization of lattice-ordered groups, Heyting algebras, and BL-algebras. When formulated with exchange, weakening, and falsum, generalized basic logic is a fragment of both Hájek's basic logic and propositional intuitionistic logic. In this formulation, the relationship between generalized basic logic and Łukasiewicz logic parallels the thoroughly-studied relationship between intuitionistic logic and classical logic. This talk explores several ways that the latter parallel manifests. First, we illustrate a relational semantics for generalized basic logic where worlds are valued in MV-algebras (analogous to the usual, Boolean-valued Kripke semantics for intuitionistic logic). Second, we present a translation of generalized basic logic into a modal Łukasiewicz logic that is analogous to the Gödel-McKinsey-Tarski translation of intuitionistic logic into the classical modal logic S4. All of these results are obtained with the help of the algebraic theory of GBL-algebras, and we also provide a brief survey of the latter.

#### A maximality result for bi-intuitionistic propositional logic

Grigory Olkhovikov

I will report on a recent Lindström theorem for bi-intuitionistic propositional logic (joint work with Guillermo Badia) showing that this logic is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under bi-asimulations.

The result constitutes an extension of previous work done for the propositional intuitionistic logic in: G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020).

However, due to the presence of a backwards-looking connective in bi-intuitionistic logic, the current result features a number of non-trivial modifications of the techniques and ideas employed in the previous work.