Autumn Semester HS2021

Unless stated otherwise, all talks will take place in room A97 of the Exakte Wissenschaft Building (ExWi) at the University of Bern. Please note that the university requires all seminar attendees have a Covid certificate and to wear a mask.

Schedule

Abstracts

Faroldi argues that deontic modals are hyperintensional and thus traditional modal logic cannot provide an appropriate formalization of deontic situations. To overcome this issue, we introduce novel justification logics as hyperintensional analogues to non-normal modal logics. We establish soundness and completness with respect to various models and we study the problem of realization.

The story of (orthomodular) quantum logic, as motivated by the work of Birkoff and von Neumann, has been a story of varied success. It is well known that the observables of a quantum system can be modeled by complex separable Hilbert space, where the lattice of projection operators upon it form the canonical example of what is known as an orthomodular lattice. Orthomodular lattices form a variety denoted OML.

While the 1-assertional logic of OML is regularly algebraizable with the variety of OML's, many questions have remained famously open; such as decidability of the provability (or even the deducibility) relation. Some sequent calculi have been proposed in the literature (cf. Nishimura 1980, Cutland & Gibbons 1982), however such calculi have not been shown to be (Gentzen-)algebraizable with OML, and have not proved fruitful in answering these questions. Likewise, these issues are mirrored algebraically in OML; it remains unknown whether the equational theory of OML is decidable (or even the word problem), or whether OML admits any form of completions.

Recent attempts have been made to approach these problems under the umbrella of residuated structures, structures which have had great success in answering classical problems in many nonclassical logics. While no (term-definable) operation in OML satisfies the (two-sided) law of residuation, it has been folklore that the Sasaki operations form a one-sided residuated pair; moreover, OML is term definable with the variety OG of orthomodular groupoids (Chajda & Länger 2017). Such structures form a subvariety in the variety PLRG of pointed left-residuated lattice order groupoids (introduced in Fazio, Ledda, & Paoli 2021), perhaps the first "algebraic-umbrella" containing both quantum structures and residuated lattices.

In this talk we will present and motivate an algebraizable sequent calculus whose equivalent algebraic semantics is PLRG, following the traditional "substructural" roadmap (e.g., as in Galatos & Ono 2010). Within this framework, we also present an algebraizable calculus whose equivalent algebraic semantics is OG; realizing orthomodular quantum logic via structural rules. We also discuss the difficulties that naturally arise, and perhaps, are inherent-to such an approach. While we cannot provide any definitive answers to the classical problems, such a marriage of substructural and quantum logics seems fruitful. This is ongoing work in collaboration with Davide Fazio, Antonio Ledda, and Francesco Paoli at the University of Cagliari.

Agents may use their own, distinct vocabularies to reason and talk about the world, structured into knowledge representations, also called ontologies. In order to communicate, they use alignments: translations between terms of their ontologies. However, ontologies may be subjected to change, when agents learn new terms from their environment or from their peers, requiring their alignments to evolve accordingly. Experimental cultural knowledge evolution aims at studying the mechanisms that agents use to evolve their knowledge and has been applied to the evolution of alignments in the Alignment Repair Game (ARG). Experiments have shown that, through ARG, agents improve their alignments and reach successful communication. Yet, they are not sufficient to understand the formal properties of cultural knowledge evolution.

In this talk, I will present a modeling of ARG in Dynamic Epistemic Logic to assess its formal properties. I will show that all but one adaptation operator are correct, they are incomplete and partially redundant. These properties are, of course, of the game and of the modeling. I will discuss the differences between the two, which inspires to introduce an independent model of awareness based on partial valuations and weakly reflexive relations. This is used to define an alternative modeling of ARG under which the formal properties are re-examined, therefore showing that DEL is insufficient to model cultural knowledge evolution.

Time warps are join-preserving endofunctions on the ordinal ω + 1. The set of time warps gives rise to a residuated lattice, the time warp algebra, where multiplication is composition and join and meet are defined point-wise. Time warps were first studied as graded modalities for a type system in (Guatto 2018) to quantify the growth of information in the course of program execution. For potential real-world applications of the time warp algebra it is important to have a decidable equational theory. In this talk I will present an overview of a proof for the decidability of the equational theory of the time warp algebra.

This is joint work with Sam van Gool, Adrien Guatto, and George Metcalfe.

A powerful theory of uncertainty is that of coherent sets of desirable gambles (or desirability). It encompasses, in a uniform way, the Bayesian theory of probability as well as Bayesian robustness and many other theories of uncertainty. Several attempts have been carried out to explicitly formulate desirability as a logical system; for our purposes, we follow the approach by Gillett, Scherl and Shafer (for another perspective: recently Kohlas, Casanova and Zaffalon have shown that coherent sets of gambles can be embedded into the algebraic structure of information algebras). After briefly introducing the deductive structure of the theory, and its link with (lexicographic) probabilities, we discuss some intriguing issues related to two possible modifications of the theory: (1) the case when one allows the operation of (explicitly) rejecting gambles, and (2) the case in which we put constraints on axioms so to capture some form of “bounded rationality”. In particular, we will show that, in the latter case, connections with quantum theory can be established.

In search of simple and convincing first principles on which quantum mechanics is based, numerous types of algebraic, relational, or categorical structures have been proposed and studied. The so-called quantum structures have one thing in common: they are all equipped, in one form or the other, with an orthogonality relation. Orthogonality expresses in the physical model the mutual exclusiveness of measurement results. It was the idea of David Foulis to investigate how far we might get in a framework that is based on this single aspect.

An orthoset (or orthogonality space) is a set equipped with a symmetric, irreflexive binary relation. The prototypical example is the (projective) Hilbert space together with the usual orthogonality relation. In this talk we will review to which extent orthosets characterise inner-product spaces. We will see that a quite simple combinatorial property characterises orthosets of finite rank that arise from orthomodular spaces (generalised Hilbert spaces). Moreover, group theory plays in physics a predominant role. We shall argue that the most convincing way of constructing the standard model of physics from orthosets is based on hypotheses regarding symmetries.

The reduction of personal power consumption can play an important role to overcome the future energy challenges of our society. With the rise of smart meters there are more sophisticated methods of interacting with power consumption than just a regular energy bill. This reduction can be achieved by confronting the end-user with his own power consuming behaviour. It is essential, that feedback on this behaviour is received in real-time and that the user can link his behaviour to activities in the household. In this talk I will present the framework to achieve the aforementioned points and present the hypothesis that are examined.

In philosophical logic, a lot of work has gone into dealing with paradoxes. "Classical solutions" usually constrain language in some way. Tarski is a famous proponent of such a classical approach: By introducing the distinction between object- and metalanguage he ruled out paradoxes like the liar paradox which rely on simple diagonalization. However, why do we want to rule out paradoxes in the first place? The classical answer is obvious: paradoxes yield contradictions and contradictions are false. Even worse, if a theory is strong enough to derive paradoxical sentences, this theory is trivial due to the principle of explosion. In my talk I will argue that there is another interesting way to deal with paradoxes. Instead of weakening our formal language until it cannot express paradoxical statements anymore, we could try to develop a logical system which can appropriately model paradoxes and help us reason about them. I will talk about the requirements such a system must meet and present a concrete proposal, the logic LP developed by Priest.