Meeting on May 3rd, 2019

On May 3rd, 2019, the first meeting of the Swiss Logic Gathering will take place. All necessary information can be found below.

Program

Location

The meeting will take place at

Abstracts

We will discuss a new semantics for justification logic based on subset relations, where justifications are sets of possible worlds. A whole family of logics will be presented of which each member is sound and complete w.r.t. the presented semantics. Furthermore the new semantics will be used to subsume Artemov's approach to aggregating probabilistic evidence.

In the first part of this talk I will briefly introduce permutation models. As an application we will then discuss the relation between two weak choice principles, namely

\(\operatorname{ACF}^-\) := For every infinite family \(\mathcal{F}\) of finite sets there is an infinite subfamily \(\mathcal{G}\subseteq\mathcal{F}\) with a choice function.

and

\(\operatorname{KWF}^-\) := For every infinite family \(\mathcal{F}\) of finite sets of size at least two there is an infinite subfamily \(\mathcal{G}\subseteq\mathcal{F}\) with a Kinna-Wagner selection function. I.e. there is a function \(g:\mathcal{G}\to\mathcal{P}\left(\bigcup\mathcal{G}\right)\) with \(\emptyset\not =g(G)\subsetneq G\) for all \(G\in\mathcal{G}\).

The Wadge order of a topological space is the quasi-order induced by continuous reductions on the subsets of the space. Under determinacy, this order is well understood for Polish spaces. For example, it is a well-quasi-order if and only if the space has a basis consisting of clopen subsets. Recently, de Brecht introduced the class of quasi-Polish spaces, a generalization of the Polish spaces that includes non-metrizable spaces. In this talk, we show that the Wadge order on the Scott domain - a typical quasi-Polish but non-Polish space - is not a well-quasi-order.

Two obvious properties of the Lebesgue measure on the real unit interval [0,1] are that it is full (every nonempty open set has positive measure) and nonatomic (countable subsets have measure zero). Conversely, every full nonatomic Borel probability measure on [0,1] is homeomorphic to the Lebesgue measure. This is the elementary 1-dimensional case of the fundamental Homeomorphic Measures Theorem by Oxtoby and Ulam [4]. The situation for the Cantor space is radically different. The Cantor space—unique up to homeomorphism—is a nonempty, perfect, zero-dimensional, compact metrisable space. Equivalently, it is the Stone dual of any non-trivial countable Boolean algebra without atoms. The problem of classifying full nonatomic measures on the Cantor space up to homeomorphism has interested researchers for the last 20 years, and their joint effort to solve it has resulted in several impressive papers (see, e.g., [1, 2, 3]). In this talk, we are going to use tools from duality theory to reinterpret some of the results from [1, 2], in an attempt to shed new light on the problem. In particular, we are going to classify full nonatomic measures up to order-homeomorphism (cf. [1]). Good measures are full nonatomic measures with a certain homogeneity property identified in [2]. We show that, up to order-homeomorphism, full nonatomic measures are precisely the push-forwards of good measures along continuous onto almost one-to-one maps. Further, we are going to show that there is a one-to-one correspondence between non-trivial countable Archimedean totally ordered groups with strong unit (equivalently, countably infinite simple MV-algebras), and homeomorphism classes of good measures.

[1] Akin, Ethan. Measures on Cantor space. Proceedings of the 14th Summer Conference on General Topology and its Applications (Brookville, NY, 1999). Topology Proc. 24 (1999), Summer, 1–34 (2001).

[2] Akin, Ethan. Good measures on Cantor space. Trans. Amer. Math. Soc. 357 (2005), no. 7, 2681–2722.

[3] Akin, Ethan; Dougherty, Randall; Mauldin, R. Daniel; Yingst, Andrew. Which Bernoulli measures are good measures? Colloq. Math. 110 (2008), no. 2, 243–291.

[4] Oxtoby, John; Ulam, Stanisław. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42, (1941). 874–920.