Jouko Väänänen

We bring an abstract model theory perspective to interpolation. We ask, what is the role of interpolation in the study of extensions of first order logic, such as infinitary logics, generalized quantifiers and higher order logics? The abstract model theory approach reveals the basic connections between various interpolation properties in isolation, on their own, as well as with respect to other model theoretic properties, such as compactness.

We study dependence and independence concepts found in quantum physics, especially those related to hidden variables and non-locality, through the lens of team semantics and probabilistic team semantics, adapting a relational framework introduced in [1]. This also leads to new developments in independence logic and probabilistic independence logic.

We introduce a new inner model [ ] arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively PFA, the regular uncountable cardinals of [ ] are measurable in the inner model [ ] and [ ] satisfies CH. Moreover, assuming a proper class of Woodin cardinals, the theory of [ ] is (set) forcing absolute. We introduce an auxiliary concept that we call Club Determinacy, which simplifies the construction of [ ] greatly but may have also independent interest. Based on Club Determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model [ ].

We propose a systematic study of the so-called flattening operator in team semantics. This operator was first introduced by Hodges in 1997, and has not been studied in more detail since. We begin a systematic study of the expressive power this operator adds to the most well-known team-based logics, such as dependence logic, anonymity logic, inclusion logic and exclusion logic.

Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this does not exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with two leading contemporary writers, Charles Parsons and the co-authors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case we ask: what does the author set out to accomplish, philosophically?; what do they actually do (or what can be done), mathematically?; and does what is done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.

I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.