Let M be the weak set theory obtained from ZF by removing the Collection Scheme, restricting the Separation Scheme to bounded formulae, and adding an axiom asserting that every set is contained transitive set. In this talk I will consider two formulations of the set-theoretic Collection Scheme restricted formulae that are Π_n in the Levy hierarchy: Π_n-Collection and Strong Π_n-Collection. It is known that, over M, Strong Π_n-Collection is equivalent to Π_n-Collection plus Σ_n+1-Separation, and Strong Π_n-Collection proves the consistency of M + Π_n-Collection. In this talk I will show that for all n > 0, every well-founded model of M + Π_n-Collection satisfies Strong Π_n-Collection. In particular, for n > 0, M + Strong Π_n-Collection does not prove the existence of a transitive model of M + Π_n-Collection. And, for n > 0, the minimum model of M + Strong Π_n-Collection and M + Π_n-Collection coincide. If time permits, I will indicate how the assumption that that the model is well-founded can be replaced with the assumption that the model instead satisfies a fragment of Foundation. This reveals new equivalences between subsystems of ZF that include the Powerset axiom.

We explain some of the connections between two research areas: cyclic proof theory and proof assistants. This talk is based on joint work-in-progress with Lide Grotenhuis.

In cyclic proof theory, we replace induction axioms with circular reasoning. By carefully placing restrictions on cycles, we can make sure that we do not prove more than with induction. Such cyclic proofs can be interpreted as programs using the Curry-Howard correspondence: cycles correspond to recursive calls, while the restrictions make sure that the program always terminates. Using this perspective, we show how the type theory implemented by proof assistants can be seen as a cyclic proof system: one where programs/proofs are defined using (co)pattern matching. We show some of the difficulties that come up specifically in dependent type theory: why unification becomes important, and how one can translate cyclic proofs to inductive proofs while preserving computational behavior. In addition, we show how some of the techniques from cyclic proof theory might be used to extend proof assistants.

Voting on two alternatives appears unproblematic compared to voting on three (or more). When faced with only two alternatives, many arguments show that Majority Rule distinguishes itself from all other ways of making a group decision. For three or more alternatives, one faces the so-called “Paradox of Voting”: there may be elections with a majority cycle in which a majority of voters prefer A to B, a majority of voters prefer B to C, and a majority of voters prefer C to A. In this talk, I will explain a series of results that axiomatically characterize rules for resolving majority cycles in elections. These rules avoid the “Strong No Show Paradox” by responding properly to the addition of new voters to an election and mitigate spoiler effects by responding properly to the addition of new candidates to an election.

This talk is based on joint work with Wes Holliday.

The 2024 Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, will be held in Gothenburg on 24-28 June, 2024 and hosted by the Faculty of Humanities at the University of Gothenburg. The colloquium comprises 3 tutorials, 7 plenary lectures and 6 special sessions as well as contributed talks. In addition, the 2024 Gödel Lecture was delivered at the meeting.

See lc2024.se for full information.

Inquisitive modal logic is a generalization of standard modal logic where the language also contains questions, and modal operators that can be applied to them. In this talk, I will provide an introductory overview of inquisitive modal logic. I will review some motivations for the approach, present some prominent examples of inquisitive modal logics, mention some results about them, and outline directions for future work.