Archive of events from 2026

Codd’s Theorem, a fundamental result of database theory, asserts that relational algebra and relational calculus have the same expressive power on relational databases. We explore Codd’s Theorem for databases over semirings and establish two different versions of this result for such databases: the first version involves the five basic operations of relational algebra, while in the second version the division operation is added to the five basic operations of relational algebra. In both versions, the difference operation of relations is given semantics using semirings with monus, while on the side of relational calculus a limited form of negation is used. The reason for considering these two different versions of Codd’s theorem is that, unlike the case of ordinary relational databases, the division operation need not be expressible in terms of the five basic operations of relational algebra for databases over an arbitrary positive semiring; in fact, we show that this inexpressibility result holds even for bag databases.

The Logic Group at the University of Gothenburg hosts its annual World Logic Day Pub Quiz. For more information on this other World Logic Day events around the world, see http://wld.cipsh.international/wld2026.html.

Throughout proof theory, proofs are often taken to be well-founded (often finite) trees of inferences. Theories of inductive definitions, among other theories, bump up against the limitations of this perspective, and a variety of formalisms have been used to push beyond this – Girard’s beta-proofs, non-well-founded proofs, proofs-as-functions. We briefly describe some features of these approaches, the way these perspectives are essentially equivalent, and the way well-foundedness reasserts itself as a core property of proofs.

In 1981 Kaufmann proved the set-theoretic analogue of a result due to Paris and Kirby in the context of arithmetic that connects fragments of the collection scheme to the existence of proper partially elementary end extensions. In particular, he shows that a countable resolvable model of a weak set theory has a proper \(\Sigma_{n+1}\)-elementary end extension if and only if it satisfies collection for all \(\Pi_n\)-formulae. In the same paper Kaufmann asks if every \(L_\alpha\) (the \(\alpha\)-th level of Goedel’s constructible hierarchy) that has a proper \(\Sigma_2\)-elementary end extension necessarily has one that satisfies bounded collection. Clote (1983) asks a generalised version of Kaufmann’s question for models of fragments of PA: Does every countable model of \(B\Sigma_{n+2}\) have a proper \(\Sigma_{n+2}\) elementary end extension that satisfies \(B\Sigma_{n+1}\)? Sun Mengzhou has recently (2025) shown that Clote’s question has a positive answer. In this talk I will describe recent work showing that, in contrast, the set-theoretic analogue of Clote’s question (the generalisation of Kaufmann’s original question) has a negative answer

Team semantics is a semantic framework originally introduced by Hodges (1997) for the study of dependence and independence concepts, and later systematically developed by Väänänen (2007). It is also independently adopted in inquisitive logic by Ciardelli and Roelofsen (2011). In team semantics, formulas are evaluated with respect to sets of evaluation points, called teams, rather than single evaluation points as in standard semantics.

Logics based on team semantics are typically extensions of classical logic and thus inherit classical implication over classical formulas. However, the earliest versions of team-based logic, such as independence-friendly logic and dependence logic, do not include a conditional connective for arbitrary formulas. An adequate conditional, known as intuitionistic implication, was proposed for dependence logic by Abramsky and Väänänen (2009). This connective is also part of the syntax of inquisitive logic. Intuitionistic implication behaves well in these downward closed logics, in the sene that it preserves downward closure and satisfies both Modus Ponens and the Deduction Theorem.

In recent years, many variants of dependence logic with different closure properties have been introduced, including union closed and convex logics. In these settings, the intuitionistic implication no longer behaves well, as it either fails to preserve the relevant closure property or fails to satisfy the Deduction Theorem. In this talk, we show that this failure is unavoidable: these logics cannot be enriched with any conditional connective that simultaneously preserves the closure property and satisfies both Modus Ponens and the Deduction Theorem.

This is joint work with Fausto Barbero.

The claim is that intensions (or meanings) can be modeled usefully by algorithms which compute truth values, proofs (in various systems), denotations and implementations of programs (in various programming languages), etc.. I will discuss the nature of these ‘algorithms’ and present some unpublished results by me and others on this topic. Most of what I will say is in the book [1] which contains many unpublished results by Lou van den Dries, Vaughn Pratt, Anush Tserunyan and others.

1. Yiannis Moschovakis. Abstract recursion and intrinsic complexity, Cambridge University Press, Volume 48 in the Lecture Notes in Logic, Association for Symbolic Logic, 2019. (See YM’s homepage).