Seminars

Disquotational truth can play a justificatory role for new mathematical knowledge. In particular, we focus on the known case of consistently extending a theory of first-order arithmetic (PA) with a truth predicate governed by disquotational principles for positive sentences. One philosophically interesting feature is that the addition of this truth predicate is conservative over the arithmetical theory, giving disquotational truth the same epistemic status of a mathematical definition. Following Feferman and McGee, we argue in favour of the open-ended nature of the induction schema: once a new mathematical concept is legitimately defined the schema should hold for the extended language. For truth, this results in more justificatory power as one can prove the consistency statement of the arithmetical theory. Moreover, the exact nature of the justificatory power of positive disquotational truth can be pinned down via an ordinal analysis of the theory, as done by Halbach.

We explore the possibility of extending this disquotational definition of truth via a principle of Generalised Induction introduced by Cantini. The principle is conservative over PA, meaning it preserves the epistemic status of a definition. Furthermore, we argue that the adoption of this principle is particularly motivated in the case of positive disquotational truth. Once this new truth predicate is allowed in the induction schema, we obtain significantly more justificatory power than the original disquotational theory. I conclude by sketching a possible strategy via a collapsing procedure to show that this result is sharp.

Non-classical logics have been proposed in a number of domains, including constructive mathematics and quantum mechanics. In this talk, I will identify a base logic beneath some of these non-classical logics that I suggest has a certain fundamental status. I will give an introduction to the proof theory and semantics of this “Fundamental Logic.”

An associated paper is available at https://arxiv.org/abs/2207.06993.

Previous works by Goré, Postniece and Tiu have provided sound and cut-free complete proof systems for modal logics extended with path axioms using the formalism of nested sequent. Our aim is to provide (i) a constructive cut-elimination procedure and (ii) alternative modular formulations for these systems. We present our methodology to achieve these two goals on a subclass of path axioms, namely quasi-transitivity axioms.

The axioms of quantum mechanics present the state of a quantum system as a unit vector in complex Hilbert space. However, when Dirac [1] presented his bra- and ket-vectors, he had a more general space in mind. Schwartz [4] later gave a rigorous account for Dirac’s “vectors” as distributions, but in elementary physics books one still encounters presentations where ket-vectors are presented just as elements of a Hilbert space, and treated with methods from finite-dimensional linear algebra.

During the last years Tapani Hyttinen and I [2,3] have been looking at various models justifying the finite dimensional approaches from such textbooks. Our approaches are based on various embeddings of a Hilbert space into metric ultraproducts of finite-dimensional Hilbert spaces. In this talk I will present the basic ideas, their benefits and limitations. Time permitting, I will also contrast our approach to Boris Zilber’s work on the same questions, that was the original inspiration for us.

1. P.A.M. Dirac. The principles of Quantum Mechanics, 3rd ed, Clarendon Press, Oxford, 1947.
2. Å. Hirvonen, T. Hyttinen, On eigenvectors, approximations and the Feynman propagator, Ann. Pure Appl. Logic 170 (2019).
3. Å. Hirvonen, T. Hyttinen, On Ultraproducts, the Spectral Theorem and Rigged Hilbert Spaces, to appear in J. Symb. Log.
4. L. Schwartz, Théory de Distributions, Hermann, Paris, 1950.

This dissertation concerns the properties of and relationship between positive truth and fixpoints over intuitionistic arithmetic in three respects: the strength of these theories relative to the arithmetic base theories, relationships between theories of strictly positive fixpoints and compositional and disquotational truth for strictly truth-positive sentences, and a comparison with the classical case.

Opponent: Gerhard Jäger, Professor Emiratus University of Bern

Natural languages vary in their quantity expressions, but the variation seems to be constrained by general properties, so-called universals. Explanations thereof have been sought among constraints of human cognition, communication, complexity, and pragmatics. In this work, we examine whether perceptual constraints and coordination dynamics contribute to the development of two universals: monotonicity and convexity. Using a state-of-the-art multi-agent language coordination model (originally applied to colour terms) we evolve communicatively usable quantity terminologies. We compare the degrees of convexity and monotonicity of languages evolving in populations of agents with and without approximate number sense (ANS). The results suggest that ANS supports the development of monotonicity and, to a lesser extent, convexity. We relate our results to some classical observations about generalised quantifiers in mathematical logic and to the research on conceptual spaces.

This is joint work with Dariusz Kalociński, Franek Rakowski and Jakub Uszyński.

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.