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# The Logic Seminar 2019

## Spring 2019 Logic Seminar Schedule

## Mailing list

The logic seminar takes place on alternate Thursdays between 13:15 and 15:00 at Olof Wijksgatan 6 in room T340 (unless otherwise stated).

The archive of older seminars (starting 2005) is divided into two web pages: 2005–2013 Logic Seminars and 2014–2018.

24 January (Room 112,** Dicksonsgatan 4**)

Speaker: **Graham E. Leigh**Title:

7 February (Room T340): *Informal meeting*

21 February: (Room 112, **Dicksonsgatan 4**)

7 March (**T219**)

Speaker: **Peter Pagin** (Stockholm)

Title: **Indexicality and time**

Abstarct: In this talk, I shall briefly present the *Switcher Semantics *framework and the related idea of *general compositionality*. I shall then move on to David Kaplan's arguments that we need to postulate *temporal propositions*, true/false at world-time pairs, as the meanings-in-context of some natural language sentences, more precisely sentences with temporal context dependence. Kaplan's arguments concern the rejection of vacuous temporal operations and the compositionality of content, that is, meaning-in-context.

I shall present the arguments, some of the reactions to them in the literature (if there is time), and the Switcher Semantics solution. I will finally consider the question whether validity in Kaplan's Logic of Demonstratives is preserved in the switcher semantics for these sentences.

21 March (T340)

Speaker: **Devdatt Dubhashi** (Chalmers)

Title: **Independence Results in Machine Learning**

Abstract: The celebrated results in the foundations of Mathematics by Goedel and Cohen showed that statements such as the Axiom of Choice (AC) and the Continuum Hypothesis (CH) were independent of standard (ZF) Set Theory. Recent results have shown that surprisingly such results also bedevil machine learning. Although machine learning is identified in the popular imagination as a practical engineering field, there is in fact a well developed underlying statistical theory which also guides practical applications. Some basic statements on learnability in this framework turn out to be independent of set theory just as AC and CH are. We will give an introductory survey of independence results in general and on these recent results in machine learning in particular.

4 April (T340)

Speaker: **Graham E Leigh**

Title: **When proofs become grammars**

Abstract: I will present some recent results from structural proof theory linking automated theorem proving to formal language theory. Using this idea, the existence of (simple) proofs by induction is equivalent to the existence of tree grammars covering particular term languages. This seminar is intended as an informal follow-up on some applications of learning theory brought up in Devdatt Dubhashi's talk two weeks ago. The results I will be presenting are due to the research group in mathematics at TU Wien, specifically recent work of Stefan Hetzl and Sebastian Eberhard.

~~18 April (T340)~~ postponed to 2nd May

2 May (**T346**)

Speaker: **Paul Gorbow** (FloV)

Title: **The reflective multiverse of set theory**

Abstract: I present a construction of a model of an expansion of set theory. It embodies a conception of a multiverse of universes of set theory, with an untyped notion of truth-relative-to-a-universe. The construction is partly motivated by the philosophical concern of the significance of having proofs from the axioms of a set theory such as ZF. Here are three potential answers to this question:

1. (Universe) The conclusions of proofs from ZF are true about sets.

2. (Multiverse) The conclusions of proofs from ZF are true in every structure/universe satisfying ZF.

3. (Formal foundation) For a proof p from ZF to gain significance it needs to be interpreted in a particular mathematical theory or structure founded in ZF, e.g. as establishing a truth about the real numbers.

I will argue that the these three viewpoints can coexist and, together, motivate a construction I call the Reflective Multiverse, an expansion of ZF by a primitive notion of truth-relative-to-a-universe. In this talk I will explain the construction and detail some of the philosophical story behind it. As this is work in progress, I'm keen to receive feedback.

16 May: **no seminar** (departmental conference).

23 May (Room 112, **Dicksonsgatan 4**)

Speaker: **Anders Lundstedt** (Stockholm)

Title: **Necessarily non-analytic induction proofs**

Abstract: Sometimes when trying to prove a fact F by induction one gets "stuck" when trying to prove the induction step. The solution is sometimes to instead prove a "stronger" fact S by induction. This proof method is usually called something like "strengthening of the induction hypothesis". However, there need not always be a precise sense in which the fact S is "stronger". Thus, following Hetzl and Wong (2018), we use the more general terminology "non-analytic induction proofs" for such proofs. A natural question for such proofs is whether the non-analyticity is necessary—that is, whether one could prove F without the "detour" via proving S. Hetzl and Wong have made precise sense of this question for first-order theories and sentences of arithmetic. Based on this, we investigate whether some particular induction proofs are necessarily non-analytic. This is joint work with Eric Johannesson.

Reference: Stefan Hetzl and Tin Lok Wong (2018). "Some observations on the logical foundations of inductive theorem proving". Logical Methods in Computer Science 13.4, pp. 1–26.

30 May: **no seminar** (public holiday).

13 June (T340)

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