The DVMLG organizes a special colloquium for recent PhD graduates as part of the Colloquium Logicum. It honours excellent junior researchers for outstanding work in their doctoral thesis.

**Speakers**

Juan P. Aguilera (Vienna, Ghent)

Lorenzo Galeotti (Amsterdam)

Zoe McConaughey (Lille)

Lothar Sebastian Krapp (Konstanz)

**Titles and Abstracts**

**Juan P. Aguilera (Vienna, Ghent)**

Title: sigma-projective sets, cut elimination, and Woodin cardinals.

Abstract: The sigma-projective sets are the smallest sigma-algebra on the reals which contains the open sets and is closed under continuous images. We study the principle asserting that all sigma-projective sets are determined and state equivalent formulations of it in terms of games of transfinite length, a cut-elimination theorem, and the existence of an uncountable sequence of certain inner models of set theory with finitely many Woodin cardinals. This will include joint work with S. Müller and P. Schlicht.

**Lorenzo Galeotti (Amsterdam)**

Title: Computing Over Higher Order Reals

Abstract: In classical computability theory computations are thought of as finite and discrete processes carried out by idealised machines on a finite amount of data. Although these assumptions are quite natural, since the beginning of the research in this area, researchers have been developing theories in which these assumptions are weakened.

Particularly interesting are those notions of computability in which the finiteness of the

process and of the data are relaxed. Prominent in this area are models of computability over the real line. These machines do indeed work on (sometimes representations of) real numbers and are in some cases allowed to run for infinitely many steps.

An extreme weakening of the finiteness restrictions on computability led to the notion of transfinite computability. Models of transfinite computability are generalisations of classical models of computability to the transfinite. Particularly important in this area are the Infinite Time Turing Machines (ITTMs) introduced by Hamkins and Lewis, and Ordinal Turing Machines (OTMs) introduced by Koepke.

In the context of transfinite computability, it is natural to ask whether classical notions of computability over the reals can be generalised to the transfinite. In this talk we will present an overview of different approaches to solve this problem.

**Zoe McConaughey (Lille)**

Title: Reconstructing Aristotle’s syllogistic with the modern formal framework of “dialogical logic”

Abstract: My PhD dissertation “Aristotle, science and the dialectician’s activity. A dialogical approach to Aristotle’s logic” defends the thesis that Aristotle’s logic is best understood from a dialogical perspective. I will here be presenting the formal part of my dissertation in which I provide a full formal reconstruction of Aristotle’s assertoric syllogistic using the modern, formal framework of “dialogical logic” (first developed by Paul Lorenzen and Kuno Lorenz). I thus present a pragmatic, interaction-based reconstruction of all of Aristotle’s assertoric syllogistic, encompassing the three figures of the syllogism, ecthesis, direct and indirect deductions, reductions to the first figure, as well as the inconclusive moods. All of Aristotle’s syllogistic is thus reconstructed using “dialogical logic”.

The dialogical framework explicits the dialogical context of the practice of dialectical debates which underlies Aristotle’s logic as a whole and which has been stressed by modern scholars such as Michel Crubellier. What is more, in order to achieve the formal reconstruction, I use the variant of the dialogical framework called “Immanent Reasoning” (Rahman, McConaughey, Clerbout, Klev 2018) which incorporates notation from Per Martin-Löf’s Type Theory. All of syllogistic can thus be built out of Aristotle’s meaning explanation of quantification spelled out in the “dictum de omni” (APr. I 1), a bridge between the syllogistic of the Prior Analytics and the dialectic of the Topics (Marion & Rückert 2016). Aristotelian scholarship and modern logical tools are thus combined to express all of Aristotle’s assertoric syllogistic in a dialogical framework.

References:

• Lorenzen, Paul and Kuno Lorenz, 1978, Dialogische Logik, Darmstadt: Wissenschaftliche Buchgesellschaft.

• Crubellier, Michel, 2011, “Du Sullogismos au syllogisme”, Revue philosophique de la France et de l’étranger, 136(1): 17–36

• Marion, Mathieu and Helge Rückert, 2016, “Aristotle on Universal Quantification: A Study from the Point of View of Game Semantics”, History and Philosophy of Logic, 37(3): 201–229. doi:10.1080/01445340.2015.1089043

• Martin-Löf, Per, 1984, Intuitionistic Type Theory. Notes by Giovanni Sambin of a Series of Lectures given in Padua, June 1980, Naples: Bibliopolis.

• Rahman, Shahid, Zoe McConaughey, Ansten Klev, and Nicolas Clerbout, 2018, Immanent Reasoning or Equality in Action: A Plaidoyer for the Play Level, (Logic, Argumentation & Reasoning 18), Cham: Springer International Publishing. doi:10.1007/978-3-319-91149-6

**Lothar Sebastian Krapp (Konstanz)**

Title: O-minimal Exponential Fields and Peano Arithmetic

Abstract:

In his much celebrated work “A decision method for elementary algebra and geometry” (1951), Tarski presented a procedure determining whether a first-order sentence in the language {+,⋅,0,1,<} is true or false over the field of real numbers. His question on the decidability of the first-order theory T_exp of the field of real numbers in the language that additionally contains a unary function symbol exp (interpreted as the standard exponential function e^x) remains open to the date.

A major breakthrough was made in 1996 by Macintyre and Wilkie: under the assumption of Schanuel’s Conjecture, they provided a recursive axiomatisation of T_exp, thus showing the decidability of this theory. Another candidate for a recursive axiomatisation of T_exp is the first-order theory of o-minimal exponential fields whose exponential satisfies the differential equation exp^’=exp with initial condition exp(0)=1. Studying algebraic and model theoretic properties of models of this theory was the main subject of my doctoral thesis, under the supervision of Salma Kuhlmann.

In my talk, I will firstly motivate and outline the major concepts and results of my dissertation project with particular focus on Schanuel’s Conjecture and Tarski’s decidability question. Secondly, I will report on joint work with Merlin Carl concerning models of Peano Arithmetic as integer parts of o-minimal exponential fields.