UHH > Fakultäten > MIN-Fakultät > Mathematik > Bereiche > Mathematische Logik > Colloquium Logicum 2016 STiNE |  KUS-Portal |

## Abstracts and Slides

### Invited Lectures and PhD Colloquium talks

• Dana Bartošová
Dual Ramsey Theorem and its Applications in Topological Dynamics

The Dual Ramsey Theorem by Graham and Rothschild is a powerful partition principle with many applications. We will present its recent applications to dynam- ics of the homeomorphism group of the Lelek fan, of the group of linear isometries of the Gurarij space and of the group of affine homeomorphisms of the Poulsen simplex.

• Elisabeth Bouscaren
Model Theory and its applications to algebraic geometry

We will try to present some of the model theoretic notions and tools which are important in the now classical applications of model theory to Diophantine geometry, as well as one of the main concerns of modern model theory, how to characterize classical mathematical structures by abstract conditions (the Zilber trichotomy priniciple, Zariski structures). We will try to keep the talk as elementary as the subject allows.

• Ekaterina Fokina
Equivalence Relations and Computable Structures

Equivalence relations formalize the idea of resemblance between math- ematical objects. In this talk we discuss several approaches to study the role of equivalence relations in computable structure theory. Within one approach, equivalence relations are used to measure the complexity of clas- sification of computable structures. Another approach is to employ degrees of atomic diagrams of structures to characterize the inherent complexity of equivalence classes of structures, up to various equivalence relations. Yet an- other direction so to use equivalence relations to compare the informational content of universes for effectively given structures. For each approach, we explain the context, give an overview of the most important results and list some open questions.

• Laura Fontanella
Reflection of stationary sets, tree property and square

One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, groups...) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. There is a tension between large cardinals axioms and the axiom of constructibility V = L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut.

• Philipp Hieronymi
Diophantine approximation, scalar multiplication and decidability

It has long been known that the first order theory of the expansion (ℝ,<,+,ℤ) of the ordered additive group of real numbers by just a predicate for the set of integers is decidable. Arguably due to Skolem, the result can be deduced easily from Büchi's theorem on the decidability of monadic second order theory of one successor, and was later rediscovered independently by Weispfenning and Miller. However, a consequence of Gödel's famous first incompleteness theorem states that when expanding this structure by a symbol for multiplication, the theory of the resulting structure (ℝ,<,+,*,ℤ) becomes undecidable. This observation gives rise to the following natural and surprisingly still open question: How many traces of multiplication can be added to (ℝ,<,+,ℤ) without making the first order theory undecidable? We will give a complete answer to this question when "traces of multiplication" is taken to mean scalar multiplication by certain irrational numbers. To make this statement precise: for b in ℝ, let fb: ℝ → ℝ be the function that takes x to bx. I will show that the theory of (ℝ,<,+,ℤ,fb) is decidable if and only if b is quadratic. The proof rests on the observation that many of the Diophantine properties (in the sense of Diophantine approximation) of b can be coded in these structures.

• Rosalie Iemhoff
Logics and Rules

As is well-known, logics can be defined in various ways: via models, via proof systems, and in more exotic ways as well. In the approach via proof systems rules of inference are the central notion. There are numerous proof systems for a given logic, natural deduction and sequent calculi are two of the most famous examples. But even if one considers one particular proof system, there are countless ways to describe a given logic. To describe these possibilities in a structured way, the admissible rules of a logic, the rules under which the logic is closed, have to be established. This talk discusses admissible rules, explains their connection with unification theory, treats examples of logics where a full description of their admissible rules is known and draws conclusions about possible proof systems for such logics.

• Dominik Klein
Social interaction -- a formal exploration

Abstract: TBA

• Daniel Körnlein
Quantitative analysis of iterative algorithms in fixed point theory and convex optimisation

"What more do we know if we have proved a theorem by restricted means than if we merely know that it is true?" In proof-mining practice, the additional information is usually in the form of convergence rates or so-called rates of metastability. Numerous results in this vein have appeared in the last 15 years, see e.g. [1,2]. The talk will focus on proof mining results obtained from proofs in fixed point theory and optimization, particulary those using weak sequential compactness.

[1] U. Kohlenbach. Effective uniform bounds from proofs in abstract functional analysis. In S. B. Cooper, B. Löwe, and A. Sorbi, editors, New computational paradigms, pages 223-258. Springer, New York, 2008.

[2] U. Kohlenbach. Recent progress in proof mining in nonlinear analysis. To appear in a forthcoming book with invited articles by recipients of a Gödel Centenary Research Prize Fellowship, 2016. Available from http://www.mathematik.tu-darmstadt.de/~kohlenbach/

• Gyesik Lee
Formal proofs, variable binding, and program extraction from proofs

We talk about some issues related to making use of formal proofs, including variable binding and program extraction from proofs. First, we give a short introduction to formal proofs and explain why variable binding matters in doing formal proofs. Then, we demonstrate how normalization by evaluation can be realized by program extraction from proofs. The example which will be used for the demonstration is related to the cut elimination process of the intuitionistic first-order predicate logic.

• Norbert Preining
Gödel Logics - a short survey

Gödel logics can be characterized in a rough-and-ready way as follows: The language is a standard (propositional, quantified-propositional, first-order) language. The logics are many-valued, and the sets of truth values considered are (closed) subsets of [0,1] which contain both 0 and 1. 1 is the designated value,' i.e., a formula is valid if it receives the value 1 in every interpretation. The truth functions of conjunction and disjunction are minimum and maximum, respectively, and in the first-order case quantifiers are defined by infimum and supremum over subsets of the set of truth values. The characteristic operator of Gödel logics, the Gödel conditional, is defined by A → B = 1 if A ≤ B, and B otherwise. Because the truth values are ordered, the semantics of Gödel logics is suitable for formalizing comparisons. It is related in this respect to a more widely known many-valued logic, Lukasiewicz (or fuzzy') logic. In contrast to Lukasiewicz logic, which might be considered a logic of "absolute" or "metric comparison", Gödel logics are logics of "relative comparison".

Approaching from a different angle, Gödel logic is one of the three basic t-norm based logics which have received increasing attention in the last 15 or so years (the others are Lukasiewicz and product logic). Yet Gödel logic is also closely related to intuitionistic logic: it is the logic of linearly-ordered Heyting algebras. In the propositional case, infinite-valued Gödel logic can be axiomatized by the intuitionistic propositional calculus extended by the axiom schema (A -> B) V (B -> A). This connection extends also to Kripke semantics for intuitionistic logic: Gödel logics can also be characterized as logics of (classes of) linearly ordered and countable intuitionistic Kripke structures with constant domains. Furthermore, the infinitely valued propositional Gödel logic can be embedded into the box fragment of LTL in the same way as intuitionistic propositional logic can be embedded into S4.

We will give a short overview of this family, presenting propositional, quantified propositional (weak second order), and first order logics, including proof-theory, relation to intuitionistic logic, and recent results on fragments.

• Szymon Toruńczyk
Computation with Atoms

The framework of Sets with Atoms has its roots in the works of Fraenkel and Mostowski, and has been recently rediscovered in the Computer Science community. I will show how this framework can be used in order to represent infinite structures in a finite way which can be processed by algorithms. Other notions from model theory, notably countable-categoricity, play a role in the complexity analysis of such algorithms. I will present some applications to verification of infinite state systems. I will also present two working tools which effectively manipulate such infinite structures.

• Charlotte Werndl
Calibration and Confirmation in Climate Science

A hotly debated issue on confirmation in climate science (as well as in philosophy) is the requirement of use-novelty (i.e. that data can only confirm models if they have not already been used before, e.g. for calibrating parameters). This paper investigates the issue of use-novelty in the context of the mathematical methods provided by model selection theory. We will first argue that model selection theory provides a good method for evaluating many climate models. Then we will show that the picture model selection theory presents us with about use-novelty is more subtle and nuanced than the commonly endorsed positions by climate scientists and philosophers. More specifically, we will argue that there are two main cases in model selection theory. On the one hand, there are the methods such as cross-validation where the data are required to be use-novel. On the other hand, there are the methods such as Bayesian confirmation or the Akaike Information Criterion (AIC) for which the data cannot be use-novel. Still, for some of these methods (like AIC) certain intuitions behind the use-novelty approach are preserved: there is a penalty term in the expression for the degree of confirmation by the data because the data have already been used for calibration. The common positions arguedience and philosophy are either that data should always be use-novel or that the use-novelty criterion is irrelevant. According to model selection theory these positions are too simple: whether or not data should be use-novel depends on the specific method used. For certain methods data should be use-novel, but for others they cannot and thus need not be use-novel.

### Contributed Talks

Truth in Fiction via Non-Standard Belief Revision

Is the sentence Sherlock Holmes lives in Baker Street 221b' true? Most people would agree, or would give the proviso 'well, he does, in the fiction'. In literary studies, many claims of this form are made: Romeo loves Juliet. They both live in Verona. And so forth. All these sentences, so some hold, are not literally true but true in the fiction. Thus, one prefixes sentences like the mentioned ones by a fiction operator In the fiction f', which we represent by writing 'In f', and then evaluates whether the resulting sentence of the form 'In f: p' is true. The question then is: what are the truth conditions for a sentence of this form?

Fiction operators such as `In the fiction f' (In f:) have seen applications particularly in philosophy of fiction, but more broadly in any ontological/metaphysical debate. For example there are fiction operator approaches towards modality, mathematics and morality. Giving a suitable analysis for when a sentence of the form 'In f: p' is true, is hence of importance. The most famous approach has been David Lewis's analysis. However, it has certain shortcomings, especially when applied to inconsistent fictions in which not everything is true.

We start by taking Lewis's Analysis 2 and give it a formal interpretation that takes into account impossible worlds and ideas from belief revision theory. Our formal framework comprises multi-agent plausibility models with a domain of possible and impossible worlds, ordered by a group plausibility ordering. This gives rise to Grove-style sphere models which are known to be models for the AGM axioms. We extend these models to an impossible world setting.

Then, a sentence of the form 'In f: p' is true under our interpretation of Analysis 2 iff. for any world that is, after revising with the explicit content of the fiction, at least as plausible as any common belief world and that makes the explicit content of the fiction true, it also makes p true.

On our models, in inconsistent fictions not everything is true. Moreover, if F is the explicit content of the fiction, we have that In f: F is a logical truth, that is, in the fiction its explicit content is true, which is desirable. Moreover, import of background beliefs and background knowledge is accounted for by the plausibility ordering. However, within the scope of the operator 'In f:', we can invalidate almost any logical inference. This accounts for the anarchy of fiction with respect to logic: for any logic L, it seems, we can write a fiction where L fails.

• Bernd Buldt
Mathematical practice and human cognition. A critical appraisal of Frank Quinn's Science of Mathematics

Frank Quinn (of Jaffe-Quinn fame, see [1]) worked out the basics of his own account of mathematical practice, an account that is informed by an analysis of contemporary mathematics and its pedagogy (see [2]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects and which, second, proceeds according to the following three-pronged approach:

(i) sever as many ties to ordinary language as possible and limit ordinary language explanations to an absolute minimum;
(ii) introduce axiomatic definitions and bundle them up with a sufficient number of examples, lemmata, propositions, etc. into small cognitive packages;
(iii) practice hard with one new cognitive package at a time.

Drawing on research in cognitive science, and especially in mathematics education, I will then show how cognitive science provides supporting evidence for the effectiveness of this mathematical practice. Time permitting, I will then complement these findings from a phenomenological perspective by exhibiting a fruitful convergence between mathematics, cognitive science, and phenomeno- logical analysis.

The full paper is divided into the following sections: Remark on mathematical practice − Frank Quinns Contributions − Mathematical practice: defining concepts − Mathematical practice: acquisition of concepts − Mathematical practice: corroboration by cognitive science − Mathematical practice: convergence with phenomenology.

References
[1] Jaffe, Arthur, Quinn, Frank."Theoretical Mathematics: Towards a synthesis of mathematics and theoretical physics," Bulletin of the American Mathematical Society NS 29:1 (1993), 113.
[2] Quinn, Frank. Contributions to a Science of Mathematics, manuscript (October 2011), 98pp.

• Bernd Buldt
On fixed points, diagonalization, and self-reference

We clarify the respective role fixed points, diagonalization, and self- reference play in proofs of Gödel's first incompleteness theorem. We first show that the usual fixed-point construction can be reduced to a double diagonalization; this is done to address widely held views such as that fixed-point are "paradoxical" (Priest), or work by "black magic" (Soare), or that their construction is "intuitively unclear" (Kotlarski). We then discuss three notions of self-reference; this can be seen an extension of a recent study by Halbach and Visser and is meant to show that we do not (yet?) have a robust theory that would allow us to establish a firm link between fixed points and self-reference.

• Huimin Dong and Norbert Gratzl

This talk proposes a new solution to the well-know Free Choice Permission Paradoxes, combining ideas from substructural logic and non-monotonic reasoning. Free choice permission is intuitively understood as "if it is permitted to do A or B then it is permitted to do A and it is permitted to do B." This is usually formalized as follows:

P (A ∨ B) ⊢ P A ∧ P B. (FCP)

There are many well-known problems associated with FCP. In this paper we focus on three of them. First, in many deontic systems adding FCP allows for a form of conjunctive inference which seems clearly unacceptable: if it is permitted to order a lunch then it is permitted to order a lunch and not pay for it. Second, many deontic logics become resource-insensitive in the presence of FCP. They validate inferences of the form "if the patient with stomach trouble is allowed to eat one cookie then he is allowed to eat more than one," which are also counter-intuitive. Third, in its classical form FCP entails that the classically equivalent formulas can be substituted to the scope of a permission operator. This is also implausible: It is permitted to eat an apple or not iff it is permitted to sell a house or not. The challenge for a logic of free choice permission is to exclude such counter-intuitive consequences while not giving up too much deductive power. We need a suitable non-classical calculus for free choice permission. We ce permission. We suggest that the right way to do so is using a family of substructural logics augmented with a principle borrowed from non-monotonic reasoning.

The solution that we put forward in this talk is to build the theory of FCP on top of a plausible calculus for action types. This calculus must, in our view, be at the same time relevant, non-monotonic and resource-sensitive. So the present talk can be seen as a follow-up to a suggestion made in [1]. It is structured as follows. We first clarify various background concepts related to free choice permission, and apply the semantic strategy suggested in [1] to analyze those three problems. In this section we emphasize that one of the main goals of this talk is to balance the counter-intuitive consequences of free choice permission with the deductive power of its logic. Our solution is to use the substructural "normality" logics which, we argue, preserve enough deductive power while keeping counter-intuitive FCP inferences at bay. These logics are presented in the third section, where it is shown that they are sounds and complete and satisfy cut elimination.

References
[1] Albert J.J. Anglberger, Huimin Dong, and Olivier Roy. Open reading without free choice. In Fabrizio Cariani, Davide Grossi, Joke Meheus, and Xavier Parent, editors, Deontic Logic and Normative Systems, volume 8554 of Lecture Notes in Computer Science, pages 19-32. Springer International Publishing, 2014.

• Bernhard Fisseni.
Perspectives as Vector Spaces

Perspective is an important concept in the informal modeling of narrative. E.g., Schmid distinguishes five dimensions of perspective (Perception, Ideology, Space, Time, Language). Formal modeling of perspective has concentrated on identifying 'subjective statements' (Wiebe). We discuss a conceptualization of perspective which builds on the metaphor of a mathematical (vector) *space*. The main claim is that such a notion of perspective allows to compare perspectives, merge and unify perspectivized information by operationalizing the metaphor: projecting between different semantic spaces, e.g. the 'objective' space and the spaces of individual and temporal stances. (The 'objective' space is also just a perspective in this framework.)

The conceptualization builds on the assumption that for two descriptions to be considered different perspectives on the same state of affairs, they have to have some facts and descriptive aspects in common, which can be captured as (applications of) a common semantic subspace, but may also have conflicting information and categorizations. What sets this concept of perspective apart from other notions of semantic spaces or mental spaces is that (a) the space is not just a container that is different from other spaces, but that its internal structure and the 'co-ordinate system' or basis are of prime importance for the comparison and merging of different perspectives, and that (b) the space that is modeled is not the space of a concrete application of the semantic inventories, but the inventories themselves.

• Christine Gaßner
BSS RAM's with ν-Oracle and the Axiom of Choice

We deal with some questions arising from the comparison of two different models of computation over mathematical structures. One of these models is our BSS RAM model that results from a generalization of the Turing machine and the BSS model of computation and allows to describe algorithms on a high level of abstraction. The other model is a machine-independent model of abstract computation over first-order structures. Moschovakis defined it by generalizing the classical concept of μ-recursive functions. In this connection the μ-operator was replaced by a nondeterministic ν-operator. In order to better understand the similarities and differences between both models, we extend our BSS RAM's by introducing such a \nu-operator and study the new machines for several structures. Finally, we discuss the consequences in particular for well-ordered structures, but also for unordered structures.

• Lev Gordeev
Proof compression and NP = PSPACE

The conjecture NP = PSPACE seems provable by elementary dag-like proof compression techniques in standard natural deduction proof system for propositional minimal logic, while using intermediate sequent calculus of Joerg Hudelmaier.

• Walter Hoering
Über die Notwendigkeit des Zufalls (The author would be pleased to send his paper if asked by e-mail: hoering(at)lmu(dot)de)

Wir stellen eine vermutlich neue Art von formalen Systemen dar, in denen Zufallsentscheidungen eine wesentliche Rolle spielen.

Definability of safisfaction in outer models

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞, ω, Barwise's results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. We also review some unpublished results of Mack Stanley which are directly related to our topic.

The work is joint with Sy-David Friedman.

• Nick de Hoog
Empirical investigations of mathematical styles

Every mathematician has their own style of doing mathematics. Such styles are influenced by many factors, e.g. familiarity with research areas, proof techniques and ideas, but also expectations and valuation of peers. In developing one's personal style, local circumstances seem to play an important role. Anecdotal evidence seems to suggest correlations between non-mathematical cultural factors (e.g. nationality, gender, etc.) and distinct mathematical research cultures. How might these claims be made more precise and substantiated?

A preliminary report on two questionnaire studies on cultural influences in mathematical research will be presented. These studies were conducted during the DMV-Jahrestagung in Hamburg (September 2015) and the 7ECM in Berlin (July 2016) and form the cornerstone for a study on national styles in mathematical practice.

• Mirjana Ilic
A natural deduction system for positive contractionless relevant logic

We give a normalizing system of natural deduction for positive contraction--less relevant logic RW+o. Unlike other known natural deduction systems of relevant logics, including RW+o, constructed by Brady, Dunn and Urquhart, this one has a certain, translational, relationship to a particular sequent calculus of RW+o. Namely, due to the fact that our system is based on general, instead of standard (special) elimination rules, it is possible to define translations from one system to another, such that normal natural deduction derivations correspond to cut-free sequent calculus derivations and vice versa. Based on those translations, normalization and closure under composition can be proved, with the help of cut-elimination.

• Guram Bezhanishvili, Nick Bezhanishvili and Julia Ilin
Superintuitionistic logics and extensions of lax logic

Propositional lax logic (PLL) is an intuitionistic modal logic. The modality of PLL, called the lax modality, is quite peculiar, as it has features of both, the box and the diamond modality from classical modal logic. The lax modality arises in several contexts. From an algebraic point of view, it corresponds to a nucleus on a Heyting algebra. The axioms of the lax modality also describe the logic of Grothendiek topologies which was studied by Goldblatt. Fairtlough and Mendler use the lax modality in formal verification of computer hardware. Wolter and Zakharyaschev studied PLL in the context of preservation results between superintuitionistic to intuitionistic modal logics. It is well-known that PLL is Kripke complete, has the finite model property and is decidable.

In this paper, we make a first attempt to study extensions of PLL. We will develop the machinery of Zakharyaschev's canonical formulas for PLL. The canonical formulas allow us to axiomtize the extensions of PLL in a uniform way. We are taking an algebraic approach to this problem. Our main result concerns extensions of PLL that arise by equipping a superintuitionistic logic L (i.e., an extension of the intuitionistic propositional calculus (IPC)) with the lax modality. These are extensions of the shape PLL+A, were A is a set of formulas in the language of intuitionistic logic. We show that if L=IPC+A has the finite model property, is tabular or Kripke complete, then PLL+A enjoys the same property. Moreover, if the logic L is Kripke complete and decidable, then also PLL+A is decidable. As a corollary we obtain (an alternative proof of the fact) that PLL has the fmp and is decidable.

Similar preservation results for extensions of the basic intuitionistic modal logic IntK have been obtained by Wolter and Zakharyaschev. Their proofs are based on embedding intuitionistic modal logics into fusions of classical modal logics. Our proofs on the other hand rely on the canonical formulas for PLL.

• Maciej Kleczek.
A Fully Compositional Semantics for First-Order Languages

Our aim is to provide a semantics for first-order languages (with no individual constants and function symbols), which takes into account meanings of subatomic constituents of formulas.

The issue of the meaning of a first-order variable is a vexed one [2], [4]. However, we shall identify the meaning of each xi; ∈ Varα in ℑ with the i-th projection function. The meaning of a formula φ in ℑ is identified with the set of satisfying sequences. A look at the existing literature reveals that concepts of the meaning of a first-order variable and a first-order formula in ℑ have been developed independently [5], [6], [7]. The meaning of an atomic formula, as a set of satisfying assignments, is taken as primitive. This enables to doubt whether the standard meaning assignment to first-order formulas in ℑ is fully compositional.

In order to rectify it we claim that atomic formulas are both syntactically and semantically complex. This idea is not a novelty. It circulates in the algebraic literature [1], [8], [9]. Therein an atomic formula is regarded as syntactically and semantically complex just in case the sequence of variables succeeding a predicate symbol is not the proper initial segment of the canonical well-order on Varα. Nevertheless, contrary to the algebraic approach, we shall regard each atomic formula as complex.

Consequently we propose the constituent analysis with following types of expressions: (a) an n-ary sequences of variables for each n ∈ ω (b) predicate symbols (c) boolean operators (d) indecomposable quantifiers prefixes of the form ∀xi and ∃xi for each i<α and (e) formulas. Next we define the class of σ-structures Strσ. Its elements are tuples of the form ℑ' := 〈 Dℑ', Iℑ' 〉 where Iℑ' (Pn): (Dℑ'n) → 2 for each Pn ∈ σ. The meaning of each predicate symbol Pn (denoted by μℑ'(Pn)) is identified with the interpretation function. Furthermore we define by the induction on the length of a sequence, the meaning assignment to each member of Varα+:
(a) μℑ(xi) := xi
(b) μℑ 〈x0, ... , xn-1, xn 〉 := μ ℑ 〈x0, ..., xn-1 〉 × μℑ (xn).
Crucially, the product operation is non-commutative. This allows to explain non-synonymy of R(v2,v3) and R(v3,v2) where R is interpreted as an asymmetric relation. Next we define inductively the satisfaction function ⊨' : ℑ × Lσ × D ℑn → 2 with the atomic clause ⊨'(ℑ', α, P(x0, ... , xn-1)) := 1 iff μℑ'(Pn) o (μℑ 〈 x0, ..., xn-1 〉)[α] := 1.

Moreover we extract the algebraic content of the satisfaction function. Given Varα, we define for each ℑ' ∈ Strσ the tuple Cf (ℑ')α := 〈Xf, ∧, ∨, *, ci, di,j, o, ×〉 i,j < α. Members of the resulting class of algebras are called partial cylindric function algebras of dimension α with product and composition. The carrier of Cf(ℑ')α consists of: (1) functions corresponding to meanings of : (a) n-ary sequence of variables for each n ∈ ω (b) predicate symbols and (2) set of functions satisfying formulas. Next we define an embedding f: Cs(ℑ)α → Cf(ℑ')α. Cs(ℑ)α is a cylindric set algebra of dimension α generated by the standard first-order structure ℑ, which corresponds to ℑ'. Hence the subalgebra of Cf(ℑ')α generated by the image of h is locally finite dimensional and satisfies the analogue of regularity. Furthermore via this embedding we recover algebraic axiomatization of first-order logic.

Finally, relatively to this class of algebraic structures we provide a fully compositional meaning assignment in the manner of [4].

References:
[1] Andreka Hajnal, et.al. Algebraic Logic. In Handbook of Philosophical Logic vol2, ed. Dov Gabbay, et.al. Kluwer, 2001.
[2] Fine Kit, Semantic Relationism, Blackwell Publishing, 2007.
[3] Hodges Wilfrid, Formal Features of Compositionality. Journal of Logic, Language and Information. vol. 10. 2001
[4] Hodges Wilfrid, Truth in a Structure. Proceedings of the Aristotelian Society, Vol.86, 1986
[5] Janssen Theo, Compositionality. In Handbook of Logic and Language, ed. Johan van Benthem, et.al, MIT Press. 1997
[6] Kracht Marcus, Interpreted Languages and Compositonality, Springer Verlag, 2011
[7] Monk Donald, Mathematical Logic, Springer Verlag, 1976
[8] Monk Donald, Substitutionless Predicate Logic with Identity, Archiv fur Mathematische Logik und Grundlagenforschung, Vol.7, 1965
[9] Tarski Alfred, A Simplified Formalization of Predicate Logic with Identity, Archiv fur Mathematische Logik und Grundlagenforschung, Vol.7, 1964
[10] Visser Albert, Contexts in Dynamic Predicate Logic, Journal of Logic, Language and Information. vol 7, 1998.

• Tomer Kotek and Johann Makowsky
Integer Sequences arising from Graph Polynomials: An application of a Theorem by C. Blatter and E. Specker

A. Mani and R. Stones in their paper from 2016 (SIAM-Journal of Discrete Mathematics) study the sequence of integers tc a,b (n) = T(Kn,a,b), where T(G,X,Y)\$is the Tutte polynomial and a,b ∈ ℤ. For the case a=1 this gives a sequence C(n,b) which for b=2 counts the number of labeled connected graphs of order n.

They give a description of the modular behavior of C(n,b) modulo prime powers pk for b ≠ 1 mod p. Their tools for the analysis of the modular behaviour of C(n,b) are elementary number theory and group actions. Similar methods have been employed in 1981 by C. Blatter and E. Specker using additinally tools from logic to analyze the modular of a very large class of combinatorial counting functions. In fact it follows from their work that the sequence C(n,2) is ultimately periodic modulo every m ∈ ℕ.

A. Mani and R. Stone also formulate a conjecture concerning the modular behavior of tc(n,a,b). In this talk we study the modular behaviour of tc(n,a,b), for a,b ∈ ℤ and a modulus m and prove a modified form of their conjecture. We also study the modular behaviour of P(Kn,a,b) for a wide class of graph polynomials. Our main technical contribution consists in showing how the Specker-Blatter can be applied in this new context.

• Stephan Krämer.
The Logic of Ground

Many philosophers believe that there is an important relation of objective priority that can obtain between some facts Γ and another fact P, which they call ground. Roughly, the relation is supposed to obtain when the latter fact obtains in virtue of the former facts. Standard examples include: the fact that the ball is scarlet grounds the fact that the ball is red; the fact that Kant was unmarried and the fact that Kant was male together ground the fact that Kant was a bachelor, and the fact that Socrates was a philosopher grounds the fact that Socrates was a philosopher or a pirate. Some philosophers then put the notion of ground to heavy philosophical work by claiming, for instance, that all normative facts are grounded by non-normative facts, or that all mental facts are grounded by physical facts.

If the notion of ground is to play this key role in metaphysical theorizing, we had better have a good understanding of its logic. As it stands, however, the study of the logic of ground is very much in its infancy. The talk will provide a sketch of the state of the art in this debate, and a brief overview of my own work on the topic, focusing on the so-called pure and propositional logics of ground.

The pure logic of ground describes the purely structural properties of ground. The orthodox view on these may be summarized by saying that ground is irreflexive, transitive, and satisfies an amalgamation principle according to which any the union of any two strict grounds of a fact is also a strict ground. There are two semantics for pure logic of ground for which soundness and completeness results have been obtained. One, by Louis de Rosset, uses acyclic directed hypergraphs. The other, Kit Fine's truthmaker semantics, is a form of situation semantics distinguished by the fact that verification of a sentence by a situation is not subject to monotonicity.

The propositional logic of ground attempts to also capture the interaction between ground and conjunction, disjunction, and negation. The guiding idea behind all existing proposals is that truthfunctionally complex facts are grounded by their constituents. Thus, any fact grounds any disjunction in which it occurs, and any two facts jointly ground their conjunction. As yet, no adequate semantics for such systems has been formulated, but some of my own work suggests that one can develop such a semantics by extending Fine's truthmaker semantics using a suitably understood notion of a mode of verification. The idea is that if P is grounded by Γ, then any situation verifying P does so by verifying Γ, so Γ represents one way, or mode, in which P may be verified. By interpreting a sentence not by a mere set of verifying situations, but by its modes of verifications, we may obtain an adequate semantics for a plausible propositional logic of ground.

• Philipp Lücke.
Forcings that characterize large cardinals

The relative consistency of many combinatorial principles in set theory can be established by collapsing a large cardinal to be the successor a smaller cardinal.
In many important cases, the resulting principle implies that the relevant successor cardinal has the same large cardinal properties in some canonical inner model and therefore the large cardinal assumption was necessary for the consistency proof. We will consider the question whether certain collapse forcings characterize large cardinal properties through the validity of combinatorial principles in their forcing extensions, in the sense that the collapse forces the principle to hold if and only if the collapsed cardinal possess the corresponding large cardinal property in the ground model.

It is easy to see that the Levy collapse cannot characterize inaccessible cardinals in this way. In contrast, other canonical collapses can characterize many important types of large cardinals. For example, the forcing that first adds a Cohen real and then Levy collapse a cardinal to be ω2 can characterize inaccessible, Mahlo, weakly compact, indescribable, measurable, supercompact and huge cardinals. In the case of larger large cardinals, these characterizations make use of results of Apter and Hamkins on induced measures in ground models. The characterization of smaller large cardinals relies on classical combinatorial characterizations of these cardinals. Since there are no suitable combinatorial characterizations for indescribable cardinals, the forcing characterizations of these cardinals rely on new results that identify these cardinals as the images of the critical points of certain elementary embeddings.

This is joint work in progress with Peter Holy (Bonn).

• Luca San Mauro
Classifying Relations via Computable Reducibility

Computable reducibility provides a natural way of ranking binary relations on positive integers according to their complexity. Such reducibility is defined as follows: let R and S be two binary relations, we say that R is computably reducible to S iff there is a computable function f such that, for all x and y, the following holds: xRy if and only if f(x)Sf(y).

Computable reducibility has been object of study for decades, being mostly applied to the case of equivalence relations. In particular, a prominent problem in the area has been that of characterizing universal equivalence relations, i.e. relations to which all others relations, of a given complexity, can be computably reduced (see [1]).

In this talk, we address the problem of universality for a more general context than that of equivalence relations. First, we prove that, contrary to the case of equivalence relations and preorders (see [2]), for each level of the arithmetical hierarchy there is a universal binary relation. Then, we define natural examples of universal Σ0n binary relations and of universal Π0n binary relations, obtained by fairly simple manipolations of the most fundamental set-theoretic relations.

References
[1] Uri Andrews, Steffen Lempp, Joseph S. Miller, Keng Meng Ng, Luca San Mauro, Andrea Sorbi, Universal computably enumerable equivalence relations, The Journal of Symbolic Logic, vol. 79 (2014), no. 1, pp. 60 −88.
[2] Egor Ianovski, Russell Miller, Keng Meng Ng, Andre Nies, Complexity of equivalence relations and preorders from computability theory, The Journal of Symbolic Logic, vol. 79 (2015), no. 3, pp. 859 − 881.
[3] Luca San Mauro, Infomal Proofs and Computability, PhD Thesis, SNS, Pisa (2015).

• Klaus Meer
Real interactive proofs

Interactive proofs provide a powerful tool in complexity theory to verify statements. An almighty prover interacts in several rounds with a verifier to convince the latter of a statement. The verifier usually is a randomized polynomial time machine and should detect erroneous proofs with high probability. Shamir's famous result characterizes the class IP of problems that can be accepted by such interactive proofs as PSPACE or, equivalently, by the class PAR of problems decidable in parallel polynomial time.

• We study interactive proofs in the framework of real number complexity theory as introduced by Blum, Shub, and Smale. Since space resources alone are known not to make much sense in real number computations the question arises whether a real version of IP can be similarly characterized by real number complexity classes. We shall derive both an upper and a lower bound for the power of real interactive proofs. In particular, we show that Shamir like techniques can be applied to give interactive protocols for problems that can be solved in real parallel polynomial time. This is joint work with M. Baartse.

• Hugo Nobrega
Tree games for Baire classes and partition classes

We report on ongoing work for characterizing subclasses of Borel functions in Baire space by games.

Among previously known results are Wadge's game for continuous functions, Duparc's game for Baire class 1 functions, Andretta's game for the class of functions preserving Σ02 under preimages, and Semmes's games for Baire class 2 functions, for the class of functions preserving Σ03 under preimages, for the class of functions pulling back Σ02 to Σ03 under preimages, and for the class of Borel functions itself.

In this work, we characterize the Baire class alpha functions---i.e., those functions which pull back Σ01 to Σ0α + 1 under preimages---via a game for each fixed countable ordinal alpha. This game is obtained by first relaxing Semmes's labeled tree game for Borel functions via a notion of bisimilarity of labeled trees common in computer science. We then restrict the relaxed labeled tree game in terms of two measures of complexity of trees which we introduce, called the pruning rank and the finitely branching rank of a tree. Wadge's and Duparc's games can be seen as particular cases of our games, but interestingly Semmes's game for Baire class 2 is significantly different from the one we obtain.

Furthermore, given a subclass Gamma of Borel functions which is characterized by a restriction of the labeled tree game and which satisfies a certain weak assumption, we obtain a game characterization of the class of functions which are piecewise Gamma on a countable Σ0α partition of Baire space---i.e., the class of functions f for which there exists a partition 〈A_n ; n natural number〉 of Baire space into Π0α sets such that the restriction of f to An is in Γ, for each n. Again, this game is obtained by a suitable restriction of the labeled tree game, and Andretta's game and Semmes's games for the class of functions preserving Σ03 under preimages and for the class of functions pulling back Σ02 to Σ03 under preimages are particular cases of our construction.

• Stephane Le Roux, Arno Pauly and Jean-Francois Raskin
On the decidability of Minkowski-games

Motivated by control theory, we introduce Minkowski-games as a notion of games played in a Euclidean vector space. The game state is a vector. The protagonist choses from a finite list of subsets. Then the antagonist picks a the new state from the Minkowski-sum of the previous state and the chosen subset.

We consider two types of winning conditions, and show that while both are determined, the winner is decidable for one but not for the other.

• Benjamin Rin
Realizability Semantics for Quantified Modal Logic

In this talk, a non-Kripkean semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis together with Shapiro's system of first-order modal arithmetic (known as Epistemic Arithmetic). While the bulk of the talk is devoted to developing the details of the semantics, to illustrate the scope of this approach we observe that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. Of these, the talk will focus primarily on (i).

• Luis Rosa
A framework for testing the relevance of logic to reasoning

Much has been discussed lately about the relevance of logic to the normativity of human reasoning (both, in the epistemology literature and in the cognitive psychology literature). In this presentation, we will flesh out a semantic framework that allows us to test whether and to what extent certain logical systems are normatively binding with respect to human reasoning. The framework uses families of models that represent the cognitive situations of reasoners. It is then possible to change the reasoner's inferential accessibility relations in accordance with different logics, which will give rise to alternative verdicts as to what is rational for the reasoner to believe or doubt. As we expect to show, logic can be relevant to the normativity of human reasoning after all -- but only if we accept a certain form of logical pluralism.

• Philipp Schlicht
Random reals and infinite time Turing machines

We prove an analogue to Sacks' theorem for infinite time Turing machines: if a real is infinite time Turing computable relative to all reals in a set of positive measure, then it is infinite time Turing computable. To this end, we prove results about random forcing over admissible sets and several other results about infinite time machines. This is joint work with Merlin Carl.

• Julian Schlöder
Classical Negation and Weak Rejections

It is an old debate in logic whether we should admit the principle of double negation elimination (DNE) or not -- whether we can be classical logicians or should be intuitionists. Gentzen (1934) proposed to explain the meaning of the logical connectives through (harmonious) introduction and elimination rules. Dummett (1991) argues that this licenses intuitionism, but not classical logic. A defence of classical negation due to Smiley (1996) and Rumfitt (2000) goes as follows. Classical negation can be axiomatised in bilateralist logics that track both asserted and rejected content; they write +A for an assertion of A and −A for a rejection. Bilateralism faces some challenges:

(a) Dummett (2002) finds Rumfitt's defense of DNE almost question-begging, since he uses − as a sign of falsity and stipulates to eliminate rejected negations in the rule (− ¬ E.): − ¬A +A; (b) Dickie (2010) criticises that bilateralists infer negated assertions from rejections; she provides examples of so-called weak rejections to show that this is unwarranted in general; (c) the Frege-Geach embedding problem, first mentioned in Frege 1919: negation can embed, but rejection cannot.

I present a weak rejection logic (WRL) that faces these problems head on (joint work with Luca Incurvati). All rules from Rumfitt 2000 that are invalid for weak rejections can be removed or appropriately weakened. To solve the Frege-Geach problem, I show that a single principle is sufficient: the Smilean inference +(¬ A → ¬ B), − A ⊢ − B. This rule has been defended by Smiley (1996) and I provide further linguistic evidence supporting it, building on Grice's (1991) notion of unassertability. I show that, given the other bilateralist inference rules, Smilean inference is equivalent to a weakened version of Rumfitt's (− ¬ E.) that is still sufficient to derive +(¬ ¬ A → A). Thus, it is a natural alternative to the rule Dummett finds objectionable.

WRL has a number of interesting metalogical properties. First, it justifies classical logic in the following sense. For each classical tautology φ, it verifies that +φ is a WRL-tautology and that for a classically valid inference φ ⊢ ψ it holds that +φ ⊢ +ψ. Thus, WRL restricted to + is classical logic. Second, however, this does not hold for rejections; most notably, a rejected premise breaks the Deduction theorem. Third, WRL embeds into KD45 modal logic as follows: there is a natural mapping f of the bilateralist language into the language of public commitment modal logic (Asher and Lascarides, 2008). Then, WRL is sound and complete for KD45 in the following sense: Γ ⊢ ± φ in WRL iff f [Γ] ⊨ f (± φ) in KD45.

References:

• Nicholas Asher and Alex Lascarides. Commitments, beliefs and intentions in dialogue. In SemDial 2008, 2008.
• Imogen Dickie. Negation, anti-realism, and the denial defence. Philosophical Studies, 150(2):161-185, 2010.
• Michael Dummett. 'Yes', 'No' and 'Can't say'. Mind, 111(442):289-295, 2002.
• Michael AE Dummett. The logical basis of metaphysics, volume 5. Harvard UP, 1991.
• Gottlob Frege. Die Verneinung: Eine logische Untersuchung. Beiträge zur Philosophie des deutschen Idealismus, 1:143-157, 1919.
• Gerhard Gentzen. Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 1934.
• H Paul Grice. Studies in the Way of Words. Harvard University Press, 1991.
• Ian Rumfitt. 'Yes' and 'No'. Mind, 109(436):781-823, 2000.
• Timothy Smiley. Rejection. Analysis, 56(1):1-9, 1996.

• David Schrittesser
Maximal cofinitary groups

A group of permutations of the set of natural numbers is cofinitary iff the only group element with infinitely many fixed points is the identity. A cofinitary group is maximal iff it is maximal under inclusion among such groups. Can a maximal cofinitary group be definable? Kastermans showed that in L, the answer is yes. In recent joint work with V. Fischer and A. Törnquist, we showed that the same holds after adding arbitrarily many Cohen reals to L.

• Andrei Sipos
Proof mining and positive-bounded logic

Proof mining is a research program introduced by U. Kohlenbach in the 1990s, which aims to obtain explicit quantitative information (witnesses and bounds) from proofs of an apparently ineffective nature. This paradigm in applied logic has successfully led so far to obtaining some previously unknown effective bounds, primarily in nonlinear analysis and ergodic theory. A large number of these are guaranteed to exist by a series of logical metatheorems which cover general classes of bounded or unbounded metric structures.

In order to apply these metatheorems, the structures are typically formalized in higher-order systems of arithmetic and analysis, using appropriate codings of real numbers and related operations. The classes for which metatheorems have already been proven include normed spaces and hyperbolic spaces. Recently, Günzel and Kohlenbach have shown that, in principle, one could obtain metatheorems for a plethora of classes of structures, provided that they are formalized in positive-bounded logic (in the sense of Henson and Iovino) and that some preparation of the axioms is undertaken beforehand. We aim to extend this kind of results to other clases of spaces formalizable in positive-bounded logic and to derive concrete applications from existing and relevant prima facie noneffective theorems for those spaces.

• Florian Steinberg
Representations of spaces of integrable functions

We consider different ways to compute on spaces of integrable functions. To make elements of function spaces accessible to digital computers, they can be encoded by string functions. Carrying out computations on the function spaces can then be done by operating on the codes, that is, computing functions on Baire space.

The framework of second order representations is a formalization of this devised by Kawamura and Cook. The name "second order representation" is due to the use of second order complexity theory on Baire space. Computationally it is equivalent to the TTE, the accepted model of computation over continuous structures. In particular Kawamura and Cook investigated the space of continuous functions on the unit interval and succeeded to show that there is a weakest second order representation such that evaluation is possible in polynomial time. We use this result as a starting point for generalizations.

In a first step we specify the weakest representation of the space of integrable functions on the unit interval that renders integration polynomial time computable and show that it is discontinuous with respect to both the norm and the weak topology.

This is in contrast to the continuous functions where the norm is polynomial space computable. We proceed to strengthen the representation enough to make it possible to compute the norm in polynomial space. This necessitates to talk about integral moduli. Integral moduli can be understood to classify integrable functions by their approximizability by dyadic step functions just like the modulus of continuity does for continuous functions. This is reflected in the Frechet-Kolmogorov-Theorem, a Arezela-Ascoli style classification Theorem of the compact subsets of spaces of integrable functions.

Finally we use the strong counter examples provided by Friedman and Ko to show that certain operators can not be expected to be rendered polynomial time computable by any reasonable representation. Furthermore we point out a surprising connection between the complexity of integrating a continuous function and multiplying integrable functions whose product is well defined.

• Sarka Stejskalova
The tree property and the continuum function

We will review some of the more recent results we have obtained jointly with Sy-David Friedman and Radek Honzik regarding the relationship between the tree property and the continuum function. We will discuss the following key areas:

(a) The possibility of obtaining a strong limit cardinal κ with 2κ (arbitrarily) large and with the tree property at κ++. The possibility of having κ = ℵω in the previous result.
(b) The possibility of obtaining the results in (a) from more optimal large-cardinal assumptions (supercompacts vs. strong cardinals of low degree).
(c) The possibility of obtaining a model where the continuum function below a strong limit ℵω is as arbitrary as possible with the tree property holding at (some/all) cardinals ℵn , 1 < n < ω

This work is a part of my PhD project.

• Jan Walker
Axiomatizing Finitist Truth

Let S be a theory in a language L. Define R(S) to be the theory in L + {T} which is obtained by adding to S: firstly, for all formulae in which the new predicate symbol T occurs, the corresponding substitution instances of axiom schemata and rules of S; secondly, axioms to the effect that T denotes partial truth as outlined by Saul Kripke [2]. In a seminal article, Solomon Feferman [1] proposed to identify the statements in L which are derivable in Ref(S) as the statements in L one ought to accept if one has accepted S.

My talk is inspired by Feferman's approach, but takes a different direction. It starts off with the assumption that only theorems of finitist (primitive recursive) arithmetic ought to be accepted and aims at answering the question: To what extend can one reason about truth under this general assumption? To be more specific, I will address truth-theoretic questions such as: How to adapt the axioms for Kripke's notion of partial truth to what is acceptable from the finitist perspective? How to devise finitist variants of other predominant axiomatizations of truth to be found in the literature? Is there a core feature shared by all these finitist variants of axiomatizing truth?

In the sequel of my talk, I will turn to a proof-theoretic analysis of (Turing-like) progressions in which reflection (or formal soundness) principles are successively added to theories in a language of typed truth, where each type level has its own truth predicate axiomatized in accordance with finitist standards.

References
[1] Solomon Feferman, Reflecting on incompleteness, The Journal of Symbolic Logic, vol. 56 (1991), no. 1, pp. 1-49.
[2] Saul Kripke, Outline of a theory of truth, The Journal of Philosophy, vol. 72 (1975), no. 19, pp. 690-716.

• Gunnar Wilken.
Gödel's T as a case study for ordinal assignment

The variant of Gödel's system T of primitive recursive functionals of finite type which is based on typed lambda-calculus can be seen as a paradigm for higher order rewrite systems. In an article joint with A.Weiermann (LMCS 2012) T was analyzed using an assignment technique extending Howard's original assignment from 1970.

In my talk I am going to explain this technique under two aspects: the non-unique assignment needed in the argument, which goes back to Howard's original treatment, and Weiermann's collapsing technique, which allowed for a derivation lengths classification of T.

• Wolfgang Wohofsky and Jörg Brendle.
Borel's conjecture for the Marczewski ideal

We will show the following ZFC theorem which is recently finished work joint with Joerg Brendle: There is no set of size continuum which is "s0-shiftable", i.e., which can be translated away from every set in the Marczewski ideals0 (where a set of reals is in s0 if for every perfect set there is a perfect subset disjoint from it). We will concentrate on regular continuum, since the proof is easier in this case.

The theorem is very much in contrast to the respective situation when s0 is replaced by the meager ideal: there are models (e.g., all models that satisfy CH) with large meager-shiftable (i.e., strong measure zero) sets.

Our original proof dealt with the reals in the sense of the Cantor space 2ω. However, it can be generalized to other Polish groups.

 Impressum 2016-10-07, ACB