In order to give some illustrative examples of the many techniques and
problems involved in working in these subsystems, we will sketch some
results related to Lebesgue spaces (every open covering has a Lebesgue
number) and Atsuji spaces (every continuous function defined on them is
uniformly
continuous) within subsystems of second order arithmetic. The most
interesting result is ``every Atsuji space is Lebesgue''. It is known that
it is
provable in ACA0,
however we do not know yet if it is equivalent to ACA0.
As improvement of such known result, we prove that the known proof needs
ACA0, and we conjecture that the statement is actually equivalent to
ACA0.
Another example is from Combinatorics: it is related to the Free Set
Theorem and Ramsey's Theorem and uses techniques from computability theory.
This example shows how some theorems of ordinary mathematics may not fit
in one of the subsystems mentioned above.
We will also comment on a couple of interesting results: the first one is
related to Set Theory. While Cantor's proof that every
countable closed set is a set of uniqueness makes essential use of
ATR0, the result itself can be proved in ACA by
different means (result due to J. Humphreys).
The second one is again related to Combinatorics, in particular König's
duality theorem.
In this talk we will examine aspects of the latter issue.
First, we will consider results of generalized fixed point accounts that
originated from the classical work of Saul Kripke. We will see that in such theories
truth predicates are essentially inductively defined. Second, we will consider
Gupta-Belnap's revision theory of truth. According to a result of Antonelli and
Kremer it can be shown that the complexity of validity of the semantical systems
S* and S# is Pi12. Third, we shall examine the
modeling of truth in a situation theoretic account, using Peter Aczel's theory of
non-well-founded sets. Using the theory of hypersets instead of classical ZFC
enables us to model circular propositions in a situation theoretic framework.
In particular the coalgebraic features of this representation is of special interest
in this context.
All these frameworks will be informally discussed with a stress of the mathematical
techniques used in such theories. Furthermore, we will mention a number of
open problems. Examples are the following ones.
Finally it should be discussed how some well-known methods of empirical
science can be explicated within structuralism. In the centre of these
considerations are methods which describe some changing of theories.
Special types of changing can be represented as intertheoretical
relations. In this metatheoretical frame it will be argued that "Analogy"
could be comprehended as such an intertheoretical relation.
In this talk we discuss:
NOTE: This presentation reports on joint work with David Makinson,
Unesco, Paris
Modal Logics for Games:
tools for
studying the dynamics of knowledge and belief
Alexandru Baltag
top
back to main page
The diversity of models in statistical mechanics: Views about the structure of scientific theories
Anouk Barberousse
top
back to main page
The logical architecture of natural language
Johan van Benthem
top
back to main page
Topics in Reverse Mathematics
Mariagnese Giusto
Slides of this talk: DVI-File
top
back to main page
Leon Horsten
We consider a language containing partial predicates for subjective
knowability and truth. For this language, inductive hierarchy rules are
proposed which build up the extension and anti-extension of these partial
predicates in stages. The logical interaction between the extension of the
truth predicate and the anti-extension of the knowability predicate is
investigated.
top
back to main page
Marcus Kracht
The emergence of attribute value formalisms in linguistic theory has
provoked a debate about the nature of representations. The painfully
explicit notation of GPSG and HPSG, for example, was criticized by
transformationalists as not psychologically adequate. In their view,
these notations conflate inherently descriptive with computational
content. However, it is not clear at the outset where to draw the
boundary between these two notions. In order to decide this issue,
one needs a neutral platform where these two positions can be nego-
tiated. We shall argue that logic and model theory is such a plat-
form. This talk will present a tour d'horizon of the logical theory
of syntactic (and phonological) structures. In particular, we shall
present a definition of naturalness for features that will allow us
to distinguish between features which are in some sense essential
and those which are only virtual ("ghost features") and which seem
to be proxy for something else that is not yet understood. Using
this dichotomy, the nature of certain representations can actually
be better motivated.
top
back to main page
Kai-Uwe Kühnberger
Theories of truth are a highly discussed topic in philosophy. In the last twenty-five
years several different frameworks were developed in order to provide a modeling
of classical paradoxes in natural language. In a wider context these frameworks can be
used to model circular phenomena in general. Whereas, most of these frameworks
are well-examined concerning their empirical properties (for example concerning
their properties of modeling Liar-like sentences) less attention was paid to questions
concerning complexity issues of these frameworks.
What are necessary and sufficient conditions for the existence of fixed points in
generalized versions of Kripke's framework?
Is there a coalgebraic representation of revision theories?
What are the properties of a second-order version of revision theories?
What can be said about the complexity of coalgebraic theories?
top
back to main page
Mario Piazza
In this talk, we tackle the philosophical problem of the
applicability of mathematics, also known as the problem of
"the unreasonable effectiveness of mathematics": how can facts
about mathematical structures be relevant to the empirical world?
For instance, how do topological concepts such as that of boundary
apply to the ordinary world around us? Why, generally speaking,
observations of the natural world seem to fit so well into logical
structures? (Are logical entities the sine qua non of mental
activity?) It may be argued that mathematics is essential to
physics, i.e. it has a causal counterpart, because we select for
study only those features of the natural world that can be
framed into mathematical form. Mathematics is effective since
it is generative. Several examples are provided in order to explain
this phenomenon.
top
back to main page
Techniques and Methods of Science from a Structuralist Point of View
Martin Rotter
At a first stage it will be explained what are the main metatheoretical
aspects of Structuralist Theory of Science. It will be illustrated which
technical tools within the Structuralists' program are developed which
make possible a lot of applications to empirical sciences. One of the main
applications are reconstructions of empirical theories within a semantic
approach founded on informal set theory. These techniques will be
illustrated with Quantum Theories which deliver a couple of examples.
top
back to main page
Hans-Jörg Tiede
In a series of articles, Lambek laid out a program advocating a proof
theoretical approach to grammar. Current interest in proof theoretical grammars
stems from their applicability to natural language syntax and semantics, their
relation to linear logic, and studies of their formal properties - which often
combine proof theoretic and recursion theoretic arguments.
This talk will focus on formal properties of proof theoretical grammars - in
particular formal language theoretic and complexity theoretic aspects. Recent
work on the formal properties of grammar formalisms has paid considerable
attention to the "strong generative capacity," i.e. the structures or derivation
trees assigned to the generated strings. We will discuss what could take the
role of structures or derivation trees in the proof theoretical framework and
the relationship of these structures to the derivation trees of context-free
grammars.
top
back to main page
Input-output logics and their applications
Leon van der Torre
In a range of contexts, one comes across processes resembling inference,
but where input propositions are not in general included among outputs,
and the operation is not in any way reversible. Examples arise in
contexts of conditional obligations, goals, ideals, preferences,
actions, and beliefs. Our purpose is to develop a theory of such
input/output operations. Four are singled out: simple-minded, basic
(making intelligent use of disjunctive inputs), simple-minded reusable
(in which outputs may be recycled as inputs), and basic reusable. They
are defined semantically and characterised by derivation rules, as well
as in terms of relabeling procedures and modal operators. Their
behaviour is studied on both semantic and syntactic levels.
top
back to main page