Literaturhinweise

"It seems that in order to understand common knowledge (a crucial feature of communication), circular propositions, various aspects of perceptual knowledge and self-awareness, we had to admit that there are situations that are not wellfounded under the "constituent of" relation. This meant that the most natural route to modeling situations was blocked by the axiom of foundation. As a result, we either had to give up the tools of set theory which are so well loved in mathematical logic, or we had to enrich the conception of set, finding one that admits of circular sets, at least. I wresled with this dilemma for well over a year before I argued for the latter move in (Barwise 1986). It was just this point that Aczel visited CSLI and gave the seminar which formed the basis of this book. Since then, I have found several applications of Aczel's set theory, far removed from the problems in computer science that originally motivated Aczel."

     - Appendix B: Background set theory [340 KB, pdf]
 

"Given the fruitfulness of diagonal arguments in the rest of logic, one wonders whether the path followed in model theory was really the most productive reaction to the [Liar] paradox. In this book, we present an account of the Liar that shows it to be a true diagonal argument, one with profound consequences for our understanding of the most basic semantical mechanism found in ordinary language. Indeed, we think the Liar is every bit as significant for the foundations of semantics as the set-theoretic paradoxes were for the foundations of set theory."

          - Chapter 3:   The Universe of Hypersets  [640 KB, pdf]
 

"The subject of non-wellfounded sets came to prominence with the 1988 publication of Peter Aczel’s book on the subject. Since then, a number of researchers in widely differing fields have used non-well-founded sets (also called “hypersets”) in modeling many types of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and programming languages.
    Vicrous Circles offers an introduction to this fascinating and timely topic. Written as a book to learn from, theoretical points are always illustrated by examples from the applications and by exercises whose
solutions are also presented. The text is suitable for use in a classroom, seminar, or for individual study.
    In addition to presenting the basic material on hypersets and their applicatrons, this volume thoroughly develops the mathematics behind solving systems of set equations, greatest fixed points, coinduction,
and corecursion. Much of this material has not appeared before. The application chapters also contain new material on modal logic and new explorations of paradoxes from semantics and game theory."

Alexandru Baltag:  STS: A structural theory of sets. PhD  Indiana University, June 1998. [800 KB, pdf]

"This paper is an attempt to build a set theory on a purely structural view on the concept of set. I make a distinction between a potential structure and its actualization into a set (via decoration or closure). I propose an analytical picture, in which objects are analyzed in stages and all we can know about them are their unfoldings or partial descriptions. A set is what is left from this process of analysis: it is the trace of unfolding of some possible object, its pattern of analytical behavior. I have a notion of observational equivalence between structures, definied as identity of analytical behavior. Sets can be understood as arbitrary structures modulo observational equivalence. As collections, sets are closed, completed classes, which are as large as their pattern of unfolding allows them. They contain every object which cannot be separated from all their elements at any stage of unfolding. This gives them well-defined boundaries and a clear-cut identity.
    I explore the connection between this notion of set and modal logic. Sets can be identified with the maximally consistent theories that characterize them. Sets can also be understood as modally definable classes. This provides a proof (and so a justification) for the Power Set axiom on different grounds
than the ones of the classical conceptions.
    The universe of sets described has nice fixed-point and closure propoties.  Recursion and corecursion are related in a simpler manner over this universe than over Aczel's hyperset universe. Some  category-theory notions can be stated as objects (sets), not just as classes. The topological aspect comes from the underlying presence of a notion of observational approximation (structures can be "almost bisimilar" up to any ordinal depth). This universe provides models for many recursive and corecursive domains, which could be used as uniform frameworks for giving denotational semantics. This universe of sets seems also to be a good candidate for a general framework to study semantical paradoxes."
 

"This thesis contains three logical investigations into dynamic semantics. The subjects of these three investigations are:

An application of dynamic semantics to the Problem of the Liar Paradox and other circular propositions (Chapter 2).

 A theoretical investigation of notions of logical consequence in dynamic semantics (Chapter 3).

An extension of dynamic semantics to various systems of dynamic epistemic logic that deal with changes of higher-order information (Chapter 4)."

"This paper is the result of combining two traditions in formal logic: epistemic logic and dynamic semantics.
    Dynamic semantics is a branch of formal semantics that is concerned with change, and more in particular with change of information. The main idea in dynamic semantics is that the meaning of a syntactic unit|be it a sentence of natural language or a computer program|is best described as the change it brings about in the state of a human being or a computer. The motivation for, and applications of this `paradigm-shift' can be found in areas such as semantics of programming languages (cf. Harel, 1984), default logic (Veltman, 1996), pragmatics of natural language (Stalnaker, 1972) and of man-computer interaction, theory of anaphora (Groenendijk and Stokhof, 1991) and presupposition theory (Beaver, 1995). Van Benthem (1996) provides a survey.
    This paper is firmly rooted in this paradigm, but at the same time it is much in influenced by another tradition: that of the analysis of epistemic logic in terms of multi-modal Kripke models.
    This paper is the result of combining these two traditions. It contains a semantics and a deduction system for a multi-agent modal language extended with a repertoire of programs that describe information change. The language is designed in such a way that everything that is expressible in the object language can be known or learned by each of the agents. The possible use of this system is twofold: it might be used as a tool for reasoning agents in computer science and it might be used as a logic for formalizing certain parts of pragmatics and discourse theory.
    The paper is organized as follows. The next section contains a short description of classical modal logic and introduces models based on non-well-founded sets as an alternative to Kripke semantics. In the section after that I introduce programs and their interpretation and I give a sound and complete axioma-
tization of the resulting logic in section 4. The last section is devoted to a comparison with update semantics of Veltman (1996).
    Finally, I would like to mention the dissertations of Groeneveld (1995), Jaspars (1994) and de Rijke (1992) and the book by Fagin, Halpern, Moses and Vardi (1995) as precursors and sources of inspiration. The article by Willem Groeneveld and me (to appear) contains some ideas similar to those presented here."
 

  "In this paper, we have combined techniques from epistemic and dynamic logic to arrive at a logic for describing multi-agent information change. The key concept of dynamic semantics is that the meaning of an assertion is the way in which the assertion changes the information of the hearer. Thus a dynamic epistemic semantics consist in a explicit formal de nition of the information change potential of a sentence. We used these ideas to arrive at the system of Dynamic Epistemic Semantics, which is semantics for a language describing information change in a multi-agent setting. This semantics proved useful for analyzing the Muddy Children paradox, and also for giving a semantics for knowledge programs, since it enabled us to model knowledge change by giving an explicit semantics to the triggers of the information change (the latter being the assertions made, or the messages sent). We feel that this is an important extension, since standard approaches to for example the Muddy Children (e.g. Fagin et al. 1995) generally use static epistemic logics like S5 to describe the situation before and after a certain epistemic event, leaving the transition between `before' and`after' to considerations in the meta-language. In contrast, in dynamic epistemic logic, epistemic actions like updates are firrst class citizens of the object language of DES. For one thing, this opens the possibility of making artificial agents a bit more intelligent, by giving them an axiomatics for DEL as their tool for reasoning about knowledge change."