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Definable Naming Relations in Metalevel Systems
 Proceedings of the Third Workshop on Metaprogramming in Logic (META'92), volume 649 of Lecture Notes in Computer Science
, 1992
"... . Metalevel architectures are always, implicitly or explicitly, equipped with a component that establishes a relation between their object and metalevel layers. This socalled naming relation has been a neglected part of the architecture of metalevel systems. This paper argues that the naming re ..."
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Cited by 13 (5 self)
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. Metalevel architectures are always, implicitly or explicitly, equipped with a component that establishes a relation between their object and metalevel layers. This socalled naming relation has been a neglected part of the architecture of metalevel systems. This paper argues that the naming relation can be employed to increase the expressiveness and efficiency of metalevel architectures, while preserving known logical properties. We argue that the naming relation should not be a fixed part of a metalevel architecture, but that it should be definable to allow suitable encoding of syntactic information. Once the naming relation is definable, we can also make it meaningful. That is, it can also be used to encode pragmatic and semantic information, allowing for more compact and efficient metatheories. We explore the formal constraints that such a definable naming relation must satisfy, and we describe a definition mechanism for naming relations which is based on term rewriting s...
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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Cited by 8 (2 self)
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Knowledgelevel Reflection
, 1992
"... This paper presents an overview of the REFLECT project. It defines the notion of knowledge level reflection that has been central to the project, it compares this notion with existing approaches to reflection in related fields, and investigates some of the consequences of the concept of knowledge le ..."
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Cited by 4 (2 self)
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This paper presents an overview of the REFLECT project. It defines the notion of knowledge level reflection that has been central to the project, it compares this notion with existing approaches to reflection in related fields, and investigates some of the consequences of the concept of knowledge level reflection: what is a general architecture for knowledge level reflection, how to model the object component in such an architecture, what is the nature of reflective theories, how can we design such architectures, and what are the results of our actual experiments with such systems?
On Knowledge, Strings, and Paradoxes
, 1998
"... . A powerful syntactic theory as well as expressive modal logics have to deal with selfreferentiality. Selfreferentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logica ..."
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Cited by 2 (2 self)
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. A powerful syntactic theory as well as expressive modal logics have to deal with selfreferentiality. Selfreferentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logical system. There is a large amount of different attempts to tackle these problems. Some of them are compared in this paper, futhermore a simple approach based on a threevalued logic is advocated. In this approach paradoxes may occur and are treated formally. However, it is necessary to be very careful, otherwise a system built on such an attempt trivialises as well. In order to be able to formally deal with such a system, the reason for selfreferential paradoxes is studied in more detail and a semantical condition on the connectives is given such that paradoxes are excluded. Keywords: Knowledge representation, selfreferentiality, paradoxe, Kleene logic Ich habe manche Zeit damit verloren, ...
Living with Paradoxes
"... A good knowledge representation system has to nd a balance between expressive power on the one hand and ecient reasoning on the other. Furthermore it is necessary to understand its limitations and problems. A logic which contains strings is very expressive and allows for very natural representation ..."
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A good knowledge representation system has to nd a balance between expressive power on the one hand and ecient reasoning on the other. Furthermore it is necessary to understand its limitations and problems. A logic which contains strings is very expressive and allows for very natural representations, which in turn allow for appropriate reasoning patterns. However, such a system has the feature that it is possible to formulate selfreferential paradoxes in it. This can be considered as a strength and as a weakness at the same time. On the one hand it is a positive aspect that it is possible to represent paradoxes, which can be formulated in natural language. On the other hand it is necessary to be careful and not to trivialise the logical system. In the paper dierent aspects of knowledge representation which allows selfreferentiality will be discussed. A system will be presented which is a pragmatic compromise between expressive power on the one hand and simplicity and eciency of the reasoning process on the other hand. It is built on a threevalued system that makes it possible to use reasoning techniques from classical rstorder logic.
On Arithmetical FirstOrder Theories allowing Encoding and Decoding of Lists
, 1998
"... In Computer Science, ntuples and lists are usual tools; we investigate both notions in the framework of firstorder logic within the set of nonnegative integers. Gödel had firstly shown that the objects which can be defined by primitive recursion schema, also can be defined at firstorder, using na ..."
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In Computer Science, ntuples and lists are usual tools; we investigate both notions in the framework of firstorder logic within the set of nonnegative integers. Gödel had firstly shown that the objects which can be defined by primitive recursion schema, also can be defined at firstorder, using natural order and some coding devices for lists. Secondly he had proved that this encoding can be defined from addition and multiplication. We show this can be also done with addition and a weaker predicate, namely the coprimeness predicate. The theory of integers equipped with a pairing function can be decidable or not. The theory of decoding of lists (under some natural condition) is always undecidable. We distinguish the notions encoding of ntuples and encoding of lists via some properties of decidabilityundecidability. At last, we prove it is possible in some structure to encode lists although neither addition nor multiplication are definable in this structure. Résumé On utilise couramment en informatique les nuplets et les listes sur un ensemble donné; nous étudions ces deux notions dans le cadre de la logique du premier ordre et pour l’ensemble des entiers naturels. Gödel a montré que les objets définis par un schéma de récurrence primitive sont définissables au premier ordre avec la relation d’ordre et le codage des listes, euxmêmes définissable avec l’addition et la multiplication; nous montrerons que ce codage peut également s’effectuer avec l’addition et un prédicat plus faible que la multiplication, à savoir la coprimarité. On montre aussi que les notions de nuplets et de listes se distinguent par des arguments de décidabilitéindécidabilité. La théorie des entiers munis d’une fonction de couplage peutêtreou non décidable. Par contre la théorie du décodage des listes, soumise à une certaine condition naturelle, est toujours indécidable. On montre enfin qu’il existe des structures dans lesquelles on peut coder les listes sans pour autant que l’addition (et donc l’ordre) et la multiplication ne soient définissables.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
On Truth, Knowledge, Strings, and Paradoxes
"... uction it is possible to define syntactic and semantical predicates about the language itself. In particular it is possible to define the syntax of the language within the language itself. Details of such a construction can be found in [GN87, Chapter 10]. Although it is intended that strings and the ..."
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uction it is possible to define syntactic and semantical predicates about the language itself. In particular it is possible to define the syntax of the language within the language itself. Details of such a construction can be found in [GN87, Chapter 10]. Although it is intended that strings and the objects they stand for are closely related, we must carefully distinguish between them. There is an intuitive difference between long(John) and long("John") (where long is assumed to be a polymorphic predicate symbol that is true for persons over 195cm and strings consisting of at least eight characters.) In such a setting it is easy to express a problematic concept like the liar sentence: ffl This sentence is false. L :j :True("L") If we assume a standard firstorder semantics there is no problem with the sentence above, since strings like "L" are ordinary ground terms in the language and have nothing to do with their counterparts like L. However, as such they are not particularly usef
Lesniewski's Early Liar, Tarski and Natural Language
"... This paper is a contribution to the reconstruction of Tarski's semantic background and to the history of metalanguage and truth. Although in his 1933 monograph Tarski credits his master, Stanisl/aw Le#niewski, with crucial negative results on the semantics of natural language, the conceptual relatio ..."
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This paper is a contribution to the reconstruction of Tarski's semantic background and to the history of metalanguage and truth. Although in his 1933 monograph Tarski credits his master, Stanisl/aw Le#niewski, with crucial negative results on the semantics of natural language, the conceptual relationship between the two logicians has never been investigated in a thorough manner. This paper shows that it was not Tarski, but Le#niewski who first avowed the impossibility of giving a satisfactory theory of truth for ordinary language, and the necessity of sanitation of the latter for scientific purposes. In an early article (1913) Le#niewski gave an interesting solution to the Liar Paradox, which, although different from Tarski's in detail, is nevertheless important to Tarski's semantic background. To illustrate this I give an analysis of Le#niewski's solution and of some related aspects of Le#niewski's later thought.