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20
The Proper Treatment of Events
, 2004
"... this paper was developed by Murray Shanahan (building upon earlier work by Kowalski and Sergot [64]) in a series of papers [102], [101], [103] and [100]. Shanahan's discussion of the frame problem and his proposed solution can also be found in the book [104] ..."
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Cited by 33 (5 self)
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this paper was developed by Murray Shanahan (building upon earlier work by Kowalski and Sergot [64]) in a series of papers [102], [101], [103] and [100]. Shanahan's discussion of the frame problem and his proposed solution can also be found in the book [104]
The Strength of Some MartinLöf Type Theories
 Arch. Math. Logic
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely ..."
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Cited by 25 (5 self)
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One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLof intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. 0 Introduction MartinLof's intuitionistic theory of types was originally introduce...
Logical Patterns in Space
 University of Amsterdam
, 1999
"... In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra ..."
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Cited by 17 (5 self)
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In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra iss'e style games between patterns in space. Finally, we consider spatial languages of increased logical power in the direction of geometry. Also, Intelligent Sensory Information Systems, University of Amsterdam 1 Contents 1 Reasoning about Space 3 2 Topological Structure: a Modal Approach 4 2.1 The topological view of space . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Special properties of topological spaces . . . . . . . . . . . 6 2.1.3 Structure preserving mappings . . . . . . . . . . . . . . . 7 3 Basic Modal Logic of Space 8 3.1 Topological language and semantics . . . . . . . . . . . . . . . . 8 3.2 Topologi...
Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics
 University of Utrecht, Dept of Philosophy
, 1992
"... At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intui ..."
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Cited by 16 (2 self)
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At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader. 1.
Un Calcul De Constructions Infinies Et Son Application A La Verification De Systemes Communicants
, 1996
"... m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to ..."
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Cited by 16 (0 self)
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m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w
The Strength of Some MartinLöf Type Theories
 ARCHIVE FOR MATHEMATICAL LOGIC
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely th ..."
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Cited by 14 (10 self)
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One objective of this paper is the determination of the prooftheoretic strength of MartinLöf's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLöf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.
Logic, context and valid inference. Or: Can there be a logic of law? Legal Knowledge Based Systems
 JURIX 1999: The Twelfth Conference
, 1999
"... The question is addressed whether it makes sense to speak of a logic of law. It is shown that what counts as valid inference depends to a large extent on contextdependent choices. This suggests that our question has a simple answer, namely that a logic of law can exist. After noticing that one logi ..."
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Cited by 8 (4 self)
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The question is addressed whether it makes sense to speak of a logic of law. It is shown that what counts as valid inference depends to a large extent on contextdependent choices. This suggests that our question has a simple answer, namely that a logic of law can exist. After noticing that one logic can serve as the background of another, it is explicated that a more subtle answer can be given. On the one hand a logic of law can exist, and on the other hand it can be possible to reduce such a logic to a set of legal premises in a more abstract logic. It is posited how a ‘contextual logic ’ approach and an ‘abstract logic ’ approach can lead to different priorities in the formalization of legal reasoning.
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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Cited by 8 (3 self)
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Cited by 6 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 5 (2 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1