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Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 183 (21 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
MSL: A Model for W3C XML Schema
, 2001
"... some of the core idea in XML Schema. The benefits of a formal description is that it is both concise and precise. MSL has already proved helpful in work on the design of XML Query. We expect that similar techniques can be used to extend MSL to include most or all of XML Schema. ..."
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Cited by 28 (1 self)
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some of the core idea in XML Schema. The benefits of a formal description is that it is both concise and precise. MSL has already proved helpful in work on the design of XML Query. We expect that similar techniques can be used to extend MSL to include most or all of XML Schema.
The GirardReynolds isomorphism
 Proc. of 4th Int. Symp. on Theoretical Aspects of Computer Science, TACS 2001
, 2001
"... Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented ..."
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Cited by 6 (1 self)
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Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented in F2. Reynolds additionally proved an abstraction theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. The Girard projection discards all firstorder quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embedding should also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions corresponding to inductive definitions of naturals, products, sums, and fixpoint types. 1
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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Cited by 5 (0 self)
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.
Presenting Proofs Using Logicographic Symbols
 In Proc. of the Workshop on Proof Transformation and Presentation and Proof Complexities (PTP01
, 2001
"... Abstract. Mathematics has a rich tradition in creating symbols and notation that is soundly integrated into the syntax of the underlying formal language and, at the same time, conveys the intuition behind the concepts described by the symbols and notation. Continuing this idea, in the Theorema syste ..."
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Cited by 4 (1 self)
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Abstract. Mathematics has a rich tradition in creating symbols and notation that is soundly integrated into the syntax of the underlying formal language and, at the same time, conveys the intuition behind the concepts described by the symbols and notation. Continuing this idea, in the Theorema system, with the new feature of logicographic symbols, we now provide a means to invent arbitrary new symbols and notation. In this paper we describe how logicographic symbols can be created, declared, and afterwards used as a part of the formal language of Theorema with an example. Also with logicographic symbols, formal proof text automatically generated by Theorema provers can become easy to read in a way that resembles telling a pictorial story about the mathematical concepts involved. 1
Predicative Fragments of Frege Arithmetic
, 2003
"... Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply al ..."
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Cited by 3 (0 self)
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Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1
Frege’s Judgement Stroke and the Conception of Logic as the Study of Inference not Consequence
"... One of the most striking differences between Frege’s Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted, that is, which are being used to express judgemen ..."
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Cited by 3 (0 self)
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One of the most striking differences between Frege’s Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted, that is, which are being used to express judgements. There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement, rather than—as in the typical contemporary view—of consequence relations amongst certain objects (propositions or wellformed formulae). The paper also explains why Frege’s use of the judgement stroke is not in conflict with his avowed antipsychologism, and why Wittgenstein’s criticism of the judgement stroke as “logically quite meaningless ” is unfounded. The key point here is that while the judgement stroke has no content, its use in logic and mathematics is subject to a very stringent norm of assertion. A notable feature of Frege’s logic is the presence therein of the judgement stroke—the vertical line ‘’: a symbol which marks those propositions which are being asserted. For Frege, assertion is the external act corresponding to the inner act of judgement: “we distinguish: (1) the grasp of a thought—thinking, (2) the acknowledgement of the truth of a thought—the act of judgement, (3) the manifestation of this judgement—assertion ” [Frege 1918–26, pp. 355–6]. After the twodimensional graphical nature of his symbolism, the judgement stroke is, to our eyes, the next most striking thing about Frege’s logical system(s), 1 because standard current systems of logic employ no analogue of it: that is, they give us no way of asserting a proposition—of putting it forward as being true—as opposed to presenting or displaying a proposition so that its content may be considered. The judgement stroke was equally noteworthy both for Frege himself, and for his contemporary readers. In a review of Frege’s
The formal method known as B and a sketch for its implementation
, 2002
"... This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also conside ..."
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Cited by 2 (2 self)
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This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also considered: both its development and important points relevant to formal methods. Automated reasoning, particularly its theoretical limits as well as unification and resolution, is discussed. The main part of this thesis is a systematic reconstruction of the Bmethod, starting from its version of untyped predicate calculus and typed set theory, continuing with the Generalized Substitution Language (GSL) and finishing with the Abstract Machine Notation (AMN). Specification, refinement and implementation of a simple example using the Bmethod is presented. Both validation and verification of specifications, refinements and implementations using the Bmethod is discussed. The thesis concludes with a report of the current state of the effort (by the author) to implement the tool support of the Bmethod, as the Ebba Toolset. The main design decisions are discussed. The use of Unicode as the primary input encoding of AMN and GSL in Ebba is described.
Book Review: Building Large KnowledgeBased Systems: Representation and Inference in the CYC Project
 Representation and Inference in the Cyc Project (D.B. Lenat and R.V. Guha). Artificial Intelligence
, 1993
"... ical reasoning facility that would allow them, for example, to use information about treating one disease to help determine how to treat another. The Cyc project, under the leadership of Douglas Lenat and R. V. Guha, is an attempt to build an AI system that is neither narrow nor brittle. Lenat and G ..."
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Cited by 2 (1 self)
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ical reasoning facility that would allow them, for example, to use information about treating one disease to help determine how to treat another. The Cyc project, under the leadership of Douglas Lenat and R. V. Guha, is an attempt to build an AI system that is neither narrow nor brittle. Lenat and Guha believe that the key to Supported by a grant from the Powell foundation and by the National Science Foundation under Award No. IRI9110813. Both authors thank Devika Subramanian, Jan Wiebe and the reviewers for their comments on drafts of this review. The authors are listed alphabetically. y Much of this work was performed at the Department of Computer Science of the University of Toronto, where it was supported by the Institute for Robotics and Intelligent Systems and by an operating grant from Canada's Natural Science and Engineering Research Council. 1 The discussion is based on the description appearing in the book; th