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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 219 (28 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
Reasoning with Individuals in Concept Languages
 Data and Knowledge Engineering
, 1994
"... One of the main characteristics of knowledge representation systems based on the description of concepts is the clear distinction between terminological and assertional knowledge. Although this characteristic leads to several computational and representational advantages, it usually limits the expre ..."
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Cited by 87 (2 self)
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One of the main characteristics of knowledge representation systems based on the description of concepts is the clear distinction between terminological and assertional knowledge. Although this characteristic leads to several computational and representational advantages, it usually limits the expressive power of the system. For this reason, some attempts have been done, allowing for a limited form of amalgamation between the two components and a more complex interaction between them. In particular, one of these attempts is based on letting the individuals to be referenced in the concept expressions. This is generally performed by admitting a constructor for building a concept from a set of enumerated individuals. In this paper we investigate on the consequences of introducing constructors of this type in the concept description language. We also provide a complete reasoning procedure to deal with these constructors and we obtain some complexity results on it. 1 Introduction The ide...
Logical Systems for Structured Specifications
, 2000
"... We study proof systems for reasoning about logical consequences and refinement of structured specifications, based on similar systems proposed earlier in the literature [ST 88, Wir 91]. Following Goguen and Burstall, the notion of an underlying logical system over which we build specifications is fo ..."
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Cited by 45 (1 self)
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We study proof systems for reasoning about logical consequences and refinement of structured specifications, based on similar systems proposed earlier in the literature [ST 88, Wir 91]. Following Goguen and Burstall, the notion of an underlying logical system over which we build specifications is formalized as an institution and extended to a more general notion, called (D, T )institution. We show that under simple assumptions (essentially: amalgamation and interpolation) the proposed proof systems are sound and complete. The completeness proofs are inspired by proofs due to M. V. Cengarle (see [Cen 94]) for specifications in firstorder logic and the logical systems for reasoning about them. We then propose a methodology for reusing proof systems built over institutions rich enough to satisfy the properties required for the completeness results for specifications built over poorer institutions where these properties need not hold.
A General Mathematics of Names
 Information and Computation
, 2007
"... We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying sy ..."
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Cited by 20 (15 self)
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We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying syntax — have a relation generalising to all sets and not only sets of syntax trees. We also give syntaxfree accounts of Barendregt representatives, scope extrusion, and other phenomena associated to αequivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.
On the representability of actions in a semiabelian category
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... of actions of the object G on the object X, in the sense of the theory of semidirect products in V. We investigate the representability of the functor Act(−,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of ..."
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Cited by 18 (1 self)
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of actions of the object G on the object X, in the sense of the theory of semidirect products in V. We investigate the representability of the functor Act(−,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semiabelian theory in a Grothendieck topos, thus in particular all semiabelian varieties of universal algebra. For such categories, we prove first that the representability of Act(−,X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(−,X) in a general semiabelian category. Finally, we exhibit the precise form of the more involved “if and only if ” amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of “normalization of a morphism”. We provide also a wide supply of algebraic examples and counterexamples, giving in particular evidence of the relevance of the object representing Act(−,X), when it turns out to exist. 1. Actions and split exact sequences A semiabelian category is a Barrexact, Bournprotomodular, finitely complete and finitely cocomplete category with a zero object 0. The existence of finite limits and a zero object implies that Bournprotomodularity is equivalent to, and so can be replaced with, the following split version of the short five lemma: K K given a commutative diagram of “kernels of split epimorphisms” qisi =1Q, ki = Ker qi, i =1, 2 k1 k2
Proof Systems for Structured Specifications and Their Refinements
, 1999
"... Reasoning about specifications is one of the fundamental activities in the process of formal program development. This ranges from proving the consequences of a specification, during the prototyping or testing phase for a requirements speci cation, to proving the correctness of refinements (or imple ..."
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Cited by 13 (6 self)
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Reasoning about specifications is one of the fundamental activities in the process of formal program development. This ranges from proving the consequences of a specification, during the prototyping or testing phase for a requirements speci cation, to proving the correctness of refinements (or implementations) of specifications. The main proof techniques for algebraic specifications have their origin in equational Horn logic and term rewriting. These proof methods have been well studied in the case of nonstructured speci cations (see Chapters 9 and 10). For large systems of specifications built using the structuring operators of speci cation languages, relatively few proof techniques have been developed yet; for such proof systems, see [SB83, HST94, Wir91, Far92, Cen94, HWB97]. In this chapter we focus on proof systems designed particularly for modular specifications. The aim is to concentrate on the structuring concepts, while abstracting as much as possible from the par...
A Universal Scale of Comparison
 Linguistics and Philosophy
, 2008
"... Abstract. Comparative constructions form two classes, those that permit direct comparisons (comparisons of measurements as in Seymour is taller than he is wide) and those that only allow indirect comparisons (comparisons of relative positions on separate scales as in Esme is more beautiful than Eins ..."
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Cited by 11 (0 self)
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Abstract. Comparative constructions form two classes, those that permit direct comparisons (comparisons of measurements as in Seymour is taller than he is wide) and those that only allow indirect comparisons (comparisons of relative positions on separate scales as in Esme is more beautiful than Einstein is intelligent). In contrast with other semantic theories, this paper proposes that the interpretation of the comparative morpheme remains the same whether it appears in sentences that compare individuals directly or indirectly. To develop a unified account, I suggest that all comparisons (wether in terms of height, intelligence or beauty) involve a scale of universal degrees that are isomorphic to the rational (fractional) numbers between 0 and 1. Crucial to a unified treatment, the connection between the individuals being compared and universal degrees involves two steps. First individuals are mapped to a value on a primary scale that ranks individuals with respect to the gradable property (whether it be height, beauty or intelligence). Second, the value on the primary scale is mapped to a universal degree that encodes the value’s relative position on the primary scale. Direct comparison results if measurements such as seven feet participate in the primary scale (as in Seven feet is tall). Otherwise the result is an indirect comparison.
Dynamic Belief Analysis
 Intelligent Agents III  Proceedings of the Third International Workshop on Agent Theories, Architectures, and Languages (ATAL96), Lecture Notes in Artificial Intelligence
, 1997
"... The process of rational inquiry can be defined as the evolution of the beliefs of a rational agent as a consequence of its internal inference procedures and its interaction with the environment. These beliefs can be modelled in a formal way using belief logics. The possible worlds model and its asso ..."
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Cited by 7 (4 self)
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The process of rational inquiry can be defined as the evolution of the beliefs of a rational agent as a consequence of its internal inference procedures and its interaction with the environment. These beliefs can be modelled in a formal way using belief logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they commit us to model agents that are logically omniscient and perfect reasoners. These problems can be avoided with a syntactic view of possible worlds, defining them as arbitrary sets of sentences in a propositional belief logic. In this article this syntactic view of possible worlds is taken, and a dynamic analysis of the beliefs of the agent is suggested in order to model the process of rational inquiry in which the agent is permanently engaged. 1 INTRODUCTION The aim of this work 1 is to model the process of rational inquiry, i.e. the (rationally controlled) transformation of the beliefs of an intelligent...
Fresh logic: Prooftheory and semantics for FM and nominal . . .
, 2005
"... In this paper we introduce Fresh Logic, a natural deduction style firstorder logic extended with termformers and quantifiers derived from the FMsets model of names and binding in abstract syntax. Fresh Logic can be classical or intuitionistic depending on whether we include a law of excluded mi ..."
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Cited by 5 (0 self)
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In this paper we introduce Fresh Logic, a natural deduction style firstorder logic extended with termformers and quantifiers derived from the FMsets model of names and binding in abstract syntax. Fresh Logic can be classical or intuitionistic depending on whether we include a law of excluded middle; we present a proofnormalisation procedure for the intuitionistic case and a semantics based on Kripke models in FMsets for which it is sound and
Identity, Indiscernibility, and Philosophical Claims
, 2002
"... The standard ways classical logic and mathematics deal with the concept of indiscernibility (indistinguishability), with special emphasis to the concept of indiscernibility in a structure are considered. ..."
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Cited by 4 (1 self)
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The standard ways classical logic and mathematics deal with the concept of indiscernibility (indistinguishability), with special emphasis to the concept of indiscernibility in a structure are considered.