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49
Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics)
, 1994
"... MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to ..."
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Cited by 178 (47 self)
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MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical, epistemological and implementational. From a technical point of view, we prove, among other things, that the set of theorems of the most common modal logics can be embedded (under the obvious bijective mapping between a modal and a first order language) into that of the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. This claim is motivated by the study of how ML systems can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. Finally, from an implementation point of view, we argue that ML systems resemble closely the current practice in the computer representation of propositional attitudes and metatheoretic theorem proving.
Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Cited by 32 (9 self)
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
FirstOrder Logic of Proofs
, 2011
"... The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capa ..."
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Cited by 20 (9 self)
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The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capable of realizing firstorder modal logic S4 and, therefore, the firstorder intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for firstorder S4 and HPC compliant with BrouwerHeytingKolmogorov requirements. FOLP opens the door to a general theory of firstorder justification.
A SetTheoretic Translation Method for Polymodal Logics
, 1995
"... The paper presents a settheoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we propos ..."
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Cited by 19 (12 self)
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The paper presents a settheoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we proposed applies to any normal complete finitelyaxiomatizable polymodal logic, regardless if it is firstorder complete or if an explicit semantics is available for it. Moreover, the finite axiomatizability of\Omega makes it possible to implement mechanical proof search procedures via the deduction theorem or more specialized and efficient techniques. In the last part of the paper, we briefly discuss the application of set T resolution to support automated derivability in (a suitable extension of) \Omega\Gamma This work has been supported by funds MURST 40% and 60%. The second author was supported by a grant from the Italian Consiglio Nazionale delle Ricerche (CNR). 1 Introduction The paper...
A Foundation for Metareasoning, Part I: The Proof Theory
, 1997
"... We propose a framework, called OM pairs, for the formalization of metareasoning. OM pairs allow us to generate deductively the object theory and/or the meta theory. This is done by imposing, via appropriate reflection rules, the relation we want to hold between the object theory and the meta theory. ..."
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Cited by 13 (5 self)
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We propose a framework, called OM pairs, for the formalization of metareasoning. OM pairs allow us to generate deductively the object theory and/or the meta theory. This is done by imposing, via appropriate reflection rules, the relation we want to hold between the object theory and the meta theory. In this paper we concentrate on the proof theory of OM pairs. We study them from three different points of view: we compare the strength of the object and meta theories generated by different OM pairs; for each OM pair we study the precise form of the object theory and meta theory; and, finally, we study three important case studies.
Using Reflection to Explain and Enhance Type Theory
 Proof and Computation, volume 139 of NATO Advanced Study Institute, International Summer School held in Marktoberdorf, Germany, July 20August 1, NATO Series F
, 1994
"... The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to ..."
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Cited by 11 (5 self)
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The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to safely add new decision procedures to the Nuprl prover. It relies on a reflection mechanism and is applicable to any tacticoriented prover with sufficient reflection. The lectures explain reflection in the setting of constructive type theory, the core logic of Nuprl.
SelfReferential Justifications in Epistemic Logic
, 2009
"... This paper is devoted to the study of selfreferential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such selfreferential justifications are present in the reasoning described by standard modal epistemic logics suc ..."
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Cited by 10 (6 self)
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This paper is devoted to the study of selfreferential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such selfreferential justifications are present in the reasoning described by standard modal epistemic logics such as S4. We argue that the modal language by itself is too coarse to capture this concept of selfreferentiality and that the language of justification logic can serve as an adequate refinement. We consider wellknown modal logics of knowledge/belief and show, using explicit justifications, that S4, D4, K4, and T with their respective justification counterparts LP, JD4, J4, and JT describe knowledge that is selfreferential in some strong sense. We also demonstrate that selfreferentiality can be avoided for K and D. In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspection.
Staged Computation with Names and Necessity
, 2005
"... Staging is a programming technique for dividing the computation in order to exploit the early availability of some arguments. In the early stages the program uses the available arguments to generate, at run time, the code for the late stages. The late stages may then be explicitly evaluated when app ..."
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Cited by 9 (1 self)
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Staging is a programming technique for dividing the computation in order to exploit the early availability of some arguments. In the early stages the program uses the available arguments to generate, at run time, the code for the late stages. The late stages may then be explicitly evaluated when appropriate. A type system for staging should ensure that only welltyped expressions are generated, and that only expressions with no free variables are permitted for evaluation.
Referential logic of proofs
 Theoretical Computer Science
"... We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness ..."
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Cited by 7 (0 self)
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We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness and completeness with respect to the intended provability semantics. We show that FLPref provides a complete test of admissibility of inference rules in a sound extension of arithmetic. Key words: proof theory, explicit modal logic, single conclusion logic of proofs, proof term, reference, unification, admissible inference rule. 1