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A Cube of Proof Systems for the Intuitionistic Predicate mu,nuLogic
 Dept. of Informatics, Univ. of Oslo
, 1997
"... This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight pr ..."
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Cited by 6 (5 self)
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This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight prooftheoretically interesting naturaldeduction calculi for this logic and propose a classification of these into a cube on the basis of the embeddibility relationships between these. 1 Introduction ¯,logics, i.e. logics with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers, have turned out to be a useful formalism in a number of computer science areas. The classical 1storder predicate ¯,logic can been used as a logic of (nondeterministic) imperative programs and as a database query language. It is also one of the relation description languages studied in descriptive complexity theory (finite model theory) (for a survey on this hi...
Tool Support for Logics of Programs
 Mathematical Methods in Program Development: Summer School Marktoberdorf 1996, NATO ASI Series F
, 1996
"... Proof tools must be well designed if they... ..."
Validity concepts in prooftheoretic semantics
 ProofTheoretic Semantics. Special issue of Synthese
"... Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in det ..."
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Cited by 5 (4 self)
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Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in detail various notions of prooftheoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. 1. Introduction: Prooftheoretic
Structured induction proofs in Isabelle/Isar
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM 2006), LNAI
, 2006
"... Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support ..."
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Cited by 4 (1 self)
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Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out wellstructured and readable. Our techniques can be applied to nonstandard variants of induction as well, such as coinduction and nominal induction. This demonstrates that Isar provides a viable platform for building domainspecific tools that support fullyformal mathematical proof composition.
Focusing in linear metalogic: Extended report. Available from http://hal.inria.fr/inria00281631
, 2008
"... Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for suc ..."
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Cited by 4 (4 self)
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Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and nonnormal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics. 1
Hypothetical reasoning and definitional reflection in logic programming
 Extensions of Logic Programming
, 1990
"... This paper describes the logical and philosophical background of an extension of logic programming which uses a general schema for introducing assumptions and thus presents a new view of hypothetical reasoning. The detailed proof theory of this system is given in [7], matters of implementation and c ..."
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Cited by 3 (2 self)
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This paper describes the logical and philosophical background of an extension of logic programming which uses a general schema for introducing assumptions and thus presents a new view of hypothetical reasoning. The detailed proof theory of this system is given in [7], matters of implementation and control of the corresponding programming language GCLA with detailed examples can be found in [1, 2]. In Section 1 we consider the local rulebased approach to a notion of atomic consequence as opposed to the global logical approach. Section 2 describes our system and characterises the inference schema of definitional reflection which is central for our approach. In Section 3 we motivate the computational interpretation of this system. Finally, Section 4 relates our approach to the idea of logical frameworks and the way elimination inferences for logical constants are treated therein, and thus to the notions of logic and structure. It shows that from a certain perspective, logical reasoning is nothing but a special case of reasoning in our system. 1 Local and global consequence
Implementing Modal and Relevance Logics in a Logical Framework
, 1996
"... We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, ..."
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Cited by 2 (2 self)
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We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework. 1 INTRODUCTION The origins of natural deduction (ND) are both philosophical and practical. In philosophy, it arises from an analysis of deductive inference in an attempt to provide a theory of meaning for the logical connectives [24, 33]. Practically, it provides a language for building proofs, which can be seen as providing the deduction theorem directly, rather than as a derived result. Our interest is on this practical side, and a development of our work on ap...
2004).On the notion of assumption in logical systems
 In R
"... When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either ass ..."
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Cited by 2 (2 self)
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When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either assumptions or axioms, formulas of the sort (ii) are conclusions of (proper) inference rules. Axioms may be viewed as conclusions of (improper) inference rules, viz. inference rules without premisses. In what follows I refer to conclusions of proper or improper inference rules as assertions. 1 In natural deduction systems, inference rules deal both with assumptions and assertions, as the assumptions on which the conclusion of an inference rule depends, are not necessarily given by the collection of all assumptions on which the premisses depend, in case the rule permits the discharging of assumptions. For example, the rule of implication introduction
Generalized definitional reflection and the inversion principle
 Logica Universalis
, 2007
"... Abstract. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the inver ..."
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Abstract. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction,. It is closely related to definitional reflection, which is a principle for reasoning on the basis of rulebased atomic definitions, proposed by Hallnäs and SchroederHeister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ωprinciple, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself.