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A Brief History of Natural Deduction
 HISTORY AND PHILOSOPHY OF LOGIC
, 1999
"... Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaskowski in 1934. This article traces the development of natural dedu ..."
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Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaskowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks  the standard conduits of the method to a generation of philosophers  with an eye to determining what the `essential characteristics’ of natural deduction are.
Are Tableaux an Improvement on TruthTables? CutFree proofs and Bivalence
, 1992
"... We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutf ..."
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We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutfree proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableaulike method without affecting its "analytic" nature. 1 Introduction The truthtable method, introduced by Wittgenstein in his Tractatus LogicoPhilosophicus, provides a decision procedure for propositional logic which is immediately implementable on a machine. However this timehonoured method is usually mentioned only to be immediately dismissed because of its incurable inefficiency. The wellknown tableau method (which is closely related to Gentzen's cutfree sequent calculus) is commonly regarded as a "shortcut" in testing the logical validity of complex propositions...
Occurrences in Debugger Specifications
 In Proceedings of the ACM SIGPLAN'91 Conference on Programming Language Design and Implementation
, 1991
"... We describe formal manipulations of programming language semantics that permit execution animation for interpreters. We first study the use of occurrences in the calculus and we describe an implementation of the notion of residuals. We then describe applications in the development of interpreters f ..."
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We describe formal manipulations of programming language semantics that permit execution animation for interpreters. We first study the use of occurrences in the calculus and we describe an implementation of the notion of residuals. We then describe applications in the development of interpreters for the lazy calculus and the parallel language Occam. 1. Introduction Formal descriptions of programming language semantics have already been shown to yield executable specifications of interpreters for these languages [MiniML], [Esterel]. However, while the obtained interpreters have the clear advantage of being "certified" implementations, they lack a nice user interface for the very reason that the definition only deals with semantic values. An interpreter can be tranformed into a debugging tool by adding tracing, profiling, or control of execution functionalities. For us, an execution trace is a list of basic instruction calls that describes a history of execution, a profile is a list...
C.: Principles of Superdeduction
 In: Proc. of the 22nd Annual IEEE Symposium on Logic in Computer Science (LICS
, 2007
"... In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, Deduction modulo allows us to make use of the computational part of ..."
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In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, Deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focussing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system “superdeduction”. We introduce a proofterm language and a cutelimination procedure both based on Christian Urban’s work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently show how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants. We finally present lemuridæ, our current implementation of superdeduction modulo. 1
The Deduction Rule and Linear and Nearlinear Proof Simulations
"... ... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n). ..."
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... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n).
A Certified Compiler for an Imperative Language
, 1998
"... This paper describes the process of mechanically certifying a compiler with respect to the semantic specification of the source and target languages. The proofs are performed in type theory using the Coq system. These proofs introduce specific theoretical tools: fragmentation theorems and general in ..."
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This paper describes the process of mechanically certifying a compiler with respect to the semantic specification of the source and target languages. The proofs are performed in type theory using the Coq system. These proofs introduce specific theoretical tools: fragmentation theorems and general induction principles.
Semantic cut elimination in the intuitionistic sequent calculus
 Typed Lambda Calculi and Applications, number 3461 in Lectures
, 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to ..."
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Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.
Semantic Verification of Web Sites Using Natural Semantics
 PROCEEDINGS OF THE 6TH RIAO CONFERENCE  CONTENTBASED MULTIMEDIA INFORMATION ACCESS
, 2000
"... The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this pap ..."
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The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this paper we explore a different approach inherited from software engineering: specifying the semantics of Web sites, allowing semantic verifications that will help both the conception and the maintenance of Web sites. To achieve this goal, we have experimented with the application of Natural Semantics (traditionally used to specify the semantics of programming languages) to Web sites specification and verification.
On the Logic and Learning of Language
, 2002
"... algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of la ..."
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algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of languages . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 The algebra of formulae . . . . . . . . . . . . . . . . . . . 38 3.2.2 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Associated algebras . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.5 LindenbaumTarski quotient algebras . . . . . . . . . . . . 42 3.3 Algebras of deductive systems . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Determining a class of algebras . . . . . . . . . . . . . . . 45 3.3.2 Algebra of a sequent calculus . . . . . . . . . . . . . . . . . 46 3.3.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Subsuming special cases: an example . . . . . . . . . . . . . . . . 49 3.4.1 The sequent system GL . . . . . . . . . . . . . . . . . . . . 49 3.4.2 The equivalent system t(GL) . . . . . . . . . . . . . . . . . 51 3.4.3 Algebraic models for GL . . . . . . . . . . . . . . . . . . . 52 3.5 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Categorial type logics 61 4.1 The typed lambda calculus . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Categorial grammar . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Forms of Lambek's calculus . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Classical CG revisited . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 The nonassociative productfree system . . . . . . . . . . . 70 4.3.3 Addin...
Combining Derivations and Refutations for Cutfree Completeness in BiIntuitionistic Logic
, 2008
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1