86,604 research outputs found
Hyper Natural Deduction
Paper introduces a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system, by adding additional rules providing means for communication between derivations. It is shown that the Hyper Natural Deduction system is sound and complete for infinite-valued propositional GoÌdel Logic, by giving translations to and from Avron's Hyper sequent Calculus. The paper also provides conversions for normalisation and prove the existence of normal forms for the Hyper Natural Deduction system
Structural rules in natural deduction with alternatives
Natural deduction with alternatives extends GentzenâPrawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proofsystem in which the connective rules are unmodified from the usual Prawitz introduction and elimination rules â the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules. The key result of this paper is that the two principles that introduce kindsofirrelevanceto natural deduction proofs: (a) the rule of explosion (from acontradiction, anything follows); and (b) the structural rule of vacuous discharge;are shown to be two sides of a single coin, in the same way that they correspond tothe structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.Publisher PDFPeer reviewe
Hyper Natural Deduction for Gödel LogicâA natural deduction system for parallel reasoning
We introduce a system of Hyper Natural Deduction for GoÌdel Logic as an extension of Gentzenâs system of Natural Deduction. A deduction in this system consists of a finite set of derivations which uses the typical rules of Natural Deduction, plus additional rules providing means for communication between derivations. We show that our system is sound and complete for infinite-valued propositional GoÌdel Logic, by giving translations to and from Avronâs Hypersequent Calculus. We provide conversions for normalization extending usual conversions for Natural Deduction and prove the existence of normal forms for Hyper Natural Deduction for GoÌdel Logic. We show that normal deductions satisfy the subformula property
Relevant Connexive Logic
In this paper, a connexive extension of the Relevance logic Râ was presented. It is defined by means of a natural deduction system, and a deductively equivalent axiomatic system is presented too. The goal of such an extension is to produce a logic with stronger connection between the antecedent and the consequent of an implication
Definitions by Rewriting in the Calculus of Constructions
The main novelty of this paper is to consider an extension of the Calculus of
Constructions where predicates can be defined with a general form of rewrite
rules. We prove the strong normalization of the reduction relation generated by
the beta-rule and the user-defined rules under some general syntactic
conditions including confluence. As examples, we show that two important
systems satisfy these conditions: a sub-system of the Calculus of Inductive
Constructions which is the basis of the proof assistant Coq, and the Natural
Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award
A New Type Assignment for Strongly Normalizable Terms
We consider an operator definable in the intuitionistic theory of monadic predicates and we axiomatize some of its properties in a definitional extension of that monadic logic. The axiomatization lends itself to a natural deduction formulation to which the Curry-Howard isomorphism can be applied. The resulting Church style type system has the property that an untyped term is typable if and only if it is strongly normalizable
Normalisation for Some Quite Interesting Many-Valued Logics
In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3â. Also, we present a detailed version of Prawitzâs proof of Nelsonâs logic N4 and its extension by intuitionist negation
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