3,909 research outputs found
An Alternative Natural Deduction for the Intuitionistic Propositional Logic
A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.This work is supported by the Ministary of Science and Technology of Serbia, grant number ON174026
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
Rasiowa–Harrop disjunction property
We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (IPC), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of IPC into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of IPC (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.info:eu-repo/semantics/publishedVersio
Characterisation of Strongly Normalising lambda-mu-Terms
We provide a characterisation of strongly normalising terms of the
lambda-mu-calculus by means of a type system that uses intersection and product
types. The presence of the latter and a restricted use of the type omega enable
us to represent the particular notion of continuation used in the literature
for the definition of semantics for the lambda-mu-calculus. This makes it
possible to lift the well-known characterisation property for
strongly-normalising lambda-terms - that uses intersection types - to the
lambda-mu-calculus. From this result an alternative proof of strong
normalisation for terms typeable in Parigot's propositional logical system
follows, by means of an interpretation of that system into ours.Comment: In Proceedings ITRS 2012, arXiv:1307.784
A Meta-Logic of Inference Rules: Syntax
This work was intended to be an attempt to introduce the meta-language for
working with multiple-conclusion inference rules that admit asserted
propositions along with the rejected propositions. The presence of rejected
propositions, and especially the presence of the rule of reverse substitution,
requires certain change the definition of structurality
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