2,030 research outputs found
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
Proof-graphs for Minimal Implicational Logic
It is well-known that the size of propositional classical proofs can be huge.
Proof theoretical studies discovered exponential gaps between normal or cut
free proofs and their respective non-normal proofs. The aim of this work is to
study how to reduce the weight of propositional deductions. We present the
formalism of proof-graphs for purely implicational logic, which are graphs of a
specific shape that are intended to capture the logical structure of a
deduction. The advantage of this formalism is that formulas can be shared in
the reduced proof.
In the present paper we give a precise definition of proof-graphs for the
minimal implicational logic, together with a normalization procedure for these
proof-graphs. In contrast to standard tree-like formalisms, our normalization
does not increase the number of nodes, when applied to the corresponding
minimal proof-graph representations.Comment: In Proceedings DCM 2013, arXiv:1403.768
Strong normalization of lambda-Sym-Prop- and lambda-bar-mu-mu-tilde-star- calculi
In this paper we give an arithmetical proof of the strong normalization of
lambda-Sym-Prop of Berardi and Barbanera [1], which can be considered as a
formulae-as-types translation of classical propositional logic in natural
deduction style. Then we give a translation between the
lambda-Sym-Prop-calculus and the lambda-bar-mu-mu-tilde-star-calculus, which is
the implicational part of the lambda-bar-mu-mu-tilde-calculus invented by
Curien and Herbelin [3] extended with negation. In this paper we adapt the
method of David and Nour [4] for proving strong normalization. The novelty in
our proof is the notion of zoom-in sequences of redexes, which leads us
directly to the proof of the main theorem
The First-Order Hypothetical Logic of Proofs
The Propositional Logic of Proofs (LP) is a modal logic in which the modality â–ˇA is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de ComputaciĂłn; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y TecnologĂa; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentin
Natural deduction systems for some non-commutative logics
Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
- …