Towards an Efficient Prover for the C1 Paraconsistent Logic (original) (raw)

A reasoning method for a paraconsistent logic

Studia Logica - An International Journal for Symbolic Logic, 1993

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational efficiency. A prototype based on this second system was effectively implemented.

Analytical Tableaux for da Costa's Hierarchy of Paraconsistent Logics

Electronic Notes in Theoretical Computer Science, 2006

In this paper we present a new hierarchy of analytical tableaux systems TNDCn, 1 ≤ n < ω, for da Costa's hierarchy of propositional paraconsistent logics Cn, 1 ≤ n < ω. In our tableaux formulation, we introduce da Costa's "ball" operator "•", the generalized operators "k" and "(k)", and the negations "∼ k ", for k ≥ 1, as primitive operators, differently to what has been done in the literature, where these operators are usually defined operators. We prove a version of Cut Rule for the TNDCn, 1 ≤ n < ω, and also prove that these systems are logically equivalent to the corresponding systems Cn, 1 ≤ n < ω. The systems TNDCn constitute completely automated theorem proving systems for the systems of da Costa's hierarchy Cn, 1 ≤ n < ω. 3

Matrix proof method in annotated paraconsistent logic

1995

The matrix connection method (MCM) is an alternative procedure for theorem proving than the usual resolution technique. We already have used the MCM for finding models in a real-time knowledge-based system generator. In this paper, we adapt the MCM to the particular case of sorne annotated propositional paraconsistent logics. Further developments related to these ideas are also outlined.

Paraconsistent Logic: A Proof-Theoretical Approach*

2006

A logic is paraconsistent if it allows for non-trivial inconsistent theories. Given the usual definition of inconsistency, the notion of paraconsistent logic seems to rely upon the interpretation of the sign ‘¬’. As paraconsistent logic challenges properties of negation taken to be basic in other contexts, it is disputable that an operator lacking those properties will count as real negation. The conclusion is that there cannot be truly paraconsistent logics. This objection can be met from a substructural perspective since paraconsistent sequent calculi can be built with the same operational rules as classical logic but with slightly different structural rules.

How to build your own paraconsistent logic: an introduction to the Logics of Formal (In) Consistency

J. Marcos, D. Batens, and WA Carnielli, organizers, …

The logics of formal inconsistency (LFIs) are logics that allow to explicitly formalize the concepts of consistency and inconsistency by means of formulas of their language. Contradictoriness, on the other hand, can always be expressed in any logic, provided its la nguage includes a symbol for negation. Besides being able to represent the distinction between contradiction and inconsistency, LFIs are non-explosive logics, in the sense that a contradiction does not entail arbitrary statements, but yet are gently explosive, in the sense that, adjoining the additional requirement of consistency, then contradictoriness does cause explosion. Several logics can be seen as LFIs, among them the great majority of paraconsistent logics developed under the Brazilian tradition, as well as the sytems developed under the Polish tradition. We present here their semantical interpretations by way of possible-translations semantics, stressing their significance and applications to human reasoning and machine reasoning. We also give tableaux systems for some important LFIs: bC, Ci and LFI1.

Tableaux for First Order Logic of Proofs

arXiv: Logic, 2016

In this paper we present a tableau proof system for first order logic of proofs FOLP. We show that the tableau system is sound and complete with respect to Mkrtychev models of FOLP.

Herod and Pilate: Two Tableau Provers for Basic Hybrid Logic

Lecture Notes in Computer Science, 2010

This work presents two provers for basic hybrid logic HL(@), which have been implemented with the aim of comparing the internalised tableau calculi independently proposed, respectively, by Bolander and Blackburn [3] and Cerrito and Cialdea Mayer [5]. Experimental results are reported, evaluating, from the practical point of view, the different treatment of nominal equalities of the two calculi.

A Generalisation of a Refutation-related Method in Paraconsistent Logics

Logic and Logical Philosophy, 2018

This article describes a refutation method of proving the maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experiment that confirms the theoretical results.

Analytical tableaux for da Costa ’ s paraconsistent logics 1

2008

In 1963 da Costa (see da Costa (1974)) introduces his hierarchies of logical calculi for the study of inconsistent but non-trivial theories: the hierarchy of propositional calculi Cn, 1≤n≤ω, the hierarchy of predicate calculi Cn*, 1≤n≤ω, the hierarchy of predicate calculi with equality Cn, 1≤n≤ω, and the hierarchy of calculi of descriptions Dn, 1≤n≤ω. Marconi (1980) introduces a variant of semantical tableaux systems, à la Beth (see Beth (1959)), in order to prove the completeness and decidability of da Costa’s propositional system C1. He also claims that his method can be expanded for the systems Cn, 2≤n<ω. The system introduced by Marconi is based on the same intuitions underlying the definition of quasi-matrices introduced by da Costa and Alves (1976), simplifying the verification process of validity of the formulae. In Marconi’s tableaux system the rules for the connectives &, ∨ and ⊃ are the standard ones, and two special rules are added to operate with the paraconsistent ne...