Dual Erotetic Calculi and the Minimal $${\mathsf{LFI}}$$ LFI

Abstract

Abstract.An erotetic calculus for a given logic constitutes a sequent-style prooftheoretical formalization of the logic grounded in Inferential Erotetic Logic (IEL). In this paper, a new erotetic calculus for Classical Propositional Logic (CPL), dual with respect to the existing ones, is given. We modify the calculus to obtain complete proof systems for the propositional part of paraconsistent logic CLuN and its extensions CLuNs and mbC. The method is based on dual resolution. Moreover, the resolution rule is non-clausal. According to the authors knowledge, this is the first account of resolution for mbC. Last but not least, as the method is grounded in IEL, it constitutes an important tool for the so-called question-processing.

Rasiowa–Sikorski Deduction Systems with the Rule of Cut: A Case Study

“…We are interested in (a) relations between different deduction systems, (b) duality of proof-procedures, (c) relationship between efficiency of proof-procedures and the rule of cut. The choice of logics was motivated by our previous research presented in [14], but also by the fact that the logics we have chosen allow for a neat generalisation-by the use of an extended uniform notation the various logics are characterized by only four schemas of rules. What is more, the general unifying treatment extends to the proofs of soundness and completeness.…”

“…In [14,40] the authors have considered the propositional fragments of logics CLuN and CLuNs expressed, however, in a linguistic extension of the original logics, containing both paraconsistent and classical negation. We shall follow this approach in this paper.…”

“…We shall follow this approach in this paper. As in [14,40], for the sake of simplicity, we will use the names CLuN and CLuNs for the propositional logics considered here.…”

“…The idea is to think of '∼ A' as a disjunction of a classically negated formula '¬A' and a semantically atomic expression 'χ ∼ A', where 'χ' is introduced to the language in order to express syntactically the fact that a paraconsistently negated formula '∼ A' can get a direct assignment of a logical value, and thus can be true even though A is also true. The idea comes from [5], and have been used successfully in [40] and later in [14], where the consistency operator '•' is treated in a similar manner. 2 The uniform notation is introduced for the richer language with 'χ'.…”

“…The Logics of Formal Inconsistency have various proof-theoretical descriptions-there is a tableau method for mbC [10], the KE tableau method, which is also implemented [29], and there is also a sequent system for mbC [15]. In [14] the authors have also introduced resolution systems for the three logics, which are also grounded in Inferential Erotetic Logic (see Section 6 for more information concerning the logic of questions and its connection with the proof methods analysed in [14] and in this paper). We do not know, however, if there is any description of CLuN and CLuNs in terms of KE-like tableaux.…”