Towards an efficient prover for the C1 paraconsistent logic

The KE inference system is a tableau method developed by Marco Mondadori which was presented as an improvement, in the computational efficiency sense, over Analytic Tableaux. In the literature, there is no description of a theorem prover based on the KE method for the C1 paraconsistent logic. Paraconsistent logics have several applications, such as in robot control and medicine. These applications could benefit from the existence of such a prover. We present a sound and complete KE system for C1, an informal specification of a strategy for the C1 prover as well as problem families that can be used to evaluate provers for C1. The C1 KE system and the strategy described in this paper will be used to implement a KE based prover for C1, which will be useful for those who study and apply paraconsistent logics.Comment: 16 page

Refutation Systems : An Overview and Some Applications to Philosophical Logics

Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics

Investigations in Belnap's Logic of Inconsistent and Unknown Information

Nuel Belnap schlug 1977 eine vierwertige Logik vor, die -- im Gegensatz zur klassischen Logik -- die Faehigkeit haben sollte, sowohl mit widerspruechlicher als auch mit fehlender Information umzugehen. Diese Logik hat jedoch den Nachteil, dass sie Saetze der Form 'wenn ..., dann ...' nicht ausdruecken kann. Ausgehend von dieser Beobachtung analysieren wir die beiden nichtklassischen Aspekte, Widerspruechlichkeit und fehlende Information, indem wir eine dreiwertige Logik entwickeln, die mit widerspruechlicher Information umgehen kann und eine Modallogik, die mit fehlender Information umgehen kann. Beide Logiken sind nicht monoton. Wir untersuchen Eigenschaften, wie z.B. Kompaktheit, Entscheidbarkeit, Deduktionstheoreme und Berechnungkomplexitaet dieser Logiken. Es stellt sich heraus, dass die dreiwertige Logik, nicht kompakt und ihre Folgerungsmenge im Allgemeinen nicht rekursiv aufzaehlbar ist. Beschraenkt man sich hingegen auf endliche Formelmengen, so ist die Folgerungsmenge rekursiv entscheidbar, liegt in der Klasse $\Sigma_2^P$ der polynomiellen Zeithierarchie und ist DIFFP-schwer. Wir geben ein auf semantischen Tableaux basierendes, korrektes und vollstaendiges Berechnungsverfahren fuer endliche Praemissenmengen an. Darueberhinaus untersuchen wir Abschwaechungen der Kompaktheitseigenschaft. Die nichtmonotone auf S5-Modellen basierende Modallogik stellt sich als nicht minder komplex heraus. Auch hier untersuchen wir eine sinnvolle Abschwaechung der Kompaktheitseigenschaft. Desweiteren studieren wir den Zusammenhang zu anderen nichtmonotonen Modallogiken wie Moores autoepistemischer Logik (AEL) und McDermotts NML-2. Wir zeigen, dass unsere Logik zwischen AEL und NML-2 liegt. Schliesslich koppeln wir die entworfene Modallogik mit der dreiwertigen Logik. Die dabei enstehende Logik MKT ist eine Erweiterung des nichtmonotonen Fragments von Belnaps Logik. Wir schliessen unsere Betrachtungen mit einem Vergleich von MKT und verschiedenen informationstheoretischen Logiken, wie z.B. Nelsons N und Heytings intuitionistischer Logik ab

Paraconsistency and analyticity

William Parry conceived in the early thirties a theory of entailment, the theory of analytic implication, intended to give a formal expression to the idea that the content of the conclusion of a valid argument must be included in the content of its premises. This paper introduces a system of analytic, paraconsistent and quasi-classical propositional logic that does not validate the paradoxes of Parry’s analytic implication. The interpretation of the expressions of this logic will be given in terms of a four-valued semantics, and its proof theory will be provided by a system of signed semantic tableaux that incorporates the techniques developed to improve the efficiency of the tableaux method for many-valued logics

Two-layered logics for probabilities and belief functions over Belnap--Dunn logic

This paper is an extended version of an earlier submission to WoLLIC 2023. We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic. We consider two probabilistic logics that present two perspectives on the probabilities in the Belnap--Dunn logic: $\pm$-probabilities and $\mathbf{4}$-probabilities. In the first case, every event $\phi$ has independent positive and negative measures that denote the likelihoods of $\phi$ and $\neg\phi$, respectively. In the second case, the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $\phi$. In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions. They equip events with two measures (positive and negative) with their main difference being whether the negative measure of $\phi$ is defined as the \emph{belief in $\neg\phi$} or treated independently as \emph{the plausibility of $\neg\phi$}. We provide a sound and complete Hilbert-style axiomatisation of the logic of $\mathbf{4}$-probabilities and establish faithful translations between it and the logic of $\pm$-probabilities. We also show that the satisfiability problem in all the logics is $\mathsf{NP}$-complete.Comment: arXiv admin note: text overlap with arXiv:2303.0456

From Inconsistency to Incompatibility

The aim of this article is to generalize logics of formal inconsistency (LFIs) to systems dealing with the concept of incompatibility, expressed by means of a binary connective. The basic idea is that having two incompatible formulas to hold trivializes a deduction, and as a special case, a formula becomes consistent (in the sense of LFIs) when it is incompatible with its own negation. We show how this notion extends that of consistency in a non-trivial way, presenting conservative translations for many simple LFIs into some of the most basic logics of incompatibility, thereby evidencing in a precise way how the notion of incompatibility generalizes that of consistency. We provide semantics for the new logics, as well as decision procedures, based on restricted non-deterministic matrices. The use of non-deterministic semantics with restrictions is justified by the fact that, as proved here, these systems are not algebraizable according to Blok-Pigozzi nor are they characterizable by finite Nmatrices. Finally, we briefly compare our logics to other systems focused on treating incompatibility, specially those pioneered by Brandom and further developed by Peregrin

Two-layered logics for paraconsistent probabilities

We discuss two two-layered logics formalising reasoning with paraconsistent probabilities that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic