Minimal inconsistency-tolerant logics: a quantitative approach

The Logics of Formal Inconsistency (LFIs) are paraconsistent logics which internalize the notions of consistency and inconsistency by means of connectives. Based on that idea, in this paper we propose two deontic systems in which contradictory obligations are allowed, without trivializing the system. Thus, from conflicting obligations Oϕ and O¬ϕ contained in (or derived from) an information set, it can be derived that the sentence ϕ is deontically inconsistent. This avoids the logic collapse, and, on the other hand, this allows to "repair" or to refine the given information set. This approach can be used, for instance, for analyzing paradoxes based on contrary-toduty obligations. KEYWORDS: Logics of formal inconsistency, deontic logic, contrary-to-duty obligations.

In some cases one is provided with inconsistent information and has to reason about various consistent scenarios contained in that information. Our goal is to argue that filtered paraconsistent logics are not the right tool to handle such cases and that the problems generalize to a large class of paraconsistent logics. A wide class of paraconsistent (inconsistency-tolerant) logics is obtained by filtration: adding conditions on the classical consequence operation (one example is weak Rescher-Manor consequence --- which bears $\Gamma$ to $\phi$ just in case $\phi$ follows classicaly from at least one maximally consistent subset of $\Gamma$). We start with surveying the most promising candidates and comparing their strength. Then we discuss the mainstream views on how non-classical logics should be chosen for an application and argue that none of these allows us to chose any of the filtered logics for action-guiding reasoning with inconsistent information, roughly because such a reasoning has to start with selecting possible scenarios and such a process does not correspond to any of the mathematical models offered by filtered paraconsistent logics. Finally, we criticize a recent attempt to defend explorative hypothetical reasoning by means of weak Rescher-Manor consequence operation by Meheus et al.

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.

Should the presence of contradictions make it impossible to derive anything sensible from a theory or a logic where such contradictions appear, as the classical logician would maintain? Or are there maybe situations in which contradictions are at least temporarily admissible, if only their wild behavior can somehow be controlled? The theoretical and practical relevance of such questions shows paraconsistency to be a bold programme in the foundations of formal sciences. As time goes by, the problems and methods of formal logic, traditionally connected to mathematics and philosophy, can more and more be seen to affect and influence several other areas of knowledge, such as computer science, information systems, formal philosophy, theoretical linguistics, and so forth. In such areas, certainly more than in mathematics, contradictions are presumably unavoidable: If contradictory theories appear only by mistake, or are due to some kind of resource-boundedness on computers, or depend on an altered state of reality, contradictions can hardly be prevented from at least being taken into consideration, as they often show up as gatecrashers. The pragmatic point thus is not whether contradictory theories exist, but how to deal with them.

2019

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In partic- ular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to be identified with an infinite sequence of conse- quence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting con- sequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences.

Manuscrito, 2019

In this review I briefly analyse the main elements of each chapter of the book centred in the general areas of logic, epistemology, philosophy and history of science. Most of them are developed around a fine-grained investigation on the principle of non-contradiction and the concept of consistency, inquired mainly into the broad area of paraconsistent logics. The book itself is the result of a work that was initiated on the Studia Logica conference "Trends in Logic XVI: Consistency, Contradiction, Paraconsistency and Reasoning-40 years of CLE", held at the State University of Campinas (Unicamp), Brazil, between September 12-15, 2016.

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.

2014

This paper proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this paper we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.

In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of noncontradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formal inconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ¬A does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A.

Journal of Logic and Computation, 2021

The aim of this paper is to develop an algebraic and logical study of certain paraconsistent systems, from the family of the logics of formal inconsistency (LFIs), which are definable from the degree-preserving companions of logics of distributive involutive residuated lattices ($\textrm {dIRL}$s) with a consistency operator, the latter including as particular cases, Nelson logic ($\textsf {NL}$), involutive monoidal t-norm based logic ($\textsf {IMTL}$) or nilpotent minimum ($\textsf {NM}$) logic. To this end, we first algebraically study enriched dIRLs with suitable consistency operators. In fact, we consider three classes of consistency operators, leading respectively to three subquasivarieties of such expanded residuated lattices. We characterize the simple and subdirectly irreducible members of these quasivarieties, and we extend Sendlewski’s representation results for the case of Nelson lattices with consistency operators. Finally, we define and axiomatize the logics of three ...