This paper introduces the logic of evidence and truth \ as an extension of the Belnap–Dunn four-valued logic \. \ is a slightly modified version of the logic \, presented in Carnielli and Rodrigues. While \ is equipped only with a classicality operator \, \ is equipped with a non-classicality operator \ as well, dual to \. Both \ and \ are logics of formal inconsistency and undeterminedness in which the operator \ recovers classical logic for propositions in its scope. (...) Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition \ even if \ is not true. As well as \, \ is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for \ where statements \\) and \\) express, respectively, the amount of evidence available for \ and the degree to which the evidence for \ is expected to behave classically—or non-classically for \ \). A probabilistic scenario is paracomplete when \ + P 1\), and in both cases, \ < 1\). If \ = 1\), or \ = 0\), classical probability is recovered for \. The proposition \, a theorem of \, partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory. (shrink)

In the last two decades modal logic has undergone an explosive growth, to thepointthatacompletebibliographyofthisbranchoflogic,supposingthat someone were capable to compile it, would?ll itself a ponderous volume. What is impressive in the growth of modal logic has not been so much the quick accumulation of results but the richness of its thematic dev- opments. In the 1960s, when Kripke semantics gave new credibility to the logic of modalities? which was already known and appreciated in the Ancient and Medieval times? no one could (...) have foreseen that in a short time modal logic would become a lively source of ideas and methods for analytical philosophers,historians of philosophy,linguists, epistemologists and computer scientists. The aim which oriented the composition of this book was not to write a new manual of modal logic buttoo?ertoeveryreader,evenwithnospeci?cbackground in logic, a conceptually linear path in the labyrinth of the current panorama of modal logic. The notion which in our opinion looked suitable to work as a compass in this enterprise was the notion of multimodality, or, more speci?cally, the basic idea of grounding systems on languages admitting more than one primitive modal operator. (shrink)

We propose a new approach to positive modal logics, hereby called anodic modal logics. Our treatment is completely positive since the language has neither negation nor any falsum or minimal particle. The elimination of the minimal particle of the language requires introducing the new concept of factual sets and factual deductions which permit us to talk about deductions in the actual world. We start from a positive fragment of the standard system K, denoted by K⊃, ∧, ◊, which is a (...) bimodal system with □ and ◊ as primitive. This system is then extended to a class of fragments of the Lemmon and Scott systems (cf. (Lemmon et al., 1977)), denoted by K⊃, ∧, ◊ + Gk;l;m;n + Gm;n;k;l. It is shown that such classes of systems are characterized with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories as introduced in the modal context by J. M. Dunn in (Dunn, 1995). We also obtain a surprising limiting result showing that the incompleteness phenomenon in modal logic is independent of negation. (shrink)

This paper discusses logical accounts of the notions of consistency and negation, and in particular explores some potential means of defining consistency and negation when expressed in modal terms. Although this can be done with interesting consequences when starting from classical normal modal logics, some intriguing cases arise when starting from paraconsistent modalities and negations, as in the hierarchy of the so-called cathodic modal paraconsistent systems (cf. Bueno-Soler, Log Univers 4(1):137–160, 2010). The paper also takes some first steps in exploring (...) the philosophical significance of such logical tools, comparing the notions of consistency and negation modally defined with the primitive notions of consistency and negation in the family of Logics of Formal Inconsistency (LFIs), suggesting some experiments on their expressive power. (shrink)

A system is classified as multimodal if its language has more than one modal operator as primitive, and such operators are not interdefinable. We extend the anodic and cathodic modal systems, introduced in Bueno-Soler and Bueno-Soler , to a class of the so-called basilar multimodal systems generating, in this way, the classes of anodic and cathodic multimodal logics. The cathodic multimodal systems are defined as extensions of positive multimodal systems by adding degrees of negation plus consistency operators. In this way, (...) cathodic multimodal systems are logics of formal inconsistency [the paraconsistent LFIs, as treated in Carnielli et al. ] enriched with multimodal operators. We focus the attention on models for such classes of systems and discuss how modal possible-translation semantics, as well as possible-worlds , can be defined to interpret basilar cathodic multimodal systems. While anodic systems are modeled by Kripke models only, we introduce the modal possible-translation models for cathodic systems. Such models, given by combinations of three-valued modal logics, besides their own interest, explain the role of non-trivializing contradictions in multimodal environment. (shrink)

In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291–310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency , introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic . These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper (...) is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics . Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al. (Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and many-valued logics, defining new blends of logics whose relevance we intend to show. (shrink)

This paper defends the use of possibility and necessity models based on the Logics of Formal Inconsistency, taking advantage of their expressivity in terms of the notions of consistency and inconsistency. The present proposal directly generalizes the approach of Besnard and Lang, whose main guidelines we borrow here. Some basic properties of possibility and necessity functions over the Logics of Formal Inconsistency are obtained and it is shown, by revisiting a paradigmatic example, how paraconsistent possibility and necessity reasoning can, in (...) general, attain realistic models for artificial judgement. We will call such models credal calculi, emphasizing some of their appealing consequences. (shrink)

This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.

“Trends in Logic XVI: Consistency, Contradiction, Paraconsistency, and Reasoning - 40 years of CLE” is being organized by the Centre for Logic, Epistemology and the History of Science at the State University of Campinas (CLEUnicamp) from September 12th to 15th, 2016, with the auspices of the Brazilian Logic Society, Studia Logica and the Polish Academy of Sciences. The conference is intended to celebrate the 40th anniversary of CLE, and is centered around the areas of logic, epistemology, philosophy and history of (...) science, while bringing together scholars in the fields of philosophy, logic, mathematics, computer science and other disciplines who have contributed significantly to what Studia Logica is today and to what CLE has achieved in its four decades of existence. It intends to celebrate CLE’s strong influence in Brazil and Latin America and the tradition of investigating formal methods inspired by, and devoted to, philosophical views, as well as philosophical problems approached by means of formal methods. The title of the event commemorates one of the three main areas of CLE, what has been called the “Brazilian school of paraconsistency”, combining such a pluralist view about logic and reasoning. (shrink)