Abilio Rodrigues

This book is a collection of essays and research articles honoring Walter Carnielli and his work. It brings together contributions from renowned scholars from all over the world whose work draws on or is somehow related to the several important researches Carnielli has conducted throughout his career. It includes advanced work in paraconsistent and non-classical logics, computability, probability theory, philosophy of logic, and philosophy of science. Carnielli’s contributions to logic are widely acknowledged by the scientific community, and encompass very diverse areas – ranging from critical thinking to probability theory. During his career he has published several important research papers, as well as widely adopted text-books on logic and related topics. This book reflects this aspect of Carnielli’s work and is thus of interest to logicians, philosophers of science and mathematics, mathematicians, and computer scientists. Chapter "Beyond Paraconsistency: A Plea for a Radical Breach with the Aristotelean Orthodoxy in Logic" is available open access under a Creative Commons Attribution 4.0 International License via link. springer. com.

This paper has two parts. In the first I discuss two claims made by Priest in Some Comments and Replies (DOI: 10.1007/978-3-030-25365-3 27, 2019): (i) that the idea of ‘contradictions in reality’ lacks textual support in his work, and (ii) that such contradictions would require a theory of truth as correspondence with entities such as facts or states of affairs. With respect to (i), I will recognize that, after a closer reading of his texts, in particular In Contradiction (2nd edition, OUP, 2006), I have not been able to find clear examples of dialetheias related to real contradictions. As far as I could see, for Priest, dialetheism is mainly a thesis about our conceptual framework and mental processes. As for (ii), in opposition to him, I will argue that all we need to make sense of the idea of contradictions in reality is the apparatus provided by Tarski’s definition of truth, and an object a and a predicate P such that a satisfies both P and ¬P. In the second part, I discuss what one must accept to be a dialetheist and what is required for dialetheism to be a plausible and interesting thesis. My conclusion is that dialetheism ends up being either a strong but implausible thesis about contradictions in extra-mental and extra-linguistic phenomena, or a weaker but interesting claim about contradictions produced by thought and language.

This paper extends the three-valued paraconsistent logic LFI1 to a class of multimodal systems, generating an infinite hierarchy of three-valued paraconsistent multimodal logics. LFI1, a member of the family of Logics of Formal Inconsistency, is now enriched with multimodal operators. A system is classified as multimodal if its language has more than one modal operator as primitive and such operators are not interdefinable. We provide possible-worlds (or Kripke semantics) characterizations for all such multimodal logics by means of natural canonical models ruled by a relation algebra, and discuss some aspects connected to the interpretation and uses of such logics.

This paper discusses the approaches to paraconsistency proposed by Jaśkowski in 1948 and independently by da Costa in 1963, which are historically important and highly influential in the context of non-classical logics. In their seminal works, they initiated extensive investigations into logics with non-explosive negations, later baptized paraconsistent logics, thus distinguishing the concepts of triviality and inconsistency, which are classically equivalent. The focus will be on Jaśkowski’s logic $$\textit{D}_\textit{2}$$ and da Costa’s $$C_n$$ hierarchy. We will discuss the properties of these formal systems and their further developments, as well as the main motivations and the historical context.

The aim of this paper is to introduce the logics $$\textit{FFDE}$$ and $$\textit{FN}{4}$$, which are universally free versions of Belnap-Dunn’s four-valued logic, also known as the logic of first-degree entailment ($$\textit{FDE}$$), and Nelson’s paraconsistent logic $$N^{-}$$ (a.k.a. $$Q\!N {4}$$). Both $$\textit{FDE}$$ and $$Q\!N {4}$$ are suitable to be interpreted as information-based logics, that is, logics that are capable of representing the deductive behavior of possibly inconsistent and incomplete information in a database. Like $$Q\!N {4}$$ and some non-free first-order extensions of $$\textit{FDE}$$, $$\textit{FFDE}$$ and $$\textit{FN}{4}$$ are endowed with Kripke-style variable domain semantics, which allows representing the dynamic aspect of information processing, that is, how a database receives new information over time, including information about new individuals. We argue, however, that $$\textit{FFDE}$$ and $$\textit{FN}{4}$$ can better represent the development of inconsistent and incomplete information states (i.e., configurations of a database) over time than their non-free versions. First, because they allow for empty domains, which corresponds to the idea that a database may acknowledge no individual at all at an early stage of its development. Second, because they allow for empty names, which get interpreted as information about new individuals is inserted into the database. Also, both systems include an identity predicate that is interpreted along the same lines of the other logical operators, viz., in terms of independent positive and negative rules.

This paper contains proofs of the strong adequacy, with respect to variable domain semantics, of the first-order version QvFDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{v}F\!D\!E$$\end{document} of the logic of first-degree entailment (FDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\!D\!E$$\end{document}) and some of its extensions—viz., variable domain versions of strong Kleene K3 and the logic of paradox LP, as well as Nelson’s logics QN3 and QN4. The paper also introduces the logic QNP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QN\!P$$\end{document}, obtained by extending QvFDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{v}F\!D\!E$$\end{document} with excluded middle and a constructive implication. QNP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QN\!P$$\end{document} combines constructive and non-constructive features in an interesting way, and we argue that it can be interpreted analogously to the three-valued logic of formal inconsistency LFI1∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LF\!I\! 1 \!^{*}$$\end{document} as a logic suitable for closed world databases. The paper also examines whether the logics investigated here are suitable for an interpretation in terms of information.

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. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account of paraconsistency.

The main aim of this paper is to introduce the logics of evidence and truth $$LET_{K}^+$$ and $$LET_{F}^+$$ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics $$LET_{K}$$ and $$LET_{F}^-$$ with rules of propagation of classicality, which are inferences that express how the classicality operator $${\circ }$$ is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for $$LET_{K}$$ is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for $$LET_{K}^+$$ is then obtained by imposing restrictions on the semantics of $$LET_{K}$$. These restrictions correspond exactly to the rules of propagation of classicality that extend $$LET_{K}$$. The logic $$LET_{F}^+$$ is obtained as the implication-free fragment of $$LET_{K}^+$$. We also show that the 6 values of $$LET_{K}^+$$ and $$LET_{F}^+$$ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that $$LET_{K}^+$$ is Blok-Pigozzi algebraizable and that its implication-free fragment $$LET_{F}^+$$ coincides with the degree-preserving logic of the involutive Stone algebras.

This paper introduces the logic _Q__L__E__T_ _F_, a quantified extension of the logic of evidence and truth _L__E__T_ _F_, together with a corresponding sound and complete first-order non-deterministic valuation semantics. _L__E__T_ _F_ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (_FDE_) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘_A_ entails that _A_ behaves classically, ∙_A_ follows from _A_’s violating some classically valid inferences. The semantics of _Q__L__E__T_ _F_ combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of _FDE_, _K3_, and _LP_, we show how these tools, which we call here the method of _anti-extensions + valuations_, can be naturally applied to a number of non-classical logics.

In his thesis 'Para uma Teoria Geral dos Homomorfismos' (1944) the Portuguese mathematician José Sebastião e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein’s Erlangen Program and that foreshadows some notions and results of today’s model theory; an analogous theory was independently worked out by M. Krasner in 1938. In this paper, we present a version of the theory making use of tools which were not at Silva’s disposal. At the same time, we tried to keep in mind, so much as possible, the gist of his standpoint.

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.