In this article, I discuss Alain Badiou’s 2008 address titled “The Three Negations.” Though the text was originally presented in a symposium concerning the relationship of law to Badiou’s theory of the event, I discuss the way this brief address offers an introduction to the broad sweep of Badiou’s metaphysics, outlining his accounts of being, appearing, and transformation. To do so, Badiou calls on the resources of three paradigms of negation: from classical Aristotelian logic, from Brouwer’s intuitionist logic, and in (...) paraconsistent logics developed by DaCosta. I explain Badiou’s use of negation in the three primary areas of his metaphysics, as well as to diagnose the degrees of transformation that may have occurred in a situation. My analysis of Badiou’s use of negation in this text is aided by examples from his broader ontological oeuvre. I also explain the underlying requirement in Badiou’s work that formal considerations - mathematical or logical - get their sense by being tethered to readily-identifiable political, aesthetic, scientific, or interpersonal concerns. I conclude by addressing the foundation Badiou’s work establishes for pursuing a new metaphysics, and by discussing certain of the liabilities that remain in the wake of his account. (shrink)

Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that $\neg(a\wedge\neg a)$ is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique.

Wansing’s extended intuitionistic linear logic with strong negation, called WILL, is regarded as a resource-conscious refinment of Nelson’s constructive logics with strong negation. In this paper, (1) the completeness theorem with respect to phase semantics is proved for WILL using a method that simultaneously derives the cut-elimination theorem, (2) a simple correspondence between the class of Petri nets with inhibitor arcs and a fragment of WILL is obtained using a Kripke semantics, (3) a cut-free sequent calculus for WILL, called twist (...) calculus, is presented, (4) a strongly normalizable typed λ-calculus is obtained for a fragment of WILL, and (5) new applications of WILL in medical diagnosis and electric circuit theory are proposed. Strong negation in WILL is found to be expressible as a resource-conscious refutability, and is shown to correspond to inhibitor arcs in Petri net theory. (shrink)

The work is the presentation of a logical theory - Logic in Reality (LIR) - and of applications of that theory in natural science and philosophy, including ...

Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new negation as $\neg \varphi =_{Def} \sim \Box \varphi$. We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from (...) most other C-systems in having the important replacement property. We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5. (shrink)

Arenhart_Krause_Uma-logica-da-indistinguibilidade.

In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations as formalizing that particular kind of opposition, one needs not worry with issues about the meaning of true contradictions and the like, given that “true contradictions” are not involved in these paraconsistent logics. Our strategy will consist in showing that, on the one hand, the (...) natural translation for subcontrariety in formal languages is not a contradiction in natural language, and on the other, translating alleged cases of contradiction in natural language to paraconsistent formal systems works only provided we transform them into a subcontrariety. Transforming contradictions into subcontrariety shall provide for an intuitive interpretation for paraconsistent negation, which we also discuss here. By putting all those pieces together, we hope a clearer sense of paraconsistency can be made, one which may liberate us from the need to tame contradictions. (shrink)

In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (_LFI_) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such _LFI_s. Here, we intend to make a first step in this direction. On the other hand, the logic _Ciore_ was developed to provide new logical systems in the study of inconsistent databases from the point of view of _LFI_s. (...) An interesting fact about _Ciore_ is that it has a _strong_ consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of _Ciore_, namely _QCiore_, was defined preserving the spirit of _Ciore_, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both _Ciore_ and _QCiore_ respectively. In first place, we introduce a two-sided sequent system for _Ciore_. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique. (shrink)