An encompassing framework for Paraconsistent Logic Programs

We propose a framework which extends Antitonic Logic Programs [Damásio and Pereira, in: Proc. 6th Int. Conf. on Logic Programming and Nonmonotonic Reasoning, Springer, 2001, p. 748] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting's bilattice approaches, this framework allows a precise definition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [Pereira and Alferes, in: European Conference on Artificial Intelligence, 1992, p. 102], according to which explicit negation entails default negation. We then define Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalizing many paraconsistent semantics for logic programs. In particular, Paraconsistent Well-Founded Semantics with eXplicit negation (WFSX_p) [Alferes et al., J. Automated Reas. 14 (1) (1995) 93–147; Damásio, PhD thesis, 1996]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

AI,ME AND LEWIS (ABELIAN IMPLICATION,MATERIAL EQUIVALENCE AND C I LEWIS 1920)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). Meyer was supported in this work as a Visiting Fellow in the College of Engineering and Computer Science, ANU.     

Understanding Negation Implicationally in the Relevant Logic R

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.     

All Brutes are Subhuman: Aristotle and Ockham on Private Negation

The mediaeval logic of Aristotelian privation, represented by Ockham's expositionof All A is non-P as All S is of a type T that is naturally P and no S is P, iscritically evaluated as an account of privative negation. It is argued that there aretwo senses of privative negation: (1) an intensifier (as in subhuman), the dualof Neoplatonic hypernegation (superhuman), which is studied in linguistics asan operator on scalar adjectives, and (2) a (often lexicalized) Boolean complementrelative to the extension of a privative negation in sense (1) (e.g., Brute). Thissecond sense, which is the privative negation discussed in modern linguistics, isshown to be Aristotle's. It is argued that Ockham's exposition fails to capture muchof the logic of Aristotelian privation due to limitations in the expressive power of thesyllogistic.     

A First Order Nonmonotonic Extension of Constructive Logic

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QN^c₅, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QN^c₅, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference.     

Negation: A theory of its meaning,representation, and use

This article presents a model-based theory of what negation means, how it is mentally represented, and how it is understood. The theory postulates that negation takes a single argument that refers to a set of possibilities and returns the complement of that set. Individuals therefore tend to assign a small scope to negation in order to minimize the number of models of possibilities that they have to consider. Individuals untrained in logic do not know the possibilities corresponding to the negation of compound assertions formed with if, or, and and, and have to infer the possibilities one by one. It follows that negations are easier to understand, and to formulate, when individuals already have in mind the possibilities to be negated. The paper shows that the evidence, including the results of recent studies, corroborates the theory.     

Dwelling on the hill: Impressions of residents of two favelas in Rio de Janeiro regarding religion and public space

Drawing on fieldwork and interviews with residents in two favelas in Rio de Janeiro, this paper argues that the concept of ‘the divided city’ (fixed and nomadic, planned and negotiated, rich and poor) allows us to make sense of these residents' views of religion and public space. Life ‘on the hill’ is characterised by a greater degree of negotiation and improvisation in all its social relations than is life below in ‘the asphalt’. This offers important insights into the success of Evangelicals [Apart from here in the abstract, references to ‘evangelicals’ in the text follow Brazilian usage, i.e., a broad term including three groups of Protestants: mainstream historical churches or “evangélicos de missão” (Lutheran, Presbyterian, Methodist, etc); Pentecostals (Assembléia de Deus, Congregação Cristã, Deus é Amor, etc.); and Neo-pentecostals (Igreja Universal do Reino de Deus, Igreja da Graça de Deus, Igreja Apostólica Renascer em Cristo, etc).(S.E./C.M.)] and Pentecostals, and the relative lack of success by Catholics, in taking a lead in public activities within the community. This finding may well have broader application. However, the paper makes a theoretical and methodological qualification: because the place of religion depends on the particular history and context of the location studied, the most pressing need is for more local studies.     

Fuzzy intuitionistic quantum logics

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL ³) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

Rumfitt on truth-grounds,negation, and vagueness

In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.     

Paraconsistent logic from a modal viewpoint

In this paper we study paraconsistent negation as a modal operator, considering the fact that the classical negation of necessity has a paraconsistent behavior. We examine this operator on the one hand in the modal logic S5 and on the other hand in some new four-valued modal logics.