Minimal Negation in the Ternary Relational Semantics

Minimal Negation is defined within the basic positive relevance logic in the relational ternary semantics: B+. Thus, by defining a number of subminimal negations in the B+ context, principles of weak negation are shown to be isolable. Complete ternary semantics are offered for minimal negation in B+. Certain forms of reductio are conjectured to be undefinable (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive logic are offered

Proof Theory for Positive Logic with Weak Negation

Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete

Tipos de negación en la lógica contemporánea

Uno de los problemas más interesantes en la teoría lógica actual radica en la determinación de la caracterización formal de la negación y su relación con el lenguaje natural. En el presente trabajo nos proponemos presentar, analizar y comparar las propuestas de H. Curry en Foundations of Mathematical Logic y la de M. Dunn en The Kite of Negations, a los efectos de determinar si dan cuenta de los mismos sistemas lógicos.Ponencia presentada en la Comisión G - Lógica.Departamento de Filosofí

A semantic analysis of some distributive logics with negation

In this paper we shall study some extensions of the semilattice based deductive systems S (N) and S (N, 1), where N is the variety of bounded distributive lattices with a negation operator. We shall prove that S (N) and S (N, 1) are the deductive systems generated by the local consequence relation and the global consequence relation associated with ¬-frames, respectively. Using algebraic and relational methods we will prove that S (N) and some of its extensions are canonical and frame complete.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires.facultad de Ciencias Exactas; Argentin

Negation in natural language

Negation is ubiquitous in natural language, and philosophers have developed plenty of different theories of the semantics of negation. Despite this, linguistic theorizing about negation typically assumes that classical logic's semantics for negation---a simple truth-functional toggle---is adequate to negation in natural language, and philosophical discussions of negation typically ignore vital linguistic data. The present document is thus something of an attempt to fill a gap, to show that careful attention to linguistic data actually militates {\\em against} using a classical semantics for negation, and to demonstrate the philosophical payoff that comes from a nonclassical semantics for natural-language negation. I present a compositional semantics for natural language in which these questions can be posed and addressed, and argue that propositional attitudes fit into this semantics best when we use a nonclassical semantics for negation. I go on to explore several options that have been proposed by logicians of various stripes for the semantics of negation, providing a general framework in which the options can be evaluated. Finally, I show how taking non-classical negations seriously opens new doors in the philosophy of vagueness