The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collections of inferences, and thus substructural logics can be regarded as those logics which have fewer valid metainferences that Classical Logic. In order to investigate duality in substructural logics, we will focus on the case study of the logics ST and TS, the former lacking Cut, the latter Reflexivity. The sense in which these logics, and these metainferences, are dual has yet to be explained in the context of a thorough and detailed exposition of duality for frameworks of this sort. Thus, our intent here is to try to elucidate whether or not this way of talking holds some ground-specially generalizing one notion of duality available in the specialized literature, the so-called notion of negation duality. In doing so, we hope to hint at broader points that might need to be addressed when studying duality in relation to substructural logics.

The aim of this article is to study the notion of derivability and its semantic counterpart in the context of non-transitive and non-reflexive substructural logics. For this purpose we focus on the study cases of the logics ST and TS. In this respect, we show that this notion doesn't coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the notion of local validity. Following this, and building on some previous work by Humberstone, we prove that in these systems derivability can be characterized in terms of a notion we call absolute global validity. However, arriving at these results doesn't lead us to disregard local validity. First, because we discuss the conditions under which local, and also global validity, can be expected to coincide with derivability. Secondly, because we show how taking into account certain families of valuations can be useful to describe derivability for different calculi used to present ST and TS.

ResumenEn el presente artículo presentaremos un panorama sencillo de las principales lógicas no clásicas que se han propuesto para lidiar con la paradoja de Sorites, esto es, las lógicas débilmente paracompletas, las débilmente paraconsistentes y las difusas de tipo 1. Notaremos algunas ventajas y problemas de estos sistemas, y finalmente propondremos una cuarta solución -basada en una lógica difusa de tipo 2-que permite superar algunas de las dificultades planteadas.PALABRAS CLAVE: Sorites, vaguedad, lógica difusa, supervaluacionismo, subvaluacionismo. AbstractIn this paper, we present a simple overview of the main non classical logics proposed for dealing with the Sorites paradox, that is, weakly paracomplete logics, weakly paraconsistent, and type 1 fuzzy logics. We note some of their advantages and problems and we suggest that the problems can be at least partially overcome by adopting a solution which relies on interval based type 2 fuzzy logics.

In this paper, I present two presumed alternative definitions of metavalidity for metainferences: Local and Global. I defend the latter, first, by arguing that it is not too weak with respect to metainference-cases, and that local metavalidity is in fact too strong with respect to types. Second, I show that although regarding metainference-schemas Local metavalidity is always stable, Global metavalidity is also stable when the language satisfies reasonable expressibility criteria (and that in fact, both concepts collapse in those cases).

Los intentos de solucionar la paradoja de Sorites han generado una multiplicidad muy grande de propuestas. Tantas, que es importante detenerse a reflexionar acerca de qué criterios deberíamos usar para compararlas. En el presente trabajo voy a, en primer lugar, ofrecer una taxonomía gruesa de las teorías de la vaguedad que clasifica a las propuestas en dos grandes grupos: aquellas que consideran que Sorites se trata de un problema lógico y aquellas que no. Luego, voy a proponer un criterio general de comparación entre las teorías que pertenecen al primer grupo. Por último, voy a aplicar ese criterio a la comparación de dos casos particulares: una versión del Supervaluacionismo y una de la Teoría de la revisión aplicada a términos vagos. There are already too many attempts to solve the Sorites paradox. So many, that it becomes crucial to stop and reflect on which criteria should we use to compare them. On the present paper, I will offer, on the first place, a coarse taxonomy of theories of vagueness, that classifies them into two big groups: those which consider Sorites to be a logical problem, and those which do not. Then, I will propose a general criterion for comparing theories of the first type. Lastly, I will show how it applies to two particular cases: one version of Supervaluationism and a Revision Theory for vague terms.

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Cook (forthcoming) presents a paradox which he says is not circular. I see no reasons to doubt the non-circularity claim, but I do have some concerns regarding its paradoxicality. My point will be that his proposal succeeds in offering a formalization, but fails in providing a formal paradox, at least of the same type and strength as the Liar.

ResumenEn el presente artículo presentaremos un panorama sencillo de las principales lógicas no clásicas que se han propuesto para lidiar con la paradoja de Sorites, esto es, las lógicas débilmente paracompletas, las débilmente paraconsistentes y las difusas de tipo 1. Notaremos algunas ventajas y problemas de estos sistemas, y finalmente propondremos una cuarta solución -basada en una lógica difusa de tipo 2-que permite superar algunas de las dificultades planteadas.PALABRAS CLAVE: Sorites, vaguedad, lógica difusa, supervaluacionismo, subvaluacionismo. AbstractIn this paper, we present a simple overview of the main non classical logics proposed for dealing with the Sorites paradox, that is, weakly paracomplete logics, weakly paraconsistent, and type 1 fuzzy logics. We note some of their advantages and problems and we suggest that the problems can be at least partially overcome by adopting a solution which relies on interval based type 2 fuzzy logics.