Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...) tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep’s clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference. (shrink)

One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter (...) we present and discuss two theories for vague predicates based on this idea: supervaluationism and subvaluationism. The chapter is structured in three parts. The first presents the super- and subvaluationist theories: their similarities and differences in the semantics, the resulting logics with their most characteristic features. The second reviews the super- and subvaluationist solutions to the Sorites Paradox and provides discussion on several controversies surrounding these theories. The third part introduces proof procedures for s’valuationist logics. (shrink)

This article deals with supervaluationism and the failure of truth-functionality. It draws some distinctions that may contribute to a better understanding of this semantic framework.

Building on early work by Girard ( 1987 ) and using closely related techniques from the proof theory of many-valued logics, we propose a sequent calculus capturing a hierarchy of notions of satisfaction based on the Strong Kleene matrices introduced by Barrio et al. (Journal of Philosophical Logic 49:93–120, 2020 ) and others. The calculus allows one to establish and generalize in a very natural manner several recent results, such as the coincidence of some of these notions with their classical (...) counterparts, and the possibility of expressing some notions of satisfaction for higher-level inferences using notions of satisfaction for inferences of lower level. We also show that at each level all notions of satisfaction considered are pairwise distinct and we address some remarks on the possible significance of this (huge) number of notions of consequence. (shrink)

Aunque irreductibles entre sí, los actos del entendimiento y de la voluntad constituyen el dinamismo operativo propio del viviente racional. Ahora bien, al estudiar la relación o interacción entre ambas potencias, se suele decir que el entendimiento tiene prioridad en cuanto a la especificación de la voluntad, mientras que la voluntad lo tiene con respecto al ejercicio. En este artículo, sin embargo, quisiéramos invertir el problema y analizar, desde principios tomasianos, en qué sentido se podría decir que el ejercicio de (...) la voluntad depende del entendimiento, por una parte, y cómo puede la inclinación afectar el conocimiento de un objeto. (shrink)

In this article, we shed light on the debate about the financial performance of socially responsible investment (SRI) mutual funds by separately analyzing the contributions of before-fee performance and fees to SRI funds' performance, and by investigating the role played by fund management companies in the determination of those variables. We apply the matching estimator methodology to obtain our results and find that in the period 1997–2005, US SRI funds had better beforeand after-fee performance than conventional funds with similar characteristics. (...) The differences, however, are driven exclusively by SRI funds run by management companies specialized in SRI. While these funds significantly outperform similar conventional funds, funds run by companies not specialized in SRI underperform their matched conventional funds. We find no significant differences in fees between SRI and conventional funds except in one case: SRI funds are cheaper than conventional funds run by the same management company. (shrink)

Agustín Echavarría y Rubén Pereda (Eds.). Providencia, libertad y mal. Estudios en Teología filosófica analítica. Granada, Comares, 2021, 169 pp. ISBN 978-84-1369-2098.

El Programa de Termalismo del IMSERSO, perteneciente al Instituto de Mayores y Servicios Sociales del Gobierno de España, se encuentra englobado dentro de de las políticas y actuaciones públicas de carácter social y económico que tienen como objetivo favorecer la salud e integración social de las personas mayores y su envejecimiento activo. Este artículo estudia, desde el año 2010 hasta el 2013, la evolución de las plazas adscritas al programa y hace una prospección para 2014-2015. Los resultados revelan una disminución (...) generalizada del número de plazas, tendencia que se mantendrá en los próximos años. (shrink)

8 March, now known as International Women’s Day, is a day for feminist claims where demonstrations are organized in over 150 countries, with the participation of millions of women all around the world. These demonstrations can be viewed as collective rituals and thus focus attention on the processes that facilitate different psychosocial effects. This work aims to explore the mechanisms involved in participation in the demonstrations of 8 March 2020, collective and ritualized feminist actions, and their correlates associated with personal (...) well-being and collective well-being, collective efficacy and collective growth, and behavioral intention to support the fight for women’s rights. To this end, a cross-cultural study was conducted with the participation of 2,854 people from countries in Latin America and Europe, with a retrospective correlational cross-sectional design and a convenience sample. Participants were divided between demonstration participants and non-demonstrators or followers who monitored participants through the media and social networks. Compared with non-demonstrators and with males, female and non-binary gender respondents had greater scores in mechanisms and criterion variables. Further random-effects model meta-analyses revealed that the perceived emotional synchrony was consistently associated with more proximal mechanisms, as well as with criterion variables. Finally, sequential moderation analyses showed that proposed mechanisms successfully mediated the effects of participation on every criterion variable. These results indicate that participation in 8M marches and demonstrations can be analyzed through the literature on collective rituals. As such, collective participation implies positive outcomes both individually and collectively, which are further reinforced through key psychological mechanisms, in line with a Durkheimian approach to collective rituals. (shrink)

This paper discusses partial results of an ongoing project focused on analysing the current usefulness and implications of developing research on agent-based social simulation models beyond academic, hobbyist or educational purposes. Design, development and testing phases of such modelling are discussed along with common issues evidence-driven modellers often face whilst collecting, analysing and modelling quantitative and qualitative data into social simulations. It also includes a discussion on the evidence gathered in published literature and structured interviews with researchers that have lead (...) mid- to long-term (3–5 years) projects on social simulation in Europe and the United States. Finally, good-practice recommendations are put forward by presenting a development methodology fully guided by evidence. (shrink)

Objectives To explore how factors relating to grades and grading affect the correctness of choices that grant-review panels make among submitted proposals. To identify interventions in panel design that may be expected to increase the correctness of choices. -/- Method Experimentation with an empirically-calibrated computer simulation model of panel review. Model parameters are set in accordance with procedures at a national science funding agency. Correctness of choices among research proposals is operationalized as agreement with the choices of an elite panel. (...) -/- Conclusions The simulation model generates several hypotheses to guide further research. Increasing the number of grades used by panel members increases the correctness of simulated choices among submitted proposals. Collective decision procedures giving panels a greater capacity for discriminating among proposals also increase correctness. Surprisingly, differences in grading standards among panel members do not appreciably decrease correctness. (shrink)

A partir de los términos que indican “vivir de lo propio” y/o “casa alquilada” en Hch 28,16.30, el artículo se propone descubrir el valor que el autor les ha otorgado en el contexto de la conclusión de los Hechos y, particularmente, los sumarios de la estancia y predicación de Pablo en Roma. Para ello se analizará la imagen lucana de Pablo, como héroe victorioso y como mártir por la fe. Se comparará el fin de los Hechos, particularmente el (...) sumario de 28,30-31, con paralelos en la literatura judeohelenista de corte y en Josefo. Y se finalizará dando una explicación de la visión política de Lucas y la situación de su comunidad: la casa permanente de Pablo implica la esperanza de tener un espacio en la ecumene romana y de seguridad intraeclesial. (shrink)

We say that a sentence A is a permissive consequence of a set X of premises whenever, if all the premises of X hold up to some standard, then A holds to some weaker standard. In this paper, we focus on a three-valued version of this notion, which we call strict-to-tolerant consequence, and discuss its fruitfulness toward a unified treatment of the paradoxes of vagueness and self-referential truth. For vagueness, st-consequence supports the principle of tolerance; for truth, it supports the (...) requisite of transparency. Permissive consequence is non-transitive, however, but this feature is argued to be an essential component to the understanding of paradoxical reasoning in cases involving vagueness or self-reference. (shrink)

El discurso que San Pablo dirige a los atenienses en el Areópago constituye el más claro ejemplo de inculturación que aparece en el Nuevo Testamento. Siendo el precedente inmediato de todo el esfuerzo que los Padres de la Iglesia realizarán por establecer un diálogo fecundo entre la filosofía y la revelación, entre la fe y la razón. Bajo un esquema histórico de «predicación kerigmática a los gentiles», San Lucas crea un relato donde vierte todo su saber filosófico y (...) retórico. Aspecto este último que puede ser percibido con claridad a la luz del De Doctrina Christiana de San Agustín. (shrink)

RésuméAu cœur de la théorie de Sophia Moreau sur la discrimination répréhensible réside le devoir moral de traiter les autres comme des êtres égaux. Cet essai soulève quelques questions sur les contours de ce devoir et suggère des moyens de renforcer la théorie. En particulier, il suggère d'intégrer une vision cosmopolite de la portée de ce devoir, d’éclairer les caractéristiques à la base du statut moral égal des personnes afin d'en identifier les fondements, ainsi que d'identifier certaines considérations relatives à (...) des intérêts d'importance afin d'en préciser le contenu. La relation entre la théorie de Moreau et le discours sur les droits humains est également brièvement examinée, et une discussion plus approfondie de ce dernier est recommandée. (shrink)

Hilden and Gammelgaard's comments on the evaluation and application of statistical trials and on informed consent lead to the same conclusion that there are special circumstances when clinical trials are performed in emergency situations.The decision to comment on the work that H&G responded to was motivated by the wish to manifest the need to reach a consensus on modifying the conditions in which clinical trials should be performed in special circumstances.One unquestionable fact must first be addressed: clinical trials are necessary (...) for acquiring new knowledge and consequent improvements in health care services. At the same time, patient participation should be encouraged under conditions that respect patients' decision making capability.Are these conditions being met at present? An answer to this is given below. (shrink)

The problem of non-compliance with treatment and its repercussions on the clinical evolution of different conditions has been widely investigated.1–4 Non-compliance has also been shown to have significant economic implications, not only as a result of product loss but also indirectly through the complication of disease management and its subsequent healthcare and social costs.5–7Non-compliance as a health problemThe term “non-compliance” might be taken to refer both to the failure to follow a drug regimen and to the failure to adopt other (...) measures that contribute to improvement in health—for example, changes in lifestyle or diet. This letter focuses on the former.Non-compliance with a drug regimen can be the result of a number of different factors9–11 and a variety of techniques have been developed in an attempt to control it.12,13 Of these, the few techniques that have been shown to be effective have only managed to solve the problem in specific situations over short periods of time. The use of such techniques to control non-compliance, particularly where these are effective, raises interesting ethical questions about the extent to which their application constitutes an infringement of the patient’s right to decide on how to manage their own health.8 Here we suggest that in some cases one factor that leads to non-compliance is the tendency to provide extensive and exhaustive information on side effects in …. (shrink)

Advances in medicine are closely linked to clinical research, but certain study procedures may be in conflict with the fundamental principles of ethics and codes of conduct in medicine. Following an analysis of two studies involving treatments for acute myocardial infarction (AMI), the admissibility of continuing a study was questioned after the initial results for two types of treatment showed that one was significantly better than the other. Also considered doubtful was the information provided to patients with the object of (...) obtaining their informed consent. (shrink)

Translation of Aristotle's Posterior Analytics I into Portuguese, with a few notes, experimental glossary and introduction. The translation, which was made at 2003/4, was preliminary and its publication was intended to provide a didactic tool for courses as well as a provisional resource in research seminars. It needs some revision. I am currently working (slowly...) on the revision of the translation and a new revised one will surely appear at some point.

In Posterior Analytics 71b9 12, we find Aristotle’s definition of scientific knowledge. The definiens is taken to have only two informative parts: scientific knowledge must be knowledge of the cause and its object must be necessary. However, there is also a contrast between the definiendum and a sophistic way of knowing, which is marked by the expression “kata sumbebekos”. Not much attention has been paid to this contrast. In this paper, I discuss Aristotle’s definition paying due attention to this contrast (...) and to the way it interacts with the two conditions presented in the definiens. I claim that the “necessity” condition ammounts to explanatory appropriateness of the cause. (shrink)

In this paper, we develop an impure somatic theory of emotion, according to which emotions are constituted by the integration of bodily perceptions with representations of external objects, events, or states of affairs. We put forward our theory by contrasting it with Prinz's pure somatic theory, according to which emotions are entirely constituted by bodily perceptions. After illustrating Prinz's theory and discussing the evidence in its favor, we show that it is beset by serious problems—i.e., it gets the neural correlates (...) of emotion wrong, it isn't able to distinguish emotions from bodily perceptions that aren't emotions, it cannot account for emotions being directed towards particular objects, and it mischaracterizes emotion phenomenology. We argue that our theory accounts for the empirical evidence considered by Prinz and solves the problems faced by his theory. In particular, we maintain that our theory gives a unified and principled account of the relation between emotions and bodily perceptions, the intentionality of emotions, and emotion phenomenology. (shrink)

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph (...) conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

I discuss what Aristotle means when he say that scientific demonstration must proceed from necessary principles. I argue that, for Aristotle, scientific demonstration should not be reduced to sound deduction with necessary premises. Scientific demonstration ultimately depends on the fully appropriate explanatory factor for a given explanandum. This explanatory factor is what makes the explanandum what it is. Consequently, this factor is also unique. When Aristotle says that demonstration must proceed from necessary principles, he means that each demonstration requires the (...) principle that is the necessary one for the fully appropriate explanation of its explanandum. This picture also provides a key to understand Aristotle's thesis that scientific explanation depends on essences: it is the essence of the attribute to be explained that should be stated as the fully appropriate explanatory factor. (shrink)

Experiences like pains, pleasures, and emotions have affective phenomenal character: they feel pleasant or unpleasant. Imperativism proposes to explain affective phenomenal character by appeal to imperative content, a kind of intentional content that directs rather than describes. We argue that imperativism is on the right track, but has been developed in the wrong way. There are two varieties of imperativism on the market: first-order and higher-order. We show that neither is successful, and offer in their place a new theory: reflexive (...) imperativism. Our proposal is that an experience P feels pleasant in virtue of being constituted by a Command with reflexive imperative content, while an experience U feels unpleasant in virtue of being constituted by a Command with reflexive imperative content : More of P!Less of U! If you need a slogan: experiences have affective phenomenal character in virtue of commanding us Get more of me! Get less of me! (shrink)

Analyzing geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural network structure, implementation attributes, and learning performance. In this work, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are (...) rarely correctly identified as such in expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e., the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable to correctly identify saddle information, we propose to study projections along dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to about 7×10^6 parameters. (shrink)

Aristotle contrasts episteme and doxa through the key notions of universal and necessary. These notions have played a central role in Aristotle’s characterization of scientific knowledge in the previous chapters of APo. They are not spelled out in APo I.33, but work as a sort of reminder that packs an adequate characterization of scientific knowledge and thereby gives a highly specified context for Aristotle’s contrast between episteme and doxa. I will try to show that this context introduces a contrast in (...) terms of explanatory claims: on the one hand, episteme covers those claims which capture explanatory connections that are universal and necessary and thereby deliver scientific understanding; on the other hand, doxa covers the explanatory attempts that fail at doing so. (shrink)

I examine Aristotle’s definition of scientific knowledge in Posterior Analytics 71b 9-12 and try to understand how it relates to the sophistical way of knowing and to "kata sumbebekos knowledge". I claim that scientific knowledge of p requires knowing p by its appropriate cause, and that this appropriate cause is a universal (katholou) in the restricted sense Aristotle proposes in 73b 26-27 ff., i.e., an attribute coextensive with the subject (an extensional feature) and predicated of the subject in itself (an (...) intensional feature). Kata sumbebekos knowledge, on the other hand, can occur even when the predicate of a conclusion is coextensive with its subject and is proved by a convertible meson which is not the most appropriate from an explanatory standpoint. (shrink)

I discuss the exact meaning of the thesis according to which the object of scientific knowledge is necessary. The thesis is expressed by Aristotle in the Posterior Analytics, in his definition of scientific knowledge. The traditional interpretation understands this definition as depending on two parallel and independent requirements, the causality requirement and the necessity requirement. Against this interpretation, I try to show, through the examination of several passages that refer to the definition of scientific knowledge, that the necessity requirement specifies (...) more exactly the causality requirement: what cannot be otherwise is the explanatory relation between the explanandum and the cause by which it is what it is. (shrink)

This book uses mathematical models of language to explain why there are certain gaps in language: things that we might expect to be able to say but can't. For instance, why can we say I ran for five minutes but not *I ran to the store for five minutes? Why is five pounds of books acceptable, but *five pounds of book not acceptable? What prevents us from saying *sixty degrees of water to express the temperature of the water in a (...) swimming pool when sixty inches of water can express its depth? And why can we not say *all the ants in my kitchen are numerous? The constraints on these constructions involve concepts that are generally studied separately: aspect, plural and mass reference, measurement, and distributivity. In this book, Lucas Champollion provides a unified perspective on these domains, connects them formally within the framework of algebraic semantics and mereology, and uses this connection to transfer insights across unrelated bodies of literature and formulate a single constraint that explains each of the judgments above. (shrink)

I discuss the methodological passage in the begin- ning of Ethica Eudemia I.6 (1216b26-35), which has received attention in connection with Aristotle’s notion of dialectic and his methodology in Ethics. My central focus is not to discuss whether Aristotle is prescribing and using what has been called the method of endoxa. I will focus on how this passage coheres with the remaining parts of the same chapter, which also are advancing methodological remarks. My claim is that the meth- od of (...) Ethica Eudemia I.6 is in agreement with many features of Aristotle’s theory of explanation as presented in the Posterior Analytics: Aristotle’s main concern is a warning against misuses of explanatory arguments. (shrink)

Mindreading (or folk psychology, Theory of Mind, mentalizing) is the capacity to represent and reason about others’ mental states. The Simulation Theory (ST) is one of the main approaches to mindreading. ST draws on the common-sense idea that we represent and reason about others’ mental states by putting ourselves in their shoes. More precisely, we typically arrive at representing others’ mental states by simulating their mental states in our own mind. This entry offers a detailed analysis of ST, considers theoretical (...) arguments and empirical data in favour of and against it, discusses its philosophical implications, and illustrates some alternatives to it. (shrink)

Translation of passages of Aristotle's De Anima into Portuguese. The passages are these: I.1, I.4 (the 'Rylean passage'); II.1-6; III.1-8. The translation is preliminary.

In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is a (...) strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G LJ*\c in IPC*\c. (shrink)

In this paper we obtain a finite Hilbert-style axiomatization of the implicationless fragment of the intuitionistic propositional calculus. As a consequence we obtain finite axiomatizations of all structural closure operators on the algebra of {–}-formulas containing this fragment.

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the strict-tolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external logic and (...) argue that while ST+’s nonclassicality can be granted, its self-dual character does not tie it to LP+ more closely than to K3+. (shrink)

This paper examines Aristotle’s notion of priority with the specific aim of capturing the sort of priority that characterizes the primacy of substances in his metaphysics. I reject the traditional interpretation, which understands the ontological priority of substances in terms of independent existence. But there are rather two sorts of priority: the ontological priority of substances should be understood in terms of completeness, whereas the ontological priority of “substances-of-something” (the essences) is a causal-explanatory priority. Furthermore, an important piece of Aristotle’s (...) argument against Platonism is that these two sorts of priority – namely, the completeness priority and the causal-explanatory priority – should be kept distinct. (shrink)

This paper discusses some issues concerning the definition of moral virtue in Nicomachean Ethics 1106b 36- 1107a 2. It is reasonable to expect from a definition the complete enumeration of the relevant features of its definiendum, but the definition of moral virtue seems to fail in doing this task. One might be tempted to infer that this definition is intended by Aristotle as a mere preliminary account that should be replaced by a more precise one. The context of the argument (...) Aristotle develops in Book II of his NE give us some help. I argue that the definition of moral virtue, once considered in the light of its context, is far from being an incomplete and provisional account: it rather introduces coherently the same notion of moral virtue that Aristotle employs in other texts (as in Nicomachean Ethics VI 13). My main proposal is that the way in which "hexis" is understood in the context of previous chapters allows Aristotle to encode in it the notion of an ability to do the right things regularly. Thus, moral virtue is a "hexis prohairetike etc.", but the ability to do the right things regularly is already encoded in the occurrence of "hexis" in the definiens account of moral virtue, as if Aristotle meant "hexis [praktike] prohairetike". (shrink)

Joint attention customarily refers to the coordinated focus of attention between two or more individuals on a common object or event, where it is mutually “open” to all attenders that they are so engaged. We identify two broad approaches to analyse joint attention, one in terms of cognitive notions like common knowledge and common awareness, and one according to which joint attention is fundamentally a primitive phenomenon of sensory experience. John Campbell’s relational theory is a prominent representative of the latter (...) approach, and the main focus of this paper. We argue that Campbell’s theory is problematic for a variety of reasons, through which runs a common thread: most of the problems that the theory is faced with arise from the relational view of perception that he endorses, and, more generally, they suggest that perceptual experience is not sufficient for an analysis of joint attention. (shrink)

This paper discusses the contrast between scientific knowledge and opinion as it is presented by Aristotle in Posterior Analytics A.33. Aristotle's contrast is formulated in terms of understanding or not understanding some "necessary items". I claim that the contrast can only be understood in terms of explanatory relevance. The "necessary items" are middle terms (or explanatory factors) that are necessary for the fully appropriate explanation. This approach gives a coherent interpretation of each step in the chapter.

Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...) to get rapidly into the logical issues. This paper is relatively self-contained. A modest background on propositional modal logic is, though not strictly necessary, advisable. The reader might find useful the Compass papers Kracht (2011) and Negri (2011) (though these papers cover issues of more complexity than what is demanded to follow this paper). (shrink)

These are commentaries to the translation into Portuguese of Nicomachean Ethics VI, found in the same volume of Dissertatio.