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En On What Matters, Derek Parfit trata, según nos dice él mismo, de solucionar lo que Sidgwick denominó “el dualismo de la razón práctica”. Según Sidgwick, el intuicionismo y el consecuencialismo son fácilmente compatibles dentro de un utilitarismo sensato, si bien no se pueden hacer compatibles con el egoísmo ético. Parfit, quien considera a Kant y al propio Sidgwick como sus dos influencias más importantes, cree que es posible encontrar una solución a través de lo que denomina The Triple Theory. Una vez rechazado el egoísmo ético, el intuicionismo objetivista puede hacerse compatible con el universalismo kantiano a través del consecuencialismo de Thomas Scanlon. La articulación y defensa de esta propuesta conforman los tres gruesos volúmenes de On What Matters. En este trabajo trataré de argumentar que, además de tener que enfrentar importantes problemas teóricos ya clásicos, el proyecto de unificación de las diferentes propuestas éticas puede dejar sin contestar importantes cuestiones prácticas que eran tratadas por las distintas teorías por separado., The main topic in On What Matters is, in Parfit’s own words, trying to solve what Henry Sidgwick called “the practical reason´s dualism”. Following Sidgwick, intuitionism and consequentialism are easily compatible related to a reasonable utilitarianism, but both became strongly incompatible with ethic egoism. Parfit, who considers Kant and Sidgwick as his most important influence, believes that it is possible to find a solution through what he calls “The Triple Theory”. Once rejected ethical egoism in his several forms, he thinks that objectivist intuitionism can be compatible with Kantian universalism through Thomas Scanlon consequentialism in his work What We Owe Each Other. The structure and defense of this thesis will make up most of the three volumes of On What Matters. In this essay I will try to argue that, in addition to having to face several classical criticisms, the aim of unifying different ethical theories can leave unanswered important practical questions that where dealt with by the different theories separately.

The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. This paper presents a class of intuitionistic theories that features stateful computations, such as reference cells, and shows that these theories can be extended with continuity axioms. The modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled in the theory.

From a legal point of view, the question about killing may seem straightforward. In all jurisdiction in well-ordered states murder is criminalized. The legal case is not so clearcut, however. It is true that murder is typically legally prohibited, but in certain circumstances you are legally required to kill other people (such as in war or when capital punishment is practiced). On the other hand, even if a person suffering in a terminal state requires help to die, it is in many countries forbidden for a physician even to help this person to commit suicide (physician assisted suicide), let alone actively and intentionally to kill this person at request (euthanasia). All this means that it is of interest to view the subject matter of killing from a moral point of view. In particular, it is of interest to try to find out what different moral theories tell us about when we should, and when we should not kill, and why. Both moral guidance and moral explanation are needed.

Jonathan Haidt has exhibited in several publications a strong influence by Hume as his inspiration to build up his Intuitionist Social Model especially in reference to the emotional and intuitive roots of morals. So, this article aims to go deeper into that influence to analyze, review and compare their concepts and hypotheses in order to establish common features in both Philosophers´ programs. By demonstrating some similarities between both authors -both critical to rationalism, and compelled to investigate the roots of morals in Psychology-, their proposals will be compared and analyzed; and taking these critics into account the intuitionist assumptions that both Philosophers assume or could assume will be revised. In order to focus on the motivation of moral statements as a priority in Haidt research, the core of this article is dedicated to that issue. I will also explore Hume’s difficulty to conciliate his Emotivism with the rightness of moral statements that he considered necessary. Finally, my conclusions are that both positions sustain normative consequences ruled by an Ethics of Virtue., Jonathan Haidt ha puesto de manifiesto en diversas publicaciones la influencia que tuvo Hume como inspirador de su modelo intuicionista social, en especial en lo referente a la raíz emocional e intuitiva que da origen al juicio moral. Este artículo se propone indagar en dicha influencia con el objetivo de analizar, revisar y comparar aquellos conceptos e hipótesis que puedan tener en común los programas los dos autores. A partir de mostrar las semejanzas de ambos en el objetivo de investigar las raíces de la moralidad en la psicología, se analizan y comparan sus respectivas críticas al racionalismo y, en función de ellas, los supuestos intuicionistas que ambos asumen o podrían asumir. Dado que la motivación de los juicios morales es el tema prioritario en las investigaciones de Haidt, la parte central del trabajo está dedicada a él; al respecto se muestran las dificultades de Hume para conciliar su emotivismo con la corrección del juicio moral, a la que considera necesaria. Por último, se concluye que ambas posiciones poseen consecuencias normativas regidas por una ética de la virtud.

Time-progressing expressions, i.e., expressions that compute to different values over time such as Brouwerian choice sequences or reference cells, are a common feature in many frameworks. For type theories to support such elements, they usually employ sheaf models. In this paper, we provide a general framework in the form of an extensional type theory incorporating various time-progressing elements along with a general possible-worlds forcing interpretation parameterized by modalities. The modalities can, in turn, be instantiated with topological spaces of bars, leading to a general sheaf model. This parameterized construction allows us to capture a distinction between theories that are "agnostic", i.e., compatible with classical reasoning in the sense that classical axioms can be validated, and those that are "intuitionistic", i.e., incompatible with classical reasoning in the sense that classical axioms can be proven false. This distinction is made via properties of the modalities selected to model the theory and consequently via the space of bars instantiating the modalities. We further identify a class of time-progressing elements that allows deriving "intuitionistic" theories that include not only choice sequences but also simpler operators, namely reference cells.

Este trabajo examina la concepción de la justificación en la que se inscribe el método al que Rawls apela para justificar sus principios de justicia, una de las cuestiones metodológicas que más reparos ha generado. En este sentido, se propone explicar por qué el equilibrio reflexivo supone una explicación ampliamente coherentista de la justificación, así como de qué modo la visión que ofrece de esta trata de evitar cualquier elemento de naturaleza fundacionalista. Puesto que, en cuanto teoría de la justificación, el fundacionalismo es el enfoque que más abiertamente parece oponerse al coherentismo, pero algunas de las críticas que se han formulado del equilibrio reflexivo han tratado de presentarlo como una forma encubierta de intuicionismo –la versión más importante de fundacionalismo moral–, el trabajo intenta poner de manifiesto algunas de las diferencias más relevantes que cabe trazar entre esos dos modos de concebir la justificación. Tomando esto como base, pretende evidenciar por qué los intentos de presentar la justicia como equidad como participando de esa clase de fundacionalismo son poco razonables, pero, sobre todo, ofrecer algunas respuestas a propósito de ese tipo de reproches, subrayando las cuestiones en las que debe ponerse el foco a la hora de evaluarlos., This paper examines the conception of justification that the method to which Rawls appeals to justify his principles of justice falls within, one of the methodological questions that more objections has generated. In this sense, it sets out to explain why reflective equilibrium supposes a broadly coherentist account of justification, as well as how the vision that it offers of this seeks to avoid any element of a foundationalist nature. Since foundationalism is the approach to justification that it seems to oppose coherentism most straightly, but some of the criticisms of reflective equilibrium have attempted to present it as a disguised form of intuitionism –the most important version of moral foundationalism–, the paper tries to highlight some of the most significant differences that can be drawn between these two ways of seeing justification. Taking this as a basis, it aims to show why attempts to present justice as fairness as participating in that kind of foundationalism are unreasonable, but, above all, to offer some answers regarding these objections, emphasizing the questions that must be put the focus when evaluating them.

One of the differences between Brouwerian intuitionistic logic and classical logic is their treatment of time. In classical logic truth is atemporal, whereas in intuitionistic logic it is time-relative. Thus, in intuitionistic logic it is possible to acquire new knowledge as time progresses, whereas the classical Law of Excluded Middle (LEM) is essentially flattening the notion of time stating that it is possible to decide whether or not some knowledge will ever be acquired. This paper demonstrates that, nonetheless, the two approaches are not necessarily incompatible by introducing an intuitionistic type theory along with a Beth-like model for it that provide some middle ground. On one hand they incorporate a notion of progressing time and include evolving mathematical entities in the form of choice sequences, and on the other hand they are consistent with a variant of the classical LEM. Accordingly, this new type theory provides the basis for a more classically inclined Brouwerian intuitionistic type theory.

To make evaluations about the morally relevant impacts of algorithms, transparency is needed. This paper lays out discussion of algorithms and transparency in an explicitly moral analysis with a special focus on the domain of justice and security. The paper provides an account of the moral import of transparency, defined itself as an instrumental value denoting a state of affairs conducive to acquisition of knowledge about some X. A normative account of transparency is outlined relying on an intuitionist framework rooted in the works of Ross and Robert Audi. It will be argued that transparency can be derived as a subsidiary (prima facie) principle from other duties including beneficence and justice and that it is groundable in the value of knowledge. Building on this foundation, the paper examines transparency and duty conflict with a special focus on algorithms in justice and security, recognising that complete transparency can be impossible where duties conflict. It is argued that as a subsidiary (prima facie) principle, transparency is overridable but ineradicable, which is to say that sufficiently justifiable reasons for secrecy or opacity can licence limiting transparency, that is, there may be occasion where full transparency is not our final duty., Ethics & Philosophy of Technology

The contribution pursues two interrelated aims: the first one is to interpret the transition from Morality to Ethical Life in the Elements of the Philosophy of Right as a theory of moral justification; the second one is to show the meaning of this theory for contemporary philosophy. The essay thus seeks to turn to Hegel’s conception in order to shed light on the limits of the current discussion in moral epistemology and then to present a possible alternative. To this end, Hegel’s text is confronted with the main theses of intuitionism, on the one hand, and Kantian constructivism, on the other, as the most prominent contemporary forms, respectively, of ethical foundationalism and coherentism., O artigo tem dois objetivos mutuamente interligados: o primeiro é interpretar a transição da moralidade para a ética nas Linhas fundamentais da filosofia do direito como uma teoria da justificação moral; o segundo é mostrar o significado dessa teoria para a filosofia contemporânea. Dessa forma, o artigo procura retomar a reflexão hegeliana para lançar luz sobre alguns limites do debate contemporâneo acerca da epistemologia moral e, a seguir, apresentar uma possível alternativa. Para tanto, o texto hegeliano é posto em diálogo com as teses centrais do intuicionismo, de um lado, e do construtivismo kantiano, de outro, na medida em que representam as formas contemporâneas que têm mais autoridade, respectivamente, a do fundacionalismo e a do coerentismo ético.

The present text deals with Charles Larmore's theory, more precisely, of the relations between the conceptions of reasonable disagreement, pluralism and liberalism. Such a theory has as its characteristics rational intuitionism, contextualism and defends a realistic position of morality., O presente texto trata da teoria de Charles Larmore, mais precisamente, das relações entre os conceitos de desacordo razoável, pluralismo e liberalismo. Tal teoria tem como características intuicionismo racional, contextualismo e defende uma posição realista da moralidade.

The present text deals with Charles Larmore’s theory, more precisely, of the relations between the conceptions of reasonable disagreement, pluralism and liberalism. Such a theory has as its characteristics rational intuitionism, contextualism and defends a realistic position of morality., El presente texto trata de la teoría de Charles Larmore, más precisamente, de las relaciones entre las concepciones de desacuerdo razonable, de pluralismo y de liberalismo. Tal teoría tiene como características el intuicionismo racional, contextualismo y defiende una posición realista de la moralidad.

The present work tries to analyze the moral philosophy of William David Ross. First, Ross's ethics precedents are exposed, focusing on three authors against whom he acknowledges entering into discussion: Kant, the utilitarians and Moore. This is the aim of the second chapter. Next, Ross's proposal is explained through three axes: 1) the theory of moral knowledge, where his well-known intuitionist theory will appear; 2) the doctrine of correction, composed of deontological pluralism; 3) and the doctrine of good, in which its axiology appears. The following three chapters, respectively, will be dedicated to all this. Finally, we provide the critical vision of some Ross followers and propose half a dozen conclusions., El presente trabajo procura analizar la filosofía moral de William David Ross. En primer lugar, se exponen los precedentes de la ética de Ross, centrándose en tres autores contra los que él mismo reconoce entrar en discusión: Kant, los utilitaristas y Moore. A ello se dedicará el segundo capítulo. Seguidamente, se explica la propuesta de Ross a través de tres ejes: 1) la teoría del conocimiento moral, en donde aparecerá su conocida teoría intuicionista; 2) la doctrina de la corrección, compuesta por el pluralismo deontológico; 3) y la doctrina del bien, en donde se sitúa su axiología. A todo ello se dedicarán los tres siguientes capítulos, respectivamente. Por último, se proponen media docena de conclusiones.

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting's view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference.

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen's remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed in answering what a proof is. Gentzen's ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier.

Es presenten les motivacions pedagògiques per incloure l'algorísmica com a part del currículum i la importància perquè aquesta presència tingui més pes en un futur pròxim, l'evolució d'aquests estudis en l'educació secundària a Catalunya, i breu referència de la situació referent a altres països de l'entorn.

The aim of this chapter is to propose strategies for improving ethical decision-making in organizations by taking into consideration the contributions of neuroethics research. Our hypothesis is that results from research in neuroscience and cognitive science offer new insight and perspectives on ethical decision-making in organizations. This chapter has three main parts. The first part is descriptive, and we present the main factors affecting unethical decision-making in organizations. Understanding why good people make or contribute to unethical decisions in organizations is a first step. In the second part, we discuss the contribution of neuroscience research to ethical decision-making. Finally, we propose strategies and actions that managers should take into account to develop a more consistent ethical decision-making model for individuals and organizations.

This paper analyzes the linkage between Husserl’s phenomenology and Brouwer’s and Heyting’s intuitionism. The linkages are presented from some theses of Intuitionism and Phenomenology, namely, the role of our subjectivity in the constitution of the mathematical world; the notion of fulfilment and the study of the concept of negation in classical logic., El objetivo de este trabajo es resaltar el vínculo entre la fenomenología de Husserl y el intuicionismo de Brouwer y Heyting. Los vínculos se presentaran siguiendo algunas de las tesis principales del Intuicionismo y de la Fenomenología, a saber, el papel que tiene nuestra subjetividad en la constitución-construcción del mundo; a la noción de cumplimiento y en el estudio del concepto de negación en la lógica clásica.

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent (Frege’s) logicism is compatible with (Dehaene’s) intuitionism., En Numbers and Arithmetic Frege escribe que kindergarten-numbers y «un modo a priori de cognición» pueden tener «un origen geométrico». Esto se asemeja a algunos descubrimientos recientes sobre cognición aritmética. En mi artículo, exploro la semejanza entre la útlima posición de Gottlob Frege acerca del origen geométrico del conocimiento aritmético, y algunas posiciones actuales en la bibliografía sobre cognición aritmética, especialmente la de Stanislas Dehaene. En mi análisis, intento principalmente determinar hasta qué punto el logicismo (de Frege) es compatible con el intuicionismo (de Dehaene).

In this thesis I defend a view in intuitionist moral epistemology on which some basic moral beliefs are justified by perception. Moral intuitionists, I argue, ought to hold that we have some justified particular moral beliefs. Since the paradigm non-moral case of a justified particular belief is perceptual, the possibility of justified moral perception ought to be of interest to moral intuitionists. Drawing upon the work of Susanna Siegel, I argue that moral concepts can cognitively penetrate our perceptual content in the same manner as complex non-moral concepts. Like non-moral perceptual beliefs, moral perceptual beliefs arrive as perceptual seemings, and become fully-fledged beliefs only when we endorse their content. I argue that forming a conceptually rich moral perceptual belief entails taking on a commitment to certain core claims implicit in the associated moral concept: claims which, if false, render the associated moral concept defective. I tackle two major problems with my account of intuitionist moral perception. First, that cognitive penetration, especially by a complex concept, entails that the penetrated belief is epistemically dependent on the relevant complex concept, and thus is inferentially justified if it is justified at all. Second, that my account of intuitionist moral perception would entail that very many bigoted moral perceptions are justified basic beliefs. In response to the first, I rely upon Robert Audi’s work on epistemic dependence. While cognitive penetration entails epistemic dependence, it only entails negative epistemic dependence and not positive epistemic dependence. Further, I argue that only positive epistemic dependence would require a justified belief to be inferentially justified. I distinguish between several ways in which a belief can be inferential and conclude that my view only entails negative epistemic dependence. In response to the second, I outline a further criterion that must be met for a moral perception to be ju

How to Say 'Yes' or 'No': Logical Approaches to Modes of Assertion and Denial. Book of Abstracts.

We naturally take for granted that by performing inferences we can obtain evidence or grounds for assertions that we make. But logic should explain how this comes about. Why do some inferences give us grounds for their conclusions? Not all inferences have that power. My first aim here is to draw attention to this fundamental but quite neglected question. It seems not to be easily answered without reconsidering or reconstructing the main concepts involved, that is, the concepts of ground and inference. Secondly, I suggest such a reconstruction, the main idea of which is that to make an inference is not only to assert a conclusion claiming that it is supported by a number of premisses, but is also to operate on the grounds that one assumes or takes oneself to have for the premisses. An inference is thus individuated not only by its premisses and conclusion but also by a particular operation. A valid inference can then be defined as one where the involved operation results in a ground for the conclusion when applied to grounds for the premisses. It then becomes a conceptual truth that a valid inference does give a ground for the conclusion provided that one has grounds for the premisses.

After introducing semantic anti-realism and the paradox of knowability, the paper offers a reconstruction of the anti-realist argument from the theory of understanding. The proposed reconstruction validates an unrestricted principle to the effect that truth requires the existence of a certain kind of “demonstration”. The paper shows that the principle fails to imply the problematic instances of the original unrestricted knowability principle but that the overall view still has unrestricted epistemic consequences. Appealing precisely to the paradox of knowability, the paper also argues, against BHK semantics, for the non-constructive character of the demonstrations envisaged by anti-realists, and contends that, in such a setting, one of the most natural arguments in favour of a revision of classical logic loses all its force.

In 2001 Lee, Jones and Ben-Amram introduced the notion of size-change termination (SCT) for first order functional programs, a sufficient condition for termination. They proved that a program is size-change terminating if and only if it has a certain property which can be statically verified from the recursive definition of the program. Their proof of the size-change termination theorem used Ramsey's Theorem for pairs, which is a purely classical result. In 2012 Vytiniotis, Coquand and Wahlsteldt intuitionistically proved a classical variant of the size-change termination theorem by using the Almost-Full Theorem instead of Ramsey's Theorem for pairs. In this paper we provide an intuitionistic proof of another classical variant of the SCT theorem: our goal is to provide a statement and a proof very similar to the original ones. This can be done by using the H-closure Theorem, which differs from Ramsey's Theorem for pairs only by a contrapositive step. As a side result we obtain another proof of the characterization of the functions computed by a tail-recursive SCT program, by relating the SCT Theorem with the Termination Theorem by Podelski and Rybalchenko. Finally, by investigating the relationship between them, we provide a property in the "language" of size-change termination which is equivalent to Podelski and Rybalchenko's termination.

Assume that we may prove in Classical Functional Analysis that a primitive recursive relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that such a proof of well-foundation may be made intuitionistic. We conclude that if we are able to formulate any mathematical problem as the inductive well-foundation of some primitive recursive relation, then intuitionistic and classical provability coincide, and for such a statement of well-foundation we may always find an intuitionistic proof if we may find a proof at all. The core of intuitionism are the methods for computing out data with given properties from input data with given properties: these are the results we are looking for when we do constructive mathematics. Proving that a primitive recursive relation R is inductively well-founded is a more abstract kind of result, but it is crucial as well, because once we proved that R is inductively well-founded, then we may write programs by induction over R. This is the way inductive relation are currently used in intuitionism and in proof assistants based on intuitionism, like Coq. In the paper we introduce the comprehension axiom for Functional Analysis in the form of introduction and elimination rules for predicates of types Prop, Nat->Prop, ..., in order to use Girard's method of candidates for impredicative arithmetic.

[EN] Two common beliefs in historical linguistics will be reviewed here: (1) that a once successful change which is now fully integrated in the grammar must have followed a pattern of sustained growth (a “constant rate” S-curve) all along its history; and (2) that only increasing frequencies are associated with grammaticalization processes in any significant or symptomatic way. The first belief is based on the linguist’s intuitions about contemporary use, since a gramatically unsurprising phenomenon is not likely to be considered obsolescent or on the verge of abandonment. The second belief draws on an equivalence (also of intuitive nature, though observable in many particular changes) between grammaticalization as a process resulting in the emergence of processing routines and increased use of thus grammaticalized elements and constructions, a view that, albeit not necessarily wrong itself, has left in the shadows the role of failed changes in grammaticalization continua. Quantitative analysis of large amounts of data allows us to challenge both generalizations. By looking closely at the grammaticalization and diffusion of the definite article el before completives headed by que (‘that’) in Spanish, we will show that a clearly recessive phenomenon might go perfectly unnoticed to both speakers and linguists, thus revealing the potential frailty of intuition and introspective judgment as sources of access to historically conditioned grammatical facts. We will also claim that a failed evolution might in turn motivate the failure of a related (in fact, derived) construction at the far end of a grammaticalization continuum, which is tantamount to assuming that frequency drops can be both significant and symptomatic in the last stages of secondary grammaticalization, and possibly as well herald the reuse or exaptation of highly grammaticalized forms., [ES] Este trabajo pasa revista a dos creencias comunes en la lingüística histórica: que el cambio que tuvo éxito en algún segmento del pasado y es de uso común en nuestros días ha debido llegar hasta nosotros según un patrón de crecimiento constante; y que solo las frecuencias crecientes resultan esperables, y por tanto informativas y sintomáticas, cuando se abordan los cambios por gramaticalización. La primera idea está asentada en la intuición que el lingüista posee acerca de la variedad actual, pues nada invita a considerar caduco o en abandono un fenómeno que no se juzga gramaticalmente extraño. La segunda idea está basada en una equivalencia (igualmente intuitiva, aunque observable en multitud de evoluciones) entre la creación de rutinas de procesamiento que supone una gramaticalización y el mayor uso de las construcciones y elementos gramaticalizados, lo que lleva a desatender el papel de los cambios fracasados en los continuos de gramaticalización. El análisis cuantitativo de grandes masas de datos a través de corpus electrónicos permite hoy la enmienda de estas dos generalizaciones excesivas. A través del estudio de los orígenes y la di- fusión del artículo ante subordinadas completivas con que, mostraremos que un fenómeno clara- mente recesivo puede no ser percibido como anómalo, lo que revela los peligros de la intuición gramatical propia como fuente exclusiva de acceso a los datos. Sugeriremos, además, que una evolución en declive puede motivar el fracaso de otra derivada en los últimos eslabones de una cadena evolutiva, de modo que también los descensos en frecuencia pueden resultar esperables en procesos de retracción característicos de la gramaticalización secundaria y, a su vez, actuar como heraldos de la reutilización o exaptación de las formas gramaticalizadas.

It is proposed a formalization of the basic concepts of socionics with algebraic models of intuitionistic logic., Предложена формализация основных понятий соционики при помощи алгебраических моделей интуиционистской логики.

This paper explores Antonio Casos Christian thought as a form of religiosity that opposed Catholicism in Mexico. This Christianity com bined with liberal ideas and certain philosophies from the first haif of the 20th century: such as personalism and intuitionism, resulted in a faith that wasquite separate from the insti- tutionality defended by the Church. The author highlights the survival of Christianity despite che secularization process in nne of the most representative members of the Ateneo de la Juventud and Mexican intellectual life of the past century., El artículo aborda el pensamiento cristiano de Antonio Caso como una religiosidad opositora al catolicismo en México. Este cristianismo conjugado con las ideas liberales y algunas filosofías de la primera mitad del siglo xx, como el personalismo y el intuicionismo, dieron como resultado una fe apartada de la institucionalidad defendida por la Iglesia. Se destaca la pervivencia del cristianismo ante el proceso secularizador en uno de los miembros más representativos del Ateneo de la Juventud y de la intelectualidad mexicana en el siglo pasado.

In his paper “Framing Moral Intuitions” (Sinnott-Armstrong 2008), Sinnott-Armstrong argues that three studies which he cites provide evidence of unreliability in all moral intuitions in all circumstances. Therefore, he argues that all of one’s moral intuitions require inferential confirmation for justification. I argue that the studies do not strike me as evidence for this conclusion. Thus, anyone who reads the studies and is not struck that they are evidence for this conclusion can have justified moral intuitions without inferential confirmation. Further, arguments are given for why Sinnott-Armstrong’s arguments fail to show that all moral intuitions are unreliable in all circumstances. Finally an argument is given which shows that even if one accepts the master argument, it need not lead to moral scepticism.

The connection between the concepts of constructive building in mathematics and a special role to play for mental and cognitive processes in children is detected. It is given the illustration of construction of various objects in folklore, including fairy tales., Обнаружена связь между понятиями конструктивного построения в математике и особой ролью игры для развития психических и познавательных процессов у детей. Дана иллюстрация построения различных объектов в фольклоре, в том числе — в сказках.

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?

What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless sense. Would the latter alternative amount to realism with respect to proofs or grounds in a way that would be contrary to the supposedly anti-realistic standpoint underlying the epistemic understanding of linguistic expressions? Before discussing this question, I shall consider reasons for construing linguistic meaning epistemically and relations between such reasons and reasons for taking an anti-realist point of view towards the discourse in question., AuthorCount:1

El filósofo marxista G. A. Cohen dirigió un ataque muy severo contra el liberalismo igualitarista de J. Rawls. A su juicio, algunos argumentos de la obra de Rawls, en especial el argumento de los incentivos al talento, justifican desigualdades que pueden llegar a ser muy acusadas. De hecho, el argumento de los incentivos se ha constituido en una de las principales coartadas a la hora de legitimar las políticas de desnivelación social que han acometido gobiernos de diverso signo. Este ensayo trata de mostrar que, en su legítima denuncia contra el fomento de la desigualdad, Cohen erró el blanco de su ataque: la filosofía política de Rawls no sólo dispone de argumentos suficientes para denunciar las desigualdades acusadas, sino que los posee en mayor medida que la propia teoría de Cohen. Por último, la resistencia de la teoría de Rawls a este tipo de críticas empujó a Cohen hacia tesis morales y metaéticas radicales, extremas e incompatibles con cualquier interpretación del marxismo., The Marxist philosopher G. A. Cohen launched a harsh attack against J. Rawls’ egalitarian Liberalism. In his view, some arguments of Rawls’ theory, especially his approval of incentives for talent, justify broad and considerable inequalities. Moreover, the incentives for talent became the main alibi for all the policies orientated to foster inequality and implemented by governments with different political badges. This essay attempts to prove that, in his legitimate denunciation of the growing inequality, Cohen chose the wrong target: Rawls’ political philosophy owns enough arguments to condemn broad inequalities, including some inequalities which would be inevitable from Cohen’s theory. Finally, Rawls’ resilience against these sorts of criticisms pushed Cohen’s theory towards some moral and metaethical positions which can be judged as radical, extreme and incompatible with any interpretation of what we may call «Marxism».

In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus). Algorithms based in our de?nitions yields decision methods for these calculi.

One of Da Costa's motives when he constructed the paraconsistent logic Cw was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to CWo Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. The paper then investigates dualising the intuitionistic conditional in the same way. This establishes various connections between the logic, and a logic called in the literature 'Brouwerian logic' or 'closed-set logic'., http://dx.doi.org/10.5007/1808-1711.2009v13n2p165Uma das razões de da Costa para erigir a lógica paraconsistente Cw foi a dualização da negaço intuicionista. Neste artigo eu exploro urn modo alternativo de chegar a este resultado. Uma lógica é definida a partir de uma semântica de Kripke para a lógica intuicionista, e então dualizando as condições de verdade para a negação. Várias propriedades desta lógica são estabelecidas, incluindo suas relações com Cw. O Sistemas de tableau e de dedução natural são apresentados, bem como as estruturas algébricas apropriadas. 0 artigo investiga então a dualização do condicional intuicionista seguindo o mesmo procedimento. Isso estabelece várias conexões entre a lógica, e uma lógica denominada na literatura de 'lógica Brouweriana' ou lógica 'closed-set'.

Dag Prawitz has argued (Prawitz 1998) that it is possible intuitionist- ically to prove the validity of ‘A → there is a proof of [A]’ by induction over formula complexity, provided we observe an ob ject language/meta- language distinction. In the present paper I mainly argue that if the ob ject language with its axioms and rules can be represented as a formal system, then the proof fails. I also argue that if this restriction is lifted, at each level of the language hierarchy, then the proof can go through, but at the expense of virtually reducing the concept of a proof to that of truth in a non-constructive sense., PAVA

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s. The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. The volume will be of interest primarily to researchers and graduate students of philosophy, logic, mathematics and theoretical computer science. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields.

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s. The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.

Three topics, namely, computer science, philosophical logic, and mathematics, meet in intuitionistic type theory, which thus simultaneously is a programming language, a philosophy of language, and a foundation of mathematics. The present thesis compares, relates, and equates two concepts, one from philosophical logic and one from computer science, viz., reference and computation. In mathematical practice, we are used to viewing mathematical expressions as referring to their value or object, as 2 + 2 refers to 4: it is the responsibility of the foundations of mathematics to explain exactly what these objects are and how the expressions refer to them. The nature of mathematics, and related issues, such as reference and computation, have been debated a long time, in particular around the turn of the last century. The position defended in this thesis is that intuitionism, i.e., the philosophy behind intuitionistic type theory, provides a satisfactory answer to these questions, with the additional benefit of increasing the applicability of mathematics. Of course, these benefits do not come for free: some modern mathematical practices have to be given up, most notably the nonconstructive existence proofs. In addition to critical discussions of the topics mentioned above, the main contributions of this thesis are that computations are brought into the language of intuitionistic type theory, and that intuitionistic type theory is adopted to eager evaluation. The latter contribution is of particular importance in computer science.

In this paper we contribute to the terminological debate about Atanassov's use of the term "Intuitionistic" in defining his structure based on ortho-pairs of fuzzy sets. In particular, we stress that it is defined as "intuitionistic" a negation which from one side does not satisfy a standard property of the intuitionistic Brouwer negation (contradiction law) and on the contrary asserts some principles rejected by intuitionism (strong double negation law and one of the de Morgan laws). An algebraic Brouwer negation is studied in the context of IFS showing that it can be induced from a Heyting implication. A similar situation occurs in the case of standard fuzzy sets (FS). Some conditions which allow one to distinguish from the algebraic point of view FS from IFS are treated. Finally, a particular subclass of IFS consisting of ortho-pairs of crisp sets (denoted by ICS) is studied, showing that shadowed sets can be algebraically identified with ICSs. (c) 2006 Elsevier B.V. All rights reserved.

An exposition and interpretation of Antonio Caso’s concept of intuition. Intuitionism is the basis of Caso’s thought and the path he followed to solve his philosophical concerns: the critic of positivism, the problem of transcendence, the ethical values, the refoundation of metaphysics. This essay seeks to prove that an evolution of intuition’s idea is the main issue of the Casian thought. The different concepts of intuition explored by Caso based on three authors, Schopenhauer, Bergson and Husserl, are examined. From the latter and against him, he makes his original contribution: the analogical intuition applied to the knowledge of human existence and to ethics., Esta es una exposición e interpretación del concepto de intuición de Antonio Caso. El intuicionismo es la base del pensamiento de Caso y el camino que siguió para resolver sus preocupaciones filosóficas: la crítica al positivismo, el problema de la trascendencia, los valores éticos, la refundación de la metafísica. En este ensayo pretendemos demostrar que la evolución de la idea de intuición es el hilo conductor de la obra casiana. Revisamos los diferentes conceptos de intuición que progresivamente fue desarrollando el filósofo en sus obras, basándose especialmente en tres autores: Schopenhauer, Bergson y Husserl. A partir del último y contra él, hace su aportación original: la intuición analógica aplicada al conocimiento de la existencia humana y a la ética.

The lack of a systematic structuring of its principles is one of the greatest difficulties of the explanatory paradigm that stems from the late developments of Wittgenstein's thought. When we try to build that architectural frame, however, we need to have recourse to "molecular" positions which have an obvious fregean origin. The anti-realistic programme of the English philosopher M. Dummett is without any doubt one of the most interesting attempts of conciliating both types of ideas: Wittgenstein's influence on the basic assumptions on which Dummett builds the guidelines that a theory of meaning has to follow seems clear. But, at the same time, he does not seem to be willing accept that explanation can interpreted restrictively as a specification of the different linguistic "uses", as this would be equal to putting an end to the notion of an objective meaning and would lead us to admitting the impossibility of a systematic theory of language performance. The problems that arise form his proposal are the topic we discus in this paper., Una de las mayores dificultades del paradigma explicativo que arranca del último Wittgenstein es la ausencia de una estructuración sistemática de sus principios. Cuando se trata de construir ese marco arquitectónico parece necesario acudir, sin embargo, a posiciones moleculares de origen claramente fregeano, El programa anti-realista del filósofo inglés M. Dummett constituye sin duda uno de los intentos más interesantes de conjugar estos dos tipos de ideas: si, por una parte, parece clara la influencia de Wittgenstein en los presupuestos básicos sobre los que Dummett construye las líneas que ha de seguir una teoría del significado, por otra, no parece estar dispuesto a admitir que tal explicación pueda agotarse completamente en una especificación de los diferentes usos lingüísticos, porque eso supondría acabar con la noción misma de un significado objetivo y, por tanto, nos llevaría a tener que admitir la imposibilidad de una teoría sistemática del funcionamiento del lenguaje. Los problemas que plantea su propuesta constituyen el objeto de discusión de este artículo.

This review goes into Nicola Grana's contributions to the study of logic, especially his comparative examination of different systems of paraconsistent logics, including transitive logic (i.e. contradictorial gradualism). Grana's scope is broad, ranging from quantum logic to intuitionism.