Comparative valuation of different policy interventions often requires interpersonal comparability of benefit. In the field of health economics, the metric commonly used for such comparison, quality adjusted life years (QALYs) gained, has been criticized for failing to respect the equality of all persons’ intrinsic worth, including particularly those with disabilities. A methodology is proposed that interprets ‘full quality of life’ as the best health prospect that is achievable for the particular individual within the relevant budget constraint. This calibration is challenging (...) both conceptually and operationally as it shifts dramatically when technology or budget developments alter what can be achieved for incapacitated individuals. The proposal nevertheless ensures that the maximal achievable satisfaction of one person’s preferences can carry no more intrinsic value than that of another. This approach, which can be applied to other domains of social valuation, thus prevents implicit discrimination against the elderly and those with irremediable incapacities. (shrink)

An essay on the compatibility of Jewish and Muslim aspirations for the Land that argues that a vision of peaceful coexistence to provide a spur to the settling of grievances, but that this vision must respect the deeply felt aspirations of both communities, including in particular those underpinned by religious lore. Existing formulations of the aims of each community are blatantly inconsistent; it is time to attempt to characterise the just and religiously authentic goals of each community in ways that (...) are compatible. I argue that there religious and personal dimensions of the conflict that have been displaced by less deep-seated political goals, but which can be addressed more readily. I outline a tiered constitutional settlement that could address legitimate aspirations by avoiding the anathema of partition. (shrink)

Even where willingness-to-pay as a measure of welfare impact is adjusted for diminishing marginal utility, welfare economics is shown to favour policies that add to the life expectancy or that enhance the quality of life of persons who are already better-off. I propose an alternative, Equal Respect methodology, under an axiomatic claim that at the point of decision the prospective life years of all individuals are of equal intrinsic social value. This justifies equal valuation of risk mitigation across all persons; (...) similarly, all appraised impacts should be scaled to accord equal respect to difficult but no-less-valuable lives. (shrink)

_Groups in Conflict_ addresses the conflict and tensions that exist between impartiality and partiality in political philosophy, ordinary thought, and practice by setting theoretical arguments in the context of contemporary issues such as immigration and public policy. Donald Franklin asserts that two camps of ethicists—those concerned with political philosophy and those concerned with personal morality—have been ignoring the implications of inconsistency in their mutual approaches. Far more than just exposing these irreconcilable differences, Franklin also proposes the modifications (...) necessary to approach the nature of human equality. (shrink)

In 1947 Donald Cary Williams claimed in The Ground of Induction to have solved the Humean problem of induction, by means of an adaptation of reasoning first advanced by Bernoulli in 1713. Later on David Stove defended and improved upon Williams’ argument in The Rational- ity of Induction (1986). We call this proposed solution of induction the ‘Williams-Stove sampling thesis’. There has been no lack of objections raised to the sampling thesis, and it has not been widely accepted. In (...) our opinion, though, none of these objections has the slightest force, and, moreover, the sampling thesis is undoubtedly true. What we will argue in this paper is that one particular objection that has been raised on numerous occasions is misguided. This concerns the randomness of the sample on which the inductive extrapolation is based. (shrink)

Whether or not quantum physics can account for molecular structure is a matter of considerable controversy. Three of the problems raised in this regard are the problems of molecular structure. We argue that these problems are just special cases of the measurement problem of quantum mechanics: insofar as the measurement problem is solved, the problems of molecular structure are resolved as well. In addition, we explore one consequence of our argument: that claims about the reduction or emergence of molecular structure (...) cannot be settled independently of the choice of a particular resolution to the measurement problem. Specifically, we consider how three standard putative solutions to the measurement problem inform our under- standing of a molecule in isolation, as well as of chemistry’s relation to quantum physics. (shrink)

A polemical account of Australian philosophy up to 2003, emphasising its unique aspects (such as commitment to realism) and the connections between philosophers' views and their lives. Topics include early idealism, the dominance of John Anderson in Sydney, the Orr case, Catholic scholasticism, Melbourne Wittgensteinianism, philosophy of science, the Sydney disturbances of the 1970s, Francofeminism, environmental philosophy, the philosophy of law and Mabo, ethics and Peter Singer. Realist theories especially praised are David Armstrong's on universals, David Stove's on logical probability (...) and the ethical realism of Rai Gaita and Catholic philosophers. In addition to strict philosophy, the book treats non-religious moral traditions to train virtue, such as Freemasonry, civics education and the Greek and Roman classics. (shrink)

Hume's Treatise arguments concerning space, time, and geometry, especially ones involving his denial of infinite divisibility; have suffered harsh criticism. I show that in the section "Of the ideas of space and time," Hume gives important characterizations of his skeptical approach, in some respects Pyrrhonian, that will be developed in the rest of the Treatise. When that approach is better understood, the force of Hume's arguments can be appreciated, and the influential criticisms of them can be seen to miss the (...) mark. (shrink)

Understanding Hume’s theory of space and time requires suspending our own. When theorizing, we think of space as one huge array of locations, which external objects might or might not occupy. Time adds another dimension to this vast array. For Hume, in contrast, space is extension in general, where being extended is having parts arranged one right next to the other like the pearls on a necklace. Time is duration in general, where having duration is having parts occurring one aft (...) er another like the notes of a song. Hume’s diff erent view stems from his empiricism, his reliance on experience and observation as the foundation of our concepts. Nothing in our experience suggests a single vast array of locations. Rather, we simply notice that bodies are similar insofar as they have lengths that can be compared. Likewise, nothing in our experience suggests a single dimension of time. Rather, we simply notice that diff erent successions are similar insofar as they have durations that can be compared. Th eorizing that these observations show there to be a single multidimensional array goes well beyond the evidence for Hume. As a skeptic, he fi nds himself unable to assent to theories that stray too far beyond the deliverances of the senses. For Hume, the ideas of space and time are each a general idea of simple—partless— objects arrayed in a certain manner. He argues that the structures of the ideas of space and time refl ect the structures of space and time. Th erefore, space and time are not infi - nitely divisible, and they are ways simple objects are arrayed. Consequently, there is no such thing as empty space nor time without change. (shrink)

Explains Aristotle's views on the possibility of continuous variation between biological species. While the Porphyrean/Linnean classification of species by a tree suggests species are distributed discretely, Aristotle admitted continuous variation between species among lower life forms.

I argue that an individual has aspects numerically identical with it and each other that nonetheless qualitatively differ from it and each other. This discernibility of identicals does not violate Leibniz's Law, however, which concerns only individuals and is silent about their aspects. They are not in its domain of quantification. To argue that there are aspects I will appeal to the internal conflicts of conscious beings. I do not mean to imply that aspects are confined to such cases, but (...) the best way to start is to recognize them experientially. We can feel the conflicts within ourselves. In doing so we can feel some of our aspects. I will try to enhance our understanding of the concept of aspect by listing and formalizing some principles for its use. After that I will argue that all sorts of individual things have aspects, not just people who are conflicted. (shrink)

This chapter offers a new criticism of the Mind argument that is both decisive and instructive. It introduces a plausible principle (γ) that places a requirement on one’s having a choice about an event whose causal history includes only other events. Depending on γ’s truth-value, the Mind argument fails in such a way that one or the other of the two main species of libertarianism is the best approach to the metaphysics of freedom. Libertarians argue the compatibility of freedom and (...) indeterminism, and their biggest obstacle is the Mind argument that argues an incompatibility between the two. The chapter aims to build a case for the truth of γ, and so for nonreductive libertarianism. This is achieved by defending it from the best objections that have been brought to light, thereby emphasizing γ’s prima facie plausibility. (shrink)

Confucius gave counsel that is notoriously hard to follow: "What you do not wish for yourself, do not impose on others" (Huang 1997: 15.24). People tend to be concerned with themselves and to be indifferent to most others. We are distinct from others so our self-concern does not include them, or so it seems. Were we to realize this distinctness is merely apparent--that our true self includes others--Confucius's counsel would be easier to follow. Concern for our true self would extend (...) concern beyond the narrow selves we appear to be. The neo-Confucians held just such a view. They espoused an identity with the universe and everything in it, arguing that this identity explains a natural concern for everyone and everything, not just for our narrow selves. However, many things in the universe differ from each other, that is, some have qualities others lack. If they are all one and the same thing then that one thing differs from itself. I will suggest that the objection can be answered with some metaphysical innovation. I will address the objection by sketching a theory--call it the theory of aspects--that explains how numerically identical things can differ qualitatively. (shrink)

This essay interprets Butler’s distinction between identity in the loose and popular sense and in the strict and philosophical sense. Suppose there are different standards for counting the same things. Then what are two distinct things counting strictly may be one and the same thing counting loosely. Within a given standard identity is one-one. But across standards it is many-one. An alternative interpretation using the parts-whole relation fails, because that relation should be understood as many-one identity. Another alternative making identity (...) relative to sort fails, because whole and parts can be of the same sort. (shrink)

ABSTRACT According to David Lewis, alteration is "qualitative difference between temporal parts of something." It follows that moments, since they are simple and lack temporal parts, cannot alter from future to present to past. Here then is another way to put McTaggart's paradox about change in tense. I will appeal to my theory of Aspects to rebut the thought behind this rendition of McTaggart. On my theory, it is possible that qualitatively differing things be numerically identical. I call these differing, (...) numerically identical things "aspects." I will argue that alteration can be a qualitative difference between temporal aspects of something that lacks temporal parts. So a moment can alter in tense. By rejecting Lewis's assumption my theory can solve this version of McTaggart's paradox. (shrink)

The concept of metaphor as primarily a vehicle for conveying ideas, even if unusual ones, seems to me as wrong as the parent idea that a metaphor has a special meaning. I agree with the view that metaphors cannot be paraphrased, but I think this is not because metaphors say something too novel for literal expression but because there is nothing there to paraphrase. Paraphrase, whether possible or not, inappropriate to what is said: we try, in paraphrase, to say it (...) another way. But if I am right, a metaphor doesn't say anything beyond its literal meaning. This is not, of course, to deny that a metaphor has a point, nor that that point can be brought out by using further words.... My disagreement is with the explanation of how metaphor works its wonders. To anticipate: I depend on the distinction between what words mean and what they are used to do. I think metaphor belongs exclusively to the domain of use. It is something brought off by the imaginative employment of words and sentences and depends entirely on the ordinary meanings of those words and hence on the ordinary meanings of the sentences they comprise. Donald Davidson is University Professor of philosophy at the University of Chicago. He is the author of many important essays, including "Actions, Reasons and Causes," "Causal Relations," and "Truth and Meaning," coauthor of Decision-Making: An Experimental Approach, and coeditor of Words and Objections, Semantics of Natural Language, and The Logic of Grammar. (shrink)

There is more than one way to say that composition is identity. Yi has distinguished the Weak Composition thesis from the Strong Composition thesis and attributed the former to David Lewis while noting that Lewis associates something like the latter with me. Weak Composition is the thesis that the relation between the parts collectively and their whole is closely analogous to identity. Strong Composition is the thesis that the relation between the parts collectively and their whole is identity. Yi is (...) right that Strong Composition does not fully reflect my view. We must recognize further what, borrowing Lewis's characterization, could be called the "stronger, and stranger" Composition thesis, or what I'll call the Stronger Composition thesis for short. On Strong Composition, it is only collectively that the parts are identical with the whole. On Stronger Composition, they are individually identical with it as well. I will explain and motivate this thesis. (shrink)

Construing the instantiation of a universal by a particular in terms of my theory of aspects resolves the basic mystery of this "non-relational tie", and gives theoretical unity to the four characteristics of instantiation discerned by Armstrong. Taking aspects as distinct in a way akin to Scotus's formal distinction, I suggest that instantiation is the sharing of an aspect by a universal and a particular--a kind of partial identity. This approach allows me to address Plato's multiple location and One over (...) Many problems, Bradley's problem concerning the instantiation of relations, and the problem of change. (shrink)

How is it possible for David Hume to be both withering skeptic and constructive theorist? I recommend an answer like the Pyrrhonian answer to the question how it is possible to suspend all judgment yet engage in active daily life. Sextus Empiricus distinguishes two kinds of assent: one suspended across the board and one involved with daily living. The first is an act of will based on appreciation of reasons; the second is a causal effect of appearances. Hume makes the (...) same distinction, only he extends the sort of assent involved in daily life to theoretical matters as well. He is a skeptic both in finding no reason to grant the first sort of assent and in being subject to the second. (shrink)

In the correspondence with Arnauld, Leibniz contends that each corporeal substance has a substantial form. In support he argues that to be real a corporeal substance must be one and indivisible, a true unity. I will show how this argument precludes a tempting interpretation of corporeal substances as composite unities. Rather it mandates the interpretation that each corporeal substance is a single monad.

El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...) Dedekind siguiendo el texto “Set Theory” de Jech, y otras ideas propias del autor de este artículo. Adicionalmente, se formulan algunas versiones débiles del axioma de elección relacionadas con las álgebras booleanas, las cuales son también de gran importancia para la investigación en teoría de conjuntos y teoría de modelos, pues estas son poderosas técnicas de construcción de modelos, como por ejemplo, el teorema de compacidad (permite construir modelos no estándar, etc) y el teorema del ultrafiltro, que permite construir ultraproductos (pueden ser usados para investigar problemas de cardinales grandes, etc). Se presentan algunas referencias de problemas abiertos sobre el tema. -/- The objective of this paper is to present a demonstration of a classical theorem on boolean algebras and partial orders of current relevance in set theory, as for example, for applications of model construction method called forcing" (with boolean algebras complete or with partial orders). The theorem to be proved is as follows: Any partial order can be extended to a single complete boolean algebra (up to isomorphism)". Where to extend means embed densely". Such a demonstration is done using Dedekind's cuts following the text Set Theory" of Jech, and other ideas of the author of this article. In addition, some weak versions of the axiom of choice related to boolean algebras are formulated, which are also of great importance for the research in set theory and model theory, since this are powerful model construction techniques, such as the compactness theorem (allows the construction of non-standard models, etc.) and the ultralter theorem, which allows the construction of ultraproducts (can be used to investigate problems of large cardinals, etc). Some references of open problems on the subject are presented. (shrink)

Eshel Ben-Jacob’s manuscript entitled ‘Bacterial wisdom, Gödel’s theorem and creative genomic webs’ summarizes decades of work demonstrating adaptive mutagenesis in bacterial genomes. Bacterial genomes, each an essential part of a Kantian whole that is a single bacterium, are thus not independent of the environment as sensed; and a single bacterium is therefore a semiotic entity. Ben-Jacob suggests this but errs in 1) assigning autonomy to the genome, and 2) analogizing through computation without making clear whether he is doing so for (...) illustrative purposes or making committed ontological propositions. We reinterpret adaptive mutagenesis and related phenomena in ways both metaphysically rigorous and revealing. We conclude that bacteria are much farther removed from the ‘self-organizing’ world of inanimate process than from the Peircian world of signs; and a critical reappraisal of existing knowledge can enhance our understanding of selfhood, semiosis, and the roots of subjective experience. (shrink)

I argue via examples that there are cases in which things that are not two distinct things qualitatively differ without contradiction. In other words, there are cases in which something differs from itself. Standard responses to such cases are to divide the thing into distinct parts, or to conceive of the thing under different descriptions, or to appeal to different times, or to deny that the property had is the property lacked. I show these responses to be unsatisfactory. I then (...) gather and systematize available ways of talking about such cases with phrases like ‘insofar as’, ‘qua’, ‘to the extent that’, ‘in some respect’, etc., while paying special attention to the scope of ‘not’ when used with these phrases. This allows me to show how we can speak of self-differing without contradiction. (shrink)

The ancient theory of substance and accident is supposed to make sense of complex unities in a way that respects both their unity and their complexity. On Hume’s view such complex unities are only fictitiously unities. This result follows from his thoroughgoing critique of the theory of substance. I will characterize the theory Hume is critiquing as it is presented in Locke, presupposing what Bennett calls the “Leibnizian interpretation.” Locke uses the word ‘substance’ in two senses. Call substance in the (...) first sense “individual substance” and in the second sense “pure substance.” I will discuss the seven main parts of Hume’s view: (I) that we have no idea of pure substance, (II) that there is no complex individual substance, except in a loose sense, (III) that the fiction of complex individual substance arises in a way parallel to that of the fiction of identity through time, and (IV) results in the fiction of pure substance, (V) that simple qualities and perceptions satisfy the definition of individual substance, (VI) that there is no such thing as inherence, and (VII) that there is no such thing as pure substance. Hume’s views on substance are often mentioned without being discussed in detail. Kemp Smith, Stroud, and Garrett, for example, mostly summarize various claims of Hume in the course of expounding on his theory of the idea of personal identity. In contrast, I will attempt to present a systematic treatment of Hume on substance as a refutation of Locke. (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

Hume discusses the distinction of reason to explain how we distinguish things inseparable, and so identical, e.g., the color and figure of a white globe. He says we note the respect in which the globe is similar to a white cube and dissimilar to a black sphere, and the respect in which it is dissimilar to the first and similar to the second. Unfortunately, Hume takes these differing respects of resemblance to be identical with the white globe itself. Contradiction results, (...) undermining his theory of abstraction. The way out is apparently to admit an intrinsic complexity in even simple things. (shrink)

El objetivo principal de este artículo es presentar la demostración directa del Teorema de compacidad de la Lógica de primer orden (Gama tiene un modelo si y sólo si cada subconjunto finito de Gama tiene un modelo) que se realiza utilizando el Método de construcción de modelos llamado "Ultraproductos" que, a su vez, usa "Ultrafiltros". Actualmente es más común demostrar el Teorema de compacidad como un corolario del Teorema de completitud de Gödel y usar el método de reducción al absurdo (...) para probarlo. Sin embargo, vale la pena estudiar también esta prueba directa que usa Ultraproductos porque dicha técnica tiene importantes aplicaciones en investigaciones contemporáneas de matemáticas, por ejemplo en la Teoría de conjuntos y en el Análisis. Al final del artículo se realiza un breve comentario sobre Compacidad, Ultraproductos, Cardinales grandes y Modelos no estándar. (shrink)

Is a social complex identical to many united people or is it a group entity in addition to the people? For specificity, I will assume that a social complex is a plural subject in Margaret Gilbert’s sense. By appeal to my theory of Aspects, according to which there can be qualitative difference without numerical difference, I give an answer that is a middle way between metaphysical individualism and metaphysical holism. This answer will enable answers to two additional metaphysical questions: (i) (...) how can two social complexes have all the same members and (ii) how can there be a social complex of social complexes? (shrink)

The divide between oneself and others has made altruism seem irrational to some thinkers, as Sidgwick points out. I use characterizations of grief, especially by St. Augustine, to question the divide, and use a composition‐as‐identity metaphysics of parts and wholes to make literal sense of those characterizations.

Ordinary usage gives a way to think of identity through time: the Pittsburgh of 1946 was the same city as the Pittsburgh of today is--namely Pittsburgh. Problem: The Pittsburgh of 1946 does not exist; Pittsburgh still does. How can they have been identical? I reject the temporal parts view on which they were not but we may speak as though they were. Rather I argue that claiming their identity is not contradictory. I interpret ‘the Pittsburgh of 1946’ as ‘Pittsburgh as (...) it was in 1946’ and suggest that the apparent contradiction results from an ambiguity in the scope of ‘as’. (shrink)

According to the Weak Supplementation Principle (WSP)—a widely received principle of mereology—an object with a proper part, p , has another distinct proper part that doesn't overlap p . In a recent article in this journal, Nikk Effingham and Jon Robson employ WSP in an objection to endurantism. I defend endurantism in a way that bears on mereology in general. First, I argue that denying WSP can be motivated apart from the truth of endurantism. I then go on to offer (...) an explanation of WSP's initial appeal, argue that denying WSP fails to have untoward consequences for the rest of mereology, and show that the falsity of WSP is consistent with a primary guiding thought behind it. (shrink)

To the question whether Hume believed in mind-independent physical objects (or as he would put it, bodies), the answer is Yes and No. It is Yes when Hume writes “We may well ask, What causes induce us to believe in the existence of body? but ’tis in vain to ask, Whether there be body or not? That is a point, which we must take for granted in all our reasonings.” However the answer is No after inquiring into the causes of (...) that belief, when he writes, “What then can we look for from this confusion of groundless and extraordinary opinions but error and falsehood? And how can we justify to ourselves any belief we repose in them?” My view is that Hume’s Yes and No answers can best be accounted for by seeing him as a Pyrrhonian skeptic in the tradition of Sextus Empiricus. This interpretation, while finding considerable truth in the various positions canvassed, is closest to those of Norton, Fogelin, and especially Popkin. (shrink)

Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in combination) to topology, Measure Theory, algebra, combinatorial infinite, set theory and first-order logic, (...) also formulating some updated open problems of set theory that refer to non-principal ultrafilters on N, the Mathias’s model and the Solovay’s model. -/- Resumen: Los ultrafiltros son objetos matemáticos muy importantes en la investigación matemática [6, 22, 23]. Existen una gran variedad de teoremas clásicos en diversas ramas de la matemática donde se aplican ultrafiltros en su demostración, y otros teoremas clásicos que tratan directamente sobre ultrafiltros. El objetivo de este artículo es contribuir (de una manera divulgativa) con la investigación sobre ultrafiltros describiendo las demostraciones de algunos de tales teoremas relacionados (de manera única o combinada) con topología, teoría de la medida, álgebra, combinatoria infinita, teoría de conjuntos y lógica de primer orden, formulando además algunos problemas abiertos actuales de la teoría de conjuntos que se refieren a ultrafiltros no principales sobre N, al Modelo de Mathias y al Modelo de Solovay. (shrink)

Para matemáticos interesados en problemas de fundamentos, lógico-matemáticos y filósofos de la matemática, el axioma de elección es centro obligado de reflexión, pues ha sido considerado esencial en el debate dentro de las posiciones consideradas clásicas en filosofía de la matemática (intuicionismo, formalismo, logicismo, platonismo), pero también ha tenido una presencia fundamental para el desarrollo de la matemática y metamatemática contemporánea. Desde una posición que privilegia el quehacer matemático, nos proponemos mostrar los aportes que ha tenido el axioma en varias (...) áreas fundamentales de la matemática, su aplicación en la lógica de primer orden, así como una breve descripción de las pruebas de consistencia relativa debidas a Gödel y Cohen, las cuales establecieron su independencia del sistema axiomático Zermelo-Fraenkel (ZF). Con todo lo anterior mostraremos cómo el quehacer matemático contemporáneo se adscribe al platonismo matemático en los términos de Bernays y Ferreirós. Revisaremos también los argumentos de Zermelo y Cantor para permitir el uso de asunciones en la matemática, los cuales se acercan a los planteamientos de la investigación científica y esbozan relaciones con la filosofía de la práctica matemática. Finalmente, justificamos el uso del axioma de elección en la contemporaneidad, abogando por unas relaciones de equidad entre la matemática y la filosofía, presentando además su plena vigencia, a través de la referencia a algunos problemas abiertos en la actualidad que vinculan el axioma de elección con la teoría de Ramsey. (shrink)

En este artículo se presentan dos demostraciones del teorema de interpolación: Una para la lógica proposicional y otra para la lógica de primer orden. Ambas se realizan en el contexto de la teoría de modelos. El teorema de interpolación afirma que si A y B son fórmulas, donde A no es una contradicción, B no es válida, y B es una consecuencia lógica de A, entonces existe una fórmula C que esta escrita en el lenguaje común al de A y (...) de B, tal que C es una consecuencia lógica de A y B es una consecuencia lógica de C. El teorema de interpolación fue demostrado por primera vez para la lógica de primer orden por William Craig en 1957, y desde entonces se ha investigado la posibilidad de generalizarlo o aplicarlo. Dicho teorema tiene generalizaciones o aplicaciones en teoría de la demostración, teoría de modelos abstracta, ciencias de la computación, lógica modal, lógica intuicionista, etc. Se presentan ejemplos de aplicaciones o generalizaciones de la propiedad de interpolación relacionados con lógicas infinitarias, cuantificadores generalizados, segundo orden, no clásicas, abstractas, etc. También se ofrecen referencias de problemas abiertos sobre interpolación en el contexto de la teoría de modelos abstracta. (shrink)

The objective of this document is to present three introductory notes on set theory: The first note presents an overview of this discipline from its origins to the present, in the second note some considerations are made about the evaluation of reasoning applying the first-order Logic and Löwenheim's theorems, Church Indecidibility, Completeness and Incompleteness of Gödel, it is known that the axiomatic theories of most commonly used sets are written in a specific first-order language, that is, they are developed within (...) the framework of first-order logic, so this note is relevant; and the third note refers to the presence of mathematical platonism in the axioms of ZFC and in the axioms of "complete ordered field", it is known that the last axioms mentioned characterize (except isomorphism) the real number system and are currently used to develop the real Analysis in the context of set theory. It is hoped that this article will be of pedagogical utility for students interested in set theory and in the philosophy of mathematics (that are beginning in the subject). (shrink)

En este artículo realizamos una reconstrucción del Programa original de Hilbert antes del surgimiento de los teoremas limitativos de la tercera década del siglo pasado. Para tal reconstrucción empezaremos por mostrar lo que Torretti llama los primeros titubeos formales de Hilbert, es decir, la defensa por el método axiomático como enfoque fundamentante. Seguidamente, mostraremos como estos titubeos formales se establecen como un verdadero programa de investigación lógico-matemático y como dentro de dicho programa la inquietud por la decidibilidad de los problemas (...) matemáticos y en específico la decidibilidad de la Lógica de primer orden cobra peso. Luego pasamos a analizar como la inquietud por la decibilidad toma lugar dentro del pensamiento filosófico-matemático de Hilbert presentándose como uno de los grandes problemas a los cuales la metamatemática debe encontrar una solución, esto lo hacemos mostrando un contraste con autores, como John von Neumann y Roberto Torretti, quienes de alguna u otra manera no interpretan el problema de la decidibilidad de la Lógica de primer orden como un problema de peso dentro del programa original de Hilbert. Finalmente argumentamos que el resultado meta-teórico de Church puede entenderse como una refutación del optimismo intelectual que permea a todo el programa original de Hilbert. (shrink)

Armstrong has loose identity be an equivalence relation, yet in cases of something becoming something else, loose identity is not transitive. My alternate account has an attribution of loose identity be really two: a true attribution of an underlying relation (perhaps not transitive) and a false attribution--a Humean feigning-of strict identity. The feigning may become less appropriate as the underlying relation grows more distant. What makes it appropriate initially is that the underlying relation supports a predictable change in some collective. (...) The importance of the predictably changing collective is signaled by regarding it as a single thing. (shrink)

El estudio de los "cardinales grandes" es uno de los principales temas de investigación de la teoría de conjuntos y de la teoría de modelos que ha contribuido con el desarrollo de dichas disciplinas. Existe una gran variedad de tales cardinales, por ejemplo cardinales inaccesibles, débilmente compactos, Ramsey, medibles, supercompactos, etc. Tres valiosos teoremas clásicos sobre cardinales medibles son los siguientes: (i) compacidad débil, (ii) Si κ es un cardinal medible, entonces κ es un cardinal inaccesible y existen κ cardinales (...) inaccesibles menores que κ , y (iii) Si existe un cardinal medible, entonces el axioma de constructibilidad (V=L) es falso. El objetivo de este artículo es presentar una demostración de cada uno de estos tres teoremas en el contexto de la teoría de modelos usando ideas del texto de Chang y Keisler (Model Theory). Tales demostraciones tienen en común el uso del método de construcción de modelos llamado ultraproductos, de lógicas infinitarias o fragmentos de la lógica de segundo orden, y del axioma de elección. Cardinales grandes y/o ultraproductos son importantes en teoría de conjuntos, teoría de modelos, análisis matemático, teoría de la medida, probabilidades, topología, análisis funcional, física, teoría de números, finanzas, etc. (shrink)

Resumen: En la actualidad uno de los libros más usados para dar lógica elemental es el de Irving Copi y Carl Cohen (Introducción a la lógica, 2001), allí se presentan unas reglas para decidir la validez de los silogismos categóricos de forma estándar. Pero en tal texto ni en ninguno que nosotros conozcamos se ofrece una fundamentación de las mismas. Es decir, una demostración de que ellas son realmente una condición necesaria y suficiente de la validez de un silogismo categórico (...) de forma estándar, lo único que hemos leído es su motivación como condición necesaria. El objetivo de este trabajo es ofrecer una fundamentación de las mismas usando la noción de conjunto (o clase) y la técnica de diagramas de Venn, un concepto y una técnica que están expuestos en los principales textos de lógica elemental, incluyendo el referido anteriormente. -/- Abstract: Currently, one of the most widely used textbooks in elementary logic is the one by Irving Copi and Carl Cohen (Introduction to Logic, 2001). They offer rules for deciding the validity of standard categorical syllogisms. But neither that textbook, nor any other we know of, provides the foundations of these rules; that is to say, the demonstration that they are a necessary and sufficient condition for the validity of a standard categorical syllogism. We have only found motivations for the necessary condition. The main goal of this work is to provide a foundation of these rules using the notion of set as well as Venn diagrams, a concept and a technique exposed in the main textbooks of logic, including the aforementioned. (shrink)

We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

This article’s objetive is to present four application examples of Herbrand’s theorem to decide the validity of reasoning on first order language, in accordance whit Church’s Undecidability’s theorem. Also, to tell which is the principal problem around it. The logical resolution calculus will be worked on this article, which is a method used in artificial intelligence.

The work argues that engaging Africa's cultural and epistemic resources as social imaginaries, and not as metaphysical or ontological “essences,” could help practitioners of African philosophy overcome the cluster of shortcomings and undesirable features associated with “ethnophilosophy.” A number of points are outlined to buttress this claim. First, the framework of social imaginaries does not operate with the false assumption that Africa's cultural forms and epistemic resources are static and immutable. Second, this framework does not lend itself to sweeping generalizations (...) about Africa or large swathes of it. Third, the framework of social imaginaries remedies ethnophilosophy's problem of collectivism. Fourth, unlike ethnophilosophy, it does not romanticize and canonize a supposedly “idyllic” African past, which militates against a realistic and forward‐looking philosophizing. Finally, with the framework of social imaginaries, Africa's indigenous cultural and epistemic resources become amenable to being engaged with critical philosophical rigor. This not only enhances their potential for cross‐cultural philosophical conversation but also enhances their usefulness for addressing current sociopolitical issues affecting Africa. The discussion in this paper somewhat touches upon the question of method in African philosophy, using the ethnophilosophy problem to navigate the vast gamut of issues involved. (shrink)

El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...) ellas. (shrink)

¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...) Matemática y Fundamentos de la Matemática (ver). También en la entrada "The Continuum Hypothesis" de la web de Enciclopedia de Filosofía de la Universidad de Stanford existe información importante y actualizada al respecto. En esta breve nota se comenta sobre el tema de una manera divulgativa. (shrink)

The objective of this article is to present an original proof of the following theorem: Thereis a generic extension of the Solovay’s model L(R) where there is a linear order of P(N)/fin that extends to the partial order (P(N)/f in), ≤*). Linear orders of P(N)/fin are important because, among other reasons, they allow constructing non-measurable sets, moreover they are applied in Ramsey's Theory .