Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica Gustavo Fernández Díez Sets and Pluralities Revista Colombiana de Filosofía de la Ciencia, vol. IX, núm. 19, 2009, pp. 5-22, Universidad El Bosque Colombia Available in: http://www.redalyc.org/articulo.oa?id=41418349001 Revista Colombiana de Filosofía de la Ciencia, ISSN (Printed Version): 0124-4620 [email protected] Universidad El Bosque Colombia How to cite Complete issue More information about this article Journal's homepage www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative Sets and Pluralities1 Gustavo Fernández Díez2 Resumen En este artículo estudio el trasfondo filosófico del sistema de lógica conocido como “lógica plural”, o “lógica de cuantificadores plurales”, de aparición relativamente reciente (y en alza notable en los últimos años). En particular, comparo la noción de “conjunto” emanada de la teoría axiomática de conjuntos, con la noción de “plura- lidad” que se encuentra detrás de este nuevo sistema. Mi conclusión es que los dos son completamente diferentes en su alcance y sus límites, y que la diferencia proviene de las diferentes motivaciones que han dado lugar a cada uno. Mientras que la teoría de conjuntos es una teoría genuinamente matemática, que tiene el interés matemático como ingrediente principal, la lógica plural ha aparecido como respuesta a considera- ciones lingüísticas, relacionadas con la estructura lógica de los enunciados plurales del inglés y el resto de los lenguajes naturales. Palabras clave: conjunto, teoría de conjuntos, pluralidad, cuantificación plural, lógica plural. Abstract In this paper I study the philosophical background of the relatively recent (and in the last few years increasingly flourishing) system of logic called “plural logic”, or “logic of plural quantifiers”. In particular, I compare the notion of “set” emanated from axio- matic set theory, with the notion of “plurality” which lies behind this newly proposed logical system. I conclude that the two are utterly different in scope and limits, and that the difference comes from the distinct motivations that have led to each of them. While set theory is a genuine mathematical theory, with a primary mathematical inte- rest, plural logic has arisen from linguistic considerations, which have to do with the adequate representation of the logical structure of plural statements of English and most other natural languages. Keywords: set, set theory, plurality, plural quantification, plural logic. 1 Research supported by Spanish Government Grant FIS2007–63830. 2 Doctor en Filosofía – London School of Economic and Political Science. Profesor Departamento de Filosofía. Universidad de Murcia – España. Correo electrónico: [email protected]. Sets and Pluralities - Gustavo Fernández Díez Introduction A set is a collection of objects which, regarded as a whole (or as a unity), can be considered in turn as a new object. This simple and apparently innocuous idea, gave rise to one of the greatest revolutions in logic and the foundations of mathematics, by the end of the 19th century. Few times in history such a simple insight has led to so many consequences. The idea was put forward by the Russian–born German mathematician and philosopher Georg Cantor. In Cantor’s own words, “A set is a collection into a whole of definite distinct objects of our intuition or of our thought. The objects are called the elements (members) of the set” (Cantor 1895 481). In turn, the new object obtained, i.e., the set, can itself be considered as a member of other collections of objects, thus being used for the construction of other sets. So we will have sets, some of whose members are themselves other sets. And in turn these larger sets will themselves be members of other sets, even larger than them. And so on, leading to sets which are larger and larger, and more and more complex. In a series of seminal papers between 1879 and 1884, Georg Cantor published his first treatise, in six parts, on what soon was to become known as “set theory”: a completely new discipline, invented by him, in which, starting from the notion of set and the membership relation, Cantor managed to build up a full autono- mous branch of mathematics in its own right. With the creation of set theory, Cantor fulfilled two major achievements. In the first place, he conducted, for the first time in the history of mathematics, a truly rigorous and systematic study of mathematical infinite: he pointed out the existence of different sizes among infinite sets, and devised a precise method to compare them; later on, a full scale would be provided for the exact dimension measurement of infi- nite sets of all sizes (von Neumann’s theory of ordinals, in 1923). In the second place, Cantor’s set theory positioned itself as the main essential discipline for the exploration of the foundations of mathematics, as it started to become clear that most (indeed, virtually all) mathematical concepts could be defined, and the respective theorems proved, in the framework of set theory. Since then, the importance that set theory, and the notion of “set”, have had for mathematics and philosophy, is difficult to overestimate. In particular, in the philosophical field, it has been a central topic of discussion in philosophy of mathematics: the ontological commitments of that notion and of the axioms governing it, have been scrutinized in detail, as well as the very philosophical nature of the notion itself, and its efficacy in providing a foundational ground for mathematical science. Moreover, set theory has also had an impact in philo- sophy through formal logic, and through those branches of philosophy which [6] Revista Colombiana de Filosofía de la Ciencia • Vol. IX ∙ No. 19 Suplemento • 2009 • Págs. 5-22 use formal logic. Thus, formal logic is standardly developed today in a set–theo- retic framework, and all its basic notions, such as formal language, deduction, model, logical consequence, etc, are defined in set–theoretical terms. In philo- sophy of language, an increasing number of theories of meaning for natural languages (from Quine onwards), make use of formal logic and set–theoretical notions. And the same happens within some trends of philosophy of science, such as the so–called structuralist approach, and others. For all these reasons, it has been highly surprising the appearance, a decades ago, and the rapid proliferation in the last few years, of a completely new logical discipline called “plural logic” (or, sometimes, “plural quantification logic”, or “logic of plural quantifiers”). A new logical discipline that attempts to describe a logical regimentation in which a plurality of objects is accounted for without treating it as a logical object in itself, but only as a collective way of referring to the members that constitute it. Hence, within plural logic —so it is alleged— we can refer to a plurality of objects by way of a plural (combined, collective) refe- rence to the objects that constitute it, bypassing the ontological commitment to the plurality as an entity in itself. Likewise, within plural logic we are suppo- sedly able to predicate a property over a bunch of objects collectively taken, and to make existential and universal quantification over such pluralities of objects, without committing ourselves to the existence of the pluralities as such, but only of the objects that are contained within them. Plural logic, hence, is a trial to give a logical treatment of pluralities of objects, not as objects in themselves, but only as collective representations of the objects which form each plurality. In a certain way, this initial stan- dpoint is quite the opposite as that of set theory, where the idea of gathering a collection of different objects into a new object (a whole, a unity, a new thing), was essential. And yet, as plural logic theoreticians point out, this new logical system has been used in attempts “to account for set theory, and to elimi- nate ontological commitments to mathematical objects and complex objects” (Linnebo 2004, Introduction). The reason why plural logic might have an impact on the philosophical implications of set theory lies in the fact that, if it turns out that some of the role that sets play in standard set theory, can be covered by this new logical treatment of pluralities, then the ontological commitments that are usually attributed to sets, both in set theory and in the applications of set theory to other disciplines, might drastically vary. In other words, if there is a possibility that plural logic provides a new way of looking at the sets of set theory, discharging some of the ontological commitments that they carry within them, then, in the light of this new logical system, some of those ontological commitments might not longer be considered inevitable. [7] Sets and Pluralities - Gustavo Fernández Díez Whether these perspectives are finally confirmed or not, the fact is that the appearance of plural logic on the scene, the growing interest that it is attracting, and the utter importance of the notions involved, demands that a serious philo- sophical debate is opened up to discuss all theoretical background of the matter. In this respect, the purpose of the present paper is to outline the general terms in which such a debate could be conducted. I will start by giving a brief exposition of set theory, plural logic, and the contrast between them, in a way as clear and accesible as I can make it, avoiding technicalities. I hope that the basic conceptual features of both theories can be got conveniently reflected in a direct, colloquial language. Then, I will go on to sketch the main lines of what would be a philoso- phical assessment of the new theory at issue, the motivations behind it, its scope and limits (judging from the formulations that have so far been devised), and indeed the main implications of the theory in its present form.