The 1900 Turn in Bertrand Russell’s Logic, the Emergence of his Paradox, and the Way Out

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process

On Leśniewski’s Characteristica Universalis

Leśniewski's systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski's work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz's characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is that Leśniewski built his characteristica universalis following the conditions of de Jong and Betti's Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski's systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance. © The Author(s) 2008

Philosophical and Mathematical Correspondence between Gottlob Frege and Bertrand Russell in the years 1902-1904 : Some Uninvestigated Topics

Although the connections between Frege’s and Russell’s investigations are commonly known (Hylton 2010), there are some topics in their letters which do not seem to have been analysed until now: 1. Paradoxes formulated by Russell on the basis of Frege’s rules: a) „»ξ can never take the place of a proper name« is a false proposition when ξ is a proposition”; b) “A function never takes the place of a subject.” A solution of this problem was based on the reference/sense theory and on the distinction between the first- and second-level names (Frege). 2. The inconsistency in Frege’s system may be avoided by the introduction of: a) a new kind of objects called quasi-objects (Frege); b) logical types (Frege and Russell); c) mathematics without classes (Russell); d) some restrictions on the domain of function (Frege). 3. Since the inconsistency is connected with a class, what is class? In one of the letters, Frege compared a class to a chair composed of atoms. This approach seems to be similar to the collective understanding of a set (Stanisław Leśniewski). 4. Russell doubted that the difference between sense and reference of expressions was essential. Hence, Frege found some additional reasons to distinguish between them: semiotic, epistemological, from identity, and from mathematical practice. This discussion can be seen as a next step in developing the theory of descriptions by Bertrand Russell

On the Phases of Reism

Kotarbiński is one of the leading figures in the Lvov-Warsaw school of Polish philosophy. We summarize the development of Kotarbiński’s thought from his early nominalism and ‘pansomatistic reism’ to the later doctrine of ‘temporal phases’. We show that the surface clarity and simplicity of Kotarbiński’s writings mask a number of profound philosophical difficulties, connected above all with the problem of giving an adequate account of the truth of contingent (tensed) predications. The paper will examine in particular the attempts to resolve these difficulties on the part of Leśniewski. It will continue with an account of the relations of Kotarbińskian reism to the ontology of things or entia realia defended by the later Brentano. Kotarbiński’s identification of Brentano as a precursor of reism is, it will be suggested, at least questionable, and the paper will conclude with a more careful attempt to situate the Brentanian and Kotarbińskian ontologies within the spectrum of competing ontological views

Ontologie et théorie des ensembles

L’ontologie de Leśniewski est un calcul général des noms. Elle fut créée par Leśniewski pour apporter une solution naturelle au paradoxe de Russell en théorie naïve des ensembles. L’ontologie a été perçue par ses défenseurs et par ses adversaires comme une théorie incompatible avec la théorie des ensembles. Dans le présent texte, nous montrons que l’ontologie de Leśniewski permet, au contraire, de définir une théorie des ensembles qui coïncide avec la théorie de Zermelo- Fraenkel

Поворот 1900 года в логике Бертрана Рассела, возникновение парадокс а и сп особ его разрешения

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called “denoting phrase”. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s longelaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the “substitutional theory”, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process.Исходный философский проект Рассела (1898) состоял в том, чтобы сделать математику строгой, сведя её к логике. До августа 1900 г., однако, логика Рассела представляла собой не более чем мереологию. В августе 1900 г. он познакомился с идеями Пеано, что привело его к отказу от логики части и целого и принятию вместо неё своего рода интенсиональной логики предикатов. Среди прочего, логика предикатов помогла Расселу найти способ справиться с парадоксом бесконечных чисел с помощью одного-единственного понятия — того, что он назвал «обозначающим выражением». К сожалению, вскоре обнаружился новый парадокс, на этот раз связанный с классами. Основной тезис данной статьи состоит в том, что новая концепция Рассела лишь перенесла парадокс бесконечности из области бесконечных чисел в сферу отношений включения на множествах. Решение парадокса, над которым Рассел долго работал в период между 1905 и 1908 гг., заключалось не в чём ином, как в том, чтобы отказаться от некоторых идей, принятых им во время «поворота» в августе 1900 г. (i) В теории дескрипций была отчасти восстановлена мереология сложных и простых объектов в духе периода, предшествовавшего августу 1900 г. (ii) Элиминация классов посредством «подстановочной теории» и устранение пропозиций за счёт теории суждения как множественного отношения завершили этот процесс

Some remarks of Jan Śleszyński regarding foundations of mathematics of Stanisław Leśniewski

Jan Śleszyński, a great mathematician, is considered a pioneer of Polish logic; however, he was not connected with the famous Warsaw School of Logic (WSL). He believed that his mission was a critical evaluation of work of other logicians in the field of foundations of mathematics and proof theory. Among his writings we find several notes regarding the work of Stanisław Leśniewski (the co-founder of the WSL) and his collective set theory. These remarks are the subject of investigation of the presented paper

Mereology then and now

This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field

La no-class theory de Stanisław Leśniewski

Insatisfait du calcul des classes et des relations de Whitehead et Russell, Leśniewski élabora en 1919-20 une théorie extensionnelle des noms qu’il nomma Ontologie. Sans entrer dans une description technique du formalisme de Leśniewski, nous montrons dans cet article que l’Ontologie permet un traitement général du distributif qui ne s’appuie à aucun moment sur une notion de classe. Nous illustrons enfin cette particularité importante du système de Leśniewski en proposant une définition logiciste de la notion de cardinalité qui répond d’une manière radicale aux impératifs d’une no-class theory.Leśniewski was not satisfied by Whitehead and Russell’s calculus of classes and relations. In 1919-20, he elaborated an extensional theory of names he called Ontology. Without a description of the full technical apparatus of Leśniewski’s formalism, I show here that Ontology gives rise to a general treatment of distributive predication which makes no use of the notion of class. In order to illustrate the importance of this peculiarity, I will give a logicist definition of cardinality which is radically conform with the requierments of a no-class theory