Leśniewski's systems of logic and foundations of mathematics
From the author's introduction: ``[M]y intention is to provide a survey that covers many aspects of what Leśniewski published, make a clear distinction between what views can be assigned to him based on textual evidence and what he is reported or believed to have claimed, and, when his philosophical views are discussed, assesses them critically.'' ``[T]he intended audience of this book includes philosophy majors, graduate students and professional philosophers interested in logic, mathematics and their philosophy and history.'' Also very short comments on other surveys of Leśniewski's work, a brief biography, and a survey of primary sources are given in the introduction. Chapter 2 is a discussion of Leśniewski's early philosophical views. Chapter 3 concerns Protothetic. The author describes here the historical context of the development of Protothetic, explains the notorious notation used by Leśniewski and his motivations for Protothetic. Also, all the gradually developed systems of Protothetic are described, and examples of proofs are given. In Chapter 4, devoted to Ontology, the differences of Ontology from classical logic are pointed out, and the language and the original axiomatization of the former are described. Also, some philosophical moments and relations between Ontology and Russell's theory of descriptions are discussed here. The subject of Chapter 5 is Mereology. First, Leśniewski's motivation -- a nominalistically acceptable alternative to the Cantorian set theory as a foundational system -- is analyzed in detail. Then the author compares four successive axiomatizations of Mereology (given by Leśniewski himself) and analyzes the potential role of Mereology in the foundations of mathematics; his assessment is rather negative. The goal of Chapter 6 is to trace the origin and development of a popular misunderstanding that the consistency and conservativeness requirements on definition go back to Leśniewski's work. It is explained why Leśniewski's definitions are creative, and the role of Łukasiewicz and Ajdukewich in the development of the theory of definitions is stressed. In Chapter 7, the author discusses the question to what extent some set theory could be emulated within Leśniewski's systems. He analyzes, from this viewpoint, distinguishing collective and distributive classes, Słupecki's generalized mereology, and Ontology. Chapter 8 is mostly philosophical in character. Its subject is nominalism and higher-order quantification, including substitutional reading of quantifiers, nominalistic semantics for Ontology, Rickey's models of Ontology, plural quantification and Boolos' semantics for it. Each chapter is accompanied with an annotated guide to the literature ``for further reading'' and a list of relevant references.
scientific article; zbMATH DE number 6176496