| Stanislaw Lesniewski (1886-1939) developed his system of foundations of mathematics as an alternative to the system of Principia Mathematica. He constructed three systems: Protothetic, which is his version of a generalized propositional calculus, his own logic of predication called Ontology, and a theory of parthood called Mereology.;The first part (chapters 1-5) is historical. Chapter 1 contains a brief biography of Lesniewski and a short guide to primary sources. Chapter 2 is devoted to a detailed discussion of Lesniewski's views on the philosophical problems he wrote about. These include existential propositions, principles of contradiction and excluded middle, the eternity of truth, existence of universals, and certain paradoxes (most importantly, Russell's paradox). Chapters 3-5 are concerned with a fairly detailed presentation of Lesniewski's systems in a modern framework. Each of them is provided with a guide to secondary literature. The second part (chapters 6 and 7) is more theoretical. In chapter 6 Lesniewski's and Sobocinski's (Sobocinski 1949b) solution of Russell's paradox is studied and argued to be unsuccessful. Next, a strengthening of Mereology (Slupecki 1958) that was meant to improve the situation is discussed and difficulties that it copes with are indicated. In chapter 7 a modal factor to provide a nominalist semantics for a language very similar to that of Elementary Ontology is introduced. The idea here is that higher-order quantification ranges over 'ways names could be'. A Kripkean semantics is provided which is intended to capture that notion and prove that the system thus obtained is equivalent to its standard set-theoretic version. Certain philosophical issues surrounding this approach are discussed. |
