University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies Legacy Theses 2008 Leśniewski’s Systems of Logic and Mereology; History and Re-evaluation. Urbaniak, Rafal Urbaniak, R. (2008). Leśniewski’s Systems of Logic and Mereology; History and Re-evaluation. (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/15194 http://hdl.handle.net/1880/46697 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Le´sniewski’s Systems of Logic and Mereology; History and Re-evaluation. by Rafal Urbaniak A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHILOSOPHY CALGARY, ALBERTA April, 2008 �c Rafal Urbaniak 2008 The author of this thesis has granted the University of Calgary a non-exclusive license to reproduce and distribute copies of this thesis to users of the University of Calgary Archives. Copyright remains with the author. Theses and dissertations available in the University of Calgary Institutional Repository are solely for the purpose of private study and research. They may not be copied or reproduced, except as permitted by copyright laws, without written authority of the copyright owner. Any commercial use or publication is strictly prohibited. The original Partial Copyright License attesting to these terms and signed by the author of this thesis may be found in the original print version of the thesis, held by the University of Calgary Archives. The thesis approval page signed by the examining committee may also be found in the original print version of the thesis held in the University of Calgary Archives. Please contact the University of Calgary Archives for further information, E-mail: [email protected] Telephone: (403) 220-7271 Website: http://www.ucalgary.ca/archives/ Abstract Stanis�law Le´sniewski (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 Le´sniewski and a short guide to primary sources. Chapter 2 is devoted to a detailed discussion of Le´sniewski’s views on the philosophical problems he wrote about. These include existential propositions, principles of contradiction and excluded middle, the eter­ nity of truth, existence of universals, and certain paradoxes (most importantly, Russell’s paradox). Chapters 3–5 are concerned with a fairly detailed presentation of Le´sniewski’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 Le´sniewski’s and Sobocin´ski’s (Sobocinski´ 1949b) solution of Russell’s paradox is studied and argued to be unsuccessful. Next, a strengthening of Mereology (S�lupecki 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 quan­ tification 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. ii Table of Contents Abstract . ii Table of Contents . iii Introduction . 1 Part I. Historical aspects 9 1 Stanis�law Le´sniewski: life and work . 10 1.1 Life and philosophical development . 10 1.2 Primary sources . 15 2 Le´sniewski’s early philosophical views . 19 2.1 Introduction . 19 2.2 Linguistic conventions . 20 2.3 Existential propositions . 32 2.4 The principle of contradiction . 37 2.5 The principle of excluded middle . 39 2.6 Eternity of truth . 42 2.7 Paradoxes . 53 2.7.1 The Nelson-Grelling paradox . 53 2.7.2 Meinong’s paradox . 54 2.7.3 Epimenides’ paradox (Liar) . 56 2.7.4 Russell’s paradox . 59 2.8 On universals . 65 2.9 Further readings . 68 3 Le´sniewski’s Protothetic . 71 3.1 Introduction . 71 3.2 Semantic categories . 74 3.3 Le´sniewski’s notation . 77 3.4 Remarks on Protothetic . 83 3.5 Axiomatic formulations . 92 3.6 (Annotated) further readings . 99 4 Le´sniewski’s Ontology . 105 4.1 Ontology — basic intuitions . 105 4.2 The language of Ontology . 107 4.3 The axiomatic system . 109 4.4 Some definitions . 111 4.5 Against universals . 114 4.6 Le´sniewski’s original style of presentation . 118 4.7 (Annotated) further readings . 132 5 Le´sniewski’s Mereology . 140 5.1 Why Mereology? . 140 5.2 Axiomatizations . 148 iii iv 5.2.1 1916 axiomatization . 149 5.2.2 1918 axiomatization . 152 5.2.3 1920 axiomatization . 154 5.2.4 1921 axiomatization . 155 5.2.5 Some other necessary and sufficient conditions of being a class . 157 5.3 Mereology as a replacement for Set Theory . 158 5.4 (Annotated) further readings . 169 Part II. Re-evaluation (sets and ontological commitment) 175 6 Le´sniewski’s approach to Russell’s paradox and set theory . 176 6.1 The problem . 176 6.2 Mereological classes . 179 6.3 Distributive classes . 184 6.4 Some more difficulties . 186 6.5 Proof sketches . 188 6.6 Higher-order epsilon . 193 6.6.1 What is higher-order epsilon? . 193 6.6.2 HO " and Russell’s paradox (outlined) . 197 6.6.3 HO " and Russell’s paradox (technically speaking) . 198 6.7 S�lupecki’s generalized mereology . 206 6.7.1 System M . 206 6.7.2 Difficulties . 211 7 Modal interpretation of plural quantifiers . 214 7.1 Introduction . 214 7.2 Quantified name logic . 216 7.2.1 The language . 216 7.2.2 Set-theoretic semantics . 218 7.2.3 QNL " and Boolos’ logic of plurals . 219 7.3 Plural quantification and ontological commitment . 227 7.4 A modal alternative . 242 7.4.1 Simons on combinatorial semantics . 242 7.4.2 Rickey’s natural models . 247 7.4.3 Kripke semantics for QNL " . 249 7.4.4 Modal logic of naming . 257 7.4.5 Remarks . 259 7.4.6 Summary of the chapter . 271 Introduction The Lvov-Warsaw school of logic and analytic philosophy was one of the most important schools of philosophical thought in twentieth century. In early 1910s its members already discussed the validity of the principles of excluded middle and contradiction. Among ideas developed in this school one might count Luk� asiewicz’s view that contradictions may be believed to be true and that certain sentences can be neither true nor false which led to the construction of his three-valued logic, Ajdukiewicz’s conventionalism about meaning and his formal work on definitions (it seems that it was Ajdukiewicz who first put forward the consistency, translatability and non-creativity conditions on definitions), Ja´skowski’s approach to natural deduction and his work on discussive logics, Lindenbaum’s lemma on maximally consistent sets of formulas, Presburger’s work on arithmetic, Kotarbinski´ semantical reism, and Tarski’s work on formal semantics and truth, just to give a few examples. One of the representatives of this school was Stanis�law Le´sniewski (1886–1939) (Alfred Tarski, whose importance in twentieth century logic it is hard to overestimate, was his only PhD student). Le´sniewski 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. Le´sniewski’s work is interesting for a few reasons. • If one is interested in history of logic in general, it is hard to deny that Le´sniewski was one of the key figures in one of the most important schools of logic in twentieth century, who also seriously tried to develop an alternative to the system of Principia Mathematica, and hence he is worth studying in his own right. 1 2 • If one is interested in the development of Tarski’s thought it might be useful to learn what his PhD supervisor’s views were and how Le´sniewski’s work and Tarski’s ideas are (or are not) related. • Philosophical discussions in which Le´sniewski participated pertained to issues which are discussed quite lively even today: his approach to semantical and set-theoretic paradoxes and his views on the validity of the principle of excluded middle and of the principle of contradiction are quite interesting philosophically. • Le´sniewski was a nominalist and his systems were a nominalistic attempt to provide a system of foundations of mathematics. It is a major attempt of this sort and as such is worth an examination. • His metalogic is also quite specific. Nominalist as he was, he wanted to develop a purely inscriptional syntactic description of his systems in a way that did not make any reference to expression types. It is interesting to see how he proceeded and whether this approach works. • His logical systems are also interesting in their own right.