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Xerox University Microfilms INFORMATION TO USERS This material was produced from a microfilm copy of the orignal document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. 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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 74-24,324 FOSTER, Thomas Rowland, 1942- THE RUSSELL-LEIBNIZ DEFINITION OF IDENTITY: SOME PROBLEMS. The Ohio State University, Ph.D., 1974 Philosophy University Microfilms, A XEROX Company, Ann Arbor, Michigan © Copyright by Thomas Eowland Poster 1974 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. THE HUSSELL-LEIBHIZ DEFIIîITIOH OE IDEHTITT: SOME PROBLEMS DISSEETÂTIOH Presented in Partial Pulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Thomas Rowland Poster, B.A. The Ohio State University 1974 Reading Committee: Approved By Robert Turnbull Lee Brown Adviser Department of Philosophy ACKJJOWLEDGMENTS I wish, to make my acknowledgments brief. Alain Hausman of Ohio State, advisor and co-author of the last chapter, has carefully read and helpfully criticized severail drafts of this thesis. Silvano Miracchi of Ball State, friend and colleague, has discussed many of the issues involved. Lastly, I wish to thank my wife, my friends, and the Department of Philosophy at Ball State for their patience during the completion of this task. 7 I T A July 21, 194-2 . Born— jBvanston, Illinois 1955 ........... B.A., Oberlin College, Oberlin, Ohio 1957-1971 ....... Teaching Assistant, Department of Philosophy, The Ohio State University, Columbus, Ohio 1958 ........... National Defense Education Act Fellowship, Title IV 1971-1974-....... Assistant Professor, Department of Philosophy, Ball State University, Munci e, Indiana FIELDS OF STUDY Major Fields: Metaphysics and Logic TART. t; 0? COMJENTS Page ACKITOWLEDGMENTS................................ ii 7 I T A ......................................... iii Cliapter I. INTEODUCÎDIOII TO THE P R O B L E M .............. 1 I. The Definition of Identity II. The Notion of Adequacy III. Challenges to the Definition II. THE PRINCIPLE OF SDBSTÜTIVITY............. 18 III. SYMMETRICAL DNI7ER8E8..................... 30 f: I. On Possibility II. The Distinguishing Property III. Weak and Strong Symmetry IV. Logically Proper Names V. Other Distinguishing Properties VI. On Showing the II True VII. The Property Difference IV. THE NECESSITY OF THE I I ................... 66 I. Unreasonable Demands II. Different Senses of Difference III. Arguments Attempting to Establish the Non-Necessity of the II IV. Differing Senses of Individuation V. The Notion of Property Difference V. THE IDENTITY OF INDISCERNIBLES AND KINDS OF T H I N G S ................................ 103 I. Criteria of Difference II. Cartwright's Argument III. A Distinction Between Classes and Attributes BIBLIOGRAPHY .................................. 159 INTEODUCTIOIî TO THE PHOBLEM This chapter will introduce and provide a back­ ground for the problems to which the next four chapters address themselves. This chapter, then, serves as an introduction to the facets of identity and difference that this dissertation will discuss in detail. In general, the aim of this thesis is to provide a defense of Russell's definition of identity.^ For the Russellian definition has come under attack from various quarters. The purpose of this chapter is to outline several of these attacks on Russell's definition, while the following chapters serve to answer each of the attacks in detail. In preparation for both the outline of the alleged problems and the defense of Russell's definition, I will, first, introduce Russell's definition and consider some of its important consequences, and how such consequences are related to the identity of indiscernibles as well as some other principles which involve identity. The question about the Russellian definition is over its adequacy. Bertrand Russell and Alfred North Whitehead, Principia Mathematica (Cambridge: Cambridge University Press, 1914), p. 158. 2 Since this notion is crucial, the second section of this chapter shall he devoted to a brief discussion of the notion of what it is for a definition to be adequate, as well as some demands that are too restrictive. The last and most important section of this chapter will be an outline of several problems which will be discussed in detail in later chapters. I. The Definition of Identity One may introduce the sign into a formalism either as defined or as primitive. If the sign is introduced as a defined sign— in the manner that Russell does— then certain crucial sentences, such as (x) (y) (x/y=(3f) (fx--fy) ) are rendered necessary or analytic. They are necessary by virtue of the definition. In a system where the notion of sameness or difference is introduced as a primitive or undefined, similar sentences are not necessary or analytic. Often, however, the introduction of the "=” sign as primi­ tive in a formalism is accompanied by a set of rules and/or axioms. These axioms, such as ”(x)(x=x)," are derivable from the Russellian definition. The reluctance to intro­ duce "=” as defined probably stems from a desire to avoid quantifying over predicate variables, thereby avoiding commitment to properties. 3 Irrespective of whether the "='* sign is introduced as primitive or defined, questions may he raised as to the adequacy of the rules and/or axioms in the one case, and the definition in the other. Since the charges of inade­ quacy in either case are the same sort of charges, I shall introduce the "=" sign as defined in the Eussellian manner. Suppose, then, one does introduce the '*=” sign in the manner of Russell, namely, (1) x=y=df(f)(fx2fy) The motivation for defining identity in such a fashion is that one often thinks of x and y as being the same if, and only if, "they” share all properties in common. Conversely, one thinks of x and y as different, if, and only if, one has a property that the other lacks. The range of "f" in the definition is meant to include at least undefined predicates. It has been the subject of much debate as to whether the "f" of the definition should range over rela­ tional predicates such as "being-to-the-left-of-a," modal predicates such as "being-necessarily-f," metalinguistic predicates such as "being-designated-by-'a'" and predicates which involve identity, such as "...=a." 1 shall discuss neither modal predicates nor ones involving identity. 1 shall take relational predicates as within the range of "f" of the Eussellian definition. There will also be occasion to ^eak of "metalinguistic" predicates. 4 To begin with, it follows from the above definition that (2) (x) (y) (x=ys(f ) (f xsfy) ) Prom (2) it follows that both (3) (x)(y)(x=7 »(f)(fx=fy)) as well as (4) (x) (y) ( (f ) (fx=fy)^x=y) Equivalent to (5) and (4) respectively are: (5) (x)(y)((3f)(fx*-fy>x/y) and (5) (x)(y)(x?^y3(af)(fx--fy)) Notice that (2) - (6) all follow from the definition of "=" and are thus trivially true. Notice, further, that there is a rule which seems to be the metalinguistic counterpart of (3), namely, (7) x=y In questioning the adequacy of the Eussellian definition, the crucial sentences and/or rules under consideration will be (5), (5), and (7), respectively. The sorts of questions raised about the adequacy of (3) and (5), on the one hand, and (7) on the other will differ. Eor (7) is a rule of inference, while (3) and (6) are sentences. Rules are either valid or invalid, but neither true nor false as sentences are. 5 Notice that corresponding to sentence (3)? which is within the Eussellian formalism, there is another sen­ tence, not within the Eussellian formalism, as follows: (5)' If "two” things are one and the same, then "they" share all properties in common. Further, corresponding to sentence (6), there is the sentence (6)' For any pair of things, one has a property that the other lacks. Call (5)' and (6)' the principle of identity (PI) and the identity of indiscernibles (II) respectively. There is also an English counterpart of (7) which will be called the principle of substutivity (PS).
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