University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations 1896 - February 2014 1-1-1981 An introduction to a theory of abstract objects. Edward N. Zalta University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_1 Recommended Citation Zalta, Edward N., "An introduction to a theory of abstract objects." (1981). Doctoral Dissertations 1896 - February 2014. 2187. https://scholarworks.umass.edu/dissertations_1/2187 This Open Access Dissertation is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations 1896 - February 2014 by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. AN INTRODUCTION TO A THEORY OF ABSTRACT OBJECTS A Dissertation Presented By EDWARD NOURI ZALTA Submitted to the Graduate School of the University of Massachusetts in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 1981 Philosophy P ERRATA The corrected line appears after the page and line indication. p. 19, 1. 14: being abstract ("Al") = tAx ~E!x] ^f n n n n : + p. 24, 1. 16: P(P )("the power set of P ") , i.e., £.xt R P(V ). vK n n We call e.xX^(A ) the exemplification exten-/ p. 25, 1. 4,5: PLUG maps (R^UR^U...) X V into (R^UR^U...). PLLiGy ^ for each j, j>l, maps (R.UR U...) X V into (R. UR.U...). 3 J+l 3-1 3 PLUG is subject to the condition:/ ^ PROJ p. 25, 1. 17,18: PR0J maps (R^R^...) into (R^R^...). , 1 (R^UR_. ^U. into (R^_^UR^.U . ) . PR0J_^ for each j, j>l, maps + . ) is subject to the condition:/ 1 A £ R_ into P(V) i.e., 27, 1. 18: function, ext., which maps each , p. L ext : R - P(V). A x propositional formula <t>, the/ p. 35, 1. 11: A-EQUIVALENCE : for any is governed by p. 68, 1. 16,17: The behavior of our new descriptions the following proper axiom schema ("DESCRIPTIONS"):/ the definition of satis- p. 76, 1. 19-22: The first two clauses in faction must be redefined. For example, clause one should read . (3o (3 a satisfies iff (3o. ) . ) ) If <p = • .o , then $ p\. n •*- u n n c ,(o & A =cL ( ) & d (0 )& " . & o =d ) / (< 7 i 11 i n » q T‘ i,rf i n , q 1 ext^V )) RELATIONS - you Also, we need to restrict A-EQUIVALENCE and descriptions./ can't use formulas $ which contain ("Al") = [AxD~E!x] p. 80, 1. 29: D being abstract x dfn I .. I n III p. 83, 1 .2: X into P(V ) , where n>l. and which maps Rq X W n into {T,F}. ext (4. ) / IAJ p. 86, 1 . 10: member of I, ext » is a function which maps R^ into A P(V). is/ n ] .. 3-6: (j) = o • • • satisfies with respect to W p. 91, If p , & j 0^ 11 . (3-i ° =d & iff (3o^) . (30^) ) (o^=dj '(° ) ^(°^) n j n n n n = (-i tL d ^(P ) & <0 ,...,0 > £ ext^ )) j L n IV iff (3o)(34.) If cj) = op'*", satisfies $ with respect to 1 1 1 (o & n & o e ( o=d 7 / ) =d 1 Ap ) eU,^ )) 1 A i.a > o p. 92, L. 23,24: (This is a proper axiom) (j) which p. 92, L. 25: X- EQUIVALENCE: for any propositional formula has no descriptions, the/ the universal p. 93, 1. 23: isn't free and (j) has no descriptions, closure of the following is/ propositional formula p. 94, 1. 17: PROPOSITIONS: where cp is any descriptions, the/ where F^* isn't free and which has no = of z at w ("Cor(x,z,w) ) p. 112, 1. 18: x is a correlate d £ correlate of z at w iff x exemplifies p. 112, 1. 20: That is, x is a F at w iff z/ teacher of Alexander is a teacher p. 129 1. 8: (4) Necessarily, the t//p t/,p P ") = lAx D~E! x] 142 1. 26: D being abstract ("A! p. 1 df X W - P(P UP. U...UP ) 1. 16: (a) ext,:: RK t. t p. 148 , t, 'u> (t,,...,t )/p n' * 1 2 assigns each type t, ext maps R into P(P ) . p. 152 1. 19: A t/p t higher order/ I I p. 158, 1. 11-14: (Redefine these clauses as in correction for page 91 above) p. 159, 1. 13-15: (DESCRIPTIONS is a proper axiom) p. 159, 1. 16,17: A-EQUIVALENCE: for any propositional formula which has no descriptions, the universal closure of the following is an axiom:/ 1 EDWARD NOURI ZALTA 1980 All Rights Reserved ii AN INTRODUCTION TO A THEORY OF ABSTRACT OBJECTS A Dissertation Presented By EDWARD NOURI ZALTA Approved as to style and content by: Terence Parsons, Chairperson of Committee r X- 'f- L / h - y* ,K 'y/'AA / Edmund Gettier, Member v> • vl v " -~ 1 •* / Gary Hardegree, Member Barbara Partee, Outside Member / ' A : yf / yt c Head Edmund Gettier, Acting Department Department of Philosophy iii s PREFACE Alexius Meinong and his student, Ernst Mally, were the two most influential members of a school of philosophers and psychologists working in Graz in the early part of the twentieth century. They in- vestigated psychological, abstract, and nonexistent objects — a realm of objects which weren’t being taken seriously by Anglo-American philosophers in the Russell tradition. I first took the views of Meinong and Mally seriously in a course on metaphysics taught by Terence Parsons in the Fall of '78. Parsons had developed an axiomatic version of Meinong' s naive theory of objects. The theory with which I was confronted in the penultimate draft of Parsons' book. Nonexistent Objects , had a profound impact upon me. I was convinced that Parsons' work would serve as a new paradigm for philosophical investigations. While canvassing the literature during my research for Parsons' course, I discovered, indirectly, that Mally, who had originated the nuclear /extranuclear distinction among properties (a seminal distinc- tion adopted by both Meinong and Parsons) , had had another idea which could be developed into an alternative axiomatic theory. This dis- covery was the result of reading both a brief description of Mally 's theory in J.N. Findlay's book, Meinong' s Theory of Objects and Values (pp. 110-112) and what appeared to be an attempt to reconstruct Mally' theory by W. Rapaport in his paper "Meinongian Theories and a Rus- sellian Paradox." With the logical devices Parsons had used in his IV book, plus others that I had learned from my colleagues, I began elaborating and applying the alternative theory in a series of un- published papers written between November 1978 and August 1979. These papers were then assimilated into the first draft of this work in Fall 1979. The entire project could not have been carried off without the inspiration and aid of teachers and colleagues. Throughout the project, Parsons served as a sharp critic. Our conversations every couple of weeks always left me with an idea for improving what I had done or with an outline of a problem which had to be tackled and solved. It is to his credit that he was such a great help despite the fact that our theories offered rival explanations to certain pieces of data. Barbara Hall Partee graciously gave of her time in weekly dis- cussions during the writing of the first draft. Her enthusiasm, en- couragement, and suggestions were invaluable. My colleague, Alan McMichael, also deserves special mention. Besides teaching me the techniques of algebraic semantics, and discover- ing a paradox within the theory, McMichael served as my first critic. Whenever I discovered a new application of the theory or got stuck on a point of logic, I frequently presented it to Alan. His criticisms and suggestions helped me to sharpen up many of the intricate details. I'd also like to thank Mark Aronszajn, Blake Barley, Cynthia Heidelberger Larry Freeland, Edmund Gettier, Gary Hardegree, Herbert , Hohm, Michael Jubien, and Robert Sleigh. Spirited discussions with these individuals forced me to think deeply about a variety of v issues. They were some of the many people who helped to make the philosophy department here such a stimulating one. Finally, thanks goes to Nancy Scott for her dedication in typing unfriendly looking manuscripts Ed Zalta October, 1980 University of Massachusetts/Amherst vi . ABSTRACT An Introduction to a Theory of Abstract Objects (February 1981) Edward Nouri Zalta, B.A., Rice University, Ph.D., University of Massachusetts Directed by: Professor Terence Parsons An axiomatic theory of abstract objects is developed and used to construct models of Plato’s Forms, Leibniz's Monads, Possible Worlds, Frege's Senses, stories, and fictional characters. The theory takes six primitive metaphysical notions: object (x,y,...); n-place n n J-^gticms (F , G x 1 ,..,x_ exemplifv F ("F x, ...x ") : x -1- n In x s E s ( E!x ) ; it is necessary (" - — that <J) 04>") and ; x encodes F^ ("xF "). Properties and propositions are one place and zero place relations, respectively. Abstract objects ("A!x") are objects which necessarily fail to exist ("0~E!x"). The two most important proper axioms are that (1) no possibly existing object encodes any properties ((x)(C> E!x -* ~(3F)xF)), and (2) for every expressible condition on properties, there is an abstract object which encodes just the proper- ties satisfying the condition E ((3x)(A!x & (F) (xF $)), where <J> has no free x's). Semantically, an abstract object encodes a property iff the property is an element of the set of properties correlated with the object.