Arts, Faculty of
Philosophy, Department of
DSpace
UBCV
Rein, Andrew Paul
Master of Arts - MA
University of British Columbia
There are two contentious ontological claims that Frege makes. Firstly, that predicates have reference and secondly, that their reference is 'unsaturated'. It is the primary aim of this essay to examine the latter claim. Following an introductory exposition of the Fregean concepts crucial to this essay, the claim that the references of predicates are 'unsaturated' is considered as a response to the problem of the unity of the proposition. As a special case of this problem, F.H. Bradley's attack on relations is discussed. It is argued that once one understands that relations are unsaturated, Bradley's problem, and by extension the problem of the unity of the proposition, disappears. But a crucial semantic principle is shown to emerge from this attempted solution to these problems: Any language capable of talking about relations and concepts cannot refer to these by means of proper names. This principle, it then appears, renders us incapable of specifying in a given instance which concept a predicate refers to. Two recent attempts to avoid this problem are discussed and rejected. Insight gained from this discussion leads to a third, purely Fregean solution to the problem. This solution, which is forced upon Frege by his own principles, is seen to involve him in a vicious circle. It appears that Frege's ontological doctrines are ill-founded - that they cannot solve the very problems for which they were designed.
( FREGE'S ONTOLOGY: A C R I T I C A L EXAMINATION OF ITS FOUNDATIONS by ANDREW PAUL REIN B . A . The U n i v e r s i t y o f Cape Town, 197^ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS ^ f i n THE FACULTY OF GRADUATE STUDIES (Depar tment o f P h i l o s o p h y ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA September 1979 (c) Andrew P a u l R e i n , 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f jMJAo£0£Ji^ The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e fir Sefrennet mi i i Abstract There are two contentious ontological claims that Frege makes. Firstly, that predicates have reference and secondly, that their reference is 'unsaturated'. It is the primary aim of this essay to examine the latter claim. Following an introductory exposition of the Fregean concepts crucial to this essay, the claim that the references of predicates are 'unsaturated' is considered?! as a response to the problem of the unity of the proposition. As a special case of this problem, F.H. Bradley's attack on relations is discussed. It i s argued that once one understands that relat-ions are unsaturated, Bradley's problem, and by extension the problem of the unity of the proposition, disappears. But a crucial semantic principle i s shown to emerge from this attempted solution to these problems: Any language capable of talking about relations and concepts cannot refer to these by means of proper names. This principle, i t then appears, renders us incapable of specifying in a given instance which concept a predicate refers to. Two recent attempts to avoid this problem are discussed and rejected. Insight gained from this discussion leads to a third, purely Fregean solution to the problem. This solution, which is forced upon Frege by his own principles, is seen to involve him in a vicious c i r c l e . It appears that Frege's ontological doctrines are ill-founded - that they cannot solve the very problems for which they were designed. i i i Contents A b s t r a c t P i i Acknowledgements p i v Chapter 1 I n t r o d u c t i o n . . . p l j Proper names . . . p2; Sense/Reference thereof . . . p3» Function names . . . p6{ Denotation thereof . . . p9i Concepts . . . plO? Senses of f u n c t i o n names . . . p l 2 ; Second l e v e l concepts . . . p l 3 i R e l a t i o n s . . . p l 4 ; I d e n t i t y c o n d i t i o n s f o r fu n c t i o n s . . . pl5» Problems i n the e x p o s i t i o n of Frege's d o c t r i n e s . . . p l 8 . Chapter 2 I n t r o d u c t i o n . . . p22j Frege's views on the r e l a t i o n s h i p between language and thoughts the r o l e of sense . . . p26s These views support the view that every component of a sentence expressing a thought has a sense . . . p27; The problem of the u n i t y of the p r o p o s i t i o n . , . p31s Bradley's argument aga i n s t r e l a t i o n s . . . p33» R u s s e l l ' s views on t h i s problem . . . P35? Frege's proposed s o l u t i o n . . . p38j Frege's s o l u t i o n viewed from w i t h i n h i s ontology . . . p ^ l ; A c r u c i a l semantic p r i n c -i p l e emerges from t h i s proposal . . . p^3. Chapter 3_ I n t r o d u c t i o n . . . p46; D i f f i c u l t i e s i n speaking of concepts . . . p48j M i l t o n F i s k ' s paradox . . . p51; A suggestion of Montague, Kaplan et a l . . . p55» Dummett's proposal . . . p58{ C r i t i c i s m of Dummett's proposal . . . p 6 l ; A Fregean r e s o l u t i o n of the problem: the 'prenote* r e l a t i o n . . . p6?» Why t h i s won't work , . . p69? Summary and conc l u s i o n . . . p71. Bi b l i o g r a p h y P72 iv Acknowledgements My supervisor, Howard Jackson, read an i n i t i a l draft of this thesis and made many detailed and valuable comments. A subsequent draft was read by Tom ,Patton and his comments too, were most useful. I had several rewarding conversations on the topic with Robert Bunn to whom I am also indebted for allowing me the use of his translation of Frege's "Ausfuhrungen uber Sinn und Bedeutung". 1 Chapter 1 Introduction The main purpose of this opening chapter is to familiarize the reader with those Fregean concepts which feature in the remainder of this essay. Only the briefest introduction to those concepts is intended. A more detailed and extensive survey of Frege's doctrines can be found elsewhere (for example, Furth's introduction to Bib. (20)). Most of the issues dealt with in this chapter are developed in greater depth there. A reader who is reasonably well acquainted with Frege's ontolog-i c a l and semantic theories may omit the expository portion of the f i r s t chapter (ie. a l l but section 1.45). A system of cross reference has been employed such that each key idea developed in this introductory chapter i s sequentially numbered. Where that idea features crucially in later chapters, the reader is referred by these numbers to the appropriate section in the f i r s t chapter. Thus i f a claim made in later chapters appears to be based on an unusual interpretation of Frege's doctrines, the reader w i l l be able to refer immediately to that portion of the f i r s t chapter where that unusual interpretation is defended. 2 ( 1 . 0 1 ) F r e g e ' s f u n d a m e n t a l o n t o l o g i c a l p r i n c i p l e i s t h i s i E v e r y t h i n g i s e i t h e r a f u n c t i o n o r an o b j e c t and n o t h i n g i s b o t h . Bu t F rege r e p e a t e d l y c l a i m s t h a t the t e rms ' f u n c t i o n ' and ' o b j e c t ' a r e l o g i c a l l y s i m p l e and t h e r e f o r e i m p o s s i b l e t o d e f i n e . He w r i t e s t ( 1 . 0 2 ) Now s o m e t h i n g l o g i c a l l y s i m p l e i s no more g i v e n us a t the o u t s e t t h a n most o f the c h e m i c a l e l e m e n t s a r e j i t i s r e a c h e d o n l y by means o f s c i e n t i f i c w o r k . I f s o m e t h i n g has been d i s c o v e r e d t h a t i s s i m p l e , o r a t l e a s t must c o u n t as s i m p l e f o r t h e t ime b e i n g , we s h a l l have t o c o i n a t e rm f o r i t , s i n c e l anguage w i l l n o t o r i g o n a l l y c o n t a i n a n e x p r e s s i o n t h a t e x a c t l y a n s w e r s . On the i n t r o d u c t i o n o f a name f o r s o m e t h i n g l o g i c a l l y s i m p l e , a d e f i n i t i o n i s n o t p o s s i b l e ; t h e r e i s n o t h i n g f o r i t b u t t o l e a d the r e a d e r o r h e a r e r , by means o f h i n t s , t o u n d e r s t a n d the words as i s i n t e n d e d . ( "Concep t and O b j e c t " B i b . (28Hp43)/ ( 1 . 0 3 ) I t t u r n s ou t t h a t we c a n r e a c h a c l e a r e r u n d e r -s t a n d i n g o f t h i s o n t o l o g i c a l p r i n c i p l e v i a two o t h e r F r e g e a n d i s t i n c t i o n s - t he p r o p e r n a m e / f u n c t i o n name d i s t i n c t i o n and the s e n s e / r e f e r e n c e d i s t i n c t i o n . These f u r t h e r d i s t i n c t i o n s i n t u r n a r e pe rhaps b e s t e x p l a i n e d by f i r s t c o n s i d e r i n g F r e g e ' s c a t e g o r y o f p r o p e r names. ( 1 . 0 4 ) P r o p e r Names Frege e x t e n d s c o n s i d e r a b l y the g r a m m a r i a n ' s n o t i o n o f p r o p e r names. Thus f o r e x a m p l e , a p a r t f rom s u c h o b v i o u s p r o p e r names a s ' P e g a s u s ' , • J o h n S m i t h ' and ' M o s e s ' , he i n c l u d e s d e f i n i t e d e s c r i p t i o n s such a s 'The winged h o r s e t h a t was c a p t u r e d by B e l l e r o p h o n * 1 1 Due t o Q u i n e . 3 and 'The a u t h o r o f W a v e r l e y ' . S e n t e n c e s t o o f a l l i n t o the c l a s s o f p r o p e r names. F r ege nowhere a t t e m p t s a f o r m a l c l a s s i f i c a t i o n o f t h i s c l a s s o f names t h o u g h he f r e q u e n t l y r emarks t h a t a s a r u l e , t he d e f i n i t e a r t i c l e f o l l o w e d by a common noun i n t h e s i n g u l a r , forms a p r o p e r name. As a sample o f o t h e r k i n d s o f p r o p e r names c o n s i d e r : ' C a e s a r ' s d e a t h * , ' H u m a n i t y ' , ' C h a r i t y ' , ' K i l l i n g * , and ' H i s l a t e a r r i v a l ' . (1.05) L e t us now see how t h e s e n s e / r e f e r e n c e ( ' S i n n * / ' B e d e u t u n g ' ) d i s t i n c t i o n a p p l i e s t o p r o p e r names. A n e x p r e s s i o n i s s a i d t o denote i t s r e f e r e n c e and e x p r e s s i t s s e n s e . I n the case o f p r o p e r names, t h e r e f e r e n c e o f a p r o p e r name i s s i m p l y t h a t e n t i t y r e f e r r e d t o by the p r o p e r name. S o , f o r a d e f i n i t e d e s c r i p t i o n , t he e n t i t y r e f e r r e d to i s j u s t t h a t e n t i t y w h i c h f i t s t he d e f i n i t e d e s c r i p t i o n . Thus ' t h e a u t h o r o f W a v e r l e y ' r e f e r s to S c o t t , t he a u t h o r o f t he W a v e r l e y n o v e l s . (1 . 0 6 ) C r u d e l y p u t , t he sense o f a n e x p r e s s i o n i s what one needs t o g r a s p i n o r d e r t o l o c a t e i t s d e n o t a t i o n . - ^ I n the case o f what g rammar ians c a l l p r o p e r names, i t m igh t be t h o u g h t d o u b t f u l w h e t h e r t h e y have senses a t a l l so i t i s b e s t a t t h i s s t a g e t o c o n s i d e r d e f i n i t e d e s c r i p t i o n s . Thus t h e sense o f the e x p r e s s i o n ' t h e a u t h o r o f W a v e r l e y ' i s r o u g h l y , what one needs t o g r a s p i n o r d e r t o d e t e r m i n e t h a t i t s r e f e r e n c e i s S c o t t h i m s e l f . 2 Due t o R u s s e l l . 3_ I use ' D e n o t a t i o n ' and ' R e f e r e n c e ' i n t e r c h a n g e a b l y . 4 ( 1 . 0 7 ) I n a d d i t i o n t o i t s sense and r e f e r e n c e , a n e x p r e s s i o n c a n have a n i d e a a s s o c i a t e d w i t h i t . F r ege i n t r o d u c e s the n o t i o n o f the a s s o c i a t e d i d e a s o f e x p -r e s s i o n s p r i m a r i l y t o c o n t r a s t a s s o c i a t e d i d e a s o f e x p r e s s i o n s w i t h t h e i r s e n s e s . A n i d e a a s s o c i a t e d w i t h a n e x p r e s s i o n c o n s i s t s r o u g h l y i n the images and pe rhaps f e e l i n g s wh ich the e x p r e s s i o n c o n j u r e s up i n the r e a d e r o r h e a r e r . I t i s e n t i r e l y s u b j e c t i v e - n e v e r the same f o r two p e o p l e . On the o t h e r hand the sense o f an e x p r e s s i o n i s an o b j e c t i v e e x i s t e n t e n t i t y w h i c h c a n be g r a s p e d by any s k i l l e d u s e r o f the l anguage i n w h i c h t h a t e x p r e s s i o n i s e m p l o y e d . Any two s k i l l e d l anguage u s e r s w i l l g r a s p the sense o f an e x p r e s s i o n ( a s suming the e x p r e s s i o n t o be u n a m b i g u o u s ) . F rege i n v o k e s t h e a n a l o g y o f a t e l e s c o p e t o e l u c i d a t e the r e l a t i o n s h i p be tween the s e n s e , r e f e r e n c e and a s s o c i a t e d i d e a s o f an e x p r e s s i o n . I f we t r a i n a t e l e s c o p e o n t o the moon, a n image o f the moon a p p e a r s w i t h i n the t e l e s c o p e w h i c h r e f l e c t s on to the r e t i n a o f t he o b s e r v e r ' s e y e . The moon i t s e l f i s l i k e n e d t o the r e f e r e n c e o f a n e x p r e s s i o n , the image w i t h i n t h e t e l e s c o p e t o t h e sense ( i t c a n be p e r c e i v e d by any number o f o b s e r v e r s ) , and the r e t i n a l image t o a n a s s o c i a t e d i d e a . F o r t h i s image i s n e v e r the same f o r two o b s e r v e r s . ( 1 . 0 8 ) S e n t e n c e s a r e somewhat s p e c i a l p r o p e r names f o r F r e g e . He coun t enances two t r u t h v a l u e s , w h i c h he 5 c a l l e d 'The True* and 'The F a l s e ' and a s en t ence i s s a i d t o denote one o f t h e s e t r u t h v a l u e s a c c o r d i n g l y as i t i s t r u e o r f a l s e . ( 1 . 0 9 ) The sense o f a s e n t e n c e i s a p r o p o s i t i o n o r t h o u g h t . ( 1 . 1 0 ) F r e g e c o n s i d e r e d i t a d e f e c t o f n a t u r a l l a n g -uages that t h e r e a r e a ) names w h i c h do n o t d e n o t e , b ) names w h i c h denote more t h a n one e n t i t y , and c ) names w h i c h a r e ambiguous - have more t h a n one s e n s e . He s t i p u l a t e d t h a n i n a l o g i c a l l y p e r f e c t l anguage each name must e x p r e s s e x a c t l y one sense and denote e x a c t l y one r e f e r e n c e . ( 1 . 1 1 ) F r e g e i n t r o d u c e d t h e s e n s e / r e f e r e n c e d i s t i n c t i o n i n o r d e r t o e x p l a i n the i n f o r m a t i v e n e s s o f t r u e i d e n t i t y s t a t e m e n t s s u c h a s ' H e s p e r u s = P h o s p h o r u s ' . The q u e s t i o n a r i s i n g i s why s u c h s t a t e m e n t s a r e i n f o r m a t i v e w h i l e s t a t e m e n t s s u c h as ' H e s p e r u s = H e s p e r u s ' a r e n o t . The s o l u t i o n F rege p r o p o s e d was t h a t w h i l e ' H e s p e r u s ' and ' P h o s p h o r u s ' have t h e same d e n o t a t i o n t h e y do n o t e x p r e s s the same s e n s e . ( 1 . 1 2 ) A n e n t i t y may be named by s e v e r a l e x p r e s s i o n s and the s ense s o f t h e s e e x p r e s s i o n s may d i f f e r . F u r t h e r m o r e one and the same sense may be e x p r e s s e d b y d i f f e r e n t e x p r e s s i o n s ( e i t h e r o f t he same l anguage o r o f d i f f e r e n t l a n g u a g e s ) . B u t F rege r e g a r d e d i t a s a f a u l t o f n a t u r a l l a n g u a g e s t h a t a n e x p r e s s i o n k_ F r e g e ' s t e r m ' G e d a n k e ' has been v a r i o u s l y t r a n s l a t e d a s e i t h e r ' p r o p o s i t i o n ' o r ' t h o u g h t ' . I s h a l l f rom he re on use ' t h o u g h t ' . 6 may e x p r e s s a sense w i t h o u t d e n o t i n g a n y t h i n g . (Though o f c o u r s e , i f i t does denote s o m e t h i n g t h e n i t must e x p r e s s a s e n s e . ) ( 1 . 1 3 ) B e f o r e t u r n i n g t o the d i s c u s s i o n o f f u n c t i o n names, i t i s w o r t h e n d i n g the d i s c u s s i o n o f sense and r e f e r e n c e by n o t i n g F r e g e ' s c o m p o s i t i o n a l t h e o r y o f sense and re ference* . i ) I f a component o f a d e n o t i n g e x p r e s s i o n i s r e p l a c e d by an e x p r e s s i o n h a v i n g the same d e n o t a t i o n as t h a t component , the d e n o t a t i o n o f t h e e n t i r e e x p r e s s i o n i s unchanged . F u r t h e r m o r e , i f any component o f a n e x p r e s s i o n l a c k s a r e f e r e n c e t h e n the e n t i r e e x p r e s s i o n l a c k s a r e f e r e n c e . i i ) I f a component o f a n e x p r e s s i o n i s r e p l a c e d by a n e x p r e s s i o n h a v i n g the same sense a s t h a t component , t he sense o f t he e n t i r e e x p r e s s i o n i s u n c h a n g e d . F u r t h e r m o r e , i f any component o f an e x p r e s s i o n f a i l s t o e x p r e s s a sense t h e n the e n t i r e e x p r e s s i o n a l s o f a i l s t o e x p r e s s a s e n s e . ( 1 . 1 4 ) F u n c t i o n Names G i v e n the g e n e r a l n o t i o n o f a p r o p e r name, i t i s p o s s i b l e t o i n t r o d u c e the n o t i o n o f a f u n c t i o n name. F rege c l a i m e d t o have a d o p t e d t h e n o t i o n f rom ma thema t i c s and t o have e x t e n d e d i t c o n -s i d e r a b l y f o r h i s own p u r p o s e s by a p p l y i n g i t t o t h e n a t u r a l l a n g u a g e s . I t seems a p p r o p r i a t e t h e n t o t r a c e 7 this development. Consider the expressions: •2.13 + 1' •2.43 + 4' •2.53 + 5' Each one of these expressions is for Frege a proper name denoting a number. Yet we recognise a common structure insofar as each expression can be seen as formed by successively placing '1', '4' and '5* into the gaps of the expression •2.( ) 3 + ( )' where in this case we indicate the position of the gaps by means of parentheses. Frege calls the expression *2.( ) 3 + ( )' a function,.;name. In general, the removal of a proper name from a proper name containing that name as a proper part results in a function name. Applying this now to a natural language, consider the proper name 'The author of Waverley'. If we remove from this expression the proper name 'Waverley' and mark the position from which i t is removed with parentheses, the resulting expression, •The author of ( )' is a function name. (1.15) The result of inserting 8 the proper name 'Waverley' into the gap of this function name (Waverley' then being the name of an argument of the function) yields a proper name whose denotation ie . Scott is said to be the value of the function for the argument Waverley. (1.16) Frege frequently employs the term 'ungesattigt* (translated variously as 'unsaturated' or 'incomplete') to describe function names. For example, in "What is a Function?", writing about mathematical functions, he notes* " . . . the sign for a function is unsaturated; i t needs to be completed with a numeral, which we then c a l l the argument-sign." (Bib. (23) see(28) pll3A) (1.17) Instead of brackets, Frege usually employs small Greek letters to indicate the gaps in function names. Thus he would write the last mentioned function name asi 'The author of ^ ' . We w i l l see shortly that there are function names of higher levels than the above which require a different kind of argument name for their completion and different Greek letters serve to mark the appropriate type of argument name. (1.18) Now just as proper names express their sense and denote their reference, so too with function names. The senses of function names w i l l be discussed a l i t t l e 9 l a t e r . ( 1 . 1 9 ) F o r t h e moment l e t us r e s t r i c t o u r s e l v e s t o c o n s i d e r i n g the d e n o t a t i o n o f f u n c t i o n names, i e . f u n c t i o n s . F rege p r o v i d e s a c l e a r s t a t e m e n t o f t h e c o n d i t i o n s unde r w h i c h a f u n c t i o n name d e n o t e s , and a l t h o u g h t h e r e i s a c i r c u l a r i t y i n v o l v e d i n the com-b i n a t i o n o f t h i s s t a t e m e n t and h i s s t a t e m e n t o f t he c o n -d i t i o n s u n d e r w h i c h a p r o p e r name deno te s (we c a n o n l y know w h e t h e r a p r o p e r name d e n o t e s i f we a l r e a d y know w h i c h f u n c t i o n names denote and we c a n o n l y know w h e t h e r a f u n c t i o n name deno tes i f we a l r e a d y know w h i c h p r o p e r names d e n o t e ) , i t i s w o r t h n o t i n g . F rege w r i t e s : "A name o f a f i r s t - l e v e l f u n c t i o n o f one argument has a d e n o t a t i o n (deno te s s o m e t h i n g , s u c c e e d s i n d e n o t i n g ) i f t he p r o p e r name t h a t r e s u l t s f rom t h i s f u n c t i o n name b y i t s argument p l a c e b e i n g f i l l e d by a p r o p e r name a l w a y s has a d e n o t a t i o n i f t h e name s u b s t i t u t e d d e n o t e s s o m e t h i n g . " ( B i b . (20) p84) ( 1 . 2 0 ) What k i n d o f e n t i t y a f u n c t i o n i s , o f c o u r s e , i s n o t y e t c l e a r . Bu t pe rhaps i f the n o t i o n o f a f u n c t i o n name i s c l e a r and i f t he r e l a t i o n s h i p be tween a p r o p e r name and i t s d e n o t a t i o n i s r e a s o n a b l y c l e a r , t h e n by v i e w i n g a f u n c t i o n as s t a n d i n g i n a n a n a l o g o u s r e l a t i o n s h i p t o i t s name, some i n s i g h t i n t o i t s n a t u r e i s g a i n e d . F rege h i m s e l f r e s o r t s t o me taphor . A t t e m p t i n g t o e x p l a i n m a t h e m a t i c a l f u n c t i o n s , he w r i t e s : ( 1 . 2 1 ) "The p e c u l i a r i t y o f f u n c t i o n a l s i g n s , w h i c h we he re c a l l e d ' u n s a t u r a t e d n e s s ' , n a t u r a l l y has s o m e t h i n g a n s w e r i n g t o i t i n the f u n c t i o n s t h e m s e l v e s . 10 They t o o may be c a l l e d ' u n s a t u r a t e d ' and i n t h i s way we mark them ou t as f u n d a m e n t a l l y d i f f e r e n t f rom numbers . Of c o u r s e t h i s i s n o t a d e f i n i t i o n ; b u t l i k e -w i s e none he re i s p o s s i b l e . I must c o n f i n e m y s e l f t o h i n t i n g a t what I have i n mind by means o f m e t a p h o r i c a l e x p r e s s i o n , and here I r e l y on ray r e a d e r ' s a g r e e i n g to meet me h a l f - w a y . " ( B i b . ( 2 3 ) see (28) p l l 5 ) (1 .22) We a r e now b e t t e r p o i s e d t o a p p r e c i a t e F r e g e ' s d i s t i n c t i o n be tween f u n c t i o n and o b j e c t . P r o p e r names denote o b j e c t s i e , s a t u r a t e d e n t i t i e s w h i l e f u n c t i o n names denote f u n c t i o n s i e . u n s a t u r a t e d e n t i t i e s . E v e r y -t h i n g i s s u c h t h a t i t i s a p p r o p r i a t e l y named e i t h e r by a p r o p e r name o r by a f u n c t i o n name. N o t h i n g i s a p p -r o p r i a t e l y named by b o t h a p r o p e r name and a f u n c t i o n name. Thus the o n t o l o g i c a l d i v i s i o n be tween f u n c t i o n and o b j e c t i s p r e c i s e l y t i e d t o the l i n g u i s t i c d i v i s i o n be tween p r o p e r names and f u n c t i o n names. (1.23) C o n c e p t s F u n c t i o n s ^ o f one argument w h i c h a l w a y s t a k e t r u t h v a l u e s a s t h e i r v a l u e s a r e c a l l e d ' c o n c e p t s ' . S i n c e s e n t e n c e s denote t r u t h v a l u e s , a c o n c e p t name c a n be formed by o m i t t i n g a p r o p e r name f rom a d e c l a r a t i v e s e n t e n c e . Thus* ' £l w r o t e Wave r l ey* names a c o n c e p t whose v a l u e i s t he True f o r S c o t t as argument and the F a l s e f o r a n y t h i n g e l s e a s a rgumen t . (1 .24) Concep t names a r e f o r F rege p r e d i c a t e s and he r e p e a t e d l y s t r e s s e s the p r e d i c a t i v e n a t u r e o f c o n c e p t s -11 t h a t t h e y , l i k e f u n c t i o n s , a r e t h e r e f e r e n c e s o f i n c o m -p l e t e e x p r e s s i o n s . He w r i t e s t "What I c a l l he re the p r e d i c a t i v e n a t u r e o f t h e c o n c e p t i s j u s t a s p e c i a l case o f t he need o f s u p p l e m e n t a t i o n , the * u n s a t u r a t e d n e s s ' t h a t I gave a s the e s s e n t i a l f e a t u r e o f a f u n c t i o n . " ( B i b . ( 2 1 ) see (28) p^7n) ( 1 . 2 5 ) I f t he v a l u e f o r a g i v e n o b j e c t as argument o f a c o n c e p t i s the T r u e , t h a t o b j e c t i s s a i d t o f a l l unde r the c o n c e p t . ( 1 . 2 6 ) F r e g e a l s o s t r e s s e d t h a t "a c o n c e p t must be eve rywhere d e f i n e d " $ " I t must be d e t e r m i n a t e f o r e v e r y o b j e c t , whe the r i t f a l l s u n d e r the c o n c e p t o r n o t ; a c o n c e p t word w h i c h does no t s a t i s f y t h i s r e q u i r e m e n t on i t s r e f e r e n c e , i s d e n o t a t i o n l e s s . . . " ( B i b . ( 1 3 ) p l 3 3 ) C l e a r l y t h i s r e q u i r e m e n t i s j u s t F r e g e ' s c o n d i t i o n f o r a f i r s t l e v e l f u n c t i o n name h a v i n g a d e n o t a t i o n a p p l i e d t o c o n c e p t s . ( 1 .27) I t i s w o r t h n o t i n g two c r i t e r i a F r ege m e n t i o n e d f o r d i s t i n g u i s h i n g c o n c e p t names. The i n d e f i n i t e a r t i c l e ' a * f o l l o w e d by a common noun e g . ' a hor se* i n ' a h o r s e i s a n a n i m a l ' u s u a l l y i n d i c a t e s a c o n c e p t name. P l u r a l forms o f common nouns e g . ' h o r s e s ' i n ' h o r s e s a r e a n i m a l s ' a l s o u s u a l l y i n d i c a t e c o n c e p t names. ( 1 . 2 8 ) I t i s a l s o w o r t h n o t i n g t h a t F r e g e h e l d t h a t the d i s t i n c t i o n be tween p r o p e r names and p r e d i c a t e s and t h u s the d i s t i n c t i o n be tween o b j e c t s and c o n c e p t s a p p l y u n i v e r s a l l y t o a l l l a n g u a g e s . 12 ,. ( 1 . 2 9 ) B e f o r e c o n s i d e r i n g s e c o n d l e v e l c o n c e p t s , i t i s c o n v e n i e n t he re t o f u l f i l a n e a r l i e r p r o m i s e o f s a y i n g s o m e t h i n g abou t the s ense s o f p r e d i c a t e s and f u n c t i o n names i n g e n e r a l . ( 1 . 3 0 ) I n the e a r l y s t a g e s o f commentary on F r e g e ' s o n t o l o g y , a c o n t r o v e r s y a r o s e a s t o w h e t h e r F rege i n t -ended h i s s e n s e / r e f e r e n c e d i s t i n c t i o n t o a p p l y t o f u n c t i o n names. Some commenta tors ( e g . M a r s h a l l - see B i b . ( 3 1 ) ) a r g u e d t h a t F r e g e ' s use o f ' B e d e u t u n g ' i n c o n n e c t i o n w i t h f u n c t i o n names was n o t t o be t a k e n i n t h e s t r i c t t e c h n i c a l sense w h i c h i t has i n c o n n e c t i o n w i t h p r o p e r names b u t was i n t e n d e d t o mean s o m e t h i n g l i k e t he E n g l i s h ' m e a n i n g * . I n w h i c h case the ' B e d e u t u n g ' o f a f u n c t i o n names was much l i k e the ' S i n n ' o f a p r o p e r name?and t h e n o t i o n o f t h e ' S i n n ' o f a f u n c t i o n name s i m p l y d i d no t a p p l y . The p u b l i c a t i o n o f F r e g e ' s ' N a c h g e l a s s e n e S c h r i f t e n ' ( B i b . (13) ) ended t h i s d e b a t e . T h e r e i n he w r o t e , r e g a r d i n g the d i s t i n c t i o n w h i c h he had drawn i n " S i n n und B e d e u t u n g " be tween the sense and r e f e r e n c e o f a p r o p e r name* "The same d i s t i n c t i o n c a n a l s o be made w i t h r e g a r d t o c o n c e p t w o r d s . There i s some danger t h a t t he c l a s s i f i c a t i o n i n t o c o n c e p t s and o b j e c t s w i l l be m i x e d up w i t h the d i s t i n c t i o n be tween sense and r e f e r e n c e , so t h a t sense and c o n c e p t on t h e one hand and r e f e r e n c e and o b j e c t on the o t h e r w i l l be f u s e d . To e v e r y c o n c e p t word o r p r o p e r name c o r r e s p o n d s , as a r u l e , a sense and a r e f e r e n c e . . . ( B i b . ( 1 3 ) p l 2 8 ) F u r t h e r r e s e a r c h r e v e a l e d t h a t t he s enses o f f u n c t i o n names 13 a r e a l s o u n s a t u r a t e d . Thus t h e symmetry o f u n s a t u r a t e d n e s s be tween a s i g n and i t s r e f e r e n c e i s p r e s e r v e d be tween a s i g n and i t s s e n s e . The sense o f a f u n c t i o n name i s u n s a t u r a t e d w h i l e t he sense o f a p r o p e r name i s s a t -u r a t e d . (1.31) Second l e v e l c o n c e p t s name w h i c h i s a p r o p e r p a r t o f t h a t s en t ence i s o m i t t e d , t he r e s u l t i n g e x p r e s s i o n i s , a s we have s e e n , a c o n c e p t name o r p r e d i c a t e . The c o n c e p t named by s u c h a n e x p r e s s i o n , F r ege c a l l s a f i r s t l e v e l c o n c e p t . I f however the g r a m m a t i c a l p r e d i c a t e i s d ropped f rom a s e n t e n c e , t he r e s u l t i n g e x p r e s s i o n i s a name o f a s e c o n d l e v e l c o n c e p t w h i c h t a k e s f i r s t l e v e l c o n c e p t s as a r g u m e n t s . ( P r o v i d e d o f c o u r s e t h a t t he c o n d i t i o n s f o r b e i n g a c o n c e p t a r e met b y the e n t i t y so named.) F o r example , deno te s a s e c o n d l e v e l c o n c e p t whose v a l u e i s the True f i r s t l e v e l f u n c t i o n names. ( 1 . 3 2 ) The most u b i q u i t o u s s e c o n d l e v e l c o n c e p t i n F r e g e ' s s y s t e m i s t he u n i v e r s a l q u a n t i f i e r , The e x p r e s s i o n fl see H . J a c k s o n B i b . ( 2 9 ) I f f rom a s e n t e n c e a p r o p e r • S c o t t <f> • 14 u s e d t o denote t h i s c o n c e p t i s » T h i s e x p r e s s i o n names a s econd l e v e l c o n c e p t whose v a l u e f o r a p a r t i c u l a r f i r s t l e v e l f u n c t i o n as argument i s the True i f and o n l y i f the v a l u e o f t h a t f u n c t i o n i s t he True f o r e v e r y a rgumen t . (1.34) I f t h e r e s u l t o f com-p l e t i n g a s e c o n d l e v e l c o n c e p t w i t h a f i r s t l e v e l c o n c e p t y i e l d s the T r u e , the f i r s t l e v e l c o n c e p t i s s a i d t o f a l l w i t h i n t he s e c o n d l e v e l f u n c t i o n . (1.35) R e l a t i o n s F u n c t i o n s o f two arguments w h i c h a l w a y s t a k e t r u t h v a l u e s a s t h e i r v a l u e s a r e c a l l e d r e l a t i o n s . I f f o r example we d r o p t h e p r o p e r name ' W a v e r l e y * f rom the c o n c e p t name ' ^ w r o t e W a v e r l e y ' t h e r e s u l t i n g e x p r e s s i o n ' ^ wro t e C, ' d e n o t e s a r e l a t i o n . I w i l l c a l l t he a rguments . :o f a r e l a t i o n i t s - t e r m s ' . R e l a t i o n s , l i k e c o n c e p t s , c a n be o f f i r s t o r s e c o n d l e v e l i e . some a r e c o m p l e t e d by o b j e c t s and some by f i r s t l e v e l f u n c t i o n s . (1.36) B u t some r e l a t i o n s have the f u r t h e r p r o p e r t y o f r e l a t i n g t e r m s o f d i f f e r e n t l e v e l s e g . a r e l a t i o n may t a k e a n o b j e c t a s one o f i t s t e rms and a f i r s t l e v e l f u n c t i o n a s the o t h e r . Such r e l a t i o n s a r e c a l l e d u n e q u a l l e v e l l e d r e l a t i o n s . 15 R e l a t i o n s , b o t h o f whosevterms a r e o b j e c t s o r f u n c t i o n s o f t he same l e v e l a r e c a l l e d e q u a l l e v e l l e d r e l a t i o n s . The l e v e l o f a n u n e q u a l l e v e l l e d r e l a t i o n i s one h i g h e r t h a n i t s h i g h e s t l e v e l l e d a rgumen t . (1 .37) I d e n t i t y c o n d i t i o n s f o r f u n c t i o n s We now t u r n t o a c r u c i a l q u e s t i o n s How a r e F r e g e ' s f u n c t i o n s i n d i v i d u a t e d ? Or c o n s i d e r i n g j u s t f i r s t l e v e l c o n c e p t s ? A r e c o n c e p t s and Tj. i d e n t i c a l i f e x a c t l y the same o b j e c t s f a l l u n d e r each?^ Commentators have d i s a g r e e d . R u l o n W e l l s f o r example , c l a i m s Frege " . . . i s p r e p a r e d t o a d m i t t h a t t h e r e c a n be f u n c t i o n s t h a t a r e . d i f f e r e n t f rom e a c h o t h e r e v e n t hough t h e y have p r e c i s e l y the same v a l u e s f o r the same a rguments . . . T h i s i s t an tamount t o r e j e c t i n g s o - c a l l e d e x t e n -s i o n a l i t y o f f u n c t i o n s . We c a n o n l y s ay t h a t F r ege has n o t l a i d down f o r us c o n d i t i o n s u n d e r w h i c h d i f f e r e n t e x p r e s s i o n s i d e n t i f y the same f u n c t i o n s . " ( B i b . ( 4 5 ) s e e ( 3 0 ) p l l ) On the o t h e r hand , F u r t h m a i n t a i n s t h a t : " . . . i n F r e g e ' s t h e o r y , f u n c t i o n s and c o n c e p t s a r e a l s o e x t e n s i o n a l , t h e y a r e n o t d i s t i n g u i s h e d f rom c o u r s e s o f v a l u e s and the e x t e n s i o n o f c o n c e p t s on t h i s g r o u n d , b u t on the g r o u n d t h a t c o u r s e s o f v a l u e s and the e x t e n s i o n o f c o n c e p t s a r e o b j e c t s whereas f u n c t i o n s and c o n c e p t s a r e n o t . " ( B i b . (20) p x x x i x ) (1 .38) I h a s t e n t o add an e x p l a n a t i o n o f F r e g e ' s n o t i o n o f c o u r s e - o f - v a l u e s o f f u n c t i o n s . C o n s i d e r a n example s t a t e d i n the t e rms o f s e t t h e o r y . The c o u r s e -6 F rege u s e s c a p i t a l Greek l e t t e r s where modern a u t h o r s m i g h t use m e t a l i n g u i s t i c v a r i a b l e s r a n g i n g o v e r f i r s t l e v e l f u n c t i o n s . 16 of-values of the concept ^ = 4 (Frege uses the expression '€,(€, = 4)* to denote this course-of-values) can be seen as the set of ordered pairs (x,y) such that y i s the value of this concept for x as argument. The course-of-values of a relation can s imilarly be viewed as the set of ordered tr ip les (x,y,z) where z is the value of the relat ion for x and y as arguments. In the case of concepts/relations, the second/third member of each member of the set of ordered pa irs / tr ip les constituting the course of values of the concept/relation, is a truth value. (1 .39) The question of the identity conditions of concepts can now be restated. Are two concepts the same when they have the same course-of-values? The answer is a qualif ied yes. Qualified, because as Frege noted, the relat ion of equality or sameness cannot s t r i c t l y be said to hold between concepts. He wrotet (1.40) ". . . the relat ion of equality, by which I mean complete coincidence, identity, i s also conceivable only in connection with objects, not with concepts . . . " (Bib. (13) P130/1) and again, "We have now recognised that the relat ion of equality between objects cannot also be conceived between concepts, and that there is however a corresponding re lat ion. The word 'same*, which is used for the desig-nation of the former relat ion between objects, cannot consequently properly serve also for the designation of the lat ter ." (Bib. (13) P132) 17 Perhaps t h i s p o i n t i s b e t t e r e x p r e s s e d i f we d e p i c t t h e r e l a t i o n o f i d e n t i t y thus* C l e a r l y , t h i s r e l a t i o n can o n l y be c o m p l e t e d by o b j e c t s . Any f i l l i n g o f the argument p l a c e s o f t h i s r e l a t i o n name by c o n c e p t names y i e l d s n o t t h e name o f a t r u t h v a l u e b u t t he name o f a new r e l a t i o n . (1.41) I f we w i s h t o e x p r e s s the t h o u g h t t h a t the c o n c e p t s ^ and 1p s t a n d i n t h a t r e l a t i o n (men t ioned above by F r e g e ) c o r r e s p o n d i n g t o t h e r e l a t i o n o f i d e n t i t y be tween o b j e c t s , we may s a y i Abou t t h i s r e l a t i o n , F r ege i s q u i t e e x p l i c i t . He sums u p i "Bu t ;even i f the r e l a t i o n o f e q u a l i t y i s o n l y c o n c e i v a b l e i n c o n n e c t i o n w i t h o b j e c t s , a s i m i l a r one o c c u r s i n c o n n e c t i o n w i t h c o n c e p t s w h i c h , a s a r e l a t i o n be tween c o n c e p t s I c a l l a r e l a t i o n o f s e c o n d o r d e r w h i l e I c a l l the f o r m e r e q u a l i t y a f i r s t o r d e r r e l a t i o n . We s a y , a n o b j e c t a i s e q u a l t o an o b j e c t b ( i n the sense o f comple t e c o i n c i d e n c e ) , i f a f a l l s u n d e r e v e r y c o n c e p t u n d e r w h i c h b f a l l s , and c o n v e r s e l y . We o b t a i n s o m e t h i n g c o r r e s p o n d i n g t o t h i s f o r c o n c e p t s , i f we l e t the c o n c e p t and o b j e c t i n t e r c h a n g e t h e i r r o l e s . We c o u l d s ay t h e n , t he r e l a t i o n m e n t i o n e d above o c c u r s be tween t h e c o n c e p t a) and the concep t "X, i f e v e r y o b j e c t w h i c h f a l l s u n d e r ^ a l s o f a l l s u n d e r X " a n d c o n v e r s e l y . " ( B i b . (13) P13D (1.42) I d e n t i t y o f c o u r s e s - o f - v a l u e i s f o r F r e g e , a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h i s r e l a t i o n h o l d i n g be tween c o n c e p t s . (1.43) B u t c o n c e p t s a r e s t i l l t o be d i s t i n g u i s h e d f rom t h e i r c o u r s e s - o f - v a l u e s - t he 18 former are unsaturated, the latter saturated. This principle naturally extends to relations and to functions in general. (1.44) In the course of this essay I shall have occasion to enquire whether relation A is the same as relation B. Owing to the above considerations, the reader should understand this to mean,'Does relation A have that relation to B which corresponds to the relation of identity for objects? Obscure though this question may seem, the answer i s readily available once we know the courses-of-values of A and B. (1.45) This is the end of the brief exposition of those Fregean notions with which this essay i s concerned. But the reader may already have noticed a problem which has infected this exposition <„-and which w i l l in fact plague anyone attempting such an exposition. During the course of my exposition, I have said such things as» "£ wrote Waverley has the truth value the True for Scott as argument," But this expression i s not a sentence by Frege's c r i t e r i a , for i t is incomplete in the same way that the f i r s t level predicate * ^ wrote Waverley' i s incomplete. If anything i t denotes a f i r s t level concept. The d i f f i c u l t y is that a concept (and in general a function) can only be referred to within the context of a complete name. But i t seems that in order to say the things about concepts which any expo-19 sition of Frege's doctrines requires saying, we need to refer to concepts on their own, so to speak - not in the context of complete names. Frege was well aware of this d i f f i c u l t y . He commenteds "It is clear that we cannot present a concept as independent, like an object* rather i t can occur only in connection. One may say that i t can be d i s t i n -guished within, but that i t cannot be separated from the context in which i t occurs. A l l apparent contra-dictions that one may encounter here derive from the fact that we are tempted to treat a concept like an ob-ject, contrary to i t s unsaturated nature. This is some-times forced upon us by the nature of our language. Nevertheless i t is merely a linguistic necessity." (Bib (19) P 34n) But this problem is really twofold and goes deeper than Frege w i l l allow. Predicates, according to Frege, exhibit that feature corresponding to concepts them-selves - they are unsaturated. We attempted to exhibit this feature by indicating the argument place by means of Greek letters or parentheses. But our standard way of referring to these expressions - by enclosing the en-tir e expression in quotation marks - refers to just those signs within the quotation marks and these signs notice-ably bear no trace of unsaturatedness. Thus for example, the quotation mark name, ' £ wrote Waverley' refers to an object - the group of signs within the quotation marks. Another way of seeing this is that quotation mark names are complete names and thus on Frege's principles cannot denote an unsaturated entity. 20 And this problem remains no matter how one chooses to i n -dicate the argument place of a predicate. Worse s t i l l , even i f we could form names of predicates which were themselves incomplete, they would be no use whatsoever. For suppose, '! ^ wrote Waverley!' were such a name. Then, •! £ wrote Waverley! denotes a function of one . argument' is exactly on par with, ' £ wrote Waverley has the truth value the True for Scott as argument.' insofar as i t is incomplete and hence not a sentence. Perhaps i t is considerations such as these which have prompted Dummett to claim that predicates for Frege, are not linguistic expressions at a l l - contrary to the way we have been treating them throughout this exposition. Dummett writesi "The incompleteness of a predicate . . . resides in the fact that i t i s not, in general, a separate piece of the sentence but i s rather, a feature of the way in which the sentence is constructed. A predicate is thus something which l i t e r a l l y cannot be exhibited separately." (Bib (10)vol 3 , P 230/1) Thus there appears to be an insuperable obstacle to explaining Frege's theories. Clearly, the unsaturated/ saturated division is at the heart of this obstacle. Why 21 did Frege employ such a distinction which effectively prevented him from stating his own theory? It is the purpose of this essay to answer this question and u l t i -mately to show that the function/object or saturated/ unsaturated distinction i s ill-founded - that i t cannot solve the very problems for which i t was designed. For this purpose however, I must set aside for the time being the problems just discussed and I shall continue refer-ring to concepts and predicates in the same way as I have been referring to them throughout this exposition. 22 Chapter 2 Introduction There are two contentious ontological claims that Frege makes. F i r s t l y that predicates have reference and secondly that their reference is 'unsaturated* or 'incomplete1. This chapter concerns i t s e l f with the nature and justification of these claims. From Frege*s analysis of the relationship between language and thought, an argument is produced in support of the view that every component of a sentence must have a sense. An analogous argument is considered in support of the claim that every component of a sentence (and hence predicates) must have a reference. But this argument i s shown to be based on an assumption which Frege himself later repudiated. Frege's claim that the reference of a pred-icate is 'unsaturated* is then considered as a response to the problem of the unity of the proposition. As a special case of this problem, we consider Bradley's problem of relations and construct a Fregean response to this problem in some detail. It i s argued that once relations are seen to be explained as that which relation-a l expressions refer to, Bradley's problem and the general problem of which i t is a special case, disappear. It is characteristic of relations that they can only be explained in the above manner and i t is this characteristic which 23 best elucidates the Fregean metaphors of 'unsaturatedness* and 'incompleteness'. But - i t i s argued - a crucial semantic principle emerges from this discussions We cannot refer to a concept or relation (and in general a function) by using a proper name. Furthermore, any language capable of talking about relations and concepts at a l l , must be governed by this principle. 2h Dummett has p o i n t e d ou t t h a t nowhere i n F r e g e ' s p u b l i s h e d w r i t i n g s does F rege e x p l i c i t l y a rgue f o r h i s p o s i t i o n t h a t f u n c t i o n words have ' B e d e u t u n g ' . ( 1 . 1 8 , 1 . 1 9 , 8 1.30) However , a m a n u s c r i p t o f 1906, r e p o r t s Dummett, c o n t a i n s two s t a t e m e n t s a p p r o a c h i n g s u c h an a rgumen t . Frege w r i t e s 1 " I t i s a l t o g e t h e r i m p r o b a b l e t h a t a p r o p e r name s h o u l d be so d i f f e r e n t f rom the r e m a i n i n g p a r t o f a s i n g -u l a r s e n t e n c e t h a t i t s h o u l d be i m p o r t a n t f o r i t a l o n e t o have ' B e d e u t u n g ' . . . I t i s u n t h i n k a b l e t h a t t h e r e c o u l d be a ' B e d e u t u n g ' o n l y i n the case o f p r o p e r names and n o t i n the r e m a i n i n g p a r t o f the sentence*; However , t h i s s t a t e m e n t i s j u s t t e n d e n t i o u s u n l e s s we assume t h a t a s en t ence d e n o t e s a t r u t h v a l u e and we c a n adduce some s u p p o r t f o r t he p r i n c i p l e t h a t i f p a r t o f an e x p r e s s i o n l a c k s ' B e d e u t u n g ' t h e n so does the e n t i r e e x p r e s s i o n . F r e g e ' s s econd s t a t e m e n t i s n o t much more c o n v i n c i n g . I t p r o c e e d s a l o n g t h e s e l i n e s » C o n s i d e r t h e r e l a t i o n a l s en t ence ' J u p i t e r i s l a r g e r t h a n M a r s ' . T h i s s t a t e s t h a t a c e r t a i n r e l a t i o n h o l d s be tween J u p i t e r and M a r s . Bu t s u r e l y s u c h a r e l a t i o n must s u b s i s t i n the same r e a l m a s J u p i t e r and Mars - namely the r e a l m o f r e f e r e n c e . F o r the b e i n g - l a r g e r - t h a n , i f i t i s t o h o l d be tween two 2 Of c o u r s e , F rege p r o b a b l y a r r i v e d a t h i s p o s i t i o n f rom h i s a n a l y s i s o f number. However R e s n i k has a r g u e d c o n -v i n c i n g l y ( B i b . (36) P 3 3 3 A ) t h a t one c a n n o t a p p e a l t o t h i s a n a l y s i s i n s u p p o r t o f t h e v i e w t h a t f u n c t i o n words have 'Bedeu tung* w i t h o u t b e g g i n g the q u e s t i o n . 8 see B i b . (9) p205/6 3$ e n t i t i e s o f the o b j e c t i v e w o r l d , must i t s e l f be a p a r t o f t h a t w o r l d . T h i s argument o n l y b e g i n s t o be c o n v i n c i n g i f we f i n d the move f rom ' J u p i t e r i s l a r g e r t h a n M a r s ' t o •There i s a r e l a t i o n w h i c h h o l d s be tween J u p i t e r and Mars* u n o b j e c t i o n a b l e . B u t i t i s p r e c i s e l y t h i s move w h i c h anyone who doub t s t h a t r e l a t i o n words have r e f e r e n c e i s l i k e l y t o f i n d o b j e c t i o n a b l e . The move i s f rom a n o n t o l o g i c a l l y i n n o c e n t r e l a t i o n a l s t a t e m e n t t o a s t a t e m e n t e x p l i c i t l y a s s e r t i n g the e x i s t e n c e o f a r e l a t i o n . I f we a c c e p t t h i s move we w i l l much more r e a d i l y a c c e p t t h a t r e l a t i o n words have ' B e d e u t u n g ' . B u t f o r the p r e s e n t i t i s n o t o b v i o u s t h a t we must a c c e p t t he move. There a r e two c o n t e n t i o u s c l a i m s t h a t F rege makes r e g e r d i n g f u n c t i o n w o r d s . F i r s t l y t h a t t h e y have r e f -e rence and s e c o n d l y t h a t t h e i r r e f e r e n c e i s ' u n s a t u r a t e d * , • i n c o m p l e t e * . ( 1 . 1 4 , 1 . 1 6 , 1 . 1 8 , 1 . 1 9 , 1 . 2 1 ) T h i s s e c o n d c l a i m I p ropose r e c e i v e s i t s s t r o n g e s t s u p p o r t f rom F r e g e ' s a n a l y s i s o f the r e l a t i o n s h i p be tween l anguage and the w o r l d . P r o b l e m s i n v o l v i n g t h i s r e l a t i o n s h i p a r i s e w h i c h a r e s een t o r e q u i r e f o r t h e i r s o l u t i o n t h e v i e w t h a t t h e r e f e r e n c e s o f c o n n e c t i v e s a r e u n s a t u r a t e d . R e s n i k has a r g u e d ^ t h a t t he f i r s t c l a i m t o o i s g rounded upon s i m i l a r c o n c e r n s . I n t h i s s e c t i o n I d i s c u s s the m e t a p h y s i c s w h i c h p u r p o r t e d l y s u p p o r t s t h e s e c l a i m s . 9_ see B i b . (36) 26 I t h i n k t h e two c l a i m s c a n p r o f i t a b l y be s e p a r a t e d a l t h o u g h the o v e r a l l t h e o r y i s c l e a r l y l i n k e d . B u t I s h a l l i n d i c a t e i n due c o u r s e where such l i n k a g e o c c u r s . R e s n i k ' s argument i s b a s e d upon a n a n a l o g y w i t h F r e g e ' s a n a l y s i s o f t h o u g h t s and the s e n t e n c e s whose s e n s e s t h e y a r e . (1 .09) We b e g i n t h e n by c o n s i d e r i n g t h i s a n a l y s i s . Some p r o b l e m s w i t h w h i c h F r e g e ' s t h e o r y o f sense e x p l i c i t l y d e a l s a r e the f o l l o w i n g ! 1) How do we s u c c e e d i n c o m m u n i c a t i n g new t h o u g h t s ? 2) How w i t h o n l y a few s y l l a b l e s i n o u r l anguage a r e we c a p a b l e o f e x p r e s s i n g a huge number o f t h o u g h t s ? 3) How do we s u c c e e d i n c o m m u n i c a t i n g t h o u g h t s a t a l l ? The answers t o a l l t h e s e q u e s t i o n s w i l l become a p p a r e n t i f we s u c c e e d i n a n s w e r i n g j u s t the f i r s t . L e t us t h e n c o n c e n t r a t e on t h i s one . F r e g e ' s a p p r o a c h t o t h e s e q u e s t i o n s i s i n t h e s t y l e o f the d e s c r i p t i v e m e t a p h y s i c i a n - we know t h a t new t h o u g h t s a r e communica ted , what t h e n a r e the c o n d i t i o n s w h i c h make t h i s p o s s i b l e ? He w r i t e s » " I t i s a s t o n i s h i n g what l anguage c a n d o . W i t h a few s y l l a b l e s i t c a n e x p r e s s a n i n c a l c u l a b l e number o f t h o u g h t s , so t h a t even a t h o u g h t g r a s p e d by a human b e i n g f o r the v e r y f i r s t t i m e c a n be pu t i n t o a fo rm o f words w h i c h w i l l be u n d e r s t o o d by someone t o whom t h e t h o u g h t i s e n t i r e l y new. T h i s w o u l d be i m p o s s i b l e , were we no t a b l e t o d i s t i n g u i s h p a r t s i n the t h o u g h t c o r r e s p o n d i n g t o the p a r t s o f a s e n t e n c e , so t h a t t he s t r u c t u r e o f t he s e n t e n c e s e r v e s a s an image o f the s t r u c t u r e o f t h e t h o u g h t . . . I f t h e n , weJLook upon t h o u g h t s as composed o f s i m p l e r p a r t s , and t ake t h e s e , i n t u r n t o c o r r e s p o n d t o t h e s i m p l e p a r t s o f s e n t e n c e s , we c a n u n d e r s t a n d how a few p a r t s o f 27 sentences can go to make up a great multitude of thoughts." (Bib.(15) see (30) P537/8) Let us attempt a reconstruction in some detail of the conditions which make the transmission of a new thought possible. Suppose I one day grasp a thought never before grasped and I wish to communicate i t to another. I perceive the structure of the thought, that i s I perceive i t s components and the way in which these components combine to constitute the whole thought. (This last I shall c a l l the ordering of the components of the thought.) Now the structure of the thought corresponds to the structure of a piece of language, in particular a sentence whose sense the thought i s . ("The world of thoughts has a model in the world of sentences, expressions, words, signs. To the structure of the thought there corresponds the compounding of words into a sentence and here the order i s in general not indifferent."^) I then locate those expressions in my language the senses of which constitute the components of the thought. I then arrange these expressions in that ordering corresponding to the ordering of the components of the thought. I have now formed a complete sentence whose sense is the thought I wish to communicate. It remains to be seen how the sense of this sentence is grasped by another individual who has never before grasped this sense. This individual is presented with a series of expressions a l l of whose 10 Bib. (17) see (28) pl23 28 senses we may assume he g r a s p s . A l l t h a t i s new to the o t h e r p e r s o n i s t he o r d e r i n g o f t he s ense s o f t h e components o f my sen tence and he i s p r e s e n t e d w i t h a model o f t h i s i n the o r d e r i n g o f t h e component e x p r e s s i o n s o f my s e n t e n c e s i n c e as we have s e e n , t h i s o r d e r i n g m i r r o r s t he o r d e r i n g o f t he t h o u g h t components . He i s t h u s i n a p o s i t i o n t o g r a s p the new t h o u g h t . L o o k i n g back on t h i s now we see what i s e s s e n t i a l t o the c o m m u n i c a t i n g o f a new t h o u g h t . 1) A s k i l l e d l anguage u s e r c a n l o c a t e , g i v e n a component sense o f a t h o u g h t , an e x p r e s s i o n whose sense i t i s . 2) There.are o b j e c t i v e s ense s o f e x p r e s s i o n s w h i c h can be g r a s p e d by a s k i l l e d l anguage u s e r . (1 .07) 3) Thoughts have a p e r c e b t i b l e s t r u c t u r e w h i c h i s m i r r o r e d i n the s t r u c t u r e o f the sen tences- e x p r e s s i n g t h o s e t h o u g h t s . The i m p o r t a n t p o i n t he re i s (3). S u c c e s s i n communic-a t i n g new t h o u g h t s r e q u i r e s t h a t " the s t r u c t u r e o f t he s en t ence s e r v e s as a n image o f the s t r u c t u r e o f the t h o u g h t " 11 There i s n o t h i n g odd abou t t h i s a s s u m p t i o n . We migh t have assumed t h a t t h e r e i s no e x p r e s s i o n i n my v o c a b -u l a r y w h i c h i s n o t i n t h a t o f t h e i n d i v i d u a l and t h i s i s q u i t e c o m p a t i b l e w i t h my g r a s p i n g a t h o u g h t w h i c h has n e v e r been g r a s p e d by t h i s i n d i v i d u a l . C e r t a i n l y , i f t h e r e i s a component o f my n e w l y formed s e n t e n c e w h i c h i s n o t u n d e r s t o o d by t h e o t h e r p e r s o n , t h e n he w i l l n o t u n d e r -s t a n d the sense o f the e n t i r e s e n t e n c e . F o r t h i s r e a s o n we assume t h a t he g r a s p s t h e s enses o f a l l components o f t he s e n t e n c e . 12 T h i s s t a t e m e n t i s i n c o m p a t i b l e w i t h F r e g e ' s s t a t e d v i e w t h a t t h e s e n t e n c e s ' A ' and ' A & A ' e x p r e s s t h e same s e n s e . F o r t h e s e s e n t e n c e s c l e a r l y have d i f f e r e n t s t r u c t -u r e s and the s enses t h e y e x p r e s s w o u l d need t o be d i f f e r e n t i f the s t r u c t u r e s o f the s e n t e n c e s were t o m i r r o r the s t r u c t u re o f t he t h o u g h t s t h e y e x p r e s s . I i g n o r e t h i s p r o b l e m h e r e 2 9 so that to each component of the sentence there corresponds a component of the thought. (If this were not so then presenting the ordering of expressions by say, uttering a sentence, would not present the hearer with a model of the ordering of the components of the thought we wish him to grasp.) Now this entails that every component of every sentence expressing a thought has a sense and thus we derive Frege's principle that i f a part of a sentence lacks a sense then so does the entire sentence. (1 .13) It i s interesting to speculate whether Frege might at one time have thought that an analogous argument dealing with reference could be adduced in support of his view that functional expressions have reference. For he once held the view (which he later abandoned) that to the structure of a definite description there corr-esponds the structure of the object i t names and i f we ask the question 'How do we succeed in forming definite descriptions of things never before named?', we might be tempted to give the following response: 'We perceive the structure of the object to be named ie. i t s components and their ordering. We then locate those expressions whose references are the components of the object to be named and arrange these expressions in that ordering which models the ordering of the compon-ents of the object. Crucial to our success in this is that the structure of the definite description corresponds to the structure of i t s reference and this entails that 30 each component o f o u r d e f i n i t e d e s c r i p t i o n names a com-ponent o f the named o b j e c t . S i n c e d e f i n i t e d e s c r i p t i o n s c o n t a i n f u n c t i o n a l e x p r e s s i o n s , t h i s f u r t h e r e n t a i l s t h a t f u n c t i o n a l e x p r e s s i o n s have r e f e r e n c e . ' However , F rege abandoned the v i e w t h a t o b j e c t s were composed o f p a r t s w h i c h a r e deno ted by p a r t s o f the d e f -i n i t e d e s c r i p t i o n d e n o t i n g the o b j e c t , when f a c e d w i t h h a v i n g t o d e f e n d the l u d i c r o u s p o s i t i o n t h a t Denmark i s a p a r t o f Copenhagen . F o r ' D e n m a r k ' i s a component o f ' t h e c a p i t a l o f Denmark ' and the l a t t e r e x p r e s s i o n deno te s Copenhagen . The u p s h o t o f t h i s i s t h a t w h i l e t he t h e o r y we e x p l o r e d i n r e l a t i o n t o sense and t h o u g h t c a n c o h e r e n t l y be h e l d t o s u p p o r t t he v i e w t h a t e v e r y component o f a s en t ence e x p r e s s i n g a t h o u g h t has a s e n s e , the same s t y l e t h e o r y r e l a t i n g t o r e f e r e n c e canno t c o h e r e n t l y be employed t o s u p p o r t t he v i e w t h a t f u n c t i o n a l e x p r e s s i o n s have r e f e r e n c e . We t u r n now t o c o n s i d e r the a d v a n t a g e s . o f p o s i t i n g i n c o m p l e t e o r u n s a t u r a t e d e n t i t i e s . The p rob l ems c o n c e r n i n g c o m m u n i c a t i o n o f t h o u g h t , we have a r g u e d , a r e e f f e c t i v e l y r e s o l v e d by F r e g e ' s v i e w s on the r e l a t i o n s h i p be tween l anguage and t h o u g h t f rom w h i c h emerge c e r t a i n o f h i s s e m a n t i c p r i n c i p l e s . C e n t r a l t o t he se v i e w s i s h i s b e l i e f t h a t s e n t e n c e s have a s t r u c t -u r e , t h a t t h e y c a n be decomposed i n t o components s u c h t h a t t he manner o f d e c o m p o s i t i o n m i r r o r s the manner i n w h i c h the t h o u g h t w h i c h the s e n t e n c e e x p r e s s e s i s decomposed i n t o i t s componen t s . Tha t t he t h o u g h t c a n be so decomp-31 posed provides the basis for Frege's solution to the abovementioned problems. But how from i t s components a single thought is formed gives rise to another range of problems. This second type of problem Frege attempted to solve by postulating that the sense of a predicate is unsaturated or incomplete. Similar problems arising at the level of reference induce Frege to postulate that the reference of predicates i s likewise unsaturated. The problems referred to here, exercised some of the greatest minds of this century including Russell, Moore and Wittgenstein and cannot indeed be neatly divided into those which relate to sense and those which relate to reference. This division of these problems unmistakeably bears Frege's stamp and of course those who worked with these problems usually did not adhere to such Fregean distinctions as sense and reference. A Russellian proposition of 1903 for example, in no way resembles a Fregean thought. But to appreciate the force of Frege's attempted solution to these problems i t i s useful to view them from the perspective of a non-Fregean ontology (in particular, an ontology corrsponding roughly to Russell's of 1903) and once we have done this we can then divide the problem into i t s two Fregean components and see how Frege's attempted solution applies to the problems so divided. Consider the relational proposition expressed by 'Brutus k i l l e d Caesar' 3 2 I f we a t t e m p t t o l i s t the c o n s t i t u e n t s o f t h i s R u s s e l l i a n p r o p o s i t i o n we wou ld m e n t i o n B r u t u s , the r e l a t i o n o f k i l l i n g and C a e s a r . B u t t he se c o n s t i t u e n t s a re a l s o p r e s e n t i n the p r o p o s i t i o n e x p r e s s e d by ' C a e s a r k i l l e d B r u t u s ' and the two p r o p o s i t i o n s a r e c l e a r l y d i s t i n c t . I n o r d e r t o c a p t u r e the d i f f e r e n c e , we migh t s a y (as R u s s e l l a t one t ime s a i d ) t h a t a f u r t h e r c o n s t i t u e n t o f the f i r s t p r o p o s i t i o n i s a p a r t i c u l a r t h r e e t e rmed r e l a t i o n r e l a t -i n g B r u t u s , k i l l i n g and C a e s a r and t h a t t h i s r e l a t i o n i s n o t e x p r e s s e d by any o f t he words o f the s e n t e n c e b u t by a r e l a t i o n among the words o f the s e n t e n c e . B u t -so a c o u n t e r - a r g u m e n t r u n s - i f the r e l a t i o n k i l l i n g i s n o t s u f f i c i e n t t o r e l a t e B r u t u s and C a e s a r i n the p r o p o s i t i o n and a s econd r e l a t i o n i s r e q u i r e d , t h e n t h i s r e l a t i o n t o o w i l l n o t be s u f f i c i e n t t o r e l a t e B r u t u s , k i l l i n g and C a e s a r and a t h i r d r e l a t i o n w i l l be r e q u i r e d . B u t t h i s p r o c e s s i s e n d l e s s . S i m p l e r e l a t i o n a l p r o p o s -i t i o n s i t i s a r g u e d , b r e a k down on a n a l y s i s i n t o a n i n f -i n i t e number o f r e l a t i o n s . C h i e f among the p r o p o n e n t s o f t h i s t y p e o f argument was F . H . B r a d l e y who i n v o k e d i t i n h i s a t t a c k a g a i n s t p l u r a l i s m . I f the w o r l d i s composed o f many t h i n g s , t h e n t h e s e t h i n g s must be i n some r e l a t i o n t o e a c h o t h e r o r e l s e we c o u l d know n o t h i n g o f the w o r l d . I f we c a n know n o t h i n g o f the w o r l d t h e n p l u r a l i s m i s s e l f - d e f e a t i n g . But Bradley's argument purports to show that the notion of relations is self-contradictory and thus too is the notion of a p l u r a l i s t i c universe. Before considering the details of his argument, i t should be pointed out that the problem of the unity of the proposition, the problem of how the elements of a proposition create a unity which is greater than the sum of i t s parts, i s of course not restricted to relational propositions. The Russellian proposition expressed by the simple subject/predicate sentence •Cassius i s i l l * has as i t s constituents, Cassius, the property of being i l l and some relation between them. But the mere l i s t i n g of these entities f a i l s to attribute illness to Cassius -a crucial feature of the proposition. And no matter how many more relations one might l i s t as constituents of the proposition, this essential ingredient of the proposition w i l l be missing from the l i s t . With this in mind we turn to Bradley's argument against relations. His central argument is as follows: "But how the relations can stand to the qualities i s , on the other side, unintelligible. If i t i s nothing to the qualities, then they are not related at a l l , and i f so, as we saw, they have ceased to be qualities, and their relation i s a-non-entity. But i f i t is to be something to them, then clearly we shall require a new connecting relation. For the relation hardly can be the mere adjective of one or both of i t s terms; or at least, as such i t seems indefensible. And being something i t s e l f , i f i t does not i t s e l f bear a relation to the terms, in what i n t e l l i g i b l e way w i l l i t succeed in being anything to them? But here again 3^ we are hurried off into the eddy of a hopeless process, since we are forced to go on finding new relations without end. The links are united by a link, and this bond of union is a link which also has two endsj and these require each a fresh link to connect them with the old. The problem i s to find how the relations can stand to i t s qualities and this problem i s insoluble. If you take " the connection as a solid thing, you have got to show and you cannot show, how the other solids are joined to i t . And i f you take i t as a kind of medium or unsubstantial atmosphere, i t i s a connection no longer." (Bib. (3) P32/3) To modern eyes there i s much that i s vague in this passage. What for example does i t mean for a relation to "succeed in being something to" i t s terms? What is meaait by a"solid thing"? S t i l l , perhaps the crux of the argument is clear enoughs If a relation i s an entity connecting two terms then there must be a further relation linking that relation with each of i t s terms or else the sentence w i l l not say anything. But the same d i f f i c u l t y arises for the new relation and so on ad infinitum and we are led into an infinite regress. Interesingly, this argument recurs in more modern 13 authors, notably Dummett. J He argues that Bradley's infinite regress results from viewing relations as special kinds of objects ie. as nameable by proper names. I paraphrase: If we try to consider the relational expression in an atomic sentence as standing for an object, we shall be at aloss to account for the unity of the sentence. 11 Bib. (9) P256 35 We shall have to think of the sentence as saying that the objects named by the two proper names stand in a particular relation to the object referred to by the relational expression. We shall be in a quandary how to explain the character of this peculiar relationship, which a l l our assertions involve. If we regard this relationship as an object we embark on an inf i n i t e regress. If we regard i t as a function in Frege's sense then we might as well have explained the i n i t i a l relation as such a function. In any case we are trapped in a c i r c l e . Any explanation of a relation must involve reference to the mysterious relat-ion which relates objects to the (object-) relation. But we w i l l be unable to explain what this mysterious relation is for this reason: The explanation of any relation must presuppose knowledge of what kinds of entities can serve as terms of the relation, and this cannot be presupposed in this case since one of the terms of the relation to be explained i s i t s e l f only explained with reference to this relation. Russell discussed the problem of the unity of the proposition in The Principles of Mathematics. Therein he discussed the difference between the propositions expressed by 'Socrates is human* and 'Humanity belongs to Socrates', the difference between verbs and their nominals (eg. ' k i l l s ' and 'killing*) and the relationship between declarative sentences and their nominals (eg. 'Caesar died' and 'the death of Caesar'). He argued that any attempt to 36 distinguish the references of verbs or adjectives used in predicates from the references of the corresponding nominals of these expressions results in a contradictory state of aff a i r s . He noted» "It might be thought that a distinction ought to be made between a concept as such and a concept used as a term, between eg. such pairs as is and being, human and humanity, one in such a proposition as "This i s one" and 1 in "1 is a number". But inextricable d i f f i c u l t i e s w i l l envelop us i f we allow such a view," (Bib. (41) p45) The d i f f i c u l t i e s Russell i s referring to are akin to those discussed in 1.45. In order to deny that human is the same as humanity one i s forced to use the expression •human' not in a predicate position but in a subject position and how then could i t s reference differ from the reference of 'humanity'? Such considerations were sufficient to lead Russell to his view that each pair of underlined expressions above simply contains different names of the same entity. But this view also has i t s d i f f i c u l t i e s . For one thing one cannot then explain why a l i s t i n g of the elements of a proposition f a i l s to express that proposition i t s e l f . Russell was well aware of this. He remarked* "A proposition, in fact, is essentially a unity and when analysis has destroyed the unity, no enumeration of constituents w i l l restore the proposition. The verb when used as a verb embodies the unity of the proposition, and is thus distinguishable from the verb considered as a term, though I do not know how to give a clear account of the precise nature of this distinction." (Bib. (41) P50) C37 Now Frege o f c o u r s e t o o k e x a c t l y t h a t p a t h w h i c h R u s s e l l c l a i m e d was c o n t r a d i c t o r y . The r e f e r e n c e o f • h u m a n i t y ' i s f o r F rege a n o b j e c t - s o m e t h i n g q u i t e d i s t i n c t f rom the r e f e r e n c e o f the p r e d i c a t e ' i s h u m a n ' . ( 1 . 2 2 ) A r e l a t i o n i s s o m e t h i n g q u i t e d i s t i n c t f rom i t s t e rms where i t s t e rms a r e o b j e c t s . (1.35) A l l d i f f i c u l t i e s w h i c h r e s u l t f rom t h i s p o s i t i o n a r e p u r e l y l i n g u i s t i c , F rege d e c l a r e d . They a r e r o o t e d i n o u r g r a m m a t i c a l s t r u c t u r e s and n o t founded on the n a t u r e o f t h i n g s . He n o t e d : " . . . t he o b s t a c l e i s e s s e n t i a l , and founded on the n a t u r e o f o u r l a n g u a g e ; . . . w e canno t a v o i d a c e r t a i n i n a p p r o p r i a t e n e s s o f l i n g u i s t i c e x p r e s s i o n ; and t h e r e i s n o t h i n g f o r i t b u t t o r e a l i z e t h i s and a l w a y s t a k e i t i n t o a c c o u n t . " ( B i b . (21) see (28) p55) I t i s i n t e r e s t i n g t h a t R u s s e l l h i m s e l f , w e l l a f t e r The P r i n c i p l e s o f M a t h e m a t i c s , e x p r e s s e d a s i m i l a r p o i n t o f v i e w . C o n c e r n i n g the s e n t e n c e ' C a e s a r l o v e s B r u t u s ' he w r o t e : "There i s i n the above s e n t e n c e a r e l a t i o n w h i c h i s s y m b o l i s e d by a r e l a t i o n , n o t by a w o r d ; t h i s i s the t h r e e t e rmed r e l a t i o n o f l o v e t o C a e s a r and B r u t u s . T h i s i s s y m b o l i s e d by the o r d e r o f t he words i e . by a t h r e e t e r m r e l a t i o n . B u t i n o r d e r t o m e n t i o n t h i s r e l a t i o n , i t i s n e c e s s a r y t o t r e a t ' l o v e ' g r a m m a t i c a l l y as a s u b s t a n t i v e w h i c h t e n d s t o confuse the d i s t i n c t i o n be tween a s u b s t a n c e and a r e l a t i o n . However i t i s n o t v e r y d i f f i c u l t t o a v o i d the f a l s e s u g g e s t i o n s due t o t h i s p e c u l i a r i t y o f l anguage when once the danger o f them has been p o i n t e d o u t . " ( B i b . (40) p243) No doubt more c o u l d be s a i d abou t Dummet t ' s and B r a d l e y ' s a rgumen t s , b u t o u r main purpose he re i s m e r e l y t o s k e t c h t h e p r o b l e m s w h i c h l e d F rege t o h i s v i e w o f u n s a t u r a t e d 38 entities and to enhance our understanding of the crucial metaphor. Suffice i t to note for the moment that Bradley partly anticipates a Fregean response to his argument. The infinite regress develops from taking relations to be entities of the same kind as the terms they relate. Frege's reply, briefly, is that relations are entities of a completely different kind from the objects they relate. Bradley ant-icipates this by arguing that they are then 'unsubstantial' and hence incapable of performing the function we attribute to them. This argument appears to be based on the pre-supposition that i f a relation is to *;be anything' to i t s terms, i t must be something solid ie. of the same kind as i t s terms. Frege's answer to this is that i t is simply a mistake. For a relation to relate two objects indeed implies that i t i t s e l f must belong to the realm of reference. It is not necessary however, that this realm be uniform ie. consist only of entities of the same kind. Rather this realm consists of two distinct kinds of entities, the saturated and the unsaturated. Let us now view in detail Frege's attempted solution to the problems under discussion. We have already seen a crucial part of i t - that the references of predicates are absolutely distinct from the references of proper names. Precisely because one constituent of a Russellian proposition is a concept (or relation), i t is impossible to l i s t a l l the constituents of such a proposition; for a concept or relation can only be referred to within the context of a sentence. (1.45) 3 9 One c a n u n d e r s t a n d what the c o n s t i t u e n t s o f s u c h a p r o p o s i t i o n a r e bu t one canno t l i s t them. Nor i s t h e r e any d i f f i c u l t y u n d e r s t a n d i n g how the e l e m e n t s o f a p r o p o s i t i o n hang t o g e t h e r once we u n d e r s t a n d a c o n c e p t / r e l a t i o n t o be t h a t w h i c h a c o n c e p t w o r d / r e l a t i o n word r e f e r s t o . I n g e n e r a l , by t y i n g the o n t o l o g i c a l d i s t i n c t i o n be tween f u n c t i o n s and o b j e c t s t o the l i n g u i s t i c d i s t i n c t i o n be tween f u n c t i o n names and p r o p e r names ( 1 . 2 2 ) , Frege i s a b l e t o a v o i d such p r o b l e m s a s the u n i t y o f the p r o p o s i t i o n . F o r once we u n d e r s t a n d what p r o p e r names a r e , we c a n u n d e r s t a n d the w o r k i n g s o f f u n c t i o n a l e x p r e s s i o n ( 1 . 1 4 ) and once f u n c t i o n s a r e e x p l a i n e d as t h a t which f u n c t i o n a l e x p r e s s i o n s r e f e r t o , ( 1 . 1 8 , 1 . 1 9 , , 1 . 2 0 ) our u n d e r s t a n d i n g o f how f u n c t i o n s and o b j e c t s j o i n up t o fo rm comple t e e n t i t i e s i s t i e d t o o u r u n d e r s t a n d i n g o f how f u n c t i o n a l e x p r e s s i o n s and p r o p e r names j o i n up t o fo rm comple t e e x p r e s s i o n s , and i n t h i s t h e r e i s no d i f f i c u l t y . B r a d l e y ' s p r o b l e m can now be seen as a r i s i n g f rom the same s o u r c e o f c o n f u s i o n as R u s s e l l ' s - v i e w i n g r e l a t i o n s as s p e c i a l k i n d s o f o b j e c t s . Once one u n d e r s t a n d s r e l a t i o n s as b e i n g what r e l a t i o n a l e x p r e s s i o n s r e f e r t o , ( 1 . 3 5 ) one u n d e r s t a n d s t h a t a r e l a t i o n r e q u i r e s a l i n k t o c o n n e c t i t w i t h i t s t e rms no more t h a n a r e l a t i o n a l e x p r e s s i o n r e q u i r e s a l i n k t o c o n n e c t i t w i t h the e x p r e s s i o n s f i l l i n g i t s 14 argument p l a c e s . As Dummett e x p r e s s e s t h i s p o i n t » 14 My d i s c u s s i o n o f F r e g e ' s p r o p o s e d s o l u t i o n t o the p r o b l e m s unde r d i s c u s s i o n owes much t o Dummett. See e s p e c i a l l y B i b . ( 9 ) » 40 "A concept and an object, or a relation and two objects, need no glue to f i t them together; they f i t together naturally, in a way we can think of as anal-ogous to that in which a predicate and a proper name, or a relational expression and two proper names, f i t together to form a sentence. And this w i l l seem to us natural and unproblematic as soon as we grasp that we can think of a con-cept only as the referent of a predicate, of a relation . only as the referent of a relational expression." (Bib. (9) pl75) What the unsaturatedness of concepts and relations does for Frege then is to eliminate the need for a further relation connecting concept (or relation) and subsumed object(s) in a proposition.-;;. Although Frege frequently writes as though the relation of subsumption is a genuine relation between for example, a concept and an object, in his more careful treatment of the subject he urges that we are deceived in thinking that there i s such a relation by the inaccuracy of our grammatical expressions by means of which we attempt to analyse sentences. It is this inaccuracy whichs " . . . erweckt auch den Anschein, als ob die Beziehung der Subsumption ein Drittes ware, was zu dem Gegenstande und dem Begriffe^hinzukomme. Das i s t nicht der F a l l , sondern die Ungesattigheit des Begriffes bewirkt, dass der Gegenstand, indem er die Sattigung bewirkt, unmittelbar an dem Begriffe haftet, ohne dass es eines besonderen Bindesmittels bedurfte." 1^ Once this relation is abolished, the f i r s t step into Bradley's regress i s blocked. Yet the difference between 15 My renditions " . . . creates the impression that the relation of subsumption is a third thing which attaches to the object and concept. That is not the case, but rather the unsaturatedness of the concept brings i t about that the object in which i t effects the saturation adheres directly to the concept, without needing a special binding agent." 41 the proposition expressed by 'Brutus k i l l e d Caesar* and •Caesar k i l l e d Brutus' can s t i l l be explained as the result of completing a relation in two different ways though indeed with the same objects in both cases. In a word then, Frege•s attempted solution to the problem of the unity of the proposition, and to the problem of Bradley's infinite regress is that concepts and relations are unsaturated - they can be explained only as the referents of predicates and relational expressions. It is this attempted solution which w i l l be the main concern of the remainder of this essay. But let us now return to a Fregean ontology and s p l i t • the problems we have discussed above into two in order to see how Frege deals with them f i r s t at the level of sense and then at the level of reference. At the level of sense, we are faced with this problems A sentence expresses a thought and each of i t s components a sense. (1 .13) But this thought i s not just a mere series of senses - i t i s a single communicable thought. How do the components of a thought combine to produce such a unity? Or as Frege puts its "But. the question now arises how the thought comes to be constructed, and how i t s parts are so combined together that the whole amounts to something more than the parts taken separately . . ." (Bib. (15) see (30) P537/8) The answer is that the sense of the predicate in a sentence i s unsaturated ( 1 . 3 0 ) in just the sense explained 42 a b o v e . I t i s t h i s u n s a t u r a t e d sense w h i c h p r o v i d e s the bonds , so t o speak , w h i c h e n a b l e s t h e s enses t o c o a l e s c e i n t o a s i n g l e t h o u g h t . The sense o f a s i m p l e s en t ence i s n o t j u s t a s e r i e s o f senses but a s e r i e s o f s enses w h i c h a t t a i n s a u n i t y by v i r t u e o f i t s h a v i n g a n u n s a t u r a t e d component w h i c h i s s a t u r a t e d by s a t u r a t e d c o m p o n e n t ( s ) . F rege w r i t e s * " . . . n o t a l l t he pa r t s o f a t h o u g h t c a n be comple te} a t l e a s t one must be ' u n s a t u r a t e d * o r p r e d i c a t i v e * o t h e r w i s e t h e y w o u l d n o t h o l d t o g e t h e r . F o r e x a m p l e , the sense o f the ph ra se ' t h e number 2* does n o t h o l d t o g e t h e r w i t h t h a t o f the e x p r e s s i o n ' t h e c o n c e p t p r ime number ' w i t h o u t a l i n k . We s u p p l y s u c h a l i n k i n the s e n t e n c e •The number 2 f a l l s under the c o n c e p t p r ime number ' t i t i s c o n t a i n e d i n the words ' f a l l s u n d e r * , w h i c h need t o be c o m p l e t e d i n two ways - by a s u b j e c t and a n a c c -u s a t i v e s and o n l y because t h e i r sense i s t h u s ' u n s a t -u r a t e d ' a r e t h e y c a p a b l e o f s e r v i n g a s a l i n k . O n l y when t h e y have been supp l emen ted i n t h i s t w o f o l d r e s p e c t do we g e t a comple t e s e n s e , a t h o u g h t . I s ay t h a t such words o r p h r a s e s s t a n d f o r a r e l a t i o n . " ( B i b . (18) s e e ( 2 8 ) p 5 4 / 5 ) The r e f e r e n c e o f a s e n t e n c e i s f o r F r e g e , as we have seen, a t r u t h v a l u e . ( 1 . 0 8 ) S i n c e f u n c t i o n a l - , p r e d i c a t i v e -and r e l a t i o n a l e x p r e s s i o n s a r e a l l d e n o t i n g e x p r e s s i o n s f o r F r e g e , i t i s incumbent upon h im t o e x p l a i n how f o r example , by r e f e r r i n g t o two o b j e c t s and a r e l a t i o n , we have succeeded i n r e f e r r i n g t o a t r u t h v a l u e . How do we s u c c e e d f o r e x a m p l e , i n naming the True by u t t e r i n g • C a i n i s t he b r o t h e r o f A b e l ' when a c c o r d i n g t o F r e g e ' s a n a l y s i s we have named C a i n , a c e r t a i n r e l a t i o n and A b e l i n t h a t o r d e r . On t h i s i s s u e , F rege w r o t ei 43 "Take the proposition 'Two i s a prime number'. . . The f i r s t constituent, 'two', is a proper name of a certain number; i t designates an object, a whole that no longer requires completion. The predicative constituent 'is a prime number', on the other hand, does require completion and does not designate an object. I also c a l l the f i r s t constituent saturated; the second, unsaturated. To this difference in the signs there of course corresponds an analogous one in the realm of references: to the proper name there corresponds the object; to the predicative part, something I c a l l a concept. This i s not supposed to be a definition; for the decomposition into a saturated and an unsaturated part must be considered a logically primitive phenomenon which must simply be accepted and cannot be reduced to something simpler. I am well aware that expressions like 'saturated* and 'unsaturated' are metaphorical and only serve to indicate what is meant - whereby one must always count on the cooperative understanding of the reader. Nevertheless, i t may perhaps be made a l i t t l e clearer why these parts must be different. An object, eg. the number 2, cannot logically ; adhere to another object, eg. Julius Caesar, without some means of connection. This, in turn, cannot be an object but rather must be unsaturated. A logical connection into a whole can come about only through this, that an unsaturated part is saturated or completed by one or more parts." (Bib. (19) P32/3) Once again the answer is the same. Once one realizes that a sentence names a truth value, and that concepts and relations are that which are referred to by predicates and relational expressions, the problem disappears. It is the very nature of concepts to adhere to objects to form truth values and similarly for f i r s t level relations -the relation i s the mapping of objects, onto truth values. From Frege's attempted solution to the problems we have been discussing, a crucial semantic principle emerges: A) We cannot refer to a concept (or indeed a relation or any function) by.means of a proper name. The problems discussed above are avoided by explaining a concept and f i r s t level relation as that entity 44 which a predicate refers to, a relation as that which a relat ional expression refers to. By tying the notion of a concept direct ly to the notion 'denoted by a pred-icate' , the problem of the unity of subject/predicate sentences disappears. Again, by tying the notion of a relat ion direct ly to the notion 'denoted by a relat ional expression', Bradley's inf ini te regress cannot be generated. As Furth notes, these notions are* " . . . related by that str ictest of equivalences of meaning which connects explicandum and explicans." (Bib. (24) p22) Relating this to functions in general, the principle emerges that for a given languages B) If X denotes a function then X is an incomplete expression. Clearly, one cannot retain this analysis and allow that a proper name can denote a function. To allow a proper name to denote a function is to reject that analysis which provides the escape from Bradley's inf ini te regress and the larger problem of the unity of the proposition'; Nor is principle (A) restricted to the language in which this discussion has taken place. For Frege, as we have seen (1 .28), the dist inct ion between proper names and predicates applies universally to a l l languages. Since we can only understand a function as being that which a functional expression refers to, any language capable of talking about (in particular) concepts and relations, 15 I use 'talk about* in a wide sense, eg. in the sense in which 'Socrates is wise' is as much about the concept re-ferred to by ' is wise' as i t is about Socrates. 45 must be governed by principle (B) and hence by the principle that concepts and relations cannot be referred to by using proper names. To put this in another wayi Since the notions of a concept and a relation are directly tied to the notions 'denoted by a predicate' and'denoted by a relational expression' respectively, we simply could not conceive of a language in which concepts and relations were referred to by some other means. It is not that every language must talk about relations and concepts - though a l i t t l e reflection w i l l be sufficient to see that a language which cannot talk about relations and concepts must be a most impoverished language indeed -but that i f a language does talk about relations and concepts, i t can only employ relational- and predicated expressions to refer to them. We shall see now that this result has dire consequences for Frege's ontology. J*6 Chapter 3 Introduction Milton Fisk's attempt to derive a paradox from Frege's principle that a concept can only be referred to by a predicate is discussed and c r i t i c i z e d . Even though Fisk's paradox is ill-formed as i t stands, i t seems that the maxim derived in the last chapter - that any language capable of talking about relations and concepts cannot employ proper names to refer to them - renders us in principle incapable of specifying in a given instance which concept a predicate refers to. Two recent attempts to manoeuvre Frege out of this d i f f i c u l t y are discussed and c r i t i c i z e d in some detail. It is shown that the problem of specifying predicate reference remains. Nevertheless, from the second attempt (Dummett's), valuable insights are salvaged. The relation of denotation between a predicate and a concept i s one of second level and i t is necessary to coin a new r e l -ational expression to refer to this relation. Accordingly I coin the expression 'prenote* and exhibit i t s usage. Once this expression is in our vocabulary, there i s no longer any d i f f i c u l t y in specifying predicate reference. Unfortunately however, our attempt to escape Bradley's problem of relations and the larger problem of the unity of the proposition, has come f u l l , vicious-: c i r c l e . Bradley's problem was avoided in the f i r s t 47 place by viewing relations as 'unsaturated' - as explained as that which relational expressions denote. But the use of 'denote' here i s inappropriate - a relational express-ion prenotes. But we can understand what this means only i f we can understand what this prenoting relation i s . Now crucial to understanding any relation, i t i s argued, i s a knowledge of the kind of entities which.can serve as terms of the relation. But one of the terms of the pre-noting relation is i t s e l f a relation - precisely that entity which we explained as being that which a relat-ional expression prenotes. We are blocked in principle from explaining the very relation which was to pave the way for the escape from Bradley's regress. Thus i t appears that the better we understand Frege's ontological doctrines, the clearer i t becomes that they are ill-founded - that they cannot solve the very problems for which they were designed. ^8 I f one w i s h e s t o s ay o f an o b j e c t - s ay V e s u v i u s - t h a t i t i s a n o b j e c t , one n a t u r a l way i s t o fo rm the f o l l o w i n g s en t ence t •The o b j e c t V e s u v i u s i s a n o b j e c t * • A n a l o g o u s l y , i f one w i s h e s t o s a y o f a c o n c e p t t h a t i t i s a c o n c e p t one n a t u r a l l y forms a s en tence s u c h a s i •The c o n c e p t h o r s e i s a c o n c e p t ; ' U n f o r t u n a t e l y t h i s w i l l n o t d o . The t h r e e words ' t h e c o n c e p t h o r s e ' c l e a r l y do n o t c o n s t i t u t e a g r a m m a t i c a l p r e d i c a t e b u t a p r o p e r name ( 1 . 0 4 ) and hence canno t on F r e g e ' s p r i n c i p l e s denote a c o n c e p t . (1.22) I f i n f a c t • the c o n c e p t h o r s e ' d e n o t e s a t a l l t h e n 'The c o n c e p t h o r s e i s n o t a c o n c e p t b u t an o b j e c t . ' i s a t r u e t h o u g h a d m i t t e d l y p a r a d o x i c a l s e n t e n c e . F rege r e c o g n i s e d t h i s d i f f i c u l t y . He n o t e d « "The es sence o f t he c o n c e p t i s a g r e a t h i n d r a n c e f o r a p p r o p r i a t e e x p r e s s i o n and f o r u n d e r s t a n d i n g . I f I want>to speak o f a c o n c e p t , l anguage f o r c e s a n u n s u i t a b l e e x p r e s s i o n on me w i t h a l m o s t i n e s c a p a b l e m igh t . . . I f I s ay ' t h e c o n c e p t e q u i l a t e r a l t r i a n g l e ' t h e n one s h o u l d assume by g r a m m a t i c a l a n a l o g y t h a t I t h e r e b y d e s i g n a t e a c o n c e p t , j u s t as I s u r e l y name a p l a n e t , when I s ay ' t h e p l a n e t N e p t u n e ' . Bu t t h i s i s no t the c a s e ; f o r t he p r e d i c a t i v e n a t u r e i s l a c k i n g . T h e r e f o r e the r e f -e r e n c e o f t he e x p r e s s i o n ' t h e c o n c e p t e q u i l a t e r a l t r i a n g l e ' ( so f a r a s one e x i s t s ) i s a n o b j e c t . " ( B i b . (13) P130) When F rege f i r s t n o t e d t h i s d i f f i c u l t y i n h i s p u b -l i s h e d w r i t i n g s , ( i n "Concep t and O b j e c t " ) he r e g a r d e d i t a s a n u n a v o i d a b l e q u i r k o f l a n g u a g e . The i n a p p r o p r i a t e -ne s s o f e x p r e s s i o n i s embedded i n o u r l anguage and " t h e r e 49 is nothing for i t but to realize this and always take i t into account." One cannot allow that concepts can be referred to by proper names for this would undermine their unsaturated nature and i t is this feature of concepts which as we saw in the preceeding chapter provides an escape from Bradley's infinite regress and attendant problems. Now Erege's theory of functions and-objects can be seen as previding a set of formation rules for a language. We could choose to provide the same rules purely formally. Thus for example, instead of speaking of the denotation of ' ^ is a horse' as determining how this predicate is used, we could just say, " * ^ i s a horse' is a f i r s t level predicate" and f i x the behaviour of the expression by forming rules governing f i r s t level predicates. Present-ing the formation rules thus, we would avoid such para-doxical sentences as •The concept horse i s not a concept.' but we have seen in the previous chapter some of the considerations which may have led Frege to express his theory in the material mode. It is interesting to note however that Frege seemed to consider that a formal treatment failed to avoid the above-mentioned d i f f i c u l t i e s . In a footnote to his admission of these d i f f i c u l t i e s , he wrote« "A similar thing happens when we say as regards the sentence 'this rose i s red't The grammatical predicate 59 ' i s r e d ' b e l o n g s t o the s u b j e c t ' t h i s r o s e ' . Here the words " the g r a m m a t i c a l p r e d i c a t e ' i s r e d ' " a r e n o t a a g r a m m a t i c a l p r e d i c a t e b u t a s u b j e c t . By t h e v e r y a c t o f e x p l i c i t l y c a l l i n g i t a p r e d i c a t e , we d e p r i v e i t o f t h i s p r o p e r t y . " ( B i b . (21) s e e ( 2 8 ) p46n) B u t F rege i s wrong h e r e . There i s n o t h i n g p a r a d o x i c a l abou t a g r a m m a t i c a l p r e d i c a t e ( e g . ' i s r e d ' ) b e i n g t h e r e f e r e n c e o f t he g r a m m a t i c a l s u b j e c t o f a s e n t e n c e . Now I t h i n k t h a t F rege was c o r r e c t i n r e g a r d i n g • the c o n c e p t h o r s e i s n o t a c o n c e p t ' a s a mere p e c c a d i l l o . I t i s t r u e t h a t i t goes a g a i n s t a n a c c e p t e d p r i n c i p l e g o v e r n i n g d e f i n i t e d e s c r i p t i o n s 1 ^ b u t t h e r e i s no r e a s o n why we c o u l d n o t t o l e r a t e a n e x c e p t i o n t o t h i s p r i n c i p l e . We c o u l d f o r example c o n s i d e r the u n d e r l i n e d e x p r e s s i o n f o l l o w i n g ' t h e c o n c e p t ' a s f o r m i n g a s p e c i a l c o n t e x t t o w h i c h the p r i n c i p l e i n q u e s t i o n s h o u l d n o t be a p p l i e d , and r e t a i n t h e p r i n c i p l e f o r a l l o t h e r c o n t e x t s . B u t u n f o r t u n a t e l y we c a n n o t r e s t w i t h t h i s . What above has a p p e a r e d a s j u s t a q u i r k o f l anguage has deepe r r o o t s a n d , i t has been w i d e l y a r g u e d , b l o s s o m s i n t o a genu ine p a r a d o x . The p a r a d o x w h i c h I s h a l l d i s c u s s he re 17 i s t he one p r o p o s e d b y M i l t o n F i s k . There a r e v a r i a n t s o f t h i s pa r adox and i n d e e d t h e y have been t h o r o u g h l y d i s c u s s e d i n the s e c o n d a r y l i t e r a t u r e . N e v e r t h e l e s s , 16 Namely t h a t p r i n c i p l e w h i c h i n f o r m s us t h a t f o r any common noun B and name ©< , r t h e r e f e r e n c e o f r t h e fio< n must be a/sT i s t r u e . ' l i B i b . (12) ;51 t he one t o be d i s c u s s e d he re p r o v i d e s as u s e f u l a s t a r t i n g p o i n t a s any f o r d e l v i n g t o the h e a r t o f t h e m a t t e r . F i s k ' s argument i s as f o l l o w s « 1) ' t h e c o n c e p t h o r s e ' i s a p r o p e r name. 2) ' i s a h o r s e ' i s a p r e d i c a t e . 3) I f ' t h e c o n c e p t h o r s e ' i s a p r o p e r name, the c o n c e p t h o r s e i s an o b j e c t . 4 ) I f ' i s a h o r s e ' i s a p r e d i c a t e , ' i s a h o r s e ' r e f e r s t o the c o n c e p t h o r s e . 5) I f ' i s a h o r s e ' i s a p r e d i c a t e , ' i s a h o r s e ' does n o t r e f e r t o a n . ' o b j e c t . 6) ' i s a ho r se* does n o t r e f e r t o an o b j e c t , ( f rom 2 & 5) 7) The c o n c e p t h o r s e i s a n o b j e c t , ( f rom 1 & 3) 8) ' i s a h o r s e ' r e f e r s t o the c o n c e p t h o r s e . ( f rom 2 & 4) 9) ' i s a h o r s e ' r e f e r s t o a n o b j e c t , ( f rom 7 & 8) (1) f o l l o w s f rom F r e g e ' s c r i t e r i o n t h a t the d e f i n i t e a r t i c l e f o l l o w e d by a s i n g u l a r t e r m a l w a y s i n t r o d u c e s a p r o p e r name. (1 .04) (2) i s u n c o n t r o v e r s i a l . (3) t o (5) F i s k c l a i m s , f o l l o w f rom F r e g e ' s p r i n c i p l e s t h a t the r e f e r e n c e o f a p r o p e r name i s an o b j e c t , t h a t t he r e f e r e n c e o f a p r e d i c a t e i s a c o n c e p t and t h a t no c o n c e p t i s an o b j e c t . (1.22,1.23,1.24) Now (6) c l e a r l y c o n t r a d i c t s (9) w h i c h s u g g e s t s t h a t the a b o v e - m e n t i o n e d p r i n c i p l e s a r e i n c o n -s i s t e n t . F i s k h i m s e l f t a k e s t h i s pa r adox as c o n t r a d i c t i n g F r e g e ' s v i e w o f " the p r e d i c a t i v e n a t u r e o f t h e c o n c e p t " -the v i e w t h a t c o n c e p t s c a n o n l y be r e f e r r e d t o by p r e d -i c a t i v e e x p r e s s i o n s and n o t b y p r o p e r names. I n "On Concep t and O b j e c t " F rege n o t e d r e g a r d i n g the s e n t e n c e ' t h e c o n c e p t h o r s e i s n o t av ;concep t .' J "One w o u l d e x p e c t t h a t t he r e f e r e n c e o f t he g r a m m a t i c a l s u b j e c t w o u l d be t h e c o n c e p t , b u t t he c o n c e p t as such c a n n o t p l a y t h i s p a r t , i n v i e w o f i t s p r e d i c a t i v e B i b . (20) see (30) p46) 52 If we reject "the predicative nature of the concept", we reject accordingly step ( 3 ) of the argument without which the paradox cannot be derived. For to reject the view that concepts can only be referred to by predicates is to accept the view that they may be referred to by proper names (1.22) and hence even though 'the concept horse' i s a proper name, i t no longer follows from this that the concept horse is an object. But we need not be quite so hasty in rejecting what we have seen to be so crucial to Frege's metaphysics. Fisk's argument i s ill-formed as i t stands. Step (4) is just false, 'is a horse' does not refer to the concept horse because the concept horse is an object and 'is a horse' refers to a concept since i t is a predicate. Nevertheless we have here a fundamental d i f f i c u l t y . Whatever name we substitute for 'the concept horse' in, (k) w i l l , i f i t is a proper name, refer to an object. In which case (4) w i l l be false. What we need is a name of a concept ie. a predicate. But substituting a predicate for 'the concept horse' in the consequent of (4) yields not a true sentence but a senseless one. eg. " 'is a horse* refers to is a horse ." It i s apparent that a fundamental asymmetry exists between the relationship between a proper name and the object i t denotes on the one hand and a predicate and the concept i t denotes on the other. In the case of proper 53 names, i t i s always possible meaningfully and truly to complete the schema 'A denotes x' where quoted proper names substitute for 'A' and names of the referents of these proper names substitute for *x*. eg. "'Charles Dickens' denotes Charles Dickens." If we take seriously Frege*s view that function names must always carry with them their argument places, we see that completion truly of the above schema for predicate expressions i s always impossible. Considers "'F(^)' denotes x." If in place of 'x' we put a proper name we get a false sentence. If in place of 'x' we put a function name, we get an incomplete and hence senseless sentence. What made (1) true in the above argument was Frege's criterion that the definite article followed by a singular term always introduces a proper name. We see now that d i f f -iculties beset Frege's doctrines independently of the validity of this criterion. Indeed, these doctrines render us in principle incapable of specifying in a given instance which concept a predicate refers to. Could we nevertheless adhere to the view that pred-icates refer to concepts and admit the impossibility of specifying in a given instance which concept a predicate refers to? This last resort' i s indeed consistent but depicts 54 a highly improbable state of affairs, for the following reason: We arrived at the notion of the relation of denotation between proper names and objects by observing the use of the predicate 'denotes' in such sentences as: "•Charles Dickens' denotes Charles Dickens," This i s not to say that :this is the only way we could have arrived at the idea of such a relation, only that as a matter of fact we did arrive at i t in this way. Now i f we cannot in principle form such sentences as the above for predicates then we w i l l be faced with the d i f f -iculty of providing a justification for supposing that the relation of denotation holds between predicates and concepts. What the ina b i l i t y of specifying predicate reference does then, is to cast doubt upon the very notion of a predicate denoting a concept, a notion which we have seen is crucial to Frege's ontology and metaphysics. We are nearer now to reaching an understanding of the causes of the d i f f i c u l t i e s which we have been dis-cussing. Before presenting what I consider to be the resolution of these d i f f i c u l t i e s - a resolution which is fa t a l to Frege's doctrine - i t i s instructive to consider two suggestions for manoeuvring out of these d i f f i c u l t i e s . Both suggestions provide valuable insight into the problem and w i l l render my proposed resolution almost obvious. I intend f i r s t to consider a suggestion attributed by 18 Furth to, among others, Richard Montague and David Kaplan. 18 Bib (24) The suggestion is this: We specify denotation of an object language expression, say ' ^ i s a horse' in a metalanguage. Thus for example, " • ^ i s a horse' denotes the concept horse.» is a sentence of the metalanguage. To argue that this sentence i s false because 'the concept horse' i s a proper name and henc;e denotes an object i s to make the straight-forward mistake of applying rules governing the object language to expressions*in the metalanguage. 'the concept horse' in the above sentence is an expression of the metalanguage and by the rules governing this language, i t may well denote a concept. 'the concept horse' may not occur at a l l as an expression in the object language or i f i t does i t could have a completely different den-otation from i t s metalinguistic counterpart. Before considering the merits of this argument, i t is interesting to see what i t implies for Fisk's argument. We rejected Fisk's argument because of the faulty premise (4) but this suggestion would seem to restore that premise. Is then the paradox Fisk attempts to derive genuine? No, because (7) i s a statement in the object language while (8) is a statement in the metalanguage and so (9) cannot be derived from (7) and (8). For i t i s clear that (9) follows from (7) and (8) only i f the exp-ression 'the concept horse' denotes the same entity in (7) as i t does in (8). But 'the concept horse' in (8) i s an expression of the metalanguage and the same phrase •56 may occur in the metalanguage and the object language with different denotations. Since the expression may denote dist inct entities in i t s occurrences in (7) and (8), the truth of (7) and (8) does not guarantee the truth of (9). One merit of the suggestion we are here discussing then, is that i t permits the premises of Fisk's argument but effectively blocks the conclusion. But d i f f i cu l t i e s remain. Whatever comes after *»• * ^ is a horse* denotes . . . " in a metalinguistic specification of denotation must either 19 be a proper name or a predicate expression. 7 Thus for example we might let 'the concept horse' be a proper name of the metalanguage and complete the above specification of denotation thust '" ' ^ is a horse* denotes the concept horse." Or we might let ' ^ is a horse' be a predicate of the metalanguage and complete i t thust " '^ i s a horse' denotes is a horse. " The f i r s t completion is obviously unsatisfactory. We saw in the previous chapter that any language capable of talking about relations (and the metalanguage in question must have this capability for i t to talk about the relat ion of denotation) cannot employ proper names to refer to such relations. The second completion is less obviously 19 As we have seen, Frege held that these categories apply universally to a l l languages. (1.28) 57 unsatisfactory. The chief d i f f i c u l t y is making sense of i t . If the metalanguage is English, i t i s clearly sense-less. If the metalanguage is some other language, one way of making sense of i t would be to translate i t into English. This however seems like a thankless undertaking given that the object language in this case i s English and the problem arose precisely because of the imposs-i b i l i t y of specifying the denotation of a predicate in this object language. The second suggestion I wish to consider is due to 20 Dummett, and I wish to consider i t in some detail. We have seen that Frege•s semantic and ontological principles lead to the intolerable view that we cannot specify for any given predicate which concept i t refers to. Dummett does not accept this conclusion and attempts to provide a mode of expression for specifying predicate reference which is f u l l y in accord with Frege's doctrines. On the other hand, Dummett accepts as genuinely para-doxical the sentence ' 'The concept horse i s not a concept which we regarded as tolerably irksome. Dummett's proposed solution then, attempts to solve both these d i f f i c u l t i e s . 20 Bib (9) P211-222 58 Dummett's proposal The problem arises, he argues, with the predicate ' ^ i s a concept'. The word 'concept' is a common noun and hence, argues Dummett, the predicate formed from i t is one of f i r s t level. Since according to Frege's formation rules only object names (proper names) can f i l l the argument places of f i r s t level predicates, every appropriate completion of * ^ is a concept*will yield a false sentence. The predicate ' £ is a concept* is totally unsuited for the work we want i t to do ie. say of some concept that i t is a concept. What we need is a predicate that w i l l operate with respect to concepts as the predicate • £ is an object* operates with respect to objects ie. yields a true sentence upon every appropriate completion. What we need then is a second level predicate (1.31) which is appropriately completed only by f i r s t level predicates and such that every such completion yields a true sentence. In the natural language, relative pronouns and such words as * something','someone', have a double use. We may characterise this use loosely by saying that such words introduce clauses which sometimes refer to objects and sometimes to properties, eg. 'what you gave me yesterday* and 'what you are and I have just become'. The former refers to some object while the latter refers to some property eg. impecuniousness. Clauses such as the latter can be converted into f i r s t level predicates by 59 prefixing a copula to them. eg. 'John i s what you are and I have just become.' But these clauses have a further use. They can be converted into second level predicates in the same way. When thus used they are completed by what Dummett calls a •predicative expression', eg. •Impecunious is what you are and I have just become;'.: The predicative expression here is 'impecunious'. To see generally which expressions Dummett takes to be predicative expressions, consider the simplest subject/ predicate sentences which are formed by completing a f i r s t level predicate with a proper name. Now, says Dummett t ". . . i f the grammatical predicate, consists of the copula together with an adjectival phrase or one whose main constituent is a common noun preceeded by the indef-inite a r t i c l e , the predicative expression is formed by simply dropping the copula; i f the main verb is anything other than the copula, the predicative expression i s formed by converting the main verb into the p a r t i c i p i a l form of the same tense." (Ibid. p215) What we need then as a replacement for the1 i l l i c i t predicate • ^ is a concept' i s a predicate which i s completed by predicative expressions and such that every completion yields a true sentence. Taking advantage of Frege's requirement that a concept must be everywhere defined, (1.26) Dummett proposes the following* " is something which everything either is or is not." 60 (This should be understood as 'For a l l /a , /a is (f> or /a, is not p ' where '^ ' represents the argument place. cf.(1.33) This solves half the problem. We s t i l l need a predicate expression which w i l l refer to the concept horse. Oue natural choice in referring to the concept in question was 'the concept horse' but we w i l l have to choose again because 'the concept horse' is not a predicative expression. Dummett proposess "what ' ^ i s a horse' refers to;" He argues that we are misled into viewing this as a proper name by the double use of r.elative pronouns such as 'what' but that we must view this expression as standing for a concept not an object. I w i l l examine this argument in due course but for the moment let us accept i t s conclusion. Thus we view "Kentucky Kid is what ' £ is a horse' stands for." as parallel in construction to 'Kentucky Kid i s four legged.' We are now in a position to express the thought we found ourselves incapable of expressing with the sentence 'The concept horse i s a concept.* We now say: "What ' £ is a horse * refers to i s something which everything either i s or is not." By eliminating the false predicate ' ^ is a concept' we have succeeded in eliminating such paradoxical 61 constructions as 'The concept horse is not a concept.' We have however yet to see how a l l this enables us to specify which concept a particular predicate refers to. Dumraett concludes the above discussion thus* "Once pseudo-predicates like ' £ is a concept' and ' £ is a relation' have been extruded, there is no longer any means of constructing paradoxical sentences like 'The concept horse is not a concept.' nor is there any d i f f i c u l t y in 2?aying what in particular any given pred-icate stands for. We can, for example, say ' A philosopher is what " *; is a philosopher" stands for' , or more inform-atively, 'What " *| is a philosopher" stands for is what Socrates and Plato both were.' " (p217) It is at this point that doubts arise about Dummett's proposed solution. His f i r s t offering t 1) "A philosopher is what ' M s a philosopher' stands for." is a permissable construction only i f we view "is what • ^ is a philosopher' stands for" as a second level predicate and (1) as the completion of this predicate by the predicative expression 'a philosopher'. But in that case, a l l that (1) asserts is that one concept (namely the concept philosopher) f a l l s within another (namely the concept what ' $ is a philosopher' stands for). Loosely, we may say that (1) asserts that the property of being a philosopher has the property of being referred to by the expression * ^ is a philosopher'. But this is hardly a specification of the reference of ' £ is a 21 Dummett uses 'stands for' where I have been using 'denotes*. 62 philosopher'. Certainly (1) can be seen as a statement about the concept what ' £ i s a philosopher' stands for in the same way that 'Socrates i s wise' can be viewed as a statement about the concept wise but i t cannot be construed as informing us which concept ' £ i s a philosopher' denotes. Nor is 2) "What ' 4 is a philosopher' stands for is what Socrates and Plato both were." more informative. This would be more informative i f Socrates and Plato had just one property in common but this is hardly the case. This objection may be put more formally as followss A minimum requirement of success-f u l l y specifying reference in the case of predicates i s that in the event that two predicates denote the same concept we be able to recognise this fact from identical specifications of their references. This requirement phrased for proper names is clearly met. For example, from "The reference of 'Cicero' is the man who did "y<*" and "The reference of 'Tully is the man who did °y » we can conclude that 'Cicero* denotes the same object as 'Tully'. But from "What ' £ i s a philosopher' stands for i s what Socrates and Plato both were." and 63 "What ' £ is a Greek' stands for is what Socrates and Plato both were." we cannot conclude that ' £ is a philosopher' and • £ is a Greek' stand for the same concept since they obviously do not. Dummett's solution to this part of the d i f f i c u l t i e s clearly doesn't work. The intolerability of not being able to specify a predicate's reference remains. There is more that i s dubious about Dummett's proposal. Much of his enterprise depends on our construing the clause "What ' £ is a.horse' stands for" as a predicative expression. This construal is probably inspired by the following remark of Frege'st "Strictly speaking the expression 'the reference of the concept word A' is to be rejected, because the definite article before 'reference' relates to an object and revokes the predicative nature of the concept. It would be better no doubt to say 'what the concept word A refers to' for this is. in any case to be used pred-icative ly« 'Jesus is what the concept word 'man' refers to (bedeutet)' in the sense of 'Jesus is a man.' " (Bib. (13) P133) Now i t should be noticed that in a l l the contexts in which the expression "what ' £ is a horse' stands for" or others like i t are to feature, we could substitute a proper name without loss of sense. Thus consider the sentence t "Jesus is what the concept word 'man' signifies." It makes perfectly good sense to sayi "Jesus is the Son of God." As another example considert "What * 4 is a philosopher' refers to is what Plato and Socrates both are." We have already seen that the predicate ' is what Plato and Socrates both are' is ambiguous between f i r s t and second level construal. Thus i t is meaningful to say: •Aristotle is what Plato and Socrates both are.' Now we can normally distinguish predicate expressions from proper names precisely because the latter cannot be substituted^ for the former in certain contexts, eg. 'Swimming is my favorite pastime.' Here 'swimming' is not meaningfully substituted for by some proper name except perhaps such a r t i f i c i a l ones such as 'the activity of swimming'. What grounds do we have for regarding "What ' ^ is a horse' stands for" as a predic-ative expression? Perhaps appreciating these d i f f i c u l t i e s , Dummett feels constrained to offer an argument for his view: "The two expressions 'Mount Everest' and 'what "Mount Everest" stands for' are completely interchangeable: the conditions for their having a"reference are the same, and i f they have one, i t must be the same for each. This' is just to say that the expression 'what "Mount Everest" stands for' must, i f i t stands for anything, stand for what •Mount Everest' stands for; . . . By analogy, i t follows that "what •• *• is a horse" stands for' ought, i f i t stands for anything, to stand for what ' £ is a horse' stands for. Hence in particular, i t cannot be construed as a proper name (singular term), for then i t would stand for an object, and hence could not have the same reference as ' £ i s a horse'." (P213A) 65: This argument is unconvincing. Exactly the same considerations w i l l show that 'the reference of " ^ is a horse" * is not a proper name, which contradicts Frege's formal criterion regarding the use of the definite a r t i c l e . (1.0-4) This points to the error in the above argument. One cannot argue by analogy that because a particular mode of expression adapted to proper names can be used to refer to objects, that same mode of expression adapted to predicates must refer to concepts. In particular, one cannot argue that because 'what "Mount Everest" refers to' refers to what 'Mount Everest' refers to, * so must 'what " ^ is a horse" refers to' refer to what ' ^ i s a horse' refers to. One cannot argue this because i t is precisely due to a disanalogy between the way we can refer to objects and the way we can refer to concepts that the seeming paradox arose in the f i r s t place. We attempted to refer to the reference of • ^ is a horse' by the means most natural to referring to the reference of a proper name ie. 'the concept horse'. It seems that we must accept that we simply do not have the means for picking out particular concepts which we have for picking out particular objects and i f we attempt to pick out concepts with the same type of expression we employ to pick out objects, we w i l l f a i l in our task. This i t seems, must conclude this discussion unless independent grounds can be advanced for regarding such expressions as "What ' M s a horse' stands for" as predicative. 66 Nevertheless we can salvage some valuable insights from Dummett's discussion of these problems. The pred-icate ' ^ i s a concept' is at best useless, as Dummett remarks, for we can never obtain a true sentence by com-pleting i t . But just as this predicate is useless so is the relational expression ' £ denotes ' when i t s f i r s t argument place is f i l l e d with a name of a predicate expression. For the relation of denotation between a proper name and i t s referent is not supposed by Frege to be the same relation as that between a predicate and i t s referent. The terms of the f i r s t mentioned relation are both objects while the terms of the last mentioned relation 22 are one an object and the other an unsaturated entity. Two relations are the 'same' for Frege i f they have the same values for the same arguments. (1.37-1.42) Now since the two relations in question cannot take the same values, they cannot be the 'same * relation. No wonder then that i t is in principle impossible to complete 'is a horse' denotes ' for we have used the name of an equal-, f i r s t levelled relation where we needed a name of an unequal-, second levelled relation taking as f i r s t argument an object and as second argument a concept, something unsaturated. (1.36) What we need then is a new relational expression which 22 This is perhaps not s t r i c t l y true in view of (1.45) but since the conclusion of this argument holds irresp-ectively, (that the relations under discussion are not the same) I w i l l continue to ignore such complications. 67 names this unequal levelled relation. That there is no such expression in our vocabulary should not surprise us, since before Frege, no such relation was thought of. We have dis-covered a new relation and there is nothing for i t but to coin a new expression. By doing so, we make a useful addition to our natural language. Let us use the expression 'prenote* to refer to this relation. We can now sayt " • £ is a horse' prenotes is a horse" and any failure to see this as a well formed sentence of our language is a failure to understand the use of the relational expression 'prenotes'. J We can illustrate i t s usage thus* ' £ prenotes ( c, )' In fact, the realization that this new relational expression is required, spawns an entire range of new expressions i f we wish to extend the f a c i l i t y of speech which we have for talking about objects to talking about functions. We saw that a quite general obstacle stood in our way i f we wanted to talk about a specific concept. For example, in saying that the sentence 'Blue Peter is a horse* says some-thing about both Blue Peter and the concept horse we miss the thought we wish to express. We can now view this failure as the result of using the inappropriate expression 'says something 23 Here the reader may balk at the insouciant use of the expression 'prenotes' - which clearly contravenes con-ventional syntactic rules. But as w i l l soon become apparent, I use this expression merely to illustrate a point and do not advocate i t s employment as serious reform. 68 about'. This expression denotes an equal levelled relation when we required an unequal levelled relation. As regards the oddity, we can either retain the useless predicate 'is a concept' J and regard 'the concept horse' as embedded in an except-ional context or we can devise a new second level pred-icate which replaces 'is a concept', and which takes names of concepts eg. * £ is a horse' in i t s argument place, or both. The expressions 'the reference of A' or 'the prenot-ation of A' w i l l have to be banned and Frege in fact 2k insisted on this too in a letter to Russell. But we can get by without this expression simply by using the name of the concept when we wish to say something about that concept. Just as the expression "the reference of •Vesuvius'" is redundant once we have the expression •Vesuvius', so too is the corresponding expression for quoted predicates. Of course, once we have posited such new relations we s t i l l have to provide some justification for thinking them to exist, as was suggested earlier. We would also need to investigate their properties. For?example we would need 2:k After a l l , this predicate does ..on Frege's principles denote a concept since this concept w i l l be everywhere defined. (1.26) 2 5 Dated June 29th, 1902. This is mentioned by Resnik (see Bib. (37) p2k6) The concept horse i s not a concept • 69 to compare the properties of the prenoting relation to the denoting relation to see i f these properties were sufficient-ly alike to even warrant talking of functions, concepts and relations as we talk of objects. But for the moment let i t suffice to seeehow the positing of such new relations enhances our understanding of Bradley's infinite regress. We can now see that Bradley's problem arose from the mistaken view that relational expressions denote. If we view relational expressions as denoting then s t r i c t l y , we must view the entities they denote as objects. And once we do this we are beset immediately with the well known d i f f i c u l t i e s explored in the preceeding chapter. But once we understand that relations prenote, then we can have no d i f f i c u l t y in understanding how relations connect their terms and a l l the d i f f i c u l t i e s which hinged upon this disappear. Unfortunately however, we have come f u l l , vicious;, 2*5 c i r c l e . J To recapitulate: Bradley's problem arose in the f i r s t place from viewing relations as objects. This we now see, is equivalent to viewing relational expressions as denoting. I n i t i a l l y we avoided Bradley's problem by explaining a relation as that which a relational expression denotes. We argued that once a relation is understood by way of this explanation, there can be no greater d i f f i c u l t y in understanding how a relation connects i t s terms than 26 The reader should notice the similarity of the argument which follows with that of Dummett's para-phrased on p35/6. 70 there is in understanding how a relational expression functions in a sentence. But our enhanced understanding of the denotation relation; forces us to modify this argument for we can no longer explain a relation as that which a relational expression denotes for this would imply that relations are objects. We now need to argue that Bradley's problem disappears once a relation i s understood to be that which a relational expression prenotes. But we also need to explain what this prenoting relation i s . Indeed our success in explaining what a relation is w i l l depend entirely on the success of this explanation. But a minimum requirement for being able to explain what any given relation i s , is an a b i l i t y to explain what kind of entities can serve as terms of the relation ie. what kind of things the relation relates. For example, in order to explain what the grandfather relat-ion is we would begin by explaining that i t is a relation which relates people, and then proceed to explain specif-i c a l l y which people are related in this way. But i t is obvious that this explanation is useless unless i t is already understood what people are. Now one of the terms of the prenoting relation i s i t s e l f a relation and thus i f our explanation of the prenoting relation i s to succeed we must assume a prior understanding of what a relation i s . But - and this is the crucial point - we cannot assume such prior understanding for as we have seen, an understanding of what a relation i s , i s reached only 71 via an explanation that i t i s what a relational expression prenotes. Assuming a prior understanding of what a relation i s , is assuming a prior understanding of what the prenoting relation*;is. We are caught in a trap. We cannot explain a relation as that entity which a relational expression denotes and we cannot explain a relation as that entity which a relational expression prenotes because this latter explanation makes reference to a relation which we cannot understand unless we already know what a relation i s . The explanans presupposes knowledge of the explanandum. Thus i t is that the better we understand Frege's doctrines, the less convincing i s his proposed solution to those problems which prompted the. formation of those doctrines in the f i r s t place. The problem of the unity of the proposition, Frege would have argued, results from viewing concepts and relations as special kinds of objects. Frege attempted to avoid this and related problems by arguing that concepts and relations are unsaturated - that is that they can only be explained as that which predicate- and relational expressions denote. But we see now that this explanation is incorrect as i t stands and the required revision invokes a notion whose explication presupposes a prior understanding of the notion. The ontological division between saturated and unsaturated simply cannot be made out and hence cannot solve the problems for which i t was designed. 72 Bibliography 1) Birjukov, B.V, 2) Black, M. 3) Bradley, F.H, 4) Carnap, R. 5) Caton, C E . 6) Church, A. 7) Copi, I.M. 8) Dummett, M. 9) 10) 11) 12) ". Fisk, M. 13) Frege, G. Two Soviet Studies on Frege Translated and edited by I. Angelelli. D. Reidel Publishing Co. 1964 "Frege on Functions" in Max Blacki Problems of Analysis Reprinted in (30) Appearance and Reality Allen and Unwin 1940 Meaning and Necessityi A study in Semantics and Modal Logic Univ. of Chicago Press 1947 "An apparent Difficulty in Frege's Ontology" The Philosophical Review LXXX1 (19627~p462-475. Reprinted in (30) Introduction to Mathematical Logic Vol. 1 Princeton Univ. Press 1956 "Objects Properties and Relations in the 'Tractatus'" in Essays on Wittgenstein's Tractatus eds. Irving M. Copi and Robert W. Beard. Rout-ledge & Kegan Paul 1966 "Frege on Functions* A Reply" The Philosophical Review LX1V 1955 P96-107. Reprinted in (30) Fregei Philosophy of Language Duckworth and Co. 1973 "Gottlob Frege" in Encyclopedia of Philosophy ed. Paul Edwards "Note: Frege on Functions" The Philosophical Review LX1V 1955 P561-570. Reprinted in (30) "A Paradox in Frege's Semantics" Philosophical Studies XIV I963 P 5 6 - 6 3 . R e p r i n t e d in (30) "Ausfuhrungen uber Sinn und Bedeutung" in Nachgelassene Schriften Felix Meiner Hamburg I 9 6 9 . English translation courtesy of Robert Bunn 73 14) Frege G. 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) Furth, M. 25) Geach, P. Die Grundlagen der Arithmetic Breslau 1884 Translated as "The Foundations of A r i t h -metic" by J . L . Austin. Basi l Blackwell Oxford 1974 "Gedankengef'uge". F irs t Published 1923. Translated as "Compound Thoughts" by R.H. Stoothof. Reprinted in (30) "Der Gedanke" F ir s t Published 1918. Translated as "The Thought" by A.M. and M. Quinton in (30) "Die Verneinung" Firs t Published 1919. Translated as "Negation" in (28) "Funktion und Begriff" An address delivered by Frege in 1891. Translated as "Function and Concept" in (28) On the Foundations of Geometry and Formal Theories of Arithmetic Translated and with an introduction by Eike-Henner W. Kluge, Yale Univ. 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B a s i l Blackwell Oxford 1970 "Frege on Sense Functions" Analysis XX111 1962-3 p84-87 Reprinted i n (30) ) Essays on Frege Univ. of I l l i n o i s Press 19o"8 "Frege's Theory of Functions and Objects" The Philosophical Review LX11 1953 P374-390 Reprinted i n (30) 0« Frege t Logische Untersuchungen Vanden Hoeck and Ruprecht i n Gottingen I966 From a Logical Point of View Harper and Row 1963 Methods of Logic Holt Rinehart and Winston Inc. 1950 Philosophy of Logic Prentice H a l l Inc. 1970 "Frege's Theory of Incomplete E n t i t i e s " i n Philosophy of Science I965 P329-341 Frege's Methodologyt A C r i t i c a l Study Ph.D. Diss. Harvard Univ. Cambridge Mass. 1963 An Inquiry into Meaning and Truth A l l e n and Unwin 1940 An Outline of Philosophy Meridian i960 The Analysis of Matter Publications 1954 Dover The P r i n c i p l e s of Mathematics A l l e n and Unwin 1903 75 Taylor, A.E. Elements of Metaphysics Methuen and Co. Ltd. 1903 Thiel, C. Sense and Reference in Frege's Log D. Reidel Publishing Co. I968 Walker, J.D.B. A Study of Frege Oxford . TBlackwell) 1965 Wells, R.S. "Frege's Ontology" The Review of Metaphysics IV 1951 P537-573 Reprinted in (30) Wollheim, R. F.H. Bradley Penguin 1959
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Frege's ontology : a critical examination of its foundations
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