University of Sussen

This seems t o me t o be one o f t h e most d i f f i c u l t problems in repres en t a t ion theory a t present . The only way I can imagine handling substances is by regarding each substance a s a ( spec i a l s o r t o f ) indivi dual, t o which such p rope r t i e s a s hardness, dens i ty , e t c . a r e a t t r i bu t ed . These individuals can be regarded as p la ton ic i d e a l s , o r a l t e r n a t i v e l y a s t h e physica l t o t a l i t y o f a l l samples of t h e substance: you can take your nominalism o r leave it. We have t h e naPve axiom (e.g. : a lump o f hard s t u f f is hard). which t ransmits p rope r t i e s f n r ; substances t o p ieces of them, (Caw i s needed: s t e e l sh ips f l o a t , f o r example; a f a c t which of ten amazes young chi ldren . ) Eiotice t h i s axiom is f i r s t o r d e r ( i n a sugared syntax). Quantity is now a funct ion from (p i eces )X(s tu f f ) t o some sca l e of measwenent , so we can express conservation of quant i ty through some physical a l t e r a t i o n Q by: And so on. This works up t o a po in t , but seems t o F& t o be e s s e n t i a l l y unsat i s fac tory . There i s a c lose analogy between being made of a substance, and 3eing made u of a number of pa r t s . And a corresponding analogy betueen uant i ty & and number (of p-s). Sand and p i l e s of small pebbles :re intermediate cases: and we often t r e a t an assembly of individuals a s a f l u i d , e.g. a s i n " t r a f f i c flow". The major d i f ference seems t o be t h a t d i f f e r e n t s c a l e s of measurement a r e used i n conunon-sense reasoning (but not i n physics. where quant i ty i s number of atoms), a s t h e "paradox of t he heap" shows. "Pire runs a s follows: a heap with cane atone in it is m a l l , I f you atid j u s t one s tone t o a slag11 heap, i t ' s s t i l l a small heap. hence by mtheme t i ca l induction a l l heaps a r e senall. "re @'paradoxt cones by switching from the i n f o m a 1 quant i ty s c a k o f 'small-large' t o t he p r e c i s e number sca le . Induction is not va l id i n t he former, which ( f o r a x a ~ l e ) exh ib i t s hys t e re s i s . Things rn of t en made up of p a r t s joined o r r e l a t e d in s o w way. Obvious exglnples a r e physica l ob jec t s m d e of p i eces glued o r assembled together : cups, caps, steam engines, animals. But t h e m are other3: prourn= made up o f subprocesses; t imein tervals made up o f times. The idea of organised co l l ec t ions o f e n t i t i e s being regarded themselves a s e n t i t i e s penneates OW th inking, Now t h i s f a c t s t r i k e s a t t h e roo t of an ' individual-based' ontology i n t h e same s o r t of way t h a t substances do. The only way of handling co l l ec t ions i s t o count = t h e co l l ec t ion and its p a r t s as individuals , r e l a t e d by some s o r t of made o f o r has-as-part r e l a t i on , But then these assembled individuals behave i n odd ways: they sometimes merge (two heaps make one heap) l i k e p ieces of s t u f f : sometimes they can be disassembled, cease t o e x i s t f o r a time and then perhaps be reassembled: i s it t h e same individual? (Our i n t u i t i o n says: yes , i n most cases). Modal l og i c i ans now have very e legant semantic t heo r i e s which can accommodate such odd behaviour in individuals . But these allow an 2 pa t t e rn of vanishing, reappearing and changing proper'ties. The polnt i s t o f i n d a way of represent ing the f a c t t h a t composite individuals have t h i s spec i a l way of vanishing (being taken a p a r t ) , and t o d i s t i ngu i sh , f o r example, those composites which cannot be reassembled (animals, cups) from those t h a t can ( ca r s , steam engines): and t o do a l l t h i s i n a framework which assumes t h a t t h ings , by and l a rge , don' t j u s t vanish and reappear spontaneously. Composites a r e thus a d i f f e r e n t 5 of individual , in a very deep sense. A r e l a t e d i s sue is how t o s t a t e c r i t e r i a upon which we r e i f y a co l l ec t ion i n t o a composite individual . Physical compactness is sometimes s u f f i c i e n t ( a heap), but not always necessary ( the wiring system of a house), f o r example. Of course, one does not expect a s ing le general answer, but I do not know o f any reasonable answers a t a l l , even f o r spec i a l cases. I have already remarked on the s i m i l a r i t i e s between being made of ( s t u f f ) and being made up o f ( p a r t s ) . I s t h i s anything more t h a d f a c n i l e analogy? Is the re some common framework in which the fundamental ontologic a l not ion , r a t h e r than exis tence , is space-occupancy? I t might be useful t o s t r i v e f o r a representa t ion which allowed t h e simultaneous expression in d i f f e r e n t schemes of both ' ex is tence ' and 'space-occupancy'. (The schemes would, I believe,ve t o be e s s e n t i a l l y d i f f e r e n t . ) Indeed, i n a crude way one can see how i t might be done d i r e c t l y by "arrays of fac ts" : t he ar ray subsc r ip t s g ive one access v ia s p a t i a l r e l a t i o r s h i p s t o the l o c a l presence of ob jec t s , which :artake of = l a t ionsh ips ( represented by a networn, s a y ) between themselves and o the r , non-space-fi l l ing, indivi d . ~ a l s (such a s colours) . Decomposabili-ty i s indica ted i?, the ar ray a l s o by 'break l i n e s ' wriicii separa te the space i n t o regions: d i f f e r en t s c r t s of connection could be f a i r i y ea s i ly handied (glued, detachab1.e .., ). 3ut t h i s i s ve ry crude and has s eve ra l c r u c i a l drawbacks (norably p l a s t i c i t y : imagine moving an ob jec t tnrougn the space, preserving i z s shape.) Huch hear Ls p c n e ~ a t e d $Y disputes baaed o r~ c C % a s ~ i f f ~ a t i ~ ? b i b wh ch do not comesp8~1d wieh the f a c t s , n r ~ h i c h a t l e a s t k v r outlivaaf. t h e i r usef&ness, Tow sueb sm tha '"enerality vs, ewk;iertPsat"deata m ~ d t he s o w recen t '$mccdwra vo, msemiona" debate. @oth of these aabae f r o m a r e v a s i o n agains t a g a n ' t i c ~ A w s m l y naive idea &out how 2s orgasrise i n t e l l i g e n t p~a&r;nm, which m e could (perhaps u n f a i r l y ) c a l l t h e genera?. problem-solver fal?iacy, C%pw Bap.pe9.t. c a l l s it, t h e a l i t d i n g white ' l ight theory. This was the early SnstTence that p m b l e ~ s o l v i n g nsthods nad t o be wrapped up i n lack boxes ca l l ed problem-solvers, whose (canly) lnpuC was a problem and whose (only] cutput a so lu t ion , ?x.ohlem-solvers were supposed t o be a s md as a s poss ib le , One had nor: t o "cheate' by "givin bBm-s so lu t ion i n any sense, o,g, by e p r o g l a m b g it o r c l e v w l y ceding t h e problem i n some we>r ( t h i s is made e x p l i c i t i n [79 1, U ~ B r t u n a t e l y , of course, t h i s col?ect ion of r u l e s means that t he re is of g e t t i n g sa jecr-mat ter -dependant knowledge i n t o t he ulack b it eannat be t he re ( v i o l a t e s gene ra l i t y ) , and it cannot be put i n t o t he pmblerr, ( c and the re a r e n ' t any o t h e r inputs . This is a ca r i ca tu re , but not much of a cmice t u re . Mucn work i n automatic theorem~proving wsls done with t h e imp l i c i t idea t h a t t h e theorem-provers were t o be regarded a s problem-solvcss i n t h i s sense (c.f . t h e widely P e l t 'need1 f o r adequate cr i tcsr ia of r e l a t i v e e f f i c i ency o f theorem-provers: "my problem-solver is more powerful than yours". (See [2,10] f o r a f a l l e r d iscuss ion) , The HIT school have now succeeded admhably i n des tmying t h i s idea, but unfor tunate ly have got ten it confuaed with some others , Stmely we need both gene ra l i t y and expe r t i s e : t h e f a l l a c y is not t h e mphas i s un g e n e r m Y , but t h e i x s t e n c e upon t h e black box and t h e "no c t~ea t ing" ru l e s . The general mechanism o f means-end ana lys i s , h e u r i s t i c search and computational l o g i c should not be r e j ec t ed , but r a t h e r incorporated i n t o more f l e x i b l e systems, s a t h e r than wrapped up i n closed 'pmhlerns a lv ing subrout ines ' o r 'methods' o r whatever. Thus, t o r e j e c t ccmven'tional uniform theorem-proving s y s t e m because they work with a s s e e i o n a l r a t h e r than 'procedural ' languages, i s t o miss t he point. (Whether sl language k considered t o be a programing language c~r. not , i s l a rge ly a matrer of t a s t e , i n any case. LISP can be regarded as (an incomplete) higher-order predica te ca lculus , o r a s a --order appl ied predica te c s l cu lus : predica te ca lculus can be regarded a s a programing language, alt\ough bv i t s e l f not a very good one.) The fo rce of t h e MIT c r i t i c i s n o f computat i o n a l l og i c is d i r ec t ed agains t t h e 'problem-solverq view a d i t s consequences, e spec i a l ly t h e lack of any access ib le and manipulable ( p r o p a m a b l e ) con t ro l s t u c t u r e i n conventional tneorem-proving systems. The G O M X system -r e f e r r e d t o e a r l i e r is an attempt t o f i l l t h i s lack d i r e c t l y with an e spec i a l ly devised con t ro l Language. A more recent a t t ack on conventional theorem-~rcving 573 1s t ?a? it i s t oo concerned with "machine or iented" log ic , and not e n o a h with "numan or iented" logic . I confess t o being qu i t e unable t o understand w:hat t h i s could possibly man.