62
obje object cts, s, that that is that that '(gP',
D. F0LL F0LLES ESDA DA
eg
caus causal al cont contex exts ts on shou should ld avoi avoi cont contrar raryy-to to-f -fact act cond condit itio iona nals ls scie scient nt lawlaw-st stat atem emen ents ts conf confir irma mati tion on stat statem emen ents ts an many many type type of preb prebab abil ilit it stat statem emen ents ts an disp disp siti sition on term termss-if if Onewan Onewan to ma sens sense. e. SAUL
A.
KRIPKE
aper aper ives ives an ex siti siti some some feat featur ures es sema sema tica tica he ry moda moda logi logics cs.' .' Fo cert certai ai quan quanti tifi fied ed exte extens nsio io of S5 this this theo theory ry wa ente ente in 'A Comp Comple lete te es he re in odal odal Lo ic an it as been been al al en er ec anti antifi fier erss-an an itwillres itwillrestr tric ic itse itselfin lfin th main main to on meth method od ofachie ofachievi ving ng hasi he pa er will will pure purely ly sema semant ntic ical al an ence ence 's en Th em hasi willom willom th se sema sema ti able ableau aux, x, whic whic is esse esse tial tial to fu re tati tation on of th theo theory ry," ," Proo Proofs fs also also will largel largel be suppre suppresse ssed. d. cons cons de ou mo al syst system ems. s. Form Form la usin usin th conn connec ecti tive ve P, Q, ha th foll follow owin in axio axio sche scheme me an rule rules: s:
HI
AD
Truth-fun Truth-function ctional al tautologi tautologies es B)
Rl. A, R2 AIDA
DA
DB
BIB
weadd weadd th foll follow owin in axio axio
From Acta Acta Phil Philoso osoph phic ic ho an th publ publis ishe hers rs
sche scheme me we ge S4
19 63 63 ) 8 33- 94 94 . Fenni Fennica ca 1 6 ( 19 ocie ocieta ta
Phil Philos osop ophi hica ca
e pr pr in in te te d b y p er er mi mi ss ss io io n o f t h enni ennica ca Hels Helsin inki ki
o gi gi c' c' , e it it s h ri ri f f u M at at he he 63 -9 J. Hintikka, -2 h e a ut ut h r s l os os es es t t o t h th or ea to i nt nt i an an r. r es es e t re re at at me me n ua ti i on on , h ow ow ev ev er er , i s u n iq iq u a s f a a s k no no w a lt lt ho ho u i t d er er iv iv e s om om e i ns ns pi pi ra ra ti ti o f ro ro m a in in ta ta nc nc e w it it h t h v er er y d if if fe fe re re n m et et ho ho d o f r io io r a n H in in ti ti kk kk a (1959), 9), 1-15. 1-15. Jour Journa na of Symb Symbol olic ic Logi Logic, c, 24 (195 Ibid Ibid., ., pp 323323- (Abstr bstrac act) t) Journa na of Symb Symbol olic ic h es es e s e ' A l et et e e s eo em in al c ' Jour 95 - 1 a n ' Se ma ma nt nt ic ic a al si of al i c' c' , Zeit ogle, Zeitsc schr hrif if fu athem athemati atisc sche he Logi Logi un Grun Grundla dlage ge de Math Mathem emati atik, k, 9, 67-96 67-96 se
h e L og og i
r ip ip ke ke , ' Se ma ma nt nt ic ic a
un
n al al ys ys i
o f M od od a
G ru ru nd nd la la ge ge n d e M at at he he ma ma titi k ality ality and Quan Quantif tifica icatio tion' n' Theoria,
64
IP
Brouwersche
st
we
MA
65
M:
This This comp comple lete tene ness ss .the .theor orem em equa equate te th synt syntac acti tica ca noti notion on of provability moda moda syst syste: e:TI TIwi with th sema semant ntic ical al noti notion on of validity. A : : : > D<)A Th rest rest of ~hIS ~hISpa pape pe conc concer erns ns with with th exce except ptio io of some some conc conclu ludi ding ng 5, if we add: add: re:n re:nar arks ks th intr intr duct ductio io quan quanti tifi fier ers. s. To this this we must must asso associ ciat at D<)A <)A:::> ch ai ex that that worl worl Form Formal ally ly we defi defi quan quanti tifi fica cati tion onal al mode mode stru struct ctur ur (q.m.s.) sy em eo ms cl ed mo R), toge togeth ther er with with func functi tion on lj! which which assign assign cl em ar al e a .a set cal~edthe of H. Intu In tuit itiv ivel el lj!(H) is th ~( domain '!pCI;I),. deve develo lope pe theo theory ry whic whic appl applie ie to such such nonnon-no norm rmal al syst system em as Lewi Lewis' s' setof al indi in divi vidu duals als exis ex isti ting ng Noti No tice ce of cour co urse se that th at lj!( lj !(H)n H)nee ee no H. S2 an S3 we will will rest restri rict ct urse urselv lves es here here to norm normal al syst system ems. s. be th same sa me se fo diff di ffer eren en ar umen um ents ts H, just ju st as intu in tu tive ti vely ly in worl wo rlds ds To ge sema semant ntic ic fo moda moda logi logic, c, weintro weintrodu duce ce th noti notion on ofa (nor (norma ma othe ot he than th an th real re al one, on e, some so me actu ac tual ally ly exis ex isti ting ng indi in divi vidu dual al ma be abse ab sent nt el ct .s G, model model struc structur ture. e. enwi eP is refl reflex exiv iv rela relati ti on K, an K. Intuitivel Intuitively, y,
Y~
individua dua vanabl vanables es x, y, an fo each each nonn nonneg egat ativ iv inte inte er world'. If HI an H2 ar tw worl worlds ds HIR H2 mean mean intu intuit itiv ivel el that that H2 indivi list list ~f n-ad n-adic ic pred predic icat at lett letter er P», wher wher th supe supers rscr crip ipts ts will will 'pos 'possi sibl bl relati relative ve to HI i.e., tha every every propos propositi ition on true possible someti time me ~e unde unders rsto tood od from from th cont contex ext. t. We coun coun prop propos osit itio iona na vari vari HI Clea Clearl rly, y, then then th rela relati tion on shou should ld inde indeed ed be refl reflex exiv ive; e; ever ever worl worl some able able (a omic omic form formul ula~ a~ as 'O-a 'O-a ic pred pred cate cate ette etters rs We then then uild uild is possible rela relati tive ve to itse itself lf sinc sinc ever ever prop propos osit itio io true well-formed well-f ormed formulae formu lae ossi ossi le in H. Refl Reflex ex vity vity is thus thus an intu intuit itiv ivel el natu natura ra requ requir ir fortiori, selvesto selvesto define define quanti quantific ficati ationa ona model. ment ment We ma impo impose se addi additi tion onal al requ requir irem emen ents ts corr corres espo pond ndin in to vari vari ~o defi de fin. n.ea ea quan qu anti tifi fica cati tion onal al mode model, l, we must must exte extend nd th orig origin inal al noti notion on 'red 'red ct on axio axioms ms of moda moda ogic ogic If si al G, an S4-m S4-m.s .s.; .; if is symm symmet etri ric, c, (G K, R) is Brouwersche Anal Analog ogou ousl sly: y: we must must s~pp s~ppos os that that in each each worl worl give give n-ad n-adic ic pred predic icat at is an equi equiva vale lenc nc rela relati tion on we call call (G K, R) an S5-m S5-m.s .s mode mode stru struct ct letterdet letterdeterm ermin ines es cert certai ai setof orde ordere re n-tu n-tupl ples, es, it extension in that that worl world. d. with withou ou rest restri rict ctio io is also also call called ed an M-mo M-mode de stru struct ctur ure. e. Cons Consid ider er fo exam exampl ple, e, th case case of mo adic adic pred predic icat at lett letter er P(x). We et model. wo at e d a t P(x) stru struct ctur ur (G K, R), model assig assigns ns to eac atomi atomi formu formula la (pro (propo posi siti ti a ls l s e ay variable) K. orma ormall lly, y, mo rela re lati tive ve to cert ce rtai ai assi as sign gnme ment nt of elem el emen ents ts of lj!( lj !(H)to H)to H) x , r p (P ( P (x ( x ) ctio rp(P, H) wher wher varies rp R) is bi ar fu ctio rela re lati tive ve to othe ot hers rs H) F. Th se of al indi in divi vidu dual al of whic wh ic rp(P(x), atom atomic ic form formul ulae ae an ar es ve elem elemen ents ts K, wh se rang rang is he a l e d extension {T }. Gi en mo el we ca defi defi th assi assi nmen nments ts trut trut -val -val es give trut truthh-va valu lu when when is assi assign gned ed valu valu in th doma domain in (P(x), H) be give nonnon-at atom omic ic form formul ulae ae by indu induct ctio ion. n. Assu Assume me rp(A, rp(B, of som m a e l other H) H) rp(A, rp(B, (x) means 'x is al '-ar '-ar we assi assign gn trut trut -val -val to he subs substi titu tuti tion on define H) T; otherwi otherwise, se, H) F. rp ,-A, H) isdefin isdefin inst instan ance ce 'S erlo erlock ck Holm Holmes es is bald bald'? '? Ho me does does no ex st ut in ther ther H) T; othe otherw rwis ise, e, H) al stat states es of affa affair irs, s, he woul woul have have exis existe ted. d. Shou Should ld we assi assign gn defi defini nite te trut truthhe ve ve r '; r p ( D A , H) rp(A, H') wise, r p ( D A , H) F. Intu Intuit itiv ivel ely, y, this this says says that that ss is tr al worl worl H' ossi ossibl bl re at ve to H. Comp Comple lete tene ness ss theo theore rem. m. Brouwersche system) hiI,OSOfi IOSOP et s~ rp(A, G) rp (1 92 -5 li t ra ra n l at at i e ac ac h l ac ac k TranslaM-(S4-, S5-, Brou phische Kritik, l io io n f ro ro n t h P hi hi lo lo so so ph ph ic ic a W ri ri ti ti ~g ~g s ~ f G ~t ~t tltl o l !r !r eg eg e (Oxf struct ctur ur (G K, R).5 wersche) mo el stru (Oxfor ord: d: Blac Blackw kw~l ~l1, 1, 1952 1952), ), Zeit Zeitsc schr hrif if
.•
,9.
Philosophic hical al Analys Analysis is (New and Feig Feig an Sell Sellar ar (eds (eds.) .) Readings InPhilosop (New York York Cent Centur ur Crof Crofts ts 1949 1949). ). P. F. Stra Straws wson on 'O refe referr rrin ing' g' Mind, n.s. n.s. 59 (195 (1950) 0) 320320--4 -44. 4.
Appl Applet eton on
0, rp(pn, H) repr repres esent ent
alter alternat nativ iv
et
conventions. an
Prio Priorr-PP-
ow de
e,
:2::
ct el
or ev
ul
rp(A, H),
truth-value
A.
), where P" lett letter er an
A(x) c o a i
va
bl
ed at
an ), H) rp(pn,
a,,)
x,
di
:;:: :;:::: :: 1,
define ne Xn, we defi formula
1, rp(pn, H) is 13 K,
),
); otherwise,
F,
,Y where
H)
value F re re ge ge -S -S tr tr aw aw so so n is true
c ho ho ic ic e i nv nv ol ol ve ve s
ee
S ho ho ul ul d w e t ak ak e DA (in H) H), o r j u n o t f a ls ls e
'L q ue ue st st io io n a ri ri se se s f o c on on ju ju nc nc ti ti on on : i f isfalse isfalse an t ak ak e t o b e a ls ls e o r t ru ru th th -v -v al al ue ue le le ss ss ?
ar
A(x,
es
Yn). h e
H) (whe (where re the b,
de b. to Yn), H) resp respect ective ively ly wher wher
In),
'NMN'. h a n o t ru ru th th -v -v al al ue ue ,
s im im il il a h ou ou l
rp«(A(x,
assi assign gnme ment nt of a, otherwise, rp«x)A(x, en at
x)DA(x)
::::>
, b to 13
D(x)A( D(x)A(x): x):::: ::: an
uc
2. Clearly
mode modell
R)
bi ar di at tt
ct
t ru ru c
by
ni
7p(G)
{a}, 'IJ!(H)
(x)DA( (x)DA(x). x).
For H,
is refl reflexi exive, ve, trans transit itiv ive, e, R) to
el
7p(H);
7p(H)
R), 'IJ!(H).
13
Yn), H)
D(x)A(x)
pl
Yn ev
{a b}
quan quanti tifi fica cati tion onal al where and
P, m od od e rp {a}, {a}, rp(P rp(P H) {a}. Then DP(x) Then clea clearl rl Bertran Bertran Rnssel Rnssell, l, 'On denotin denoting', g', Mind, n.s., 14 (1905) when is assi assign gned ed an (1905) 479-9 479-93. 3. (for rp(P(x), (x) DP(x). DP(x). But, (x)P(x) Prior, Time (Oxford: d: Clarendo Clarendo Press, Press, 1957, VIII 148 pp.) pp.) when when is assi assign.ed gn.ed b), n c D(x)P(x) 10 Time an Moda Modali lity ty (Oxfor u ra ra l um ha predicat cat shoul shoul be false or 11 atomic predi o f a l l t ho ho s i nd nd iv iv id id ua ua l n o e xi xi st st in in g i n t h a w or or ld ld ; t ha ha t i s t ha ha t t h e xt xt en en si si o o f a p re re d c ou ou nt nt er er ex ex am am pl pl e t o t h B a c a f o m ul ul a o ti ti c t ha ha t t hi hi s c ou ou nt nt er er ex ex am am pl pl e c at at e l et et te te r m us us t c on on si si s o f a ct ct ua ua ll ll y e xi xi st st in in g i nd nd iv iv id id ua ua ls ls . W e c a d o t hi hi s b y r eq eq ui ui ri ri n i s q ui ui t i nd nd ep ep en en de de n o f h et et he he r P(x) semanti semanticall call that rp(pn, [7p(H)]n; th seman semanti tical cal treat treatr: r:1en 1en below below when when woul woul othe otherw rwis is suff suffic icewit ewitho hout ut chan change ge We woul woul have have to ad to th axiom system be os es ul of th A(xt) :: : av ch no hi ca ofsu on ns at eq od uc co io at (1 n). longerhold longerhold theor theorem em woul woul hold hold fo atomi atomi form formul ulae ae whichwou whichwould ld no hold hold when when th K) S; 'IJ! a to to mi mi c f or or mu mu la la e a r r ep ep la la ce ce d b y a rb rb it it ra ra r f or or mu mu la la e ( Th Th i a ns ns we we r q ue ue st st io io n o f. f. t e.. {a b } {a}, r t e c e B 'IJ!(G) J!H)
a ri ri a
ng
ov
n -a -a d
ar
n,
in which which rp(P, G)
IP
68
{ a b } p ( H) {a}, where where where agai agai where Pi i= b. Define cp(P, G) give give mona monadi di pred predic icat at lett letter er Then Then clea clearl rl (x)P(x) ol in bo and cp(P(x), H) when is assign assigned ed H, so ha cp(D(x)P(x), G) b, so that cp(DP(x), x), G) F. Hence cp«x)DP(x), G) that when when is assign assigned ed b, cp(DP( F, an wehav th desi desire re coun counte tere rexa xamp mple le to th conv conver erse se ofthe Barc Barcan an form formul ula. a. This This coun counte tere rexa xamp mple le howe howeve ver, r, depe depend nd on asse assert rtin in that that in H, actuall false when is assigne assigne b; it migh migh hu disa disapp ppea ea if fo P(x) is actuall thi assignm assignment ent P(x) were were decl declar ared ed to lack lack trut truthh-va valu lu in H. In this this case case we wi stil stil av coun counte tere re ampl ampl if we re uire uire eces ecessa sary ry stat statem emen en to be true in al poss possib ible le worl worlds ds (Pri (Prior or's 's 'L'), me el eq re that that it neve neve be fals fals (Pri (Prior or's 's 'NMN'). On ou pres presen en co vent vent on we ca e xa xa m eq ac s. 1 J ! ( H ) S; 1 J ! ( H ' ) whenever HRH'. e xa xa m e s e a ec ar av co nter ntermo mode dels ls quan quanti tifi fied ed S5 to both both th Barc Barcan an form form la an ts concon-
er ab quantified
ed by th foll follow owin in argum argumen ent: t:
em
(A) (x)A(x) :::> A(y) (by quanti quantific ficati ation on theory theory (B) D«x) necessitation) n) D«x)A( A(x) x):: :::> :> A(y» A(y» (by necessitatio (Axiom om A2 D«x) D«x)A( A(x) x):: :::> :> A(y» A(y» :::> D(x)A(x) :::> DA(y) (Axi (D) D(x)A(x) :::> DA(y) (fro (fro (B an (C) (y)(D( D(x) x)A( A(x) x):: :::> :> DA(y DA(y (E) (y)( (gen (gener eral alizi izing ng on (D (F) D(x) quanti tifi fica cati tion on theo theory ry an (E D(x)A( A(x) x):: :::> :> (y)D (y)DA( A(y) y) (b quan (C)
We seem seem to have have deri derive ve th conc conclu lusi sion on usin usin prin princi cipl ples es that that shou should ld al vali vali he mo el-t el-the heor or Actu Actual ally ly th flaw flaw lies lies in th appl appl cati cati of eces ecessi sita tati ti to (A). (A). In form form la like like (A), (A), we give give th free free aria aria le th
generality generality interpretat interpretation ion :13 en A) se viate viate asser asserti tion on of it ordi ordina nary ry univ univer ersal sal clos closur ur (N (y)«x)A(x)
No
A(y»
:::>
if we appl applie ie
to (N), we woul woul ge A(y» On th ot er and, and, (B) itse itself lf is inte interp rpre rete te as asser asserti ting ng (B") (y)D«x)A(x) :::> A(y» (B') D(y)«x)A(x)
nece necess ssit itat atio io
:::>
infer (B") (y)DC(y),
D(y)C(y)
tryi tryi to pr ve In fact fact it isrea il chec chec ed that that (B") fail fail in th coun counte terrmo el iven iven ab ve fo he co vers vers Barc Barcan an form formul ula, a, if we re lace lace A(x) by P(x). We ca av id this this sort sort of diff diffic icul ulty ty if foll follow owin in Quin Quine= e= we form formul ulat at quan quanti tifi fica cati tion on theo theory ry so that that only only closed formu formula la ar asser asserte ted. d. Asser Asserti tion on ofform offormul ulae ae cont contai aini ning ng free free vari variab able le is at best best conv conven enie ienc nce; e; asse assert rtio io of A(x) with with free free ca alwa alwa repl replac aced ed by asse assert rtio io of (x)A(x). mu ee ar es closure of If to be an form formul ul with with ut free free aria aria le obta obtain ined ed refi refixi xi ni ersa ersa quan quanti tifi fier er an eces ecessi si sign signs, s, in an rder rder to A. axio axioms ms of quan quanti tifi fied ed c he he m a : (0) TruthTruth-fun functi ctiona ona
tautol tautologi ogies es
:::>
DA :::> B) (x)A, where ( 4 ( x) x) ( :::> B) (x)A
:::>
DB
:::>
(5 (y)«x (y)«x)A )A(x (x
:::>
:::>
(x)B
A(y»
Th rule rule of infe infere renc nc is deta detach chme ment nt fo mate materi rial al impl implic icat atio ion. n. Nece Necess ssit itaaio ca be obta obta ne as deri derive ve rule rule Brouwersche system, simp si mplyadd lyadd to th axio ax io sche sc hema mata ta al clos cl osur ures es of th appr ap prop opri riat at redu reduct ctio io J ou ou rn rn a o f S ym ym bo bo lili c L og og ic ic , 21 (195 (1956) 6) 12 axiom. 60-2. asse assert rted ed that that th gene genera rali lity ty inte interp rpre reta tati tion on of theo theore rems ms with with free free vari variab able le 13 It is no ms ed av wi s: isth os bl e. ne mi ht is f or or m l a p ro ro va va b f f f o e ac ac h o de de l stra straig ight htfo forw rwar ar exte extens nsio io of th moda moda prop propos osit itio iona na logi logics cs with withou ou cp,cp(A,G) A. B u t he he n (x)A(x) ar il ot e or or e f ac ac t i n t h uner de a bo bo v t o B a c a o rm rm ul ul a :::> A(y) th modi modifi fica cati tion on of Prio Prior' r' Q; st cp«x)P(x) :::> P(y), G) if is assign assigned ed b. q u n t f ic ic a i o e or or y ld av rest restri rict ctio ion, n, unli unlike ke Hint Hintik ikka ka's 's pres presen enta tati tion on an neve nevert rthe hele less ss neit neithe he th r ev ev i e d l o t h l i e s r o o se se d inik ( i ' Ex Ex i t en en t a l P r s u s i i o nd Existe Existenti ntial al Commi Commitme tments nts', ', Jour Journa na of Phil Philos osop ophy hy Philoso Philosophic phical al Review Review eban n d T . a ilil pe pe r (i 'N nd si at S i u la la r e r s ' ( 1 ) , 9 -4 -4 3) 3) . h i r oc oc ed ed ur ur e h a ch r ec ec o m e av a do do p e d it sinc sinc we wish wished ed to show show that that th diff diffic icul ulty ty ca be solv solved ed with withou ou revi revisi sing ng quan quanti tifi fica ca i o t he he or or y o da da l r o i titi o a l l o i c
at 14
eo
W. Quin Quine, e, Mathe Mathemat matic ical al e d . , e v. v. , 1 , x II II +3 +3 4
ed Logic Logic ( C m br br id id g .)
ma Ma .:
a rv rv ar ar d
v,
re
0;
KRIPKE
70
Th sema semant ntic ical al comp comple lete tene ness ss theo theore re we gave gave fo moda moda prop propos osit itio iona na lo ic ca be exte exte de to th ew syst system ems. s. We ca intr introd oduc uc e xi se xi st st en en c e a s p re re di di ca ca t e like like Sema Semant ntic ical ally ly exis existe tenc nc is mona monadi di pred predic icat at E(x) satisf satisfyin ying, g, for R) th iden 'IjJ(H) fo ever each each mode mode rp identi tity ty rp(E, H) ever ma an at K. clos closur ures es of form formul ulae ae of th form form A(y), and (x)E(x). :: The predic predicate ate used used abov abov in th coun counte terr-ex exam ampl pl to th conv conver erse se Barc Barcan an form form la ca ow reco recogn gniz ized ed as simp simply ly exis existe te ce Th fact fact show show ho existen existence ce differ differ fro the tautol tautologi ogical cal predic predicate ate eve though though D(x)E(x) (x)DE(x)
(x)D(A (x)D(A(x) (x)v,. v,.......- A(x
isnot alth althou ough gh it isneces isnecessa sary ry that that ever ever thin thin exis exists ts it does does no foll follow ow that that ever everyt ythi hing ng ha th prop proper erty ty of nece necess ssar ar exis existe tenc nce. e. We ca intr introd oduc uc iden identi tity ty sema semant ntic ical ally ly in th mode mode theo theory ry by defi defini ning ng an ss me an ther therwi wise se fals false; e; exis existe tenc nc coul coul then then be efin efined ed in term term of iden identi tity ty y). or reas by stip stipul ulat atin in that that E(x) means (3y)(x reas ns no gi en ere, ere, broa broade de theo theory ry of iden identi tity ty coul coul be obta obtain ined ed if we comp compli lica cate te th noti notion on of quan quanti tifi ficat catio iona na mode mode struc structu ture re We conc conclu lu with with some some brie brie an sket sketch ch rema remark rk on th 'pro 'pro abil abilit ity' y' nter nterpr pret etat atio io mo al lo ics, ics, whic whic we iv in each each case case fo ro osiositi na calc calc lu nl Th read reader er will will av btai btai ed th main main poin poin of this this pape pape if omit omit this this sect sectio ion. n. Prov Provab abil ilit it in er reta retati ti ns ar base base on
willbe willbe inte interp rpre rete te as true true if is prov provab able le in th syst system em ha been been argu argued ed that that such such 'pro 'prova vabi bili lity ty inte interp rpre reta tati tion on of mode mode oper operat ator or ar disp dispen ensa sabl bl in favo favour ur of prov provab abil ilit it predicate, atta attach chin in to th Gode Gode umbe umbe A; es ag en me st leas leas some some doub doub on this this view viewpo poin int. t. Le us cons consid ider er th form formal al syst system em PA of Pean Pean arit arithm hmet etic ic as form formal aliz ized ed ee e. at r-«, (the (the conj conjun unct ctio io an nega negati tion on adjo adjoin ined ed ar to be dist distin inct ct from from thos thos of th orig origin inal al syst system em), ), oper operat atin in on clos closed ed form formul ulae ae only only In th mode mode theo theory ry we av abov above, e, we took took atom atomic ic form form la be pr posi positi tion onal al aria aria les, les, or pred predic icat at lett letter er foll follow owed ed by pare parent nthe hesi size ze indi indivi vidu dual al vari variab able les; s; here here cl ed mu ae P A (not st at mi mu A) mo R), al st co ab mo 15 C. Klee Kleene ne 1952 1952 x+55 x+55 pp.) pp.)
Intro Introdu duct ction ion
to Metam Metamat athe hema matic tic
ew
o rk rk :
an
o st st ra ra nd nd ,
PA,G PA,G isth stan standa dard rd mo el th natu natura ra umbe umbers rs an is th Cart Cartesi esian an product 2. We defi define ne mode mode rp by requ requir irin in that that fo an atom atomic ic form formul ul T(F) T(F) if is ru (fal (false se in th mode mode H. (Rem (Remem em K, rp(P, H) ber, up he eval eval atio atio fo comp comp un form formul ulae ae as before.!" is tr isto sa it istrue istrue th real real worl worl G; an fo an atom atomic ic P , p (D (D P , G) is pr vabl vabl in PA (Not (Notic ic that that rpCP, G) Tiff intu intuit itiv iv sens sense) e) Sinc Sinc (G K, R) vali vali on this this nter nterpr pret etat atio io an we ca sh that that only al is Gode Godel' l' ndec ndecid idab able le form formul ula, a, ep(DPv ep(DPv ,....,....P, G) ,....,A.)
An ther ther pr vabi vabili li in er reta retati ti is th foll follow owin in Agai Agai we take take th atom atomic ic form formul ulae ae to be th clos closed ed wffsof wffsof PA, an then then buil buil up ne form formuuaeus ed ct ,, .. et et orde ordere re pair pair (E e x ) , s is is t si PA ex is (c unta unta le mo el th syst system em (PA, e x o ) , where s ta ta n mo ay (E', ex'), wher wher (E ex an (E', ex') P, define rp(P, (E, e x ) ) T( if e) an a t m i P, that ep(DP, (E, ex)) is rova rova le in E; in ar ic lar, lar, rp(DP, G) iff isprov isprovab able le inPA. Sinc Sinc (G K, R) isan S4-m S4-m.s .s., ., al th laws laws of S4hold S4hold is Gode Godel' l' unde undeci cida dabl bl form formul ul DP:::> :> D,.. D,.... .. DP), DP), G) ep((""'- DP:: !f(D,....- D«)A A, !f(D,....A), G) eo em Ki ey ." Bysuit Bysuitab able le modi modifi fica cati tion on this this diff diffic icul ulty ty coul coul be remo remove ved; d; bu we do no into into he matt matter er here here Simi Simila la inte interp rpre reta tati tion on of co Brouwersche s ta ta t es es thos thos give give abov above. e. We ment mentio io on more more clas clas of prov provab abil ilit it inte interp rpre reta ta tion tions, s, th 'ref 'refle lexi xive ve exte extens nsio ions ns of PA. Le be form formal al syst system em cont contai aini ning ng
<>,_,
."6 m a b e p ro ro te te st st e t ha ha t a lr lr ea ea d c on on ta ta i s ym ym bo bo l f o c on on ju ju nc nc ti ti o an ne non, non, sa '& ns we we r i s '_ T h a ns t ha ha t i f and a r a to to mi mi c f or or mu mu la la e t he he n is also a to to mi mi c i n t h p re re se se n s en en s /\ the s in in c i t i s w el el ll- fo fo rm rm e i n PA but p r v io io u t he he or or y i n w hi hi c t h c on on ju ju nc nc ti ti o o f a to to mi mi c f or or mu mu la la e i s n o a to to mi mi c w e n ee ee d 't;'. rp(P/\ Q, H), rt ss fo and Q, rp(P&Q, H) '& soth o nf nf us us i it se no rm ra im rk pl t o n e t io io n a n t o t h p ro ro va va bi bi li li t i nt nt er er pr pr et et at at io io n o f s 4 i n t h n e p ar ar a r ap ap h 17 ee M c i ns ns e 'O th n ta ta ct ct ic ic a o ns ns tr tr uc uc ti ti o o f y st st em em s o f M od od a Journa na of Symb Symbol olic ic Logi Logic, c, 10(1945), Logic', Jour 10(1945), 83-94. 83-94.
IP
72
PA el me mu ae ar me mula mula of PA us th conn connec ecti tive ve &, ---. ---. s: an~ ca at me ct an .as PA itse itself lf no intr introd oduc ucin in ne ones ones Se foot footno note te 16 p. 71.) 71.) Then Then IS call called ed refl reflex exiv iv exte extens nsio io of PA iff: iff: (1) is an ines inesse sent ntia ia exte exte sion sion of PA (2 DA isprova isprovabl bl if is (3 ther ther is valu valuat atio io .IX, mappi mapping ng t~ clos closed ed form formul ulae ae fE into into th se {T,F},su {T,F},such ch that that co ju ctio ctio an egat egatio io obey obey cl mu al isprova isprova le in E, an al he theo theore rems ms of et ~h~v ~h~val al cx(DA) T. It ca be show show that that ther ther ar refl reflex exiv iv exte extens nsio ions ns of PA cont contam ammg mg th axio axioms ms of or even even 4.1, 4.1, ut on co tain tainin in 5. al we ma at al ma logi logi into into 4, we ca ge mode mode heor heor fo th in~u in~uit itio ioni nist st pred predi~ i~at at calcu calculu lus. s. We willnot willnot give give this this mode mode theo theory ry here, here, bu~Inst bu~Instead ead w~ll w~llme men~ n~lO lOn, n, fo prop propos osit itio iona na calc calcul ulus us only only part partic icul ular ar usef useful ul inte interp rpre reta tati tion on of.I of.Int ntuuitio itioni nist stic ic logi logi that that resu result lt from from th mode mode theo theory ry Le be an cons consis iste tent nt exte extens nsio io of PA We sa form formul ul provab able le verified in if it is prov in We take take he cl se wffs wffs a t m ic ic , of them them usin usin th intu intuit itio ioni nist stic ic conn connec ecti tive ve 1\ V , and We the~ the~ stipul stipulate ate induct inductive ively: ly: an Bare Bare 1\ IS veri verifi fied ed in if or is ---. ---. is veri verifi fied ed in if ther ther IS no cons consist isten en ex tens tensio io veri verify fyin in A; e ve ve r en ::> exte extens nsio io E' of veri verify fyin in also verifi verifies es ed but, but, e.g., e.g., ---. ---. is th GOde GOde unde undeci cida dabl bl formu formula la ~nf~tur ~nf~tur work work we will will exte extend nd this this inte interp rpre reta tati tion on furt furthe her, r, an show show that that usmg usmg ~e ca find find an inte interp rpre reta tati tion on fo Krei Kreise sel' l' syst system em FC of abso absolu lute te~yfr ~yfree ee choi choice ce sequences." Itis clear, clear, incide incidenta ntally lly thatPa can be repl~ repl~ced ced II th pr~v pr~vaae r e ta ta t an sy ., s te te m el et ch se e) wh er at ap es an ma system whatsoever. whatsoever. 18
plete pletenes ness' s'
Proofs Proofs
Jour Journa na of Symb Symbol olic ic Logi Logic, c, 23 (195 (1958) 8) 369369-88 88
VI
involv lvin in essen essenti tial alis is ar no recei receivi ving ng grea grea deal deal of atten attenti tion on PROBLEMS invo fr moda moda lo icia icia an hilo hiloso so hers hers Even Even curs curs ry la ce at work work inthis inthis fiel field, d, howe howe er soon soon re eals eals ha ther ther ar many many doct doctri rine ne wh ch by this this titl title. e. will will isol isolat at an disc discus us on such such octr octrin ine. e. In part partic icul ular ar ter isolat isolating ing one versio versio of essenti essentialis alis (Secti (Sections ons and II), will will argu argu that that workin workin quan quanti tifi fied ed moda moda logi logi ca be an is inde indepe pend nden en of th acce accept ptan ance ce he tr th of this this octr octrin in (Sec (Secti ti ns III-V). In th as sect sectio io (Sec (Secti ti I) wi em si ct st ec wh this this part partic icul ular ar form form of esse esse ti~l ti~lis is is phil phil soph sophic ical al III-V, us al at an needno needno even even pres presup uppo pose se th meaningfulness of essen essenti tial alis is claim claim i n
an (b
part partia ia vind vindic icat atio io 1.
of quan quanti tifi fied ed moda moda logi logic. c.
IMIN IMIN RY
CL RI IC TI
et se al st ct eki as c a individual esse essenc nces es an th othe othe withwh withwhat at shal shal call call general esse essenc nces es Th form former er doct doctri rine ne make make some some ai ct al ar er erti erties es wh ch ar so inti intima mate te asso associ ciat ated ed with with th bjec bjec that that noth noth ~g else could (wit (wit emph emphas asis is on th 'cou 'could ld') ') have have prec precis isel el thos thos char charac acte teri risscsw ec me stro strong nger er thesi thesi than than th Iden Identi tity ty of Indi Indisc scer erni nibl bles es whic whic hold hold mere merely ly that that no tw obje object ct ca imul imulta tane neou ousl sl exis exis whil whil shar sharin in al prop proper erti ties es ays: ays: (1 it proh prohib ibit it th simu simult ltan aneo eou' u'Se Sexi xist sten ence ce of tw obje objecs cs which which share share same same indi indi id al esse esse ce (e en when when they they coul coul diff differ er in ot er of thei thei had From Th Philos Philosop ophic hical al Revie Review, w, Philoso sophi phica ca er is on th a n Th Philo
refere refere of Th Philo Philoso sophi phica ca
Revi Review ew f o
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