Stalnaker s Argument

Transcription

1 Stlker s Argumet (This is supplemet to Coutble Additiviy, Dutch Books d the Sleepig Beuty roblem ) Stlker (008) suggests rgumet tht c be stted thus: Let t be the time t which Beuty wkes o Mody morig. Upo wkeig, Beuty lers somethig tht is relevt to Heds: while she does t kow whe t is, she does ler, of t, tht she is wke the. Hvig lered this, she kows tht oe of the followig three possibilities obtis: T*: Tils d t is o Mody. T*: Tils d t is o Tuesdy. H*: Heds d t is o Mody. But if Beuty is to be represeted s lerig, o Mody morig, tht she is wke t t, the she must be represeted s ot yet kowig this o Sudy eveig, d hece s hvig ot yet ruled out the followig possibility: H*: Heds d t is o Tuesdy. O Sudy eveig, Beuty should regrd ech of the four bove possibilities s eqully probble. Ad o Mody morig, she should coditiolize o the egtio of the fourth possibility, with the result tht her credece i H*, d hece i Heds, is oe-third. Ufortutely, while Stlker sys tht ll prties to the debte should be ble to gree bout how Beuty s degrees of belief should be pportioed o Sudy mog these four possibilities, he does t idicte how this cosesus should be reched. However, elsewhere (008b) he offers clue bout the kid of priciple he wts to ivoke: [Beuty] is i positio tht is like [tht of] perso who is sked to mke probbility judgmet bout the followig situtio: Two people re wlkig together i Cetrl rk. Cll them A d B. How probble do you thik it is tht B is werig ruig shoes, give tht A is? [H]owever she swers the questio, if she is rtiol, she hd better swer this follow-up questio i the sme wy: Now cosider the coditiol degree of belief tht it is A who is werig ruig shoes, o the coditio tht B is. The reso the questios hve to be swered the sme wy is tht they re relly the sme questio, sice A d B re just the theorist s lbels for two geericlly idetified people... The oly idifferece priciple tht I wt to rely o is the priciple used i this exmple, requirig tht the two questios bout coditiol degree of belief be swered i the sme wy for geericlly described situtios. I the SB cse, this priciple pplies o Sudy to the possibilities tht will be distiguished o Mody d Tuesdy, but tht c be described oly geericlly o Sudy. O some wys of formlizig the idifferece priciple Stlker lludes to, this priciple would, by itself, coflict with CA. The iterestig questio, however, is whether CA coflicts with restricted,

2 fiitistic versio of this idifferece priciple, together with the other ssumptios eeded to derive the oe-third solutio. We c formulte restricted versio of the idifferece priciple s follows: Arbitrry Time Idifferece. Suppose T is fiite set cosistig of times, t, t, t, such tht Beuty represets the elemets of T simply s rbitrry times. Ad for some propositio, p, d some predicte, A, suppose tht, coditiol o p, the set of propositios {A(t ), A(t ), A(t )} costitutes prtitio. I this cse, Beuty should hve equl credece, coditiol o p, i ech of the propositios i T. We c derive the oe-third solutio from the bove priciple, together with Fiite Additivity, the Restricted ricipl riciple, d the followig three ssumptios: rior Admissibility. I y Sleepig Beuty problem, o the eveig prior to the first experimetl wkeig, Beuty kows tht she hs o iformtio tht is idmissible i reltio to y of the hypotheses i S. De Re Coditioliztio. I y Sleepig Beuty problem, if t is the time t which Beuty first wkes, the her credeces t t should be the sme s they would be if she hd iitilly represeted t s rbitrry time, d the coditiolized o the propositio tht she udergoes experimetl wkeig t t. Arbitrry Time Idepedece. I y Sleepig Beuty problem, for y hypothesis h i S, where propositio p is specifictio of the loctios of times tht Beuty represets simply s rbitrry times, Beuty s credece i h give p should be equl to her ucoditiol credece i h. I order to derive the oe-third solutio, let t represet the time whe Beuty wkes o Mody morig, d let t' represet y other time which Beuty represets, o both o Sudy eveig d o Mody morig, simply s rbitrry time. Let be Beuty s credece fuctio whe she wkes o Mody morig, d let be the credece fuctio she would hve o Sudy eveig, ssumig tht t tht time she lso represeted t simply s rbitrry time. Let the propositios H' d X be defied s follows: H'. Heds d t' is o Mody. X. t d t' re the two times whe Beuty would udergo experimetl wkeigs give Tils. Ad let Y be the followig propositio, which is etiled by X: Y. t' is oe of the times whe Beuty would udergo experimetl wkeigs give Tils, d t' does ot coicide with t.

3 Coditiol o (Heds & X), the set {H*, H'} costitutes prtitio. Ad sice the propositios i this set differ oly with respect to times which, reltive to, re represeted simply s rbitrry times, it follows from Arbitrry Time Idifferece tht H * Heds& X H Heds& X Ad coditiol o (Heds & X), H' is equivlet to H*. Hece, H* Heds& X H* Heds& X Sice, coditiol o X, Heds is equivlet to the disjuctio of H* d H*, we my ifer H * X H * X Heds X But sice X is specifictio of the loctios of times which, reltive to, re represeted simply s rbitrry times, it follows from Arbitrry Time Idepedece tht Heds X Heds Ad from the ricipl riciple, together with rior Admissibility Hece, from the lst three lies, Heds ChHeds / H * X H * X () 4 We c ow determie Beuty s Mody morig credece i Heds. Sice Y is specifictio of the loctios of time which, o Mody morig, Beuty represets simply s rbitrry time, it follows from Arbitrry Time Idifferece tht Heds HedsY Ad where W is the propositio tht Beuty udergoes experimetl wkeig t t, it follows from De Re Coditioliztio tht Ad sice (W & Y) etils X, HedsY HedsW & Y

4 Heds W & Y Heds W & X & Y Heds & W X & Y W X & Y But coditiol o (X & Y), (Heds & W) is equivlet to H*, d W is equivlet to the egtio of H*. Hece, from (), Heds & W X & Y W X & Y Ad so it follows, from the lst four lies, tht Heds = /3. H* X / 4 H* X 3/ 4 3 I will ow rgue tht the premises of the bove rgumet coflict with CA. I y Sleepig Beuty problem defied by prtitio, S, let i d j be y two hypotheses i S. If N(j) N(i), the let A deote i, d let B deote j; otherwise, let A deote j d let B deote i. Thus, N(B) N(A). Let t 0 be the time of Beuty s first experimetl wkeig. Ad let t, t 3, t N(B) be y other times such tht, both upo first wkeig d o the previous eveig, Beuty represets these times simply s rbitrry times. Let be Beuty s credece fuctio whe she first wkes, d let be the credece fuctio she would hve the previous eveig, ssumig tht t tht time she lso represeted t simply s rbitrry time. Let propositios X' d Y' be defied s follows: X'. Y'. The times t through t N(B) re the N(B) times whe Beuty would udergo experimetl wkeig give B. The times t through t N(B) re ll distict times t which Beuty would udergo experimetl wkeig give B, d oe of these times coicides with t. Ad for y two itegers, m d, betwee d N(B), let us defie the propositio s follows: m. Hypothesis A is true, d t m m is the time of the th experimetl wkeig. Now cosider the followig mtrix: N ( B) We c ow determie the probbilities of the propositios i this row, reltive to. Coditiol o (A & X'), every row, d every colum, of this mtrix costitutes prtitio. Ad the propositios i y give colum differ from oe other oly with respect to times which, reltive to, re represeted

5 simply s rbitrry times. Ad so it follows from Arbitrry Time Idifferece tht, coditiol o (A & X'), ech of the propositios i y give colum must hve the sme probbility, d these probbilities must sum to oe. Therefore, every propositio i y give row must likewise hve the sme probbility, coditiol o (A & X'). Cosider the first row. As we hve see, coditiol o (A & X'), ll the propositios i this row hve the sme probbility. Ad, coditiol o X', the disjuctio of these propositios is equivlet to A. Hece, coditiol o X', Beuty s credeces i these propositios must be equl, d must sum to her credece i A. Thus, sice there re N(B) propositios i this row, For ll s.t. N( B), X NB A X Applyig Arbitrry Time Idepedece, the ricipl riciple, d rior Admissibility, we obti For ll s.t. N( j), X N A B Ch N A B (') We c ow determie wht Beuty s credeces should be whe she first wkes. Sice Y' is specifictio of the loctios of times which, upo first wkeig, Beuty represets simply s rbitrry times, it follows from Arbitrry Time Idepedece tht A AY. Hece, where W' is the propositio tht Beuty udergoes experimetl wkeig t t, it follows from De Re Coditioliztio tht A AW Y &. By similr resoig, B BW Y A BW & Y B AW & Y Ad sice (A & W' & Y') d (B & W' & Y') ech etils X', A B & W X & Y B A& W X & Y &. Hece, But coditiol o (X' & Y'), (B & W') is equivlet to B, d (A & W') is equivlet to the disjuctio N. Hece, Ad so, from ('), by Fiite Additivity, N A B X B A B X BNAChA NB / Ad by Arbitrry Time Idepedece, the Restricted ricipl riciple, d rior Admissibility, X

6 B X B X ChB Hece, from the lst two lies ANBChB BNAChA wheever Ni, Ni Z Now recll tht A d B re simply the rbitrry hypotheses, i d j, belogig to S. Becuse of the symmetry of the bove equtio, we c ifer tht, regrdless of whether A = i d B = j, or vice vers, in jch j jni Chi wheever Ni, Ni Z Thus, the premises of the De Re Coditioliztio rgumet etil the GT. Refereces Stlker, R. (008). Our Kowledge of the Iterl World. Oxford: Oxford Uiversity ress. Stlker, R. (008b). Commet o Stlker o Sleepig Beuty Bri is: group blog i the philosophy of mid. (ccessed Oct., 009).