ARB THERE TImRB COUNTEREXAMPLES TO THE TIm CU)SURE CLOSURE PRINCIPLE?


 Meredith Lang
 2 years ago
 Views:
Transcription
1 JONATHAN VOGEL ARB THERE TmRB COUNTEREXAMPLES TO THE Tm CU)SURE CLOSURE PRNCPLE? PRNCPLB7 Very ten, person cn't know proposition propositon without wilhout knowing vrious logicl consequences tht ht proposition. So, for fur instnce, if know ht tht r friend wering yellow tie, cn't fil to know tht lht r friend wering tie. tie, period. n th cse, he reltion logicl consequence obvious. When he reltion n't obvi obvious, proposition know my hve logicl consequence don't know  for exmple, suitbly obscure mmticl orem. n light se considertions. considertions, it seems plusible to hold tht if person knows given proposition, tht ht person must lso know ny logicl consequence tht proposition which he or she nxognizts recognizes s such. such, Putting it differently, we might sy tht ht knowledge closed under known logicl impli impli ction.' 1 The problem skepticm bout he externl world gives th eptemic principle (herefter, he "Closure Principle") specil interest. When skeptic rgues tht ht we hve no knowledge world cuse we don't know tht ht we ren't mssively de deceived in some wy, he or she ppers to ssume tht ht knowledge hs closure prop pr0perty. But if it possible 10 to find cler exmples demonstrting tht closure sometimes sometims fils, fils. crucil piece support for skeplicm skepticm will removed. The purpose th h pper to show tht even he strongest pprent counterexmples to closure don't hold up under scrutiny. To tht extent. t problem skepticm still with us. us, DRETSKES DRETSKE'S ZEBRA CASE n widely red pper, pper. Fred Dretske fered n intriguing iniguing exmple which ment to show ht tht he Closure Principle invlid. t worthwhile to 10 quote Dretske's dcussion tlenglh: length: You tke r son 10 to zoo, ux,, see severl zebrs, Wid nd when questioned quesuoncd by r son, scm, lell tell him y lhcy re zebrs. Do know kmw y re rc zebrs? Well, mos most us would hve little ble hesittion hesirdon in sying tht we did know th. We know wht zebrs look like, nd, sides, sides. th che city ZOO ux, nd nimls re in pen clerly mrked "Zebrs: "Zebrs." Yet, Y, something's ing zebr implies tht it not nol mule Wid, nd. in pniculm, prticulr, not mule muk cleverly ckverly dgued by ZOO ux, uthorities 10 to look like.zein. zebn. Do know tht se nimls re rc not mules cleverly dgued by he zoo uthorities 10 to look lmk lke O 7.ebrs? 7ebr97 f re rc empled lcmpted 10 to sy Yes" "Yes" 10 to th question, quesuon, think moment bout wht re!ons RBSDR~ hve, whl whr evidence cn produce in fvor hv thi., th clim. The evidence hd hrd (or for thinking m zebrs hs en cn effectively eltwtively DCulllil.ed, neutrliml, since it tioes clws not countlowllfd nwrd ir thlr not ing mules muler cleverly dgued 10 to look like zebrs. zebm. You hve some generl uniformities uniformiues on which rely, rdy. regulrities 10 to which yoo give expression by such rellllllb remrlcs s "Th "Thr n't very vy likely" or "Why 'Why should zoo uthorilies uthorities do tht?" Grnled, Grnted. hypos (if we my cll it thi) tht) not very plusible, plusible. given wht we know bout people pcople nd zoos. But question he'c here not wher whew th lerntive lterntive plusible, plusible. not wher it more or less plusible thwi thn tht re re rc rel zebrs zebm in lhc pen, pen. but wher know brow tht th lterntive lmdve hypos flse. don't think cto2. doz. 13
2 14 14 JONATHAN VOGEL ARE THERE COUNTEREXAMPLES TO THE CLOSURE PRNCPLE? _ 15 S According to Dretske, Zebr Cse counterexmple to closure cuse know () nimls in in pen re zebms, zebrs, but don't know cler logicl consequence (), nmely,(b) nmely.(b) nimls in in pen ren't cleverly dgued mules. find th description sitution implusible. Given wht Dretske hs sid in in lying out exmple, think it it more resonble to conclude tht if if know () know (b) s well, nd closure preserved fter ll. The reson know tht n niml in in pen not dgued mule (if do know it's zebm) zebr) tht hve true lief to tht effect bcked up by good evi evi dedence. Tht evidence includes bckground informtion bout nture nd function zoos. You know tht zoos genemlly generlly exhibit genuine specimens, nd tht it it would gret del trouble to to dgue mule nd to to substitute it it for zebm. zebr. Only under most unlikely nd bizrre bizm circumstnces, if if t t ll, would such substitution mde, nd re no reson whtsoever to to think tht ny such circumstnces obtin. f f did feel re ws chnce tht switch hd en mde, would hve reson to doubt lht tht niml see zebt. zebr. You would nol, not, n, n. know th tht it it zebr, contmry trmy to to wht ws wr ssumed. ssctmed. con Dretske's motivtions for denying tht know ren't seeing dgued mule re not fully cler. He himself grnts tht "hypos" tht niml relly mule "not very plusible", yet dds BUl But question here nol not wher whc~hcr lh th llcmlive llcmtive plusible, plusible. nol not whelher wher il it more or or less plusible thn thl cht lhere re re rel zebr~ zebrs in in pen, pen. bul but whelher wher /cnow know thl cht lh th llcmlive llcmlive rlse. rlsc.3 3 One might hve thought tht if if lief much more plusible thn its its denil, person would justified in in ccepting tht lief. And, And. n, brring Gettierlike complictions, complictions. tht person's lief, if if true, would knowledge. kn~wledge.~ 4 Perhps Dretske's point th: When look t t pen where niml,, hve evidence tht re zebm zebr re, re. nmely tht niml looks like zebm. zebr. Your vul evidence does not, though, give ny support to to r lief tht niml re seeing n't dgued mule. For, For. if if it it were dgued mule, mule. r vul experience pcrience would just s s it it.. As Dretske sys, sys. "The evidence hd for thinking hem m zebrs hs en effectively neutrlized, since it it does not count towrd ir not  exing mules cleverly dgued to to look like zebrs".s zebrs".5 The upshot tht do know re zebm, zebr. since hve true lief to to tht effect supported by evidence. You do not nor know tht niml n't dgued mule, since r lief in in th cse true but not supported by by vilble evidence. So, know first proposition, but don't know its its cler logicl consequence. indicted bove why think th nlys incorrect. Your bckground knowledge does give justifiction for denying tht niml mule, so so know tht it it n't one. Still, it it my pper tht possibility filure for Closure Principle res out sitution s s descrid it. it. t t seems tht usul dequte evidence for clim "t's zebr" (i.e. vul evidence) different from bckground evidence which supports "t's not cleverly dgued mule." f f so, so. could conceivbly in in position where hd vul evidence nd knew re ws bckzebr, but lcked bckground knowledge, nd hence didn't know re wsn't dgued mule. n such circumstnces, Closure Principle would fce counterexmple. To my mind, th pprl bsed on n overly tomtic conception evidence nd justifiction. Your lief tht niml t t zoo zebm zebr justified in in prt by r vul evidence, but it it lso supported by bckground informtion tht counts ginst niml's ing dgued mule. By itself, vul evidence wouldn't sufficient to give knowledge tht re zebr. To see th, consider cse where proper bckground knowledge lcking. mgine tht re driving through rnchlnd out West nd for some reson or or stop by rodside, rodside. Across wy see blck nd white striped equine creture tmnquilly trnquilly grzing gzing in in its pen. n n situ situ counterextion th sort, it it seems to to me, it it fr from cler tht could know niml  fore to zebm, zebr, even though it it looks just s much like zebr s s niml in in zoo does. The difference here tht hve no pplicble bckground informtion which mkes it it more likely tht zebmlike zebrlike niml relly zebr mr rr thn n oddly colored mule. So, even bck t t zoo, r justifiction tht wht see zebr depends on bckground bnckground informntion inform~~tion just s ns justifiction j~~stifiction for r denil tht it's dgued mule would so depend."l'here h There no dcrepncy dcrepnncy here which provides grounds for thinking tht Closure Principle flse. One might object tht defense closure just given mkes unrelticlly high demnds so fr s s evidence concemed. concerned. A ng child t t zoo, zoo. seeing n niml tht resembles n illustrtion in in picture book might point nd hppily sy "Zebm''', "Zebr!". Despite fct tht child knows nothing bout how zoos work, doesn't tht child know niml zebr? The sues here re complex, but re re vrious resons not to to tke th objection s s decive. First, even if if it it grnted tht child knows in in full sense tht niml zebr, zebr. if if he or or she n't cpble dmwing drwing inference bout dgued mules, child's cse doesn't r on vlidity Closure Principle. Moreover, it's uncler tht, under circumstnces, child relly ought to descrid s s knowing tht niml zebr. Suppose tht child cn't concejr concep infertully dtinguh tween 'looks like n n zebr' nd ' ' zebr'. Perhps child knows only tht niml it it sees looks like zebr, nd wouldn't know tht niml zebr without cquiring furr conceptul resources nd informtion.' inf~rmtion.~ CAR THEFT THET CASES hve mintined tht Dretske's Zebr Cse does not fumh furnh counterexmple to to Closure Principle. But wht hve sid so so fr fr rs lrgely on on prticulr detils cse s s Dretske sets it it up. H remrks point towrds formultion exmples which cnnot treted so so strightforwrdly. cll se "Cr Theft Cses", for resons which will come cler in in moment. t t my,. in in fct, tht Zebr Cse properly understood one se. Suppose own cr which prked few hours go on on side street smt in in mjor metropolitn re. You rememr clerly where left it. it. Do know where r cr? We re inclined to to sy tht do. Now it it true tht every dy hundreds crs re stolen in in mjor cities United Sttes. Do know tht r cr hs not en stolen? Mny people hve intuition tht would not know tht. f f th intuition combined with previous one, n it it seems tht closure principle
3 16 JONATHAN VOGEL VCGEL ARE THERE COUNTEREXAMPLES TO HE THE CLOSURE PRNCPLE? violted. Tht : You know proposition 'My cr now prked on (sy) Avenue A'. You lso know tht tht thnt proposition entils (q) (9) 'My cr hs not en stolen nd driven wy from where it ws prked'. Yet, it seems. seems, do not know q. 9, despite fct tht it for cler logicl consequence p. p, which do know. Since, in th instnce:, instnce, (pprently) fil to know cler logicl consequence proposition do know, know. Oosure Closure Principle (pprently) violted. Th exmple turns on rr unusul feture cler logicl consequence q. 9. Given r evidence. evidence, tht proposition much more probble thn not, nd it t lest s likely to true s p. To tht extent, it seems s though should s justified in lieving q s re in lieving p. Neverless, Neverless. even though r lief tht p, if true, me, my knowledge, r lief tht q, 9, if true, not. You do not know tht r cr hsn't en stolen by someone nd driven wy, despite high probbility tht r lief to tht effect true. n th respect. respect, r lief tht q resembles someone's lief tht ticket, which he holds, holds. will not win fir lonery. lottery. No mtter how high odds tht ticket will not win. it strikes us tht ticketholder doesn't know tht h ticket will w~ll not win. n fct, fct. nlogy tween subject's lief bout holding losing lottery ticket nd one's lief tht one's cr hs not en stolen goes even furr thn th nd quite illumint illumint ing. A numr fetures lottery sitution re especilly relevnt here. First, lthough winning lottery on pniculr prticulr ticket unlikely or improbble, it would not bnorml in some intuitive sense, for it it to tum turn out tht ticket one holds hppens to winner. Second, Second. even though weight evidence certinly ginst ny prticulr ticket's winning, re still some sttticl evidence in fvor proposition tht certin pniculr prticulr ticket will win. i. i. e. re some (smll) reson to think pr pr prticulr ticketholder will win.s 8 A third importnt considertion tht, with respect to its chnces winning lottery. ech ticket indtinguhble from every or one. So, ny reson hve for thinking tht r pniculr prticulr ticket will lose would n eqully good reson for lieving ny or ticket in lottery tht it. it, too. too, will lose. Under se circumstnces, circumstnces. it it would rbitrry to lieve some tickets (including r own) but not ors tht y will not win. So, So. if re constent rr thn rbitrry, nd do conclude on bs evidence vilble tht r ticket will not win, will conclude sme every or lottery ticket. Neverless, hold lief tht some ticket or or will win. On pin rbitrriness. rbitrriness, n. n, it it seems tht cn't justifibly hold both tht r ticket will lose nd tht some ticket will win. A fortiori, cn't know tht r ticket will lose nd tht some ticket will win. win? 9 Now, Now. in certin importnt wys, one's eptemic sitution with respect to lottery like one's eptemic sitution in Cr Theft Cseyl C~e.~" n effect. when prk r cr in n re with n pprecible rte uto ft, enter lottery in which crs nre re picked, essentilly t rndom, to stolen nd driven wy. Hving r cr slolen stolen unfortunte counterprt counterpn to winning lottery. And, just s one doesn'l doesn't lotknow tht one will not hve one's numr come up in lottery, it it seems one doesn't know tht one's numr won't come up. up, so to spek, for cr heft. ft. To more prticulr, lieving tht r cr won't stolen like lieving won't win lottery, in wys just cnvssed. () f prk r cr in n re with high rte cr ft, n re where it virtully certin tht some cr like rs will stolen, it would not bnorml for r cr to stolen. (2) n Cr Theft Cse, r knowledge tht re considerble mount uto ft gives some rel sttticl reson to think cr will stolen.1 11 (3) t would rbitrry rbitmy to lieve tht r cr, but nol not ll ors relevntly similr to it, won't stolen. n generl, if person fils to know proposition cuse considertions like se, will wilicll proposition not known louery lotteryproposition. The point th extended compron lottery nd Cr Theft Cse hs en to try to chrcterize fmily pprent counterexmples to Closure Principle. The essentil feture se exmples tht y re cses in which cler logicl consequence known proposition itself lottery proposition meeting criteri just dcussed. Wht mkes Zebr Cse, in my opinion, weker potentil counterexmple to Closure Principle thn Cr Theft Cse, just fct tht. ~e c1e~ cler log: log counterexicl consequence 7.ebr &br Cse hrder to see s lottery proposllon. proposition. First, First. t it would bnorml for dgued mule to in zoo uw, enclosure mrked "Zebrs". "Zebrns". Second, s Dretske descris exmple, exnrple, it n't pprent tht hve ny reson (sttticl or orwe) to think that tht re might dgued mule in zebr pen. These two weknesses re rc relted to third: it difficult to see presence dgued mule in zebr pen s outcome ny lotterylike process. Tht, it not s though know tht dgued mule hs en plced in some zebr pen in some zoo chosen t rndom. n tht cse, ny reson hd for thinking tht ni ni dml hppen to see n't dgued mule would pply in every or sitution. You would, n, hve to conclude tht no zoo_ zoo hd dgued mule running round  in contrdiction with wht know to cse, viz. re dgued mule in some zoo m somewhere. However, th kind lottery element n't present in Zebr Cse s Dretske descrid it. So, it uncler why, s Dretske mintins, do not know tht striped niml fore n't dgued mule.12 ll CAR THEFT CASES AND SKEPTCSM would like to tum turn now to implictions Cr Theft Cse. Tht cse supposed to COUnt count s counterexmple to Closure Principle. For, For. in Cr Theft Cse, seem to know proposition proposition bout where r cr,. but pprently fil to know nor proposition which cler logicl consequence first one. f will mintin low tht tking Cr Theft Cse in th fshion, fshion. s counterexmple to closure, not only, only. or st wy, to understnd it. But, suppose tht Cr Theft Cse does stnd s counterexmple to closure; does tht relly help us with problem skepticm? The thought ws tht C,lr Ci~r Theft Cse would show tht closure n't vlid vr~lid in generl. Then skeptic's relince on tht principle in course rgument from deception would illegitimte, nd nrgument rgument wouldn't go through. However, wht Cr Theft Cse relly shows bout Closure Principle, if if it it shows nything t ll, ll. tht tht principle invlid when cler logicl consequence involved lottery proposition with fetures mentioned bove. The Cr Theft Cse gives us no reson to think tht closure fils to hold for cler logicl consequences which don't stfy those criteri.
4 18 JONATHAN VOGEL ARE THERE mere COUNTEREXAMPLES TO THE CLOSURE PRNCPLE? 19 The question t th point wher cler logicl consequence in skeptic's rgument lottery proposition in specified sense. The cler logicl consequence skeptic invokes something like ' m not brin in vt thoroughly deceived by sinter neurophysiologts'. And th clerly not lottery proposition stfying three criteri hving to do with bnormlity, relince on sttticl evidence. evidence, nd nonr nonrbitrriness. Let me tke se out order. (1) f skeptic's logicl consequence were lottery proposition. proposition, would hve to n indtinguhble memr clss sub subjects which it known tht t lest one memr brin in vt (mking it rbitrry for me to lieve tht 'm not such brin). Th hrdly cse, since 1 don't know tht re rt re ny brins in vts nywhere. The lotterylike element which ws crucil to smcture structure Cr Theft Cse refore lcking here. (2) Moreover, since re no reson to think tht some brins re put into vts s mtter course, it might well bnorml. bnonnl, in n intuitive sense, for someone to turn tum out to brin in vt. (3) Finlly, given (), re no bs for ssigning rel, positive sttticl probbility to proposition tht someone brin in vt. The force se observtions tht sitution in which skeptic invokes closure cnnot esily ssimilted to situtions like Cr Theft Cse, in which re some reson to think closure fils. Hence, Cr Theft Cse s such gives little support to clim tht Closure Principle fils when skeptic ppels to it. Th mens tht Cr Theft Cse provides no convincing bs for rejecting Deceiver Argument. t my tht, if Crtesin skepticm sue, no more needs to sid bout Zebr Cse or Cr Theft Cse. will, however, pursue question wher Cr Theft Cse genuine counterexmple to Closure Principle. Aside from whtever intrinsic inmnsic interest tht question my hve, it worth seeing tht results strengn, rr thn weken, conclusion tht se exmples do not undercut skejr skepticm. V 1V THE NTERPRETATON OF NTUTONS NTUTTONS ABOlT ABOUT me THE PROBLEM CASES The Cr Theft Cse nd its nlogues provide counterexmples to Closure Principle if we tke our intuitions bout such cses t fcevlue. fctvlue. For, n, it seems tht in circumstnces descrid, person my know some proposition (e.g. 'My cr on Avenue A, where prked it') yet not know cler logicl consequence tht proposition (e.g. 'My cr hsn't en stolen nd driven wy from where it ws prked'). t's worth noting, though, tht some dditionl rections people hve suggest tht closure preserved in se situtions fter ll. Often, when fced with possibility tht ir crs might hve en stolen, people withdrw, t lest temporrily, ir initil clims to know where ir crs re. Such response just wht Closure Principle would require. Now, think it must dmitted tht intuitions we hve here re wek. t would difficult to find decive support for closure in tendency people hve to cllnge chnge ir minds in wy just mentioned. Still, fct tht Closure Principle seems to respected to extent tht it provides motivtion for nlyzing tht cse in wy tht doesn't presuppose filure closure. possi The problem fcing ny such nlys to ccommodte or dcredit intu intuitions tht produce impression closure filure in first plce. Those rt re intuitions which led us to sy, first, tht person, under certin circumstnces, would know some proposition, nd, second, tht person doesn't know cler logicl con consequence tht proposition. One wy trying to reconcile se intuitions with closure to rgue tht some kind shift tkes plce tween se responses. The clim would n tht, for no fixed set circumstnces, do we regrd subject s knowing proposition while filing to know one its cler logicl consequences. Certin psychologicl studies provide independent resons to lieve tht shift th kind tkes plce. These studies concern people's ttitudes towrds improbble events. They re relevnt to Cr Theft Cse cuse essentil role plyed in 1 i tht cse by unlikely possibility tht r cr hs en stolen. f closure does fil 1 here, it cuse possibility ft, though highly improbble, undercuts clim! tht know tht r cr hsn't en stolen, even while tht possibility somehow leves intct r knowing tht r cr t certin spot. n studies mentioned, it hs en found tht people my mt tret improbble events eir s likelier thn y relly rc re or s hving essentilly no chnce occurring. Moreover, se ssessments rc re unstble, nd subjects cn esily influenced to grnt possibility more weight thn orwe, if tht possibility mde slient to m.13 mp Such psychologicl considertions provide n explntion for our intuitions bout Cr Theft Cse. nitilly nd generlly, in evluting knowledge clims in tht cse, we tret chnce r cr's ing stolen s essentilly zero. You cn, n, n. s sure s need to tht r cr where left it; re rt fully justified in tht lief. Thus, we re rc likely to sy without hesittion tht in sitution descrid know where r cr. Lter, however, when we dwell on rte cr ft, ft. chnce r cr's hving en stolen lent more weight. Given (now) significnt possibility tht my wrong in lieving tht r cr hsn't en stolen, we re rc no longer prepred to sy tht know it hsn't en stolen. And, viewing sitution in th light, giving weight to chnce tht cr n't where left it, we my inclined to go on to sy tht don't know where cr fter ll. Tht, re seems to motivtion to deny r initil knowledge clim in set circum circum signiftnces where cnnot clim to know cler logicl consequence wht thought knew. n tht wy, wy. Closure Principle respected. n short, fct tht t one time we would sy tht know loction r cr, nd tht shortly refter we might sy tht don't know r cr hsn't en stolen, does not estblh invlidity Closure Principle. For, it my tht t no one time do we ffrrm ffirm tht know something yet fil to know one its cler logicl consequences. t doubtful, n, tht Cr Theft Cse, when properly understood, provides counterexmple to Closure Principle. hve suggested tht nomlous non>lous chrcter our intuitions bout Cr Theft Cse my due to some kind eptemiclly importnt shift rr thn to clo clo unsure filure. My conjecture hs en tht shift chnge in probbility ssignment, but or mechnms my t work insted. An lterntive explntion our intuitions tht we re somehow induced to shift our sense degree ssurnce knowledge requires. Thus, our estimtion chnce subject could wrong  ssigncuse cr ft would remin constnt, but we would chnge our minds s to wher
5 20 JONATHAN VOGEL knowledge constent with tht level eptemic rk. There re still or forms shift could tke. t might even tht thnt movement in Cr Thefttype situtions tween wholly dtinct notions knowledge embodying different sets necessry nd sufficient conditions. For my purposes, detils wht ctully occurs re reltively unimponnt. unimportnt. The min point wh to mke tht re re explntions or thn closure filure for our intuitions bout Cr Theft Cses}4 Cses.14 Or, to put it differently, strightforwrd ppel to those intuitions insufficient to estblh tht Closure Principle does not hold without restriction. V TlE THE PROBLEM OF SEMSKEPTCSM SEMSKEmlCSM hve just rgued tht simple inspection our intuitions bout Cr Theft Cse does not conclusively refute Closure Principle. The dvocte closure cn clim tht Closure Principle only ppers to fil, s result n eptemiclly imponnt importnt switch tht tkes plce in course our thinking bout exmple. However, clim th son leves open wht subject, in fct, does nd doesn't know in Cr Thefttype situtions. The Closure Principle fces strong objection to effect tht it incomptible with ny cceptble ccount wht known in Cr Theft Cses. f closure holds, nd some uniform stndrd knowledge pplies cross bord, bord. eir don't know where r cr, or do know tht it hsn't en stolen. The ltter clim seems hrd to sustin. Th impression strengned by similrity tween Cr Theft Cse nd rel lottery sitution. Knowing tht r cr hsn't en stolen would, in wys 've mentioned, like knowing someone will lose fir lottery. And tht seems like son thing one doesn't know. So, So. given untenbility sying tht know r cr hsn't en stolen, stolen. Closure Principle will require tht, contrry to wht we might hve thought, don't know where r cr. Th result seems unwelcome, nd things worsen quickly. t turns out tht, propositions bout externl world which we tke ourselves to know, gret mny entil lottery propositions s in Cr Theft Cse. (The propositions with se con con Princequences re, specificlly, propositions bout current stte world yond our immedite environments). To see rnge Cr Thefttype cses consider some orexmples: exmples: Bush Cse: Q. Do know who current President United Sttes? A. Yes, it's George Bush. Q. Do know tht Bush hsn't hd ftl hen ttck in lst five minutes? A. No. LuncheonetteCse: Luncheonettecse: Q. Do know where cn get good hmburger? i t ARE mere THERE COUNfEREXAMPLES COUNTEREXAMPLES TO THE CLOSURE PRNCPLE? 21 A. Yes, Yes. re's luncheonette severl blocks from here. Q. Do know tht thnt fire hsn't just broken out re? n? A. No. Meteorite Cse: Q. Do know wht stnds t mouth Sn Frncco By? A. Yes, By spnned by Golden Gte Bridge. Q. Do know tht Bridge wsn't just demolhed by flling meteorite? A. No. t's pprent tht vritions vuritions on se cses cn constructed for ny numr propositions bout people, things, or ctivities. Tht to sy, ll propositions bout such mtters, which we tke ourselves to know, entil lottery propositions which, it seems, we do not know. f closure holds, long with intuition tht we do not in fct know cler logicl consequences in question, question. result tht we hve gret del less knowledge world thn we hd supposed. n or words, Qosure Closure me Principle leds, even without rgument from deception, to firly strong nd unpltble semiskepticm. The cse ginst closure ppers tht much stronger. But does thret semiskepticm relly count ginst Closure Principle? The key ide here tht re supposed to some feture which lottery proposi proposi untions in Cr Theft Cses shre with propositions bout genuine lotteries, in virtue which we cn't correctly descrid s knowing those propositions. Wht tht feture? One nswer tht, cuse sttticl probbility tht r ticket my win in genuine lottery, re "rel" possibility error in lieving tht will lose. n or words, crucil lief in se circumstnces lcks kind certinty, nd hence cn't count s knowledge}s kno~ledge.~~ Similrly, lottery propositions which figure in Cr Theft Cses re sllch such tht "rel" possibility exts tht y re flse. Since, refore, refore. subject cn't cenin certin truth se lottery propositions, subject cn't know m. By Closure Principle, it would follow tht subject cn't hve knowledge propositions which he knows to entil those lottery propositions. Th would result, resull fes we hve seen, in pervsive semiskepticm. The imponnt thing to relize bout th wy viewing mtters tht it doesn't relly justify concluding tht Closure Principle invlid. For, ccording to objection, jection. lesson genuine lottery exmples tht lief cn't knowledge if re "rel", "rel". nd not merely logicl, possibility tht subject wrong bout it. f th correct, n semiskepticm follows without Closure Principle. After ll, ll. obre "rel" possibility tht, e. g. my wrong in lieving tht r cr t cenin certin spot; it possible tht r cr hs en stolen. The sme point pplies, mutt ltmutnd, to ny or Cr Theft Cse. So, perhps, re legitimte eptemo epteme mulogicl problem in thret semiskepticm derived from certinty requirement for knowledge. However, since rejecting closure won't void tht problem, tht problem doesn't provide reson for denying Closure Principle's vlidity. On nor wy nlyzing lottery exmples, exn~ples, unknowbility in se contexts propositions like 'My ticket will lose' due to rbitrriness ccepting
6 22 JONATAN JONATHAN VOOEL VOGEL ARE THERE COUNTEREXAMPlES COUNTEREXAMPLES TO THE CLOSURE PRNCPLE? 23 ny proposition tht fonn. form. By nlogy. nlogy, in in Cr Theft Cse. Cse, wouldn't know proposition 'My cr hs not en stolen'; re reson to to think tht some cr or crs similr to to rs will stolen, nd hve no nonrbitrry nonrbiry ground for lieving tht r cr in in prticulr won't one (or one ones) stolen. Once more, it it looks s though ll dl knowledge clims bout lottery propositions in in or Cr Theft cses would undercut by similr considertions. Then, semiskepticm will inevitble if if closure holds. Here gin, though, though. m inclined to to think tht re no rgument to to found ginst Closure Principle s s such. The nlys lottery effect now ing entertined mkes following ssumption: ll ll or things ing equl, it it unjustified to to enterccept ny memr set propositions L. L, such tht memrs L re equiprob ble nd subject knows (or hs good reson to to lieve) tht t t lest one memr L flse.16 t t turns out tht th principle sufficient to to estblh semiskepticm regrdless vlidity Closure Principle. To see why th might so, let's tke Cr Theft Cse s s bsic model. The present ttempt to to uch ttch burden semiskepticm to to Closure Principle mounts to to clim tht nonrbitrriness requirement just stted defets r clim to to know lottery proposition tht r cr hsn't en stolen  while it it leves intct r clim to to know proposition (i.e. 'My cr on Avenue A, A, where prked it') clerly entiling tht lottery proposition. But entiling proposition itself memr set equiprobble propositions which, which. hve good reson to to lieve, contins t t lest one flsehood. Tht set contins, long with 'My cr on on Avenue A, A. where prked it', propositions like 'My neighbor's cr semiskeptiwhere he he prked it', it'. 'The postmn's cr where he he prked it', nd so so on. You my not ble to to stte ll ll memrs set explicitly. explicitly, but still hve very good reson to to think tht re such set L. L. By nonrbitrriness requirement, it it would follow tht don't know Originl originl proposition 'My cr on on Avenue A, A, where prked it'p it'.17 The sme line thought would seem to to pply to to ny cse Cr Thefttype where knowledge lottery proposition blocked by by nonrbitrriness constrint. So, if if nonrbitrriness condition strong enough to to estblh ignornce cross bord for for lottery propositions, it it lso strong smng enough to to estblh ignornce propositions which. in in Cr Theft cses, cses. entil lottery propositions. Tht to to sy, if if nonrbitrriness condition plus closure genertes semiskepticm, so so too does nonrbitrriness condition lone. Therefore, opponent closure cnnot use tht condition s s bs for for n n rgument tht Closure Principle invlid cuse it it would led to to semiskepticm. The preceding dcussion mkes clerer wht would required in in order to to mke cse ginst closure work. The critic Closure Principle hs to to identify some wy in in which liefs in in lottery propositions re re eptemiclly defective. defective, nd th defect must not shred by by mundne liefs whose contents, in in Cr Theft cses, re known to to entil those lottery propositions. t t n't esy to to see wht such defect would,, if if not not ones just considered.18 S n n th section, 1 hve tried to to show tht our nomlous intuitions bout Cr Theft Cses nd relted thret semiskepticm relly hve little to to do do with closure. No NO ttempt hs en mde here to to give fully cceptble positive ccount wht relly known in se cses, nd suspect tht such n ccount my not vilble t ll. For it it my tht Cr Theft Cses toger with problem semiskepticm reflect deepseted, unresolved conflicts in in wy we think bout knowledge.19 V CAR THEFf THEFT CASES AND RELEVANT ALTERNATVES TVES t t tempting to to think tht omsion positive ccount wht we know could mde good by dopting version relevnt lterntives pproch to to knowledge.20 Th pproch promes ll ll dvntges, without defects, tretment just given. n n my view, view. tum turn to to relevnt lterntives pproch not dvble, but proposl interesting nd deserves considertion. According to to relevnt lternntives ltem~ives ort, ort. demnds demiincls for knowledge re restricted nd contextul. On one version ory, ory. S knows thut tht p just in in cse S possesses evidence which COunts counts ginst ll ll relevnt ltemtives ltenltives to to p; p; on on nor fonnu formu reltion, S knows tht p just in in cse S would right bout p over some clss relevnt lterntive situtions. A mjor problem for for relevnt lterntives pproch to to explicte crucil notion relevnce it it invokes. Relevnce lterntives will vry c c explicording to to subject's sitution; it it my lso (depending on on detils ory) detennined determined by by content subject's lief nd context ttribution thibution for for knowledge clim. f f stndrd relevnce oys certin constrints, relevnt lterntives ory my used to to explin intuitions bout Cr Theft Cses in in wy tht doesn't deny vlidity Closure Principle. How would th go? Suppose fcts re s s descrid in in Cr Theft Cse. nitilly, we we operte with stndrd relevnce ccording to to which possibility l Cr Theft too remote to to considered. At At th point, fct tht would wrong bout loction r cr, cr, hd hd it it en stolen.21 stolen>l doesn't impir clim tht know where r cr cr.. Moreover, since possibility cr cr ft remote, tht tht possibility doesn't undercut clim tht know r cr hsn't en stolen. Closure sure mintined. Wht produces impression to to contrry? When possibility cr cr ft explicitly red, somehow new, more generous stndrd relevnce Cloinstted, ccording to to which possibility cr cr ft relevnt. By By th th stndrd, stndrd. know neir where r cr nor tht it it hsn't hilsn't en stolen. Closure still preserved. s s fore.22 There re re severl drwbcks to to nlyzing Cr Cr Theft Cses in in th th fshion. First, supposed vinue virtue nlys tht tht it it provides n n CCount ccount wht would nd wouldn't know in in circumstnces given. But in in giving such n n ccount, relevnt lterntives ort must must sy sy tht, tht, in in some sense or or from some stndpoint, ~ou would know tht r cr hsn't en stolen. Th seems plinly wrong, nd tntuition intuition tht tht it it wrong just just wht wht mkes it it so so hrd hrd to to give give n n dequte tretment Cr Cr Theft Cse Cse nd nd its its nlogues. The The relevnt lterntives pproch relly doesn't ccommodte body body our our intuitions in in n n unforced, convincing wy, wy, contrry to to wht one might hve hoped. Let Let me me tum turn to to furr point. The The relevnt lterntives ort hypolhesizes hyposizes tht, tht. in in problem cses, re shift in in stndrd eptemic relevnce. n n Cr Theft Theft Cse Cse specificlly, possibility cr cr ft ft supposed to to,, lterntively, too too remote nd nd not not too too remote to to relevnt. t t nturl to to presume tht tht "remoteness" here here
7 JONArnANVOOEL JONATHAN VOGEL to understood in probbiltic terms. Thus. Thus, t t one time. time, chnce cr ft treted s smll enough to ignored; lter. in more scrupulous frme mind. mind, we find even tht iule little probbility pmbbility error sufficient to undercut knowledge. Relevnce, n, n. function n lterntive's probbility. Th probbiltic criterion relevnce seems ttrctive, but it it leds to trouble, trouble. especilly if if knowledge requires hving evidence tht excludes relevnt lterntives. Suppose know proposition k, k. Let 1 n n lterntive probble enough to to relevnt to k k, nd let m ny or lterntive to k which should count s irrelevnt. imlevnt. Consider, Consider. in ddition, djunction (v ( m). m), which logiclly incomptible with k. Th djunction t t lest s s probble s s its its djunct., so so it it probble enough to to relevnt to to dr knowing k. k. Now, Now. since (v v m) relevnt to to r knowing k, k, hve to to hve good evidence ginst it. it. Tht to to sy, hve to to hve good evidence for for negtion (v v m), m), nmely conjunction (ootl (not/ & notm). Why th problem? f f hve good evidence for (not (not4& notm), presumbly hve good evidence for notm /onep lone.* Thus, r ing in in th fvorble position with respect to to notm condition for r knowing k. k. So, m n't irrelevnt imlevnt negto to r knowing k, k, contrry to to wht we originlly supposed, supposed. nd re thret contrdiction.z4 24 n n fce th objection, relevnt lterntives ort my eschew probbiltic criterion relevnce s s such. Yet, it's it's hrd to to see wht lterntive, nd orwe stfctory, stndrd relevnce would yield desired conclusions bout Cr Theft Cses, nd vlue relevnt lterntives pproch in in deling with such cses seems questionble, questionble. es An importnt motivtion for pursuing tht pproch hope tht th would contribute, down line, to to solution problems red by by Crtesin skepticm, skepticm. Typiclly, relevnt lterntives ort tkes position tht we we cn hve knowledge externl world even though we we my victims mssive sensory deception. On On th view, possibility such deception leves our knowledge world iolct intct cuse, with respect to to such knowledge, possibility deception n n irrelevnt imlevnt lter lter ntive. Of Of course, it it won't help just to to declre skepticl lterntives irrelevnt imlevnt  tht evlution hs hs to to mde in in principled wy. Now, Now. suppose tht relevnt lterntives pproch relly did provide n n cceptble ccount Cr Theft Cses. Such success would men tht reltively pedestrin possibilities like cr cr ft re, in in some con con lterntexts t t lest, eptemiclly irrelevnt. imlevnt. All All more reson, n, to to hold tht outlndh possibilities red by by skeptics re re irrelevnt s s well. The envioned ntiskepticl strtegy to to try try to to ssimilte problem skepticm to to problem knowledge in in Cr Theft Cses. Such n n ttempt seems mguided, in in light considertions red bove. The The sues ring in in Cr Cr Theft Cses hve to to do do with knowledge on on bs sttticl evidence nd, nd, perhps, outrequirement nonrbitrriness in in forming justified liefs. As As hve rgued, se re re not not sues red by by Crtesin skepticm, nd nd re no no reson to to expect tht solution to to one set set problems will hve ny ring on on or set. To To more spe spe socific, let's imgine tht tht prepondernce sttticl evidence cn cn crete situtions in in which some lterntives re re irrelevnt. Th nol not sitution in in which we we confront skeptic (i.e., (i.e., it's it's not not s s though we we know, ntecedently, tht tht just just hndful sentient cretures in in universe re re mssively deceived). So, So, it it n't esy to to see see here ny ny bs ARE ARE THERE THERE COUNTEREXAMPLES TO TO THE THE CLOSURE PRNCPLE?.. 2S 25 for clim tht possibility red by by skeptic, for us us now, n n irrelevnt imlevnt lterntive. V CONCLUSONS hve rgued for numr points concerning Closure Principle. First, Dretske's Zebr Cse does docs not, on on my my view, provide genuine counterexmple to to Closure PrinCiple. Principle. t t seems more plusible tht re violtion closure in in exmples like Cr Theft Cse. However, However. even if if thc Closure Clowre Principle does fil in in cses tht sort, son, lterre,. mintin, mintin. no no reson to to lieve licve tht such 1 n fuilure fitilure crries over to to contexts where skeptic my my ppel to to closure. Finlly, in in my my view, serious questions my my red s s to to wher Cr Cr Theft Cses relly do do demonstrte ny ny filure Closure PrinCiple Principle t t ll. l1.25 Amherst College NOTES 1 Th Th fonnultion formultion Slnds slnds in in need need furlher furher refinemenls. refincmcnls. For, For. suppose suppose someone someone knows knows both both pnd pnd q); q); if if tht tht person person doesn't doesn't put put se hese things hings toger, togethcr, he he or or s she might might fil fil to to infer, infer, nd nd hence hence not not mow, know, q. q. Th Th kind kind compliction compliction doesn't doesn't ffect fit wbt wht wnt wnt to to sy sy low, bclow, so so will will dregrd dregrd it. it. Where When logicl logicl consequence consequence properly properly recognized mmgnized s s such, such. will will cll cll it it "cler" "cler" logicl logicl consequence. consquence. 2 Fred Fred DreLSke, Drerske. [31, [31. p. p DreLSke Dreuke lso lso employs employs hc exmple cxmplc in in h h more more recent rccenl 15J, S]. p. p Dretske, Dnuke. [31, [31. p. p The The problem problem cn't cn't hc tht h1 ren't ren't cenin ccrlin tht tht wht wht see see n't n't mule. mule. For, For. ny ny chnce chnce or or possibility possibility tht tht he niml niml mule mule chnce chnce tht ht it's it's not not zebr. zcbr. f f th h chnce chnce mkes mkcs uncenin uncenin '!'s 't's no not mule' mule' it it should should mke mke eqully uncenin uncemin 't's 'ts zebr'. zebr'. 55 Drctske. Drcuke. [3J p. p Someone Sornmne might might minlin minlnin tht ht yoll don't don't need nccd th h son son hckground bckground in in fonntion formtion t t he zoo; zoo; such such informtion informtion required required out out West Wcst only only cuse cuse re re hve hve infonntion informtion which which conflicls conflicls with wih clim clim tht tht niml miml zebr zebr (viz. (viz. zebrs xbrs ren't ren't generlly gencrlly found found on on Western Western rnchlnd). rnchl;md). My My first first response response would would tht tht t~. zoo zoo nd nd rnchlnd rnchtnd silutions situtions re re slill slill n(ogolls. mlogous. f f hppen hppen to to t, t. sy, sy, he Brool Bmnx Zoo, Zoo. hve hve evdence evidence tht ht conflicts conllicu with with clim clim tht ht lhe he niml niml in in pen pen zebr, zebr, nmely. nmely, lhe he infonntion informtion tht ht zebrs z c bm ~'t.ntive ren't ntive to to New New York York City. City. n n ny ny cse, cse. exmple exmple collld could furr furr modified. modified. Suppose Suppose re re in sitution where men to identify n niml by sight. but hvc no informtion t ll bout n stluuon where men to identify n niml by sight, but bve no informtion t ll bout ~helher whcr such such onim~ls mimls rc re found found in in r r loction, loction, nor nor bout bout presence prcscncc or or bsence bscnce similr similr looking looking but but dfferent diffuenl erelures cretures m in re. rc. Under Under those those circumslllnccs, circumstnces, think, think, couldn't couldn't know know tht ht lhe. niml niml. SOrt son would would tke tke it it 10 to. bc. m m indebted indebled here herc to to ROrt Robcrt Audi. Audi. 7 ' For For dcussion dcussion se hcse sues, sucs. see sce Rort Rohcrt Sllllnker, Sulnkcr. [101. [lo], especilly especilly pp. pp When When sy sy th~ ht re thcre sltticl sttinicl reson reson or or sttlicl sttticl evidence cvidcncc in in fvor fvor proposilion, proposition. men men roughly roughly lhe he follow following. mg. Let Let us us sy sy tht h1 sllticl slt6cl probbility probbility n n A's A's ing king B B one one th tht ssigned ssigned on on he bs bs reltive rcltive frequencies, frequencies. counting cses, cscs, nd nd so so forth. forth. On On bs bs such such sttticl probbilities. probbilities, sttticl sttticl probbility probbility my my ssigned ssigned by by direct direct inference inference to to he proposition pmposition Th Th A A s', B: f f th h sttticl sttticl probbility. probbility, in in tum, turn. not not zero, zcro. we wc bve, hve. or or hings things ing ing equl, equl. some some resonperhps resonperhps very wry smllto smllto th.nk hink tht tht A A in in question question S, B. m m clling clling such such reson reson sttticl sttticl reson. reson. (My (My usge usge re hen: follows follows John John Pollock,0J, Pollock p. p ), ).
8 26 JONATHAN VOGEL ARE mere THERE COUNTEREXAMPLES TO THE CLOSURE PRNCPLE? 27 9 Th nlys will seem mbulded mguided to those who douht doubt tht justified cceptnce closed under conjunction. However. it might sli still tht extence he relevntly similr tickets. ticke. one which known 10 lo win. somehow undercuts underculs justifiction (nd knowledge) knowledgc) regrdless rcgrdlcss how things stnd with conjunction. For such view. see Lurence Bonjour.. [). The role mle!he nonrbitrriness constrint consmint in situtions like th lso clouded by!he fct tht someone somwne my fil 10 to know tht h or her ticket lickel will lose in 10Ueries lollcries in which!he winning chnces tickets tickels re uneven. 1 hope 10 to pursue se sues in furlher funhcr pper; pper: for now. now, il it would sufficient sumcient for my purposes if nothing yond stlticl sltticl probbilily probbility nd bnormlity enters into!he proper chrcteriztion se exmples. My conclusions low should remin rcmin unffected by dropping ny ssumptions bout!he significnce nonrbilmriness nonrbimriness in se thcse conte~ls. conxu. conjunc 10 The connection tween lotlerylike situtions nd situtions where whcrc closure (pprently) fils hs lso en noriced noticed by Jeffrey Olen Olcn in 181. p m indebled to Dvid Shtz Shcz for th reference. 1 Compre th lh.o;el sc! circumslllnces circumslnces with hose thox crimefree crimefrcc smll Own. town. n locle loclc where whcrc crs re never slolen. stolen. would hve no reson rcqon t ll to think tht r cr in prticulr hs en cn stolen. nd cn know tht it's where left il. it. Notice. Notice, 100, too. tht lht in such circumstnees circumstnces r cr's ing llken lkcn would bnorml. 12 nterestingly lntereslingly enough, enough. Zebr Cse cn mde more mom convincing by filling il it out so th ht louery elemenl ment introduced. invoduced. The exmple could developed in th wy:. elo Q. Do know wht niml in pen? A. Sure, Surc. i's it's 7.ebr. wbr. Q. Do know for fct th lht memrs mcmrs some college frlernily didn't stel zebr lst night s prnk, prnk. leving hind dgued mule? The reson one might hesilllle hesirle 10 lo clim 10 to know tht such prnk wsn't crried crricd oul out my tht re some reson to think tht lht successful, temporrily undelected undetected college pr.ldks prnks re brought f from time 10 to time. Then, Thcn, in tum. turn, my not entitled entitlcd to sy ht tht know tht re thcrc n't cleverly dgued mule fore. So, it my tht, tht. properly propcrly understood or properly propcrly filled out, oul. Dretske's Drettc's Zebr Cse should lken s memr fmily cses for which Cr Theft Thcft Cse ws wns thc prdigm. 13 These findings re summrized nd dcussed by Dniel Dnicl Khnemn Khncmn nd Amos Tversky,161. Tvcrsky. [ l4 Which nol not 10 to sy. sy, course, coursc, tht lht Rltemmive dternnlive e'plntions, enplnlions. involving closure closurc filure, filurc. cn't lso olso hc deved. lm indebted here to Richrd Feldmn. Fcldmn. dc 15 l5 By "rel" possibilily, possibility. men just one for which hcre thcre positive, positive. even if smll, sltticl probbility; th richer nolion notion thn plin logicl possibility. The ssociled ssocid nolion notion cerlinty ceminty bsence ny rel possibility error. Th notion cerlinly certinty weker thn conception cerlinty ceminty ccording 10 to probbilwhich one must hve evidence ht tht cnlils enlils hc truth truh lief for tht lief to cerlllin. cerlin. t queslionble questionble wher he stronge! svongw sljlndrd stndrd cerlinty represents represenu condilion condition for knowledge, since il it ipso fcto Bclo rules mlcs OU out possibility knowledge knowlcdgc by induction. should mke mkc il it cler clcr here, hcre. though, though. lhl tht don'l don't inlend intend se thcse glosses 10 to serve s substntive ccount rel rcl pos.,ibility possibility or cerlinty. ccninty. 16 The Sllltement sltemen1 h th principle rough, rough. since it it doesn't rule out tht memrs mcmrs L L could entirely unrelted unrelled in content. Some stipultion nceded needed 10 to ensure tht L suilbly nturl or pproprile; pproprite; th lh problem,. course, closely relled relted 10 lo tht choosing n pproprite reference rcfercnce clss for direct inference infcrcnce boul bout probbilities. 17 l7 A similr poinl point mde by Bonjour, Bonjour. [], []. p.73n. 18 S Jeffrey Jcffrey Olen suggests th tht know mundne proposition =use bccuse thcre "nomic connection" conncction" tween stte slte ffirs picked out by propositions which rc r evidence cvidcncc nd Slte slte ffirs lieve 10 to obtin; oblin; in cse r helief helicf in lhe cler logicl consequence, consequence. however, howcvcr, thc connection merely probbil(ic prohbillic nd nol not nomic, nd don'l don't know. NOlice, Noricc, though, tht in!he Cr Theft Thcft Cse, Cse. it it nomologiclly nornologiclly possible for to hve evjdcn~'c evidence hve hvc nd yet ycl wrong in r lief licf bout both boh lhe initil proposition nd he cler clcr logicl consequcnce. conscqucnce. So, So. it it uticsl t lst obscure obscurc exclly cxcdy how Olen Olcn mens mcns to drw crucil dlinction. dtinction. See SCC Olen, Olcn (81. Anor Anothcr explntion closure filure filurc tht would fit Cr Theft Co;cs Cw ht tht "trck", "vck", in!he sense scnsc dcussed dcusxd by Nozick. Nozick, he truth inilil initil proposition proposilion but nol not tht lht cler logicl consequence, wnsequencc. Nozick's ccount presented prcscntcd in h Philrophic} Philosophicl Explntions (Cmbridge: Hrvrd University Press. Prcss. 1981); however howcvcr dcussion d~scussion Nozick's work lies lics oulside!he ouuidc hc scope =Ope th essy, essy For more mom dcussion th possibility. possibili~y. see scc my docloml dtrtoml dserttion dscrllion "Crtc"in "Crtesin SkeptiCm Skepticm nd Eptcmic Eptemic Principles" (Yle University. Univcrsity. 1986). Chpler Chplcr mportnt mporlnt erly sllltements sllcmcnu!he relevnt rclevnt lterntives lmntivcs ory thcory rc found in Fred Frcd Dretske. Drctskc.! nd 141, 141. nd in Alvin Goldmn lsi. [S. 21 Or, llerntively: lrntively:!he fct fcl th tht r evidence cvidcncc doosn't dncsn't e,clude cxclutlc possihility possibility cr ft. 22 A sophticted sophticled version th line linc thoughl hought hs en kcn developed dcvclopcd by Stewrt Stcwrt Cohen Cohcn in [ The relevrullernlives rclcvntollcrntivcs orl thcort cn'l cn't blk hlk t th point, pint, since we're wc'rc ssuming sswninp tht he or she endorses endoms hc Closure Closurc Principle. 24 A relted relled rgument rgumcnt my given 10 to show th tht pmb"bilislic pmb;rbiltic crilcrion criterion 01' relcvnce rclcvncc lln8cceplble uncceptble when relevnl rclevnt lterntives ltcrntivcs ory thcory couched in terms lcnns relibility rclibilily over rnge ~'OunterfclUl counterfctul silwltions. lions. sin 25 'm grteful gmful 10 to mny people pcoplc for help in hinking thinking bout boul sues sucs red rcd herc: hcrc: Rort Audi, Audi. Pilip Phillip Bricker, Bricker. Anthony Brueckner. Brueckncr. Fred Frcd DrelSke, Drclskc. Richrd Feldmn, Fcldmn. John Mrtin Fcher, Fchcr. Hrry Hny Frnkfurt. Frnkfun. nd Dvid Shl1., Shu. Recenlly, Rcccndy. hve hvc ne ncfitcd filed grelly gretly from fmm converstions convcrslions with wilh Stewrt Stcwnn Cohen. Cohcn. BBLOGRAPHY ( [] Bonjour. Lurence. Lurcnce. "The Externlt Exrnll Conception Knowledgc" Knowlcdpc" in P. French. Frcnch, T. Uehling, Uchling. nd H. Wettslein, Wctuin. e(/s, &. Midwcst Midwest Studies in Philosophy. Philosophy, Vol. V: Studie.s Sludic.~ in Eptemology Eptcmoogy (Minnepol: University Univcrsity Minnesot Minncsol Press, Prcss. 1980). 12] 121 Cohen, Cohcn. Stewrt Stcwrl. "How To Be Bc A Fllibilt", Fllihilt", in J. Tomrlin, Totnkrlin. ed. cd. P/Jilosopllicl Pl1il(1~ophicl ftrspcctivcs. Pcrspcclivcs. Vol, VOL : k Eptcrnology Eplcmology (Atscdero: (Alscdcro: Ridgeview Ridgcvicw Puhlhing Publhing Comp(my, Compny. (988). 1988). 131 DreL,ke, Drc~skc. Fred. Frcd. "Eplemic "Eptemic Opcmtors", Opcrntors". 171e ntc )ollnwl Jolmrel Pllilsop/>' N~ilosoplry 69 (1970), (1970). p, p OS ] 141 Dretske. Drcukc. Fred. Frcd. "The Prgmulic Prugmulic Dimens;o" Di~nct~siot~ or Knowledge", Kt~owlulgc". P/ilosopllicl Phil~~(Jp/i~d Studic,f Sludics 40 (1981). p, p [Si [5] DrClskc, Drctskc. Fred. Frcd. Knowledge Knowbdge ndhc tile Flow lnfonntion(cmhridge: lnfonnlron (Cmbridgc: Brdford Books. 1981). [ Goldmn, Goldmn. Alvin. "Dcriminlion "Dcrimintion ond Pcrceptul Pcrccptul Knowledge", Knowlcdgc". 17lc nrc Journl PhilosCfJhy73 Philosophy (1976), (1976). p (71 [71 Khnemn. Khncmn. Dniel Dnicl nd Tversky, Tvcrsky. Amos. "Choices, "Choiccs, Vlues, Vlues. nd Frmes", Frmcs". Americn A~ncricr~ Psychologt39 Rychologt (1984), (1984). p [ [Rl Nozick, Rort. Ron. Philosophicl Philosophic81 Explntions (Cmbridge: (Cmhridgc: Hrvrd Hrviud University Univcrsity Press), Prcss) [9] [91 Olen, Olcn. Jeffrey. Jcffrey. "Knowledge, "Knowlcdgc. Probbilily, Probbility. nd nnd Nomic Connections". Conncc~~ons". 17lc 77s SOllthcm Soirlhcrn Journl Philosophy Philsophy S 5 (1977), (1977). p [10] 1101 Pollock, John. "Eptemology nd Probbility", Pmbbilily", SYllthcse Syrrrhcsc S5 55 (1983), (1983). p [ 1 Sllnkcr, Sdnkcr. Rort. Ron. Jnqui)'(Cmbridge: nquiry (Cmbridgc: MT Press. Prcss. 1984) 19x4)
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationSmall Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic sechnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
More informationCcrcs Cognitive  Counselling Research & Conference Services (eissn: 23012358)
Ccrcs Cognitive  Counselling Reserch & Conference Services (eissn: 23012358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationSTA 2023 Test #3 Practice Multiple Choice
STA 223 Test #3 Prctice Multiple Choice 1. A newspper conducted sttewide survey concerning the 1998 rce for stte sentor. The newspper took rndom smple (ssume it is n SRS) of 12 registered voters nd found
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More information1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if
Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should
More informationNetwork Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3010323 36 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationAccording to Webster s, the
dt modeling Universl Dt Models nd P tterns By Len Silversn According Webster s, term universl cn be defined s generlly pplicble s well s pplying whole. There re some very common ptterns tht cn be generlly
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationSmall Business Networking
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationProtocol Analysis. 17654/17764 Analysis of Software Artifacts Kevin Bierhoff
Protocol Anlysis 17654/17764 Anlysis of Softwre Artifcts Kevin Bierhoff TkeAwys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention
More informationJaERM SoftwareasaSolution Package
JERM SoftwresSolution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationbaby on the way, quit today
for mumstobe bby on the wy, quit tody WHAT YOU NEED TO KNOW bout smoking nd pregnncy uitting smoking is the best thing you cn do for your bby We know tht it cn be difficult to quit smoking. But we lso
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationUnleashing the Power of Cloud
Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
More informationAssessing authentically in the Graduate Diploma of Education
Assessing uthenticlly in the Grdute Diplom of Eduction Dr Mree DinnThompson Dr Ruth Hickey Dr Michelle Lsen WIL Seminr JCU Nov 12 2009 Key ides plnning process tht embeds uthentic ssessment, workintegrted
More informationEuropean Convention on Civil Liability for Damage caused by Motor Vehicles
Europen Trety Series  No. 79 Europen Convention on Civil Liility for Dmge cused y Motor Vehicles Strsourg, 14.V.1973 Premle The memer Sttes of the Council of Europe, signtories of this Convention, Considering
More informationSmall Businesses Decisions to Offer Health Insurance to Employees
Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employersponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults
More informationMechanics Cycle 1 Chapter 5. Chapter 5
Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies
More informationChapter 8  Practice Problems 1
Chpter 8  Prctice Problems 1 MULTIPLE CHOICE. Choose the one lterntive tht best completes the sttement or nswers the question. A hypothesis test is to be performed. Determine the null nd lterntive hypotheses.
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationA Composite of Peculiar Patience
Dy 1 A Divine Hope Dy 2 Be Still, Be Filled Dy 3 Lest You Be Judged Dy 4 Hve Mercy Dy 5 A Clen Slte W eek 6 A Composite of Peculir Ptience Hve you ever known serene person who never pssed judgment on others?
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationNOTES. Cohasset Associates, Inc. 2015 Managing Electronic Records Conference 8.1
Cohsset Assocites, Inc. Expnding Your Skill Set: How to Apply the Right Serch Methods to Your Big Dt Problems Juli L. Brickell H5 Generl Counsel MER Conference My 18, 2015 H5 POWERING YOUR DISCOVERY GLOBALLY
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationPay over time with low monthly payments. Types of Promotional Options that may be available: *, ** See Page 10 for details
With CreCredit... Strt cre immeditely Py over time with low monthly pyments For yourself nd your fmily Types of Promotionl Options tht my be vilble: Not ll enrolled helthcre prctices offer ll specil finncing
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationChromebook Parent/Student Information
Chromebook Prent/Student Informtion 1 Receiving Your Chromebook Student Distribution Students will receive their Chromebooks nd cses during school. Students nd prents must sign the School City of Hmmond
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationObject Semantics. 6.170 Lecture 2
Object Semntics 6.170 Lecture 2 The objectives of this lecture re to: to help you become fmilir with the bsic runtime mechnism common to ll objectoriented lnguges (but with prticulr focus on Jv): vribles,
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics  Everything
More informationRecognition Scheme Forensic Science Content Within Educational Programmes
Recognition Scheme Forensic Science Content Within Eductionl Progrmmes one Introduction The Chrtered Society of Forensic Sciences (CSoFS) hs been ccrediting the forensic content of full degree courses
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationWell say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid?
Chpter 9 Buffers Problems 2, 5, 7, 8, 9, 12, 15, 17,19 A Buffer is solution tht resists chnges in ph when cids or bses re dded or when the solution is diluted. Buffers re importnt in Biochemistry becuse
More informationVendor Rating for Service Desk Selection
Vendor Presented By DATE Using the scores of 0, 1, 2, or 3, plese rte the vendor's presenttion on how well they demonstrted the functionl requirements in the res below. Also consider how efficient nd functionl
More information2. Transaction Cost Economics
3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More information