%&5&3.*/*4. Barry Loewer
%FUFSNJOJTNJTBDPOUJOHFOUNFUBQIZTJDBMDMBJNBCPVUUIFGVOEBNFOUBMOBUVSBMMBXT UIBUIPMEJOUIFVOJWFSTF*UTBZT
The natural laws and the way things are at time t determine the way things will be at later times.
5IF NBUIFNBUJDJBO 1JFSSF4JNPO -BQMBDF FYQSFTTFE IJT CFMJFG UIBU EFUFS- minism is true this way:
8F PVHIU UP SFHBSE UIF QSFTFOU TUBUF PG UIF VOJWFSTF BT UIF FGGFDU PG JUT antecedent state and as the cause of the state that is to follow. An intel- MJHFODF LOPXJOH BMM UIF GPSDFT BDUJOH JO OBUVSF BU B HJWFO JOTUBOU BT XFMM as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were suf!ciently QPXFSGVMUPTVCKFDUBMMEBUBUPBOBMZTJT UPJUOPUIJOHXPVMECFVODFSUBJO UIF future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such intelligence.
5IF QIZTJDT PG -BQMBDFT EBZ UIF àSTU EFDBEFT PG UIF OJOFUFFOUI DFOUVSZ XBT /FXUPOJBO DMBTTJDBM NFDIBOJDT*TBBD/FXUPOGPSNVMBUFEQSJODJQMFTUIBUIFUIPVHIU FYQSFTTUIFMBXTEFTDSJCJOHIPXGPSDFTEFUFSNJOFUIFNPUJPOTPGCPEJFT F ma) and IPXUIFQPTJUJPOTPGCPEJFTBOEPUIFSGBDUPSTEFUFSNJOFHSBWJUBUJPOBMBOEPUIFSLJOET PGGPSDFT6TJOHUIFTFQSJODJQMFT /FXUPOBOEQIZTJDJTUTGPMMPXJOHIJNXFSFBCMFUP QSFEJDUBOEFYQMBJOUIFNPUJPOTPGDFMFTUJBMBOEUFSSFTUSJBMCPEJFT'PSFYBNQMF UIFTF laws account for the orbits of the planets, the trajectories of cannon balls, and the QFSJPET PG QFOEVMVNT-JLF/FXUPO -BQMBDFEJE OPULOPX BMM UIFGPSDFTUIFSF BSF but he envisioned that, once those forces (and the corresponding force laws) were LOPXO /FXUPOJBO QIZTJDT XPVME CF B complete physical theory. That is, its laws would account for the motions of all material particles. And since he thought that FWFSZUIJOH UIBU FYJTUT JO TQBDF JT DPNQPTFE PG WBSJPVT LJOET PG WFSZ TNBMM NBUFSJBM #"33:-0&8&3
QBSUJDMFT PSBUPNT IFUIPVHIUUIBU/FXUPOJBONFDIBOJDT PODFBMMUIFGPSDFTXFSF LOPXO XPVMECFXIBUUPEBZXFXPVMEDBMMthe theory of everything. *UTFFNFEDMFBS UP IJN UIBU UIF DPNQMFUFE /FXUPOJBO UIFPSZ XPVME CF EFUFSNJOJTUJD BOE UIBU JU would thus be in principle possible accurately to predict the future (and retrodict UIFQBTU GSPNDPNQMFUFLOPXMFEHFPGUIFQSFTFOU*UTIPVMECFOPUFE IPXFWFS UIBU UIFSFBSFTVCUMFUJFTDPODFSOJOHXIFUIFS/FXUPOJBONFDIBOJDTJTEFUFSNJOJTUJDJOUIF XBZ-BQMBDFJNBHJOFEJUUPCF*UIBTCFFOTIPXOUIBUUIFSFBSFJOJUJBMDPOEJUJPOT compatible with the laws for which the laws do not determine all future positions. )PXFWFS UIPTFDPOEJUJPOTBSFVOVTVBMBOEJUJTQMBVTJCMFUIBUUIFZDBOCFSVMFEPVU as obtaining in our world. .BOZQFPQMFàOEUIFJEFBPGEFUFSNJOJTNBCIPSSFOUBOEJODSFEJCMF*UJTGFMUUP CF BCIPSSFOU CZ UIPTF XIP UIJOL UIBU EFUFSNJOJTN JT JODPNQBUJCMF XJUI GSFF XJMM BOE IVNBO EJHOJUZ *U NBZ TFFN UIBU JG EFUFSNJOJTN PCUBJOT UIFO QFPQMF BSF MJLF NBSJPOFUUFTXIPTFNPWFNFOUTBSFVOEFSUIFDPOUSPMPGJNQFSTPOBMMBXTPGOBUVSF*U BMTPTUSJLFTNBOZBTJODSFEJCMFCFDBVTFJUTFFNTUIBUTPNVDIPGXIBUIBQQFOTmOPU KVTUEFMJCFSBUFIVNBOBDUJPO CVUBMTPUIFXFBUIFS UIFTUPDLNBSLFU GBMMJOHJOMPWF BOETPPOmJTJSSFNFEJBCMZVOQSFEJDUBCMFBOETP UIFZUIJOL DPOTUJUVUFTQSPPGUIBU determinism is false. 0OUIFPUIFSIBOE TPNFQFPQMFàOEEFUFSNJOJTNUPCFBOBUUSBDUJWFBOEFWFO JOTQJSJOHNFUBQIZTJDBMWJFX*UTFFNTUPJNQMZUIBUFWFSZFWFOU FYDFQUQFSIBQTUIF !rst event, if there is one) has a scienti!c explanation. And while it is granted that we cannot predict much of the future it might be argued that the reason is not that deter- NJOJTNJTGBMTF CVU BT-BQMBDFTVHHFTUT UIBUPVSJOUFMMFDUJTUPPGFFCMFUPBDRVJSFUIF SFMFWBOUJOGPSNBUJPOBOENBLFUIFSFRVJSFEDBMDVMBUJPOT 8IBUFWFS WJTDFSBM SFBDUJPO POF IBT UP EFUFSNJOJTN JU JT XJEFMZ CFMJFWFE UIBU EFCBUFTDPODFSOJOHJUCFMPOHUPBQSFWJPVTFSBTJODFJUJTOPXLOPXOUIBU/FXUPOJBO NFDIBOJDTJTGBMTFBOEUIFUIFPSJFTUIBUSFQMBDFJUmJOQBSUJDVMBSRVBOUVNNFDIBOJDT m BSFOPUEFUFSNJOJTUJD#VU BTXFXJMM TFF UIFTJUVBUJPOJT NPSFDPNQMJDBUFEBOE interesting.
Clarifying determinism *OUIFGPSNVMBUJPOPGEFUFSNJOJTN iEFUFSNJOFuNFBOTiMPHJDBMMZOFDFTTJUBUFTu5IF /FXUPOJBO MBXT BSF NPEVMP UIF SFNBSL BCPVU VOVTVBM JOJUJBM DPOEJUJPOT BCPWF two-way deterministic because they and the state at t logically necessitates both the future and the past of t4PNFQIJMPTPQIFSTIBWFTPNFUIJOHTUSPOHFSJONJOECZ iEFUFSNJOFTu5IFJSJEFBJTUIBUUIFQSFTFOU BOEUIFMBXT EPOPUKVTUMPHJDBMMZJNQMZ the future but that they bring about GVUVSFTUBUFT0OUIJTVOEFSTUBOEJOH BUFNQPSBM EJSFDUJPOJTCVJMUJOUPUIFDIBSBDUFSJ[BUJPOPGEFUFSNJOJTNTJODFXFUIJOLPGUIFQBTU BT CSJOHJOH BCPVU UIF GVUVSF CVU OPU UIF PUIFS XBZ BSPVOE * XJMM TBZ NPSF BCPVU iCSJOHJOHBCPVUuXIFOEJTDVTTJOHlaws. The state at tJTFYQMBJOFEJOUFSNTPGUIFTQBDFmUJNFBOEUIFGVOEBNFOUBMPOUPMPHZ BOENBHOJUVEFT5IFFYJTUFODFPGUIFTUBUFBUtQSFTVQQPTFTBWJFXBCPVUTQBDFmUJNF and fundamental ontology on which there is a complete temporal ordering of all
%&5&3.*/*4.
FWFOUTBOEUIFGVOEBNFOUBMNBHOJUVEFTBSFFYFNQMJàFEJOTUBOUBOFPVTMZ5IFWBMVFT of all these quantities specify the state at t*O/FXUPOJBONFDIBOJDTUIFTUBUFBUt is TQFDJàFEJOUFSNTPGUIFQPTJUJPOT NPNFOUVNBOEJOUSJOTJDRVBOUJUJFT MJLFNBTTBOE charge, of each particle at time t*OàFMEUIFPSJFTUIFTUBUFBUt is speci!ed in terms of the !eld values (which can be vectors) at all spatial points at time t. There are GVOEBNFOUBMUIFPSJFTUIBUQPTJUTQBDFmUJNFTBOEPOUPMPHJFTUIBUEPOPUTIBSFUIPTF QSFTVQQPTJUJPOT'PSFYBNQMF JOUIFTQBDFmUJNFTPG&JOTUFJOTUIFPSZPGTQFDJBMBOE HFOFSBMSFMBUJWJUZUIFSFBSFFWFOUTUIBUBSFOPUUFNQPSBMMZDPNQBSBCMF/FWFSUIFMFTT WFSTJPOTPGEFUFSNJOJTNDBOCFGPSNVMBUFEGPSNBOZPGUIPTFTQBDFmUJNFTCZàOEJOH something that plays the role of the state at a time such that it and the laws determine UIFFWFOUTUISPVHIPVUBMMPGUIFTQBDFmUJNF &BSNBO The most controversial and philosophically signi!cant concept in the charac- terization of determinism is that of law of nature. The idea that there are laws of nature and that it is the job of the sciences to discover them developed during the seventeenth and eighteenth centuries with the rise of classical mechanics. An overly TJNQMFTVHHFTUJPOUIBUNBZIBWFBHSBJOPGUSVUIJTUIBUMBXTBTUIFCBTJTPGFYQMBOBUJPO DBNFUPCFTFFOBTBOJOUFSNFEJBSZCFUXFFO(PETXJMMBOEIJTDSFBUJPOPSFWFOBT BSFQMBDFNFOUGPSUIFPMPHJDBMFYQMBOBUJPO*UCFDBNFBDFOUSBMUFOFUPGQIZTJDT BOE NBOZPGUIFPUIFSTDJFODFT UIBULOPXMFEHFPGUIFMBXTPGOBUVSFJTUIFLFZUPTDJFO- UJàDFYQMBOBUJPOBOESFMJBCMFQSFEJDUJPO/PUFWFSZUSVFHFOFSBMJ[BUJPO FRVBUJPOPS GVODUJPOUIBUNBQTFBDITUBUFPOUPJUTGVUVSF JTPSJTBTTPDJBUFEXJUIBMBX*GJUXFSF UIFOEFUFSNJOJTNXPVMECFUSJWJBM4PUIFRVFTUJPOJT 8IBUNBLFTBHFOFSBMJ[BUJPO PSFRVBUJPOMBXGVM 1BSUPGUIFBOTXFSJTQSPWJEFECZUIFDPOOFDUJPOTCFUXFFOMBXT BOEPUIFSDFOUSBMOPUJPOTJOUIFTDJFODFT JOQBSUJDVMBSFYQMBOBUJPO DPVOUFSGBDUVBMT DBVTBUJPO BOE DPOàSNBUJPO &YQMBOBUJPOT PGUFO JOWPMWF TQFDJGZJOH IPX B MBX BOE JOJUJBMDPOEJUJPOTFOUBJMUIFFWFOUUPCFFYQMBJOFE-BXTTVQQPSUDPVOUFSGBDUVBMTUBUF- NFOUTGPSFYBNQMF JGUIFEJTUBODFCFUXFFOUIFFBSUIBOEUIFTVOXFSFr meters then 2 the gravitational force between them would be F Gmems/r . Further, propositions UIBUBSFBQUGPSFYQSFTTJOHMBXGVMHFOFSBMJ[BUJPOTBSFDPOàSNFECZUIFJSJOTUBODFT 8IJMFUIFGFBUVSFTKVTUNFOUJPOFEIFMQUPJEFOUJGZMBXT UIFSFJTTUJMMBRVFTUJPOPG what laws are. There are two main philosophical positions concerning the metaphysics PGMBXT XIJDI*XJMMDBMMi)VNFBOuBOEiNFUBQIZTJDBMuBDDPVOUT5IFNPTUTPQIJT- UJDBUFE WFSTJPO PG UIF )VNFBO WJFX JT EVF UP %BWJE -FXJT BOE UIF NPTU TPQIJTUJDBUFEWFSTJPOPGUIFNFUBQIZTJDBMWJFXJTEVFUP5JN.BVEMJO TFF.BVEMJO 2007). 0O -FXJTT BDDPVOU UIF MBXT BSF DPOUJOHFOU HFOFSBMJ[BUJPOT JNQMJFE CZ UIF best systematization PGUIFEJTUSJCVUJPOPGGVOEBNFOUBMFOUJUJFT NBHOJUVEFT FUD)FSFJTUIF JEFB-FULCFBMBOHVBHFXIPTFBUPNJDQSFEJDBUFTFYQSFTTPOMZGVOEBNFOUBMNBHOJ- tudes and relations and mathematical notions and let W be the set of all truths of L. 5IFMBXT DBMMUIFNi-MBXTu BSFEFàOFEBTGPMMPXT
5BLF BMM EFEVDUJWF TZTUFNT XIPTF UIFPSFNT BSF USVF 4PNF BSF TJNQMFS CFUUFSTZTUFNBUJ[FEUIBOPUIFST4PNFBSFTUSPOHFS NPSFJOGPSNBUJWF UIBO others. These virtues compete: an uninformative system can be very simple,
#"33:-0&8&3
an unsystematized compendium of miscellaneous information can be very JOGPSNBUJWF5IFCFTUTZTUFNJTUIFPOFUIBUTUSJLFTBTHPPEBCBMBODFBTUSVUI XJMMBMMPXCFUXFFOTJNQMJDJUZBOETUSFOHUI)PXHPPEBCBMBODFUIBUJTXJMM EFQFOEPOIPXLJOEOBUVSFJT"SFHVMBSJUZJTBMBXJGGJUJTB
"DDPSEJOHUP.BVEMJOTNFUBQIZTJDBMBDDPVOU MBXT DBMMUIFNi.MBXTu BSFOPU themselves generalizations or regularities but rather fundamental elements of the XPSMETPOUPMPHZUIBUproduce UIFMBXGVMSFHVMBSJUJFT.BVEMJOTBZTMJUUMFNPSFBCPVU XIBUMBXTBSFBOEFYBDUMZIPXBMBX produces SFHVMBSJUZ)JTJEFBTFFNTUPCFUIBUMBXT are described by dynamical equations (e.g., F ma). Given the state of the universe at t the laws evolve that state into subsequent states, producing a regularity satisfying the equation. 5IFRVFTUJPOJTXIFUIFSUIFGVOEBNFOUBMMBXTPGPVSXPSMEBSF-MBXTPS.MBXT PSTPNFPUIFSBDDPVOU 0O-FXJTTBDDPVOUUIFCFTUTZTUFNPGBXPSMEJTEFUFSNJOFE CZUIFFOUJSFIJTUPSZPGTUBUFTPGUIFVOJWFSTF*UGPMMPXTUIBUUIF-MBXTsupervene on UIFUPUBMJUZPGTUBUFT*ODPOUSBTU .MBXT JGUIFSFBSFBOZ EPOPUTVQFSWFOFPOUIF totality of states since different laws can produce the same total histories. For some BEWPDBUFT PG .MBXT UIJT DPOUSBTU JT FOPVHI UP FTUBCMJTI UIBU -MBXT BSF UPP XFBL UPEPUIFXPSLUIBUMBXTBSFTVQQPTFEUPEP5IFZTBZUIBU-MBXTBSFJODBQBCMFPG FYQMBJOJOHTUBUFFWPMVUJPOTJODFUIFZBSFEFUFSNJOFECZUIFTUBUFT#VUUIFJTTVFJT NPSFTVCUMFTJODF-MBXTBOEUIFTUBUFEPentail TVCTFRVFOUTUBUFT"EWPDBUFTPG-MBXT HPPOUPTBZUIBUXFIBWFOPJEFBPGIPX.MBXTproduce TUBUFT8FDBOOPUTFUUMFUIF issue here but will note some other differences between the two accounts. The two accounts of laws may render different verdicts concerning determinism TJODFUIFHFOFSBMJ[BUJPOTFOUBJMFECZUIFXPSMETCFTUUIFPSZ JGUIFSFJTPOF NBZCF EJGGFSFOUGSPNUIFHFOFSBMJ[BUJPOTCSPVHIUBCPVUCZUIFXPSMET.MBXT JGUIFSFBSF BOZ 5IFUXPBDDPVOUTBMTPEJGGFSXJUISFTQFDUUPUIFDPOOFDUJPOTUIFZNBLFCFUXFFO laws and time. The metaphysical account presupposes a temporal direction since the MBXTFWPMWFUIFXPSMEUPXBSEUIFGVUVSF5IF-WJFXEPFTOPUQSFTVQQPTFBOZJOUSJOTJD temporal direction but attempts to account for temporal direction in terms of the distribution of the structure of the totality of states. *UIBTCFFOTVHHFTUFEUIBUWJFXTBCPVUMBXTIBWFDPOTFRVFODFTGPSUIFUISFBUUIBU EFUFSNJOJTNQPTFTUPUIFFYJTUFODFPGGSFFXJMM*UIBTCFFOBSHVFEBTGPMMPXTXFIBWF OPDPOUSPMPWFSUIFQBTUBOEPSUIFMBXT BOEJGEFUFSNJOJTNJTUSVFJUTFFNTUPGPMMPX UIBUXFIBWFOPDPOUSPMPWFSUIFGVUVSFFJUIFS4PNFQIJMPTPQIFSTIBWFSFTQPOEFEUP UIJTBSHVNFOUCZPCTFSWJOHUIBUXIJMFUIFBSHVNFOUNBZCFTPVOEJGMBXTBSF.MBXT JUGBJMTJGMBXTBSF-MBXT5IFSFBTPOJTUIBUUIF-MBXTBSFEFUFSNJOFECZUIFUPUBMJUZ PGGBDUTJODMVEJOHGBDUTBCPVUXIBUXFDIPTF TP SBUIFSUIBODPOTUSBJOJOHPVSDIPJDFT UIFZBSFQBSUMZEFUFSNJOFECZUIFN .FMFBOE#FFCFF )PFGFS *EPOPU BTTFTTUIFTUSFOHUIPGUIJTSFTQPOTFIFSF FYDFQUUPOPUFUIBUJGJUQSPWFTBHPPESFTQPOTF JUXPVMEDBTUEPVCUPOUIFDMBJNUIBU-MBXTDBOFYQMBJOBOETVQQPSUDPVOUFSGBDUVBMT The belief that determinism entails predictability is a reason why some people !nd determinism abhorrent. They might fear that if determinism is true, then others (or
%&5&3.*/*4.
a superior intelligence) would be able to calculate what they will do and thus thwart UIFJSQMBOT#VUEFUFSNJOJTNBOEQSFEJDUBCJMJUZBSFRVJUFEJGGFSFOUDMBJNT BOEOFJUIFS FOUBJMTUIFPUIFS%FUFSNJOJTNJTBmetaphysical claim about the fundamental laws of UIFVOJWFSTF QSFEJDUBCJMJUZJTBOepistemic claim about what we can know about the future. There are a number of considerations that show why determinism does not entail predictability. First it may be impossible (because of our natures and the laws UIFNTFMWFT GPSVTUPLOPXXIBUUIFMBXTBSF&WFOJGXFLOFXUIFMBXTXFNJHIU OPUCFBCMFUPVTFUIFNUPHBJOLOPXMFEHFPGDFSUBJOGVUVSFFWFOUTCFDBVTFBDDVSBUF QSFEJDUJPOT SFRVJSF LOPXJOH BO FOPSNPVT BNPVOU m QPTTJCMZ BO JOàOJUF BNPVOU mBCPVUUIFQSFTFOU*OUIFDBTFPG/FXUPOJBONFDIBOJDT QFSGFDUMZSFMJBCMFQSFEJD- tions of the exactGVUVSFNPUJPOTPGQBSUJDMFTSFRVJSFLOPXMFEHFPGUIFFYBDUQSFTFOU QPTJUJPOTBOENPUJPOTPGBMMUIFQBSUJDMFTJOUIFVOJWFSTF BOEUIFFYBDUQPTJUJPOPG BQBSUJDMFXJMMUZQJDBMMZCFSFQSFTFOUFECZBOJOàOJUFMZMPOHEFDJNBM*UNBZUVSOPVU UIBUUIFMBXTUIFNTFMWFTFOUBJMUIBUUIFLOPXMFEHFSFRVJSFEUPNBLFDFSUBJOQSFEJD- UJPOTJTJNQPTTJCMFUPPCUBJO'VSUIFS TNBMMEJGGFSFODFTBUPOFUJNFDBONBLFGPSWFSZ big differences a short time later with respect to matters that concern us. Another PCTUBDMFUPQSFEJDUJPOJTUIBUUIFNBUIFNBUJDBMFRVBUJPOTFYQSFTTJOHUIFMBXTNBZOPU CFTPMWBCMFFYDFQUBQQSPYJNBUFMZ5IJT JOGBDU JTUIFDBTFGPSUIFTJNQMF/FXUPOJBO XPSMEXIFOUISFFPSNPSFQBSUJDMFTBSFJOWPMWFE-BQMBDFXBTJEFBMJ[JOHFOPSNPVTMZ XIFOIFTVHHFTUFEUIBUBOiJOUFMMJHFODFuDPVMEQSFEJDUGVUVSFTUBUFTGSPNUIFQSFTFOU state and the laws. 0OUIFPUIFSIBOE UIFGBJMVSFPGEFUFSNJOJTNEPFTOPUQSFDMVEFUIFQPTTJCJMJUZPG SFMJBCMFQSFEJDUJPOTBCPVUUIFGVUVSF0GDPVSTF UIFFYUFOUUPXIJDIXFDBOSFMJBCMZ QSFEJDUUIFGVUVSFEFQFOETPOFYBDUMZXIBUUIFMBXTBSF*GUIFMBXTBSFQSPCBCJMJTUJD it may turn out that, given the state or even a partial description of the state at t, the MBXTTQFDJGZQSPCBCJMJUJFTWFSZDMPTFUPGPSTPNFGVUVSFFWFOUT5IVT FWFOJGDPJO UPTTFTBSFGVOEBNFOUBMMZSBOEPN XFDBOQSFUUZBDDVSBUFMZQSFEJDUUIBU UPTTFTPG BOPSEJOBSZDPJOXJMMSFTVMUJOCFUXFFOBOEIFBET5IFNPSBMPGBMMUIJTJTUIBU XFTIPVMELFFQJONJOEUIBUEFUFSNJOJTNJTBNFUBQIZTJDBMDMBJNBCPVUUIFMBXTXIJMF predictability is an epistemic claim about what we can reliably predict, and neither entails the other.
Determinism and quantum theory -BQMBDF DPOTJEFSFE EFUFSNJOJTN UP CF USVF CFDBVTF IF BDDFQUFE UIBU /FXUPOJBO NFDIBOJDT JT UIF USVF UIFPSZ PG FWFSZUIJOH BOE UIBU JU FOUBJMT EFUFSNJOJTN #VU /FXUPOJBONFDIBOJDTIBTCFFOTVQFSTFEFECZRVBOUVNNFDIBOJDT 2. BOETPUIF question arises of its consequences for determinism. *O OPOSFMBUJWJTUJD 2. UIF TUBUF PG BO JTPMBUFE TZTUFN JT TQFDJàFE OPU CZ UIF QPTJUJPOT BOE NPNFOUB PG QBSUJDMFT BT JO /FXUPOJBO NFDIBOJDT CVU CZ B WFDUPS valued wave function Y(t) that speci!es the probabilities of the values of measurements made at t of the observable quantities of the system. The observable quantities, corre- sponding to position, momentum, total energy, spin, and so on, are the properties of
#"33:-0&8&3
RVBOUVNTZTUFNT5IFZOFFEOPUMJUFSBMMZCFPCTFSWBCMF/PTUBUFY assigns a proba- CJMJUZPGGPSFWFSZPCTFSWBCMF*OQBSUJDVMBS OPYBTTJHOTBQSPCBCJMJUZPGUPWBMVFT PGCPUIUIFNPNFOUVNBOEUIFQPTJUJPOPCTFSWBCMFTBTTPDJBUFEXJUI GPSFYBNQMF BO FMFDUSPO5IJTJTBOJOTUBODFPG)FJTFOCFSHTuncertainty principle0OUIFPSUIPEPY PS i$PQFOIBHFO uJOUFSQSFUBUJPOPG2. BOPCTFSWBCMF0 FH BQBSUJDMFTNPNFOUVN is said to have a determinate value if and only if YBTTJHOTBQSPCBCJMJUZUPBQBSUJDVMBS WBMVFPGUIBUPCTFSWBCMF 5IFi$PQFOIBHFOuJOUFSQSFUBUJPOSFGFSTUPBDPMMFDUJPOPG XBZTPGUIJOLJOHBCPVU2.BTTPDJBUFEXJUI/JFMT#PISBOE8FSOFSWPO)FJTFOCFSH UIBU DBNF UP CF BDDFQUFE BT UIF PSUIPEPY XBZ PG VOEFSTUBOEJOH 2. " HPPE EJTDVTTJPODBOCFGPVOEJO$VTIJOH *UGPMMPXTUIBUOPFMFDUSPO PSBOZPUIFS 2.TZTUFN IBTCPUIBEFUFSNJOBUFQPTJUJPOBOEBEFUFSNJOBUFWFMPDJUZ*OGBDU GPS typical states of elementary particles, neither position nor momentum, nor any other GBNJMJBSRVBOUJUJFT QPTTFTTEFUFSNJOBUFWBMVFT2.BMTPJODMVEFTBEZOBNJDBMMBXm 4DISÕEJOHFSTequationmEFTDSJCJOHYTFWPMVUJPO4DISÕEJOHFSTMBXJTEFUFSNJOJTUJD BOEMJOFBS4PUIFRVFTUJPOOBUVSBMMZBSJTFTPGIPXQSPCBCJMJUJFTDPNFJOUPUIFQJDUVSF 0OUIFPSUIPEPYBDDPVOU UIFBOTXFSJTUIBUYPCFZT4DISÕEJOHFSTEFUFSNJOJTUJDMBX except XIFOBTZTUFNJTCFJOHNFBTVSFE PSPCTFSWFE 8IFOBNFBTVSFNFOUPG0JT made, the system randomly jumps JOUPBTUBUFJOXIJDI0IBTUIFEFUFSNJOBUFWBMVF with the probabilities speci!ed by Y. 5IFSFBSFBOVNCFSPGOPWFMBOEQFDVMJBSGFBUVSFTPG2.5IFNPTUTUSJLJOHJTUIF DMBJNUIBURVBOUJUJFTMJLFQPTJUJPONBZOPUCFEFUFSNJOBUF5IJTMBDLPGEFUFSNJOBUFOFTT is different from a failure of determinism since it says that at a given time a certain RVBOUJUZ GPSJOTUBODFQPTJUJPO IBTOPTQFDJàDWBMVF6OEFSMZJOHUIJTJTUIF2.QSJODJQMF
PGTVQFSQPTJUJPO*GYJTBTUBUFDPSSFTQPOEJOHUPBQBSUJDMFCFJOHMPDBUFEJOSFHJPO and Y2 corresponds to the particle being located in a distinct region 2, then there are superpositions of these states, aY + bY2, that correspond to the particle being located somewhere in the union of the two regions but at no speci!c place within the union of the two regions. The coef!cients a and b determine the probabilities of the outcomes PGQPTJUJPONFBTVSFNFOUTJOUIFSFTQFDUJWFSFHJPOT0OUIFPSUIPEPYJOUFSQSFUBUJPO JU is not just that we do not know UIFFYBDUMPDBUJPOPGUIFQBSUJDMFCVUUIBUJUTMPDBUJPO is indeterminate. Another peculiar feature is the role of measurement (or observation) in UIFGPSNVMBUJPOPGUIFMBXT5IJTTFFNTUPNBLF2.QFDVMJBSMZTVCKFDUJWFBOEDFSUBJOMZ NBLFTJUJOFYBDU XJUIPVUBQSFDJTFDIBSBDUFSJ[BUJPOPGNFBTVSFNFOUT"UIJSEQFDVMJ- arity is non-locality*UUVSOTPVUUIBUUIFSFBSFTUBUFTPG GPSFYBNQMF BTQBUJBMMZTFQBSBUFE pair of electrons for which, when a measurement of one of the electrons is made, the state of it and the other electron KVNQTJOUPBOFXTUBUF #FMM "MCFSU 5IFQFDVMJBSJUZPGUIFTFGFBUVSFTFODPVSBHFENBOZQIZTJDJTUTUPUBLFBOJOTUSVNFO- UBMJTUBUUJUVEFUPXBSETUIFUIFPSZ*OTUSVNFOUBMJTUTUIJOLPG2.BTmerely providing SVMFTGPSQSFEJDUJOHUIFPVUDPNFTPGNFBTVSFNFOUT4PVOEFSTUPPE 2.JTTJMFOUBCPVU the ontology and the laws, whatever they might be, that lie behind its predictions. 4PNF QIZTJDJTUT CFMJFWFE JU UP CF JNQPTTJCMF UP TVQQMFNFOU PS NPEJGZ 2. XIJMF preserving its predictions and impossible to remove the notion of observation from UIFUIFPSZ*GUIJTXFSFUIFMBTUXPSEBCPVU2.UIFO2.XPVMECFTJMFOUPOXIFUIFS determinism is true.
%&5&3.*/*4.
)PXFWFS UIFSFBSFSFBMJTUXBZTBOEBMTPEFUFSNJOJTUJDXBZTPGVOEFSTUBOEJOH2. UIBU BSF OPX CFHJOOJOH UP CF UBLFO TFSJPVTMZ CZ TPNF QIZTJDJTUT BOE QIJMPTPQIFST 5IFNPTUJNQPSUBOUEFUFSNJOJTUJDBDDPVOUJTUIFTPDBMMFEiIJEEFOWBSJBCMFTUIFPSZu EFWJTFECZ%BWJE#PINJO TFF#FMM "MCFSU $VTIJOH 5IF POUPMPHZ PG #PINJBO NFDIBOJDT DPOTJTUT PG QBSUJDMFT UIBU BMXBZT QPTTFTT EFàOJUF positions) and a quantum !eld that corresponds to the wave function. The state of a system at t is determined by the positions of the particles at t and the values of the quantum !eld at t5IFEZOBNJDBMMBXTBSF4DISÕEJOHFSTMBXBOEBMBX UIFiHVJEBODF FRVBUJPOu UIBU TQFDJàFT UIF WFMPDJUJFT PG UIF QBSUJDMFT 5IFTF MBXT BSF UIPSPVHIMZ EFUFSNJOJTUJD1SPCBCJMJUJFTDPNFJOUPUIFQJDUVSFUISPVHIBQSPCBCJMJUZEJTUSJCVUJPO that is posited to hold over initial positions of particles of a system compatible with its XBWFGVODUJPO.FBTVSFNFOUTBSFTJNQMZJOUFSBDUJPOTCFUXFFOUXPTZTUFNTUIBUSFTVMU in the value of a quantity of the measured system being correlated with a macro-state of the measurement instrument. The predictions of the results of measurements on #PINTUIFPSZBSFFYBDUMZUIFTBNFBTUIPTFPGPSUIPEPY2.*OQBSUJDVMBS #PINJBO mechanics entails the uncertainty principles and all the other probabilistic predictions PG2.5IFVODFSUBJOUZJTJSSFNFEJBCMFTJODFJUGPMMPXTGSPNUIFMBXTBOEUIFJOJUJBM QSPCBCJMJUZEJTUSJCVUJPOUIBUJUJTJNQPTTJCMFUPLOPXUIFDPNQMFUFTUBUFPGBTZTUFN 5IFSFBSFBMTPSFBMJTUWFSTJPOTPG2.XIPTFEZOBNJDBMMBXTBSFJOEFUFSNJOJTUJD 5IFNPTUGVMMZXPSLFEPVUPGUIFTFJTUIF(38UIFPSZ TPDBMMFEBGUFSJUTGPSNVMBUPST (IJSBSEJ 3JNJOJ BOE 8FCFS TFF "MCFSU (IJSBSEJ 5IF (38 UIFPSZ SFQMBDFTUIFEFUFSNJOJTUJD4DISÕEJOHFSMBXXJUIBOJOEFUFSNJOJTUJDMBXUIBUTQFDJàFT the probabilities of the state at tiKVNQJOHuJOUPWBSJPVTQPTTJCMFTUBUFTBUTVCTFRVFOU times. The law has the consequence that for a system whose quantum state involves few degrees of freedom (with respect to particle position) the evolution will be as TQFDJàFECZ4DISÕEJOHFSTFRVBUJPO FYDFQUGPSWFSZSBSFiKVNQTuUIBUMPDBMJ[FQBSUJDMF QPTJUJPO 5IFCBTJDJEFBPGUIF(38UIFPSZJTUIBUUIFRVBOUVNTUBUFPGBTZTUFN FWPMWFTJOBDDPSEXJUI4DISÕEJOHFSTMBX FYDFQUUIBUUIFSFJTBQSPCBCJMJUZQFSVOJU time of the wave function of the state being multiplied by a very narrow Gaussian: see (IJSBSEJ #VUJONBDSPTDPQJDTZTUFNT FH BNFBTVSJOHEFWJDFUIBUDPOTJTUTPG NBOZQBSUJDMFT JUJTWFSZMJLFMZUIBUBUMFBTUPOFPGUIPTFQBSUJDMFTXJMMVOEFSHPBKVNQ JOBGSBDUJPOPGBTFDPOE4JODFUIFQPTJUJPOTPGUIFQBSUJDMFTBSFDPSSFMBUFE XIFOPOF jumps into a localized position state, the rest must follow. The consequence is that measurements and other macroscopic interactions result in quantum states in which macroscopic objects have determinate positions. There is no need to introduce the OPUJPOTPGiNFBTVSFNFOUuPSiPCTFSWFSuJOUPUIFGPSNVMBUJPOPGUIFUIFPSZ 5IF NPTU JNQPSUBOU QPJOU GPS PVS EJTDVTTJPO JT UIBU PSUIPEPY RVBOUVN UIFPSZ (38 BOE#PINJBONFDIBOJDTBSF GPSBMMQSBDUJDBMQVSQPTFT FNQJSJDBMMZFRVJWBMFOU even though the !rst two are incompatible with determinism and the latter entails it. (There are, in principle, empirical differences between theories with collapses, MJLFUIFPSUIPEPYUIFPSZBOE(38 BOEOPDPMMBQTFUIFPSJFTMJLF#PINT)PXFWFS it is plausible that they are empirically equivalent for all practical purposes, since it JT VOMJLFMZ UIBU JU XJMM FWFS CF QPTTJCMF UP DPOEVDU BO FYQFSJNFOU XIPTF PVUDPNFT discriminate among these theories.) This is a dramatic case of the underdetermination
#"33:-0&8&3 of theory by all possible evidence. Although neither of these theories is true (since they GBJMUPUBLFJOUPBDDPVOUSFMBUJWJUZ JUJTWFSZQMBVTJCMFUIBUJGUIFSFJTBUIFPSZPGFWFSZ- thing, there will be also be empirically equivalent theories that are deterministic and JOEFUFSNJOJTUJD4PJUJTWFSZMJLFMZUIBUUIFRVFTUJPOPGXIFUIFSPSOPUEFUFSNJOJTNJT true is plausibly something that we will never be in a position to answer.
Determinism and statistical mechanics &WFOJGUIFEZOBNJDBMMBXTBSFEFUFSNJOJTUJD BTJO/FXUPOJBONFDIBOJDTBOE#PINT UIFPSZ QSPCBCJMJUJFT BSF SFRVJSFE GPS FYQMBOBUJPO BOE QSFEJDUJPO 4VQQPTF UIBU BT -BQMBDF UIPVHIU UIF XPSME DPOTJTUT PG QPJOU QBSUJDMFT BOE UIF MBXT BSF HJWFO CZ classical mechanics. The macroscopic state of a system (even the universe) at a time is TQFDJàFECZUIFWBMVFTPGNBDSPTDPQJDRVBOUJUJFTMJLFUFNQFSBUVSF BWFSBHFGSFRVFODZ of radiation, average mass, and charge density, in small, but not too small, volumes PG TQBDF 5IF NBDSPTDPQJD TUBUF JT UZQJDBMMZ JOTVGàDJFOU UP QJO EPXO GPS FYBNQMF XIFUIFSPSOPUUIFSFJTBOJDFDVCFáPBUJOHJOBQBJMPGXBSNXBUFSJOTPNFQBSUJDVMBS room (or whether a room is full of people and other macroscopic features). For a given macroscopic state of a system at t there are in!nitely many possible micro-states (states characterized by precise positions and momenta of all the particles that compose the system) only one of which actually composes the system at t*O/FXUPOJBONFDIBOJDT XJUIBQBSUJDMFPOUPMPHZ TJNJMBSSFNBSLTBQQMZUPRVBOUVNUIFPSJFT the macroscopic state of the universe (or an isolated system) at t and the deterministic dynamical laws determine very little about the macroscopic TUBUFT BU PUIFS UJNFT 'PS FYBNQMF UIF NBDSPTUBUFPGBOJDFDVCFJOXBSNXBUFSJTDPNQBUJCMFXJUIiNBWFSJDLuNJDSPTUBUFT whose futures (as entailed by the deterministic laws) involve the ice-cube growing CJHHFSPSFWFOGPSNJOHUIFTIBQFPG+JNNZ%VSBOUFTOPTFBOEKVNQJOHPVUPGUIF XBUFS4P JGXFKVTULOPXUIFNBDSPTUBUFPGUIFTZTUFN UIBUJUJTBOJDFDVCFáPBUJOH in warm water), the deterministic laws are not suf!cient to predict that the ice-cube will melt. The same point applies to the prediction of the motions of the planets and FWFSZPUIFSBQQMJDBUJPOPG/FXUPOTMBXT JGXFUIJOLPGQMBOFUT BT-BQMBDFEJE BT DPNQPTFEPGBUPNTUIBUPCFZ/FXUPOTMBXT -VEXJH#PMU[NBOOGBDFEUIJTQSPCMFNXIFOIFUSJFEUPFYQMBJOIPXUIFMBXTPG thermodynamics are related to the fundamental dynamical laws. Thermodynamics includes laws that are temporally asymmetric and that reliably and deterministically QSFEJDUIPXBTZTUFNFWPMWFT'PSFYBNQMF UIFsecond law of thermodynamics says that the entropy of an isolated system never decreases. The entropy of a system is, roughly, UIFTJ[FPGUIFDPMMFDUJPOPGNJDSPTUBUFTUIBUBSFDPNQBUJCMFXJUIUIFTZTUFNTNBDSP state. The increase in entropy of the ice-cube in warm water corresponds to the JDFDVCFTNFMUJOH4P UIFQSPCMFN#PMU[NBOOGBDFEXBTIPXUPTRVBSFUIFUFNQPSBMMZ EJSFDUFE TFDPOE MBX XJUI UIF UFNQPSBMMZ TZNNFUSJD GVOEBNFOUBM MBXT #PMU[NBOOT solution is based on the observation that micro-states which the laws evolve to states SFBMJ[JOHNBDSPTUBUFTXJUIHSFBUFSFOUSPQZmNBWFSJDLNJDSPTUBUFTmBSF JOBDFSUBJO TFOTF iSBSFu5IFTFOTFJOXIJDINBWFSJDLTUBUFTBSFSBSFJTOPUUIBUUIFSFBSFGFXFS PGUIFNmUIFSFBSFJOàOJUFMZNBOZmCVUUIBUBWFSZOBUVSBMNFBTVSFPOUIFTFUPG
%&5&3.*/*4.
NJDSPTUBUFTBTTJHOTUIFTFUPGNBWFSJDLTUBUFTBNFBTVSFDMPTFUP 4FF4LMBS and Albert 2000 for philosophical discussions of statistical mechanics.) #PMU[NBOO DPOTUSVFE UIJT NFBTVSF BT B QSPCBCJMJUZ EJTUSJCVUJPO PWFS UIF NJDSP states that are compatible with a given macro-state, and this has the consequence UIBUNBWFSJDLNJDSPTUBUFT FH UIPTFUIBUTQPOUBOFPVTMZGPSNJOUPUIFTIBQFPGB OPTF BSFFYDFFEJOHMZVOMJLFMZ*UUVSOTPVU BHBJOOPUTVSQSJTJOHMZTJODFUIFEZOBNJDBM laws are temporally symmetric) that the uniform distribution over the micro-states DPNQBUJCMF XJUI UIF JDFDVCF JO XBSN XBUFS FOUBJMT UIBU JU JT IJHIMZ MJLFMZ UIBU JO the past (just as in the future) the pail contained water at a uniform temperature. A way of avoiding this consequence while preserving the good consequences is to posit the uniform distribution over micro-states compatible with the macro-state of the VOJWFSTFJNNFEJBUFMZBGUFSUIF#JH#BOHBOEUPQPTJUUIBUUIFFOUSPQZPGUIJTJTWFSZ WFSZMPX5IJTIBTUIFDPOTFRVFODFUIBUJUJTWFSZMJLFMZUIBUUIFFOUSPQZPGUIFFOUJSF universe (and its relatively isolated subsystems) increases over time. (JWFOUIFEZOBNJDBMMBXTBOEUIFJOJUJBMNJDSPTUBUF UIFTUBUJTUJDBMmNFDIBOJDBM probability distribution implies that the evolution at a macroscopic level appears to CF JOEFUFSNJOJTUJD7FSZ TNBMM EJGGFSFODFT JO UIF NJDSPTUBUFT UIBU SFBMJ[F B NBDSP TUBUFFOUBJMWFSZEJGGFSFOUGVUVSFFWPMVUJPOT&WFOJGBEFNPOLOPXTBWFSZEFUBJMFE NBDSPEFTDSJQUJPOPGUIFSPVMFUUFXIFFMBOEUIFNPUJPOTPGUIFDSPVQJFSTIBOE BOE TPPO BOELOPXTUIFEZOBNJDBMMBXTBOEDPVMEQFSGPSNUIFSFMFWBOUDBMDVMBUJPOT IF DPVMEOPUQSFEJDUUIFPVUDPNFPGBUVSOPGUIFXIFFM0VSXPSMEJTBQQBSFOUMZGVMMPG NBDSPTDPQJDQIFOPNFOB TPDBMMFEiDIBPUJDTZTUFNTu XIPTFGVUVSFFWPMVUJPOJTWFSZ sensitive to the initial micro-states that realize their macro-states. 5IFSF JT DPOUSPWFSTZ DPODFSOJOH FYBDUMZ XIBU iQSPCBCJMJUZu NFBOT JO TUBUJTUJDBM mechanics since the dynamical laws are deterministic. The same issue arises in #PINJBONFDIBOJDT BTJUTEZOBNJDBMMBXTBSFBMTPEFUFSNJOJTUJD4JODFUIFPVUDPNFPG a turn of the roulette wheel is strictly determined by the laws and the complete micro- state of the world prior to the turn of the wheel, it is often said that the probabilities JOWPMWFEJOEFUFSNJOJTUJDUIFPSJFTNVTUSFáFDUmerely TVCKFDUJWFJHOPSBODF#VUUIJT EPFTOUTFFNRVJUFSJHIU TJODFUIFTFQSPCBCJMJUJFTBSFCBTFEPOPCKFDUJWFGBDUTBCPVU PVSXPSMEBOEBSFTVQQPTFEUPFYQMBJOUIFTFDPOEMBX'PSUIBUSFBTPOJUJTQMBVTJCMF to consider them objective and lawful. (There are proposals for how to understand QSPCBCJMJUJFTPCKFDUJWFMZJGEFUFSNJOJTNPCUBJOT JODMVEJOHBHFOFSBMJ[BUJPOPG-FXJTT CFTUTZTUFNBDDPVOUPGMBXTEJTDVTTFEFBSMJFSTFF-PFXFSBOE
Conclusion "UUIFUVSOPGUIFUXFOUZàSTUDFOUVSZ QIZTJDJTUTIBWFOPUSFBMJ[FE-BQMBDFTESFBN PGBUIFPSZPGFWFSZUIJOHBOEJGUIFSFJTTVDIUIFPSZ JUJTOPULOPXOXIFUIFSJUJT EFUFSNJOJTUJD/POFUIFMFTT UIFTVDDFTTPG2.BOETUBUJTUJDBMNFDIBOJDT XIJDINVTU be accounted for by any complete theory) provides very strong reason to believe that scienti!c account of the universe will involve probabilities either in indeterministic EZOBNJDBMMBXT BTJOPSUIPEPY2.BOE(38 PSBTJOJUJBMDPOEJUJPOQSPCBCJMJUJFT BT JOTUBUJTUJDBMNFDIBOJDTBOE#PINJBONFDIBOJDT 'VSUIFS JUJTWFSZMJLFMZUIBUJGUIFSF
#"33:-0&8&3 is an empirically adequate proposal for a complete theory whose dynamical laws are probabilistic, there will also be an empirically equivalent account in which the funda- NFOUBMMBXTBSFEFUFSNJOJTUJD5IFVQTIPUJTUIBUJUJTMJLFMZUIBUXFXJMMOFWFSLOPX XIFUIFSPSOPUEFUFSNJOJTNJTUSVF CVUJUJTDFSUBJOUIBUJGJUJTUSVFUIFOUIFSFDBOCF no predicting the future with certainty. This conclusion will doubtlessly be frustrating UPUIPTFXIPUIJOLUIBUXIFUIFSPSOPUEFUFSNJOJTNPCUBJOTIBTWBTUDPOTFRVFODFT for free will and other philosophical issues.
See also-BXTPGOBUVSF 1IZTJDT 1SPCBCJMJUZ 6OEFSEFUFSNJOBUJPO
References "SNTUSPOH %BWJE What Is a Law of Nature? $BNCSJEHF$BNCSJEHF6OJWFSTJUZ1SFTT "MCFSU %BWJE Quantum Mechanics and Experience, $BNCSJEHF .")BSWBSE6OJWFSTJUZ1SFTT mmmm Time and Chance $BNCSJEHF .")BSWBSE6OJWFSTJUZ1SFTT #FMM + 4 Speakable and Unspeakable in Quantum Mechanics $BNCSJEHF $BNCSJEHF 6OJWFSTJUZ 1SFTT $BSSPMM +PIO FE Readings on Laws of Nature 1JUUTCVSHI 1"6OJWFSTJUZPG1JUUTCVSHI1SFTT $VTIJOH +5 Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony $IJDBHP 6OJWFSTJUZPG$IJDBHP1SFTT &BSNBO +PIO A Primer on Determinism, %PSESFDIU3FJEFM (IJSBSEJ (JBODBSMP i$PMMBQTF 5IFPSJFT u JO &EXBSE /;BMUB FE The Stanford Encyclopedia of Philosophy (sQSJOH FEJUJPO BWBJMBCMF IUUQQMBUPTUBOGPSEFEVBSDIJWFTTQSFOUSJFT qm-collapse. )PFGFS $BSM i$BVTBM%FUFSNJOJTN uJO&EXBSE/;BMUB FE The Stanford Encyclopedia of Philosophy (sVNNFS FEJUJPO BWBJMBCMF IUUQQMBUPTUBOGPSEFEVBSDIJWFTTVNFOUSJFTEFUFSNJOJTN causal. ,BOF 3PCFSU The Signi!cance of Free Will, 0YGPSE0YGPSE6OJWFSTJUZ1SFTT -BQMBDF 1 Essai philosophique sur les probabilités,GPSNJOHUIF*OUSPEVDUJPOUPIJTThéorie analytique des probabilités 1BSJT 7 $PVSDJFS USBOT ' 8 5SVTDPUU BOE ' - &NPSZ BT A Philosophical Essay on Probabilities /FX:PSL%PWFS -FXJT % i$IBODFBOE$SFEFODF)VNFBO4VQFSWFOJFODF%FCVHHFE uMindm -PFXFS #BSSZ i%FUFSNJOJTNBOE$IBODF uStudies in the History and Philosophy of Modern Physics m mmmm i%BWJE-FXJTT)VNFBO5IFPSZPG0CKFDUJWF$IBODF uPhilosophy of Sciencem .BVEMJO 5JN The Metaphysics Within Physics0YGPSE0YGPSE6OJWFSTJUZ1SFTT .FMF "MGSFEBOE#FFCFF )FMFO i)VNFBO$PNQBUJCJMJTN uMind m 4LMBS -BSSZ Physics and Chance, $BNCSJEHF$BNCSJEHF6OJWFSTJUZ1SFTT 5PPMFZ .JDIBFM Causation 0YGPSE$MBSFOEPO1SFTT
Further reading 5IFCFTUCPPLMFOHUIEJTDVTTJPOPGEFUFSNJOJTNJT&BSNBO 'PSWBSJPVTWJFXTBCPVUUIFOBUVSF PGMBXT TFF$BSSPMM BOE-BOHFTDPOUSJCVUJPOUPUIJTDPMMFDUJPO'PS.BVEMJOTWJFXPGMBXT TFF .BVEMJO 'PS FMFNFOUBSZ CVU QIJMPTPQIJDBMMZ TPQIJTUJDBUFE EJTDVTTJPOT PG RVBOUVN NFDIBOJDT BOETUBUJTUJDBMNFDIBOJDT TFF"MCFSU BOE 'PSBEWBODFEEJTDVTTJPOTPGUIFQIJMPTPQIZPG RVBOUVNNFDIBOJDTTFF#FMM 'PSBEWBODFEEJTDVTTJPOTPGUIFQIJMPTPQIZPGTUBUJTUJDBMNFDIBOJDT TFF4LMBS