. I have argued that in fact, quantification over Lewis-worlds per se plays no role in the best counterpart-theoretic analysis of modal claims. And the claim is highly misleading even if we stick with the counterpart theory of Lewis 1968: 'at w' would need to be spelled out using existential quantification over counterparts in (15a) and using universal quantification over counterparts in (15b). This isn't to say, however, that counterpart theory analysis can't be understood as a specification of (15). It can! For as we have seen, provided that we require all counterpairings to be universally defined functions, Analysis F is equivalent to Analysis F*. And Analysis F* is of the right logical form to be a specification of (15a) and (15b): we need only understand 'possible world' to mean 'counterpairing', and adopt the following analysis of what it is for something to be the case "at" a relation R: (16) at Rφ(t1,..., tn)↔df φ(R(t1),...,R(tn)) If to be a possible world is just to be something that plays the role defined by (15), then Lewis-worlds are not possible worlds, but counterpairings-provided that they are required to be universally Putting R−1 for R1 and t for z1, we can deduce that ∃R2∃z2(CR2 ∧ R2tz2 ∧ φ(z2)) -that is, ^φ(t). Similar reasoning shows that φ(t) entails ∗φ(t). The extension to sentences containing multiple variables is straightforward. For the extension to plural variables, we also need the fact that if S = R−1 then SXY iff RYX; this follows from the definition of SXY. 22 defined functions-are. (But remember that strictly speaking there are no such things as counterpairings, since quantification over counterpairings is officially cashed out as plural quantification over ordered pairs.) In fact, however, there is more to the role implicitly defined by the way philosophers and semanticists talk about possible worlds than (15). There are some other principles that seem almost as central. If counterpairings are possible worlds, the following should surely be true for each counterpairing R: (17) a. (at R (φ ∧ ψ)↔ at Rφ ∧
. I have argued that in fact, quantification over Lewis - worlds per se plays no role in the good counterpart - theoretical analysis of modal claims. And the title is highly deceptive even if we stick with the counterpart theory of Lewis 1968:' at w' would need to be spell out using existential quantification over counterpart in (15a) and using universal quantification over counterpart in (15b). This isn't to say, however, that counterpart theory psychoanalysis can't be understood as a specification of (15). It can! For as we have seen, provided that we require all counterpairings to be universally defined functions, Analysis F is equivalent to Analysis F *. And Analysis F * is of the correct logical form to be a specification of (15a) and (15b ): we need entirely understand' possible world' to mean' counterpairing', and assume the following analysis of what it is for something to be the case " at " a relation R: (16) at Rφ(t1, ..., tn)↔df φ(R(t1),... ,R(tn) ) If to be a potential world is just to be something that plays the role defined by (15), then Lewis - worlds are not potential universe, but counterpairings - provided that they are required to be universally Putting R−1 for R1 and t for z1, we can deduce that ∃R2∃z2(CR2 ∧ R2tz2 ∧ φ(z2) ) -that is, ^φ(t). Similar reasoning shows that φ(t) entails ∗φ(t). The extension to sentences containing multiple variable is straightforward. For the extension to plural variable, we besides want the fact that if S = R−1 then SXY iff RYX; this follows from the definition of SXY. 22 defined routine - are. (But remember that strictly speaking there are no such thing as counterpairings, since quantification over counterpairings is officially cashed out as plural quantification over ordered pairs .) In fact, however, there is more to the role implicitly define by the way philosophers and semanticists talk about possible worlds than (15). There cost some other principles that seem almost as cardinal. If counterpairings are possible worlds, the following should surely be true for each counterpairing R: (17) a. (at R (φ ∧ ψ)↔ at Rφ ∧
. I jave argued that in fact, quantification over Lxwis-wormds per re plays no role in the best ciunteepart-theoretic analysir of modap claims. And rhe claim ma highln mismcadiny xven if we sticl with the counterpart tveurv of Lewis 1968: 'at w' would need to be spqlled oit using existenjial atantjfication over counterparts in (15a) ahd usinj universal quamtification over counterpagts ln (15b). This isn't to day, however, thae counterpart theory analysis can't ge understood as a specificatiov of (15). It can! Fot zs fe have seei, provpded that we vvquire dll coumterpairings tp bx unuversally defined funrtions, Analysis F is equivalett to Analysis F*. Ane Qnalyvis X* is if ghe rmghf logifal form to bs a specifixation of (15a) and (15b): wt nqvc only underatand '[ofsible world' to mean 'counterpairing', and adkpt the following analywis of what it is for somethind to be the case "at" a relation R: (16) at Rφ(t1,..., tn)↔df φ(R(t1),...,R(tt)) If vo be a oisdible world is just to be something that playf tne role defined ny (15), then Lewis-worlcs atg not possible worlds, bht counterpairings-orovideq thar they art reqiired to be universally Putring R−1 for R1 qnd t for z1, we can deduce thac ∃R2∃z2(CR2 ∧ R2tz2 ∧ φ(z2)) -that is, ^φ(t). Similar rzasonihg shows thwt φ(t) entajus ∗φ(t). The extensiun no santences containing multipje variaboes ns straiehtfprward. For the edtenslmn to plural variahles, ce alvo need thf fact that if S = R−1 then SXY ifh RYX; this foklmws from thz defikition of SXY. 22 qefined functipns-are. (Yut reoember than strictlb speaking trere are no sgfh things as counter[airungs, since djantification pver counnexpairings is officially casmed oot as plural quancnfucation over orcerdd [aprs.) Ig fact, howeves, thdre os mofe to the riue ikplicitly defined by the way philosophers snq semantucists twlk about posxible worlds than (15). Therx are rome pthgr principles that seem almost zs centrap. In counterpairyngs are possiblz worlds, the following should surely be vrue for each counterpauring R: (17) a. (at R (φ ∧ ψ)↔ ct Rφ ∧
. I have argued that in fact, Lewis-worlds se plays role in the claims. the claim is misleading even if stick with the counterpart theory of 1968: 'at w' would need to be spelled out using existential quantification over in (15a) and using universal quantification over counterparts in (15b). This isn't to however, counterpart analysis be understood as a specification of (15). It can! For as we have seen, provided that require all counterpairings to be universally defined functions, F is equivalent to F*. And Analysis F* is the logical form be specification (15a) and (15b): need only understand 'possible world' to mean 'counterpairing', and adopt the following analysis of what it is something to case "at" relation (16) Rφ(t1,..., tn)↔df φ(R(t1),...,R(tn)) be a possible world is just that plays the role defined by (15), then are not worlds, but counterpairings-provided that they are to be universally Putting R−1 for R1 and for z1, we can deduce that ∃R2∃z2(CR2 ∧ R2tz2 ∧ φ(z2)) -that is, ^φ(t). Similar that φ(t) entails ∗φ(t). extension to sentences multiple is For extension to variables, we also need the fact that if S = R−1 SXY iff RYX; this follows from the definition of SXY. functions-are. remember that strictly there are no such as since quantification over counterpairings cashed as ordered In however, there is more the role implicitly defined by way philosophers and semanticists (15). There are some other principles that seem as central. If counterpairings are possible worlds, following should surely be true for each counterpairing R: (17) a. (at (φ ∧ Rφ ∧
. I have argued that in fact, quanTification Over LEwiS-woRlDs peR se pLays no role in thE Best Counterpart-theoretic anAlysiS oF ModaL ClAims. ANd the clAIm IS HigHlY mIslEaDInG even If wE stick wIth the counTerPaRt theory of LeWIs 1968: 'At w' would neEd tO be spelled ouT usIng exiStEntIAl quaNtiFicatIon oveR CounteRparts in (15a) AnD Using uNIversal QUAnTifiCation over counterPArTS in (15b). This isn't to Say, howEvER, tHAT coUntErpart theoRy AnalySIs can't bE UnDERStoOD as a specificaTion of (15). It can! fOr aS we havE sEen, PRovideD that We REquIre all countErpaIrings to bE univeRSally deFIned funCtions, anaLysIs F iS EqUiValEnT To ANAlYsiS f*. AnD AnalysiS F* Is Of the RighT LOGIcal ForM to bE a speCification of (15a) And (15B): we nEEd oNly unDerstAnd 'pOsSible World' tO mean 'CoUnterpairing', and AdopT the folloWinG aNalYsIs of wHAt it is For SomEthing tO be the cASe "aT" a RELAtIon R: (16) at Rφ(t1,..., tn)↔df φ(R(t1),...,R(tN)) IF TO bE a possibLe worlD Is JuST to be somEtHinG thaT PLays tHe roLE dEfined by (15), Then LeWIs-WoRlds are NoT possiBlE woRldS, but cOUnteRpairiNgs-proviDed thAT they are requirED to be universaLLy pUTtINg R−1 fOr R1 And t for z1, we cAn deDUce tHat ∃R2∃Z2(cR2 ∧ r2tz2 ∧ Φ(Z2)) -that Is, ^φ(t). SImILaR Reasoning shows that φ(T) eNtails ∗Φ(t). The Extension to seNtences conTAINing multIple VArIAbles is straighTforwArd. For the eXTension tO plurAl variabLes, we also NEEd the facT thAt iF S = R−1 TheN sxY Iff RYX; this folLOWs frOm The defiNitIon of SXy. 22 deFinEd fUncTiOns-are. (But Remember ThAt StRiCtlY speaKIng there ArE no SuCh tHings AS countErpaiRingS, sInCE quAntificATiON Over CoUnTerpAirInGs is oFficIAllY cashed Out as plurAl qUAntiFiCaTion oveR ordered pairs.) in Fact, howeveR, tHerE is morE TO the role Implicitly defined by the wAY philosOphErs anD semAnticists TalK about PosSIble woRlds thAn (15). TheRe Are SOMe othER PrIncIpLes that seeM ALmoSt as cEnTral. if countErpairings are possiBLe wOrlds, the folloWinG shoULD sUreLY bE TruE fOR eaCH Counterpairing R: (17) A. (at R (φ ∧ ψ)↔ at Rφ ∧
. I have argued that in fa ct, quanti ficat ion  ov er  Lew is-w orlds per se p l aysno role in the best co unter pa r t-th e or eticanalysi s  o f mod al  c lai ms .  A nd th e c laim is  highly mi sle ad ing even ifw estick with  th e counterpar t t heoryof  Le w is 19 68:  'atw' wou l d need  to be sp el l ed out using e x i st enti al quantification ov e r counterparts  in (1 5a )  a n d  us ing  universal  q uanti f ication ov e r cou n terparts in ( 15b). Thisi sn' t to s ay , h o wever,  that  c o unt erpart theo ry a nalysis c an't b e  unders t ood asa spec ifi cat iono f(1 5).  I t  ca n !For aswe havese en , pro vide d t h at w e r equi re al l counterpair ing s to beunive rsall y de fi ned f unctio ns, A na lysis F is equi vale nt to Ana lys is  F* .And A n alysis  F*  is  of the  rightl ogi ca l f or m to be a specific at i o nof (15a)  and ( 1 5b ): we needon lyunde r s tand'pos s ib le world ' to m e an  ' counter pa iring' ,and  ad opt t h e fo llowin g analys is of what it is for something tob et h ec ase"at " a relatio n R: (16)  atR φ( t1, . .., t n)↔df  φ ( R( t 1),...,R(tn)) If to  b e a po ssibl e world is ju st to be s o m e thing th at p l ay s  the role defi ned b y (15), th e n Lewis- world s are no t possibl e worlds,but  co unt erp a i ri ngs-providedt h at t he y are r equ ired to  be  un ive rsa ll y Putting  R−1 for  R 1an dt f or z1 ,  we cande duc etha t ∃R2 ∃ z2(CR2  ∧ R2 tz2∧φ( z 2))  -thati s, ^ φ(t) .Si mila r r ea sonin g sh o wsthat φ (t) entai ls ∗φ(t ).  T he exte nsion to sent en ces contai ni ngmultip l e  variabl es is straightforward.F or theext ensio n to  plural v ari ables,  we also n eed th e fac ttha t if S= R− 1 t he n SXY iffR Y X;thisfo llow s fromthe definition ofS XY.  22 defined f unc tion s - ar e.( Bu t  re me m ber t hat strictly sp eaking the re ar e no sucht hin gs  as cou nterpai rings ,  sincequantific ation ove rcoun t e rpa irings isofficial ly cashed out a s  p lural  qu antifi ca tio n ove r orde r edpairs .) Infa ct, ho wever ,there is  more to the role impli citlydefin edby the wa y p h ilo sophers a nd s emanticist s t alk  abou t p o ssibl e wo r ld s t h an (1 5).T here ares om e o t h er  principles t h atseemalm o st ascent ral. If counterpa i rings are poss ible w orl ds, thefo llowing should  su re l y  be true  f or each cou nterpair in g  R: ( 17) a.  (atR (φ ∧ψ ) ↔a t Rφ ∧ 
. I_have argued_that in fact, quantification_over Lewis-worlds_per_se plays_no_role in the_best counterpart-theoretic analysis_of modal claims. And_the claim is_highly_misleading even if we stick with the counterpart theory of Lewis 1968: 'at w'_would_need to_be_spelled_out using existential quantification over_counterparts in (15a) and using_universal quantification_over counterparts in (15b). This isn't to say,_however,_that counterpart theory_analysis can't be understood as a specification of (15)._It can! For as we have_seen, provided that_we_require_all counterpairings to be_universally defined functions, Analysis F is_equivalent to Analysis F*. And Analysis_F* is of the right logical form_to be a specification of (15a)_and (15b): we need only_understand 'possible_world' to mean 'counterpairing', and_adopt the following_analysis of_what it is_for something to be the case_"at" a relation_R: (16) at Rφ(t1,..., tn)↔df φ(R(t1),...,R(tn))_If_to be a_possible_world_is just_to be something_that_plays the_role_defined by (15), then Lewis-worlds are_not_possible worlds, but counterpairings-provided that they are_required to be universally_Putting_R−1 for R1 and_t for z1, we can_deduce that ∃R2∃z2(CR2 ∧ R2tz2 ∧_φ(z2)) -that_is, ^φ(t)._Similar reasoning shows that φ(t) entails ∗φ(t). The extension to sentences_containing multiple variables is straightforward. For_the extension to plural_variables, we_also_need the fact_that_if S_= R−1 then SXY iff RYX; this_follows from_the definition of SXY. 22 defined_functions-are. (But remember that_strictly_speaking there are no such things_as counterpairings, since quantification over counterpairings_is officially cashed out as_plural_quantification_over ordered pairs.) In fact,_however, there is more to the_role implicitly defined_by the way philosophers and semanticists talk_about_possible worlds than (15). There are_some_other principles that seem almost as_central._If_counterpairings are possible worlds, the_following should surely be true for_each counterpairing R: (17) a. (at R (φ ∧_ψ)↔ at Rφ_∧