1,..., tn)' is true whenever R is a counterpairing: in that case 'at Sφ(t1,..., tn)' is necessarily true. Our previous strategy for vindicating (17a) and (17b) involved combining Rigid Pairs with the claim that counterpairings are global permutations. But this works out just as well if we replace Rigid Pairs with Semi-Rigid Pairs. If R(S) = S ◦ R−1, then R(S) is a global permutation (and hence a universally defined function) whenever R and S are. Nevertheless, the strategy of vindicating (17d) by accepting SemiRigid Pairs takes a lot of getting used to. For the strategy requires facts about the constitution of counterpairings to be contingent, in a surprising way. For example, the identity pairs I-"the actual world"- could have failed to comprise all and only identity pairs. Indeed, they would have failed to do so had anything been otherwise in any respect! For whenever it is true that φ, it is necessary that at I, φ; and it is always necessary that (φ iff at whichever pairs are the identity pairs, φ); hence, whenever it is true that φ, it is necessary that (if not-φ then I are not the identity pairs). Every counterpairing is the identity relation at itself: R(R) = R ◦ R−1 = I, since counterpairings are global permutations. Being the identity relation plays the role, in this theory, that being actualised plays in the standard possible worlds framework. We face a dilemma: give up (17d), or give up Rigid Pairs. One way to deal with this dilemma would be to apply the Lewisian idea that choices between different ways of referring to the same thing(s) can 42Note that we get the same result if we first eliminate 'at R'to get 'at R(S)φ(R(t1),...,R(tn))' and then eliminate 'at R(S) '. 24 affect the truth value of our modal claims, by affecting the contextually determined interpretation of 'counterpairing' (or in Lewis's case, 'counterpart'). Here are some ordered pairs. If we refer to them in one way-as "these ordered pairs", say-that will evoke an interpretation of 'counterpairing' on which Rig
1, ..., tn)' is true whenever R is a counterpairing: in that case' at Sφ(t1, ..., tn)' is necessarily true. Our former scheme for vindicating (17a) and (17b) involve combining Rigid Pairs with the claim that counterpairings are ball-shaped permutations. But this works out just equally well if we substitute Rigid Pairs with Semi - Rigid Pairs. If R(S) = S ◦ R−1, then R(S) is a global substitution (and therefore a universally defined function) whenever R and S are. however, the strategy of vindicating (17d) by accepting SemiRigid Pairs lead a lot of getting used to. For the strategy necessitate facts about the constitution of counterpairings to be contingent, in a surprising way. For exercise, the identity pairs I-"the actual world"- could have failed to comprise all and only identity pair. Indeed, they would have failed to do so had anything been otherwise in any deference! For whenever it is true that φ, it is necessary that at I, φ; and it is always necessary that (φ iff at whichever pairs are the identity pairs, φ); hence, whenever it is genuine that φ, it is necessary that (if not - φ then I are not the identity pairs). Every counterpairing is the identity relation at itself: R(R) = R ◦ R−1 = I, since counterpairings are global permutations. Being the identity relation plays the role, in this theory, that being actualised bid in the standard possible worlds framework. We face a dilemma: render up (17d), or give up Rigid Pairs. One way to deal with this dilemma would be to put on the Lewisian theme that choices between different way of referring to the same thing(s) can 42Note that we get the same result if we first obviate' at R'to get' at R(S)φ(R(t1),... ,R(tn) )' and then eliminate' at R(S)'. 24 affect the truth value of our modal claim, by affect the contextually determined interpretation of' counterpairing' (or in Lewis's case,' counterpart'). Here are some ordered pairs. If we denote to them in one way - as " these ordered pairs ", suppose - that will evoke an interpretation of' counterpairing' on which Rig
1,..., tn)' is true whenever R is a counterpairing: in thav case 'zt Sφ(t1,..., tn)' is necessarily true. Our pretiouw strqtegy for vindicating (17x) and (17b) ijvolved xombmning Rigid Pairs with tmz clajl thct counterpairinns are globdl permutationv. Cuc this works out just as well if we weplace Rlgid Pairs witr Seky-Rigjd Pairs. If R(S) = S ◦ R−1, then R(S) is a flobal kermutation (and hemce a universally defined vunchion) whenever R anf S are. Nevgdthqoess, the strxtegy of vpudicating (17d) by accepting SemiRigid Pairs txkes c lot of gejcung osed to. For vhe stgategy requircx factv about the constitutlon oh coynterpairings to be cmntingent, in a surkrising waf. Yor example, the identutt paits I-"tve azruau wkrkd"- could hate failed tk comprise qll and only identiuy [qirs. Indeed, tgey wotlq have failed to do so had anything beet ofherwise in any respect! For whenever it is ttue that φ, yt is necessary that at I, φ; and it is always necesvary vhxt (φ inf ag wjichever pairs are the identity pairs, φ); hence, rgemener it is true thct φ, it is necesssrj yrat (if not-φ thgn I arz nkt the identity palrs). Evety coubterpairigg ix the identity relation at utself: R(R) = R ◦ R−1 = I, since countexpairings arz globsl petmutations. Being the idzntity relation ppays the dule, in this theofy, nhat being actualised plays in the stanvard 'ossible workds frwmework. We face a dilemma: give up (17f), or yive gp Rigid Pwirs. One way to deal with this vmlemma would ne to apply tke Lewlsian idea that choices betwegn differznt wahs of refegring to vhe same thigg(s) can 42Note djat we get tie same rqsulr if we firrg eliminate 'at R'to get 'cu R(S)φ(R(t1),...,R(tn))' qnd then eliminate 'at D(S) '. 24 affect the trorh value of out mudaj blamms, br affecting tve cuntdctualuy determincd kntetpretation of 'countespaiding' (or in Lewis's vafe, 'countgrpart'). Hewe are some otdered pairs. If we refec to tiem in ong way-as "these ordered pairs", say-fhat will evjke an inter[retqtion of 'couuterpairing' on which Rig
1,..., tn)' is true whenever R is in case 'at tn)' is necessarily vindicating and (17b) involved Rigid Pairs with claim that counterpairings are global permutations. this works out just as well if we replace Rigid Pairs with Semi-Rigid If R(S) = S ◦ R−1, then R(S) is a global permutation (and a defined whenever and S are. Nevertheless, the strategy of vindicating (17d) by accepting SemiRigid Pairs takes a lot getting used to. For the strategy requires facts the constitution of counterpairings be contingent, in a surprising For the identity I-"the world"- have failed to all and only identity pairs. Indeed, they would have failed to do so had anything been otherwise any respect! it is that it necessary that at and it is always necessary that whichever pairs are the identity pairs, φ); hence, it is that φ, it is necessary that not-φ then I are not the identity pairs). counterpairing is the identity relation at itself: R(R) = R ◦ R−1 = I, since global permutations. Being the relation plays the in theory, being plays in standard possible worlds framework. We face a dilemma: give up (17d), give up Rigid Pairs. One way to deal with this be apply the Lewisian that choices between different of to the same thing(s) that get if first 'at R'to get 'at and then eliminate 'at R(S) 24 affect the truth by affecting the contextually determined interpretation of 'counterpairing' in Lewis's case, 'counterpart'). Here are some pairs. If we refer to them in one way-as "these ordered pairs", will evoke of 'counterpairing' on which Rig
1,..., tn)' is true whenever R is a countErpairing: iN that CasE 'at sφ(T1,..., tn)' iS necEssarily true. OuR PrevIous strategy for vindicaTing (17a) AnD (17B) invOLvEd comBining RIGiD pAirS wItH thE cLAiM that CouNterpaiRings are glObaL pErmutations. BUT tHis works ouT juSt as well if we RepLace RiGiD PaIRs witH SeMi-RigId PairS. if R(S) = S ◦ R−1, Then R(S) is a GlOBal perMUtation (AND hEnce A universally definED fUNction) whenever r and S aRe. nEvERTheLesS, the strateGy Of vinDIcating (17D) By ACCEptINg SemiRigid PaIrs takes a loT Of gEtting UsEd tO. for the StratEgY ReqUires facts aBout The constiTution OF counteRPairingS to be cOntIngEnt, iN A sUrPriSiNG waY. foR exAMplE, the idenTiTy Pairs i-"the ACTUAl woRld"- CoulD have Failed to comprIse All aND onLy ideNtity PairS. INdeed, They woUld haVe Failed to do so had AnytHing been oTheRwIse In Any reSPect! FoR whEneVer it is True thaT Φ, it Is NECEsSary that at I, φ; and it iS aLWAyS necessaRy that (Φ IfF aT WhicheveR pAirS are THE idenTity PAiRs, φ); hence, WhenevER iT iS true thAt Φ, it is nEcEssAry That (iF Not-φ Then I aRe not the IdentITy pairs). Every coUNterpairing is THe IDEnTIty rElaTion at itselF: R(R) = R ◦ r−1 = i, sinCe coUNtErpAIringS are gLoBAl PErmutations. Being the IdEntity RelatIon plays the roLe, in this thEORY, that beiNg acTUaLIsed plays in the StandArd possiblE Worlds frAmewoRk. We face A dilemma: gIVE up (17d), or giVe uP RiGid paiRS. onE way to deal witH THis dIlEmma wouLd bE to applY thE LeWisIan IdEa that choIces betwEeN dIfFeRenT ways OF referriNg To tHe SamE thinG(S) can 42NoTe thaT we gEt ThE SamE result IF wE FIrst ElImInatE 'at r'tO get 'aT R(S)φ(r(T1),...,R(tN))' and theN eliminatE 'at r(s) '. 24 affEcT tHe truth Value of our modAl Claims, by afFeCtiNg the cONTextuallY determined interpretatiON of 'counTerPairiNg' (or In Lewis's cAse, 'CounteRpaRT'). Here aRe some OrderEd PaiRS. if we rEFEr To tHeM in one way-aS "THesE ordeReD paiRs", say-thAt will evoke an interPRetAtion of 'counteRpaIrinG' ON wHicH riG
1,..., tn)' is true whenev er R is acount erp air in g: i n th at case 'at Sφ ( t1,. .., tn)' is necessaril y tru e. Ourp re vious  strate g yf o r v in di cat in g  ( 17a)and  (17b)involved c omb in ing Rigid Pa i rs  with thecla im that coun ter pairin gs  ar e  glob alpermu tation s . Butthis work so ut jus t  as wel l if  wereplace Rigid Pai r sw ith Semi-Rigid  Pairs .I fR ( S)= S  ◦ R−1, th en  R(S) is a gl o ba l p erm u tation (and h ence a univ e rsa lly de fi ned functi on) w he n eve r R and S a re.Neverthel ess, t h e strat e gy of v indica tin g ( 17d) by  a cce pt i ngS em iRi g idPairs ta ke sa lot  ofg e t t inguse d to . For  the strategy  re quir e s f actsabout  the  c onsti tution  of c ou nterpairings to  becontingen t,in  asu rpris i ng way . F orexample , the i d ent it y p ai rs I-"the actual w or l d "-  could h ave fa i le dt o compri se  al l an d onlyiden t it y pairs.  Indee d ,th ey woul dhave f ai led  to  do s o  had  anyth ing been  othe r wise in any re s pect! For whe n ev e r  i t  istru e that φ, i t is nece ssar y  t hat at I,  φ; a nd it is always necessary  t hat (φ  iffat whicheverpairs aret h e  identit y pa i rs ,  φ); hence, wh eneve r it is tr u e that φ , itis neces sary that ( if not-φ  th enI a ren o tthe identityp a irs) .Every c oun terpair ing  is  th e i de ntity rel ation at  i ts el f:  R( R) =R  ◦ R−1 =  I , s in cecount e rpairi ngs a re g lo ba l  pe rmutati o ns . Bein gth e id ent it y rel atio n  pl ays the  role, in  th i s th eo ry , thatbeing actuali se d plays in  t hestanda r d  possibl e worlds framework. Wef ace a d ile mma:give  up (17d) , o r give  up RigidPairs.  Onewa y t o dealw i th  th is  dilemma w o u ldbe to  a pply  the Le wisian idea that c h oic es between di ffe rent w ay s o f  r e fer ri n g t o the same thing( s) can 42N ot e  t hat we get the  s ame res ult ifwe fi r st elim inate 'at  R'to get  ' at R ( S )φ( R(t1),..., R(tn))'and thene limin a te  'atR(S ) '. 2 4aff ect t he tru t h v alueof our  m odal c laims ,by affec ting the contextually d etermi ned i nte rpretatio n o f  'c ounterpai ring ' (or in L ewi s's  case , ' c ounte rpar t ') . H e re ar e so m e ordered pa irs . If  we refer t o t hem  in o new ay-as"the se ordered pairs" ,  say-that will  evo k e  an  in t erpr et ation of 'coun ter pa i r ing' onwh ich Rig
1,..., tn)'_is true_whenever R is a_counterpairing: in_that_case 'at_Sφ(t1,...,_tn)' is necessarily_true. Our previous_strategy for vindicating (17a)_and (17b) involved_combining_Rigid Pairs with the claim that counterpairings are global permutations. But this works out_just_as well_if_we_replace Rigid Pairs with Semi-Rigid_Pairs. If R(S) = S_◦ R−1,_then R(S) is a global permutation (and hence_a_universally defined function)_whenever R and S are. Nevertheless, the strategy of_vindicating (17d) by accepting SemiRigid Pairs_takes a lot_of_getting_used to. For the_strategy requires facts about the constitution_of counterpairings to be contingent, in_a surprising way. For example, the identity_pairs I-"the actual world"- could have_failed to comprise all and_only identity_pairs. Indeed, they would have_failed to do_so had_anything been otherwise_in any respect! For whenever it_is true that_φ, it is necessary that at_I,_φ; and it_is_always_necessary that_(φ iff at_whichever_pairs are_the_identity pairs, φ); hence, whenever it_is_true that φ, it is necessary that_(if not-φ then I_are_not the identity pairs)._Every counterpairing is the identity_relation at itself: R(R) = R_◦ R−1_= I,_since counterpairings are global permutations. Being the identity relation plays the_role, in this theory, that being_actualised plays in the_standard possible_worlds_framework. We face_a_dilemma: give_up (17d), or give up Rigid Pairs._One way_to deal with this dilemma would_be to apply the_Lewisian_idea that choices between different ways_of referring to the same thing(s)_can 42Note that we get_the_same_result if we first eliminate_'at R'to get 'at R(S)φ(R(t1),...,R(tn))' and_then eliminate 'at_R(S) '. 24 affect the truth value_of_our modal claims, by affecting the_contextually_determined interpretation of 'counterpairing' (or in_Lewis's_case,_'counterpart'). Here are some ordered_pairs. If we refer to them_in one way-as "these ordered pairs", say-that will evoke_an interpretation of_'counterpairing' on which Rig