 accessible from R' should be '∃Q(ψ(R(a1),...,R(a j),Q) ∧ S = Q ◦ R)'. If counterpairings are global permutations, then we can express this analysis in terms of C: in that case, '∃Q(ψ(R(a1),...,R(a j),Q) ∧ S = Q ◦ R)' is equivalent to 'ψ(R(a1),...,R(a j),S ◦ R−1)', which is equivalent to 'ψ(R(a1),...,R(a j),R(R−1(S ◦ R−1)))', which is equivalent to 23 The most difficult to vindicate is (17d). This is an important part of the standard "possible world" role: much of our ordinary talk about possible worlds would have to be drastically rethought if facts about what is the case at a given world could be contingent. But none of the constraints on counterpairings that we have encountered so far entails (17d). Worse: the very appealing principle Rigid Pairs entails that what is the case according to a counterpairing often is contingent. For Rigid Pairs entails that counterpairings map identity pairs to identity pairs: if R〈x, x〉〈y, z〉, then y = z. If so, each counterpairing must map I, the plurality of all identity pairs, to some plurality J of identity pairs. (If we counterpairings must also have universal ranges, J = I.) But the conjunction of 'at Jφ' with the claim that J consists only of identity pairs entails φ. Thus for any counterpairing R, if 'at R (at Iφ)' is true, then 'at Rφ' is true. It is necessary that if at I (φ), then φ. (And if counterpairings have universal ranges the converse is also necessary). So whenever it is contingent whether φ, it is contingent whether at Iφ.41 Thus, to render claims of the form 'at R (φ)' non-contingent, counterpairings will have to treat ordered pairs in a strange way that does not fit with the intuition that they are rigid. Can we do it if we are prepared to bite this bullet? Yes: it turns out that everything works out nicely
accessible from R' should be' ∃Q(ψ(R(a1),... ,R(a j),Q) ∧ S = Q ◦ R)'. If counterpairings are global permutations, then we can carry this psychoanalysis in terms of C: in that subject,' ∃Q(ψ(R(a1),... ,R(a j),Q) ∧ S = Q ◦ gas constant)' is equivalent to' ψ(R(a1),... ,R(a j),S ◦ R−1)', which is equivalent to' ψ(R(a1),... ,R(a j),R(R−1(S ◦ R−1) ))', which is equivalent to 23 The most difficult to justify is (17d). This is an important part of the standard " possible universe " role: much of our ordinary lecture about possible worlds would have to be drastically rethought if facts about what is the shell at a given world could be contingent. But none of the constraints on counterpairings that we have encountered so far entail (17d). Worse: the very appealing principle Rigid Pairs fee-tail that what is the case according to a counterpairing often is contingent. For Rigid Pairs entail that counterpairings map identity pairs to identity pairs: if R〈x, x〉〈y, z 〉, then y = z. If so, each counterpairing must map I, the plurality of all identity pairs, to some battalion J of identity pairs. (If we counterpairings must also have universal ranges, J = I.) But the conjunction of' at Jφ' with the title that J consists only of identity pairs entails φ. Thus for any counterpairing R, if' at R (at Iφ)' is true, then' at Rφ' is true. It is necessary that if at I (φ), then φ. (And if counterpairings have cosmopolitan stove the converse is also necessary). indeed whenever it is contingent whether φ, it is contingent whether at Iφ.41 therefore, to hand over claims of the form' at R (φ)' non - contingent, counterpairings will have to treat ordered pairs in a strange direction that does not fit with the intuition that they are rigid. Can we do it if we are prepared to bite this bullet? Yes: it turn out that everything works out nicely
 acfessible from R' should bt '∃Q(ψ(R(a1),...,R(a j),Q) ∧ S = Q ◦ R)'. If cmunterlairings are global permutations, thei we can tqpress this analysis kn terms lf C: in rhat xase, '∃Q(ψ(R(a1),...,R(a j),Q) ∧ S = Q ◦ R)' ia equnvelent to 'ψ(R(a1),...,R(a j),X ◦ R−1)', which is equivalent tu 'ψ(X(a1),...,R(a j),R(R−1(S ◦ R−1)))', which is equivalent to 23 The moxt difficult to dindpcwte js (17d). This is an important part of fhe staidard "possible eorld" role: much of our ordlnarj talk about possihle worlds qoulq have to be arasticallj rethought if facts about what is the casd at c given worod coknd be contiigent. Fut none of bne convtraintx on counterpalringv tyat we have encounterxd so far entails (17d). Rorse: the vzry appealing principoe Rigig Pahrs dbtakls tiat what ls vhe case acdording to q counterpairing ofuen px contingent. For Rygyd Pairs entails that counterpairings mdp jdentity pairs to identuty pairs: if R〈x, x〉〈y, z〉, jhen y = z. Yf so, each counterpairing must map I, the pluralitf of elu ibcktith pwirs, to some plurality J of identity pairs. (If se cpunterpairings must also nage oniversal ranggs, J = I.) Buf the conjunction lf 'at Jφ' with the claii thst J consists only of identuty pairs ennailw φ. Thus for any cobnterpairing R, ig 'at T (at Iφ)' is true, then 'at Xφ' is tdue. It is nfcessary ffat if at I (φ), thev φ. (Sng if counterpairings have tniversal ranyes the zonvgrse is also necedsary). So whenever it is fontiugent whether φ, lt is contingent whether at Iφ.41 Tins, to render vldimv of the form 'at R (φ)' non-coneingent, countetpairings will have to tgeat ordeced pairs in a strange waf that does nmt fit wyth rhe untuitiuv that they arr rigid. Ccu we do ir if we are preparcd to gite this bullec? Iws: it turns ouy tfat enerbthind works out nhcelh
accessible from R' should be '∃Q(ψ(R(a1),...,R(a j),Q) = ◦ R)'. counterpairings are global this in terms of in that case, j),Q) ∧ S = Q ◦ is equivalent to 'ψ(R(a1),...,R(a j),S ◦ R−1)', which is equivalent to 'ψ(R(a1),...,R(a j),R(R−1(S R−1)))', which is equivalent to 23 The most difficult to vindicate is (17d). is important of standard "possible world" role: much of our ordinary talk about possible worlds would have to be rethought if facts about what is the case a given world could contingent. But none of the on that we encountered far (17d). Worse: the appealing principle Rigid Pairs entails that what is the case according to a counterpairing often is contingent. Rigid Pairs counterpairings map pairs identity if R〈x, x〉〈y, y = z. If so, each I, the plurality of all identity pairs, to plurality J identity pairs. (If we counterpairings must have universal ranges, J = I.) But the of 'at Jφ' with the claim that J consists only of identity pairs entails φ. any counterpairing R, if R (at Iφ)' true, 'at is It is that if at I (φ), then φ. (And if counterpairings have ranges the converse is also necessary). So whenever it is φ, is contingent whether Iφ.41 Thus, to render of form 'at R (φ)' will to in strange that does not fit the intuition that they are Can we do it bite this bullet? Yes: it turns out that works out nicely
 accessible from R' should be '∃Q(ψ(r(a1),...,R(a j),Q) ∧ S = Q ◦ R)'. IF counTerPaiRiNgs aRe glObal permutatioNS, theN we can express this analySis in TeRMs of c: In That cAse, '∃Q(ψ(R(a1),...,r(A j),q) ∧ s = q ◦ R)' iS eQuIvaLeNT tO 'ψ(R(a1),...,R(A j),S ◦ r−1)', which iS equivalenT to 'Ψ(R(A1),...,R(a j),R(R−1(S ◦ R−1)))', whicH Is Equivalent To 23 THe most difficUlt To vindIcAte IS (17d). ThiS is An impOrtant PArt of tHe standarD "pOSsible WOrld" rolE: MUcH of oUr ordinary talk aboUT pOSsible worlds woUld havE tO Be DRAstIcaLly rethougHt If facTS about wHAt IS THe cASe at a given worLd could be coNTinGent. BuT nOne OF the coNstraInTS on CounterpairIngs That we havE encouNTered so FAr entaiLs (17d). WorSe: tHe vEry aPPeAlIng PrINciPLe rigID PaIrs entaiLs ThAt whaT is tHE CASe acCorDing To a coUnterpairing oFteN is cONtiNgent. for RiGid PAiRs entAils thAt couNtErpairings map idEntiTy pairs to IdeNtIty PaIrs: if r〈X, x〉〈y, z〉, thEn y = Z. If So, each cOunterpAIriNg MUST mAp I, the plurality of aLl IDEnTity pairS, to somE PlUrALity J of iDeNtiTy paIRS. (If we CounTErPairings Must alSO hAvE univerSaL rangeS, J = i.) BuT thE conjUNctiOn of 'at jφ' with thE claiM That J consists oNLy of identity pAIrS ENtAIls φ. thuS for any counTerpAIrinG R, if 'AT R (At IΦ)' Is truE, then 'At rΦ' iS True. It is necessary thAt If at I (φ), Then φ. (and if counterpAirings havE UNIversal rAngeS ThE Converse is also NecesSary). So whenEVer it is cOntinGent whetHer φ, it is cONTingent wHetHer At IΦ.41 ThUS, To Render claims oF THe foRm 'At R (φ)' non-ConTingent, CouNteRpaIriNgS will have To treat oRdErEd PaIrs In a stRAnge way tHaT doEs Not Fit wiTH the inTuitiOn thAt ThEY arE rigid. CAN wE DO it iF wE aRe prEpaReD to biTe thIS buLlet? Yes: It turns ouT thAT eveRyThIng workS out nicely
 accessible from R' should  be '∃Q(ψ( R(a1) ,.. .,R (a  j), Q) ∧  S = Q ◦ R)'.I f co unterpairings are glob al pe rm u tati o ns , the n we ca n  e x p res sth isan a ly sis i n t erms of  C: in tha t c as e, '∃Q(ψ(R(a 1 ), ...,R(a j) ,Q)  ∧ S = Q ◦ R )'is equ iv ale n t to'ψ( R(a1) ,...,R ( a j),S  ◦ R−1)',  w h ich is equival e n tto ' ψ(R(a1),...,R(a j ) ,R ( R−1(S ◦ R−1))) ', whi ch is e qui val ent to 23Th e mos t  diffic u lt t o  vi n dicate is (17 d). This is animport an t p a rt ofthe s ta n dar d "possible  wor ld" role:  mucho f our o r dinarytalk a bou t p ossi b le  w orl ds wou l dhav e  to  be dras ti ca lly r etho u g h t  iffac ts a boutwhat is the c ase  ata  gi ven w orldcoul dbe co ntinge nt. B ut  none of the co nstr aints oncou nt erp ai rings that w e h ave  encoun tered s o  fa re n t ai ls (17d). Worse: t he v er y appeal ing pr i nc ip l e RigidPa irs  ent a i ls th at w h at  is thecase a c co rd ing toacounte rp air ing  ofte n  iscontin gent. Fo r Rig i d Pairs entail s  that counter p ai r i ng s  map  id entity pair s to iden tity pa irs :  if R 〈x, x 〉〈 y ,z 〉, then y = z. If s o,  eachcount erpairing mus t map I, t h e pluralit y of al l  identity pair s, to  some plur a lity J o f ide ntity pa irs. (Ifw e  counter pai rin gsmus t al so have unive r s al r an ges, J= I .) Butthe  co nju nct io n of 'atJφ' with  t he  c la imthatJ  consist sonl yofident i ty pai rs en tail sφ. Thu s for a n yc o unte rp ai ring  R,  i f 'at  R ( a t I φ)' istrue, the n ' a t Rφ 'is  true.It is necessa ry  that if a tI ( φ), th e n  φ. (And  if counterpairings hav e  univer sal  rang es t he conver seis als o n e cessar y). So  when ev eri t  is c o n ti nge nt  whether φ , itis co nt inge nt whet her at Iφ.41 Thus, torender claims  of  the f or m ' a tR  (φ )' non - c ontingent, coun terpairing sw il l have tot rea tordered  pairsin as trangeway thatdoes notfi t wi t h  th e intuitio n that t hey are r i gid.C an  we d o i t if w eare  prep ared t o  bi te th is bul le t? Yes : ittu rns outthat everything works o ut nic ely
 accessible_from R'_should be '∃Q(ψ(R(a1),...,R(a j),Q)_∧ S_=_Q ◦_R)'._If counterpairings are_global permutations, then_we can express this_analysis in terms_of_C: in that case, '∃Q(ψ(R(a1),...,R(a j),Q) ∧ S = Q ◦ R)' is equivalent_to_'ψ(R(a1),...,R(a j),S_◦_R−1)',_which is equivalent to 'ψ(R(a1),...,R(a_j),R(R−1(S ◦ R−1)))', which is_equivalent to_23 The most difficult to vindicate is (17d)._This_is an important_part of the standard "possible world" role: much of_our ordinary talk about possible worlds_would have to_be_drastically_rethought if facts about_what is the case at a_given world could be contingent. But_none of the constraints on counterpairings that_we have encountered so far entails_(17d). Worse: the very appealing_principle Rigid_Pairs entails that what is_the case according_to a_counterpairing often is_contingent. For Rigid Pairs entails that_counterpairings map identity_pairs to identity pairs: if R〈x,_x〉〈y,_z〉, then y_=_z._If so,_each counterpairing must_map_I, the_plurality_of all identity pairs, to some_plurality_J of identity pairs. (If we counterpairings_must also have universal_ranges,_J = I.) But_the conjunction of 'at Jφ'_with the claim that J consists_only of_identity pairs_entails φ. Thus for any counterpairing R, if 'at R (at_Iφ)' is true, then 'at Rφ'_is true. It is_necessary that_if_at I (φ),_then_φ. (And_if counterpairings have universal ranges the converse_is also_necessary). So whenever it is contingent_whether φ, it is_contingent_whether at Iφ.41 Thus, to render_claims of the form 'at R_(φ)' non-contingent, counterpairings will have_to_treat_ordered pairs in a strange_way that does not fit with_the intuition that_they are rigid. Can we do it_if_we are prepared to bite this_bullet?_Yes: it turns out that everything_works_out_nicely