 trickery using plural quantification over ordered triples. Thus: R is a counterpairing in the extended sense ↔df there is a number n and some ordered triples X such that (i) the first element of each of X is a number less than n; (ii) whenever 0 ≤ i < n, the ordered pairs 〈x, y〉 such that 〈i, x, y〉 is one of X are a counterpairing in the old sense; (iii) any ordered pair 〈x, y〉 is one of R iff for some function f, f (0) = x and f (n) = y and whenever 0 ≤ i < n, 〈i, f (i), f (i + 1)〉 is one of X. 39Here is the proof for B: the others are similar. Let C∗R be defined as CR ∨CR−1; let ^∗ and ∗ be defined as in Analysis F but using C∗ instead of C. It is obvious from the definition that for any formula φ, ^φ entails ^∗φ and ∗φ entails φ; what we need to show is that the converse entailments also hold. For simplicity, suppose φ contains just one singular term t and no plural terms. Assume ^∗φ(t): that is, ∃R∃x((CR ∨ CR−1) ∧ Rtx ∧ φ(x)). If CR, then evidently ^φ(t); so suppose CR−1 ∧ Rtx ∧ φ(x). Since B is valid for  and ^, it follows that ^φ(x): that is, ∀R1∀z1((CR1 ∧ R1 yz1)→ ∃R2∃z2(CR2 ∧ R2z1z2 ∧ φ(z2))) 10 "Possible worlds" Lewis often writes in OPW as if the following schematic equivalences were common ground between him and his opponents: (15) a. Possibly φ iff φ at some possible world. b. Necessarily φ iff φ at every possible world. Lewis presents his analysis as the result of analysing 'possible world' in these schemas as 'Lewis-world', and analysing 'at w, φ' (also written 'φ according to w' and 'w represents that φ') in terms of counterparts
trickery using plural quantification over ordered triples. Thus: radius is a counterpairing in the extensive sense ↔df there is a number n and some order triples X such that (one) the beginning element of each of X is a act less than n; (ii) whenever 0 ≤ i < n, the ordered pairs 〈 x, y 〉 such that 〈 i, x, y 〉 is one of adam are a counterpairing in the old sense; (iii) any order pair 〈 x, y 〉 is one of R iff for some function f, f (0) = x and f (n) = y and whenever 0 ≤ i < n, 〈 i, f (i), f (i + 1) 〉 is one of X. 39Here is the proof for B: the others are similar. permit C∗R be defined as CR ∨CR−1; let ^∗ and ∗ be defined as in Analysis F but use C∗ instead of C. It is obvious from the definition that for any formula φ, ^φ entails ^∗φ and ∗φ entails φ; what we necessitate to show is that the converse entailments also hold. For simplicity, presuppose φ contains just one singular term t and no plural terms. Assume ^∗φ(t ): that is, ∃R∃x((CR ∨ CR−1) ∧ Rtx ∧ φ(x) ). If CR, then evidently ^φ(t); so suppose CR−1 ∧ Rtx ∧ φ(x). Since B is valid for  and ^, it follows that ^φ(x ): that is, ∀R1∀z1((CR1 ∧ R1 yz1)→ ∃R2∃z2(CR2 ∧ R2z1z2 ∧ φ(z2) )) 10 " Possible worlds " Lewis often writes in OPW as if the following schematic equivalences were coarse ground between him and his opponents: (15) a. Possibly φ iff φ at some potential populace. b. inevitably φ iff φ at every possible world. Lewis presents his analysis as the result of analysing' possible world' in these schema as' Lewis - world', and analysing' at w, φ' (also written' φ according to w' and' w represents that φ') in term of counterparts
 trlckery using plural quanuification over otdwred tciples. Fhus: R ir a counterpairing in the exvendwd sebse ↔df there is a numbdr n and dome ordwred rriples X such thab (i) tgc firvv element of eagh of X is d number less dhxn n; (ii) whenever 0 ≤ i < n, the ordered pwirs 〈x, u〉 duch that 〈i, x, i〉 is pge or X are a counterpairing in the ols sense; (iii) any ordeted pair 〈x, y〉 is one of R ivf flr some function f, f (0) = x and d (n) = t and whenevdr 0 ≤ i < n, 〈p, f (i), f (i + 1)〉 is one of X. 39Here is the proof wor B: the otherw qre vimilar. Let C∗R bv defined as GG ∨CR−1; led ^∗ and ∗ be defined as in Anqlysis F but using C∗ mnstead of C. It is ofvious frmm the definition tyar for any foroyla φ, ^φ tnteila ^∗φ and ∗φ eitails φ; whaf we need ti show is that the vognrrse entailmsnts ajsj hold. For simplicity, suppose φ contains juat one singular term t qnd no plural terms. Adsume ^∗φ(t): trat is, ∃R∃x((CR ∨ CR−1) ∧ Rtx ∧ φ(x)). If CR, then evidently ^φ(t); so suppkre EV−1 ∧ Etd ∧ φ(x). Since B is valid for  and ^, it follows thwf ^φ(c): nhat is, ∀R1∀z1((CR1 ∧ R1 yd1)→ ∃R2∃z2(CR2 ∧ R2z1z2 ∧ φ(z2))) 10 "Poxslbkg worlds" Lewis often wrjtes in OPW as if hhe foljowint schematyc eauivalences were common groynd between kim and his opponents: (15) a. Possiblv φ iff φ at xome possible world. b. Nzcessadily φ iff φ wt every lussible world. Ledis psesents his analysis as thq result if aualysing 'posxible rorld' in tjese schemas as 'Lewis-worpd', anb ananysing 'at a, φ' (also written 'φ according to w' and 'w represemtv tvat φ') in cerms pf counterpares
trickery using plural quantification over ordered triples. is counterpairing in extended sense ↔df and ordered triples X that (i) the element of each of X is number less than n; (ii) whenever 0 ≤ i < n, the ordered 〈x, y〉 such that 〈i, x, y〉 is one of X are a in old (iii) ordered pair 〈x, y〉 is one of R iff for some function f, f (0) = and f (n) = y and whenever 0 i < n, 〈i, (i), f (i + 1)〉 one X. 39Here the for the others are Let C∗R be defined as CR ∨CR−1; let ^∗ and ∗ be defined as in Analysis F using C∗ C. It obvious the that for any ^φ entails ^∗φ and ∗φ entails need to show is that the converse entailments hold. For suppose φ contains just one singular t and no plural terms. Assume ^∗φ(t): that ∃R∃x((CR ∨ CR−1) ∧ Rtx ∧ φ(x)). If CR, then evidently ^φ(t); so suppose CR−1 ∧ φ(x). Since B valid for  ^, follows ^φ(x): is, ∀R1∀z1((CR1 R1 yz1)→ ∃R2∃z2(CR2 ∧ R2z1z2 ∧ φ(z2))) 10 "Possible worlds" Lewis writes in OPW as if the following schematic equivalences were between and his opponents: a. Possibly φ iff at possible world. b. Necessarily φ every presents analysis the result of analysing world' in these schemas as and analysing 'at w, to w' and 'w represents that φ') in of counterparts
 trickery using plural quantiFication ovEr ordEreD trIpLes. THus: R Is a counterpairINg in The extended sense ↔df therE is a nUmBEr n aND sOme orDered trIPlES x suCh ThAt (i) ThE FiRst elEmeNt of eacH of X is a numBer LeSs than n; (ii) wheNEvEr 0 ≤ i < n, the ordEreD pairs 〈x, y〉 such ThaT 〈i, x, y〉 is OnE of x Are a cOunTerpaIring iN The old Sense; (iii) aNy ORdered PAir 〈x, y〉 is ONE oF R ifF for some function f, F (0) = X aND f (n) = y and wheneveR 0 ≤ i < n, 〈i, f (i), F (i + 1)〉 IS oNE Of X. 39herE is the prooF fOr B: thE Others aRE sIMILar. lEt C∗R be defined As CR ∨CR−1; let ^∗ anD ∗ Be dEfined As In ANAlysis f but uSiNG C∗ iNstead of C. It Is obVious from The defINition tHAt for anY formuLa φ, ^Φ enTailS ^∗Φ aNd ∗Φ enTaILs φ; WHaT we NEed To show is ThAt The coNverSE ENTailMenTs alSo holD. For simplicitY, suPposE Φ coNtainS just One sInGular Term t aNd no pLuRal terms. Assume ^∗φ(T): thaT is, ∃R∃x((CR ∨ CR−1) ∧ rtx ∧ Φ(x)). if Cr, tHen evIDently ^Φ(t); sO suPpose CR−1 ∧ rtx ∧ φ(x). SiNCe B Is VALId For  and ^, it follows thaT ^φ(X): THaT is, ∀R1∀z1((CR1 ∧ R1 Yz1)→ ∃R2∃z2(CR2 ∧ r2Z1z2 ∧ Φ(z2))) 10 "pOssible wOrLds" lewiS OFten wRiteS In oPW as if tHe follOWiNg SchematIc EquivaLeNceS weRe comMOn grOund beTween him And hiS Opponents: (15) a. PossIBly φ iff φ at somE PoSSIbLE worLd. b. necessarily Φ iff Φ At evEry pOSsIblE World. lewis PrESeNTs his analysis as the rEsUlt of aNalysIng 'possible woRld' in these SCHEmas as 'LeWis-wORlD', And analysing 'at W, φ' (alsO written 'φ aCCording tO w' and 'W represeNts that φ') iN TErms of coUntErpArtS
 trickery using plural qua ntificatio n ove r o rde re d tr iple s. Thus: R isa  cou nterpairing in the ext ended  s e nse↔ df  ther e is an um b e r n  a nd  so me or dered  tr iples X  such that  (i )the first el e me nt of each  of  X is a numb erless t ha n n ;  (ii)  wh eneve r 0 ≤i  < n,the order ed pairs〈 x, y〉 s u c hthat  〈i, x, y〉 is one of X are a counte rpairi ng in t heold  sense; (i ii ) any ordered pa i r 〈x, y〉 is one ofR iff for s o mefuncti on  f, f (0)= x a nd f ( n) = y andwhen ever 0 ≤i < n, 〈i, f ( i ), f (i  + 1)〉  is  on e of X.  3 9He re ist he  pr o offor B: t he  o thers  are s i m ilar . L et C ∗R be  defined as C R ∨ CR−1 ;  le t ^∗and  ∗ be  d efine d as i n Ana ly sis F but using  C∗instead o f C .Itis  obvi o us fro m t hedefinit ion tha t  fo ra n y  f ormula φ, ^φ entai ls ^ ∗φ  and ∗φ  entai l sφ ;  what we  n eed  tos h ow is  tha t  t he conve rse en t ai lm ents al so  hold.  F orsim plici t y, s uppose  φ conta ins j u st one singula r  term t and n o  p l u ra l  ter ms.  Assume ^∗φ (t): that  is, ∃R ∃x( ( CR ∨CR−1)  ∧ Rt x  ∧ φ(x)). If CR, th en  evide ntly^φ(t); so sup pose CR−1∧ R tx ∧ φ(x ). S i nc e  B is valid fo r  a nd ^, it f o llows th at ^ φ(x): th at is, ∀R 1 ∀ z1((CR1∧ R 1 y z1) → ∃ R 2 ∃z 2(CR2 ∧ R2z1z 2 ∧ φ( z2 ))) 10"Po ssiblewor lds " L ewi soften wri tes in O PW  a sif  th e fol l owing sc he mat ic  eq uival e nces w ere c ommo ngr o und  betwee n  h i m  and  h is  opp one nt s: (1 5) a .  Po ssiblyφ iff φ a t s o me p os si ble wor ld. b. Necess ar ily φ iffφateveryp o ssible w orld. Lewis presents hi s  analys isas th e re sult of a nal ysing'po s sibleworld'  in t he ses c hemas a s'Le wi s-world',a n d a nalys in g 'a t w, φ'  (also written 'φa cco rding to w' a nd'w r e p re sen t st hat  φ ' ) i n terms of counte rparts
 trickery_using plural_quantification over ordered triples._Thus: R_is_a counterpairing_in_the extended sense_↔df there is_a number n and_some ordered triples_X_such that (i) the first element of each of X is a number less_than_n; (ii)_whenever_0_≤ i < n, the_ordered pairs 〈x, y〉 such_that 〈i,_x, y〉 is one of X are a_counterpairing_in the old_sense; (iii) any ordered pair 〈x, y〉 is one_of R iff for some function_f, f (0)_=_x_and f (n) =_y and whenever 0 ≤ i_< n, 〈i, f (i), f_(i + 1)〉 is one of X._39Here is the proof for B:_the others are similar. Let_C∗R be_defined as CR ∨CR−1; let_^∗ and ∗_be defined_as in Analysis_F but using C∗ instead of_C. It is_obvious from the definition that for_any_formula φ, ^φ_entails_^∗φ_and ∗φ_entails φ; what_we_need to_show_is that the converse entailments also_hold._For simplicity, suppose φ contains just one_singular term t and_no_plural terms. Assume ^∗φ(t):_that is, ∃R∃x((CR ∨ CR−1)_∧ Rtx ∧ φ(x)). If CR,_then evidently_^φ(t); so_suppose CR−1 ∧ Rtx ∧ φ(x). Since B is valid for_ and ^, it follows that_^φ(x): that is, ∀R1∀z1((CR1_∧ R1_yz1)→_∃R2∃z2(CR2 ∧ R2z1z2_∧_φ(z2))) 10_"Possible worlds" Lewis often writes in OPW_as if_the following schematic equivalences were common_ground between him and_his_opponents: (15) a. Possibly φ iff_φ at some possible world. b._Necessarily φ iff φ at_every_possible_world. Lewis presents his analysis_as the result of analysing 'possible_world' in these_schemas as 'Lewis-world', and analysing 'at w,_φ'_(also written 'φ according to w'_and_'w represents that φ') in terms_of_counterparts