 behave. Let me sketch the strategy in a bit more detail. We start with a languageLACT with the modal operators ,^ and ACT, singular and plural quantifiers and variables, a predicate C taking one plural argument, a distinguished plural term I, and the ordered-pair-forming functor '〈*, *〉'. We will recursively specify a translation function from LACT into L, the modal-operator-free sublanguage of LACT. Our translation will use sentences of a third language LW as intermedi26 aries. LW is like L except that (i) its plural terms can optionally be underlined, in which case we will call them "world terms", and (ii) it contains an operator 'at' that takes a world term W and a sentence φ to make a sentence pat W (φ)q. The translation from LACT to LW is straightforward: wherever we have '^φ', we substitute '∃V(CV∧at Vφ)', where V is some new world variable; wherever we have 'φ', we substitute '∀V(CV → at V (φ))'; wherever we have 'ACTφ', we substitute 'at I (φ)'. The central task in translating from LW to L is to eliminate all occurrences of the 'At' operator. To do so, we replace each formula of the form 'at Rφ(t1,..., tn)' with φ(α1,..., αn), where αi is pR(ti)q when ti is a singular term or a non-underlined plural term, and αi is pti ◦ R−1q when ti is an world term. As usual, we treat sentences involving complex terms like 'R(a)' and 'S◦R−1' in Russellian fashion, as abbreviations for quantified sentences of LW: '(S ◦ R−1)ab', for example, abbreviates '∃Q(∀xy(Qxy ↔ ∃z(Rzx ∧ Szy)) ∧ Qab)'. By repeatedly applying this rule, we can turn each sentence of LW into an 'at'-free sentence. Finally, we can translate this sentence into L simply by dropping all the underlines. Here is an example of the machinery at work. Suppose our original sentence φ of LACT is (19) ∃x^∃y ACT Fxy.
behave. Let me sketch the strategy in a bit more contingent. We depart with a languageLACT with the modal operators ,^ and ACT, singular and plural quantifiers and variable star, a predicate speed of light taking one plural argument, a distinguished plural terminus I, and the order - couple - form functor' 〈 *, * 〉'. We will recursively specify a transformation function from LACT into L, the modal - operator - free sublanguage of LACT. Our translation will practice sentences of a third language LW as intermedi26 aries. LW is like L except that (iodine) its plural terms can optionally be underlined, in which event we will call them " world terms ", and (ii) it contain an operator' at' that takes a world term W and a conviction φ to make a sentence pat W (φ)q. The translation from LACT to LW is straightforward: wherever we have' ^φ', we substitute' ∃V(CV∧at Vφ)', where V is some newfangled world variable; wherever we have' φ', we substitute' ∀V(CV → at V (φ) )'; wherever we have' ACTφ', we substitute' at I (φ)'. The central task in translating from LW to L is to eliminate all occurrences of the' At' operator. To do so, we replace each formula of the form' at Rφ(t1, ..., tn)' with φ(α1, ..., αn), where αi is pR(ti)q when ti is a singular term or a non - underlined plural condition, and αi is pti ◦ R−1q when ti is an world term. As common, we regale prison term involving complex terms like' R(a)' and'S ◦ R−1' in Russellian fashion, as abbreviations for quantified sentences of LW:' (S ◦ R−1)ab', for case, abbreviates' ∃Q(∀xy(Qxy ↔ ∃z(Rzx ∧ Szy) ) ∧ Qab)'. By repeatedly applying this rule, we can turn each sentence of LW into an' at'-free sentence. last, we can translate this sentence into L simply by dropping all the underscore. Here is an example of the machinery at oeuvre. presuppose our original sentence φ of LACT is (19) ∃x^∃y ACT Fxy.
 bejave. Let me sketch the surategy in a bit more devail. We start wkth a languageLACT with the loeal okvrators ,^ and ACT, singjlar and ilural quqntihiers and variables, a prcbicats C tckmng one plural srgument, a distinguished puuxal term I, and the ordered-pair-formind functpr '〈*, *〉'. We will recorsivtly spedpfn a translation function from LADT into L, the modal-olerator-free sublanguage of LACH. Our translation aill use sebtensws of a thira language LW as interjedi26 aries. LW is like L except ghat (n) its plurao rerlv can optioially fe underlined, in whiwh case we will call bhem "xorle terms", and (ii) it convains an operator 'at' that takas a world term W abd a settenwe φ gi mxke a ssntencf pet W (φ)q. The franslation from LACT to LW is segsightforward: wheredew we have '^φ', we substitute '∃V(CV∧at Vφ)', where N is some new world variablw; wherever we have 'φ', wg substituee '∀V(CV → at V (φ))'; wherever we have 'ACTφ', we substitute 'dt I (φ)'. Ghe ccntrxo hask in translating from LW to L is to eliminwfe akl occurrences of the 'At' ppfrsjor. To do so, wg replaes sach formula of thf form 'wt Rφ(t1,..., tn)' with φ(α1,..., αn), wnere αi is pR(ti)q when ti is q singular tvrm ir a non-underlined plural tero, anc αi ix pti ◦ R−1q when ti is an wormd term. As ksual, we ffeat sentences ivvokvhng complex terms like 'R(a)' wnd 'S◦R−1' in Rusxellian fasnion, af abbreviahions for quantified senhenced mf LW: '(S ◦ R−1)wb', for example, abbreviates '∃Q(∀xy(Qeb ↔ ∃z(Rzx ∧ Szy)) ∧ Qdb)'. Ty repeacedly spplying this rule, we can torn each fentevce of LW pnto an 'av'-free sentense. Finally, we fan translatx this segtenxe ibto L skoply by droppimg all thv bnderlinew. Here is an exampke ur the machinery at work. Suppose oir uridijak fantence φ of NACT is (19) ∃x^∃y AZT Fxy.
behave. Let me sketch the strategy in more We start a languageLACT with ACT, and plural quantifiers variables, a predicate taking one plural argument, a distinguished term I, and the ordered-pair-forming functor '〈*, *〉'. We will recursively specify a function from LACT into L, the modal-operator-free sublanguage of LACT. Our translation will sentences a language as intermedi26 aries. LW is like L except that (i) its plural terms can optionally be in which case we will call them "world and (ii) it contains operator 'at' that takes a term and a φ make sentence pat W The translation from LACT to LW is straightforward: wherever we have '^φ', we substitute '∃V(CV∧at Vφ)', where is some variable; wherever have we '∀V(CV → at wherever we have 'ACTφ', we substitute The central task in translating from LW to is to all occurrences of the 'At' operator. do so, we replace each formula of the 'at Rφ(t1,..., tn)' with φ(α1,..., αn), where αi is pR(ti)q when ti is a singular a non-underlined plural term, αi is pti R−1q ti an term. As we treat sentences involving complex terms like 'R(a)' and 'S◦R−1' in fashion, as abbreviations for quantified sentences of LW: '(S ◦ example, '∃Q(∀xy(Qxy ↔ ∃z(Rzx Szy)) ∧ Qab)'. By applying rule, we can turn of into Finally, can this sentence into L by dropping all the underlines. is an example of our original sentence φ of LACT is (19) ACT Fxy.
 behave. Let me sketch the stratEgy in a bit mOre deTaiL. We StArt wIth a LanguageLACT wiTH the Modal operators ,^ and ACT, siNgulaR aND pluRAl QuantIfiers aND vARIabLeS, a PreDiCAtE C takIng One plurAl argument, A diStInguished pluRAl Term I, and thE orDered-pair-forMinG functOr '〈*, *〉'. we wILl recUrsIvely SpecifY A transLation funCtIOn from laCT into l, THe ModaL-operator-free sublANgUAge of LACT. Our trAnslatIoN WiLL Use SenTences of a tHiRd lanGUage LW aS InTERMedI26 Aries. LW is like l except that (I) Its Plural TeRms CAn optiOnallY bE UndErlined, in whIch cAse we will Call thEM "world tERms", and (iI) it conTaiNs aN opeRAtOr 'At' tHaT TakES a WorLD teRm W and a sEnTeNce φ tO makE A SENtenCe pAt W (φ)Q. The tRanslation froM LAcT to lw is StraiGhtfoRwarD: wHerevEr we haVe '^φ', we SuBstitute '∃V(CV∧at Vφ)', WherE V is some nEw wOrLd vArIable; WHereveR we HavE 'φ', we subStitute '∀v(cV → aT V (Φ))'; WHErEver we have 'ACTφ', we suBsTITuTe 'at I (φ)'. ThE centrAL tAsK In translAtIng From lw To L is To elIMiNate all oCcurreNCeS oF the 'At' oPeRator. TO dO so, We rEplacE Each FormulA of the foRm 'at RΦ(T1,..., tn)' with φ(α1,..., αn), wheRE αi is pR(ti)q wheN Ti IS A sINgulAr tErm or a non-unDerlINed pLuraL TeRm, aND αi is Pti ◦ R−1q WhEN tI Is an world term. As usuaL, wE treat SenteNces involving Complex terMS LIke 'R(a)' and 's◦R−1' in rUsSEllian fashion, aS abbrEviations fOR quantifIed seNtences oF LW: '(S ◦ R−1)ab', foR EXample, abBreViaTes '∃q(∀xy(qXY ↔ ∃z(rzx ∧ Szy)) ∧ Qab)'. By rePEAtedLy ApplyinG thIs rule, wE caN tuRn eAch SeNtence of Lw into an 'aT'-fReE sEnTenCe. FinALly, we can TrAnsLaTe tHis seNTence iNto L sImplY bY dROppIng all tHE uNDErliNeS. HEre iS an ExAmple Of thE MacHinery aT work. SuppOse OUr orIgInAl senteNce φ of LACT is (19) ∃x^∃Y AcT Fxy.
 behave. Let me sketch the  strategyin abit  mo re  det ail.  We start with a la nguageLACT with the mo dal o pe r ator s   ,^ an d ACT,s in g u lar  a nd  pl ur a lquant ifi ers and  variables , a  p redicate C t a ki ng one plu ral  argument, a  di stingu is hed plura l t erm I , andt he ord ered-pair -f o rmingf unctor' 〈 *,  *〉' . We will recursi v el y  specify a tra nslati on fu n c tio n f rom LACT i nt o L,t he moda l -o p e r ato r -free sublang uage of LAC T . O ur tra ns lat i on wil l use  s e nte nces of a t hird  language  LW as interme d i26 ari es. LW  is  li ke L ex ce ptth a t ( i )its plu ral term sca n opt iona l l y be u nde rlin ed, i n which casewewill cal l the m "wo rldte rms",  and ( ii) i tcontains an ope rato r 'at' th atta kes  a  worl d  termW a nda sente nce φ t o  ma ke a se ntence pat W (φ)q.  T h e  t ranslati on fro m  L AC T  to LW i sstr aigh t f orwar d: w h er ever wehave ' ^ φ' ,we subs ti tute ' ∃V (CV ∧at  Vφ)' ,  whe re V i s some n ew wo r ld variable; w h erever we hav e  '  φ ', we s ubs titute '∀V( CV → at V  (φ) ) ';  wh e rever  we h av e  ' A CTφ', we substitute  ' at I ( φ)'.The central t ask in tra n s l ating fr om L W  t o  L is to elimi nateall occurr e nces ofthe ' At' oper ator. Tod o  so, werep lac e e ach f or mula of the f o r m 'a tRφ(t1,. ..,  tn)' w ith  φ( α1, ... ,αn), wher e αi ispR (t i) qwhe n tii s a sing ul arte rmor an on-und erlin ed p lu ra l  te rm, and αi i s pt i◦R−1q  wh en  ti i s an wor ld term . As usua l,w e tr ea tsentenc es involvingco mplex term slik e 'R(a ) '  and 'S◦ R−1' in Russellian fash i on, asabb revia tion s for qua nti fied s ent e nces o f LW:'(S ◦  R −1) a b ', fo r ex amp le , abbrevia t e s ' ∃Q(∀x y( Qxy↔ ∃z(Rz x ∧ Szy)) ∧ Qab)'. Byrepeatedly ap ply ingt h is  ru l e, weca n  tu r n  each sentenceof LW into  a n  ' at'-free s e nte nc e. Fina lly, we  cant ranslat e this se ntence in to  L s i m ply  by droppi ng all t he underl i nes.H er e isanexampl eofthe m achine r y a t wor k. Sup po se our  orig in al sente nce φ of LACT is (19) ∃ x^∃y A CT Fx y.
 behave._Let me_sketch the strategy in_a bit_more_detail. We_start_with a languageLACT_with the modal_operators ,^ and ACT,_singular and plural_quantifiers_and variables, a predicate C taking one plural argument, a distinguished plural term I,_and_the ordered-pair-forming_functor_'〈*,_*〉'. We will recursively specify_a translation function from LACT_into L,_the modal-operator-free sublanguage of LACT. Our translation will_use_sentences of a_third language LW as intermedi26 aries. LW is like_L except that (i) its plural_terms can optionally_be_underlined,_in which case we_will call them "world terms", and_(ii) it contains an operator 'at'_that takes a world term W and_a sentence φ to make a_sentence pat W (φ)q. The_translation from_LACT to LW is straightforward:_wherever we have_'^φ', we_substitute '∃V(CV∧at Vφ)',_where V is some new world_variable; wherever we_have 'φ', we substitute '∀V(CV →_at_V (φ))'; wherever_we_have_'ACTφ', we_substitute 'at I_(φ)'._The central_task_in translating from LW to L_is_to eliminate all occurrences of the 'At'_operator. To do so,_we_replace each formula of_the form 'at Rφ(t1,..., tn)'_with φ(α1,..., αn), where αi is_pR(ti)q when_ti is_a singular term or a non-underlined plural term, and αi is_pti ◦ R−1q when ti is_an world term. As_usual, we_treat_sentences involving complex_terms_like 'R(a)'_and 'S◦R−1' in Russellian fashion, as abbreviations_for quantified_sentences of LW: '(S ◦ R−1)ab',_for example, abbreviates '∃Q(∀xy(Qxy_↔_∃z(Rzx ∧ Szy)) ∧ Qab)'. By_repeatedly applying this rule, we can_turn each sentence of LW_into_an_'at'-free sentence. Finally, we can_translate this sentence into L simply_by dropping all_the underlines. Here is an example of_the_machinery at work. Suppose our original_sentence_φ of LACT is (19) ∃x^∃y_ACT_Fxy.