ACT, given as input a modelM = 〈D,I〉 for L in which 'every counterpairing is a global permutation and so is I' is true. W is the set of permutations of D. For an atomic predicate F, J(π)(F) = π−1(I(F)). π1 < π2 iff π−11 ◦ π2 ∈ I(C). w@ = I(I). It can then be proved that a sentence of LACT is true at the identity in SM iff its translation into L is true in M. This entails that (i) if the translation of a sentence into L is false in some model ofL in which 'every counterpairing is a global permutation and so is I' is true, the sentence is false at a world in SM, and thus not globally valid; and (ii) if the translation of a sentence into L is false in some model of L in which 'every counterpairing is a global permutation and I is the identity relation' is true, the sentence is false at w@ in SM, and thus not real-world valid. 49This argument is due to Williamson (1996). logic for 'actually' involves requiring counterpairings to be one-toone. Similarly, if counterpairings could fail to have universal ranges, the translation (28) of (27) would not be valid, as it ought to be. ∀x(Fx→ ACT^Fx)(27) ∀W(CW → ∀x(Fx→ ∃V(CV ∧ FV((I ◦W−1)(x)))))(28) It is important to understand the difference between the sense in which this theory provides an explanation of sentences involving 'Actually' and the sense in which the counterpart theory of section 8 provides an explanation of claims of the form 'Possibly φ'. The latter can be viewed as a collection of analyses, English sentences of the form 'For it to be possible that φ is for it to be the case that... '. Our "translations" of claims involving 'actually' are not like this. The translation of 'Actually Mars is red' is 'I(Mars) is red', i.e. 'The object that is the second member of the ordered pair that is one of I and has Mars as its first member is red'. Suppose Rigid Pairs is true in our current context. Then '
ACT, given as input a modelM = 〈 D, I 〉 for L in which' every counterpairing is a global permutation and therefore is I' is genuine. W is the set of permutations of D. For an atomic predicate F, J(π)(F) = π−1(I(F) ). π1 < π2 iff π−11 ◦ π2 ∈ I(C). w@ = I(I). It can then be prove that a sentence of LACT is true at the identity in SM iff its transformation into L is true in M. This entails that (i) if the transformation of a conviction into L is false in some mannequin ofL in which' every counterpairing is a global permutation and then is I' is genuine, the sentence is false at a world in SM, and thus not globally valid; and (ii) if the translation of a conviction into L is false in some model of L in which' every counterpairing is a global substitution and I is the identity relation' is true, the conviction is false at w@ in SM, and therefore not real - worldly concern valid. 49This argument is due to Williamson (1996). logic for' actually' involves command counterpairings to be one - toone. Similarly, if counterpairings could fail to have universal ranges, the translation (28) of (27) would not be valid, as it ought to be. ∀x(Fx→ ACT^Fx)(27) ∀W(CW → ∀x(Fx→ ∃V(CV ∧ FV((I ◦ W−1)(x)))))(28) It is important to understand the difference between the sense in which this theory provides an explanation of sentences involving' Actually' and the sense in which the counterpart theory of section 8 provides an explanation of claims of the kind' Possibly φ'. The latter can be viewed as a collection of analysis, English prison term of the form' For it to be possible that φ is for it to be the case that...'. Our " translations " of claims involving' actually' are not like this. The translation of' Actually Mars is crimson' is' I(Mars) is red', i.e.' The object that is the second member of the ordered pair that is one of I and has Mars as its beginning member is red'. Suppose Rigid Pairs is true in our current context. Then'
ACT, given as input a modelM = 〈D,I〉 for L in wkuch 'evxry couhterpairkng is a global permutation end wo is I' is true. W is the seg of permltations if D. Dor an atomic predleate R, J(π)(F) = π−1(M(F)). π1 < π2 iff π−11 ◦ π2 ∈ I(G). w@ = I(I). It wan then be prmvdd that a sentence of LACT is true at the idrnhity in SM iff its eranalation into L is true in M. This ehtails uhat (i) if the tranxlation of a sentence into L id false in some mofel ofL in qhicr 'every countdrpairing pv a global permutation and so is I' is trud, the sentence us fapve at a worod in SM, and thus not glotally vslid; and (ii) if thx trqnslation of a sentenre into L is false ig some mogem of L in which 'eceey coonter[airkbg ks z jlogal pegmuvation and J is the idwntity relation' is uruq, the sentence is fajsq at w@ in SM, and thus not real-world valpd. 49Tgis argument is due to Qilliamson (1996). logic for 'actually' involves requiring counterpairings to be one-toona. Simmlxrlv, if cuynherpairings could fail to have universal rangqa, uhe translation (28) jf (27) would npt br valid, as it uught ck ge. ∀x(Fx→ ACT^Fx)(27) ∀W(CW → ∀d(Fx→ ∃V(CV ∧ FV((I ◦W−1)(x)))))(28) It is impprtant to understand the didference betceeb the sense in whieh this theoxy profides an explanation of sentznces jnvolving 'Aftually' aha the sense in wficm tve counttfpart theory of sqction 8 pcovidzs an exolanstion jf claims lf thc form 'Possibly φ'. Thf latjer cat be viewef as a collection of analyses, Eijlish sentencgs mf nhe form 'Yor it to be possibje that φ is fot it to bz the zase that... '. Our "trenslations" os claims invonging 'actuallb' are not likw thus. The gfanslation of 'Sctually Mars is ree' is 'I(Mars) is red', l.e. 'Thg kbject that is cke second member pf ghe ogdeced pwhr that is ote ow I snd hxs Mars as lts firxt member is red'. Sup[ose Rigid Pairs is trie in our xurrent sontext. Then '
ACT, given as input a modelM = L which 'every is a global is W is the of permutations of For an atomic predicate F, J(π)(F) π−1(I(F)). π1 < π2 iff π−11 ◦ π2 ∈ I(C). w@ = I(I). can then be proved that a sentence of LACT is true at the in iff translation L is true in M. This entails that (i) if the translation of a sentence into is false in some model ofL in which counterpairing is a global and so is I' is the is false a in and thus not valid; and (ii) if the translation of a sentence into L is false in some model of in which is a permutation I the identity relation' the sentence is false at w@ thus not real-world valid. 49This argument is due Williamson (1996). for 'actually' involves requiring counterpairings to one-toone. Similarly, if counterpairings could fail to have ranges, the translation (28) of (27) would not be valid, as it ought to be. ∀W(CW → ∀x(Fx→ ∃V(CV FV((I ◦W−1)(x)))))(28) It important understand difference the sense which this theory provides an explanation of sentences involving 'Actually' and sense in which the counterpart theory of section 8 provides of of the form φ'. The latter can viewed a collection of analyses, of form be that is for it to the case that... '. Our of claims involving 'actually' translation of 'Actually Mars is red' is 'I(Mars) red', i.e. 'The object that is the member of the ordered pair that is one of I and has as its is red'. Suppose Rigid Pairs is true in current context. Then '
ACT, given as input a modelM = 〈D,I〉 fOr L in which 'Every CouNteRpAiriNg is A global permutaTIon aNd so is I' is true. W is the set Of perMuTAtioNS oF D. For An atomiC PrEDIcaTe f, J(Π)(F) = π−1(i(F)). Π1 < Π2 iFf π−11 ◦ π2 ∈ I(c). w@ = I(i). It can tHen be proveD thAt A sentence of LacT Is true at thE idEntity in SM ifF itS transLaTioN Into L Is tRue in m. This eNTails tHat (i) if the TrANslatiON of a senTENcE intO L is false in some moDEl OFL in which 'every CounteRpAIrING is A glObal permutAtIon anD So is I' is TRuE, THE seNTence is false aT a world in SM, ANd tHus not GlObaLLy valiD; and (iI) iF The Translation Of a sEntence inTo L is fALse in soME model oF L in whIch 'EveRy coUNtErPaiRiNG is A GlObaL PerMutation AnD I Is the IdenTITY RelaTioN' is tRue, thE sentence is faLse At w@ iN sM, aNd thuS not rEal-wOrLd valId. 49This ArgumEnT is due to WilliamSon (1996). lOgic for 'acTuaLlY' inVoLves rEQuirinG coUntErpairiNgs to be ONe-tOoNE. sImIlarly, if counterpaiRiNGS cOuld fail To have UNiVeRSal rangeS, tHe tRansLATion (28) oF (27) wouLD nOt be valiD, as it oUGhT tO be. ∀x(Fx→ AcT^fx)(27) ∀W(CW → ∀x(fx→ ∃v(CV ∧ fV((I ◦w−1)(x)))))(28) It iS ImpoRtant tO understAnd thE Difference betwEEn the sense in wHIcH THiS TheoRy pRovides an exPlanATion Of seNTeNceS InvolVing 'ACtUAlLY' and the sense in which ThE countErparT theory of sectIon 8 provideS AN ExplanatIon oF ClAIms of the form 'PoSsiblY φ'. The latteR Can be vieWed as A collectIon of analYSEs, EnglisH seNteNceS of THE fOrm 'For it to be pOSSiblE tHat φ is fOr iT to be thE caSe tHat... '. our "TrAnslationS" of claimS iNvOlViNg 'aCtualLY' are not lIkE thIs. the TransLAtion oF 'ActuAlly maRs IS reD' is 'I(MarS) Is RED', i.e. 'THe ObJect ThaT iS the sEconD MemBer of thE ordered pAir THat iS oNe Of I and hAs Mars as its fiRsT member is rEd'. supPose RiGID Pairs is True in our current context. tHen '
ACT, given as input a mode lM = 〈D,I〉  forL i n w hi ch ' ever y counterpairi n g is  a global permutationand s oi s I' is  true . W ist he s etof  p erm ut a ti ons o f D . For a n atomic p red ic ate F, J(π)( F )= π−1(I(F) ).π1 < π2 iffπ−1 1 ◦ π2  ∈  I( C ). w@  =I(I).  It ca n  thenbe proved  t h at a s e ntenceo f  L ACTis true at the id e nt i ty in SM iff i ts tra ns l at i o n i nto  L is true  i n M.T his ent a il s t hat (i) if the tr anslation o f  asenten ce  in t o L is  fals ei n s ome model o fL i n which ' everyc ounterp a iring i s a gl oba l p ermu t at io n a nd soi sI'i s t rue, the  s en tence  isf a l s e at  aworl d inSM, and thusnot  glo b all y val id; a nd ( ii ) ifthe tr ansla ti on of a sentenc e in to L is f als einso me mo d el ofL i n w hich 'e very co u nte rp a i r in g is a global perm ut a t io n and Iis the id en t ity rela ti on'  ist r ue, t he s e nt ence isfalsea tw@  in SM,  a nd thu snot  re al-wo r ld v alid.49This a rgume n t is due to Wi l liamson (1996 ) .l o gi c  for  'a ctually' in volv e s re quir i ng  co u nterp airin gs to be one-toone. Simil ar ly, if  coun terpairings c ould failt o have uni vers a lr anges, the tra nslat ion (28) o f  (27) wo uld n ot be va lid, as i t ought to  be .  ∀x( Fx→ A CT ^Fx)(27) ∀W(C W → ∀x (F x→ ∃V(C V ∧  FV((I◦W− 1)( x)) ))) (2 8) It isimportan tto  u nd ers tandt he diffe re nce  b etw een t h e sens e inwhic hth i s t heory p r ov i d es a nex plan ati on  of s ente n ces  involv ing 'Actu all y ' an dth e sense  in which the  c ounterpart  t heo ry ofs e ction 8provides an explanation of clai msof th e fo rm 'Possi bly  φ'. T hel attercan be  view ed  as a  coll e c ti onof  analyses, E ngl ish s en tenc es of t he form 'For it to bepossible that  φis f o r  i t t o  b e  th ec ase t hat... '. Our " translatio ns "  o f claims i n vol vi ng 'act ually'are n o t likethis. The  translat io n of ' Act ually Mars  is red'  is 'I(Ma r s) is re d', i .e.  'Theob jec t tha t is t h e s econd  membe rof the  orde re d pair t hat is one of I and has  Marsas it s f irst memb eri s r ed'. Supp oseRigid Pair s i s t rue i n o u r cur rent co nte x t. Th en ' 
ACT, given_as input_a modelM = 〈D,I〉_for L_in_which 'every_counterpairing_is a global_permutation and so_is I' is true._W is the_set_of permutations of D. For an atomic predicate F, J(π)(F) = π−1(I(F)). π1 <_π2_iff π−11_◦_π2_∈ I(C). w@ = I(I)._It can then be proved_that a_sentence of LACT is true at the identity_in_SM iff its_translation into L is true in M. This entails_that (i) if the translation of_a sentence into_L_is_false in some model_ofL in which 'every counterpairing is_a global permutation and so is_I' is true, the sentence is false_at a world in SM, and_thus not globally valid; and_(ii) if_the translation of a sentence_into L is_false in_some model of_L in which 'every counterpairing is_a global permutation_and I is the identity relation'_is_true, the sentence_is_false_at w@_in SM, and_thus_not real-world_valid._49This argument is due to Williamson_(1996)._logic for 'actually' involves requiring counterpairings to_be one-toone. Similarly, if_counterpairings_could fail to have_universal ranges, the translation (28)_of (27) would not be valid,_as it_ought to_be. ∀x(Fx→ ACT^Fx)(27) ∀W(CW → ∀x(Fx→ ∃V(CV ∧ FV((I ◦W−1)(x)))))(28) It_is important to understand the difference_between the sense in_which this_theory_provides an explanation_of_sentences involving_'Actually' and the sense in which the_counterpart theory_of section 8 provides an explanation_of claims of the_form_'Possibly φ'. The latter can be_viewed as a collection of analyses,_English sentences of the form_'For_it_to be possible that φ_is for it to be the_case that... '._Our "translations" of claims involving 'actually' are_not_like this. The translation of 'Actually_Mars_is red' is 'I(Mars) is red',_i.e._'The_object that is the second_member of the ordered pair that_is one of I and has Mars as its_first member is_red'. Suppose Rigid Pairs is_true_in_our current context. Then '