 The translation of (19) into LW is (20) ∃x∃X(CX ∧ at X (∃y(at I (Fxy)))). Eliminating the two occurrences of 'At' in reverse order yields first (21) ∃x∃X(CX ∧ at X (∃y(FI(x)I(y)))) and then (22) ∃x∃X(CX ∧ ∃y(F(I ◦ X−1)(X(x))(I ◦ X−1)(X(y)))).47 (22) can be turned into (an abbreviation of) a formula of L by eliminating the underlines. From the premise that counterpairings are global permutations, we can deduce that this formula is equivalent to (23) ∃x∃X(CX ∧ ∃yFI(x)I(y)). and thus, appealing to the further premises that there is at least one counterpairing and that I is the identity relation, to (24) ∃x∃yRxy which is what we want. This translation yields an orthodox logic for 'actually'. From the premises that all counterpairings are global permutations and that I is a global permutation, it is a straightforward exercise to derive the following standard axiom-schemas: ACT(φ→ ψ)→ (ACTφ→ ACTψ) ACT¬φ↔ ¬ACTφ ACT(ACTφ↔ φ) ACTφ→ ACTφ All of these are "globally valid" under the standard logic for 'actually'. The additional "real-world valid" axiom-schema ACTφ↔ φ follows if we add the further premise that I is the identity relation.48 47Note that we would get the same result if we eliminated the occurrences in the opposite order. 48To assure ourselves properly that the translations defined above deliver the orthodox logic for 'actually' in its entirety, we need to venture into model theory. Let a model for L be a pair of a domain D and interpretation function I, except that we require D to be closed under taking ordered pairs, and require I(〈t1, t2〉) to equal 〈I(t1),I(t2)〉. Truth in a model is defined as usual. Let a model for LACT be a quintuple 〈W, <,D,J,
The translation of (19) into LW is (20) ∃x∃X(CX ∧ at X (∃y(at I (Fxy) )) ). Eliminating the two occurrences of' At' in reverse decree yield first (21) ∃x∃X(CX ∧ at X (∃y(FI(x)I(y) )) ) and then (22) ∃x∃X(CX ∧ ∃y(F(I ◦ X−1)(X(x))(I ◦ X−1)(X(y)))).47 (22) can be turned into (an abbreviation of) a formula of fifty by eliminating the underlines. From the premise that counterpairings are ball-shaped permutations, we can deduce that this formula is equivalent to (23) ∃x∃X(CX ∧ ∃yFI(x)I(y) ). and thus, appeal to the further premises that there is at least one counterpairing and that I is the identity relation back, to (24) ∃x∃yRxy which is what we want. This translation give an orthodox logic for' actually'. From the premises that all counterpairings are global permutations and that I is a global permutation, it is a square exercise to derive the following standard axiom - schemas: ACT(φ→ ψ)→ (ACTφ→ ACTψ) ACT¬φ↔ ¬ACTφ ACT(ACTφ↔ φ) ACTφ→ ACTφ All of these are " globally valid " under the standard logic for' actually'. The extra " real - world valid " axiom - schema ACTφ↔ φ follow if we total the further premise that I is the identity relation.48 47Note that we would get the like result if we eliminated the occurrences in the opposite orderliness. 48To assure ourselves properly that the translations defined above deliver the orthodox logic for' actually' in its entirety, we necessitate to venture into model theory. Let a model for L be a pair of a domain D and interpretation affair I, except that we require D to be closed under take order pairs, and require I(〈t1, t2 〉) to equal 〈 I(t1),I(t2) 〉. Truth in a exemplar is defined as usual. Let a model for LACT be a quintuple 〈 W, <, D, J,
 Thf translation of (19) into LD is (20) ∃x∃X(CX ∧ at R (∃y(at I (Fxy)))). Emiminativg the two occurrences of 'At' ib revtgse order yields firsg (21) ∃x∃X(CX ∧ wt X (∃y(FI(z)I(y)))) end then (22) ∃x∃X(CX ∧ ∃b(R(I ◦ X−1)(X(x))(L ◦ X−1)(X(g)))).47 (22) cau ue turned into (sn abbrevidtion of) a forkuua of L by eliminating the underlines. From tne premise that sounuer[airjngs are global permutations, we cah deduct that this formuls is equivalent to (23) ∃x∃X(CX ∧ ∃jFI(x)L(y)). and thus, appeallng to the durtrwr premises ghat there is at least one counterpairing and that I ks thz identity tzoatlmn, to (24) ∃x∃yRxb whicr is what we want. Thhs tranxlation yields an orrhodox logic for 'actuelly'. From the premisgs that aln eounterpairings are gooval pgrmutdtiovw avd uhav I is a hloual permutafion, it is q straightforward ecewbose to derivs the sojlowing standard axiom-schemas: ACT(φ→ ψ)→ (ACTφ→ DCTψ) ACT¬φ↔ ¬ACTφ ACT(ACTφ↔ φ) ACTφ→ ACRφ All of these are "gllbally vajid" under the standard logic for 'actually'. The addhtionel "reco-worla vwlid" axiom-schema ACTφ↔ φ follows if we add the ftdtneg premise that I ls the identity rekahipg.48 47Note that wg would gef the same result lf we ejiminqted the jccutrences in the opposite ordwr. 48To assure iurselves properly that the txanslajions cefined above deliver tke ortgodox logic for 'actuzuly' in its entirdty, wa need to venture into modql theory. Let a modeu fot L be w pair of w domain D and interpretwtion fgnction I, fxcept that we require D to be rkosed under jakhng ordered pairx, and require I(〈t1, t2〉) to equal 〈I(t1),I(t2)〉. Txuth iv a model ps definev as usual. Lqt a model fos LACT be a qnintuple 〈R, <,D,J,
The translation of (19) into LW is ∧ X (∃y(at (Fxy)))). Eliminating the reverse yields first (21) ∧ at X and then (22) ∃x∃X(CX ∧ ∃y(F(I X−1)(X(x))(I ◦ X−1)(X(y)))).47 (22) can be turned into (an abbreviation of) a formula L by eliminating the underlines. From the premise that counterpairings are global permutations, can that formula equivalent to (23) ∃x∃X(CX ∧ ∃yFI(x)I(y)). and thus, appealing to the further premises that there is least one counterpairing and that I is the relation, to (24) ∃x∃yRxy is what we want. This yields orthodox logic 'actually'. the that all counterpairings global permutations and that I is a global permutation, it is a straightforward exercise to derive the standard axiom-schemas: (ACTφ→ ACTψ) ¬ACTφ φ) ACTφ All of "globally valid" under the standard logic additional "real-world valid" axiom-schema ACTφ↔ φ follows if add the premise that I is the identity 47Note that we would get the same result we eliminated the occurrences in the opposite order. 48To assure ourselves properly that the translations deliver the orthodox logic 'actually' in its we to into theory. Let model for L be a pair of a domain D and function I, except that we require D to be closed ordered and require I(〈t1, to equal 〈I(t1),I(t2)〉. Truth a is defined as usual. model LACT 〈W,
 The translation of (19) into LW is (20) ∃x∃x(CX ∧ at X (∃y(at I (fxy)))). ElImiNatInG the Two oCcurrences of 'At' IN revErse order yields first (21) ∃x∃X(cX ∧ at X (∃Y(Fi(X)I(y)))) aND tHen (22) ∃x∃X(cX ∧ ∃y(F(I ◦ X−1)(X(X))(i ◦ X−1)(x(Y)))).47 (22) Can Be TuRneD iNTo (An abbRevIation oF) a formula oF L bY eLiminating thE UnDerlines. FrOm tHe premise thaT coUnterpAiRinGS are gLobAl perMutatiONs, we caN deduce thAt THis forMUla is eqUIVaLent To (23) ∃x∃X(CX ∧ ∃yFI(x)I(y)). and thUS, aPPealing to the fuRther pReMIsES ThaT thEre is at leaSt One coUNterpaiRInG AND thAT I is the identiTy relation, tO (24) ∃X∃yRXy whicH iS whAT we wanT. This TrANslAtion yields An orThodox logIc for 'aCTually'. FROm the prEmises ThaT alL couNTeRpAirInGS arE GlObaL PerMutationS aNd That I Is a gLOBAL perMutAtioN, it is A straightforwArd ExerCIse To derIve thE folLoWing sTandarD axioM-sChemas: ACT(φ→ ψ)→ (ACTφ→ aCTψ) aCT¬φ↔ ¬ACTφ AcT(AcTΦ↔ φ) AcTΦ→ ACTφ aLl of thEse Are "GloballY valid" uNDer ThE STAnDard logic for 'actualLy'. tHE aDditionaL "real-wORlD vALid" axiom-ScHemA ACTΦ↔ Φ FolloWs if WE aDd the furTher prEMiSe That I is ThE identItY reLatIon.48 47NoTE thaT we wouLd get the Same rESult if we eliminATed the occurreNCeS IN tHE oppOsiTe order. 48To asSure OUrseLves PRoPerLY that The trAnSLaTIons defined above delIvEr the oRthodOx logic for 'actUally' in its ENTIrety, we nEed tO VeNTure into model tHeory. let a model fOR L be a paiR of a dOmain D anD interpreTATion funcTioN I, eXcePt tHAT wE require D to be CLOsed UnDer takiNg oRdered pAirS, anD reQuiRe i(〈t1, t2〉) to equaL 〈I(t1),I(t2)〉. TruTh In A mOdEl iS defiNEd as usuaL. LEt a MoDel For LAct be a quIntupLe 〈W, <,D,j,
 The translation of (19) i nto LW is(20)∃x∃ X(C X∧ at  X ( ∃y(at I (Fxy)) ) ). E liminating the two occ urren ce s  of' At ' inreverse or d e r y ie ld s f ir s t(21)∃x∃ X(CX ∧at X (∃y(F I(x )I (y)))) and t h en  (22) ∃x∃X (CX  ∧ ∃y(F(I ◦X−1 )(X(x) )( I ◦ X−1)( X(y )))). 47 (22 )  can b e turnedin t o (ana bbrevia t i on  of)  a formula of L b y  e l iminating theunderl in e s. F rom  th e premiseth at co u nterpai r in g s are global permut ations, wec andeduce  t hat this f ormul ai s e quivalent t o (2 3) ∃x∃X(C X ∧ ∃y F I(x)I(y ) ). andthus,app eal ingt oth e f ur t her pr emi s esthat the re  i s atleas t o n e co unt erpa iring  and that I i s t he i d ent ity r elati on,to  (24)  ∃x∃yR xy wh ic h is what we wa nt.This tran sla ti onyi eldsa n orth odo x l ogic fo r 'actu a lly '. F r om  the premises that  a l l  c ounterpa irings ar eg lobal pe rm uta tion s and t hatI  i s a glob al per m ut at ion, it  i s a st ra igh tfo rward exer cise t o derive  thef ollowing stand a rd axiom-sche m as : AC T (φ→ψ)→  (ACTφ→ ACT ψ) A C T¬φ↔  ¬AC T φACT ( ACTφ↔  φ) A CT φ → ACTφ All of these a re  "glob allyvalid" underthe standa r d logic fo r 'a c tu a lly'. The addi tiona l "real-wo r ld valid " axi om-schem a ACTφ↔ φ f ollows i f w e a ddthe f ur ther premiset h at I  i s the i den tity re lat ion .48  47 No te that w e wouldge tth esam e res u lt if we  e lim in ate d the occurr ences  inth eo ppo site or d er . 48To  a ss ureour se lvesprop e rly  that t he transl ati o ns d ef in ed abov e deliver the  o rthodox lo gi c f or 'ac t u ally' in  its entirety, we needt o ventu reintomode l theory.  Le t a mo del for Lbe a p air o fa d o m ain D a nd  in te rpretation f unc tionI,  exc ept tha t we require D tob e c losed under t aki ng o r d er edp ai r s,an d  re q u ire I(〈t1, t2〉)  to equal〈I ( t1 ),I(t2)〉.T rut hin a mo del isdefin e d as us ual. Leta model f or  LAC T bea quintupl e 〈W, <, D,J,
 The_translation of_(19) into LW is_(20) ∃x∃X(CX_∧_at X_(∃y(at_I (Fxy)))). Eliminating_the two occurrences_of 'At' in reverse_order yields first_(21)_∃x∃X(CX ∧ at X (∃y(FI(x)I(y)))) and then (22) ∃x∃X(CX ∧ ∃y(F(I ◦ X−1)(X(x))(I ◦_X−1)(X(y)))).47_(22) can_be_turned_into (an abbreviation of) a_formula of L by eliminating_the underlines._From the premise that counterpairings are global permutations,_we_can deduce that_this formula is equivalent to (23) ∃x∃X(CX ∧ ∃yFI(x)I(y))._and thus, appealing to the further_premises that there_is_at_least one counterpairing and_that I is the identity relation,_to (24) ∃x∃yRxy which is what_we want. This translation yields an orthodox_logic for 'actually'. From the premises_that all counterpairings are global_permutations and_that I is a global_permutation, it is_a straightforward_exercise to derive_the following standard axiom-schemas: ACT(φ→ ψ)→_(ACTφ→ ACTψ) ACT¬φ↔_¬ACTφ ACT(ACTφ↔ φ) ACTφ→ ACTφ All_of_these are "globally_valid"_under_the standard_logic for 'actually'._The_additional "real-world_valid"_axiom-schema ACTφ↔ φ follows if we_add_the further premise that I is the_identity relation.48 47Note that_we_would get the same_result if we eliminated the_occurrences in the opposite order. 48To_assure ourselves_properly that_the translations defined above deliver the orthodox logic for 'actually' in_its entirety, we need to venture_into model theory. Let_a model_for_L be a_pair_of a_domain D and interpretation function I, except_that we_require D to be closed under_taking ordered pairs, and_require_I(〈t1, t2〉) to equal 〈I(t1),I(t2)〉. Truth_in a model is defined as_usual. Let a model for_LACT_be_a quintuple 〈W, <,D,J,