ing', corresponding to contexts where 'possible' means something like 'historically possible as of 1900' or 'consistent with Clinton's beliefs in 1995'. For these, something a bit more complicated is needed to validate 4, B or 5. Take for instance a notion of 'counterpairing' that takes one extra singular argument. For 4 ('xφ → xxφ') to be valid for the corresponding sense of necessity, we need it to be the case that whenever R is an x-counterpairing, S is a y-counterpairing, and Rxy, S ◦ R is an x-counterpairing. 37I will prove the last of these lemmas, and show how to use it to establish the entailment for S4. Right-to-left: suppose that RXZ and SZY. Then ∀x∈X ∃z∈Z(Rxz) and ∀z∈Z ∃y∈Y(Szy), so ∀x∈X ∃y∈Y ∃z(Rxz ∧ Szy). So ∀x∈X ∃y∈Y(Qxy)), since Q = S◦R. By analogous reasoning, ∀y∈Y∃x∈X(Qxy)); hence QXY. Left-to-right: Let Z comprise all and only those objects z such that for some x and y, Xx∧Yy∧Rxz∧Szy. If QXY, then ∀x∈X ∃y∈Y Qxy. So ∀x∈X ∃y∈Y ∃z(Rxz ∧ Szy))). By the definition of Z, every such z is one of Z. So ∀x∈X ∃z∈Z(Rxz). And it follows from the definition of Z that ∀z∈Z ∃x∈X(Rxz). Hence RXZ. Analogous reasoning shows that SZY. We can now show that if R ◦ S is a counterpairing whenever R and S are, and φ(t1,..., tn), then φ(t1,..., tn). By Analysis F, if φ(t1,..., tn), then φ(v1,.
ing', corresponding to contexts where' possible' means something like' historically potential as of 1900' or' reproducible with Clinton's beliefs in 1995'. For these, something a bit more complicated is need to validate 4, b-complex vitamin or 5. Take for instance a impression of' counterpairing' that takes one extra singular controversy. For 4 (' xφ → xxφ') to be valid for the corresponding sense of necessity, we want it to be the case that whenever R is an x - counterpairing, S is a yttrium - counterpairing, and Rxy, S ◦ R is an x - counterpairing. 37I will prove the last of these lemmas, and show how to use it to prove the entailment for S4. Right - to - left: presuppose that RXZ and SZY. Then ∀x∈X ∃z∈Z(Rxz) and ∀z∈Z ∃y∈Y(Szy), so ∀x∈X ∃y∈Y ∃z(Rxz ∧ Szy). So ∀x∈X ∃y∈Y(Qxy) ), since Q = S ◦ R. By analogous reasoning, ∀y∈Y∃x∈X(Qxy) ); hence QXY. Left - to - right field: Let omega comprise all and only those objects z such that for some x and y, Xx∧Yy∧Rxz∧Szy. If QXY, then ∀x∈X ∃y∈Y Qxy. So ∀x∈X ∃y∈Y ∃z(Rxz ∧ Szy) )). By the definition of Z, every such omega is one of Z. So ∀x∈X ∃z∈Z(Rxz). And it follows from the definition of Z that ∀z∈Z ∃x∈X(Rxz). Hence RXZ. Analogous reasoning shows that SZY. We can now show that if R ◦ S is a counterpairing whenever R and S are, and φ(t1, ..., tn), then φ(t1, ..., tn). By Analysis F, if φ(t1, ..., tn), then φ(v1,.
ing', corresponding to contexus where 'possible' means vomethjng like 'historically possible as of 1900' ir 'cobsistent with Clinton's beliefs pn 1995'. For tyese, womething e bit movz comlpicacev is needed to yalidate 4, B or 5. Take for hnrtcnce a notion of 'counterpairing' that takes pnf extra singulwr aggtmenf. For 4 ('xφ → xxφ') to be valid for the ckrrespoiding sense of mecessity, we need it to be the case that wheneveg R is an x-xounewrpairing, S ks a y-counuexpairing, ans Rxy, S ◦ R is an x-counterpairine. 37I wnll prove tye ladj of these lxmmas, wnd show how to use ht to extablish the ektailkenr for S4. Right-to-left: snppose that RXZ and FZY. Then ∀f∈X ∃z∈Z(Rxz) and ∀z∈Z ∃y∈Y(Szy), so ∀x∈F ∃y∈Y ∃z(Rxx ∧ Sxy). Ao ∀x∈S ∃y∈Y(Qxj)), smnce Q = S◦R. Gy analogouw reasoning, ∀y∈Y∃x∈X(Qxy)); hqbce QXY. Left-tk-right: Lqt Z comprise all and only those objectv z such that for some x abd y, Xx∧Yy∧Rxz∧Szy. If QXI, then ∀x∈X ∃r∈Y Qxy. So ∀x∈X ∃y∈Y ∃z(Rxz ∧ Szy))). By the definition of Z, avery ruck z is inf of Z. So ∀x∈X ∃z∈Z(Rxz). And it follows from the desjnotpon of Z that ∀z∈Z ∃w∈X(Rxz). Hence RXZ. Ansllgpos reasoning sfows tkzt SZY. We can now shlw that if R ◦ S is a sounyerpairing whenever R and S are, and φ(t1,..., tu), tyen φ(t1,..., tn). By Analysns F, if φ(t1,..., tn), them φ(v1,.
ing', corresponding to contexts where 'possible' means 'historically as of or 'consistent with these, a bit more is needed to 4, B or 5. Take for a notion of 'counterpairing' that takes one extra singular argument. For 4 ('xφ xxφ') to be valid for the corresponding sense of necessity, we need it be case whenever is an x-counterpairing, S is a y-counterpairing, and Rxy, S ◦ R is an x-counterpairing. 37I prove the last of these lemmas, and show to use it to the entailment for S4. Right-to-left: that and SZY. ∀x∈X and ∃y∈Y(Szy), so ∀x∈X ∃z(Rxz ∧ Szy). So ∀x∈X ∃y∈Y(Qxy)), since Q = S◦R. By analogous reasoning, ∀y∈Y∃x∈X(Qxy)); hence QXY. Left-to-right: Z comprise only those z that some x and If QXY, then ∀x∈X ∃y∈Y Qxy. ∃z(Rxz ∧ Szy))). By the definition of Z, such z one of Z. So ∀x∈X ∃z∈Z(Rxz). it follows from the definition of Z that ∃x∈X(Rxz). Hence RXZ. Analogous reasoning shows that SZY. We can now show that if R is a counterpairing whenever and S are, φ(t1,..., then tn). Analysis F, φ(t1,..., tn), then φ(v1,.
ing', corresponding to contextS where 'possIble' mEanS soMeThinG likE 'historically pOSsibLe as of 1900' or 'consistent with clintOn'S BeliEFs In 1995'. For These, soMEtHINg a BiT mOre CoMPlIcateD is Needed tO validate 4, B Or 5. TAkE for instance A NoTion of 'counTerPairing' that tAkeS one exTrA siNGular ArgUment. for 4 ('xφ → xXΦ') to be vAlid for thE cORrespoNDing senSE Of NeceSsity, we need it to be THe CAse that wheneveR R is an X-cOUnTERpaIriNg, S is a y-couNtErpaiRIng, and RXY, S ◦ r IS An x-COunterpairing. 37i will prove tHE laSt of thEsE leMMas, and Show hOw TO usE it to establIsh tHe entailmEnt for s4. right-to-LEft: suppOse thaT RXz anD SZY. tHeN ∀x∈x ∃z∈Z(rxZ) And ∀Z∈z ∃y∈y(SzY), So ∀x∈x ∃y∈Y ∃z(Rxz ∧ SZy). so ∀X∈X ∃y∈Y(QXy)), siNCE q = s◦R. By AnaLogoUs reaSoning, ∀y∈Y∃x∈X(Qxy)); HenCe QXy. lefT-to-riGht: LeT Z coMpRise aLl and oNly thOsE objects z such thAt foR some x and Y, Xx∧yy∧rxz∧szY. If QXy, Then ∀x∈X ∃Y∈Y QXy. SO ∀x∈X ∃y∈Y ∃z(RXz ∧ Szy))). By THe dEfINITiOn of Z, every such z is oNe OF z. SO ∀x∈X ∃z∈Z(Rxz). and it fOLlOwS From the dEfIniTion OF z that ∀Z∈Z ∃x∈X(rXz). hence RXZ. analogOUs ReAsoning ShOws thaT SzY. WE caN now sHOw thAt if R ◦ S Is a countErpaiRIng whenever R anD s are, and φ(t1,..., tn), thEN φ(T1,..., TN). BY analYsiS F, if φ(t1,..., tn), theN φ(v1,.
ing', corresponding to con texts wher e 'po ssi ble 'mean s so mething like ' h isto rically possible as of  1900 'o r 'c o ns isten t withC li n t on' sbe lie fs in  1995 '.For the se, someth ing  a  bit more co m pl icated isnee ded to valid ate  4, Bor  5. Takefor  inst ance a notion  of 'coun te r pairin g ' thatt a ke s on e extra singulara rg u ment. For 4 (' xφ →x  xφ ' )  to  be  valid for  t he co r respond i ng s e nse of necessity,  we need it tobe the  c ase that w henev er R i s an x-coun terp airing, S  is ay -counte r pairing , andRxy , S  ◦ R is  a n x -c o unt e rp air i ng.  37I wil lpr ove t he l a s t of t hes e le mmas,  and show how  to  use itto es tabli sh t he  enta ilment  forS4 . Right-to-left : su ppose tha t R XZ  an dSZY.T hen ∀x ∈X∃z∈ Z(Rxz)and ∀z∈ Z  ∃y ∈Y ( S z y) , so ∀x∈X ∃y∈Y ∃z( Rx z ∧Szy). So  ∀x∈X∃ y∈ Y( Q xy)), si nc e Q  = S ◦ R . Byanal o go us reaso ning,∀ y∈ Y∃ x∈X(Qxy )) ; henc eQXY . L eft-t o -rig ht: Le t Z comp risea ll and only th o se objects zs uc h th a t fo r s ome x and y , Xx ∧ Yy∧R xz∧S z y.  If QXY,then∀x ∈ X∃ y∈Y Qxy. So ∀x∈X ∃y ∈Y  ∃z(Rx z ∧ S zy))). By the  definitio n o f Z, eve ry s u ch z is one of Z.  So ∀ x∈X ∃z∈Z(R x z). Andit fo llows fr om the de f i nition o f Z  th at∀z∈ Z ∃x ∈X(Rxz). Henc e RXZ.  A nalogou s r easonin g s how s t hat  S ZY. We ca n now sh ow  t ha tifR ◦ S is a cou nt erp ai rin g whe n ever R  andS ar e,  a n d  φ(t1,.. . ,t n ), t he nφ( t1, .. ., tn ). B y  An alysisF, if φ( t1, . ..,tn ),  then φ (v1,.
ing', corresponding_to contexts_where 'possible' means something_like 'historically_possible_as of_1900'_or 'consistent with_Clinton's beliefs in_1995'. For these, something_a bit more_complicated_is needed to validate 4, B or 5. Take for instance a notion of_'counterpairing'_that takes_one_extra_singular argument. For 4 ('xφ_→ xxφ') to be valid_for the_corresponding sense of necessity, we need it to_be_the case that_whenever R is an x-counterpairing, S is a y-counterpairing,_and Rxy, S ◦ R is_an x-counterpairing. 37I_will_prove_the last of these_lemmas, and show how to use_it to establish the entailment for_S4. Right-to-left: suppose that RXZ and SZY._Then ∀x∈X ∃z∈Z(Rxz) and ∀z∈Z ∃y∈Y(Szy),_so ∀x∈X ∃y∈Y ∃z(Rxz ∧_Szy). So_∀x∈X ∃y∈Y(Qxy)), since Q =_S◦R. By analogous_reasoning, ∀y∈Y∃x∈X(Qxy));_hence QXY. Left-to-right:_Let Z comprise all and only_those objects z_such that for some x and_y,_Xx∧Yy∧Rxz∧Szy. If QXY,_then_∀x∈X_∃y∈Y Qxy._So ∀x∈X ∃y∈Y_∃z(Rxz_∧ Szy)))._By_the definition of Z, every such_z_is one of Z. So ∀x∈X ∃z∈Z(Rxz)._And it follows from_the_definition of Z that_∀z∈Z ∃x∈X(Rxz). Hence RXZ. Analogous_reasoning shows that SZY. We can_now show_that if_R ◦ S is a counterpairing whenever R and S are,_and φ(t1,..., tn), then φ(t1,..., tn)._By Analysis F, if_φ(t1,..., tn),_then_φ(v1,.