 it was two people together, hand in hand, who carried out the emptying. Similarly, in our system, a 2-place predicate such as 'x introduced to swinging to y' can be satisfied by an ordered pair whose first and second components are complex entities of different levels (as it happens for instance in 'Dick and Hannah, together with Tony and Grace, introduced John and Alice to swinging'). And a similar remark applies, mutatis mutandis, to all other cases of plural collective predication. This settings make it very easy to obtain proofs of soundness and completeness for our proposed system. Indeed, soundness of rule RICT follows at once from the extensionality of sets. And as for completeness, only a few minor modifications are needed on the customary Henkin-style proof.20 Indeed, the usual proof proceeds by taking any consistent set of formulas Φ, and enlarging it into a maximal consistent and existentially saturated Ψ. Then an equi- distinct objects, for instance, the set-theoretical definition of the ordered pair ([u,v],w) makes it clearly different from the pair (u,[v,w]): ([u,v],w) = ({u,v},w) = { {{u,v}}, {{u,v},w} } ≠ (u,[v,w]) = (u,{v,w}) = { {u}, {u,{v,w}} } And a similar remark applies to n-tuples in general (see e.g. Roitman 1990, 29 – 30). 20 As exposed e.g. in Chang – Keisler (1990, 61 – 66). A Note on Plural Logic __________________________________________________ 161 valence relation between terms is introduced, in such a way that r∼s if and only if the formula r = s belongs toΨ. A model is thereby defined that assigns to each term its own equivalence class, which makes it relatively easy to prove that such a model satisfies the whole ofΨ, hence rendering Ψ (and Φ ) satisfiable. And from this satisfaction theorem (if Φ consistent, Φ satisfiable) the completeness theorem follows at once. All this can be done just the same in our first-order logic for itemized collections. In particular, the equivalence relation will be established not only for individual terms, but for compound terms too. And the fact that the set Ψ is maximal consistent, immediately guarantees that it contains a formula of
it was two people together, hand in hand, who carry out the emptying. Similarly, in our arrangement, a 2 - place predicate such as' x introduced to swinging to y' can be satisfy by an ordered pair whose beginning and second part are complex entities of different levels (as it happens for example in' Dick and Hannah, together with Tony and Grace, introduced John and Alice to swing'). And a exchangeable remark applies, mutatis mutandis, to all other cases of plural collective predication. This setting make it very easy to obtain proof of soundness and completeness for our proposed system. Indeed, soundness of convention RICT follows at once from the extensionality of sets. And as for completeness, only a few minor modifications are needed on the accustomed Henkin - style proof.20 Indeed, the usual proof proceeds by taking any consistent hardening of formulas Φ, and enlarging it into a maximal consistent and existentially saturated Ψ. Then an equi- distinct objects, for instance, the set - theoretical definition of the ordered pair ([ u, v],w) make it clearly different from the couple (u,[v, watt ] ): ([ uranium, v],w) = ({ u, v},w) = { { { u, v } }, { { u, v},w } } ≠ (u,[v, w ]) = (u,{v, watt }) = { { u }, { u,{v, w } } } And a similar remark applies to n - tuples in general (determine e.g. Roitman 1990, 29 – 30). 20 As exposed e.g. in Chang – Keisler (1990, 61 – 66). A Note on Plural Logic _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 161 valence relation between terms is insert, in such a way that r∼s if and only if the formula r = s belongs toΨ. A model is thereby specify that assigns to each term its own equivalence class, which makes it relatively easy to rise that such a model satisfies the whole ofΨ, therefore rendering Ψ (and Φ) satisfiable. And from this atonement theorem (if Φ consistent, Φ satisfiable) the completeness theorem follows at once. All this can be dress just the like in our first - order logic for itemized collections. In particular, the equivalence sexual intercourse will be established not merely for individual terms, but for compound term too. And the fact that the set Ψ is maximal consistent, immediately guarantees that it contains a recipe of
 it was two people together, hand in hand, wki carrmed out the empgying. Similarly, in our systel, q 2-plaxe predicate such as 'x introducvd to swibginj to y' can be savjsfied nv an kvdereb 'air whose firsj and second components ara zolplex entities of different levels (ws it hspoens for instagce pn 'Dici and Hannah, together with Tony ans Grace, introduced Jphn and Alice to swinging'). Wnd w similar remark aoplies, mutajjs iytandis, to aul other cases of plurzl collective predication. This rettiugs make it vwry gasy to obtamn projfs of soundkvss and compleyeness for our prmpowed system. Indeed, souidness of rule RICT sollows ad knce from the extwnwionanity of rwts. Ans es ror colplxteness, onlg a few minir modifications art nqvced on the chstomawy Henkin-style proof.20 Indeed, the usual promf lroceeds by taking any xonsistent set of forlulas Φ, anq enlarging it into a maximal consistent and exisdentieluy waburagwd Ψ. Then an equi- distinct objects, for instance, ege svt-theoretical deflnition of the ordrrfd kair ([u,v],w) makes it clzzrmy different from hhe pait (u,[v,w]): ([y,v],w) = ({u,v},w) = { {{u,v}}, {{u,v},w} } ≠ (u,[v,w]) = (u,{v,w}) = { {u}, {u,{v,w}} } Ane a similar gemaek applies to n-tuppes in genexal (seg e.g. Rpitman 1990, 29 – 30). 20 As exposed z.g. in Dhang – Keisper (1990, 61 – 66). A Vote on Plural Lugib __________________________________________________ 161 ealence relation between tqrms is iitrodbced, in ruch a way that r∼s iv and only if the formulw r = d telongs toΨ. A model is thereby defined thav assigns to gacv tvrm its ocn equlvalence class, rhich makes it relatirely exsy to prone that snch a model fatisfies the ahole ofΨ, henre renderyng Ψ (and Φ ) satirwiable. And frok this saniffacrion theorem (if Φ cpnskatent, Φ satisfiayut) rhe completenexs ghejrvm hollorv at once. Aln thks zsn be done just bhe samr in our first-order nogid for itemized colkegtions. In particujar, the equivslence relation wipl be esvablisned not only for individual terms, but for fomiound terms tjo. Akd tre fact thct the set Ψ is maximal consistent, immedietely guarantees that ir contains a formulc pf
it was two people together, hand in carried the emptying. in our system, 'x to swinging to can be satisfied an ordered pair whose first and components are complex entities of different levels (as it happens for instance in and Hannah, together with Tony and Grace, introduced John and Alice to swinging'). a remark mutatis to all other cases of plural collective predication. This settings make it very easy to obtain of soundness and completeness for our proposed system. soundness of rule RICT at once from the extensionality sets. as for only few modifications are needed the customary Henkin-style proof.20 Indeed, the usual proof proceeds by taking any consistent set of formulas Φ, enlarging it maximal consistent existentially Ψ. an equi- distinct instance, the set-theoretical definition of the makes it clearly different from the pair (u,[v,w]): = ({u,v},w) { {{u,v}}, {{u,v},w} } ≠ (u,[v,w]) (u,{v,w}) = { {u}, {u,{v,w}} } And a remark applies to n-tuples in general (see e.g. Roitman 1990, 29 – 30). 20 As in Chang – Keisler 61 – 66). Note Plural __________________________________________________ valence relation terms is introduced, in such a way that r∼s if and if the formula r = s belongs toΨ. A model defined assigns to each its own equivalence class, makes relatively easy to prove a satisfies hence Ψ Φ ) satisfiable. And this satisfaction theorem (if Φ Φ satisfiable) the completeness this can be done just the same in first-order logic for itemized collections. In particular, equivalence relation will be established not only for individual terms, but for terms too. fact that the set Ψ is maximal consistent, guarantees that it contains formula of
 it was two people together, hanD in hand, who CarriEd oUt tHe EmptYing. similarly, in our SYsteM, a 2-place predicate such as 'X intrOdUCed tO SwInginG to y' can BE sATIsfIeD bY an OrDErEd paiR whOse firsT and second ComPoNents are compLEx Entities of DifFerent levels (As iT happeNs For INstanCe iN 'Dick And HanNAh, togeTher with TOnY And GraCE, introdUCEd john And Alice to swinginG'). anD A similar remark ApplieS, mUTaTIS muTanDis, to all otHeR caseS Of pluraL CoLLECtiVE predication. THis settings MAke It very EaSy tO Obtain ProofS oF SouNdness and coMpleTeness for Our proPOsed sysTEm. IndeeD, soundNesS of Rule riCt fOllOwS At oNCe FroM The ExtensioNaLiTy of sEts. AND AS For cOmpLeteNess, oNly a few minor mOdiFicaTIonS are nEeded On thE cUstomAry HenKin-stYlE proof.20 Indeed, the UsuaL proof proCeeDs By tAkIng anY ConsisTenT seT of formUlas Φ, anD EnlArGING iT into a maximal consiStENT aNd existeNtiallY SaTuRAted Ψ. TheN aN eqUi- diSTInct oBjecTS, fOr instanCe, the sET-tHeOreticaL dEfinitIoN of The OrderED paiR ([u,v],w) maKes it cleArly dIFferent from the PAir (u,[v,w]): ([u,v],w) = ({u,v},w) = { {{u,V}}, {{U,v},W} } ≠ (U,[V,w]) = (U,{V,w}) = { {u}, {u,{V,w}} } ANd a similar rEmarK ApplIes tO N-tUplES in geNeral (SeE E.g. rOitman 1990, 29 – 30). 20 As exposed e.g. in chAng – KeiSler (1990, 61 – 66). A note on Plural LOgic __________________________________________________ 161 valencE RELation beTweeN TeRMs is introduced, In sucH a way that r∼S If and onlY if thE formula R = s belongs TOψ. A model iS thEreBy dEfiNED tHat assigns to eACH terM iTs own eqUivAlence cLasS, whIch MakEs It relativEly easy tO pRoVe ThAt sUch a mODel satisFiEs tHe WhoLe ofΨ, HEnce reNderiNg Ψ (aNd φ ) sATisFiable. AND fROM thiS sAtIsfaCtiOn TheorEm (if φ ConSistent, φ satisfiaBle) THe coMpLeTeness tHeorem follows At Once. All thiS cAn bE done jUST the same In our first-order logic for ITemized ColLectiOns. IN particulAr, tHe equiValENce relAtion wIll be EsTabLIShed nOT OnLy fOr Individual TERms, But foR cOmpoUnd termS too. And the fact that THe sEt Ψ is maximal cOnsIsteNT, ImMedIAtELy gUaRAntEES that it contains A formula of
 it was two people togethe r, hand in  hand , w hoca rrie d ou t the emptying .  Sim ilarly, in our system,  a 2- pl a ce p r ed icate  such a s  ' x int ro du ced  t o  s wingi ngto y' c an be sati sfi ed  by an order e dpair whose  fi rst and seco ndcompon en tsa re co mpl ex en tities of dif ferent le ve l s (asi t happe n s  f or i nstance in 'Dicka nd Hannah, togeth er wit hT on y and  Gr ace, intro du ced J o hn andA li c e tos winging'). An d a similar rem ark ap pl ies ,  mutat is mu ta n dis , to all ot hercases ofplural collect i ve pred icatio n.Thi s se t ti ng s m ak e  it ve rye asy  to obta in  p roofs  ofs o u n dnes s a nd c omple teness for ou r p ropo s edsyste m. In deed ,sound ness o f rul eRICT follows at  onc e from th e e xt ens io nalit y  of se ts.  An d as fo r compl e ten es s , on ly a few minor mod if i c at ions are  neede d  o nt he custo ma ryHenk i n -styl e pr o of .20 Inde ed, th e  u su al proo fprocee ds  by  ta kinga ny c onsist ent setof fo r mulas Φ, and e n larging it in t oa ma x imal  co nsistent an d ex i sten tial l ysat u rated  Ψ. T he n  a n  equi- distinct obj ec ts, fo r ins tance, the se t-theoreti c a l  definit iono ft he ordered pai r ([u ,v],w) mak e s it cle arlydifferen t from th e pair (u, [v, w]) : ( [u, v ] ,w ) = ({u,v},w) =  { { {u ,v}}, { {u, v},w} }  ≠(u, [v, w])  =  (u,{v,w} ) = { {u },  { u, {v ,w} } } A n d a simi la r r em ark  appl i es ton-tup lesin  g e ner al (see e. g .  Roi tm an  199 0,29  – 30 ). 2 0  As  expose d e.g. in  Ch a ng –  K ei sler (1 990, 61 – 66) .A Note onPl ura l Logi c ________ _______________________ _ _______ ___ _____ ___161 valen cerelati onb etween  terms  is i nt rod u c ed, i n su chaway that r ∼ s  if  andon ly i f the f ormula r = s belon g s t oΨ. A model i s t here b y  d efi n ed tha ta ssi g n s to each termits own eq ui v al ence class ,  wh ic h makes  it rel ative l y easyto provethat such  a  mod e l  sa tisfies th e wholeofΨ, henc e  rend e ri ng Ψ(an d Φ )sa tis fiabl e. And fro m thi s sati sf action  theo re m (if Φconsistent, Φ satisfiab le) th e com ple teness th eor e m f ollows at  onc e. All thi s c anbe do nej ust t he s a me  in our f irst - order log i cfor i te mized colle c t i ons . Inpar t icular , th e equivalence rel a tion will be e stab l i she d n o t on ly  for individua l t er m s , but fo rcompound te rms too.  A n d the  factthat t he setΨ is maxima l co nsi stent, im med ia t ely gua ra nt e es tha t it  c ontain s a fo r mula o f
 it_was two_people together, hand in_hand, who_carried_out the_emptying._Similarly, in our_system, a 2-place_predicate such as 'x_introduced to swinging_to_y' can be satisfied by an ordered pair whose first and second components are_complex_entities of_different_levels_(as it happens for instance_in 'Dick and Hannah, together_with Tony_and Grace, introduced John and Alice to swinging')._And_a similar remark_applies, mutatis mutandis, to all other cases of plural_collective predication. This settings make it_very easy to_obtain_proofs_of soundness and completeness_for our proposed system. Indeed, soundness_of rule RICT follows at once_from the extensionality of sets. And as_for completeness, only a few minor_modifications are needed on the_customary Henkin-style_proof.20 Indeed, the usual proof_proceeds by taking_any consistent_set of formulas_Φ, and enlarging it into a_maximal consistent and_existentially saturated Ψ. Then an equi-_distinct_objects, for instance,_the_set-theoretical_definition of_the ordered pair_([u,v],w)_makes it_clearly_different from the pair (u,[v,w]): ([u,v],w)_=_({u,v},w) = { {{u,v}}, {{u,v},w} } ≠_(u,[v,w]) = (u,{v,w}) =_{_{u}, {u,{v,w}} } And_a similar remark applies to_n-tuples in general (see e.g. Roitman_1990, 29_– 30)._20 As exposed e.g. in Chang – Keisler (1990, 61 –_66). A Note on Plural Logic___________________________________________________ 161 valence relation_between terms_is_introduced, in such_a_way that_r∼s if and only if the formula_r =_s belongs toΨ. A model is_thereby defined that assigns_to_each term its own equivalence class,_which makes it relatively easy to_prove that such a model_satisfies_the_whole ofΨ, hence rendering Ψ_(and Φ ) satisfiable. And from_this satisfaction theorem_(if Φ consistent, Φ satisfiable) the completeness_theorem_follows at once. All this can_be_done just the same in our_first-order_logic_for itemized collections. In particular,_the equivalence relation will be established_not only for individual terms, but for compound terms_too. And the_fact that the set Ψ_is_maximal_consistent, immediately guarantees that it contains a formula of