 } Thus, itemized collections are represented as sets.17 16 So we have finally shown (1) to be a first-order inference, as we set out to do in §1. 17 Strictly speaking, the clause has to be spelt out by induction on the degree of complexity of the term. Thus [r1,..., rn] has complexity 1 when all r1,..., rn are individual terms (i.e., terms of complexity 0), and in that case the values r1I,..., rn I are already A Note on Plural Logic __________________________________________________ 159 This leads us to deal with two kinds of 'entities' made up from the individual members of the interpretation I: on the one hand, simple entities, which will just be the individuals of U (where U is the universe of I); and on the other hand, complex entities, which will be finite sets of individuals of U, or finite sets of finite sets of individuals of U, or combinations of these, etc. etc. In more precise terms (by induction): every individual of U will be a simple entity (entity of complexity 0) of U; and if u1,..., un are entities of U, the more complex of which has complexity k, then {u1,..., un} will be a complex entity of U of complexity k+1. No need to say that I will assign simple entities to individual terms, and complex entities to compound terms.18 Likewise, we shall consider that a property defined on U is a set of entities of U, and more in general, that an n-place relation defined on U is a set of n-tuples of entities of U. Then the interpretation of an n-place predicate letter P under I (say PI) will be an n-place relation on U, i.e., a set of n-tuples of entities of U. So that an n-tuple of entities of U will be said to satisfy P under I, if and only if it belongs to the relation (set) PI. Thus, if P is for example a 2-place predicate letter, its interpretation PI will be a set of ordered pairs, pairs whose components do not necessarily have to be plain individuals of the universe U (as it occurs in the standard first-order logic treatment), but that can be complex entities, as long as they are built up starting from the individuals of U only.19 given. And [r1,..., rn] has complexity
} Thus, itemized collections are represented as sets.17 16 So we have ultimately usher (1) to be a first - order inference, as we set out to perform in § 1. 17 Strictly speaking, the article has to be spelt out by induction on the academic degree of complexity of the condition. Thus [ r1, ..., rn ] suffer complexity 1 when all r1, ..., rn are individual terms (i.e., term of complexity 0), and in that case the values r1I, ..., rn I are already A Note on Plural Logic _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 159 This leads us to deal with two kinds of' entity' made up from the individual members of the interpretation one: on the one hand, simple entity, which will just be the individual of U (where U is the universe of one); and on the other hand, complex entities, which will be finite set of individuals of U, or finite sets of finite sets of individuals of U, or combination of these, etc. etc. In more precise terms (by induction ): every individual of U will be a simple entity (entity of complexity 0) of U; and if u1, ..., un are entities of U, the more complex of which has complexity k, then { u1, ..., un } will be a complex entity of U of complexity k+1. No need to say that I will assign elementary entities to individual terms, and complex entity to compound terms.18 Likewise, we shall think that a place defined on U is a set of entities of U, and more in general, that an n - place relation defined on U is a set of newton - tuples of entities of U. Then the interpretation of an n - place predicate letter P under I (say PI) will be an n - place relation on U, i.e., a bent of n - tuples of entities of U. So that an n - tuple of entity of U will be said to meet P under I, if and only if it belongs to the relation (set) PI. Thus, if P is for case a 2 - place predicate letter, its interpretation PI will be a set of order couple, pairs whose components do not inevitably have to be plain individuals of the universe U (as it occurs in the standard inaugural - order logic discussion), but that can be complex entities, as long as they are built up starting from the individual of U only.19 given. And [ r1, ..., rn ] has complexity
 } Tjus, itemized collections are representeb as sevs.17 16 So se have winally shown (1) to be a first-lreer ibference, as we set out to do in §1. 17 Stricrly wpeaking, thx clause has tk be v'elt out by indoction on tha degree of cokpuerity of the term. Thus [r1,..., rn] has complqxity 1 ehfn all r1,..., rn arg indpvyduam terms (i.e., terms of complexity 0), ans in thet case the valies r1I,..., rn I are already A Jote on Plural Logic __________________________________________________ 159 This leads us ei deal with gwo kinds of 'entities' jade up from the individual memcers pf the intgxprehdtion I: on vhe onv hand, simple entitiev, which will just be bhe iidivuduals of U (where U iv the universe of Y); and on dhz other hand, complex wnritiev, whhch dull be fmnife setd oh individuams of U, or dinite sets of finiue fvys of indivisuals jf U, or combinations of these, etc. etc. In kors precise terms (by induxtion): every individuap of U wijl be a simple entity (entity of complexity 0) of U; dnd ih j1,..., uu are dbtlties of U, the more complex of which has compjsxotj k, then {u1,..., un} wilj be a compked rgtity of U of complzsify k+1. No need to saj that Y wilo assign fimpke entities to individual twrms, and comilex entities to compobnd terms.18 Liyewixe, we shall consider that a 'roperfy defined ln U is a ret of entities uf L, ang more in general, that an g-place reoatipn defived pn U if a set of n-tuples of entities of K. Theu the interpretwtion of an n-place predicate levver P under I (sdy II) will bz an n-ilace relation jn U, i.e., a set pf n-tupjes ow entities of U. Sm that an n-euple of entidles of U wiln be saiq to satusfy P jvder I, if and pnly if in yelongs ti the relation (set) PI. Fhus, if P is fox txqmple a 2-place lreaicwtv lxtter, hts interpredatiun OO wilu be a set iw orcered pairs, pairs whmse domponents do not megessarily have to be plain indoviduals of the unlversx U (as it ovcuts in the standard first-order lkgic treahmekt), but that cwn bc coiplex enticies, as long as they are built up startiig from the individuals of U only.19 given. Anb [v1,..., rn] has com'lexitr
} Thus, itemized collections are represented as So have finally (1) to be set to do in 17 Strictly speaking, clause has to be spelt out induction on the degree of complexity of the term. Thus [r1,..., rn] has 1 when all r1,..., rn are individual terms (i.e., terms of complexity 0), in case values rn I are already A Note on Plural Logic __________________________________________________ 159 This leads us to deal two kinds of 'entities' made up from the members of the interpretation on the one hand, simple which just be individuals U U is the of I); and on the other hand, complex entities, which will be finite sets of individuals of or finite finite sets individuals U, combinations of these, In more precise terms (by induction): U will be a simple entity (entity of 0) of and if u1,..., un are entities U, the more complex of which has complexity then {u1,..., un} will be a complex entity of U of complexity k+1. No need that I will assign entities to individual and entities compound Likewise, we consider that a property defined on U is a set of of U, and more in general, that an n-place relation U a set of of entities of U. the of an n-place predicate under (say an relation U, i.e., a set n-tuples of entities of U. that an n-tuple of said to satisfy P under I, if and if it belongs to the relation (set) Thus, if P is for example a 2-place predicate letter, its interpretation will be of ordered pairs, pairs whose components do not have to be plain of the universe U (as it occurs in the first-order treatment), but can be complex as long as are built up the individuals U given. has complexity
 } Thus, itemized collections arE representEd as sEts.17 16 so wE hAve fInalLy shown (1) to be a fiRSt-orDer inference, as we set out To do iN §1. 17 STRictLY sPeakiNg, the clAUsE HAs tO bE sPelT oUT bY induCtiOn on the Degree of coMplExIty of the term. tHuS [r1,..., rn] has comPleXity 1 when all r1,..., Rn aRe indiViDuaL Terms (I.e., tErms oF complEXity 0), anD in that caSe THe valuES r1I,..., rn I aRE AlReadY A Note on Plural LogIC __________________________________________________ 159 THIs leads us to deaL with tWo KInDS Of 'eNtiTies' made up FrOm the INdividuAL mEMBErs OF the interpretAtion I: on the ONe hAnd, simPlE enTIties, wHich wIlL JusT be the indivIduaLs of U (wherE U is thE UniversE Of I); and oN the otHer HanD, comPLeX eNtiTiES, whICh WilL Be fInite setS oF iNdiviDualS OF u, Or fiNitE setS of fiNite sets of indIviDualS Of U, Or comBinatIons Of These, Etc. etc. in morE pRecise terms (by inDuctIon): every iNdiViDuaL oF U wilL Be a simPle EntIty (entiTy of comPLexItY 0) OF u; aNd if u1,..., un are entities Of u, THe More compLex of wHIcH hAS complexItY k, tHen {u1,..., UN} Will bE a coMPlEx entity Of U of cOMpLeXity k+1. No NeEd to saY tHat i wiLl assIGn siMple enTities to IndivIDual terms, and coMPlex entities tO CoMPOuND terMs.18 LIkewise, we shAll cONsidEr thAT a ProPErty dEfineD oN u iS A set of entities of U, anD mOre in gEneraL, that an n-place Relation deFINEd on U is a Set oF N-tUPles of entities Of U. ThEn the interPRetation Of an n-Place preDicate letTER P under I (Say pI) wIll Be aN N-PlAce relation on u, I.E., a seT oF n-tupleS of EntitieS of u. So ThaT an N-tUple of entIties of U WiLl Be SaId tO satiSFy P under i, iF anD oNly If it bELongs tO the rElatIoN (sET) PI. thus, if P IS fOR ExamPlE a 2-PlacE prEdIcate LettER, itS interpRetation Pi wiLL be a SeT oF ordereD pairs, pairs whOsE componentS dO noT necesSARily have To be plain individuals of tHE univerSe U (As it oCcurS in the staNdaRd firsT-orDEr logiC treatMent), bUt ThaT CAn be cOMPlEx eNtIties, as lonG AS thEy are BuIlt uP startiNg from the individuaLS of u only.19 given. And [R1,..., rn] Has cOMPlExiTY
 } Thus, itemized collecti ons are re prese nte d a ssets .1716 So we havef inal ly shown (1) to be a f irst- or d er i n fe rence , as we se t out  t odoin §1 . 17Str ictly s peaking, t hecl ause has tob espelt outbyinduction on  th e degr ee  of compl exi ty of  the t e rm. Th us [r1,.. ., rn] ha s  comple x i ty  1 w hen all r1,..., r n  a r e individual t erms ( i. e ., t erm s o f complexi ty  0),a nd in t h at c a set he values r1I ,..., rn Ia realread yA N o te onPlura lL ogi c _________ ____ _________ ______ _ _______ _ _______ ______  15 9 T hisl ea ds  us  t o  de a lwit h  tw o kindsof  ' entit ies' m a d e up  fr om t he in dividual memb ers  oft heinter preta tion  I : onthe on e han d,  simple entitie s, w hich will  ju st  be  t he in d ividua lsofU (wher e U ist heun i v e rs e of I); and on th eo t he r hand,comple x  e nt i ties, wh ic h w illb e  fini te s e ts  of indi vidual s  o fU, or f in ite se ts  of  fi nites etsof ind ividuals  of U ,  or combinatio n s of these, e t c. e tc .  Inmor e precise t erms (byindu c ti on) :  ever y ind iv i du a l of U will be a si mp le ent ity ( entity of com plexity 0) o f  U; andif u 1 ,. . ., un are enti tiesof U, them ore comp lex o f whichhas compl e x ity k, t hen  {u 1,. .., u n}  will be a co m p lexen tity of  Uof comp lex ity  k+ 1.No  need tosay that  I  w il lass ign s i mple ent it ies  t o i ndivi d ual te rms,andco mp l exentitie s  t o comp ou nd  ter ms. 18  Like wise ,  we  shallconsidertha t  a p ro pe rty def ined on U isaset of ent it ies  of U, a nd morein general, that an n-p l ace rel ati on de fine d on U is  aset of  n- t uplesof ent ities  o f U . Thent h eint er pretationo f  an  n-pl ac e pr edicate  letter P under I( say  PI) will beann-pl a c erel a ti o n o nU , i . e ., a set of n-t uples of e nt i ti es of U. S o  th at  an n-t uple of  enti t ies ofU will be  said tosa tisf y P u nder I, if  and onl y if it b e longs to  therel ation(s et)  PI.Thus,i f P  is f or exa mp le a 2 -plac epredicat e letter, its interpret ationPI wi llbe a setofo rde red pairs , pa irs whosecom pon entsdon ot ne cess a ri lyh ave t o be plain ind i vi dua l s  o f the unive r s e  U(as i t o c curs i n th e standard first- o rder logic tre atme n t ),but that  c an be complexent it i e s, as lo ng  as they ar e builtup start ing fr om the  indivi d u al s  of Uonly .19  given. A nd[r 1 ,..., r n]  h a s comp lexi ty 
 }_Thus, itemized_collections are represented as_sets.17 16_So_we have_finally_shown (1) to_be a first-order_inference, as we set_out to do_in_§1. 17 Strictly speaking, the clause has to be spelt out by induction on_the_degree of_complexity_of_the term. Thus [r1,..., rn]_has complexity 1 when all_r1,..., rn_are individual terms (i.e., terms of complexity 0),_and_in that case_the values r1I,..., rn I are already A Note_on Plural Logic __________________________________________________ 159 This_leads us to_deal_with_two kinds of 'entities'_made up from the individual members_of the interpretation I: on the_one hand, simple entities, which will just_be the individuals of U (where_U is the universe of_I); and_on the other hand, complex_entities, which will_be finite_sets of individuals_of U, or finite sets of_finite sets of_individuals of U, or combinations of_these,_etc. etc. In_more_precise_terms (by_induction): every individual_of_U will_be_a simple entity (entity of complexity_0)_of U; and if u1,..., un are_entities of U, the_more_complex of which has_complexity k, then {u1,..., un}_will be a complex entity of_U of_complexity k+1._No need to say that I will assign simple entities to_individual terms, and complex entities to_compound terms.18 Likewise, we_shall consider_that_a property defined_on_U is_a set of entities of U, and_more in_general, that an n-place relation defined_on U is a_set_of n-tuples of entities of U._Then the interpretation of an n-place_predicate letter P under I_(say_PI)_will be an n-place relation_on U, i.e., a set of_n-tuples of entities_of U. So that an n-tuple of_entities_of U will be said to_satisfy_P under I, if and only_if_it_belongs to the relation (set)_PI. Thus, if P is for_example a 2-place predicate letter, its interpretation PI will_be a set_of ordered pairs, pairs whose_components_do_not necessarily have to be plain individuals of the universe_U (as_it occurs in_the standard first-order logic treatment), but that can be complex_entities, as long as they are built_up starting from the individuals of U only.19 given. And [r1,...,_rn] has complexity