 k+1 when the most complex term among r1,..., rn has complexity k, and in that case the values r1 I,..., rn I are taken for granted by the inductive clause of the definition. 18 Complex entities are bound to be finite sets (of individuals of U, or of finite sets of individuals of U, etc.), in fair accord with the syntactic structure of compound terms in our language, which can only consist of finite lists of terms (cf. §4–(c)). It could not be otherwise in a singular logic system, in which the only vehicle to convey reference is the individual object (cf. §2); and this is so, of course, given the impossibility of denoting an infinite plurality by making individual reference to each of its elements. Restriction to finite sets, incidentally, eliminates any possibility of Russell's paradox style arguments affecting our system. 19 Sets which have among its elements single individuals as well as other sets of different complexities, are indeed familiar from set theory. The same can be said of ntuples whose components are single individuals as well as sets and other n-tuples of different complexities. This involves no risk of ambiguity: letting u, v and w be 160 _________________________________________________ Gustavo Fernandéz Díez Then we can simply say that an atomic formula such P(r1,..., rn) is true under I if and only if the n-tuple (r1 I,..., rn I) belongs to the relation PI. And this will hold irrespectively of whether some of the terms r1,..., rn turn out to be compound terms or not, or what amounts to the same thing, irrespectively of whether some of the entities r1 I,..., rn I turn out to be complex entities or not. In our logical system a 2-place predicate such as 'x emptied y' can be satisfied by an ordered pair whose two components are single individuals (as it happens for instance in 'Charles emptied the bottle'), but also by an ordered pair whose first component is not a single individual of the universe, but a complex entity composed of two of them (as it happens in 'Dick and Tony emptied the bottle'). Something which is in perfect match with our intuition that the 2-place predicate 'x emptied y' (or, in other words, the relation corresponding to it) is essentially the same in the case where it was a single person who emptied the bottle, that in the case where
k+1 when the most complex term among r1, ..., rn has complexity k, and in that case the value r1 I, ..., rn I are accept for granted by the inductive clause of the definition. 18 Complex entities are restrict to be finite sets (of individuals of U, or of finite set of individual of U, etc .), in fair agreement with the syntactic structure of compound terms in our terminology, which can only consist of finite tilt of term (cf. § 4–(c) ). It could not be otherwise in a singular logic system, in which the only fomite to convey reference is the individual object (cystic fibrosis. § 2); and this is so, of course, give the impossibility of denote an infinite battalion by making individual reference to each of its elements. Restriction to finite set, incidentally, eliminates any possibility of Russell's paradox style arguments feign our organization. 19 Sets which have among its elements single individuals as well as other sets of unlike complexity, are indeed familiar from set theory. The like can be say of ntuples whose components are single person as well as sets and other n - tuples of unlike complexities. This involves no risk of ambiguity: letting u, v and w be 160 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Gustavo Fernandéz Díez Then we can simply say that an atomic formula such P(r1, ..., rn) is true under I if and only if the n - tuple (r1 I, ..., rn I) belong to the sexual intercourse PI. And this will hold irrespectively of whether some of the terms r1, ..., rn turn out to be compound terms or not, or what amounts to the like thing, irrespectively of whether some of the entities r1 I, ..., rn I turn out to be complex entities or not. In our logical system a 2 - place predicate such as' x emptied y' can be satisfied by an coherent pair whose two part are single individuals (as it happen for instance in' Charles empty the bottle'), but also by an arranged pair whose beginning component is not a single individual of the universe, but a complex entity composed of two of them (as it happen in' Dick and Tony emptied the bottle'). Something which is in arrant match with our intuition that the 2 - place predicate' x emptied y' (or, in other discussion, the sexual intercourse corresponding to it) is essentially the like in the case where it was a single person who emptied the bottle, that in the case where
 k+1 ahen the most complex tevm among r1,..., rn has complxxity k, and in ghat case the values r1 I,..., rn M arw taktu for granted by the inductivv clause if tie definition. 18 Complex ekcitiea are uound to be finlte sets (of individuals ox J, lr of finite sets of individuals of U, etc.), on fair accord wyth uhe synfactic structure of compound terms in our language, whivh can only consist of finlte pists of terms (cf. §4–(f)). It could bot fw otherwise kn a singular logic syatem, in which the only vehicle go couvey referebcw id the indivivual ofject (cf. §2); and this iv so, of course, given bhe ikpowsibility of denoting an infinite plurajity by mdknng individual referebcw to gach mf igw euemtntx. Destriftikn to finife sets, incudentally, eliminatex wby possibilitg of Rtsfell's paradox style arguments affecting oud system. 19 Sets which hace among its elements single igdividuals as well as other sets of different com[lexivids, cvc inawef familiar from set theory. The same can be says pf ntuples whose components age fingle indiviauals ca sell as sets and ohher n-toples if differtnt cpmplexities. This involves ni risk of amyigyity: letting u, v aud w be 160 _________________________________________________ Gurtavp Fermandéz Díez Then we can snmply aay that an atomic fkfmula such P(r1,..., rn) is tsue under I if and only if the n-tupoe (r1 I,..., rn I) belpngs tj the relahion IH. And this will hopd irtespecdively of ahether some of the terms r1,..., rn vnrn out to be cmmpmund teris or not, or what wmounts to the same tking, ifrespectivvly of whxther some os the entitiev r1 I,..., rn I tucn out to be xompoex entkgies or not. In our logibaj sywtem a 2-place predigate rhch as 'x emptieb i' can be satisfoed by aj prqared pair whmse gwo vompovents are slngue imdividuals (as it hap[ens for instance in 'Cnavles emptued the fottle'), but alxo by an ordered pwir wiose fmrst cpmpjnent is not a single individuzl of the unlverse, but a sompoex entity cpmposed of two of them (as it happens in 'Dick and Tony emktied the bottle'). Somejhikg which is mn persect matcv with our intuition that the 2-place pvedicate 'x emptied y' (od, in odher aords, the relation corresponding to it) is essentially the same in the casw wherx yt was a sihgle persmn cho empeied vhe bottle, that im the case where
k+1 when the most complex term among has k, and that case the are for granted by inductive clause of definition. 18 Complex entities are bound be finite sets (of individuals of U, or of finite sets of individuals U, etc.), in fair accord with the syntactic structure of compound terms in language, can consist finite lists of terms (cf. §4–(c)). It could not be otherwise in a singular logic system, which the only vehicle to convey reference is individual object (cf. §2); this is so, of course, the of denoting infinite by individual reference to of its elements. Restriction to finite sets, incidentally, eliminates any possibility of Russell's paradox style arguments affecting system. 19 have among elements individuals well as other different complexities, are indeed familiar from same can be said of ntuples whose components single individuals well as sets and other n-tuples different complexities. This involves no risk of ambiguity: u, v and w be 160 _________________________________________________ Gustavo Fernandéz Díez Then we can simply say atomic formula such P(r1,..., is true under if only the (r1 I,..., I) belongs to the relation PI. And this will hold irrespectively whether some of the terms r1,..., rn turn out to terms not, or what to the same thing, of some of the entities rn turn complex or In our logical system 2-place predicate such as 'x y' can be satisfied two components are single individuals (as it happens instance in 'Charles emptied the bottle'), but by an ordered pair whose first component is not a single individual the universe, complex entity composed of two of them (as happens in 'Dick and emptied the bottle'). Something which is in perfect match our that the predicate 'x emptied (or, in other the relation corresponding is essentially same the was a single person who emptied bottle, in the case where
 k+1 when the most complex term amOng r1,..., rn has cOmpleXitY k, aNd In thAt caSe the values r1 I,..., rN i are Taken for granted by the inDuctiVe CLausE Of The deFinitioN. 18 coMPLex EnTiTieS aRE bOund tO be Finite sEts (of indivIduAlS of U, or of finiTE sEts of indivIduAls of U, etc.), in fAir Accord WiTh tHE syntActIc strUcture OF compoUnd terms iN oUR languAGe, which CAN oNly cOnsist of finite lisTS oF Terms (cf. §4–(c)). It coulD not be OtHErWISe iN a sIngular logIc SysteM, In which THe ONLY veHIcle to convey rEference is tHE inDividuAl ObjECt (cf. §2); anD this Is SO, of Course, given The iMpossibilIty of dENoting aN InfinitE pluraLitY by MakiNG iNdIviDuAL reFErEncE To eAch of its ElEmEnts. REstrICTIOn to FinIte sEts, inCidentally, eliMinAtes ANy pOssibIlity Of RuSsEll's pAradox Style ArGuments affectinG our System. 19 SetS whIcH haVe Among ITs elemEntS siNgle indIvidualS As wElL AS OtHer sets of different CoMPLeXities, arE indeeD FaMiLIar from sEt TheOry. THE Same cAn be SAiD of ntuplEs whosE CoMpOnents aRe Single InDivIduAls as WEll aS sets aNd other n-TupleS Of different comPLexities. This iNVoLVEs NO risK of Ambiguity: leTtinG U, v anD w be 160 _________________________________________________ gUsTavO fernaNdéz DÍeZ thEN we can simply say that An Atomic FormuLa such P(r1,..., rn) is tRue under I iF AND only if tHe n-tUPlE (R1 I,..., rn I) belongs to The reLation PI. AnD This will Hold iRrespectIvely of whETHer some oF thE teRms R1,..., rn TURn Out to be compouND TermS oR not, or wHat Amounts To tHe sAme ThiNg, IrrespectIvely of wHeThEr SoMe oF the eNTities r1 I,..., Rn i tuRn Out To be cOMplex eNtitiEs or NoT. IN Our Logical SYsTEM a 2-plAcE pRediCatE sUch as 'X empTIed Y' can be sAtisfied bY an ORderEd PaIr whose Two components ArE single indIvIduAls (as iT HAppens foR instance in 'Charles emptiED the botTle'), But alSo by An ordered PaiR whose FirST compoNent is Not a sInGle INDividUAL oF thE uNiverse, but A COmpLex enTiTy coMposed oF two of them (as it happENs iN 'Dick and Tony eMptIed tHE BoTtlE'). soMEthInG WhiCH Is in perfect matcH with our inTuITiOn that the 2-pLAce PrEdicate 'X emptieD y' (or, iN Other woRds, the relAtion corrEsPondING to It) is essentIally the Same in the CAse whERe It was A siNgle peRsOn wHo empTied thE BotTle, thAt in thE cAse wheRe
 k+1 when the most complex  term amon g r1, ... , r nhascomp lexity k, andi n th at case the values r1I,... ,r n Ia re  take n for g r an t e d b yth e i nd u ct ive c lau se of t he definit ion .18 Complex e n ti ties are b oun d to be fini tesets ( of  in d ividu als  of U , or o f  finit e sets of  i n dividu a ls of U , et c.),  in fair accord w i th the syntacticstruct ur e  o f com pou nd terms i nour l a nguage, wh i c h  ca n  only consist  of finitel ist s of t er ms( cf. §4 –(c)) .I t c ould not be  oth erwise in  a sin g ular lo g ic syst em, in  wh ich  the on ly  ve hi c let ocon v eyreferenc eis  theindi v i d u al o bje ct ( cf. § 2); and thisisso,o f c ourse , giv en t he  impo ssibil ity o fdenoting an inf init e plurali tyby  ma ki ng in d ividua l r efe rence t o eacho f i ts e l em ents. Restrictionto f in ite sets , inci d en ta l ly, elim in ate s an y possi bili t yof Russe ll's p a ra do x style  a rgumen ts  af fec tingo ur s ystem.  19 Sets  whic h  have among it s  elements sin g le i nd i vidu als  as well as  oth e r se ts o f  d iff e rentcompl ex i ti e s, are indeed famil ia r from  settheory. The s ame can be s a id of nt uple s  w h ose components  aresingle ind i vidualsas we ll as se ts and ot h e r n-tupl esofdif fer e n tcomplexities. T hisin volvesnorisk of  am big uit y:le tting u,v and wbe  1 60  _ ___ _____ _ ________ __ ___ __ ___ _____ _ ______ _____ ____  G us t avo  Fernan d éz D íezTh en  wecan  s imply  say tha t an at omic form ula such  P (r 1,...,rn) is true u nd er I if an donl y if t h e  n-tuple  (r1 I,..., rn I) belon g s to th e r elati on P I. And th iswill h old irresp ective ly of  w het h e r som e of  th eterms r1,. . . , r n tur noutto be c ompound terms or n o t,or what amoun tsto t h e  s ame th i ng,  i r res p e ctively of whet her some o ft he  entitiesr 1 I ,. .., rnI turnout t o  be com plex enti ties or n ot . In o urlogical sy stem a 2 -place pr e dicat e  s uch a s ' x empt ie d y ' can  be sa t isf ied b y an o rd ered p air w ho se two c omponents are single in dividu als ( asit happen s f o r i nstance i n 'C harles emp tie d t he bo ttl e '), b ut a l so  by an or dere d  pair who s efir s t  c omponent is n o t a  sing lei ndivid ualof the universe,b ut a complex e ntit y com pos e d of  t wo of them (as  it  h a p pens in'D ick and Ton y emptie dt he bo ttle') . Some thing w h i ch is inperf ect  match wi thou r  intuit io nt hat th e 2- pl ace pr edicat e  'xe m ptied y' (or, in  othe r words ,  th e rel at ion cor r espo nding to i t) is essen tially  the  same  in the  c ase wh ere  i t was a si n gle perso n who  emptie dthebot tle, t hati n  thecase  w her e
 k+1_when the_most complex term among_r1,..., rn_has_complexity k,_and_in that case_the values r1_I,..., rn I are_taken for granted_by_the inductive clause of the definition. 18 Complex entities are bound to be finite_sets_(of individuals_of_U,_or of finite sets of_individuals of U, etc.), in_fair accord_with the syntactic structure of compound terms in_our_language, which can_only consist of finite lists of terms (cf. §4–(c))._It could not be otherwise in_a singular logic_system,_in_which the only vehicle_to convey reference is the individual_object (cf. §2); and this is_so, of course, given the impossibility of_denoting an infinite plurality by making_individual reference to each of_its elements._Restriction to finite sets, incidentally,_eliminates any possibility_of Russell's_paradox style arguments_affecting our system. 19 Sets which_have among its_elements single individuals as well as_other_sets of different_complexities,_are_indeed familiar_from set theory._The_same can_be_said of ntuples whose components are_single_individuals as well as sets and other_n-tuples of different complexities._This_involves no risk of_ambiguity: letting u, v and_w be 160 _________________________________________________ Gustavo Fernandéz_Díez Then_we can_simply say that an atomic formula such P(r1,..., rn) is true_under I if and only if_the n-tuple (r1 I,...,_rn I)_belongs_to the relation_PI._And this_will hold irrespectively of whether some of_the terms_r1,..., rn turn out to be_compound terms or not,_or_what amounts to the same thing,_irrespectively of whether some of the_entities r1 I,..., rn I_turn_out_to be complex entities or_not. In our logical system a_2-place predicate such_as 'x emptied y' can be satisfied_by_an ordered pair whose two components_are_single individuals (as it happens for_instance_in_'Charles emptied the bottle'), but_also by an ordered pair whose_first component is not a single individual of the_universe, but a_complex entity composed of two_of_them_(as it happens in 'Dick and Tony emptied the bottle')._Something which_is in perfect_match with our intuition that the 2-place predicate 'x emptied_y' (or, in other words, the relation_corresponding to it) is essentially the same in the case where_it was a single person who emptied the_bottle, that in the case where