order logic, we add compound terms. A compound term will be a discrete and linear list of terms between square brackets, which behaving like a unit, acts itself as a new term.10 Hence we will have: (a) an individual variable is a term; (b) an individual constant is a term; (c) if r1,..., rn are terms, then [r1,..., rn] is a term.11 Terms formed by clauses (a) and (b) are the usual individual terms; terms formed by clause (c) will be called 'compound terms'. The inductive structure of the definition enables us to accommodate nested structures of itemized collections. Like for example: 'Dick and Hannah, together with Tony and Grace, introduced John and Alice to swinging': (9) I ( [[d,h],[t,g]], [j,a] )12 Next, the usual definition of formula does not need to be changed. In particular, an atomic formula P(r1,..., rn) can be simply defined as 10 Just as in 'Dick and Tony emptied the bottle', the phrase 'Dick and Tony' takes the argument place of the one who empties, with respect to the 2-place relation 'x emptied y'. 11 I take a language without descriptions and function symbols, for simplicity. 12 Where predicate letter I stands for the 2-place relation 'x introduced to swinging to y'. In this example both places of the relation are filled up by compound terms, the first of which is made up itself of two compound terms. Such nested pluralities are not normally accepted in plural logic treatments, although some authors have advocated for them (Rayo 2006; Linnebo – Nicolas 2008). Our treatment covers them, but only in the context of itemized collections. A Note on Plural Logic __________________________________________________ 157 an n-place predicate letter P followed by n terms. And quantification will be allowed – as usual – over individual variables only.13 As for the deductive system, the usual quantification rules (or axioms) will enable us to validate inference (7): 'Dick and Tony emptied the bottle. Hence, Dick and someone else emptied the bottle': E ( [d,t], b ) ∴ ∃x ( [d,x], b ) And in order to cover the other odd inferences of our list (that is, inferences (1), (3) and (5)), we introduce a Rule for the Indis
order logic, we add compound terms. A compound term will be a discrete and analogue tilt of term between square bracket, which behave like a unit, acts itself as a modern term.10 therefore we will have: (a) an individual variable is a term; (b) an individual constant is a term; (c) if r1, ..., rn are terms, then [ r1, ..., rn ] is a term.11 term formed by clauses (a) and (b) are the usual individual terms; terms formed by article (c) will be called' compound terms'. The inductive social organization of the definition enables us to accommodate nested structures of itemized collections. Like for model:' Dick and Hannah, together with Tony and Grace, introduced John and Alice to swinging': (9) I ([ [ d, h],[t, g ] ], [ j, a ]) 12 Next, the usual definition of formula does not necessitate to be changed. In particular, an atomic formula P(r1, ..., rn) can be simply defined as 10 Just as in' Dick and Tony emptied the bottle', the phrase' Dick and Tony' takes the argument place of the one who empties, with respect to the 2 - place relation' x emptied y'. 11 I consider a language without descriptions and function symbol, for simplicity. 12 Where predicate letter I stands for the 2 - place relation back' x introduced to swinging to y'. In this example both places of the relation back are filled up by compound terms, the first of which is made up itself of two compound terms. Such nested pluralities are not normally accept in plural logic treatments, although some authors have advocated for them (Rayo 2006; Linnebo – Nicolas 2008). Our treatment covers them, but only in the context of itemized collection. A Note on Plural Logic _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 157 an n - place predicate letter P followed by n term. And quantification will be allowed – as usual – over individual variables only.13 As for the deductive arrangement, the usual quantification rules (or axioms) will enable us to validate inference (7 ):' Dick and Tony emptied the bottle. therefore, Dick and someone else emptied the bottle': E ([ d, t ], b) ∴ ∃x ([ d, x ], b) And in order to cover the other odd inference of our list (that is, inferences (1), (3) and (5) ), we introduce a Rule for the Indis
ordfr logic, we add compound terms. A compouue term will ge a diszrete and linear list of terls betwtvn square brackets, whkch behavpng like q unmt, acts itself as a new bzrm.10 Hskce wz xill have: (a) an lndividual eariable is a defm; (b) an individual constant is a term; (c) if r1,..., rj are terms, thgn [r1,..., gn] is z term.11 Terms formed by clauses (a) ahd (b) art the usual indivicual terms; terms formed by clakse (c) will be callfd 'compound teriw'. The inductkve structlxe of the dgfinition enables us to accommodxte nzsted strucjbees mf itemized collvctions. Like npr exakple: 'Divk and Hannah, bogetier qith Tony and Grace, iitroduced John and Ajice to sfiuging': (9) I ( [[d,h],[t,g]], [j,a] )12 Nezt, the osual defkbitkon oh fkrmula doxs not need to be chanted. In particular, am wnpmic formula P(r1,..., rn) cwn be simply defined as 10 Just as in 'Dicn ahd Tony emptied the botrle', the phrase 'Dick ajd Tony' twkes the argument place of the one who empties, widh reaoecu bj rhf 2-place relation 'x emptied y'. 11 I take a languads einhout descriptionf and functooj ximbols, for simklicity. 12 Wgere predicate lether I sjands dor the 2-pjace relation 'x introduced to sqinging to y'. Un this example boch places of the relayion are filled up by cumpohnd terms, tje first kw which is made jp ptsenf of two compound terms. Stch nestev pluxalities are not njrmally acfepted in plural logic tgeatmgnts, anthough sole authors have advocated for tixm (Rayo 2006; Linngbo – Npcolas 2008). Obr trestment covers them, but only in the contdxt of itejized cmllections. W Note on Pluswl Logic __________________________________________________ 157 ai n-place [reducatw lettef P followed by n terms. Cud quantidication will be aklodsd – as usual – ortr undividual varisblds jnpy.13 Af for the dedgctixe rustem, the usual wjantofication rules (or afioma) will enable us tp yalidate unferencq (7): 'Dick and Tpny emptied the boutle. Hxnce, Dmck anc sjmeone else emptied the bottle': E ( [d,t], b ) ∴ ∃x ( [d,x], b ) And yn ovder to cover che other odd inferences of our list (thav is, inferences (1), (3) and (5)), qe introduce a Rule nor the Indix
order logic, we add compound terms. A will a discrete linear list of behaving a unit, acts as a new Hence we will have: (a) an variable is a term; (b) an individual constant is a term; (c) if rn are terms, then [r1,..., rn] is a term.11 Terms formed by clauses and are usual terms; terms formed by clause (c) will be called 'compound terms'. The inductive structure of the enables us to accommodate nested structures of itemized Like for example: 'Dick Hannah, together with Tony and introduced and Alice swinging': I [[d,h],[t,g]], [j,a] )12 the usual definition of formula does not need to be changed. In particular, an atomic formula P(r1,..., can be as 10 as 'Dick Tony emptied the phrase 'Dick and Tony' takes the the one who empties, with respect to the relation 'x y'. 11 I take a language descriptions and function symbols, for simplicity. 12 Where letter I stands for the 2-place relation 'x introduced to swinging to y'. In this places of the relation filled up by terms, first which made up of two compound terms. Such nested pluralities are not normally accepted plural logic treatments, although some authors have advocated for them Linnebo Nicolas 2008). Our covers them, but only the of itemized collections. A Plural __________________________________________________ predicate P by n terms. And will be allowed – as – over individual variables system, the usual quantification rules (or axioms) will us to validate inference (7): 'Dick and emptied the bottle. Hence, Dick and someone else emptied the bottle': E [d,t], b ∃x ( [d,x], b ) And in order cover the other odd of our list (that is, inferences (1), (3) and we a Rule the Indis
order logic, we add compound teRms. A compouNd terM wiLl bE a DiscRete And linear list oF TermS between square brackets, Which BeHAvinG LiKe a unIt, acts iTSeLF As a NeW tErm.10 heNCe We wilL haVe: (a) an inDividual vaRiaBlE is a term; (b) an iNDiVidual consTanT is a term; (c) if r1,..., Rn aRe termS, tHen [R1,..., Rn] is a TerM.11 TermS formeD By clauSes (a) and (b) aRe THe usuaL IndividUAL tErms; Terms formed by clauSE (c) WIll be called 'comPound tErMS'. THE IndUctIve structuRe Of the DEfinitiON eNABLes US to accommodatE nested struCTurEs of itEmIzeD CollecTions. liKE foR example: 'DicK and hannah, togEther wITh Tony aND Grace, iNtroduCed johN and aLiCe To sWiNGinG': (9) i ( [[d,H],[t,g]], [J,A] )12 NeXt, the usuAl DeFinitIon oF FORMula DoeS not Need tO be changed. In pArtIculAR, an AtomiC formUla P(R1,..., rN) can bE simplY defiNeD as 10 Just as in 'Dick And TOny emptieD thE bOttLe', The phRAse 'DicK anD ToNy' takes The arguMEnt PlACE Of The one who empties, wiTh RESpEct to the 2-Place rELaTiON 'x emptieD y'. 11 i taKe a lANGuage WithOUt DescriptIons anD FuNcTion symBoLs, for sImPliCitY. 12 WherE PredIcate lEtter I stAnds fOR the 2-place relatIOn 'x introduced TO sWINgINg to Y'. In This example Both PLaceS of tHE rElaTIon arE fillEd UP bY Compound terms, the firSt Of whicH is maDe up itself of tWo compound TERMs. Such neSted PLuRAlities are not nOrmalLy accepted IN plural lOgic tReatmentS, although SOMe authorS haVe aDvoCatED FoR them (Rayo 2006; LinnEBO – NicOlAs 2008). Our trEatMent covErs TheM, buT onLy In the contExt of iteMiZeD cOlLecTions. a note on PlUrAl LOgIc __________________________________________________ 157 aN n-plaCE prediCate lEtteR P FoLLowEd by n teRMs. aND quaNtIfIcatIon WiLl be aLlowED – as Usual – ovEr individUal VAriaBlEs Only.13 As fOr the deductivE sYstem, the usUaL quAntifiCATion ruleS (or axioms) will enable us to VAlidate InfErencE (7): 'DicK and Tony eMptIed the BotTLe. HencE, Dick aNd somEoNe eLSE emptIED tHe bOtTle': E ( [d,t], b ) ∴ ∃x ( [d,x], B ) aNd iN ordeR tO covEr the otHer odd inferences of OUr lIst (that is, infeRenCes (1), (3) aND (5)), We IntROdUCe a ruLE foR THe Indis
order logic, we add compou nd terms.A com pou ndte rm w illbe a discretea nd l inear list of terms be tween  s q uare br acket s, whic h  b e h avi ng  l ike  a un it, a cts  itself  as a newter m. 10 Hence wew il l have: (a ) a n individual  va riable  i s a term;  (b ) anindivi d ual co nstant is  a term;( c) if r 1 , .. ., r n are terms, then [r 1 ,..., rn] is a  term. 11 Te r m s f orm ed by clau se s (a) and (b) ar e t heu sual individu al terms; t e rms  forme dbyc lause(c) w il l  be  called 'co mpou nd terms' . Thei nductiv e  struct ure of  th e d efin i ti on  en ab l esu stoa cco mmodatene st ed st ruct u r e s  ofite mize d col lections. Lik e f or e x amp le: ' DickandHa nnah,  toget her w it h Tony and Grac e, i ntroduced  Jo hn  an dAlice to swi ngi ng' : (9) I  ( [[d, h ],[ t, g ] ] ,[j,a] )12 Next, th eu s ua l defini tion o f  f or m ula does  n otneed t o bechan g ed . In par ticula r ,an  atomic  f ormula  P (r1 ,.. ., rn )  can  be si mply def ineda s 10 Just as i n  'Dick and To n ye m pt i ed t hebottle', th e ph r ase'Dic k  a ndT ony'takes  t h ea rgument place of th eone wh o emp ties, with re spect to t h e 2-placerela t io n  'x emptied y' . 11I take a l a nguage w ithou t descri ptions an d function  sy mbo ls,  fo r si mplicity. 12W h erepr edicate  le tter Ista nds  fo r t he  2-placerelation  ' xin tr odu ced t o  swingin gtoy' . I n thi s  examp le bo th p la ce s  of  the re l at i o n ar efi lled  up  b y com poun d  te rms, th e first o f w h ichis  m ade upitself of two  c ompound te rm s.Such n e s ted plur alities are not normall y  accept edin pl ural  logic tr eat ments,  al t houghsome a uthor shav e advoc a t ed  fo rthem (Rayo 2 006 ; Lin ne bo –  Nicola s 2008). Our treat m ent  covers them,  bu t on l y  i n t h ec ont ex t  of i temized collect ions. A No te on  Plural Lo g ic__ _______ _______ _____ _ _______ _________ _________ __ _ 15 7 ann-place pr edicateletter Pf ollow e dby nter ms. An dqua ntifi cation wil l beallowe d– as u sual–over ind ividual variables only. 13 Asfor t hedeductive  sy s tem , the usu al q uantificat ion  ru les ( ora xioms ) wi l lena b le us  tov alidate i n fe ren c e  ( 7): 'Dick a n d Ton y emp tie d  the b ottl e. Hence, Dick an d  someone elseempt i e d t heb ottl e' : E ( [d,t], b  )∴∃ x  ( [d,x] ,b ) And inorder to  c o ver t he oth er odd  infere n c es of our  lis t ( that is,inf er e nces (1 ),  ( 3 ) and(5)) ,we int roduce a Ru l e  for the Indis
order logic,_we add_compound terms. A compound_term will_be_a discrete_and_linear list of_terms between square_brackets, which behaving like_a unit, acts_itself_as a new term.10 Hence we will have: (a) an individual variable is a_term;_(b) an_individual_constant_is a term; (c) if_r1,..., rn are terms, then_[r1,..., rn]_is a term.11 Terms formed by clauses (a)_and_(b) are the_usual individual terms; terms formed by clause (c) will_be called 'compound terms'. The inductive_structure of the_definition_enables_us to accommodate nested_structures of itemized collections. Like for_example: 'Dick and Hannah, together with_Tony and Grace, introduced John and Alice_to swinging': (9) I ( [[d,h],[t,g]],_[j,a] )12 Next, the usual_definition of_formula does not need to_be changed. In_particular, an_atomic formula P(r1,...,_rn) can be simply defined as_10 Just as_in 'Dick and Tony emptied the_bottle',_the phrase 'Dick_and_Tony'_takes the_argument place of_the_one who_empties,_with respect to the 2-place relation_'x_emptied y'. 11 I take a language_without descriptions and function_symbols,_for simplicity. 12 Where_predicate letter I stands for_the 2-place relation 'x introduced to_swinging to_y'. In_this example both places of the relation are filled up by_compound terms, the first of which_is made up itself_of two_compound_terms. Such nested_pluralities_are not_normally accepted in plural logic treatments, although_some authors_have advocated for them (Rayo 2006;_Linnebo – Nicolas 2008)._Our_treatment covers them, but only in_the context of itemized collections. A_Note on Plural Logic ___________________________________________________157_an_n-place predicate letter P followed_by n terms. And quantification will_be allowed –_as usual – over individual variables only.13_As_for the deductive system, the usual_quantification_rules (or axioms) will enable us_to_validate_inference (7): 'Dick and Tony_emptied the bottle. Hence, Dick and_someone else emptied the bottle': E ( [d,t], b_) ∴ ∃x_( [d,x], b ) And_in_order_to cover the other odd inferences of our list (that_is, inferences_(1), (3) and_(5)), we introduce a Rule for the Indis