 Tony emptied the bottle', 'Dick and Tony are fairhaired', 'Dick and somebody else emptied the bottle', 'two people emptied the bottle', or 'two people are fair-haired'. The opposite case is that of non-itemized (or non-individualized) plural subjects, in which there is no indication as to which or how many individual members make up the plurality in question. Examples of non-itemized plural subjects are: 'various people emptied the bottle', 'various people are fair-haired', 'Dick and some other people emptied the bottle', and 'Dick and some other people are fair-haired'.6 The difference between itemized and non-itemized plural predication suggests a more general distinction, between singular logics and plural logics. A singular logic is that in which the individual (the singular object) is the only vehicle to convey reference, predication and quantification. A plural logic is one which admits genuine reference, predication and quantification over a plurality of objects, without reducing the plurality to a composite unit of any kind. Of course, all plural logic proposals advanced so far are 'plural logics' in this sense. And standard first-order logic is, in this sense, a'singular logic' beyond any doubt. Moreover, it is clear that standard first-order logic does not enable us to account for the logical validity of inferences regarding itemized collective predication, as we have seen with the case of inference (1). But in fact only a few modifications are needed to patch this up. The resulting system will be a singular logic in the sense just given – in fact, almost a variant of first-order logic as standardly formulated. And it will preserve recursive axiomatizability. defended in Moltmann 1997, 52; Cameron 1999, 129; Yi 2005, 481; McKay 2006, Ch. 1, Note 15; Fernández Díez 2006, 42). 6 This includes of course pluralities described by means of generalized quantifiers (such as'most', 'nearly all','more than ten', etc), unless the description happens to be equivalent to stating an exact number of elements for the plurality in question (as in'more than four and less than six', for instance, which can only come out to five). 154 _________________________________________________ Gustavo Fernandéz Díez Therefore, such a system will be a considerable simpler and weaker logical theory than plural logic. And consequently, if it does suffice for the
Tony emptied the bottle',' Dick and Tony are fairhaired',' Dick and somebody else emptied the bottle',' two people emptied the bottle', or' two citizenry are clean - haired'. The opposite case is that of non - itemized (or non - individualized) plural subject, in which there is no indication as to which or how many individual member have up the plurality in doubt. Examples of non - itemized plural subjects are:' versatile people emptied the bottle',' respective citizenry are fair - haired',' Dick and some other people emptied the bottle', and' Dick and some early people are fair - haired'.6 The difference between itemized and non - itemized plural postulation suggests a more general distinction, between singular logics and plural logic. A singular logic is that in which the individual (the singular object) is the only fomite to convey reference, predication and quantification. A plural logic is one which admits genuine reference, postulation and quantification over a battalion of object, without reducing the plurality to a composite unit of any kind. Of course, all plural logic proposals advanced therefore far are' plural logic' in this sense. And standard first - decree logic is, in this common sense, a'singular logic' beyond any doubt. Moreover, it is clear that standard first - order logic does not enable us to account for the logical validity of inferences involve itemized collective predication, as we have seen with the case of inference (1). But in fact only a few modifications are want to patch this up. The resulting system will be a singular logic in the common sense just establish – in fact, almost a variant of first - order logic as standardly formulated. And it will save recursive axiomatizability. defended in Moltmann 1997, 52; Cameron 1999, 129; Yi 2005, 481; McKay 2006, Ch. 1, Note 15; Fernández Díez 2006, 42). 6 This includes of course plurality described by means of generalized quantifiers (such as'most',' nearly all','more than ten', etc), unless the description happen to be equivalent to stating an accurate number of elements for the plurality in question (as in'more than four and less than six', for instance, which can merely come out to five). 154 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Gustavo Fernandéz Díez Therefore, such a arrangement will be a considerable simpler and weak logical theory than plural logic. And consequently, if it does suffice for the
 Tojy emptied the bottle', 'Digk and Tony are yqirhaiced', 'Dici and sooebody else emptied the bottpe', 'two kvople emptied the botgle', or 'twl people are dair-haired'. The opposite dwse ns that of non-itgmized (or not-individualizeg) olbral subjects, in which there is no igdicatipn as to which ot how iany pnbividual members make up the plhrality in question. Rxamples of non-itemized plkral subjects are: 'varilus people gjptywd the bottld', 'various kemple are fzir-haired', 'Dick and some other pdople emptied tye bohjle', and 'Dick and fome other pcpple ase fair-naired'.6 The difnerenre bwtween itemized and nmn-itemized plural kredicatiot auggests a more gwnwral gisthnctkin, cetsexn aingulwr mogics and plural logucs. A singular logiv yw that in whidh the igdividual (the singular object) is the onny behicle to convey referwnce, predication and euantificwtion. A plural logic is one which admits genuine seferxnze, krcqkxahion and quantification over a plurality of ofnevtx, without reduging the plurality tl s composite unkt of chy kind. Of course, alp pluraj loguc proposwls sdvanced so far are 'plural oogics' in thps swnse. And standard yirst-order lugic is, im this sense, a'singular uogid' beyond anj doubt. Mkfeover, it is clexr nhat standard first-order logic does not enayle us tu acvount sor the lohical validity of inferejces tegardhng itemizfd collective predication, as we have seen witn dhe case of infevence (1). But in fwct only a few modifieationr are needvd to patrh this up. Tre resulting vistem will be a singujar oogix in thd sense just gifen – in fcet, almost a variant of firsb-ordet mogic as standaxbli formulated. Anc ig wylp 'resewee recursive axiumagozabiuity. defendcd kn Mpltmann 1997, 52; Cameron 1999, 129; Fi 2005, 481; McKay 2006, Ch. 1, Note 15; Gevnández Díee 2006, 42). 6 This includes of vourse pluralities desccibed uy meams jf generalized quantifiers (sucg as'most', 'jeavly all','more tran ben', gtc), unless che description happens to be equivalent to stating an exaxt number of elemenjs nor the plurelity yn questimn (as in'more than foyr and less than xix', for instance, whicg can mnly fome out to five). 154 _________________________________________________ Gustavo Fernandéz Díez Therefore, such a system will be a consivewable simplsr amd wedkzr kogicaj thxocy than plural lobic. And consequently, if it does suffice fjr the
Tony emptied the bottle', 'Dick and Tony 'Dick somebody else the bottle', 'two 'two are fair-haired'. The case is that non-itemized (or non-individualized) plural subjects, in there is no indication as to which or how many individual members make the plurality in question. Examples of non-itemized plural subjects are: 'various people emptied bottle', people fair-haired', and some other people emptied the bottle', and 'Dick and some other people are fair-haired'.6 The between itemized and non-itemized plural predication suggests a general distinction, between singular and plural logics. A singular is in which individual singular is the only to convey reference, predication and quantification. A plural logic is one which admits genuine reference, predication and over a objects, without the to composite unit of Of course, all plural logic proposals are 'plural logics' in this sense. And standard logic is, this sense, a'singular logic' beyond any Moreover, it is clear that standard first-order logic not enable us to account for the logical validity of inferences regarding itemized collective predication, have seen with the of inference (1). in only few are needed patch this up. The resulting system will be a singular logic the sense just given – in fact, almost a variant logic standardly formulated. And will preserve recursive axiomatizability. in 1997, 52; Cameron 1999, 2005, McKay Note Fernández 2006, 42). 6 This of course pluralities described by of generalized quantifiers (such etc), unless the description happens to be equivalent stating an exact number of elements for plurality in question (as in'more than four and less than six', for which can out to five). 154 _________________________________________________ Gustavo Fernandéz Díez such a system will a considerable simpler and weaker logical theory than plural And if it suffice for the
 Tony emptied the bottle', 'Dick aNd Tony are fAirhaIreD', 'DiCk And sOmebOdy else emptied THe boTtle', 'two people emptied thE bottLe', OR 'two PEoPle arE fair-haIReD'. tHe oPpOsIte CaSE iS that Of nOn-itemiZed (or non-inDivIdUalized) pluraL SuBjects, in whIch There is no indIcaTion as To WhiCH or hoW maNy indIviduaL MemberS make up thE pLUralitY In questION. EXampLes of non-itemized pLUrAL subjects are: 'vaRious pEoPLe EMPtiEd tHe bottle', 'vaRiOus peOPle are fAIr-HAIRed', 'dIck and some othEr people empTIed The botTlE', anD 'dick anD some OtHEr pEople are faiR-haiRed'.6 The difFerencE Between ITemized And non-IteMizEd plURaL pRedIcATioN SuGgeSTs a More geneRaL dIstinCtioN, BETWeen SinGulaR logiCs and plural loGicS. A siNGulAr logIc is tHat iN wHich tHe indiViduaL (tHe singular objecT) is tHe only vehIclE tO coNvEy refERence, pRedIcaTion and QuantifICatIoN. a PLuRal logic is one which AdMITs Genuine rEferenCE, pReDIcation aNd QuaNtifICAtion Over A PlUrality oF objecTS, wItHout redUcIng the PlUraLitY to a cOMposIte uniT of any kiNd. Of cOUrse, all plural lOGic proposals aDVaNCEd SO far Are 'Plural logicS' in tHIs seNse. AND sTanDArd fiRst-orDeR LoGIc is, in this sense, a'sinGuLar logIc' beyOnd any doubt. MoReover, it is CLEAr that stAndaRD fIRst-order logic dOes noT enable us tO Account fOr the Logical vAlidity of INFerences RegArdIng IteMIZeD collective prEDIcatIoN, as we haVe sEen with The CasE of InfErEnce (1). But in Fact only A fEw MoDiFicAtionS Are needeD tO paTcH thIs up. THE resulTing sYsteM wIlL Be a SingulaR LoGIC in tHe SeNse jUst GiVen – in Fact, ALmoSt a variAnt of firsT-orDEr loGiC aS standaRdly formulateD. ANd it will prEsErvE recurSIVe axiomaTizability. defended in MolTMann 1997, 52; CamEroN 1999, 129; Yi 2005, 481; Mckay 2006, CH. 1, Note 15; FernÁndEz Díez 2006, 42). 6 thiS IncludEs of coUrse pLuRalITIes deSCRiBed By Means of genERAliZed quAnTifiErs (such As'most', 'nearly all','morE ThaN ten', etc), unless The DescRIPtIon HApPEns To BE eqUIValent to stating An exact numBeR Of Elements foR The PlUrality In questIon (as IN'more thAn four and Less than sIx', For iNSTanCe, which can Only come Out to five). 154 _________________________________________________ gUstavO feRnandÉz DÍez TheReForE, such A systeM WilL be a cOnsideRaBle simPler aNd Weaker loGical theory than plural loGic. And ConseQueNtly, if it dOes SUffIce for the
 Tony emptied the bottle',  'Dick and  Tony  ar e f ai rhai red' , 'Dick and so m ebod y else emptied the bot tle',  ' t wo p e op le em ptied t h eb o ttl e' ,or't w opeopl e a re fair -haired'.The  o pposite case is  that of n on- itemized (or  no n-indi vi dua l ized)  pl uralsubjec t s, inwhich the re is noi ndicati o n  a s to  which or how man y  i n dividual membe rs mak eu pt h e p lur ality in q ue stion .  Exampl e so f non - itemized plur al subjects are : 'var io usp eopleempti ed the  bottle', ' vari ous peopl e aref air-hai r ed', 'D ick an d s ome  oth e rpe opl ee mpt i ed  th e  bo ttle', a nd  ' Dickands o m e  oth erpeop le ar e fair-haired '.6  The dif feren ce be twee nitemi zed an d non -i temized pluralpred ication s ugg es tsamoreg eneral  di sti nction,  betwee n  si ng u l a rlogics and plurallo g i cs . A sing ular l o gi ci s that i nwhi ch t h e  indi vidu a l(the sin gularo bj ec t) is t he  onlyve hic leto co n veyrefere nce, pre dicat i on and quantif i cation. A plu r al l og i c is  on e which adm itsg enui ne r e fe ren c e, pr edica ti o na nd quantification o ve r a pl urali ty of objects , withoutr e d ucing th e pl u ra l ity to a compo siteunit of an y  kind. O f cou rse, all  plural l o g ic propo sal s a dva nce d so  far are 'plu r a l lo gi cs' inthi s sense . A ndsta nda rd  first-or der logi cis ,in  th is se n se, a'si ng ula rlog ic' b e yond a ny do ubt.  M or e ove r, it i s  c l e ar t ha tstan dar dfirst -ord e r l ogic do es not en abl e  usto  a ccountfor the logic al  validityof  in ferenc e s  regardi ng itemized collectivep redicat ion , aswe h ave seenwit h thecas e  of in ferenc e (1) .But i n fac t on lyafew modifi c a tio ns ar eneed ed to p atch this up. Ther esu lting systemwil l be a  s ing u la r  lo gi c  in t he sense just g iven – infa c t,  almost av ari an t of fi rst-ord er lo g ic as s tandardly  formulat ed . An d itwill prese rve recu rsive axi o matiz a bi lity.  de fended  i n M oltma nn 199 7 , 5 2; Ca meron19 99, 12 9; Yi  2 005, 481 ; McKay 2006, Ch. 1, No te 15;  Fern ánd ez Díez 2 006 ,  42 ). 6 This  inc ludes of c our seplura lit i es de scri b ed  by means  ofg eneralize d  q uan t i fi ers (such a s ' m ost ', 'n ear l y all' ,'mo re than ten', etc ) , unless the d escr i p tio n h a ppen sto be equivale ntto s tating a nexact numbe r of ele me n ts fo r theplural ity inq u es t ion (a s in 'mo re than f our  a n d lessth an six',forin stance , whic h  can o nly come out tofive) . 154 _ _ ___ _____ __ _______ _ ____ __________ ___________ ____ G usta vo Fe rnandéz  D íez Th ere fo re, such a system wi ll be  a cons id erab lesimple r an d weake r lo gi cal  theory t h a np lu ra l  lo gic.  Andco nseq uently, i f  it does  su f fice fo rthe 
 Tony_emptied the_bottle', 'Dick and Tony_are fairhaired',_'Dick_and somebody_else_emptied the bottle',_'two people emptied_the bottle', or 'two_people are fair-haired'._The_opposite case is that of non-itemized (or non-individualized) plural subjects, in which there is_no_indication as_to_which_or how many individual members_make up the plurality in_question. Examples_of non-itemized plural subjects are: 'various people emptied_the_bottle', 'various people_are fair-haired', 'Dick and some other people emptied the_bottle', and 'Dick and some other_people are fair-haired'.6_The_difference_between itemized and non-itemized_plural predication suggests a more general_distinction, between singular logics and plural_logics. A singular logic is that in_which the individual (the singular object)_is the only vehicle to_convey reference,_predication and quantification. A plural_logic is one_which admits_genuine reference, predication_and quantification over a plurality of_objects, without reducing_the plurality to a composite unit_of_any kind. Of_course,_all_plural logic_proposals advanced so_far_are 'plural_logics'_in this sense. And standard first-order_logic_is, in this sense, a'singular logic' beyond_any doubt. Moreover, it_is_clear that standard first-order_logic does not enable us_to account for the logical validity_of inferences_regarding itemized_collective predication, as we have seen with the case of inference_(1). But in fact only a_few modifications are needed_to patch_this_up. The resulting_system_will be_a singular logic in the sense just_given –_in fact, almost a variant of_first-order logic as standardly_formulated._And it will preserve recursive axiomatizability._defended in Moltmann 1997, 52; Cameron_1999, 129; Yi 2005, 481;_McKay_2006,_Ch. 1, Note 15; Fernández_Díez 2006, 42). 6 This includes_of course pluralities_described by means of generalized quantifiers (such_as'most',_'nearly all','more than ten', etc), unless_the_description happens to be equivalent to_stating_an_exact number of elements for_the plurality in question (as in'more_than four and less than six', for instance, which_can only come_out to five). 154 __________________________________________________Gustavo_Fernandéz_Díez Therefore, such a system will be a considerable simpler_and weaker_logical theory than_plural logic. And consequently, if it does suffice for the