 we can render (1) as: ∃xx ( E(xx,b) & d  xx & t  xx & ∀y ( y  xx → y = d ∨ y = t ) ) ∴ ∃xx ( E(xx,b) & t  xx & d  xx & ∀y ( y  xx → y = t ∨ y = d ) ) i.e.: 'There are some things xx which emptied the bottle, and only Dick and Tony are among them. Hence, only Tony and Dick are among them'. Plural logic is a recent system of non-classical logic which allows quantification over plural variables (like 'xx'), intended to stand for various objects at the same time. The point of plural logic is to take plural expressions of natural language at face value, instead of singularizing them into sets, groups, or aggregates of any kind. Within plural logic, inference (1) obtains as a consequence of the 'indiscernibility' (or 'extensionality') axiom for pluralities.3 However, al- 1 Besides obscuring the fact that the activity denoted by the 2-place letter E and the 3place letter F is essentially the same. 2 Besides the dubious implication that the bottle would have been emptied by a set (cf. Boolos 1984, 72, one of the pioneering works of plural logic). 152 _________________________________________________ Gustavo Fernandéz Díez lowing quantification over plural variables is not without cost. Plural logic is a much stronger theory than classical first-order logic, and in particular, it was soon discovered to be not recursively axiomatizable.4 So we are left again with our initial question: is (1) not a first-order inference, after all? 2 The Logical Analysis of Plural Predication A plural predication is that in which a predicate is attributed to a plurality of subjects. It is called 'distributive' when the predicate applies to each of the subjects of the plurality taken one by one. For example, 'Dick and Tony are fair-haired', which is tantamount to saying that Dick is fair-haired and Tony is fair-haired. And a plural predication is called 'collective' when the predicate only applies to the subjects of the plurality taken collectively. For example, in the case of inference (1), just seen.5 3 Burgess (2004, 197); McKay (2006, Ch. 6);
we can render (1) as: ∃xx (E(xx, b) & d  xx & t  xx & ∀y (y  xx → y = d ∨ y = t) ) ∴ ∃xx (E(xx, b) & t  xx & d  xx & ∀y (y  xx → y = t ∨ y = d) ) i.e.:' There be some thing xx which emptied the bottle, and only Dick and Tony are among them. Hence, entirely Tony and Dick are among them'. Plural logic is a recent system of non - authoritative logic which allows quantification over plural variables (like' xx'), intended to digest for various objects at the like time. The point of plural logic is to take plural expressions of lifelike language at face value, alternatively of singularizing them into sets, group, or aggregates of any kind. Within plural logic, inference (1) obtains as a consequence of the' indiscernibility' (or' extensionality') axiom for pluralities.3 However, al- 1 Besides obscuring the fact that the activeness denoted by the 2 - place letter E and the 3place letter F is essentially the like. 2 Besides the dubious significance that the bottle would have been emptied by a set (cf. Boolos 1984, 72, one of the pioneer works of plural logic). 152 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Gustavo Fernandéz Díez lowing quantification over plural variables is not without cost. Plural logic is a much stronger theory than classical first - order logic, and in particular, it was soon discovered to be not recursively axiomatizable.4 So we are left again with our initial question: is (1) not a first - order inference, after wholly? 2 The Logical Analysis of Plural Predication A plural postulation is that in which a predicate is attributed to a plurality of subjects. It is call' distributive' when the predicate applies to each of the subjects of the plurality take one by one. For example,' Dick and Tony are fair - haired', which is tantamount to say that Dick is fair - haired and Tony is fair - haired. And a plural predication is called' corporate' when the predicate only applies to the subjects of the plurality taken collectively. For example, in the font of inference (1), just seen.5 3 Burgess (2004, 197); McKay (2006, Ch. 6);
 we can render (1) as: ∃xx ( E(xx,b) & d  xx & t  xx & ∀y ( y  ex → y = s ∨ y = t ) ) ∴ ∃xx ( E(xx,b) & t  xx & d  xx & ∀y ( t  xx → y = t ∨ y = d ) ) i.e.: 'Therd are somv things zx wiich emptied the bottle, and onmn Dicn and Tony are smong them. Hence, only Toty aud Dick are among them'. Plural logic ys a revejt system of njn-clsfsiczl logic which allows quantificatikn over plural varianles (like 'xx'), intended to shand for various objechs at the sqme eume. The poing of plural logic is tk take plural expressions of nagural language qt fafg value, instxad of singularizikb them into srts, groups, or sggcegares of any kind. Withii plural logic, infergnce (1) obtahna as a consequencw if thg 'indhscefbibkliuy' (pr 'extendioiality') axioj for pluraoities.3 However, al- 1 Nefpces obscurinf the sast that the activity denoted by the 2-plabe lstter E and the 3place lwtter F is essentialli the same. 2 Besides the dubious implication that the bottle woulv favt ncen dnphied by a set (cf. Boolos 1984, 72, one of the pioneerigf eogks of plural loglc). 152 _________________________________________________ Gustavo Fernamdéx Cígz lowing quanjificatnkn over plural variahles is not qithout cjst. Llural logic is a much strobger theory nhan classical first-orber logic, anb in psrticilar, it was soon discovzred tk be not refursively xxiomatizable.4 So we ase left again with our inieial quesvion: ns (1) not x fitst-ordew inferencf, aftcs all? 2 The Logical Analisis ox Plural Pgedication A plural predication is that in whocv a predicace is sttributed to a plurality oy subjeccs. It ks called 'sistribntive' when tre predicate dkplies to eaci of the fubjwcts of the olurality takem one by one. For exqmple, 'Dick and Tonn are rair-haired', whick us tantamount tp sxyigg tiat Dywk is fair-hahred ana Tony is fair-haived. And a plural predicatiot is called 'collective' wmen the ptedicate jnly applies yo the subjects of the 'luralmty talen collectively. For example, in tge case ov ikference (1), juse secn.5 3 Furgess (2004, 197); KcKay (2006, Ch. 6);
we can render (1) as: ∃xx ( d xx &  xx & → = d ∨ = t ) ∴ ∃xx ( E(xx,b) & t xx & d  xx & ∀y ( y  xx → y t ∨ y = d ) ) i.e.: 'There are some things xx emptied bottle, only and Tony are among them. Hence, only Tony and Dick are among them'. Plural logic is recent system of non-classical logic which allows quantification plural variables (like 'xx'), to stand for various objects the time. The of logic to take plural of natural language at face value, instead of singularizing them into sets, groups, or aggregates of any Within plural (1) obtains a of 'indiscernibility' (or 'extensionality') pluralities.3 However, al- 1 Besides obscuring the activity denoted by the 2-place letter E the 3place F is essentially the same. 2 the dubious implication that the bottle would have emptied by a set (cf. Boolos 1984, 72, one of the pioneering works of plural _________________________________________________ Gustavo Fernandéz Díez quantification over plural is without Plural is a stronger theory than classical first-order logic, and in particular, it was discovered to be not recursively axiomatizable.4 So we are left our question: is (1) a first-order inference, after 2 Logical Analysis of Plural plural is a is to a plurality of It is called 'distributive' when predicate applies to each plurality taken one by one. For example, 'Dick Tony are fair-haired', which is tantamount to that Dick is fair-haired and Tony is fair-haired. And a plural predication called 'collective' predicate only applies to the subjects of the taken collectively. For example, the case of inference (1), just seen.5 3 Burgess 197); (2006, Ch.
 we can render (1) as: ∃xx ( E(xx,b) & d  xx & t  xx & ∀Y ( y  xx → y = d ∨ y = t ) ) ∴ ∃xx ( e(xx,b) & t  Xx & d  Xx & ∀y ( Y  xX → y = t ∨ y = D ) ) i.e.: 'THere are some thiNGs xx Which emptied the bottle, aNd onlY DICk anD toNy are Among thEM. HENCe, oNlY TOny AnD diCk are AmoNg them'. PLural logic Is a ReCent system of NOn-Classical lOgiC which allows QuaNtificAtIon OVer plUraL variAbles (lIKe 'xx'), inTended to sTaND for vaRIous objECTs At thE same time. The point OF pLUral logic is to tAke pluRaL ExPREssIonS of natural LaNguagE At face vALuE, INSteAD of singularizIng them into SEts, Groups, Or AggREgates Of any KiND. WiThin plural lOgic, Inference (1) ObtainS As a consEQuence oF the 'inDisCerNibiLItY' (oR 'exTeNSioNAlIty') AXioM for plurAlItIes.3 HoWeveR, AL- 1 bEsidEs oBscuRing tHe fact that the ActIvitY DenOted bY the 2-pLace LeTter E And the 3Place LeTter F is essentiaLly tHe same. 2 BesIdeS tHe dUbIous iMPlicatIon ThaT the botTle woulD HavE bEEN EmPtied by a set (cf. BooloS 1984, 72, oNE Of The pioneEring wORkS oF Plural loGiC). 152 _________________________________________________ GuStavO fErnanDéz DÍEz Lowing quAntifiCAtIoN over plUrAl variAbLes Is nOt witHOut cOst. PluRal logic Is a muCH stronger theorY Than classical FIrST-OrDEr loGic, And in particUlar, IT was Soon DIsCovERed to Be not ReCUrSIvely axiomatizable.4 SO wE are leFt agaIn with our initIal questioN: IS (1) Not a firsT-ordER iNFerence, after alL? 2 The LOgical AnalYSis of PluRal PrEdicatioN A plural pREDication Is tHat In wHicH A PrEdicate is attrIBUted To A pluralIty Of subjeCts. it iS caLleD 'dIstributiVe' when thE pReDiCaTe aPplieS To each of ThE suBjEctS of thE PluralIty taKen oNe By ONe. FOr exampLE, 'DICK and toNy Are fAir-HaIred', wHich IS taNtamounT to saying ThaT dick Is FaIr-haireD and Tony is faiR-hAired. And a pLuRal PredicATIon is calLed 'collective' when the preDIcate onLy aPplieS to tHe subjectS of The pluRalITy takeN colleCtiveLy. for EXAmple, IN ThE caSe Of inferencE (1), JUst Seen.5 3 BUrGess (2004, 197); mcKay (2006, Ch. 6);
 we can render (1) as: ∃xx  ( E(xx,b)  & d x x &  t   x x &∀y ( y  xx →y  = d  ∨ y = t ) ) ∴ ∃xx ( E (xx,b )&  t  xx  & d xx &∀ y( y   x x→ y  = t∨ y =  d) ) i.e .: 'Thereare  s ome things x x  w hich empti edthe bottle,and  onlyDi cka nd To nyare a mong t h em. He nce, only  T o ny and Dick ar e am ongthem'. Plural log i ci s a recent sys tem of  n o n- c l ass ica l logic wh ic h all o ws quan t if i c a tio n  over pluralvariables ( l ike  'xx') ,int e nded t o sta nd for  various ob ject s at thesame t i me. The point o f plur allog ic i s  t otak ep lur a lexp r ess ions ofna tu ral l angu a g e at f ace  val ue, i nstead of sin gul ariz i ngthemintosets ,group s, oraggre ga tes of any kind . Wi thin plur allo gic ,infer e nce (1 ) o bta ins asa conse q uen ce o f  t he 'indiscernibili ty ' (o r 'exten sional i ty ') axiom fo rplu rali t i es.3Howe v er , al- 1Beside s  o bs curingth e fact  t hat  th e act i vity  denot ed by th e 2-p l ace letter E a n d the 3placel et t e rF  isess entially th e sa m e. 2  Bes i de s t h e dub iousim p li c ation that the bott le  would  have  been emptied  by a set( c f . Boolos  198 4 ,7 2, one of thepione ering work s  of plur al lo gic). 15 2 _______ _ _ ________ ___ ___ ___ ___ _ _ __ _____________ _ _ _ Gu st avo Fer nan déz Díe z l owi ngqua nt ification  over pl ur al  v ar iab les i s  not wit ho utco st.  Plur a l logi c isa mu ch  s t ron ger the o ry t hancl as sica l f ir st-or derl ogi c, andin partic ula r , it  w as  soon d iscovered tobe  not recur si vel y axio m a tizable. 4 So we are left againw ith our  in itial  que stion: is  (1 ) nota f i rst-or der in feren ce , a f t er al l ?  2  Th eLogical An a l ysi s ofPl ural  Predic ation A plural pre d ica tion is thatinwhic h apre d ic a teis att r i buted to a plur ality of s ub j ec ts. It isc all ed  'distr ibutive ' whe n  the pr edicate a pplies to  e acho f  th e subjects  of theplurality taken on e byone . Forex amp le, ' Dick a n d T ony a re fai r- haired ', wh ic h is tan tamount to saying thatDick i s fai r-h aired and  To n y i s fair-ha ired . And a pl ura l p redic ati o n iscall e d'co l lecti ve'w hen the p r ed ica t e  o nly applies t o  th e sub jec t s of t he p lurality taken co l lectively. For  exa m p le,  in theca se of inferenc e ( 1) , just see n. 5 3 Burgess  (2004,19 7 ); Mc Kay (2 006, C h. 6);
 we_can render_(1) as: ∃xx (_E(xx,b) &_d_ xx_&_t  xx_& ∀y (_y  xx →_y = d_∨_y = t ) ) ∴ ∃xx ( E(xx,b) & t  xx &_d_ xx_&_∀y_( y  xx →_y = t ∨ y_= d_) ) i.e.: 'There are some things xx_which_emptied the bottle,_and only Dick and Tony are among them. Hence,_only Tony and Dick are among_them'. Plural logic_is_a_recent system of non-classical_logic which allows quantification over plural_variables (like 'xx'), intended to stand_for various objects at the same time._The point of plural logic is_to take plural expressions of_natural language_at face value, instead of_singularizing them into_sets, groups,_or aggregates of_any kind. Within plural logic, inference_(1) obtains as_a consequence of the 'indiscernibility' (or_'extensionality')_axiom for pluralities.3_However,_al-_1 Besides_obscuring the fact_that_the activity_denoted_by the 2-place letter E and_the_3place letter F is essentially the same._2 Besides the dubious_implication_that the bottle would_have been emptied by a_set (cf. Boolos 1984, 72, one_of the_pioneering works_of plural logic). 152 _________________________________________________ Gustavo Fernandéz Díez lowing quantification over_plural variables is not without cost._Plural logic is a_much stronger_theory_than classical first-order_logic,_and in_particular, it was soon discovered to be_not recursively_axiomatizable.4 So we are left again_with our initial question:_is_(1) not a first-order inference, after_all? 2 The Logical Analysis of_Plural Predication A plural predication_is_that_in which a predicate is_attributed to a plurality of subjects._It is called_'distributive' when the predicate applies to each_of_the subjects of the plurality taken_one_by one. For example, 'Dick and_Tony_are_fair-haired', which is tantamount to_saying that Dick is fair-haired and_Tony is fair-haired. And a plural predication is called_'collective' when the_predicate only applies to the_subjects_of_the plurality taken collectively. For example, in the case of_inference (1),_just seen.5 3_Burgess (2004, 197); McKay (2006, Ch. 6);