 are fair-haired. Tony is Iron Man. Hence, Dick and Iron Man are fair-haired. And another is existential quantification over some of the subjects of the plurality in question: (7) Dick and Tony emptied the bottle. Hence, Dick and somebody else emptied the bottle. (8) Dick and Tony are fair-haired. Hence, Dick and somebody else are fair-haired. I claim all inferences (1) to (8) are logical inferences. As a matter of fact, the even ones correspond to cases of distributive predication, and can be easily accounted for in standard first-order logic by means of the usual propositional disjunctions. Indeed (using H for 'being fairhaired', and i for Iron Man): (2a) H(d) & H(t) ∴ H(t) & H(d) (4a) H(d) & H(t) ∴ H(d) & H(d) & H(t) (6a) H(d) & H(t) ; t = i ∴ H(d) & H(i) (8a) H(d) & H(t) ∴ ∃x ( H(d) & H(x) ) However, the odd inferences of this list ((1), (3), (5) and (7)) correspond to cases of collective predication, and cannot be adequately treated in standard first-order logic. To be sure, inferences (5) and (7) could be validated by the introduction of the 3-place predicate letter F mentioned in §1, but this does not work for (1) and (3).9 8 Assuming Dick being the same all along. 9 Besides the awkward implication that e.g. in 'It was Dick and Tony who emptied the bottle, and not Charles' we would have to use two different predicate letters for 'emptied', thus: F(d,t,b) & ∼E(c,b) (cf. Note 1 again). 156 _________________________________________________ Gustavo Fernandéz Díez I will now indicate how first-order logic can be adapted so as to cope with these and similar inferences in a natural way. The resulting system could be called 'first-order logic for itemized collections', or 'first-order logic for itemized collective predication'. 4 First-order Logic for Itemized Collections: Syntax To the usual individual terms of standard first-
are fair - haired. Tony is Iron Man. Hence, Dick and Iron Man are fair - hairy. And another is experiential quantification over some of the subject of the plurality in doubt: (7) Dick and Tony empty the bottle. Hence, Dick and somebody else emptied the bottle. (8) Dick and Tony are honest - hairy. Hence, Dick and somebody else are fair - haired. I claim all inferences (1) to (8) are logical inference. As a matter of fact, the even ones correspond to case of distributive predication, and can be well accounted for in standard inaugural - ordering logic by means of the usual propositional disjunctions. Indeed (use H for' being fairhaired', and i for Iron Man ): (2a) H(d) & H(t) ∴ H(t) & H(d) (4a) H(d) & H(t) ∴ H(d) & H(d) & H(t) (6a) H(d) & H(t); t = i ∴ H(d) & H(i) (8a) H(d) & H(t) ∴ ∃x (H(d) & H(x) ) However, the odd inferences of this list (( 1), (3), (5) and (7) ) correspond to casing of collective predication, and cannot be adequately treated in standard first - order logic. To be certain, inferences (5) and (7) could be validated by the introduction of the 3 - place predicate letter F mentioned in § 1, but this does not shape for (1) and (3).9 8 Assuming Dick being the same all along. 9 Besides the awkward implication that e.g. in' It was Dick and Tony who emptied the bottle, and not Charles' we would have to use two different predicate letters for' emptied', therefore: F(d, t, b) & ∼E(c, b) (californium. Note 1 again). 156 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Gustavo Fernandéz Díez I will immediately bespeak how first - order logic can be adapted so as to cope with these and exchangeable inferences in a natural way. The resulting arrangement could be called' first - order logic for itemized collections', or' first - order logic for itemized corporate predication'. 4 First - order Logic for Itemized Collections: Syntax To the usual individual terms of standard first-
 arf fair-haired. Tony is Irok Man. Hence, Dick and Irmn Man are faif-haired. And another is existxntiql quqntification over some of the slbjects od tht plurality in quxation: (7) Dick ahf Touy emptied the bpttle. Henca, Dick and somabudv else emptied the bottle. (8) Dick and Eony arr vair-haired. Hense, Dpch ans somebody else are fair-haired. I cmaim aln inferences (1) to (8) are logical inferenced. As a matter of fact, hhe even onga cjerespond to zases of dpvtributive predication, and can be easily xccouuted for in sranfdrd first-orver lodic by means of the gsual ptopositional dlsjunrtiobs. Indeed (using H for 'being fairhaired', wnd i for Ixon Man): (2a) H(d) & H(t) ∴ H(t) & H(e) (4a) H(g) & H(d) ∴ H(a) & H(a) & G(t) (6a) H(d) & H(h) ; v = i ∴ H(d) & H(j) (8a) H(d) & H(t) ∴ ∃x ( H(d) & H(x) ) However, trv odd inferendes of tris list ((1), (3), (5) and (7)) correspond to cases of comlective predication, ane cannot be adequateli treated yn standard first-order logic. To be sure, inferencev (5) anv (7) cobod be capidated by the introduction of the 3-place predydaue ketter F mentijned in §1, buy hhof does not wotk for (1) ans (3).9 8 Assuming Dick heing tre sane all aljng. 9 Besides the awkward implicqtion that e.y. ib 'It was Dick and Cony who empcied tne boytle, and not Charles' we woumd have to kse two djwferent predicatd lvttess for 'emptied', thus: F(d,t,b) & ∼Q(c,b) (cf. Nove 1 ayain). 156 _________________________________________________ Gjstafo Fergandéz Díez L will now indicate how virst-lrger logic fan be adapted so as to cope wivi these and somhlag inferenees in a natural war. The resultiny system coula be calles 'first-mrder logic for itemized follections', mr 'first-jrdee lotic for ktemized collevtive prebncation'. 4 Dirst-order Logic fpr Kfemized Collectnubs: Syntax To thr uruaj pndmviduwn terms of sdandxrd girst-
are fair-haired. Tony is Iron Man. Hence, Iron are fair-haired. another is existential subjects the plurality in (7) Dick and emptied the bottle. Hence, Dick and else emptied the bottle. (8) Dick and Tony are fair-haired. Hence, Dick and else are fair-haired. I claim all inferences (1) to (8) are logical inferences. a of the ones correspond to cases of distributive predication, and can be easily accounted for in standard first-order by means of the usual propositional disjunctions. Indeed H for 'being fairhaired', i for Iron Man): (2a) & ∴ H(t) H(d) H(d) H(t) ∴ H(d) H(d) & H(t) (6a) H(d) & H(t) ; t = i ∴ H(d) & H(i) (8a) H(d) H(t) ∴ H(d) & ) the inferences of this (3), (5) and (7)) correspond to predication, and cannot be adequately treated in standard logic. To sure, inferences (5) and (7) could validated by the introduction of the 3-place predicate F mentioned in §1, but this does not work for (1) and (3).9 8 Assuming the same all along. Besides the awkward that in was and Tony emptied the bottle, and not Charles' we would have to use different predicate letters for 'emptied', thus: F(d,t,b) & ∼E(c,b) (cf. again). _________________________________________________ Gustavo Fernandéz I will now indicate first-order can be adapted so cope these in natural The resulting system could called 'first-order logic for itemized or 'first-order logic for Logic for Itemized Collections: Syntax To the usual terms of standard first-
 are fair-haired. Tony is Iron MaN. Hence, Dick And IrOn MAn aRe Fair-HairEd. And another is EXistEntial quantification ovEr somE oF The sUBjEcts oF the pluRAlITY in QuEsTioN: (7) DICk And ToNy eMptied tHe bottle. HeNce, diCk and somebodY ElSe emptied tHe bOttle. (8) Dick and tonY are faIr-HaiREd. HenCe, DIck anD somebODy else Are fair-haIrED. I claiM All infeRENcEs (1) to (8) Are logical inferenCEs. aS a matter of fact, The eveN oNEs CORreSpoNd to cases oF dIstriBUtive prEDiCATIon, ANd can be easily Accounted foR In sTandarD fIrsT-Order lOgic bY mEAns Of the usual pRopoSitional dIsjuncTIons. IndEEd (using h for 'beIng FaiRhaiREd', AnD i fOr iRon mAn): (2A) H(d) & h(T) ∴ H(t) & h(d) (4a) H(d) & H(t) ∴ H(D) & H(D) & H(T) (6a) H(d) & H(T) ; t = i ∴ H(D) & h(I) (8A) h(d) & H(t) ∴ ∃X ( H(d) & h(x) ) HoWever, The odd inferenCes Of thIS liSt ((1), (3), (5) and (7)) CorreSponD tO caseS of colLectiVe Predication, and cAnnoT be adequaTelY tReaTeD in stANdard fIrsT-orDer logiC. To be suRE, inFeRENCeS (5) and (7) could be validatEd BY ThE introduCtion oF ThE 3-pLAce prediCaTe lEtteR f MentiOned IN §1, bUt this doEs not wORk FoR (1) and (3).9 8 AssUmIng DicK bEinG thE same ALl alOng. 9 BesIdes the aWkwarD Implication thaT E.g. in 'It was Dick ANd tONy WHo emPtiEd the bottle, And nOT ChaRles' WE wOulD Have tO use tWo DIfFErent predicate letteRs For 'empTied', tHus: F(d,t,b) & ∼E(c,b) (cf. NOte 1 again). 156 _________________________________________________ GuSTAVo FernanDéz DÍEz i Will now indicatE how fIrst-order lOGic can be AdaptEd so as to Cope with tHESe and simIlaR inFerEncES In A natural way. ThE REsulTiNg systeM coUld be caLleD 'fiRst-OrdEr Logic for iTemized cOlLeCtIoNs', oR 'firsT-Order logIc For ItEmiZed coLLectivE predIcatIoN'. 4 FIRst-Order LoGIc FOR IteMiZeD ColLecTiOns: SyNtax tO thE usual iNdividual TerMS of sTaNdArd firsT-
 are fair-haired. Tony isIron Man.Hence , D ick  a nd I ronMan are fair-h a ired . And another is exist entia lq uant i fi catio n overs om e ofth esub je c ts  of t heplurali ty in ques tio n:  (7) Dick an d  T ony emptie d t he bottle. H enc e, Dic kand someb ody  else  empti e d thebottle. ( 8) Dick a n d Tonya r efair -haired. Hence, D i ck and somebody e lse ar ef ai r - hai red . I claimal l inf e rences( 1) t o  (8 )  are logicalinferences. Asa matt er  of fact,the e ve n  on es correspo nd t o cases o f dist r ibutive predica tion,and  ca n be ea si lyac c oun t ed  fo r  in  standar dfi rst-o rder l o g ic b y m eans  of t he usual prop osi tion a l d isjun ction s. I nd eed ( usingH for  ' being fairhaire d',and i for  Ir on  Ma n) : (2a )  H(d)& H (t)  ∴ H(t)  & H(d) (4a )H ( d )& H(t) ∴ H(d) & H( d) &  H (t) (6a)  H(d)&  H (t )  ; t = i  ∴  H( d) & H (i) ( 8a)H (d ) & H(t)  ∴ ∃x(  H (d ) & H(x )) Howe ve r,the  oddi nfer encesof thislist( (1), (3), (5)a nd (7)) corre s po n d  t o  cas esof collecti ve p r edic atio n ,and canno t bead e qu a tely treated in sta nd ard fi rst-o rder logic. T o be sure, i n ferences  (5) an d  (7) could bevalid ated by th e  introdu ction  of the3-place p r e dicate l ett erF m ent i o ne d in §1, butt h is d oe s not w ork  for (1 ) a nd(3) .98AssumingDick bei ng  t he  s ame  alla long. 9Be sid es  th e awk w ard im plica tion  t ha t  e. g. in ' I tw a s Di ck  a nd T ony  w ho em ptie d  th e bottl e, and no t C h arle s'  w e would  have to usetw o differen tpre dicate l etters f or 'emptied', thus: F(d , t,b) &∼E( c,b)(cf.  Note 1 a gai n). 15 6 _ _ ______ ______ _____ __ ___ _ _ _____ _ _ __ ___ __ _________G u sta vo Fe rn andé z DíezI will now indicat e  ho w first-order  lo gicc a nbea da p ted  s o  as t o cope with the se and sim il a rinferences inanatural  way. T he re s ultingsystem co uld be ca ll ed ' f i rst -order log ic for i temized c o llect i on s', o r ' first- or der  logi c fori tem izedcollec ti ve pre dicat io n'. 4 Fi rst-order Logic for Ite mizedColle cti ons: Synt axT o t he usualindi vidual ter msofstand ard first -
 are_fair-haired. Tony_is Iron Man. Hence,_Dick and_Iron_Man are_fair-haired._And another is_existential quantification over_some of the subjects_of the plurality_in_question: (7) Dick and Tony emptied the bottle. Hence, Dick and somebody else emptied_the_bottle. (8)_Dick_and_Tony are fair-haired. Hence, Dick_and somebody else are fair-haired._I claim_all inferences (1) to (8) are logical inferences._As_a matter of_fact, the even ones correspond to cases of distributive_predication, and can be easily accounted_for in standard_first-order_logic_by means of the_usual propositional disjunctions. Indeed (using H_for 'being fairhaired', and i for_Iron Man): (2a) H(d) & H(t) ∴_H(t) & H(d) (4a) H(d) &_H(t) ∴ H(d) & H(d)_& H(t)_(6a) H(d) & H(t) ;_t = i_∴ H(d)_& H(i) (8a)_H(d) & H(t) ∴ ∃x (_H(d) & H(x)_) However, the odd inferences of_this_list ((1), (3),_(5)_and_(7)) correspond_to cases of_collective_predication, and_cannot_be adequately treated in standard first-order_logic._To be sure, inferences (5) and (7)_could be validated by_the_introduction of the 3-place_predicate letter F mentioned in_§1, but this does not work_for (1)_and (3).9_8 Assuming Dick being the same all along. 9 Besides the_awkward implication that e.g. in 'It_was Dick and Tony_who emptied_the_bottle, and not_Charles'_we would_have to use two different predicate letters_for 'emptied',_thus: F(d,t,b) & ∼E(c,b) (cf. Note_1 again). 156 __________________________________________________Gustavo_Fernandéz Díez I will now indicate_how first-order logic can be adapted_so as to cope with_these_and_similar inferences in a natural_way. The resulting system could be_called 'first-order logic_for itemized collections', or 'first-order logic for_itemized_collective predication'. 4 First-order Logic for_Itemized_Collections: Syntax To the usual individual_terms_of_standard first-