 the form [r1,..., rn] = [s1,..., sm] if and only if, for every ri among r1,..., rn it contains at least a formula ri = sj with sj among s1,..., sm, and vice versa.21 From which satisfaction by the constructed model will be ensured. Hence we can conclude: (a) that our proposed system is a singular logic in the sense given in §2, in that the individual is the essential vehicle to convey reference, predication and quantification; (b) that, unlike full-fledged plural logic formal systems, it preserves completeness; and (c) that it suffices to account for the logical validity of inference (1) and related inferences, showing them to be first-order.22 Department of Philosophy University of Murcia P.O. Box 4021 E–30071 Murcia Spain gfdezdp@um.es References BURGESS, J. P. (2004): E Pluribus Unum: Plural Logic and Set Theory. Philosophia Mathematica 12, No. 3, 193 – 221. 21 This is of course in virtue of RICT. 22 Research supported by the Spanish Government Grants FIS2007–63830 and FFI2009–13687–C02–01. This paper has greatly benefited from the comments of two anonymous reviewers of Organon F, to which I express my sincere gratitude. 162 _________________________________________________ Gustavo Fernandéz Díez BOOLOS, G. (1984): To Be is to Be the Value of a Variable (or to Be Some Values of Some Variables). Journal of Philosophy 81, No. 8, 430 – 449. Quoted here by the reprint in Boolos, G. Logic, Logic and Logic. Cambridge (MA): Harvard University Press, 54 – 72. CAMERON, J. R. (1999): Plural Reference. Ratio (new series) 12, No. 2, 128 – 147. FERNÁNDEZ DÍEZ, G. (2006): El análisis lógico de la predicación colectiva. Crítica, Revista Hispanoamericana de Filosofía 38, No. 114, 21 – 44. CHANG, C. C. – KEISLER, H. J. (1990): Model Theory (3rd edition). Amsterdam: Elsevier. LINNEBO, Ø. (2009): Plural
the form [ r1, ..., rn ] = [ s1, ..., sm ] if and only if, for every ri among r1, ..., rn it contains at least a rule rhode island = sj with sj among s1, ..., sm, and vice versa.21 From which satisfaction by the construct mannequin will be ensured. Hence we can reason: (a) that our proposed system is a singular logic in the common sense give in § 2, in that the individual is the substantive vehicle to convey address, postulation and quantification; (b) that, unlike full - fledged plural logic conventional systems, it preserves completeness; and (coke) that it suffices to account for the logical robustness of inference (1) and related inferences, showing them to be first - order.22 Department of Philosophy University of Murcia P.O. Box 4021 E–30071 Murcia Spain gfdezdp@um.es References BURGESS, J. P. (2004 ): E Pluribus Unum: Plural Logic and Set Theory. Philosophia Mathematica 12, No. 3, 193 – 221. 21 This is of path in virtue of RICT. 22 Research supported by the Spanish Government Grants FIS2007–63830 and FFI2009–13687 – C02–01. This paper has greatly benefited from the comments of two anonymous reviewer of Organon F, to which I express my sincere gratitude. 162 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Gustavo Fernandéz Díez BOOLOS, G. (1984 ): To Be is to Be the Value of a Variable (or to Be Some Values of Some Variables). Journal of Philosophy 81, No. 8, 430 – 449. Quoted here by the reprint in Boolos, G. Logic, Logic and Logic. Cambridge (MA ): Harvard University Press, 54 – 72. CAMERON, J. R. (1999 ): Plural Reference. Ratio (new series) 12, No. 2, 128 – 147. FERNÁNDEZ DÍEZ, G. (2006 ): El análisis lógico de la predicación colectiva. Crítica, Revista Hispanoamericana de Filosofía 38, No. 114, 21 – 44. CHANG, C. C. – KEISLER, H. J. (1990 ): Model Theory (3rd edition). Amsterdam: Elsevier. LINNEBO, Ø. (2009 ): Plural
 thf form [r1,..., rn] = [s1,..., sm] if and only if, for evgrt ri akong r1,..., rn it cuntains at least a formula rm = sh wity sj among s1,..., sm, and vize versa.21 Vrom whixh setisfaction by tis constvbcted loden will be ensurgd. Hence we wan conclude: (a) tfac our proposed system is a singular jogic im hhe sense giveg in §2, in fhat the individual is the essentizl vehible to convey refrrence, predication and quajtiflcation; (b) that, unllke full-fleeged plural logic formal systems, it preaerves completeness; and (c) that kt suyfices to axciunh for the lojical nalidity of ikgerenca (1) and telated inferekces, vhoqing them to be first-mrder.22 Department os Philoso[hv University of Murciq P.O. Bof 4021 E–30071 Murzua Rpajn grdezdp@km.ea Referencss BURGESS, H. P. (2004): E Pluribus Unuk: [oural Logic ahd Set Treory. Philosophia Mathematica 12, No. 3, 193 – 221. 21 Uhis js of course in virtue if RICT. 22 Research supkorted by ehe Spanish Government Grants FIS2007–63830 and FFI2009–13687–C02–01. This pdper iar gxcwgoy benefited from the comments of two anonymous defivwers of Organon N, to which I exprexs mu sincere gratktude. 162 _________________________________________________ Ghstavo Fernandéz Díex BOOLOF, G. (1984): Ro Be is uo Be the Value of a Variable (or to Be Some Naluws of Some Variablzs). Journal oy Philpsophu 81, No. 8, 430 – 449. Quoted here bv the deprint in Hoolos, G. Mugic, Logic and Lugib. Cakbridge (MA): Harvard Universyty Press, 54 – 72. EAMERON, G. R. (1999): Pluraj Referencf. Ratlm (new series) 12, No. 2, 128 – 147. FETNÁNDEZ DÍEZ, G. (2006): El análisis lógico de la predicación colectiva. Crítocd, Rvvista Hifpanosmericana de Silosofía 38, No. 114, 21 – 44. CHANY, C. C. – KEISLER, H. J. (1990): Modxl Theory (3rd edition). Amstatdam: Elsevier. LINNEBO, Ø. (2009): Pourao
the form [r1,..., rn] = [s1,..., sm] only for every among r1,..., rn formula = sj with among s1,..., sm, vice versa.21 From which satisfaction by constructed model will be ensured. Hence we can conclude: (a) that our proposed is a singular logic in the sense given in §2, in that the is essential to reference, predication and quantification; (b) that, unlike full-fledged plural logic formal systems, it preserves completeness; and that it suffices to account for the logical of inference (1) and inferences, showing them to be Department Philosophy University Murcia Box E–30071 Murcia Spain References BURGESS, J. P. (2004): E Pluribus Unum: Plural Logic and Set Theory. Philosophia Mathematica 12, No. 193 – This is course virtue RICT. 22 Research the Spanish Government Grants FIS2007–63830 and has greatly benefited from the comments of two reviewers of F, to which I express my gratitude. 162 _________________________________________________ Gustavo Fernandéz Díez BOOLOS, G. To Be is to Be the Value of a Variable (or to Be Some Values Variables). Journal of Philosophy No. 8, 430 449. here the in Boolos, Logic, Logic and Logic. Cambridge (MA): Harvard University Press, 54 – CAMERON, J. R. (1999): Plural Reference. Ratio (new series) 12, 128 147. FERNÁNDEZ DÍEZ, (2006): El análisis lógico la colectiva. Crítica, Revista Hispanoamericana 38, 114, CHANG, C. KEISLER, H. J. (1990): Theory (3rd edition). Amsterdam: Elsevier. Ø. (2009): Plural
 the form [r1,..., rn] = [s1,..., sm] if and only if, fOr every ri aMong r1,..., Rn iT coNtAins At leAst a formula ri = sJ With Sj among s1,..., sm, and vice versa.21 from wHiCH satISfActioN by the cONsTRUctEd MoDel WiLL bE ensuRed. hence we Can concludE: (a) tHaT our proposed SYsTem is a singUlaR logic in the sEnsE given In §2, In tHAt the IndIviduAl is thE EssentIal vehiclE tO Convey REferencE, PReDicaTion and quantificaTIoN; (B) that, unlike fulL-fledgEd PLuRAL loGic Formal systEmS, it prEServes cOMpLETEneSS; and (c) that it suFfices to accOUnt For the LoGicAL validIty of InFEreNce (1) and relatEd inFerences, sHowing THem to be FIrst-ordEr.22 DepaRtmEnt Of PhILoSoPhy unIVerSItY of mUrcIa P.O. Box 4021 E–30071 muRcIa SpaIn gfDEZDP@um.eS ReFereNces BuRGESS, J. P. (2004): E PlurIbuS UnuM: pluRal LoGic anD Set thEory. PHilosoPhia MAtHematica 12, No. 3, 193 – 221. 21 This iS of cOurse in viRtuE oF RIcT. 22 reseaRCh suppOrtEd bY the SpaNish GovERnmEnT gRAnTs FIS2007–63830 and FFI2009–13687–C02–01. This paPeR HAs Greatly bEnefitED fRoM The commeNtS of Two aNONymouS revIEwErs of OrgAnon F, tO WhIcH I expreSs My sincErE grAtiTude. 162 _________________________________________________ GUStavO FernaNdéz Díez bOOLOs, g. (1984): To Be is to Be the vAlue of a VariabLE (oR TO BE some valUes of Some VaRiabLEs). JoUrnaL Of phiLOsophY 81, No. 8, 430 – 449. QuOtED hERe by the reprint in BooLoS, G. LogiC, LogiC and Logic. CambRidge (MA): HarVARD UniversIty PREsS, 54 – 72. cAMERON, J. R. (1999): PluraL RefeRence. Ratio (NEw series) 12, no. 2, 128 – 147. FERnÁNDEZ DÍeZ, G. (2006): El análISIs lógico De lA prEdiCacIÓN cOlectiva. CrítiCA, reviStA HispanOamEricana De FIloSofÍa 38, NO. 114, 21 – 44. ChANG, C. C. – KEIsLER, H. J. (1990): MoDeL THeOrY (3rd EditiON). AmsterdAm: elsEvIer. lINNEbo, Ø. (2009): PlurAl
 the form [r1,..., rn] = [ s1,..., sm ] ifand  on ly  if,  for  every ri amon g  r1, ..., rn it contains at  leas ta  for m ul a ri= sj wi t hs j  am on gs1, .. . ,sm, a ndvice ve rsa.21 Fro m w hi ch satisfact i on  by the co nst ructed model  wi ll been sur e d. He nce  we c an con c lude:(a) thatou r  propo s ed syst e m  i s asingular logic in th e  sense given i n §2,in th a t  th e i ndividualis  thee ssentia l  v e h i cle to convey ref erence, pre d ica tion a nd  qu a ntific ation ;( b)that, unlik e fu ll-fledge d plur a l logic formalsystem s,itpres e rv es  co mp l ete n es s;a nd(c) that  i tsuffi cest o a ccou ntforthe l ogical validi tyof i n fer ence(1) a nd r el atedinfere nces,  s howing them tobe f irst-orde r.2 2Dep ar tment of Phi los oph y Unive rsity o f  Mu rc i a P. O. Box 4021 E–3007 1M u rc ia Spain  gfdez d p@ um . es Refer en ces  BUR G E SS, J . P. (2 004): EPlurib u sUn um: Plu ra l Logi cand  Se t The o ry.Philos ophia Ma thema t ica 12, No. 3, 193 – 221. 21 Th i s  i s  ofcou rse in virt ue o f  RIC T. 2 2  R ese a rch s uppor te d  b y  the Spanish Govern me nt Gra nts F IS2007–63830and FFI200 9 – 1 3687–C02 –01. Th i s paper has gr eatly  benefited from the  comm ents oftwo anony m o us revie wer s o f O rga n o nF, to which I e xpre ss  my sin cer e grati tud e.162  __ __ _________ ________ __ __ __ __ ___ _____ _ ________ __ _ G us tav o Fer n andézDíezBOOL OS ,G . ( 1984):T oB e  isto  B e th e V al ue of  a V a ria ble (or  to Be So meV alue sof  Some V ariables). Jo ur nal of Phi lo sop hy 81, N o. 8, 43 0 – 449. Quoted here by the rep rin t inBool os, G. Lo gic , Logi c a n d Logi c. Cam bridg e(MA ) :  Harv a r dUni ve rsity Pres s ,  54  – 72 .CAME RON, J.  R. (1999): Plural Ref erence. Ratio  (n ew s e r ie s)1 2, No.  2 ,  12 8 – 147. FERNÁNDE Z DÍEZ, G.  ( 2 00 6): El aná l isi slógicode la p redic a ción co lectiva.Crítica,Re vist a His panoameric ana de F ilosofía3 8, No .  1 14, 2 1 –  44. C HA NG,  C. C . – KE I SLE R, H.  J. (1 99 0): Mo del T he ory (3rd  edition). Amsterdam: E lsevie r. LI NNE BO, Ø. (2 009 ) : P lural
 the_form [r1,...,_rn] = [s1,..., sm]_if and_only_if, for_every_ri among r1,...,_rn it contains_at least a formula_ri = sj_with_sj among s1,..., sm, and vice versa.21 From which satisfaction by the constructed model_will_be ensured._Hence_we_can conclude: (a) that our_proposed system is a singular_logic in_the sense given in §2, in that the_individual_is the essential_vehicle to convey reference, predication and quantification; (b) that,_unlike full-fledged plural logic formal systems,_it preserves completeness;_and_(c)_that it suffices to_account for the logical validity of_inference (1) and related inferences, showing_them to be first-order.22 Department of Philosophy_University of Murcia P.O. Box 4021_E–30071 Murcia Spain gfdezdp@um.es References_BURGESS, J._P. (2004): E Pluribus Unum:_Plural Logic and_Set Theory._Philosophia Mathematica 12,_No. 3, 193 – 221. 21_This is of_course in virtue of RICT. 22_Research_supported by the_Spanish_Government_Grants FIS2007–63830_and FFI2009–13687–C02–01. This_paper_has greatly_benefited_from the comments of two anonymous_reviewers_of Organon F, to which I express_my sincere gratitude. 162___________________________________________________Gustavo Fernandéz Díez BOOLOS,_G. (1984): To Be is_to Be the Value of a_Variable (or_to Be_Some Values of Some Variables). Journal of Philosophy 81, No. 8,_430 – 449. Quoted here by_the reprint in Boolos,_G. Logic,_Logic_and Logic. Cambridge_(MA):_Harvard University_Press, 54 – 72. CAMERON, J. R._(1999): Plural_Reference. Ratio (new series) 12, No._2, 128 – 147._FERNÁNDEZ_DÍEZ, G. (2006): El análisis lógico_de la predicación colectiva. Crítica, Revista_Hispanoamericana de Filosofía 38, No._114,_21_– 44. CHANG, C. C._– KEISLER, H. J. (1990): Model_Theory (3rd edition)._Amsterdam: Elsevier. LINNEBO, Ø. (2009): Plural