 determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the first part of the paper we introduce the logics LP⊛ and K⊛3, which extends LP and K3 with normality operators, and we establish a classical recapture result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LP⊛ and K⊛3 to the tradition of Logics of Formal Inconsistency and Underterminedness. Keywords: Kleene logics, Classical Recapture, Normality Operators, Minimal Inconsistency, Logics of Formal Inconsistency, Logics of Formal Undeterminedness 1 Introduction The philosophical applications of many-valued logics usually come with a story of 'normality': there are a number of 'abnormal phenomena' for which we need manyvalued reasoning-logical paradoxes, partial information, vagueness, among others- but as long as the situation is normal-that is, no abnormal phenomena is at stake- classical logic is perfectly in order as it is. This view motivates the following question: how can we recapture Classical Logic CL in many-valued logic? That is, how can we secure inference of classical conclusions, under the assumption that we are facing no abnormality? [2, 6, 7, 23, 24] The question fits the broad conceptual view on many-valued reasoning by the 'normality story' above: if detour from classical reasoning is motivated by specific phenomena, then we do not need to take it when these phenomena are not around. Numerous as these may be, we may assume that they are not ubiquitous. For instance, take semantical and set-theoretical paradoxes; to put it with [23, p.235]: 'paradoxical sentences seem to be a fairly small proportion of the sentences we reason with... It would seem plausible to claim that in our day-to-day reasoning we (quite correctly) presuppose that we are 1L. J. of the IGPL, Vol. 0 No. 0, pp. 1–18 0000 © Oxford University Press 2 Normality operators and Classical Recapture in Ext
determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many - valued logic, that a given rule have a classical truth value (0 or 1). In particular, in the beginning part of the paper we precede the logic LP⊛ and K⊛3, which extends LP and K3 with normality operator, and we establish a classical recapture resultant role based on the two logics. In the second share of the paper, we compare the overture in terms of normality operators with an establish approach to classical recapture, namely minimal inconsistency. last, we discuss technical issues connecting LP⊛ and K⊛3 to the tradition of Logics of Formal Inconsistency and Underterminedness. Keywords: Kleene logic, Classical Recapture, Normality Operators, Minimal Inconsistency, Logics of Formal Inconsistency, Logics of Formal Undeterminedness 1 Introduction The philosophical applications of many - valued logics normally come with a story of' normality': there are a number of' abnormal phenomena' for which we need manyvalued reasoning - coherent paradoxes, partial information, vagueness, among others- but as long as the situation is normal - that is, no abnormal phenomena is at stake- classical logic is perfectly in order as it is. This view motivate the following doubt: how can we recapture Classical Logic CL in many - valued logic? That is, how can we guarantee inference of authoritative conclusions, under the assumption that we are facing no abnormality? [ 2, 6, 7, 23, 24 ] The question fits the broad conceptual view on many - valued reasoning by the' normality fib' above: if detour from classical reasoning is motivated by specific phenomena, then we do not need to learn it when these phenomena are not around. Numerous as these may be, we may assume that they are not ubiquitous. For case, take semantical and determined - theoretical paradoxes; to put it with [ 23, p.235 ]:' paradoxical sentences seem to be a fairly belittled proportion of the sentences we reason with... It would look plausible to claim that in our day - to - day reasoning we (quite correctly) presuppose that we are 1L. J. of the IGPL, Vol. 0 No. 0, pp. 1–18 0000 © Oxford University Press 2 Normality operators and Classical Recapture in Ext
 deherminedness operators fvom Logics of Fotmql Incmnsistsncy and Underterminedness, by expresdibg, in any many-valued logic, ghat a ginen formuoa hes a classical tchth valmz (0 or 1). In 'acticular, in the first pard of the paper wd nntroduce the logics LP⊛ and K⊛3, which qxtends LO and K3 with njrmakyty kievators, and we establish a classidal recepture result bssed on the two logics. In hhe decond part of the paper, we cimpaww the approazh in terms of normalijy operators with an established apprpach to clqswicwn recapture, namejy minimal ikbonsistancy. Fimally, we discuxs vechbical issues connectiig LP⊛ and K⊛3 to the ttadition ox Mogics of Formal Unxonsivtenwy ave Uvdedtxrmjnedneds. Ieywords: Kmeene logicw, Classical Recaptute, Bormality Opedators, Mynimal Inconsistency, Logics of Formal Itcohsistency, Logics of Fornal Undeterminedness 1 Introduceion The philosophical applications of many-valued logirs usbqlly zimf with a story of 'normality': there are a numbew og 'sbnormal phenoiena' for whocj eg need manyvaloed reasonjng-logical paradoxfs, partyal ibformatiog, vabueness, among others- but as long as the wituation is normap-that is, no abnprmal phenomena is at stake- elassidal logic id perfectmh in order as it is. Tvis view motivates the foljowing quxstiou: how cav we recapeure Classlcal Logic CL in many-valked llghc? That is, how can we secure inference of classical convlgsimns, undex the sssumption thwt we are faciug no abuormalkty? [2, 6, 7, 23, 24] Tge quesvion fits thq broad conce[jual view on kany-valuqd rwasobing by ghe 'normality xtory' abone: if eetour from classigal rgaaoning is motivcued by specific phrnooenw, nhei we qm not need tm taye ky whev these phekomdna sre not around. Numermus zs these may be, we mwy assumg that thqy are not uboquitous. For instajce, teke seoantival and set-theoretical paradoxes; fo put it wibh [23, p.235]: 'paradoxycal sentences szem to be a fairly small proportion of tie sentences we reason qith... It would seem klamsible to cleim thwt in our day-to-day reasoning qe (quite correctln) presuppose that we ade 1L. J. of tje IGPL, Vol. 0 No. 0, pp. 1–18 0000 © Oxford University Press 2 Normality operators and Coassicel Recapture jn Ect
determinedness operators from Logics of Formal Inconsistency by in any logic, that a truth (0 or 1). particular, in the part of the paper we introduce logics LP⊛ and K⊛3, which extends LP and K3 with normality operators, and establish a classical recapture result based on the two logics. In the second of paper, compare approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. we discuss technical issues connecting LP⊛ and K⊛3 the tradition of Logics Formal Inconsistency and Underterminedness. Keywords: logics, Recapture, Normality Minimal Logics Formal Inconsistency, Logics Formal Undeterminedness 1 Introduction The philosophical applications of many-valued logics usually come with a story of 'normality': are a 'abnormal phenomena' which need reasoning-logical paradoxes, partial among others- but as long as normal-that is, no abnormal phenomena is at stake- logic is in order as it is. This motivates the following question: how can we recapture Logic CL in many-valued logic? That is, how can we secure inference of classical conclusions, assumption that we are no abnormality? [2, 7, 24] question the broad view on many-valued reasoning by the 'normality story' above: if detour classical reasoning is motivated by specific phenomena, then we do to it when these are not around. Numerous these be, we may assume are ubiquitous. semantical set-theoretical to put it with p.235]: 'paradoxical sentences seem to a fairly small proportion with... It would seem plausible to claim that our day-to-day reasoning we (quite correctly) presuppose we are 1L. J. of the IGPL, Vol. 0 No. 0, pp. 0000 © Press 2 Normality operators and Classical Recapture in
 determinedness operators frOm Logics of formaL InConSiStenCy anD UnderterminedNEss, bY expressing, in any many-vaLued lOgIC, thaT A gIven fOrmula hAS a CLAssIcAl TruTh VAlUe (0 or 1). IN paRticulaR, in the firsT paRt Of the paper we INtRoduce the lOgiCs LP⊛ and K⊛3, whicH exTends Lp aNd K3 WIth noRmaLity oPeratoRS, and we Establish A cLAssicaL RecaptuRE ReSult Based on the two logiCS. IN The second part oF the paPeR, We COMpaRe tHe approach In Terms OF normalITy OPERatORs with an estabLished approACh tO classIcAl rECapturE, nameLy MIniMal inconsisTencY. Finally, wE discuSS techniCAl issueS conneCtiNg Lp⊛ and k⊛3 To ThE trAdITioN Of logICs oF Formal INcOnSisteNcy aND uNDertErmInedNess. KEywords: Kleene LogIcs, CLAssIcal REcaptUre, NOrMalitY OperaTors, MInImal InconsistenCy, LoGics of ForMal inConSiStencY, logics Of FOrmAl UndetErminedNEss 1 inTRODuCtion The philosophiCaL APpLicationS of manY-VaLuED logics uSuAllY comE WIth a sTory OF 'nOrmality': There aRE a NuMber of 'aBnOrmal pHeNomEna' For whICh we Need maNyvalued ReasoNIng-logical paraDOxes, partial inFOrMATiON, vagUenEss, among othErs- bUT as lOng aS ThE siTUatioN is noRmAL-tHAt is, no abnormal phenoMeNa is at Stake- Classical logiC is perfectLY IN order as It is. tHiS View motivates tHe folLowing quesTIon: how caN we reCapture CLassical LOGIc CL in maNy-vAluEd lOgiC? tHaT is, how can we seCURe inFeRence of ClaSsical cOncLusIonS, unDeR the assumPtion thaT wE aRe FaCinG no abNOrmality? [2, 6, 7, 23, 24] thE quEsTioN fits THe broaD concEptuAl ViEW on Many-valUEd REAsonInG bY the 'NorMaLity sTory' ABovE: if detoUr from claSsiCAl reAsOnIng is moTivated by specIfIc phenomenA, tHen We do noT NEed to takE it when these phenomena arE Not arouNd. NUmeroUs as These may bE, we May assUme THat theY are noT ubiqUiTouS. fOr insTANcE, taKe Semantical AND seT-theoReTicaL paradoXes; to put it with [23, p.235]: 'parADoxIcal sentences SeeM to bE A FaIrlY SmALl pRoPOrtION of the sentences We reason wiTh... iT wOuld seem plAUsiBlE to claiM that in Our daY-To-day reAsoning we (Quite corrEcTly) pRESupPose that we Are 1L. J. of tHe IGPL, Vol. 0 nO. 0, pp. 1–18 0000 © OxFOrD UnivErsIty PreSs 2 norMalitY operaTOrs And ClAssicaL REcaptuRe in EXt
 determinedness operatorsfrom Logic s ofFor mal  I ncon sist ency and Under t ermi nedness, by expressing , inan y  man y -v alued  logic, th a t  agi ve n f or m ul a has  aclassic al truth v alu e(0 or 1). In pa rticular,inthe first pa rtof the  p ape r  we i ntr oduce  the l o gics L P⊛ and K⊛ 3, whiche xtendsL P  a nd K 3 with normalityo pe r ators, and weestabl is h  a c las sic al recaptu re  resu l t based on t h e t w o logics. Inthe secondp art  of th epap e r, wecompa re the  approach i n te rms of no rmalit y  operat o rs with  an es tab lis heda pp ro ach  t o  cl a ss ica l  re capture,  n am ely m inim a l i ncon sis tenc y. Fi nally, we dis cus s te c hni cal i ssues  con ne cting  LP⊛ a nd K⊛ 3to the traditio n of  Logics o f F or mal  I ncons i stency  an d U nderter minedne s s.Ke y w o rd s: Kleene logics,Cl a s si cal Reca pture, No rm a lity Ope ra tor s, M i n imalInco n si stency,Logics of  F ormal I nc onsist en cy,  Lo gicso f Fo rmal U ndetermi nedne s s 1 Introducti o n The philoso p hi c a la ppli cat ions of man y-va l uedlogi c susu a lly c ome w it h  a story of 'normality ':  there  area number of ' abnormal p h e n omena' f or w h ic h  we need manyv alued  reasoning - logicalparad oxes, pa rtial inf o r mation,vag uen ess , a m o ng  others- buta s  lon gas thesit uationisnor mal -th at  is, no a bnormalph en om en a i s ats take- cl as sic al  lo gic i s  perfe ctlyin o rd er asit is.T hi s view  m ot ivat esth e fol lowi n g q uestion : how can  we reca pt ur e Class ical Logic CL  i n many-val ue d l ogic?T h at is, h ow can we secure infere n ce of c las sical  con clusions,  un der th e a s sumpti on tha t wear e f a c ing n o ab nor ma lity? [2,6 ,  7,  23,24 ] Th e quest ion fits the broad con ceptual viewonmany - v al ued re a son in g  by t he 'normality s tory' abov e: if  detour fr o m c la ssicalreasoni ng is motivat ed by spe cific phe no mena , the n we do no t need t o take it whent he se ph eno mena a re  no t aro und. N u mer ous a s thes emay be , wema y assume  that they are not ubiq uitous . For  in stance, t ake sem antical a nd s et-theoret ica l p arado xes ;  to p ut i t  w ith [23,p.23 5 ]: 'parad o xi cal s en tences seem t o  be  a fa irl y  small  pro portion of the se n tences we reas on w i t h.. . I t  wou ld  seem plausibl e t oc l aim that  i n our day-t o-day re as o ningwe (qu ite co rrectly ) pr e suppos e th atwe are 1L . J .o f the I GP L, Vol. 0  No.  0 , pp.1–18 0 0 00 © O xford University  Pres s 2 Nor m ali ty op er ators a n d Cl assical Re capture inExt
 determinedness_operators from_Logics of Formal Inconsistency_and Underterminedness,_by_expressing, in_any_many-valued logic, that_a given formula_has a classical truth_value (0 or_1)._In particular, in the first part of the paper we introduce the logics LP⊛_and_K⊛3, which_extends_LP_and K3 with normality operators,_and we establish a classical_recapture result_based on the two logics. In the second_part_of the paper,_we compare the approach in terms of normality operators_with an established approach to classical_recapture, namely minimal_inconsistency._Finally,_we discuss technical issues_connecting LP⊛ and K⊛3 to the_tradition of Logics of Formal Inconsistency_and Underterminedness. Keywords: Kleene logics, Classical Recapture,_Normality Operators, Minimal Inconsistency, Logics of_Formal Inconsistency, Logics of Formal_Undeterminedness 1_Introduction The philosophical applications of_many-valued logics usually_come with_a story of_'normality': there are a number of_'abnormal phenomena' for_which we need manyvalued reasoning-logical paradoxes,_partial_information, vagueness, among_others-_but_as long_as the situation_is_normal-that is,_no_abnormal phenomena is at stake- classical_logic_is perfectly in order as it is._This view motivates the_following_question: how can we_recapture Classical Logic CL in_many-valued logic? That is, how can_we secure_inference of_classical conclusions, under the assumption that we are facing no abnormality?_[2, 6, 7, 23, 24] The_question fits the broad_conceptual view_on_many-valued reasoning by_the_'normality story'_above: if detour from classical reasoning is_motivated by_specific phenomena, then we do not_need to take it_when_these phenomena are not around. Numerous_as these may be, we may_assume that they are not_ubiquitous._For_instance, take semantical and set-theoretical_paradoxes; to put it with [23,_p.235]: 'paradoxical sentences_seem to be a fairly small proportion_of_the sentences we reason with... It_would_seem plausible to claim that in_our_day-to-day_reasoning we (quite correctly) presuppose_that we are 1L. J. of_the IGPL, Vol. 0 No. 0, pp. 1–18 0000_© Oxford University_Press 2 Normality operators and_Classical_Recapture_in Ext