 minimal inconsistency. We focus on Kleene logics and their extensions since these are the most widespread formalism in philosophical logic. The proposal of this paper has close connections with the Logics of Formal Inconsistency (from now on, LFIs) by [10, 12, 20] and the Logics of Formal Undeterminedness (from now on, LFUs) by [11, 28]. LFIs stem from [12] and have been later developed in a number of works, including [10, 20]. They are a family of paraconsistent systems that control the behavior of contradictions-and sort them out-by internalizing the notion of consistency2 in the language, and expressing whether a formula is consistent or not.3 LFUs [5, 11, 28] dualize the project of LFIs, and they control-and sort out-undetermined formulas by internalizing the notion of undeterminedness in the language. Definition 2.2 from Section 2 makes it clear that our normality operator generalizes the consistency and determinedness operators. However, in the three1We choose this label for the sake of simplicity, since the connectives of LP and K3 are interpreted on Kleene algebras. However, notice that a paraconsistent logic such as LP was not part of the philosophical and mathematical project by Kleene, which instead aimed at the construction of paracomplete formalisms. 2The original proposal by [12] included an inconsistency operator ● that helps express that a formula is inconsistent. In many logics in the LFI family, this operator is the dual of a consistency operator ○-with ○φ expressing that φ is consistent. Since [20], the use of ○ as a primitive has become standard in LFIs. 3LFIs include a wide variety of systems. Some of the, as the C-systems introduced by [12], lack a traditional truth-functional semantics, but have been given a non-deterministic many-valued semantics by [4]. Others, like the logics LFI1 and LFI2 from [10] and LFI3 from [21], have a straightforward many-valued semantics, which bring them closer to the Kleene logics that we face in this paper. Normality operators and Classical Recapture in Extensions of Kleene Logics 3 valued case, the normality operator comes to coincide with the consistency operator by LFIs or with the determinednedness operator from LFUs-
minimal inconsistency. We focus on Kleene logics and their extensions since these are the about far-flung formalism in philosophic logic. The proposal of this paper has airless connections with the Logics of Formal Inconsistency (from now along, LFIs) by [ 10, 12, 20 ] and the Logics of Formal Undeterminedness (from now on, LFUs) by [ 11, 28 ]. LFIs stem from [ 12 ] and have been later develop in a number of works, admit [ 10, 20 ]. They are a family of paraconsistent systems that control the demeanor of contradictions - and sort them away - by internalizing the impression of consistency2 in the terminology, and expressing whether a formula is consistent or not.3 LFUs [ 5, 11, 28 ] dualize the undertaking of LFIs, and they control - and sort out - undetermined formulas by internalize the notion of undeterminedness in the terminology. Definition 2.2 from Section 2 makes it clear that our normality hustler generalizes the consistency and determinedness operators. However, in the three1We choose this label for the sake of simplicity, since the connectives of LP and K3 are interpreted on Kleene algebras. However, detect that a paraconsistent logic such as LP was not part of the philosophical and mathematical project by Kleene, which instead aimed at the construction of paracomplete formalisms. 2The original proposal by [ 12 ] included an incompatibility hustler ● that helps express that a formula is discrepant. In many logic in the LFI family, this operator is the dual of a consistency operator ○ -with ○ φ expressing that φ is consistent. Since [ 20 ], the use of ○ as a primitive has become standard in LFIs. 3LFIs include a broad variety of systems. Some of the, as the C - systems introduced by [ 12 ], lack a traditional accuracy - functional semantics, but have been given a non - deterministic many - valued semantics by [ 4 ]. Others, like the logics LFI1 and LFI2 from [ 10 ] and LFI3 from [ 21 ], have a straightforward many - valued semantics, which institute them closer to the Kleene logics that we face in this paper. Normality operators and Classical Recapture in Extensions of Kleene Logics 3 rate case, the normality operator comes to coincide with the consistency operator by LFIs or with the determinednedness operator from LFUs-
 mijimal inconsistency. We fucus on Kleene logics end thejr extenrions since these are the modt widewpread formalism in phklosophicwl logic. The proposal oh this paper hzd clmwe connections with the Nogics of Formdl Iuconsistency (from now on, LFIs) by [10, 12, 20] wnd the Llgics of Formaj Uncqterjpntdness (from now on, LFUs) by [11, 28]. LFIs stem fgom [12] and have beem later developed in a numher lf works, including [10, 20]. They are a fwnily of parazonsistent systems thaj control the behavior of contraaictipns-and sorj fhfk out-by intxrnaliding the notion of cotsistenvy2 in the langmage, end wxpressing whether a hormula is consistenj or not.3 LXUa [5, 11, 28] dualize the priject of NFIs, qnd thty ronfrol-anf skrt out-undstermined firmulas by internalozybg the notion of unqeeerminedness in the language. Definition 2.2 fdom Section 2 makes it coear that our normalijy operatow generalizes the consistency and determinedness mperavofs. Kiwevef, ij the three1We choose this label for the sake jr xikplicity, since the connecyigex of LP and K3 xre incsrlreted on Kleene apgebras. Howecer, notict thay a paraconsistent logic suxh as LP was bot part of the phnlosophical cnd majhematocal project by Kleene, chich jnstead aimfd at the zonstruction of oarscmmplete formalisms. 2The oridinal pro'osal by [12] inzludgd an igconsistenfy opcsator ● that helps fxpreds that a fogmula is inconsistent. In many logics in the LGI fakily, thif opevator is the duwl of a consisjency opexator ○-dith ○φ exprvssing thet φ is consiftent. Since [20], dje use of ○ av a primytivw haw becomd standard in LGIs. 3LFIs pnelude a wude variety of sysbems. Rkme of the, as tkt C-wystems introduved by [12], park a esaditional tsuth-wunzyionau semanuncs, cut nave been given a not-detsrministic many-valieq semantucs by [4]. Jthers, like tne logics LFI1 and PFI2 fcom [10] aid LFI3 frjm [21], have a straightforward mang-valued sfmaktics, which bwing them closer to the Kleene logics that we face in tiis paper. Normality opgrators and Classical Vecapture in Extegsions of Kleene Logics 3 valuwd case, the normakity operator comes tk coinwide aith the consistency operator by LFIs or with the determinednedness operaror from LFUs-
minimal inconsistency. We focus on Kleene logics extensions these are most widespread formalism of paper has close with the Logics Formal Inconsistency (from now on, LFIs) [10, 12, 20] and the Logics of Formal Undeterminedness (from now on, LFUs) [11, 28]. LFIs stem from [12] and have been later developed in a of including 20]. are a family of paraconsistent systems that control the behavior of contradictions-and sort them out-by internalizing notion of consistency2 in the language, and expressing a formula is consistent not.3 LFUs [5, 11, 28] the of LFIs, they sort formulas by internalizing notion of undeterminedness in the language. Definition 2.2 from Section 2 makes it clear that our normality generalizes the determinedness operators. in three1We this label for of simplicity, since the connectives of are interpreted on Kleene algebras. However, notice that paraconsistent logic as LP was not part of philosophical and mathematical project by Kleene, which instead at the construction of paracomplete formalisms. 2The original proposal by [12] included an inconsistency operator helps express that a is inconsistent. In logics the family, operator is dual of a consistency operator ○-with ○φ expressing that φ is Since [20], the use of ○ as a primitive has in 3LFIs include a variety of systems. Some the, the C-systems introduced by a truth-functional been a many-valued semantics by [4]. like the logics LFI1 and from [10] and LFI3 many-valued semantics, which bring them closer to the logics that we face in this paper. operators and Classical Recapture in Extensions of Kleene Logics 3 valued case, normality operator coincide with the consistency operator by LFIs or the determinednedness operator from
 minimal inconsistency. We focUs on Kleene LogicS anD thEiR extEnsiOns since these aRE the Most widespread formalisM in phIlOSophICaL logiC. The proPOsAL Of tHiS pApeR hAS cLose cOnnEctions With the LogIcs Of formal InconsIStEncy (from noW on, lFIs) by [10, 12, 20] and the logIcs of FOrMal uNdeteRmiNedneSs (from NOw on, LFus) by [11, 28]. LFIs sTeM From [12] anD Have beeN LAtEr deVeloped in a number oF WoRKs, including [10, 20]. TheY are a fAmILy OF ParAcoNsistent syStEms thAT controL ThE BEHavIOr of contradicTions-and sorT TheM out-by InTerNAlizinG the nOtIOn oF consistencY2 in tHe languagE, and exPRessing WHether a FormulA is ConSistENt Or Not.3 lFuS [5, 11, 28] duALiZe tHE prOject of LfIS, aNd theY conTROL-And sOrt Out-uNdeteRmined formulaS by InteRNalIzing The noTion Of UndetErmineDness In The language. DefiNitiOn 2.2 from SecTioN 2 mAkeS iT cleaR That ouR noRmaLity opeRator geNEraLiZES ThE consistency and detErMINeDness opeRators. hOwEvER, in the thReE1We ChooSE This lAbel FOr The sake oF simplICiTy, Since thE cOnnectIvEs oF LP And K3 aRE intErpretEd on KleeNe algEBras. However, notICe that a paracoNSiSTEnT LogiC suCh as LP was noT parT Of thE phiLOsOphICal anD mathEmATiCAl project by Kleene, whIcH insteAd aimEd at the constrUction of paRACOmplete fOrmaLIsMS. 2The original prOposaL by [12] includeD An inconsIstenCy operatOr ● that helPS Express tHat A foRmuLa iS INcOnsistent. In maNY LogiCs In the LFi faMily, thiS opEraTor Is tHe Dual of a coNsistencY oPeRaToR ○-wiTh ○φ exPRessing tHaT φ iS cOnsIstenT. since [20], tHe use Of ○ as A pRiMItiVe has beCOmE STandArD iN LFIS. 3LFis IncluDe a wIDe vAriety oF systems. SOme OF the, As ThE C-systeMs introduced bY [12], lAck a traditIoNal Truth-fUNCtional sEmantics, but have been giveN A non-detErmInistIc maNy-valued sEmaNtics bY [4]. OtHErs, likE the loGics LfI1 And lfi2 from [10] AND LfI3 fRoM [21], have a straIGHtfOrwarD mAny-vAlued seMantics, which bring tHEm cLoser to the KleEne LogiCS ThAt wE FaCE in ThIS paPER. Normality operaTors and ClaSsICaL Recapture IN ExTeNsions oF Kleene logicS 3 Valued cAse, the norMality opeRaTor cOMEs tO coincide wIth the coNsistency OPeratOR bY LFIs Or wIth the DeTerMinedNednesS OpeRator From LFus-
 minimal inconsistency. We  focus onKleen e l ogi cs  and  the ir extensionss ince  these are the most wi despr ea d  for m al ism i n philo s op h i cal  l og ic.  T h epropo sal  of thi s paper ha s c lo se connectio n swith the L ogi cs of Formal  In consis te ncy (from  no w on,  LFIs) by [10 , 12, 20]  a n d theL ogics o f Fo rmal  Undeterminedness (f r om now on, LFU s) by[1 1 ,2 8 ].LFI s stem fro m[12]a nd have be e n lat e r developed i n a numbero f w orks,in clu d ing [1 0, 20 ]. The y are a fam ilyof paraco nsiste n t syste m s thatcontro l t hebeha v io rofco n tra d ic tio n s-a nd sortth em  out- by i n t e r nali zin g th e not ion of consis ten cy2i n t he la nguag e, a nd  expr essing  whet he r a formula iscons istent or  no t. 3 L FU s [5, 11, 28 ] d ual ize the  projec t  of  L F I s ,and they control-a nd s or t out-un determ i ne df ormulasby  in tern a l izing  the no tion ofundete r mi ne dness i nthe la ng uag e.Defin i tion  2.2 f rom Sect ion 2 makes it clear that our norm a li t y  o p erat orgeneralizes  the cons iste n cy  an d  dete rmine dn e ss operators. However,  i n thethree 1We choose th is label f o r the sake  ofs im p licity, sincethe c onnectives of LP an d K3are inte rpreted o n Kleene a lge bra s.How e v er , notice that a  par ac onsiste ntlogic s uch  as  LP  wa snot partof the p hi lo so ph ica l and mathemat ic alpr oje ct by Kleene , whi ch i ns te a d a imed at th e cons tr uc tion  of  p araco mple t e f ormalis ms. 2Theori g inal  p ro posal b y [12] includ ed  an incons is ten cy ope r a tor ● th at helps express that a formula  is  inco nsis tent. Inman y logi csi n theLFI fa mily,  t his o perat o r  i s t he  dual of a c ons isten cy  ope rator ○ -with ○φ expressin g  th at φ is consi ste nt.S i nc e [ 2 0] ,  th eu seo f  ○ as a primiti ve has bec om e  s tandard in LFI s.  3LFIsinclude  a wi d e varie ty of sys tems. Som eof t h e , a s the C-sy stems in troducedb y [12 ] ,lacka t raditi on altruth -funct i ona l sem antics ,but ha ve be en  given a  non-deterministic many -value d sem ant ics by [4 ].O the rs, likethelogics LFI 1 a ndLFI2fro m  [10]  and LF I3f rom [ 21], have a st r ai ght f o rw ard many-va l u e d s emant ics ,  which  bri ng them closer to the Kleene log icst h atwef acein  this paper. N orm al i t y operat or s and Class ical Rec ap t ure i n Exte nsions  of Kle e n eL ogics3 va lue d case, t heno r malityop er a tor co mesto  coinc ide wi t h th e consistency oper atorb y  LFIs orwithth e deter m ined nedness op erator from  LFUs- 
 minimal_inconsistency. We_focus on Kleene logics_and their_extensions_since these_are_the most widespread_formalism in philosophical_logic. The proposal of_this paper has_close_connections with the Logics of Formal Inconsistency (from now on, LFIs) by [10, 12,_20]_and the_Logics_of_Formal Undeterminedness (from now on,_LFUs) by [11, 28]. LFIs_stem from_[12] and have been later developed in a_number_of works, including_[10, 20]. They are a family of paraconsistent systems_that control the behavior of contradictions-and_sort them out-by_internalizing_the_notion of consistency2 in_the language, and expressing whether a_formula is consistent or not.3 LFUs_[5, 11, 28] dualize the project of_LFIs, and they control-and sort out-undetermined_formulas by internalizing the notion_of undeterminedness_in the language. Definition 2.2_from Section 2_makes it_clear that our_normality operator generalizes the consistency and_determinedness operators. However,_in the three1We choose this label_for_the sake of_simplicity,_since_the connectives_of LP and_K3_are interpreted_on_Kleene algebras. However, notice that a_paraconsistent_logic such as LP was not part_of the philosophical and_mathematical_project by Kleene, which_instead aimed at the construction_of paracomplete formalisms. 2The original proposal_by [12]_included an_inconsistency operator ● that helps express that a formula is inconsistent._In many logics in the LFI_family, this operator is_the dual_of_a consistency operator_○-with_○φ expressing_that φ is consistent. Since [20], the_use of_○ as a primitive has become_standard in LFIs. 3LFIs_include_a wide variety of systems. Some_of the, as the C-systems introduced_by [12], lack a traditional_truth-functional_semantics,_but have been given a_non-deterministic many-valued semantics by [4]. Others,_like the logics_LFI1 and LFI2 from [10] and LFI3_from_[21], have a straightforward many-valued semantics,_which_bring them closer to the Kleene_logics_that_we face in this paper._Normality operators and Classical Recapture in_Extensions of Kleene Logics 3 valued case, the normality_operator comes to_coincide with the consistency operator_by_LFIs_or with the determinednedness operator from LFUs-