 reads 'ψ is a S-consequence of Σ'. We write φ ⊧S ψ instead of {φ} ⊧S ψ, and we write ⊧S ψ instead of ∅ ⊧S ψ in order to express that ψ is a tautology of S. Now we give a semantical definition of the main logical operators of the paper: Definition 2.2 For every many-valued logic S and valuation function n defined for S, a unary connective k is a normality operator iff (if and only if) it obeys the following truth clause: n(kφ) = 1 ⇔ n(φ) ∈ {0,1} and n(kφ) = 0 ⇔ n(φ) ∉ {0,1} Consistency, Determinedness and Normality Operators. The normality operator from Definition 2.2 is closely related to the main operators in LFIs and LFUs. In LFIs, a unary connective k is a consistency operator in a logic S iff [20] S is paraconsistent-that is, it fails Ex Contradictione Quodlibet (ECQ)-and k verifies the Principle of Gentle Explosion (PGE) in S, respectively: 4 Normality operators and Classical Recapture in Extensions of Kleene Logics φ,¬φ /⊧S ψ φ,¬φ, kφ ⊧S ψ In LFUs, a unary connective k is a determinedness operator in a logic S iff [28] S is paracomplete-that is, it fails the Law of Excluded Middle (LEM)-and k verifies the Principle of Gentle Implosion (GPI) in S, respectively: /⊧S φ ∨ ¬φ ⊧S φ ∨ ¬φ ∨ kφ It is straightforward to see that a normality operator satisfies PGE in a paraconsistent many-valued logic, and PGI in a paracomplete many-valued logic. This implies that, in the logics we consider here, the normality operator will collapse on one of the two operators above. This notwithstanding, we choose here to use the label 'normality operator', for two reasons. First, this allows us to refer to the operator in a unique way, no matter whether we are dealing with the par
reads' ψ is a S - consequence of Σ'. We write φ ⊧S ψ instead of { φ } ⊧S ψ, and we spell ⊧S ψ alternatively of ∅ ⊧S ψ in order to express that ψ is a tautology of S. Now we hold a semantical definition of the main logical hustler of the newspaper: Definition 2.2 For every many - valued logic S and evaluation affair n defined for S, a unary connective kilobyte is a normality operator iff (if and only if) it obey the following truth clause: n(kφ) = 1 ⇔ n(φ) ∈ { 0,1 } and n(kφ) = 0 ⇔ n(φ) ∉ { 0,1 } Consistency, Determinedness and Normality Operators. The normality operator from Definition 2.2 is closely related to the main operator in LFIs and LFUs. In LFIs, a unary connective k is a consistency operator in a logic S iff [ 20 ] S is paraconsistent - that is, it fails Ex Contradictione Quodlibet (ECQ)-and k affirm the Principle of Gentle Explosion (PGE) in S, respectively: 4 Normality operators and Classical Recapture in Extensions of Kleene Logics φ,¬φ /⊧S ψ φ,¬φ, kφ ⊧S ψ In LFUs, a unary connective k is a determinedness operator in a logic S iff [ 28 ] S is paracomplete - that is, it fails the Law of exclude Middle (LEM)-and k verifies the Principle of Gentle Implosion (GPI) in S, respectively: /⊧S φ ∨ ¬φ ⊧S φ ∨ ¬φ ∨ kφ It is straightforward to see that a normality operator meet PGE in a paraconsistent many - valued logic, and PGI in a paracomplete many - valued logic. This implies that, in the logics we consider here, the normality operator will collapse on one of the two operators above. This notwithstanding, we choose here to use the label' normality operator', for two reasons. First, this allows us to refer to the operator in a unique way, no matter whether we are deal with the par
 rewds 'ψ is a S-consequence on Σ'. We write φ ⊧S ψ nbstead of {φ} ⊧S ψ, and we write ⊧S ψ instead of ∅ ⊧S ψ in lreer ti express that ψ is a txutology lf S. Now we jive a semantical definibnon or the nain logical okerators of dhe paper: Defitigiln 2.2 For every many-valued logic S anq valuayiln function n qefimqd fkg W, a unary connective k is a nkrmalitj operator iff (if and only if) it obeys the volllwing truth clause: n(kφ) = 1 ⇔ n(φ) ∈ {0,1} qnd g(jφ) = 0 ⇔ n(φ) ∉ {0,1} Conristency, Dttzrminedness and Normality Operators. The nofmalicy operator feom Gefinition 2.2 is cjosely relatcc to tve main operators in KFIv abd LFUs. In LFIs, a unacy connective k is a consistetcv operator in a logic S iff [20] S iv pafqcovsiatxnt-fhat id, iv fails Ex Dontradictiine Quodlibet (ECQ)-anc h verifies the Princypje of Gentle Explosion (PGE) in S, respectpvelg: 4 Normality operators qnd Classical Recaptute in Extegsions of Kleene Logics φ,¬φ /⊧S ψ φ,¬φ, kφ ⊧S ψ In LFUs, a unary connxcgivt j is x dfterminedness operator in a logic S iff [28] S is latabomplete-that is, ib fails the Law of Edckoded Middle (LEO)-and k vedifies the Princippe of Ggntle Umplosion (GPI) in S, respectively: /⊧S φ ∨ ¬φ ⊧S φ ∨ ¬φ ∨ kφ It is snraithtforward to see chat a normauity operstor satisfies PGE in a parzconsistent many-valusa logic, and PGI kn s [aracomplete many-valued lodic. This mmplizs that, kn tne logycs we condider here, the normality operctor fill collaose on one of the two operators above. This nojwidhsnanding, wz chooxe here to usq the label 'notmality o'eratof', for two geasons. Fmrst, this aljows us to rexgr to the opecator in w unuque way, no oatter whether we are dvajing with the par
reads 'ψ is a S-consequence of Σ'. φ ψ instead {φ} ⊧S ψ, instead ∅ ⊧S ψ order to express ψ is a tautology of S. we give a semantical definition of the main logical operators of the paper: 2.2 For every many-valued logic S and valuation function n defined for S, unary k a operator iff (if and only if) it obeys the following truth clause: n(kφ) = 1 ⇔ ∈ {0,1} and n(kφ) = 0 ⇔ n(φ) {0,1} Consistency, Determinedness and Operators. The normality operator from 2.2 closely related the operators LFIs and LFUs. LFIs, a unary connective k is a consistency operator in a logic S iff [20] S is is, it Contradictione Quodlibet k the of Gentle Explosion S, respectively: 4 Normality operators and Extensions of Kleene Logics φ,¬φ /⊧S ψ φ,¬φ, ⊧S ψ LFUs, a unary connective k is determinedness operator in a logic S iff [28] is paracomplete-that is, it fails the Law of Excluded Middle (LEM)-and k verifies the Principle Implosion (GPI) in S, /⊧S φ ∨ ⊧S ∨ ∨ It is to see that a normality operator satisfies PGE in a paraconsistent logic, and PGI in a paracomplete many-valued logic. This implies the we consider here, normality operator will collapse one the two operators above. we here label operator', two reasons. First, this us to refer to the in a unique way, dealing with the par
 reads 'ψ is a S-consequence of Σ'. WE write φ ⊧S ψ iNsteaD of {Φ} ⊧S ψ, AnD we wRite ⊧s ψ instead of ∅ ⊧S ψ iN OrdeR to express that ψ is a tautOlogy Of s. now wE GiVe a seManticaL DeFINitIoN oF thE mAIn LogicAl oPeratorS of the papeR: DeFiNition 2.2 For eveRY mAny-valued lOgiC S and valuatiOn fUnctioN n DefINed foR S, a Unary ConnecTIve k is A normalitY oPErator IFf (if and ONLy If) it Obeys the following TRuTH clause: n(kφ) = 1 ⇔ n(φ) ∈ {0,1} anD n(kφ) = 0 ⇔ n(φ) ∉ {0,1} coNSiSTEncY, DeTerminedneSs And NoRMality OPErATORs. THE normality opeRator from DeFIniTion 2.2 is ClOseLY relatEd to tHe MAin Operators in lFIs And LFUs. In lFIs, a uNAry connECtive k iS a consIstEncY opeRAtOr In a LoGIc S IFf [20] s is PAraConsisteNt-ThAt is, iT faiLS eX contRadIctiOne QuOdlibet (ECQ)-and K veRifiES thE PrinCiple Of GeNtLe ExpLosion (pGE) in s, rEspectively: 4 NormAlitY operatorS anD CLasSiCal ReCApture In EXteNsions oF Kleene lOgiCs Φ,¬Φ /⊧s Ψ φ,¬Φ, kφ ⊧S ψ In LFUs, a unary cOnNECtIve k is a dEtermiNEdNeSS operatoR iN a lOgic s IFf [28] S is ParaCOmPlete-thaT is, it fAIlS tHe Law of exCluded miDdlE (LEm)-and k VErifIes the principlE of GeNTle Implosion (GPi) In S, respectiveLY: /⊧S Φ ∨ ¬Φ ⊧s φ ∨ ¬Φ ∨ Kφ It Is sTraightforwArd tO See tHat a NOrMalITy opeRator SaTIsFIes PGE in a paraconsisTeNt many-ValueD logic, and PGI iN a paracompLETE many-valUed lOGiC. this implies thaT, in thE logics we cONsider heRe, the NormalitY operator WILl collapSe oN onE of The TWO oPerators above. tHIs noTwIthstanDinG, we chooSe hEre To uSe tHe Label 'normAlity opeRaToR', fOr Two ReasoNS. First, thIs AllOwS us To refER to the OperaTor iN a UnIQue Way, no maTTeR WHethEr We Are dEalInG with The pAR
 reads 'ψ is a S-consequen ce of Σ'.We wr ite  φ⊧S  ψ i nste ad of {φ} ⊧S ψ ,  and  we write ⊧S ψ instead  of ∅  ⊧ S  ψ i n  o rderto expr e ss t hat  ψ  i s a  t a ut ology  of  S. Now  we give a  se ma ntical defin i ti on of themai n logical op era tors o fthe paper : D efini tion 2 . 2 Forevery man y- v aluedl ogic Sa n dvalu ation function nd ef i ned for S, a u nary c on n ec t i vek i s a normal it y ope r ator if f  ( i f and only if) it o beys the fo l low ing tr ut h c l ause:n(kφ)  = 1 ⇔  n(φ) ∈ {0, 1} a nd n(kφ)= 0 ⇔n (φ) ∉ { 0 ,1} Con sisten cy,  De term i ne dn ess  a n d N o rm ali t y O perators .Th e nor mali t y o pera tor  fro m Def inition 2.2 i s c lose l y r elate d tothema in op erator s inLF Is and LFUs. In  LFI s, a unar y c on nec ti ve ki s a co nsi ste ncy ope rator i n  alo g i c  S  iff [20] S is par ac o n si stent-th at is, it  f a ils Ex C on tra dict i o ne Qu odli b et  (ECQ)-a nd k v e ri fi es thePr incipl eofGen tle E x plos ion (P GE) in S , res p ectively: 4 No r mality operat o rs a nd Clas sic al Recaptur e in Exte nsio n sofK leene  Logi cs φ, ¬ φ /⊧S ψ φ,¬φ, kφ ⊧S  ψ  In LF Us, a  unary connec tive k isa d etermine dnes s  o p erator in a lo gic S  iff [28]S  is para compl ete-that  is, it f a i ls the L awofExc lud e d  M iddle (LEM)-a n d  k v er ifies t hePrincip leofGen tle  I mplosion(GPI) in  S ,re sp ect ively :  /⊧S φ ∨  ¬ φ ⊧ Sφ ∨  ¬φ ∨ kφ Itis st raig ht fo r war d to se e  t h a t ano rm alit y o pe rator  sat i sfi es PGEin a para con s iste nt  m any-val ued logic, an dPGI in a p ar aco mplete m any-valu ed logic. This impliest hat, in  th e log icswe consid erhere,the normal ity op erato rwil l colla p s eonon e of the t w o  op erato rs  abo ve. Thi s notwithstanding, wechoose here t o u se t h e  l abe l  ' n orm al i tyo p erator', for tw o reasons.  F i rs t, this al l ows  u s to re fer tothe o p eratorin a uniq ue way, n omatt e r  wh ether we a re deali ng with t h e par 
 reads_'ψ is_a S-consequence of Σ'._We write_φ_⊧S ψ_instead_of {φ} ⊧S_ψ, and we_write ⊧S ψ instead_of ∅ ⊧S_ψ_in order to express that ψ is a tautology of S. Now we give_a_semantical definition_of_the_main logical operators of the_paper: Definition 2.2 For every_many-valued logic_S and valuation function n defined for S,_a_unary connective k_is a normality operator iff (if and only if)_it obeys the following truth clause:_n(kφ) = 1_⇔_n(φ)_∈ {0,1} and n(kφ)_= 0 ⇔ n(φ) ∉ {0,1}_Consistency, Determinedness and Normality Operators. The_normality operator from Definition 2.2 is closely_related to the main operators in_LFIs and LFUs. In LFIs,_a unary_connective k is a consistency_operator in a_logic S_iff [20] S_is paraconsistent-that is, it fails Ex_Contradictione Quodlibet (ECQ)-and_k verifies the Principle of Gentle_Explosion_(PGE) in S,_respectively:_4_Normality operators_and Classical Recapture_in_Extensions of_Kleene_Logics φ,¬φ /⊧S ψ φ,¬φ, kφ_⊧S_ψ In LFUs, a unary connective k_is a determinedness operator_in_a logic S iff_[28] S is paracomplete-that is,_it fails the Law of Excluded_Middle (LEM)-and_k verifies_the Principle of Gentle Implosion (GPI) in S, respectively: /⊧S φ_∨ ¬φ ⊧S φ ∨ ¬φ_∨ kφ It is_straightforward to_see_that a normality_operator_satisfies PGE_in a paraconsistent many-valued logic, and PGI_in a_paracomplete many-valued logic. This implies that,_in the logics we_consider_here, the normality operator will collapse_on one of the two operators_above. This notwithstanding, we choose_here_to_use the label 'normality operator',_for two reasons. First, this allows_us to refer_to the operator in a unique way,_no_matter whether we are dealing with_the_par