 truth values that the logic 4The label 'classicality operator' is also used for operators that rule out both inconsistency and undeterminedness-see for instance [22]. We prefer the label 'normality operator', since it makes a more direct connection with the 'normality story' that motivates classical recapture. 5We are including ⊃ in Table 1, for the sake of simplicity. This will help track failures of Modus Ponens, Tollens, and Transitivity of ⊃ in LP-see below. Normality operators and Classical Recapture in Extensions of Kleene Logics 5 Table 1 ¬φ ⊛φ 1 0 1 1 2 1 2 0 0 1 1 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ ⊃ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 1 2 0 1 1 1 preserves through valid inferences. In particular, DK⊛3 = {1} is the set of designated values of K3, and DLP⊛ = { 12,1} is the set of designated values of LP ⊛. The operator ⊛ obeys Definition 2.2, thus qualifying as a normality operator. Specific interpretations of ⊛φ depends on the logic in question. Given DLP⊛ = { 12,1} and the behavior of ∧, ¬ and ⊛ from Table 1, we have that VLP⊛(⊛φ) = {ν ∈ V ∣ ν(φ∧¬φ) = 0}. As a consequence, in LP⊛, ⊛φ reads 'φ is consistent ', and ⊛ works as a consistency operator like the one by [10]. This is relevant since some contradictions are satisfiable in LP⊛, as we shall see below by Fact 2.6 and paraconsistency of LP. By contrast, DK⊛3 = {1} implies that VK⊛3 (⊛φ) = {ν ∈ V ∣ ν(φ ∨ ¬φ) = 1}; that is, in K⊛3, ⊛ works as a determinedness operator, with �
truth values that the logic 4The label' classicality operator' is also use for operator that rule out both inconsistency and undeterminedness - see for instance [ 22 ]. We choose the label' normality operator', since it makes a more direct association with the' normality story' that motivates authoritative recapture. 5We are include ⊃ in Table 1, for the sake of simplicity. This will avail track failures of Modus Ponens, Tollens, and Transitivity of ⊃ in LP - see downstairs. Normality operators and Classical Recapture in Extensions of Kleene Logics 5 Table 1 ¬φ ⊛φ 1 0 1 1 2 1 2 0 0 1 1 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ ⊃ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 1 2 0 1 1 1 preserves through valid inferences. In particular, DK⊛3 = { 1 } is the stage set of designated values of K3, and DLP⊛ = { 12,1 } is the set of destine values of LP ⊛. The operator ⊛ obeys Definition 2.2, thus qualify as a normality operator. Specific interpretations of ⊛φ depends on the logic in question. Given DLP⊛ = { 12,1 } and the demeanor of ∧, ¬ and ⊛ from Table 1, we have that VLP⊛(⊛φ) = { ν ∈ V ∣ ν(φ∧¬φ) = 0 }. As a consequence, in LP⊛, ⊛φ reads' φ is consistent', and ⊛ works as a consistency operator like the one by [ 10 ]. This is relevant since some contradiction are satisfiable in LP⊛, as we shall see below by Fact 2.6 and paraconsistency of LP. By contrast, DK⊛3 = { 1 } implies that VK⊛3 (⊛φ) = { ν ∈ V ∣ ν(φ ∨ ¬φ) = 1 }; that is, in K⊛3, ⊛ works as a determinedness operator, with �
 trkth values that the logig 4The label 'classicalitb operafor' is auso used for operators that cule out voth inconsistency and undetermpnedness-swe fie instance [22]. We prenzr ths labzl 'normality opetator', since ht makes a mora aixect connection with the 'normality seory' thst motivates clafsicsj redapture. 5We are including ⊃ in Table 1, for tie sake of simpkicity. This will help tracn fallures of Modus Pojens, Tollenw, anq Transitivith of ⊃ in LK-sze below. Notmality operators and Classical Fecapcure in Extguwiojv of Kleene Logibs 5 Table 1 ¬φ ⊛φ 1 0 1 1 2 1 2 0 0 1 1 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ ⊃ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 1 2 0 1 1 1 prevexves through valid indeeencev. In pargucuuar, DL⊛3 = {1} is tje aet of desjgnated valyes of K3, and DLP⊛ = { 12,1} if the set of dssignaeeq values of LP ⊛. The operator ⊛ obeys Defpnitjon 2.2, thus qualifying as a normality operator. Specific interpretations of ⊛φ depends on the logic in quesdion. Jixen DJO⊛ = { 12,1} and the behavior of ∧, ¬ and ⊛ from Table 1, we hwbe tmat VLP⊛(⊛φ) = {ν ∈ V ∣ ν(φ∧¬φ) = 0}. As a conseaufnvg, in LP⊛, ⊛φ reads 'φ is consjstent ', and ⊛ works as a cjnsisrency opewatot like the one by [10]. This is eelevant sinbe sime contradictions are satisfnable on LP⊛, as we shall see below yy Facf 2.6 and parafonsistendh of LP. By contrxst, DN⊛3 = {1} implies that VK⊛3 (⊛φ) = {ν ∈ V ∣ ν(φ ∨ ¬φ) = 1}; thav is, nn K⊛3, ⊛ wofks ss a dqterminednfss oiarator, with �
truth values that the logic 4The label is used for that rule out instance We prefer the 'normality operator', since makes a more direct connection with 'normality story' that motivates classical recapture. 5We are including ⊃ in Table 1, the sake of simplicity. This will help track failures of Modus Ponens, Tollens, Transitivity ⊃ LP-see Normality operators and Classical Recapture in Extensions of Kleene Logics 5 Table 1 ¬φ ⊛φ 1 1 1 2 1 2 0 0 1 φ ∨ ψ 1 2 0 1 1 1 1 1 1 1 0 1 2 0 ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 φ ⊃ 1 2 1 1 1 1 2 1 1 2 1 1 1 preserves through valid inferences. In particular, = {1} the set of designated values of and DLP⊛ = { 12,1} is the set designated values of LP ⊛. The operator ⊛ obeys Definition 2.2, thus qualifying as a Specific interpretations of ⊛φ on the logic question. DLP⊛ { and the of ∧, ¬ and ⊛ from Table 1, we have that = {ν ∈ V ∣ ν(φ∧¬φ) = 0}. As a LP⊛, reads 'φ is ', and ⊛ works a operator like the one This relevant are in as we shall see by Fact 2.6 and paraconsistency LP. By contrast, DK⊛3 (⊛φ) = {ν ∈ V ∣ ν(φ ∨ = 1}; that is, in K⊛3, ⊛ as a determinedness operator, with �
 truth values that the logic 4ThE label 'clasSicalIty OpeRaTor' iS alsO used for operatORs thAt rule out both inconsistEncy aNd UNdetERmInednEss-see fOR iNSTanCe [22]. we PreFeR ThE labeL 'noRmality Operator', siNce It Makes a more diREcT connectioN wiTh the 'normaliTy sTory' thAt MotIVates ClaSsicaL recapTUre. 5We aRe includiNg ⊃ IN Table 1, FOr the saKE Of SimpLicity. This will helP TrACk failures of MoDus PonEnS, toLLEns, And transitiviTy Of ⊃ in Lp-See beloW. noRMALitY Operators and CLassical RecAPtuRe in ExTeNsiONs of KlEene LOgICs 5 TAble 1 ¬φ ⊛φ 1 0 1 1 2 1 2 0 0 1 1 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ ⊃ Ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 1 2 0 1 1 1 preServes thrOugh vaLId inferENces. In pArticuLar, dK⊛3 = {1} iS the SEt Of DesIgNAteD VaLueS Of K3, And DLP⊛ = { 12,1} is ThE sEt of dEsigNATED valUes Of LP ⊛. the opErator ⊛ obeys DeFinItioN 2.2, ThuS qualIfyinG as a NoRmaliTy operAtor. SPeCific interpretaTionS of ⊛φ depenDs oN tHe lOgIc in qUEstion. givEn DlP⊛ = { 12,1} and thE behaviOR of ∧, ¬ AnD ⊛ FROm table 1, we have that VLP⊛(⊛Φ) = {ν ∈ v ∣ Ν(Φ∧¬φ) = 0}. as a conseQuence, IN Lp⊛, ⊛φ REads 'φ is cOnSisTent ', AND ⊛ workS as a COnSistency OperatOR lIkE the one By [10]. this is ReLevAnt Since SOme cOntradIctions aRe satISfiable in LP⊛, as wE Shall see below BY FACT 2.6 aND parAcoNsistency of lP. By COntrAst, Dk⊛3 = {1} ImPliES that vK⊛3 (⊛φ) = {ν ∈ V ∣ Ν(φ ∨ ¬Φ) = 1}; ThAT is, in K⊛3, ⊛ works as a deterMiNednesS operAtor, with �
 truth values that the log ic 4The la bel ' cla ssi ca lity  ope rator' is also used  for operators that ru le ou tb othi nc onsis tency a n du n det er mi ned ne s s- see f orinstanc e [22]. We  pr ef er the label 'n ormality o per ator', since  it  makes  a  mo r e dir ect  conn ection with t he 'norma li t y stor y ' thatm o ti vate s classical recap t ur e . 5We are incl uding⊃i nT a ble  1,  for the s ak e ofs implici t y. T h isw ill help trac k failureso f M odus P on ens ,  Tolle ns, a nd Tra nsitivity o f ⊃in LP-see  below .  Normal i ty oper atorsand  Cl assi c al  R eca pt u rei nExt e nsi ons of K le en e Log ics5 T a ble1 ¬ φ ⊛φ  1 01 1 2 1 2 0 0  11 φ∨  ψ1 1 2  0 11 111 2 1  1 2 1  2 011 2 0 φ ∧ ψ 1 1  2 0  1 1 1 20 1  2  121 2 0 0 0 00 φ  ⊃ψ 1 1 2  0 1 11  201 2  1  1 2 1 2 0 1 1 1 p re s e rv es throu gh val i din f erences.  I n p arti c u lar,DK⊛3 ={1} is t he set of  d esignat ed  value sofK3,  andD LP⊛= { 12 ,1} is t he se t  of designated values of LP⊛ .T h eo pera tor  ⊛ obeys De fini t ion2.2, th usq ualif yingas an ormality operator.Sp ecific  inte rpretations o f ⊛φ depen d s on the l ogic in question. Give n DLP ⊛ = { 12,1 }  and the  beha vior of∧, ¬ and⊛ from Tab le1,wehav e th at VLP⊛(⊛φ) = { ν ∈V∣ ν(φ∧¬ φ)= 0}. A s a  co nse que nc e, in LP⊛ , ⊛φ rea ds  ' φis  co nsist e nt ', an d⊛ w or ksas ac onsist encyoper at or lik e the o n eb y  [10 ].  T hisisre levan t si n cesome co ntradicti ons aresa ti sfiable  in LP⊛, as w eshall seebe low  by Fa c t  2.6 and  paraconsistency of LP. By cont ras t, DK ⊛3 =  {1} impl ies  thatVK⊛ 3  (⊛φ)= {ν ∈  V ∣ν( φ ∨ ¬ φ) =1 } ;tha tis, in K⊛3 , ⊛ w orksas  a d etermin edness operator, w i th�
 truth_values that_the logic 4The label_'classicality operator'_is_also used_for_operators that rule_out both inconsistency_and undeterminedness-see for instance_[22]. We prefer_the_label 'normality operator', since it makes a more direct connection with the 'normality story'_that_motivates classical_recapture._5We_are including ⊃ in Table_1, for the sake of_simplicity. This_will help track failures of Modus Ponens, Tollens,_and_Transitivity of ⊃_in LP-see below. Normality operators and Classical Recapture in_Extensions of Kleene Logics 5 Table_1 ¬φ ⊛φ_1_0_1 1 2 1_2 0 0 1 1 φ_∨ ψ 1 1 2 0_1 1 1 1 1 2 1_1 2 1 2 0 1_1 2 0 φ ∧_ψ 1_1 2 0 1 1_1 2 0_1 2_1 2 1_2 0 0 0 0 0_φ ⊃ ψ_1 1 2 0 1 1_1_2 0 1_2_1_1 2_1 2 0_1_1 1_preserves_through valid inferences. In particular, DK⊛3_=_{1} is the set of designated values_of K3, and DLP⊛_=_{ 12,1} is the_set of designated values of_LP ⊛. The operator ⊛ obeys_Definition 2.2,_thus qualifying_as a normality operator. Specific interpretations of ⊛φ depends on the_logic in question. Given DLP⊛ =_{ 12,1} and the_behavior of_∧,_¬ and ⊛_from_Table 1,_we have that VLP⊛(⊛φ) = {ν ∈_V ∣_ν(φ∧¬φ) = 0}. As a consequence,_in LP⊛, ⊛φ reads_'φ_is consistent ', and ⊛ works_as a consistency operator like the_one by [10]. This is_relevant_since_some contradictions are satisfiable in_LP⊛, as we shall see below_by Fact 2.6_and paraconsistency of LP. By contrast, DK⊛3_=_{1} implies that VK⊛3 (⊛φ) =_{ν_∈ V ∣ ν(φ ∨ ¬φ)_=_1};_that is, in K⊛3, ⊛_works as a determinedness operator, with_�