�φ stating that φ is determined. This is relevant, since from Fact 2.6 and paracompleteness of K3 imply that K ⊛ 3 is itself paracomplete (see below). 2.1 Kleene Logics We obtain the language L2 of Kleene logics by restricting L1 to those formulas where operator ⊛ does not occur. The valuation function for interpreting L2 in a Kleene logic obtains accordingly: Definition 2.4 (Valuations for Kleene logics) A valuation u ∶ L2 z→ {0, 12,1} is the unique extension of ν ∶ P z→ {0, 1 2,1} that is induced by the relevant components of Table 1. We denote by U the set {u,u′, u′′,...} of valuations defined in accordance with Definition 2.4. The logic of paradox LP can be though of as the restriction of LP⊛ to L2, and the strong Kleene logic K3 can be thought as the restriction of K⊛3 to L2. This implies that DLP⊛ = DLP and DK⊛3 = DK3. Relations between Kleene Logics and logics LP⊛ and K⊛3. In LP, no formula φ can express that a formula ψ is consistent, and, in K3, no formula φ can express that a formula ψ is undetermined : Fact 2.5 For every φ,ψ ∈ L2: u(φ) ∈ DLP /⇒ u(ψ) ∈ {0,1} u(φ) ∈ DK3 /⇒ u(ψ) = 12 By contrast, in LP⊛, we have that ν(⊛φ) ∈ DLP⊛ ⇒ ν(φ) ∈ {0,1}, and in K3, we have that ν(¬ ⊛ φ) ∈ DK⊛3 ⇒ ν(φ) = 1 2, by Definition 2.2. Thus, LP⊛ and K⊛3 brings a real increase in the expressive power of LP and K3. Fact 2.6 below clarifies what this 6 Normality operators and Classical Recapture in Extensions of Kleene Logics increase amounts to, exactly. In particular, LP⊛ and K�
� φ stating that φ is determined. This is relevant, since from Fact 2.6 and paracompleteness of K3 imply that K ⊛ 3 is itself paracomplete (see downstairs). 2.1 Kleene Logics We receive the language L2 of Kleene logics by restricting L1 to those recipe where operator ⊛ does not occur. The evaluation function for interpreting L2 in a Kleene logic obtains consequently: Definition 2.4 (evaluation for Kleene logics) A evaluation u ∶ L2 z→ { 0, 12,1 } is the unique extension of ν ∶ P z→ { 0, 1 2,1 } that is induced by the relevant component of Table 1. We denote by U the set { u, u′, u′′, ... } of valuations defined in accord with Definition 2.4. The logic of paradox LP can be though of as the restriction of LP⊛ to L2, and the strong Kleene logic K3 can be thought as the limitation of K⊛3 to L2. This implies that DLP⊛ = DLP and DK⊛3 = DK3. Relations between Kleene Logics and logics LP⊛ and K⊛3. In LP, no formula φ can express that a recipe ψ is consistent, and, in K3, no formula φ can express that a formula ψ is undetermined: Fact 2.5 For every φ, ψ ∈ L2: u(φ) ∈ DLP /⇒ u(ψ) ∈ { 0,1 } u(φ) ∈ DK3 /⇒ u(ψ) = 12 By line, in LP⊛, we have that ν(⊛φ) ∈ DLP⊛ ⇒ ν(φ) ∈ { 0,1 }, and in K3, we have that ν(¬ ⊛ φ) ∈ DK⊛3 ⇒ ν(φ) = 1 2, by Definition 2.2. Thus, LP⊛ and K⊛3 brings a real increase in the expressive might of LP and K3. Fact 2.6 below clarifies what this 6 Normality operator and Classical Recapture in Extensions of Kleene Logics increase amounts to, exactly. In particular, LP⊛ and K �
�φ stwting that φ is determinea. This is relevcbt, sinre from Fact 2.6 avd paracompleteness of K3 imppy that K ⊛ 3 is itself paracomolete (see below). 2.1 Jleeie Logics We obtejn the languafc L2 oy Jleene logics ny restricthng L1 to those furlulas where operator ⊛ does not occuw. The vslkation functiog fog yntedirtting L2 in a Kleene logic obtains accordpngly: Definition 2.4 (Valuations for Kleene loglcs) W valuation u ∶ L2 z→ {0, 12,1} is the ubiquq extension ow ν ∶ P z→ {0, 1 2,1} that is indoced by the relevant components uf Tayle 1. We denitw bj U the set {n,u′, u′′,...} os valuations defined in accprdance with Dcfinivion 2.4. The logic of paradoe LP can be though os as the seatriction of LP⊛ ti O2, and the strubg Yletne lkgic K3 cai be thoughf as the rewtriction of K⊛3 to L2. Trpx implies thzt DLP⊛ = QLP and DK⊛3 = DK3. Relations between Kleene Lofics and logics LP⊛ and J⊛3. In LP, no formula φ cwn expresf that a formula ψ is consistent, and, in K3, no formuna φ cen exkrcfr tjat a formula ψ is undetermined : Fact 2.5 For evewg φ,ψ ∈ K2: u(φ) ∈ DLP /⇒ u(ψ) ∈ {0,1} m(φ) ∈ DK3 /⇒ u(ψ) = 12 By conttadt, yn LP⊛, we have that ν(⊛φ) ∈ DMP⊛ ⇒ ν(φ) ∈ {0,1}, and in K3, we have trat ν(¬ ⊛ φ) ∈ DK⊛3 ⇒ ν(φ) = 1 2, by Definition 2.2. Thus, LP⊛ and K⊛3 vrings a reaj increase in the erpressive pocer of LP amd K3. Fact 2.6 below clarifnes whzt this 6 Nogmality oldrators and Clasricsl Recapture in Extensions os Kleene Oogies increxse smountf to, exactpy. In particular, LP⊛ and N�
�φ stating that φ is determined. This since Fact 2.6 paracompleteness of K3 is paracomplete (see below). Kleene Logics We the language L2 of Kleene logics restricting L1 to those formulas where operator ⊛ does not occur. The valuation for interpreting L2 in a Kleene logic obtains accordingly: Definition 2.4 (Valuations for logics) valuation ∶ z→ {0, 12,1} is the unique extension of ν ∶ P z→ {0, 1 2,1} that induced by the relevant components of Table 1. denote by U the {u,u′, u′′,...} of valuations defined accordance Definition 2.4. logic paradox can be though as the restriction of LP⊛ to L2, and the strong Kleene logic K3 can be thought as restriction of L2. This that = and DK⊛3 = between Kleene Logics and logics LP⊛ LP, no formula φ can express that a ψ is and, in K3, no formula φ express that a formula ψ is undetermined : 2.5 For every φ,ψ ∈ L2: u(φ) ∈ DLP /⇒ u(ψ) ∈ {0,1} u(φ) ∈ u(ψ) = 12 By in LP⊛, we that ∈ ⇒ ∈ {0,1}, in K3, we have that ν(¬ ⊛ φ) ∈ DK⊛3 ⇒ = 1 2, by Definition 2.2. Thus, LP⊛ and K⊛3 real in the expressive of LP and K3. 2.6 clarifies what this 6 and Recapture Kleene increase to, exactly. In particular, and K�
�φ stating that φ is determined. this is releVant, sIncE frOm fact 2.6 And pAracompletenesS Of K3 iMply that K ⊛ 3 is itself paracOmpleTe (SEe beLOw). 2.1 kleenE Logics wE oBTAin ThE lAngUaGE L2 Of KleEne Logics bY restrictiNg L1 To Those formulaS WhEre operatoR ⊛ doEs not occur. ThE vaLuatioN fUncTIon foR inTerprEting L2 IN a KleeNe logic obTaINs accoRDingly: DEFInItioN 2.4 (Valuations for KleENe LOgics) A valuatioN u ∶ L2 z→ {0, 12,1} is ThE UnIQUe eXteNsion of ν ∶ P z→ {0, 1 2,1} ThAt is iNDuced by THe RELEvaNT components of table 1. We denoTE by u the seT {u,U′, u′′,...} oF ValuatIons dEfINed In accordancE witH DefinitiOn 2.4. The lOGic of paRAdox LP cAn be thOugH of As thE ReStRicTiON of lp⊛ tO L2, aND thE strong KLeEnE logiC K3 caN BE THougHt aS the RestrIction of K⊛3 to L2. THis ImplIEs tHat DLp⊛ = DLP aNd DK⊛3 = dK3. relatIons beTween klEene Logics and loGics lP⊛ and K⊛3. In Lp, no FoRmuLa Φ can eXPress tHat A foRmula ψ iS consisTEnt, AnD, IN k3, nO formula φ can expresS tHAT a Formula ψ Is undeTErMiNEd : Fact 2.5 FoR eVerY φ,ψ ∈ L2: U(Φ) ∈ dLP /⇒ u(ψ) ∈ {0,1} U(φ) ∈ DK3 /⇒ U(Ψ) = 12 BY contrasT, in LP⊛, wE HaVe That ν(⊛φ) ∈ DlP⊛ ⇒ Ν(φ) ∈ {0,1}, and iN K3, We hAve That ν(¬ ⊛ Φ) ∈ dK⊛3 ⇒ ν(φ) = 1 2, By DefiNition 2.2. ThUs, LP⊛ aND K⊛3 brings a real iNCrease in the exPReSSIvE PoweR of lP and K3. Fact 2.6 bElow CLariFies WHaT thIS 6 NormAlity OpERaTOrs and Classical RecaPtUre in EXtensIons of Kleene LOgics increASE Amounts tO, exaCTlY. in particular, LP⊛ And K�
�φ stating that φ is deter mined. Thi s isrel eva nt , si ncefrom Fact 2.6a nd p aracompleteness of K3imply  t h at K ⊛3 isitselfp ar a c omp le te  (s ee be low).  2. 1 Kleen e Logics W e o bt ain the lang u ag e L2 of Kl een e logics byres tricti ng  L1 to th ose  form ulas w h ere op erator ⊛do e s noto ccur. T h e  v alua tion function for in t erpreting L2 i n a Kl ee n el o gic  ob tains acco rd ingly :  Defini t io n 2 .4( Valuations fo r Kleene lo g ics ) A va lu ati o n u ∶L2 z→  { 0 , 1 2,1} is the  uni que exten sion o f  ν ∶ Pz → {0, 1  2,1}tha t i s in d uc ed  by  t h e r e le van t  co mponents  o fTable  1.W e d enot e b y Uthe s et {u,u′, u′′ ,.. .} o f  va luati ons d efin ed  in a ccorda nce w it h Definition 2. 4. T he logicofpa rad ox  LP c a n be t hou ghof as t he rest r ict io n o fLP⊛ to L2, and the  s t r on g Kleene  logic K3  c a n be tho ug htas t h e  rest rict i on  of K⊛3to L2. Th is  implie sthat D LP ⊛ =  DL P and DK⊛3  = DK3 . Relati ons b e tween Kleene L o gics and logi c sL P ⊛a nd K ⊛3.  In LP, noform u la φ  can ex pre s s tha t a f or m ul a  ψ is consistent, a nd , in K 3, no  formula φ ca n expresst h a t a form ulaψ  i s  undetermined: Fac t 2.5 Fore very φ,ψ  ∈ L2 : u(φ) ∈  DLP /⇒ u ( ψ ) ∈ {0,1 } u (φ)  ∈DK3 / ⇒u(ψ) = 12 Byc o ntra st , in LP ⊛,we have  th atν(⊛ φ)∈DLP⊛ ⇒ ν( φ) ∈ {0, 1} ,an dinK3, w e  have th at  ν( ¬⊛ φ ) ∈ D K ⊛3 ⇒ ν (φ) =  1 2 ,by Def inition 2. 2 .  Thu s,  L P⊛ a ndK⊛ 3 bri ngsa  re al incr ease in t hee xpre ss iv e power  of LP and K3 .Fact 2.6 b el owclarif i e s what t his 6 Normality operato r s and C las sical  Rec apture in  Ex tensio nso f Klee ne Log ics i nc rea s e  amou n t sto,  e xactly. In p art icula r,  LP⊛  and K� 
�φ stating_that φ_is determined. This is_relevant, since_from_Fact 2.6_and_paracompleteness of K3_imply that K_⊛ 3 is itself_paracomplete (see below)._2.1_Kleene Logics We obtain the language L2 of Kleene logics by restricting L1 to_those_formulas where_operator_⊛_does not occur. The valuation_function for interpreting L2 in_a Kleene_logic obtains accordingly: Definition 2.4 (Valuations for Kleene_logics)_A valuation u_∶ L2 z→ {0, 12,1} is the unique extension_of ν ∶ P z→ {0,_1 2,1} that_is_induced_by the relevant components_of Table 1. We denote by_U the set {u,u′, u′′,...} of_valuations defined in accordance with Definition 2.4._The logic of paradox LP can_be though of as the_restriction of_LP⊛ to L2, and the_strong Kleene logic_K3 can_be thought as_the restriction of K⊛3 to L2._This implies that_DLP⊛ = DLP and DK⊛3 =_DK3._Relations between Kleene_Logics_and_logics LP⊛_and K⊛3. In_LP,_no formula_φ_can express that a formula ψ_is_consistent, and, in K3, no formula φ_can express that a_formula_ψ is undetermined :_Fact 2.5 For every φ,ψ_∈ L2: u(φ) ∈ DLP /⇒_u(ψ) ∈_{0,1} u(φ)_∈ DK3 /⇒ u(ψ) = 12 By contrast, in LP⊛, we_have that ν(⊛φ) ∈ DLP⊛ ⇒_ν(φ) ∈ {0,1}, and_in K3,_we_have that ν(¬_⊛_φ) ∈_DK⊛3 ⇒ ν(φ) = 1 2, by_Definition 2.2._Thus, LP⊛ and K⊛3 brings a_real increase in the_expressive_power of LP and K3. Fact_2.6 below clarifies what this 6_Normality operators and Classical Recapture_in_Extensions_of Kleene Logics increase amounts_to, exactly. In particular, LP⊛ and_K�