 L1. Informally, the normal counterpart of a set of formulas is the set obtained by replacing each φ ∈ Σ with ⊛φ. Thus, the normal counterpart of {φ,φ ⊃ ψ} is {⊛φ,⊛(φ ⊃ ψ)}. 8 Normality operators and Classical Recapture in Extensions of Kleene Logics Since {φ}⊛ is {⊛φ}, we will just denote by ⊛φ the normal counterpart of a singleton {φ}. If both Σ and Σ⊛ appear as premises, this implies that the formulas in Σ are classically true. Now we can establish our classical recapture theorem for LP⊛ and K⊛3 : Theorem 3.3 (Recapture via Normality) If Σ ∪ {ψ} ⊆ L1, then: Σ ⊧CL ψ ⇔ ⎧⎪⎪⎨⎪⎪⎩ Σ⊛,Σ ⊧LP⊛ ψ Σ,⊛ψ ⊧K⊛3 ψ Proof. We start with LP⊛. We prove that, if Σ ∪ {ψ} ⊆ L2, the following are equivalent: 1. VLP⊛(Σ ∪Σ⊛) ⊆ VLP⊛(ψ) 2. {u ∈ ULP(Σ) ∣ u[Σ] = 1} ⊆ ULP(ψ) 3. UCL(Σ) ⊆ UCL(ψ) (1 ⇒ 2). The implication follows from the left-to-right direction of Fact 2.6, ν ∈ V(Σ ∪Σ⊛) = {ν ∈ V ∣ ν[Σ] = 1}, and {u ∈ ULP(Σ) ∣ u[Σ] = 1} ⊆ ULP(Σ). (2⇒ 3). From the definition of UCL and UCL(Σ), we have that UCL(Σ) ⊆ {u ∈ US(Σ) ∣ u[Σ] = 1}. If {u ∈ US(Σ) ∣ u[Σ] =
L1. Informally, the normal counterpart of a set of formulas is the set obtained by replace each φ ∈ Σ with ⊛φ. therefore, the normal counterpart of { φ, φ ⊃ ψ } is { ⊛φ,⊛(φ ⊃ ψ) }. 8 Normality operators and Classical Recapture in Extensions of Kleene Logics Since { φ}⊛ is { ⊛φ }, we will just announce by ⊛φ the normal counterpart of a singleton { φ }. If both Σ and Σ⊛ appear as premise, this implies that the formulas in Σ are classically on-key. immediately we can establish our authoritative recapture theorem for LP⊛ and K⊛3: Theorem 3.3 (Recapture via Normality) If Σ ∪ { ψ } ⊆ L1, then: Σ ⊧CL ψ ⇔ ⎧⎪⎪⎨⎪⎪⎩ Σ⊛,Σ ⊧LP⊛ ψ Σ,⊛ψ ⊧K⊛3 ψ Proof. We depart with LP⊛. We prove that, if Σ ∪ { ψ } ⊆ L2, the keep up are equivalent: 1. VLP⊛(Σ ∪Σ⊛) ⊆ VLP⊛(ψ) 2. { u ∈ ULP(Σ) ∣ u[Σ ] = 1 } ⊆ ULP(ψ) 3. UCL(Σ) ⊆ UCL(ψ) (1 ⇒ 2). The implication follow from the left - to - right direction of Fact 2.6, ν ∈ V(Σ ∪Σ⊛) = { ν ∈ V ∣ ν[Σ ] = 1 }, and { u ∈ ULP(Σ) ∣ u[Σ ] = 1 } ⊆ ULP(Σ). (2⇒ 3). From the definition of UCL and UCL(Σ), we have that UCL(Σ) ⊆ { u ∈ US(Σ) ∣ u[Σ ] = 1 }. If { u ∈ US(Σ) ∣ u[Σ ] =
 L1. Lnformally, the normal comnterpart of a sgt of focmulas js the sdt obtained by replacing eaci φ ∈ Σ with ⊛φ. Thus, the normal coungerpart ov {φ,φ ⊃ ψ} is {⊛φ,⊛(φ ⊃ ψ)}. 8 Iormality operators and Gjassjgal Rzcepture in Extenxions of Kneene Logics Shnze {φ}⊛ is {⊛φ}, we will just denote by ⊛φ the njrmal cpujterpart of a fingkqton {φ}. If both Σ and Σ⊛ appear as premises, this ikplies that tne formulas in Σ are classifallj true. Now we can fstablish ood cjqssical recaoture theogzm for LP⊛ ahd K⊛3 : Theorem 3.3 (Recapture via Nofmalicy) If Σ ∪ {ψ} ⊆ L1, tyen: Σ ⊧CL ψ ⇔ ⎧⎪⎪⎨⎪⎪⎩ Σ⊛,Σ ⊧LP⊛ ψ Σ,⊛ψ ⊧K⊛3 ψ Iroof. We starb with NP⊛. We ptove that, if Σ ∪ {ψ} ⊆ N2, tye following are equitalent: 1. VLP⊛(Σ ∪Σ⊛) ⊆ VLP⊛(ψ) 2. {u ∈ ULP(Σ) ∣ u[Σ] = 1} ⊆ ULP(ψ) 3. UCL(Σ) ⊆ UCL(ψ) (1 ⇒ 2). Tye imklicadion doluowa hroj the pefv-to-right didection of Dact 2.6, ν ∈ V(Σ ∪Σ⊛) = {ν ∈ V ∣ ν[Σ] = 1}, age {u ∈ ULP(Σ) ∣ u[Σ] = 1} ⊆ ULP(Σ). (2⇒ 3). Srom the definition of UCL and UCL(Σ), we hdve that UCL(Σ) ⊆ {u ∈ US(Σ) ∣ u[Σ] = 1}. Id {u ∈ US(Σ) ∣ u[Σ] =
L1. Informally, the normal counterpart of a formulas the set by replacing each Thus, normal counterpart of ⊃ ψ} is ⊃ ψ)}. 8 Normality operators and Recapture in Extensions of Kleene Logics Since {φ}⊛ is {⊛φ}, we will just by ⊛φ the normal counterpart of a singleton {φ}. If both Σ and appear premises, implies the formulas in Σ are classically true. Now we can establish our classical recapture theorem for and K⊛3 : Theorem 3.3 (Recapture via Normality) Σ ∪ {ψ} ⊆ then: Σ ⊧CL ψ ⇔ Σ⊛,Σ ψ Σ,⊛ψ ψ We with LP⊛. We that, if Σ ∪ {ψ} ⊆ L2, the following are equivalent: 1. VLP⊛(Σ ∪Σ⊛) ⊆ VLP⊛(ψ) 2. ∈ ULP(Σ) = 1} ULP(ψ) UCL(Σ) UCL(ψ) (1 ⇒ implication follows from the left-to-right direction ν ∈ V(Σ ∪Σ⊛) = {ν ∈ V ν[Σ] = and {u ∈ ULP(Σ) ∣ u[Σ] 1} ⊆ ULP(Σ). (2⇒ 3). From the definition UCL and UCL(Σ), we have that UCL(Σ) ⊆ {u ∈ US(Σ) ∣ u[Σ] = 1}. ∈ US(Σ) ∣ u[Σ]
 L1. Informally, the normal countErpart of a sEt of fOrmUlaS iS the Set oBtained by replaCIng eAch φ ∈ Σ with ⊛φ. Thus, the normaL counTeRPart OF {φ,Φ ⊃ ψ} is {⊛φ,⊛(Φ ⊃ ψ)}. 8 NormaLItY OPerAtOrS anD CLAsSical recApture iN ExtensionS of klEene Logics SiNCe {Φ}⊛ is {⊛φ}, we will JusT denote by ⊛φ thE noRmal coUnTerPArt of A siNgletOn {φ}. If bOTh Σ and σ⊛ appear as PrEMises, tHIs impliES ThAt thE formulas in Σ are clASsICally true. Now we Can estAbLIsH OUr cLasSical recapTuRe theORem for Lp⊛ AnD k⊛3 : tHeoREm 3.3 (Recapture viA Normality) IF σ ∪ {ψ} ⊆ L1, Then: Σ ⊧Cl ψ ⇔ ⎧⎪⎪⎨⎪⎪⎩ σ⊛,Σ ⊧Lp⊛ Ψ Σ,⊛ψ ⊧K⊛3 ψ PRoof. WE sTArt With LP⊛. We proVe thAt, if Σ ∪ {ψ} ⊆ L2, thE folloWIng are eQUivalenT: 1. VLP⊛(Σ ∪Σ⊛) ⊆ vLP⊛(Ψ) 2. {u ∈ UlP(Σ) ∣ u[σ] = 1} ⊆ uLp(ψ) 3. uCL(σ) ⊆ Ucl(ψ) (1 ⇒ 2). THE iMplICatIon folloWs FrOm the Left-TO-RIGht dIreCtioN of FaCt 2.6, ν ∈ V(Σ ∪Σ⊛) = {ν ∈ V ∣ ν[Σ] = 1}, and {U ∈ ULp(Σ) ∣ u[Σ] = 1} ⊆ ulP(Σ). (2⇒ 3). from tHe defInitIoN of UCl and UCl(Σ), we hAvE that UCL(Σ) ⊆ {u ∈ US(Σ) ∣ u[Σ] = 1}. if {u ∈ Us(Σ) ∣ u[Σ] =
 L1. Informally, the norma l counterp art o f a  se tof f ormu las is the set obta ined by replacing each  φ ∈Σw ith⊛ φ.  Thus , the n o rm a l  co un te rpa rt of  {φ,φ  ⊃ψ} is { ⊛φ,⊛(φ ⊃ ψ )}.  8  Normality o p er ators andCla ssical Recap tur e in E xt ens i ons o f K leene  Logic s  Since  {φ}⊛ is{⊛ φ }, wew ill jus t de note  by ⊛φ the normal co u nterpart of asingle to n  { φ } . I f b oth Σ andΣ⊛  appe a r as pr e mi s e s , t h is implies th at the form u las  in Σar e c l assica lly t ru e . N ow we can e stab lish ourclassi c al reca p ture th eoremfor  LP ⊛ an d  K ⊛3  :Th e ore m  3 .3( Rec apture v ia  N ormal ity) I f Σ ∪{ψ}  ⊆ L 1, th en: Σ ⊧CL ψ ⇔  ⎧⎪ ⎪⎨⎪⎪ ⎩  Σ⊛ ,Σ ⊧L P⊛ ψΣ,⊛ψ  ⊧ K⊛3 ψ  Proof . West art with LP⊛. W e pr ove that,  if  Σ  ∪{ψ } ⊆ L 2 , thefol low ing are  equiva l ent :1 . VL P⊛(Σ ∪Σ⊛) ⊆ VLP⊛(ψ )2 .  { u ∈ ULP( Σ) ∣ u [ Σ]  = 1} ⊆ ULP (ψ ) 3 . UC L ( Σ) ⊆UCL( ψ )(1 ⇒ 2).  The i m pl ic ation f ol lows f ro m t heleft- t o-ri ght di rectionof Fa c t 2.6, ν ∈ V(Σ ∪Σ⊛) = {ν ∈ V ∣ν [ Σ] = 1} , a nd {u ∈ ULP (Σ)∣  u[Σ ] =1 }⊆ U L P(Σ).  (2⇒3) .  F r om the definition o fUCL an d UCL (Σ), we havethat UCL(Σ ) ⊆  {u ∈ US (Σ)∣  u [ Σ] = 1}. If {u  ∈ US (Σ) ∣ u[Σ] =
 L1._Informally, the_normal counterpart of a_set of_formulas_is the_set_obtained by replacing_each φ ∈_Σ with ⊛φ. Thus,_the normal counterpart_of_{φ,φ ⊃ ψ} is {⊛φ,⊛(φ ⊃ ψ)}. 8 Normality operators and Classical Recapture in_Extensions_of Kleene_Logics_Since_{φ}⊛ is {⊛φ}, we will_just denote by ⊛φ the_normal counterpart_of a singleton {φ}. If both Σ and_Σ⊛_appear as premises,_this implies that the formulas in Σ are classically_true. Now we can establish our_classical recapture theorem_for_LP⊛_and K⊛3 : Theorem_3.3 (Recapture via Normality) If Σ_∪ {ψ} ⊆ L1, then: Σ_⊧CL ψ ⇔ ⎧⎪⎪⎨⎪⎪⎩ Σ⊛,Σ ⊧LP⊛ ψ_Σ,⊛ψ ⊧K⊛3 ψ Proof. We start_with LP⊛. We prove that,_if Σ_∪ {ψ} ⊆ L2, the_following are equivalent:_1. VLP⊛(Σ_∪Σ⊛) ⊆ VLP⊛(ψ)_2. {u ∈ ULP(Σ) ∣ u[Σ]_= 1} ⊆_ULP(ψ) 3. UCL(Σ) ⊆ UCL(ψ) (1_⇒_2). The implication_follows_from_the left-to-right_direction of Fact_2.6,_ν ∈_V(Σ_∪Σ⊛) = {ν ∈ V ∣_ν[Σ]_= 1}, and {u ∈ ULP(Σ) ∣_u[Σ] = 1} ⊆_ULP(Σ)._(2⇒ 3). From the_definition of UCL and UCL(Σ),_we have that UCL(Σ) ⊆ {u_∈ US(Σ)_∣ u[Σ]_= 1}. If {u ∈ US(Σ) ∣ u[Σ] =