 u(p) = 12 ⇔ p ∈ IΣ} ⊆ ULP(ψ). From this and Corollary 4.8, it follows that {u ∈ ULP(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′} ⊆ ULP(ψ). Indeed, take a u ∈ ULPmi(Σ) such that p ≠ 12 for every p ∈ var(Σ ′)∖ var(IΣ), and take any u′ ∈ U where u′(p) = 1 2 for every p ∈ IΣ∪Σ′ and u′(q) = u(q) for every q ∉ IΣ∪Σ′. Clearly, u′ ∈ ULPmi(Σ∪Σ′). Also, u′ ∈Xu,ψ. From this and Corollary 4.8, u′(ψ) ∈ DLP. Since u′ is arbitrary, we have {u ∈ ULP(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′} ⊆ ULP(ψ) and, as a consequence, Σ,Σ′ ⊧LPmi ψ. Finally, the next result individuates a condition at which mi-consequence coincides with LP-consequence: Fact 4.11 var(Σ) ⊆ var(ψ) ⇒ (Σ ⊧LPmi ψ ⇔ Σ ⊧LP ψ) Proof. Assume var(Σ) ⊆ var(ψ) and Σ ⊧LPmi ψ as the initial hypothesis. (⇐) It follows immediately from Fact 4.4. (⇒) Remind the definition IΣ = {p ∈ var(Σ) ∣ u(p) ≠ 1 2 ⇒ u(φ) = 0 for some φ ∈ Σ} of the set of the variables from Σ whose inconsistency is Normality operators and Classical Recapture in Extensions of Kleene Logics 13 indispensable for Σ to be designated, and take the class KΣ = {u ∈ ULP(Σ) ∣ u(q) = 12 for some q �
u(p) = 12 ⇔ p ∈ IΣ } ⊆ ULP(ψ). From this and Corollary 4.8, it follows that { u ∈ ULP(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′ } ⊆ ULP(ψ). Indeed, take a u ∈ ULPmi(Σ) such that p ≠ 12 for every p ∈ var(Σ ′)∖ var(IΣ), and subscribe any u′ ∈ uracil where u′(p) = 1 2 for every p ∈ IΣ∪Σ′ and u′(q) = u(q) for every q ∉ IΣ∪Σ′. Clearly, u′ ∈ ULPmi(Σ∪Σ′). Also, u′ ∈Xu, ψ. From this and Corollary 4.8, u′(ψ) ∈ DLP. Since u′ is arbitrary, we own { u ∈ ULP(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′ } ⊆ ULP(ψ) and, as a consequence, Σ, Σ′ ⊧LPmi ψ. Finally, the next consequence individuates a condition at which mi - consequence coincide with LP - consequence: Fact 4.11 var(Σ) ⊆ var(ψ) ⇒ (Σ ⊧LPmi ψ ⇔ Σ ⊧LP ψ) Proof. Assume var(Σ) ⊆ var(ψ) and Σ ⊧LPmi ψ as the initial hypothesis. (⇐) It follows immediately from Fact 4.4. (⇒) prompt the definition IΣ = { p ∈ var(Σ) ∣ u(p) ≠ 1 2 ⇒ u(φ) = 0 for some φ ∈ Σ } of the set of the variables from Σ whose inconsistency is Normality operators and Classical Recapture in Extensions of Kleene Logics 13 essential for Σ to be designated, and take the course KΣ = { u ∈ ULP(Σ) ∣ u(q) = 12 for some q �
 u(p) = 12 ⇔ p ∈ IΣ} ⊆ ULP(ψ). From this xnd Corollary 4.8, nr follmws thzt {u ∈ ULO(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′} ⊆ ULP(ψ). Indeed, vake a u ∈ ULPmi(Σ) such that p ≠ 12 fur every i ∈ var(Σ ′)∖ vqr(IΣ), end take any u′ ∈ N where m′(') = 1 2 rlr erecy p ∈ IΣ∪Σ′ and u′(q) = u(q) for evary q ∉ IΣ∪Σ′. Clearny, u′ ∈ ULPmi(Σ∪Σ′). Also, u′ ∈Xu,ψ. From this and Corjllary 4.8, u′(ψ) ∈ DLP. Since u′ ys agbytradj, ce have {u ∈ ULP(Σ ∪ Σ′) ∣ u(p) = 12 ⇔ p ∈ IΣ∪Σ′} ⊆ UMP(ψ) and, es a consequencr, Σ,Σ′ ⊧LPmi ψ. Finally, the next gesupt individuates a fondition aj whyxh mi-consequdnce coincpbes with LP-donsequence: Fact 4.11 var(Σ) ⊆ var(ψ) ⇒ (Σ ⊧LPoi ψ ⇔ Σ ⊧LP ψ) Proof. Awsulg var(Σ) ⊆ var(ψ) aid Σ ⊧LPii ψ as the ikptial hfpothesos. (⇐) It follows imkeduately from Fact 4.4. (⇒) Rekind the definitiog IΣ = {p ∈ vdr(Σ) ∣ u(p) ≠ 1 2 ⇒ u(φ) = 0 for wone φ ∈ Σ} of dhe rwt uf uhe vzriablfs hrom Σ whose inconsistebcy is Normality optraeirs and Classjcal Rqcwpture in Extensions of Kleene Logics 13 pndiapensable for Σ to be dewignated, and take the class KΣ = {u ∈ ULP(Σ) ∣ u(q) = 12 for some q �
u(p) = 12 ⇔ p ∈ IΣ} From and Corollary it follows that ∣ = 12 ⇔ ∈ IΣ∪Σ′} ⊆ Indeed, take a u ∈ ULPmi(Σ) that p ≠ 12 for every p ∈ var(Σ ′)∖ var(IΣ), and take u′ ∈ U where u′(p) = 1 2 for every p ∈ IΣ∪Σ′ u′(q) u(q) every ∉ IΣ∪Σ′. Clearly, u′ ∈ ULPmi(Σ∪Σ′). Also, u′ ∈Xu,ψ. From this and Corollary 4.8, u′(ψ) ∈ Since u′ is arbitrary, we have {u ∈ ∪ Σ′) ∣ u(p) 12 ⇔ p ∈ IΣ∪Σ′} ULP(ψ) as a Σ,Σ′ ψ. the next result a condition at which mi-consequence coincides with LP-consequence: Fact 4.11 var(Σ) ⊆ var(ψ) ⇒ (Σ ⊧LPmi ψ Σ ⊧LP Assume var(Σ) var(ψ) Σ ψ as the (⇐) It follows immediately from Fact the definition IΣ = {p ∈ var(Σ) ∣ ≠ 1 ⇒ u(φ) = 0 for some ∈ Σ} of the set of the variables Σ whose inconsistency is Normality operators and Classical Recapture in Extensions of Kleene Logics 13 Σ to be designated, take the class = ∈ ∣ = 12 some q �
 u(p) = 12 ⇔ p ∈ IΣ} ⊆ ULP(ψ). From this and CorolLary 4.8, it follOws thAt {u ∈ uLP(σ ∪ Σ′) ∣ U(p) = 12 ⇔ p ∈ Iσ∪Σ′} ⊆ ULp(ψ). Indeed, take a u ∈ ulPmi(σ) such that p ≠ 12 for every p ∈ var(σ ′)∖ var(Iσ), aND takE AnY u′ ∈ U whEre u′(p) = 1 2 foR EvERY p ∈ Iσ∪Σ′ AnD u′(q) = U(q) FOr Every Q ∉ IΣ∪σ′. ClearlY, u′ ∈ ULPmi(Σ∪Σ′). ALso, U′ ∈XU,ψ. From this anD coRollary 4.8, u′(ψ) ∈ DlP. SInce u′ is arbitRarY, we havE {u ∈ uLP(σ ∪ σ′) ∣ u(p) = 12 ⇔ p ∈ Iσ∪Σ′} ⊆ UlP(ψ) anD, as a coNSequenCe, Σ,Σ′ ⊧LPmi ψ. fiNAlly, thE Next resULT iNdivIduates a condition AT wHIch mi-consequenCe coinCiDEs WITh Lp-coNsequence: FAcT 4.11 var(Σ) ⊆ VAr(ψ) ⇒ (Σ ⊧LPmI Ψ ⇔ Σ ⊧lp Ψ) proOF. Assume var(Σ) ⊆ vaR(ψ) and Σ ⊧LPmi ψ AS thE initiAl HypOThesis. (⇐) it folLoWS imMediately frOm FaCt 4.4. (⇒) Remind tHe defiNItion IΣ = {P ∈ Var(Σ) ∣ u(p) ≠ 1 2 ⇒ u(Φ) = 0 for soMe φ ∈ σ} of The sET oF tHe vArIAblES fRom σ WhoSe inconsIsTeNcy is normALITY opeRatOrs aNd ClaSsical RecaptuRe iN ExtENsiOns of kleenE LogIcS 13 indiSpensaBle foR Σ To be designated, aNd taKe the clasS KΣ = {U ∈ UlP(Σ) ∣ U(q) = 12 For soME q �
 u(p) = 12 ⇔ p ∈ IΣ} ⊆ ULP (ψ). Fromthisand  Co ro llar y 4. 8, it followst hat{u ∈ ULP(Σ ∪ Σ′) ∣ u(p ) = 1 2⇔  p ∈ IΣ ∪Σ′}⊆ ULP(ψ ) .I n dee d,  t ake  a u∈ ULP mi( Σ) such  that p ≠12fo r every p ∈v ar (Σ ′)∖ var (IΣ ), and takeany  u′ ∈Uwhe r e u′( p)= 1 2  for e v ery p∈ IΣ∪Σ′ a nd u′(q)=  u(q) f o r  e very  q ∉ IΣ∪Σ′. Clear l y, u′ ∈ ULPmi(Σ∪Σ ′). Al so ,  u ′ ∈Xu ,ψ.  From this  a nd Co r ollary4 .8 , u ′(ψ )  ∈ DLP. Since  u′ is arbi t rar y, weha ve{ u ∈ UL P(Σ ∪  Σ ′ ) ∣  u(p) = 12⇔ p∈ IΣ∪Σ′}⊆ ULP( ψ ) and,a s a con sequen ce,  Σ, Σ′ ⊧ L Pm iψ.Fi n all y ,the nex t result  i nd ividu ates a c ondi tio n at  whic h mi-conseque nce  coi n cid es wi th LP -con se quenc e: Fac t 4.1 1var(Σ) ⊆ var(ψ)  ⇒ ( Σ ⊧LPmi ψ  ⇔Σ⊧LP  ψ ) Pro o f. Ass ume  va r(Σ) ⊆var(ψ)a ndΣ⊧ L P mi  ψ as the initialhy p o th esis. (⇐ ) It f o ll ow s  immedia te lyfrom F act 4 .4.( ⇒)  Remindthe de f in it ion IΣ={p ∈ v ar (Σ)  ∣u(p)≠  1 2  ⇒ u(φ ) = 0 fo r som e  φ ∈ Σ} of the set of the va r ia b l es from  Σwhose incon sist e ncyis N o rm ali t y ope rator sa nd Classical Recapture  i n Exte nsion s of Kleene L ogics 13 i n d i spensabl e fo r  Σ to be designat ed, a nd take th e  class K Σ = { u ∈ ULP( Σ) ∣ u(q) =  12 forsom e q  �
 u(p)_= 12_⇔ p ∈ IΣ}_⊆ ULP(ψ)._From_this and_Corollary_4.8, it follows_that {u ∈_ULP(Σ ∪ Σ′) ∣_u(p) = 12_⇔_p ∈ IΣ∪Σ′} ⊆ ULP(ψ). Indeed, take a u ∈ ULPmi(Σ) such that p_≠_12 for_every_p_∈ var(Σ ′)∖ var(IΣ), and_take any u′ ∈ U_where u′(p)_= 1 2 for every p ∈ IΣ∪Σ′_and_u′(q) = u(q)_for every q ∉ IΣ∪Σ′. Clearly, u′ ∈ ULPmi(Σ∪Σ′)._Also, u′ ∈Xu,ψ. From this and_Corollary 4.8, u′(ψ)_∈_DLP._Since u′ is arbitrary,_we have {u ∈ ULP(Σ ∪_Σ′) ∣ u(p) = 12 ⇔_p ∈ IΣ∪Σ′} ⊆ ULP(ψ) and, as_a consequence, Σ,Σ′ ⊧LPmi ψ. Finally,_the next result individuates a_condition at_which mi-consequence coincides with LP-consequence:_Fact 4.11 var(Σ)_⊆ var(ψ)_⇒ (Σ ⊧LPmi_ψ ⇔ Σ ⊧LP ψ) Proof._Assume var(Σ) ⊆_var(ψ) and Σ ⊧LPmi ψ as_the_initial hypothesis. (⇐)_It_follows_immediately from_Fact 4.4. (⇒)_Remind_the definition_IΣ_= {p ∈ var(Σ) ∣ u(p)_≠_1 2 ⇒ u(φ) = 0 for_some φ ∈ Σ}_of_the set of the_variables from Σ whose inconsistency_is Normality operators and Classical Recapture_in Extensions_of Kleene_Logics 13 indispensable for Σ to be designated, and take the_class KΣ = {u ∈ ULP(Σ)_∣ u(q) = 12_for some_q_�