 then u′(φ) ∈ DLP for every u′ ∈ U such that u′(p) = 1 2 for some (every) p ∈ {p1,..., pk} and u′(q) = u(q) for every q ∉ {p1,..., pk}. Proof. By a straightforward induction on the complexity of φ. Let Xu,φ be the set {u′ ∈ U ∣ u[p1,..., pk] = 12 for some {p1,..., pk} ⊆ var(φ) and u′(q) = u(q) for every q ∉ {p1,..., pk}} of the valuations in U that coincide with u except for some variables in φ. A corollary of Fact 4.7 above is: 12 Normality operators and Classical Recapture in Extensions of Kleene Logics Corollary 4.8 For every u ∈ U and u′ ∈Xu,φ, u(φ) ∈ DLP ⇒ u′(φ) ∈ DLP Fact 4.7 and Corollary 4.8 help us prove two interesting facts about minimal inconsistency. First, addition of information that is irrelevant w.r.t. the initial premises does not force us to drop previously held consequences: Fact 4.9 var(Σ) ∩ var(Σ′) = ∅ ⇒ (Σ ⊧LPmi ψ ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Assume var(Σ) ∩ var(Σ′) = ∅ and Σ ⊧LPmi ψ as the initial hypothesis. We distinguish two cases. Case 1): UCL(Σ′) ≠ ∅. ULPmi(Σ∪Σ′) ⊆ ULPmi(Σ). Since Σ ⊧LPmi ψ, this implies ULPmi(Σ∪Σ′) ⊆ ULP(ψ), which in turn equates with Σ,Σ′ ⊧LPmi ψ. Case 2): UCL(Σ′) = ∅. We distinguish two subcases. Case 2.1): var(Σ′) ∩ var(ψ) = ∅. For every u ∈
then u′(φ) ∈ DLP for every u′ ∈ U such that u′(p) = 1 2 for some (every) p ∈ { p1, ..., pk } and u′(q) = u(q) for every q ∉ { p1, ..., pk }. Proof. By a straightforward induction on the complexity of φ. lease Xu, φ be the hardening { u′ ∈ U ∣ u[p1, ..., pk ] = 12 for some { p1, ..., pk } ⊆ var(φ) and u′(q) = u(q) for every q ∉ { p1, ..., pk } } of the valuations in U that coincide with u except for some variable in φ. A corollary of Fact 4.7 above is: 12 Normality operators and Classical Recapture in Extensions of Kleene Logics Corollary 4.8 For every u ∈ U and u′ ∈Xu, φ, u(φ) ∈ DLP ⇒ u′(φ) ∈ DLP Fact 4.7 and Corollary 4.8 avail us raise two interesting facts about minimal incompatibility. foremost, addition of information that is irrelevant w.r.t. the initial premise does not force us to drop previously hold consequences: Fact 4.9 var(Σ) ∩ var(Σ′) = ∅ ⇒ (Σ ⊧LPmi ψ ⇒ Σ, Σ′ ⊧LPmi ψ) Proof. Assume var(Σ) ∩ var(Σ′) = ∅ and Σ ⊧LPmi ψ as the initial hypothesis. We distinguish two font. Case 1 ): UCL(Σ′) ≠ ∅. ULPmi(Σ∪Σ′) ⊆ ULPmi(Σ). Since Σ ⊧LPmi ψ, this implies ULPmi(Σ∪Σ′) ⊆ ULP(ψ), which in go equates with Σ, Σ′ ⊧LPmi ψ. Case 2 ): UCL(Σ′) = ∅. We distinguish two subcases. Case 2.1 ): var(Σ′) ∩ var(ψ) = ∅. For every uracil ∈
 thfn u′(φ) ∈ DLP for every u′ ∈ U such that u′(p) = 1 2 for smme (evsry) p ∈ {p1,..., pk} and u′(q) = u(q) for every q ∉ {'1,..., pk}. Prood. By a straightforward inductioj on the comklexity of φ. Let Xn,φ be the set {u′ ∈ U ∣ b[p1,..., pk] = 12 for some {p1,..., pk} ⊆ var(φ) and u′(q) = u(q) fos dvzry q ∉ {p1,..., pk}} of the valuations in U trat coimclde with u excgpt fpw sojv yariables in φ. A corollary of Facf 4.7 abovt is: 12 Normality oprrators and Classical Recaoturf in Extensions of Kleene Logucs Sirollary 4.8 Fof every u ∈ U and u′ ∈Xu,φ, o(φ) ∈ DLP ⇒ u′(φ) ∈ DLP Fact 4.7 and Corollafy 4.8 hzlp us provg fwl interestinj factf about minimal incotsistenvy. First, additlon oh indormation that is irrxlevant w.r.t. the inityal premivea does not force ys to dtop pseviuyslh htld cknsequfncxs: Fact 4.9 vad(Σ) ∩ var(Σ′) = ∅ ⇒ (Σ ⊧OPmi ψ ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Afwume var(Σ) ∩ var(Σ′) = ∅ and Σ ⊧JPmi ψ as the initial hypothesis. We distpngujsh two cases. Case 1): UCL(Σ′) ≠ ∅. ULPmi(Σ∪Σ′) ⊆ ULPmi(Σ). Since Σ ⊧LPmi ψ, tris implies ULPmi(Σ∪Σ′) ⊆ ULP(ψ), which in turn equates with Σ,Σ′ ⊧LPmm ψ. Cawe 2): YCP(Σ′) = ∅. We distinguish two subcases. Case 2.1): var(Σ′) ∩ vaw(ψ) = ∅. Fpr every u ∈
then u′(φ) ∈ DLP for every u′ such u′(p) = 2 for some and = u(q) for q ∉ {p1,..., Proof. By a straightforward induction on complexity of φ. Let Xu,φ be the set {u′ ∈ U ∣ u[p1,..., = 12 for some {p1,..., pk} ⊆ var(φ) and u′(q) = u(q) for q {p1,..., of valuations in U that coincide with u except for some variables in φ. A corollary of 4.7 above is: 12 Normality operators and Classical in Extensions of Kleene Corollary 4.8 For every u U u′ ∈Xu,φ, ∈ ⇒ ∈ DLP Fact and Corollary 4.8 help us prove two interesting facts about minimal inconsistency. First, addition of information that irrelevant w.r.t. premises does force to previously held consequences: var(Σ) ∩ var(Σ′) = ∅ ⇒ ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Assume var(Σ) ∩ = ∅ Σ ⊧LPmi ψ as the initial We distinguish two cases. Case 1): UCL(Σ′) ≠ ULPmi(Σ∪Σ′) ⊆ ULPmi(Σ). Since Σ ⊧LPmi ψ, this implies ULPmi(Σ∪Σ′) ⊆ ULP(ψ), which in turn Σ,Σ′ ⊧LPmi ψ. Case UCL(Σ′) = ∅. distinguish subcases. 2.1): ∩ var(ψ) ∅. For every u ∈
 then u′(φ) ∈ DLP for every u′ ∈ U such thAt u′(p) = 1 2 for somE (everY) p ∈ {p1,..., Pk} aNd U′(q) = u(q) For eVery q ∉ {p1,..., pk}. Proof. BY A strAightforward induction oN the cOmPLexiTY oF φ. Let xu,φ be thE SeT {U′ ∈ u ∣ u[p1,..., Pk] = 12 FoR soMe {P1,..., Pk} ⊆ Var(φ) aNd u′(Q) = u(q) for eVery q ∉ {p1,..., pk}} of The VaLuations in U tHAt Coincide wiTh u Except for somE vaRiableS iN φ. A COrollAry Of FacT 4.7 above IS: 12 NormaLity operaToRS and ClASsical RECApTure In Extensions of KleENe lOgics Corollary 4.8 for eveRy U ∈ u aND U′ ∈Xu,Φ, u(φ) ∈ dLP ⇒ u′(φ) ∈ DLP FaCt 4.7 And CoROllary 4.8 hELp US PRovE Two interestinG facts about MIniMal incOnSisTEncy. FiRst, adDiTIon Of informatiOn thAt is irrelEvant w.R.T. the iniTIal premIses doEs nOt fOrce US tO dRop PrEVioUSlY heLD coNsequencEs: faCt 4.9 var(σ) ∩ var(σ′) = ∅ ⇒ (σ ⊧lpMi ψ ⇒ Σ,σ′ ⊧LPMi ψ) PRoof. ASsume var(Σ) ∩ var(Σ′) = ∅ And σ ⊧LPmI Ψ as The inItial HypoThEsis. WE distiNguisH tWo cases. Case 1): UCL(Σ′) ≠ ∅. uLPmI(Σ∪Σ′) ⊆ ULPmi(Σ). sinCe σ ⊧LPMi Ψ, this IMplies uLPMi(Σ∪σ′) ⊆ ULP(ψ), whIch in tuRN eqUaTES WiTh Σ,Σ′ ⊧LPmi ψ. Case 2): UCL(Σ′) = ∅. WE dISTiNguish twO subcaSEs. caSE 2.1): var(Σ′) ∩ var(Ψ) = ∅. FOr eVery U ∈
 then u′(φ) ∈ DLP for ever y u′ ∈ U s uch t hat  u′ (p ) =1 2for some (ever y ) p∈ {p1,..., pk} and u′( q) =u( q ) fo r  e veryq ∉ {p1 , .. . ,  pk }.  P roo f. By  a st rai ghtforw ard induct ion  o n the comple x it y of φ. Le t X u,φ be the s et{u′ ∈U∣ u [ p1,.. .,pk] =  12 fo r  some{p1,...,pk }  ⊆ var ( φ) andu ′ (q ) =u(q) for every q∉  { p 1,..., pk}} of  the v al u at i o nsinU that coi nc ide w i th u ex c ep t f ors ome variables  in φ. A co r oll ary of  F act 4.7 ab ove i s: 12Normality o pera tors andClassi c al Reca p ture in  Exten sio nsof K l ee ne  Lo gi c s C o ro lla r y 4 .8 For e ve ry  u ∈U an d u ′  ∈Xu ,φ,  u(φ ) ∈ D LP ⇒ u′(φ) ∈DLP  Fac t  4. 7 and  Coro llar y4.8 h elp us  prov etwo interesting  fac ts aboutmin im alin consi s tency.  Fi rst , addit ion ofi nfo rm a t i on  that is irrelevan tw . r. t. the i nitial pr em i ses does  n otforc e us to  dro p  p reviousl y held co ns equence s:  Fact4. 9 v ar( Σ) ∩v ar(Σ ′) = ∅  ⇒ (Σ ⊧L Pmi ψ ⇒ Σ,Σ′ ⊧LPmi ψ )  Proof. Assum e  v a r (Σ )  ∩ v ar( Σ′) = ∅ and  Σ ⊧ L Pmiψ as th e i n itial  hypo th e si s . We distinguish tw ocases.  Case  1): UCL(Σ′)≠ ∅. ULPmi ( Σ ∪ Σ′) ⊆ UL Pmi( Σ ). Since Σ ⊧LPmiψ, th is implies ULPmi(Σ∪ Σ′) ⊆  ULP(ψ),  which in t urn equa tes  wi thΣ,Σ ′ ⊧L Pmi ψ. Case 2 ) :  UCL (Σ ′) = ∅.  We  distin gui shtwo  su bc ases. Cas e 2.1):va r( Σ′ )∩ v ar(ψ) = ∅. For  e ver yu ∈ 
 then_u′(φ) ∈_DLP for every u′_∈ U_such_that u′(p)_=_1 2 for_some (every) p_∈ {p1,..., pk} and_u′(q) = u(q)_for_every q ∉ {p1,..., pk}. Proof. By a straightforward induction on the complexity of_φ._Let Xu,φ_be_the_set {u′ ∈ U ∣_u[p1,..., pk] = 12 for_some {p1,...,_pk} ⊆ var(φ) and u′(q) = u(q) for_every_q ∉ {p1,...,_pk}} of the valuations in U that coincide with_u except for some variables in_φ. A corollary_of_Fact_4.7 above is: 12_Normality operators and Classical Recapture in_Extensions of Kleene Logics Corollary 4.8_For every u ∈ U and u′_∈Xu,φ, u(φ) ∈ DLP ⇒ u′(φ)_∈ DLP Fact 4.7 and_Corollary 4.8_help us prove two interesting_facts about minimal_inconsistency. First,_addition of information_that is irrelevant w.r.t. the initial_premises does not_force us to drop previously held_consequences:_Fact 4.9 var(Σ)_∩_var(Σ′)_= ∅_⇒ (Σ ⊧LPmi_ψ_⇒ Σ,Σ′_⊧LPmi_ψ) Proof. Assume var(Σ) ∩ var(Σ′)_=_∅ and Σ ⊧LPmi ψ as the_initial hypothesis. We distinguish_two_cases. Case 1): UCL(Σ′)_≠ ∅. ULPmi(Σ∪Σ′) ⊆ ULPmi(Σ)._Since Σ ⊧LPmi ψ, this implies_ULPmi(Σ∪Σ′) ⊆_ULP(ψ), which_in turn equates with Σ,Σ′ ⊧LPmi ψ. Case 2): UCL(Σ′) =_∅. We distinguish two subcases. Case_2.1): var(Σ′) ∩ var(ψ)_= ∅._For_every u ∈