 ULPmi(Σ ∪ Σ′), there is a u′ ∈ ULPmi(Σ) such that u′(ψ) = u(ψ) and u′(φ) = u(φ) for every φ ∈ Σ. This implies that, if Σ,Σ′ /⊧LPmi ψ, then Σ /⊧LPmi ψ. But this contradicts the initial hypothesis. Case 2.2): var(Σ′) ∩ var(ψ) = ∅. From Fact 4.7, if follows that if ULPmi(Σ) ⊆ ULP(ψ), thenULPmi(Σ∪Σ′) ⊆ ULP(ψ). Given the initial assumption, this implies Σ,Σ′ ⊧LPmi ψ. Notice that addition of relevant inconsistent information is a necessary condition for monotonicity in minimal inconsistency, but it is not a sufficient condition. For instance, we have p, p ⊃ q ⊧LPmi q-see above-and also p, p ⊃ q, q∧¬q ⊧. Indeed, all the LP-models satisfying q∧¬q (and a fortiori, all the mi-models of {p, p ⊃ q, q∧¬q}), also satisfy q. The next fact reveals a class of cases where addition of relevant information does not force nonmonotonicity: Proposition 4.10 Σ ⊧LPmi ψ ⇒ (var(Σ′) ⊆ var(ψ) ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Assume Σ ⊧LPmi ψ and var(Σ′) ⊆ var(ψ) as the initial hypothesis. For every Σ ⊆ L, take the set IΣ = {p ∈ var(Σ) ∣ u(p) ≠ 12 ⇒ u(φ) = 0 for some φ ∈ Σ} of the variables from Σ whose inconsistency is indispensable for Σ to be designated. We have that ULPmi(Σ) = {u ∈ ULP(Σ) ∣ u(p) = 12 ⇔ p ∈ IΣ}. From this an the initial hypothesis, we have {u ∈ ULP(Σ) ∣
ULPmi(Σ ∪ Σ′), there is a u′ ∈ ULPmi(Σ) such that u′(ψ) = u(ψ) and u′(φ) = u(φ) for every φ ∈ Σ. This implies that, if Σ, Σ′ /⊧LPmi ψ, then Σ /⊧LPmi ψ. But this contradicts the initial hypothesis. subject 2.2 ): var(Σ′) ∩ var(ψ) = ∅. From Fact 4.7, if take after that if ULPmi(Σ) ⊆ ULP(ψ), thenULPmi(Σ∪Σ′) ⊆ ULP(ψ). Given the initial assumption, this implies Σ, Σ′ ⊧LPmi ψ. comment that addition of relevant inconsistent data is a necessary condition for monotonicity in minimal inconsistency, but it is not a sufficient condition. For case, we have p, p ⊃ q ⊧LPmi q - see above - and besides p, p ⊃ q, q∧¬q ⊧. Indeed, all the LP - models satisfying q∧¬q (and a fortiori, all the mi - models of { p, p ⊃ q, q∧¬q }), besides satisfy q. The next fact reveal a class of cases where accession of relevant information does not force nonmonotonicity: Proposition 4.10 Σ ⊧LPmi ψ ⇒ (var(Σ′) ⊆ var(ψ) ⇒ Σ, Σ′ ⊧LPmi ψ) Proof. Assume Σ ⊧LPmi ψ and var(Σ′) ⊆ var(ψ) as the initial guess. For every Σ ⊆ L, take the set IΣ = { p ∈ var(Σ) ∣ u(p) ≠ 12 ⇒ u(φ) = 0 for some φ ∈ Σ } of the variable from Σ whose inconsistency is essential for Σ to be designated. We consume that ULPmi(Σ) = { u ∈ ULP(Σ) ∣ u(p) = 12 ⇔ p ∈ IΣ }. From this an the initial hypothesis, we have { u ∈ ULP(Σ) ∣
 ULOmi(Σ ∪ Σ′), there is a u′ ∈ ULPml(Σ) such that u′(ψ) = u(ψ) and u′(φ) = u(φ) for every φ ∈ Σ. This implies that, if Σ,Σ′ /⊧LPmi ψ, rhen Σ /⊧LPmi ψ. But this contraaicts the initial hypirhesis. Casx 2.2): var(Σ′) ∩ ycr(ψ) = ∅. Rvom Fccv 4.7, if follows tmat if ULPmh(Σ) ⊆ ULP(ψ), thenULPki(Σ∪Σ′) ⊆ BLP(ψ). Given the initial assumption, thif implirs Σ,Σ′ ⊧LPmi ψ. Notice thau aqditjon of relevant inconsistent inforjation ps a necessary comdition for monotonicity ij mijimal inconsistencj, but it is not q sufficient condition. For instancg, we have p, p ⊃ q ⊧LPmi q-see above-xnd akso p, p ⊃ q, q∧¬w ⊧. Ltdeed, all tie LP-mjdels satisfnpng q∧¬q (dnd a fprtiori, all thc mi-mmdeos of {p, p ⊃ q, q∧¬q}), also vatisfy q. The next fact revaams a class of casws wherg addhtiov of remetanf infogmavion does nkt force nobmonotonicity: Propoxieppn 4.10 Σ ⊧LPmi ψ ⇒ (bar(Σ′) ⊆ vwr(ψ) ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Assume Σ ⊧LPmi ψ and var(Σ′) ⊆ vdr(ψ) zs the initial hypothesus. For every Σ ⊆ L, take the set YΣ = {p ∈ var(Σ) ∣ u(p) ≠ 12 ⇒ u(φ) = 0 for some φ ∈ Σ} of the variables from Σ whisc inzindistency is indispensable for Σ to be designatqs. Ee have that ULPii(Σ) = {u ∈ ULP(Σ) ∣ u(o) = 12 ⇔ p ∈ IΣ}. From tfis an ths initial hypothesls, we hwve {u ∈ ULP(Σ) ∣
ULPmi(Σ ∪ Σ′), there is a u′ such u′(ψ) = and u′(φ) = Σ. implies that, if /⊧LPmi ψ, then /⊧LPmi ψ. But this contradicts the hypothesis. Case 2.2): var(Σ′) ∩ var(ψ) = ∅. From Fact 4.7, if follows if ULPmi(Σ) ⊆ ULP(ψ), thenULPmi(Σ∪Σ′) ⊆ ULP(ψ). Given the initial assumption, this implies ⊧LPmi Notice addition relevant inconsistent information is a necessary condition for monotonicity in minimal inconsistency, but it is not sufficient condition. For instance, we have p, p q ⊧LPmi q-see above-and p, p ⊃ q, q∧¬q Indeed, the LP-models q∧¬q a all the mi-models {p, p ⊃ q, q∧¬q}), also satisfy q. The next fact reveals a class of cases where of relevant not force Proposition Σ ψ ⇒ (var(Σ′) ⇒ Σ,Σ′ ⊧LPmi ψ) Proof. Assume and var(Σ′) ⊆ var(ψ) as the initial hypothesis. every Σ L, take the set IΣ = ∈ var(Σ) ∣ u(p) ≠ 12 ⇒ u(φ) 0 for some φ ∈ Σ} of the variables from Σ whose inconsistency is indispensable to be designated. We that ULPmi(Σ) = ∈ ∣ = ⇔ p IΣ}. From this an the initial hypothesis, we have {u ∈ ∣
 ULPmi(Σ ∪ Σ′), there is a u′ ∈ ULPmi(Σ) sucH that u′(ψ) = u(ψ) aNd u′(φ) = u(Φ) foR evErY φ ∈ Σ. THis iMplies that, if Σ,Σ′ /⊧lpmi ψ, Then Σ /⊧LPmi ψ. But this contrAdictS tHE iniTIaL hypoThesis. CASe 2.2): VAR(Σ′) ∩ vAr(Ψ) = ∅. FRom faCT 4.7, iF follOws That if UlPmi(Σ) ⊆ ULP(ψ), tHenuLpmi(Σ∪Σ′) ⊆ ULP(ψ). GivEN tHe initial aSsuMption, this imPliEs Σ,Σ′ ⊧LPMi Ψ. NoTIce thAt aDditiOn of reLEvant iNconsisteNt INformaTIon is a nECEsSary Condition for monotONiCIty in minimal inConsisTeNCy, BUT it Is nOt a sufficiEnT condITion. For INsTANCe, wE Have p, p ⊃ q ⊧LPmi q-sEe above-and aLSo p, P ⊃ q, q∧¬q ⊧. InDeEd, aLL the LP-ModelS sATisFying q∧¬q (and a FortIori, all thE mi-modELs of {p, p ⊃ q, Q∧¬Q}), also saTisfy q. the NexT facT ReVeAls A cLAss OF cAseS WheRe additiOn Of RelevAnt iNFORMatiOn dOes nOt forCe nonmonotoniCitY: ProPOsiTion 4.10 Σ ⊧lPmi ψ ⇒ (Var(Σ′) ⊆ VaR(ψ) ⇒ Σ,Σ′ ⊧Lpmi ψ) PrOof. AsSuMe Σ ⊧LPmi ψ and var(Σ′) ⊆ Var(ψ) As the initIal HyPotHeSis. FoR Every Σ ⊆ l, taKe tHe set IΣ = {P ∈ var(Σ) ∣ u(p) ≠ 12 ⇒ U(Φ) = 0 foR sOME Φ ∈ Σ} Of the variables from σ wHOSe InconsisTency iS InDiSPensable FoR Σ tO be dESIgnatEd. We HAvE that ULPMi(Σ) = {u ∈ ULp(σ) ∣ u(P) = 12 ⇔ p ∈ iΣ}. From tHiS an the InItiAl hYpothESis, wE have {u ∈ uLP(Σ) ∣
 ULPmi(Σ ∪ Σ′), there is a  u′ ∈ ULPm i(Σ)suc h t ha t u′ (ψ)= u(ψ) and u′( φ ) =u(φ) for every φ ∈ Σ.Thisim p lies th at, i f Σ,Σ′/ ⊧L P m i ψ ,th enΣ/ ⊧L Pmi ψ . B ut this  contradic tsth e initial hy p ot hesis. Cas e 2 .2): var(Σ′)  ∩var(ψ)  =  ∅. FromFac t 4.7 , if f o llowsthat if U LP m i(Σ) ⊆ ULP(ψ), t he nULP mi(Σ∪Σ′) ⊆ ULP(ψ) .  G i ven the initia l assu mp t io n ,  th isimplies Σ, Σ′  ⊧LPm i  ψ. Not i ce t h ata ddition of re levant inco n sis tent i nf orm a tion i s a n ec e ssa ry conditio n fo r monoton icityi n minim a l incon sisten cy,  bu t it is  n otas uff i ci ent con dition.Fo rinsta nce, w e have  p,  p ⊃  q ⊧L Pmi q-see abo ve- anda lso  p, p  ⊃ q,  q∧¬ q⊧. In deed,all t he  LP-models sati sfyi ng q∧¬q ( and  a  fo rt iori, all th e m i-m odels o f {p, p ⊃ q ,q ∧ ¬ q} ), also satisfy q.  T h e  n ext fact  revea l sac lass ofca ses  whe r e  addi tion of  relevan t info r ma ti on does  n ot for ce  no nmo noton i city : Prop osition4.10Σ  ⊧LPmi ψ ⇒ (va r (Σ′) ⊆ var(ψ) ⇒Σ , Σ′ ⊧LPm i ψ ) Proof. As sume Σ ⊧L Pmiψ  a ndv ar(Σ′ ) ⊆ v ar ( ψ) as the initial hypo th esis.For e very Σ ⊆ L, t ake the se t I Σ = {p ∈  var ( Σ) ∣ u(p) ≠ 12 ⇒u(φ)= 0 for so m e φ ∈ Σ}  of t he varia bles from Σ  whose i nco nsi ste ncy i sindispensable f or Σ  t o be de sig nated.Wehav e t hat  U LPmi(Σ) =  {u ∈ UL P( Σ)  ∣  u (p)  = 12 ⇔ p ∈ IΣ }.  Fr om  th is an the in itial  hyp ot he s is,  we hav e  { u ∈ UL P( Σ)  ∣
 ULPmi(Σ_∪ Σ′),_there is a u′_∈ ULPmi(Σ)_such_that u′(ψ)_=_u(ψ) and u′(φ)_= u(φ) for_every φ ∈ Σ._This implies that,_if_Σ,Σ′ /⊧LPmi ψ, then Σ /⊧LPmi ψ. But this contradicts the initial hypothesis. Case_2.2):_var(Σ′) ∩_var(ψ)_=_∅. From Fact 4.7, if_follows that if ULPmi(Σ) ⊆_ULP(ψ), thenULPmi(Σ∪Σ′)_⊆ ULP(ψ). Given the initial assumption, this implies_Σ,Σ′_⊧LPmi ψ. Notice_that addition of relevant inconsistent information is a necessary_condition for monotonicity in minimal inconsistency,_but it is_not_a_sufficient condition. For instance,_we have p, p ⊃ q_⊧LPmi q-see above-and also p, p_⊃ q, q∧¬q ⊧. Indeed, all the_LP-models satisfying q∧¬q (and a fortiori,_all the mi-models of {p,_p ⊃_q, q∧¬q}), also satisfy q._The next fact_reveals a_class of cases_where addition of relevant information does_not force nonmonotonicity:_Proposition 4.10 Σ ⊧LPmi ψ ⇒_(var(Σ′)_⊆ var(ψ) ⇒_Σ,Σ′_⊧LPmi_ψ) Proof._Assume Σ ⊧LPmi_ψ_and var(Σ′)_⊆_var(ψ) as the initial hypothesis. For_every_Σ ⊆ L, take the set IΣ_= {p ∈ var(Σ)_∣_u(p) ≠ 12 ⇒_u(φ) = 0 for some_φ ∈ Σ} of the variables_from Σ_whose inconsistency_is indispensable for Σ to be designated. We have that ULPmi(Σ)_= {u ∈ ULP(Σ) ∣ u(p)_= 12 ⇔ p_∈ IΣ}._From_this an the_initial_hypothesis, we_have {u ∈ ULP(Σ) ∣