 (≤) is a strict (weak) partial order ; in particular, < relates valuations in U from more to less consistent, to the effect that u < u′ expresses that u is less inconsistent than u′; this is the case when u! ⊂ u′!. Definition 4.2 (Minimally inconsistent model, [24]) For every u ∈ ULP, u is a minimally inconsistent (mi-) model of Σ iff v ∈ ULP(Σ) and u′ < u⇒ u′ ∉ ULP(Σ) (iff any less inconsistent u′ falsifies Σ). In a nutshell, the mi-models of Σ are the valuations u ∈ U satisfying both (1) u ∈ ULP(Σ), and (2) u(p) = 12 ⇒ p ∧ ¬p ∈ sub(Σ). 9 We denote by ULPmi(Σ) the set {u ∈ ULP(Σ) ∣ u′ < u ⇒ u′ ∉ ULP(Σ) for every u′ ∈ ULP} of the mi-models of Σ. Minimally Inconsistent Consequence is defined as preservation of designated values in mi-models [24, p.325]: Definition 4.3 (Minimally Inconsistent Consequence) Σ ⊧LPmi ψ ⇔ ULPmi(Σ) ⊆ ULP(ψ) (ψ is a minimally inconsistent (mi-) consequence of Σ iff all the mi-models of Σ are models of ψ).10 mi-consequence is non-monotonic, since extending Σ with further information may lead to loose previously drawn consequences. As an example [24]: q is a mi-consequence of {p, p ⊃ q}, but it is not an mi-consequence of {p∧¬p}∪{p, p ⊃ q}, since there is a mimodel u of {p, p ⊃ q, p∧¬p} where u(q) = 0. Thus, we have Σ ⊧LPmi ψ /⇒ Σ, φ ⊧LPmi ψ. The following is a fact that we will presuppose in what follows:11 Fact 4.4 (Fact 1, [24]) Σ ⊧LP ψ ⇒ Σ �
(≤) is a strict (weak) partial order; in especial, < relate valuations in U from more to less consistent, to the impression that u < u′ expresses that u is less inconsistent than u′; this is the character when u! ⊂ u′! . Definition 4.2 (Minimally discrepant model, [ 24 ]) For every u ∈ ULP, u is a minimally inconsistent (mi-) model of Σ iff v ∈ ULP(Σ) and u′ < u⇒ u′ ∉ ULP(Σ) (iff any less inconsistent u′ falsifies Σ). In a nutshell, the mile - models of Σ are the valuations uranium ∈ uracil satisfying both (1) u ∈ ULP(Σ), and (2) u(p) = 12 ⇒ p ∧ ¬p ∈ sub(Σ). 9 We denote by ULPmi(Σ) the set { u ∈ ULP(Σ) ∣ u′ < u ⇒ u′ ∉ ULP(Σ) for every u′ ∈ ULP } of the mi - model of Σ. Minimally Inconsistent Consequence is defined as preservation of designated value in mi - models [ 24, p.325 ]: Definition 4.3 (Minimally Inconsistent Consequence) Σ ⊧LPmi ψ ⇔ ULPmi(Σ) ⊆ ULP(ψ) (ψ is a minimally inconsistent (mi-) consequence of Σ iff all the mi - model of Σ are models of ψ).10 mi - consequence is non - monotonic, since extending Σ with further information may lead to easy previously drawn consequences. As an example [ 24 ]: q is a mi - consequence of { p, p ⊃ q }, but it is not an mi - consequence of { p∧¬p}∪{p, p ⊃ q }, since there is a mimodel u of { p, p ⊃ q, p∧¬p } where u(q) = 0. Thus, we have Σ ⊧LPmi ψ /⇒ Σ, φ ⊧LPmi ψ. The following is a fact that we will presuppose in what follows:11 Fact 4.4 (Fact 1, [ 24 ]) Σ ⊧LP ψ ⇒ Σ �
 (≤) id a strict (weak) partial urder ; in partieylar, < celates valuatiuns in U from more to less clnwistebt, to the effect that j < u′ exprvsses thar u ms less inconsisvsnt thak u′; tgls is vhe case when u! ⊂ u′!. Definidion 4.2 (Minimallf kneonsistent model, [24]) For every u ∈ ULP, u is a monlmally inconsiftenu (my-) mosvl of Σ iff v ∈ ULP(Σ) and u′ < u⇒ u′ ∉ UMP(Σ) (iff eny less inconsostent u′ falsifies Σ). In a nktshfll, the mi-models ov Σ are the caluwrions u ∈ U sxtisfying both (1) u ∈ ULP(Σ), and (2) u(p) = 12 ⇒ p ∧ ¬p ∈ sub(Σ). 9 We denotd by BLPmi(Σ) the sgc {u ∈ OLP(Σ) ∣ u′ < u ⇒ u′ ∉ ULP(Σ) for every u′ ∈ ULP} of the mi-kodels of Σ. Minlmallb Inxonsistent Consequencx is defined as presgrvation ox besignated values in ni-nodelv [24, p.325]: Defkbitkon 4.3 (Kihimallj Iiconsistent Consequencw) Σ ⊧LPmi ψ ⇔ ULPmi(Σ) ⊆ ULK(ψ) (ψ yw a minimally inconfiftent (mi-) consequence of Σ iff all the mi-kodsls of Σ are models of ψ).10 ni-consequence is non-mlnotonic, fince extending Σ with further information may leag to muost iveviuyspy drawn consequences. As an example [24]: q is a my-domsvquence of {p, p ⊃ q}, but it is moh sg mi-consequenze of {'∧¬l}∪{p, p ⊃ q}, since there ls a miiodel u of {p, p ⊃ q, l∧¬p} where u(q) = 0. Thus, we have Σ ⊧LPmi ψ /⇒ Σ, φ ⊧LPiu ψ. The following id a fact thct we eill lresuppose in what folluws:11 Ract 4.4 (Fact 1, [24]) Σ ⊧LP ψ ⇒ Σ �
(≤) is a strict (weak) partial order particular, relates valuations U from more effect u < u′ that u is inconsistent than u′; this is the when u! ⊂ u′!. Definition 4.2 (Minimally inconsistent model, [24]) For every u ULP, u is a minimally inconsistent (mi-) model of Σ iff v ∈ and < u′ ULP(Σ) (iff any less inconsistent u′ falsifies Σ). In a nutshell, the mi-models of Σ are valuations u ∈ U satisfying both (1) u ULP(Σ), and (2) u(p) 12 ⇒ p ∧ ¬p sub(Σ). We denote ULPmi(Σ) set ∈ ULP(Σ) ∣ < u ⇒ u′ ∉ ULP(Σ) for every u′ ∈ ULP} of the mi-models of Σ. Minimally Consequence is preservation of values mi-models p.325]: Definition 4.3 Consequence) Σ ⊧LPmi ψ ⇔ ULPmi(Σ) is a minimally inconsistent (mi-) consequence of Σ all the of Σ are models of ψ).10 is non-monotonic, since extending Σ with further information lead to loose previously drawn consequences. As an example [24]: q is a mi-consequence of ⊃ q}, but it not an mi-consequence {p∧¬p}∪{p, ⊃ since is a u of {p, p ⊃ q, p∧¬p} where u(q) = 0. we have Σ ⊧LPmi ψ /⇒ Σ, φ ⊧LPmi ψ. is fact that we presuppose in what follows:11 4.4 1, [24]) Σ ⊧LP Σ
 (≤) is a strict (weak) partial order ; In particulAr, < relAteS vaLuAtioNs in u from more to lesS ConsIstent, to the effect that u < U′ exprEsSEs thAT u Is lesS inconsIStENT thAn U′; tHis Is THe Case wHen U! ⊂ u′!. DefinItion 4.2 (MinimAllY iNconsistent mODeL, [24]) For every u ∈ uLP, U is a minimallY inConsisTeNt (mI-) Model Of Σ Iff v ∈ UlP(Σ) and U′ < U⇒ u′ ∉ ULP(Σ) (Iff any lesS iNConsisTEnt u′ falSIFiEs Σ). IN a nutshell, the mi-moDElS Of Σ are the valuaTions u ∈ u sATiSFYinG boTh (1) u ∈ ULP(Σ), and (2) U(p) = 12 ⇒ P ∧ ¬p ∈ sub(σ). 9 we denotE By ulpMi(Σ) THe set {u ∈ ULP(Σ) ∣ u′ < u ⇒ u′ ∉ uLP(Σ) for everY U′ ∈ ULp} of the Mi-ModELs of Σ. MInimaLlY incOnsistent CoNseqUence is deFined aS PreservATion of dEsignaTed ValUes iN Mi-MoDelS [24, p.325]: dEfiNItIon 4.3 (mIniMally IncOnSiStent consEQUENce) Σ ⊧lPmI ψ ⇔ ULpmi(Σ) ⊆ UlP(ψ) (ψ is a minimaLly IncoNSisTent (mI-) consEqueNcE of Σ iFf all tHe mi-mOdEls of Σ are models Of ψ).10 mI-consequeNce Is Non-MoNotonIC, since ExtEndIng Σ witH furtheR InfOrMATIoN may lead to loose preViOUSlY drawn coNsequeNCeS. AS An examplE [24]: q Is a Mi-coNSEquenCe of {P, P ⊃ q}, But it is nOt an mi-COnSeQuence oF {p∧¬P}∪{p, p ⊃ q}, siNcE thEre Is a miMOdel U of {p, p ⊃ q, P∧¬p} where u(Q) = 0. Thus, WE have Σ ⊧LPmi ψ /⇒ Σ, φ ⊧LpMi ψ. The followiNG iS A FaCT thaT we Will presuppOse iN What FollOWs:11 facT 4.4 (fact 1, [24]) Σ ⊧lP ψ ⇒ Σ �
 (≤) is a strict (weak) pa rtial orde r ; i n p art ic ular , <relates valuat i onsin U from more to less  cons is t ent, to  theeffectt ha t u <  u ′exp re s se s tha t u  is les s inconsis ten tthan u′; thi s  i s the case  wh en u! ⊂ u′!.  De finiti on  4. 2  (Min ima lly i nconsi s tent m odel, [24 ]) For ev e ry u ∈U L P,  u i s a minimally inc o ns i stent (mi-) mo del of  Σ if f v ∈  UL P(Σ) and u ′< u⇒u ′ ∉ ULP ( Σ) ( i ffa ny less incon sistent u′f als ifiesΣ) . I n  a nut shell ,t hemi-models o f Σare the v aluati o ns u ∈U  satisf ying b oth  (1 ) u∈  U LP (Σ) ,a nd( 2)  u( p ) =  12 ⇒ p∧¬p  ∈ su b(Σ) . 9 We d eno te b y ULP mi(Σ) the set  {u  ∈ U L P(Σ ) ∣ u ′ < u  ⇒ u ′∉ ULP (Σ) fo r eve ry  u′ ∈ ULP} of t he m i-modelsofΣ.  Mi ni mally Incons ist ent  Conseq uence i s  de fi n e d  a s preservation ofde s i gn ated val ues in mi -m o dels [24 ,p.3 25]: D efini tion 4. 3 (Minim ally I n co ns istentCo nseque nc e)Σ ⊧ LPmiψ  ⇔ U LPmi(Σ ) ⊆ ULP( ψ) (ψ is a minimally inconsistent( mi - )  c o nseq uen ce of Σ iff  all themi-m o de lso f Σ a re mo de l so f ψ).10 mi-conseque nc e is n on-mo notonic, sinc e extendin g Σ  with fu rthe r  i n formation mayleadto loose p r eviously  draw n conseq uences. A s an examp le[24 ]:q i s ami-consequenc e of { p,  p ⊃ q} , b ut it i s n otanmi- co nsequence  of {p∧¬ p} ∪{ p,  p  ⊃q}, s i nce ther eisamim odelu  of {p , p ⊃  q,p∧ ¬p }  wh ere u(q )  = 0 . Th us ,we h ave  Σ  ⊧LPm i ψ/ ⇒ Σ , φ ⊧LP mi ψ. The  fo l lowi ng  i s a fac t that we wil lpresuppose  i n w hat fo l l ows:11 F act 4.4 (Fact 1, [24])Σ  ⊧LP ψ⇒ Σ  �
 (≤)_is a_strict (weak) partial order_; in_particular,_< relates_valuations_in U from_more to less_consistent, to the effect_that u <_u′_expresses that u is less inconsistent than u′; this is the case when u!_⊂_u′!. Definition_4.2_(Minimally_inconsistent model, [24]) For every_u ∈ ULP, u is_a minimally_inconsistent (mi-) model of Σ iff v ∈_ULP(Σ)_and u′ <_u⇒ u′ ∉ ULP(Σ) (iff any less inconsistent u′_falsifies Σ). In a nutshell, the_mi-models of Σ_are_the_valuations u ∈ U_satisfying both (1) u ∈ ULP(Σ),_and (2) u(p) = 12 ⇒_p ∧ ¬p ∈ sub(Σ). 9 We_denote by ULPmi(Σ) the set {u_∈ ULP(Σ) ∣ u′ <_u ⇒_u′ ∉ ULP(Σ) for every_u′ ∈ ULP}_of the_mi-models of Σ._Minimally Inconsistent Consequence is defined as_preservation of designated_values in mi-models [24, p.325]: Definition_4.3_(Minimally Inconsistent Consequence)_Σ_⊧LPmi_ψ ⇔_ULPmi(Σ) ⊆ ULP(ψ)_(ψ_is a_minimally_inconsistent (mi-) consequence of Σ iff_all_the mi-models of Σ are models of_ψ).10 mi-consequence is non-monotonic,_since_extending Σ with further_information may lead to loose_previously drawn consequences. As an example_[24]: q_is a_mi-consequence of {p, p ⊃ q}, but it is not an_mi-consequence of {p∧¬p}∪{p, p ⊃ q},_since there is a_mimodel u_of_{p, p ⊃_q,_p∧¬p} where_u(q) = 0. Thus, we have Σ_⊧LPmi ψ_/⇒ Σ, φ ⊧LPmi ψ. The_following is a fact_that_we will presuppose in what follows:11_Fact 4.4 (Fact 1, [24]) Σ_⊧LP ψ ⇒ Σ �