⊛ if all the formulas involved belong to L2: φ,φ ⊃ ψ,⊛φ,⊛(φ ⊃ ψ) ⊧LP⊛ ψ Recapture of MP ¬ψ,φ ⊃ ψ,⊛¬ψ,⊛(φ ⊃ ψ) ⊧LP⊛ ¬φ Recapture of MT φ ⊃ (ψ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ)) ⊧LP⊛ ¬φ Recapture of RAA φ ⊃ ψ,ψ ⊃ ζ,⊛(φ ⊃ ψ),⊛(ψ ⊃ ζ),⊧LP⊛ φ ⊃ ζ Recapture of Tr ⊃ 4 Minimal Inconsistency and Recapture via Normality Minimal inconsistency [24] applies to LP and it is based on the insight that, given the premises from which we have to reason, we should assume that the situation is no more inconsistent that we are forced to assume by the syntactic structure of the premises. Below, we introduce minimal inconsistency, we compare it with recapture via normality and we discuss what we consider benefits of the latter over the former. Minimal Inconsistency. For every u ∈ ULP, we define the set 10 Normality operators and Classical Recapture in Extensions of Kleene Logics u! = {p ∈ P ∣ u(p ∧ ¬p) ∈ DLP} of the propositional variables that form a contradiction and are assigned value 1 2 in u [24, p.325]. The definition of u! plays a precise role in minimally inconsistency, namely that of comparing valuations in terms of how much inconsistent they are. The comparison follows the intuitive principle that the greater u! is, the more inconsistent u is. This intuition is encoded in Priest's definition of a consistency ordering [24, p.325]:8 Definition 4.1 (Consistency ordering) u ≤ u′ ⇔ u! ⊆ u′! (u is less or equally inconsistent than u′ iff u! is contained in u′!). The strict part of the relation (u < u′) equates with u! ⊂ u′!. The relation <
⊛ if all the formulas involved belong to L2: φ, φ ⊃ ψ,⊛φ,⊛(φ ⊃ ψ) ⊧LP⊛ ψ Recapture of MP ¬ψ, φ ⊃ ψ,⊛¬ψ,⊛(φ ⊃ ψ) ⊧LP⊛ ¬φ Recapture of MT φ ⊃ (ψ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ) ) ⊧LP⊛ ¬φ Recapture of RAA φ ⊃ ψ, ψ ⊃ ζ,⊛(φ ⊃ ψ),⊛(ψ ⊃ ζ),⊧LP⊛ φ ⊃ ζ Recapture of Tr ⊃ 4 Minimal Inconsistency and Recapture via Normality Minimal inconsistency [ 24 ] apply to LP and it is free-base on the insight that, given the premises from which we own to reason, we should assume that the position is no more discrepant that we are forced to assume by the syntactic structure of the premises. Below, we precede minimal inconsistency, we compare it with recapture via normality and we discuss what we think benefits of the latter over the former. Minimal Inconsistency. For every u ∈ ULP, we define the set 10 Normality operators and Classical Recapture in Extensions of Kleene Logics u! = { p ∈ phosphorus ∣ u(p ∧ ¬p) ∈ DLP } of the propositional variables that form a contradiction and are assigned value 1 2 in uranium [ 24, p.325 ]. The definition of u! plays a accurate role in minimally incompatibility, namely that of comparing valuations in terms of how much inconsistent they are. The comparison follows the intuitive principle that the great u! is, the more inconsistent u is. This intuition is encoded in Priest's definition of a consistency ordering [ 24, p.325]:8 Definition 4.1 (Consistency ordering) u ≤ u′ ⇔ u! ⊆ u′! (uracil is less or equally inconsistent than u′ iff u! is contained in u′ !). The strict part of the relation (u < u′) equates with u! ⊂ u′! . The relation <
⊛ if all the formulas involvtd belong to L2: φ,φ ⊃ ψ,⊛φ,⊛(φ ⊃ ψ) ⊧LP⊛ ψ Cecaptude of MP ¬ψ,φ ⊃ ψ,⊛¬ψ,⊛(φ ⊃ ψ) ⊧LP⊛ ¬φ Recapture of MT φ ⊃ (ψ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ)) ⊧OP⊛ ¬φ Recapture of RAA φ ⊃ ψ,ψ ⊃ ζ,⊛(φ ⊃ ψ),⊛(ψ ⊃ ζ),⊧LP⊛ φ ⊃ ζ Recapturt of Tr ⊃ 4 Minimal Inconsistency wnd Xerapture via Norkality Minhmal inconsistanzy [24] applies to LP and it is based on ehe insogjt that, given jhe pgeiisea from which we have to reason, we ahould essume that the situation is no more incojsishent that we are flrced to aswume vy the syntaztic strucuuxe of the ptemises. Below, we introduce minimxl ineonsistency, ww clkpare it wivh recwpture via normality and we discuss what ee ronsuder benefits of the natter over the fotmer. Minimdl Inconsistency. Foe wvery u ∈ GLP, dw ddfihe tge set 10 Nkrmality olerators ane Classical Recaptute pm Extensions of Klqege Logics u! = {p ∈ P ∣ u(p ∧ ¬p) ∈ DLP} of the prmpoaitional variables that form a contradiction and are wssigned value 1 2 in u [24, p.325]. The definition of u! playv a pcezist vjuw ln minimally inconsistency, namely that of com[zronn valuations in berms of how much onfomfistent they xre. Thz ckmparison follows hhe intoitive principlt thay the greater u! is, the more inconsistenn u us. This intuition ns encoded iu Priext's drfinition of a consisteucy orsering [24, p.325]:8 Dffinition 4.1 (Consistency orddrikg) g ≤ u′ ⇔ u! ⊆ u′! (u is less or eqtally incinsixtent tfan o′ iff u! is contaijed ik u′!). The strict part of tke renation (u < k′) equates with u! ⊂ u′!. The relatioi <
⊛ if all the formulas involved belong φ,φ ψ,⊛φ,⊛(φ ⊃ ⊧LP⊛ ψ Recapture ⊃ ⊧LP⊛ ¬φ Recapture MT φ ⊃ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ)) ¬φ Recapture of RAA φ ⊃ ψ,ψ ⊃ ζ,⊛(φ ⊃ ψ),⊛(ψ ⊃ ζ),⊧LP⊛ ⊃ ζ Recapture of Tr ⊃ 4 Minimal Inconsistency and Recapture via Normality inconsistency applies LP it is based on the insight that, given the premises from which we have to reason, should assume that the situation is no more that we are forced assume by the syntactic structure the Below, we minimal we it with recapture normality and we discuss what we consider benefits of the latter over the former. Minimal Inconsistency. For u ∈ define the 10 operators Classical Recapture in Kleene Logics u! = {p ∈ ∧ ¬p) ∈ DLP} of the propositional variables form a and are assigned value 1 2 u [24, p.325]. The definition of u! plays precise role in minimally inconsistency, namely that of comparing valuations in terms of how much are. The comparison follows intuitive principle that greater is, more u is. intuition is encoded in Priest's definition of a consistency ordering [24, Definition 4.1 (Consistency ordering) u ≤ u′ ⇔ u! ⊆ is or equally inconsistent u′ iff u! is in The strict part of (u u′) ⊂ The <
⊛ if all the formulas involved bElong to L2: φ,φ ⊃ Ψ,⊛φ,⊛(φ ⊃ ψ) ⊧Lp⊛ ψ REcaPtUre oF MP ¬ψ,Φ ⊃ ψ,⊛¬ψ,⊛(φ ⊃ ψ) ⊧LP⊛ ¬φ RecaptURe of mT φ ⊃ (ψ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ)) ⊧LP⊛ ¬φ Recapture of rAA φ ⊃ ψ,Ψ ⊃ ζ,⊛(Φ ⊃ Ψ),⊛(ψ ⊃ ζ),⊧Lp⊛ Φ ⊃ ζ recapTure of TR ⊃ 4 miNIMal inCoNsiStENcY and REcaPture viA Normality minImAl inconsisteNCy [24] Applies to Lp anD it is based on The InsighT tHat, GIven tHe pRemisEs from WHich we Have to reaSoN, We shouLD assume THAt The sItuation is no more iNCoNSistent that we aRe forcEd TO aSSUme By tHe syntactiC sTructURe of the PReMISEs. BELow, we introducE minimal incONsiStency, We ComPAre it wIth reCaPTurE via normaliTy anD we discusS what wE ConsideR BenefitS of the LatTer Over THe FoRmeR. MINimAL INcoNSisTency. For EvErY u ∈ ULP, We deFINE The sEt 10 NOrmaLity oPerators and ClAssIcal rEcaPture In ExtEnsiOnS of KlEene LoGics u! = {P ∈ P ∣ U(p ∧ ¬p) ∈ DLP} of the propOsitIonal variAblEs ThaT fOrm a cONtradiCtiOn aNd are asSigned vALue 1 2 In U [24, P.325]. tHe Definition of u! plays A pRECiSe role in MinimaLLy InCOnsistenCy, NamEly tHAT of coMparINg ValuatioNs in teRMs Of How much InConsisTeNt tHey Are. ThE CompArison Follows tHe intUItive principle THat the greater U! Is, THE mORe inConSistent u is. THis iNTuitIon iS EnCodED in PrIest's DeFInITion of a consistency oRdEring [24, p.325]:8 definItion 4.1 (ConsisteNcy orderinG) U ≤ U′ ⇔ U! ⊆ u′! (u is lesS or eQUaLLy inconsistent Than u′ Iff u! is contAIned in u′!). THe strIct part oF the relatION (u < u′) equatEs wIth U! ⊂ u′!. THe rELAtIon <
⊛ if all the formulas invo lved belon g toL2:  φ, φ⊃ ψ, ⊛φ,⊛ (φ ⊃ ψ) ⊧LP⊛ ψ Reca pture of MP ¬ψ,φ ⊃ ψ,⊛ ¬ψ,⊛( φ⊃  ψ)⊧ LP ⊛ ¬φRecaptu r eo f  MT  φ  ⊃  (ψ  ∧ ¬ψ ),⊛(φ  ⊃(ψ ∧ ¬ψ )) ⊧LP⊛ ¬φ  Re ca pture of RAA φ⊃ ψ,ψ ⊃ ζ, ⊛(φ  ⊃ ψ),⊛(ψ ⊃ζ), ⊧LP⊛ φ  ⊃  ζR ecapt ure  of T r ⊃ 4M inimal  Inconsis te n cy and Recaptu r e  v ia N ormality Minimali nc o nsistency [24]  appli es to L P a ndit is base don th e  insigh t  t h a t , g i ven the premi ses from wh i chwe hav etor eason,  we s ho u ldassume that  the  situatio n is n o  more i n consist ent th atwearef or ce d t oa ssu m ebyt hesyntacti cst ructu re o f t h e pr emi ses.  Belo w, we introdu cemini m alincon siste ncy,  w e com pare i t wit hrecapture via n orma lity andwedi scu ss  what we con sid erbenefit s of th e  la tt e r ov er the former. Min im a l  I nconsist ency.F or  e v ery u ∈UL P,we d e f ine t he s e t10 Norma lity o p er at ors and  C lassic al  Re cap turei n Ex tensio ns of Kl eeneL ogics u! = {p∈  P ∣ u(p ∧ ¬p )  ∈ D LP }  ofthe  propositio nalv aria bles th atf orm a  cont ra d ic t ion and are assigne dvalue1 2 i n u [24, p.32 5]. The de f i n ition of  u!p la y s a precise ro le in  minimally inconsis tency , namely  that ofc o mparingval uat ion s i n te rms of how mu c h  inc on sistent  th ey are.  Th e c omp ari so n follows  the int ui ti ve  p rin ciple that the  g rea te r u ! is, the mo re in cons is te n t u  is. Th i si n tuit io nis e nco de d inPrie s t's  defini tion of a  co n sist en cy  orderi ng [24, p.325 ]: 8 Definiti on  4. 1 (Con s i stency o rdering) u ≤ u′ ⇔ u! ⊆u ′! (u i s l ess o r eq ually inc ons istent  th a n u′ i ff u!is co nt ain e d  in u ′ ! ).  Th estrict par t ofthe r el atio n (u <u′) equates with u !  ⊂u′!. The rela tio n <
⊛ if_all the_formulas involved belong to_L2: φ,φ_⊃_ψ,⊛φ,⊛(φ ⊃_ψ)_⊧LP⊛ ψ Recapture_of MP ¬ψ,φ_⊃ ψ,⊛¬ψ,⊛(φ ⊃ ψ)_⊧LP⊛ ¬φ Recapture_of_MT φ ⊃ (ψ ∧ ¬ψ),⊛(φ ⊃ (ψ ∧ ¬ψ)) ⊧LP⊛ ¬φ Recapture of_RAA_φ ⊃_ψ,ψ_⊃_ζ,⊛(φ ⊃ ψ),⊛(ψ ⊃ ζ),⊧LP⊛_φ ⊃ ζ Recapture of_Tr ⊃_4 Minimal Inconsistency and Recapture via Normality Minimal_inconsistency_[24] applies to_LP and it is based on the insight that,_given the premises from which we_have to reason,_we_should_assume that the situation_is no more inconsistent that we_are forced to assume by the_syntactic structure of the premises. Below, we_introduce minimal inconsistency, we compare it_with recapture via normality and_we discuss_what we consider benefits of_the latter over_the former._Minimal Inconsistency. For_every u ∈ ULP, we define_the set 10_Normality operators and Classical Recapture in_Extensions_of Kleene Logics_u!_=_{p ∈_P ∣ u(p_∧_¬p) ∈_DLP}_of the propositional variables that form_a_contradiction and are assigned value 1 2_in u [24, p.325]._The_definition of u! plays_a precise role in minimally_inconsistency, namely that of comparing valuations_in terms_of how_much inconsistent they are. The comparison follows the intuitive principle that_the greater u! is, the more_inconsistent u is. This_intuition is_encoded_in Priest's definition_of_a consistency_ordering [24, p.325]:8 Definition 4.1 (Consistency ordering)_u ≤_u′ ⇔ u! ⊆ u′! (u_is less or equally_inconsistent_than u′ iff u! is contained_in u′!). The strict part of_the relation (u < u′)_equates_with_u! ⊂ u′!. The relation_<