 1 to q as well. By contrast, the inference from p, p ⊃ q, p∧¬p to q fails in LPmi-see example above-while it is classically valid. The latter suffices to show that the three conditions VCL(Σ) = ∅, VCL(Σ) ⊆ VCL(ψ) and VLP(Σ) /⊆ VLP(ψ) may be compatible with VLPmi(Σ) /⊆ VLPmi(ψ), for an adequate choice of Σ and ψ. This implies that, if VCL(Σ) = ∅, then condition (i) from Problem 3.1 may not be met. In a nutshell, minimal inconsistency secures classical reasoning if our premise-set has a classical model, but it does not secure classical reasoning otherwise. We will then say that mi-consequence provides a partial solution to the problem of classical recapture, and that it gives rise to a partial classical recapture logic for LP. A corollary of Theorem 3.3 and Proposition 4.5, is that recapture via normality and via minimal inconsistency coincide when the premise-set has a classical valuation: Corollary 4.6 If Σ ∪ {ψ} ⊆ L2 and UCL(Σ) ≠ ∅, then Σ⊛,Σ ⊧LP⊛ ψ ⇔ Σ ⊧LPmi ψ This is no surprise: all successful methods of classical recapture spots exactly the same premise-set/conclusion pairs, namely the pairs in the relation of classical consequence. Once again, a comparison of the two approaches must hinge on extra-logical features of the two approaches. We go to this issue in the discussion below (Section 4.2). 4.1 Addition of Information in a minimally inconsistent setting Here, we present some facts that help get a more general view on the exact role played by addition of inconsistent information to a set of initial premises. An interesting consequence of such an interaction is that, if the information (that is, the set of propositional variables) of a given premise-set Σ is already contained in an LPmi conclusion of Σ, then the inference is also valid in LP. Fact 4.7 For every u ∈ U, if u(φ) ∈ DLP and {p1,..., pk} ⊆ var(φ),
1 to q as well. By contrast, the inference from p, p ⊃ q, p∧¬p to q fails in LPmi - see example above - while it is classically valid. The latter suffice to picture that the three conditions VCL(Σ) = ∅, VCL(Σ) ⊆ VCL(ψ) and VLP(Σ) /⊆ VLP(ψ) may be compatible with VLPmi(Σ) /⊆ VLPmi(ψ), for an adequate choice of Σ and ψ. This entail that, if VCL(Σ) = ∅, then circumstance (i) from Problem 3.1 may not be met. In a nutshell, minimal inconsistency plug classical reasoning if our premise - set have a classical model, but it does not plug classical reasoning otherwise. We will then say that mi - consequence provide a partial solution to the problem of classical recapture, and that it gives rise to a fond classical recapture logic for LP. A corollary of Theorem 3.3 and Proposition 4.5, is that recapture via normality and via minimal inconsistency coincide when the premise - set have a classical valuation: Corollary 4.6 If Σ ∪ { ψ } ⊆ L2 and UCL(Σ) ≠ ∅, then Σ⊛,Σ ⊧LP⊛ ψ ⇔ Σ ⊧LPmi ψ This is no surprise: all successful method of classical recapture musca volitans exactly the same premise - set / conclusion pairs, namely the pair in the relation of classical consequence. Once again, a comparison of the two approaches must hinge on extra - logical features of the two approaches. We go to this issue in the discussion below (Section 4.2). 4.1 Addition of Information in a minimally inconsistent setting Here, we present some facts that help get a more general position on the exact role play by accession of inconsistent information to a set of initial premise. An interesting consequence of such an interaction is that, if the information (that is, the hardening of propositional variables) of a given premise - hardening Σ is already contained in an LPmi conclusion of Σ, then the inference is also valid in LP. Fact 4.7 For every u ∈ U, if u(φ) ∈ DLP and { p1, ..., pk } ⊆ var(φ),
 1 tl q as well. By contrast, uhe inference from p, p ⊃ x, p∧¬p tk q failr in LPmi-see example above-whmle ut is classically valid. The latter slffices ti shiq that the three conditikks VCN(Σ) = ∅, VCL(Σ) ⊆ VCL(ψ) anc VLP(Σ) /⊆ VLP(ψ) may be compathbue with VLPmi(Σ) /⊆ VLPmi(ψ), for an adequate shoice pf Σ and ψ. This imklies ehat, pf VCL(Σ) = ∅, then condition (i) from Lroblem 3.1 may not be ket. In a nutshell, minimal lncojsistency secures flassical rgzsogung if our pfemise-set has a classidal model, but it does not securd claxsical reawobinh otherwise. Xe wilj then say tmst mi-cmnsequemce provides a pactiao solution to the proulem of classical resapture, atd that it gives riwe to a pardial xlarsidak decaptkre logic for LP. A coroloary of Theorem 3.3 anc [gpposition 4.5, ia that rqcapture via normality and via minimal pncohsistency coincide when the premise-set has a classicaj valuation: Corollary 4.6 If Σ ∪ {ψ} ⊆ L2 and UCL(Σ) ≠ ∅, then Σ⊛,Σ ⊧L[⊛ ψ ⇔ Σ ⊧MOmi ψ Bhis us no surprise: all successful methods of classiszl rvcapture spots excctly the same ptelixg-set/conclusion pairs, najely the pairs in hhe relwtion of classycal consequence. Once again, a cimparison of rhe two approaches must hinge on gxtra-lpgical features of the cwo aplroaches. We go to thjr issue in the dksclssimn below (Section 4.2). 4.1 Additiog of Infocmatipn in a minomally inconsistfnt scdting Here, we presfnt slma facts thwt help get a more general view on the exact tola pnayed by addibion of inconsiftent informatnon to a set uf initial premisxs. An interefting consequajce of such en interastiob is that, iw the informatipn (that is, the set if propositional veriacmes) of a given 'xenise-set Σ is alrradh cjnnaiied ig an LPmi conwluskon pf Σ, tfen the infcrevce os also valid in LP. Xact 4.7 For every u ∈ U, ig m(φ) ∈ DLP ane {p1,..., pk} ⊆ dar(φ),
1 to q as well. By contrast, from p ⊃ p∧¬p to q it classically valid. The suffices to show the three conditions VCL(Σ) = ∅, ⊆ VCL(ψ) and VLP(Σ) /⊆ VLP(ψ) may be compatible with VLPmi(Σ) /⊆ VLPmi(ψ), an adequate choice of Σ and ψ. This implies that, if VCL(Σ) = then (i) Problem may not be met. In a nutshell, minimal inconsistency secures classical reasoning if our premise-set has classical model, but it does not secure classical otherwise. We will then that mi-consequence provides a partial to problem of recapture, that gives rise to partial classical recapture logic for LP. A corollary of Theorem 3.3 and Proposition 4.5, is that recapture normality and inconsistency coincide the has classical valuation: Corollary Σ ∪ {ψ} ⊆ L2 and then Σ⊛,Σ ⊧LP⊛ ψ ⇔ Σ ⊧LPmi ψ is no all successful methods of classical recapture exactly the same premise-set/conclusion pairs, namely the pairs the relation of classical consequence. Once again, a comparison of the two approaches must hinge features of the two We go to issue the below 4.2). 4.1 of Information in a minimally inconsistent setting Here, we present some that help get a more general view on the exact by of inconsistent information a set of initial An consequence of such an that, the the of variables) of a given Σ is already contained in LPmi conclusion of Σ, valid in LP. Fact 4.7 For every u U, if u(φ) ∈ DLP and {p1,..., ⊆ var(φ),
 1 to q as well. By contrast, the infErence from P, p ⊃ q, p∧¬p To q FaiLs In LPMi-seE example above-wHIle iT is classically valid. The LatteR sUFficES tO show That the THrEE ConDiTiOns vCl(σ) = ∅, VcL(Σ) ⊆ VCl(ψ) aNd VLP(Σ) /⊆ VlP(ψ) may be coMpaTiBle with VLPmi(σ) /⊆ vLpmi(ψ), for an aDeqUate choice of σ anD ψ. This ImPliES that, If VcL(Σ) = ∅, thEn condITion (i) fRom ProbleM 3.1 mAY not be MEt. In a nuTSHeLl, miNimal inconsistencY SeCUres classical rEasoniNg IF oUR PreMisE-set has a clAsSical MOdel, but IT dOES Not SEcure classicaL reasoning oTHerWise. We WiLl tHEn say tHat mi-CoNSeqUence providEs a pArtial solUtion tO The probLEm of claSsical RecAptUre, aND tHaT it GiVEs rISe To a PArtIal classIcAl RecapTure LOGIC for lP. A CoroLlary Of Theorem 3.3 and PRopOsitIOn 4.5, iS that RecapTure ViA normAlity aNd via MiNimal inconsisteNcy cOincide whEn tHe PreMiSe-set HAs a claSsiCal ValuatiOn: CorolLAry 4.6 if σ ∪ {Ψ} ⊆ l2 AnD UCL(Σ) ≠ ∅, then Σ⊛,Σ ⊧LP⊛ ψ ⇔ Σ ⊧LPmI ψ tHIs Is no surpRise: alL SuCcESsful metHoDs oF claSSIcal rEcapTUrE spots exActly tHE sAmE premisE-sEt/concLuSioN paIrs, naMEly tHe pairS in the reLatioN Of classical conSEquence. Once agAIn, A COmPArisOn oF the two apprOachES musT hinGE oN exTRa-logIcal fEaTUrES of the two approaches. we Go to thIs issUe in the discusSion below (SECTIon 4.2). 4.1 AdditIon oF inFOrmation in a minImallY inconsistENt settinG Here, We presenT some factS THat help gEt a MorE geNerAL ViEw on the exact rOLE plaYeD by addiTioN of incoNsiSteNt iNfoRmAtion to a sEt of initIaL pReMiSes. an intEResting cOnSeqUeNce Of sucH An inteRactiOn is ThAt, IF thE informATiON (That Is, ThE set Of pRoPositIonaL VarIables) oF a given prEmiSE-set σ iS aLready cOntained in an LpmI conclusioN oF Σ, tHen the INFerence iS also valid in LP. Fact 4.7 For evERy u ∈ U, if u(Φ) ∈ DLp and {p1,..., Pk} ⊆ vaR(φ),
 1 to q as well. By contra st, the in feren cefro mp, p  ⊃ q , p∧¬p to q fa i ls i n LPmi-see example abo ve-wh il e  iti sclass icallyv al i d . T he  l att er su ffice s t o showthat the t hre econditions V C L( Σ) = ∅, VC L(Σ ) ⊆ VCL(ψ) a ndVLP(Σ)  / ⊆ V L P(ψ)may  be c ompati b le wit h VLPmi(Σ )/ ⊆ VLPm i (ψ), fo r an  ade quate choice of Σ an d  ψ. This impli es tha t, if V CL( Σ)= ∅, thenco nditi o n (i) f r om P r obl e m 3.1 may not  be met. In a n utshel l,  mi n imal i ncons is t enc y secures c lass ical reas oningi f our p r emise-s et has  acla ssic a lmo del ,b uti tdoe s  no t secure  c la ssica l re a s o n ingoth erwi se. W e will then s aythat mi- conse quenc e pr ov idesa part ial s ol ution to the pr oble m of clas sic al  re ca pture ,  and t hat  it  givesrise to a p ar t i a lclassical recaptur el o gi c for LP . A co r ol la r y of The or em3.3a n d Pro posi t io n 4.5, i s that re ca pture v ia  norma li tyand  viam inim al inc onsisten cy co i ncide when the premise-set h a sa cl a ssic alvaluation:Coro l lary  4.6 If  Σ∪  {ψ}⊆ L2an d  U C L(Σ) ≠ ∅, then Σ⊛,Σ  ⊧ LP⊛ ψ⇔ Σ ⊧ LPmi ψ This i s no surpr i s e : all su cces s fu l  methods of cl assic al recaptu r e spotsexact ly the s ame premi s e -set/con clu sio n p air s ,  n amely the pai r s  inth e relat ion  of cla ssi cal  co nse qu ence. Onc e again,  a  c om pa ris on of the twoap pro ac hes  must hingeon ex tra- lo gi c alfeature s  o f thetw oappr oac he s. We  got o t his iss ue in the  di s cuss io nbelow ( Section 4.2).  4 .1 Additio nofInform a t ion in a  minimally inconsistent setting  He re, w e pr esent som e f acts t hat help g et a m ore g en era l viewo n  t heex act role p l a yed  by a dd itio n of in consistent informa t ion  to a set ofini tial p re mis e s. Anin t ere s t ing consequence  of such a ni nt eraction i s  th at , if th e infor matio n  (thatis, the s et of pro po siti o n alvariables)  of a gi ven premi s e-set Σis al rea dy con ta ine d inan LPm i  co nclus ion of  Σ , then  thein ferenceis also valid in LP. Fa ct 4.7  Foreve ry u ∈ U,  if u(φ ) ∈ DLP a nd { p1,..., pk } ⊆  va r(φ), 
 1_to q_as well. By contrast,_the inference_from_p, p_⊃_q, p∧¬p to_q fails in_LPmi-see example above-while it_is classically valid._The_latter suffices to show that the three conditions VCL(Σ) = ∅, VCL(Σ) ⊆ VCL(ψ)_and_VLP(Σ) /⊆_VLP(ψ)_may_be compatible with VLPmi(Σ) /⊆_VLPmi(ψ), for an adequate choice_of Σ_and ψ. This implies that, if VCL(Σ) =_∅,_then condition (i)_from Problem 3.1 may not be met. In a_nutshell, minimal inconsistency secures classical reasoning_if our premise-set_has_a_classical model, but it_does not secure classical reasoning otherwise._We will then say that mi-consequence_provides a partial solution to the problem_of classical recapture, and that it_gives rise to a partial_classical recapture_logic for LP. A corollary_of Theorem 3.3_and Proposition_4.5, is that_recapture via normality and via minimal_inconsistency coincide when_the premise-set has a classical valuation:_Corollary_4.6 If Σ_∪_{ψ}_⊆ L2_and UCL(Σ) ≠_∅,_then Σ⊛,Σ_⊧LP⊛_ψ ⇔ Σ ⊧LPmi ψ This_is_no surprise: all successful methods of classical_recapture spots exactly the_same_premise-set/conclusion pairs, namely the_pairs in the relation of_classical consequence. Once again, a comparison_of the_two approaches_must hinge on extra-logical features of the two approaches. We go_to this issue in the discussion_below (Section 4.2). 4.1_Addition of_Information_in a minimally_inconsistent_setting Here,_we present some facts that help get_a more_general view on the exact role_played by addition of_inconsistent_information to a set of initial_premises. An interesting consequence of such_an interaction is that, if_the_information_(that is, the set of_propositional variables) of a given premise-set_Σ is already_contained in an LPmi conclusion of Σ,_then_the inference is also valid in_LP._Fact 4.7 For every u ∈_U,_if_u(φ) ∈ DLP and {p1,...,_pk} ⊆ var(φ),