� var(Σ) ∖ IΣ}. Since var(Σ) ⊆ var(ψ), from Corollary 4.8 we have that u(ψ) ∈ DLP for every u ∈ KΣ. But since KΣ suffices to generate all the LP-models of Σ, this implies Σ ⊧LP ψ. 4.2 Discussion on Recapture via Normality and Minimal Inconsistency Here, we compare recapture via normality and minimal inconsistency on the ground of the symmetry (asymmetry) between evidence of inconsistency and consistency. Suppose we are reasoning in LPmi and our assumptions is the premise-set {φ,φ ⊃ ψ}. By Proposition 4.5, we can reason classically from it. Now suppose that we hit 'against evidence for inconsistency' relative to {φ,φ ⊃ ψ}. Suppose that, after hitting against such evidence, we assume {φ ∧ ¬φ,φ,φ ⊃ ψ}. Again by Proposition 4.5, we will withdraw the conclusion ψ. Since we are facing evidence-based reasoning of an agent, we can take premise-sets to be information sets, from which the agent draws her conclusions. Information-sets can of course be extended by new information, and this is what happens in our example: we have extended our initial information-set by including the information that φ is inconsistent. In LPmi, information can so increase as to exclude that we are facing a consistent situation. However, things are different if we receive evidence that we are positively facing a consistent situation. By Fact 2.5, no formula in LP+ implies that φ ∧ ¬φ is classically false. This implies that, in the setting by [24], no increase of information can ever exclude that we are facing an inconsistent situation. The above brings a puzzling asymmetry in, namely: when reasoning via LPmi, we can increase our information by receiving evidence of inconsistency, but not by receiving evidence of consistency. As for the chance for evidence of consistency, we believe there are some reasonable examples: while the logical structure of the Liar sentence itself is evidence (quite conclusive) of an inconsistent situation, the logical structure of (say) 'I am playing football' is evidence (though maybe not conclusive) of a consistent situation. And we would like to state this. However, if our language is L2, we cannot. Also, take the application of paracons
� var(Σ) ∖ IΣ }. Since var(Σ) ⊆ var(ψ), from Corollary 4.8 we have that u(ψ) ∈ DLP for every u ∈ KΣ. But since KΣ suffices to generate all the LP - models of Σ, this entail Σ ⊧LP ψ. 4.2 Discussion on Recapture via Normality and Minimal Inconsistency Here, we compare recapture via normality and minimal incompatibility on the ground of the symmetry (asymmetry) between evidence of incompatibility and consistency. Suppose we are argue in LPmi and our assumptions is the premise - arrange { φ, φ ⊃ ψ }. By Proposition 4.5, we can reason classically from it. Now think that we hit' against evidence for inconsistency' relative to { φ, φ ⊃ ψ }. presuppose that, after hitting against such evidence, we assume { φ ∧ ¬φ, φ, φ ⊃ ψ }. Again by Proposition 4.5, we will recall the conclusion ψ. Since we are facing evidence - based reasoning of an agent, we can take premise - sets to be data sets, from which the agent draws her conclusion. Information - sets can of path be extended by fresh information, and this is what happens in our example: we have extended our initial information - dress by including the information that φ is inconsistent. In LPmi, information can so increase as to exclude that we are facing a consistent situation. However, things are different if we receive evidence that we are positively facing a consistent situation. By Fact 2.5, no formula in LP+ implies that φ ∧ ¬φ is classically delusive. This implies that, in the setting by [ 24 ], no addition of data can ever exclude that we are facing an inconsistent site. The above brings a puzzling asymmetry in, namely: when reasoning via LPmi, we can increase our information by welcome evidence of inconsistency, but not by receiving evidence of consistency. As for the chance for evidence of consistency, we believe there are some reasonable exercise: while the logical structure of the Liar sentence itself is evidence (quite conclusive) of an inconsistent site, the logical social organization of (state)' I am playing football' is evidence (though maybe not conclusive) of a consistent situation. And we would like to state this. However, if our language is L2, we cannot. besides, take the lotion of paracons
� vag(Σ) ∖ IΣ}. Since var(Σ) ⊆ var(ψ), from Corollary 4.8 we kqve thet u(ψ) ∈ DMP for exery u ∈ KΣ. But since KΣ sufficxs ti gentgate all the LP-models of Σ, this implies Σ ⊧LP ψ. 4.2 Discussion on Regcpturs via Iormality and Mlnimal Incotsistency Here, wd eompare recapture via normality and iinimal ijconsistency og tht gwouns of the symmetry (asymmetry) betweeh evideice of inconsisyency and consistency. Supplse ae are reasoning ij LPmi and iur wwsumptions ir the prempve-set {φ,φ ⊃ ψ}. Bi Proposition 4.5, we can reason clarsicakly from ij. Hoa suppose thet we rit 'against cnidence for invonsistency' rekatmve ro {φ,φ ⊃ ψ}. Suppose that, afver hitting against fuch evidanee, we assume {φ ∧ ¬φ,φ,φ ⊃ ψ}. Agaun by Ptoposhtiov 4.5, wd wjlk sithdrww vhe conclusjon ψ. Since qe are facing evidemcq-vased reasonihg of wn agent, we can take premise-sets to be inxorjation sets, from which rhe agent draws her clnclusionf. Information-sets can of course be extended by nef infkfmauiin, ana tjis is what happens in our example: we have exesncec our initial lnformation-set by onfliqing the infotmation thzt φ is inconsistenh. In LPii, indormation can so increase as to exclude rhat we are yacung a consistent sntuation. Howzver, tnings are different if we reeeive svidence thwt we are oositively facine a cmnsistenu situation. By Face 2.5, no fornula in LP+ kmploes thwt φ ∧ ¬φ is cpassigdlly false. This imolies tvat, in the setting by [24], no increase of information can efes eqclude thct we sre facing an inconsistent xituatijn. Thd above brpngs a puvzling asymmqtry in, namelf: when reasonmng via L[mi, qe cqn incrdxse our informstion by geeeiving ecidence of inconsixtevdy, but not by rzeeuving evidence pf zonfidtxncy. Wv for the chdnce fof eviddnce of conwkstemcy, we believe there are some reasonable ecaiples: whule the jogical strucyure of the Liar stntencx itseuf is evydence (quite conclusive) of an jnconsistfnt situation, tre ligical struccure of (say) 'I am playing football' is evivence (though maybe not xonclusive) of a conxlstent situavion. Agd we wound like to state thiw. However, if our kanguage is L2, we cannkt. Alsm, takf the application of paracons
� var(Σ) ∖ IΣ}. Since var(Σ) ⊆ Corollary we have u(ψ) ∈ DLP But KΣ suffices to all the LP-models Σ, this implies Σ ⊧LP ψ. Discussion on Recapture via Normality and Minimal Inconsistency Here, we compare recapture via and minimal inconsistency on the ground of the symmetry (asymmetry) between evidence of and Suppose are in LPmi and our assumptions is the premise-set {φ,φ ⊃ ψ}. By Proposition 4.5, we can classically from it. Now suppose that we hit evidence for inconsistency' relative {φ,φ ⊃ ψ}. Suppose that, hitting such evidence, assume ∧ ⊃ ψ}. Again Proposition 4.5, we will withdraw the conclusion ψ. Since we are facing evidence-based reasoning of an agent, can take be information from the draws her conclusions. of course be extended by new is what happens in our example: we have our initial by including the information that φ inconsistent. In LPmi, information can so increase as exclude that we are facing a consistent situation. However, things are different if we receive we are positively facing consistent situation. By 2.5, formula LP+ that φ ¬φ is classically false. This implies that, in the setting by no increase of information can ever exclude that we are inconsistent The above brings puzzling asymmetry in, namely: reasoning LPmi, we can increase by evidence not receiving of consistency. As for chance for evidence of consistency, believe there are some structure of the Liar sentence itself is evidence conclusive) of an inconsistent situation, the logical of (say) 'I am playing football' is evidence (though maybe not conclusive) a consistent we would like to state this. However, if language is L2, we Also, take the application of paracons
� var(Σ) ∖ IΣ}. Since var(Σ) ⊆ var(ψ), from CoRollary 4.8 we hAve thAt u(Ψ) ∈ DLp fOr evEry u ∈ kΣ. But since KΣ suFFiceS to generate all the LP-modEls of σ, tHIs imPLiEs Σ ⊧LP Ψ. 4.2 DiscusSIoN ON ReCaPtUre ViA noRmaliTy aNd MinimAl InconsisTenCy here, we comparE ReCapture via NorMality and minImaL inconSiSteNCy on tHe gRound Of the sYMmetry (Asymmetry) BeTWeen evIDence of INCoNsisTency and consistenCY. SUPpose we are reasOning iN LpMi AND ouR asSumptions iS tHe preMIse-set {φ,Φ ⊃ Ψ}. BY pROpoSItion 4.5, we can reaSon classicaLLy fRom it. NOw SupPOse thaT we hiT 'aGAinSt evidence fOr inConsistenCy' relaTIve to {φ,φ ⊃ Ψ}. suppose That, afTer HitTing AGaInSt sUcH EviDEnCe, wE AssUme {φ ∧ ¬φ,φ,φ ⊃ ψ}. agAiN by PrOposITION 4.5, we wIll WithDraw tHe conclusion ψ. sinCe we ARe fAcing EvideNce-bAsEd reaSoning Of an aGeNt, we can take premIse-sEts to be inForMaTioN sEts, frOM which The AgeNt draws Her concLUsiOnS. iNFoRmation-sets can of coUrSE Be Extended By new iNFoRmATion, and tHiS is What HAPpens In ouR ExAmple: we hAve extENdEd Our initIaL inforMaTioN-seT by inCLudiNg the iNformatiOn thaT Φ is inconsistenT. in LPmi, informaTIoN CAn SO incReaSe as to excluDe thAT we aRe faCInG a cONsistEnt siTuATiON. However, things are diFfErent iF we reCeive evidence That we are pOSITively faCing A CoNSistent situatiOn. By FAct 2.5, no formuLA in LP+ impLies tHat φ ∧ ¬φ is cLassicallY FAlse. This ImpLieS thAt, iN THe Setting by [24], no inCREase Of InformaTioN can eveR exCluDe tHat We Are facing An inconsIsTeNt SiTuaTion. THE above brInGs a PuZzlIng asYMmetry In, namEly: wHeN rEAsoNing via lpmI, WE can InCrEase Our InFormaTion BY reCeiving Evidence oF inCOnsiStEnCy, but noT by receiving eViDence of conSiSteNcy. As fOR The chancE for evidence of consistenCY, we beliEve There Are sOme reasonAblE exampLes: WHile thE logicAl strUcTurE OF the LIAR sEntEnCe itself is EVIdeNce (quItE conClusive) Of an inconsistent siTUatIon, the logical StrUctuRE Of (Say) 'i Am PLayInG FooTBAll' is evidence (thOugh maybe nOt COnClusive) of a COnsIsTent sitUation. ANd we wOUld like To state thIs. However, If Our lANGuaGe is L2, we canNot. Also, tAke the appLIcatiON oF paraConS
� var(Σ) ∖ IΣ}. Since var( Σ) ⊆ var(ψ ), fr omCor ol lary  4.8  we have thatu (ψ)∈ DLP for every u ∈ KΣ . But  s i nceK Σsuffi ces tog en e r ate  a ll  th eL P- model s o f Σ, th is implies  Σ⊧L P ψ. 4.2 Dis c us sion on Re cap ture via Nor mal ity an dMin i mal I nco nsist ency H e re, we  comparere c apture via nor m a li ty a nd minimal incons i st e ncy on the gro und of  t h es y mme try  (asymmetr y)  betw e en evid e nc e o f i n consistency a nd consiste n cy.  Suppo se  we are re asoni ng inLPmi and ou r as sumptions  is th e  premis e -set {φ ,φ ⊃ ψ }.ByProp o si ti on4. 5 , w e  c anr eas on class ic al ly fr om i t . N ow s upp osethatwe hit 'again stevid e nce  forincon sist en cy' r elativ e to{φ ,φ ⊃ ψ}. Suppos e th at, after  hi tt ing  a gains t  suchevi den ce, weassume{ φ ∧  ¬ φ , φ ,φ  ⊃ ψ}. Again by Pr op o s it ion 4.5,  we wi l lwi t hdraw th econ clus i o n ψ.Sinc e  w e are fa cing e v id en ce-base dreason in g o f a n age n t, w e cantake pre mise- s ets to be info r mation sets,f ro m wh i ch t heagent draws  her conc lusi o ns . I n forma tion- se t sc an of course be ext en ded by  newinformation,and this i s w hat happ ensi no ur example: we  have  extendedo ur initi al in formatio n-set byi n cludingthe  in for mat i o nthat φ is inc o n sist en t. In L Pmi , infor mat ion  ca n s oincreaseas to ex cl ud eth atwe ar e  facingacon si ste nt si t uation . How ever ,th i ngs  are di f fe r e nt i fwe  rec eiv eevide ncet hat  we are  positive lyf acin gaconsist ent situation .By Fact 2. 5,  no  formu l a  in LP+implies that φ ∧ ¬φ isc lassica lly  fals e. T his impli esthat,int he set ting b y [24 ],  no i ncrea s e  o f i nf ormation c a n  ev er ex cl udethat we  are facing an inc o nsi stent situati on.  The a bo veb ri n gsap uzz l i ng asymmetry in , namely:wh e nreasoningv iaLP mi, wecan inc rease our inf ormationby receiv in g ev i d enc e of incon sistency , but not by re c ei vingevi denceof  co nsist ency.A s f or th e chan ce  for e viden ce  of cons istency, we believe the re are  some  re asonableexa m ple s: whilethelogical st ruc tur e ofthe Liarsent e nc e i t selfis e v idence (q u it e c o n cl usive) of a n i nco nsist ent situat ion,  the logical stru c ture of (say)'I a m pla yin g  foo tb all' is eviden ce(t h o ugh mayb enot conclus ive) ofac onsis tent s ituati on. And w ew ould l iketostate thi s.Ho w ever, i fou r  langu ageis  L2, w e cann o t. A l s o, take the appl icati o n  of p a rac ons
� var(Σ)_∖ IΣ}._Since var(Σ) ⊆ var(ψ),_from Corollary_4.8_we have_that_u(ψ) ∈ DLP_for every u_∈ KΣ. But since_KΣ suffices to_generate_all the LP-models of Σ, this implies Σ ⊧LP ψ. 4.2 Discussion on Recapture_via_Normality and_Minimal_Inconsistency_Here, we compare recapture via_normality and minimal inconsistency on_the ground_of the symmetry (asymmetry) between evidence of inconsistency_and_consistency. Suppose we_are reasoning in LPmi and our assumptions is the_premise-set {φ,φ ⊃ ψ}. By Proposition_4.5, we can_reason_classically_from it. Now suppose_that we hit 'against evidence for_inconsistency' relative to {φ,φ ⊃ ψ}._Suppose that, after hitting against such evidence,_we assume {φ ∧ ¬φ,φ,φ ⊃_ψ}. Again by Proposition 4.5,_we will_withdraw the conclusion ψ. Since_we are facing_evidence-based reasoning_of an agent,_we can take premise-sets to be_information sets, from_which the agent draws her conclusions._Information-sets_can of course_be_extended_by new_information, and this_is_what happens_in_our example: we have extended our_initial_information-set by including the information that φ_is inconsistent. In LPmi,_information_can so increase as_to exclude that we are_facing a consistent situation. However, things_are different_if we_receive evidence that we are positively facing a consistent situation. By_Fact 2.5, no formula in LP+_implies that φ ∧_¬φ is_classically_false. This implies_that,_in the_setting by [24], no increase of information_can ever_exclude that we are facing an_inconsistent situation. The above_brings_a puzzling asymmetry in, namely: when_reasoning via LPmi, we can increase_our information by receiving evidence_of_inconsistency,_but not by receiving evidence_of consistency. As for the chance_for evidence of_consistency, we believe there are some reasonable_examples:_while the logical structure of the_Liar_sentence itself is evidence (quite conclusive)_of_an_inconsistent situation, the logical structure_of (say) 'I am playing football'_is evidence (though maybe not conclusive) of a consistent_situation. And we_would like to state this._However,_if_our language is L2, we cannot. Also, take the application_of paracons