istent logic to vagueness. Empirical information could give evidence of the fact that 'Jude Law is bald' (j) expresses a borderline case of baldness and is, then, an inconsistent sentence. However, suitable empirical information would also give evidence that 'Ben Kingsley is bald' (b) does not express a borderline case, and in this case, we would like to state that that sentence is consistent-that is, either classically true, or classically false. The import of the above fact can be better grasped with an epistemic analogy: suppose that reasoners only make given assumptions (i.e., assert the formulas in a premise-set) if they have received persuasive information-that is, information that leads you to attribute probability 1 to the formulas in question. Then, the asymmetry above equates with the fact that reasoners can be sure that an inconsistency is at stake, but they cannot be sure that the situation is perfectly consistent. By contrast, recapture via normality does not bring any asymmetry in. Again, we can add φ ∧ ¬φ after receiving evidence of inconsistency of φ. However, we can also add ⊛φ,⊛(φ ⊃ ψ) if we get evidence that we are facing a normal case. In particular, in LP⊛ we can use j ∧ ¬j to state that 'Jude Law is bald and not bald' (hence, a 14 Normality operators and Classical Recapture in Extensions of Kleene Logics borderline case of baldness) and b ∧ ⊛b to state that 'Ben Kingsley is bald and is not a borderline case of baldness.' By running the epistemic analogy again: with recapture via normality, reasoners can be sure that an inconsistency is at stake, but they can also be sure that the situation is perfectly consistent. Indeed, information can increase both because evidence of inconsistency and evidence of consistency.12 5 Extensions of Kleene Logics, Normality Operators, Logics of Formal Inconsistency and Undeterminedness In this section, we go through the connections between LP⊛ and LFIs, Theorem 3.3 and Derivability Adjustment Theorems [11, 20], definability issues and semantical closure. Theorem 3.3 and Derivability Adjustment Theorems. Results similar to Theorem 3.3 appear in the tradition of LFIs, where they are usually referred to as Derivability Adjust
istent logic to vagueness. Empirical information could give evidence of the fact that' Jude Law is bald' (j) expresses a boundary line subject of baldness and is, then, an inconsistent sentence. However, desirable empirical data would also give evidence that' Ben Kingsley is bald' (bel) does not press out a borderline event, and in this case, we would like to express that that sentence is consistent - that is, either classically dependable, or classically assumed. The import of the above fact can be better grasped with an epistemic analogy: suppose that reasoner only make given assumption (i.e., assert the formulas in a premise - stage set) if they have received persuasive data - that is, information that lead you to attribute probability 1 to the formulas in question. Then, the asymmetry above equal with the fact that reasoners can be sure that an inconsistency is at stake, but they cannot be sure that the situation is perfectly consistent. By contrast, recapture via normality does not bring any asymmetry in. Again, we can add φ ∧ ¬φ after receiving evidence of inconsistency of φ. However, we can also add ⊛φ,⊛(φ ⊃ ψ) if we get evidence that we are facing a normal event. In particular, in LP⊛ we can use j ∧ ¬j to express that' Jude Law is bald and not bald' (therefore, a 14 Normality operator and Classical Recapture in Extensions of Kleene Logics borderline case of baldness) and b ∧ ⊛b to state that' Ben Kingsley is bald and is not a borderline case of baldness.' By running the epistemic analogy again: with recapture via normality, reasoner can be sure that an inconsistency is at stake, but they can also be sure that the situation is perfectly consistent. Indeed, information can increase both because evidence of inconsistency and evidence of consistency.12 5 Extensions of Kleene Logics, Normality Operators, Logics of Formal Inconsistency and Undeterminedness In this section, we go through the connection between LP⊛ and LFIs, Theorem 3.3 and Derivability Adjustment Theorems [ 11, 20 ], definability consequence and semantical closure. Theorem 3.3 and Derivability Adjustment Theorems. Results similar to Theorem 3.3 look in the tradition of LFIs, where they are usually referred to as Derivability Adjust
istfnt logic to vagueness. Eopirical informcrion cmuld gjve eviddnce of the fact that 'Jude Lew iw bale' (j) expresses a borderuine case of baldbess qnd is, thei, an inconsistskt seutxnce. However, sultable empisical informathov could also give evidence that 'Ben Kiggsley os bald' (b) does njt eqpwess a borderline case, and in this cass, we wolld like to state that that sentence is condistfnt-that is, either flassically truq, or classicauly false. Uhz import of the above fact can be better gfaspeb with an eknwtelhc analogy: wuppofe that reasoners onny make given assumptlons (m.e., awsert the formulas in a premise-set) if trey have seeeived persuasive infirnatiot-thad is, unfurmztmon that peavs you to aftribute pribability 1 to the fprilkas in questjon. Thqn, the asymmetry above equates with the fdct that reasoners can be wure that an inconsisjency is ae stake, but they cannot be sure that the situatiot is 'effeebjh clnsistent. By contrast, recapture via normality sots kot bring any asnmmetry in. Again, wr fam add φ ∧ ¬φ after receirjnf evidence of incojsistensy of φ. However, we van also add ⊛φ,⊛(φ ⊃ ψ) if we get ecidence that qe are facing a noxmal case. In parjiculat, in LP⊛ we can use j ∧ ¬j to atate that 'Uude Law jr bald and not bxld' (hance, a 14 Normality operatorf and Clawsiccl Recapgure in Exeensions ov Klecte Logics borderlije cade of baldneds) and b ∧ ⊛b to state that 'Ben Kmigsley is balc dnd is not c bordcrline case of faldness.' By ruuning thz episgemic analkgy agamn: with reca[ture via norkwlity, reasonxrs can bq suee tyat an kvconsistency ix at stakv, yut they xan also be sure tmat tfs situation is 'trftcrly consistent. Inaeeq, pnfpridtion can inwreare cpth bdcause tridcncd of inconsistency and eeidehce of consistency.12 5 Cxtensionw of Kleqne Logics, Notmality Operators, Pogicv oh Formsl Ynconsistency and Undeterminedhess In tjis section, we do tmroudh the conuections between LP⊛ and LFIs, Theorem 3.3 anv Derivability Adjustmgnt Theorems [11, 20], definayikity issues and femantican closure. Theorem 3.3 abd Derivability Acjustment Theorems. Reaults vimilwr to Theorem 3.3 appear in the tradition of LFIs, where they are usually refwrred vo as Derivabjlitu Adjlst
istent logic to vagueness. Empirical information could of fact that Law is bald' of and is, then, inconsistent sentence. However, empirical information would also give evidence 'Ben Kingsley is bald' (b) does not express a borderline case, and in case, we would like to state that that sentence is consistent-that is, either true, classically The of the above fact can be better grasped with an epistemic analogy: suppose that reasoners only given assumptions (i.e., assert the formulas in a if they have received information-that is, information that leads to probability 1 the in Then, the asymmetry equates with the fact that reasoners can be sure that an inconsistency is at stake, but they be sure situation is consistent. contrast, via normality does any asymmetry in. Again, we can ¬φ after receiving evidence of inconsistency of φ. we can add ⊛φ,⊛(φ ⊃ ψ) if we evidence that we are facing a normal case. particular, in LP⊛ we can use j ∧ ¬j to state that 'Jude Law is not bald' (hence, a Normality operators and Recapture Extensions Kleene borderline case baldness) and b ∧ ⊛b to state that 'Ben Kingsley is and is not a borderline case of baldness.' By running analogy with recapture via reasoners can be sure an is at stake, but also sure is consistent. information can increase both evidence of inconsistency and evidence consistency.12 5 Extensions of of Formal Inconsistency and Undeterminedness In this section, go through the connections between LP⊛ and Theorem 3.3 and Derivability Adjustment Theorems [11, 20], definability issues and semantical Theorem 3.3 Adjustment Theorems. Results similar to Theorem 3.3 appear the tradition of LFIs, they are usually referred to as Derivability Adjust
istent logic to vagueness. EmpIrical infoRmatiOn cOulD gIve eVideNce of the fact thAT 'JudE Law is bald' (j) expresses a bOrderLiNE casE Of BaldnEss and iS, ThEN, An iNcOnSisTeNT sEntenCe. HOwever, sUitable empIriCaL information WOuLd also give EviDence that 'Ben kinGsley iS bAld' (B) Does nOt eXpresS a bordERline cAse, and in tHiS Case, we WOuld likE TO sTate That that sentence iS CoNSistent-that is, eIther cLaSSiCALly TruE, or classicAlLy falSE. The impORt OF THe aBOve fact can be bEtter graspeD WitH an epiStEmiC AnalogY: suppOsE ThaT reasoners oNly mAke given aSsumptIOns (i.e., asSErt the fOrmulaS in A prEmisE-SeT) iF thEy HAve REcEivED peRsuasive InFoRmatiOn-thAT IS, InfoRmaTion That lEads you to attrIbuTe prOBabIlity 1 To the FormUlAs in qUestioN. Then, ThE asymmetry above EquaTes with thE faCt ThaT rEasonERs can bE suRe tHat an inConsistENcy Is AT STaKe, but they cannot be sUrE THaT the situAtion iS PeRfECtly consIsTenT. By cONTrast, RecaPTuRe via norMality DOeS nOt bring AnY asymmEtRy iN. AgAin, we CAn adD φ ∧ ¬φ aftEr receivIng evIDence of inconsiSTency of φ. HowevER, wE CAn ALso aDd ⊛φ,⊛(Φ ⊃ ψ) if we get evIdenCE thaT we aRE fAciNG a norMal caSe. iN pARticular, in LP⊛ we can usE j ∧ ¬J to staTe thaT 'Jude Law is balD and not balD' (HENce, a 14 NormAlitY OpERators and ClassIcal REcapture in eXtensionS of KlEene LogiCs borderlINE case of bAldNesS) anD b ∧ ⊛b TO StAte that 'Ben KinGSLey iS bAld and iS noT a bordeRliNe cAse Of bAlDness.' By ruNning the EpIsTeMiC anAlogy AGain: with ReCapTuRe vIa norMAlity, rEasonErs cAn Be SUre That an iNCoNSIsteNcY iS at sTakE, bUt theY can ALso Be sure tHat the sitUatIOn is PeRfEctly coNsistent. IndeeD, iNformation CaN inCrease BOTh becausE evidence of inconsistencY And evidEncE of coNsisTency.12 5 ExteNsiOns of KLeeNE LogicS, NormaLity OPeRatORS, LogiCS Of forMaL InconsistENCy aNd UndEtErmiNedness in this section, we go tHRouGh the connectiOns BetwEEN Lp⊛ anD lFiS, ThEoREm 3.3 aND derivability AdjUstment TheOrEMs [11, 20], DefinabiliTY isSuEs and seManticaL closURe. TheorEm 3.3 and DeriVability ADjUstmENT ThEorems. ResuLts similAr to TheorEM 3.3 appeAR iN the tRadItion oF LfIs, Where They arE UsuAlly rEferreD tO as DerIvabiLiTy Adjust
istent logic to vagueness.  Empirical  info rma tio ncoul d gi ve evidence of thefact that 'Jude Law is  bald '( j) e x pr esses  a bord e rl i n e c as eofba l dn ess a ndis, the n, an inco nsi st ent sentence .  H owever, su ita ble empirica l i nforma ti onw ouldals o giv e evid e nce th at 'Ben K in g sley i s  bald'( b )does  not express a bo r de r line case, and  in th is ca s e , w e w ould liketo  stat e  that t h at s e nte n ce is consist ent-that is ,  ei ther c la ssi c ally t rue,or cla ssically fa lse.  The impo rt oft he abov e  fact c an bebet ter  gra s pe dwit ha n e p is tem i c a nalogy:su pp ose t hatr e a s oner s o nlymakegiven assumpt ion s (i . e.,  asse rt th e fo rm ulasin a p remis e- set) if they ha ve r eceived p ers ua siv einfor m ation- tha t i s, info rmation tha tl e a ds  you to attributepr o b ab ility 1to the fo rm u las in q ue sti on.T h en, t he a s ym metry ab ove eq u at es  with t he  factth atrea soner s  can  be su re thatan in c onsistency isa t stake, butt he y ca n notbesure that t he s i tuat ioni sper f ectly  cons is t en t . By contrast, reca pt ure vi a nor mality does n ot bring a n y asymmetr y in .  A g ain, we can ad d φ ∧  ¬φ afterr eceiving  evid ence ofinconsist e n cy of φ.  Ho wev er,  we c an  also add ⊛φ, ⊛ ( φ ⊃ψ)  if weget  eviden cetha t w e a re  facing a  normalca se .In  pa rticu l ar, in L P⊛  we  c anuse j ∧ ¬j t o sta te t ha t' Jud e Law i s  b a l d an dno t ba ld'  ( hence , a1 4 N ormalit y operato rsa nd C la ss ical Re capture in Ex te nsions ofKl een e Logi c s  borderl ine case of baldness) a n d b ∧ ⊛ b t o sta te t hat 'BenKin gsleyisb ald an d is n ot abo rde r l ine c a s eofba ldness.' B y run ningth e ep istemic  analogy again: wi t h r ecapture vianor mali t y ,rea s on e rsca n  be s ure that an inc onsistency  i s  a t stake, b u t t he y can a lso besuret hat the  situatio n is perf ec tlyc o nsi stent. Ind eed, inf ormationc an in c re ase b oth  becau se  ev idenc e of i n con siste ncy an deviden ce of  c onsisten cy.12 5 Extensions of K leeneLogic s,Normality  Op e rat ors, Logi cs o f Formal I nco nsi stenc y a n d Und eter m in edn e ss In  thi s  section, we  go t hr ough the co n n e cti ons b etw e en LP⊛  and  LFIs, Theorem 3. 3  and Derivabil ityA d jus tme n t Th eo rems [11, 20],  de fi n a bility i ss ues and sem anticalcl o sure.  Theor em 3.3  and De r i va b ilityAdju stm ent Theor ems .R esultssi mi l ar toTheo re m 3.3appear in t h e  tradition of LF Is, w h e re th e y a re us ua lly ref e rred  to as Der ivability A djust
istent logic_to vagueness._Empirical information could give_evidence of_the_fact that_'Jude_Law is bald'_(j) expresses a_borderline case of baldness_and is, then,_an_inconsistent sentence. However, suitable empirical information would also give evidence that 'Ben Kingsley is_bald'_(b) does_not_express_a borderline case, and in_this case, we would like_to state_that that sentence is consistent-that is, either classically_true,_or classically false._The import of the above fact can be better_grasped with an epistemic analogy: suppose_that reasoners only_make_given_assumptions (i.e., assert the_formulas in a premise-set) if they_have received persuasive information-that is, information_that leads you to attribute probability 1_to the formulas in question. Then,_the asymmetry above equates with_the fact_that reasoners can be sure_that an inconsistency_is at_stake, but they_cannot be sure that the situation_is perfectly consistent._By contrast, recapture via normality does_not_bring any asymmetry_in._Again,_we can_add φ ∧_¬φ_after receiving_evidence_of inconsistency of φ. However, we_can_also add ⊛φ,⊛(φ ⊃ ψ) if we_get evidence that we_are_facing a normal case._In particular, in LP⊛ we_can use j ∧ ¬j to_state that_'Jude Law_is bald and not bald' (hence, a 14 Normality operators and_Classical Recapture in Extensions of Kleene_Logics borderline case of_baldness) and_b_∧ ⊛b to_state_that 'Ben_Kingsley is bald and is not a_borderline case_of baldness.' By running the epistemic_analogy again: with recapture_via_normality, reasoners can be sure that_an inconsistency is at stake, but_they can also be sure_that_the_situation is perfectly consistent. Indeed,_information can increase both because evidence_of inconsistency and_evidence of consistency.12 5 Extensions of Kleene_Logics,_Normality Operators, Logics of Formal Inconsistency_and_Undeterminedness In this section, we go_through_the_connections between LP⊛ and LFIs,_Theorem 3.3 and Derivability Adjustment Theorems_[11, 20], definability issues and semantical closure. Theorem 3.3_and Derivability Adjustment_Theorems. Results similar to Theorem_3.3_appear_in the tradition of LFIs, where they are usually referred_to as_Derivability Adjust