ment Theorems. [20, Theorem 3.11] establishes a similar classical recapture theorem for the paraconsistent logic bC [9], and [20, Theorem 3.46] proves a particular recapture that involves a satisfaction-preserving translation τ from CL to the paraconsistent logic Ci [20]. We briefly comment on the differences between Theorem 3.3 and these results. First, Derivability Adjustment Theorems involve the derivability relation-whence the name-and their proofs display a proof-theoretical approach, while the proof of Theorem 3.3 relies on semantical considerations. Second, Theorem 3.11 from [20] relies on adding the normal counterpart of some set ∆ of well-formed formulas, and not necessarily the normal counterpart of the premise-set Σ itself. Third, Theorem 3.46 from [20] relies on translating CL into Ci. By contrast, no translation is needed for our purposes. Fourth, the wide majority of Derivability Adjustment Theorems above apply to paraconsistent logics, and give no recipe for recapture in paracomplete logics, while Theorem 3.3 does. One exception is [11, Theorem 7.2.20], which applies to the paracomplete logic Kw3 from [8]. That result differs from Theorem 3.3 since it focuses on propositional variables from the conclusion, not on the conclusion itself. This focus is not necessary for Theorem 3.3. Fifth and final, none of the Derivability Adjustment Theorems comes with a comparison with other current recapture techniques-such as minimal inconsistency. We believe that these differences speak for the relevance of our approach and results. In particular, the semantical methodology that we follow in this paper allows for an easier comparison with other approaches to recapture, and especially with minimal inconsistency, which has a strong semantical focus, and a relatively undeveloped proof 12Notice that LP⊛ does not provide, per se, any belief-revision (or better, information-revision) procedure, contrary to LP⊛ (Graham Priest brought this to our attention in private conversation. However, it is easy to extend the semantical and linguistic setting of LP⊛ with operations analogous to expansion, contraction from AGM theory [3]. Once this is done, it is possible to use the resulting extension of LP⊛ in order to capture belief-revision. In particular, the
ment Theorems. [ 20, Theorem 3.11 ] establishes a similar classical recapture theorem for the paraconsistent logic bC [ 9 ], and [ 20, Theorem 3.46 ] proves a especial recapture that imply a satisfaction - preserving translation τ from CL to the paraconsistent logic Ci [ 20 ]. We concisely comment on the differences between Theorem 3.3 and these results. foremost, Derivability Adjustment Theorems necessitate the derivability relation - whence the name - and their proof display a proof - theoretical approach, while the validation of Theorem 3.3 relies on semantical considerations. Second, Theorem 3.11 from [ 20 ] relies on adding the normal counterpart of some set ∆ of well - form formulas, and not necessarily the normal counterpart of the premise - set Σ itself. Third, Theorem 3.46 from [ 20 ] relies on translating CL into Ci. By line, no translation is needed for our purposes. fourthly, the wide majority of Derivability Adjustment Theorems above apply to paraconsistent logics, and give no recipe for recapture in paracomplete logics, while Theorem 3.3 make. One exception is [ 11, Theorem 7.2.20 ], which applies to the paracomplete logic Kw3 from [ 8 ]. That result differs from Theorem 3.3 since it focuses on propositional variables from the stopping point, not on the conclusion itself. This focus is not necessary for Theorem 3.3. Fifth and final, none of the Derivability Adjustment Theorems comes with a comparison with other current recapture technique - such as minimal inconsistency. We believe that these differences speak for the relevance of our approach and results. In particular, the semantical methodology that we follow in this paper allows for an easy comparison with other approaches to recapture, and specially with minimal incompatibility, which hold a strong semantical focus, and a relatively unexploited proof 12Notice that LP⊛ does not leave, per se, any belief - revision (or good, information - revision) procedure, contrary to LP⊛ (Graham Priest brought this to our attention in secret conversation. However, it is easy to extend the semantical and linguistic setting of LP⊛ with operations analogous to expansion, compression from AGM theory [ 3 ]. Once this is dress, it is potential to practice the resulting extension of LP⊛ in order to capture belief - revision. In particular, the
menh Theorems. [20, Theorem 3.11] estxblishes a similar clavsical recaptufe theorem for the paraconsidtwnt ligic bC [9], and [20, Theorem 3.46] proves a particuoar cecapture that iibolves a satianactimi-preserving trakslation τ fsom CL to the [afaeonsistent logic Ci [20]. We briefly commqnt on yhf differences fetwten Thekgen 3.3 and these results. First, Dedivabilpty Adjustment Throrems involve the derivabllitj relation-whence tje name-and jgeiw proofs dispuay a proof-theoretical approach, while the proof of Thdorem 3.3 relies ob wemwttical consmderatpons. Second, Tmvorem 3.11 xrom [20] rrlies on addinn the noemal counterpart of smme set ∆ of well-fotmed formunaa, and not necessaeioy thg norkal ziungerkarv or the orejise-set Σ ifself. Third, Theorem 3.46 from [20] reloef on translatihg CL yneo Ci. By contrast, no translation is neeged for our purposes. Fourty, the wide majority ov Derivabylity Adjustment Theorems above apply to paraconshstenv uogngf, qnf give no recipe for recapture in paracompletq lpgpcs, while Theorem 3.3 does. One rxfeljion is [11, Theorgm 7.2.20], whieg zpplies to the parwcompleje loguc Kw3 froi [8]. Tnat result differs from Theirem 3.3 since pt ficuses on propositnonal variabues grom yhe conclusion, not on tke condlusion itsflf. This rucus is not necersagy fmr Theorto 3.3. Fifth and finaj, none of the Derivacilijy Adjuftment Thelrems comes with a compagison whth other furrent recapture techniques-suci as minimal onwonvistency. We bclieve that thefe differences speak yor thd relevancv of our epproach and results. In pdtticular, the vemanticwl mwthoeology gfat we follow on this pcker allows for an easier comiarisuh with other ap'xoqches to recaptire, anq vspxcialjf with minimdl ivcovxistevcy, which hcs a syrong semantical focgs, ahd a relatively unceyeloped ptoof 12Notise that LP⊛ dors not provide, per se, aiy belmef-revosijn (or better, information-revisikn) procedkre, contrary to LP⊛ (Nrahwm Priest yrought this to our attention in private conversation. Howgver, it is easy to exjenc the semanvical wnd linguhstic setting of LP⊛ qith operations akalogous to expansion, dontrawtion from AGM theory [3]. Once this is done, it is possible to use the resulting eztension of LP⊛ in krdet to bapcurz belies-revmsmon. In particular, the
ment Theorems. [20, Theorem 3.11] establishes a recapture for the logic bC [9], a recapture that involves satisfaction-preserving translation τ CL to the paraconsistent logic Ci We briefly comment on the differences between Theorem 3.3 and these results. First, Adjustment Theorems involve the derivability relation-whence the name-and their proofs display a proof-theoretical while proof Theorem relies on semantical considerations. Second, Theorem 3.11 from [20] relies on adding the normal counterpart of set ∆ of well-formed formulas, and not necessarily normal counterpart of the Σ itself. Third, Theorem 3.46 [20] on translating into By no translation is for our purposes. Fourth, the wide majority of Derivability Adjustment Theorems above apply to paraconsistent logics, and no recipe in paracomplete while 3.3 One exception is 7.2.20], which applies to the paracomplete [8]. That result differs from Theorem 3.3 since focuses on variables from the conclusion, not on conclusion itself. This focus is not necessary for 3.3. Fifth and final, none of the Derivability Adjustment Theorems comes with a comparison with recapture techniques-such as minimal We believe that differences for relevance our approach results. In particular, the semantical methodology that we follow in this allows for an easier comparison with other approaches to recapture, with inconsistency, which has strong semantical focus, and relatively proof 12Notice that LP⊛ provide, se, better, procedure, to LP⊛ (Graham Priest this to our attention in conversation. However, it is and linguistic setting of LP⊛ with operations analogous expansion, contraction from AGM theory [3]. Once is done, it is possible to use the resulting extension of LP⊛ order to In particular, the
ment Theorems. [20, Theorem 3.11] establIshes a simiLar clAssIcaL rEcapTure Theorem for the pARacoNsistent logic bC [9], and [20, TheoRem 3.46] prOvES a paRTiCular RecaptuRE tHAT inVoLvEs a SaTIsFactiOn-pReserviNg translatIon Τ fRom CL to the paRAcOnsistent lOgiC Ci [20]. We briefly ComMent on ThE diFFerenCes BetweEn TheoREm 3.3 and tHese resulTs. fIrst, DeRIvabiliTY adJustMent Theorems involVE tHE derivability rElatioN-wHEnCE The NamE-and their pRoOfs diSPlay a prOOf-THEOreTIcal approach, wHile the prooF Of THeorem 3.3 ReLieS On semaNticaL cONsiDerations. SeCond, theorem 3.11 frOm [20] reliES on addiNG the norMal couNteRpaRt of SOmE sEt ∆ oF wELl-fORmEd fORmuLas, and noT nEcEssarIly tHE NORmal CouNterPart oF the premise-seT Σ iTselF. thiRd, TheOrem 3.46 fRom [20] rElIes on TranslAting cL Into Ci. By contrasT, no tRanslatioN is NeEdeD fOr our PUrposeS. FoUrtH, the widE majoriTY of deRIVAbIlity Adjustment TheOrEMS aBove applY to parACoNsIStent logIcS, anD givE NO reciPe foR ReCapture iN paracOMpLeTe logicS, wHile ThEoRem 3.3 DoeS. One eXCeptIon is [11, THeorem 7.2.20], whIch apPLies to the paracOMplete logic Kw3 FRoM [8]. tHaT ResuLt dIffers from THeorEM 3.3 sinCe it FOcUseS On proPositIoNAl VAriables from the concLuSion, noT on thE conclusion itSelf. This foCUS Is not necEssaRY fOR Theorem 3.3. Fifth aNd finAl, none of thE derivabiLity ADjustmenT Theorems COMes with a ComParIsoN wiTH OtHer current recAPTure TeChniqueS-suCh as minImaL inConSisTeNcy. We beliEve that tHeSe DiFfEreNces sPEak for thE rEleVaNce Of our APproacH and rEsulTs. in PArtIcular, tHE sEMAntiCaL mEthoDolOgY that We foLLow In this pAper allowS foR An eaSiEr CompariSon with other aPpRoaches to rEcAptUre, and ESPecially With minimal inconsistencY, Which haS a sTrong SemaNtical focUs, aNd a relAtiVEly undEvelopEd proOf 12notICE that lp⊛ DoEs nOt Provide, per SE, Any BelieF-rEvisIon (or beTter, information-revISioN) procedure, conTraRy to lp⊛ (grAhaM prIEst BrOUghT THis to our attentiOn in privatE cONvErsation. HoWEveR, iT is easy To extenD the sEManticaL and linguIstic settInG of Lp⊛ WIth Operations AnalogouS to expansIOn, conTRaCtion FroM AGM thEoRy [3]. ONce thIs is doNE, it Is posSible tO uSe the rEsultInG extensiOn of LP⊛ in order to capture bElief-rEvisiOn. IN particulAr, tHE
ment Theorems. [20, Theore m 3.11] es tabli she s a  s imil ar c lassical recap t uretheorem for the paraco nsist en t  log i cbC [9 ], and[ 20 , The or em  3. 46 ]  p roves  aparticu lar recapt ure  t hat involves asatisfacti on- preserving t ran slatio nτ f r om CL  to  theparaco n sisten t logic C i[ 20]. W e  briefl y co mmen t on the differen c es between Theore m 3.3an d  t h e seres ults. Firs t,  Deri v ability Ad j u s tme n t Theorems in volve the d e riv abilit yrel a tion-w hence  t h e n ame-and the ir p roofs dis play a proof-t h eoretic al app roa ch,  whi l eth e p ro o f o f  T heo r em3.3 reli es  o n sem anti c a l cons ide rati ons.Second, Theor em3.11 fro m [20 ] rel ieson  addi ng the  norm al  counterpart of  som e set ∆ o f w el l-f or med f o rmulas , a ndnot nec essaril y  th en o r ma l counterpart of t he p re mise-set  Σ its e lf .T hird, Th eo rem  3.4 6 from[20] re lies ontransl a ti ng  CL int oCi. By  c ont ras t, no tran slatio n is nee ded f o r our purposes .  Fourth, thew id e ma j orit y o f Derivabil ityA djus tmen t  T heo r ems a boveap p ly to paraconsistent l og ics, a nd gi ve no recipefor recapt u r e  in para comp l et e  logics, while  Theo rem 3.3 do e s. One e xcept ion is [ 11, Theor e m  7.2.20] , w hic h a ppl i e sto the paraco m p lete  l ogic Kw 3 f rom [8] . T hat  re sul tdiffers f rom Theo re m3. 3sin ce it focuseson  pr op osi tiona l  varia blesfrom  t he con clusion ,  n o t  onth econc lus io n its elf. Thi s focus  is not n ece s sary  f or  Theore m 3.3. Fifthan d final, n on e o f theD e rivabili ty Adjustment Theoremsc omes wi tha com pari son withoth er cur ren t  recap ture t echni qu es- s u ch as m in ima linconsiste n c y.We be li evethat th ese differences sp e akfor the relev anc e of o ur  ap p ro a chan d  re s u lts. In particu lar, the s em a nt ical metho d olo gy  that w e follo w int his pap er allows  for an e as ierc o mpa rison with  other a pproaches to re c ap ture,  an d espe ci all y wit h mini m alincon sisten cy , whic h has  a  strongsemantical focus, and a  relat ively  un developed  pr o of12Noticethat  LP⊛ doesnot  pr ovide , p e r se,  any be lie f -revi sion (or bette r ,inf o r ma tion-revisi o n )  pr ocedu re, contra ry t o LP⊛ (Graham Pri e st brought thi s to o uratt e ntio nin private con ver sa t i on. Howe ve r, it is ea sy to ex te n d the  seman ticaland lin g u is t ic set ting  of  LP⊛ with  op er a tions a na lo g ous to  exp an sion,contra c tion f rom AGM theory [ 3]. O n c e thi s  is  done ,it is p o ssib le to usethe resulti ng ext ensi on of  LP⊛ in  o rder t o c ap ture belie f -revision . Inparticu la r, t he
ment Theorems._[20, Theorem_3.11] establishes a similar_classical recapture_theorem_for the_paraconsistent_logic bC [9],_and [20, Theorem_3.46] proves a particular_recapture that involves_a_satisfaction-preserving translation τ from CL to the paraconsistent logic Ci [20]. We briefly comment_on_the differences_between_Theorem_3.3 and these results. First,_Derivability Adjustment Theorems involve the_derivability relation-whence_the name-and their proofs display a proof-theoretical approach,_while_the proof of_Theorem 3.3 relies on semantical considerations. Second, Theorem 3.11_from [20] relies on adding the_normal counterpart of_some_set_∆ of well-formed formulas,_and not necessarily the normal counterpart_of the premise-set Σ itself. Third,_Theorem 3.46 from [20] relies on translating_CL into Ci. By contrast, no_translation is needed for our_purposes. Fourth,_the wide majority of Derivability_Adjustment Theorems above_apply to_paraconsistent logics, and_give no recipe for recapture in_paracomplete logics, while_Theorem 3.3 does. One exception is_[11,_Theorem 7.2.20], which_applies_to_the paracomplete_logic Kw3 from_[8]._That result_differs_from Theorem 3.3 since it focuses_on_propositional variables from the conclusion, not on_the conclusion itself. This_focus_is not necessary for_Theorem 3.3. Fifth and final,_none of the Derivability Adjustment Theorems_comes with_a comparison_with other current recapture techniques-such as minimal inconsistency. We believe that_these differences speak for the relevance_of our approach and_results. In_particular,_the semantical methodology_that_we follow_in this paper allows for an easier_comparison with_other approaches to recapture, and especially_with minimal inconsistency, which_has_a strong semantical focus, and a_relatively undeveloped proof 12Notice that LP⊛_does not provide, per se,_any_belief-revision_(or better, information-revision) procedure, contrary_to LP⊛ (Graham Priest brought this_to our attention_in private conversation. However, it is easy_to_extend the semantical and linguistic setting_of_LP⊛ with operations analogous to expansion,_contraction_from_AGM theory [3]. Once this_is done, it is possible to_use the resulting extension of LP⊛ in order to_capture belief-revision. In_particular, the