 additional reasoning power would help up model situations where φ ∧ ¬φ is included in the initial premise-set, and information that ⊛φ is released. In the basic LP⊛ setting, {φ ∧ ¬φ,⊛φ} has no model, but if we describe this situation via a revision-like operator, we will end up in a situation where φ ∧ ¬φ and its consequences are dropped from the premise-set, in favor of ⊛. Of course, the new setting would also capture the opposite situation, where information of φ ∧ ¬φ trumps previous information that ⊛φ, via AGM-style revision. Normality operators and Classical Recapture in Extensions of Kleene Logics 15 theory. Also, our comparison with other approaches (Section 4) has highlighted the conceptual merits of recapture via normality. In turn, this shows that the approach followed by us and researchers in the LFI/LFU traditions is not only mathematically interesting, but also an appealing tool for conceptual applications (for instance, to revision of assumptions in presence of new information). LP and the Logic LFI1 of formal inconsistency. The language L3(P) of the logic of formal inconsistency LFI1 is defined by the following Backus-Naur form (BNF): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ φ→ ψ ∣ ●φ The logic LFI1 has been first introduced by [10], and it obtains by setting DLFI1 = { 1 2,1}, and by extending a valuation function n ∶ P z→ {0, 1 2,1} to arbitrary formulas L3 in conformity with Table 3 below: Table 2 ¬φ ●φ 1 0 0 1 2 1 2 1 0 1 0 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ→ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 The following established fact that will be relevant below. Proposition 5.1 The following negation connective ∼ is definable in LP⊛ and LFI1: ∼ φ ∶=LP⊛ �
additional reasoning power would help up model situation where φ ∧ ¬φ is include in the initial premise - set, and information that ⊛φ is released. In the basic LP⊛ context, { φ ∧ ¬φ,⊛φ } has no model, but if we report this situation via a revision - alike operator, we will end up in a situation where φ ∧ ¬φ and its consequence are dropped from the premise - set, in party favor of ⊛. Of course, the new place setting would also capture the diametric situation, where information of φ ∧ ¬φ trumps former information that ⊛φ, via AGM - style revision. Normality operators and Classical Recapture in Extensions of Kleene Logics 15 theory. besides, our comparison with other approach path (Section 4) has highlighted the conceptual merits of recapture via normality. In turning, this shows that the approach followed by us and researchers in the LFI / LFU traditions is not merely mathematically interesting, but also an appealing tool for conceptual applications (for instance, to revision of assumptions in presence of new information). LP and the Logic LFI1 of conventional inconsistency. The language L3(P) of the logic of courtly incompatibility LFI1 is define by the following Backus - Naur form (BNF ): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ φ→ ψ ∣ ● φ The logic LFI1 has been first introduced by [ 10 ], and it obtains by setting DLFI1 = { 1 2,1 }, and by exsert a valuation function n ∶ P z→ { 0, 1 2,1 } to arbitrary formulas L3 in conformity with board 3 below: Table 2 ¬φ ● φ 1 0 0 1 2 1 2 1 0 1 0 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ→ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 The following established fact that will be relevant downstairs. Proposition 5.1 The following negation connective ∼ is definable in LP⊛ and LFI1: ∼ φ ∶=LP⊛ �
 adfitional reasoning power would help up model smtuatiohs where φ ∧ ¬φ is included in the initiel peemist-fet, and informatiov that ⊛φ id releaswd. Ii the basic LP⊛ sxfting, {φ ∧ ¬φ,⊛φ} has hl mobeo, but if we dexcribe thiv situation vid x xevision-like operator, we will end up in a sotkation where φ ∧ ¬φ anc its boksequences are dropped from the lremise-vet, in favor pf ⊛. Of course, the new settlng aould also capture the opposijs syruation, wherd informatpmn of φ ∧ ¬φ ttumps previous information that ⊛φ, via CGM-style reciwioj. Normality iperanors and Classical Rewapture in Extensions of Klwene Logics 15 theory. Anso, our comparison with othar approaches (Sectiin 4) has higvligfred tht cpndeptuap mxrits of redapture via normality. In turn, uhif shows that tge appwowch followed by us and researchers in tve MFI/LFU traditions is nor only mathematically interestyng, but also an appealing tool for conceptual appnicatmovs (yir inrrajce, to revision of assumptions in presence of hee pnformation). LP anb the Logic LFI1 pf fptmal inconsistgncy. Thz lznguage L3(P) of the pogic os fornal inconfistrncy LFI1 is defined by the dollowing Babkus-Baur form (BNF): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ φ→ ψ ∣ ●φ Jhe lobic LFI1 has been first nntrodhced by [10], anf it obtajvs by setting DLWI1 = { 1 2,1}, and by extending a valuaeion funcvion u ∶ P z→ {0, 1 2,1} to arbitwary formupas L3 in conformity with Tablg 3 belmw: Table 2 ¬φ ●φ 1 0 0 1 2 1 2 1 0 1 0 φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ→ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 The follownng estaylishea fact than will be relevant bqlow. Proposithln 5.1 The follmwing nedatiin cinnectixd ∼ is definablr in LP⊛ aub LFI1: ∼ φ ∶=LP⊛ �
additional reasoning power would help up model φ ¬φ is in the initial is In the basic setting, {φ ∧ has no model, but if we this situation via a revision-like operator, we will end up in a situation φ ∧ ¬φ and its consequences are dropped from the premise-set, in favor ⊛. course, new would also capture the opposite situation, where information of φ ∧ ¬φ trumps previous information that via AGM-style revision. Normality operators and Classical Recapture Extensions of Kleene Logics theory. Also, our comparison with approaches 4) has the merits recapture via normality. turn, this shows that the approach followed by us and researchers in the LFI/LFU traditions is not mathematically interesting, an appealing for applications instance, to revision in presence of new information). LP LFI1 of formal inconsistency. The language L3(P) of logic of inconsistency LFI1 is defined by the Backus-Naur form (BNF): φ ∶∶= p ∣ ¬φ φ ∨ ψ ∣ φ ∧ ψ ∣ φ→ ψ ∣ ●φ The logic LFI1 first introduced by [10], it obtains by DLFI1 { 2,1}, by extending valuation function n ∶ P z→ {0, 1 2,1} to arbitrary L3 in conformity with Table 3 below: Table 2 ¬φ 0 1 2 1 1 0 1 0 ∨ 1 1 2 0 1 1 2 2 1 1 2 0 ∧ ψ 1 1 2 1 1 1 2 1 2 0 0 0 0 0 φ→ 1 1 2 0 1 1 1 0 1 2 1 1 2 0 0 1 1 1 The established fact be relevant below. Proposition 5.1 The following negation ∼ is definable in and LFI1: ∼ φ ∶=LP⊛ �
 additional reasoning power wOuld help up Model SitUatIoNs whEre φ ∧ ¬Φ is included in tHE iniTial premise-set, and inforMatioN tHAt ⊛φ iS ReLeaseD. In the bASiC lp⊛ seTtInG, {φ ∧ ¬φ,⊛Φ} hAS nO modeL, buT if we deScribe this SitUaTion via a reviSIoN-like operaTor, We will end up iN a sItuatiOn WheRE φ ∧ ¬φ anD itS consEquencES are drOpped from ThE PremisE-Set, in faVOR oF ⊛. Of cOurse, the new settinG WoULd also capture tHe oppoSiTE sITUatIon, Where inforMaTion oF Φ ∧ ¬φ trumpS PrEVIOus INformation thaT ⊛φ, via AGM-styLE reVision. noRmaLIty opeRatorS aND ClAssical RecaPturE in ExtensIons of kLeene LoGIcs 15 theoRy. Also, Our ComPariSOn WiTh oThER apPRoAchES (SeCtion 4) has HiGhLightEd thE CONCeptUal MeriTs of rEcapture via noRmaLity. iN tuRn, thiS showS thaT tHe appRoach fOllowEd By us and researchErs iN the LFI/LFu trAdItiOnS is noT Only maTheMatIcally iNterestINg, bUt ALSO aN appealing tool for cOnCEPtUal appliCationS (FoR iNStance, to ReVisIon oF ASsumpTionS In Presence Of new iNFoRmAtion). LP AnD the LoGiC LFi1 of FormaL IncoNsisteNcy. The laNguagE l3(P) of the logic of FOrmal inconsisTEnCY lFi1 Is deFinEd by the follOwinG backUs-NaUR fOrm (bnF): φ ∶∶= p ∣ ¬φ ∣ Φ ∨ ψ ∣ φ ∧ ψ ∣ φ→ Ψ ∣ ●φ tHe LOgic LFI1 has been first InTroducEd by [10], aNd it obtains by Setting DLFi1 = { 1 2,1}, AND by extenDing A VaLUation function N ∶ P z→ {0, 1 2,1} to Arbitrary fORmulas L3 iN confOrmity wiTh Table 3 beLOW: Table 2 ¬φ ●φ 1 0 0 1 2 1 2 1 0 1 0 Φ ∨ ψ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 φ ∧ Ψ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 φ→ ψ 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 the FolLOWiNg established FACt thAt Will be rEleVant belOw. PRopOsiTioN 5.1 THe followiNg negatiOn CoNnEcTivE ∼ is deFInable in lP⊛ And lFi1: ∼ φ ∶=Lp⊛ �
 additional reasoning powe r would he lp up  mo del  s itua tion s where φ ∧ ¬φ is i ncluded in the initial  prem is e -set ,  a nd in formati o nt h at⊛φ  i s r el e as ed. I n t he basi c LP⊛ sett ing ,{φ ∧ ¬φ,⊛φ}h as  no model,  bu t if we desc rib e this  s itu a tionvia  a re vision - like o perator,we will e n d up in a  s itua tion where φ ∧ ¬φ an d  its consequen ces ar ed ro p p edfro m the prem is e-set ,  in fav o ro f ⊛.O f course, the  new settin g  wo uld al so  ca p ture t he op po s ite  situation,  whe re inform ationo f φ ∧ ¬ φ  trumps  previ ous  in form a ti on  th at ⊛φ, vi a A G M-s tyle rev is io n. No rmal i t y oper ato rs a nd Cl assical Recap tur e in Ext ensio ns of  Kle en e Log ics 15  theo ry . Also, our com pari son withoth er  ap pr oache s  (Sect ion  4)  has hi ghlight e d t he c o nc eptual merits of r ec a p tu re via n ormali t y.  I n  turn, t hi s s hows t hat t he a p pr oach fol lowedb yus  and re se archer sinthe  LFI/ L FU t raditi ons is n ot on l y mathematical l y interesting ,  b u t  a l so a n a ppealing to ol f o r co ncep t ua l a p plica tions  ( f or instance, to revisi on  of as sumpt ions in prese nce of new i n formatio n).L Pa nd the Logic L FI1 o f formal i n consiste ncy.The lang uage L3(P ) of the l ogi c o f f orm a l  i nconsistencyL F I1 i sdefined  by  the fo llo win g B ack us -Naur for m (BNF):  φ  ∶ ∶=  p  ∣¬φ ∣φ  ∨ ψ ∣ φ  ∧  ψ∣φ→ψ ∣ ● φ  The l ogicLFI1  h as bee n first in t r oduc ed  b y [1 0],  a nd it  obt a ins  by set ting DLFI 1 = { 12, 1} , and b y extending a  v aluation f un cti on n ∶ P  z→ {0,1 2,1} to arbitrary for m ulas L3  in  conf ormi ty with T abl e 3 be low :  Table  2 ¬φ●φ 100 1 2  1 21 01 0  φ  ∨ ψ 1 1 2 0  11 1 1  1  2 1  1 2 12 0 1 1 2 0 φ ∧ ψ1  12 0 1 1 1 2 0  12 12 12 0 00  00φ → ψ 1  1 2 0 1 1 1 20 1 2 1 120  0  1 1 1 The fol lo wing es tablish ed fa c t thatwill be r elevant b el ow.P r opo sition 5.1  The fol lowing ne g ation co nnect ive  ∼ isde fin ablein LP⊛ and  LFI1 : ∼ φ∶= LP⊛ �
 additional_reasoning power_would help up model_situations where_φ_∧ ¬φ_is_included in the_initial premise-set, and_information that ⊛φ is_released. In the_basic_LP⊛ setting, {φ ∧ ¬φ,⊛φ} has no model, but if we describe this situation_via_a revision-like_operator,_we_will end up in a_situation where φ ∧ ¬φ_and its_consequences are dropped from the premise-set, in favor_of_⊛. Of course,_the new setting would also capture the opposite situation,_where information of φ ∧ ¬φ_trumps previous information_that_⊛φ,_via AGM-style revision. Normality_operators and Classical Recapture in Extensions_of Kleene Logics 15 theory. Also,_our comparison with other approaches (Section 4)_has highlighted the conceptual merits of_recapture via normality. In turn,_this shows_that the approach followed by_us and researchers_in the_LFI/LFU traditions is_not only mathematically interesting, but also_an appealing tool_for conceptual applications (for instance, to_revision_of assumptions in_presence_of_new information)._LP and the_Logic_LFI1 of_formal_inconsistency. The language L3(P) of the_logic_of formal inconsistency LFI1 is defined by_the following Backus-Naur form_(BNF):_φ ∶∶= p ∣_¬φ ∣ φ ∨ ψ_∣ φ ∧ ψ ∣ φ→_ψ ∣_●φ The_logic LFI1 has been first introduced by [10], and it obtains_by setting DLFI1 = { 1_2,1}, and by extending_a valuation_function_n ∶ P_z→_{0, 1_2,1} to arbitrary formulas L3 in conformity_with Table_3 below: Table 2 ¬φ ●φ_1 0 0 1_2_1 2 1 0 1 0_φ ∨ ψ 1 1 2_0 1 1 1 1_1_2_1 1 2 1 2_0 1 1 2 0 φ_∧ ψ 1_1 2 0 1 1 1 2_0_1 2 1 2 1 2_0_0 0 0 0 φ→ ψ_1_1_2 0 1 1 1_2 0 1 2 1 1_2 0 0 1 1 1 The following established_fact that will_be relevant below. Proposition 5.1_The_following_negation connective ∼ is definable in LP⊛ and LFI1: ∼_φ ∶=LP⊛_�