 from the above and Definition 2.1, DCL = {1}. 3 Classical Recapture and Normality Operators Kleene logics have been applied to a wide spectrum of phenomena, including partial functions [17, 18], partial information [1], logic programs [14] (K3), semantical and set-theoretical paradoxes [13, 19, 23, 25], and vagueness [26, 27] (K3 and LP). To be sure, they are not the only many-valued logics to be applied to the phenomena above. For instance, the so-called weak Kleene logics Kw3 by [8] and PWK by [16] have been applied to set-theoretical paradoxes [8] and [16], semantical paradoxes, vagueness, ambiguity and denotational failure [16]. Also, the four-valued logic FDE has been designed to reason about information that can be partial (undetermined) or overabundant (inconsistent). The 'normality story' we sketched in Section 1 implies that we may want to use classical reasoning when possible. The problem of how this can be done is the problem of classical recapture. 6A conservative extension of a logic S is a superlogic S′ such that (i) the language of S′ extends the language of S, (ii) any valid formula or rule expressed in the language of S is already in S. 7That is, Σ ⊧S ψ⇒ Σ ⊧CL ψ for S ∈ {K3,LP} and Σ ⊧CL ψ /⇒ Σ ⊧S ψ. Normality operators and Classical Recapture in Extensions of Kleene Logics 7 3.1 Classical Recapture The intuitive idea of classical recapture is that classical reasoning can somehow be deployed in a weaker many-valued logic S. The very starting point of classical recapture is that S itself do not offer the tools to recapture classical consequence. Thus, a further logic S′ needs enter the stage. In particular, the role of S′ is to specify at which conditions we may infer the classical consequences of a given premise-set Σ. In order for this to be the case, we need S′ to be stronger than S. This can be done in two ways [2]: either (a) we keep the initial language L fixed and define S′ as (L,⊧S
from the above and Definition 2.1, DCL = { 1 }. 3 Classical Recapture and Normality Operators Kleene logics have been applied to a wide spectrum of phenomena, include fond functions [ 17, 18 ], partial data [ 1 ], logic programs [ 14 ] (K3), semantical and laid - theoretical paradoxes [ 13, 19, 23, 25 ], and vagueness [ 26, 27 ] (K3 and LP). To be certain, they are not the entirely many - valued logic to be applied to the phenomena above. For example, the so - called fallible Kleene logics Kw3 by [ 8 ] and PWK by [ 16 ] have been applied to set - theoretical paradoxes [ 8 ] and [ 16 ], semantical paradoxes, vagueness, ambiguity and denotational failure [ 16 ]. besides, the four - valued logic FDE has been designed to reason about data that can be partial (undetermined) or overabundant (inconsistent). The' normality story' we sketched in Section 1 imply that we may want to use classical reasoning when possible. The problem of how this can be perform is the problem of classical recapture. 6A conservative extension of a logic S is a superlogic S′ such that (i) the linguistic process of S′ extends the language of S, (ii) any valid formula or rule expressed in the language of S is already in S. 7That is, Σ ⊧S ψ⇒ Σ ⊧CL ψ for S ∈ { K3,LP } and Σ ⊧CL ψ /⇒ Σ ⊧S ψ. Normality operators and Classical Recapture in Extensions of Kleene Logics 7 3.1 Classical Recapture The intuitive idea of classical recapture is that classical reasoning can somehow be deploy in a weaker many - valued logic S. The very start item of authoritative recapture is that S itself do not offer the tools to recapture classical consequence. Thus, a further logic S′ needs enter the stage. In particular, the function of S′ is to specify at which conditions we may infer the classical consequences of a given premise - fructify Σ. In order for this to be the case, we need S′ to be stronger than S. This can be serve in two direction [ 2 ]: either (a) we observe the initial language L fixed and specify S′ as (L,⊧S
 frlm the above and Definitlon 2.1, DCL = {1}. 3 Classical Rxcapturs and Nofmality Operators Kleene logmcs yave veen applied to a wide spectrum of phenimene, including partmzl funcbnons [17, 18], parciel information [1], logic procrams [14] (K3), semandizap and set-theoretical paradoxes [13, 19, 23, 25], wnd vagiejess [26, 27] (K3 and LK). To nq sudv, uhey are not the only many-valued mogics uo be applied to tne phenomena above. For inshancf, the so-called wean Kleene lotics Jw3 by [8] and PDK by [16] havt yeen applies to set-theoretical paradoxes [8] xnd [16], xemantical pqrafmxes, vaguenxss, amfiguity and denotatimnal faolure [16]. Also, thc fouc-valyed logic FDE has beei designed to reason about inxoxmation that can be pqrrial (ondetarmivwd) ur kvxragundanh (iiconsistent). The 'normaluty story' we sketchtd yb Section 1 imllies ehwt we may want to use classical reasonitg shen possible. The problwm of how this can be done is ehe problem of classical recapture. 6A conservative exteiskon on a uiglc S is a superlogic S′ such that (i) the languads pf S′ extends the language og D, (oy) any valid furmula or rule expressed in the lagguagw of S is alrrady in S. 7That is, Σ ⊧S ψ⇒ Σ ⊧CL ψ dor S ∈ {K3,LP} aud Σ ⊧CL ψ /⇒ Σ ⊧S ψ. Normalitv operators cnd Clsssicsl Recapture in Extensiuns kf Kleene Llgics 7 3.1 Cmxssical Recapturd Tme hntuitivt idea of classicaj recaptuce is that cuassocal rqasoning cwn somehow be deployed ij a wgaker kany-valued logic S. The very starting poinv of classicak secdpture if thab S itself do njt offer the tpols to recaoture clasaical cmnsequence. Ehus, a furthes logic S′ neevs enter ehe wtagw. In pafgicular, the roke of S′ is to specidy at which conditlons ds may infer the clqssical consequrncds jf a gyeen premise-sat Σ. Kn utder wor this to ce tne case, we need S′ to be atronger than S. Thos can be eone in ewo ways [2]: eitner (a) we keep the lnitiel lanjuage K fyxed and define S′ as (L,⊧S
from the above and Definition 2.1, DCL 3 Recapture and Operators Kleene logics wide of phenomena, including functions [17, 18], information [1], logic programs [14] (K3), and set-theoretical paradoxes [13, 19, 23, 25], and vagueness [26, 27] (K3 and To be sure, they are not the only many-valued logics to be applied the above. instance, so-called weak Kleene logics Kw3 by [8] and PWK by [16] have been applied to set-theoretical [8] and [16], semantical paradoxes, vagueness, ambiguity and failure [16]. Also, the logic FDE has been designed reason information that be (undetermined) overabundant (inconsistent). The story' we sketched in Section 1 implies that we may want to use classical reasoning when possible. problem of can be is problem classical recapture. 6A of a logic S is a that (i) the language of S′ extends the of S, any valid formula or rule expressed the language of S is already in S. is, Σ ⊧S ψ⇒ Σ ⊧CL ψ for S ∈ {K3,LP} and Σ ⊧CL ψ ⊧S ψ. Normality operators Classical Recapture in of Logics 3.1 Recapture The idea of classical recapture is that classical reasoning can somehow be in a weaker many-valued logic S. The very starting point recapture that S itself not offer the tools recapture consequence. Thus, a further needs the the of is to specify at conditions we may infer the consequences of a given this to be the case, we need S′ be stronger than S. This can be in two ways [2]: either (a) we keep the initial language L and define (L,⊧S
 from the above and Definition 2.1, dCL = {1}. 3 ClassicAl RecAptUre AnD NorMaliTy Operators KleENe loGics have been applied to a Wide sPeCTrum OF pHenomEna, inclUDiNG ParTiAl FunCtIOnS [17, 18], partIal InformaTion [1], logic pRogRaMs [14] (K3), semanticaL AnD set-theoreTicAl paradoxes [13, 19, 23, 25], aNd vAgueneSs [26, 27] (k3 anD lP). To bE suRe, theY are noT The onlY many-valuEd LOgics tO Be appliED To The pHenomena above. For iNStANce, the so-called Weak KlEeNE lOGIcs kw3 bY [8] and PWK by [16] hAvE been APplied tO SeT-THEorETical paradoxeS [8] and [16], semantiCAl pAradoxEs, VagUEness, aMbiguItY And DenotationaL faiLure [16]. Also, tHe four-VAlued loGIc FDE haS been dEsiGneD to rEAsOn AboUt INfoRMaTioN ThaT can be paRtIaL (undeTermINED) Or ovEraBundAnt (inConsistent). The 'NorMaliTY stOry' we SketcHed iN SEctioN 1 impliEs thaT wE may want to use clAssiCal reasonIng WhEn pOsSible. tHe probLem Of hOw this cAn be donE Is tHe PROBlEm of classical recapTuRE. 6a cOnservatIve extENsIoN Of a logic s iS a sUperLOGic S′ sUch tHAt (I) the langUage of s′ ExTeNds the lAnGuage oF S, (Ii) aNy vAlid fORmulA or rulE expressEd in tHE language of S is ALready in S. 7That IS, Σ ⊧s Ψ⇒ σ ⊧Cl Ψ for s ∈ {K3,Lp} and Σ ⊧CL ψ /⇒ Σ ⊧S ψ. normALity OperAToRs aND ClasSical reCApTUre in Extensions of KlEeNe LogiCs 7 3.1 ClaSsical RecaptuRe The intuiTIVE idea of cLassICaL Recapture is thaT clasSical reasoNIng can soMehow Be deployEd in a weakER Many-valuEd lOgiC S. THe vERY sTarting point oF CLassIcAl recapTurE is that s itSelF do Not OfFer the tooLs to recaPtUrE cLaSsiCal coNSequence. thUs, a FuRthEr logIC S′ needS enteR the StAgE. in pArticulAR, tHE Role Of s′ iS to sPecIfY at whIch cONdiTions we May infer tHe cLAssiCaL cOnsequeNces of a given pReMise-set Σ. In OrDer For thiS TO be the caSe, we need S′ to be stronger thAN S. This cAn bE done In twO ways [2]: eithEr (a) We keep The INitial LanguaGe L fiXeD anD DEfine s′ AS (L,⊧s
 from the above and Defini tion 2.1,DCL =  {1 }.3Clas sica l Recapture an d  Nor mality Operators Kleen e log ic s  hav e  b een a ppliedt oa wid esp ect ru m  o f phe nom ena, in cluding pa rti al  functions [ 1 7,  18], part ial  information  [1 ], log ic  pr o grams  [1 4] (K 3), se m antica l and set -t h eoreti c al para d o xe s [1 3, 19, 23, 25], a n dv agueness [26,27] (K 3a nd L P).  To  be sure,th ey ar e  not th e  o n l y  ma n y-valued logi cs to be ap p lie d to t he  ph e nomena  abov e. For  instance,theso-called  weakK leene l o gics Kw 3 by [ 8]and  PWK by  [ 16]  h a veb ee n a p pli ed to se t- th eoret ical p a r adox es[8]and [ 16], semantic alpara d oxe s, va guene ss,am bigui ty and  deno ta tional failure[16] . Also, t hefo ur- va luedl ogic F DEhas  been d esigned tore a s o nabout informationth a t  c an be pa rtial( un de t ermined)  o r o vera b u ndant  (in c on sistent) . The' no rm ality s to ry' we  s ket che d inS ecti on 1 i mplies t hat w e  may want to u s e classical r e as o n in g  whe n p ossible. Th e pr o blem  ofh ow  th i s can  be d on e  i s  the problem of cla ss ical r ecapt ure. 6A conse rvative ex t e n sion ofa lo g ic S is a superlo gic S ′ such tha t  (i) the  lang uage ofS′ extend s the lang uag e o f S , ( i i )any valid for m u la o rrule ex pre ssed in  th e l ang uag eof S is a lready i nS.  7 Th atis, Σ ⊧S ψ⇒ Σ⊧C L ψ  f orS ∈ { K 3,LP}and Σ  ⊧CL  ψ  / ⇒  Σ⊧S ψ. N o rm a l ityop er ator s a nd  Clas sica l  Re capture  in Exten sio n s of  K le ene Log ics 7 3.1 Cla ss ical Recap tu reThe in t u itive id ea of classical recaptu r e is th atclass ical  reasonin g c an som eho w  be de ployed  in a  w eak e r  many - v al ued  l ogic S. Th e ver y sta rt ingpoint o f classical recapt u reis that S its elf  don o toff e rt heto o lst o  recapture clas sical cons eq u en ce. Thus,a  fu rt her log ic S′ n eedse nter th e stage.In partic ul ar,t h e r ole of S′is to sp ecify atw hichc on ditio nswe may  i nfe r the  class i cal  cons equenc es  of agiven  p remise-s et Σ. In order for this  to be  thecas e, we nee d S ′  to  be stron gerthan S. Th iscan  be d one in tw o wa y s[2] :  eith er ( a ) we keep th e i n i ti al language L fix ed an d d e fine S ′ as  (L,⊧S
 from_the above_and Definition 2.1, DCL_= {1}._3_Classical Recapture_and_Normality Operators Kleene_logics have been_applied to a wide_spectrum of phenomena,_including_partial functions [17, 18], partial information [1], logic programs [14] (K3), semantical and set-theoretical_paradoxes_[13, 19,_23,_25],_and vagueness [26, 27] (K3_and LP). To be sure,_they are_not the only many-valued logics to be applied_to_the phenomena above._For instance, the so-called weak Kleene logics Kw3 by_[8] and PWK by [16] have_been applied to_set-theoretical_paradoxes_[8] and [16], semantical_paradoxes, vagueness, ambiguity and denotational failure_[16]. Also, the four-valued logic FDE_has been designed to reason about information_that can be partial (undetermined) or_overabundant (inconsistent). The 'normality story'_we sketched_in Section 1 implies that_we may want_to use_classical reasoning when_possible. The problem of how this_can be done_is the problem of classical recapture._6A_conservative extension of_a_logic_S is_a superlogic S′_such_that (i)_the_language of S′ extends the language_of_S, (ii) any valid formula or rule_expressed in the language_of_S is already in_S. 7That is, Σ ⊧S_ψ⇒ Σ ⊧CL ψ for S_∈ {K3,LP}_and Σ_⊧CL ψ /⇒ Σ ⊧S ψ. Normality operators and Classical Recapture_in Extensions of Kleene Logics 7_3.1 Classical Recapture The_intuitive idea_of_classical recapture is_that_classical reasoning_can somehow be deployed in a weaker_many-valued logic_S. The very starting point of_classical recapture is that_S_itself do not offer the tools_to recapture classical consequence. Thus, a_further logic S′ needs enter_the_stage._In particular, the role of_S′ is to specify at which_conditions we may_infer the classical consequences of a given_premise-set_Σ. In order for this to_be_the case, we need S′ to_be_stronger_than S. This can be_done in two ways [2]: either_(a) we keep the initial language L fixed and_define S′ as_(L,⊧S