′), or (b) we increase the expressive power of S and define S′ as a conservative extension (L′,⊧S′) of S. The two options naturally suggest a formal definition of classical recapture: Problem 3.1 (Classical Recapture) For every many-valued logic S weaker than CL and set Σ ∪ {ψ} ⊆ L of formulas, the problem of classical recapture is the problem of finding a consequence relation ⊧S′ and a set Σ′ of formulas such that: (i) Σ ⊧CL ψ ⇔ Σ′ ⊧S′ ψ; (ii) Σ ⊆ Σ′; (iii) Σ ⊧S ψ⇒ Σ ⊧S′ ψ; (iv) Σ′ ⊆ L or Σ′ ⊆ L′ for some extension L′ of L. Accordingly, every solution to Problem 3.1 will have the form imposed by condition (i) above, and will fulfill conditions (ii)-(iv). If S′ satisfies condition (i)-(iv), we will call it a classical recapture logic for S. Also, we might want to recapture the inference of a many-valued logic S within a weaker logic S′. This can be easily done by a natural adjustment of the definition of Problem 3.1, thus yielding a more general problem of S-recapture. We will face one such case in Section 4. To the best of our knowledge, none has given a formal definition of the problem of classical recapture before. With our present definition, we hope we can help settle a reasonable package of formal requirements to impose on proposals that aim at providing techniques for classical recapture. The approach by [24] fall under option (a) above; extension of a logic S with normality operators fall under option (b). We go to this approach now, and we show that LP⊛ and K⊛3 are classical recapture logics for LP and K3. 3.2 Recapture via Normality Here we establish a recapture result based of LP⊛ and K⊛3. First, we need an auxiliary notion: Definition 3.2 (Normal Counterpart) Given a set Σ ⊆ L2, we say that the set Σ⊛ = {⊛φ ∈ L1 ∣ φ ∈ Σ} is the normal counterpart of Σ in
′), or (b) we increase the expressive power of S and define S′ as a conservative extension (L′,⊧S′) of S. The two choice naturally indicate a formal definition of classical recapture: Problem 3.1 (Classical Recapture) For every many - prize logic S weaker than CL and set Σ ∪ { ψ } ⊆ L of formula, the trouble of classical recapture is the problem of finding a consequence relation ⊧S′ and a set Σ′ of formula such that: (i) Σ ⊧CL ψ ⇔ Σ′ ⊧S′ ψ; (ii) Σ ⊆ Σ′; (iii) Σ ⊧S ψ⇒ Σ ⊧S′ ψ; (iv) Σ′ ⊆ L or Σ′ ⊆ L′ for some extension L′ of L. Accordingly, every solution to Problem 3.1 will get the form imposed by condition (i) above, and will fulfill conditions (ii)-(iv). If S′ satisfies circumstance (i)-(iv), we will call it a classical recapture logic for S. Also, we might desire to recapture the inference of a many - valued logic S within a weaker logic S′. This can be easily done by a natural alteration of the definition of Problem 3.1, thus yielding a more general problem of S - recapture. We will front one such case in Section 4. To the best of our knowledge, none has given a formal definition of the problem of classical recapture before. With our present definition, we hope we can help settle a reasonable package of formal requirements to impose on proposals that drive at providing techniques for authoritative recapture. The overture by [ 24 ] decrease under option (a) above; extension of a logic S with normality operators fall under option (b). We go to this overture now, and we show that LP⊛ and K⊛3 are classical recapture logics for LP and K3. 3.2 Recapture via Normality Here we establish a recapture result base of LP⊛ and K⊛3. First, we need an auxiliary notion: Definition 3.2 (Normal Counterpart) give a set Σ ⊆ L2, we order that the set Σ⊛ = { ⊛φ ∈ L1 ∣ φ ∈ Σ } is the normal counterpart of Σ in
′), or (b) we increase the exprersive power of S and dxfine S′ as a covservative extension (L′,⊧S′) of S. Tye twi options naturally suegest a flrmal dediniuion of classical recaptuvz: Progpem 3.1 (Rlassical Recapjure) For evesy many-valued noeie S weaker than CL and set Σ ∪ {ψ} ⊆ L of sormulax, hhe problem of claxficam recapture is the problem of findjng a cmnsequence rekation ⊧S′ and a set Σ′ of forlulad such that: (i) Σ ⊧CL ψ ⇔ Σ′ ⊧S′ ψ; (ii) Σ ⊆ Σ′; (iii) Σ ⊧S ψ⇒ Σ ⊧S′ ψ; (iv) Σ′ ⊆ L or Σ′ ⊆ L′ for some extgnsion L′ of L. Accordingly, every rolutnon to Proboen 3.1 ahll have thx form imposed by gpnditimn (i) abpve, and will fmlfiln cinditions (ii)-(iv). If S′ setisfies condition (i)-(yv), we wiln eall it a classical rwcqpturg loghc fue S. Alao, ws mighh went to recalture the ibference of a many-vsltvc logic S wifhin a wqaker logic S′. This can be easily done bj a hatural adjustment of tye definition of Probpem 3.1, thus yielding a more general problem of S-recapture. We will wact ine sjxh case in Section 4. To the best of our knowledgq, npnv has given a forial definitooj ps the problem of clcasjcal recapture beflre. Witr our present qefimition, we hope we can help wettle a reafinable package of yormal requixementx to ompose on proposals thac aim zt providinh techniqhds for classical rebaptgre. The approach by [24] fall tnder optmon (a) above; dxtemsion jf a logic S wibv normality operatlrs fcll utder optioj (b). We go to this approach now, eid we show thst LP⊛ and K⊛3 axe claxsical recapttre logics for LP and K3. 3.2 Rdcapture vpa Normalmty Here we qstablish a rafapture resunt based of OP⊛ abd K⊛3. Fifrt, we need an suxiliary notion: Dedinition 3.2 (Normal Cpunjedpart) Given a szu Σ ⊆ L2, we say that yhe see Σ⊛ = {⊛φ ∈ J1 ∣ φ ∈ Σ} is the tormxl zpuntefpart of Σ ik
′), or (b) we increase the expressive S define S′ a conservative extension options suggest a formal of classical recapture: 3.1 (Classical Recapture) For every many-valued S weaker than CL and set Σ ∪ {ψ} ⊆ L of formulas, problem of classical recapture is the problem of finding a consequence relation ⊧S′ a Σ′ formulas that: (i) Σ ⊧CL ψ ⇔ Σ′ ⊧S′ ψ; (ii) Σ ⊆ Σ′; (iii) Σ ⊧S Σ ⊧S′ ψ; (iv) Σ′ ⊆ L or ⊆ L′ for some L′ of L. Accordingly, every to 3.1 will the imposed condition (i) above, will fulfill conditions (ii)-(iv). If S′ satisfies condition (i)-(iv), we will call it a classical recapture logic S. Also, want to the of many-valued logic S weaker logic S′. This can be a natural adjustment of the definition of Problem thus yielding more general problem of S-recapture. We face one such case in Section 4. To best of our knowledge, none has given a formal definition of the problem of classical With our present definition, hope we can settle reasonable of requirements to on proposals that aim at providing techniques for classical recapture. The by [24] fall under option (a) above; extension of a with operators fall under (b). We go to approach and we show that K⊛3 classical LP K3. Recapture via Normality Here establish a recapture result based LP⊛ and K⊛3. First, Definition 3.2 (Normal Counterpart) Given a set Σ L2, we say that the set Σ⊛ {⊛φ ∈ L1 ∣ φ ∈ Σ} is the normal counterpart of in
′), or (b) we increase the expressivE power of S aNd defIne s′ as A cOnseRvatIve extension (L′,⊧S′) OF S. ThE two options naturally suGgest A fORmal DEfInitiOn of claSSiCAL reCaPtUre: prOBlEm 3.1 (ClaSsiCal RecaPture) For evEry MaNy-valued logiC s wEaker than Cl anD set Σ ∪ {ψ} ⊆ L of forMulAs, the pRoBleM Of claSsiCal reCapturE Is the pRoblem of fInDIng a coNSequencE RElAtioN ⊧S′ and a set Σ′ of formuLAs SUch that: (i) Σ ⊧CL ψ ⇔ Σ′ ⊧S′ Ψ; (ii) Σ ⊆ Σ′; (iIi) σ ⊧s ψ⇒ σ ⊧s′ Ψ; (iv) σ′ ⊆ L oR Σ′ ⊆ L′ for some ExTensiON L′ of L. AcCOrDINGly, EVery solution tO Problem 3.1 wilL HavE the foRm ImpOSed by cOnditIoN (I) abOve, and will fUlfiLl conditiOns (ii)-(iV). if S′ satiSFies conDition (I)-(iv), We wIll cALl It A clAsSIcaL ReCapTUre Logic for s. ALsO, we miGht wANT TO recAptUre tHe infErence of a many-ValUed lOGic s withIn a weAker LoGic S′. THis can Be easIlY done by a natural AdjuStment of tHe dEfIniTiOn of PROblem 3.1, tHus YieLding a mOre geneRAl pRoBLEM oF S-recapture. We will fAcE ONe Such case In SectIOn 4. to THe best of OuR knOwleDGE, none Has gIVeN a formal DefiniTIoN oF the proBlEm of clAsSicAl rEcaptURe beFore. WiTh our preSent dEFinition, we hope WE can help settlE A rEASoNAble PacKage of formaL reqUIremEnts TO iMpoSE on prOposaLs THaT Aim at providing technIqUes for ClassIcal recapture. the approacH BY [24] Fall undeR optIOn (A) Above; extension Of a loGic S with noRMality opEratoRs fall unDer option (B). wE go to thiS apProAch Now, AND wE show that LP⊛ anD k⊛3 Are cLaSsical rEcaPture loGicS foR LP And k3. 3.2 REcapture vIa NormalItY HErE wE esTabliSH a recaptUrE reSuLt bAsed oF lP⊛ and K⊛3. first, We neEd An AUxiLiary noTIoN: dEfinItIoN 3.2 (NorMal coUnterPart) gIveN a set Σ ⊆ L2, We say that The SEt Σ⊛ = {⊛φ ∈ l1 ∣ φ ∈ σ} iS the norMal counterparT oF Σ in
′), or (b) we increase the  expressiv e pow erofSanddefi ne S′ as a con s erva tive extension (L′,⊧S′ ) ofS. Thet wo  opti ons nat u ra l l y s ug ge staf or mal d efi nitionof classic alre capture: Pro b le m 3.1 (Cla ssi cal Recaptur e)For ev er y m a ny-va lue d log ic S w e aker t han CL an ds et Σ ∪ {ψ} ⊆ L o fform ulas, the problem of classical reca ptureis th e pro ble m of findi ng  a co n sequenc e  r e l a tio n  ⊧S′ and a se t Σ′ of for m ula s such  t hat :  (i) Σ  ⊧CLψ⇔  Σ′  ⊧S′ ψ; (ii ) Σ⊆ Σ′; (ii i) Σ ⊧ S  ψ⇒ Σ ⊧ S ′ ψ; (i v) Σ′⊆ L  or  Σ′⊆  L ′for  s o mee xt ens i onL′ of L.  A cc ordin gly, e v e ry s olu tion  to P roblem 3.1 wi llhave the  form  impo sedby  cond ition(i) a bo ve, and will fu lfil l conditi ons  ( ii) -( iv).I f S′ s ati sfi es cond ition ( i )-( iv ) , we  will call it a cl as s i ca l recapt ure lo g ic  f o r S. Als o,  we  mig h t  want  tor ec apture t he inf e re nc e of ama ny-val ue d l ogi c S w i thin  a wea ker logi c S′. This can be ea s ily done by a na t u ra l  adj ust ment of the  def i niti on o f  P rob l em 3. 1, th us yi e lding a more genera lproble m ofS-recapture.We will fa c e one such  cas e  i n  Section 4. To  thebest of ou r  knowled ge, n one hasgiven a f o r mal defi nit ion  of  th e pr oblem of clas s i calre capture  be fore. W ith  ou r p res en t definit ion, weho pe  w ecan  help settle a  r eas on abl e pac k age of  form al r eq ui r eme nts toi mp o s e on  p ro posa lsth at ai m at pro vidingtechnique s f o r cl as si cal rec apture. The a pp roach by [ 24 ] f all un d e r option  (a) above; extension o f  a logi c S  with  nor mality op era tors f all underoption  (b).  W e g o to th i s  a ppr oa ch now, an d weshowth at L P⊛ andK⊛3 are classicalr eca pture logicsfor  LPa n dK3. 3. 2  Re ca p tur e via Normality H ere we est ab l is h a recapt u rere sult ba sed ofLP⊛ a n d K⊛3.First, we  need anau xili a r y n otion: Def inition3.2 (Norm a l Cou n te rpart ) G iven a  s etΣ ⊆ L 2, wes aythatthe se tΣ⊛ = { ⊛φ ∈L1  ∣ φ ∈ Σ } is the normal counter part o f Σ i n
′), or_(b) we_increase the expressive power_of S_and_define S′_as_a conservative extension_(L′,⊧S′) of S._The two options naturally_suggest a formal_definition_of classical recapture: Problem 3.1 (Classical Recapture) For every many-valued logic S weaker than_CL_and set_Σ_∪_{ψ} ⊆ L of formulas,_the problem of classical recapture_is the_problem of finding a consequence relation ⊧S′ and_a_set Σ′ of_formulas such that: (i) Σ ⊧CL ψ ⇔ Σ′_⊧S′ ψ; (ii) Σ ⊆ Σ′;_(iii) Σ ⊧S_ψ⇒_Σ_⊧S′ ψ; (iv) Σ′_⊆ L or Σ′ ⊆ L′_for some extension L′ of L._Accordingly, every solution to Problem 3.1 will_have the form imposed by condition_(i) above, and will fulfill_conditions (ii)-(iv)._If S′ satisfies condition (i)-(iv),_we will call_it a_classical recapture logic_for S. Also, we might want_to recapture the_inference of a many-valued logic S_within_a weaker logic_S′._This_can be_easily done by_a_natural adjustment_of_the definition of Problem 3.1, thus_yielding_a more general problem of S-recapture. We_will face one such_case_in Section 4. To_the best of our knowledge,_none has given a formal definition_of the_problem of_classical recapture before. With our present definition, we hope we can_help settle a reasonable package of_formal requirements to impose_on proposals_that_aim at providing_techniques_for classical_recapture. The approach by [24] fall under_option (a)_above; extension of a logic S_with normality operators fall_under_option (b). We go to this_approach now, and we show that_LP⊛ and K⊛3 are classical_recapture_logics_for LP and K3. 3.2_Recapture via Normality Here we establish_a recapture result_based of LP⊛ and K⊛3. First, we_need_an auxiliary notion: Definition 3.2 (Normal_Counterpart)_Given a set Σ ⊆ L2,_we_say_that the set Σ⊛ =_{⊛φ ∈ L1 ∣ φ ∈_Σ} is the normal counterpart of Σ in