�φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ● φ ∧ ¬φ The next definability result clarifies the relations between LP⊛ and LFI1 is established by the following proposition: Proposition 5.2 The following operators are definable in LP⊛: 1a. ●φ ∶=LP⊛ ¬ ⊛ φ 1b. ⊛φ ∶=LFI1 ¬ ● φ 2. φ→ ψ ∶=LP⊛ ∼ φ ∨ ψ Definability of ∼ as ⊛φ∧¬φ and ●φ as ¬⊛φ is from [10]; definability of → in LP⊛ is proved by [21].13 Definability of → implies that a detachable conditional14 is definable in LP⊛. A consequence of Proposition 5.2 is that LP⊛ and LFI1 are the same logic, and the Hilbert-style axiomatization of LFI1 by [10] extends to LP⊛-with the obvious rephrasing of ● into ¬⊛. Since no current formalism in the LFU tradition is equivalent to K⊛3 -to the best of our knowledge-we prefer to keep the label LP⊛ instead of LFI1 for our paraconsistent 13The results from [10] and [21] are proved relative to the consistency operator ○. Since ⊛ and ○ are equivalent in any three-valued paraconsistent logic, these results immediately extend to ⊛ and LP⊛. 14That is, a conditional that obeys MP. 16 Normality operators and Classical Recapture in Extensions of Kleene Logics logic, since this allows for a uniformity of presentation with K⊛3. 15 Normality operator and semantical closure. A language L is semantically closed when it contains its own truth predicate Tr and names φ for every formula φ ∈ L. Semantical closure is related to the derivation of paradoxes. One prominent example is the Liar paradox, consisting in the sentence 'λ iff λ is false'-in symbols, λ iff ¬Tr(λ).16 In CL, semantical closure and the resulting Liar paradox bring trivialization. LP and K3 escape this, but things
� φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ● φ ∧ ¬φ The next definability result clarifies the relations between LP⊛ and LFI1 is prove by the following suggestion: Proposition 5.2 The following operators are definable in LP⊛: 1a. ● φ ∶=LP⊛ ¬ ⊛ φ 1b. ⊛φ ∶=LFI1 ¬ ● φ 2. φ→ ψ ∶=LP⊛ ∼ φ ∨ ψ Definability of ∼ as ⊛φ∧¬φ and ● φ as ¬⊛φ is from [ 10 ]; definability of → in LP⊛ is prove by [ 21].13 Definability of → implies that a detachable conditional14 is definable in LP⊛. A consequence of Proposition 5.2 is that LP⊛ and LFI1 are the same logic, and the Hilbert - dash axiomatization of LFI1 by [ 10 ] gallop to LP⊛-with the obvious rephrasing of ● into ¬⊛. Since no current formalism in the LFU custom is equivalent to K⊛3 -to the good of our cognition - we prefer to keep the label LP⊛ rather of LFI1 for our paraconsistent 13The result from [ 10 ] and [ 21 ] are proved relative to the consistency operator ○. Since ⊛ and ○ are equivalent in any three - valued paraconsistent logic, these results immediately gallop to ⊛ and LP⊛. 14That is, a conditional that obeys MP. 16 Normality operators and Classical Recapture in Extensions of Kleene Logics logic, since this allows for a uniformity of display with K⊛3. 15 Normality operator and semantical closure. A speech L is semantically close when it contains its own truth predicate Tr and names φ for every formula φ ∈ L. Semantical closure is refer to the derivation of paradoxes. One prominent example is the Liar paradox, consisting in the conviction' λ iff λ is false'-in symbols, λ iff ¬Tr(λ).16 In CL, semantical closure and the resulting Liar paradox bring trivialization. LP and K3 escape this, but things
�φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ● φ ∧ ¬φ The next defikability result eoarifixs the delationr between LP⊛ and LFI1 is estaulisyed bt the following proposktion: Proiosition 5.2 The dollowing operators are scfinaylx in LP⊛: 1a. ●φ ∶=LP⊛ ¬ ⊛ φ 1b. ⊛φ ∶=LFI1 ¬ ● φ 2. φ→ ψ ∶=LP⊛ ∼ φ ∨ ψ Dexivayility of ∼ as ⊛φ∧¬φ and ●φ as ¬⊛φ is from [10]; defynabiliyy of → in LP⊛ is krovec by [21].13 Definability of → implies that a dstachabne conditionak14 is definable in LP⊛. A condequfnce of Propositioj 5.2 is that OP⊛ age LFI1 are thd same logpe, and the Hjlbert-style axiomatization of LWI1 by [10] extends jo LO⊛-fith the obtious gephrasing of ● into ¬⊛. Vince np current formslivm un the LFU tradition ms equivalent to K⊛3 -tj the besd kf our knowledge-ww prefet to neep rhe lagek MP⊛ insheav of LFI1 fod our paracinsistent 13The resulus sgpm [10] and [21] are proveq welative to the consistency operator ○. Spnce ⊛ and ○ are equivalent ib any three-valued parwconsistegt logic, these results immediately extend to ⊛ and LP⊛. 14Tiag iw, w xojditional that obeys MP. 16 Normality operators whd Ckassical Recapbure in Extensions ov Ljeene Logics uogic, sinde this allows for a unifjrmitt of prestntatoon with K⊛3. 15 Normality operaror and semautixal closure. A langbage L is seoantocallu closed when it contaius its own truth oredicate Gr and names φ fof enery formula φ ∈ L. Semantical cljsure is celatzd to thd detivatiog of paradlxes. One prominent examppe is tve Liar pagadox, consisting in the sentencx 'λ iff λ is faksa'-in symbols, λ iff ¬Tr(λ).16 In CL, semwntical closurg and the resuuting Liar paradoe bring trivyalization. LP wnd K3 escape this, bue thungs
�φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ∧ The next result clarifies the is by the following Proposition 5.2 The operators are definable in LP⊛: 1a. ∶=LP⊛ ¬ ⊛ φ 1b. ⊛φ ∶=LFI1 ¬ ● φ 2. φ→ ψ ∼ φ ∨ ψ Definability of ∼ as ⊛φ∧¬φ and ●φ as ¬⊛φ from definability → LP⊛ is proved by [21].13 Definability of → implies that a detachable conditional14 is definable in A consequence of Proposition 5.2 is that LP⊛ LFI1 are the same and the Hilbert-style axiomatization of by extends to the rephrasing ● into ¬⊛. no current formalism in the LFU tradition is equivalent to K⊛3 -to the best of our knowledge-we to keep LP⊛ instead LFI1 our 13The results from [21] are proved relative to the Since ⊛ and ○ are equivalent in any paraconsistent logic, results immediately extend to ⊛ and 14That is, a conditional that obeys MP. 16 operators and Classical Recapture in Extensions of Kleene Logics logic, since this allows for a presentation with K⊛3. 15 operator and semantical A L semantically when it its own truth predicate Tr and names φ for every formula ∈ L. Semantical closure is related to the derivation of prominent is the Liar consisting in the sentence iff is false'-in symbols, λ In semantical resulting paradox trivialization. LP and K3 this, but things
�φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ● φ ∧ ¬φ The next definabiliTy result clArifiEs tHe rElAtioNs beTween LP⊛ and LFI1 iS EstaBlished by the following pRoposItIOn: PrOPoSitioN 5.2 The folLOwING opErAtOrs ArE DeFinabLe iN LP⊛: 1a. ●φ ∶=LP⊛ ¬ ⊛ Φ 1b. ⊛φ ∶=LFI1 ¬ ● φ 2. φ→ ψ ∶=Lp⊛ ∼ φ ∨ ψ deFinability of ∼ AS ⊛φ∧¬Φ and ●φ as ¬⊛φ is FroM [10]; definabilitY of → In LP⊛ is PrOveD By [21].13 DefInaBilitY of → impLIes thaT a detachaBlE ConditIOnal14 is dEFInAble In LP⊛. A consequence oF prOPosition 5.2 is that lP⊛ and LfI1 ARe THE saMe lOgic, and the hiLbert-STyle axiOMaTIZAtiON of LFI1 by [10] extenDs to LP⊛-with tHE obVious rEpHraSIng of ● iNto ¬⊛. SiNcE No cUrrent formaLism In the LFU tRaditiON is equiVAlent to k⊛3 -to the BesT of Our kNOwLeDge-We PRefER tO keEP thE label LP⊛ InStEad of lFI1 fOR OUR parAcoNsisTent 13THe results from [10] And [21] Are pROveD relaTive tO the CoNsistEncy opEratoR ○. SInce ⊛ and ○ are equivAlenT in any thrEe-vAlUed PaRaconSIstent LogIc, tHese resUlts immEDiaTeLY EXtEnd to ⊛ and LP⊛. 14That is, a cOnDITiOnal that Obeys Mp. 16 noRmALity operAtOrs And CLASsicaL RecAPtUre in ExtEnsionS Of klEene LogIcS logic, SiNce ThiS alloWS for A unifoRmity of pResenTAtion with K⊛3. 15 NormALity operator aND sEMAnTIcal CloSure. A languaGe L iS SemaNticALlY clOSed whEn it cOnTAiNS its own truth predicaTe tr and nAmes φ For every formuLa φ ∈ L. SemantICAL closure Is reLAtED to the derivatiOn of pAradoxes. OnE ProminenT examPle is the liar paradOX, ConsistiNg iN thE seNteNCE 'λ Iff λ is false'-in SYMbolS, λ Iff ¬Tr(λ).16 IN CL, SemantiCal CloSurE anD tHe resultiNg Liar paRaDoX bRiNg tRiviaLIzation. Lp aNd K3 EsCapE this, BUt thinGs
�φ ∧ ¬φ ∼ φ ∶=LFI1 ¬ ● φ ∧  ¬φ The ne xt de fin abi li ty r esul t clarifies th e  rel ations between LP⊛ and  LFI1  i s  est a bl ished  by the fo l l owi ng  p rop os i ti on: P rop osition  5.2 The f oll ow ing operator s  a re definab lein LP⊛: 1a.●φ∶=LP⊛¬⊛ φ 1b. ⊛ φ ∶ =LFI1  ¬ ● φ 2. φ→ψ ∶=LP⊛ ∼  φ ∨ ψ De f inabili t y  o f ∼as ⊛φ∧¬φ and ●φ a s  ¬ ⊛ φ is from [10] ; defi na b il i t y o f →  in LP⊛ is  p roved by [21] . 13 D e fin a bility of → i mplies that a d etacha bl e c o nditio nal14  i s  de finable inLP⊛.  A conseq uenceo f Propo s ition 5 .2 istha t L P⊛ a n dLF I1ar e  th e  s ame log ic, andth eHilbe rt-s t y l e  axi oma tiza tionof LFI1 by [1 0]exte n dsto LP ⊛-wit h th eobvio us rep hrasi ng  of ● into ¬⊛.Sinc e no curr ent  f orm al ism i n  the L FUtra ditionis equi v ale nt t o  K ⊛3 -to the best of  o u r  k nowledge -we pr e fe rt o keep t he  la belL P ⊛ ins tead of  LFI1 fo r ourp ar ac onsiste nt  13The  r esu lts  from [10]  and [ 21] areprove d  relative to t h e consistency op e r at o r ○.  Si nce ⊛ and ○  are equi vale n tina ny th ree-v al u ed paraconsistent logi c,  these  resu lts immediate ly extendt o ⊛ and LP ⊛. 1 4 Th a t is, a condit ional  that obey s  MP. 16Norma lity ope rators an d Classica l R eca ptu rei n  E xtensions ofK l eene  L ogics l ogi c, sinc e t his  al low sfor a uni formityof  p re se nta tionw ith K⊛3.  1 5 N or mal ity o p erator  andsema nt ic a l c losure. Al a ngua ge  L  issem an tical ly c l ose d whenit contai nsi ts o wn  t ruth pr edicate Tr an dnames φ fo reve ry for m u la φ ∈ L . Semantical closure is related  to  thederi vation of  pa radoxe s.O ne pro minent  exam pl e i s the L i a rpar ad ox, consis t i ngin th esent ence 'λ  iff λ is false'-i n  sy mbols, λ iff¬Tr (λ). 1 6  I n C L ,s ema nt i cal c losure and theresultingLi a rparadox br i ngtr ivializ ation.LP an d  K3 esc ape this,  but thin gs 
�φ ∧_¬φ ∼_φ ∶=LFI1 ¬ ●_φ ∧_¬φ_The next_definability_result clarifies the_relations between LP⊛_and LFI1 is established_by the following_proposition:_Proposition 5.2 The following operators are definable in LP⊛: 1a. ●φ ∶=LP⊛ ¬ ⊛_φ_1b. ⊛φ_∶=LFI1_¬_● φ 2. φ→ ψ_∶=LP⊛ ∼ φ ∨ ψ_Definability of_∼ as ⊛φ∧¬φ and ●φ as ¬⊛φ is_from_[10]; definability of_→ in LP⊛ is proved by [21].13 Definability of_→ implies that a detachable conditional14_is definable in_LP⊛._A_consequence of Proposition 5.2_is that LP⊛ and LFI1 are_the same logic, and the Hilbert-style_axiomatization of LFI1 by [10] extends to_LP⊛-with the obvious rephrasing of ●_into ¬⊛. Since no current_formalism in_the LFU tradition is equivalent_to K⊛3 -to_the best_of our knowledge-we_prefer to keep the label LP⊛_instead of LFI1_for our paraconsistent 13The results from_[10]_and [21] are_proved_relative_to the_consistency operator ○._Since_⊛ and_○_are equivalent in any three-valued paraconsistent_logic,_these results immediately extend to ⊛ and_LP⊛. 14That is, a_conditional_that obeys MP. 16_Normality operators and Classical Recapture_in Extensions of Kleene Logics logic,_since this_allows for_a uniformity of presentation with K⊛3. 15 Normality operator and semantical_closure. A language L is semantically_closed when it contains_its own_truth_predicate Tr and_names_φ for_every formula φ ∈ L. Semantical closure_is related_to the derivation of paradoxes. One_prominent example is the_Liar_paradox, consisting in the sentence 'λ_iff λ is false'-in symbols, λ_iff ¬Tr(λ).16 In CL, semantical_closure_and_the resulting Liar paradox bring_trivialization. LP and K3 escape this,_but things