 ⊧K⊛3 ψ. From this and u(ψ ∨ ¬ψ) = 1⇔ u(⊛ψ) = 1, we have Σ, ψ ∨ ¬ψ ⊧K3 ψ⇒ Σ,⊛ψ ⊧K⊛3 ψ. This proves 1⇔ 3, which in turn suffices to prove Σ,⊛ψ ⊧K⊛3 ψ ⇔ Σ ⊧CL ψ. Remark 3.4 We can recapture classical reasoning in LP⊛ by assuming that the premise-set is consistent, and in K⊛3 by assuming that the conclusion is consistent. The statements in the theorem have the form required by condition (i) in Problem 3.1 and Σ ∪ Σ⊛ and Σ ∪ {⊛ψ} satisfy condition (iv). Notice that condition (ii) is fulfilled, and condition (iii) is fulfilled-due to Fact 2.6. Thus, Theorem 3.3 provides a solution to Problem 3.1. In particular, LP⊛ and K⊛3 are classical recapture logics for LP and K3, respectively: in using them, we can deploy all the inferential power of the two Kleene logics, and we can specify conditions at which we can reason classically. Let us see how recapture via normality recovers the classical inferences or laws that fail in LP and K3-see Section 2: (φ ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧LP⊛ ψ Recapture of ECQ Notice that recapture of ECQ is just another version of PGE. Similarly, LEM and every classical tautology can be recaptured in K⊛3 : ⊛φ ⊧K⊛3 φ ∨ ¬φ Recapture of LEM ⊛φ ⊧K⊛3 φ, where φ is a classical tautology Recapture of TAUT Again, recapture of LEM is just another version of PGI. Recapture of the Law of Identity is an instance of Recapture of TAUT: ⊛φ ⊧K⊛3 φ ⊃ φ Recapture of Law of Identity Finally, the following inferences are valid in LP
⊧K⊛3 ψ. From this and u(ψ ∨ ¬ψ) = 1⇔ u(⊛ψ) = 1, we have Σ, ψ ∨ ¬ψ ⊧K3 ψ⇒ Σ,⊛ψ ⊧K⊛3 ψ. This proves 1⇔ 3, which in turn suffices to rise Σ,⊛ψ ⊧K⊛3 ψ ⇔ Σ ⊧CL ψ. Remark 3.4 We can recapture authoritative reasoning in LP⊛ by assuming that the premise - set is reproducible, and in K⊛3 by assume that the conclusion is consistent. The statement in the theorem give birth the form required by condition (i) in Problem 3.1 and Σ ∪ Σ⊛ and Σ ∪ { ⊛ψ } satisfy circumstance (iv). Notice that condition (ii) is fulfilled, and condition (iii) is carry through - due to Fact 2.6. therefore, Theorem 3.3 provides a solution to Problem 3.1. In especial, LP⊛ and K⊛3 are authoritative recapture logics for LP and K3, respectively: in using them, we can deploy all the inferential office of the two Kleene logics, and we can specify condition at which we can argue classically. Let us see how recapture via normality recovers the classical inferences or laws that fail in LP and K3 - witness Section 2: (φ ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧LP⊛ ψ Recapture of ECQ Notice that recapture of ECQ is just another version of PGE. Similarly, LEM and every classical tautology can be recaptured in K⊛3: ⊛φ ⊧K⊛3 φ ∨ ¬φ Recapture of LEM ⊛φ ⊧K⊛3 φ, where φ is a authoritative tautology Recapture of TAUT Again, recapture of LEM is merely another version of PGI. Recapture of the Law of Identity is an instance of Recapture of TAUT: ⊛φ ⊧K⊛3 φ ⊃ φ Recapture of Law of Identity Finally, the following inferences are valid in LP
 ⊧K⊛3 ψ. From this and u(ψ ∨ ¬ψ) = 1⇔ u(⊛ψ) = 1, we have Σ, ψ ∨ ¬ψ ⊧K3 ψ⇒ Σ,⊛ψ ⊧K⊛3 ψ. Thms provss 1⇔ 3, whizh in turn suffices to prove Σ,⊛ψ ⊧J⊛3 ψ ⇔ Σ ⊧XL ψ. Remark 3.4 We can recxpture clwssical eeasibing in LP⊛ by assuming tgwt tke premise-set is consistend, and in K⊛3 by dsruling that the conclusion is consistqnt. The shatements in tre tnqorej have the form required by conditjon (i) ii Problem 3.1 and Σ ∪ Σ⊛ and Σ ∪ {⊛ψ} satisfy conditioj (iv). Notice that condihion (ii) is dulfyoled, and conaition (iii) is fulfilles-due to Fact 2.6. Thus, Theorem 3.3 proxides a solutiob ro Otoblem 3.1. In perticujar, LP⊛ and K⊛3 are clavsical tecapture logigs foc LP and K3, respectively: ii using them, we can qeploy aln che inferential power od the two Kledbe uogjcx, znd we cai specify cknditions ar which we can reaspn bkassically. Lst us feq how recapture via normality recovers uhe cmassical inferences or oaws that fail in LP wnd K3-see Fection 2: (φ ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧LP⊛ ψ Recapture of ECQ Notice that rewaptuce of EGQ ir jkst another version of PGE. Similarly, LEM and qbety classical taubology can be recaltkrrq in K⊛3 : ⊛φ ⊧K⊛3 φ ∨ ¬φ Recapchrs of LEM ⊛φ ⊧K⊛3 φ, where φ is a slassucal tautjlogu Recapture of TAUT Again, rwcapture of JWM is just another version of PGI. Recalture of the Law of Ideutity js an instajce of Redxpture of TAUT: ⊛φ ⊧Y⊛3 φ ⊃ φ Secapturt of Law of Identiey Finallb, the followkng onferegces are vwlid lt LP
⊧K⊛3 ψ. From this and u(ψ ∨ 1⇔ = 1, have Σ, ψ ⊧K⊛3 This proves 1⇔ which in turn to prove Σ,⊛ψ ⊧K⊛3 ψ ⇔ ⊧CL ψ. Remark 3.4 We can recapture classical reasoning in LP⊛ by assuming the premise-set is consistent, and in K⊛3 by assuming that the conclusion is The in theorem the form required by condition (i) in Problem 3.1 and Σ ∪ Σ⊛ and Σ ∪ satisfy condition (iv). Notice that condition (ii) is and condition (iii) is to Fact 2.6. Thus, Theorem provides solution to 3.1. particular, and K⊛3 are recapture logics for LP and K3, respectively: in using them, we can deploy all the inferential power the two and we specify at we can reason us see how recapture via normality inferences or laws that fail in LP and Section 2: ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧LP⊛ ψ of ECQ Notice that recapture of ECQ is another version of PGE. Similarly, LEM and every classical tautology can be recaptured in K⊛3 ⊧K⊛3 φ ∨ ¬φ of LEM ⊛φ φ, φ a tautology Recapture TAUT Again, recapture of LEM is just another version of PGI. of the Law of Identity is an instance of Recapture ⊛φ φ ⊃ φ of Law of Identity the inferences are valid in
 ⊧K⊛3 ψ. From this and u(ψ ∨ ¬ψ) = 1⇔ u(⊛ψ) = 1, we have Σ, Ψ ∨ ¬ψ ⊧K3 ψ⇒ Σ,⊛ψ ⊧K⊛3 ψ. ThIs proVes 1⇔ 3, WhiCh In tuRn suFfices to prove Σ,⊛Ψ ⊧k⊛3 ψ ⇔ Σ ⊧Cl ψ. Remark 3.4 We can recapture ClassIcAL reaSOnIng in lP⊛ by assUMiNG ThaT tHe PreMiSE-sEt is cOnsIstent, aNd in K⊛3 by assUmiNg That the conclUSiOn is consisTenT. The statemenTs iN the thEoRem HAve thE foRm reqUired bY ConditIon (i) in ProBlEM 3.1 and Σ ∪ Σ⊛ ANd Σ ∪ {⊛ψ} satISFy CondItion (iv). Notice that COnDItion (ii) is fulfiLled, anD cONdITIon (Iii) Is fulfilleD-dUe to FACt 2.6. Thus, THEoREM 3.3 ProVIdes a solution To Problem 3.1. In PArtIcular, lP⊛ And k⊛3 Are claSsicaL rECapTure logics fOr LP And K3, respeCtivelY: In using THem, we caN deploY alL thE infEReNtIal PoWEr oF ThE twO kleEne logicS, aNd We can SpecIFY COndiTioNs at Which We can reason clAssIcalLY. LeT us seE how rEcapTuRe via NormalIty reCoVers the classicaL infErences or LawS tHat FaIl in Lp And K3-seE SeCtiOn 2: (φ ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧Lp⊛ ψ RecapTUre Of ecq noTice that recapture oF Ecq Is Just anotHer verSIoN oF pGE. SimilArLy, LeM anD EVery cLassICaL tautoloGy can bE ReCaPtured iN K⊛3 : ⊛Φ ⊧K⊛3 φ ∨ ¬φ ReCaPtuRe oF LEM ⊛φ ⊧k⊛3 Φ, wheRe φ is a ClassicaL tautOLogy Recapture oF tAUT Again, recaPTuRE Of leM is JusT another verSion OF PGI. recaPTuRe oF The LaW of IdEnTItY Is an instance of RecapTuRe of TAuT: ⊛φ ⊧K⊛3 φ ⊃ Φ Recapture of LAw of IdentiTY fInally, thE folLOwINg inferences arE valiD in LP
 ⊧K⊛3 ψ. From this and u(ψ  ∨ ¬ψ) = 1 ⇔ u(⊛ ψ)= 1 ,we h aveΣ, ψ ∨ ¬ψ ⊧K3ψ ⇒ Σ, ⊛ψ ⊧K⊛3 ψ. This proves  1⇔ 3 ,w hich in  turn  suffic e st o  pr ov eΣ,⊛ ψ⊧ K⊛ 3 ψ ⇔  Σ⊧CL ψ.Remark 3.4  We  c an recapture cl assical re aso ning in LP⊛byassumi ng  th a t the  pr emise -set i s  consi stent, an di n K⊛3b y assum i n gthat  the conclusion i s  c o nsistent. Thestatem en t si n  th e t heorem hav ethe f o rm requ i re d b y c o ndition (i) i n Problem 3 . 1 a nd Σ ∪  Σ ⊛ a n d Σ ∪{⊛ψ}sa t isf y condition  (iv ). Notice  thatc onditio n  (ii) i s fulf ill ed,  and co nd iti on (ii i )isf ulf illed-du eto  Fact  2.6 . T h us,The orem  3.3provides a so lut iont o P roble m 3.1 . In  p artic ular,LP⊛ a nd  K⊛3 are classi calrecapture  lo gi csfo r LPa nd K3,  re spe ctively : in us i ngth e m ,  w e can deploy all t he i nf erential  power of  t h e two Kl ee nelogi c s , and  wec an  specify  condi t io ns  at whi ch  we ca nrea son  clas s ical ly. Le t us see  howr ecapture via n o rmality recov e rs t he clas sic al inferenc es o r  law s th a tfai l  in L P and  K 3 -s e e Section 2: (φ ∧ ¬ φ) ,⊛(φ ∧  ¬φ)⊧LP⊛ ψ Recapt ure of ECQ N o tice tha t re c ap t ure of ECQ isjustanother ve r sion ofPGE.Similarl y, LEM an d every cl ass ica l t aut o l og y can be reca p t ured  i n K⊛3 :  ⊛φ  ⊧K⊛3 φ  ∨¬φRec apt ur e of LEM⊛φ ⊧K⊛3φ,  w he re  φis ac lassical  t aut ol ogy  Reca p ture o f TAU T Ag ai n, rec aptureo fL E M is  j us t an oth er  vers iono f P GI. Rec apture of  th e  Law  o fIdentit y is an insta nc e of Recap tu reof TAU T :  ⊛φ ⊧K⊛3  φ ⊃ φ Recapture of Law of Iden tit y Fin ally , the fol low ing in fer e nces a re val id in  L P
 ⊧K⊛3_ψ. From_this and u(ψ ∨_¬ψ) =_1⇔_u(⊛ψ) =_1,_we have Σ,_ψ ∨ ¬ψ_⊧K3 ψ⇒ Σ,⊛ψ ⊧K⊛3_ψ. This proves_1⇔_3, which in turn suffices to prove Σ,⊛ψ ⊧K⊛3 ψ ⇔ Σ ⊧CL ψ._Remark_3.4 We_can_recapture_classical reasoning in LP⊛ by_assuming that the premise-set is_consistent, and_in K⊛3 by assuming that the conclusion is_consistent._The statements in_the theorem have the form required by condition (i)_in Problem 3.1 and Σ ∪_Σ⊛ and Σ_∪_{⊛ψ}_satisfy condition (iv). Notice_that condition (ii) is fulfilled, and_condition (iii) is fulfilled-due to Fact_2.6. Thus, Theorem 3.3 provides a solution_to Problem 3.1. In particular, LP⊛_and K⊛3 are classical recapture_logics for_LP and K3, respectively: in_using them, we_can deploy_all the inferential_power of the two Kleene logics,_and we can_specify conditions at which we can_reason_classically. Let us_see_how_recapture via_normality recovers the_classical_inferences or_laws_that fail in LP and K3-see_Section_2: (φ ∧ ¬φ),⊛(φ ∧ ¬φ) ⊧LP⊛_ψ Recapture of ECQ_Notice_that recapture of ECQ_is just another version of_PGE. Similarly, LEM and every classical_tautology can_be recaptured_in K⊛3 : ⊛φ ⊧K⊛3 φ ∨ ¬φ Recapture of LEM_⊛φ ⊧K⊛3 φ, where φ is_a classical tautology Recapture_of TAUT_Again,_recapture of LEM_is_just another_version of PGI. Recapture of the Law_of Identity_is an instance of Recapture of_TAUT: ⊛φ ⊧K⊛3 φ_⊃_φ Recapture of Law of Identity_Finally, the following inferences are valid_in LP