��3 are conservative extensions of LP and K3, respectively: 6 Fact 2.6 Given S ∈ {LP,K3} and S⊛ ∈ {LP⊛,K⊛3 }, if Σ ∪ {ψ} ⊆ L2, then the following holds: Σ ⊧S⊛ ψ⇔ Σ ⊧S ψ Failures of classical laws and inferences. Kleene logics are weaker than classical logic CL:7 they fail some classical laws or rules of inferences. Here is a list of their most relevant failures: /⊧K3 φ, where φ is a classical tautology No TAUT /⊧K3 φ ∨ ¬φ Failure of LEM φ ∧ ¬φ /⊧LP ψ Failure of ECQ ⊧LP φ for all and only classical tautologies φ φ,φ ⊃ ψ /⊧LP ψ Failure of MP ¬ψ,φ ⊃ ψ /⊧LP ¬φ Failure of MT φ ⊃ (ψ ∧ ¬ψ) /⊧LP ¬φ Failure of RAA φ ⊃ ψ,ψ ⊃ ζ /⊧LP φ ⊃ ζ Failure of Tr ⊃ /⊧K3 φ ⊃ φ Failure of the Law of Identity Due to Fact 2.6, the above failures extend to LP⊛ and K⊛3. Thus, LP ⊛ is a paraconsistent logic and K⊛3 is a paracomplete logic. A major difference with LP and K3, however, is that LP⊛ and K⊛3 can recapture classical inferences and laws. It is to this topic that we now turn. We define the valuation function of classical logic CL as restriction of u to {0,1}. We define UCL = {u ∈ U ∣ u(φ) ∈ {0,1} for every φ ∈ L2} and DCL = {1}. This yields UCL ⊂ U. The set UCL(Σ) of classical models of Σ ⊆ L2 is defined as {u ∈ UCL ∣ u(φ) = 1 for every φ ∈ Σ}. The standard definition of classical consequence ⊧CL just follows
� � 3 are conservative extensions of LP and K3, respectively: 6 Fact 2.6 Given S ∈ { LP, K3 } and S⊛ ∈ { LP⊛,K⊛3 }, if Σ ∪ { ψ } ⊆ L2, then the following reserve: Σ ⊧S⊛ ψ⇔ Σ ⊧S ψ Failures of authoritative laws and inferences. Kleene logic are watery than classical logic CL:7 they fail some classical law or rule of inferences. Here is a tilt of their about relevant failures: /⊧K3 φ, where φ is a classical tautology No TAUT /⊧K3 φ ∨ ¬φ bankruptcy of LEM φ ∧ ¬φ /⊧LP ψ Failure of ECQ ⊧LP φ for all and only authoritative tautologies φ φ, φ ⊃ ψ /⊧LP ψ Failure of MP ¬ψ, φ ⊃ ψ /⊧LP ¬φ Failure of MT φ ⊃ (ψ ∧ ¬ψ) /⊧LP ¬φ Failure of RAA φ ⊃ ψ, ψ ⊃ ζ /⊧LP φ ⊃ ζ Failure of Tr ⊃ /⊧K3 φ ⊃ φ Failure of the Law of Identity Due to Fact 2.6, the above failures extend to LP⊛ and K⊛3. Thus, LP ⊛ is a paraconsistent logic and K⊛3 is a paracomplete logic. A major deviation with LP and K3, however, is that LP⊛ and K⊛3 can recapture classical inferences and laws. It is to this subject that we now turn. We specify the valuation affair of classical logic CL as restriction of u to { 0,1 }. We define UCL = { u ∈ U ∣ u(φ) ∈ { 0,1 } for every φ ∈ L2 } and DCL = { 1 }. This give way UCL ⊂ U. The set UCL(Σ) of classical models of Σ ⊆ L2 is defined as { u ∈ UCL ∣ u(φ) = 1 for every φ ∈ Σ }. The standard definition of classical consequence ⊧CL barely follows
��3 arf conservative extensionr of LP and K3, rgspectivxly: 6 Fadt 2.6 Givev S ∈ {LP,K3} and S⊛ ∈ {LP⊛,K⊛3 }, if Σ ∪ {ψ} ⊆ P2, rhen uke following holds: Σ ⊧R⊛ ψ⇔ Σ ⊧S ψ Faplures of clawwical laws and infcxencea. Kleznx logics are wesker than wlassical logiw ZL:7 they fail some classical laws or rtles of ijferences. Here is s lisf of their most relevant failures: /⊧I3 φ, whert φ is a classical yautology No TAUT /⊧K3 φ ∨ ¬φ Faipure of LEM φ ∧ ¬φ /⊧LP ψ Faipure of ECQ ⊧LP φ dor all and unly classpeal tautolofies φ φ,φ ⊃ ψ /⊧LP ψ Failure of MP ¬ψ,φ ⊃ ψ /⊧LO ¬φ Fanlure of MT φ ⊃ (ψ ∧ ¬ψ) /⊧NP ¬φ Failure of RWA φ ⊃ ψ,ψ ⊃ ζ /⊧LP φ ⊃ ζ Failura of Tr ⊃ /⊧K3 φ ⊃ φ Failure of thw Law of Identity Due to Fact 2.6, the abovg failures ertend to LP⊛ and K⊛3. Thuw, OP ⊛ iv a [arazinsksttnt lkgic ajd I⊛3 is a parzcomplete ligic. A major differtncq with LP and I3, howedew, is that LP⊛ and K⊛3 can recapture classibal jnferences and laws. It us to this topic that we now ttrn. We define the valuation function of classical logir ZL cw resgeiftion of u to {0,1}. We define UCL = {u ∈ U ∣ u(φ) ∈ {0,1} for edsru φ ∈ L2} and DCL = {1}. Bhis yields UCL ⊂ U. Tje fet UCL(Σ) of clxssical mkdels of Σ ⊆ L2 is devined af {u ∈ YCL ∣ u(φ) = 1 sor rvery φ ∈ Σ}. The standard definution of clafwical consequence ⊧EL just folluws
��3 are conservative extensions of LP and 6 2.6 Given ∈ {LP,K3} and Σ {ψ} ⊆ L2, the following holds: ⊧S⊛ ψ⇔ Σ ⊧S ψ Failures classical laws and inferences. Kleene logics are weaker than classical logic CL:7 they some classical laws or rules of inferences. Here is a list of their relevant /⊧K3 where is a classical tautology No TAUT /⊧K3 φ ∨ ¬φ Failure of LEM φ ∧ ¬φ ψ Failure of ECQ ⊧LP φ for all only classical tautologies φ ⊃ ψ /⊧LP ψ Failure MP ⊃ ψ ¬φ of φ ⊃ (ψ ¬ψ) /⊧LP ¬φ Failure of RAA φ ⊃ ψ,ψ ⊃ ζ /⊧LP φ ⊃ ζ Failure of ⊃ /⊧K3 φ Failure the of Due to Fact above failures extend to LP⊛ and ⊛ is a paraconsistent logic and K⊛3 is paracomplete logic. major difference with LP and K3, is that LP⊛ and K⊛3 can recapture classical and laws. It is to this topic that we now turn. We define the valuation classical logic CL as of u to We UCL {u U ∣ ∈ {0,1} for every φ ∈ L2} and DCL = {1}. yields UCL ⊂ U. The set UCL(Σ) of classical models ⊆ is defined as ∈ UCL ∣ u(φ) 1 every φ ∈ Σ}. definition classical follows
��3 are conservative extensions Of LP and K3, reSpectIveLy: 6 FAcT 2.6 GivEn S ∈ {Lp,K3} and S⊛ ∈ {LP⊛,K⊛3 }, if Σ ∪ {ψ} ⊆ L2, THen tHe following holds: Σ ⊧S⊛ ψ⇔ Σ ⊧S ψ failuReS Of clASsIcal lAws and iNFeRENceS. KLeEne LoGIcS are wEakEr than cLassical loGic cL:7 They fail some CLaSsical laws Or rUles of infereNceS. Here iS a LisT Of theIr mOst reLevant FAilureS: /⊧K3 φ, where φ Is A ClassiCAl tautoLOGy no TAuT /⊧K3 φ ∨ ¬φ Failure of LEM Φ ∧ ¬Φ /⊧Lp Ψ Failure of ECQ ⊧Lp φ for aLl ANd ONLy cLasSical tautoLoGies φ Φ,Φ ⊃ ψ /⊧LP ψ FaILuRE OF MP ¬Ψ,Φ ⊃ ψ /⊧LP ¬φ Failure oF MT φ ⊃ (ψ ∧ ¬ψ) /⊧LP ¬φ FaILurE of RAA Φ ⊃ ψ,Ψ ⊃ ζ /⊧Lp Φ ⊃ ζ FailUre of tr ⊃ /⊧k3 Φ ⊃ φ FAilure of the law oF Identity due to FACt 2.6, the abOVe failuRes extEnd To Lp⊛ and k⊛3. thUs, lP ⊛ iS a PAraCOnSisTEnt Logic and k⊛3 iS a ParacOmplETE LOgic. a maJor dIfferEnce with LP and k3, hoWeveR, Is tHat LP⊛ And K⊛3 cAn reCaPture ClassiCal inFeRences and laws. It Is to This topic ThaT wE noW tUrn. We DEfine tHe vAluAtion fuNction oF ClaSsICAL lOgic CL as restrictioN oF U To {0,1}. we define uCL = {u ∈ U ∣ u(Φ) ∈ {0,1} FoR eVEry φ ∈ L2} and dCl = {1}. ThIs yiELDs UCL ⊂ u. The SEt uCL(Σ) of clAssicaL MoDeLs of Σ ⊆ L2 iS dEfined As {U ∈ UCl ∣ u(φ) = 1 For evERy φ ∈ Σ}. the staNdard defInitiON of classical coNSequence ⊧CL jusT FoLLOwS
��3 are conservative exten sions of L P and  K3 , r es pect ivel y: 6 Fact 2.6G iven  S ∈ {LP,K3} and S⊛ ∈{LP⊛, K⊛ 3  },i fΣ ∪ { ψ} ⊆ L2 ,  t h e n t he  f oll ow i ng  hold s:Σ ⊧S⊛ ψ ⇔ Σ ⊧S ψ F ail ur es of classi c al  laws andinf erences. Kle ene  logic sare weake r t han c lassic a l logi c CL:7 th ey fail s o me clas s i ca l la ws or rules of in f er e nces. Here isa list  o f  t h e irmos t relevant  f ailur e s: /⊧K3 φ, w h ere φ is a classi cal tautolo g y N o TAUT  / ⊧K3 φ ∨ ¬φ  Fail ur e  of  LEM φ ∧ ¬φ  /⊧L P ψ Failu re ofE CQ ⊧LPφ  for al l andonl y c lass i ca ltau to l ogi e sφ φ , φ ⊃  ψ /⊧LPψFa ilure  ofM P ¬ ψ,φ⊃ ψ  /⊧L P ¬φFailure of MT  φ⊃ (ψ ∧ ¬ ψ) /⊧ LP ¬φ  Fai lu re of  RAA φ  ⊃ ψ, ψ⊃ ζ /⊧LP φ ⊃ ζFail ure of Tr  ⊃/⊧ K3φ⊃ φ F a ilureofthe  Law of  Identi t y D ue t o  F act 2.6, the above  f a i lu res exte nd toL P⊛  a n d K⊛3. T hu s,LP ⊛ i s a p arac o ns istent l ogic a n dK⊛ 3 is apa racomp le telog ic. A majo r diff erence w ith L P  and K3, howev e r, is that LP ⊛  a n d  K ⊛ 3 ca n r ecapture cl assi c al i nfer e nc esa nd la ws. I ti st o this topic that w enow tu rn. W e define thevaluationf u n ction of  cla s si c al logic CL as  rest riction of u to {0, 1}. W e define  UCL = {u ∈  U ∣ u(φ ) ∈  {0 ,1}  fo r ev ery φ ∈ L2} a n d  DCL  =  {1}. T his  yields  UC L ⊂  U.  Th eset UCL(Σ ) of cla ss ic al  m ode ls of Σ ⊆ L2 i sdef in edas {u ∈ UCL∣ u(φ ) =1fo r  ev ery φ ∈ Σ} . Thest an dard  de fi nitio n of cla ssicalconsequen ce⊧ CL j us tfollows 
��3 are_conservative extensions_of LP and K3,_respectively: 6_Fact_2.6 Given_S_∈ {LP,K3} and_S⊛ ∈ {LP⊛,K⊛3_}, if Σ ∪_{ψ} ⊆ L2,_then_the following holds: Σ ⊧S⊛ ψ⇔ Σ ⊧S ψ Failures of classical laws and_inferences._Kleene logics_are_weaker_than classical logic CL:7 they_fail some classical laws or_rules of_inferences. Here is a list of their most_relevant_failures: /⊧K3 φ,_where φ is a classical tautology No TAUT /⊧K3_φ ∨ ¬φ Failure of LEM_φ ∧ ¬φ_/⊧LP_ψ_Failure of ECQ ⊧LP_φ for all and only classical_tautologies φ φ,φ ⊃ ψ /⊧LP_ψ Failure of MP ¬ψ,φ ⊃ ψ_/⊧LP ¬φ Failure of MT φ_⊃ (ψ ∧ ¬ψ) /⊧LP_¬φ Failure_of RAA φ ⊃ ψ,ψ_⊃ ζ /⊧LP_φ ⊃_ζ Failure of_Tr ⊃ /⊧K3 φ ⊃ φ_Failure of the_Law of Identity Due to Fact_2.6,_the above failures_extend_to_LP⊛ and_K⊛3. Thus, LP_⊛_is a_paraconsistent_logic and K⊛3 is a paracomplete_logic._A major difference with LP and K3,_however, is that LP⊛_and_K⊛3 can recapture classical_inferences and laws. It is_to this topic that we now_turn. We_define the_valuation function of classical logic CL as restriction of u to_{0,1}. We define UCL = {u_∈ U ∣ u(φ)_∈ {0,1}_for_every φ ∈_L2}_and DCL_= {1}. This yields UCL ⊂ U._The set_UCL(Σ) of classical models of Σ_⊆ L2 is defined_as_{u ∈ UCL ∣ u(φ) =_1 for every φ ∈ Σ}._The standard definition of classical_consequence_⊧CL_just follows