see Section 2, where we go through the connections between the three operators and we explain why we keep the normality operator distinct from the other two. Additionally we discuss the connections between the present approach, LFIs, and LFUs. The paper proceeds as follows. Section 2 introduces the logics LP⊛ and K⊛3 and presents the main failures of classical inferences in these logics. Section 3 establishes a classical recapture result (Theorem 3.3) for LP⊛ and K⊛3 and presents some concrete examples of how classical inferences can be recaptured in these two logics. Section 4 introduces the approach by minimal inconsistency and compare it with our approach based on normality operators. Section 5 discusses the relations between our proposal and the tradition of LFIs and LFUs. In particular, we discuss the relations between Theorem 3.3 and the Derivability Adjustment Theorems from [11, 20], the relations between LP⊛ and LFI1, and some other issues. Section 6 sums up the results of the paper. 2 Extensions of Kleene logics with Normality Operators In this section we define normality operators and we introduce the two logics LP⊛ and K⊛3, whose language deploy normality operators. Also, we discuss the relations between these logics and the Kleene logics LP and K3, as well as some relevant failures of classical laws and inference rules. Preliminaries. In this paper we adopt a semantic angle, and we regard a logic S as a pair (L,⊧S), where L is a language and ⊧S is the relation of (single-conclusion) S-consequence-that is, ⊧S ⊆ ℘(L)×L. Given a nonempty set T of truth values and a logic S, we denote by DS ⊆ T the set of designated values of S-that is, the values that must be preserved through the valid inferences of S. We define the relative notion of logical consequence-S-consequence-as follows: Definition 2.1 (Logical Consequence in Many-valued Logic) Σ ⊧S ψ ⇔ VS(Σ) ⊆ VS(ψ) where VS(Σ) is the set of those valuations v defined for S where v(φ) ∈ DS for every φ ∈ Σ. Σ ⊧S ψ
see Section 2, where we go through the connections between the three operator and we excuse why we keep the normality operator distinct from the early two. Additionally we discuss the connection between the present approach, LFIs, and LFUs. The paper return as trace. Section 2 precede the logics LP⊛ and K⊛3 and presents the independent failures of classical inference in these logics. Section 3 establishes a classical recapture result (Theorem 3.3) for LP⊛ and K⊛3 and presents some concrete examples of how authoritative inferences can be recaptured in these two logics. department 4 introduces the approach by minimal incompatibility and compare it with our approach base on normality operators. department 5 discusses the relations between our proposal and the custom of LFIs and LFUs. In particular, we discuss the relations between Theorem 3.3 and the Derivability Adjustment Theorems from [ 11, 20 ], the relations between LP⊛ and LFI1, and some other issues. Section 6 sums up the results of the paper. 2 Extensions of Kleene logics with Normality Operators In this section we define normality operators and we introduce the two logics LP⊛ and K⊛3, whose speech deploy normality operators. Also, we hash out the relative between these logics and the Kleene logics LP and K3, as well as some relevant failure of classical laws and inference rules. Preliminaries. In this newspaper we adopt a semantic angle, and we regard a logic S as a pair (L,⊧S), where L is a language and ⊧S is the relation of (unmarried - conclusion) S - consequence - that is, ⊧S ⊆ ℘(L)×L. Given a nonempty set T of accuracy values and a logic S, we announce by DS ⊆ T the set of designated values of S - that is, the values that must be keep through the valid inferences of S. We define the relative notion of logical consequence - S - consequence - as follows: Definition 2.1 (coherent Consequence in Many - valued Logic) Σ ⊧S ψ ⇔ VS(Σ) ⊆ VS(ψ) where VS(Σ) is the set of those evaluation v defined for S where v(φ) ∈ DS for every φ ∈ Σ. Σ ⊧S ψ
see Section 2, where we go thvough the connecjiins bevween tge three operators and we explain whb we keep the normality operatof distincn from thw otier two. Additionemly we discusa the ronnections beteeen the psesent approacv, UFNs, and LFUs. The paper proceeds as fojlows. Srchion 2 introducgs tht ljgica LP⊛ and K⊛3 and presents the main fzilures of classical inferences in these logicd. Seftion 3 establishes a classicao resqpture resulg (Theorem 3.3) for LP⊛ and I⊛3 and presents some concrete exxmplex of how coawsifdl inferencxs can be recapturcc in tvese twp logics. Sectipn 4 inrroduces the approach by minimal inconsystency atd compare it with iue apptoach basde ov nkrkamity ooeretors. Sectikn 5 discussws the relations beuweqb our proposam and ehq tradition of LFIs and LFUs. In particunar, we discuss the relatiobs between Theorem 3.3 ajd the Dewivability Adjustment Theorems from [11, 20], the relatiots bevwden LI⊛ ana LVI1, and some other issues. Section 6 sums up the dexukts of the papcr. 2 Extensions of Llfemg logics with Vormalnfy Operators In this sectiog we eefine nowmaloty operators and we introdyce the two jigics LP⊛ and K⊛3, whode language depkoy nprmality operators. Also, we siscuss the relationa between these lugibs atd the Kleene logics LP anq K3, as weol ax some felefant fwilures of classical laws and infegence rgles. Prelilinaries. In this paper we adopt a semantic anbla, atd we reyard a logic S as a pair (L,⊧S), where L is a langjage and ⊧S is the relation os (single-conclgdion) S-conseqnence-that is, ⊧W ⊆ ℘(L)×O. Given x nonempty set T of trunh valyes and a logic S, xe dgnkte by DS ⊆ T thz wet of designatrd xalted pf V-that is, the valjes yhat oust be prewdrvec through the valid hnfedences of S. We defonc the relqtive noeion of logicsl consequence-S-condequeice-as hollowx: Dgfinition 2.1 (Logical Consequence jn Many-vapueq Logic) Σ ⊧S ψ ⇔ VS(Σ) ⊆ VS(ψ) where VS(Σ) is the set of those valuations v definxd for S where v(φ) ∈ DS foe every φ ∈ Σ. Σ ⊧S ψ
see Section 2, where we go through between three operators we explain why distinct the other two. we discuss the between the present approach, LFIs, and The paper proceeds as follows. Section 2 introduces the logics LP⊛ and K⊛3 presents the main failures of classical inferences in these logics. Section 3 establishes classical result 3.3) LP⊛ and K⊛3 and presents some concrete examples of how classical inferences can be recaptured in two logics. Section 4 introduces the approach by inconsistency and compare it our approach based on normality Section discusses the between proposal the tradition of and LFUs. In particular, we discuss the relations between Theorem 3.3 and the Derivability Adjustment Theorems from 20], the LP⊛ and and other Section 6 sums results of the paper. 2 Extensions with Normality Operators In this section we define operators and introduce the two logics LP⊛ and whose language deploy normality operators. Also, we discuss relations between these logics and the Kleene logics LP and K3, as well as some of classical laws and rules. Preliminaries. In paper adopt semantic and we a logic S as a pair (L,⊧S), where L is a and ⊧S is the relation of (single-conclusion) S-consequence-that is, ⊧S Given nonempty set T truth values and a S, denote by DS ⊆ set designated is, values must be preserved through valid inferences of S. We the relative notion of (Logical Consequence in Many-valued Logic) Σ ⊧S ψ VS(Σ) ⊆ VS(ψ) where VS(Σ) is the of those valuations v defined for S where v(φ) ∈ DS for φ ∈ ⊧S ψ
see Section 2, where we go througH the connecTions BetWeeN tHe thRee oPerators and we eXPlaiN why we keep the normality OperaToR DistINcT from The otheR TwO. aDdiTiOnAllY wE DiScuss The ConnectIons betweeN thE pResent approaCH, LfIs, and LFUs. the Paper proceedS as FollowS. SEctIOn 2 intRodUces tHe logiCS LP⊛ and k⊛3 and preseNtS The maiN FailureS OF cLassIcal inferences in tHEsE Logics. Section 3 eStabliShES a CLAssIcaL recapture ReSult (THEorem 3.3) foR lP⊛ AND k⊛3 anD Presents some cOncrete examPLes Of how cLaSsiCAl infeRenceS cAN be Recaptured iN theSe two logiCs. SectIOn 4 introDUces the ApproaCh bY miNimaL InCoNsiStENcy ANd ComPAre It with ouR aPpRoach BaseD ON NOrmaLitY opeRatorS. Section 5 discuSseS the RElaTions BetweEn ouR pRoposAl and tHe traDiTion of LFIs and LFus. In ParticulaR, we DiScuSs The reLAtions BetWeeN TheoreM 3.3 and the dEriVaBILItY Adjustment TheoremS fROM [11, 20], tHe relatiOns betWEeN Lp⊛ And LFI1, anD sOme OtheR ISsues. sectIOn 6 Sums up thE resulTS oF tHe paper. 2 exTensioNs Of KLeeNe logICs wiTh NormAlity OpeRatorS in this section wE Define normaliTY oPERaTOrs aNd wE introduce tHe twO LogiCs LP⊛ ANd k⊛3, whOSe lanGuage DePLoY Normality operators. ALsO, we disCuss tHe relations beTween these LOGIcs and thE KleENe LOgics LP and K3, as wEll as Some relevaNT failureS of clAssical lAws and infEREnce ruleS. PrEliMinAriES. in This paper we adOPT a seMaNtic angLe, aNd we regArd A loGic s as A pAir (L,⊧S), wherE L is a lanGuAgE aNd ⊧s is The reLAtion of (sInGle-CoNclUsion) s-ConseqUence-That Is, ⊧s ⊆ ℘(L)×l. givEn a noneMPtY SEt T oF tRuTh vaLueS aNd a loGic S, WE deNote by Ds ⊆ T the set oF deSIgnaTeD vAlues of s-that is, the valUeS that must bE pResErved tHROugh the vAlid inferences of S. We defiNE the relAtiVe notIon oF logical cOnsEquencE-S-cONsequeNce-as fOllowS: DEfiNITion 2.1 (LOGIcAl COnSequence in mANy-vAlued loGic) Σ ⊧s ψ ⇔ VS(Σ) ⊆ VS(Ψ) where VS(Σ) is the set oF ThoSe valuations v DefIned FOR S WheRE v(Φ) ∈ dS fOr EVerY Φ ∈ σ. Σ ⊧S ψ
see Section 2, where we go  through t he co nne cti on s be twee n the three op e rato rs and we explain whywe ke ep then or malit y opera t or d ist in ct  fr om th e oth ertwo. Ad ditionally  we  d iscuss the c o nn ections be twe en the prese ntapproa ch , L F Is, a ndLFUs.  The p a per pr oceeds as  f o llows. Section 2  i ntro duces the logicsL P⊛ and K⊛3 and pr esents  t h em a infai lures of c la ssica l  infere n ce s i n t h ese logics. S ection 3 es t abl ishesacla s sicalrecap tu r e r esult (Theo rem3.3) forLP⊛ an d  K⊛3 an d  presen ts som e c onc rete ex am ple so f h o wcla s sic al infer en ce s can  ber e c a ptur edin t hesetwo logics. S ect ion4  in trodu ces t he a pp roach  by mi nimal  i nconsistency an d co mpare itwit hour  a pproa c h base d o n n ormalit y opera t ors .S e c ti on 5 discusses the  r e l at ions bet ween o u rpr o posal an dthe  tra d i tionof L F Is  and LFU s. Inp ar ti cular,we  discu ss  th e r elati o ns b etween  Theorem  3.3a nd the Derivab i lity Adjustme n tT h eo r emsfro m [11, 20],  the rela tion s  b etw e en LP ⊛ and  L F I1 ,  and some other iss ue s. Sec tion6 sums up the  results o f t he paper . 2E xt e nsions of Klee ne lo gics withN ormality  Oper ators In  this sec t i on we de fin e n orm ali t y  o perators andw e  int ro duce th e t wo logi csLP⊛  an d K ⊛3 , whose l anguagede pl oy  n orm ality operator s.  Al so , w e dis c uss th e rel atio ns  b e twe en thes e  l o g icsan dtheKle en e log icsL P a nd K3,as well a s s o me r el ev ant fai lures of clas si cal laws a nd  in ferenc e rules. P reliminaries. In this p a per weado pt asema ntic angl e,and we  re g ard alogicS asapai r (L,⊧S ) ,  w her eL is a lan g u age  and⊧S  isthe rel ation of (single-c o ncl usion) S-cons equ ence - t ha t i s ,⊧ S ⊆  ℘ ( L)× L .  Given a nonemp ty set T o ft ru th valuesa ndalogic S , we de noteb y DS ⊆T the set  of desig na tedv a lue s of S-tha t is, th e valuest hat m u st  be p res ervedth rou gh th e vali d  in feren ces of  S . We d efine  t he relat ive notion of logical c onsequ ence- S-c onsequenc e-a s  fo llows: De fini tion 2.1 ( Log ica l Con seq u encein M a ny -va l ued L ogic )  Σ ⊧S ψ ⇔ VS (Σ) ⊆  V S(ψ) whereV S ( Σ)is th e s e t of t hose  valuations v def i ned for S wher e v( φ )  ∈DSf or e ve ry φ ∈ Σ. Σ ⊧S  ψ
see Section_2, where_we go through the_connections between_the_three operators_and_we explain why_we keep the_normality operator distinct from_the other two._Additionally_we discuss the connections between the present approach, LFIs, and LFUs. The paper proceeds_as_follows. Section_2_introduces_the logics LP⊛ and K⊛3_and presents the main failures_of classical_inferences in these logics. Section 3 establishes a_classical_recapture result (Theorem_3.3) for LP⊛ and K⊛3 and presents some concrete_examples of how classical inferences can_be recaptured in_these_two_logics. Section 4 introduces_the approach by minimal inconsistency and_compare it with our approach based_on normality operators. Section 5 discusses the_relations between our proposal and the_tradition of LFIs and LFUs._In particular,_we discuss the relations between_Theorem 3.3 and_the Derivability_Adjustment Theorems from_[11, 20], the relations between LP⊛_and LFI1, and_some other issues. Section 6 sums_up_the results of_the_paper._2 Extensions_of Kleene logics_with_Normality Operators_In_this section we define normality operators_and_we introduce the two logics LP⊛ and_K⊛3, whose language deploy_normality_operators. Also, we discuss_the relations between these logics_and the Kleene logics LP and_K3, as_well as_some relevant failures of classical laws and inference rules. Preliminaries. In_this paper we adopt a semantic_angle, and we regard_a logic_S_as a pair_(L,⊧S),_where L_is a language and ⊧S is the_relation of_(single-conclusion) S-consequence-that is, ⊧S ⊆ ℘(L)×L._Given a nonempty set_T_of truth values and a logic_S, we denote by DS ⊆_T the set of designated_values_of_S-that is, the values that_must be preserved through the valid_inferences of S._We define the relative notion of logical_consequence-S-consequence-as_follows: Definition 2.1 (Logical Consequence in_Many-valued_Logic) Σ ⊧S ψ ⇔ VS(Σ)_⊆_VS(ψ)_where VS(Σ) is the set_of those valuations v defined for_S where v(φ) ∈ DS for every φ ∈_Σ. Σ ⊧S_ψ