aconsistent or paracomplete case. This has practical advantages, since here we face both cases. Second, the coincidence between the three operators is lost if more than three values are at stake. In particular, the consistency operator ○′ by [22] and the determinedness operator by [5] are defined in four-valued settings that are paraconsistent due to a non-classical designated value (b), and paracomplete due to a non-classical undesignated value (n). Thus, although the normality operator collapses on the consistency (or determinedness) operator is three-values are considered, it proves a distinct operator if more than three values are at stake.4 Syntax. Given a nonempty denumerable set P = {p, q, r...} of propositional variables, the language L1(P) of LP⊛ and K⊛3 is defined by the following Backus-Naur form (BNF): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ ⊛φ where p ∈ P, connectives ¬, ∨, ∧ receive their usual informal readings (negation, disjunction, conjunction), and ⊛φ reads 'φ has a classical truth value'. We define φ ⊃ ψ as an abbreviation for ¬φ ∨ ψ. We denote sets of arbitrary formulas in L1(P) by ∆, Γ, Σ,... and we omit reference to P when possible. var(φ) is the set of propositional variables occurring in φ ∈ L1; we use var(Σ) as short for ⋃φ∈Σ var(φ). Semantics. Variables in P are assigned a truth value from the triple {0, 1 2,1} by a valuation function ν ∶ P z→ {0, 1 2,1}. The function is generalized to arbitrary formulas as follows: Definition 2.3 A valuation ν ∶ L1 z→ {0, 12,1} is the unique extension of ν ∶ P z→ {0, 1 2,1} that is induced by Table 1.5 We denote by V the set {ν, ν′, ν′′,...} of valuations defined in accordance with Definition 2.3. The designated values of a logic S are the
aconsistent or paracomplete case. This has practical advantages, since here we front both case. Second, the coincidence between the three operators is lost if more than three value are at stake. In particular, the consistency hustler ○ ′ by [ 22 ] and the determinedness operator by [ 5 ] are defined in four - valued setting that are paraconsistent due to a non - classical designated value (bel), and paracomplete due to a non - classical undesignated value (n). therefore, although the normality operator collapses on the consistency (or determinedness) operator is three - values are considered, it proves a distinct hustler if more than three values are at stake.4 Syntax. Given a nonempty denumerable bent P = { p, q, r... } of propositional variable, the language L1(P) of LP⊛ and K⊛3 is specify by the following Backus - Naur form (BNF ): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ ⊛φ where p ∈ P, connectives ¬, ∨, ∧ receive their usual cozy readings (negation, disjunction, conjunction), and ⊛φ reads' φ has a classical accuracy value'. We specify φ ⊃ ψ as an abbreviation for ¬φ ∨ ψ. We denote sets of arbitrary formulas in L1(P) by ∆, Γ, Σ, ... and we omit reference to P when possible. var(φ) is the set of propositional variables happen in φ ∈ L1; we use var(Σ) as short for ⋃φ∈Σ var(φ). Semantics. variable in phosphorus are assign a truth value from the triple { 0, 1 2,1 } by a evaluation function ν ∶ P z→ { 0, 1 2,1 }. The affair is generalized to arbitrary formulas as follow: Definition 2.3 A valuation ν ∶ L1 z→ { 0, 12,1 } is the unique extension of ν ∶ P z→ { 0, 1 2,1 } that is induced by Table 1.5 We denote by V the set { ν, ν′, ν′′, ... } of valuation defined in accordance with Definition 2.3. The designated values of a logic sulfur are the
acojsistent or paracomplete case. This has krqctican advahtages, sknce here we face both cases. Swcond, the coincidence betwedn the thgee operarors us lost if more than thrsc valbew are at stake. In particglar, the consivtdney operator ○′ by [22] and the determinednqss opetahor by [5] are desinec in rour-valued settings that are paracknsisteit due to a non-vlassical designated value (b), ajd paracomplete duf to a non-coassyxal undesignxted value (n). Thus, althkugh the normality operator coluapsex on the cinwishgncy (or detecminedgess) operatov is thsee-valurs are consideved, iv prives a distinct operavor if more than thrge values drz at stake.4 Syntax. Givwn a notempdy ddbumdraglx sst P = {o, q, r...} of propksitional vqriables, the languabe O1(P) of LP⊛ and I⊛3 is dqfyned by the following Backus-Naur form (BTF): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ∣ ⊛φ where p ∈ P, connectives ¬, ∨, ∧ refeive theyr usual informal readings (negation, disjunction, cmnjunrtkon), akd ⊛φ fwafs 'φ has a classical truth value'. We define φ ⊃ ψ wa sn abbreviation nor ¬φ ∨ ψ. We denote srtd ps arbitrary furmulas ih L1(P) by ∆, Γ, Σ,... and we lmit reserenxe to P wren lossible. var(φ) is the set of propositionaj variables occurriug in φ ∈ L1; we use var(Σ) ss short for ⋃φ∈Σ var(φ). Semancics. Vzriables in P are asakgned a truth vauue fsom the ufiple {0, 1 2,1} by a valtation fuictiou ν ∶ P z→ {0, 1 2,1}. Tne funstion is gfneralized to arbitrary vormupav as folloas: Definition 2.3 A valuation ν ∶ L1 v→ {0, 12,1} is the unoqge vxtension of ν ∶ P z→ {0, 1 2,1} that ys induced by Jable 1.5 We denoge by V thv set {ν, ν′, ν′′,...} mf valuatiogs defined in wccordance wmth Definytiob 2.3. Tye desievated values og a logic S are the
aconsistent or paracomplete case. This has practical here face both Second, the coincidence lost more than three are at stake. particular, the consistency operator ○′ by and the determinedness operator by [5] are defined in four-valued settings that are due to a non-classical designated value (b), and paracomplete due to a non-classical value Thus, the operator collapses on the consistency (or determinedness) operator is three-values are considered, it proves a distinct if more than three values are at stake.4 Given a nonempty denumerable P = {p, q, r...} propositional the language of and is defined by following Backus-Naur form (BNF): φ ∶∶= p ∣ ¬φ ∣ φ ∨ ψ ∣ φ ∧ ψ ⊛φ where P, connectives ∨, receive usual informal readings conjunction), and ⊛φ reads 'φ has value'. We define φ ⊃ ψ as an for ¬φ ψ. We denote sets of arbitrary in L1(P) by ∆, Γ, Σ,... and we reference to P when possible. var(φ) is the set of propositional variables occurring in φ we use var(Σ) as for ⋃φ∈Σ var(φ). Variables P assigned truth value the triple {0, 1 2,1} by a valuation function ν ∶ z→ {0, 1 2,1}. The function is generalized to arbitrary follows: 2.3 A valuation ∶ L1 z→ {0, is unique extension of ν z→ 1 induced Table We denote by V set {ν, ν′, ν′′,...} of defined in accordance with of a logic S are the
aconsistent or paracomplete Case. This haS pracTicAl aDvAntaGes, sInce here we face BOth cAses. Second, the coincidenCe betWeEN the THrEe opeRators iS LoST If mOrE tHan ThREe ValueS arE at stakE. In particuLar, ThE consistency OPeRator ○′ by [22] and The DeterminedneSs oPeratoR bY [5] arE DefinEd iN four-Valued SEttingS that are pArAConsisTEnt due tO A NoN-claSsical designated vALuE (B), and paracompleTe due tO a NOn-CLAssIcaL undesignaTeD valuE (N). Thus, alTHoUGH The NOrmality operaTor collapseS On tHe consIsTenCY (or detErminEdNEss) Operator is tHree-Values are ConsidERed, it prOVes a disTinct oPerAtoR if mORe ThAn tHrEE vaLUeS arE At sTake.4 SyntAx. giVen a nOnemPTY DEnumEraBle sEt P = {p, q, R...} of propositioNal VariABleS, the lAnguaGe L1(P) Of lP⊛ and k⊛3 is defIned bY tHe following BackUs-NaUr form (BNF): Φ ∶∶= p ∣ ¬φ ∣ Φ ∨ ψ ∣ Φ ∧ ψ ∣ ⊛φ WhEre p ∈ P, COnnectIveS ¬, ∨, ∧ reCeive thEir usuaL InfOrMAL ReAdings (negation, disjUnCTIoN, conjuncTion), anD ⊛Φ rEaDS 'φ has a clAsSicAl trUTH valuE'. We dEFiNe φ ⊃ ψ as an AbbrevIAtIoN for ¬φ ∨ ψ. WE dEnote sEtS of ArbItrarY FormUlas in l1(P) by ∆, Γ, Σ,... anD we omIT reference to P wHEn possible. var(Φ) Is THE sET of pRopOsitional vaRiabLEs ocCurrINg In φ ∈ l1; We use Var(Σ) aS sHOrT For ⋃φ∈Σ var(φ). Semantics. VArIables In P arE assigned a truTh value froM THE triple {0, 1 2,1} bY a vaLUaTIon function ν ∶ P z→ {0, 1 2,1}. the fuNction is geNEralized To arbItrary foRmulas as fOLLows: DefiNitIon 2.3 a vaLuaTIOn Ν ∶ L1 z→ {0, 12,1} is the uniquE EXtenSiOn of ν ∶ P z→ {0, 1 2,1} ThaT is induCed By TAblE 1.5 We DeNote by V thE set {ν, ν′, ν′′,...} oF vAlUaTiOns DefinED in accorDaNce WiTh DEfiniTIon 2.3. The DesigNateD vAlUEs oF a logic s ArE THe
aconsistent or paracomplet e case. Th is ha s p rac ti caladva ntages, sinceh erewe face both cases. Se cond,  t h e co i nc idenc e betwe e nt h e t hr ee  op er a to rs is  lo st if m ore than t hre evalues are a t  s take. In p art icular, thecon sisten cy  op e rator  ○′  by [ 22] an d  the d etermined ne s s oper a tor by[ 5 ]aredefined in four-v a lu e d settings tha t arepa r ac o n sis ten t due to a  n on-cl a ssicald es i g n ate d  value (b), a nd paracomp l ete  due t oa n o n-clas sical  u n des ignated val ue ( n). Thus,  altho u gh then ormalit y oper ato r c olla p se sonth e  co n si ste n cy(or dete rm in ednes s) o p e r a toristhre e-val ues are consi der ed,i t p roves  a di stin ct  oper ator i f mor ethan three valu es a re at sta ke. 4Syn ta x. Gi v en a n one mpt y denum erables etP= { p,  q, r...} of propo si t i on al varia bles,t he  l a nguage L 1( P)of L P ⊛  andK⊛3i sdefinedby the fo ll owing B ac kus-Na ur  fo rm(BNF) :  φ ∶ ∶= p ∣  ¬φ ∣ φ∨ ψ ∣ φ ∧ ψ ∣ ⊛φ whe r e p ∈ P, conn e ct i v es ¬, ∨ , ∧  receive th eiru sual  inf o rm alr eadin gs (n eg a ti o n, disjunction, con ju nction ), an d ⊛φ reads 'φ  has a cla s s i cal trut h va l ue ' . We define φ⊃ ψ a s an abbre v iation f or ¬φ  ∨ ψ. We  denote s e t s of arb itr ary  fo rmu l a sin L1(P) by ∆ , Γ, Σ ,. .. andweomit re fer enc e t o P  w hen possi ble. var (φ )is  t heset o f  proposi ti ona lvar iable s  occur ringin φ  ∈  L 1 ; w e use v a r( Σ )  assh or t fo r ⋃ φ∈ Σ var (φ). Sem antics.  Variable s i n  P a re  a ssigned  a truth valu efrom the t ri ple  {0, 1 2 ,1} by a  valuation function ν ∶ P z→ {0 , 1  2,1} . Th e functio n i s gene ral i zed to  arbit raryfo rmu l a s asf o ll ows :Definition 2 .3A val ua tion  ν ∶ L1  z→ {0, 12,1} is t h e u nique extensi onof ν ∶  P  z→ {0 ,  12, 1 } t h a t is induced by  Table 1.5  W e  d enote by V the  s et {ν,ν′, ν′′ ,...} of valu ations de fined inac cord a n cewith Defin ition 2. 3. The de s ignat e dvalue s o f a lo gi c S  arethe
aconsistent or_paracomplete case._This has practical advantages,_since here_we_face both_cases._Second, the coincidence_between the three_operators is lost if_more than three_values_are at stake. In particular, the consistency operator ○′ by [22] and the determinedness_operator_by [5]_are_defined_in four-valued settings that are_paraconsistent due to a non-classical_designated value_(b), and paracomplete due to a non-classical undesignated_value_(n). Thus, although_the normality operator collapses on the consistency (or determinedness)_operator is three-values are considered, it_proves a distinct_operator_if_more than three values_are at stake.4 Syntax. Given a_nonempty denumerable set P = {p,_q, r...} of propositional variables, the language_L1(P) of LP⊛ and K⊛3 is_defined by the following Backus-Naur_form (BNF):_φ ∶∶= p ∣ ¬φ_∣ φ ∨_ψ ∣_φ ∧ ψ_∣ ⊛φ where p ∈ P,_connectives ¬, ∨,_∧ receive their usual informal readings_(negation,_disjunction, conjunction), and_⊛φ_reads_'φ has_a classical truth_value'._We define_φ_⊃ ψ as an abbreviation for_¬φ_∨ ψ. We denote sets of arbitrary_formulas in L1(P) by_∆,_Γ, Σ,... and we_omit reference to P when_possible. var(φ) is the set of_propositional variables_occurring in_φ ∈ L1; we use var(Σ) as short for ⋃φ∈Σ var(φ)._Semantics. Variables in P are assigned_a truth value from_the triple_{0,_1 2,1} by_a_valuation function_ν ∶ P z→ {0, 1 2,1}._The function_is generalized to arbitrary formulas as_follows: Definition 2.3 A_valuation_ν ∶ L1 z→ {0, 12,1}_is the unique extension of ν_∶ P z→ {0, 1_2,1}_that_is induced by Table 1.5_We denote by V the set_{ν, ν′, ν′′,...}_of valuations defined in accordance with Definition_2.3._The designated values of a logic_S_are the