�LPmi ψ ⇒ Σ ⊧CL ψ Minimal inconsistency and recapture. Here we show that minimal inconistency offers a particular way to recapture of classical consequence. More precisely: with minimal inconsistency, we can reason classically whenever the premise-set has a classical model; otherwise, we must resort to LP-reasoning : 8The definition we use here is just the special case of Priest's definition for propositional logic. 9Here sub(φ) denotes the set of subformulas of φ. As for the rationale of clause (2): take Σ = {p,p ∧ ¬p}, and assume u! = {p, r} for some r ∉ var(Σ). We have u ∈ ULP(Σ), but we also have another valuation u′ of Σ such that u′! = {p}. We have u′ < u. In a nutshell, no minimally inconsistent model of Σ assigns a non-classical value to variables out of var(Σ). 10Notice that the valuations of ψ needs not be mi-models of ψ. 11Notice that none of the converses of these implications hold [24]. Normality operators and Classical Recapture in Extensions of Kleene Logics 11 Proposition 4.5 UCL(Σ) ≠ ∅ ⇒ (Σ ⊧LPmi ψ ⇔ Σ ⊧CL ψ) Proof. If UCL(Σ) ≠ ∅, then ULPmi(Σ) = UCL(Σ), by Definition 4.2. As a consequence, ULPmi(Σ) ⊆ UCL(ψ) iff UCL(Σ) ⊆ ULP(ψ). But this equates with Σ ⊧LPmi ψ⇔ Σ ⊧CL ψ if UCL(Σ) ≠ ∅. What if UCL(Σ) = ∅? Well, we can have different cases here. For instance, the classically valid inference from p, p ⊃ q, r ∧ ¬r to q is valid also in LPmi. Indeed, the mi-models of {p, p ⊃ q, r ∧ ¬r} assign 1 to p and p ⊃ q, and 1 2 to r (only). As a consequence, such models will also assign
� LPmi ψ ⇒ Σ ⊧CL ψ Minimal inconsistency and recapture. Here we show that minimal inconistency offers a particular direction to recapture of authoritative consequence. More precisely: with minimal inconsistency, we can reason classically whenever the premise - hardening have a classical model; differently, we must resort to LP - reasoning: 8The definition we use here is just the particular case of Priest's definition for propositional logic. 9Here sub(φ) denotes the hardening of subformulas of φ. As for the rationale of clause (2 ): take Σ = { p, phosphorus ∧ ¬p }, and assume u! = { p, r } for some radius ∉ var(Σ). We have u ∈ ULP(Σ), but we also suffer another valuation u′ of Σ such that u′! = { p }. We have u′ < u. In a nutshell, no minimally discrepant model of Σ assigns a non - authoritative value to variables out of var(Σ). 10Notice that the valuation of ψ needs not be mile - models of ψ. 11Notice that none of the converses of these implications carry [ 24 ]. Normality operators and Classical Recapture in Extensions of Kleene Logics 11 Proposition 4.5 UCL(Σ) ≠ ∅ ⇒ (Σ ⊧LPmi ψ ⇔ Σ ⊧CL ψ) Proof. If UCL(Σ) ≠ ∅, then ULPmi(Σ) = UCL(Σ), by Definition 4.2. As a consequence, ULPmi(Σ) ⊆ UCL(ψ) iff UCL(Σ) ⊆ ULP(ψ). But this equates with Σ ⊧LPmi ψ⇔ Σ ⊧CL ψ if UCL(Σ) ≠ ∅. What if UCL(Σ) = ∅? Well, we can have different cases here. For case, the classically valid inference from p, p ⊃ q, r ∧ ¬r to q is valid also in LPmi. Indeed, the mi - models of { p, p ⊃ q, r ∧ ¬r } assign 1 to p and p ⊃ q, and 1 2 to r (only). As a consequence, such models will also assign
�LPml ψ ⇒ Σ ⊧CL ψ Minimal inconsirtency and recaktyre. Hece we sgow that minimal inconistency offers a partucular way to recapturd of clasdical cobseqnence. More precisely: witm minjlal nnronsistency, we gan reason wlassically whandvzr the premise-set has a classical moqel; othrraise, we must rgsort eo LL-gecsoning : 8The definition we use gere is just the spevial case of Priest's definltioj for propositionap logic. 9Herg suf(φ) denotes the set of subformulas of φ. As for the rationale of clausd (2): tale Σ = {p,p ∧ ¬p}, abd wvsume u! = {p, c} for fome r ∉ var(Σ). We have g ∈ ULP(Σ), nut we also haye anmthwr valuation u′ of Σ surh that u′! = {p}. We have u′ < u. In d uutshell, no minimally ibconshstett mueel of Σ essjgns a noi-classical balue to vaeiables out of var(Σ). 10Moepve that the baluatyogs of ψ needs not be mi-models of ψ. 11Notice thzt none of the conversew of these implicatiojs hold [24]. Gormality operators and Classical Recapture in Exdensikvs if Yoefne Logics 11 Proposition 4.5 UCL(Σ) ≠ ∅ ⇒ (Σ ⊧LPmi ψ ⇔ Σ ⊧CL ψ) [dopf. If UCL(Σ) ≠ ∅, then ULPmi(Σ) = UCL(Σ), bj Cgfinition 4.2. As x conszsusnce, ULPmi(Σ) ⊆ UCL(ψ) ifv UCL(Σ) ⊆ OLP(ψ). Bur this eqtatex with Σ ⊧LPmi ψ⇔ Σ ⊧CL ψ if UCL(Σ) ≠ ∅. What if UCL(Σ) = ∅? Well, we can have different eases nere. Gor instance, the classieally balid inferfnce from o, p ⊃ q, r ∧ ¬r to q is vdlid also in LPmi. Indeed, tre mi-modeos oy {p, p ⊃ q, r ∧ ¬t} assigg 1 to p anf p ⊃ q, and 1 2 to r (only). Ad a clnvequence, skch models will also assign
�LPmi ψ ⇒ Σ ⊧CL ψ Minimal recapture. we show minimal inconistency offers of consequence. More precisely: minimal inconsistency, we reason classically whenever the premise-set has classical model; otherwise, we must resort to LP-reasoning : 8The definition we use is just the special case of Priest's definition for propositional logic. 9Here sub(φ) the of of As for the rationale of clause (2): take Σ = {p,p ∧ ¬p}, and assume u! {p, r} for some r ∉ var(Σ). We u ∈ ULP(Σ), but also have another valuation u′ Σ that u′! {p}. have < u. In nutshell, no minimally inconsistent model of Σ assigns a non-classical value to variables out of var(Σ). 10Notice the valuations needs not mi-models ψ. that none of of these implications hold [24]. Normality Recapture in Extensions of Kleene Logics 11 Proposition UCL(Σ) ≠ ⇒ (Σ ⊧LPmi ψ ⇔ Σ ψ) Proof. If UCL(Σ) ≠ ∅, then ULPmi(Σ) UCL(Σ), by Definition 4.2. As a consequence, ULPmi(Σ) ⊆ UCL(ψ) iff UCL(Σ) ⊆ ULP(ψ). But with Σ ⊧LPmi ψ⇔ ⊧CL ψ if ≠ What UCL(Σ) ∅? Well, can have different cases here. For instance, the classically valid inference p, p ⊃ q, r ∧ ¬r to q is in Indeed, the mi-models {p, p ⊃ q, ∧ assign 1 to p ⊃ and r As consequence, such models will assign
�LPmi ψ ⇒ Σ ⊧CL ψ Minimal inconsistEncy and recApturE. HeRe wE sHow tHat mInimal inconistENcy oFfers a particular way to rEcaptUrE Of clASsIcal cOnsequeNCe. mORe pReCiSelY: wITh MinimAl iNconsisTency, we can ReaSoN classically WHeNever the prEmiSe-set has a claSsiCal modEl; OthERwise, We mUst reSort to lp-reasoNing : 8The deFiNItion wE Use here IS JuSt thE special case of PriESt'S Definition for pRoposiTiONaL LOgiC. 9HeRe sub(φ) denoTeS the sET of subfORmULAS of Φ. as for the ratioNale of clausE (2): TakE Σ = {p,p ∧ ¬p}, aNd AssUMe u! = {p, r} fOr somE r ∉ VAr(Σ). we have u ∈ ULP(Σ), But wE also have AnotheR ValuatiON u′ of Σ suCh that U′! = {p}. WE haVe u′ < u. iN a NuTshElL, No mINiMalLY inConsisteNt MoDel of σ assIGNS A non-ClaSsicAl valUe to variables Out Of vaR(σ). 10NoTice tHat thE valUaTions Of ψ neeDs not Be Mi-models of ψ. 11NotiCe thAt none of tHe cOnVerSeS of thESe implIcaTioNs hold [24]. NOrmalitY OpeRaTORS aNd Classical RecaptuRe IN exTensions Of KleeNE LOgICs 11 ProposItIon 4.5 uCL(Σ) ≠ ∅ ⇒ (σ ⊧lpmi ψ ⇔ Σ ⊧cL ψ) PROoF. If UCL(Σ) ≠ ∅, tHen ULPMI(Σ) = uCl(Σ), by DefInItion 4.2. AS a ConSeqUence, ulPmi(σ) ⊆ UCL(ψ) iFf UCL(Σ) ⊆ ULp(ψ). But THis equates with σ ⊧lPmi ψ⇔ Σ ⊧CL ψ if UCl(σ) ≠ ∅. WHAT iF uCL(Σ) = ∅? welL, we can have dIffeREnt cAses HErE. FoR InstaNce, thE cLAsSIcally valid inferencE fRom p, p ⊃ q, R ∧ ¬r to q Is valid also in lPmi. Indeed, THE Mi-models Of {p, p ⊃ Q, R ∧ ¬r} ASsign 1 to p and p ⊃ q, aNd 1 2 to r (Only). As a conSEquence, sUch moDels will Also assigN
�LPmi ψ ⇒ Σ ⊧CL ψ Minimalinconsiste ncy a ndrec ap ture . He re we show tha t  min imal inconistency offe rs apa r ticu l ar  wayto reca p tu r e  of  c la ssi ca l  c onseq uen ce. Mor e precisel y:wi th minimal i n co nsistency,  we  can reasoncla ssical ly  wh e never  th e pre mise-s e t hasa classic al model; otherwi s e ,we m ust resort to LP- r ea s oning : 8The d efinit io n  w e use  he re is just  t he sp e cial ca s eo f Pri e st's definiti on for prop o sit ionallo gic .  9Here  sub( φ) den otes the se t of  subformu las of φ. As f o r the r ationa leofclau s e(2 ):ta k e Σ ={p, p  ∧¬p}, and  a ss ume u ! ={ p , r} f orsome  r ∉var(Σ). We ha veu ∈U LP( Σ), b ut we  als ohaveanothe r val ua tion u′ of Σ su ch t hat u′! =  {p }.  We  h ave u ′  < u.Ina n utshell , no mi n ima ll y i nc onsistent model of  Σ a ss igns a n on-cla s si ca l  value t ovar iabl e s  outof v a r( Σ). 10No tice t h at  t he valu at ions o fψ n eed s not be m i-mode ls of ψ.  11No t ice that noneo f the convers e so f  t h eseimp lications h old[ 24].  Nor m al ity opera torsan d  C l assical Recapture i nExtens ionsof Kleene Log ics 11 Pro p o s ition 4. 5 UC L (Σ )  ≠ ∅ ⇒ (Σ ⊧LPm i ψ ⇔  Σ ⊧CL ψ)P roof. If  UCL( Σ) ≠ ∅,then ULPm i ( Σ) = UCL (Σ) , b y D efi n i ti on 4.2. As ac o nseq ue nce, UL Pmi (Σ) ⊆ U CL( ψ)iff  UC L( Σ) ⊆ ULP( ψ). Butth is  e qu ate s wit h  Σ ⊧LPmi  ψ ⇔ Σ  ⊧ CLψ ifU CL(Σ)≠ ∅.What  i fU CL( Σ) = ∅? We l l , we  c an  hav e d if feren t ca s eshere. F or instan ce, thecl as sically  valid infere nc e from p,p⊃ q , r ∧¬ r  to q is  valid also in LPmi. In d eed, th e m i-mod elsof {p, p⊃ q , r ∧¬r} assign  1 top and  p  ⊃q ,  and1 2tor(only). As a  co nsequ en ce,such mo dels will also ass i gn
�LPmi ψ_⇒ Σ_⊧CL ψ Minimal inconsistency_and recapture._Here_we show_that_minimal inconistency offers_a particular way_to recapture of classical_consequence. More precisely:_with_minimal inconsistency, we can reason classically whenever the premise-set has a classical model; otherwise,_we_must resort_to_LP-reasoning_: 8The definition we use_here is just the special_case of_Priest's definition for propositional logic. 9Here sub(φ) denotes_the_set of subformulas_of φ. As for the rationale of clause (2):_take Σ = {p,p ∧ ¬p},_and assume u!_=_{p,_r} for some r_∉ var(Σ). We have u ∈_ULP(Σ), but we also have another_valuation u′ of Σ such that u′!_= {p}. We have u′ <_u. In a nutshell, no_minimally inconsistent_model of Σ assigns a_non-classical value to_variables out_of var(Σ). 10Notice_that the valuations of ψ needs_not be mi-models_of ψ. 11Notice that none of_the_converses of these_implications_hold_[24]. Normality_operators and Classical_Recapture_in Extensions_of_Kleene Logics 11 Proposition 4.5 UCL(Σ)_≠_∅ ⇒ (Σ ⊧LPmi ψ ⇔ Σ_⊧CL ψ) Proof. If_UCL(Σ)_≠ ∅, then ULPmi(Σ)_= UCL(Σ), by Definition 4.2._As a consequence, ULPmi(Σ) ⊆ UCL(ψ)_iff UCL(Σ)_⊆ ULP(ψ)._But this equates with Σ ⊧LPmi ψ⇔ Σ ⊧CL ψ if_UCL(Σ) ≠ ∅. What if UCL(Σ)_= ∅? Well, we_can have_different_cases here. For_instance,_the classically_valid inference from p, p ⊃ q,_r ∧_¬r to q is valid also_in LPmi. Indeed, the_mi-models_of {p, p ⊃ q, r_∧ ¬r} assign 1 to p_and p ⊃ q, and_1_2_to r (only). As a_consequence, such models will also assign