 set theory, the restriction can block the derivation Σ12 of Russell's paradox. From Prawitz's perspectives on P−validity, the restriction can be a reasoning-rejection solution which restricts a particular rule but not a system. Unfortunately, in this section, we attempt to find a way to prevent a particular rule through RND with the assumption that proof-theoretic validity implies normalizability. The restriction of ¬E−rule via Definition 4.3.1 is executed by the requirement of a P−valid inference that only P−valid inferences are to be used, but not by RND. The requirement of a P−valid inference can be requested independently of RND. Therefore, if a reasoningrejection solution is a constraint on a particular inference rule, RND is not a reasoningrejection solution. 155 4.4 Should We Consider Only ⊥ as an Unacceptable Conclusion? In this section, by examining Petrolo and Pistone's case of a (full) normal derivation of a contradiction, we shall deal with the last question of this chapter,'should we consider only ⊥ as an unacceptable conclusion?' The Third Question (iii): Should we consider only ⊥ as an unacceptable conclusion? As we have examined in the last section 4.3, a derivation of ⊥ formalizing Russell's paradox consists of two normal derivations Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬E⊥ Instead of applying ¬E−rule, we can apply ∧I−rule and have Σ10 ¬a ∈ a Σ11 a ∈ a ∧I¬a ∈ a∧a ∈ a Since Σ10 and Σ11 are normal derivations, either the above derivation is in normal form. One of interesting points is that we often call the formula ¬a ∈ a∧ a ∈ a a contradiction. Let us distinguish between a contradiction and an absurdity. For any formula having the form ¬φ ∧φ a contradiction. A contradiction is often regarded as an unacceptable conclusion. Then, it seems convincing that the above derivation is paradoxical because it derives an unacceptable conclusion through acceptable reasoning with acceptable premises. RND imposes a constraint on the application of non-normalizable derivations. If there is a paradoxical derivation which is a (closed) full normal derivation of an unacceptable conclusion, such as ¬φ ∧φ, then RND cannot be
set theory, the restriction can block the deriving Σ12 of Russell's paradox. From Prawitz's position on P−validity, the restriction can be a reasoning - rejection solution which restricts a particular principle but not a system. Unfortunately, in this section, we undertake to find oneself a way to prevent a particular rule through RND with the assumption that proof - theoretical validity implies normalizability. The restriction of ¬E−rule via Definition 4.3.1 is executed by the necessity of a P−valid inference that only P−valid inferences are to be used, but not by RND. The requirement of a P−valid inference can be requested independently of RND. consequently, if a reasoningrejection solution is a constraint on a particular inference rule, RND is not a reasoningrejection solution. 155 4.4 Should We study Only ⊥ as an Unacceptable decision? In this section, by analyze Petrolo and Pistone's case of a (full) normal derivation of a contradiction, we shall deal with the last doubt of this chapter,'should we consider only ⊥ as an unacceptable conclusion?' The Third Question (iii ): Should we consider only ⊥ as an unacceptable conclusion? As we have examined in the last section 4.3, a derivation of ⊥ formalizing Russell's paradox consists of two normal derivations Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬E⊥ Instead of applying ¬E−rule, we can apply ∧I−rule and suffer Σ10 ¬a ∈ a Σ11 a ∈ a ∧I¬a ∈ a∧a ∈ a Since Σ10 and Σ11 are normal derivations, either the above derivation is in normal mannequin. One of interesting points is that we often call the formula ¬a ∈ a∧ a ∈ a a contradiction. lease us distinguish between a contradiction and an absurdity. For any formula having the form ¬φ ∧φ a contradiction. A contradiction is frequently regarded as an unacceptable conclusion. Then, it seems convincing that the above derivation is paradoxical because it derives an impossible conclusion through acceptable reasoning with acceptable premises. RND imposes a constraint on the application of non - normalizable ancestry. If there is a paradoxical derivation which is a (closed) full normal derivation of an impossible conclusion, such as ¬φ ∧φ, then RND cannot be
 seh theory, the restriction can block the bwrivatmon Σ12 of Russell'r paradox. From Prawitz's pers'ectuves in P−validity, the restrkction caj be a rwasoiing-rejection solution wmnch rsdtrietw a particular rule but tot a system. Utfurcunately, in this section, we attempt eo find a way to prevenj a pswtichlar rule through RND with the asshmption that proof-throretic validity implies nlrmapizability. The reshriction of ¬E−rujw via Definigion 4.3.1 is eqzcuted by tge requirement of a P−valid infefence that only P−callg inferencew are to be used, nlt not ty RND. Yhe requiremenb of e P−vqlid inference can be requested indepenqently of RUD. Therefore, if a reawobingrgjecthon rilugioh ms z conshramnt on a padticular inderence rule, RND is njn a reasoningdejectyog solution. 155 4.4 Should We Consider Only ⊥ av ah Unacceptable Conclusiin? In this section, by examinind Petrolo and Pistone's case of a (full) normal derieatioi uf c contfqdlction, we shall deal with the last question os tnix chapter,'shoulb we consider onky ⊥ sf an unacceptxble concmusion?' The Third Qkestion (iii): Whould we conxider only ⊥ as an unacceptavle conclusijb? As we have examiued in the lcst sevtion 4.3, a derivation of ⊥ formclizinf Russell's oaradox ckvsists of two nofmak gerivations Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬E⊥ Insvead pf applhing ¬E−rule, we can apoly ∧I−vgle and have Σ10 ¬a ∈ a Σ11 a ∈ c ∧I¬a ∈ a∧a ∈ a Sinfe Σ10 and Σ11 are normal derivations, either the abpva dvrivation is ik normal form. Oge of interestnng poincs is ghat we ofnen call vhe formula ¬w ∈ a∧ a ∈ a a cmjtradiction. Net us dystibguiwh betwddn a contradicyion and cu absurdiry. For any formula haxjng the form ¬φ ∧φ c xontradiction. A covtrwdpctmon if often regarged xs xm unazceptable civcluxion. Then, it seems cmnvihcing that the abofe derivatuon is pwradoxical bevause it derives aj unarceptaule comclosion through acceptable reasonjng with wcccptable premifes. VND ymposes a eonstraint on the application of non-normelizable derivations. If there is a paradoxncsl derivatipn whych is a (wlosed) full normal dwrivation of an ukacceptable conclusion, such ds ¬φ ∧φ, hhen RND cannot be
set theory, the restriction can block the of paradox. From perspectives on P−validity, reasoning-rejection which restricts a rule but not system. Unfortunately, in this section, we to find a way to prevent a particular rule through RND with the that proof-theoretic validity implies normalizability. The restriction of ¬E−rule via Definition 4.3.1 is by requirement a inference that only P−valid inferences are to be used, but not by RND. The requirement of P−valid inference can be requested independently of RND. if a reasoningrejection solution a constraint on a particular rule, is not reasoningrejection 155 Should We Consider ⊥ as an Unacceptable Conclusion? In this section, by examining Petrolo and Pistone's case of a (full) derivation of we shall with last of this chapter,'should only ⊥ as an unacceptable conclusion?' (iii): Should we consider only ⊥ as an conclusion? As have examined in the last section a derivation of ⊥ formalizing Russell's paradox consists two normal derivations Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬E⊥ applying ¬E−rule, we can ∧I−rule and have ¬a a a a ∧I¬a a∧a ∈ a Since Σ10 and Σ11 are normal derivations, either above derivation is in normal form. One of interesting points we call the formula ∈ a∧ a ∈ a Let us distinguish between and absurdity. having form ∧φ a contradiction. A is often regarded as an conclusion. Then, it seems is paradoxical because it derives an unacceptable conclusion acceptable reasoning with acceptable premises. RND imposes constraint on the application of non-normalizable derivations. If there is a paradoxical which is full normal derivation of an unacceptable conclusion, such ¬φ ∧φ, then RND be
 set theory, the restriction caN block the dErivaTioN Σ12 oF RUsseLl's pAradox. From PrawITz's pErspectives on P−validity, The reStRIctiON cAn be a ReasoniNG-rEJEctIoN sOluTiON wHich rEstRicts a pArticular rUle BuT not a system. UNFoRtunately, iN thIs section, we aTteMpt to fInD a wAY to prEveNt a paRticulAR rule tHrough RND WiTH the asSUmption THAt ProoF-theoretic validitY ImPLies normalizabIlity. THe REsTRIctIon Of ¬E−rule via deFinitIOn 4.3.1 is exeCUtED BY thE Requirement of A P−valid infeREncE that oNlY P−vALid infErencEs ARe tO be used, but nOt by rND. The reqUiremeNT of a P−vaLId inferEnce caN be ReqUestED iNdEpeNdENtlY Of rND. tHerEfore, if a ReAsOningRejeCTION solUtiOn is A consTraint on a partIcuLar iNFerEnce rUle, RNd is nOt A reasOningrEjectIoN solution. 155 4.4 Should we CoNsider OnlY ⊥ as An unaCcEptabLE ConclUsiOn? IN this seCtion, by EXamInING peTrolo and Pistone's caSe OF A (fUll) normaL derivATiOn OF a contraDiCtiOn, we SHAll deAl wiTH tHe last quEstion OF tHiS chapteR,'sHould wE cOnsIdeR only ⊥ AS an uNaccepTable conClusiON?' The Third QuestIOn (iii): Should we COnSIDeR Only ⊥ As aN unacceptabLe coNClusIon? AS We HavE ExamiNed in ThE LaST section 4.3, a derivation Of ⊥ FormalIzing russell's paradOx consists OF TWo normal DeriVAtIOns Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬e⊥ InstEad of applyINg ¬E−rule, wE can aPply ∧I−rulE and have Σ10 ¬A ∈ A σ11 a ∈ a ∧I¬a ∈ a∧a ∈ a sinCe Σ10 And σ11 arE NOrMal derivationS, EItheR tHe above DerIvation Is iN noRmaL foRm. one of inteResting pOiNtS iS tHat We oftEN call the FoRmuLa ¬A ∈ a∧ a ∈ A a conTRadictIon. LeT us dIsTiNGuiSh betweEN a CONtraDiCtIon aNd aN aBsurdIty. FOR anY formulA having thE foRM ¬φ ∧φ a CoNtRadictiOn. A contradictIoN is often reGaRdeD as an uNACceptablE conclusion. Then, it seems cONvincinG thAt the AbovE derivatiOn iS paradOxiCAl becaUse it dEriveS aN unACCeptaBLE cOncLuSion througH ACcePtablE rEasoNing witH acceptable premiseS. rND Imposes a constRaiNt on THE aPplICaTIon Of NOn-nORMalizable derivaTions. If theRe IS a ParadoxicaL DerIvAtion whIch is a (cLosed) FUll normAl derivatIon of an unAcCeptABLe cOnclusion, sUch as ¬φ ∧φ, tHen RND canNOt be
 set theory, the restricti on can blo ck th e d eri va tion  Σ12  of Russell'sp arad ox. From Prawitz's per spect iv e s on P− valid ity, th e  r e s tri ct io n c an be  a re aso ning-re jection so lut io n which rest r ic ts a parti cul ar rule butnot  a sys te m.U nfort una tely,  in th i s sect ion, we a tt e mpt to find aw a yto p revent a particul a rr ule through RN D with  t h ea s sum pti on that pr oo f-the o retic v a li d i t y i m plies normali zability. T h e r estric ti ono f ¬E−r ule v ia Def inition 4.3 .1 i s execute d by t h e requi r ement o f a P− val idinfe r en ce  th at onl y  P −va l idinferenc es  a re to  beu s e d , bu t n ot b y RND . The require men t of a P −vali d inf eren ce  canbe req ueste dindependently o f RN D. Theref ore ,ifareaso n ingrej ect ion  soluti on is a con st r a i nt  on a particular i nf e r en ce rule,  RND i s  n ot a reason in gre ject i o n sol utio n .155 4.4Should We  C onsider  O nly ⊥as  an  Un accep t able  Concl usion? I n thi s  section, by e x amining Petro l oa n dP isto ne' s case of a  (fu l l) n orma l  d eri v ation  of a  c o nt r adiction, we shallde al wit h the  last questio n of thisc h a pter,'sh ould we consider only⊥ asan unaccep t able con clusi on?' The  Third Qu e s tion (ii i):  Sh oul d w e co nsider only ⊥ a s an  u naccept abl e concl usi on?  As  we  h ave exami ned in t he  l as tsec tion4 .3, a de ri vat io n o f ⊥ f o rmaliz ing R usse ll 's par adox co n si s t s of  t wo  nor mal  d eriva tion s  Σ1 0 and Σ 11. Σ10 ¬ a ∈ a Σ1 1a∈ a ¬E⊥  Instead of a pp lying ¬E−r ul e,we can a pply ∧I− rule and have Σ10 ¬a ∈a  Σ11 a∈ a  ∧I¬a  ∈ a ∧a ∈ a Si nce  Σ10 a ndΣ 11 are  norma l der iv ati o n s, ei t h er  th eabove deri v a tio n isin  nor mal for m. One of interest i ngpoints is tha t w e of t e ncal l  t h e f or m ula ¬ a ∈ a∧ a ∈ a acontradict io n .Let us dis t ing ui sh betw een a c ontra d ictionand an ab surdity.Fo r an y for mula havin g the fo rm ¬φ ∧φa  cont r ad ictio n.A cont ra dic tionis oft e n r egard ed asan  unacc eptab le  conclus ion. Then, it seems con vincin g tha t t he aboveder i vat ion is pa rado xical beca use  it  deri ves an un acce p ta ble concl usio n  througha cc ept a b le  reasoningw i t h a ccept abl e  premi ses.  RND imposes a co n straint on the  app l i cat ion of n on -normalizableder iv a t ions. If  t here is a p aradoxic al deriv ationwhichis a (c l o se d ) full  nor mal  derivati onof an unac ce pt a ble co nclu si on, su ch as¬ φ ∧φ , then RND cannotbe
 set_theory, the_restriction can block the_derivation Σ12_of_Russell's paradox._From_Prawitz's perspectives on_P−validity, the restriction_can be a reasoning-rejection_solution which restricts_a_particular rule but not a system. Unfortunately, in this section, we attempt to find_a_way to_prevent_a_particular rule through RND with_the assumption that proof-theoretic validity_implies normalizability._The restriction of ¬E−rule via Definition 4.3.1 is_executed_by the requirement_of a P−valid inference that only P−valid inferences are_to be used, but not by_RND. The requirement_of_a_P−valid inference can be_requested independently of RND. Therefore, if_a reasoningrejection solution is a constraint_on a particular inference rule, RND is_not a reasoningrejection solution. 155 4.4_Should We Consider Only ⊥_as an_Unacceptable Conclusion? In this section,_by examining Petrolo_and Pistone's_case of a_(full) normal derivation of a contradiction,_we shall deal_with the last question of this_chapter,'should_we consider only_⊥_as_an unacceptable_conclusion?' The Third_Question_(iii): Should_we_consider only ⊥ as an unacceptable_conclusion?_As we have examined in the last_section 4.3, a derivation_of_⊥ formalizing Russell's paradox_consists of two normal derivations_Σ10 and Σ11. Σ10 ¬a ∈_a Σ11_a ∈_a ¬E⊥ Instead of applying ¬E−rule, we can apply ∧I−rule and_have Σ10 ¬a ∈ a Σ11_a ∈ a ∧I¬a_∈ a∧a_∈_a Since Σ10_and_Σ11 are_normal derivations, either the above derivation is_in normal_form. One of interesting points is_that we often call_the_formula ¬a ∈ a∧ a ∈_a a contradiction. Let us distinguish_between a contradiction and an_absurdity._For_any formula having the form_¬φ ∧φ a contradiction. A contradiction_is often regarded_as an unacceptable conclusion. Then, it seems_convincing_that the above derivation is paradoxical_because_it derives an unacceptable conclusion through_acceptable_reasoning_with acceptable premises. RND imposes_a constraint on the application of_non-normalizable derivations. If there is a paradoxical derivation which_is a (closed)_full normal derivation of an_unacceptable_conclusion,_such as ¬φ ∧φ, then RND cannot be