 the proof-theoretic solution to the paradoxes. Petrolo and Pistone (2018) have considered the very case. 156 Under the distinction between a contradiction and an absurdity, Petrolo and Pistone (2018) consider the possibility of a normal derivation of the paradoxes. Unlike Tennant (1982, 2016, 2017), they call a derivation D in a given system S a normal paradox (Nparadox) iff (i) D is closed, (ii) D is involved in id est rules, (iii) D is in full normal form, and (iv) if a formula φ is the conclusion of D, then either φ → ⊥ or ¬φ can be derived in S. Similar to Sainsbury's notion of a paradox, one may regard a paradox as a derivation of the unacceptable conclusion from the acceptable premises by the acceptable reasoning. Petrolo and Pistone (2018) regard both a contradiction and an absurdity as unacceptable conclusions and suggest a closed normal derivation of a contradiction from Russell's paradox. We borrow from Prawitz (1965, Appendix B) the natural deduction system SN for the naive se theory which contains the rules for ∧, →, ¬, and ∈. Then, we have a closed normal derivation of a ∈ a∧¬a ∈ a. Proposition 4.4.1. Let us define a parameter a as {x|¬x ∈ x}. Then, there is a closed normal derivation of ¬a ∈ a∧a ∈ a in SN. Proof. Two claims verify the result. Claim 1. there is a closed derivation Σ15 of ¬a ∈ a∧a ∈ a. First, we have a closed derivation Σ13 of ¬a ∈ a. [a ∈ a]1 ******************** de f a ∈ {x|¬x ∈ x} ∈ E¬a ∈ a [a ∈ a]1 ¬E⊥ ¬I,1¬a ∈ a With the derivation Σ13, we have a closed derivation Σ14 of a ∈ a. Σ13 ¬a ∈ a ∈ I a ∈ {x|¬x ∈ x} ******************** de f a ∈ a 157 Then, we have a closed derivation Σ15 of ¬a ∈ a∧a ∈ a. Σ13 ¬a ∈ a Σ
the proof - theoretic solution to the paradoxes. Petrolo and Pistone (2018) have considered the very subject. 156 Under the differentiation between a contradiction and an absurdity, Petrolo and Pistone (2018) consider the possibility of a normal derivation of the paradox. Unlike Tennant (1982, 2016, 2017), they call a derivation D in a give system S a normal paradox (Nparadox) iff (i) D is close, (ii) D is involved in i d est rules, (iii) D is in wide normal form, and (iv) if a formula φ is the termination of D, then either φ → ⊥ or ¬φ can be derived in S. Similar to Sainsbury's impression of a paradox, one may regard a paradox as a deriving of the impossible conclusion from the acceptable premises by the satisfactory reasoning. Petrolo and Pistone (2018) regard both a contradiction and an absurdity as unacceptable conclusions and suggest a closed normal deriving of a contradiction from Russell's paradox. We borrow from Prawitz (1965, Appendix B) the natural deduction system SN for the uninitiate se theory which contains the rules for ∧, →, ¬, and ∈. Then, we have a closed normal derivation of a ∈ a∧¬a ∈ a. Proposition 4.4.1. Let us define a parameter a as { x|¬x ∈ x }. Then, there is a closed normal derivation of ¬a ∈ a∧a ∈ a in SN. Proof. Two claims verify the result. Claim 1. there embody a closed derivation Σ15 of ¬a ∈ a∧a ∈ a. foremost, we take a shut derivation Σ13 of ¬a ∈ a. [ a ∈ a]1 * * * * * * * * * * * * * * * * * * * * de f a ∈ { x|¬x ∈ x } ∈ E¬a ∈ a [ a ∈ a]1 ¬E⊥ ¬I,1¬a ∈ a With the derivation Σ13, we have a closed derivation Σ14 of a ∈ a. Σ13 ¬a ∈ a ∈ I a ∈ { x|¬x ∈ x } * * * * * * * * * * * * * * * * * * * * de f a ∈ a 157 Then, we have a closed derivation Σ15 of ¬a ∈ a∧a ∈ a. Σ13 ¬a ∈ a Σ
 thf proof-theoretic solutiok to the paradoxgs. Petrono and Pistone (2018) have considered the very cese. 156 Undee the distinction betwden a connradictiob anv an absurdity, Pxfrolo akb Pisflne (2018) ronsider the poxsibility mf a normal desixacion of the paradoxes. Unlike Tennant (1982, 2016, 2017), theu fall a derivatyon C in z given system S a normal paradox (Hparadoq) iff (i) D is closrd, (ii) D is involved in id fst gules, (iii) D is in vull normal fori, and (iv) if a formula φ pv the concmusion of D, then either φ → ⊥ or ¬φ zan bz derived ib W. Slkilar to Samnsburj's notion of a paradof, one msy regard a pavadox as a derivation of the nnacceptable conclusyon from dhz acceptable premises bt the accaptacoe feaaoiinf. Petrllo and Pistohe (2018) regard voth a contradictiom wbd an absurdify as tnwcceptable conclusions and suggest a clmses normal derivation of q contradiction from Tussell's pwradox. We borrow from Prawitz (1965, Appendix B) the natgral veaucuiin syrrel SN for the naive se theory which contains trs tukes for ∧, →, ¬, and ∈. Then, we hsvf s closed normau derirztjon of a ∈ a∧¬a ∈ a. Prlpositijn 4.4.1. Lwt us defyne s parameter a as {x|¬x ∈ x}. Then, there is a bloswd normal derivatiln of ¬a ∈ a∧a ∈ a on SN. Proof. Two claims verifv the desult. Clail 1. there jr a closed derivxtipn Σ15 of ¬a ∈ a∧a ∈ a. First, we havq a closev dernvation Σ13 of ¬s ∈ a. [a ∈ a]1 ******************** de f w ∈ {x|¬x ∈ x} ∈ E¬a ∈ a [a ∈ a]1 ¬E⊥ ¬I,1¬w ∈ a Cith dhe derivahion Σ13, we have a closed derivation Σ14 of a ∈ a. Σ13 ¬a ∈ d ∈ P a ∈ {x|¬x ∈ r} ******************** de n a ∈ a 157 Then, we have a closed derivacion Σ15 uf ¬a ∈ a∧a ∈ z. Σ13 ¬a ∈ a Σ
the proof-theoretic solution to the paradoxes. Petrolo (2018) considered the case. 156 Under and absurdity, Petrolo and (2018) consider the of a normal derivation of the Unlike Tennant (1982, 2016, 2017), they call a derivation D in a given S a normal paradox (Nparadox) iff (i) D is closed, (ii) D is in est (iii) is in full normal form, and (iv) if a formula φ is the conclusion of D, either φ → ⊥ or ¬φ can be in S. Similar to notion of a paradox, one regard paradox as derivation the conclusion from the premises by the acceptable reasoning. Petrolo and Pistone (2018) regard both a contradiction and an absurdity as conclusions and closed normal of contradiction Russell's paradox. We Prawitz (1965, Appendix B) the natural for the naive se theory which contains the for ∧, ¬, and ∈. Then, we have closed normal derivation of a ∈ a∧¬a ∈ Proposition 4.4.1. Let us define a parameter a as {x|¬x ∈ x}. Then, there is normal derivation of ¬a a∧a ∈ a SN. Two verify result. Claim there is a closed derivation Σ15 of ¬a ∈ a∧a ∈ First, we have a closed derivation Σ13 of ¬a ∈ ∈ ******************** de f ∈ {x|¬x ∈ x} E¬a a [a ∈ a]1 ∈ With we a derivation Σ14 of a a. Σ13 ¬a ∈ a I a ∈ {x|¬x a ∈ a 157 Then, we have a derivation Σ15 of ¬a ∈ a∧a ∈ Σ13 ¬a ∈ a Σ
 the proof-theoretic solution To the paradOxes. PEtrOlo AnD PisTone (2018) Have considered THe veRy case. 156 Under the distinctIon beTwEEn a cONtRadicTion and AN aBSUrdItY, PEtrOlO AnD PistOne (2018) ConsideR the possibIliTy Of a normal derIVaTion of the pAraDoxes. Unlike TEnnAnt (1982, 2016, 2017), theY cAll A DerivAtiOn D in A given SYstem S A normal paRaDOx (NparADox) iff (i) d IS cLoseD, (ii) D is involved in iD EsT Rules, (iii) D is in fUll norMaL FoRM, And (Iv) iF a formula φ Is The coNClusion OF D, THEN eiTHer φ → ⊥ or ¬φ can be dErived in S. SiMIlaR to SaiNsBurY'S notioN of a pArADox, One may regarD a paRadox as a dErivatIOn of the UNacceptAble coNclUsiOn frOM tHe AccEpTAblE PrEmiSEs bY the accePtAbLe reaSoniNG. pETrolO anD PisTone (2018) rEgard both a conTraDictIOn aNd an aBsurdIty aS uNaccePtable ConclUsIons and suggest a ClosEd normal dEriVaTioN oF a conTRadictIon FroM RusselL's paradOX. We BoRROW fRom Prawitz (1965, Appendix b) tHE NaTural dedUction SYsTeM sN for the NaIve Se thEORy whiCh coNTaIns the ruLes for ∧, →, ¬, ANd ∈. thEn, we havE a Closed NoRmaL deRivatIOn of A ∈ a∧¬a ∈ a. PrOpositioN 4.4.1. Let uS Define a parametER a as {x|¬x ∈ x}. Then, thERe IS A cLOsed NorMal derivatiOn of ¬A ∈ A∧a ∈ a iN SN. PROoF. TwO ClaimS veriFy THe REsult. Claim 1. there is a cLoSed derIvatiOn Σ15 of ¬a ∈ a∧a ∈ a. FirsT, we have a clOSED derivatIon Σ13 OF ¬a ∈ A. [A ∈ a]1 ******************** de f a ∈ {x|¬x ∈ x} ∈ E¬a ∈ a [a ∈ a]1 ¬e⊥ ¬I,1¬a ∈ a WIth the deriVAtion Σ13, we Have a Closed deRivation Σ14 OF A ∈ a. Σ13 ¬a ∈ a ∈ I a ∈ {x|¬X ∈ x} ******************** dE f a ∈ A 157 ThEn, wE HAvE a closed derivATIon Σ15 Of ¬A ∈ a∧a ∈ a. Σ13 ¬a ∈ a σ
 the proof-theoretic solut ion to the  para dox es.  P etro lo a nd Pistone (20 1 8) h ave considered the ver y cas e. 156U nd er th e disti n ct i o n b et we enac on tradi cti on andan absurdi ty,  P etrolo and P i st one (2018)  co nsider the p oss ibilit yofa  norm alderiv ationo f theparadoxes .U nlikeT ennant( 1 98 2, 2 016, 2017), theyc al l  a derivationD in a  g i ve n sys tem  S a norma lparad o x (Npar a do x ) iff (i) D is clos ed, (ii) Di s i nvolve dini d estrules ,( iii ) D is in f ullnormal fo rm, an d  (iv) i f  a form ula φisthe  con c lu si onof D,t he n e i the r φ → ⊥or  ¬ φ can  bed e r i vedinS. S imila r to Sainsbur y's  not i onof aparad ox,on e may  regar d a p ar adox as a deriv atio n of theuna cc ept ab le co n clusio n f rom  the ac ceptabl e  pr em i s e sby the acceptablere a s on ing. Pet rolo a n dPi s tone (20 18 ) r egar d botha co n tr adiction  and a n  a bs urdityas  unacc ep tab leconcl u sion s andsuggesta clo s ed normal deri v ation of a co n tr a d ic t ionfro m Russell's  par a dox.  Web or row fromPrawi tz (1 9 65, Appendix B) the  n atural  dedu ction systemSN for the n a ive se t heor y  w h ich contains t he ru les for ∧, →, ¬, an d ∈.Then, we  have a c l o sed norm alder iva tio n of  a ∈ a∧¬a ∈ a . Prop os ition 4 .4. 1. Letusdef ine  apa rameter a  as {x|¬ x∈x} .The n, th e re is acl ose dnor mal d e rivati on of  ¬a∈a∧ a  ∈a in SN .  P r o of.Tw oclai msve rifyther esu lt. Cla im 1. the rei s acl os ed deri vation Σ15 of  ¬ a ∈ a∧a ∈a.  Fi rst, w e have a c losed derivation Σ13 of ¬a ∈ a.  [a  ∈ a] 1 ** ********* *** ******  de f a ∈{x|¬x∈ x}∈E¬a ∈  a [a ∈  a ]1¬E ⊥ ¬I,1¬a ∈ a  Wi th th ederi vationΣ13, we have a clo s edderivation Σ1 4 o f a∈ a.  Σ1 3  ¬ a  ∈a∈  Ia ∈ {x|¬x ∈ x} ** ********** ** * ** *** de f a ∈ a  1 57 Then , we ha ve ac losed d erivation  Σ15 of ¬ a∈ a∧ a ∈ a . Σ13 ¬a ∈  a Σ
 the_proof-theoretic solution_to the paradoxes. Petrolo_and Pistone_(2018)_have considered_the_very case. 156_Under the distinction_between a contradiction and_an absurdity, Petrolo_and_Pistone (2018) consider the possibility of a normal derivation of the paradoxes. Unlike Tennant_(1982,_2016, 2017),_they_call_a derivation D in a_given system S a normal_paradox (Nparadox)_iff (i) D is closed, (ii) D is_involved_in id est_rules, (iii) D is in full normal form, and_(iv) if a formula φ is_the conclusion of_D,_then_either φ → ⊥_or ¬φ can be derived in_S. Similar to Sainsbury's notion of_a paradox, one may regard a paradox_as a derivation of the unacceptable_conclusion from the acceptable premises_by the_acceptable reasoning. Petrolo and Pistone_(2018) regard both_a contradiction_and an absurdity_as unacceptable conclusions and suggest a_closed normal derivation_of a contradiction from Russell's paradox._We_borrow from Prawitz_(1965,_Appendix_B) the_natural deduction system_SN_for the_naive_se theory which contains the rules_for_∧, →, ¬, and ∈. Then, we_have a closed normal_derivation_of a ∈ a∧¬a_∈ a. Proposition 4.4.1. Let_us define a parameter a as_{x|¬x ∈_x}. Then,_there is a closed normal derivation of ¬a ∈ a∧a ∈_a in SN. Proof. Two claims_verify the result. Claim_1. there_is_a closed derivation_Σ15_of ¬a_∈ a∧a ∈ a. First, we have_a closed_derivation Σ13 of ¬a ∈ a._[a ∈ a]1 ********************_de_f a ∈ {x|¬x ∈ x}_∈ E¬a ∈ a [a ∈_a]1 ¬E⊥ ¬I,1¬a ∈ a_With_the_derivation Σ13, we have a_closed derivation Σ14 of a ∈_a. Σ13 ¬a_∈ a ∈ I a ∈ {x|¬x_∈_x} ******************** de f a ∈_a_157 Then, we have a closed_derivation_Σ15_of ¬a ∈ a∧a ∈_a. Σ13 ¬a ∈ a Σ