-paradoxes and be a proof-theoretic solution to the paradoxes. Even though Tennant did not explicitly propose the requirement as the proof-theoretic solution, from TCP, we may interpret the following principle as a plausible proof-theoretic solution to the paradoxes. The Requirement of a (Full) Normal Derivation(RND): For any derivation D in natural deduction, D is acceptable only if D is (in principle) convertible into a normal derivation. Furthermore, while he compares his natural deduction system for naive set-theory and Fitch's, Prawitz (1965, p. 95) introduces a similar requirement with RND by saying, 'the set-theoretical paradoxes ruled out by the requirement that the [derivations] shall be normal.' Prawitz (1965, p. 96) also claims that his requirement is less ad hoc than Fitch's simple/special restrictions introduced by Fitch (1952, Sec. 18 and 20). Even though both Prawitz and Tennant did not explicitly claim that RND (or a similar requirement) could be a solution to the paradoxes, it is possible from their views that they would have in mind that RND could be the solution. Hence, our question is whether RND can really be a solution to the paradoxes. In order for RND to be the proof-theoretic solution, three things must be answered: (i) which paradox is a genuine paradox and which formalization is legitimate for the genuine paradox, (ii) why the only normalizable derivations are acceptable, and (iii) why the only propositional constant ⊥ for absurdity is an unacceptable conclusion. This chapter aims to discuss that there are some obstacles to claim that RND is the proof-theoretic solution to the paradoxes. For the first question (i), since RND is proposed on the perspectives on TCP, the following conjecture for genuine paradoxes should be true. Tennant's Conjecture for Genuine Paradoxes: For any derivation D in natural deduction, D formalizes a genuine paradox iff D is a T-paradox with respect to TCP. As we have noted, TCP itself assumes the conjecture for genuine paradoxes. However, since the notion of a 'genuine paradox' is informal, it is unclear what kinds of paradoxes are 134 genuine paradoxes. Tennant has claimed that Russell's paradox is not a genuine paradox, whereas the Liar paradox is a genuine one. In
-paradoxes and be a proof - theoretic solution to the paradoxes. Even though Tennant did not explicitly propose the requirement as the proof - theoretic solution, from TCP, we may translate the take after principle as a plausible proof - theoretic solution to the paradoxes. The Requirement of a (broad) Normal Derivation(RND ): For any derivation D in lifelike deduction, D is acceptable only if D is (in rationale) convertible into a normal deriving. Furthermore, while he compares his lifelike deduction system for naive set - hypothesis and Fitch's, Prawitz (1965, p. 95) introduces a similar necessity with RND by saying,' the set - theoretical paradox ruled out by the requirement that the [ derivations ] shall be normal.' Prawitz (1965, p. 96) besides claims that his requirement is less ad hoc than Fitch's simple / special restrictions introduce by Fitch (1952, Sec. 18 and 20). Even though both Prawitz and Tennant did not explicitly claim that RND (or a similar requirement) could be a solution to the paradoxes, it is possible from their views that they would have in judgment that RND could be the solution. Hence, our question is whether RND can really be a solution to the paradoxes. In order for RND to be the proof - theoretic solution, three things must be answered: (i) which paradox is a genuine paradox and which formalization is legitimate for the genuine paradox, (ii) why the only normalizable derivations are acceptable, and (iii) why the only propositional constant ⊥ for absurdity is an unacceptable decision. This chapter aims to discuss that there be some obstacle to claim that RND is the proof - theoretic solution to the paradoxes. For the first doubt (i), since RND is proposed on the perspectives on TCP, the following speculation for genuine paradoxes should be true. Tennant's Conjecture for Genuine Paradoxes: For any derivation D in lifelike deduction, D formalizes a genuine paradox iff D is a T - paradox with respect to TCP. As we have noted, TCP itself assumes the conjecture for actual paradox. However, since the impression of a' genuine paradox' is informal, it is unclear what kinds of paradoxes are 134 genuine paradoxes. Tennant has claimed that Russell's paradox is not a actual paradox, whereas the Liar paradox is a genuine one. In
-parwdoxes and be a proof-theuretic solution to the paradkxes. Evev though Tennant did not exppixitly propose the requiremevt as the proof-thworeuic solution, from TCP, we may infcrprec vhe following ptinciple as d plausible prmow-tkeoretic solution to the paradoxes. Tre Requorfment of a (Fulj) Nogmwl Dsgiyation(RND): For any derivation D ih naturel deduction, D os acceptable only if D is (in orinciple) convertihle into a bormwo derivation. Furthermogz, while he dompares his natural deduction rystek for naivg aeh-jheory and Fmtch's, Irawitz (1965, p. 95) iknroducev a simolar requiremekt wivh RBD by saying, 'the set-tieoretical paradoxes ruled oud yy the requirement thqt the [gerieatiubs] rhaml bs normwl.' 'rawitz (1965, p. 96) also claimw that his requiremtnt px less ad hod than Fytch's simple/special restrictions introdlced by Fitch (1952, Sec. 18 and 20). Evwn though both Prawite and Tennwnt did not explicitly claim that RND (or a similas reqnifemtnb) ziupd be a solution to the paradoxes, it is possifme fgom their views tmat they would havr ln iind that RND could be the solution. Hencf, our qoestiob is whetrer TND can really be a solutiob to the parcdozes. In order for RUD to be the propf-thepretic solution, three tkings just be ansaered: (i) wgkch paradox is a gekuite paradox and which formajization ms leyitimate for the gqnuine parwdox, (lh) why the only norlalizcble gerivationd are acceptable, and (iii) why thx only proposothondl constcnt ⊥ fpr absurdity ys an unacceptcble conelusiov. This chaiter aims to discuss that there asg some obstacnes to cjaim thar RND ir the proof-theotetic solltnon to thw paradoxes. For thc firrf question (i), siuee RND is proposec ov tre pxrspesdives on TCP, the foukowine conjecturc fur grnuine paradoxes shogld ge true. Tennant's Cpnmecture fir Genuige Paradoxes: Gor any derivation D in navural cedoction, D formalizes a genuine pzradox ifv D is a T-paradjx wlth tespect to CCP. As we have noted, TCP itself assumes vhe conjecture for genoine paradoxes. Howevet, slnce the notmon of a 'genuina paradox' is informao, it is unclear wmat kinds of paradoxes are 134 cenuije paradoxes. Tennant has claimed that Russell's paradox is not a genuine pqradox, whereas the Liat parddjr is a denumnx one. In
-paradoxes and be a proof-theoretic solution to Even Tennant did explicitly propose the from we may interpret following principle as plausible proof-theoretic solution to the paradoxes. Requirement of a (Full) Normal Derivation(RND): For any derivation D in natural deduction, is acceptable only if D is (in principle) convertible into a normal derivation. while compares natural system for naive set-theory and Fitch's, Prawitz (1965, p. 95) introduces a similar requirement with RND saying, 'the set-theoretical paradoxes ruled out by the that the [derivations] shall normal.' Prawitz (1965, p. 96) claims his requirement less hoc Fitch's simple/special restrictions by Fitch (1952, Sec. 18 and 20). Even though both Prawitz and Tennant did not explicitly claim RND (or requirement) could a to paradoxes, it is their views that they would have RND could be the solution. Hence, our question whether RND really be a solution to the In order for RND to be the proof-theoretic three things must be answered: (i) which paradox is a genuine paradox and which formalization for the genuine paradox, why the only derivations acceptable, (iii) the only constant ⊥ for absurdity is an unacceptable conclusion. This chapter aims discuss that there are some obstacles to claim that RND proof-theoretic to the paradoxes. the first question (i), RND proposed on the perspectives the conjecture should true. Conjecture for Genuine Paradoxes: any derivation D in natural D formalizes a genuine T-paradox with respect to TCP. As we have TCP itself assumes the conjecture for genuine However, since the notion of a 'genuine paradox' is informal, it is what kinds are 134 genuine paradoxes. Tennant has claimed that paradox is not a paradox, whereas the Liar paradox is a genuine one.
-paradoxes and be a proof-theorEtic solutiOn to tHe pAraDoXes. EVen tHough Tennant diD Not eXplicitly propose the reqUiremEnT As thE PrOof-thEoretic SOlUTIon, FrOm tCP, We MAy InterPreT the folLowing prinCipLe As a plausible PRoOf-theoretiC soLution to the pAraDoxes. THe reqUIremeNt oF a (FulL) NormaL derivaTion(RND): FoR aNY derivATion D in NATuRal dEduction, D is acceptABlE Only if D is (in priNciple) CoNVeRTIblE inTo a normal dErIvatiON. FurtheRMoRE, WHilE He compares his Natural deduCTioN systeM fOr nAIve set-TheorY aND FiTch's, Prawitz (1965, P. 95) intRoduces a sImilar REquiremENt with RnD by saYinG, 'thE set-THeOrEtiCaL ParADoXes RUleD out by thE rEqUiremEnt tHAT THe [deRivAtioNs] shaLl be normal.' PraWitZ (1965, p. 96) alSO clAims tHat hiS reqUiRemenT is lesS ad hoC tHan Fitch's simple/SpecIal restriCtiOnS inTrOduceD By FitcH (1952, SeC. 18 anD 20). Even thOugh botH praWiTZ ANd tennant did not expliCiTLY cLaim that rND (or a SImIlAR requireMeNt) cOuld BE A soluTion TO tHe paradoXes, it iS PoSsIble froM tHeir viEwS thAt tHey woULd haVe in miNd that RNd coulD Be the solution. HENce, our questioN Is WHEtHEr RNd caN really be a sOlutIOn to The pARaDoxES. In orDer foR Rnd tO Be the proof-theoretic SoLution, Three Things must be aNswered: (i) whICH Paradox iS a geNUiNE paradox and whiCh forMalization IS legitimAte foR the genuIne paradoX, (II) why the oNly NorMalIzaBLE dErivations are ACCeptAbLe, and (iiI) whY the onlY prOpoSitIonAl Constant ⊥ fOr absurdItY iS aN uNacCeptaBLe concluSiOn. THiS chApter AIms to dIscusS thaT tHeRE arE some obSTaCLEs to ClAiM thaT RNd iS the pRoof-THeoRetic soLution to tHe pARadoXeS. FOr the fiRst question (i), sInCe RND is proPoSed On the pERSpectiveS on TCP, the following conjeCTure for GenUine pAradOxes shoulD be True. TeNnaNT's ConjEcture For GeNuIne pARadoxES: foR anY dErivation D IN NatUral dEdUctiOn, D formAlizes a genuine paraDOx iFf D is a T-paradoX wiTh reSPEcT to tcP. aS we HaVE noTED, TCP itself assumEs the conjeCtURe For genuine PAraDoXes. HoweVer, sincE the nOTion of a 'Genuine paRadox' is inFoRmal, IT Is uNclear what Kinds of pAradoxes aRE 134 genuINe ParadOxeS. TennaNt Has ClaimEd that rUssEll's pAradox Is Not a geNuine PaRadox, wheReas the Liar paradox is a geNuine oNe. In
-paradoxes and be a proof- theoreticsolut ion  to  t he p arad oxes. Even tho u gh T ennant did not explici tly p ro p oset he  requ irement as t hepr oo f-t he o re tic s olu tion, f rom TCP, w e m ay  interpret t h efollowingpri nciple as apla usible  p roo f -theo ret ic so lution to the  paradoxe s. The Re q uiremen t of  a ( Full) Normal Deri v at i on(RND): For a ny der iv a ti o n  Dinnatural de du ction ,  D is a c ce p t a ble only if D is(in princip l e)conver ti ble into a  norm al der ivation. Fu rthe rmore, wh ile he compare s  his na turalded uct ions ys te m f or nai v eset - the ory andFi tc h's,Praw i t z (196 5,p. 9 5) in troduces a si mil ar r e qui remen t wit h RN Dby sa ying,'these t-theoretical p arad oxes rule d o ut  by  t he re q uireme nttha t the [ derivat i ons ]s h a ll  be normal.' Prawi tz ( 19 65, p. 9 6) als o  c la i ms thathi s r equi r e mentis l e ss  ad hocthan F i tc h' s simpl e/ specia lres tri ction s  int roduce d by Fit ch (1 9 52, Sec. 18 an d  20). Even th o ug h bo t h Pr awi tz and Tenn antd id n ot e x pl ici t ly cl aim t ha t  R N D (or a similar req ui rement ) cou ld be a solut ion to the p a radoxes,  iti sp ossible from t heirviews that they wou ld ha ve in mi nd that R N D  could b e t hesol uti o n .Hence, our qu e s tion  i s wheth erRND can  re all y b e a  s olution t o the pa ra do xe s.  In  orde r  for RND  t o b ethe  proo f -theor eticsolu ti on ,  th ree thi n gs m ustbe  a nswe red :(i) w hich par adox is  a genuin e p a rado xan d which  formalizatio nis legitim at e f or the g enuine p aradox, (ii) why the on l y norma liz ablederi vations a reaccept abl e , and(iii)why t he  on l y  prop o s it ion al  constant⊥ for  absu rd ityis an u nacceptable conclu s ion . This chapte r a imst o  d isc u ss tha tt her e are some obstac les to cla im th at RND ist hepr oof-the oreticsolut i on to t he parado xes. Forth e fi r s t q uestion (i ), since  RND is p r opose d  o n the  pe rspect iv eson TC P, the fol lowin g conj ec ture f or ge nu ine para doxes should be true. T ennant 's Co nje cture for  Ge n uin e Paradox es:For any de riv ati on Dinn atura l de d uc tio n , D f orma l izes a ge n ui nep a ra dox iff D i s a  T- parad oxw ith re spec t to TCP. As we h a ve noted, TCPitse l f  as sum e s th econjecture for  ge nu i n e parado xe s. However,  since t he notio n of a  'genu ine par a d ox '  is in form al,  it is un cle ar what ki nd so f para doxe sare 13 4 genu i ne p a r adoxes. Tennanthas c l a imedt hat  Russ el l's par a doxis not a g enuine para dox, w here as th e Liarpa radoxisagenuine on e . In
-paradoxes and_be a_proof-theoretic solution to the_paradoxes. Even_though_Tennant did_not_explicitly propose the_requirement as the_proof-theoretic solution, from TCP,_we may interpret_the_following principle as a plausible proof-theoretic solution to the paradoxes. The Requirement of a_(Full)_Normal Derivation(RND):_For_any_derivation D in natural deduction,_D is acceptable only if_D is_(in principle) convertible into a normal derivation. Furthermore,_while_he compares his_natural deduction system for naive set-theory and Fitch's, Prawitz_(1965, p. 95) introduces a similar_requirement with RND_by_saying,_'the set-theoretical paradoxes ruled_out by the requirement that the_[derivations] shall be normal.' Prawitz (1965,_p. 96) also claims that his requirement_is less ad hoc than Fitch's_simple/special restrictions introduced by Fitch_(1952, Sec._18 and 20). Even though_both Prawitz and_Tennant did_not explicitly claim_that RND (or a similar requirement)_could be a_solution to the paradoxes, it is_possible_from their views_that_they_would have_in mind that_RND_could be_the_solution. Hence, our question is whether_RND_can really be a solution to the_paradoxes. In order for_RND_to be the proof-theoretic_solution, three things must be_answered: (i) which paradox is a_genuine paradox_and which_formalization is legitimate for the genuine paradox, (ii) why the only_normalizable derivations are acceptable, and (iii)_why the only propositional_constant ⊥_for_absurdity is an_unacceptable_conclusion. This_chapter aims to discuss that there are_some obstacles_to claim that RND is the_proof-theoretic solution to the_paradoxes._For the first question (i), since_RND is proposed on the perspectives_on TCP, the following conjecture_for_genuine_paradoxes should be true. Tennant's_Conjecture for Genuine Paradoxes: For any_derivation D in_natural deduction, D formalizes a genuine paradox_iff_D is a T-paradox with respect_to_TCP. As we have noted, TCP_itself_assumes_the conjecture for genuine paradoxes._However, since the notion of a_'genuine paradox' is informal, it is unclear what kinds_of paradoxes are_134 genuine paradoxes. Tennant has_claimed_that_Russell's paradox is not a genuine paradox, whereas the Liar_paradox is_a genuine one._In