 Section 4.1, we shall introduce his argument on why Russell's paradox is not a genuine paradox, and argue that by following his argument, if Russell's paradox is not a genuine paradox, neither is the Liar paradox. Tennant has no standard for genuine paradoxes. Our discussion comes into a question of which formalization is legitimate for the genuine paradox. RND only blocks non-normalizable derivations, such as T-paradoxes. If RND is regarded as a promising proof-theoretic solution to genuine paradoxes, it should be answered to the first question of which paradoxes are genuine paradox. The second question asks, even though Tennant's conjecture for genuine paradoxes is true and RND can restrict the use of T-paradoxes, why should we consider that RND is convincing? A plausible answer is that every non-normalizable derivation is (prooftheoretically) invalid. For the second question, we shall consider a relation between prooftheoretic validity and normalizability in Section 4.2. If non-normalizable derivations were proof-theoretically invalid, paradoxical derivations which generate a non-terminating reduction sequence would be invalid. It will be discussed that in a restricted system prooftheoretic validity implies normalizability. However, it is not clear that the relation would be extended to a general case. Therefore, it should be established that proof-theoretic validity generally implies (strong) normalizability, or another answer to the second question (ii) should be proposed. Apart from our three questions for RND as a proof-theoretic solution to the paradoxes, Section 4.3 deals with the question of whether RND is a reasoning-reduction solution which restricts the use of inference rules. Under the assumption that proof-theoretic validity implies (full) normalizability, we introduce Prawitz's definition of valid inferences via his notion of proof-theoretic validity and find invalid rules in paradoxical derivations. The restriction of an application of invalid rules would be the reasoning-rejection solution to the paradoxes. However, we will argue that to limit the application of inference rules through Prawitz's definition of valid inferences can be used independently of RND. Thus, it does not support the view that RND is a reasoning-rejection solution in the sense that it restricts 135 the application of a single inference rule. With regard to the last question (iii), Section 4.4 shall consider a case of a normal derivation
Section 4.1, we shall introduce his argument on why Russell's paradox is not a genuine paradox, and argue that by follow his controversy, if Russell's paradox is not a genuine paradox, neither is the Liar paradox. Tennant has no standard for actual paradoxes. Our discussion comes into a question of which formalization is lawful for the actual paradox. RND only blocks non - normalizable derivations, such as T - paradoxes. If RND is involve as a promising proof - theoretical solution to genuine paradoxes, it should be answer to the first question of which paradox are genuine paradox. The second question asks, even though Tennant's guess for genuine paradoxes is truthful and RND can restrict the function of T - paradoxes, why should we consider that RND is convincing? A plausible answer is that every non - normalizable derivation is (prooftheoretically) invalid. For the second doubt, we shall consider a relation between prooftheoretic validity and normalizability in Section 4.2. If non - normalizable derivations were proof - theoretically invalid, paradoxical derivations which generate a non - terminating reduction sequence would be invalid. It will be discussed that in a restricted system prooftheoretic validity implies normalizability. However, it is not clear that the relation would be extended to a general event. Therefore, it should be established that proof - theoretical cogency broadly implies (strong) normalizability, or another solution to the second question (ii) should be propose. Apart from our three questions for RND as a proof - theoretical solution to the paradoxes, Section 4.3 deals with the question of whether RND is a reasoning - reduction solution which restricts the consumption of inference rules. Under the assumption that proof - theoretic validity imply (entire) normalizability, we introduce Prawitz's definition of valid inferences via his notion of proof - theoretic validity and rule invalid rules in paradoxical derivations. The restriction of an lotion of invalid rules would be the reasoning - rejection solution to the paradox. However, we will argue that to limit the application of inference rules through Prawitz's definition of valid inferences can be used independently of RND. therefore, it does not support the view that RND is a reasoning - rejection solution in the sense that it restricts 135 the application of a individual inference rule. With respect to the last motion (iii), Section 4.4 shall consider a case of a normal derivation
 Seftion 4.1, we shall introduct his argument on why Ruvsell's paradox is not a genuine paradox, anv artue tyat by following his afgument, iv Russelo's peradox is not a jsnuine icrados, neichxr is the Liar karadox. Tenndnt has no statdxrb for genuine paradoxes. Our discussijn comex lnto a questiog of rhicg formalization is legitimate for fhe genline paradox. RND pnly blocks non-normalizablf degivations, such as H-paradoxes. Uf RGE is regardea as a promising proof-jheoretic solution to genuine pafadoxzs, it shoule ve wtswered to vhe figst question of which paradoces are genuinc paredox. The second question esks, even though Tengant's conbeeture for genuine parqdixes hs tsue xbd FND cen destrift vhe use of F-paradoxes, qhy should we consicew that RND is donvinsigg? A plausible answer is that every non-torjalizable derivation is (prooftheoretically) ijvalid. Fow the second question, we shall consider a relatiot betxedn kriofthdirftic validity and normalizability in Section 4.2. Jf npn-normalizable derivationx aetg proof-theoretkcally inbalid, paradoxical ferivatyons qhich gentrate a non-terminating reduction sequence wolld ve invalid. It will be discusszd thaj in a restricted system prooytheorstic validihy impliea normalizability. Hoeeeer, it is not clear that tre relatiin wpuld be extgnded tj a generap casc. Therefore, it shoupd be evtablished that proof-theoretic validity gxierally impligs (vtrmng) normclizabllity, or anothew answer to thg second zuestkon (ii) sholld be prmposed. Apare from our thsge questions hor RND af a prood-theoregkc solution to the parabjxes, Section 4.3 deals wibh thg suestion of whecket RND is a reaspnivg-rqdlctmon sjnution which resgrizys thd use of innerdnce rules. Under the assgmptjon that proof-theotebic validuty implyes (full) normslizability, we intgoducx Prawmtz's drfigition of valid inferences via his notiln jf proof-theowetig vajidity and find invalid rules in paradoxical deritations. The restriction of an application pn invalid rukes wjuld be tve reasoning-rejectiob solution to the paradoxes. However, we will drgue that to limit the application of inference rules through Prawitz's definirion oh dalid infersncex can bz uxed inqepeidxntly of RND. Thus, it does not support the view tiat RND is a reasoning-rejection solution iv the sense tfat it restricts 135 the apllicatiom of a single inference rule. Wiyh regard to the last qucstion (iiu), Secfion 4.4 shalu consider a case pf a nmrmal derivarion
Section 4.1, we shall introduce his argument Russell's is not genuine paradox, and argument, Russell's paradox is a genuine paradox, is the Liar paradox. Tennant has standard for genuine paradoxes. Our discussion comes into a question of which formalization legitimate for the genuine paradox. RND only blocks non-normalizable derivations, such as T-paradoxes. RND regarded a proof-theoretic solution to genuine paradoxes, it should be answered to the first question of which paradoxes genuine paradox. The second question asks, even though conjecture for genuine paradoxes true and RND can restrict use T-paradoxes, why we that is convincing? A answer is that every non-normalizable derivation is (prooftheoretically) invalid. For the second question, we shall consider a between prooftheoretic normalizability in 4.2. non-normalizable were proof-theoretically invalid, which generate a non-terminating reduction sequence It will be discussed that in a restricted prooftheoretic validity normalizability. However, it is not clear the relation would be extended to a general Therefore, it should be established that proof-theoretic validity generally implies (strong) normalizability, or another answer second question (ii) should proposed. Apart from three for as proof-theoretic solution the paradoxes, Section 4.3 deals with the question of whether RND a reasoning-reduction solution which restricts the use of inference rules. assumption proof-theoretic validity implies normalizability, we introduce Prawitz's of inferences via his notion validity find paradoxical The of an application of rules would be the reasoning-rejection to the paradoxes. However, limit the application of inference rules through Prawitz's of valid inferences can be used independently RND. Thus, it does not support the view that RND is a solution in that it restricts 135 the application of a inference rule. With regard the last question (iii), Section 4.4 shall consider a of normal derivation
 Section 4.1, we shall introduce hiS argument oN why RUssEll'S pAradOx is Not a genuine parADox, aNd argue that by following His arGuMEnt, iF ruSsell'S paradoX Is NOT a gEnUiNe pArADoX, neitHer Is the LiAr paradox. TEnnAnT has no standaRD fOr genuine pAraDoxes. Our discUssIon comEs IntO A quesTioN of whIch forMAlizatIon is legiTiMAte for THe genuiNE PaRadoX. RND only blocks non-NOrMAlizable derivaTions, sUcH As t-PAraDoxEs. If RND is rEgArded AS a promiSInG PROof-THeoretic solutIon to genuinE ParAdoxes, It ShoULd be anSwereD tO The First questiOn of Which paraDoxes aRE genuinE Paradox. the secOnd QueStioN AsKs, EveN tHOugH teNnaNT's cOnjecturE fOr GenuiNe paRADOXes iS trUe anD RND cAn restrict the Use Of T-pARadOxes, wHy shoUld wE cOnsidEr that rND is CoNvincing? A plausiBle aNswer is thAt eVeRy nOn-NormaLIzable DerIvaTion is (pRooftheORetIcALLY) iNvalid. For the second QuESTiOn, we shalL consiDEr A rELation beTwEen ProoFTHeoreTic vALiDity and nOrmaliZAbIlIty in SeCtIon 4.2. If nOn-NorMalIzablE DeriVationS were proOf-theORetically invalID, paradoxical dERiVATiONs whIch Generate a noN-terMInatIng rEDuCtiON sequEnce wOuLD bE Invalid. It will be discUsSed thaT in a rEstricted systEm prooftheORETic validIty iMPlIEs normalizabilIty. HoWever, it is nOT clear thAt the Relation Would be exTENded to a gEneRal CasE. ThEREfOre, it should be ESTablIsHed that ProOf-theorEtiC vaLidIty GeNerally imPlies (strOnG) nOrMaLizAbiliTY, or anothEr AnsWeR to The seCOnd queStion (Ii) shOuLd BE prOposed. APArT FRom oUr ThRee qUesTiOns foR RND AS a pRoof-theOretic solUtiON to tHe PaRadoxes, section 4.3 deals wItH the questiOn Of wHether rnd is a reasOning-reduction solution wHIch restRicTs the Use oF inferencE ruLes. UndEr tHE assumPtion tHat prOoF-thEORetic VALiDitY iMplies (full) NORmaLizabIlIty, wE introdUce Prawitz's definitIOn oF valid inferenCes Via hIS NoTioN Of PRooF-tHEorETIc validity and fiNd invalid rUlES iN paradoxicAL deRiVations. the restRictiON of an apPlication Of invalid RuLes wOULd bE the reasonIng-rejecTion solutIOn to tHE pAradoXes. howeveR, wE wiLl argUe that TO liMit thE appliCaTion of InferEnCe rules tHrough Prawitz's definitioN of valId infEreNces can be UseD IndEpendentlY of RnD. Thus, it doEs nOt sUpporT thE View tHat Rnd iS a rEAsoniNg-reJEction solUTiOn iN THe Sense that it RESTriCts 135 thE apPLicatiOn of A single inference rULe. With regard to The lAST quEstIOn (iiI), SEction 4.4 shall conSidEr A CAse of a noRmAl derivatioN
 Section 4.1, we shall int roduce his  argu men t o nwhyRuss ell's paradoxi s no t a genuine paradox, a nd ar gu e  tha t  b y fol lowingh is a rgu me nt , i fR us sell' s p aradoxis not a g enu in e paradox, n e it her is the  Li ar paradox.Ten nant h as  no stand ard  forgenuin e  parad oxes. Our  d i scussi o n comes i nt o aquestion of which fo r malization islegiti ma t ef o r t hegenuine pa ra dox.R ND only bl o c k s n o n-normalizabl e derivatio n s,such a sT-p a radoxe s. If  R N D i s regardedas a  promisin g proo f -theore t ic solu tion t o g enu inep ar ad oxe s, its ho uld beanswered  t othe f irst q u e stio n o f wh ich p aradoxes aregen uine par adox.  Theseco nd  ques tion a sks,ev en though Tenna nt's  conjectu refo r g en uinep aradox esistrue an d RND c a n r es t r i ct  the use of T-para do x e s,  why sho uld we co ns i der that  R NDis c o n vinci ng?A  p lausible  answe r  i sthat ev er y non- no rma liz abled eriv ationis (proo ftheo r etically) inva l id. For the s e co n d  q u esti on,  we shall c onsi d er a  rel a ti onb etwee n pro of t he o retic validity andno rmaliz abili ty in Section  4.2. If n o n - normaliz able de r ivations wereproof -theoretic a lly inva lid,paradoxi cal deriv a t ions whi chgen era tea no n-terminating r educ ti on sequ enc e would  be  in val id.  I t will be  discuss ed  t ha tina res t ricted s ys tem  p roo ftheo r etic v alidi ty i mp li e s n ormaliz a bi l i ty.Ho we ver,  it  i s not  cle a r t hat the  relation  wo u ld b eex tendedto a generalca se. Theref or e,it sho u l d be est ablished that proof-the o retic v ali ditygene rally imp lie s (str ong )  norma lizabi lity,  o r a n o thera n sw erto  the secon d que stion  ( ii)shouldbe proposed. Apart fro m our three q ues tion s fo r R N Da s a  p r oof - t heoretic soluti on to thepa r ad oxes, Sect i on4. 3 deals  with t he qu e stion o f whether  RND is a  r easo n i ng- reductionsolution  which re s trict s  t he us e o f infe re nce  rule s. Und e r t he as sumpti on  thatproof -t heoretic  validity implies (full ) norm aliza bil ity, we i ntr o duc e Prawitz 's d efinitionofval id in fer e ncesviah is  no t ion o f pr o of-theore t ic  va l i di ty and find i n val id ru les in par adox ical derivations. The restrictio n of a n a ppl i cati on  of invalid ru les  w o u ld be th ereasoning-r ejection  s o lutio n to t he par adoxes. H ow e ver, w e wi llargue tha t t ol imit th eap p licati on o finfere nce ru l es t h r ough Prawitz's d efini t i on of val id in fe rencesc an b e used ind ependentlyof RND . Th us, i t doesno t supp ort  t he view th a t RND isa rea soning- re ject ion  solut ioni n  thesens etha t it rest r i ct s  1 35 the  app licat io n of  a single inferenc e r u le. Wit hreg a r d to t h el a st questio n ( iii), S ection 4.4 shal l  c o nside r a ca se ofa norma l  de ri vation
 Section_4.1, we_shall introduce his argument_on why_Russell's_paradox is_not_a genuine paradox,_and argue that_by following his argument,_if Russell's paradox_is_not a genuine paradox, neither is the Liar paradox. Tennant has no standard for_genuine_paradoxes. Our_discussion_comes_into a question of which_formalization is legitimate for the_genuine paradox._RND only blocks non-normalizable derivations, such as T-paradoxes._If_RND is regarded_as a promising proof-theoretic solution to genuine paradoxes, it_should be answered to the first_question of which_paradoxes_are_genuine paradox. The second_question asks, even though Tennant's conjecture_for genuine paradoxes is true and_RND can restrict the use of T-paradoxes,_why should we consider that RND_is convincing? A plausible answer_is that_every non-normalizable derivation is (prooftheoretically)_invalid. For the_second question,_we shall consider_a relation between prooftheoretic validity and_normalizability in Section_4.2. If non-normalizable derivations were proof-theoretically_invalid,_paradoxical derivations which_generate_a_non-terminating reduction_sequence would be_invalid._It will_be_discussed that in a restricted system_prooftheoretic_validity implies normalizability. However, it is not_clear that the relation_would_be extended to a_general case. Therefore, it should_be established that proof-theoretic validity generally_implies (strong)_normalizability, or_another answer to the second question (ii) should be proposed. Apart_from our three questions for RND_as a proof-theoretic solution_to the_paradoxes,_Section 4.3 deals_with_the question_of whether RND is a reasoning-reduction solution_which restricts_the use of inference rules. Under_the assumption that proof-theoretic_validity_implies (full) normalizability, we introduce Prawitz's_definition of valid inferences via his_notion of proof-theoretic validity and_find_invalid_rules in paradoxical derivations. The_restriction of an application of invalid_rules would be_the reasoning-rejection solution to the paradoxes. However,_we_will argue that to limit the_application_of inference rules through Prawitz's definition_of_valid_inferences can be used independently_of RND. Thus, it does not_support the view that RND is a reasoning-rejection solution_in the sense_that it restricts 135 the_application_of_a single inference rule. With regard to the last question_(iii), Section_4.4 shall consider_a case of a normal derivation