 paradoxes, it should be answered to the first question of which paradoxes are genuine paradoxes. In Section 4.2, we will explore the second question of why it is desirable only to use normal derivations. One possibility is that proof-theoretic validity implies normalizability. In other words, if a paradoxical derivation is not normalizable, it can be ruled out by RND because it is not a proof-theoretically valid derivation, RND can be a solution to the paradoxes. Section 4.2 will establish that in a particular system, proof-theoretic validity implies normalizability. However, in order for RND to be a proof-theoretic solution, the result should be extended to a general case. Section 4.3 discusses that the requirement of a normal derivation is different from the reasoning-rejection solution commonly considered as a restriction of a particular inference rule. If a reasoning-rejection solution is regarded as a solution to constrains a certain inference rule, RND will not be the reasoning-rejection solution because it constrains every derivation in an intended system. Section 4.4 introduces a normal derivation of ¬φ ∧φ presented by Petrolo and Pistone (2018) and argues that RND cannot be a proof-theoretic solution if we accept a formula of the form ¬φ ∧φ as well as ⊥ as an unacceptable conclusion. All in all, only when proof-theoretic validity generally implies normalizability and any formula having the form ¬φ ∧φ is not regarded as an unacceptable conclusion, RND can be a proof-theoretic solution to the paradoxes.

Contents Abstract Contents i 1 Introduction: A Proof-Theoretic Criterion of and Solution to the Paradoxes 1 1.1 Traditional Responses to the Paradoxes and Dialetheism.......... 3 1.1.1 Traditional Approaches to Russell's and the Liar Paradox...... 4 1.1.2 Problems of Traditional Responses and Dialetheism........ 10 1.2 Preliminaries................................. 15 1.3 Tennant's Criterion for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND)...
paradoxes, it should be answered to the first question of which paradox are actual paradox. In Section 4.2, we will explore the second question of why it is desirable entirely to use normal derivations. One hypothesis is that proof - theoretic validity implies normalizability. In early son, if a paradoxical derivation is not normalizable, it can be rule out by RND because it is not a proof - theoretically valid derivation, RND can be a solution to the paradox. Section 4.2 will establish that in a particular system, proof - theoretic validity implies normalizability. However, in ordering for RND to be a proof - theoretic solution, the result should be extended to a general character. Section 4.3 discusses that the requirement of a normal derivation is unlike from the reasoning - rejection solution commonly consider as a restriction of a particular inference rule. If a reasoning - rejection solution is involve as a solution to constrains a certain inference rule, RND will not be the reasoning - rejection solution because it constrains every derivation in an intended system. Section 4.4 introduces a normal derivation of ¬φ ∧φ presented by Petrolo and Pistone (2018) and argues that RND cannot be a proof - theoretic solution if we accept a formula of the form ¬φ ∧φ as well as ⊥ as an impossible decision. All in all, entirely when proof - theoretical robustness generally implies normalizability and any rule having the form ¬φ ∧φ is not regarded as an unacceptable decision, RND can be a proof - theoretic solution to the paradoxes. 

 Contents Abstract Contents i 1 Introduction: A Proof - Theoretic Criterion of and Solution to the Paradoxes 1 1.1 Traditional Responses to the Paradoxes and Dialetheism.......... 3 1.1.1 Traditional Approaches to Russell's and the Liar Paradox...... 4 1.1.2 Problems of Traditional Responses and Dialetheism........ 10 1.2 Preliminaries................................. 15 1.3 Tennant's Criterion for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND)...
 pagadoxes, it should be ansdered to the fitsr quesvion of which pxradoxes are genuine paradoxxs. Ib Secupon 4.2, we will explore ghe seconf questiin oh why it is desiczble only to hde nmcmal derivationx. One posshbility is thad orlof-theoretic validity implies normajizabilotj. In other worqs, ig a pzgaboxical derivation is not normamizable, it can be ruked out by RND because it ls nlt a proof-theoretifally valid derycation, RND cxn be a solution to thg paradoxes. Section 4.2 will establksh tkat in a patcucupdr system, pcoof-thvoretic validlny implhes norkalizability. Hpweter, un order for RND to bx a proof-theoretic sjlution, tve result should be eztendgd to a gdberxl daxe. Sectiln 4.3 discusses that the rwquirement of a norkaj derivation ia diffqrqnt from the reasoning-rejection solutiot ckmmonly considered as a restriction of a parjicular inserence rule. If a reasoning-rejection solution is segarvea aw w wopution to constrains a certain inference rule, DNC eill not be thc reasoning-rejectipn spjution becausg it couatdains every derivahion in an ibtended srstek. Section 4.4 introduces a nornal derivatijb of ¬φ ∧φ presented bv Petrolo anb Pistpne (2018) snd argues that RND canuot be a proof-thelretic somjtion if we acceot s xormula of the form ¬φ ∧φ as wqll as ⊥ aw an unacceotabke conslusion. Alp in all, only when proof-hheorgtic vdlidity gejerally implies normalizability and any formuka haning the yorm ¬φ ∧φ is not regarqed as an unaceeptable concuusion, RND can be a proof-thejretic solutimj to the paredoxes.

Coneentw Abwtract Zuntents i 1 Inttoduction: A Proof-Thworetic Criterion pf xhd Solution to cke Paradoxes 1 1.1 Trsdigiogap Cespogves to the Pdraduxer and Aialethtnsm.......... 3 1.1.1 Trsditional Approaches to Dussell's and the Loav Paradox...... 4 1.1.2 Problqms of Traditoonal Responses anf Dianetieism........ 10 1.2 Pteliminaries................................. 15 1.3 Tennant's Criterikn for Pagadjxicality (TC[) ane the Requirzment of a Normal Derivation (RND)...
paradoxes, it should be answered to the of paradoxes are paradoxes. In Section second of why it desirable only to normal derivations. One possibility is that validity implies normalizability. In other words, if a paradoxical derivation is not normalizable, can be ruled out by RND because it is not a proof-theoretically valid RND be solution the paradoxes. Section 4.2 will establish that in a particular system, proof-theoretic validity implies normalizability. However, order for RND to be a proof-theoretic solution, result should be extended a general case. Section 4.3 that requirement of normal is from the reasoning-rejection commonly considered as a restriction of a particular inference rule. If a reasoning-rejection solution is regarded as solution to certain inference RND not the reasoning-rejection solution constrains every derivation in an intended introduces a normal derivation of ¬φ ∧φ presented Petrolo and (2018) and argues that RND cannot a proof-theoretic solution if we accept a formula the form ¬φ ∧φ as well as ⊥ as an unacceptable conclusion. All in all, proof-theoretic validity generally implies and any formula the ¬φ is regarded as unacceptable conclusion, RND can be a proof-theoretic solution to the paradoxes. Abstract Contents i 1 Introduction: A Proof-Theoretic Criterion of and the 1 1.1 Traditional to the Paradoxes and 3 Traditional Approaches to Russell's Liar 4 Traditional and 10 1.2 Preliminaries................................. 15 Tennant's Criterion for Paradoxicality (TCP) the Requirement of a
 paradoxes, it should be answerEd to the firSt queStiOn oF wHich ParaDoxes are genuinE ParaDoxes. In Section 4.2, we will exPlore ThE SecoND qUestiOn of why IT iS DEsiRaBlE onLy TO uSe norMal DerivatIons. One posSibIlIty is that proOF-tHeoretic vaLidIty implies noRmaLizabiLiTy. IN Other WorDs, if a ParadoXIcal deRivation iS nOT normaLIzable, iT CAn Be ruLed out by RND becausE It IS not a proof-theoReticaLlY VaLID deRivAtion, RND caN bE a solUTion to tHE pARADoxES. Section 4.2 will eStablish thaT In a ParticUlAr sYStem, prOof-thEoREtiC validity imPlieS normalizAbilitY. however, IN order fOr RND tO be A prOof-tHEoReTic SoLUtiON, tHe rESulT should bE eXtEnded To a gENERAl caSe. SEctiOn 4.3 disCusses that the ReqUireMEnt Of a noRmal dErivAtIon is DifferEnt frOm The reasoning-rejEctiOn solutioN coMmOnlY cOnsidERed as a ResTriCtion of A particULar InFEREnCe rule. If a reasoning-ReJECtIon solutIon is rEGaRdED as a soluTiOn tO conSTRains A cerTAiN inferenCe rule, rnD WiLl not be ThE reasoNiNg-rEjeCtion SOlutIon becAuse it coNstraINs every derivatIOn in an intendeD SySTEm. sEctiOn 4.4 iNtroduces a nOrmaL DeriVatiON oF ¬φ ∧φ PResenTed by peTRoLO and Pistone (2018) and argueS tHat RND CannoT be a proof-theoRetic solutION If we accePt a fORmULa of the form ¬φ ∧φ aS well As ⊥ as an unacCEptable cOncluSion. All iN all, only wHEN proof-thEorEtiC vaLidITY gEnerally impliES NormAlIzabiliTy aNd any foRmuLa hAviNg tHe Form ¬φ ∧φ is nOt regardEd As An UnAccEptabLE conclusIoN, RNd cAn bE a proOF-theorEtic sOlutIoN tO The ParadoxES.

CONTentS ABsTracT CoNtEnts i 1 intrODucTion: A PrOof-TheoreTic cRiteRiOn Of and SoLution to the PaRaDoxes 1 1.1 TradiTiOnaL RespoNSEs to the PAradoxes and Dialetheism.......... 3 1.1.1 TRAditionAl APproaChes To Russell'S anD the LiAr PARadox...... 4 1.1.2 PRoblemS of TrAdItiONAl ResPONsEs aNd dialetheisM........ 10 1.2 pRelIminaRiEs................................. 15 1.3 TeNnant's CRiterion for ParadoxICalIty (TCP) and the REquIremENT oF a NORmAL DeRiVAtiON (rND)...
 paradoxes, it should be a nswered to  thefir stqu esti on o f which parado x es a re genuine paradoxes.In Se ct i on 4 . 2,  we w ill exp l or e the  s ec ond  q u es tionofwhy itis desirab leon ly to use no r ma l derivati ons . One possib ili ty isth atp roof- the oreti c vali d ity im plies nor ma l izabil i ty. Ino t he r wo rds, if a paradox i ca l  derivation is  not n or m al i z abl e,it can beru led o u t by RN D  b e c a use it is not a p roof-theore t ica lly va li d d e rivati on, R ND can  be a solut ionto the pa radoxe s . Secti o n 4.2 w ill es tab lis h th a tin  apa r tic u la r s y ste m, proof -t he oreti c va l i d i ty i mpl iesnorma lizability. H owe ver, inorder  forRNDto  be a  proof -theo re tic solution, t he r esult sho uld  b e e xt ended to a g ene ral  case.Section 4.3  d i s c us ses that the requi re m e nt  of a no rmal d e ri va t ion is d if fer entf r om th e re a so ning-rej ection so lu tion co mm only c on sid ere d asa  res tricti on of aparti c ular inference rule. If a re a so n i ng - reje cti on solution  isr egar deda sa s o lutio n toco n st r ains a certain infe re nce ru le, R ND will not b e the reas o n i ng-rejec tion so l ution becauseit co nstrains e v ery deri vatio n in anintendeds y stem. Se cti on4.4  in t r od uces a normal d eriv at ion of¬φ∧φ pres ent edbyPet ro lo and Pi stone (2 01 8)  a nd  ar guest hat RNDca nno tbea pro o f-theo retic  sol ut io n  if  we acc e pt a  for mu la  ofthe  f orm ¬ φ ∧φ aswell as  ⊥ as anuna c cept ab le  conclu sion. All inal l, only wh en  pr oof-th e o retic va lidity generally implie s  normal iza bilit y an d any for mul a havi ngt he for m ¬φ ∧ φ isno t r e g arded a sanun acceptable c onc lusio n,  RND  can be  a proof-theoretic sol ution to thepar adox e s .
Co n te n tsAb s tra c t  Contents i 1 I ntroductio n: AProof-Theo r eti cCriteri on of a nd So l ution t o the Par adoxes 11. 1 Tr a d iti onal Respo nses tothe Parad o xes a n dDiale the ism... .. ... .. 31.1.1T rad ition al App ro achesto Ru ss ell's an d the Liar Paradox..... . 4 1. 1.2 P rob lems of T rad i tio nal Respo nses  and Diale the ism ..... ... 10 1. 2 Pr e li min a ries. .... . ......... . .. ... . . .. ........ 15 1 . 3 T ennan t's Criter ionfor Paradoxicalit y  (TCP) and the  Req u i rem ent of a  N ormal Derivati on(R N D )...
 paradoxes,_it should_be answered to the_first question_of_which paradoxes_are_genuine paradoxes. In_Section 4.2, we_will explore the second_question of why_it_is desirable only to use normal derivations. One possibility is that proof-theoretic validity implies_normalizability._In other_words,_if_a paradoxical derivation is not_normalizable, it can be ruled_out by_RND because it is not a proof-theoretically valid_derivation,_RND can be_a solution to the paradoxes. Section 4.2 will establish_that in a particular system, proof-theoretic_validity implies normalizability._However,_in_order for RND to_be a proof-theoretic solution, the result_should be extended to a general_case. Section 4.3 discusses that the requirement_of a normal derivation is different_from the reasoning-rejection solution commonly_considered as_a restriction of a particular_inference rule. If_a reasoning-rejection_solution is regarded_as a solution to constrains a_certain inference rule,_RND will not be the reasoning-rejection_solution_because it constrains_every_derivation_in an_intended system. Section_4.4_introduces a_normal_derivation of ¬φ ∧φ presented by_Petrolo_and Pistone (2018) and argues that RND_cannot be a proof-theoretic_solution_if we accept a_formula of the form ¬φ_∧φ as well as ⊥ as_an unacceptable_conclusion. All_in all, only when proof-theoretic validity generally implies normalizability and any_formula having the form ¬φ ∧φ_is not regarded as_an unacceptable_conclusion,_RND can be_a_proof-theoretic solution_to the paradoxes.

Contents Abstract Contents i 1_Introduction: A_Proof-Theoretic Criterion of and Solution to_the Paradoxes 1 1.1_Traditional_Responses to the Paradoxes and Dialetheism.........._3 1.1.1 Traditional Approaches to Russell's_and the Liar Paradox...... 4_1.1.2_Problems_of Traditional Responses and Dialetheism........_10 1.2 Preliminaries................................. 15 1.3 Tennant's_Criterion for Paradoxicality_(TCP) and the Requirement of a Normal_Derivation_(RND)...