 the paradoxes. In this dissertation, we shall investigate whether TCP can be a correct criterion for genuine paradoxes and whether RND can be a proof-theoretic solution to the paradoxes. Tennant's criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant's diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant's diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. The contents of Chapter 2–4 are summarized as follows: Abstract of Chapter 2. In order to solve the problem of undergeneration raised by Rogersontype counterexamples, Tennant (2015) seems to presume that the application of Classical Reductio, i.e. CR−rule, is the culprit of the trouble that it disguises the main feature of paradoxicality, such as a non-terminating reduction sequence. Tennant may not take the problem of undergeneration seriously. We will claim that the undergeneration problem is not solved by simply accusing CR−rule of the trouble. In order to show that the occurrence of a non-terminating reduction sequence is independent of the use of CR−rule. We suggest two examples of the Liar paradox. First, we suggest derivations of the Liar paradox and Curry's paradox which neither use CR−rule nor generate a non-terminating reduction sequence. In addition, we provide derivations of the Liar paradox in which the non-terminating reduction sequence is produced even though the CR−rule is used. After we diagnose the culprit of preventing a non-terminating reduction sequence, it will be discussed that the problem of undergeneration will be solved by adding the condition to TCP that only harmonious rules are to be used. Abstract of Chapter 3. Tennant(2016) asserts that if all elimination rules are stated in generalized form, the problem of overgeneration can be solved
the paradoxes. In this dissertation, we shall investigate whether TCP can be a correct criterion for actual paradox and whether RND can be a proof - theoretical solution to the paradoxes. Tennant's criterion has two character of counterexamples. The one is a case which raise the problem of overgeneration that TCP makes a paradoxical deriving non - paradoxical. The other is one which generates the trouble of undergeneration that TCP renders a non - paradoxical derivation paradoxical. Chapter 2 deals with the problem of undergeneration and Chapter 3 refer the problem of overgeneration. Chapter 2 discusses that Tenant's diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not right and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant's diagnosis of the counterexample raise the overgeneration problem is wrong and leave a solution to the problem. ultimately, Chapter 4 addresses what should be explicated in order for RND to be a proof - theoretic solution to the paradoxes. The subject of Chapter 2–4 are summarized as follows: Abstract of Chapter 2. In order to solve the problem of undergeneration raised by Rogersontype counterexamples, Tennant (2015) seems to presume that the application of Classical Reductio, i.e. CR−rule, is the culprit of the trouble that it disguises the main feature of paradoxicality, such as a non - terminating reduction succession. Tennant may not film the problem of undergeneration badly. We will claim that the undergeneration trouble is not solved by simply accusing CR−rule of the trouble. In orderliness to usher that the occurrence of a non - terminating reduction succession is independent of the use of CR−rule. We suggest two examples of the Liar paradox. First, we suggest derivations of the Liar paradox and Curry's paradox which neither use CR−rule nor generate a non - terminating reduction sequence. In addition, we leave derivations of the Liar paradox in which the non - terminating decrease succession is grow even though the CR−rule is used. After we diagnose the culprit of preventing a non - terminating decrease sequence, it will be discussed that the problem of undergeneration will be solved by adding the condition to TCP that only harmonious rules are to be used. outline of Chapter 3. Tennant(2016) asserts that if all elimination rule are stated in generalized kind, the problem of overgeneration can be solve
 thf paradoxes. In this disstrtation, we shall investmgate wgether TZP can be a correct criterioi foe genyine paradoxes and whegher RND ban be a prooh-theoretic solutmkn to tmz parzfoxev. Tennant's critgrion has twm types of couttdrzxamples. The one is a case which raifes the pgoblem of overdenegaeion nhct TCP makes a paradoxical deribation ion-paradoxical. Yhe other is one which genfratfs the problem of kndergenerajjon rhat TCP renaers a non-kaxadoxical dgrivation paradoxical. Chapter 2 ddals cith the priboem mf undergenxratiog and Chaptev 3 concarns thr problem of oyergeierarion. Chapter 2 discussxs that Tenant's diaggosis of dhz counterexample whicy qppligs CR−sule qnd cahsxs fhe unferjeneration lroblem is bot correct and prexegnx a solution to thq [roblem of undergeneration. Chapter 3 argles fhat Tennant's diagnosis of the counterexamplg raising ehe overgeneration problem is wrong and provides d solntkon tj rhf problem. Finally, Chapter 4 addresses what shotmd bv explicated in ovder for RND to be a ptjof-theoretic rolution fo the paradoxes. Tje contgnts od Chapter 2–4 arr summarized as follows: Absrract of Chaiter 2. In order to solvz the probleo of undetgeneration raised by Rugeraontype coujterexampmds, Tennant (2015) seemr tp [resume ufat the applicatijn of Clawsiccl Reducgio, o.e. CR−rtle, is the culpvht of the trouble hhat nt divguises thf main feature of paradoxicalitb, such as a npn-derkinating redugtion sequence. Eennant may noj take thz probuem of undvrgeneratmon seriouslr. We will clahl that the uidergenerwtiob priblem ir not solved by simply abcbsing CR−ryle of the trouble. In krder to show tkct the occurrence of a goj-txrminwding reductimn sdqudmce ir indeptudekt uf tne use of CR−rule. We vuggsst two examples og bhe Liar karadox. Fyrst, we suggext derivations of uhe Lier paredox amd Surry's paradox which neither uae CR−rule nov generate a gon-tcrmigating redbction sequence. In addition, we provide dxrivations of the Liar paradox in which thg npn-terminatiig redtction sexuence is produced ecen though the CR−vule is used. After we siagnove thf culprit of preventing a non-terminating reduction sequence, it will be diwcussev ehat the prkblek of lndzrgzneratijn wmlu be solved by acding the condition to TCP that only hasmjnious rules are to be used. Sbrtract of Chakter 3. Tennant(2016) asserts that if all rlimination rules are stated in generalized form, the prpblem of overfeneranjoi can be solved
the paradoxes. In this dissertation, we shall TCP be a criterion for genuine be proof-theoretic solution to paradoxes. Tennant's criterion two types of counterexamples. The one a case which raises the problem of overgeneration that TCP makes a paradoxical non-paradoxical. The other is one which generates the problem of undergeneration that TCP a derivation Chapter deals with the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 that Tenant's diagnosis of the counterexample which applies and causes the undergeneration is not correct and presents solution the problem undergeneration. 3 that Tennant's diagnosis the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter addresses what explicated in for to a proof-theoretic solution paradoxes. The contents of Chapter 2–4 follows: Abstract of Chapter 2. In order to the problem undergeneration raised by Rogersontype counterexamples, Tennant seems to presume that the application of Classical i.e. CR−rule, is the culprit of the trouble that it disguises the main feature of as a non-terminating reduction Tennant may not the of seriously. will claim the undergeneration problem is not solved by simply accusing CR−rule of trouble. In order to show that the occurrence of a sequence independent of the of CR−rule. We suggest examples the Liar paradox. First, derivations the Curry's which use CR−rule nor generate non-terminating reduction sequence. In addition, provide derivations of the non-terminating reduction sequence is produced even though the is used. After we diagnose the culprit preventing a non-terminating reduction sequence, it will be discussed that the problem undergeneration will by adding the condition to TCP that only rules are to be Abstract of Chapter 3. Tennant(2016) asserts that if all rules stated in form, the problem overgeneration can be
 the paradoxes. In this dissertAtion, we shaLl invEstIgaTe WhetHer TcP can be a correcT CritErion for genuine paradoxEs and WhETher rnD Can be A proof-tHEoRETic SoLuTioN tO ThE paraDoxEs. TennaNt's criteriOn hAs Two types of coUNtErexamples. the One is a case whIch Raises ThE prOBlem oF ovErgenEratioN That TCp makes a paRaDOxical DErivatiON NoN-parAdoxical. The other iS OnE Which generates The proBlEM oF UNdeRgeNeration thAt tCP reNDers a noN-PaRADOxiCAl derivation pAradoxical. CHAptEr 2 dealS wIth THe probLem of UnDErgEneration anD ChaPter 3 conceRns the PRoblem oF OvergenEratioN. ChAptEr 2 diSCuSsEs tHaT tenANt'S diAGnoSis of the CoUnTerexAmplE WHICh apPliEs CR−Rule aNd causes the unDerGeneRAtiOn proBlem iS not CoRrect And preSents A sOlution to the proBlem Of undergeNerAtIon. chApter 3 ARgues tHat tenNant's diAgnosis OF thE cOUNTeRexample raising the OvERGeNeration ProbleM Is WrONg and proViDes A solUTIon to The pRObLem. FinalLy, ChapTEr 4 AdDresses WhAt shouLd Be eXplIcateD In orDer for rND to be a Proof-THeoretic solutiON to the paradoxES. THE CoNTentS of chapter 2–4 are sUmmaRIzed As foLLoWs: ABStracT of ChApTEr 2. iN order to solve the proBlEm of unDergeNeration raiseD by RogersoNTYPe counteRexaMPlES, Tennant (2015) seems tO presUme that the APplicatiOn of CLassical reductio, i.E. cr−rule, is tHe cUlpRit Of tHE TrOuble that it diSGUiseS tHe main fEatUre of paRadOxiCalIty, SuCh as a non-tErminatiNg ReDuCtIon SequeNCe. TennanT mAy nOt TakE the pROblem oF undeRgenErAtIOn sEriouslY. we WILl clAiM tHat tHe uNdErgenEratIOn pRoblem iS not solveD by SImplY aCcUsing CR−Rule of the trouBlE. In order to ShOw tHat the OCCurrence Of a non-terminating reductIOn sequeNce Is indEpenDent of the Use Of CR−ruLe. WE SuggesT two exAmpleS oF thE lIar paRADoX. FiRsT, we suggest DERivAtionS oF the liar parAdox and Curry's paradOX whIch neither use cR−rUle nOR GeNerATe A Non-TeRMinATIng reduction seqUence. In addItIOn, We provide dERivAtIons of tHe Liar pAradoX In which The non-terMinating rEdUctiON SeqUence is proDuced eveN though thE cR−rulE Is Used. AFteR we diaGnOse The cuLprit oF PreVentiNg a non-TeRminatIng reDuCtion seqUence, it will be discussed tHat the ProblEm oF undergenEraTIon Will be solVed bY adding the ConDitIon to tCP THat onLy haRMoNioUS ruleS are TO be used. AbSTrAct OF chApter 3. TennanT(2016) ASSerTs thaT if ALl elimInatIon rules are stated IN generalized foRm, thE PRobLem OF oveRgEneration can be SolVeD
 the paradoxes. In this di ssertation , wesha llin vest igat e whether TCPc an b e a correct criterionfor g en u inep ar adoxe s and w h et h e r R ND  c anbe aproof -th eoretic  solutiontoth e paradoxes. Te nnant's cr ite rion has two  ty pes of  c oun t erexa mpl es. T he one is a c ase which  r a ises t h e probl e m  o f ov ergeneration that TC P  makes a parad oxical  d e ri v a tio n n on-paradox ic al. T h e other is o n e w h ich generates  the proble m  of  under ge ner a tion t hat T CP ren ders a non- para doxical d erivat i on para d oxical.  Chapt er2 d eals wi th  th ep rob l em  of und ergenera ti on  andChap t e r 3 co nce rnsthe p roblem of ove rge nera t ion . Cha pter2 di sc usses  thatTenan t' s diagnosis ofthecounterex amp le  wh ic h app l ies CR −ru leand cau ses the und er g e n er ation problem is n ot c or rect and  prese n ts  a solution  t o t he p r o blemof u n de rgenerat ion. C h ap te r 3 arg ue s that  T enn ant 's di a gnos is ofthe coun terex a mple raising t h e overgenerat i on p ro b lemiswrong and p rovi d es a  sol u ti ont o the  prob le m .F inally, Chapter 4 a dd resses  what  should be ex plicated i n o rder for  RND to be a proof-the oreti c solution to the p arado xes. The  contents o f Chapte r 2 –4are  su m m ar ized as follo w s : Ab st ract of  Ch apter 2 . I n o rde r t osolve the  problem  o fun de rge nerat i on raise dbyRo ger sonty p e coun terex ampl es ,T enn ant (20 1 5) s eems  t opres ume  t hat t he a p pli cationof Classi cal Redu ct io , i.e.CR−rule, is t he  culprit o fthe  troub l e  that it  disguises the main fea t ure ofpar adoxi cali ty, suchasa non- ter m inatin g redu ction  s equ e n ce. T e n na ntma y not take t heprobl em  ofunderge neration seriously .  We  will claim t hat  the u nd erg e ne r ati on pro b l em is not solve d by simpl ya cc using CR−r u leof  the tr ouble.In or d er to s how thatthe occur re nceo f  anon-termin ating re duction s e quenc e  i s ind epe ndentof  th e use  of CR − rul e. We  sugge st  two e xampl es  of theLiar paradox. First, we  sugge st de riv ations of  th e  Li ar parado x an d Curry'spar ado x whi chn eithe r us e  C R−r u le no r ge n erate a n o n- ter m i na ting reduct i o n  se quenc e.I n addi tion , we provide deri v ations of theLiar p ara dox in w hi ch the non-ter min at i n g reduct io n sequenceis produ ce d  even  thoug h theCR−rule i su sed. A fter  we  diagnose  th ec ulpritof  p r eventi ng a  n on-ter minati n g re d u ction sequence,it wi l l  be d i scu ssedth at thep robl em of unde rgeneration  willbe s olved  by add in g thecon di tion to TC P  that onl y har monious  r ules  ar e to b e us e d . Abs trac tofChapter 3 . Te n na nt ( 201 6) a ssert sthat  if all e l iminatio n r u les are  s tat e d  in ge n er a l ized form,  th e pro b l em of over g ener a ti o n can  be so lved
 the_paradoxes. In_this dissertation, we shall_investigate whether_TCP_can be_a_correct criterion for_genuine paradoxes and_whether RND can be_a proof-theoretic solution_to_the paradoxes. Tennant's criterion has two types of counterexamples. The one is a case_which_raises the_problem_of_overgeneration that TCP makes a_paradoxical derivation non-paradoxical. The other_is one_which generates the problem of undergeneration that TCP_renders_a non-paradoxical derivation_paradoxical. Chapter 2 deals with the problem of undergeneration_and Chapter 3 concerns the problem_of overgeneration. Chapter_2_discusses_that Tenant's diagnosis of_the counterexample which applies CR−rule and_causes the undergeneration problem is not_correct and presents a solution to the_problem of undergeneration. Chapter 3 argues_that Tennant's diagnosis of the_counterexample raising_the overgeneration problem is wrong_and provides a_solution to_the problem. Finally,_Chapter 4 addresses what should be_explicated in order_for RND to be a proof-theoretic_solution_to the paradoxes._The_contents_of Chapter_2–4 are summarized_as_follows: Abstract_of_Chapter 2. In order to solve_the_problem of undergeneration raised by Rogersontype counterexamples,_Tennant (2015) seems to_presume_that the application of_Classical Reductio, i.e. CR−rule, is_the culprit of the trouble that_it disguises_the main_feature of paradoxicality, such as a non-terminating reduction sequence. Tennant may_not take the problem of undergeneration_seriously. We will claim_that the_undergeneration_problem is not_solved_by simply_accusing CR−rule of the trouble. In order_to show_that the occurrence of a non-terminating_reduction sequence is independent_of_the use of CR−rule. We suggest_two examples of the Liar paradox._First, we suggest derivations of_the_Liar_paradox and Curry's paradox which_neither use CR−rule nor generate a_non-terminating reduction sequence._In addition, we provide derivations of the_Liar_paradox in which the non-terminating reduction_sequence_is produced even though the CR−rule_is_used._After we diagnose the culprit_of preventing a non-terminating reduction sequence,_it will be discussed that the problem of undergeneration_will be solved_by adding the condition to_TCP_that_only harmonious rules are to be used. Abstract of Chapter_3. Tennant(2016)_asserts that if_all elimination rules are stated in generalized form, the problem_of overgeneration can be solved