 nonterminating reduction sequence. Section 2.2 introduces the Togerson-type derivation of the Liar paradox proposed by Tennant (2015, pp. 585–590). Some Rogerson-type derivations can be a counterexample to TCPE which raises the problem of undergeneration that TCPE excludes Curry's and the Liar paradox in the realm of genuine paradoxes. As we have argued in Section 2.2 and 2.3, Tennant thinks that CR−rule is the culprit of the problem. However, Section 2.3 discusses that not every case causing the undergeneration problem is related to CR−rule. Section 2.4 argues that the occurrence of a non-terminating reduction sequence relies on our choice of reduction procedures and diagnoses what the culprit of the problem of undergeneration. We have seen that the use of CR−rule does not always cause the problem, and so the rejection of the use of CR−rule is unable to be a solution. This is because, with respect to TCPE, there are four types of closed derivations which formalize Curry's and the Liar paradox: (i) a T-paradox which does not use CR−rule, e.g. Proposition 2.2.1 and 2.A.1, (ii) a T-paradox using CR−rule, e.g. Proposition 2.4.1, 2.4.2, 2.4.3, and 2.A.3, (iii) a derivation of ⊥ which uses CR−rule and is not a T-paradox, e.g. Proposition 77 2.0.1, 2.2.2, and 2.A.2, and (iv) a derivation of ⊥ which neither use CR−rule nor is a Tparadox, e.g. Proposition 2.3.1 and the derivation ∆12 of Curry's paradox in Section 2.3. The occurrence of a non-terminating reduction sequence is not always dependent on the use of CR−rule. We have considered that to have an infinite reduction sequence, every major premise should have a reduction procedure to eliminate it. Derivations employing CR−rule and axioms which has no infinite reduction sequence lacks the processes to eliminate a major premise which is a conclusion of CR−rule or an axiom. Hence, we propose a possible diagnosis that a derivation formalizing a genuine paradox generates a non-terminating reduction sequence only
nonterminating reduction sequence. Section 2.2 introduces the Togerson - character deriving of the Liar paradox proposed by Tennant (2015, pp. 585–590). Some Rogerson - type derivation can be a counterexample to TCPE which raise the problem of undergeneration that TCPE excludes Curry's and the Liar paradox in the kingdom of genuine paradoxes. As we have argue in Section 2.2 and 2.3, Tennant thinks that CR−rule is the perpetrator of the problem. However, Section 2.3 discusses that not every case cause the undergeneration problem is related to CR−rule. Section 2.4 argues that the occurrence of a non - terminating decrease sequence relies on our choice of decrease procedures and diagnoses what the culprit of the problem of undergeneration. We have seen that the function of CR−rule does not always cause the problem, and so the rejection of the use of CR−rule is ineffective to be a solution. This is because, with respect to TCPE, there are four types of closed derivations which formalize Curry's and the Liar paradox: (iodine) a T - paradox which does not use CR−rule, e.g. Proposition 2.2.1 and 2.A.1, (ii) a T - paradox using CR−rule, e.g. Proposition 2.4.1, 2.4.2, 2.4.3, and 2.A.3, (iii) a derivation of ⊥ which uses CR−rule and is not a T - paradox, e.g. Proposition 77 2.0.1, 2.2.2, and 2.A.2, and (iv) a derivation of ⊥ which neither use CR−rule nor is a Tparadox, e.g. Proposition 2.3.1 and the deriving ∆12 of Curry's paradox in Section 2.3. The occurrence of a non - terminating decrease succession is not always dependent on the use of CR−rule. We have considered that to accept an infinite reduction sequence, every major premise should have a decrease procedure to eliminate it. Derivations employing CR−rule and axioms which has no infinite reduction sequence lacks the processes to eliminate a major premise which is a conclusion of CR−rule or an maxim. Hence, we project a possible diagnosis that a derivation formalizing a genuine paradox generates a non - terminating decrease sequence only
 nojterminating reduction stquence. Section 2.2 nbtrodures the Togersov-type derivation of the Liar pqradoz proposed by Tennant (2015, pp. 585–590). Some Rogersob-typt derivations can be a comutereswmplz vo TCPE which rsises the [roblem of undareeueration that TCPE excludes Curry's agd the Kiwr paradox in jhe rtali of genuine paradoxes. As we have argusd in Stction 2.2 and 2.3, Tennamt thinks that CR−rule is tje cklprit of the probpem. However, Seceuon 2.3 discussds that nou zvery case dausing the undergeneration proclem ns related jo CG−tule. Section 2.4 argles that the occurrenwe of a non-terminatinn rednctiin sequence relies on our choice of redoction prowebures and diagnoses wyar the cul[rit if ghe pcobmem of unvergeneratikn. We have ween that the use og SG−tule does nof alwars cause the problem, and so the rejection of the use of CR−rule is ubable to be a solutioj. This is because, with respect to TCPE, there are four typev of rluseb derixqtlons which formalize Curry's and the Liar paraqkx: (i) a T-paradox whlch does not use CT−rklr, e.g. Propositiun 2.2.1 anb 2.A.1, (ii) a T-paradox usijg CR−ruje, e.g. Proposityon 2.4.1, 2.4.2, 2.4.3, and 2.A.3, (iii) a derivation od ⊥ which usef CR−rule and is not a T-paradox, e.g. Kroposotion 77 2.0.1, 2.2.2, and 2.A.2, and (iv) a derjvation of ⊥ which nejgher use CR−rule vor iv a Tparadox, e.g. Propositiog 2.3.1 and thx dernvation ∆12 of Vurry's paradox ij Secbhon 2.3. The occurrencf of c non-derminatinh reduction sequence is not alwebs dependent pn thv use of ER−rule. We have consydered that to have au infivite reducnion sequxnce, every mwjor premise vjould have a reductijn peoceeure to dliminate it. Drrivations employint CR−rule and axiomx wfjch has no infiuntt eeduction sequrncd lwcns tra processes do euimkmate x major prenkse ehich is a conclusiot of CR−rule or an axiok. Mence, we kropose a possible diabnosis that a derigatioi formelizinb a genuine paradox generates a nkn-terminahinn reduction sqquekce jnly
nonterminating reduction sequence. Section 2.2 introduces the of Liar paradox by Tennant (2015, can a counterexample to which raises the of undergeneration that TCPE excludes Curry's the Liar paradox in the realm of genuine paradoxes. As we have argued Section 2.2 and 2.3, Tennant thinks that CR−rule is the culprit of the However, 2.3 that every case causing the undergeneration problem is related to CR−rule. Section 2.4 argues that the occurrence a non-terminating reduction sequence relies on our choice reduction procedures and diagnoses the culprit of the problem undergeneration. have seen the of does not always the problem, and so the rejection of the use of CR−rule is unable to be a solution. is because, to TCPE, are types closed derivations which and the Liar paradox: (i) a not use CR−rule, e.g. Proposition 2.2.1 and 2.A.1, a T-paradox CR−rule, e.g. Proposition 2.4.1, 2.4.2, 2.4.3, 2.A.3, (iii) a derivation of ⊥ which uses and is not a T-paradox, e.g. Proposition 77 2.0.1, 2.2.2, and 2.A.2, and (iv) a ⊥ which neither use nor is a e.g. 2.3.1 the ∆12 of paradox in Section 2.3. The occurrence of a non-terminating reduction sequence not always dependent on the use of CR−rule. We have to an infinite reduction every major premise should a procedure to eliminate it. CR−rule axioms infinite sequence the processes to eliminate major premise which is a of CR−rule or an possible diagnosis that a derivation formalizing a genuine generates a non-terminating reduction sequence only
 nonterminating reduction seQuence. SectIon 2.2 inTroDucEs The TOgerSon-type derivatIOn of The Liar paradox proposed By TenNaNT (2015, pp. 585–590). SOMe rogerSon-type DErIVAtiOnS cAn bE a COuNtereXamPle to TCpE which raiSes ThE problem of unDErGeneration ThaT TCPE excludeS CuRry's anD tHe LIAr parAdoX in thE realm OF genuiNe paradoxEs. aS we havE Argued iN sEcTion 2.2 And 2.3, Tennant thinks tHAt cr−rule is the culpRit of tHe PRoBLEm. HOweVer, Section 2.3 DiScussES that noT EvERY CasE Causing the undErgeneratioN ProBlem is ReLatED to CR−rUle. SeCtIOn 2.4 aRgues that thE occUrrence of A non-teRMinatinG ReductiOn sequEncE reLies ON oUr ChoIcE Of rEDuCtiON prOcedures AnD dIagnoSes wHAT THe cuLprIt of The prOblem of undergEneRatiON. We Have sEen thAt thE uSe of Cr−rule dOes noT aLways cause the prObleM, and so the RejEcTioN oF the uSE of CR−rUle Is uNable to Be a soluTIon. thIS IS bEcause, with respect tO Tcpe, tHere are fOur typES oF cLOsed deriVaTioNs whICH formAlizE cuRry's and tHe Liar PArAdOx: (i) a T-paRaDox whiCh DoeS noT use Cr−Rule, E.g. PropOsition 2.2.1 aNd 2.A.1, (ii) A t-paradox using Cr−Rule, e.g. ProposiTIoN 2.4.1, 2.4.2, 2.4.3, ANd 2.a.3, (Iii) a DerIvation of ⊥ whIch uSEs CR−Rule ANd Is nOT a T-paRadox, E.g. pRoPOsition 77 2.0.1, 2.2.2, and 2.A.2, and (iv) a deRiVation Of ⊥ whiCh neither use Cr−rule nor is A tPAradox, e.g. propOSiTIon 2.3.1 and the derivAtion ∆12 Of Curry's paRAdox in SeCtion 2.3. the occurRence of a nON-TerminatIng RedUctIon SEQuEnce is not alwaYS DepeNdEnt on thE usE of CR−ruLe. WE haVe cOnsIdEred that tO have an iNfInItE rEduCtion SEquence, eVeRy mAjOr pRemisE Should Have a ReduCtIoN ProCedure tO ElIMInatE iT. DErivAtiOnS emplOyinG cR−rUle and aXioms whicH haS No inFiNiTe reducTion sequence lAcKs the proceSsEs tO elimiNATe a major Premise which is a conclusiON of CR−ruLe oR an axIom. HEnce, we proPosE a possIblE DiagnoSis thaT a derIvAtiON FormaLIZiNg a GeNuine paradOX GenErateS a Non-tErminatIng reduction sequenCE onLy
 nonterminating reductionsequence.Secti on2.2  i ntro duce s the Togerson - type  derivation of the Lia r par ad o x pr o po sed b y Tenna n t( 2 015 ,pp . 5 85 – 59 0). S ome  Rogers on-type de riv at ions can bea  c ounterexam ple  to TCPE whi chraises  t hep roble m o f und ergene r ationthat TCPE  e x cludes Curry's a nd  the  Liar paradox int he realm of genui ne par ad o xe s .  As  we  have argu ed  in S e ction 2 . 2a n d  2. 3 , Tennant thi nks that CR − rul e is t he  cu l prit o f the  p r obl em. However , Se ction 2.3  discu s ses tha t  not ev ery ca secau sing th eund er g ene r at ion pro blem isre la ted t o CR − r u l e. S ect ion2.4 a rgues that th e o ccur r enc e ofa non -ter mi natin g redu ction  s equence relieson o ur choice  of  r edu ct ion p r ocedur esand  diagno ses wha t  th ec u l pr it of the problemof u nd ergenera tion.W eha v e seen t ha t t he u s e  of C R−ru l edoes not  alway s  c au se thepr oblem,  a ndsothe r e ject ion of  the use  of C R −rule is unabl e  to be a solu t io n .  T h is i s b ecause, wit h re s pect  toT CP E,t hereare f ou r  t y pes of closed deriv at ions w hichformalize Cur ry's and t h e Liar par adox :  ( i ) a T-paradoxwhich  does notu se CR−ru le, e .g. Prop osition 2 . 2 .1 and 2 .A. 1,(ii ) a T -p aradox usingC R −rul e,  e.g. P rop osition  2. 4.1 , 2 .4. 2,  2.4.3, a nd 2.A.3 ,(i ii )a d eriva t ion of ⊥  w hic huse s CR− r ule an d isnotaT- p ara dox, e. g .P r opos it io n 77  2. 0. 1, 2. 2.2, and  2.A.2,  and (iv)  ad eriv at io n of ⊥which neither  u se CR−rule  n oris a T p a radox, e .g. Proposition 2.3.1 a n d the d eri vatio n ∆1 2 of Curr y's  parad oxi n Sect ion 2. 3. Th eocc u r rence o fa n on -terminati n g  re ducti on  seq uence i s not always depen d ent  on the use o f C R−ru l e .Weh av e  co ns i der e d  that to have a n infinite  r e du ction sequ e nce ,every m ajor pr emise shouldhave a re duction p ro cedu r e  to  eliminate  it. Der ivationse mploy i ng  CR−r ule  and a xi oms  whic h hasn o i nfini te red uc tion s equen ce  lacks t he processes to elimina te a m ajorpre mise whic h i s  aconclusio n of  CR−rule o r a n a xiom.  He n ce, w e pr o po sea  poss ible diagnosis th ata de rivation fo r m a liz ing a  ge n uine p arad ox generates a no n -terminating r educ t i onseq u ence  o nly
 nonterminating_reduction sequence._Section 2.2 introduces the_Togerson-type derivation_of_the Liar_paradox_proposed by Tennant_(2015, pp. 585–590)._Some Rogerson-type derivations can_be a counterexample_to_TCPE which raises the problem of undergeneration that TCPE excludes Curry's and the Liar_paradox_in the_realm_of_genuine paradoxes. As we have_argued in Section 2.2 and_2.3, Tennant_thinks that CR−rule is the culprit of the_problem._However, Section 2.3_discusses that not every case causing the undergeneration problem_is related to CR−rule. Section 2.4_argues that the_occurrence_of_a non-terminating reduction sequence_relies on our choice of reduction_procedures and diagnoses what the culprit_of the problem of undergeneration. We have_seen that the use of CR−rule_does not always cause the_problem, and_so the rejection of the_use of CR−rule_is unable_to be a_solution. This is because, with respect_to TCPE, there_are four types of closed derivations_which_formalize Curry's and_the_Liar_paradox: (i)_a T-paradox which_does_not use_CR−rule,_e.g. Proposition 2.2.1 and 2.A.1, (ii)_a_T-paradox using CR−rule, e.g. Proposition 2.4.1, 2.4.2,_2.4.3, and 2.A.3, (iii)_a_derivation of ⊥ which_uses CR−rule and is not_a T-paradox, e.g. Proposition 77 2.0.1,_2.2.2, and_2.A.2, and_(iv) a derivation of ⊥ which neither use CR−rule nor is_a Tparadox, e.g. Proposition 2.3.1 and_the derivation ∆12 of_Curry's paradox_in_Section 2.3. The_occurrence_of a_non-terminating reduction sequence is not always dependent_on the_use of CR−rule. We have considered_that to have an_infinite_reduction sequence, every major premise should_have a reduction procedure to eliminate_it. Derivations employing CR−rule and_axioms_which_has no infinite reduction sequence_lacks the processes to eliminate a_major premise which_is a conclusion of CR−rule or an_axiom._Hence, we propose a possible diagnosis_that_a derivation formalizing a genuine paradox_generates_a_non-terminating reduction sequence only