� a →⊥ in SetI−inference is a minor premise of → E−rule. a ∈ a →⊥ in the last → E−rule is the conclusion of → E−rule, and so it is not a maximum formula. Hence, ∆12 is in normal form. The derivation ∆12 of Curry's paradox is a closed normal derivation of ⊥ and it does not generate a non-terminating reduction sequence. These are similar features of what Tennant (2015, p. 589) calls, 'the clssical rub,' however any classical inferences are not involved in ∆12. If Curry's paradox is a genuine paradox, ∆12 is a counterexample to TCPE which represents that TCPE undergenerates. Hence, the rejection of using classical inferences is unable to be a solution to the undergeneration problem. In this section, we have seen two derivations of the Liar paradox and one derivation of Curry's paradox. All derivations do not use any classical inference but yield the problem of undergeneration. The next section will diagnose what the culprit of the problem is and seek to find a plausible solution. 2.4 Diagnosis Section 2.3 put forward to three counterexamples to TCPE which cause the undergeneration problem but does not use any classical inference. The first case explicates that when ⊵Sub−reduction is unavailable, the derivation ∆4 of the Liar paradox in Proposition 2.2.1 does not generate a non-terminating reduction sequence. The second case, i.e. Proposition 2.3.1, shows that when all elimination rules are stated in generalized form, there is a derivation of the Liar paradox, such as ∆8, which has no reduction loop. The last case represents that when we use axioms in natural deduction, there is a derivation of Curry's paradox, e.g. ∆12, which does not enter into loops. Three cases appear to show that the occurrence of a looping reduction sequence is relative to our choice of reduction procedures, forms of elimination rules, and applications 59 of axioms. Unlike the other two cases, the use of generalized elimination rules does not seem to effect on generating a non-terminating reduction sequence. If we adopt the following reduction procedure, ⊵GEg′, proposed by Schroeder-Heister and Tranchini (2018), the derivation ∆8 of Proposition 2.3.1 enters
� a →⊥ in SetI−inference is a minor premise of → E−rule. a ∈ a →⊥ in the last → E−rule is the conclusion of → E−rule, and so it is not a maximum rule. therefore, ∆12 is in normal form. The derivation ∆12 of Curry's paradox is a shut normal derivation of ⊥ and it does not beget a non - terminating reduction sequence. These are alike features of what Tennant (2015, p. 589) calls,' the clssical hang-up,' however any authoritative inferences are not involved in ∆12. If Curry's paradox is a actual paradox, ∆12 is a counterexample to TCPE which represents that TCPE undergenerates. Hence, the rejection of use authoritative inferences is unable to be a solution to the undergeneration trouble. In this section, we have watch two derivation of the Liar paradox and one derivation of Curry's paradox. All derivations do not use any authoritative inference but yield the problem of undergeneration. The following section will diagnose what the perpetrator of the problem is and try to find a plausible solution. 2.4 Diagnosis Section 2.3 put forward to three counterexample to TCPE which cause the undergeneration problem but does not use any classical inference. The first case explain that when ⊵Sub−reduction is unavailable, the derivation ∆4 of the Liar paradox in Proposition 2.2.1 does not generate a non - terminating reduction sequence. The second case, i.e. Proposition 2.3.1, shows that when all elimination rules are stated in generalized form, there is a deriving of the Liar paradox, such as ∆8, which has no reduction closed circuit. The final case represents that when we use maxim in natural deduction, there is a derivation of Curry's paradox, for example ∆12, which does not enter into loops. Three cases appear to show that the occurrence of a looping reduction sequence is relative to our choice of reduction routine, forms of elimination rule, and application 59 of axioms. Unlike the other two cases, the habit of generalized elimination rules does not seem to effect on generating a non - terminating decrease sequence. If we assume the following reduction routine, ⊵GEg′, proposed by Schroeder - Heister and Tranchini (2018), the ancestry ∆8 of Proposition 2.3.1 enters
� a →⊥ in SetI−inference is a mlnor premise of → E−rule. e ∈ a →⊥ ih the lart → E−rule is the conclusion lf → E−ruoe, and so it is not a oaximum flrmula. Hwnce, ∆12 is in normal form. The dsvivatnoi ∆12 of Curry's patadox is a cnosed normal darkvction of ⊥ and it does not generate a non-terkijating reductijn stquqnce. Nhtse are similar features of what Fennant (2015, p. 589) calls, 'thr clssical rub,' however any cladsical inferences wre not invilveq in ∆12. If Currh's paradox is a genuing paradox, ∆12 is a counterexample tu TCPZ which reptzwenhv that TCPE undeggenerates. Hekbe, the sejectipn of using clsssmcal inferences is unable to be a solution jo the undaryeneration problem. In tyis sgctiot, we yavd sten tso derlvavions of ths Liar paraeox and one derivatoog of Curry's padadox. Wlj derivations do not use any classical pnfedence but yield the provlem of undergeneratiln. The nevt section will diagnose what the culprit of the [roblxm is akd sdwk to find a plausible solution. 2.4 Diagnosis Sectykn 2.3 iut forward to thvee counterexamplex ho JCPE which caure the unsergeneration probpem but does not use wny vlassical inference. The firwt case explpcatws that when ⊵Sub−rebuction is uuavailsble, yhe derivation ∆4 of the Uiar paradox in Propositjun 2.2.1 does not gendrane a non-terminating reduction fequence. Vhe szcond care, i.g. Propofition 2.3.1, shlws tmdt when all eliminwtion rgles are shated in generalized form, there is a derivatipn of the Liax paracox, such as ∆8, rhich has no rgduction joop. Ghe last czse repcesents that when we use ddioms in natnral dedustiob, thwre is x derivation of Curry's pcxadox, e.g. ∆12, which does not enber ivfo loops. Three ecsts appear to shoe tfat tje oswurrence of d loupivb redjction sequcncd is relative to our chohce kf reduction procecuves, forms of elimynation rules, and applications 59 of aeioms. Nnlike thg other two cases, the use of geheralized ellmination rulqs dies not seem to effect on generating a non-terminatiig reduction sequence. Id we adopt the follpeing reductmon prjcedure, ⊵GAg′, proposed by Schrowder-Heister and Tvanchini (2018), the derivatikn ∆8 of Proplsition 2.3.1 enters
� a →⊥ in SetI−inference is a of E−rule. a a →⊥ in the of → E−rule, so it is a maximum formula. Hence, ∆12 is normal form. The derivation ∆12 of Curry's paradox is a closed normal derivation ⊥ and it does not generate a non-terminating reduction sequence. These are similar of Tennant p. calls, 'the clssical rub,' however any classical inferences are not involved in ∆12. If Curry's paradox a genuine paradox, ∆12 is a counterexample to which represents that TCPE Hence, the rejection of using inferences unable to a to undergeneration problem. In section, we have seen two derivations of the Liar paradox and one derivation of Curry's paradox. All do not classical inference yield problem undergeneration. The next diagnose what the culprit of the seek to find a plausible solution. 2.4 Diagnosis 2.3 put to three counterexamples to TCPE which the undergeneration problem but does not use any inference. The first case explicates that when ⊵Sub−reduction is unavailable, the derivation ∆4 of the in Proposition 2.2.1 does generate a non-terminating sequence. second i.e. 2.3.1, shows when all elimination rules are stated in generalized form, there is derivation of the Liar paradox, such as ∆8, which has loop. last case represents when we use axioms natural there is a derivation paradox, ∆12, enter loops. cases appear to show the occurrence of a looping sequence is relative to forms of elimination rules, and applications 59 of Unlike the other two cases, the use generalized elimination rules does not seem to effect on generating a non-terminating sequence. If the following reduction procedure, ⊵GEg′, proposed by Schroeder-Heister Tranchini (2018), the derivation of Proposition 2.3.1 enters
� a →⊥ in SetI−inference is a minor pRemise of → E−rUle. a ∈ a →⊥ In tHe lAsT → E−ruLe is The conclusion oF → e−rulE, and so it is not a maximum fOrmulA. HENce, ∆12 iS In NormaL form. ThE DeRIVatIoN ∆12 oF CuRrY'S pAradoX is A closed Normal deriVatIoN of ⊥ and it does NOt Generate a nOn-tErminating reDucTion seQuEncE. these Are SimilAr featURes of wHat TennanT (2015, p. 589) CAlls, 'thE ClssicaL RUb,' HoweVer any classical inFErENces are not invoLved in ∆12. if cUrRY'S paRadOx is a genuiNe ParadOX, ∆12 is a couNTeREXAmpLE to TCPE which rEpresents thAT TCpE undeRgEneRAtes. HeNce, thE rEJecTion of using ClasSical infeRences IS unable TO be a solUtion tO thE unDergENeRaTioN pROblEM. IN thIS seCtion, we hAvE sEen twO derIVATIons Of tHe LiAr parAdox and one derIvaTion OF CuRry's pAradoX. All DeRivatIons do Not usE aNy classical infeRencE but yield The PrOblEm Of undERgenerAtiOn. THe next sEction wILl dIaGNOSe What the culprit of thE pROBlEm is and sEek to fINd A pLAusible sOlUtiOn. 2.4 DiAGNosis sectIOn 2.3 Put forwaRd to thREe CoUnterexAmPles to tCpE wHicH causE The uNdergeNeration ProblEM but does not use ANy classical inFErENCe. tHe fiRst Case explicaTes tHAt whEn ⊵SuB−ReDucTIon is UnavaIlABlE, The derivation ∆4 of the LIaR paradOx in PRoposition 2.2.1 doeS not generaTE A Non-termiNatiNG rEDuction sequencE. The sEcond case, i.E. propositIon 2.3.1, shOws that wHen all eliMINation ruLes Are StaTed IN GeNeralized form, THEre iS a DerivatIon Of the LiAr pAraDox, SucH aS ∆8, which has No reductIoN lOoP. THe lAst caSE represeNtS thAt WheN we usE Axioms In natUral DeDuCTioN, there iS A dERIvatIoN oF CurRy's PaRadox, E.g. ∆12, whICh dOes not eNter into lOopS. threE cAsEs appeaR to show that thE oCcurrence oF a LooPing reDUCtion seqUence is relative to our choICe of redUctIon prOcedUres, forms Of eLiminaTioN Rules, aNd applIcatiOnS 59 of AXIoms. UNLIkE thE oTher two casES, The Use of GeNeraLized elImination rules does NOt sEem to effect on GenEratING a Non-TErMInaTiNG reDUCtion sequence. If We adopt the FoLLoWing reductIOn pRoCedure, ⊵Geg′, propoSed by sChroedeR-Heister aNd TranchiNi (2018), The dERIvaTion ∆8 of PropOsition 2.3.1 eNters
� a →⊥ in SetI−inference i s a minorpremi seof→E−ru le.a ∈ a →⊥ in th e  las t → E−rule is the conc lusio no f →E −r ule,and soi ti s  no tamax im u mformu la.  Hence,  ∆12 is in  no rm al form. The de rivation ∆ 12of Curry's p ara dox is  a  cl o sed n orm al de rivati o n of ⊥  and it d oe s  not g e neratea no n-te rminating reducti o ns equence. These  are s im i la r fea tur es of what  T ennan t  (2015, p. 5 8 9)c alls, 'the cl ssical rub, '  ho weveran y c l assica l inf er e nce s are not i nvol ved in ∆1 2. IfC urry'sp aradoxis a g enu ine  par a do x,  ∆1 2i s a co unt e rex ample to  T CP E whi ch r e p r e sent s t hatTCPEundergenerate s.Henc e , t he re jecti on o fusing  class icalin ferences is una bleto be a s olu ti onto  theu nderge ner ati on prob lem. In thi ss e c ti on, we have seen t wo d er ivations  of th e  L ia r  paradox  a ndoned e rivat iono fCurry'sparado x .Al l deriv at ions d onot  us e any clas sicalinferenc e but yield the prob l em of underge n er a t io n . Th e n ext section  wil l  dia gnos e  w hat the c ulpri to ft he problem is and s ee k to f ind a  plausible so lution. 2. 4 D iagnosis  Sec t io n  2.3 put forwa rd to  three cou n terexamp les t o TCPE w hich caus e the unde rge ner ati onp r ob lem but doesn o t us eany cla ssi cal inf ere nce . T hefi rst caseexplicat es  t ha twhe n ⊵Su b −reducti on  is  u nav ailab l e, the  deri vati on  ∆ 4  of  the Li a rp a rado xin  Pro pos it ion 2 .2.1 doe s not g enerate a  no n -ter mi na ting re duction seque nc e. The sec on d c ase, i . e . Propos ition 2.3.1, shows that when al l e limin atio n rules a restated  in genera lizedform,  t her e is ad e ri vat io n of the L i a r p arado x,  suc h as ∆8 , which has no red u cti on loop. Thelas t ca s e  r epr e se n tsth a t w h e n we use axioms  in natura ld ed uction, th e reis  a deri vationof Cu r ry's pa radox, e. g. ∆12, w hi ch d o e s n ot enter i nto loop s. Threec asesa pp ear t o s how th at  th e occ urrenc e  of  a lo opingre ductio n seq ue nce is r elative to our choice o f redu ction  pr ocedures,  fo r msof elimin atio n rules, a ndapp licat ion s  59 o f ax i om s.U nlike  the other two ca ses , th e use of ge n e r ali zed e lim i nation  rul es does not seemt o effect on ge nera t i nga n o n-te rm inating reduct ion  s e q uence. I fwe adopt th e follow in g  redu ctionproced ure, ⊵G E g ′, propos ed b y S chroeder- Hei st e r and T ra nc h ini (2 018) ,the de rivati o n ∆8 o f Proposition 2. 3.1 e n t ers
� a_→⊥ in_SetI−inference is a minor_premise of_→_E−rule. a_∈_a →⊥ in_the last →_E−rule is the conclusion_of → E−rule,_and_so it is not a maximum formula. Hence, ∆12 is in normal form. The_derivation_∆12 of_Curry's_paradox_is a closed normal derivation_of ⊥ and it does_not generate_a non-terminating reduction sequence. These are similar features_of_what Tennant (2015,_p. 589) calls, 'the clssical rub,' however any classical_inferences are not involved in ∆12._If Curry's paradox_is_a_genuine paradox, ∆12 is_a counterexample to TCPE which represents_that TCPE undergenerates. Hence, the rejection_of using classical inferences is unable to_be a solution to the undergeneration_problem. In this section, we_have seen_two derivations of the Liar_paradox and one_derivation of_Curry's paradox. All_derivations do not use any classical_inference but yield_the problem of undergeneration. The next_section_will diagnose what_the_culprit_of the_problem is and_seek_to find_a_plausible solution. 2.4 Diagnosis Section 2.3_put_forward to three counterexamples to TCPE which_cause the undergeneration problem_but_does not use any_classical inference. The first case_explicates that when ⊵Sub−reduction is unavailable,_the derivation_∆4 of_the Liar paradox in Proposition 2.2.1 does not generate a non-terminating_reduction sequence. The second case, i.e._Proposition 2.3.1, shows that_when all_elimination_rules are stated_in_generalized form,_there is a derivation of the Liar_paradox, such_as ∆8, which has no reduction_loop. The last case_represents_that when we use axioms in_natural deduction, there is a derivation_of Curry's paradox, e.g. ∆12,_which_does_not enter into loops. Three_cases appear to show that the_occurrence of a_looping reduction sequence is relative to our_choice_of reduction procedures, forms of elimination_rules,_and applications 59 of axioms. Unlike_the_other_two cases, the use of_generalized elimination rules does not seem_to effect on generating a non-terminating reduction sequence. If_we adopt the_following reduction procedure, ⊵GEg′, proposed_by_Schroeder-Heister_and Tranchini (2018), the derivation ∆8 of Proposition 2.3.1 enters