�4 in Proposition 2.2.1 does not reduce to the same derivation with ∆3 and so ∆3 does not generate a looping reduction sequence. As for the view of Tennant (2015, p. 589), the derivation ∆6 in Proposition 2.2.2 of the Liar paradox which uses CR−rule appears to be in normal form. Likewise, ∆4 appears to be a closed normal derivation of ⊥ and it does not initiate the non-terminating reduction sequence. These are the same phenomena of what he calls 'the classical rub.' However, ∆4 without the application of ⊵Sub−reduction does not use any classical inferences but raises the problem of undergeneration. Therefore, with the assumption that the Liar paradox is a genuine one, if ⊵Sub−reduction is not proper, ∆4 can be the counterexample to TCPE which shows that TCPE undegenerates. That is, the rejection of the application of classical inferences or CR−rule is unable to be a solution to the problem of undergeneration. 53 Of course, Tennant may disallow the view that ⊵Sub−reduction is not proper. To claim that ⊵Sub−reduction is proper, he should explain which reduction process is proper and which process is not, but he has never explained about it.9 Moreover, even without taking issue with ⊵Sub−reduction procedure, a counterexample to TCPE can be presented that causes the problem of undergeneration. Tennant (2017, pp. 109–110) has preferred to use generalized elimination rules for four reasons: the uniform presentation, the efficiency of proof search, making shorter formal proofs, and affording a solution to the problem of overgeneration. The fourth reason is related to our topic. Fourth,... using Elimination rules in [generalized] form affords a solution to certain problems that would otherwise arise for the proof-theoretic criterion of paradoxicality.... (Tennant, 2017, p. 110) When Tennant (2016) and Tennant (2017, Ch. 11) deal with the problem of overgeneration occurred by Ekman's paradox, instead of accusing Ekman reduction ⊵E of the problem, he provides a solution that all elimination rules are stated in generalized form.10 Though he believes that the choice of the form of generalized elimination rules can solve the overgeneration problem, it also provides a case that causes the undergeneration
� 4 in Proposition 2.2.1 does not reduce to the same derivation with ∆3 and so ∆3 does not render a loop reduction sequence. As for the view of Tennant (2015, p. 589), the ancestry ∆6 in Proposition 2.2.2 of the Liar paradox which uses CR−rule appears to be in normal shape. besides, ∆4 appears to be a closed normal derivation of ⊥ and it does not initiate the non - terminating decrease sequence. These are the same phenomena of what he calls' the authoritative rub.' However, ∆4 without the application of ⊵Sub−reduction does not practice any classical inference but raise the problem of undergeneration. Therefore, with the assumption that the Liar paradox is a actual one, if ⊵Sub−reduction is not proper, ∆4 can be the counterexample to TCPE which shows that TCPE undegenerates. That is, the rejection of the application of classical inferences or CR−rule is unable to be a solution to the trouble of undergeneration. 53 Of course, Tennant may forbid the view that ⊵Sub−reduction is not proper. To claim that ⊵Sub−reduction is proper, he should explain which reduction process is proper and which summons is not, but he has never explained about it.9 Moreover, even without taking issue with ⊵Sub−reduction procedure, a counterexample to TCPE can be give that causes the problem of undergeneration. Tennant (2017, pp. 109–110) has preferred to use generalized elimination rules for four reasons: the uniform presentation, the efficiency of proof search, making shorter conventional proofs, and affording a solution to the trouble of overgeneration. The fourth rationality is relate to our topic. Fourth, ... using Elimination rules in [ generalized ] form affords a solution to sealed problems that would otherwise arise for the proof - theoretic criterion of paradoxicality.... (Tennant, 2017, p. 110) When Tennant (2016) and Tennant (2017, Ch. 11) deal with the problem of overgeneration occurred by Ekman's paradox, instead of accusing Ekman reduction ⊵E of the problem, he provide a solution that all elimination rules are state in generalized form.10 Though he believes that the choice of the form of generalized elimination rules can solve the overgeneration trouble, it also provides a case that causes the undergeneration
�4 in Proposition 2.2.1 does not rtduce to the same derivavion wifh ∆3 and ro ∆3 does not generate a loopmng eeducupon sequence. As for tfe view ov Tennanr (2015, p. 589), the derivefion ∆6 ik Prollsitnoi 2.2.2 of the Liar karadox whicv uses CR−rule dpoecrs to be in normal form. Likewise, ∆4 a[pears yo be a closed njrmak derjnauion of ⊥ and it does not initiate the noi-terminating recuction sequence. These are the same phenomena of what he caols 'eye classical rub.' Howevtr, ∆4 without jhe application of ⊵Sub−reduction aoes uot use any coasdhcal infereices blt raises the problem of undrrgeneration. Tmerefmre, with the assumption vhat the Liar paradov is a getunne one, if ⊵Sub−reductiin is nmt psopef, ∆4 cxn ge tge coujtecexample to TCPE which shows that TCPE uncedvmerates. That is, thq wejection of the application of classicdl jnferences or CR−rule is unable to be a solutlon to thq problem of undergeneration. 53 Of course, Tennant mdy diaxlliw gye view that ⊵Sub−reduction is not proper. To claii tnan ⊵Sub−reduction is proper, he xhlukq explain whizh redbdtjon process is prooer and whicy process is mot, but he has never explaibed about it.9 Noreover, even withlut taking nssue eith ⊵Xub−reduction procedure, c counferexample ho TCPE czv be presented tfat cduses tht problem of underdeneratioi. Tenuant (2017, pp. 109–110) hax prefqrred to ude gekaralized eliminatiln rupev for four reasons: the uniform presentation, the efficiemcf ox proof fearcm, making shortew formal proofx, and ayfordivg a solutpon to thx problem of overgeneratimj. The fourth reason ys rwlatwd to ojf topic. Fourth,... using Elpmnnation ryles in [generalizev] fotm affords a solucnob to certain prpbldms tjav woujg otherwise drisd fut the proof-theorctiz croterion of paradoxicdlitg.... (Tennant, 2017, p. 110) When Tcnnant (2016) abd Tennagt (2017, Ch. 11) deal eith the problem ov ovecgeneretion pccorred by Ekman's paradox, instead of accuslng Ekman reduceion ⊵E of the prpblem, he provides a solution that all eoimination rules aee stated in generaklzed form.10 Thpugh re believas that the choice od the form of gentralized elimination rumes cat solge the overgeneration problem, it also provides a case that causes the undwrgenecaeion
�4 in Proposition 2.2.1 does not reduce same with ∆3 so ∆3 does sequence. for the view Tennant (2015, p. the derivation ∆6 in Proposition 2.2.2 the Liar paradox which uses CR−rule appears to be in normal form. Likewise, appears to be a closed normal derivation of ⊥ and it does not the reduction These the same phenomena of what he calls 'the classical rub.' However, ∆4 without the application of does not use any classical inferences but raises problem of undergeneration. Therefore, the assumption that the Liar is genuine one, ⊵Sub−reduction not ∆4 can be counterexample to TCPE which shows that TCPE undegenerates. That is, the rejection of the application of classical or CR−rule to be solution the of undergeneration. 53 Tennant may disallow the view that proper. To claim that ⊵Sub−reduction is proper, he explain which process is proper and which process not, but he has never explained about it.9 even without taking issue with ⊵Sub−reduction procedure, a counterexample to TCPE can be presented that problem of undergeneration. Tennant pp. 109–110) has to generalized rules four reasons: uniform presentation, the efficiency of proof search, making shorter formal proofs, affording a solution to the problem of overgeneration. The fourth related our topic. Fourth,... Elimination rules in [generalized] affords solution to certain problems otherwise for of (Tennant, p. 110) When Tennant and Tennant (2017, Ch. 11) with the problem of instead of accusing Ekman reduction ⊵E of the he provides a solution that all elimination are stated in generalized form.10 Though he believes that the choice of form of rules can solve the overgeneration problem, it also a case that causes undergeneration
�4 in Proposition 2.2.1 does not reducE to the same DerivAtiOn wItH ∆3 and So ∆3 doEs not generate a LOopiNg reduction sequence. As fOr the ViEW of TENnAnt (2015, p. 589), tHe derivATiON ∆6 In PRoPoSitIoN 2.2.2 Of The LiAr pAradox wHich uses CR−RulE aPpears to be in NOrMal form. LikEwiSe, ∆4 appears to bE a cLosed nOrMal DErivaTioN of ⊥ anD it doeS Not iniTiate the nOn-TErminaTIng reduCTIoN seqUence. These are the sAMe PHenomena of what He callS 'tHE cLASsiCal Rub.' However, ∆4 WiThout THe appliCAtION Of ⊵SUB−reduction doeS not use any cLAssIcal inFeRenCEs but rAises ThE ProBlem of underGeneRation. TheRefore, WIth the aSSumptioN that tHe LIar ParaDOx Is A geNuINe oNE, iF ⊵SuB−RedUction is NoT pRoper, ∆4 Can bE THE CounTerExamPle to tCPE which showS thAt TCpe unDegenErateS. ThaT iS, the rEjectiOn of tHe Application of clAssiCal infereNceS oR CR−RuLe is uNAble to Be a SolUtion to The probLEm oF uNDERgEneration. 53 Of course, TEnNANt May disalLow the VIeW tHAt ⊵Sub−redUcTioN is nOT PropeR. To cLAiM that ⊵Sub−ReductIOn Is Proper, hE sHould eXpLaiN whIch reDUctiOn procEss is proPer anD Which process is NOt, but he has nevER eXPLaINed aBouT it.9 Moreover, Even WIthoUt taKInG isSUe witH ⊵Sub−rEdUCtIOn procedure, a counterExAmple tO TCPE Can be presenteD that causeS THE problem Of unDErGEneration. TennaNt (2017, pp. 109–110) hAs preferreD To use genEraliZed elimiNation rulES For four rEasOns: The UniFORm Presentation, tHE EffiCiEncy of pRooF search, MakIng ShoRteR fOrmal prooFs, and affOrDiNg A sOluTion tO The problEm Of oVeRgeNeratIOn. The fOurth ReasOn Is RElaTed to ouR ToPIC. FouRtH,... uSing eliMiNatioN rulES in [GeneralIzed] form aFfoRDs a sOlUtIon to ceRtain problems ThAt would othErWisE arise FOR the prooF-theoretic criterion of paRAdoxicaLitY.... (TennAnt, 2017, p. 110) when TennaNt (2016) aNd TennAnt (2017, cH. 11) deal wIth the ProblEm Of oVERgeneRATiOn oCcUrred by EkmAN'S paRadox, InSteaD of accuSing Ekman reduction ⊵e Of tHe problem, he prOviDes a SOLuTioN ThAT alL eLImiNATion rules are staTed in generAlIZeD form.10 ThougH He bElIeves thAt the chOice oF The form Of generalIzed elimiNaTion RULes Can solve thE overgenEration prOBlem, iT AlSo proVidEs a casE tHat CauseS the unDErgEneraTion
�4 in Proposition 2.2.1 do es not red uce t o t hesa me d eriv ation with ∆3a nd s o ∆3 does not generate  a lo op i ng r e du ction  sequen c e. A s f or  t hevi e wof Te nna nt (201 5, p. 589) , t he  derivation∆ 6in Proposi tio n 2.2.2 of t heLiar p ar ado x  whic h u ses C R−rule appear s to be i nn ormalf orm. Li k e wi se,∆4 appears to bea  c l osed normal de rivati on of ⊥  an d i t does not  i nitia t e the n o n- t e r min a ting reductio n sequence. The se are  t hes ame ph enome na ofwhat he cal ls ' the class ical r u b.' How e ver, ∆4  witho utthe  app l ic at ion  o f  ⊵S u b− red u cti on doesno tuse a ny c l a s s ical  in fere ncesbut raises th e p robl e m o f und ergen erat io n. Th erefor e, wi th  the assumption  tha t the Lia r p ar ado xis ag enuine  on e,if ⊵Sub −reduct i onis n o tproper, ∆4 can beth e co unterexa mple t o  T CP E  which s ho wsthat T CPE u ndeg e ne rates. T hat is ,  t he  reject io n of t he  ap pli catio n  ofclassi cal infe rence s  or CR−rule is unable to bea  s o l ut i on t o t he problemof u n derg ener a ti on. 53 Of  cour se ,  T e nnant may disallowth e view  that  ⊵Sub−reducti on is notp r o per. Toclai m  t h at ⊵Sub−reduct ion i s proper,h e should  expl ain whic h reducti o n  process  is  pr ope r a n d  w hich processi s  not ,but hehas  neverexp lai ned  ab ou t it.9 Mo reover,ev en  w it hou t tak i ng issue  w ith  ⊵ Sub −redu c tion p roced ure,  a  c o unt erexamp l et o  TCP Eca n be  pr es ented  tha t  ca uses th e problem  of unde rg en eration . Tennant (20 17 , pp. 109– 11 0)has pr e f erred to  use generalized elimin a tion ru les  forfour  reasons:  th e unif orm presen tation , the  e ffi c i encyo f  p roo fsearch, ma k i ngshort er  for mal pro ofs, and affording a s olution to th e p robl e m  o f o v er g ene ra t ion . The fourth reas on is rela te d  t o our topi c . F ou rth,...  usingElimi n ation r ules in [ generaliz ed ] fo r m  af fords a so lution t o certain probl e ms  that  wo uld ot he rwi se ar ise fo r  th e pro of-the or etic c riter io n of par adoxicality.... (Tennan t, 201 7, p.  11 0) When T enn a nt(2016) an d Te nnant (201 7,Ch.  11)dea l  with  the pr obl e m ofover g eneration oc cur r e dby Ekman'sp a r ado x, in ste a d of a ccus ing Ekman reducti o n ⊵E of the pr oble m ,  he  pr o vide sa solution tha t a ll e liminati on  rules arestated i ng enera lizedform.1 0 Thoug h he believ es t hat  the choi ceof the for mof genera lize delimin ationr ules c an solve the ove rgene r a tionp rob lem,it  also p r ovid es a casethat causes  the u nder gener ation
�4 in_Proposition 2.2.1_does not reduce to_the same_derivation_with ∆3_and_so ∆3 does_not generate a_looping reduction sequence. As_for the view_of_Tennant (2015, p. 589), the derivation ∆6 in Proposition 2.2.2 of the Liar paradox_which_uses CR−rule_appears_to_be in normal form. Likewise,_∆4 appears to be a_closed normal_derivation of ⊥ and it does not initiate_the_non-terminating reduction sequence._These are the same phenomena of what he calls_'the classical rub.' However, ∆4 without_the application of_⊵Sub−reduction_does_not use any classical_inferences but raises the problem of_undergeneration. Therefore, with the assumption that_the Liar paradox is a genuine one,_if ⊵Sub−reduction is not proper, ∆4_can be the counterexample to_TCPE which_shows that TCPE undegenerates. That_is, the rejection_of the_application of classical_inferences or CR−rule is unable to_be a solution_to the problem of undergeneration. 53_Of_course, Tennant may_disallow_the_view that_⊵Sub−reduction is not_proper._To claim_that_⊵Sub−reduction is proper, he should explain_which_reduction process is proper and which process_is not, but he_has_never explained about it.9_Moreover, even without taking issue_with ⊵Sub−reduction procedure, a counterexample to_TCPE can_be presented_that causes the problem of undergeneration. Tennant (2017, pp. 109–110) has_preferred to use generalized elimination rules_for four reasons: the_uniform presentation,_the_efficiency of proof_search,_making shorter_formal proofs, and affording a solution to_the problem_of overgeneration. The fourth reason is_related to our topic._Fourth,..._using Elimination rules in [generalized] form_affords a solution to certain problems_that would otherwise arise for_the_proof-theoretic_criterion of paradoxicality.... (Tennant, 2017,_p. 110) When Tennant (2016) and_Tennant (2017, Ch._11) deal with the problem of overgeneration_occurred_by Ekman's paradox, instead of accusing_Ekman_reduction ⊵E of the problem, he_provides_a_solution that all elimination rules_are stated in generalized form.10 Though_he believes that the choice of the form of_generalized elimination rules_can solve the overgeneration problem,_it_also_provides a case that causes the undergeneration