 just as seriously as it does the classicist. Therefore, attempted solutions to the paradoxes, if they are to be genuine solutions, must be available to the intuitionist. Nothing about an attempted solution to a paradox should imply that the trouble it reveals lies with strictly clssical moves of reasoning. (Tennant, 2015, p. 589) It is true that we do not need CR−rule to formulate many of (self-referential) paradoxes in natural deduction. His methodological conjecture seems to be true. However, independently of his methodological conjecture, there is room for discussion on whether only classical inferences cause the trouble that he mentions. Furthermore, he does not seriously 8With respect to the derivation ∆6 of Proposition 2.2.2, ∆6 appears to be in normal form but whether it is in normal form is dependent on how we deal with the axiom ⌜Φ⌝ = ⌜¬T (⌜Φ⌝)⌝. If the axiom were derived by the application of = I−rule, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ in = E−rule would be a maximum formula. Then, ∆6 would be a non-normal derivation. If we deal with the axiom in the same manner of assumptions, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ would not be a maximum formula, and so it would be in normal form. For Tennant said that his derivation appears to be in normal form, we regard ∆6 as a seeming normal derivation. 51 consider the problem of undegeneration. His expected answer against the undergeneration problem is the rejection of the use of CR−rule. However, we will see in the next section that there are cases generating the problem of undergeneration without the application of CR−rule. 2.3 The Undergeneration Problem without CR−Rule. At the beginning of this chapter, the non-normal derivation Σ2 of Curry's paradox using CR−rule is proposed and it fails to generate a looping reduction sequence. Also, in the last section, we have the seeming normal derivation ∆6 of the Liar paradox without generating an infinite reduction sequence. It is the Rogerson-type counterexample to TCPE because it raises the problem of undergeneration. As we have seen
just as seriously as it does the classicist. consequently, try solutions to the paradoxes, if they are to be actual solutions, must be available to the intuitionist. Nothing about an undertake solution to a paradox should imply that the trouble it uncover lies with strictly clssical move of reasoning. (Tennant, 2015, p. 589) It is true that we do not need CR−rule to give voice many of (self - referential) paradoxes in natural discount. His methodological conjecture seems to be true. However, independently of his methodological conjecture, there is room for discussion on whether merely classical inferences cause the worry that he mentions. Furthermore, he does not seriously 8With respect to the derivation ∆6 of Proposition 2.2.2, ∆6 look to be in normal form but whether it is in normal form is dependent on how we deal with the axiom ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝. If the axiom were derive by the application of = I−rule, ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ in = E−rule would be a maximum formula. Then, ∆6 would be a non - normal derivation. If we deal with the maxim in the like manner of assumption, ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ would not be a maximum formula, and so it would be in normal form. For Tennant said that his derivation appear to be in normal kind, we regard ∆6 as a seeming normal ancestry. 51 think the problem of undegeneration. His expected answer against the undergeneration problem is the rejection of the habit of CR−rule. However, we will see in the next section that there are case generating the problem of undergeneration without the application of CR−rule. 2.3 The Undergeneration Problem without CR−Rule. At the beginning of this chapter, the non - normal derivation Σ2 of Curry's paradox use CR−rule is proposed and it fails to generate a looping decrease sequence. besides, in the last section, we have the seeming normal derivation ∆6 of the Liar paradox without generating an countless reduction sequence. It is the Rogerson - type counterexample to TCPE because it raises the problem of undergeneration. As we have seen
 judt as seriously as it dots the classicist. Therefmre, atfempted rolutions to the paradoxes, ih thwy art to be genuine soljtions, mudt be avqilaule to the intuivjonist. Kjthihn abobt an attempted xolution tm a paradox shmuud imply that the trouble it reveals jies wiyh strictly clssycal iovea of reasoning. (Tennant, 2015, p. 589) It is tdue thau we do not need CT−rule to formulate many of (selv-referential) paradlxes in natodal eeduction. Hir methodological conjedture seems to be true. However, kndepzndently of hus lgthodologicao contecture, therc is romm for ciscussion on ehevher only classical inferxnces cause the troufle that ve mentions. Furthernoee, he doev nog sefiohsky 8With ges'ect to the derivation ∆6 of Proposition 2.2.2, ∆6 sp[vsrs to be in normaj sorm but whether it is in normal form iv dspendent on how we deal with the axiom ⌜Φ⌝ = ⌜¬T (⌜Φ⌝)⌝. Lf the axyom were derived by the application of = I−rule, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ in = X−rjle wjjod be a maximum formula. Then, ∆6 would be a non-norizl dvrivation. If we dcal with the axiom ij yre same mannet of assumltions, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ would nlt be a maxinum formuja, amd so it would be in normal form. For Teunabt said that his dzrivation ap'ears jo be on normal form, we regarb ∆6 as z seeming nlrmal derjxation. 51 consider thv prmblem of undegeneration. Hif expectev anscer agaivst jhe undqrgeneratiln problem is the rejectlon oy the use of CR−gule. However, we will see in the next section jhad tvere are casex generating ehe problem of undergzneratkon withoun the appnication of CR−rule. 2.3 The Gjdergeneratimn Problqm wuthoyt CR−Ruud. At the beginming of tkns chaptee, the non-normal devivatkkn Σ2 of Curry's pcxaeox using CR−rulr ir pwoiosxd anq it fails to gendragr a luoping redugtiun srquence. Also, in the nast section, we have tne seeming normal qerivation ∆6 og the Liar paradox withmut genetatyng an infinite reduction sequsnce. It id tme Rogerson-ty[e ciunterexamplz to TCPE because it raises the problem if undergeneration. As we have seen
just as seriously as it does the attempted to the if they are be to the intuitionist. about an attempted to a paradox should imply that trouble it reveals lies with strictly clssical moves of reasoning. (Tennant, 2015, p. It is true that we do not need CR−rule to formulate many of paradoxes natural His conjecture seems to be true. However, independently of his methodological conjecture, there is room for discussion whether only classical inferences cause the trouble that mentions. Furthermore, he does seriously 8With respect to the ∆6 Proposition 2.2.2, appears be normal form but it is in normal form is dependent on how we deal with the axiom ⌜Φ⌝ = ⌜¬T If the derived by application = ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ E−rule would be a maximum formula. be a non-normal derivation. If we deal with axiom in same manner of assumptions, ⌜Φ⌝= ⌜¬T would not be a maximum formula, and so would be in normal form. For Tennant said that his derivation appears to be in we regard ∆6 as seeming normal derivation. consider problem undegeneration. expected answer the undergeneration problem is the rejection of the use of CR−rule. we will see in the next section that there are the of undergeneration without application of CR−rule. 2.3 Undergeneration without CR−Rule. At the this the of paradox CR−rule is proposed and fails to generate a looping sequence. Also, in the seeming normal derivation ∆6 of the Liar paradox generating an infinite reduction sequence. It is Rogerson-type counterexample to TCPE because it raises the problem of undergeneration. As have seen
 just as seriously as it does thE classicisT. TherEfoRe, aTtEmptEd soLutions to the paRAdoxEs, if they are to be genuine SolutIoNS, musT Be AvailAble to tHE iNTUitIoNiSt. NOtHInG abouT an AttemptEd solution To a PaRadox should iMPlY that the trOubLe it reveals lIes With stRiCtlY ClssiCal Moves Of reasONing. (TeNnant, 2015, p. 589) It iS tRUe that WE do not nEED Cr−rulE to formulate many oF (SeLF-referential) paRadoxeS iN NaTURal DedUction. His mEtHodolOGical coNJeCTURe sEEms to be true. HoWever, indepeNDenTly of hIs MetHOdologIcal cOnJEctUre, there is rOom fOr discussIon on wHEther onLY classiCal infEreNceS cauSE tHe TroUbLE thAT hE meNTioNs. FurtheRmOrE, he doEs noT SERIousLy 8WIth rEspecT to the derivatIon ∆6 Of PrOPosItion 2.2.2, ∆6 AppeaRs to Be In norMal forM but wHeTher it is in normaL forM is dependEnt On How We Deal wITh the aXioM ⌜Φ⌝ = ⌜¬T (⌜φ⌝)⌝. If the aXiom werE DerIvED BY tHe application of = I−ruLe, ⌜φ⌝= ⌜¬t (⌜φ⌝)⌝ iN = E−rule woUld be a MAxImUM formula. thEn, ∆6 wOuld BE A non-nOrmaL DeRivation. if we deAL wItH the axiOm In the sAmE maNneR of asSUmptIons, ⌜Φ⌝= ⌜¬T (⌜φ⌝)⌝ would noT be a mAXimum formula, anD So it would be in NOrMAL fORm. FoR TeNnant said thAt hiS DeriVatiON aPpeARs to bE in noRmAL fORm, we regard ∆6 as a seeminG nOrmal dErivaTion. 51 consider tHe problem oF UNDegeneraTion. hIs EXpected answer aGainsT the undergENeration ProblEm is the rEjection oF THe use of Cr−ruLe. HOweVer, WE WiLl see in the nexT SEctiOn That theRe aRe cases GenEraTinG thE pRoblem of uNdergeneRaTiOn WiThoUt the APplicatiOn Of Cr−rUle. 2.3 the UnDErgeneRatioN ProBlEm WIthOut CR−RuLE. AT THe beGiNnIng oF thIs ChaptEr, thE Non-Normal dErivation σ2 of cUrry'S pArAdox usiNg CR−rule is proPoSed and it faIlS to GeneraTE A looping Reduction sequence. Also, in THe last sEctIon, we Have The seeminG noRmal deRivATion ∆6 of The LiaR paraDoX wiTHOut geNERaTinG aN infinite rEDUctIon seQuEnce. it is the rogerson-type counteRExaMple to TCPE becAusE it rAISeS thE PrOBleM oF UndERGeneration. As we hAve seen
 just as seriously as it d oes the cl assic ist . T he refo re,attempted solu t ions  to the paradoxes, iftheyar e  tob egenui ne solu t io n s , m us tbeav a il abletothe int uitionist.  No th ing about an at tempted so lut ion to a par ado x shou ld  im p ly th atthe t rouble it rev eals lies  w i th str i ctly cl s s ic al m oves of reasoning .  ( T ennant, 2015,p. 589 )I ti s  tr uethat we do  n ot ne e d CR−ru l et o for m ulate many of  (self-refe r ent ial) p ar ado x es innatur al ded uction. His  met hodologic al con j ectures eems to  be tr ue.  Ho weve r ,in dep en d ent l yofh ismethodol og ic al co njec t u r e , th ere  isroomfor discussio n o n wh e the r onl y cla ssic al  infe rences  caus ethe trouble tha t he  mentions . F ur the rm ore,h e does  no t s eriousl y 8With res pe c t to  the derivation ∆6  o f Pr opositio n 2.2. 2 ,∆6 appearsto  be  inn o rmalform bu t whethe r it i s  i nnormalfo rm isde pen den t onh ow w e deal  with th e axi o m ⌜Φ⌝ = ⌜¬T (⌜ Φ ⌝)⌝. If the a x io m we r e de riv ed by the a ppli c atio n of =I−r u le, ⌜ Φ⌝= ⌜ ¬T (⌜ Φ ⌝)⌝ in = E−rule wou ld  be amaxim um formula. T hen, ∆6 wo u l d  be a no n-no r ma l  derivation. I f wedeal witht he axiom  in t he samemanner of a ssumptio ns,  ⌜Φ ⌝=⌜¬T ( ⌜Φ ⌝)⌝ would not b e ama ximum f orm ula, an d s o i t w oul dbe in nor mal form .Fo rTe nna nt sa i d that h is  de ri vat ion a p pearsto be  inno rm a l f orm, we re g a rd ∆ 6as  a s eem in g nor mald eri vation.  51 consi der thepr ob lem ofundegeneratio n.  His expec te d a nswera g ainst th e undergeneration probl e m is th e r eject ionof the us e o f CR−r ule .  Howev er, we  will  s eei n  then e xt  se ct ion that t h e reare c as es g enerati ng the problem ofu nde rgeneration w ith outt h eapp l ic a tio no f C R − rule. 2.3 The U ndergenera ti o nProblem wi t hou tCR−Rule . At th e beg i nning o f this ch apter, th enon- n o rma l derivati on Σ2 of  Curry'sp arado x  u singCR− rule i spro posed  and i t  fa ils t o gene ra te a l oopin greductio n sequence. Also, in th e last  sect ion , we have  th e  se eming nor malderivation  ∆6  of  theLia r  para doxw it hou t  gene rati n g an infi n it e r e d uc tion sequen c e .  It  is t heR ogerso n-ty pe counterexample to TCPE becaus e it r ais est he p ro blem of underg ene ra t i on. As w ehave seen
 just_as seriously_as it does the_classicist. Therefore,_attempted_solutions to_the_paradoxes, if they_are to be_genuine solutions, must be_available to the_intuitionist._Nothing about an attempted solution to a paradox should imply that the trouble it_reveals_lies with_strictly_clssical_moves of reasoning. (Tennant, 2015,_p. 589) It is true_that we_do not need CR−rule to formulate many of_(self-referential)_paradoxes in natural_deduction. His methodological conjecture seems to be true. However,_independently of his methodological conjecture, there_is room for_discussion_on_whether only classical inferences_cause the trouble that he mentions._Furthermore, he does not seriously 8With_respect to the derivation ∆6 of Proposition_2.2.2, ∆6 appears to be in_normal form but whether it_is in_normal form is dependent on_how we deal_with the_axiom ⌜Φ⌝ =_⌜¬T (⌜Φ⌝)⌝. If the axiom were_derived by the_application of = I−rule, ⌜Φ⌝= ⌜¬T_(⌜Φ⌝)⌝_in = E−rule_would_be_a maximum_formula. Then, ∆6_would_be a_non-normal_derivation. If we deal with the_axiom_in the same manner of assumptions, ⌜Φ⌝=_⌜¬T (⌜Φ⌝)⌝ would not_be_a maximum formula, and_so it would be in_normal form. For Tennant said that_his derivation_appears to_be in normal form, we regard ∆6 as a seeming normal_derivation. 51 consider the problem of_undegeneration. His expected answer_against the_undergeneration_problem is the_rejection_of the_use of CR−rule. However, we will see_in the_next section that there are cases_generating the problem of_undergeneration_without the application of CR−rule. 2.3_The Undergeneration Problem without CR−Rule. At_the beginning of this chapter,_the_non-normal_derivation Σ2 of Curry's paradox_using CR−rule is proposed and it_fails to generate_a looping reduction sequence. Also, in the_last_section, we have the seeming normal_derivation_∆6 of the Liar paradox without_generating_an_infinite reduction sequence. It is_the Rogerson-type counterexample to TCPE because_it raises the problem of undergeneration. As we have_seen