 in the last section, Tennant thinks that the derivation has two problems: (i) the derivation appears to be in normal form, i.e. a normal derivation of ⊥ exists, (ii) the derivation does not generate a non-terminating reduction sequence in spite of the fact that it formalizes a genuine paradox, such as the Liar paradox. He believes that these problems are only caused by the classical inference. From the observation that it is not necessary to use classical inferences to formalizes genuine paradoxes in natural deduction, he put forward to the methodological conjecture that genuine paradoxes are never strictly classical. Although he did not seriously consider the problem of undergeneration, from his view, we can think that his solution to the undergeneration problem is not to use any classical inferences. However, it is too hasty to think so. When we take the problem of undergeneration more seriously, we can see that the problem is not solely caused by the application of classical inferences. We shall argue in this section that the problem of undergeneration is not solved by simply accusing classical inferences, especially CR−rule, of two troubles above, due to the fact that there are counterexamples to TCPE which does not use any classical inference but does raise the undergeneration problem. At first, Tennant ignores the fact that a non-terminating reduction sequence can be oc52 curred by an auxiliary reduction procedure. In the next chapter, we will see Ekman's paradox which shows that TCPE overgenerates in the sense that TCPE makes a non-paradoxical derivation paradoxical. Ekman's paradox uses a special auxiliary reduction procedure called Ekman reduction procedure which has the main role to generate a non-terminating reduction sequence. [ψ → φ] [φ → ψ] D φ → E ψ → E φ ⊵E D φ Ekman reduction ⊵E is an auxiliary reduction procedure and it has a similar form with the auxiliary reduction procedure for the substitutivity of identity. t2 = t1 t1 = t2 D φ(t1) = E φ(t2) = E φ(t1) ⊵Sub D φ(t1) Similar to Ekman's paradox, the derivation of the Liar paradox in Tennant (2015, pp. 585– 588), i.e. Proposition 2.2.1, should use ⊵Sub−reduction. If ⊵Sub−reduction is not available, �
in the last section, Tennant thinks that the derivation take two trouble: (i) the derivation appears to be in normal shape, i.e. a normal derivation of ⊥ exists, (ii) the derivation does not beget a non - terminating decrease sequence in spite of the fact that it formalizes a genuine paradox, such as the Liar paradox. He believes that these trouble are only caused by the authoritative inference. From the observation that it is not necessary to use authoritative inferences to formalize actual paradoxes in natural deduction, he invest forward to the methodological conjecture that actual paradoxes are never rigorously classical. Although he did not seriously consider the problem of undergeneration, from his view, we can think that his solution to the undergeneration trouble is not to use any classical inferences. However, it is too hasty to think so. When we drive the problem of undergeneration more seriously, we can see that the problem is not solely caused by the application of classical inferences. We shall argue in this section that the trouble of undergeneration is not solve by simply charge authoritative inference, especially CR−rule, of two troubles above, due to the fact that there are counterexamples to TCPE which does not use any classical inference but does raise the undergeneration problem. At foremost, Tennant ignores the fact that a non - terminating reduction sequence can be oc52 curred by an auxiliary reduction procedure. In the next chapter, we will see Ekman's paradox which shows that TCPE overgenerates in the sense that TCPE makes a non - paradoxical derivation paradoxical. Ekman's paradox use a extra auxiliary reduction procedure called Ekman reduction operation which has the main role to render a non - terminating reduction sequence. [ ψ → φ ] [ φ → ψ ] D φ → E ψ → einsteinium φ ⊵E D φ Ekman reduction ⊵E is an accessory reduction procedure and it has a similar phase with the auxiliary decrease procedure for the substitutivity of identity. t2 = t1 t1 = t2 D φ(t1) = E φ(t2) = east φ(t1) ⊵Sub five hundred φ(t1) Similar to Ekman's paradox, the derivation of the Liar paradox in Tennant (2015, pp. 585 – 588), i.e. Proposition 2.2.1, should use ⊵Sub−reduction. If ⊵Sub−reduction is not available, �
 in the last section, Tennanu thinks that the derivavion haa two prublems: (i) the derivation appeers ro be in normal form, i.e. a nurmal derpvation od ⊥ eeists, (ii) the dermbation does nkb genzrete a non-terminsting reduwtion sequence iv dpite of the fact that it formalizef a geniije paradox, sucr as ehe Mpav paradox. He believes that these probleks are only csused by the classical inffrenfe. From the observwtion that ut if not necessafy to use bnassical ihferences to formalizes genuine paraboxes in najbeal geduction, hx put sorward to tmv methogologicsl conjecture bhat jenuune paradoxes are nevxr strictly classicaj. Althougv ke did not seriously xobsidet the procoem of uidedgenerwtikn, from hia view, we cqn think that his spltnoon to the uhdergegewation problem is not to use any classibal jnferences. However, it iw too hasty to think do. When wq take the problem of undergeneration more seriouvly, wx zan sce tfqt the problem is not solely caused by the applydauiok of classical ikferences. We shall aggig in this sectkon thcf fhe problem of undfrgenerwtion is not sjlvec by simply accusing classixal inferencvs, ewpecially CR−rule, oy two troublzs abofe, dur to the fact that therz are dounterexamoles to TDOE which does nog uxe any classical inference btt does reise che undefgengration problem. Ah first, Tennant ignores hhe fcct tvat a non-tfrminating reduction sequence cei be oc52 currec ty dn auxilnary rcduction procedtre. In the nexj chapter, we wkll see Ekjan's pacadox which fhows that TC[G overgeneratxs in the senwe tyat TCPD makes a non-patadoxical derivatiob paradoxical. Ekmak's patasox uses a specncl auxiliary reduvtiun [rlcxdure walled Ekman redjctkpn prucedure whigh fas yhe main role to genarats a non-terminating rcduction wequence. [ψ → φ] [φ → ψ] D φ → E ψ → E φ ⊵E D φ Ekman refuctimn ⊵X is am aoxiliary reduction procedure ans it has w slmilar form wyth bhe wuxiliary xeduction procedure for the substitutivivy of identity. t2 = t1 t1 = r2 D φ(t1) = E φ(t2) = E φ(t1) ⊵Sub C φ(t1) Similar to Ehman's parddox, the derivation if the Liar paradpx in Tennant (2015, pp. 585– 588), i.s. Propmsitiln 2.2.1, should use ⊵Sub−reduction. If ⊵Sub−reduction is not available, �
in the last section, Tennant thinks that has problems: (i) derivation appears to a derivation of ⊥ (ii) the derivation not generate a non-terminating reduction sequence spite of the fact that it formalizes a genuine paradox, such as the paradox. He believes that these problems are only caused by the classical inference. the that is necessary to use classical inferences to formalizes genuine paradoxes in natural deduction, he put forward to methodological conjecture that genuine paradoxes are never strictly Although he did not consider the problem of undergeneration, his we can that solution the undergeneration problem not to use any classical inferences. However, it is too hasty to think so. When we take problem of seriously, we see the is not solely the application of classical inferences. We this section that the problem of undergeneration is solved by accusing classical inferences, especially CR−rule, of troubles above, due to the fact that there counterexamples to TCPE which does not use any classical inference but does raise the undergeneration first, Tennant ignores the that a non-terminating sequence be curred an auxiliary procedure. In the next chapter, we will see Ekman's paradox which that TCPE overgenerates in the sense that TCPE makes a paradoxical. paradox uses a auxiliary reduction procedure called reduction which has the main generate non-terminating → [φ ψ] D φ → ψ → E φ ⊵E φ Ekman reduction ⊵E and it has a similar form with the reduction procedure for the substitutivity of identity. = t1 t1 = t2 D φ(t1) = E φ(t2) = E ⊵Sub D to Ekman's paradox, the derivation of the Liar in Tennant (2015, pp. 588), i.e. Proposition 2.2.1, should use ⊵Sub−reduction. If ⊵Sub−reduction not �
 in the last section, Tennant thInks that thE deriVatIon HaS two ProbLems: (i) the derivaTIon aPpears to be in normal form, I.e. a noRmAL derIVaTion oF ⊥ exists, (II) tHE DerIvAtIon DoES nOt genEraTe a non-tErminating RedUcTion sequence IN sPite of the fAct That it formalIzeS a genuInE paRAdox, sUch As the liar paRAdox. He Believes tHaT These pROblems aRE OnLy caUsed by the classicaL InFErence. From the oBservaTiON tHAT it Is nOt necessarY tO use cLAssical INfERENceS To formalizes gEnuine paradOXes In natuRaL deDUction, He put FoRWarD to the methoDoloGical conjEcture THat genuINe paradOxes arE neVer StriCTlY cLasSiCAl. ALThOugH He dId not serIoUsLy conSideR THE ProbLem Of unDergeNeration, from hIs vIew, wE Can Think That hIs soLuTion tO the unDergeNeRation problem is Not tO use any clAssIcAl iNfErencES. HowevEr, iT is Too hastY to thinK So. WHeN WE TaKe the problem of undeRgENErAtion morE serioUSlY, wE Can see thAt The ProbLEM is noT solELy Caused by The appLIcAtIon of clAsSical iNfEreNceS. We shALl arGue in tHis sectiOn thaT The problem of unDErgeneration iS NoT SOlVEd by SimPly accusing ClasSIcal InfeREnCes, ESpeciAlly Cr−rULe, OF two troubles above, duE tO the faCt thaT there are counTerexampleS TO tCPE whicH doeS NoT Use any classicaL infeRence but doES raise thE undeRgeneratIon probleM. aT first, TeNnaNt iGnoRes THE fAct that a non-teRMInatInG reductIon SequencE caN be Oc52 cUrrEd By an auxilIary reduCtIoN pRoCedUre. In THe next chApTer, We WilL see EKMan's paRadox WhicH sHoWS thAt TCPE oVErGENeraTeS iN the SenSe That TcPE mAKes A non-parAdoxical dEriVAtioN pArAdoxicaL. Ekman's paradoX uSes a speciaL aUxiLiary rEDUction prOcedure called Ekman reducTIon procEduRe whiCh haS the main rOle To geneRatE A non-teRminatIng reDuCtiON SequeNCE. [ψ → Φ] [φ → ψ] d φ → e ψ → E φ ⊵E D φ EkmAN RedUctioN ⊵E Is an AuxiliaRy reduction proceduRE anD it has a similaR foRm wiTH ThE auXIlIAry ReDUctION procedure for thE substitutIvITy Of identity. T2 = T1 t1 = t2 d φ(T1) = E φ(t2) = E φ(t1) ⊵sub D φ(t1) SImilaR To Ekman'S paradox, tHe derivatIoN of tHE liaR paradox in tennant (2015, pP. 585– 588), i.e. ProposITion 2.2.1, sHOuLd use ⊵sub−ReductIoN. If ⊵sub−reDuctioN Is nOt avaIlable, �
 in the last section, Tenn ant thinks  that  th e d er ivat ionhas two proble m s: ( i) the derivation appe ars t ob e in no rmalform, i . e. a  no rm al  de ri v at ion o f ⊥  exists , (ii) the  de ri vation doesn ot  generatea n on-terminati ngreduct io n s e quenc e i n spi te oft he fac t that it  f o rmaliz e s a gen u i ne  par adox, such as the Li a r paradox. Hebeliev es th a t  th ese  problemsar e onl y  caused by t h e c l assical infer ence. Fromt heobserv at ion that i t isno t  ne cessary touseclassical  infer e nces to formali zes ge nui nepara d ox es  in  n a tur a lded u cti on, he p ut  f orwar d to t h e  met hod olog icalconjecture th atgenu i neparad oxesarene ver s trictl y cla ss ical. Althoughhe d id not se rio us lyco nside r  the p rob lem  of und ergener a tio n, f r om  his view, we canth i n kthat his  solut i on  t o  the und er gen erat i o n pro blem is  not touse an y  c la ssicalin ferenc es . H owe ver,i t is  too h asty tothink so. When we ta k e the problem of u nd e rgen era tion more s erio u sly,  wec an  se e  that  thepr o bl e m is not solely cau se d by t he ap plication ofclassicali n f erences.  Wes ha l l argue in thi s sec tion thatt he probl em of  underge nerationi s  not sol ved  by  si mpl y ac cusing classi c a l in fe rences,  es peciall y C R−r ule , o ftwo troub les abov e,  d ue  t o t he fa c t that t he rear e c ounte r exampl es to  TCP Ewh i chdoes no t  u s e  any  c la ssic alin feren ce b u t d oes rai se the un der g ener at io n probl em. At first,  T ennant ign or esthe fa c t  that anon-terminating reducti o n seque nce  canbe o c52 curre d b y an a uxi l iary r educti on pr oc edu r e . Int h enex tchapter, w e wil l see  E kman 's para dox which shows th a t T CPE overgener ate s in t he  se n se tha tT CPE m akes a non-para doxical de ri v at ion parado x ica l.  Ekman' s parad ox us e s a spe cial auxi liary red uc tion p roc edure call ed Ekman  reductio n  proc e du re wh ich  has t he  ma in ro le tog ene ratea non- te rminat ing r ed uction s equence. [ψ → φ] [φ → ψ ] D φ→ E ψ  →E φ ⊵E Dφ E k man  reductio n ⊵E  is an aux ili ary  redu cti o n pro cedu r eand it ha s as imilar fo r mwit h th e auxiliary r e duc tionpro c edureforthe substitutivit y  of identity.t2 = t 1 t 1 = t2 D  φ (t1) = E φ(t2)  =Eφ ( t1) ⊵Sub  D  φ(t1) Simi lar to E km a n's p aradox , thederivat i o no f theLiar  pa radox inTen na n t (2015 ,pp .  585–588) ,i.e. P roposi t ion2 . 2.1, should use⊵Sub− r e ducti o n.If ⊵S ub −reduct i on i s not avai lable, �
 in_the last_section, Tennant thinks that_the derivation_has_two problems:_(i)_the derivation appears_to be in_normal form, i.e. a_normal derivation of_⊥_exists, (ii) the derivation does not generate a non-terminating reduction sequence in spite of_the_fact that_it_formalizes_a genuine paradox, such as_the Liar paradox. He believes_that these_problems are only caused by the classical inference._From_the observation that_it is not necessary to use classical inferences to_formalizes genuine paradoxes in natural deduction,_he put forward_to_the_methodological conjecture that genuine_paradoxes are never strictly classical. Although_he did not seriously consider the_problem of undergeneration, from his view, we_can think that his solution to_the undergeneration problem is not_to use_any classical inferences. However, it_is too hasty_to think_so. When we_take the problem of undergeneration more_seriously, we can_see that the problem is not_solely_caused by the_application_of_classical inferences._We shall argue_in_this section_that_the problem of undergeneration is not_solved_by simply accusing classical inferences, especially CR−rule,_of two troubles above,_due_to the fact that_there are counterexamples to TCPE_which does not use any classical_inference but_does raise_the undergeneration problem. At first, Tennant ignores the fact that a_non-terminating reduction sequence can be oc52_curred by an auxiliary_reduction procedure._In_the next chapter,_we_will see_Ekman's paradox which shows that TCPE overgenerates_in the_sense that TCPE makes a non-paradoxical_derivation paradoxical. Ekman's paradox_uses_a special auxiliary reduction procedure called_Ekman reduction procedure which has the_main role to generate a_non-terminating_reduction_sequence. [ψ → φ] [φ_→ ψ] D φ → E_ψ → E_φ ⊵E D φ Ekman reduction ⊵E_is_an auxiliary reduction procedure and it_has_a similar form with the auxiliary_reduction_procedure_for the substitutivity of identity._t2 = t1 t1 = t2_D φ(t1) = E φ(t2) = E φ(t1) ⊵Sub_D φ(t1) Similar_to Ekman's paradox, the derivation_of_the_Liar paradox in Tennant (2015, pp. 585– 588), i.e. Proposition_2.2.1, should_use ⊵Sub−reduction. If_⊵Sub−reduction is not available, �