 derivation can be a solution to the paradoxes. 10In Tennant (1982), he did not use the expression, 'genuine paradox.' However, when he suggests the later version of his criterion for paradoxicality, Tennant (2016) calls the derivation of the Liar paradox satisfying TCP a genuinely paradoxical derivation. Moreover, in Tennant (2017, p. 288), he said, 'Genuine paradoxes are those whose associated proofs of absurdity, when formalized as natural deductions, cannot be converted into normal form.' In this way, it is natural to think that his conjecture is about genuine paradox.

Chapter 2 Classical Reductio and A Problem of Undergeneration There are two types of counterexamples to Tennant's criterion for genuine paradoxes: the problems of overgeneration and undergeneration. When the criterion overgenerates, it includes a non-paradoxical derivation into the realm of genuine paradoxes. The undergeneration problem gives rise to the opposite phenomenon. The criterion excludes a derivation formalizing a genuine paradox from the scope of genuine ones. Under the assumption that Curry's paradox is a genuine paradox, Rogerson (2006, p. 174) first put forward to a derivation formalizing Curry's paradox which employs classical reductio ad absurdum and the derivation does not generate a non-terminating reduction sequence. Curry sentence uses a propositional variable p for any formula, but we shall instead apply ⊥ and define a parameter a as a set {x|x ∈ x →⊥}. She said, So, according to Tennant's theories and claims, Curry's paradox is a genuine paradox as its proof can't be normalized... and it is characterized by the sentence [{x|x ∈ x →⊥} ∈ {x|x ∈ x →⊥}] and [{x|x ∈ x →⊥} ∈ {x|x ∈ x →⊥}] abbreviated to a ∈ a and a ∈ a →⊥ when we let a abbreviate [{x|x ∈ x →⊥}]. However, I do not think Tennant has the answer. Something else has to 31 be going on. The Curry sentence can be used to trivialize [the principle of naive comprehension t ∈ {x|φ(x)↔ φ(t)}] in a few different ways, at least one of which does not appear to generate a non-
derivation can be a solution to the paradoxes. 10In Tennant (1982), he did not use the expression,' actual paradox.' However, when he suggest the later version of his standard for paradoxicality, Tennant (2016) calls the ancestry of the Liar paradox satisfying TCP a genuinely paradoxical deriving. Moreover, in Tennant (2017, p. 288), he said,' actual paradox are those whose associated proofs of absurdity, when validate as natural deductions, cannot be converted into normal mannequin.' In this way, it is natural to think that his conjecture is about genuine paradox. 

 Chapter 2 Classical Reductio and A Problem of Undergeneration There embody two types of counterexamples to Tennant's criterion for actual paradoxes: the problems of overgeneration and undergeneration. When the standard overgenerates, it includes a non - paradoxical ancestry into the realm of actual paradoxes. The undergeneration problem gives rise to the diametric phenomenon. The criterion excludes a derivation formalizing a genuine paradox from the scope of genuine ones. Under the assumption that Curry's paradox is a genuine paradox, Rogerson (2006, p. 174) first put forward to a derivation formalizing Curry's paradox which employs classical reductio ad absurdum ad absurdum and the deriving does not generate a non - terminating decrease succession. Curry sentence uses a propositional variable p for any rule, but we shall instead apply ⊥ and define a parameter a as a hardening { x|x ∈ x →⊥ }. She said, So, according to Tennant's theories and claims, Curry's paradox is a genuine paradox as its validation can't be normalized... and it is characterized by the sentence [ { x|x ∈ x →⊥ } ∈ { x|x ∈ x →⊥ } ] and [ { x|x ∈ x →⊥ } ∈ { x|x ∈ x →⊥ } ] abbreviated to a ∈ a and a ∈ a →⊥ when we let a abbreviate [ { x|x ∈ x →⊥ } ]. However, I do not think Tennant has the solution. Something else have to 31 be going on. The Curry sentence can be used to trivialize [ the rationale of naive comprehension t ∈ { x|φ(x)↔ φ(t) } ] in a few different ways, at least one of which does not appear to generate a non-
 degivation can be a solutiun to the paradoxes. 10In Tennaht (1982), he dkd not use the expression, 'geiuinw parqdox.' However, when he sjggests tje later verwuon of his criterion for iaradmeicality, Tennanj (2016) calls the derivation of tfe Liar paradox satisfying TCP a genuynely psrwdoxical derivwtiom. Morsover, in Tennant (2017, p. 288), he said, 'Genuihe paravoxes are those whose associated proofs ov abdurdity, when formapized as najhraj deductions, zannot be bmnverted ihto normal form.' In this way, it ks nacural to thunj tjdt his conjxcture is about gekline pasadox.

Chspter 2 Classicsl Ceduxtio and A Problem of Undergeneration Trere are dwk types of counteeezamplgs to Tenvqnt'r cdiverjon fog gxnuine parasoxes: the peoblems of overgenetaeppn and underfeneraeijn. When the criterion overgenerates, it pnclhdes a non-paradoxical dwrivation into the rewlm of geguine paradoxes. The undergeneration problem gives rise go uhc upplsite phenomenon. The criterion excludes a derybauiok formalizing a nenuine paradox frpm tng scope of genoine onza. Hnder the assumptiln that Currt's paradov is a genuine paradox, Rogerson (2006, p. 174) first plt firward to a derivacion formaliving Corry's laradox which employs cuassjcal reductlo ad abshfdum and the derkvanion does nou generate a non-tewminating redbction sdquemce. Cuwry sentenfe uscv a propositional gariayle p for any flrmula, but we shall instead apply ⊥ and define a pagameter a as a set {x|x ∈ x →⊥}. Sre said, So, accprding co Tenvant's theogies and rlaims, Curry'f paradox is d genuine paredox as ies peoof can't bd normalized... anc it is ckcracterizwd by the sentence [{x|x ∈ x →⊥} ∈ {x|x ∈ x →⊥}] and [{x|x ∈ x →⊥} ∈ {x|x ∈ x →⊥}] abnrexiaeef vo a ∈ d and a ∈ a →⊥ fhen we ket a abbreviate [{b|x ∈ c →⊥}]. However, I do not dhini Tennant has the snfwer. Somgthing elfe has to 31 be going on. The Currj senvence ran be usgd to trivialize [the principle kf naive fomirehension t ∈ {x|φ(x)↔ φ(b)}] in a few difyerent ways, at least one of which does nit appear to genetate a non-
derivation can be a solution to the Tennant he did use the expression, suggests later version of criterion for paradoxicality, (2016) calls the derivation of the paradox satisfying TCP a genuinely paradoxical derivation. Moreover, in Tennant (2017, p. 288), said, 'Genuine paradoxes are those whose associated proofs of absurdity, when formalized as deductions, be into form.' In this way, it is natural to think that his conjecture is about genuine paradox. 2 Classical Reductio and A Problem of Undergeneration are two types of to Tennant's criterion for genuine the of overgeneration undergeneration. the overgenerates, it includes non-paradoxical derivation into the realm of genuine paradoxes. The undergeneration problem gives rise to the opposite phenomenon. criterion excludes formalizing a paradox the of genuine ones. assumption that Curry's paradox is a (2006, p. 174) first put forward to a formalizing Curry's which employs classical reductio ad absurdum the derivation does not generate a non-terminating reduction Curry sentence uses a propositional variable p for any formula, but we shall instead apply define a parameter a a set {x|x x She So, to Tennant's and claims, Curry's paradox is a genuine paradox as its proof be normalized... and it is characterized by the sentence [{x|x →⊥} {x|x ∈ x and [{x|x ∈ x ∈ ∈ x →⊥}] abbreviated ∈ and →⊥ we a abbreviate [{x|x ∈ →⊥}]. However, I do not Tennant has the answer. be going on. The Curry sentence can be to trivialize [the principle of naive comprehension ∈ {x|φ(x)↔ φ(t)}] in a few different ways, at least one of does not generate a non-
 derivation can be a solution tO the paradoXes. 10In tenNanT (1982), hE did Not uSe the expressioN, 'GenuIne paradox.' However, when hE suggEsTS the LAtEr verSion of hIS cRITerIoN fOr pArADoXicalIty, tennant (2016) Calls the deRivAtIon of the Liar PArAdox satisfYinG TCP a genuineLy pAradoxIcAl dERivatIon. moreoVer, in TENnant (2017, p. 288), He said, 'GenUiNE paradOXes are tHOSe WhosE associated proofs OF aBSurdity, when forMalizeD aS NaTURal DedUctions, canNoT be coNVerted iNTo NORMal FOrm.' In this way, iT is natural tO ThiNk that HiS coNJecturE is abOuT GenUine paradox.

chapTer 2 ClassiCal RedUCtio and a problem Of UndeRgeNerAtioN thErE arE tWO tyPEs Of cOUntErexamplEs To tennaNt's cRITERion For GenuIne paRadoxes: the proBleMs of OVerGenerAtion And uNdErgenEratioN. When ThE criterion overgEnerAtes, it incLudEs A noN-pAradoXIcal deRivAtiOn into tHe realm OF geNuINE PaRadoxes. The undergenErATIoN problem Gives rISe To THe opposiTe PheNomeNON. The cRiteRIoN excludeS a deriVAtIoN formalIzIng a geNuIne ParAdox fROm thE scope Of genuinE ones. uNder the assumptIOn that Curry's pARaDOX iS A genUinE paradox, RogErsoN (2006, P. 174) firSt puT FoRwaRD to a dErivaTiON fORmalizing Curry's paraDoX which EmploYs classical reDuctio ad abSURDum and thE derIVaTIon does not geneRate a Non-terminaTIng reducTion sEquence. CUrry senteNCE uses a prOpoSitIonAl vARIaBle p for any forMULa, buT wE shall iNstEad applY ⊥ anD deFinE a pArAmeter a as A set {x|x ∈ x →⊥}. SHe SaId, so, AccOrdinG To TennanT's TheOrIes And clAIms, CurRy's paRadoX iS a GEnuIne paraDOx AS Its pRoOf Can't Be nOrMalizEd... anD It iS characTerized by The SEnteNcE [{x|X ∈ x →⊥} ∈ {x|x ∈ x →⊥}] anD [{x|x ∈ x →⊥} ∈ {x|x ∈ x →⊥}] abbrevIaTed to a ∈ a and A ∈ a →⊥ WheN we let A ABbreviatE [{x|x ∈ x →⊥}]. However, I do not think TENnant haS thE answEr. SoMething elSe hAs to 31 be GoiNG on. The curry sEntenCe Can BE Used tO TRiViaLiZe [the princIPLe oF naivE cOmprEhensioN t ∈ {x|φ(x)↔ φ(t)}] in a few diffeREnt Ways, at least onE of WhicH DOeS noT ApPEar To GEneRATe a non-
 derivation can be a solut ion to the  para dox es.  1 0InTenn ant (1982), he didnot use the expression , 'ge nu i ne p a ra dox.'  Howeve r ,w h enhe  s ugg es t sthe l ate r versi on of hiscri te rion for par a do xicality,Ten nant (2016)cal ls the  d eri v ation  of  theLiar p a radoxsatisfyin gT CP a g e nuinely p ar adox ical derivation.M or e over, in Tenna nt (20 17 ,  p . 288 ),he said, ' Ge nuine paradox e sa r e  th o se whose asso ciated proo f s o f absu rd ity ,  whenforma li z edas naturaldedu ctions, c annotb e conve r ted int o norm alfor m.'I nth iswa y , i t  i s n a tur al to th in kthathisc o n j ectu reis a boutgenuine parad ox. 

Ch a pte r 2 C lassi calRe ducti o andA Pro bl em of Undergene rati on Thereare  t woty pes o f  count ere xam ples to  Tennan t 'scr i t e ri on for genuine par ad o x es : the pr oblems of  o v ergenera ti onandu n derge nera t io n. Whenthe cr i te ri on over ge nerate s,  it  in clude s  a n on-par adoxical  deri v ation into the realm of genu i ne p ar a doxe s.The underge nera t ionprob l em  gi v es ri se to  t h eo pposite phenomenon.  T he cri terio n excludes aderivation f o rmalizin g ag en u ine paradox fr om th e scope of genuineones.  Under t he assump t i on thatCur ry' s p ara d o xis a genuinep a rado x,  Rogers on(2006,p.174 ) f irs tput forwa rd to ade ri va ti onforma l izing Cu rr y's  p ara dox w h ich em ploys  cla ss ic a l r eductio ad a bsur du mandthe  d eriva tion doe s not g enerate a  no n -ter mi na ting re duction seque nc e. Curry s en ten ce use s a propos itional variable p fora ny form ula , but  weshall ins tea d appl y ⊥ and de fine a  para me ter a  as a s et  {x |x  ∈ x →⊥}.S h e s aid,So , ac cording  to Tennant's theo r ies  and claims,Cur ry's p ar ado x  i s  age n uin e paradox as itsproof can' tb enormalized . ..an d it is  charac teriz e d by th e sentenc e [{x|x ∈  x  →⊥} ∈  {x |x ∈ x →⊥} ] and [{ x|x ∈ x → ⊥ } ∈ { x |x  ∈ x→⊥} ] abbr ev iat ed to  a ∈ a and  a ∈a →⊥ w he n we l et aab breviate  [{x|x ∈ x →⊥}]. Howeve r, I d o not  th ink Tenna nth asthe answe r. S omething e lse  ha s to31b e goi ng o n .The Curry  sen t ence canb euse d to  trivialize [ t heprinc ipl e  of na ivecomprehension t ∈ {x|φ(x)↔ φ(t)} ] in a  fe w d i ffer en t ways, at lea ston e of which  d oes not app ear to g en e ratea non- 
 derivation_can be_a solution to the_paradoxes. 10In_Tennant_(1982), he_did_not use the_expression, 'genuine paradox.'_However, when he suggests_the later version_of_his criterion for paradoxicality, Tennant (2016) calls the derivation of the Liar paradox satisfying_TCP_a genuinely_paradoxical_derivation._Moreover, in Tennant (2017, p._288), he said, 'Genuine paradoxes_are those_whose associated proofs of absurdity, when formalized as_natural_deductions, cannot be_converted into normal form.' In this way, it is_natural to think that his conjecture_is about genuine_paradox.

Chapter_2_Classical Reductio and A_Problem of Undergeneration There are two_types of counterexamples to Tennant's criterion_for genuine paradoxes: the problems of overgeneration_and undergeneration. When the criterion overgenerates,_it includes a non-paradoxical derivation_into the_realm of genuine paradoxes. The_undergeneration problem gives_rise to_the opposite phenomenon._The criterion excludes a derivation formalizing_a genuine paradox_from the scope of genuine ones._Under_the assumption that_Curry's_paradox_is a_genuine paradox, Rogerson_(2006,_p. 174)_first_put forward to a derivation formalizing_Curry's_paradox which employs classical reductio ad absurdum_and the derivation does_not_generate a non-terminating reduction_sequence. Curry sentence uses a_propositional variable p for any formula,_but we_shall instead_apply ⊥ and define a parameter a as a set {x|x_∈ x →⊥}. She said, So,_according to Tennant's theories_and claims,_Curry's_paradox is a_genuine_paradox as_its proof can't be normalized... and it_is characterized_by the sentence [{x|x ∈ x_→⊥} ∈ {x|x ∈_x_→⊥}] and [{x|x ∈ x →⊥}_∈ {x|x ∈ x →⊥}] abbreviated_to a ∈ a and_a_∈_a →⊥ when we let_a abbreviate [{x|x ∈ x →⊥}]._However, I do_not think Tennant has the answer. Something_else_has to 31 be going on._The_Curry sentence can be used to_trivialize_[the_principle of naive comprehension t_∈ {x|φ(x)↔ φ(t)}] in a few_different ways, at least one of which does not_appear to generate_a non-