 genuine paradox is a derivation of an unacceptable conclusion which employs a certain form of id est inferences and generates an infinite reduction sequence. In the next section, we will see 23 Tennant's criterion for paradoxicality and the requirement of a normal derivation as a plausible solution to the paradoxes. 1.3 Tennant's Criterion for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND) Prawitz (1965) firstly investigates that a derivation of an absurdity from the set-theoretic paradox falls into a non-terminating reduction sequence. Though he did not explicitly mention that the non-terminating reduction sequence is the distinguishing feature of the paradoxes, it is often said that he would do so. For instance, Schroeder-Heister and Tranchini (2017, p. 568) said, 'Prawitz proposed [the non-terminating reduction sequence] to be the distinguished feature of Russell's paradox.' Further to Prawitz, Tennant (1982) suggests the proof-theoretic criterion for paradoxicality and it has been developed by Tennant (1995, 2016, 2017). Let us take the natural deduction system SN for the naive set theory in the same manner of Prawitz (1965, Appendix B) which only contains the rules for ∧, →, ¬, and the following additional rules: D φ[t/x] ∈ I t ∈ {x|φ(x)} t ∈ {x|φ(x)} ∈ E φ[t/x] ∈ −rules have the following standard reduction process. D φ[t/x] ∈ I t ∈ {x|φ(x)} ∈ E φ[t/x] ▷∈ D φ[t/x] Let us define a parameter a as {x|¬x ∈ x}. Then, an application of ∈ I−rule to ¬a ∈ a derives a ∈ a and an application of ∈ E−rule to a ∈ a derives ¬a ∈ a. Prawitz (1965, p. 95) investigates that a derivation of ⊥ from Russell's paradox cannot be transformed into a 24 normal derivation because it raises an infinite reduction sequence. Proposition 1.3.1. Let us define a parameter a as {x|¬x ∈ x}. Then, there is a closed derivation of ⊥ in SN which generates a non
genuine paradox is a derivation of an unacceptable conclusion which use a sealed form of i d est inference and generates an countless reduction sequence. In the following section, we will see 23 Tennant's standard for paradoxicality and the necessity of a normal derivation as a plausible solution to the paradox. 1.3 Tennant's Criterion for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND) Prawitz (1965) firstly investigates that a derivation of an absurdity from the set - theoretical paradox falls into a non - terminating reduction succession. Though he did not explicitly mention that the non - terminating decrease succession is the distinguishing feature of the paradoxes, it is often read that he would do so. For instance, Schroeder - Heister and Tranchini (2017, p. 568) said,' Prawitz propose [ the non - terminating reduction sequence ] to be the distinguished feature of Russell's paradox.' Further to Prawitz, Tennant (1982) suggests the proof - theoretical criterion for paradoxicality and it has been developed by Tennant (1995, 2016, 2017). Let us take the natural deduction arrangement SN for the naive set theory in the same manner of Prawitz (1965, Appendix B) which only contains the rules for ∧, →, ¬, and the following additional rule: D φ[t / x ] ∈ I t ∈ { x|φ(x) } t ∈ { x|φ(x) } ∈ E φ[t / x ] ∈ −rules receive the follow standard reduction process. D φ[t / x ] ∈ I t ∈ { x|φ(x) } ∈ E φ[t / x ] ▷∈ vitamin d φ[t / x ] Let us define a parameter a as { x|¬x ∈ x }. Then, an application of ∈ I−rule to ¬a ∈ a derive a ∈ a and an application of ∈ E−rule to a ∈ a derives ¬a ∈ a. Prawitz (1965, p. 95) investigates that a derivation of ⊥ from Russell's paradox cannot be transformed into a 24 normal derivation because it promote an infinite reduction sequence. Proposition 1.3.1. lease us define a argument a as { x|¬x ∈ x }. Then, there constitute a closed derivation of ⊥ in SN which generates a non
 gejuine paradox is a derivxtion of an unaexeptabne condlusion dhich employs a certain form od id tft inferences and eenerates an infibite eeduction sequence. In ths nexc wection, we wilk see 23 Tentant's criteriot wox paradoxicality and the requirement of a nprlal derivation as s plahsible solution to the paradoxes. 1.3 Fennant'v Criterion fpr Paradoxicality (TCP) and hhe Gequirement of a Nlrmal Derivqtiog (RND) Prawitz (1965) firstly puvestigates that a derivation of an absurdkty fxom the set-jkworfjic paradox halls pnto a non-terminating reductoon sequence. Tmough he did not explicitly mxntion that the non-tgrminating rzduction sequence is rhw disjinguhshivt fdathrx or the oaredoxes, it ia often saie that he would do xo. Dor instance, Achroeqew-Heister and Tranchini (2017, p. 568) said, 'Prawitz prkposed [the non-terminatibg reduction sequence] to be thq distinguished feature of Russell's paradox.' Furthar to Oraclbz, Tdbnwnt (1982) suggests the proof-theoretic criterion fow psrsdoxicality anb it has been defepolgd by Tennant (1995, 2016, 2017). Let us take the natural feductijn sywtem SN fjr tne naive set theory in the wame manner jd Prawitz (1965, Appendir B) which onuy cpntaims the rules for ∧, →, ¬, and the following wdditionam rules: D φ[t/x] ∈ I t ∈ {x|φ(q)} t ∈ {x|φ(x)} ∈ E φ[t/x] ∈ −rules have the sollowing staudard reauctoon prjcess. D φ[t/x] ∈ I t ∈ {x|φ(x)} ∈ E φ[t/x] ▷∈ D φ[t/x] Leh us befina a parameher a as {x|¬x ∈ x}. Then, an applicatmpn of ∈ I−rule tm ¬a ∈ a derires a ∈ a and an appjication of ∈ E−tule to a ∈ a ddrives ¬a ∈ z. Prawivz (1965, p. 95) inveseigates that d derivation mf ⊥ from Ruswell'w paradub cannot be trsnsformed into a 24 nirmal derivation bccausg jt raises an inynnute reduction srqudncq. Iro'ositymn 1.3.1. Let us dafind a laramdter a as {x|¬w ∈ b}. Thrn, there is a closed derjvation of ⊥ in SN ehlch generqtes a njn
genuine paradox is a derivation of an which a certain of id est reduction In the next we will see Tennant's criterion for paradoxicality and the of a normal derivation as a plausible solution to the paradoxes. 1.3 Tennant's for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND) Prawitz (1965) investigates a of absurdity from the set-theoretic paradox falls into a non-terminating reduction sequence. Though he did not explicitly that the non-terminating reduction sequence is the distinguishing of the paradoxes, it often said that he would so. instance, Schroeder-Heister Tranchini p. said, 'Prawitz proposed non-terminating reduction sequence] to be the distinguished feature of Russell's paradox.' Further to Prawitz, Tennant (1982) suggests proof-theoretic criterion and it been by (1995, 2016, 2017). take the natural deduction system SN set theory in the same manner of Prawitz Appendix B) only contains the rules for ∧, ¬, and the following additional rules: D φ[t/x] I t ∈ {x|φ(x)} t ∈ {x|φ(x)} ∈ E φ[t/x] ∈ −rules have the following process. D φ[t/x] ∈ t ∈ {x|φ(x)} E ▷∈ φ[t/x] us define parameter a as {x|¬x ∈ x}. Then, an application of ∈ to ¬a ∈ a derives a ∈ a and an ∈ to a ∈ derives ¬a ∈ a. (1965, 95) investigates that a ⊥ Russell's transformed a normal derivation because it an infinite reduction sequence. Proposition Let us define a x}. Then, there is a closed derivation of in SN which generates a non
 genuine paradox is a derivatiOn of an unacCeptaBle ConClUsioN whiCh employs a certAIn foRm of id est inferences and GenerAtES an iNFiNite rEductioN SeQUEncE. IN tHe nExT SeCtion, We wIll see 23 TEnnant's criTerIoN for paradoxiCAlIty and the rEquIrement of a noRmaL derivAtIon AS a plaUsiBle soLution TO the paRadoxes. 1.3 TeNnANt's CriTErion foR pArAdoxIcality (TCP) and the REQuIRement of a NormaL DerivAtIOn (rnd) PrAwiTz (1965) firstly iNvEstigATes that A DeRIVAtiON of an absurditY from the set-THeoRetic pArAdoX Falls iNto a nOn-TErmInating reduCtioN sequence. though HE did not EXplicitLy mentIon ThaT the NOn-TeRmiNaTIng REdUctIOn sEquence iS tHe DistiNguiSHING feaTurE of tHe parAdoxes, it is oftEn sAid tHAt hE woulD do so. for iNsTance, schroeDer-HeIsTer and Tranchini (2017, P. 568) saiD, 'Prawitz pRopOsEd [tHe Non-teRMinatiNg rEduCtion seQuence] tO Be tHe DISTiNguished feature of RUsSELl'S paradox.' furtheR To prAWitz, TennAnT (1982) suGgesTS The prOof-tHEoRetic criTerion FOr PaRadoxicAlIty and It Has BeeN deveLOped By TennAnt (1995, 2016, 2017). Let us Take tHE natural deductIOn system SN for THe NAIvE Set tHeoRy in the same MannER of PRawiTZ (1965, APpeNDix B) wHich oNlY CoNTains the rules for ∧, →, ¬, and ThE folloWing aDditional ruleS: D φ[t/x] ∈ I t ∈ {x|φ(x)} T ∈ {X|Φ(X)} ∈ E φ[t/x] ∈ −rulEs haVE tHE following stanDard rEduction prOCess. D φ[t/x] ∈ i t ∈ {x|φ(x)} ∈ e φ[t/x] ▷∈ D φ[t/x] let us defiNE A parametEr a As {x|¬X ∈ x}. THen, AN ApPlication of ∈ I−rULE to ¬a ∈ A dErives a ∈ A anD an applIcaTioN of ∈ e−ruLe To a ∈ a derivEs ¬a ∈ a. PrawItZ (1965, p. 95) InVeStiGates THat a deriVaTioN oF ⊥ frOm RusSEll's paRadox CannOt Be TRanSformed INtO A 24 NormAl DeRivaTioN bEcausE it rAIseS an infiNite reducTioN SequEnCe. proposiTion 1.3.1. Let us defiNe A parameter A aS {x|¬x ∈ X}. Then, tHERe is a cloSed derivation of ⊥ in SN whicH GeneratEs a Non
 genuine paradox is a deri vation ofan un acc ept ab le c oncl usion which em p loys  a certain form of idest i nf e renc e sand g enerate s  a n inf in it e r ed u ct ion s equ ence. I n the next  se ct ion, we will se e 23 Tenna nt' s criterionfor  parad ox ica l ity a ndthe r equire m ent of  a normal  d e rivati o n as ap l au sibl e solution to the pa r adoxes. 1.3 Te nnant' sC ri t e rio n f or Paradox ic ality (TCP) a n dt h e  Re q uirement of a  Normal Der i vat ion (R ND ) P r awitz(1965 )f irs tly investi gate s that aderiva t ion ofa n absur dity f rom  th e se t -t he ore ti c  pa r ad oxf all s into a  n on -term inat i n g redu cti on s equen ce. Though he  di d no t  ex plici tly m enti on  that  the n on-te rm inating reducti on s equence i s t he  di st ingui s hing f eat ure  of the  parado x es,  i t i soften said that he  w o u ld  do so.For in s ta nc e , Schroe de r-H eist e r  andTran c hi ni (2017 , p. 5 6 8)  s aid, 'P ra witz p ro pos ed[then on-t ermina ting red uctio n  sequence] tob e the disting u is h e df eatu reof Russell' s pa r adox .' F u rt her to Pr awitz ,T en n ant (1982) suggests  t he pro of-th eoretic crite rion for p a r a doxicali ty a n di t has been dev elope d by Tenna n t (1995,  2016 , 2017).  Let us t a k e the na tur alded uct i o nsystem SN for t he n ai ve setthe ory inthe  sa meman ne r of Praw itz (196 5,  A pp en dix  B) w h ich only  c ont ai nsthe r u les fo r ∧,→, ¬ ,an d  th e follo w in g addi ti on al r ule s:  D φ[ t/x] ∈ I  t ∈ {x |φ(x)} t∈ { x |φ(x )}  ∈  E φ[t/ x] ∈ −rules h av e the foll ow ing  stand a r d reduct ion process. D φ[t/x] ∈ I t ∈ { x|φ (x)}∈ Eφ[t/x] ▷∈  Dφ[t/x]  Le t  us de fine a  para me ter a  as { x | ¬x  ∈x} . Then, an a ppl icati on  of∈ I−rul e to ¬a ∈ a derive s  a∈ a and an ap pli cati o n  o f ∈ E− r ule  t o  a∈ a derives ¬a ∈a. Prawitz  ( 1 96 5, p. 95)i nve st igatesthat aderiv a tion of  ⊥ from R ussell'spa rado x can not be tra nsformed  into a 2 4  norm a lderiv ati on bec au seit ra ises a n  in finit e redu ct ion se quenc e.  Proposi tion 1.3.1. Let us defi ne a p arame ter  a as {x| ¬x∈  x} . Then, t here  is a clos edder ivati ono f ⊥ i n SN wh ich gener ates a non
 genuine_paradox is_a derivation of an_unacceptable conclusion_which_employs a_certain_form of id_est inferences and_generates an infinite reduction_sequence. In the_next_section, we will see 23 Tennant's criterion for paradoxicality and the requirement of a_normal_derivation as_a_plausible_solution to the paradoxes. 1.3_Tennant's Criterion for Paradoxicality (TCP)_and the_Requirement of a Normal Derivation (RND) Prawitz (1965)_firstly_investigates that a_derivation of an absurdity from the set-theoretic paradox falls_into a non-terminating reduction sequence. Though_he did not_explicitly_mention_that the non-terminating reduction_sequence is the distinguishing feature of_the paradoxes, it is often said_that he would do so. For instance,_Schroeder-Heister and Tranchini (2017, p. 568)_said, 'Prawitz proposed [the non-terminating_reduction sequence]_to be the distinguished feature_of Russell's paradox.'_Further to_Prawitz, Tennant (1982)_suggests the proof-theoretic criterion for paradoxicality_and it has_been developed by Tennant (1995, 2016,_2017)._Let us take_the_natural_deduction system_SN for the_naive_set theory_in_the same manner of Prawitz (1965,_Appendix_B) which only contains the rules for_∧, →, ¬, and_the_following additional rules: D_φ[t/x] ∈ I t ∈_{x|φ(x)} t ∈ {x|φ(x)} ∈ E_φ[t/x] ∈_−rules have_the following standard reduction process. D φ[t/x] ∈ I t ∈_{x|φ(x)} ∈ E φ[t/x] ▷∈ D_φ[t/x] Let us define_a parameter_a_as {x|¬x ∈_x}._Then, an_application of ∈ I−rule to ¬a ∈_a derives_a ∈ a and an application_of ∈ E−rule to_a_∈ a derives ¬a ∈ a._Prawitz (1965, p. 95) investigates that_a derivation of ⊥ from_Russell's_paradox_cannot be transformed into a_24 normal derivation because it raises_an infinite reduction_sequence. Proposition 1.3.1. Let us define a_parameter_a as {x|¬x ∈ x}. Then,_there_is a closed derivation of ⊥_in_SN_which generates a non