E,4⊥ ¬I,7¬∃y(Zy ∈ y) The result states that no set contains its own Rusell subset and is an acceptable conclusion in a consistent Zermelo's set theory. The result, i.e. Σ5, does not represent a proof of Russell-Zermelo's paradox, however, Σ5 contains a subderivation Σ3 which satisfies TCPL and so is a T-paradox. As Σ5 does not formulate a paradox, it should neither be a genuine paradox nor a T-paradox. Thus, Crabbé's case shows that TCPL overgenerates. Schroeder-Heister and Tranchini (2018, Sec. 8) say that Triviality test cannot solve the problem of overgeneration caused by Crabbé's case, as they said, As remarked, the phenomenon observed by Crabbé is however unaffected by our proposed constraint on reductions, thus showing that further work is required for a thorough analysis of paradoxes along the lines of the Prawitz126 Tennant analysis. It looks as if Crabbé reduction causes a non-terminating reduction sequence, the form of Crabbé reduction, ▷∈Z, relies on the form of ∈Z I−rule. Since ∈Z I−rule has the premise ¬Zb ∈ Zb whose degree is greater than that of the conclusion Zb ∈ {x ∈ b|¬x ∈ x}, the reduced derivation has the formula ¬Zb ∈ Zb by eliminating the maximum formula Zb ∈ {x ∈ b|¬x ∈ x}. Therefore, the real problem may not be Crabbé reduction itself but be the form of ∈Z I−rule. Even if Crabbé reduction is the real matter, we can solve the problem of overgeneration by applying Translation test with the requirements of a proper reduction. The translated forms of ∈Z I− and ∈Z E−rules are as follows. D′1 Γ ⇒ t ∈ s D′2 ∆ ⇒ φ[t/x] ∈Z R Γ,∆ ⇒ t ∈ {x ∈ s|φ(x)} D′3 t ∈ s,φ[t/x],Σ ⇒ ψ ∈Z Lt ∈ {x ∈ s|φ(x)},�
E,4⊥ ¬I,7¬∃y(Zy ∈ y) The result states that no set contains its own Rusell subset and is an satisfactory ending in a consistent Zermelo's set hypothesis. The result, i.e. Σ5, does not map a proof of Russell - Zermelo's paradox, however, Σ5 contains a subderivation Σ3 which satisfy TCPL and so is a T - paradox. As Σ5 does not formulate a paradox, it should neither be a actual paradox nor a T - paradox. therefore, Crabbé's case shows that TCPL overgenerates. Schroeder - Heister and Tranchini (2018, Sec. 8) read that Triviality test cannot solve the trouble of overgeneration caused by Crabbé's case, as they suppose, As remarked, the phenomenon observed by Crabbé is however insensible by our proposed constraint on reductions, therefore showing that further work is necessitate for a thorough psychoanalysis of paradoxes along the line of the Prawitz126 Tennant analysis. It looks as if Crabbé reduction cause a non - terminating reduction sequence, the form of Crabbé reduction, ▷∈Z, relies on the form of ∈Z I−rule. Since ∈Z I−rule has the premise ¬Zb ∈ Zb whose degree is greater than that of the conclusion Zb ∈ { x ∈ b|¬x ∈ x }, the reduced derivation has the formula ¬Zb ∈ Zb by eliminating the maximum formula Zb ∈ { x ∈ b|¬x ∈ x }. consequently, the actual problem may not be Crabbé decrease itself but be the human body of ∈Z I−rule. Even if Crabbé decrease is the real matter, we can solve the problem of overgeneration by applying Translation test with the requirements of a proper decrease. The translated forms of ∈Z I− and ∈Z E−rules are as follows. D′1 Γ ⇒ t ∈ s D′2 ∆ ⇒ φ[t / x ] ∈Z R Γ,∆ ⇒ t ∈ { x ∈ s|φ(x) } D′3 t ∈ s, φ[t / x],Σ ⇒ ψ ∈Z Lt ∈ { x ∈ s|φ(x) }, �
E,4⊥ ¬I,7¬∃j(Zy ∈ y) The result states that no set courains mts own Rusell rubset and is an acceptable roncousiob in a consistent Zermdlo's set nheory. Thw rewylt, i.e. Σ5, doxa not rc'resehb a pxoif of Russell-Zgrmelo's paragox, however, Σ5 cmnganns a subderivation Σ3 which satisfies TCPL amd so is a T-paraqox. Sf Σ5 dkvs not formulate a paradox, it sgould ntither be a genuinr paradox nor a T-paradox. Tjus, Frabbé's case shows hhat TCPL ocergqberates. Schrueder-Heisttr and Trancgini (2018, Sec. 8) say that Triviality gest eannot solvg fhf problem of overdeneration caused by Crabbé's case, as they xaiv, As remarked, the phenomeion observed by Crabfé is howeeex unaffected by our peoposed convtrakbt un devucfions, hhua showing fhat furthee work is required gow a thorough ahalysif jf paradoxes along the lines of the Prafita126 Tennant analysis. It lioks as if Crabbé reduftion caufes a non-terminating reduction sequence, the form mf Crebcé rtdmgtiov, ▷∈Z, relies on the form of ∈Z I−rule. Since ∈Z I−rule rzs tme premise ¬Zb ∈ Zn whose degree is brfaygr than that ow the ekndlusion Zb ∈ {x ∈ b|¬x ∈ x}, the teducee derivatyon nas the formula ¬Zb ∈ Zb by eoiminating tke naximum formula Zb ∈ {x ∈ b|¬x ∈ x}. Cherefpre, tne real problem may not be Drabbé reduchion itsemw but be the foro on ∈Z I−rule. Even if Crabbé reduceion is tie recl mattef, we can sjlve the pgoblem of overgeneration by akplyinc Translatlon test with the requirements of a proper recuwtimn. The txanslabed forms of ∈Z Y− and ∈Z E−rules are as folluws. D′1 Γ ⇒ t ∈ s D′2 ∆ ⇒ φ[v/x] ∈Z R Γ,∆ ⇒ t ∈ {v ∈ s|φ(x)} D′3 t ∈ s,φ[t/f],Σ ⇒ ψ ∈Z Lt ∈ {x ∈ v|φ(x)},�
E,4⊥ ¬I,7¬∃y(Zy ∈ y) The result states set its own subset and is consistent set theory. The i.e. Σ5, does represent a proof of Russell-Zermelo's paradox, Σ5 contains a subderivation Σ3 which satisfies TCPL and so is a T-paradox. Σ5 does not formulate a paradox, it should neither be a genuine paradox a Thus, case that TCPL overgenerates. Schroeder-Heister and Tranchini (2018, Sec. 8) say that Triviality test cannot solve the of overgeneration caused by Crabbé's case, as they As remarked, the phenomenon by Crabbé is however unaffected our constraint on thus that work is required a thorough analysis of paradoxes along the lines of the Prawitz126 Tennant analysis. It looks as if reduction causes reduction sequence, form Crabbé ▷∈Z, relies on of ∈Z I−rule. Since ∈Z I−rule ¬Zb ∈ Zb whose degree is greater than of the Zb ∈ {x ∈ b|¬x ∈ the reduced derivation has the formula ¬Zb ∈ by eliminating the maximum formula Zb ∈ {x ∈ b|¬x ∈ x}. Therefore, the real not be Crabbé reduction but be the of I−rule. if reduction is real matter, we can solve the problem of overgeneration by applying test with the requirements of a proper reduction. The translated ∈Z and ∈Z E−rules as follows. D′1 Γ t s D′2 ∆ ⇒ R ⇒ ∈ D′3 ∈ s,φ[t/x],Σ ⇒ ψ Lt ∈ {x ∈ s|φ(x)},�
E,4⊥ ¬I,7¬∃y(Zy ∈ y) The result states that No set contaIns itS owN RuSeLl suBset And is an acceptaBLe coNclusion in a consistent ZErmelO's SEt thEOrY. The rEsult, i.e. σ5, DoES Not RePrEseNt A PrOof of rusSell-ZerMelo's paradOx, hOwEver, Σ5 containS A sUbderivatiOn Σ3 Which satisfiEs TcPL and So Is a t-ParadOx. AS Σ5 doeS not foRMulate A paradox, iT sHOuld neITher be a GENuIne pAradox nor a T-paradoX. thUS, Crabbé's case shOws thaT TcpL OVErgEneRates. SchroEdEr-HeiSTer and TRAnCHINi (2018, SEC. 8) say that TriviAlity test caNNot Solve tHe ProBLem of oVergeNeRAtiOn caused by CRabbÉ's case, as tHey saiD, as remarKEd, the phEnomenOn oBseRved BY CRaBbé Is HOweVEr UnaFFecTed by our PrOpOsed cOnstRAINT on rEduCtioNs, thuS showing that fUrtHer wORk iS requIred fOr a tHoRough AnalysIs of pArAdoxes along the lInes Of the PrawItz126 teNnaNt AnalySIs. It loOks As iF Crabbé ReductiON caUsES A NoN-terminating reductIoN SEqUence, the Form of cRaBbÉ ReductioN, ▷∈Z, RelIes oN THe forM of ∈Z i−RuLe. Since ∈Z i−rule hAS tHe Premise ¬zb ∈ zb whosE dEgrEe iS greaTEr thAn that Of the conClusiON Zb ∈ {x ∈ b|¬x ∈ x}, the reduCEd derivation hAS tHE FoRMula ¬zb ∈ ZB by eliminatIng tHE maxImum FOrMulA zb ∈ {x ∈ b|¬x ∈ X}. TherEfORe, THe real problem may not Be crabbé ReducTion itself but Be the form oF ∈z i−Rule. Even If CrABbÉ Reduction is the Real mAtter, we can SOlve the pRobleM of overgEneration BY Applying traNslAtiOn tEST wIth the requireMENts oF a Proper rEduCtion. ThE trAnsLatEd fOrMs of ∈Z I− and ∈z E−rules aRe As FoLlOws. d′1 Γ ⇒ t ∈ s D′2 ∆ ⇒ Φ[T/x] ∈Z R Γ,∆ ⇒ t ∈ {x ∈ s|Φ(x)} d′3 t ∈ s,Φ[t/X],Σ ⇒ ψ ∈z Lt ∈ {x ∈ s|Φ(X)},�
E,4⊥ ¬I,7¬∃y(Zy ∈ y) The r esult stat es th atnose t co ntai ns its own Rus e ll s ubset and is an accept ableco n clus i on  in a  consis t en t Zer me lo 'sse t  t heory . T he resu lt, i.e. Σ 5,do es not repre s en t a proofofRussell-Zerm elo 's par ad ox, howev er,  Σ5 c ontain s  a sub derivatio nΣ 3 whic h  satisf i e sTCPL  and so is a T-pa r ad o x. As Σ5 doesnot fo rm u la t e  apar adox, it s ho uld n e ither b e  a g e nui n e paradox nor  a T-parado x . T hus, C ra bbé ' s case  show st hat  TCPL overg ener ates. Sch roeder - Heister and Tra nchini  (2 018 , Se c .8)  sa yt hat Tr ivi a lit y test c an no t sol ve t h e p robl emof o verge neration caus edby C r abb é's c ase,as t he y sai d, Asremar ke d, the phenomen on o bserved b y C ra bbé  i s how e ver un aff ect ed by o ur prop o sed  c o n s tr aint on reductions ,t h us  showing  thatf ur th e r work i sreq uire d for a  tho r ou gh analy sis of pa ra doxes a lo ng the  l ine s o f the Praw itz126  Tennant  anal y sis. It looksa s if Crabbé r e du c t io n  cau ses  a non-term inat i ng r educ t io n s e quenc e, th ef or m  of Crabbé reductio n,  ▷∈Z,relie s on the form  of ∈Z I−r u l e . Since∈Z I − ru l e has the prem ise ¬ Zb ∈ Zb wh o se degre e isgreaterthan that o f the co ncl usi onZb∈ {x  ∈ b|¬x ∈ x}, t he r ed uced de riv ation h asthe  fo rmu la  ¬Zb ∈ Zb  by elim in at in gthe  maxi m um formu la  Zb  ∈  {x  ∈ b| ¬ x ∈ x} . The refo re ,t hereal pr o bl e m  may  n ot  beCra bb é red ucti o n i tself b ut be the  fo r m of  ∈ ZI−rule.  Even if Crab bé  reduction  i s t he rea l matter,we can solve the proble m  of ove rge nerat ionby applyi ngTransl ati o n test  withthe r eq uir e m entso f  a  pr op er reducti o n . T he tr an slat ed form s of ∈Z I− and ∈ZE −ru les are as fo llo ws.D ′ 1Γ ⇒ t∈  sD′ 2  ∆⇒ φ[t/x] ∈Z R Γ,∆  ⇒ t ∈ {x∈s |φ (x)} D′3 t ∈ s ,φ [t/x],Σ  ⇒ ψ ∈Z  Lt ∈ {x ∈ s| φ(x)},�
E,4⊥ ¬I,7¬∃y(Zy_∈ y)_The result states that_no set_contains_its own_Rusell_subset and is_an acceptable conclusion_in a consistent Zermelo's_set theory. The_result,_i.e. Σ5, does not represent a proof of Russell-Zermelo's paradox, however, Σ5 contains a_subderivation_Σ3 which_satisfies_TCPL_and so is a T-paradox._As Σ5 does not formulate_a paradox,_it should neither be a genuine paradox nor_a_T-paradox. Thus, Crabbé's_case shows that TCPL overgenerates. Schroeder-Heister and Tranchini (2018,_Sec. 8) say that Triviality test_cannot solve the_problem_of_overgeneration caused by Crabbé's_case, as they said, As remarked,_the phenomenon observed by Crabbé is_however unaffected by our proposed constraint on_reductions, thus showing that further work_is required for a thorough_analysis of_paradoxes along the lines of_the Prawitz126 Tennant_analysis. It_looks as if_Crabbé reduction causes a non-terminating reduction_sequence, the form_of Crabbé reduction, ▷∈Z, relies on_the_form of ∈Z_I−rule._Since_∈Z I−rule_has the premise_¬Zb_∈ Zb_whose_degree is greater than that of_the_conclusion Zb ∈ {x ∈ b|¬x ∈_x}, the reduced derivation_has_the formula ¬Zb ∈_Zb by eliminating the maximum_formula Zb ∈ {x ∈ b|¬x_∈ x}._Therefore, the_real problem may not be Crabbé reduction itself but be the_form of ∈Z I−rule. Even if_Crabbé reduction is the_real matter,_we_can solve the_problem_of overgeneration_by applying Translation test with the requirements_of a_proper reduction. The translated forms of_∈Z I− and ∈Z_E−rules_are as follows. D′1 Γ ⇒_t ∈ s D′2 ∆ ⇒_φ[t/x] ∈Z R Γ,∆ ⇒_t_∈_{x ∈ s|φ(x)} D′3 t_∈ s,φ[t/x],Σ ⇒ ψ ∈Z Lt_∈ {x ∈_s|φ(x)},�