2 [⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ [φ →¬φ]4, [¬φ → φ]4 D2 φ [⊥]6 ¬E,6⊥ Since the major premise ¬φ of ¬E−rule is derived by → I−rule and so is not an assumption, ¬−reduction process produces the same derivation of D3. Therefore, D5 generates a non-terminating reduction sequence which is not normalizable. We call the result of Proposition 3.2.3 GEkman paradox. Then, GEkman paradox shows that TCPL overgenerates the scope of genuine paradoxes if GEkman reduction is allowed. Again, TCPL has a counterexample. Likewise, if GEkman reduction is accepted, the derivation D4 of ¬(φ ↔¬φ) in Proposition 3.2.2 cannot be converted into a full normal derivation. Tennant does not seem to take GEkman reduction into account. As long as GEkman reduction is used, there is an open derivation of ⊥ from [φ →¬φ] and [¬φ → φ] and a closed derivation of ¬(φ ↔¬φ) in SEG which raise infinite reduction sequences. Therefore, our mere choice of generalized forms of elimination rules does not solve the problem of Ekman's paradox because a looping reduction is relative to the set of reduction procedures we accepted. The real issue is to be which set of proper reductions we choose and Tennant did not consider it. Tennant may answer to the problem of GEkman paradox that if we apply permutation conversion, found by Gentzen (2008) and Prawitz (1965) for ∨E− and ∃E−rules, before using GEkman reduction, we would have an open full normal derivation of ⊥ from [φ →¬φ] and [¬φ → φ] because GEkman reduction is not applicable. Permutation conversion is an essential process to eliminate the major premise derived by an elimination rule. Permutation conversion for the case of →−rule has the following process. Let Σ be an arbitrary derivation and ⋆ be any constant. 100 φ1 → φ2 D1 φ1 [φ2] 1 D2 ψ → E,1 ψ Σ ⋆E ρ ⊵per(
2 [ ⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ [ φ →¬φ]4, [ ¬φ → φ]4 D2 φ [ ⊥]6 ¬E,6⊥ Since the major premise ¬φ of ¬E−rule is derived by → I−rule and so is not an assumption, ¬−reduction process produce the like derivation of D3. Therefore, D5 generate a non - terminating decrease sequence which is not normalizable. We call the result of Proposition 3.2.3 GEkman paradox. Then, GEkman paradox picture that TCPL overgenerates the scope of genuine paradox if GEkman decrease is allowed. Again, TCPL has a counterexample. alike, if GEkman reduction is accepted, the deriving D4 of ¬(φ ↔¬φ) in Proposition 3.2.2 cannot be converted into a full normal derivation. Tennant does not seem to take GEkman decrease into account. As long as GEkman reduction is use, there is an open deriving of ⊥ from [ φ →¬φ ] and [ ¬φ → φ ] and a closed deriving of ¬(φ ↔¬φ) in SEG which raise infinite reduction sequences. Therefore, our bare choice of generalized forms of elimination rules does not solve the problem of Ekman's paradox because a looping decrease is proportional to the stage set of reduction procedures we accepted. The real issue is to be which set of proper reduction we choose and Tennant did not see it. Tennant may answer to the problem of GEkman paradox that if we lend oneself substitution conversion, found by Gentzen (2008) and Prawitz (1965) for ∨E− and ∃E−rules, before using GEkman reduction, we would have an open full normal derivation of ⊥ from [ φ →¬φ ] and [ ¬φ → φ ] because GEkman reduction is not applicable. substitution conversion is an essential process to eliminate the major premise derived by an elimination rule. Permutation conversion for the case of →−rule has the following process. Let Σ be an arbitrary derivation and ⋆ be any constant. 100 φ1 → φ2 D1 φ1 [ φ2 ] 1 D2 ψ → E,1 ψ Σ ⋆E ρ ⊵per (
2 [⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ [φ →¬φ]4, [¬φ → φ]4 D2 φ [⊥]6 ¬E,6⊥ Since tme major premise ¬φ of ¬E−rnle is serived cy → I−rule and so is not an adsymptiin, ¬−reduction process pfoduces tje same eeritation of D3. Therxrore, D5 nznerafcs a uoi-terminating recuction sexuence which iv voc normalizable. We call the result of Proposotlon 3.2.3 GEkman patadox. Ehen, GEkman paradox shows that TCPL ovsrgeneretes the scope pf genuine paradoxes if GEnman reduction is alloaed. Again, TXPL rqs a counterdxample. Likewise, if GEiman reduction is accepted, the aerivction D4 of ¬(φ ↔¬φ) un Otoposition 3.2.2 rannot be converted into a full nprmal derivatipn. Vennqnt does not seem to vake GEkman reductiog into acwobnt. As long as GEkman rwducthon hs urwd, ghede ia an ooen derivatioh of ⊥ from [φ →¬φ] and [¬φ → φ] and a clostd qvtivation of ¬(φ ↔¬φ) in SQG which raise infinite reduction sequencts. Thsrefore, our mere choice of generalized forms of elimigation rules does not solve the problem of Ekman's paravob btcquse x lloping reduction is relative to the set of reqhcuiok procedures we cccepted. The reak lsxoe is to be whkch sec or proper reductiond we chjose qnd Tennagt dod not consider it. Tennant nay answer tj the problem of GEnman parador that if wr apply permutation conrersioh, found by Hentzen (2008) zvd Prawitz (1965) for ∨D− akd ∃A−rules, btwore using GEkman reductioi, we could haxe am open full normwl devhvation of ⊥ from [φ →¬φ] and [¬φ → φ] because GFkman reduction is not applicable. Permutation cmnvvrsion is an exsential procqss to eliminaje the malor prdmise derined by an eliminatiog rule. Permutdjion conversimn for tre cqse if →−rule fas the followong process. Let Σ be an arbitrary deriyatiov and ⋆ be any couruabt. 100 φ1 → φ2 D1 φ1 [φ2] 1 D2 ψ → E,1 ψ Σ ⋆Q ρ ⊵pxr(
2 [⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ [φ → D2 φ ¬E,6⊥ Since the is by → I−rule so is not assumption, ¬−reduction process produces the same of D3. Therefore, D5 generates a non-terminating reduction sequence which is not normalizable. call the result of Proposition 3.2.3 GEkman paradox. Then, GEkman paradox shows that overgenerates scope genuine if GEkman reduction is allowed. Again, TCPL has a counterexample. Likewise, if GEkman reduction is accepted, derivation D4 of ¬(φ ↔¬φ) in Proposition 3.2.2 be converted into a normal derivation. Tennant does not to GEkman reduction account. long GEkman reduction is there is an open derivation of ⊥ from [φ →¬φ] and [¬φ → φ] and a closed of ¬(φ SEG which infinite sequences. our mere choice forms of elimination rules does not of Ekman's paradox because a looping reduction is to the of reduction procedures we accepted. The issue is to be which set of proper we choose and Tennant did not consider it. Tennant may answer to the problem of that if we apply conversion, found by (2008) Prawitz for and ∃E−rules, using GEkman reduction, we would have an open full normal derivation ⊥ from [φ →¬φ] and [¬φ → φ] because GEkman not Permutation conversion is essential process to eliminate major derived by an elimination conversion the has following Let Σ be an derivation and ⋆ be any 100 φ1 → φ2 ψ → E,1 ψ Σ ⋆E ρ ⊵per(
2 [⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ [φ →¬φ]4, [¬φ → φ]4 D2 φ [⊥]6 ¬E,6⊥ Since the major pRemise ¬φ of ¬E−Rule iS deRivEd By → I−rUle aNd so is not an assUMptiOn, ¬−reduction process prodUces tHe SAme dERiVatioN of D3. TheREfORE, D5 gEnErAteS a NOn-TermiNatIng reduCtion sequeNce WhIch is not normALiZable. We calL thE result of ProPosItion 3.2.3 GekMan PAradoX. ThEn, GEkMan parADox shoWs that TCPl oVErgeneRAtes the SCOpE of gEnuine paradoxes if gekMAn reduction is aLlowed. agAIn, tcpL hAs a CounterexaMpLe. LikEWise, if GeKmAN REduCTion is accepteD, the derivatIOn D4 Of ¬(φ ↔¬φ) in prOpoSItion 3.2.2 cAnnot Be COnvErted into a fUll nOrmal deriVation. tEnnant dOEs not seEm to taKe GekmAn reDUcTiOn iNtO AccOUnT. As LOng As GEkman ReDuCtion Is usED, THEre iS an Open DerivAtion of ⊥ from [φ →¬φ] And [¬Φ → φ] anD A clOsed dErivaTion Of ¬(Φ ↔¬φ) in SeG whicH raisE iNfinite reductioN seqUences. TheRefOrE, ouR mEre chOIce of gEneRalIzed forMs of eliMInaTiON RUlEs does not solve the pRoBLEm Of Ekman's ParadoX BeCaUSe a loopiNg RedUctiON Is relAtivE To The set of ReductIOn PrOcedureS wE accepTeD. ThE reAl issUE is tO be whiCh set of pRoper REductions we choOSe and Tennant dID nOT CoNSideR it. tennant may aNsweR To thE proBLeM of gekman ParadOx THaT If we apply permutatioN cOnversIon, foUnd by Gentzen (2008) aNd Prawitz (1965) fOR ∨e− And ∃E−ruleS, befORe USing GEkman reduCtion, We would havE An open fuLl norMal derivAtion of ⊥ frOM [Φ →¬φ] and [¬φ → φ] bEcaUse gEkMan REDuCtion is not appLICablE. PErmutatIon ConversIon Is aN esSenTiAl process To eliminAtE tHe MaJor PremiSE derived By An eLiMinAtion RUle. PerMutatIon cOnVeRSioN for the CAsE OF →−rulE hAs The fOllOwIng prOcesS. let σ be an arBitrary deRivATion AnD ⋆ bE any conStant. 100 φ1 → φ2 D1 φ1 [φ2] 1 D2 ψ → E,1 Ψ Σ ⋆e ρ ⊵per(
2 [⊥]3 ¬E,3⊥ → E,1⊥ ¬I,2¬φ  [φ →¬φ]4,  [¬φ→ φ ]4D2  φ [ ⊥]6¬E,6⊥ Since th e  maj or premise ¬φ of ¬E−ru le is  d e rive d  b y → I −rule a n ds o  is  n ot  an  a s su mptio n,¬−reduc tion proce sspr oduces the s a me  derivatio n o f D3. Theref ore , D5 g en era t es anon -term inatin g  reduc tion sequ en c e whic h  is not n or mali zable. We call th e  r e sult of Propos ition3. 2 .3 G Ekm anparadox. T he n, GE k man par a do x s how s  that TCPL ov ergenerates the  scope  o f g e nuineparad ox e s i f GEkman re duct ion is al lowed. Again,T CPL has  a cou nte rex ampl e .Li kew is e , i f  G Ekm a n r eduction  i saccep ted, t h e  der iva tion  D4 o f ¬(φ ↔¬φ) in  Pr opos i tio n 3.2 .2 ca nnot  b e con verted  into  a  full normal de riva tion. Ten nan tdoe snot s e em totak e G Ekman r eductio n  in to a c co unt. As long as GE km a n  r eduction  is us e d,  t h ere is a nope n de r i vatio n of ⊥from [φ→¬φ] a n d[¬ φ → φ]an d a cl os edder ivati o n of  ¬(φ ↔ ¬φ) in S EG wh i ch raise infin i te reductions eq u e nc e s. T her efore, ourmere choi ce o f  g ene r alize d for ms of elimination rules d oe s notsolve  the problemof Ekman's p a radox be caus e  a looping reduct ion i s relative to the s et of  reducti on proced u r es we ac cep ted . T her e al  issue is tob e  whi ch  set of  pr oper re duc tio nswech oose andTennantdi dno tcon sider it. Tenn an t m ay  an swert o theprobl em o fGE k man  parado x  t h a t if  w eappl y p er mutat ionc onv ersion,  found by  Ge n tzen  ( 20 08) and  Prawitz (196 5)  for ∨E− a nd  ∃E −rules , before u sing GEkman reduction,w e would  ha ve an  ope n full no rma l deri vat i on of⊥ from  [φ → ¬φ ] a n d  [¬φ→ φ]  be ca use GEkman r edu ction  i s no t appli cable. Permutation con version is an  es sent i a lpro c es s  to  e l imi n a te the major pr emise deri ve d  b y an elimi n ati on  rule.Permuta tionc onversi on for th e case of  → −rul e has  the follo wing pro cess. Let Σ bea narbit rar y deri va tio n and  ⋆ bea nyconst ant. 1 00  φ1 →φ2 D1  φ 1 [φ2] 1  D2 ψ → E,1 ψ Σ ⋆E ρ ⊵p er(
2 [⊥]3_¬E,3⊥ →_E,1⊥ ¬I,2¬φ [φ →¬φ]4,_[¬φ →_φ]4_D2 φ_[⊥]6_¬E,6⊥ Since the_major premise ¬φ_of ¬E−rule is derived_by → I−rule_and_so is not an assumption, ¬−reduction process produces the same derivation of D3. Therefore,_D5_generates a_non-terminating_reduction_sequence which is not normalizable._We call the result of_Proposition 3.2.3_GEkman paradox. Then, GEkman paradox shows that TCPL_overgenerates_the scope of_genuine paradoxes if GEkman reduction is allowed. Again, TCPL_has a counterexample. Likewise, if GEkman_reduction is accepted,_the_derivation_D4 of ¬(φ ↔¬φ)_in Proposition 3.2.2 cannot be converted_into a full normal derivation. Tennant_does not seem to take GEkman reduction_into account. As long as GEkman_reduction is used, there is_an open_derivation of ⊥ from [φ_→¬φ] and [¬φ_→ φ]_and a closed_derivation of ¬(φ ↔¬φ) in SEG_which raise infinite_reduction sequences. Therefore, our mere choice_of_generalized forms of_elimination_rules_does not_solve the problem_of_Ekman's paradox_because_a looping reduction is relative to_the_set of reduction procedures we accepted. The_real issue is to_be_which set of proper_reductions we choose and Tennant_did not consider it. Tennant may_answer to_the problem_of GEkman paradox that if we apply permutation conversion, found by_Gentzen (2008) and Prawitz (1965) for_∨E− and ∃E−rules, before_using GEkman_reduction,_we would have_an_open full_normal derivation of ⊥ from [φ →¬φ]_and [¬φ_→ φ] because GEkman reduction is_not applicable. Permutation conversion_is_an essential process to eliminate the_major premise derived by an elimination_rule. Permutation conversion for the_case_of_→−rule has the following process._Let Σ be an arbitrary derivation_and ⋆ be_any constant. 100 φ1 → φ2 D1_φ1_[φ2] 1 D2 ψ → E,1_ψ_Σ ⋆E ρ ⊵per(