 conjecture has a proof, then Goldbach conjecture has a proof. However, as Goldbach conjecture has not yet been proven, it is not the case that it has a proof. That is, an open derivation of Goldbach conjecture from the assumption that every conjecture has a proof is different from a closed derivation of Goldbach conjecture. So, it is not a proper reduction process from the open derivation to the closed derivation. Thus. for Translation test, we will examine whether a reduction makes an open derivation closed or a closed derivation open. Translation test can discover two problems of Ekman-type reduction procedures. The one is that it makes an open (or a closed) derivation closed (or open). The other is that in the case which allows open assumptions it generates an unnecessary detour. It will be established by Translation test that Ekman-type reductions do not satisfy both requirements (T1) and (T2). First, for Translation test, we assume that GEkman reduction is closed under substitution of derivations for open assumptions. Standard reductions introduced in Section 1.2 and 2.1 allow open derivations and they do not make any open derivation closed or vice versa. In addition to Ekman reduction, GEkman reduction allows an open derivation. We consider a special case of GEkman reduction as below. [ψ → φ] [φ → ψ] D φ [ψ]1 → E,1 ψ [φ]2 → E,2 φ ⊵GE D φ Suppose that the derivation D of φ is a closed derivation. GEkman reduction makes an open derivation closed by eliminating assumptions [φ → ψ] and [ψ → φ]. The translation 121 of the above derivation also shows the phenomenon clearly. D′ Γ ⇒ φ ψ ⇒ ψ L → Γ,φ → ψ ⇒ ψ φ ⇒ φ L → φ → ψ,ψ → φ,Γ ⇒ φ reduces to D′ Γ ⇒ φ ⊵GE−reduction yields the closed derivation D of φ from the open derivation by getting rid of the assumptions [φ → ψ] and [ψ → φ]. The original derivation on the left-side says that if φ → ψ and ψ → φ are true, then φ is true. On the other hand, the reduced derivation on the right says that φ is a theorem. Of course, one may claim that the application of Wk−rule to the
conjecture has a proof, then Goldbach conjecture take a validation. However, as Goldbach conjecture has not yet been testify, it is not the lawsuit that it has a proof. That is, an open ancestry of Goldbach guess from the assumption that every speculation has a proof is different from a shut derivation of Goldbach conjecture. So, it is not a proper reduction procedure from the open derivation to the closed derivation. Thus. for transformation test, we will examine whether a reduction make an open derivation close or a shut derivation open. Translation test can discover two problems of Ekman - character reduction procedures. The one is that it makes an open (or a closed) derivation close (or open). The other is that in the case which allows open assumptions it generates an unnecessary detour. It will be established by transformation trial that Ekman - type reduction do not satisfy both requirements (T1) and (T2). First, for Translation trial, we assume that GEkman reduction is closed under substitution of derivations for clear assumptions. Standard reductions introduced in Section 1.2 and 2.1 allow open derivation and they do not make any open derivation closed or frailty versa. In addition to Ekman decrease, GEkman reduction allows an open derivation. We consider a special shell of GEkman reduction as below. [ ψ → φ ] [ φ → ψ ] vitamin d φ [ ψ]1 → E,1 ψ [ φ]2 → E,2 φ ⊵GE D φ Suppose that the derivation D of φ is a closed derivation. GEkman decrease makes an open derivation closed by eliminating premise [ φ → ψ ] and [ ψ → φ ]. The translation 121 of the above deriving also read the phenomenon clearly. D′ Γ ⇒ φ ψ ⇒ ψ lambert → Γ, φ → ψ ⇒ ψ φ ⇒ φ L → φ → ψ, ψ → φ, Γ ⇒ φ reduces to D′ Γ ⇒ φ ⊵GE−reduction yield the closed ancestry D of φ from the open derivation by getting rid of the assumptions [ φ → ψ ] and [ ψ → φ ]. The original derivation on the left - side says that if φ → ψ and ψ → φ are truthful, then φ is true. On the other hired hand, the reduced derivation on the right says that φ is a theorem. Of course, one may claim that the application of Wk−rule to the
 cojjecture has a proof, thek Goldbach conjeerure hes a prkof. Howexer, as Goldbach conjecture hes nit yeu been proven, it is not the base that it ias a proof. That is, an oizn dedlvatimi of Goldbach cpnjecture xrom the assum[tkou that every conjecture has a proof ys diffrrfnt from a clofed cqrivzniin of Goldbach conjecture. So, jt is nmt a proper rrduction process from the lpen derivation to the closed deruvatyin. Thus. for Granslation test, we wiml examine whether a reduction oakes an open dgxuvahhon closed ir a blosed derivabpon opet. Transkation test cak disrovee two problems of Ekmen-type reduction prosedures. Tve one is that it mqkws an opet (or q cuostd) verjvatioj cmosed (or olen). The othwr is that in the cssq which allows open wsfumptions it generates an unnecessary dttour. It will be established by Translation test jhat Ekman-eype reductions do not satisfy both requirements (D1) and (G2). Fnvfg, flr Translation test, we assume that GEkman redtdtook is closed undev substitution of cegifwtions for opgn assumptjons. Standard reduftions yntroeuced in Fectoon 1.2 and 2.1 allow open derivarions and thvy di not make any opeu derivation cloxed ot vice versa. In additiou to Eiman reductlon, GEkmah reduction allowr ak o[en derivation. We consider a speciao caxe of GDkmam redustion as bflow. [ψ → φ] [φ → ψ] D φ [ψ]1 → E,1 ψ [φ]2 → E,2 φ ⊵HE D φ Sgppose thah the derivation D of φ is a closed derivation. GAkmdn reduccion mskes an open qerivation cloxed by zliminxting assujptions [φ → ψ] and [ψ → φ]. Ehe translatimj 121 of the abmve deridatiin aoso shodr the phenomenpn clearlj. B′ Γ ⇒ φ ψ ⇒ ψ L → Γ,φ → ψ ⇒ ψ φ ⇒ φ L → φ → ψ,ψ → φ,Γ ⇒ φ rgdhces to D′ Γ ⇒ φ ⊵GE−xtduxtion yields thr cuosqd dxrivaehon D of φ frmm tfe ulen ddrivation bn gdttimg rid of the assumpdiona [φ → ψ] and [ψ → φ]. The oroglnal derication og the left-sidr says that if φ → ψ wnd ψ → φ ace trur, tren φ is true. On the other hand, the redufed derivation jn tme ryght says chat φ is a theorem. Of course, one may clamm that the application of Wk−rule to the
conjecture has a proof, then Goldbach conjecture proof. as Goldbach has not yet the that it has proof. That is, open derivation of Goldbach conjecture from assumption that every conjecture has a proof is different from a closed derivation Goldbach conjecture. So, it is not a proper reduction process from the open to closed Thus. Translation test, we will examine whether a reduction makes an open derivation closed or a closed open. Translation test can discover two problems of reduction procedures. The one that it makes an open a derivation closed open). other that in the which allows open assumptions it generates an unnecessary detour. It will be established by Translation test that reductions do both requirements and First, Translation test, we GEkman reduction is closed under substitution open assumptions. Standard reductions introduced in Section 1.2 2.1 allow derivations and they do not make open derivation closed or vice versa. In addition Ekman reduction, GEkman reduction allows an open derivation. We consider a special case of GEkman below. [ψ → φ] → ψ] D [ψ]1 E,1 [φ]2 E,2 φ D φ Suppose that the derivation D of φ is a derivation. GEkman reduction makes an open derivation closed by eliminating → and [ψ → The translation 121 of above also shows the phenomenon Γ φ L Γ,φ ψ ⇒ ψ φ φ L → φ → → φ,Γ ⇒ φ φ ⊵GE−reduction yields the closed derivation D of from the open derivation by getting rid the assumptions [φ → ψ] and [ψ → φ]. The original derivation the left-side if φ → ψ and ψ → φ true, then φ is On the other hand, the reduced derivation on the says φ is theorem. Of course, may claim that application of Wk−rule
 conjecture has a proof, then GoLdbach conjEcturE haS a pRoOf. HoWeveR, as Goldbach conJEctuRe has not yet been proven, iT is noT tHE casE ThAt it hAs a prooF. thAT Is, aN oPeN deRiVAtIon of golDbach coNjecture frOm tHe Assumption thAT eVery conjecTurE has a proof is DifFerent FrOm a CLosed DerIvatiOn of GoLDbach cOnjecture. so, IT is not A Proper rEDUcTion Process from the opeN DeRIvation to the clOsed deRiVAtION. ThUs. fOr TranslatIoN test, WE will exAMiNE WHetHEr a reduction mAkes an open dERivAtion cLoSed OR a closEd derIvATioN open. TranslAtioN test can dIscoveR Two probLEms of EkMan-typE reDucTion PRoCeDurEs. tHe oNE iS thAT it Makes an oPeN (oR a cloSed) dERIVAtioN clOsed (Or opeN). The other is thAt iN the CAse Which AllowS opeN aSsumpTions iT geneRaTes an unnecessarY detOur. It will Be eStAblIsHed by tRanslaTioN teSt that EKman-typE RedUcTIONs Do not satisfy both reQuIREmEnts (T1) and (t2). First, FOr trANslation TeSt, wE assUME that gEkmAN rEduction Is closED uNdEr substItUtion oF dEriVatIons fOR opeN assumPtions. StAndarD Reductions intrODuced in SectioN 1.2 AnD 2.1 ALlOW opeN deRivations anD theY Do noT makE AnY opEN deriVatioN cLOsED or vice versa. In additIoN to EkmAn redUction, GEkman rEduction alLOWS an open dErivATiON. We consider a spEcial Case of GEkmAN reductiOn as bElow. [ψ → φ] [φ → ψ] d φ [ψ]1 → E,1 ψ [φ]2 → E,2 φ ⊵Ge d Φ Suppose ThaT thE deRivATIoN D of φ is a closeD DErivAtIon. GEkmAn rEductioN maKes An oPen DeRivation cLosed by eLiMiNaTiNg aSsumpTIons [φ → ψ] anD [ψ → Φ]. ThE tRanSlatiON 121 of the Above DeriVaTiON alSo shows THe PHEnomEnOn CleaRly. d′ Γ ⇒ Φ ψ ⇒ ψ L → Γ,Φ → ψ ⇒ ψ φ ⇒ Φ l → φ → ψ,Ψ → φ,Γ ⇒ φ redUces to D′ Γ ⇒ φ ⊵gE−rEDuctIoN yIelds thE closed derivaTiOn D of φ from ThE opEn deriVATion by geTting rid of the assumptionS [Φ → ψ] and [ψ → φ]. the OrigiNal dErivation On tHe left-SidE Says thAt if φ → ψ And ψ → φ ArE trUE, Then φ IS TrUe. ON tHe other hanD, THe rEduceD dErivAtion on The right says that φ iS A thEorem. Of course, One May cLAIm ThaT ThE AppLiCAtiON Of Wk−rule to the
 conjecture has a proof, t hen Goldba ch co nje ctu re  has  a p roof. However, as G oldbach conjecture has  notye t  bee n  p roven , it is no t the  c as e t ha t  i t has  aproof.That is, a n o pe n derivation of  Goldbachcon jecture from  th e assu mp tio n  that  ev ery c onject u re has  a proofis differ e nt from a  c lose d derivation of G o ld b ach conjecture . So,it is n ota p roper redu ct ion p r ocess f r om t h e o p en derivation  to the clo s edderiva ti on. Thus.for T ra n sla tion test,we w ill exami ne whe t her a r e duction  makes  an  op en d e ri va tio nc los e dora  cl osed der iv at ion o pen. T r a nsla tio n te st ca n discover tw o p robl e msof Ek man-t ypere ducti on pro cedur es . The one is th at i t makes a n o pe n ( or  a cl o sed) d eri vat ion clo sed (or ope n) . T he  other is that inth e ca se which  allow s  o pe n  assumpt io nsit g e n erate s an un necessar y deto u r.  I t willbe  estab li she d b y Tra n slat ion te st thatEkman - type reduction s  do not satis f yb o th requ ire ments (T1)and( T2).  Fir s t,  fo r  Tran slati on te s t, we assume that G Ek man re ducti on is closedunder subs t i t ution of  der i va t ions for openassum ptions. St a ndard re ducti ons intr oduced in S ection 1 .2and  2. 1 a l l ow  open derivat i o ns a nd  they d o n ot make  an y o pen  de ri vation cl osed orvi ce  v er sa.  In a d dition t oEkm an  re ducti o n, GEk man r educ ti on all ows ano pe n deri va ti on.Weco nside r as pec ial cas e of GEkm anr educ ti on  as bel ow. [ψ → φ] [ φ→ ψ] D φ [ ψ] 1 →  E,1 ψ [ φ]2 → E, 2 φ ⊵GE D φ Suppose tha t  the de riv ation  D o f φ is aclo sed de riv a tion.GEkman  redu ct ion m akesa n  o pen  d erivationc l ose d byel imin ating a ssumptions [φ → ψ] and  [ψ → φ]. The  tr ansl a t io n 1 2 1o f t he abo v e  derivation als o shows th ep he nomenon cl e arl y.  D′ Γ ⇒  φ ψ ⇒ψ L → Γ,φ → ψ  ⇒ ψ φ ⇒φ L → φ →  ψ ,ψ → φ ,Γ⇒ φ reduce s to D′Γ ⇒ φ ⊵GE − reduc t io n yie lds  the c lo sed  deri vation D o f φ f rom th eopen d eriva ti on by ge tting rid of the assump tions[φ →ψ]and [ψ →φ]. The  original  der ivation on  th e l eft-s ide saysthat if  φ→  ψ an d ψ→  φ are tr u e,  th e n  φ  is true. O n t heother  ha n d, the  red uced derivation o n  the right say s th a t  φisa  the or em. Of course,  on em a y claimth at the appl icationof Wk−ru le tothe
 conjecture_has a_proof, then Goldbach conjecture_has a_proof._However, as_Goldbach_conjecture has not_yet been proven,_it is not the_case that it_has_a proof. That is, an open derivation of Goldbach conjecture from the assumption that_every_conjecture has_a_proof_is different from a closed_derivation of Goldbach conjecture. So,_it is_not a proper reduction process from the open_derivation_to the closed_derivation. Thus. for Translation test, we will examine whether_a reduction makes an open derivation_closed or a_closed_derivation_open. Translation test can_discover two problems of Ekman-type reduction_procedures. The one is that it_makes an open (or a closed) derivation_closed (or open). The other is_that in the case which_allows open_assumptions it generates an unnecessary_detour. It will_be established_by Translation test_that Ekman-type reductions do not satisfy_both requirements (T1)_and (T2). First, for Translation test,_we_assume that GEkman_reduction_is_closed under_substitution of derivations_for_open assumptions._Standard_reductions introduced in Section 1.2 and_2.1_allow open derivations and they do not_make any open derivation_closed_or vice versa. In_addition to Ekman reduction, GEkman_reduction allows an open derivation. We_consider a_special case_of GEkman reduction as below. [ψ → φ] [φ → ψ]_D φ [ψ]1 → E,1 ψ_[φ]2 → E,2 φ_⊵GE D_φ_Suppose that the_derivation_D of_φ is a closed derivation. GEkman reduction_makes an_open derivation closed by eliminating assumptions_[φ → ψ] and_[ψ_→ φ]. The translation 121 of_the above derivation also shows the_phenomenon clearly. D′ Γ ⇒_φ_ψ_⇒ ψ L → Γ,φ_→ ψ ⇒ ψ φ ⇒_φ L →_φ → ψ,ψ → φ,Γ ⇒ φ_reduces_to D′ Γ ⇒ φ ⊵GE−reduction_yields_the closed derivation D of φ_from_the_open derivation by getting rid_of the assumptions [φ → ψ]_and [ψ → φ]. The original derivation on the_left-side says that_if φ → ψ and_ψ_→_φ are true, then φ is true. On the other_hand, the_reduced derivation on_the right says that φ is a theorem. Of course,_one may claim that the application of_Wk−rule to the