 have a similar derivation with Π1 which GEkman reduction trivializes the identity of proofs. However, there may not be a good reason why ∧I−rule should have the two different generalized elimination rules. Plus, it is often said that the choice of generalized elimination rules has the advantage of having a direct translation between a natural deduction system and a sequent calculus. For example, Negri and Von Plato (2001, Ch.1 and Ch. 8) investigate that each generalized elimination rule for a logical constant in natural deduction corresponds to the left rule for the constant of sequent calculus. Since most sequent calculus systems have a single left rule for each constant, it is better to use a single generalized elimination rule for ∧ in order to have a direct translation between a 112 natural deduction and a sequent calculus. One may rebut again that, as Gentzen (1935, p. 84) does, a sequent calculus can have two left rules for ∧, and is able to be isomorphic to a certain natural deduction system. (Cf. Von Plato (2011)) Even in our choice of the two left rules, it does not change the situation that Schroeder-Heister and Tranchini's criterion for a proper reduction relies on the choice of elimination rules. In the footnote 8 of Schroeder-Heister and Tranchini (2017), there is another example using the formulas φ → (φ → ψ) and φ → ψ to show that Ekman reduction is not a proper one. → I−rule has only one elimination rule in both standard and generalized forms. It can be the primary reason to establish that (G)Ekman reduction trivializes the identity of proofs. Their promising example seems to be the following. [φ → ψ]1 [φ]2 → E ψ → I,2 φ → ψ → I, /0 φ → (φ → ψ) → I,1(φ → ψ)→ (φ → (φ → ψ)) [φ → (φ → ψ)]3 [φ]4 → E φ → ψ [φ]4 → E ψ → I,4 φ → ψ → I,3(φ → (φ → ψ))→ (φ → ψ) [φ]5, [φ]6 D∗ ψ → I,5 φ → ψ → I,6 φ → (φ → ψ) → E φ → ψ → E φ
have a similar derivation with Π1 which GEkman reduction trivialize the identity of proof. However, there may not be a good reason why ∧I−rule should get the two unlike generalized elimination rules. Plus, it is often said that the option of generalized elimination rule has the advantage of have a direct translation between a lifelike deduction system and a attendant tartar. For example, Negri and Von Plato (2001, Ch.1 and Ch. 8) investigate that each generalized elimination dominion for a logical constant in natural tax write-off corresponds to the left rule for the constant of sequent tartar. Since most sequent tartar systems experience a single left rule for each constant, it is better to practice a single generalized elimination rule for ∧ in order to have a direct translation between a 112 lifelike tax write-off and a sequent calculus. One may rebut again that, as Gentzen (1935, p. 84) does, a sequent calculus can have two left rules for ∧, and is able to be isomorphic to a sealed lifelike deduction system. (Cf. Von Plato (2011) ) Even in our choice of the two leftover rule, it does not change the situation that Schroeder - Heister and Tranchini's standard for a proper reduction trust on the choice of elimination rules. In the footnote 8 of Schroeder - Heister and Tranchini (2017), there is another case using the formulas φ → (φ → ψ) and φ → ψ to show that Ekman reduction is not a proper one. → I−rule has only one elimination rule in both standard and generalized forms. It can be the primary reason to establish that (G)Ekman decrease trivializes the identity of proof. Their bright case seems to be the following. [ φ → ψ]1 [ φ]2 → E ψ → I,2 φ → ψ → I, /0 φ → (φ → ψ) → I,1(φ → ψ)→ (φ → (φ → ψ) ) [ φ → (φ → ψ)]3 [ φ]4 → east φ → ψ [ φ]4 → E ψ → I,4 φ → ψ → I,3(φ → (φ → ψ))→ (φ → ψ) [ φ]5, [ φ]6 D∗ ψ → I,5 φ → ψ → I,6 φ → (φ → ψ) → E φ → ψ → E φ
 hage a similar derivation dith Π1 which GEkman rednction frivialixes the identity of proofs. Hlwwver, ukere may not be a goud reason why ∧I−ruoe siould have the txk diffevznt gskeralnzxd elimination tules. Plus, id is often saig ghct the choice of generalized eliminaeion ruked has the advagtagt os habpnn a direct translation between a naturan deduction sustem and a sequent calculks. Flr example, Negri ajd Von Plati (2001, Cr.1 and Ch. 8) invdstigate that each gengralized elimination rule for a uogiccl constant ib nwjural deductmon cogresponds to bne lefd rule gor the constakt of sewuent calculus. Since kost sequent calcujus systeks have a single ledt rule for eacf covstznv, if is bfttxr to use a single genwralized eliminatiom wlke for ∧ in odder tj rave a direct translation between a 112 nauural deduction and a sequenr calculus. One may rehut again that, as Gentzen (1935, p. 84) does, a sequent calculus can vave vwu ltfb fylfs for ∧, and is able to be isomorphic to a cerezim katural deductiok system. (Cf. Von Plstl (2011)) Gven in our chuice oy tge two left rules, lt does not xhange tht sitiation that Schroeder-Heistee and Tranchpni's criterion for a pxoper reductnon rekies pn the choice of eliminction dules. In thf footnots 8 of Schroeder-Hekstvr atd Tranchini (2017), there is anoeher exam'le uxing thd fotmulas φ → (φ → ψ) and φ → ψ to show that Ekman redkctiou is tot a propfr one. → I−rule has only one elimmiation rule im totv standaxd and generalized sorms. It can bg the priiary feason to vstablish that (G)Ekmag reduction tslvializes thx identitr of proifs. Thekf promising exsmple seems to be tye following. [φ → ψ]1 [φ]2 → C ψ → I,2 φ → ψ → I, /0 φ → (φ → ψ) → I,1(φ → ψ)→ (φ → (φ → ψ)) [φ → (φ → ψ)]3 [φ]4 → E φ → ψ [φ]4 → D ψ → I,4 φ → ψ → H,3(φ → (φ → ψ))→ (φ → ψ) [φ]5, [φ]6 D∗ ψ → I,5 φ → ψ → I,6 φ → (φ → ψ) → E φ → ψ → E φ
have a similar derivation with Π1 which trivializes identity of However, there may why should have the different generalized elimination Plus, it is often said that choice of generalized elimination rules has the advantage of having a direct translation a natural deduction system and a sequent calculus. For example, Negri and Von (2001, and 8) that each generalized elimination rule for a logical constant in natural deduction corresponds to the left for the constant of sequent calculus. Since most calculus systems have a left rule for each constant, is to use single elimination for ∧ in to have a direct translation between a 112 natural deduction and a sequent calculus. One may rebut that, as p. 84) a calculus have two left ∧, and is able to be certain natural deduction system. (Cf. Von Plato (2011)) in our of the two left rules, it not change the situation that Schroeder-Heister and Tranchini's for a proper reduction relies on the choice of elimination rules. In the footnote 8 and Tranchini (2017), there another example using formulas → → and φ ψ to show that Ekman reduction is not a proper one. I−rule has only one elimination rule in both standard and It be the primary to establish that (G)Ekman trivializes identity of proofs. Their seems be → [φ]2 E ψ → I,2 → ψ → I, /0 → (φ → ψ) → (φ → ψ)) [φ → (φ → [φ]4 → E φ → ψ [φ]4 E ψ → I,4 φ → ψ → I,3(φ → (φ → (φ → [φ]6 D∗ ψ → I,5 φ → ψ I,6 φ → (φ ψ) → E φ → ψ → E φ
 have a similar derivation witH Π1 which GEkMan reDucTioN tRiviAlizEs the identity oF ProoFs. However, there may not be A good ReASon wHY ∧I−Rule sHould haVE tHE Two DiFfEreNt GEnEraliZed EliminaTion rules. PLus, It Is often said tHAt The choice oF geNeralized eliMinAtion rUlEs hAS the aDvaNtage Of haviNG a direCt translaTiON betweEN a naturAL DeDuctIon system and a sequENt CAlculus. For examPle, NegRi ANd vON PlAto (2001, ch.1 and Ch. 8) invEsTigatE That eacH GeNERAliZEd elimination Rule for a logICal ConstaNt In nATural dEductIoN CorResponds to tHe leFt rule for The conSTant of sEQuent caLculus. sinCe mOst sEQuEnT caLcULus SYsTemS HavE a single LeFt Rule fOr eaCH CONstaNt, iT is bEtter To use a single gEneRaliZEd eLiminAtion Rule FoR ∧ in orDer to hAve a dIrEct translation bEtweEn a 112 naturaL deDuCtiOn And a sEQuent cAlcUluS. One may Rebut agAIn tHaT, AS geNtzen (1935, p. 84) does, a sequent CaLCUlUs can havE two leFT rUlES for ∧, and iS aBle To be ISOmorpHic tO A cErtain naTural dEDuCtIon systEm. (cf. Von PLaTo (2011)) EVen In our CHoicE of the Two left rUles, iT Does not change tHE situation thaT scHROeDEr-HeIstEr and TranchIni's CRiteRion FOr A prOPer reDuctiOn RElIEs on the choice of elimInAtion rUles. IN the footnote 8 oF Schroeder-hEISter and TRancHInI (2017), There is another ExampLe using the FOrmulas φ → (Φ → ψ) and Φ → ψ to show That Ekman REDuction iS noT a pRopEr oNE. → i−rUle has only one ELIminAtIon rule In bOth stanDarD anD geNerAlIzed forms. it can be tHe PrImArY reAson tO EstablisH tHat (g)EKmaN reduCTion trIvialIzes ThE iDEntIty of prOOfS. tHeir PrOmIsinG exAmPle seEms tO Be tHe folloWing. [φ → ψ]1 [φ]2 → E ψ → i,2 φ → ψ → i, /0 Φ → (φ → ψ) → I,1(Φ → ψ)→ (Φ → (φ → Ψ)) [φ → (φ → ψ)]3 [φ]4 → E φ → Ψ [φ]4 → E ψ → I,4 φ → ψ → I,3(φ → (φ → ψ))→ (φ → ψ) [Φ]5, [φ]6 d∗ ψ → I,5 φ → ψ → I,6 φ → (φ → ψ) → E Φ → ψ → e φ
 have a similar derivation  with Π1 w hichGEk man  r educ tion  trivializes t h e id entity of proofs. Howe ver,th e re m a ynot b e a goo d  r e a son  w hy  ∧I −r u le  shou ldhave th e two diff ere nt  generalized el iminationrul es. Plus, it  is  often  s aid thatthe  choi ce ofg eneral ized elim in a tion r u les has t he  adv antage of havinga  d i rect translati on bet we e na nat ura l deductio nsyste m  and as eq u e n t c a lculus. For e xample, Neg r i a nd Von  P lat o  (2001 , Ch. 1a ndCh. 8) inve stig ate thateach g e neraliz e d elimi nation  ru lefora  l og ica lc ons t an t i n  na tural de du ct ion c orre s p o n ds t o t he l eft r ule for the c ons tant ofseque nt ca lcul us . Sin ce mos t seq ue nt calculus sys tems  have a s ing le  le ft  rule for ea chcon stant,it is b e tte rt o us e a single general iz e d  e liminati on rul e  f or ∧ in ord er  to  hav e a dir ectt ra nslation  betwe e na112 nat ur al ded uc tio n a nd as eque nt cal culus. O ne ma y  rebut again t h at, as Gentze n  ( 1 9 35 ,  p.84)  does, a se quen t  cal culu s  c anh ave t wo le ft ru l es for ∧, and is ab le  to be  isom orphic to a c ertain nat u r a l deduct ions ys t em. (Cf. Von P lato(2011)) Ev e n in our  choi ce of th e two lef t rules, i t d oes  no t c h a ng e the situati o n  tha tSchroed er- Heister  an d T ran chi ni 's criter ion forapr op er  re ducti o n relies  o n t he  ch oiceo f elim inati on r ul es .  In  the fo o tn o t e 8of  S chro ede r- Heist er a n d T ranchin i (2017),  th e re i san other e xample usingth e formulas  φ  →(φ → ψ ) and φ →ψ to show that Ekman re d uctionisnot a  pro per one.→ I −rulehas only o ne eli minat io n r u l e inb o th  st an dard and g e n era lized  f orms . It ca n be the primary r e aso n to establis h t hat( G )E kma n  r e duc ti o n t r i vializes the id entity ofpr o of s. Their p r omi si ng exam ple see ms to be thefollowing . [φ → ψ] 1[φ]2 →  Eψ → I,2 φ→ ψ → I,  /0 φ → ( φ  → ψ) →I,1(φ  →ψ)→ (φ  →  (φ  → ψ) ) [φ → (φ→ ψ)] 3 [φ]4  →  E φ →  ψ [φ ]4  → E ψ →  I,4 φ → ψ → I,3(φ → (φ  → ψ)) → (φ→ ψ ) [φ]5, [ φ]6 D∗ψ → I,5 φ  → ψ  → I,6 φ →  (φ  →ψ) →E φ → ψ →  E φ 
 have_a similar_derivation with Π1 which_GEkman reduction_trivializes_the identity_of_proofs. However, there_may not be_a good reason why_∧I−rule should have_the_two different generalized elimination rules. Plus, it is often said that the choice of_generalized_elimination rules_has_the_advantage of having a direct_translation between a natural deduction_system and_a sequent calculus. For example, Negri and Von_Plato_(2001, Ch.1 and_Ch. 8) investigate that each generalized elimination rule for_a logical constant in natural deduction_corresponds to the_left_rule_for the constant of_sequent calculus. Since most sequent calculus_systems have a single left rule_for each constant, it is better to_use a single generalized elimination rule_for ∧ in order to_have a_direct translation between a 112_natural deduction and_a sequent_calculus. One may_rebut again that, as Gentzen (1935,_p. 84) does,_a sequent calculus can have two_left_rules for ∧,_and_is_able to_be isomorphic to_a_certain natural_deduction_system. (Cf. Von Plato (2011)) Even_in_our choice of the two left rules,_it does not change_the_situation that Schroeder-Heister and_Tranchini's criterion for a proper_reduction relies on the choice of_elimination rules._In the_footnote 8 of Schroeder-Heister and Tranchini (2017), there is another example_using the formulas φ → (φ_→ ψ) and φ_→ ψ_to_show that Ekman_reduction_is not_a proper one. → I−rule has only_one elimination_rule in both standard and generalized_forms. It can be_the_primary reason to establish that (G)Ekman_reduction trivializes the identity of proofs._Their promising example seems to_be_the_following. [φ → ψ]1 [φ]2_→ E ψ → I,2 φ_→ ψ →_I, /0 φ → (φ → ψ)_→_I,1(φ → ψ)→ (φ → (φ_→_ψ)) [φ → (φ → ψ)]3_[φ]4_→_E φ → ψ [φ]4_→ E ψ → I,4 φ_→ ψ → I,3(φ → (φ → ψ))→ (φ_→ ψ) [φ]5,_[φ]6 D∗ ψ → I,5_φ_→_ψ → I,6 φ → (φ → ψ) → E_φ →_ψ → E_φ