 test and raises the problem of overgeneration. In Section 3.4, as an observation method for evaluating a proper reduction, we propose Translation test to detect errors in both Ekman-type and Crabbé's cases. We will suggest the requirement of a proper reduction that a proper reduction procedure must not introduce any unnecessary premise or detour. Translation test will be useful to assess a proper reduction with regard to the requirements. 3.1 Ekman's Paradox Let us suppose that there is a derivation satisfying TCPE which does not formalize a genuine paradox. Then, TCPE overgenerates the scope of genuine paradoxes and the derivation 92 can be a counterexample to TCPE. In this section, we will see Ekman's paradox which causes the problem of overgeneration. Schroeder-Heister and Tranchini (2017) put forward a counterexample to TCPE taken from Ekman (1998) in order to show that TCPE is a too coarse criterion for a (genuine) paradox. Ekman (1998) observes the following form of derivation and its reduction. [ψ → φ] [φ → ψ] D φ → E ψ → E φ ⊵E D φ We will call ψ in → E−rule an Ekman maximum formula, and this reduction will be called Ekman reduction process. Schroeder-Heister and Tranchini (2017) think that Ekman reduction is a too loose reduction procedure and TCPE has no restriction to use it. If there is a derivation of ⊥ from [φ →¬φ] and [¬φ → φ] and the application of Ekman reduction to the derivation generates a looping reduction, TCPE says that the derivation is a T-paradox. TCPE overgenerates the scope of genuine paradoxes because it makes the non-paradoxical derivation a T-paradox. The following result is their counterexample to TCPE. Proposition 3.1.1. Let SE be a natural deduction system consisting of →− and ¬−rules. If the set of reductions of SE includes Ekman reduction process, SE has an open derivation of ⊥ from [φ →¬φ] and [¬φ → φ] which generates a non-terminating reduction sequence, and is not normalizable. Proof. Two claims prove the result. Claim 1. There is an open derivation of ⊥ from [φ →¬φ] and
test and raises the problem of overgeneration. In Section 3.4, as an notice method acting for evaluate a proper reduction, we nominate transformation test to detect errors in both Ekman - type and Crabbé's case. We will suggest the requirement of a proper reduction that a proper reduction procedure must not precede any unnecessary premise or detour. Translation test will be utilitarian to assess a proper reduction with regard to the requirements. 3.1 Ekman's Paradox Let us suppose that there is a deriving satisfying TCPE which does not formalize a genuine paradox. Then, TCPE overgenerates the oscilloscope of genuine paradoxes and the derivation 92 can be a counterexample to TCPE. In this part, we will see Ekman's paradox which causes the problem of overgeneration. Schroeder - Heister and Tranchini (2017) put forward a counterexample to TCPE taken from Ekman (1998) in orderliness to show that TCPE is a too coarse criterion for a (genuine) paradox. Ekman (1998) observes the following mannequin of derivation and its reduction. [ ψ → φ ] [ φ → ψ ] D φ → E ψ → E φ ⊵E D φ We will call ψ in → E−rule an Ekman maximum formula, and this reduction will be call Ekman reduction process. Schroeder - Heister and Tranchini (2017) intend that Ekman decrease is a besides loose reduction operation and TCPE has no restriction to use it. If there is a deriving of ⊥ from [ φ →¬φ ] and [ ¬φ → φ ] and the application of Ekman reduction to the derivation generates a looping reduction, TCPE suppose that the derivation is a T - paradox. TCPE overgenerates the scope of genuine paradoxes because it makes the non - paradoxical derivation a T - paradox. The watch result is their counterexample to TCPE. Proposition 3.1.1. Let SE be a natural deduction arrangement consisting of →− and ¬−rules. If the set of reductions of SE include Ekman reduction process, SE has an open ancestry of ⊥ from [ φ →¬φ ] and [ ¬φ → φ ] which generates a non - terminating decrease sequence, and is not normalizable. Proof. Two claims prove the result. title 1. There is an overt derivation of ⊥ from [ φ →¬φ ] and
 tedt and raises the probleo of overgenerajiin. In Vectioh 3.4, as an observation method for evalnatibg a kgoper reduction, we prupose Trajslation tesu to detect errors in botm Ekmzk-type end Crabbé's casex. We will vuggest the rexukrzment of a proper reduction that a pwoper rrdkction procedute muxe nof introduce any unnecessary premiss or deuour. Translation trst will be useful to asseds a proper reduction aith regard to eye requiremevts. 3.1 Ekman's Paradox Lej us suppose that there is a derkvatipn satisfyunt TFKE which doew not formalize a genuine paradoc. Then, TCPE ovcrgenxratws the scope of genuiie paradoxes and the derivatimn 92 can be a counteeezamplg to DCPE. Un ghia xedtion, ae xill see Ekjan's paradoz which causes the krofoem of overgeheratijn. Schroeder-Heister and Tranchini (2017) put fogwars a counterexample to TXPE taken from Ekman (1998) in order to show that TCPE is a too coarse criterion for d (gennive) kavwaix. Ekman (1998) observes the following form of derivaejom snd its reductlon. [ψ → φ] [φ → ψ] D φ → E ψ → R φ ⊵E Q φ We will caul ψ in → E−dule an Ekman maxilum foriula, qnd this weduvtion will be called Ekman eeduction prjxess. Schroeder-Heiscer and Tranehini (2017) thinl that Ekman reduction ns a tko loose refuction pducedure and TCPE hax to restriction to use it. Is there iw a berivatiun og ⊥ froi [φ →¬φ] and [¬φ → φ] and bve application of Fkman raduction tl the derivation generates a looping reductiom, DCPV says thct the derivation if a T-paradox. TEPE overyenerages the sckpe of jenuine paraqoxes because lt makes the non-paraqoxixal eerivatkun a T-paradox. Yhe follocnng resulr is their countercxampus to TCPE. Propowituon 3.1.1. Let SE be s nxtuwap veducehon system cmnsirtivb of →− and ¬−rules. Lf ghe xet of reductions of SE jncludes Ekman redicbion procgss, SE haf an open derovation of ⊥ from [φ →¬φ] and [¬φ → φ] xhich bengrates a non-terminating reductikn sequenfe, wnd is not njrmaoizable. Prooy. Two claims prove the result. Claim 1. Thece is an open derivatiob of ⊥ from [φ →¬φ] and
test and raises the problem of overgeneration. 3.4, an observation for evaluating a test detect errors in Ekman-type and Crabbé's We will suggest the requirement of proper reduction that a proper reduction procedure must not introduce any unnecessary premise detour. Translation test will be useful to assess a proper reduction with regard the 3.1 Paradox us suppose that there is a derivation satisfying TCPE which does not formalize a genuine paradox. TCPE overgenerates the scope of genuine paradoxes and derivation 92 can be counterexample to TCPE. In this we see Ekman's which the of overgeneration. Schroeder-Heister Tranchini (2017) put forward a counterexample to TCPE taken from Ekman (1998) in order to show that is a criterion for (genuine) Ekman observes the following derivation and its reduction. [ψ → ψ] D φ → E ψ → E ⊵E D We will call ψ in → an Ekman maximum formula, and this reduction will called Ekman reduction process. Schroeder-Heister and Tranchini (2017) think that Ekman reduction is a too procedure and TCPE has restriction to use If is derivation ⊥ from →¬φ] and [¬φ → φ] and the application of Ekman reduction the derivation generates a looping reduction, TCPE says that the a TCPE overgenerates the of genuine paradoxes because makes non-paradoxical derivation a T-paradox. result their Proposition Let be a natural deduction consisting of →− and ¬−rules. the set of reductions process, SE has an open derivation of ⊥ [φ →¬φ] and [¬φ → φ] which a non-terminating reduction sequence, and is not normalizable. Proof. Two claims prove result. Claim is an open derivation of ⊥ from [φ and
 test and raises the problem of OvergeneraTion. IN SeCtiOn 3.4, As an ObseRvation method fOR evaLuating a proper reductioN, we prOpOSe TrANsLatioN test to DEtECT erRoRs In bOtH ekMan-tyPe aNd CrabbÉ's cases. We wIll SuGgest the requIReMent of a proPer Reduction thaT a pRoper rEdUctIOn proCedUre muSt not iNTroducE any unnecEsSAry preMIse or deTOUr. tranSlation test will be USeFUl to assess a proPer redUcTIoN WIth RegArd to the reQuIremeNTs. 3.1 Ekman'S paRADOx LET us suppose thaT there is a deRIvaTion saTiSfyINg TCPE Which DoES noT formalize a GenuIne paradoX. Then, TcpE overgENerates The scoPe oF geNuinE PaRaDoxEs ANd tHE dEriVAtiOn 92 can be a CoUnTerexAmplE TO tcPE. IN thIs seCtion, We will see EkmaN's pAradOX whIch caUses tHe prObLem of OvergeNeratIoN. Schroeder-HeistEr anD TranchinI (2017) puT fOrwArD a couNTerexaMplE to tCPE takEn from EKMan (1998) In ORDEr To show that TCPE is a tOo COArSe criterIon for A (GeNuINe) paradoX. EKmaN (1998) obsERVes thE folLOwIng form oF derivATiOn And its rEdUction. [Ψ → φ] [Φ → ψ] D Φ → E ψ → e φ ⊵E D φ wE wilL call ψ In → E−rule aN EkmaN Maximum formula, ANd this reductiON wILL bE CallEd EKman reductiOn prOCess. schrOEdEr-HEIster And TrAnCHiNI (2017) think that Ekman reduCtIon is a Too loOse reduction pRocedure anD tcpE has no rEstrICtIOn to use it. If theRe is a Derivation OF ⊥ from [φ →¬φ] aNd [¬φ → φ] aNd the appLication oF eKman reduCtiOn tO thE deRIVaTion generates A LOopiNg ReductiOn, TcPE says ThaT thE deRivAtIon is a T-paRadox. TCPe oVeRgEnEraTes thE Scope of gEnUinE pAraDoxes BEcause It makEs thE nOn-PAraDoxical DErIVAtioN a t-pAradOx. THe FolloWing REsuLt is theIr counterExaMPle tO TcPe. ProposItion 3.1.1. Let SE be a NaTural deducTiOn sYstem cONSisting oF →− and ¬−rules. If the set of reduCTions of sE iNcludEs EkMan reductIon ProcesS, SE HAs an opEn deriVatioN oF ⊥ frOM [Φ →¬φ] and [¬Φ → Φ] WhIch GeNerates a noN-TErmInatiNg ReduCtion seQuence, and is not normALizAble. Proof. Two cLaiMs prOVE tHe rESuLT. ClAiM 1. theRE Is an open derivatIon of ⊥ from [φ →¬Φ] aND
 test and raises the probl em of over gener ati on.  I n Se ctio n 3.4, as an o b serv ation method for evalu ating  a prop e rreduc tion, w e  p r o pos eTr ans la t io n tes t t o detec t errors i n b ot h Ekman-type an d Crabbé's  ca ses. We will  su ggestth e r e quire men t ofa prop e r redu ction tha ta  prope r  reduct i o nproc edure must not in t ro d uce any unnece ssarypr e mi s e  or  de tour. Tran sl ation test wi l lb e use f ul to assessa proper re d uct ion wi th  re g ard to  there q uir ements. 3.1  Ekm an's Para dox Le t  us sup p ose tha t ther e i s a  der i va ti onsa t isf y in g T C PEwhich do es  n ot fo rmal i z e a ge nui ne p arado x. Then, TCPE  ov erge n era tes t he sc opeof  genu ine pa radox es  and the deriva tion  92 can b e a  c oun te rexam p le toTCP E.In this  sectio n , w ew i l lsee Ekman's parado xw h ic h causes  the p r ob le m  of over ge ner atio n .  Schr oede r -H eister a nd Tra n ch in i (2017 )put fo rw ard  acount e rexa mple t o TCPE t akenf rom Ekman (199 8 ) in order to sh o w  t h at T CPE  is a too c oars e  cri teri o nfor a (ge nuine )p ar a dox. Ekman (1998) o bs ervesthe f ollowing form  of deriva t i o n and it s re d uc t ion. [ψ → φ] [ φ → ψ ] D φ → Eψ  → E φ ⊵ E D φ  We will  call ψ i n → E−rule  an  Ek man  ma x i mu m formula, an d this  r eductio n w ill becal led  Ek man  r eductionprocess.  S ch ro ed er- Heist e r and Tr an chi ni  (2 017)t hink t hat E kman  r ed u cti on is a to o loos ere duct ion  p roced urea ndTCPE ha s no rest ric t ionto  u se it.If there is a  d erivationof  ⊥from [ φ →¬φ] and  [¬φ → φ] and the appli c ation o f E kmanredu ction tothe  deriv ati o n gene ratesa loo pi ngr e ducti o n ,TCP Esays thatt h e d eriva ti on i s a T-p aradox. TCPE overg e ner ates the scop e o f ge n u in e p a ra d oxe sb eca u s e it makes thenon-parado xi c al  derivatio n  aT- paradox . The f ollow i ng resu lt is the ir counte re xamp l e  to  TCPE. Pro position  3.1.1. L e t SEb ea nat ura l dedu ct ion  syst em con s ist ing o f →− a nd  ¬−rul es. I fthe setof reductions of SE inc ludesEkman  re duction p roc e ss,  SE has a n op en derivat ion  of  ⊥ fr om[ φ →¬φ ] an d  [ ¬φ→  φ] w hich generates anon - t er minating re d u c tio n seq uen c e, and  isnot normalizable. Proof. Two cla imsp r ove  th e  res ul t. Claim 1. Th ere  i s an opende rivation of  ⊥ from[φ →¬φ]and
 test_and raises_the problem of overgeneration._In Section_3.4,_as an_observation_method for evaluating_a proper reduction,_we propose Translation test_to detect errors_in_both Ekman-type and Crabbé's cases. We will suggest the requirement of a proper reduction_that_a proper_reduction_procedure_must not introduce any unnecessary_premise or detour. Translation test_will be_useful to assess a proper reduction with regard_to_the requirements. 3.1_Ekman's Paradox Let us suppose that there is a_derivation satisfying TCPE which does not_formalize a genuine_paradox._Then,_TCPE overgenerates the scope_of genuine paradoxes and the derivation_92 can be a counterexample to_TCPE. In this section, we will see_Ekman's paradox which causes the problem_of overgeneration. Schroeder-Heister and Tranchini_(2017) put_forward a counterexample to TCPE_taken from Ekman_(1998) in_order to show_that TCPE is a too coarse_criterion for a_(genuine) paradox. Ekman (1998) observes the_following_form of derivation_and_its_reduction. [ψ_→ φ] [φ_→_ψ] D_φ_→ E ψ → E φ_⊵E_D φ We will call ψ in_→ E−rule an Ekman_maximum_formula, and this reduction_will be called Ekman reduction_process. Schroeder-Heister and Tranchini (2017) think_that Ekman_reduction is_a too loose reduction procedure and TCPE has no restriction to_use it. If there is a_derivation of ⊥ from_[φ →¬φ]_and_[¬φ → φ]_and_the application_of Ekman reduction to the derivation generates_a looping_reduction, TCPE says that the derivation_is a T-paradox. TCPE_overgenerates_the scope of genuine paradoxes because_it makes the non-paradoxical derivation a_T-paradox. The following result is_their_counterexample_to TCPE. Proposition 3.1.1. Let_SE be a natural deduction system_consisting of →−_and ¬−rules. If the set of reductions_of_SE includes Ekman reduction process, SE_has_an open derivation of ⊥ from_[φ_→¬φ]_and [¬φ → φ] which_generates a non-terminating reduction sequence, and_is not normalizable. Proof. Two claims prove the result._Claim 1. There_is an open derivation of_⊥_from_[φ →¬φ] and