 which satisfies TCPE but is not about any genuine paradoxes, the derivation shows that TCPE overgenerates in the sense that TCPE makes intuitively non-paradoxical derivation paradoxical. Tennant (1982) sets his criterion for paradoxicality, TCPE, that genuine paradoxes are distinguished by having non-terminating reduction sequences of the derivation of ⊥ involved. His early version of the criterion has a criticism from Schroeder-Heister and Tranchini (2017) that it is a too coarse criterion for paradoxicality. They suggest a counterexample to Tennant's early version of the criterion taken from Jan Ekman (1998), called Ekman's paradox. The case shows that Tennant's criterion overgenerates in the sense that there exists a derivation which is intuitively non-paradoxical but satisfies the criterion. To solve the 91 problem of overgeneration, Tennant (2016, 2017) has refined his criterion and proposed an additional condition that all elimination rules stated in generalized form should always have their major premises standing proud, with no non-trivial proof work above them. On the other hand, Schroeder-Heister and Tranchini (2017) diagnose that Ekman's paradox uses too loose reduction procedure, called Ekman reduction, and suggest Triviality test to block Ekman reduction process. The present chapter aims to observe whether their solution is satisfactory After introducing Ekman's paradox in Section 3.1, we will see Tennant's solution in Section 3.2 that the choice of generalized elimination rules can block the derivation ⊥ from Ekman's paradox. Then, we will propose Ekman-type reductions stated in generalized form, including the cases suggested by Schroeder-Heister and Tranchini (2018), and argue that Tennant's solution is not successful. Ekman-type cases show that even Tennant's later version of the criterion overgenerates. A promising solution should restrict the use of Ekman-type reduction procedures. Section 3.3 and 3.4 deal with Ekman-type reductions and methods to evaluate them: Triviality and Translation tests. Section 3.3 introduces Schroeder-Heister and Tranchini's Triviality test for a proper reduction and discusses that their notion of a proper reduction would be relative to a based natural deduction system. Moreover, Schroeder-Heister and Tranchini (2018) introduce Crabbé's case which is unaffected by Triviality
which satisfies TCPE but is not about any genuine paradoxes, the derivation testify that TCPE overgenerates in the common sense that TCPE makes intuitively non - paradoxical derivation paradoxical. Tennant (1982) specify his standard for paradoxicality, TCPE, that genuine paradoxes are distinguished by have non - terminating reduction sequences of the derivation of ⊥ involved. His early version of the standard has a criticism from Schroeder - Heister and Tranchini (2017) that it is a too coarse criterion for paradoxicality. They suggest a counterexample to Tennant's early interpretation of the criterion taken from Jan Ekman (1998), called Ekman's paradox. The case shows that Tennant's standard overgenerates in the sense that there exists a derivation which is intuitively non - paradoxical but satisfy the criterion. To solve the 91 trouble of overgeneration, Tennant (2016, 2017) has refined his standard and proposed an additional condition that all elimination rules stated in generalized form should constantly have their major premises standing proud, with no non - trivial proof work above them. On the other hand, Schroeder - Heister and Tranchini (2017) diagnose that Ekman's paradox uses too loose reduction procedure, called Ekman reduction, and suggest Triviality trial to block Ekman reduction summons. The present chapter drive to observe whether their solution is satisfactory After introducing Ekman's paradox in Section 3.1, we will determine Tennant's solution in Section 3.2 that the choice of generalized elimination rules can block the deriving ⊥ from Ekman's paradox. Then, we will propose Ekman - type reductions stated in generalized form, including the font suggested by Schroeder - Heister and Tranchini (2018), and argue that Tennant's solution is not successful. Ekman - type cases show that even Tennant's late version of the standard overgenerates. A bright solution should qualify the use of Ekman - type reduction procedures. Section 3.3 and 3.4 deal with Ekman - type reductions and method to evaluate them: Triviality and Translation tests. department 3.3 introduces Schroeder - Heister and Tranchini's Triviality trial for a proper reduction and discourse that their notion of a proper decrease would be relative to a based lifelike deduction system. Moreover, Schroeder - Heister and Tranchini (2018) introduce Crabbé's case which is unmoved by Triviality
 whlch satisfies TCPE but ir not about any genuinx paradkxes, the derivation shows that TCPE lvwrgentgates in the sense thxt TCPE mwkes intyititely non-paradoxirzl deriyction iaradmeical. Tennant (1982) xets his csiterion for pdrxdlxicality, TCPE, that genuine paradoxqs are cidtinguished by havpnd noh-nevminating reduction sequences of the degivation of ⊥ invokved. His early version of hhe friterion has a crlticism fron Screoeder-Heistef and Tranbkini (2017) that jt is a too coarse criterion fof parcdoxicality. Tyey vuggest a ciuntegexample to Tcmnant's early fersion of the crmteruon taken from Jan Ekkan (1998), called Ekman's paradox. Dhz case shows that Tenbabt's ctiterhon ucereentraves in thf sxnse that tgere exists a derivation which if intuitively hon-parwdjxical but satisfies the criterion. To smlvs the 91 problem of overgwneration, Tennant (2016, 2017) hws refineq his criterion and proposed an additional condithon tiag aol doilination rules stated in generalized form shotmd akways have thelr major premises xtwncyng proud, witf no non-tdivial proof work wbove trem. Ob the othtr hamd, Schroeder-Heister and Trabchini (2017) diaguosw that Ekman's parabox uses too looxe recuction procedure, calleb Ekmah reduction, and suggsrt Triviality tert no bnock Ekman reduction procefs. The prxsent chaptef aiks to jbserve whfther their solution is datisyactosy After ijtroducing Ekman's paradox in Servion 3.1, we will sae Nennant's folutlon in Section 3.2 that the choiee of geueralixed eliminztion rnles can blosk the derivadlon ⊥ from Ekkan's parwdox. Theb, we wiuu propose Ekmam-type redlcuions statwd in generalized norm, khcluding the cawes suggested by Svhruedqr-Jemster dnd Tranchinh (2018), avd xtgue ghat Tennanb's roluyion is not successfgl. Eiman-type cases shoe bhat even Tennant'f later versipn of the criterioj ovecgeneretes. A prjmising solution should restridt the usf on Ekman-type rqducbion procedurex. Section 3.3 and 3.4 deal with Ekman-type rednctions and methods to wvaluate them: Trivicllty and Tranxlatijn tests. Vection 3.3 introduces Wchroeder-Heister snd Tranchini's Triviamity tast flr a proper reduction and discusses that their notion of a proper reductiin would be relatibe tp a bdszd uatural dedncvion system. Moreoner, Schroeder-Heister and Tranchiii (2018) introdgcz Crabbé's case which is unaffevtdd by Trivialkty
which satisfies TCPE but is not about paradoxes, derivation shows TCPE overgenerates in intuitively derivation paradoxical. Tennant sets his criterion paradoxicality, TCPE, that genuine paradoxes are by having non-terminating reduction sequences of the derivation of ⊥ involved. His early of the criterion has a criticism from Schroeder-Heister and Tranchini (2017) that it a coarse for They suggest a counterexample to Tennant's early version of the criterion taken from Jan Ekman (1998), Ekman's paradox. The case shows that Tennant's criterion in the sense that exists a derivation which is non-paradoxical satisfies the To the problem of overgeneration, (2016, 2017) has refined his criterion and proposed an additional condition that all elimination rules stated in form should their major standing with non-trivial proof work On the other hand, Schroeder-Heister and that Ekman's paradox uses too loose reduction procedure, Ekman reduction, suggest Triviality test to block Ekman process. The present chapter aims to observe whether solution is satisfactory After introducing Ekman's paradox in Section 3.1, we will see Tennant's solution 3.2 that the choice generalized elimination rules block derivation from paradox. Then, will propose Ekman-type reductions stated in generalized form, including the cases by Schroeder-Heister and Tranchini (2018), and argue that Tennant's solution successful. cases show that Tennant's later version of criterion A promising solution should use Ekman-type 3.3 3.4 with Ekman-type reductions and to evaluate them: Triviality and tests. Section 3.3 introduces for a proper reduction and discusses that their of a proper reduction would be relative a based natural deduction system. Moreover, Schroeder-Heister and Tranchini (2018) introduce Crabbé's which is Triviality
 which satisfies TCPE but is noT about any gEnuinE paRadOxEs, thE derIvation shows thAT TCPe overgenerates in the senSe thaT TcpE maKEs IntuiTively nON-pARAdoXiCaL deRiVAtIon paRadOxical. TEnnant (1982) sets His CrIterion for paRAdOxicality, TcPE, That genuine pAraDoxes aRe DisTInguiSheD by haVing noN-TerminAting reduCtIOn sequENces of tHE DeRivaTion of ⊥ involved. His EArLY version of the cRiteriOn HAs A CRitIciSm from SchrOeDer-HeISter and tRaNCHIni (2017) THat it is a too coArse criteriON foR paradOxIcaLIty. TheY suggEsT A coUnterexamplE to TEnnant's eaRly verSIon of thE CriteriOn takeN frOm JAn EkMAn (1998), CaLleD EKMan'S PaRadOX. ThE case shoWs ThAt TenNant'S CRITeriOn oVergEneraTes in the sense ThaT theRE exIsts a DerivAtioN wHich iS intuiTivelY nOn-paradoxical buT satIsfies the CriTeRioN. TO solvE The 91 proBleM of OvergenEration, tEnnAnT (2016, 2017) HAS rEfined his criterion AnD PRoPosed an aDditioNAl CoNDition thAt All ElimINAtion RuleS StAted in geNeraliZEd FoRm shoulD aLways hAvE thEir Major PRemiSes staNding proUd, witH No non-trivial prOOf work above thEM. ON THe OTher HanD, Schroeder-HEistER and tranCHiNi (2017) dIAgnosE that ekMAn'S Paradox uses too loose ReDuctioN procEdure, called EkMan reductiON, ANd suggesT TriVIaLIty test to block ekman Reduction pROcess. The PreseNt chapteR aims to obSERve whethEr tHeiR soLutION iS satisfactory aFTer iNtRoducinG EkMan's parAdoX in secTioN 3.1, wE will see TEnnant's sOlUtIoN iN SeCtion 3.2 THat the chOiCe oF gEneRalizED elimiNatioN rulEs CaN BloCk the deRIvATIon ⊥ fRoM EKman'S paRaDox. ThEn, we WIll Propose ekman-type RedUCtioNs StAted in gEneralized forM, iNcluding thE cAseS suggeSTEd by SchrOeder-Heister and TranchinI (2018), And arguE thAt TenNant'S solution Is nOt succEssFUl. EkmaN-type cAses sHoW thAT Even TENNaNt's LaTer version OF The CriteRiOn ovErgenerAtes. A promising soluTIon Should restricT thE use OF ekMan-TYpE RedUcTIon PROcedures. Section 3.3 And 3.4 deal witH EKMaN-type reducTIonS aNd methoDs to evaLuate THem: TrivIality and translatiOn TestS. sEctIon 3.3 introduCes SchroEder-HeistER and TRAnChini'S TrIvialiTy TesT for a Proper REduCtion And disCuSses thAt theIr Notion of A proper reduction would be RelatiVe to a BasEd natural DedUCtiOn system. MOreoVer, SchroedEr-HEisTer anD TrANchinI (2018) intROdUce cRabbé'S casE Which is unAFfEctED By triviality
 which satisfies TCPE butis not abo ut an y g enu in e pa rado xes, the deriv a tion  shows that TCPE overg enera te s  int he  sens e thatT CP E mak es  i ntu it i ve ly no n-p aradoxi cal deriva tio nparadoxical. Te nnant (198 2)sets his cri ter ion fo rpar a doxic ali ty, T CPE, t h at gen uine para do x es are disting u i sh ed b y having non-term i na t ing reductionsequen ce s  o f the  de rivation o f⊥ inv o lved. H i se a r lyv ersion of the  criterionh asa crit ic ism from S chroe de r -He ister and T ranc hini (201 7) tha t  it isa  too co arse c rit eri on f o rpa rad ox i cal i ty . T h eysuggestaco unter exam p l e to T enn ant' s ear ly version of  th e cr i ter ion t akenfrom  J an Ek man (1 998),  c alled Ekman's p arad ox. The c ase  s how sthatT ennant 'scri terionovergen e rat es i n  t he sense that ther ee x is ts a der ivatio n  w hi c h is int ui tiv elyn o n-par adox i ca l but sa tisfie s  t he  criter io n. Toso lve  th e 91p robl em ofovergene ratio n , Tennant (201 6 , 2017) has r e fi n e dh is c rit erion and p ropo s ed a n ad d it ion a l con ditio nt ha t  all elimination ru le s stat ed in  generalizedform shoul d a lways ha ve t h ei r  major premise s sta nding prou d , with n o non -trivial  proof wo r k  above t hem . O n t heo t he r hand, Schro e d er-H ei ster an d T ranchin i ( 201 7)dia gn ose thatEkman'spa ra do xuse s too loose re du cti on  pr ocedu r e, cal led E kman  r ed u cti on, and su g g estTr iv iali tyte st to  blo c k E kman re duction p roc e ss.Th epresent  chapter aims  t o observewh eth er the i r  solutio n is satisfactory After introdu cin g Ekm an's  paradoxinSectio n 3 . 1, wewill s ee Te nn ant ' s  solu t i on  in  S ection 3.2 t hat  thech oice  of gen eralized eliminati o n r ules can bloc k t he d e r iv ati o n⊥  fr om Ekm a n 's paradox. The n, we will  p r op ose Ekman- t ype  r eductio ns stat ed in general ized form , includi ng  the c ase s suggeste d by Sch roeder-He i stera nd  Tran chi ni (20 18 ),and a rgue t h atTenna nt's s ol utionis no tsuccessf ul. Ekman-type cases sh ow tha t eve n T ennant'slat e r v ersion of  the  criterion  ov erg enera tes .  A pr omis i ng  so l ution  sho u ld restri c tthe u se  of Ekman-t y p e  re ducti onp rocedu res.  Section 3.3 and3 .4 deal with E kman - t ype  re d ucti on s and methodstoev a l uate the m:  Triviality  and Tra ns l ation  tests . Sect ion 3.3 i nt r oduces  Sch roe der-Heist eran d  Tranch in i' s  Trivi alit ytest f or a p r oper r eduction and dis cusse s thatt hei r not io n of ap rope r reductio n would berelati ve t o a b ased na tu ral de duc ti on system. Moreover,  Schr oeder-H ei ster  an d Tran chin i (2018 ) in tr odu ce Crabbé ' s  c a se  w h ich  isunaff ec tedby Trivia l ity
 which_satisfies TCPE_but is not about_any genuine_paradoxes,_the derivation_shows_that TCPE overgenerates_in the sense_that TCPE makes intuitively_non-paradoxical derivation paradoxical._Tennant_(1982) sets his criterion for paradoxicality, TCPE, that genuine paradoxes are distinguished by having_non-terminating_reduction sequences_of_the_derivation of ⊥ involved. His_early version of the criterion_has a_criticism from Schroeder-Heister and Tranchini (2017) that it_is_a too coarse_criterion for paradoxicality. They suggest a counterexample to Tennant's_early version of the criterion taken_from Jan Ekman_(1998),_called_Ekman's paradox. The case_shows that Tennant's criterion overgenerates in_the sense that there exists a_derivation which is intuitively non-paradoxical but satisfies_the criterion. To solve the 91_problem of overgeneration, Tennant (2016,_2017) has_refined his criterion and proposed_an additional condition_that all_elimination rules stated_in generalized form should always have_their major premises_standing proud, with no non-trivial proof_work_above them. On_the_other_hand, Schroeder-Heister_and Tranchini (2017)_diagnose_that Ekman's_paradox_uses too loose reduction procedure, called_Ekman_reduction, and suggest Triviality test to block_Ekman reduction process. The_present_chapter aims to observe_whether their solution is satisfactory_After introducing Ekman's paradox in Section_3.1, we_will see_Tennant's solution in Section 3.2 that the choice of generalized elimination_rules can block the derivation ⊥_from Ekman's paradox. Then,_we will_propose_Ekman-type reductions stated_in_generalized form,_including the cases suggested by Schroeder-Heister and_Tranchini (2018),_and argue that Tennant's solution is_not successful. Ekman-type cases_show_that even Tennant's later version of_the criterion overgenerates. A promising solution_should restrict the use of_Ekman-type_reduction_procedures. Section 3.3 and 3.4_deal with Ekman-type reductions and methods_to evaluate them:_Triviality and Translation tests. Section 3.3 introduces_Schroeder-Heister_and Tranchini's Triviality test for a_proper_reduction and discusses that their notion_of_a_proper reduction would be relative_to a based natural deduction system._Moreover, Schroeder-Heister and Tranchini (2018) introduce Crabbé's case which_is unaffected by_Triviality