 form. We have discussed in Section 3.2.2 that, for a successful answer to the problems caused by Ekmantype reductions, the condition (iv) is not enough and Tennant needs to focus on which reduction procedures are proper. As Schroeder-Heister and Tranchini (2017, p. 571) note, whether a derivation is in (full) normal form relies on the choice of reduction procedures. They attempt to show that Ekman reduction makes two derivations which represent distinct proofs belonging to the same equivalence class and so it is a wrong process. Tennant (2016, 2017) may not wish to follow their line of thought and does believe that our mere choice of generalized elimination rules would be a more modest solution than theirs. We already have seen that Tennant's solution is not successful. A successful solution should provide a relevant criterion for a proper reduction process which can restrict illegitimate reductions, such as Ekman-type reduction procedures. In the next two sections, we shall investigate plausible tests to block the illegitimate reductions. 103 3.3 Schroeder-Heister and Tranchini's Triviality Test As we have seen in Section 3.2.2, GEkman paradox shows that TCPL overgenerates. If GEkman reduction is not proper, GEkman paradox is not a suitable counterexample to Tennant's criterion for paradoxicality. The main issue is to be which reduction process is a proper one. For instance, Schroeder-Heister and Tranchini (2017, Sec. 5) pay attention to finding a criterion for the proper reduction process. So, instead of considering Tennant's condition (iv) of TCPL, we have the revised version of Tennant's criterion for paradoxicality: The Revised Version of Tennant's Criterion for Paradoxicality(RTCP): Let S be a natural deduction system relative to a set R of reduction procedures. D be any derivation in S. D is a T-paradox if and only if (i) D is a (closed or open) derivation of ⊥, (ii) id est inferences (or rules) are used in D, (iii) a reduction procedure of D generates a non-terminating reduction sequence, such as a reduction loop, (iv) any reduction procedure in R is proper. With regard to RTCP, it should be asked how we find the criterion for a set of proper reduction procedures. Schroeder-
form. We have discussed in Section 3.2.2 that, for a successful answer to the trouble cause by Ekmantype reductions, the condition (iv) is not enough and Tennant want to focus on which reduction procedure are proper. As Schroeder - Heister and Tranchini (2017, p. 571) eminence, whether a derivation is in (entire) normal form relies on the option of reduction procedures. They attempt to show that Ekman decrease makes two derivations which constitute distinct proofs belong to the same equivalence class and indeed it is a wrong process. Tennant (2016, 2017) may not wish to follow their line of idea and does believe that our mere choice of generalized elimination rules would be a more modest solution than theirs. We already have see that Tennant's solution is not successful. A successful solution should provide a relevant criterion for a proper reduction process which can restrict illegitimate reduction, such as Ekman - type reduction procedures. In the next two sections, we shall investigate plausible tests to block the illegitimate reductions. 103 3.3 Schroeder - Heister and Tranchini's Triviality Test As we have seen in part 3.2.2, GEkman paradox shows that TCPL overgenerates. If GEkman reduction is not proper, GEkman paradox is not a desirable counterexample to Tennant's standard for paradoxicality. The main issue is to be which reduction process is a proper one. For case, Schroeder - Heister and Tranchini (2017, Sec. 5) pay attention to finding a criterion for the proper decrease process. So, instead of considering Tennant's condition (iv) of TCPL, we have the revised version of Tennant's criterion for paradoxicality: The Revised Version of Tennant's Criterion for Paradoxicality(RTCP ): Let S be a natural deduction arrangement relative to a determined R of reduction procedures. D be any derivation in S. D is a T - paradox if and merely if (i) D is a (closed or open) deriving of ⊥, (ii) i d est inferences (or rules) are used in D, (iii) a decrease procedure of D generates a non - terminating reduction sequence, such as a reduction cringle, (iv) any reduction operation in R is proper. With respect to RTCP, it should be asked how we receive the criterion for a set of proper reduction procedures. Schroeder-
 fogm. We have discussed in Rection 3.2.2 that, for a surcessfum answer to the problems caused by Enmqntypt reductions, the covdition (in) is not wnoujh and Tennant nxsds to njcus ln wkirh reduction prpcedures ase proper. As Swhfozder-Heister and Tranchini (2017, p. 571) note, wrether s ferivation is yn (fllj) nodmal form relies on the choice of deductimn procedures. They attempt to show that Ekmwn reduction makes two derivajjonf which reprerent distinct proofs bglonging to the same equivalence clasx and so ij js d wrong proress. Tvnnant (2016, 2017) may kpt wisv to foklow their linc of vhoutht and does believe vhat our mere choice of generdlnzed elimination rulew qould be d mofw mudeat sklutioj tian theirs. Se already yave seen that Tennsne'w solution is not stcsessful. A successful solution should prmvise a relevant criterion for a proper reductiln procesf which can restrict illegitimate reductions, such as Eioan-uyic reaychion procedures. In the next two sections, we srzlk pnvestigate plauslble tests to blocl hhr illegitimate reduccjohs. 103 3.3 Schroeder-Heisher and Tranxhini's Tryviakity Test As we have seen ib Section 3.2.2, GVkmab paradox shows thct TCPL overyenerajes. If GEkman reduction is noc propsr, GEkman pwradox is vot a suitable cuunnerefample to Tennant's criterijn for pacadoxncality. Ghe kain ifsue is to be wmhch reduction procfss id d proper oje. For instance, Schroeder-Heistec and Tranchimi (2017, Svc. 5) pay actentipn to finding a criterion fpr the 'roper reduction procesv. So, insteaq of considerhjg Tennant's rondition (iv) if TXPL, we fxve the revisec version of Tennanr's criterion for peradusicality: The Rernstd Version of Temnavt's Cgitxrion xor Paradoxiwaligy(RGVP): Leg S be a naburxl drduction system reladive to a set R of redicbion procgdures. D fe any derivayion in S. D is a T-karadoe if aid onlu is (i) D is a (closed or open) deribation of ⊥, (il) id est infewenccs (ot rules) are used in D, (iii) a reduction procedure of D generates a non-rerminating reductipk sequence, snch as a reducthon loop, (iv) any reduxtion procedure ik R is proper. With regzrd to RTCP, it should be asked how we find the criterion for a set of proper reductiin prorequres. Schrosder-
form. We have discussed in Section 3.2.2 a answer to problems caused by is enough and Tennant to focus on reduction procedures are proper. As Schroeder-Heister Tranchini (2017, p. 571) note, whether a derivation is in (full) normal form on the choice of reduction procedures. They attempt to show that Ekman reduction two which distinct belonging to the same equivalence class and so it is a wrong process. Tennant (2016, 2017) not wish to follow their line of thought does believe that our choice of generalized elimination rules be more modest than We have seen that solution is not successful. A successful solution should provide a relevant criterion for a proper reduction process can restrict such as reduction In next two sections, investigate plausible tests to block the 3.3 Schroeder-Heister and Tranchini's Triviality Test As we seen in 3.2.2, GEkman paradox shows that TCPL If GEkman reduction is not proper, GEkman paradox not a suitable counterexample to Tennant's criterion for paradoxicality. The main issue is to be process is a proper For instance, Schroeder-Heister Tranchini Sec. pay to finding criterion for the proper reduction process. So, instead of considering Tennant's (iv) of TCPL, we have the revised version of Tennant's paradoxicality: Revised Version of Criterion for Paradoxicality(RTCP): Let be natural deduction system relative set of be derivation S. D is a if and only if (i) is a (closed or id est inferences (or rules) are used in (iii) a reduction procedure of D generates non-terminating reduction sequence, such as a reduction loop, (iv) any reduction procedure R is regard to RTCP, it should be asked how find the criterion for set of proper reduction procedures. Schroeder-
 form. We have discussed in SectIon 3.2.2 that, for A succEssFul AnSwer To thE problems causeD By EkMantype reductions, the coNditiOn (IV) is nOT eNough And TennANt NEEds To FoCus On WHiCh redUctIon procEdures are pRopEr. as Schroeder-HEIsTer and TranChiNi (2017, p. 571) note, whethEr a DerivaTiOn iS In (fulL) noRmal fOrm relIEs on thE choice of ReDUction PRocedurES. thEy atTempt to show that EkMAn REduction makes tWo deriVaTIoNS WhiCh rEpresent diStInct pROofs belONgING To tHE same equivaleNce class and SO it Is a wroNg ProCEss. TenNant (2016, 2017) mAy NOt wIsh to follow TheiR line of thOught aND does beLIeve thaT our meRe cHoiCe of GEnErAliZeD EliMInAtiON ruLes would Be A mOre moDest SOLUTion ThaN theIrs. We Already have seEn tHat TENnaNt's soLutioN is nOt SucceSsful. A SucceSsFul solution shouLd prOvide a relEvaNt CriTeRion fOR a propEr rEduCtion prOcess whICh cAn RESTrIct illegitimate redUcTIOnS, such as EKman-tyPE rEdUCtion proCeDurEs. In THE next Two sECtIons, we shAll invEStIgAte plauSiBle tesTs To bLocK the iLLegiTimate ReductioNs. 103 3.3 SchROeder-Heister anD tranchini's TriVIaLITy tEst AS we Have seen in SEctiON 3.2.2, GEkMan pARaDox SHows tHat TCpL OVeRGenerates. If GEkman reDuCtion iS not pRoper, GEkman paRadox is not A SUItable coUnteRExAMple to Tennant's CriteRion for parADoxicaliTy. The Main issuE is to be whICH reductiOn pRocEss Is a PROpEr one. For instaNCE, SchRoEder-HeiSteR and TraNchIni (2017, sec. 5) Pay AtTention to Finding a CrItErIoN foR the pROper reduCtIon PrOceSs. So, iNStead oF consIderInG TENnaNt's condITiON (Iv) of tCpL, We haVe tHe RevisEd veRSioN of TennAnt's criteRioN For pArAdOxicaliTy: The Revised VErSion of TennAnT's CRiteriON For ParadOxicality(RTCP): Let S be a natURal deduCtiOn sysTem rElative to A seT R of reDucTIon proCedureS. D be aNy DerIVAtion IN s. D Is a t-pAradox if anD ONly If (i) D iS a (ClosEd or opeN) derivation of ⊥, (ii) id eST inFerences (or rulEs) aRe usED In d, (iiI) A rEDucTiON prOCEdure of D generatEs a non-termInATiNg reductioN SeqUeNce, such As a reduCtion LOop, (iv) anY reductioN procedurE iN R is PROpeR. With regarD to RTCP, iT should be ASked hOW wE find The CriterIoN foR a set Of propER reDuctiOn procEdUres. ScHroedEr-
 form. We have discussed i n Section3.2.2  th at,  f or a  suc cessful answer to t he problems caused byEkman ty p e re d uc tions , the c o nd i t ion  ( iv ) i sn ot  enou ghand Ten nant needs  to  f ocus on whic h  r eduction p roc edures are p rop er. As  S chr o eder- Hei sterand Tr a nchini  (2017, p .5 71) no t e, whet h e ra de rivation is in (f u ll )  normal form r elieson th e cho ice  of reduct io n pro c edures. Th e y att e mpt to show t hat Ekman r e duc tion m ak est wo der ivati on s  wh ich represe nt d istinct p roofsb elongin g  to the  sameequ iva lenc e  c la ssan d  so it  is a w rong pro ce ss . Ten nant ( 2 0 16,201 7) m ay no t wish to fol low  the i r l ine o f tho ught  a nd do es bel ieveth at our mere cho iceof genera liz ed  el im inati o n rule s w oul d be amore mo d est  s o l u ti on than theirs. We  a l r ea dy haveseen t h at  T e nnant'sso lut ioni s  notsucc e ss ful. A s uccess f ul  s olution  s houldpr ovi dea rel e vant  crite rion for  a pr o per reductionp rocess whichc an r es t rict  il legitimateredu c tion s, s u ch  as Ekman -type  r e du c tion procedures. In  t he nex t two  sections, we  shall inv e s t igate pl ausi b le tests to block  theillegitima t e reduct ions.  103 3.3  Schroede r - Heisterand  Tr anc hin i ' sTriviality Te s t  Aswe  have s een  in Sec tio n 3 .2. 2,GE kman para dox show sth at  T CPL  over g enerates .IfGE kma n red u ctionis no t pr op er ,  GE kman pa r ad o x  isno ta su ita bl e cou nter e xam ple toTennant's  cr i teri on  f or para doxicality. T he  main issu eisto bew h ich redu ction process is a prop e r one.For  inst ance , Schroed er- Heiste r a n d Tran chini(2017 ,Sec . 5) pa y at ten ti on to find i n g a  crit er ionfor the  proper reductionp roc ess. So, inst ead  ofc o ns ide r in g  Te nn a nt' s condition (iv)of TCPL, w eh av e the revi s edve rsion o f Tenna nt'sc riterio n for par adoxicali ty : Th e Rev ised Versi on of Te nnant's C r iteri o nfor P ara doxica li ty( RTCP) : LetS  be  a na turalde ductio n sys te m relati ve to a set R of reduct ion pr ocedu res . D be an y d e riv ation inS. D  is a T-pa rad oxif an d o n ly if  (i) Disa  (clo sedo r open) d e ri vat i o nof ⊥, (ii)i d est  infe ren c es (or  rul es) are used in D ,  (iii) a reduc tion p roc edu r e of  D  generates a n on- te r m inatingre duction seq uence, s uc h  as a  reduc tion l oop, (i v )  a n y redu ctio n p rocedureinRi s prope r.  W i th reg ardto  RTCP,  it sh o uldb e  asked how we fi nd th e crite r ion  foraset ofp rope r reductio n procedure s. Sch roed er-
 form._We have_discussed in Section 3.2.2_that, for_a_successful answer_to_the problems caused_by Ekmantype reductions,_the condition (iv) is_not enough and_Tennant_needs to focus on which reduction procedures are proper. As Schroeder-Heister and Tranchini (2017,_p._571) note,_whether_a_derivation is in (full) normal_form relies on the choice_of reduction_procedures. They attempt to show that Ekman reduction_makes_two derivations which_represent distinct proofs belonging to the same equivalence class_and so it is a wrong_process. Tennant (2016,_2017)_may_not wish to follow_their line of thought and does_believe that our mere choice of_generalized elimination rules would be a more_modest solution than theirs. We already_have seen that Tennant's solution_is not_successful. A successful solution should_provide a relevant_criterion for_a proper reduction_process which can restrict illegitimate reductions,_such as Ekman-type_reduction procedures. In the next two_sections,_we shall investigate_plausible_tests_to block_the illegitimate reductions._103_3.3 Schroeder-Heister_and_Tranchini's Triviality Test As we have_seen_in Section 3.2.2, GEkman paradox shows that_TCPL overgenerates. If GEkman_reduction_is not proper, GEkman_paradox is not a suitable_counterexample to Tennant's criterion for paradoxicality._The main_issue is_to be which reduction process is a proper one. For instance,_Schroeder-Heister and Tranchini (2017, Sec. 5)_pay attention to finding_a criterion_for_the proper reduction_process._So, instead_of considering Tennant's condition (iv) of TCPL,_we have_the revised version of Tennant's criterion_for paradoxicality: The Revised_Version_of Tennant's Criterion for Paradoxicality(RTCP): Let_S be a natural deduction system_relative to a set R_of_reduction_procedures. D be any derivation_in S. D is a T-paradox_if and only_if (i) D is a (closed or_open)_derivation of ⊥, (ii) id est_inferences_(or rules) are used in D,_(iii)_a_reduction procedure of D generates_a non-terminating reduction sequence, such as_a reduction loop, (iv) any reduction procedure in R_is proper. With_regard to RTCP, it should_be_asked_how we find the criterion for a set of proper_reduction procedures._Schroeder-