4, our diagnosis says that a non-terminating reduction sequence does not occur if a derivation in question includes (i) a major premise which has no reduction process to eliminate it or (ii) a formula having a principal constant which has no reduction procedure to get rid of it. We have suggested an additional condition to TCPE that a derivation formalizing a genuine paradox only uses harmonious rules. With the additional condition, any counterexample in Section 2.2 and 2.3 which causes the problem of undergeneration can be singled out. In Chapter 3, we have focused on the problem of overgeneration and examined (G)Ekman's paradox. Tennant (2016) claims that the overgeneration problem raised by Ekman's paradox is solved by using generalized elimination rules. However, we have argued that even when we use generalized elimination rules, there is a (G)Ekman's paradox which shows that TCPL overgenerates. He overlooks the fact that an application of an auxiliary reduction sometimes raises a non-terminating reduction sequence. So we have explored methods to evaluate a proper reduction process, such as Triviality and Translation tests. An assessment of a proper reduction via Triviality test is relative to our choice of rules and a system. 164 Since the properness of Ekman reduction through Triviality test can be dependent on our choice of natural deduction systems, Triviality test does not block every Ekman-type reduction process. For a system-independent method to evaluate a proper reduction, we have proposed Translation test. While Ekman-type reductions in natural deduction can be translated to one in sequent calculus, the test diagnoses that they are detour-making processes and so is not proper. Eventually, from our discussions in Chapter 2 and 3, we have the following criterion for paradoxicality. The Revised Version of Proof-Theoretic Criterion for Paradoxicality: 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, (v) only harmonious rules are applied in D. Chapter 4 centers on the question, '
4, our diagnosis says that a non - terminating reduction sequence does not occur if a deriving in doubt includes (i) a major premise which has no decrease process to eliminate it or (ii) a formula have a principal constant which has no reduction procedure to get rid of it. We have suggested an extra condition to TCPE that a derivation formalizing a genuine paradox merely uses harmonious rules. With the additional condition, any counterexample in Section 2.2 and 2.3 which causes the trouble of undergeneration can be singled out. In Chapter 3, we have focused on the problem of overgeneration and examine (G)Ekman's paradox. Tennant (2016) claims that the overgeneration problem lift by Ekman's paradox is resolve by using generalized elimination rule. However, we have argued that even when we use generalized elimination rules, there be a (G)Ekman's paradox which shows that TCPL overgenerates. He overlooks the fact that an application of an auxiliary reduction sometimes raise a non - terminating reduction sequence. So we have explored methods to evaluate a proper reduction process, such as Triviality and Translation trial. An assessment of a proper reduction via technicality trial is relative to our choice of rules and a arrangement. 164 Since the properness of Ekman reduction through Triviality test can be pendent on our choice of natural deduction systems, Triviality test does not stop every Ekman - type reduction process. For a system - independent method to measure a proper decrease, we have propose Translation test. While Ekman - type reductions in natural deduction can be translated to one in consecutive calculus, the test diagnoses that they are detour - making process and so is not proper. finally, from our discussions in Chapter 2 and 3, we have the following criterion for paradoxicality. The Revised Version of Proof - Theoretic Criterion for Paradoxicality: Let S be a lifelike deduction system relative to a set radius of reduction procedures. five hundred be any derivation in S. D is a T - paradox if and entirely if (i) D is a (closed or open) derivation of ⊥, (ii) i d est inferences (or rules) are used in D, (iii) a reduction operation of D generates a non - terminating reduction succession, such as a reduction loop, (iv) any reduction procedure in R is proper, (v) only harmonious rules are enforce in D. Chapter 4 centers on the question,'
4, oug diagnosis says that a kon-terminating rgdyction sequehce does not occur if a derivation ii quwstiob includes (i) a major pfemise whpch has ni revuction process vk elimikcte if or (ni) a formula havlng a princhpal constant fhkck has no reduction procedure to get wid of ot. We have suggefted wn asditional condition to TCPE that a derivauion formalizing a genuine paradox only uses harlonious rules. With the additiinal xondition, anh countereqcmple in Sedtion 2.2 and 2.3 which causes the prublem of undergguwrahhon can be winglvd out. In Chainer 3, we have fpcused on the iroblxm od overgeneration and xxamined (G)Ekman's parwdox. Tenndnc (2016) claims that the ovwrtenerdtiot pruvleo rzixes by Enmai's paradox js solved bt using generalized ejpkination rulss. Howqvqr, we have argued that even when we use geheralized elimination ryles, there is a (G)Ekmaj's paradov which shows that TCPL overgenerates. He overlookv the wacu bmat xb wpplication of an auxiliary reduction sometimqa tapses a non-terminabing reduction seqiejcr. So we have ebploreb msthods to evaluate a propgr redyction prjcesx, such as Triviality and Trqnslation tefrs. An assessment oy a proper rzductipn vis Triviality test is reuatibe to our cjoice of djles and a systeo. 164 Xitce the kfoperness of Ekmag reductiin tkrough Tfivislity eest can bf depctdent on our choicf of uaturdl deductiln systems, Triviality test does not block evety Akmdn-type rzductipn process. Fow a system-indekendent mzthod go evaluatv a propec reduction, re have propovgd Translatioi test. Whyle Wkmab-type rdauctions in nayural dedlcuion can bw translated to onc in rsquent calculus, tht rest diagnoses thxt ehvy ere dqdour-making psocerser and ro is not pvopdr. Efentually, from our dhscuasions in Chapter 2 akd 3, we hace the fjllowing critrrion for paradoxifalitb. The Cevisec Vgrsion of Proof-Theoretic Criterjon for Pwraqoxicality: Lqt S be a naturak deduction system relative to a set R if reduction procgdures. D be any derivctlon in S. D ix a T-[aradox ix and only if (i) D is a (closed or open) derivation of ⊥, (ii) id est itferejces (or rules) are used in D, (iii) a reduction procedure of D generates a nob-termiiaeing reductjon xequetcz, sbch as w revurtion loop, (iv) any reduction procedure in R is pruper, (v) otlv harmonious rules are appliec kn D. Chapter 4 centers on the question, '
4, our diagnosis says that a non-terminating does occur if derivation in question which no reduction process eliminate it or a formula having a principal constant has no reduction procedure to get rid of it. We have suggested an condition to TCPE that a derivation formalizing a genuine paradox only uses harmonious With additional any in Section 2.2 and 2.3 which causes the problem of undergeneration can be singled out. In 3, we have focused on the problem of and examined (G)Ekman's paradox. (2016) claims that the overgeneration raised Ekman's paradox solved using elimination rules. However, have argued that even when we use generalized elimination rules, there is a (G)Ekman's paradox which shows TCPL overgenerates. the fact an of auxiliary reduction sometimes non-terminating reduction sequence. So we have evaluate a proper reduction process, such as Triviality Translation tests. assessment of a proper reduction via test is relative to our choice of rules a system. 164 Since the properness of Ekman reduction through Triviality test can be dependent choice of natural deduction Triviality test does block Ekman-type process. a system-independent to evaluate a proper reduction, we have proposed Translation test. While reductions in natural deduction can be translated to one in the diagnoses that they detour-making processes and so not Eventually, from our discussions 2 3, following for The Revised Version of Criterion for Paradoxicality: Let S a natural deduction system of reduction procedures. D be any derivation in D is a T-paradox if and only (i) D is a (closed or open) derivation of ⊥, (ii) id inferences (or used in D, (iii) a reduction procedure of generates a non-terminating reduction such as a reduction loop, (iv) any reduction procedure R proper, (v) harmonious rules are in D. Chapter centers on the
4, our diagnosis says that a non-tErminating ReducTioN seQuEnce Does Not occur if a derIVatiOn in question includes (i) a Major PrEMise WHiCh has No reducTIoN PRocEsS tO elImINaTe it oR (ii) A formulA having a prIncIpAl constant whICh Has no reducTioN procedure to Get Rid of iT. WE haVE suggEstEd an aDditioNAl condItion to TCpE THat a deRIvation FORmAlizIng a genuine paradoX OnLY uses harmoniouS rules. wiTH tHE AddItiOnal conditIoN, any cOUnterexAMpLE IN SeCTion 2.2 and 2.3 which cAuses the proBLem Of undeRgEneRAtion cAn be sInGLed Out. In ChapteR 3, we hAve focuseD on the PRoblem oF OvergenEratioN anD exAminED (G)ekMan'S pARadOX. TEnnANt (2016) cLaims thaT tHe OvergEnerATION proBleM raiSed by ekman's paradox Is sOlveD By uSing gEneraLizeD eLiminAtion rUles. HOwEver, we have argueD thaT even when We uSe GenErAlizeD EliminAtiOn rUles, theRe is a (G)EKMan'S pARADoX which shows that TCPl oVERgEnerates. he overLOoKs THe fact thAt An aPpliCATion oF an aUXiLiary redUction SOmEtImes raiSeS a non-tErMinAtiNg redUCtioN sequeNce. So we hAve exPLored methods to EValuate a propeR ReDUCtIOn prOceSs, such as TriVialITy anD TraNSlAtiON testS. An asSeSSmENt of a proper reductioN vIa TrivIalitY test is relatiVe to our choICE Of rules aNd a sYStEM. 164 Since the propeRness Of Ekman redUCtion thrOugh TRivialitY test can bE DEpendent On oUr cHoiCe oF NAtUral deduction SYStemS, TRivialiTy tEst does Not BloCk eVerY EKman-type rEduction PrOcEsS. FOr a SysteM-IndependEnT meThOd tO evalUAte a prOper rEducTiOn, WE haVe propoSEd tRAnslAtIoN tesT. WhIlE EkmaN-typE RedUctions In natural DedUCtioN cAn Be transLated to one in sEqUent calculUs, The Test diAGNoses thaT they are detour-making proCEsses anD so Is not PropEr. EventuaLly, From ouR diSCussioNs in ChApter 2 AnD 3, we HAVe the FOLlOwiNg Criterion fOR ParAdoxiCaLity. the ReviSed Version of Proof-THEorEtic Criterion For paraDOXiCalITy: 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, (v) onlY haRmoNious RulES are aPpliED iN D. CHApter 4 CentERs on the quEStIon, '
4, our diagnosis says that  a non-ter minat ing  re du ctio n se quence does no t  occ ur if a derivation inquest io n  inc l ud es (i ) a maj o rp r emi se  w hic hh as  no r edu ction p rocess toeli mi nate it or ( i i)  a formula  ha ving a princ ipa l cons ta ntw hichhas  no r educti o n proc edure toge t  rid o f  it. We h av e su ggested an additi o na l  condition toTCPE t ha t  a d eri vat ion formal iz ing a genuine pa r a d oxo nly uses harm onious rule s . W ith th eadd i tional  cond it i on,  any counte rexa mple in S ection 2.2 and 2.3 whi ch cau ses  th e pr o bl em  of  u n der g en era t ion  can besi ng led o ut.I n C hapt er3, w e hav e focused onthe  pro b lem  of o verge nera ti on an d exam ined(G )Ekman's parado x. T ennant (2 016 )cla im s tha t  the o ver gen eration  proble m  ra is e d by  Ekman's paradox i ss o lv ed by us ing ge n er al i zed elim in ati on r u l es. H owev e r,  we have  argue d  t ha t evenwh en weus e g ene raliz e d el iminat ion rule s, th e re is a (G)Ekm a n's paradox w h ic h sh o ws t hat  TCPL overg ener a tes.  Heo ve rlo o ks th e fac tt ha t  an application ofan  auxil iaryreduction som etimes rai s e s  a non-t ermi n at i ng reduction s equen ce. So weh ave expl oredmethodsto evalua t e  a prope r r edu cti onp r oc ess, such asT r ivia li ty andTra nslatio n t est s.Anas sessmentof a pro pe rre du cti on vi a  Trivial it y t es t i s rel a tive t o our  cho ic eo f r ules an d  a s yste m.  1 64 S inc ethe p rope r nes s of Ek man reduc tio n  thr ou gh  Trivia lity test can  b e dependen tonour ch o i ce of na tural deduction systems ,  Trivia lit y tes t do es not bl ock  every  Ek m an-typ e redu ction  p roc e s s. Fo r asys te m-independ e n t m ethod  t o ev aluatea proper reduction ,  we  have propose d T rans l a ti ont es t . W hi l e E k m an-type reducti ons in nat ur a ldeductionc anbe  transl ated to  onei n seque nt calcul us, the t es t di a g nos es that th ey are d etour-mak i ng pr o ce ssesand  so is  n otprope r. Eve n tua lly,from o ur  discu ssion sin Chapt er 2 and 3, we have the  follo wingcri terion fo r p a rad oxicality . Th e RevisedVer sio n ofPro o f-The oret i cCri t erion  for Paradoxic a li ty: L et  S be a nat u r a l d educt ion system  rel ative to a set Ro f reduction pr oced u r es.  Db e an yderivation inS.Di s  a T-par ad ox if and o nly if ( i) D isa (clo sed or  open)d e ri v ationof ⊥ , ( ii) id es t i nf e rences(o rr ules)areus ed inD, (ii i ) ar e duction procedur e ofD gener a tes  a no n- termina t ingreductionsequence, s uch as  a r educt ion loo p,  (iv)any  r eduction p r ocedure i n R i s prope r,  (v)  on ly har moni o u s rul es a re  ap plied inD .  C h ap te r  4cent ers o nthequestion, '
4, our_diagnosis says_that a non-terminating reduction_sequence does_not_occur if_a_derivation in question_includes (i) a_major premise which has_no reduction process_to_eliminate it or (ii) a formula having a principal constant which has no reduction_procedure_to get_rid_of_it. We have suggested an_additional condition to TCPE that_a derivation_formalizing a genuine paradox only uses harmonious rules._With_the additional condition,_any counterexample in Section 2.2 and 2.3 which causes_the problem of undergeneration can be_singled out. In_Chapter_3,_we have focused on_the problem of overgeneration and examined_(G)Ekman's paradox. Tennant (2016) claims that_the overgeneration problem raised by Ekman's paradox_is solved by using generalized elimination_rules. However, we have argued_that even_when we use generalized elimination_rules, there is_a (G)Ekman's_paradox which shows_that TCPL overgenerates. He overlooks the_fact that an_application of an auxiliary reduction sometimes_raises_a non-terminating reduction_sequence._So_we have_explored methods to_evaluate_a proper_reduction_process, such as Triviality and Translation_tests._An assessment of a proper reduction via_Triviality test is relative_to_our choice of rules_and a system. 164 Since_the properness of Ekman reduction through_Triviality test_can be_dependent on our choice of natural deduction systems, Triviality test does_not block every Ekman-type reduction process._For a system-independent method_to evaluate_a_proper reduction, we_have_proposed Translation_test. While Ekman-type reductions in natural deduction_can be_translated to one in sequent calculus,_the test diagnoses that_they_are detour-making processes and so is_not proper. Eventually, from our discussions_in Chapter 2 and 3,_we_have_the following criterion for paradoxicality._The Revised Version of Proof-Theoretic Criterion_for Paradoxicality: 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, (v) only harmonious rules are applied_in D. Chapter 4 centers on the_question, '