 undergeneration, he seems to presume that the application of CR−rule has a defect that the rule conceals the main feature of the paradoxes and proposes the methodological conjecture that genuine paradoxes are never strictly classical. However, the undergeneration problem is not solved by simply accusing CR−rule of disguising the feature because there are cases which do not use CR−rule but show that TCPE undergenerates. Tennant seems not to consider seriously the problem of undergeneration. Section 2.3 proposes some counterexamples to TCPE which represent that TCPE undegenerates, however those counterexamples do not employ CR−rule. We will see that the occurrence of a non-terminating reduction sequence relies on our choice of reduction procedures. Section 2.4 deals with the question of what makes a non-terminating reduction sequence stops. With some observations, we propose a possible diagnosis 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. Then, we suggest an additional condition to TCPE that a derivation formalizing a genuine paradox only uses harmonious rules. If the suggested condition is acceptable, the condition can solve the problem of undergeneration. 35 2.1 Preliminaries: Generalized Elimination Rules and Harmony Relation. We will see in the next chapter that the later version of Tennant's criterion for paradoxicality (TCPL) accepts an additional condition that all elimination rules must be stated in the generalized form.3 In addition, the early version of Tennant's criterion TCPE does not restrict the form of elimination rules. There would be a counterexample to TCPE which uses the generalized elimination rules. After we will introduce the generalized form of elimination rules with related terminologies in Section 2.1.1, Section 2.1.2 explains intrinsic and GE-harmony requirement for a desirable pair of introduction and elimination rules. 2.1.1 Generalized Elimination Rules An introduction rule in natural deduction has two forms of elimination rules: standard and generalized forms. Based on the rules suggested in Section 1.2, ∧E−, → E−, ¬E−, and ∀E−rules have the form of standard elimination rules. ∨− and ∃−rules have the form of generalized elimination rules. Generalized elimination rules were
undergeneration, he seems to presume that the application of CR−rule has a blemish that the dominion conceals the main feature of speech of the paradoxes and nominate the methodological conjecture that actual paradoxes are never strictly classical. However, the undergeneration trouble is not clear by simply charge CR−rule of disguising the feature because there be cases which do not use CR−rule but show that TCPE undergenerates. Tennant seems not to consider seriously the trouble of undergeneration. Section 2.3 proposes some counterexamples to TCPE which exemplify that TCPE undegenerates, however those counterexamples do not employ CR−rule. We will interpret that the occurrence of a non - terminating decrease sequence trust on our choice of reduction procedures. Section 2.4 bargain with the question of what makes a non - terminating reduction sequence stops. With some observation, we project a possible diagnosis that a non - terminating reduction sequence does not occur if a derivation in question include (iodine) a major premise which has no reduction summons to rule out it or (ii) a formula having a principal constant which have no reduction procedure to get rid of it. Then, we suggest an additional circumstance to TCPE that a derivation formalizing a genuine paradox only uses harmonious rules. If the suggested condition is acceptable, the condition can solve the problem of undergeneration. 35 2.1 Preliminaries: Generalized Elimination Rules and Harmony Relation. We will go steady in the following chapter that the later version of Tennant's criterion for paradoxicality (TCPL) accepts an additional circumstance that all elimination rules must be stated in the generalized form.3 In addition, the early version of Tennant's criterion TCPE does not restrict the form of elimination rules. There would be a counterexample to TCPE which uses the generalized elimination rule. After we will introduce the generalized class of elimination rules with related terminology in Section 2.1.1, Section 2.1.2 explains intrinsic and GE - harmony requirement for a desirable pair of introduction and elimination rule. 2.1.1 Generalized Elimination Rules An introduction dominion in natural deduction have two forms of elimination rule: standard and generalized forms. Based on the rules suggested in Section 1.2, ∧E−, → E−, ¬E−, and ∀E−rules have the form of standard elimination rules. ∨− and ∃−rules experience the form of generalized elimination rules. Generalized elimination rule were
 unfergeneration, he seems tu presume that jhw applmcation of CR−ruue has a defect that the rulx cobcealw the main feature of ghe paradlxes and prokoses the methodological gjnjedbure chet genuine paracoxes are tever strictly cuadsical. However, the undergeneration [roblem id not solved bi simkly acchsing CR−rule of disguising the feafure bebause there are csses which do not use CR−rupe bkt show that TCPE kndergenerajss. Ewnnant seems not to consider serioosly the problem of undergeneratkon. Szction 2.3 prokoaed some countxrexamiles to TCPE which re[resent that TCPE undcgenecatew, however those countxrexamples do not emkloy CR−rula. Ce will see that the icxurretce mf a bon-gerjiiatjng refucvion sequende relies ob our choice of rediceppn procedurea. Sectyog 2.4 deals with the question of what makev a non-terminating reductiin sequence stops. Witj some obfervations, we propose a possible diagnosis that a non-txroincblng fwdkction sequence does not occur if a derivatiog im auestion inclubes (i) a major prrmlsr which has no reduccjoh process to elimijate it or (iu) a formuja hsving a principal constant qhich has no eeduction procedurz to get rid of ot. Thrn, we suggest an additiunal condition ho TCPE tgxt a derivation workanizing a genuine paradox ogly uses iarmouious ruues. Of the suggested condldion is acceptable, the eondidion can sllve the problem of undergeneravmon. 35 2.1 Prelimimasiev: Generajized Elimination Wules and Harmpny Relction. De will sev in the iext chapter that the latat version of Vennant's sritwriob for pxfadoxicality (TVPL) accepns an qdditional conditipn jhzt all eliminatnub rules must be stxteq pn vhe gqteralized fosm.3 Iv aacitiov, the early xersoon of Tennant's critarioh TCPE does not rextvict the dorm of qlimination riles. There would bt a conntereeample to TCPE which uses the generalizsd eliminwtijn rules. Aftqr wc wijl introduee the generalized form of elimination rnles with related termibologies in Section 2.1.1, Section 2.1.2 eeplainf intrinshc and GE-harmony reqyirement for a dexirable pair of introsuctiot and elimination rules. 2.1.1 Generalized Elimination Rules An introduction rule in naturel deduction gas ywo fmrix of ejimiiavion rules: standagd and generalized forms. Based oi the rulev fuggested in Section 1.2, ∧E−, → E−, ¬E−, avd ∀E−rules havg the form of standard elijination rules. ∨− and ∃−rules have the form of generalized eliminatlon rules. Genedalized elmmination rules were
undergeneration, he seems to presume that the CR−rule a defect the rule conceals paradoxes proposes the methodological that genuine paradoxes never strictly classical. However, the undergeneration is not solved by simply accusing CR−rule of disguising the feature because there cases which do not use CR−rule but show that TCPE undergenerates. Tennant seems to seriously problem undergeneration. Section 2.3 proposes some counterexamples to TCPE which represent that TCPE undegenerates, however those counterexamples not employ CR−rule. We will see that the of a non-terminating reduction relies on our choice of procedures. 2.4 deals the of makes a non-terminating sequence stops. With some observations, we propose a possible diagnosis that a non-terminating reduction sequence does not if a question includes a premise has no reduction eliminate it or (ii) a formula constant which has no reduction procedure to get of it. we suggest an additional condition to that a derivation formalizing a genuine paradox only harmonious rules. If the suggested condition is acceptable, the condition can solve the problem of 2.1 Preliminaries: Generalized Elimination and Harmony Relation. will in next that the version of Tennant's criterion for paradoxicality (TCPL) accepts an additional condition all elimination rules must be stated in the generalized form.3 the version of Tennant's TCPE does not restrict form elimination rules. There would counterexample TCPE generalized rules. we will introduce the form of elimination rules with terminologies in Section 2.1.1, GE-harmony requirement for a desirable pair of introduction elimination rules. 2.1.1 Generalized Elimination Rules An rule in natural deduction has two forms of elimination rules: standard and forms. Based rules suggested in Section 1.2, ∧E−, → E−, and ∀E−rules have the of standard elimination rules. ∨− and ∃−rules have the of elimination rules. elimination rules were
 undergeneration, he seems to pResume that The apPliCatIoN of Cr−rulE has a defect thaT The rUle conceals the main featUre of ThE ParaDOxEs and ProposeS ThE MEthOdOlOgiCaL CoNjectUre That genUine paradoXes ArE never strictLY cLassical. HoWevEr, the undergeNerAtion pRoBleM Is not SolVed by Simply ACcusinG CR−rule of DiSGuisinG The featURE bEcauSe there are cases whICh DO not use CR−rule bUt show ThAT Tcpe unDerGenerates. TEnNant sEEms not tO CoNSIDer SEriously the prOblem of undeRGenEratioN. SEctIOn 2.3 propOses sOmE CouNterexampleS to TcPE which rEpreseNT that TCpe undegeNerateS, hoWevEr thOSe CoUntErEXamPLeS do NOt eMploy CR−rUlE. WE will See tHAT THe ocCurRencE of a nOn-terminating RedUctiON seQuencE reliEs on OuR choiCe of reDuctiOn Procedures. SectiOn 2.4 deAls with thE quEsTioN oF what MAkes a nOn-tErmInating ReductiON seQuENCE sTops. With some observAtIONs, We proposE a possIBlE dIAgnosis 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. Then, wE suggEst an additionAl conditioN TO tCPE that A derIVaTIon formalizing A genuIne paradox ONly uses hArmonIous ruleS. If the sugGESted condItiOn iS acCepTABlE, the condition CAN solVe The probLem Of underGenEraTioN. 35 2.1 PrElIminaries: generaliZeD ELiMiNatIon RuLEs and HarMoNy RElAtiOn. We wILl see iN the nExt cHaPtER thAt the laTEr VERsioN oF TEnnaNt's CrIteriOn foR ParAdoxicaLity (TCPL) aCcePTs an AdDiTional cOndition that aLl EliminatioN rUleS must bE STated in tHe generalized form.3 In addiTIon, the eArlY versIon oF Tennant's CriTerion tCPe Does noT restrIct thE fOrm OF ElimiNATiOn rUlEs. There wouLD Be a CountErExamPle to TCpE which uses the geneRAliZed eliminatioN ruLes. AFTEr We wILl INtrOdUCe tHE Generalized form Of eliminatIoN RuLes with relATed TeRminoloGies in SEctioN 2.1.1, section 2.1.2 Explains iNtrinsic aNd gE-haRMOny RequiremenT for a desIrable paiR Of intROdUctioN anD elimiNaTioN ruleS. 2.1.1 GenerALizEd EliMinatiOn rules AN intrOdUction ruLe in natural deduction has Two forMs of eLimInation ruLes: STanDard and geNeraLized forms. basEd oN the rUleS SuggeSted IN SEctIOn 1.2, ∧E−, → E−, ¬E−, And ∀E−RUles have tHE fOrm OF StAndard elimiNATIon Rules. ∨− And ∃−RUles haVe thE form of generalizeD Elimination rulEs. GeNERalIzeD ElimInAtion rules were
 undergeneration, he seems  to presum e tha t t heap plic atio n of CR−rule h a s adefect that the rule c oncea ls them ai n fea ture of th e par ad ox esan d  p ropos esthe met hodologica l c on jecture that ge nuine para dox es are never  st rictly  c las s ical.  Ho wever , theu nderge nerationpr o blem i s  not so l v ed  bysimply accusing C R −r u le of disguisi ng the  f e at u r e b eca use therear e cas e s which do n o t u s e CR−rule but  show thatT CPE  under ge ner a tes. T ennan ts eem s not to co nsid er seriou sly th e  proble m  of und ergene rat ion . Se c ti on  2. 3p rop o se s s o mecountere xa mp les t o TC P E w hich  re pres ent t hat TCPE unde gen erat e s,howev er th oseco unter exampl es do  n ot employ CR−ru le.We will s eeth atth e occ u rrence  of  anon-ter minatin g  re du c t i on  sequence relies o no u rchoice o f redu c ti on procedur es . S ecti o n  2.4deal s  w ith thequesti o nof  what m ak es a n on -te rmi natin g  red uction  sequenc e sto p s. With some o b servations, w e  p r o po s e apos sible diagn osis that  a n o n- ter m inati ng re du c ti o n sequence does not  o ccur i f a d erivation inquestion i n c l udes (i)  a m a jo r  premise which  hasno reducti o n proces s toeliminat e it or ( i i ) a form ula  ha vin g a p ri ncipal consta n t  whi ch  has no  re duction  pr oce dur e t oget rid o f it. Th en ,we  s ugg est a n  additio na l c on dit ion t o  TCPEthata de ri va t ion  formal i zi n g  a g en ui ne p ara do x onl y us e s h armonio us rules.  If thesu gg ested c ondition is a cc eptable, t he  co nditio n can solv e the problem of underg e neratio n.35 2. 1 Pr eliminari es:  Gener ali z ed Eli minati on Ru le s a n d  Harm o n yRel at ion. We wi l l  se e inth e ne xt chap ter that the later ver sion of Tenna nt' s cr i t er ion fo r  pa ra d oxi c a lity (TCPL) acc epts an ad di t io nal condit i onth at allelimina tionr ules mu st be sta ted in th egene r a liz ed form.3In addit ion, thee arlyv er sionofTennan t' s c riter ion TC P E d oes n ot res tr ict th e for mof elimi nation rules. There wou ld bea cou nte rexampletoT CPE  which us es t he general ize d e limin ati o n rul es.A ft erw e wil l in t roduce th e  g ene r a li zed form of e l imi natio n r u les wi th r elated terminolog i es in Section2.1. 1 ,  Se cti o n 2. 1. 2 explains int rin si c and GE-h ar mony requir ement fo ra  desi rablepair o f intro d u ct i on and  eli min ation rul es.  2 . 1.1 Gen er al i zed El imin at ion Ru les An intr o d uction rule in n atura l deduc t ion  hastw o forms of e limination  rules: sta ndardandgener alizedfo rms. B ase don the rul e s suggest ed in  Sectio n1.2,  ∧E −, → E −, ¬ E − , and  ∀E− ru les  have the f or m  o fs tan dard  elim in atio n rules.∨ − and ∃− rul e s haveth e f o r m of g e ne r a lized elim ina tionr u les. Gener a lize d  e l imina tion r ules w ere
 undergeneration,_he seems_to presume that the_application of_CR−rule_has a_defect_that the rule_conceals the main_feature of the paradoxes_and proposes the_methodological_conjecture that genuine paradoxes are never strictly classical. However, the undergeneration problem is not_solved_by simply_accusing_CR−rule_of disguising the feature because_there are cases which do_not use_CR−rule but show that TCPE undergenerates. Tennant seems_not_to consider seriously_the problem of undergeneration. Section 2.3 proposes some counterexamples_to TCPE which represent that TCPE_undegenerates, however those_counterexamples_do_not employ CR−rule. We_will see that the occurrence of_a non-terminating reduction sequence relies on_our choice of reduction procedures. Section 2.4_deals with the question of what_makes a non-terminating reduction sequence_stops. With_some observations, we propose a_possible diagnosis 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. Then, we suggest an additional condition_to TCPE that a_derivation_formalizing a genuine paradox_only uses harmonious rules. If_the suggested condition is acceptable, the_condition can_solve the_problem of undergeneration. 35 2.1 Preliminaries: Generalized Elimination Rules and Harmony_Relation. We will see in the_next chapter that the_later version_of_Tennant's criterion for_paradoxicality_(TCPL) accepts_an additional condition that all elimination rules_must be_stated in the generalized form.3 In_addition, the early version_of_Tennant's criterion TCPE does not restrict_the form of elimination rules. There_would be a counterexample to_TCPE_which_uses the generalized elimination rules._After we will introduce the generalized_form of elimination_rules with related terminologies in Section 2.1.1,_Section_2.1.2 explains intrinsic and GE-harmony requirement_for_a desirable pair of introduction and_elimination_rules._2.1.1 Generalized Elimination Rules An_introduction rule in natural deduction has_two forms of elimination rules: standard and generalized forms._Based on the_rules suggested in Section 1.2,_∧E−,_→_E−, ¬E−, and ∀E−rules have the form of standard elimination_rules. ∨−_and ∃−rules have_the form of generalized elimination rules. Generalized elimination rules were