 if every major premise in the derivation has a reduction procedure to eliminate it, or every formula including a principal constant (or operator) has a reduction procedure to eliminate it. From the possible diagnosis, we have proposed an additional condition to TCPE as a plausible solution to the problem of undergeneration that a derivation formalizing a genuine paradox only uses harmonious rules. Of course, it should be discussed why a paradoxical derivation should only use harmonious rules. However, it is rather natural to say that the use of harmonious rules is desirable in natural deduction since it is the main slogan fo Gentzen who is the founder of natural deduction that an introduction rule exhaustively defines the meaning of a principal constant (or operator) and a corresponding elimination rule is exhaustively determined by the meaning. Harmony requirement is the way to satisfy his slogan. Furthermore, though we set aside the question of why the non-terminating reduction sequence is the main feature of genuine paradoxes, since the non-terminating reduction is dependent on a set of reduction procedures, it should be discussed how a proper reduction can be suitably evaluated. While we deal with the problem of overgeneration raised by Ekman's paradox in Chapter 3, we shall investigate tests to assess a proper reduction process. The other remained issue is that which paradox is a genuine one. In this chapter, we presume that Curry's and the Liar paradox are genuine paradoxes. TCPE is the prooftheoretic criterion for genuine paradoxes, but the term 'genuine paradox' is an informal one. 78 Also, even though Tennant believes that the Liar paradox is genuine, there are derivations of the Liar which do not satisfy TCPE, e.g. Proposition 2.3.1. In Chapter 4, we will accept the different formalization of the Liar suggested by Tennant (1982) and provide the case that may represent that the Liar paradox is not a genuine paradox. We shall argue that it should be explained which paradox is a genuine paradox and which formalization is legitimate for the genuine paradox. 2.A Appendix 2.A: Tennant's id est Rules for a Liar Sentence and a T-Paradox Using CR−rule In this appendix, we will investigate three closed derivations of ⊥ which formalize the Liar paradox: (i) a T-paradox which use neither CR− nor CRE−rules, (ii) a T-paradox using both CR− and CRE−rules, and
if every major premise in the derivation has a reduction procedure to rule out it, or every rule including a principal constant (or hustler) has a reduction procedure to extinguish it. From the possible diagnosis, we have proposed an extra circumstance to TCPE as a plausible solution to the problem of undergeneration that a derivation validate a genuine paradox only uses harmonious dominion. Of course, it should be discussed why a paradoxical derivation should only use harmonious rule. However, it is rather natural to say that the manipulation of harmonious rules is desirable in natural deduction since it is the main motto fo Gentzen who is the founder of lifelike deduction that an introduction rule exhaustively define the meaning of a principal constant (or operator) and a corresponding elimination rule is thoroughly determine by the meaning. Harmony requirement is the way to satisfy his slogan. Furthermore, though we set aside the question of why the non - terminating reduction sequence is the chief feature of genuine paradox, since the non - terminating decrease is dependent on a set of reduction procedures, it should be discourse how a proper reduction can be suitably evaluated. While we deal with the trouble of overgeneration raised by Ekman's paradox in Chapter 3, we shall investigate tests to assess a proper reduction process. The other remained issue is that which paradox is a genuine one. In this chapter, we presume that Curry's and the Liar paradox are genuine paradoxes. TCPE is the prooftheoretic standard for genuine paradox, but the term' genuine paradox' is an informal one. 78 Also, even though Tennant believes that the Liar paradox is genuine, there constitute derivations of the Liar which do not satisfy TCPE, e.g. Proposition 2.3.1. In Chapter 4, we will bear the different formalization of the Liar suggested by Tennant (1982) and provide the case that may defend that the Liar paradox is not a genuine paradox. We shall argue that it should be explain which paradox is a genuine paradox and which formalization is legitimate for the actual paradox. 2.A Appendix 2.A: Tennant's i d est Rules for a Liar Sentence and a T - Paradox Using CR−rule In this appendix, we will investigate three closed ancestry of ⊥ which formalize the Liar paradox: (one) a T - paradox which use neither CR− nor CRE−rules, (ii) a T - paradox using both CR− and CRE−rules, and
 if every major premise in uhe derivation has a rednction lrocedurd to eliminate it, or every flrnula uncluding a principal zonstant (lr operaror) ias a reduction 'docedurc to spimiuave it. From the kossible diacnosis, we have pfo'osed an additional condition to TCPQ as a llwusible solutijn tp the iriblem of undergeneration that a derination formalizinb a genuine paradox only udes jarmonious rules. Ov course, it shotod be discusred why a kaxadoxical dgrivation should only use harmonkous xules. Howevgx, it hs rather netural to say that the use of harkonious rules ls devirqble in natural deducvion since it is the main slocau fo Gentzen who is tye founger mf nxrurxl sevucfion tjat an introdhction rule exhaustively defints eye meaning of a prigcypal constant (or operator) and a correspmndjng elimination rule is exhaustively determijed by thq meaning. Harmony requirement is the way to satisxy hia sligwv. Fkrthermore, though we set aside the question os wny the non-termincting reduction xeeurgce is the makn feachrs of genuine paradlxes, sigce tye non-teriinaying reduction is dependent on a set of eeduction procedurzs, it should be ciscuxsed how a proper reduccion czn be suitahly evaluzged. While we deau wpth dhe problem of overgeneratyon raisev by Zkman's pxradpx in Shapter 3, wf shall investigate testd to cssesv a proper reduction process. The other remained issue ix dhan which pcradox is a genuine one. In this ckapter, wz presjme that Clrry's and the Liar pwradox are getoine paradoxev. TCPE if thw prioftheofdtic criterion for genupnt paradoxew, but the term 'genmine kadadox' is an infirmql one. 78 Also, evrn ghotgj Vennagd believes tvat ghe Kiar oaradox is nenjine, there are derivatiots or the Liar which dp kot satisdy TCPE, q.g. Propositiom 2.3.1. In Chapter 4, we aill eccept the cifserent formalization of the Lizr suggeshed by Tennant (1982) and provide the case that may represent that the Liar 'aradox is not a genuinw paradox. We shall crnue that it xhoulq be expldined which paradox us a genuine paracox and which formaliaation is lfgitimate for the genuine paradox. 2.A Appendix 2.A: Tennant's id est Rules for q Liar Sentence ans a Y-Paragor Uxing CW−rulx Mn this appendix, ee will investigate three closev derivatimnf of ⊥ which formalize the Lisr paradox: (i) a J-paradox which use neither CR− nor VRE−rules, (ii) a T-paradox using boyh CR− and CRE−rules, and
if every major premise in the derivation reduction to eliminate or every formula operator) a reduction procedure eliminate it. From possible diagnosis, we have proposed an condition to TCPE as a plausible solution to the problem of undergeneration that derivation formalizing a genuine paradox only uses harmonious rules. Of course, it should discussed a derivation only use harmonious rules. However, it is rather natural to say that the use of harmonious is desirable in natural deduction since it is main slogan fo Gentzen is the founder of natural that introduction rule defines meaning a principal constant operator) and a corresponding elimination rule is exhaustively determined by the meaning. Harmony requirement is the way satisfy his though we aside question why the non-terminating is the main feature of genuine non-terminating reduction is dependent on a set of procedures, it be discussed how a proper reduction be suitably evaluated. While we deal with the of overgeneration raised by Ekman's paradox in Chapter 3, we shall investigate tests to assess reduction process. The other issue is that paradox a one. this chapter, presume that Curry's and the Liar paradox are genuine paradoxes. TCPE the prooftheoretic criterion for genuine paradoxes, but the term 'genuine an one. 78 Also, though Tennant believes that Liar is genuine, there are the which TCPE, Proposition In Chapter 4, we accept the different formalization of Liar suggested by Tennant that may represent that the Liar paradox is a genuine paradox. We shall argue that should be explained which paradox is a genuine paradox and which formalization legitimate for paradox. 2.A Appendix 2.A: Tennant's id est Rules a Liar Sentence and T-Paradox Using CR−rule In this appendix, we will investigate closed of ⊥ formalize the Liar (i) a T-paradox use neither CR− (ii) a using CR−
 if every major premise in the dErivation hAs a reDucTioN pRoceDure To eliminate it, oR EverY formula including a prinCipal CoNStanT (Or OperaTor) has a REdUCTioN pRoCedUrE To ElimiNatE it. From The possiblE diAgNosis, we have pROpOsed an addiTioNal condition To TcPE as a PlAusIBle soLutIon to The proBLem of uNdergenerAtIOn that A DerivatION fOrmaLizing a genuine parADoX Only uses harmonIous ruLeS. of COUrsE, it Should be diScUssed WHy a paraDOxICAL deRIvation should Only use harmONioUs ruleS. HOweVEr, it is RatheR nATurAl to say that The uSe of harmoNious rULes is deSIrable iN naturAl dEduCtioN SiNcE it Is THe mAIn SloGAn fO Gentzen WhO iS the fOundER OF NatuRal DeduCtion That an introduCtiOn ruLE exHaustIvely DefiNeS the mEaning Of a prInCipal constant (or OperAtor) and a cOrrEsPonDiNg eliMInatioN ruLe iS exhausTively dETerMiNED By The meaning. Harmony rEqUIReMent is thE way to SAtIsFY his slogAn. furTherMORe, thoUgh wE SeT aside thE questIOn Of Why the nOn-TerminAtIng RedUctioN SequEnce is The main fEaturE Of genuine paradOXes, since the noN-TeRMInATing RedUction is depEndeNT on a Set oF ReDucTIon prOceduReS, It SHould be discussed how A pRoper rEductIon can be suitaBly evaluatED. wHile we deAl wiTH tHE problem of overGenerAtion raiseD By Ekman's ParadOx in ChapTer 3, we shalL INvestigaTe tEstS to AssESS a Proper reductiON ProcEsS. The othEr rEmained IssUe iS thAt wHiCh paradox Is a genuiNe OnE. IN tHis ChaptER, we presuMe ThaT CUrrY's and THe Liar ParadOx arE gEnUIne ParadoxES. Tcpe is tHe PrOoftHeoReTic crIterIOn fOr genuiNe paradoxEs, bUT the TeRm 'Genuine Paradox' is an inFoRmal one. 78 AlsO, eVen Though tENnant belIeves that the Liar paradox IS genuinE, thEre arE derIvations oF thE Liar wHicH Do not sAtisfy tCPE, e.G. PRopOSItion 2.3.1. iN chAptEr 4, We will accePT The DiffeReNt foRmalizaTion of the Liar suggeSTed By Tennant (1982) and pRovIde tHE CaSe tHAt MAy rEpREseNT That the Liar paraDox is not a gEnUInE paradox. We SHalL aRgue thaT it shouLd be eXPlained Which paraDox is a genUiNe paRADox And which foRmalizatIon is legiTImate FOr The geNuiNe paraDoX. 2.A APpendIx 2.A: TenNAnt'S id esT Rules FoR a Liar senteNcE and a T-PaRadox Using CR−rule In this aPpendiX, we wiLl iNvestigatE thREe cLosed deriVatiOns of ⊥ which ForMalIze thE LiAR paraDox: (i) A t-pAraDOx whiCh usE Neither CR− NOr cRE−RULeS, (ii) a T-paradoX USIng Both Cr− anD cRE−rulEs, anD
 if every major premise in  the deriv ation  ha s a  r educ tion  procedure toe limi nate it, or every form ula i nc l udin g  a  prin cipal c o ns t a nt(o rope ra t or ) has  areducti on procedu reto  eliminate i t .From the p oss ible diagnos is,  we ha ve  pr o posed  an  addi tional condit ion to TC PE as a p l ausible s ol utio n to the problemo fu ndergeneration  thatad er i v ati onformalizin ga gen u ine par a do x o nly uses harmonio us rules. O f  co urse,it  sh o uld be  disc us s edwhy a parad oxic al deriva tion s h ould on l y use h armoni ous  ru les. Ho we ver ,i t i s  r ath e r n atural t osa y tha t th e u s e of  ha rmon iousrules is desi rab le i n  na tural  dedu ctio nsince  it is  thema in slogan fo Ge ntze n who isthe  f oun de r ofn atural  de duc tion th at an i n tro du c t i on  rule exhaustively  d e f in es the m eaning of  a principa lcon stan t (or o pera t or ) and acorres p on di ng elim in ationru leisexhau s tive ly det erminedby th e  meaning. Harm o ny requiremen t  i s th e  way  to  satisfy hi s sl o gan.  Fur t he rmo r e, th oughwe se t  aside the question  o f whythe n on-terminatin g reductio n s equenceis t h em ain feature of  genu ine parado x es, sinc e the  non-ter minatingr e ductionisdep end ent o na set of redu c t ionpr ocedure s,it shou ldbedis cus se d how a p roper re du ct io ncan  be s u itably e va lua te d.While we dea l wit h th epr o ble m of ov e rg e n erat io nrais edby  Ekma n'sp ara dox inChapter 3 , w e  sha ll  i nvestig ate tests toas sess a pro pe r r educti o n  process . The other remained is s ue is t hat  whic h pa radox isa g enuine  on e . In t his ch apter ,wep r esume t ha t C ur ry's and t h e  Li ar pa ra doxare gen uine paradoxes. TC P E i s the proofth eor etic c ri ter i on for  g e nui n e  paradoxes, but  the term'g e nu ine parado x ' i san info rmal on e. 78 Also, e ven thoug h Tennant  b elie v e s t hat the Li ar parad ox is gen u ine,t he re ar e d erivat io nsof th e Liar whi ch do  not s at isfy T CPE,e. g. Propo sition 2.3.1. In Chapte r 4, w e wil l a ccept the  di f fer ent forma liza tion of th e L iar  sugg est e d byTenn a nt  (1 9 82) a nd p r ovide the ca set h at  may repres e n t  th at th e L i ar par adox  is not a genuine paradox. We sh alla r gue  th a t it  s hould be expla ine dw h ich para do x is a genu ine para do x  andwhichformal ization i sl egitim atefor  the genu ine  p a radox.2. AA ppendi x 2. A:  Tenna nt's i d  est R ules for a LiarSente n c e and a T -Para do x Using CR−r ule In thi s appendix,  we wi ll i nvest igate t hr ee clo sed  d erivations of ⊥ whic h for malizeth e Li arparado x: ( i )  a T- para do x w hich usen e it h er  C R − n or C RE−ru le s, ( ii) a T-p a radox us ing both CR −and C RE−rul e s, a nd
 if_every major_premise in the derivation_has a_reduction_procedure to_eliminate_it, or every_formula including a_principal constant (or operator)_has a reduction_procedure_to eliminate it. From the possible diagnosis, we have proposed an additional condition to_TCPE_as a_plausible_solution_to the problem of undergeneration_that a derivation formalizing a_genuine paradox_only uses harmonious rules. Of course, it should_be_discussed why a_paradoxical derivation should only use harmonious rules. However, it_is rather natural to say that_the use of_harmonious_rules_is desirable in natural_deduction since it is the main_slogan fo Gentzen who is the_founder of natural deduction that an introduction_rule exhaustively defines the meaning of_a principal constant (or operator)_and a_corresponding elimination rule is exhaustively_determined by the_meaning. Harmony_requirement is the_way to satisfy his slogan. Furthermore,_though we set_aside the question of why the_non-terminating_reduction sequence is_the_main_feature of_genuine paradoxes, since_the_non-terminating reduction_is_dependent on a set of reduction_procedures,_it should be discussed how a proper_reduction can be suitably_evaluated._While we deal with_the problem of overgeneration raised_by Ekman's paradox in Chapter 3,_we shall_investigate tests_to assess a proper reduction process. The other remained issue is_that which paradox is a genuine_one. In this chapter,_we presume_that_Curry's and the_Liar_paradox are_genuine paradoxes. TCPE is the prooftheoretic criterion_for genuine_paradoxes, but the term 'genuine paradox'_is an informal one._78_Also, even though Tennant believes that_the Liar paradox is genuine, there_are derivations of the Liar_which_do_not satisfy TCPE, e.g. Proposition_2.3.1. In Chapter 4, we will_accept the different_formalization of the Liar suggested by Tennant_(1982)_and provide the case that may_represent_that the Liar paradox is not_a_genuine_paradox. We shall argue that_it should be explained which paradox_is a genuine paradox and which formalization is legitimate_for the genuine_paradox. 2.A Appendix 2.A: Tennant's_id_est_Rules for a Liar Sentence and a T-Paradox Using CR−rule_In this_appendix, we will_investigate three closed derivations of ⊥ which formalize the Liar_paradox: (i) a T-paradox which use neither_CR− nor CRE−rules, (ii) a T-paradox using both CR− and CRE−rules,_and