 topics of general proof theory is the validity of an argument. In this section, we briefly introduce Prawitz's notion of 'proof-theoretic validity' and consider a possible answer to the second question that every non-normalizable derivation is not proof-theoretically valid. If proof-theoretic validity implies normalizability then RND can be a promising proof-theoretic solution to the paradoxes. So to speak, if every paradoxical derivation, i.e. a T-paradox, generates a non-terminating reduction sequence and so is not normalizable, it cannot be a proof-theoretically valid derivation. Hence, we do not need to accept non-normalizable derivation. However, we shall argue that although it can be shown that in a particular system proof-theoretic validity implies normalizability, there should be a further research to extend the result in a general case. Prawitz (1971, Appendix A) introduces a programme of defining a general notion of 'validity' based on Gentzen's idea that an introduction rule determines the meaning of a logical constant in terms of which an elimination rule is determined. That is, a derivation (or an argument) is valid if it can be built up by introduction rules. Since the programme generalizes the result of normalization theorem. He considers that proof-theoretic validity is not only a property of formal derivations in a particular system but also of more arbitrary natural deduction system. To prove the theorem, we first distinguish certain individual 145 rules and then compose derivations from the rules. On the other hand, Prawitz's prooftheoretic validity deals with derivations (or arguments) in the first place and regards rules as steps which preserve the validity of derivations. So to speak, the investigation of the relation between Prawitz's notion of proof-theoretic validity and normalizability does not seem appropriate. However, when we restrict our concern on the proof-theoretic validity as a property of formal derivations in a particular system, we can consider an expected result that every proof-theoretically valid derivation in a specific system is normalizable. Most of our examples of paradoxical derivations use standard rules for minimal logic. If proof-theoretic validity implies normalizability in a minimal natural deduction system, at least in restricted sense, one might see the possibility that RND would be a method to single out proof-theoretically invalid derivations. Since we shall regard in this chapter a normal derivation as a formal one in
topics of general proof theory is the cogency of an controversy. In this section, we briefly introduce Prawitz's impression of' proof - theoretic validity' and consider a potential solution to the second question that every non - normalizable derivation is not proof - theoretically valid. If proof - theoretic validity implies normalizability then RND can be a bright proof - theoretic solution to the paradoxes. So to address, if every paradoxical derivation, i.e. a T - paradox, generates a non - terminating reduction sequence and so is not normalizable, it cannot be a validation - theoretically valid derivation. Hence, we do not need to take non - normalizable derivation. However, we shall argue that although it can be shown that in a especial system validation - theoretic robustness implies normalizability, there should be a further research to extend the resultant role in a general case. Prawitz (1971, Appendix A) introduces a programme of defining a general notion of' validity' based on Gentzen's idea that an introduction rule determines the meaning of a logical constant in terms of which an elimination rule is determined. That is, a deriving (or an argument) is valid if it can be built up by initiation rules. Since the programme generalizes the result of standardization theorem. He considers that proof - theoretical validity is not only a property of conventional derivations in a particular system but also of more arbitrary natural deduction system. To prove the theorem, we first distinguish certain individual 145 principle and then write derivations from the rules. On the other hand, Prawitz's prooftheoretic validity manage with derivations (or arguments) in the first home and regards rules as steps which preserve the robustness of derivations. then to speak, the investigation of the sexual intercourse between Prawitz's notion of proof - theoretical validity and normalizability does not seem appropriate. However, when we restrict our concern on the proof - theoretical validity as a property of formal derivation in a particular organization, we can consider an expected result that every proof - theoretically valid derivation in a specific system is normalizable. Most of our examples of paradoxical derivations use standard rules for minimal logic. If proof - theoretical validity implies normalizability in a minimal lifelike deduction system, at least in restricted sense, one might experience the possibility that RND would be a method to single out proof - theoretically invalid derivation. Since we shall regard in this chapter a normal derivation as a formal one in
 tooics of general proof thtory is the validnry of en argujent. In ghis section, we briefly intrldyce Peawitz's notion of 'proow-theoretib validitt' anv consider a possible answer tk the wecond questiok that everf non-normalizatld berivation is not proof-theoretically valid. Of proof-theoretis vakyditg implies normalizability then RND can be a promising lroof-theoretic solution to the paradoxes. So to soeak, if evetg pweadoxical defivation, i.t. c T-paradox, fenerates a non-terminating reduztion sequence qne sl is not nornalizwble, it cannot be a [roof-throretically vakid deeivation. Hence, we do iot need to accept njn-normalisayle derivation. Howevee, qe shdll drgud thxt zlvhohgh it cai be shown fhat in a pqrticular system prpos-nneoretic valjdity ym[lies normalizability, there should be a fudther research to extene the result in a gengral case. [rawitz (1971, Appendix A) introduces a programme of defhning x gtncval vitlon of 'validity' based on Gentzen's idea that ag imtgoduction rule debermines the meanimg og a logical covstant in terms of which an eliminwtion rule is qetetmined. That is, a derivation (or an argumvnt) us valid if it can be built u' by imtrodiction rules. Since the 'rograjme generallzes the ddsult of normalixatpon dheorem. He considers that [roof-theocetic validigy ix not jnly a prooerty of formal derivatilns iu a pdrticular dystem but also of more arbitracb natural dedocthon system. Co proye the theorem, re first distiuguish czrtain individuam 145 rulev and then sompose derivdjions from thx rules. Og thw otyer hana, Prawitz's proogtheoretib ralidity eeals with derivatlons (ud arguments) in cke first place anc rdgawdd cules ds steps whiwh pfesdtve tfe validity uf drrivations. So to spedk, tge investigation og bhe relatuon betwqen Prawitz's motion of proof-thelretir valivity amd gormalizability does not seem zppropriahe. Mowever, when re rcstryct our coucern on the proof-theoretic validity as e property of formal dgrivations in a partieukar system, xe can consider an expected result rhat every proof-tmeoretically valid derjvatiot in w specific system is normalizable. Most of our examples of paradoxical deruvatiois use standadd riles xox mnnimal jogir. Mf proof-theoretic validity implies normalizabilivy in a mitiial natural deduction system, ag least in rertricted sense, one might see the possibility that RND would be s method to single out pvoof-theorwticamly innzlmd derivations. Since we shcll retard in thid chapter a nurmsl derivatiln ax c formal one in
topics of general proof theory is the an In this we briefly introduce and a possible answer the second question every non-normalizable derivation is not proof-theoretically If proof-theoretic validity implies normalizability then RND can be a promising proof-theoretic solution the paradoxes. So to speak, if every paradoxical derivation, i.e. a T-paradox, generates non-terminating sequence so not normalizable, it cannot be a proof-theoretically valid derivation. Hence, we do not need to accept derivation. However, we shall argue that although it be shown that in particular system proof-theoretic validity implies there be a research extend result in a case. Prawitz (1971, Appendix A) introduces a programme of defining a general notion of 'validity' based on idea that rule determines meaning a constant in terms an elimination rule is determined. That (or an argument) is valid if it can built up introduction rules. Since the programme generalizes result of normalization theorem. He considers that proof-theoretic is not only a property of formal derivations in a particular system but also of natural deduction system. To the theorem, we distinguish individual rules then compose from the rules. On the other hand, Prawitz's prooftheoretic validity deals derivations (or arguments) in the first place and regards rules which the validity of So to speak, the of relation between Prawitz's notion validity normalizability appropriate. when restrict our concern on proof-theoretic validity as a property formal derivations in a an expected result that every proof-theoretically valid derivation a specific system is normalizable. Most of examples of paradoxical derivations use standard rules for minimal logic. If proof-theoretic implies normalizability minimal natural deduction system, at least in restricted one might see the that RND would be a method to single out invalid Since we regard in this a normal derivation a formal one
 topics of general proof theorY is the valiDity oF an ArgUmEnt. IN thiS section, we brieFLy inTroduce Prawitz's notion oF 'prooF-tHEoreTIc ValidIty' and cONsIDEr a PoSsIblE aNSwEr to tHe sEcond quEstion that EveRy Non-normalizaBLe Derivation Is nOt proof-theorEtiCally vAlId. IF Proof-TheOretiC validITy implIes normalIzABility THen RND cAN Be A proMising proof-theoreTIc SOlution to the paRadoxeS. SO To SPEak, If eVery paradoXiCal deRIvation, I.E. a t-PARadOX, generates a noN-terminatinG RedUction SeQueNCe and sO is noT nORmaLizable, it caNnot Be a proof-tHeoretICally vaLId derivAtion. HEncE, we Do noT NeEd To aCcEPt nON-nOrmALizAble deriVaTiOn. HowEver, WE SHAll aRguE thaT althOugh it can be shOwn That IN a pArticUlar sYsteM pRoof-tHeoretIc valIdIty implies normaLizaBility, theRe sHoUld Be A furtHEr reseArcH to Extend tHe resulT In a GeNERAl Case. Prawitz (1971, AppendiX A) INTrOduces a pRogramME oF dEFining a gEnEraL notION of 'vaLidiTY' bAsed on GeNtzen's IDeA tHat an inTrOductiOn RulE deTermiNEs thE meaniNg of a logIcal cONstant in terms oF Which an eliminATiON RuLE is dEteRmined. That iS, a deRIvatIon (oR An ArgUMent) iS valiD iF It CAn be built up by introdUcTion ruLes. SiNce the programMe generaliZES The resulT of nORmALization theoreM. He coNsiders thaT Proof-theOretiC validitY is not onlY A Property Of fOrmAl dEriVATiOns in a particuLAR sysTeM but alsO of More arbItrAry NatUraL dEduction sYstem. To pRoVe ThE tHeoRem, we FIrst distInGuiSh CerTain iNDividuAl 145 rulEs anD tHeN ComPose derIVaTIOns fRoM tHe ruLes. on The otHer hANd, PRawitz's ProoftheoRetIC valIdItY deals wIth derivationS (oR arguments) In The First pLACe and regArds rules as steps which prEServe thE vaLiditY of dErivationS. So To speaK, thE InvestIgatioN of thE rElaTIOn betWEEn praWiTz's notion oF PRooF-theoReTic vAlidity And normalizability DOes Not seem appropRiaTe. HoWEVeR, whEN wE ResTrICt oUR Concern on the proOf-theoretiC vALiDity as a proPErtY oF formal DerivatIons iN A particUlar systeM, we can conSiDer aN EXpeCted result That everY proof-theOReticALlY valiD deRivatiOn In a SpeciFic sysTEm iS normAlizabLe. most of Our exAmPles of paRadoxical derivations use StandaRd rulEs fOr minimal LogIC. If Proof-theoRetiC validity iMplIes NormaLizABilitY in a MInImaL NaturAl deDUction sysTEm, At lEASt In restricteD SENse, One miGht SEe the pOssiBility that RND woulD Be a method to sinGle oUT ProOf-tHEoreTiCally invalid deRivAtIONs. Since wE sHall regard iN this chaPtER a norMal derIvatioN as a forMAL oNE in
 topics of general proof t heory is t he va lid ity  o f an  arg ument. In this sect ion, we briefly introd uce P ra w itz' s  n otion  of 'pr o of - t heo re ti c v al i di ty' a ndconside r a possib lean swer to thes ec ond questi onthat every n on- normal iz abl e  deri vat ion i s notp roof-t heoretica ll y  valid .  If pro o f -t heor etic validity imp l ie s  normalizabili ty the nR ND c anbea promisin gproof - theoret i cs o l uti o n to the para doxes. So t o  sp eak, i feve r y para doxic al der ivation, i. e. a  T-parado x, gen e rates a non-ter minati ngred ucti o nse que nc e  an d  s o i s  no t normal iz ab le, i t ca n n o t  bea p roof -theo retically val idderi v ati on. H ence,  wedo  notneed t o acc ep t non-normaliza blederivatio n.Ho wev er , wes hall a rgu e t hat alt hough i t  ca nb e sh own that in a part ic u l ar  systemproof- t he or e tic vali di tyimpl i e s nor mali z ab ility, t here s h ou ld  be a f ur ther r es ear chto ex t endthe re sult ina gen e ral case. Praw i tz (1971, App e nd i x  A )  int rod uces a prog ramm e  ofdefi n in g a gener al no ti o no f 'validity' basedon  Gentz en'sidea that anintroducti o n rule det ermi n es the meaning of  a lo gical cons t ant in t ermsof which  an elimi n a tion rul e i s d ete rmi n e d.  That is, a d e r ivat io n (or a n a rgument ) i s v ali d i fit can be  built u pby  i nt rod uctio n  rules.Si nce  t heprogr a mme ge neral izes  t he res ult ofn or m a liza ti on  the ore m.  He c onsi d ers  that p roof-theo ret i c va li di ty is n ot only a pro pe rty of for ma l d erivat i o ns in aparticular system but a l so of m ore  arbi trar y natural  de ductio n s y stem.To pro ve th ethe o r em, w e fi rst  d istinguish c ert ain i nd ivid ual 145  rules and then co m pos e derivations  fr om t h e  r ule s .O n t he oth e r  hand, Prawitz' s proofthe or e ti c validity dea ls  with d erivati ons ( o r argum ents) inthe first  p lace a ndregards ru les as s teps whic h  pres e rv e the  va lidity  o f d eriva tions. Soto sp eak, t he  inves tigat io n of the  relation between Prawi tz's n otion  of  proof-th eor e tic  validity  and  normaliza bil ity  does  no t  seem  app r op ria t e. Ho weve r , when we re str i c tour concern o n  th e pro of- t heoret ic v alidity as a prop e rty of formalderi v a tio nsi n apa rticular syste m,we c an consi de r an expect ed resul tt hat e very p roof-t heoreti c a ll y  valid  der iva tion in a  sp ec i fic sys te mi s norm aliz ab le. Mo st ofo ur e x a mples of paradox icald e rivat i ons  usest andardr ules  for minim al logic. I f proo f-th eoret ic vali di ty imp lie snormalizab i lity in a  mini mal nat ur al d edu ctionsyst e m , atleas tinrestricte d se n se ,o nemigh t see  t he p ossibilit y  that RN D w o uld beamet h o d to s i ng l e  out proof -th eoret i c ally inval i d de r iv a tions . Sinc e we s hall re g ard  i n thischa p t er a norm al deriva t ion  a s aformal o ne  in
 topics_of general_proof theory is the_validity of_an_argument. In_this_section, we briefly_introduce Prawitz's notion_of 'proof-theoretic validity' and_consider a possible_answer_to the second question that every non-normalizable derivation is not proof-theoretically valid. If proof-theoretic_validity_implies normalizability_then_RND_can be a promising proof-theoretic_solution to the paradoxes. So_to speak,_if every paradoxical derivation, i.e. a T-paradox, generates_a_non-terminating reduction sequence_and so is not normalizable, it cannot be a_proof-theoretically valid derivation. Hence, we do_not need to_accept_non-normalizable_derivation. However, we shall_argue that although it can be_shown that in a particular system_proof-theoretic validity implies normalizability, there should be_a further research to extend the_result in a general case._Prawitz (1971,_Appendix A) introduces a programme_of defining a_general notion_of 'validity' based_on Gentzen's idea that an introduction_rule determines the_meaning of a logical constant in_terms_of which an_elimination_rule_is determined._That is, a_derivation_(or an_argument)_is valid if it can be_built_up by introduction rules. Since the programme_generalizes the result of_normalization_theorem. He considers that_proof-theoretic validity is not only_a property of formal derivations in_a particular_system but_also of more arbitrary natural deduction system. To prove the theorem,_we first distinguish certain individual 145_rules and then compose_derivations from_the_rules. On the_other_hand, Prawitz's_prooftheoretic validity deals with derivations (or arguments)_in the_first place and regards rules as_steps which preserve the_validity_of derivations. So to speak, the_investigation of the relation between Prawitz's_notion of proof-theoretic validity and_normalizability_does_not seem appropriate. However, when_we restrict our concern on the_proof-theoretic validity as_a property of formal derivations in a_particular_system, we can consider an expected_result_that every proof-theoretically valid derivation in_a_specific_system is normalizable. Most of_our examples of paradoxical derivations use_standard rules for minimal logic. If proof-theoretic validity implies_normalizability in a_minimal natural deduction system, at_least_in_restricted sense, one might see the possibility that RND would_be a_method to single_out proof-theoretically invalid derivations. Since we shall regard in this_chapter a normal derivation as a formal_one in