 to the paradoxes. After having preliminary notations and natural deduction rules, Chapter 1 introduces the early version of Tennant's criterion for paradoxicality TCP and the requirement of a normal derivation RND. As a doctor treats the disease in accordance with her diagnosis, a logician solves the paradoxes on the basis of her characterization of the paradoxes. When TCP is regarded as a proof-theoretic criterion for paradoxicality, RND can be a possible proof-theoretic solution to the paradoxes. The Early Version of Tennant's Criterion for Paradoxicality(TCPE): Let D be any derivation of a given natural deduction system 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. The Requirement of a (Full) Normal Derivation(RND): For any derivation D in natural deduction, D is acceptable only if D is (in principle) convertible into a normal derivation. 163 There are two types of counterexamples to TCP. The one generates the problem of undergeneration in the sense that TCP makes a paradoxical derivation non-paradoxical. The other counterexample causes the overgeneration problem that TCP includes non-paradoxical derivations in the realm of paradoxical derivations. Chapter 2 and 3 deal with the problems of underand overgeneration. In Chapter 2, we have focuses on Rogerson-type counterexample which employs the rule for Classical Reductio, such as CR−rule, and does not generate a non-terminating reduction sequence. Rogerson-type counterexamples raise the problem of undergeneration. In order to solve the undergeneration problem, Tennant (2015) thinks that the application of CR−rule has a defect that it conceals the primary feature of the paradoxes, i.e. a nonterminating reduction sequence. However, as we have argued in Section 2.2, CR−rule is not the only culprit of the defect. We have seen in 2.3 that there exist cases which do not use CR−rule but raise the problem of undergeneration. Furthermore, we have suggested examples employing CR−rule which generate a non-terminating reduction sequence. From our observations in Section 2.3 and 2.
to the paradoxes. After having preliminary notations and natural discount rule, Chapter 1 introduces the early version of Tennant's criterion for paradoxicality transmission control protocol and the requirement of a normal derivation RND. As a doctor treat the disease in accordance with her diagnosis, a logician solves the paradox on the footing of her characterization of the paradoxes. When TCP is involve as a proof - theoretic criterion for paradoxicality, RND can be a potential proof - theoretic solution to the paradox. The Early Version of Tennant's Criterion for Paradoxicality(TCPE ): Let D be any deriving of a contribute natural deduction system S. D is a T - paradox if and entirely if (i) D is a (closed or open) derivation of ⊥, (ii) i vitamin d est inferences (or rules) are used in D, (iii) a reduction procedure of D generate a non - terminating reduction sequence, such as a reduction loop. The Requirement of a (Full) Normal Derivation(RND ): For any derivation D in natural deduction, D is acceptable only if D is (in principle) convertible into a normal derivation. 163 There are two types of counterexamples to TCP. The one generates the trouble of undergeneration in the common sense that TCP makes a paradoxical derivation non - paradoxical. The other counterexample cause the overgeneration problem that TCP includes non - paradoxical derivations in the region of paradoxical derivations. Chapter 2 and 3 deal with the problems of underand overgeneration. In Chapter 2, we have focus on Rogerson - type counterexample which employs the rule for Classical Reductio, such as CR−rule, and does not generate a non - terminating reduction sequence. Rogerson - character counterexamples raise the problem of undergeneration. In order to solve the undergeneration problem, Tennant (2015) thinks that the lotion of CR−rule have a defect that it conceals the primary feature of speech of the paradoxes, i.e. a nonterminating reduction sequence. However, as we have argued in Section 2.2, CR−rule is not the only culprit of the defect. We have seen in 2.3 that there exist case which do not practice CR−rule but raise the problem of undergeneration. Furthermore, we have suggested examples employing CR−rule which render a non - terminating reduction succession. From our observations in Section 2.3 and 2.
 to the paradoxes. After havlng preliminary uitatiois and hatural aeduction rules, Chapter 1 intcoduxes tye early version of Tevnant's crpterion fir peradoxicality TC' and thc reqhlremeut of a normal dgrivation RNG. As a doctor drdacs the disease in accordance with hew diagnpsls, a logician folvts ehe laradoxes on the basis of her charzcterizetion of the patadoxes. When TCP is regardfd ad a proof-theoretic criterion dor [qradoxicalith, RND can be a possiblg proof-theoretic solution to the paraboxes. The Eqroy Ggrsion of Teinant's Criterion for Paradmxicaliyy(TCPE): Let D bc any deeivation of a given netural deduction sysjem S. D is a T-paradox if and inoy if (i) D is x (cluses pr open) fermvation of ⊥, (ii) id est unferences (or rules) awv used in D, (iji) a rqdtction procedure of D generates a non-tegminzting reduction sequencw, such as a reduction loop. The Requirement of a (Full) Normal Derivation(RND): For aty decixatnin D kb jatural deduction, D is acceptable only if D if (im irinciple) convertlble into a normal dfrodation. 163 There are tck fypes of counterexwmples jo TCP. The one denetates the problem of undergwneration in rhe sense that TCP makes a paxadoxival drrivation non-paradoxicau. Ths other coujterexampmd causes the ovefgekerdtion problem that TCP incjudes non-'aradpxical aerifationf in the rfalm of paradoxical derigatious. Chdpter 2 and 3 deal with the problems of undxcand overgenetathon. In Chapcer 2, wc have focuses jn Rogerson-typg counterzxampld which emiloys the rule for Cjassical Reduwjio, such as CC−rule, and doew nor generxge a non-terminsting redlcuion sequebce. Rogerson-type cpunjedexamples raise tht problem of undrrgdnewanioi. In jsder to solva thd uvcergeveration priclem, Tennant (2015) thinks thad ths application of CT−rmle has a defect ehat it concesls the primary fewture of the larwdoxes, i.e. a nonterminating redhction seeuekce. However, af we have argued in Section 2.2, CR−rule is not the only cul'rit of the defect. We hqve seen in 2.3 that tkeve exist casxs whish do not use CR−rule but raisw the problem of mndergeneration. Furthedmore, fe hage suggested examples employing CR−rule which generate a non-terminating reeuctioi fequence. Frkm oir obvexvacions ig Sertmon 2.3 and 2.
to the paradoxes. After having preliminary notations deduction Chapter 1 the early version TCP the requirement of normal derivation RND. a doctor treats the disease in with her diagnosis, a logician solves the paradoxes on the basis of her of the paradoxes. When TCP is regarded as a proof-theoretic criterion for paradoxicality, can a proof-theoretic to the paradoxes. The Early Version of Tennant's Criterion for Paradoxicality(TCPE): Let D be any derivation a given natural deduction system S. D is T-paradox if and only (i) D is a (closed open) of ⊥, id inferences rules) are used D, (iii) a reduction procedure of D generates a non-terminating reduction sequence, such as a reduction loop. Requirement of Normal Derivation(RND): any D natural deduction, D only if D is (in principle) normal derivation. 163 There are two types of to TCP. one generates the problem of undergeneration the sense that TCP makes a paradoxical derivation The other counterexample causes the overgeneration problem that TCP includes non-paradoxical derivations in the realm derivations. Chapter 2 and deal with the of overgeneration. Chapter we have on Rogerson-type counterexample which employs the rule for Classical Reductio, such CR−rule, and does not generate a non-terminating reduction sequence. Rogerson-type the of undergeneration. In to solve the undergeneration Tennant thinks that the application has defect the feature the paradoxes, i.e. a reduction sequence. However, as we argued in Section 2.2, culprit of the defect. We have seen in that there exist cases which do not CR−rule but raise the problem of undergeneration. Furthermore, we have suggested examples CR−rule which non-terminating reduction sequence. From our observations in Section and 2.
 to the paradoxes. After having PreliminarY notaTioNs aNd NatuRal dEduction rules, CHApteR 1 introduces the early verSion oF TENnanT'S cRiterIon for pARaDOXicAlItY TCp aND tHe reqUirEment of A normal derIvaTiOn RND. As a doctOR tReats the diSeaSe in accordanCe wIth her DiAgnOSis, a lOgiCian sOlves tHE paradOxes on the BaSIs of heR CharactERIzAtioN of the paradoxes. WhEN Tcp is regarded as a Proof-tHeOReTIC crIteRion for parAdOxicaLIty, RND cAN bE A POssIBle proof-theorEtic solutioN To tHe paraDoXes. tHe EarlY VersIoN Of TEnnant's CritErioN for ParadOxicalITy(TCPE): LET D be any DerivaTioN of A givEN nAtUraL dEDucTIoN sySTem 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. thE rEQuIrement of a (Full) NormAl dERiVation(RNd): For anY DeRiVAtion D in NaTurAl deDUCtion, d is aCCePtable onLy if D iS (In PrInciple) CoNvertiBlE inTo a NormaL DeriVation. 163 there are Two tyPEs of counterexaMPles to TCP. The oNE gENErATes tHe pRoblem of undErgeNEratIon iN ThE seNSe thaT TCP mAkES a PAradoxical derivatioN nOn-paraDoxicAl. The other couNterexamplE CAUses the oVergENeRAtion problem thAt TCP Includes noN-ParadoxiCal deRivationS in the reaLM Of paradoXicAl dEriVatIONs. chapter 2 and 3 deaL WIth tHe ProblemS of UnderanD ovErgEneRatIoN. In ChapteR 2, we have fOcUsEs On rogErson-TYpe countErExaMpLe wHich eMPloys tHe rulE for clAsSIcaL ReductIO, sUCH as Cr−rUlE, and DoeS nOt genEratE A noN-terminAting reduCtiON seqUeNcE. RogersOn-type counterExAmples raisE tHe pRoblem OF UndergenEration. In order to solve thE UndergeNerAtion ProbLem, TennanT (2015) thInks thAt tHE appliCation Of CR−rUlE haS A DefecT THaT it CoNceals the pRIMarY featUrE of tHe paradOxes, i.e. a nonterminatINg rEduction sequeNce. howeVER, aS we HAvE ArgUeD In SECTion 2.2, CR−rule is not The only culPrIT oF the defect. wE haVe Seen in 2.3 tHat therE exisT Cases whIch do not uSe CR−rule bUt RaisE THe pRoblem of unDergenerAtion. FurtHErmorE, We Have sUggEsted eXaMplEs empLoying cr−ruLe whiCh geneRaTe a non-TermiNaTing reduCtion sequence. From our obsErvatiOns in secTion 2.3 and 2.
 to the paradoxes. After h aving prel imina rynot at ions  and  natural deduc t ionrules, Chapter 1 intro duces  t h e ea r ly  vers ion ofT en n a nt' scr ite ri o nfor p ara doxical ity TCP an d t he  requirement of  a normalder ivation RND.  As  a doc to r t r eatsthe  dise ase in accord ance with  h e r diag n osis, a l og icia n solves the para d ox e s on the basis  of he rc ha r a cte riz ation of t he  para d oxes. W h en T C P i s  regarded asa proof-the o ret ic cri te rio n  for p arado xi c ali ty, RND can  bea possibl e proo f -theore t ic solu tion t o t hepara d ox es . T he Ear l yVer s ion  of Tenn an t' s Cri teri o n f or P ara doxi calit y(TCPE): LetD b e an y  de rivat ion o f agi ven n atural  dedu ct ion system S. D  isa T-parad oxif  an donlyi f (i)D i s a  (close d or op e n)de r i v at ion of ⊥, (ii) ides t in ferences  (or r u le s) are used  i n D , (i i i ) a r educ t io n proced ure of Dge nerates  a  non-t er min ati ng re d ucti on seq uence, s uch a s  a reduction l o op. The Requi r em e n to f a(Fu ll) NormalDeri v atio n(RN D ):  Fo r  anyderiv at i on D in natural deduct io n, D i s acc eptable onlyif D is (i n p rinciple ) co n ve r tible into a n ormal  derivatio n . 163 Th ere a re two t ypes of c o u nterexam ple s t o T CP. T he  one generate s thepr oblem o f u ndergen era tio n i n t he  sense th at TCP m ak es  a  p ara doxic a l deriva ti onno n-p arado x ical.The o ther  c ou n ter example ca u s es t he  o verg ene ra tionprob l emthat TC P include s n o n-pa ra do xical d erivations in  t he realm o fpar adoxic a l  derivat ions. Chapter 2 and 3 d e al with  th e pro blem s of unde ran d over gen e ration . In C hapte r2,w e  have f oc use son Rogerso n - typ e cou nt erex ample w hich employs the r u lefor Classical  Re duct i o ,suc h  a s  CR −r u le, a nd does not gen erate a no n- t er minating r e duc ti on sequ ence. R ogers o n-typecounterex amples ra is e th e pro blem of un dergener ation. In order to  solv e t he und er gen erati on pro b lem , Ten nant ( 20 15) th inksth at the a pplication of CR−rule h as a d efect  th at it con cea l s t he primar y fe ature of t hepar adoxe s,i .e. a  non t er min a tingredu c tion sequ e nc e.H o we ver, as weh a v e a rgued  in Sectio n 2. 2, CR−rule is not the only culpr it o f the  de f ect.  W e have seen in  2. 3t h at there  e xist caseswhich do  n o t use  CR−ru le but  raiset h ep roblem  ofund ergenerat ion .F urtherm or e, we hav e su gg estedexampl e s em p l oying CR−rule wh ich g e n erate a n on-te rm inating redu ction sequ ence. Fromour ob serv ation s in Se ct ion 2. 3 a nd  2.
 to_the paradoxes._After having preliminary notations_and natural_deduction_rules, Chapter_1_introduces the early_version of Tennant's_criterion for paradoxicality TCP_and the requirement_of_a normal derivation RND. As a doctor treats the disease in accordance with her_diagnosis,_a logician_solves_the_paradoxes on the basis of_her characterization of the paradoxes._When TCP_is regarded as a proof-theoretic criterion for paradoxicality,_RND_can be a_possible proof-theoretic solution to the paradoxes. The Early Version_of Tennant's Criterion for Paradoxicality(TCPE): Let_D be any_derivation_of_a given natural deduction_system 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. The Requirement of a (Full)_Normal Derivation(RND): For_any derivation D in natural deduction,_D_is acceptable only_if_D_is (in_principle) convertible into_a_normal derivation._163_There are two types of counterexamples_to_TCP. The one generates the problem of_undergeneration in the sense_that_TCP makes a paradoxical_derivation non-paradoxical. The other counterexample_causes the overgeneration problem that TCP_includes non-paradoxical_derivations in_the realm of paradoxical derivations. Chapter 2 and 3 deal with_the problems of underand overgeneration. In_Chapter 2, we have_focuses on_Rogerson-type_counterexample which employs_the_rule for_Classical Reductio, such as CR−rule, and does_not generate_a non-terminating reduction sequence. Rogerson-type counterexamples_raise the problem of_undergeneration._In order to solve the undergeneration_problem, Tennant (2015) thinks that the_application of CR−rule has a_defect_that_it conceals the primary feature_of the paradoxes, i.e. a nonterminating_reduction sequence. However,_as we have argued in Section 2.2,_CR−rule_is not the only culprit of_the_defect. We have seen in 2.3_that_there_exist cases which do not_use CR−rule but raise the problem_of undergeneration. Furthermore, we have suggested examples employing CR−rule_which generate a_non-terminating reduction sequence. From our_observations_in_Section 2.3 and 2.