 is a set a such that a = {x|¬x ∈ x}. Since a = {x|¬x ∈ x} is often considered to be the main reason that generates Russell's paradox, he believes that Proposition 161 4.1.1 can be the ground for claiming that Russell's paradox is not a genuine paradox. In a similar way, we have shown in Proposition 4.1.3 that there is a full normal derivation of ¬(Φ ↔¬T (⌜Φ⌝)). If the full normal derivation of ¬∃y({x|¬x ∈ x}) supports the view that Russell's paradox is not genuine, either Proposition 4.1.3 supports the view that the Liar is not. These results establish that Tennant has no clear ground for genuine paradoxes. If TCP and RND are about genuine paradoxes, it should be explained which paradoxes are genuine paradoxes. For the second question, Section 4.2 considers a possible explanation that when prooftheoretic validity implies normalizability, RND can be a proof-theoretic solution to the paradoxes. Since a paradoxical derivation, such as T-paradox, initiates a non-terminating reduction sequence and is not normalizable, it is not proof-theoretically valid. Hence, RND can block the proof-theoretically invalid derivation and so it can be a solution to the paradoxes. To support this view, we have suggested Theorem 4.2.2 and 4.2.4 that in a particular system, proof-theoretic validity implies (strong) normalizability. However, it must be shown that the result is able to be extended to a general case. If RND is a plausible proof-theoretic solution to the paradoxes, it may be regarded as a reasoning-rejection solution. Section 4.3 argues that RND is a stronger restriction than other reasoning-rejection solutions due to the fact that it does not restrict a specific inference rule but restricts every derivation in an intended system. With regard to the third question, Section 4.4 investigates a posibility that there exists a normal derivation of an unacceptable conclusion and argues that it a contradiction, ¬φ ∧φ, is regarded as a proof-theoretically unacceptable conclusion, then RND cannot be a general solution to the paradoxes. 162 Chapter 5 Conclusion In this dissertation, we have investigated a proof-theoretic criterion for and solution
is a set a such that a = { x|¬x ∈ x }. Since a = { x|¬x ∈ x } is often considered to be the main rationality that generate Russell's paradox, he believes that Proposition 161 4.1.1 can be the ground for claiming that Russell's paradox is not a actual paradox. In a similar way, we have shown in Proposition 4.1.3 that there be a full normal derivation of ¬(Φ ↔¬T (⌜ Φ ⌝) ). If the entire normal derivation of ¬∃y({x|¬x ∈ x }) supports the scene that Russell's paradox is not genuine, either Proposition 4.1.3 supports the opinion that the Liar is not. These results establish that Tennant has no clear ground for genuine paradox. If TCP and RND are about genuine paradoxes, it should be explained which paradox are genuine paradoxes. For the second question, Section 4.2 considers a potential explanation that when prooftheoretic validity incriminate normalizability, RND can be a proof - theoretic solution to the paradoxes. Since a paradoxical derivation, such as T - paradox, initiate a non - terminating reduction sequence and is not normalizable, it is not proof - theoretically valid. Hence, RND can block the proof - theoretically invalid deriving and so it can be a solution to the paradoxes. To support this view, we have suggested Theorem 4.2.2 and 4.2.4 that in a particular system, proof - theoretic validity implies (potent) normalizability. However, it must be shown that the result is able to be unfold to a cosmopolitan case. If RND is a plausible proof - theoretic solution to the paradoxes, it may be regarded as a reasoning - rejection solution. Section 4.3 argues that RND is a stronger restriction than early reasoning - rejection solutions due to the fact that it does not restrict a specific inference rule but restricts every derivation in an intended system. With regard to the third question, Section 4.4 investigates a posibility that there exists a normal deriving of an unacceptable decision and argues that it a contradiction, ¬φ ∧φ, is regarded as a proof - theoretically unacceptable conclusion, then RND cannot be a cosmopolitan solution to the paradoxes. 162 Chapter 5 Conclusion In this dissertation, we have investigated a proof - theoretic criterion for and solution
 is a set a such that a = {x|¬x ∈ x}. Since a = {x|¬x ∈ x} is mften donsiderdd to be the main reason thav geberattf Russell's paradox, he belienes that Propiwition 161 4.1.1 ceh be thc grohkd fox rlaiming that Rossell's paragox is not a ganjiue paradox. In a similar way, we have fhown im Oroposition 4.1.3 trat uhewe ia a full normal derivation of ¬(Φ ↔¬T (⌜Φ⌝)). Jf the hull normal derovation of ¬∃y({x|¬x ∈ x}) supports the view that Russell'd paradox iw noe genuine, eitfer Propospcion 4.1.3 suppotts the view that the Liar is nog. Thexe results ewtahnish that Txnnant has no cleav groung for grnuine paradoxcs. If TCP and RND are about gxnuine paradoxes, it fhould be erplained which paradozew are gengine parxdosex. Ror thf sxcond questjon, Section 4.2 considers a possinlq explanation fhat wreg prooftheoretic validity implies normanizzbility, RND can be a priof-theoretic solution to the pwradoxes. Since a paradoxical derivation, such as T-[aradkb, iulbiatdw w non-terminating reduction sequence and is noe nprkalizable, it if not proof-yhfotgtically valid. Hence, RNS can block the prlof-theoteticaoly invalyd drrivation and so it can be q solution tj the paradoxes. To dupport thir vigw, we nave suggested Theorem 4.2.2 and 4.2.4 that in a particulzf system, proof-thdorvtic validity implies (strong) njrmalizabmlity. Howevef, it must fe shown tjat tma result is able tl be gxtendad to a gejeral case. If RND is a plausiblx proof-theorejic sonution tj the paradoxes, it may be regardgd as a rzasonivg-rejectioh solutmon. Section 4.3 argues that SJD is a stroiger restwictuon rhan otfdr reasoning-rekection solutions dye to the fact thab it akes not restricc q specific infetenze wupe btd restricts averh ddtivatkon in an iktevded system. With regard do tge third question, Xegtion 4.4 incestigatqs a posibiliyy that there exisus a nmrmel derovajion of an unacceptable conclusjon and aggucs that it a sontvadistion, ¬φ ∧φ, is regarded as a proof-theoretically unaccxptable conclusion, then RND cannot be a geueval solution to tre paradofes. 162 Chapter 5 Concluwion In this disstrtation, we have investjgated a prlof-theoretic criterion for and solution
is a set a such that a ∈ Since a {x|¬x ∈ x} the reason that generates paradox, he believes Proposition 161 4.1.1 can be the for claiming that Russell's paradox is not a genuine paradox. In a similar we have shown in Proposition 4.1.3 that there is a full normal derivation ¬(Φ (⌜Φ⌝)). the normal derivation of ¬∃y({x|¬x ∈ x}) supports the view that Russell's paradox is not genuine, either 4.1.3 supports the view that the Liar is These results establish that has no clear ground for paradoxes. TCP and are genuine it should be which paradoxes are genuine paradoxes. For the second question, Section 4.2 considers a possible explanation that when validity implies can be proof-theoretic to paradoxes. Since a such as T-paradox, initiates a non-terminating is not normalizable, it is not proof-theoretically valid. RND can the proof-theoretically invalid derivation and so can be a solution to the paradoxes. To this view, we have suggested Theorem 4.2.2 and 4.2.4 that in a particular system, proof-theoretic (strong) normalizability. However, it be shown that result able be to a case. If RND is a plausible proof-theoretic solution to the paradoxes, may be regarded as a reasoning-rejection solution. Section 4.3 argues is stronger restriction than reasoning-rejection solutions due to fact it does not restrict inference but in intended With regard to the question, Section 4.4 investigates a that there exists a conclusion and argues that it a contradiction, ¬φ is regarded as a proof-theoretically unacceptable conclusion, RND cannot be a general solution to the paradoxes. 162 Chapter 5 In this have investigated a proof-theoretic criterion for and solution
 is a set a such that a = {x|¬x ∈ x}. Since a = {X|¬x ∈ x} is often ConsiDerEd tO bE the Main Reason that geneRAtes russell's paradox, he belieVes thAt pRopoSItIon 161 4.1.1 caN be the gROuND For ClAiMinG tHAt russeLl's Paradox Is not a genuIne PaRadox. In a simiLAr Way, we have sHowN in PropositiOn 4.1.3 tHat theRe Is a FUll noRmaL deriVation OF ¬(Φ ↔¬T (⌜Φ⌝)). If The full noRmAL derivATion of ¬∃y({X|¬X ∈ X}) sUppoRts the view that RusSElL'S paradox is not gEnuine, EiTHeR pRopOsiTion 4.1.3 supporTs The viEW that thE liAR IS noT. these results eStablish thaT tenNant haS nO clEAr grouNd for GeNUinE paradoxes. IF TCP And RND are About gENuine paRAdoxes, iT shoulD be ExpLainED wHiCh pArADoxES aRe gENuiNe paradoXeS. FOr the SecoND QUEstiOn, SEctiOn 4.2 conSiders a possibLe eXplaNAtiOn thaT when ProoFtHeoreTic valIdity ImPlies normalizabIlitY, RND can be A prOoF-thEoRetic SOlutioN to The ParadoxEs. Since A ParAdOXICaL derivation, such as T-PaRADoX, initiatEs a non-TErMiNAting redUcTioN seqUENce anD is nOT nOrmalizaBle, it iS NoT pRoof-theOrEticalLy ValId. HEnce, Rnd can Block tHe proof-tHeoreTIcally invalid dERivation and so IT cAN Be A SoluTioN to the paradOxes. tO supPort THiS viEW, we haVe sugGeSTeD theorem 4.2.2 and 4.2.4 that in a paRtIcular SysteM, proof-theoretIc validity IMPLies (stroNg) noRMaLIzability. HowevEr, it mUst be shown THat the reSult iS able to bE extended TO A general CasE. If rND Is a PLAuSible proof-theOREtic SoLution tO thE paradoXes, It mAy bE reGaRded as a reAsoning-rEjEcTiOn SolUtion. sEction 4.3 arGuEs tHaT RNd is a sTRonger RestrIctiOn ThAN otHer reasONiNG-RejeCtIoN solUtiOnS due tO the FAct That it dOes not resTriCT a spEcIfIc inferEnce rule but reStRicts every DeRivAtion iN AN intendeD system. With regard to the tHIrd quesTioN, SectIon 4.4 iNvestigatEs a PosibiLitY That thEre exiSts a nOrMal DERivatION oF an UnAcceptable CONclUsion AnD argUes that It a contradiction, ¬φ ∧φ, IS reGarded as a prooF-thEoreTICaLly UNaCCepTaBLe cONClusion, then RND cAnnot be a geNeRAl Solution to THe pArAdoxes. 162 CHapter 5 COncluSIon In thIs dissertAtion, we haVe InveSTIgaTed a proof-tHeoretic Criterion FOr and SOlUtion
 is a set a such that a ={x|¬x ∈ x} . Sin cea =  { x|¬x  ∈ x } is often con s ider ed to be the main reas on th at gene r at es Ru ssell's pa r a dox ,he  be li e ve s tha t P roposit ion 161 4. 1.1  c an be the gr o un d for clai min g that Russe ll' s para do x i s  nota g enuin e para d ox. In  a simila rw ay, we have sh o w nin P roposition 4.1.3t ha t  there is a fu ll nor ma l  d e r iva tio n of ¬(Φ ↔ ¬T  (⌜Φ⌝ ) ). If t h ef u l l n o rmal derivati on of ¬∃y({ x |¬x  ∈ x})  s upp o rts th e vie wt hat  Russell'spara dox is no t genu i ne, eit h er Prop ositio n 4 .1. 3 su p po rt s t he vie w  t hat the  Liar is  n ot . The se r e s u l ts e sta blis h tha t Tennant has  no  cle a r g round  forgenu in e par adoxes . IfTC P and RND are a bout  genuinepar ad oxe s,  it s h ould b e e xpl ained w hich pa r ado xe s a re  genuine paradoxes .F o rthe seco nd que s ti on ,  Section  4 .2cons i d ers a  pos s ib le expla nation th at  when p ro oftheo re tic  va lidit y  imp lies n ormaliza bilit y , RND can be a proof-theoret i cs o lu t iontothe paradox es.S ince  a p a ra dox i cal d eriva ti o n, such as T-paradox,in itiate s a n on-terminatin g reductio n s equenceandi sn ot normalizabl e, it  is not pr o of-theor etica lly vali d. Hence, R ND can b loc k t hepro o f -t heoreticallyi n vali dderivat ion  and so  it  ca n b e a  s olution t o the pa ra do xe s.  To  supp o rt thisvi ew,  w e h ave s u ggeste d The orem  4 .2 . 2 a nd 4.2. 4  t h a t in  a  p arti cul ar  syst em,p roo f-theor etic vali dit y  imp li es  (stron g) normalizab il ity. Howev er , i t must b e shownthat the result is able to be e xte ndedto a  generalcas e. IfRND is a p lausib le pr oo f-t h e oreti c so lut io n to the p a r ado xes,it  may  be reg arded as a reasoni n g-r ejection solu tio n. S e c ti on4 .3 arg ue s  th a t  RND is a stron ger restri ct i on  than othe r  re as oning-r ejectio n sol u tions d ue to the  fact tha tit d o e s n ot restric t a spec ific infe r encer ul e but  re strict seve ry de rivati o n i n anintend ed  syste m. Wi th  regardto the third question,Sectio n 4.4  in vestigate s a pos ibility t hatthere exis tsa n ormal  de r ivati on o f  a n u n accep tabl e  conclusi o nand a rg ues that it a con tradi cti o n, ¬φ∧φ,is regarded as ap roof-theoretic ally u nac cep t able  c onclusion, the n R ND c annot be  a  general so lution t ot he pa radoxe s. 162  Chapte r 5C onclus ionInthis diss ert at i on, weha ve invest igat ed  a pro of-the o reti c criterion for an d sol u t ion
 is_a set_a such that a_= {x|¬x_∈_x}. Since_a_= {x|¬x ∈_x} is often_considered to be the_main reason that_generates_Russell's paradox, he believes that Proposition 161 4.1.1 can be the ground for claiming_that_Russell's paradox_is_not_a genuine paradox. In a_similar way, we have shown_in Proposition_4.1.3 that there is a full normal derivation_of_¬(Φ ↔¬T (⌜Φ⌝))._If the full normal derivation of ¬∃y({x|¬x ∈ x})_supports the view that Russell's paradox_is not genuine,_either_Proposition_4.1.3 supports the view_that the Liar is not. These_results establish that Tennant has no_clear ground for genuine paradoxes. If TCP_and RND are about genuine paradoxes,_it should be explained which_paradoxes are_genuine paradoxes. For the second_question, Section 4.2_considers a_possible explanation that_when prooftheoretic validity implies normalizability, RND_can be a_proof-theoretic solution to the paradoxes. Since_a_paradoxical derivation, such_as_T-paradox,_initiates a_non-terminating reduction sequence_and_is not_normalizable,_it is not proof-theoretically valid. Hence,_RND_can block the proof-theoretically invalid derivation and_so it can be_a_solution to the paradoxes._To support this view, we_have suggested Theorem 4.2.2 and 4.2.4_that in_a particular_system, proof-theoretic validity implies (strong) normalizability. However, it must be shown_that the result is able to_be extended to a_general case._If_RND is a_plausible_proof-theoretic solution_to the paradoxes, it may be regarded_as a_reasoning-rejection solution. Section 4.3 argues that_RND is a stronger_restriction_than other reasoning-rejection solutions due to_the fact that it does not_restrict a specific inference rule_but_restricts_every derivation in an intended_system. With regard to the third_question, Section 4.4_investigates a posibility that there exists a_normal_derivation of an unacceptable conclusion and_argues_that it a contradiction, ¬φ ∧φ,_is_regarded_as a proof-theoretically unacceptable conclusion,_then RND cannot be a general_solution to the paradoxes. 162 Chapter 5 Conclusion In_this dissertation, we_have investigated a proof-theoretic criterion_for_and_solution