 will be introduced in Section 1.2. Tennant's criterion for paradoxicality and the requirement of a normal derivation will be introduced in Section 1.3. 1.1 Traditional Responses to the Paradoxes and Dialetheism When a doctor finds a patient, he diagnoses what the illness the patient has and prescribes it in accordance with his diagnosis. Likewise, when a logician faces a problematic 3 argument (or proof), he characterizes the problem and solves it on the basis of his characterization. It is often believed that solutions to the paradoxes are closely tied with the characterization of the paradoxes. For instance, an informal characterization of a paradox proposed by Richard M. Sainsbury (2009) says, a paradox is generally conceived as 'an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises.' Since acceptable reasoning hardly draws an unacceptable conclusion from acceptable premises, there are three ways to solve the paradox. Either the premise or the reasoning is not actually acceptable, or else the conclusion is acceptable. The first response is the premise-rejection. The second is the reasoning-rejection, and the last is the conclusion-acceptance. The last response is often supported by dialetheism which is the view that there exists a true contradiction. It would be difficult to consider that the traditional approaches to the paradoxes are properly distinguished in accordance with Sainsbury's informal characterization of a 'paradox.' For instance, it is uncertain whether the premise-rejection and the reasoning-rejection can be precisely distinguished. However, it might be the simplest way to understand the informal notion of a 'paradox,' and so we investigate the traditional responses while following Sainsbury's characterization. Traditional responses to the (self-referential) paradoxes seem to embrace the premiserejection and the reasoning-rejection. Traditional approaches to the paradoxes can be found in both the set-theoretic and the semantic paradoxes. For the case of the set-theoretic paradox, we will examine Russell's paradox. The Liar paradox will be used for the semantic paradox. 1.1.1 Traditional Approaches to Russell's and the Liar Paradox. To begin with, Russell's paradox occurs in naive set theory by considering the set of all sets not members of themselves. Let us consider that Our language has constants and quantifiers, ∧, →, ⊥, ¬, ∃, ∀, ∈ for conjunction, implication, absurdity, negation, existential
will be introduced in Section 1.2. Tennant's criterion for paradoxicality and the requirement of a normal deriving will be introduce in Section 1.3. 1.1 Traditional Responses to the Paradoxes and Dialetheism When a doctor of the church finds a affected role, he diagnoses what the illness the patient have and prescribes it in accordance with his diagnosis. Likewise, when a logician faces a problematic 3 argumentation (or proof), he characterizes the trouble and solves it on the basis of his characterization. It is often believed that solutions to the paradoxes are close tied with the characterization of the paradoxes. For example, an informal characterization of a paradox propose by Richard M. Sainsbury (2009) says, a paradox is by and large conceived as' an apparently unacceptable conclusion derived by obviously acceptable reasoning from apparently acceptable premises.' Since acceptable reasoning hardly draw an unacceptable conclusion from acceptable premises, there are three ways to solve the paradox. Either the premise or the reasoning is not actually acceptable, or else the conclusion is acceptable. The inaugural response is the premise - rejection. The second is the reasoning - rejection, and the last is the ending - toleration. The last response is often supported by dialetheism which is the view that there exists a true contradiction. It would be unmanageable to consider that the traditional approaches to the paradoxes are properly distinguished in accordance with Sainsbury's informal characterization of a' paradox.' For instance, it is uncertain whether the premise - rejection and the reasoning - rejection can be precisely distinguished. However, it might be the simplest direction to understand the cozy impression of a' paradox,' and therefore we investigate the traditional responses while following Sainsbury's characterization. Traditional response to the (self - referential) paradoxes seem to embrace the premiserejection and the reasoning - rejection. Traditional access to the paradox can be found in both the set - theoretic and the semantic paradoxes. For the case of the set - theoretic paradox, we will probe Russell's paradox. The Liar paradox will be used for the semantic paradox. 1.1.1 Traditional Approaches to Russell's and the Liar Paradox. To begin with, Russell's paradox happen in naive stage set theory by consider the set of all set not members of themselves. Let us regard that Our language have constants and quantifier, ∧, →, ⊥, ¬, ∃, ∀, ∈ for conjunction, implication, absurdity, negation, existential
 wipl be introduced in Sectlon 1.2. Tennant's crnrerion for pzradoxicxlity and the requirement of a normql derivation will be kntroducef in Secrion 1.3. 1.1 Traditioizl Responses fl thz 'aradoxes and Dlaletheism Fhen a doctor xivdd a patient, he diagnoses what the ijlness yhf patient has wnd krefcrigvs it in accordance with his dizgnosis. Likewise, whem a logician faces a problfmatlc 3 argument (or prlof), he charqctewuzes the proclem and solves it on jhe basis of his characterizatiov. It ns often beoiwvef that solutmons tj the paradowvs are wlosely tied with the cheracrerization of the paredoxes. For instance, wn informdl characterization od a pdradmx pfipored bb Rjchard M. Aainsbury (2009) says, a parqdox is generally cpnsvoved as 'an alparenelr unacceptable conclusion derived by apkarenfly acceptable reasonint from apparently accgptable prqmises.' Since acceptable reasoning hardly draws an unacreotayoe covxlksion from acceptable premises, there are threq wsyx to solve the paradox. Eiyhfr jhe premise or the rzzskning is not actuaply accgptablw, or else the conclusion is acceptable. Tye first resionsw is the premise-relection. The recomd is the reasoning-rejection, and the last id the conduusion-acceptance. Thv lavt response is often suppowted by dmaletkeism whkch os the view that therc exists a true conhradietion. It would he difficult to consider that tix traditional a[prmaches tj the paradoxes arq properly disjinguisheb in azcordance sith Samnsbury's infjrmal charactatization of a 'paradox.' For insrance, ig is uncertain ehether tkt premise-rwjection and the rcasonkhg-rejection can be precisely distongjisref. Iowevqs, it might ba thd skkplesg way to unbefstamd the informal notimn or a 'paradox,' and so wc investitate the traditional tesponses while foplowiig Saiisbury'x craracterization. Traditional reaponses tl tme (self-referegtiao) paradoxes xeem to embrace the premiserejection anv the reasoning-rejectiob. Traditional approccmes to the peradoxqs can be found in both the swt-theoretic and tme semantic paradoxes. Ror tha casf of the set-theoretic paradox, we will examine Russell's paradox. The Liar pqradox will be uses fot the szmautic pawadoe. 1.1.1 Traditional Appgoaches to Russell's and the Liar Paradox. Tj begin with, Russell's paradoc uccurs in naixe set theory by consideding the set of all sets not members of themselves. Let us consicer that Our manguage hes constants and quamtifiexs, ∧, →, ⊥, ¬, ∃, ∀, ∈ for conmunction, implizatook, absurditj, nejatnon, existential
will be introduced in Section 1.2. Tennant's paradoxicality the requirement a normal derivation 1.3. Traditional Responses to Paradoxes and Dialetheism a doctor finds a patient, he what the illness the patient has and prescribes it in accordance with his Likewise, when a logician faces a problematic 3 argument (or proof), he characterizes problem solves on basis of his characterization. It is often believed that solutions to the paradoxes are closely tied the characterization of the paradoxes. For instance, an characterization of a paradox by Richard M. Sainsbury (2009) a is generally as apparently conclusion derived by acceptable reasoning from apparently acceptable premises.' Since acceptable reasoning hardly draws an unacceptable conclusion from acceptable premises, are three solve the Either premise the reasoning is acceptable, or else the conclusion is response is the premise-rejection. The second is the and the is the conclusion-acceptance. The last response often supported by dialetheism which is the view there exists a true contradiction. It would be difficult to consider that the traditional approaches paradoxes are properly distinguished accordance with Sainsbury's characterization a For it is whether the premise-rejection and the reasoning-rejection can be precisely distinguished. However, might be the simplest way to understand the informal notion 'paradox,' so we investigate traditional responses while following characterization. responses to the (self-referential) to the reasoning-rejection. approaches the paradoxes can be in both the set-theoretic and semantic paradoxes. For the we will examine Russell's paradox. The Liar paradox be used for the semantic paradox. 1.1.1 Approaches to Russell's and the Liar Paradox. To begin with, Russell's paradox in naive by considering the set of all sets not of themselves. Let us that Our language has constants and quantifiers, ∧, →, ¬, ∀, ∈ conjunction, implication, absurdity, existential
 will be introduced in Section 1.2. tennant's crIteriOn fOr pArAdoxIcalIty and the requiREmenT of a normal derivation wiLl be iNtROducED iN SectIon 1.3. 1.1 TradITiONAl REsPoNseS tO ThE ParaDoxEs and DiAletheism WHen A dOctor finds a pATiEnt, he diagnOseS what the illnEss The patIeNt hAS and pResCribeS it in aCCordanCe with his DiAGnosis. lIkewise, WHEn A logIcian faces a probleMAtIC 3 argument (or proOf), he chArACtERIzeS thE problem anD sOlves IT on the bASiS OF His CHaracterizatiOn. It is often BEliEved thAt SolUTions tO the pArADoxEs are closelY tieD with the cHaractERizatioN Of the paRadoxeS. FoR inStanCE, aN iNfoRmAL chARaCteRIzaTion of a pArAdOx proPoseD BY rIchaRd M. sainSbury (2009) Says, a paradox iS geNeraLLy cOnceiVed as 'An apPaRentlY unaccEptabLe Conclusion derivEd by ApparentlY acCePtaBlE reasONing frOm aPpaRently aCceptabLE prEmISES.' SInce acceptable reasOnING hArdly draWs an unACcEpTAble concLuSioN froM ACceptAble PReMises, theRe are tHReE wAys to soLvE the paRaDox. eitHer thE PremIse or tHe reasonIng is NOt actually accePTable, or else thE CoNCLuSIon iS acCeptable. The FirsT RespOnse IS tHe pREmise-RejecTiON. THE second is the reasoniNg-RejectIon, anD the last is the Conclusion-ACCEptance. THe laST rESponse is often sUpporTed by dialeTHeism whiCh is tHe view thAt there exISTs a true cOntRadIctIon. iT WoUld be difficulT TO conSiDer that The TraditiOnaL apProAchEs To the paraDoxes are PrOpErLy DisTinguIShed in acCoRdaNcE wiTh SaiNSbury's InforMal cHaRaCTerIzation OF a 'PARadoX.' FOr InstAncE, iT is unCertAIn wHether tHe premise-RejECtioN aNd The reasOning-rejectioN cAn be precisElY diStinguISHed. HowevEr, it might be the simplest wAY to undeRstAnd thE infOrmal notiOn oF a 'paraDox,' ANd so we InvestIgate ThE trADItionAL ReSpoNsEs while folLOWinG SainSbUry's CharactErization. TraditionAL reSponses to the (sElf-RefeRENtIal) PArADoxEs SEem TO Embrace the premiSerejectioN aND tHe reasoninG-RejEcTion. TraDitionaL apprOAches to The paradoXes can be fOuNd in BOTh tHe set-theorEtic and tHe semantiC ParadOXeS. For tHe cAse of tHe Set-TheorEtic paRAdoX, we wiLl examInE RusseLl's paRaDox. The LiAr paradox will be used for tHe semaNtic pAraDox. 1.1.1 TraditIonAL ApProaches tO RusSell's and thE LiAr PAradoX. To BEgin wIth, RUSsEll'S ParadOx ocCUrs in naivE SeT thEORy By consideriNG THe sEt of aLl sETs not mEmbeRs of themselves. Let US consider that OUr laNGUagE haS ConsTaNts and quantifiErs, ∧, →, ⊥, ¬, ∃, ∀, ∈ FoR COnjunctiOn, Implication, AbsurditY, nEGatioN, existEntial
 will be introduced in Sec tion 1.2.Tenna nt' s c ri teri on f or paradoxical i ty a nd the requirement ofa nor ma l  der i va tionwill be in t r odu ce dinSe c ti on 1. 3.1.1 Tra ditional R esp on ses to the P a ra doxes andDia letheism Whe n a  docto rfin d s a p ati ent,he dia g noseswhat theil l ness t h e patie n t  h as a nd prescribes iti na ccordance with  his d ia g no s i s.Lik ewise, whe na log i cian fa c es a pro b lematic 3 arg ument (or p r oof ), hech ara c terize s the  p r obl em and solv es i t on thebasiso f his c h aracter izatio n.Itis o f te nbel ie v edt ha t s o lut ions toth eparad oxes a r e  clo sel y ti ed wi th the charac ter izat i onof th e par adox es . For  insta nce,an  informal chara cter ization o f a  p ara do x pro p osed b y R ich ard M.Sainsbu r y ( 20 0 9 )  s ays, a paradox isge n e ra lly conc eiveda s'a n  apparen tl y u nacc e p table  con c lu sion der ived b y  a pp arently  a ccepta bl e r eas oning from  appar ently ac cepta b le premises.'S ince acceptab l er e as o ning  ha rdly drawsan u n acce ptab l econ c lusio n fro ma cc e ptable premises, th er e arethree  ways to solv e the para d o x . Either  the pr e mise or the re asoni ng is nota ctuallyaccep table, o r else th e conclusi onisacc ept a b le . The first r e s pons eis thepre mise-re jec tio n.The  s econd isthe reas on in g- re jec tion, and thela stis  th e con c lusion -acce ptan ce .T helast re s po n s e is  o ft en s upp or ted b y di a let heism w hich is t hev iewth at  thereexists a true  c ontradicti on . I t woul d be diffi cult to consider that t h e tradi tio nal a ppro aches tothe  parad oxe s  are p roperl y dis ti ngu i s hed i n ac cor da nce with S a i nsb ury's  i nfor mal cha racterization of a 'pa radox.' For i nst ance , it  is un c ert ai n  wh e t her the premise -rejection  a n dthe reason i ng- re jection  can be  prec i sely di stinguish ed. Howev er , it m igh t be the s implestway to un d ersta n dthe i nfo rmal n ot ion  of a  'para d ox, ' and  so we  i nvesti gateth e tradit ional responses while f ollowi ng Sa ins bury's ch ara c ter ization.Trad itional re spo nse s tothe (self -ref e re nti a l) pa rado x es seem t o  e mbr a c ethe premise r e j ect ion a ndt he rea soni ng-rejection. Tra d itional approa ches t o t hep arad ox es can be foun d i nb o th the s et -theoreticand these m antic  parad oxes.For the c as e  of th e se t-t heoreticpar ad o x, we w il le xamine  Rus se ll's p aradox .  The L iar paradox will  be u s e d for the  sema nt ic para d ox.1.1.1 Trad itional App roache s to  Russ ell's a nd  the L iar  P aradox. To begin wit h, Ru ssell's  p arad oxoccurs  inn a ive s et t he ory  by consi d e ri n gth e  se t of  allse ts n ot member s  of them sel v es. Let  u s c o n sidert ha t Our langua gehas c o n stants and quan t if i ers,∧, →,⊥, ¬,∃, ∀, ∈ for  c onjunct ion , implicati on, absur d ity ,nega tion, ex is tent ia l
 will_be introduced_in Section 1.2. Tennant's_criterion for_paradoxicality_and the_requirement_of a normal_derivation will be_introduced in Section 1.3._1.1 Traditional Responses_to_the Paradoxes and Dialetheism When a doctor finds a patient, he diagnoses what the_illness_the patient_has_and_prescribes it in accordance with_his diagnosis. Likewise, when a_logician faces_a problematic 3 argument (or proof), he characterizes_the_problem and solves_it on the basis of his characterization. It is_often believed that solutions to the_paradoxes are closely_tied_with_the characterization of the_paradoxes. For instance, an informal characterization_of a paradox proposed by Richard_M. Sainsbury (2009) says, a paradox is_generally conceived as 'an apparently unacceptable_conclusion derived by apparently acceptable_reasoning from_apparently acceptable premises.' Since acceptable_reasoning hardly draws_an unacceptable_conclusion from acceptable_premises, there are three ways to_solve the paradox._Either the premise or the reasoning_is_not actually acceptable,_or_else_the conclusion_is acceptable. The_first_response is_the_premise-rejection. The second is the reasoning-rejection,_and_the last is the conclusion-acceptance. The last_response is often supported_by_dialetheism which is the_view that there exists a_true contradiction. It would be difficult_to consider_that the_traditional approaches to the paradoxes are properly distinguished in accordance with_Sainsbury's informal characterization of a 'paradox.'_For instance, it is_uncertain whether_the_premise-rejection and the_reasoning-rejection_can be_precisely distinguished. However, it might be the_simplest way_to understand the informal notion of_a 'paradox,' and so_we_investigate the traditional responses while following_Sainsbury's characterization. Traditional responses to the_(self-referential) paradoxes seem to embrace_the_premiserejection_and the reasoning-rejection. Traditional approaches_to the paradoxes can be found_in both the_set-theoretic and the semantic paradoxes. For the_case_of the set-theoretic paradox, we will_examine_Russell's paradox. The Liar paradox will_be_used_for the semantic paradox. 1.1.1_Traditional Approaches to Russell's and the_Liar Paradox. To begin with, Russell's paradox occurs in_naive set theory_by considering the set of_all_sets_not members of themselves. Let us consider that Our language_has constants_and quantifiers, ∧,_→, ⊥, ¬, ∃, ∀, ∈ for conjunction, implication, absurdity,_negation, existential