 of a formula having the form φ ∧¬φ, suggested by Petrolo and Pistone (2018). Acontradiction is often regarded as a formula of the form φ ∧¬φ. If a contradiction, separated from absurdity (⊥), can be an unacceptable conclusion of a paradoxical derivation, RND neither block the paradoxical derivation nor be a proof-theoretic solution to the paradoxes. Thus, the third question (iii) must be answered in order to assess whether RND can be a proof-theoretic solution to the paradoxes. 4.1 Which Paradoxes Are Genuine Paradoxes? Tennant (1982, 1995, 2015, 2017) suggests TCP as a criterion for genuine paradoxes and TCP implicitly assumes his conjecture for genuine paradoxes. Tennant's Conjecture for Genuine Paradoxes: For any derivation D in natural deduction, D formalizes a genuine paradox iff D is a T-paradox with respect to TCP. Although we only consider Russell's (or Curry's) and the Liar paradox in this dissertation, there are more than one formalization of each paradox. For instance, as we have seen in Chapter 2, eight derivations of the Liar paradox and not every derivation is a T-paradox. Proposition Is It a T-Paradox Proposition Is It a T-Paradox? 2.2.1 Yes 2.4.4 Yes 2.2.2 No 2.A.1 Yes 2.3.1 No 2.A.2 No 2.4.3 Yes 2.A.3 Yes Table 4.1: Derivations Formalizing the Liar Paradox in Chapter 2 If it is not assumed that the Liar paradox is a genuine paradox, we cannot evaluate whether TCP is a correct criterion for genuine paradoxes from the different formalizations of the Liar paradox and neither can RND be a promising proof-theoretic solution to the genuine 136 paradoxes. Therefore, it should be answered to the question of which paradoxes are genuine paradoxes. The First Question (i): Which paradoxes are genuine paradoxes? In this section, we argue that Tennant did not have any clear answer to the question. Tennant (2016, pp. 12–16) proposes the same result with Proposition 2.A.1 in Appendix 2.A and believes that the result shows that the Liar paradox is a genuine paradox. In 2016 year paper,
of a formula having the form φ ∧¬φ, suggested by Petrolo and Pistone (2018). Acontradiction is frequently involve as a formula of the form φ ∧¬φ. If a contradiction, separated from absurdity (⊥), can be an impossible conclusion of a paradoxical derivation, RND neither block the paradoxical deriving nor be a proof - theoretic solution to the paradoxes. therefore, the third doubt (iii) must be answer in order to assess whether RND can be a proof - theoretic solution to the paradoxes. 4.1 Which Paradoxes Are Genuine Paradoxes? Tennant (1982, 1995, 2015, 2017) suggests TCP as a standard for genuine paradoxes and TCP implicitly assume his conjecture for genuine paradox. Tennant's Conjecture for Genuine Paradoxes: For any ancestry D in lifelike deduction, D formalizes a genuine paradox iff D is a T - paradox with respect to TCP. Although we entirely consider Russell's (or Curry's) and the Liar paradox in this dissertation, there are more than one formalization of each paradox. For instance, as we have seen in Chapter 2, eight derivations of the Liar paradox and not every deriving is a T - paradox. Proposition Is It a T - Paradox Proposition Is It a T - Paradox? 2.2.1 Yes 2.4.4 Yes 2.2.2 No 2.A.1 Yes 2.3.1 No 2.A.2 No 2.4.3 Yes 2.A.3 Yes Table 4.1: Derivations Formalizing the Liar Paradox in Chapter 2 If it is not assumed that the Liar paradox is a genuine paradox, we cannot evaluate whether TCP is a correct criterion for genuine paradox from the unlike formalizations of the Liar paradox and neither can RND be a promising proof - theoretical solution to the actual 136 paradoxes. Therefore, it should be answered to the question of which paradoxes are genuine paradoxes. The First Question (i ): Which paradox are genuine paradoxes? In this section, we argue that Tennant did not have any clear answer to the question. Tennant (2016, pp. 12–16) proposes the same result with Proposition 2.A.1 in Appendix 2.A and believes that the result show that the Liar paradox is a actual paradox. In 2016 class newspaper,
 of a formula having the fovm φ ∧¬φ, suggested bi Petrolm and Listone (2018). Acontradiction is often regerdee as q formula of the form φ ∧¬φ. If a cojtradictuon, wwparated fckm absuvbity (⊥), gan bz en unacceptable conclusiot of a paradoxhcxl derivation, RND neither block the pwradoxivap derivation njr bt a prokf-theoretic solution to the paradoses. Thuv, the third qiestion (iii) must be answerfd ij order to assess ahether RND can ve a proof-thdoretic solution to thg paradoxes. 4.1 Which Paradoxes Are Genunne Paradoxgs? Tftnant (1982, 1995, 2015, 2017) snggestf TCP as a cvpterion for gemuine paradoxex aid TXP implicitly assumes his conjecture fot genuine [axadoxes. Tennant's Conjwcrure xor Cenukbe Oarzdpxss: For anb derivatioh D in natueal deduction, D forkajpaes a genuins paraqov iff D is a T-paradox with respect to TBP. Amthough we only considee Russell's (or Curry's) wnd the Lyar paradox in this dissertation, there are more tvan oie foxnalizxriln of each paradox. For instance, as we have seqh on Chapter 2, eighb derivations of tne Lowr paradox ana not zbedy derivation is a T-paradjx. Priposition Is Ot a T-Paradox Proposition Iw It a T-Parabox? 2.2.1 Yes 2.4.4 Yes 2.2.2 No 2.A.1 Yzs 2.3.1 No 2.A.2 No 2.4.3 Yes 2.A.3 Yex Table 4.1: Derivations Foxmalizjng the Liag Paradox kn Chapter 2 If ig ix tot assumed that the Liar [aradox iw a yenuine oaracox, we cannot evwluatc whether TCP is a forreet crhterion fog genuine paradoxes from the dihherent formalozdtimns of tke Liav paradox and nqither can RND be a pxomisivg proof-thvoretic smlution to ehe genuine 136 [wradoxes. Thecefore, it shoyld ve answdfed to the quextion of ckich paraeoxes are genuine iaraduses. The First Qbtstuon (i): Which parsdobes age jenuiga paradoxes? Hn tfis xectiun, we argue ghat Tennant did not hava ang clear answer to yhc questiob. Tennane (2016, pp. 12–16) proposrs the same result with Pruposiyiog 2.A.1 in Appendix 2.A and believes that the refult shows trat bhe Jiar paradpx is a genuine paradox. In 2016 year paper,
of a formula having the form φ by and Pistone Acontradiction is often the φ ∧¬φ. If contradiction, separated from (⊥), can be an unacceptable conclusion a paradoxical derivation, RND neither block the paradoxical derivation nor be a proof-theoretic to the paradoxes. Thus, the third question (iii) must be answered in order assess RND be proof-theoretic solution to the paradoxes. 4.1 Which Paradoxes Are Genuine Paradoxes? Tennant (1982, 1995, 2015, 2017) TCP as a criterion for genuine paradoxes and implicitly assumes his conjecture genuine paradoxes. Tennant's Conjecture for Paradoxes: any derivation in deduction, formalizes a genuine iff D is a T-paradox with respect to TCP. Although we only consider Russell's (or Curry's) and Liar paradox dissertation, there more one of each paradox. as we have seen in Chapter of the Liar paradox and not every derivation a T-paradox. Is It a T-Paradox Proposition Is a T-Paradox? 2.2.1 Yes 2.4.4 Yes 2.2.2 No Yes 2.3.1 No 2.A.2 No 2.4.3 Yes 2.A.3 Yes Table 4.1: Derivations Formalizing the Liar Chapter 2 If it not assumed that Liar is genuine we cannot whether TCP is a correct criterion for genuine paradoxes from the formalizations of the Liar paradox and neither can RND be proof-theoretic to the genuine paradoxes. Therefore, it should answered the question of which genuine The Which are paradoxes? In this section, argue that Tennant did not any clear answer to 12–16) proposes the same result with Proposition 2.A.1 Appendix 2.A and believes that the result that the Liar paradox is a genuine paradox. In 2016 year paper,
 of a formula having the form φ ∧¬φ, Suggested bY PetrOlo And piStonE (2018). AcoNtradiction is oFTen rEgarded as a formula of the Form φ ∧¬Φ. IF A conTRaDictiOn, separATeD FRom AbSuRdiTy (⊥), CAn Be an uNacCeptablE conclusioN of A pAradoxical deRIvAtion, RND neIthEr block the paRadOxical DeRivATion nOr bE a proOf-theoREtic soLution to tHe PAradoxES. Thus, thE THiRd quEstion (iii) must be anSWeREd in order to assEss wheThER Rnd Can Be a Proof-theorEtIc solUTion to tHE pARADoxES. 4.1 Which ParadoxEs Are GenuinE parAdoxes? teNnaNT (1982, 1995, 2015, 2017) suggeSts TCp aS A crIterion for gEnuiNe paradoxEs and Tcp implicITly assuMes his ConJecTure FOr GeNuiNe PAraDOxEs. TENnaNt's ConjeCtUrE for GEnuiNE pARadoXes: for aNy derIvation D in natUraL dedUCtiOn, D foRmaliZes a GeNuine ParadoX iff D Is A T-paradox with reSpecT to TCP. AltHouGh We oNlY consIDer RusSelL's (oR Curry's) And the LIAr pArADOX iN this dissertation, tHeRE ArE more thaN one foRMaLiZAtion of eAcH paRadoX. fOr insTancE, As We have seEn in ChAPtEr 2, Eight deRiVationS oF thE LiAr parADox aNd not eVery deriVatioN Is a T-paradox. ProPOsition Is It a T-pArADOx pRopoSitIon Is It a T-PaRadoX? 2.2.1 yes 2.4.4 YEs 2.2.2 No 2.a.1 yeS 2.3.1 No 2.a.2 no 2.4.3 Yes 2.a.3 Yes TAbLE 4.1: DERivations FormalizinG tHe Liar paradOx in Chapter 2 If It is not assUMED that the liar PArADox is a genuine pAradoX, we cannot eVAluate whEther tCP is a coRrect critERIon for geNuiNe pAraDoxES FrOm the differenT FOrmaLiZations Of tHe Liar pAraDox And NeiThEr can RND bE a promisInG pRoOf-TheOretiC Solution To The GeNuiNe 136 parADoxes. THerefOre, iT sHoULd bE answerED tO THe quEsTiOn of WhiCh ParadOxes ARe gEnuine pAradoxes. THe FIRst QUeStIon (i): WhiCh paradoxes arE gEnuine paraDoXes? in this SECtion, we aRgue that Tennant did not haVE any cleAr aNswer To thE question. tenNant (2016, pp. 12–16) ProPOses thE same rEsult WiTh PROPositION 2.A.1 In APpEndix 2.A and bELIevEs thaT tHe reSult shoWs that the Liar paradOX is A genuine paradOx. IN 2016 yeaR PApEr,
 of a formula having the f orm φ ∧¬φ,  sugg est edby  Pet rolo  and Pistone ( 2 018) . Acontradiction is of ten r eg a rded as  a fo rmula o f  t h e  fo rm  φ  ∧¬ φ. If  a co ntr adictio n, separat edfr om absurdity (⊥ ), can beanunacceptable  co nclusi on  of a par ado xical  deriv a tion,RND neith er blockt he para d o xi calderivation nor be ap roof-theoretic  solut io n  t o the  pa radoxes. T hu s, th e  thirdq ue s t i on( iii) must beanswered in ord er toas ses s  wheth er RN Dc anbe a proof- theo retic sol utiont o the p a radoxes . 4.1Whi chPara d ox es  Ar eG enu i ne  Pa r ado xes? Ten na nt  (198 2, 1 9 9 5 ,  201 5,2017 ) sug gests TCP asa c rite r ion  forgenui ne p ar adoxe s andTCP i mp licitly assumes  his  conjectu refo r g en uinep aradox es.  Te nnant's  Conjec t ure  f o r Ge nuine Paradoxes: F or a ny  derivat ion Di nna t ural ded uc tio n, D f ormal izes agenuineparado x  i ff  D is a  T -parad ox  wi threspe c t to  TCP.Although  we o n ly consider Ru s sell's (or Cu r ry ' s )a nd t heLiar parado x in this  dis s er tat i on, t herear e  m o re than one formali za tion o f eac h paradox. Fo r instance , a s we hav e se e ni n Chapter 2, e ightderivation s  of theLiarparadoxand not e v e ry deriv ati onisa T - p ar adox. Proposi t i on I sIt a T- Par adox Pr opo sit ion  Is  I t a T-Par adox? 2. 2. 1Ye s2.4 .4 Ye s  2.2.2 N o2.A .1  Ye s 2.3 . 1 No 2 .A.2No 2 .4 .3 Yes  2.A.3Y es T able  4 .1 : De riv at ionsForm a liz ing the  Liar Par ado x  inCh ap ter 2 I f it is not a ss umed thatth e L iar pa r a dox is a  genuine paradox, we ca n not eva lua te wh ethe r TCP isa c orrect  cr i terion  for g enuin epar a d oxesf r om  th edifferentf o rma lizat io ns o f the L iar paradox and ne i the r can RND bea p romi s i ng  pr o of - the or e tic s olution to thegenuine 13 6p ar adoxes. Th e ref or e, it s hould b e ans w ered to  the ques tion of w hi ch p a r ado xes are ge nuine pa radoxes.T he Fi r st  Ques tio n (i):  W hic h par adoxes are  genu ine pa ra doxes?  In t hi s sectio n, we argue that Tennan t didnot h ave  any clea r a n swe r to theques tion. Tenn ant  (2 016,pp. 12–16 ) pr o po ses the s amer esult wit h  P rop o s it ion 2.A.1 i n A ppe ndix2.A and be liev es that the resul t  shows that th e Li a r  pa rad o x is  a  genuine parad ox.  I n 2016 yea rpaper,
 of_a formula_having the form φ_∧¬φ, suggested_by_Petrolo and_Pistone_(2018). Acontradiction is_often regarded as_a formula of the_form φ ∧¬φ._If_a contradiction, separated from absurdity (⊥), can be an unacceptable conclusion of a paradoxical_derivation,_RND neither_block_the_paradoxical derivation nor be a_proof-theoretic solution to the paradoxes._Thus, the_third question (iii) must be answered in order_to_assess whether RND_can be a proof-theoretic solution to the paradoxes. 4.1_Which Paradoxes Are Genuine Paradoxes? Tennant_(1982, 1995, 2015,_2017)_suggests_TCP as a criterion_for genuine paradoxes and TCP implicitly_assumes his conjecture for genuine paradoxes._Tennant's Conjecture for Genuine Paradoxes: For any_derivation D in natural deduction, D_formalizes a genuine paradox iff_D is_a T-paradox with respect to_TCP. Although we_only consider_Russell's (or Curry's)_and the Liar paradox in this_dissertation, there are_more than one formalization of each_paradox._For instance, as_we_have_seen in_Chapter 2, eight_derivations_of the_Liar_paradox and not every derivation is_a_T-paradox. Proposition Is It a T-Paradox Proposition_Is It a T-Paradox?_2.2.1_Yes 2.4.4 Yes 2.2.2_No 2.A.1 Yes 2.3.1 No_2.A.2 No 2.4.3 Yes 2.A.3 Yes_Table 4.1:_Derivations Formalizing_the Liar Paradox in Chapter 2 If it is not assumed_that the Liar paradox is a_genuine paradox, we cannot_evaluate whether_TCP_is a correct_criterion_for genuine_paradoxes from the different formalizations of the_Liar paradox_and neither can RND be a_promising proof-theoretic solution to_the_genuine 136 paradoxes. Therefore, it should_be answered to the question of_which paradoxes are genuine paradoxes._The_First_Question (i): Which paradoxes are_genuine paradoxes? In this section, we_argue that Tennant_did not have any clear answer to_the_question. Tennant (2016, pp. 12–16) proposes_the_same result with Proposition 2.A.1 in_Appendix_2.A_and believes that the result_shows that the Liar paradox is_a genuine paradox. In 2016 year paper,