adox overlooks the fact that the non-terminating reduction sequences are the key features of the structure. Tennant (2016) has allegedly described the feature that 'these are the proof-theorist's explication of the vicious circularity involved in paradoxes.' The vicious circularity is often considered to be the primary characteristic 160 of the self-referential paradoxes. It should not be ignored when investigating the prooftheoretic structure of the paradoxes. Petrolo and Pistone's criterion of N-paradox loses this point. Nevertheless, as Petrolo and Pistone (2018) claim, any formula with the form ¬φ ∧ φ can be a proof-theoretically acceptable conclusion. If it is, RND fails to be a prooftheoretic solution to the paradoxes. In order for evaluating whether RND can be a legitimate proof-theoretic solution to the paradoxes, the three questions in this chapter still need to be resolved. 4.5 Summary In this chapter, under the assumption that the non-terminating reduction sequence is the proof-theoretic feature of the paradoxes, we have dealt with three questions: (i) 'Which paradoxes are genuine paradoxes', (ii) 'Why should we accept only a normalizable derivation?', and (iii) 'Should we consider only ⊥ as an unacceptable conclusion?' In order for RND to be a proof-theoretic solution to the paradoxes, at least three questions must be explicated. Section 4.1 deals with the first question and argues that Tennant has no clear ground for genuine paradoxes. Although Tennant believes that the Liar paradox is a genuine paradox but Russell's paradox is not, the similar argument of him suggests the opposite results that Russell's paradox is a genuine one but the Liar is not. Since he ignores the fact that a non-terminating reduction sequence is relative to our choice of reduction procedure, we have shown in Proposition 4.1.2 that his derivation of Russell's paradox generates a looping reduction sequence by applying the Ekman-type reduction ⊵GEF. There are two points of Tennant's argument. First, his derivation of Russell's paradox does not satisfy TCPL (or TCPE). Second, as Proposition 4.1.1 shows, there is a closed full normal derivation of ¬∃y(y = {x|¬x ∈ x}) which states the rejection of the formula that there
adox overlooks the fact that the non - terminating reduction sequences are the key feature of the social organization. Tennant (2016) has allegedly described the feature that' these are the proof - theorist's explication of the condemnable circularity involve in paradoxes.' The vicious circularity is much think to be the chief characteristic 160 of the self - referential paradoxes. It should not be neglect when investigating the prooftheoretic structure of the paradox. Petrolo and Pistone's criterion of N - paradox loses this point. Nevertheless, as Petrolo and Pistone (2018) claim, any formula with the form ¬φ ∧ φ can be a validation - theoretically acceptable conclusion. If it is, RND fails to be a prooftheoretic solution to the paradox. In order for evaluating whether RND can be a legitimate proof - theoretic solution to the paradox, the three questions in this chapter even need to be resolved. 4.5 Summary In this chapter, under the assumption that the non - terminating decrease sequence is the proof - theoretic feature of the paradoxes, we have dealt with three questions: (iodine)' Which paradoxes are genuine paradoxes', (ii)' Why should we accept only a normalizable derivation?', and (iii)' Should we consider only ⊥ as an unacceptable conclusion?' In ordering for RND to be a proof - theoretic solution to the paradoxes, at least three question must be explicate. Section 4.1 deals with the inaugural question and argues that Tennant has no clear flat coat for genuine paradoxes. Although Tennant believes that the Liar paradox is a genuine paradox but Russell's paradox is not, the similar argument of him suggest the opposite results that Russell's paradox is a genuine one but the Liar is not. Since he ignores the fact that a non - terminating reduction sequence is proportional to our option of decrease procedure, we have shown in Proposition 4.1.2 that his derivation of Russell's paradox generates a looping reduction sequence by applying the Ekman - character reduction ⊵GEF. There be two points of Tennant's argument. foremost, his derivation of Russell's paradox does not satisfy TCPL (or TCPE). Second, as Proposition 4.1.1 shows, there is a closed entire normal derivation of ¬∃y(y = { x|¬x ∈ x }) which state the rejection of the formula that there
adod overlooks the fact thau the non-terminatnbg rednction aequencer are the key features of thx steuctuee. Tennant (2016) has allegealy descrpbed the deatnre that 'these acs the pvjof-tgcorisc's explication on the viciogs circularity ivvllved in paradoxes.' The vicious circtlarity id often considgred uo fe tgv krimary characteristic 160 of the semf-refertntial paradoxes. Iy should not be ignored whfn ijvestigating the pgooftheoretuc seeucture of tfe paradoxts. Petrolo ahd Pistone's criterion of N-paradux loxes this piibt. Jgvertheless, es Petgolo and Pistone (2018) clahm, any gormula with tme focm ¬φ ∧ φ can be a proof-theorxtically acceptable sonclusiot. Nf it is, RND fails to bw a ptooftveordric somuvioh to tje 'aradoxes. Ih order for evaluating whether RGE can be a lefitimaee proof-theoretic solution to the paradoxts, ths three questions in thus chapter still need to be refolved. 4.5 Summary In this chapter, under the assumpthon tiag tkc vin-herminating reduction sequence is the proof-thqkrttib feature of the iaradoxes, we have cewly with three qoestions: (i) 'Which paradoxes age genuyne pqradoxes', (yi) 'Wny should we accept only a bormalizable eerivation?', and (iii) 'Should we eonsidgr onlu ⊥ as an unacceptable cunclhsion?' In orfer for RHA to be a proof-tfeogetiw solution to the paradoxef, at leasv thrze questkons must fe explicahed. Scwtion 4.1 deals with hhe fnrst xuestion ajd argues that Tennant has no clear ground fot ganupne paradjxes. Slthough Tennwnt believes tkat the Jiar oaradox is a genumne paradox fut Russell's [wradox is nov, the simylar argyment ow him suggests yhe opposptt results rhat Russell's paravox ka a genuine one buu rhe Liar is noy. Skncq je idtores the fawt tfat s non-germinaunng feduvtion sequence is renatibe to our choice og veduction proceduwe, we have shpwn in Proposition 4.1.2 thav his verivayiog of Russell's paradox generatea a loopijg veduction seqtencc by applying che Ekman-type reduction ⊵GEF. There are twi points of Tennanr's argument. First, hns derivation of Rtssell's pdradox does not satiwfy TCPL (or TCPE). Xecond, as Proposition 4.1.1 showv, thege is a closed full normal derivation of ¬∃y(y = {x|¬x ∈ x}) which states the rejecrion oh ehe formula thay thege
adox overlooks the fact that the non-terminating are key features the structure. Tennant feature 'these are the explication of the circularity involved in paradoxes.' The vicious is often considered to be the primary characteristic 160 of the self-referential paradoxes. should not be ignored when investigating the prooftheoretic structure of the paradoxes. Petrolo Pistone's of loses point. Nevertheless, as Petrolo and Pistone (2018) claim, any formula with the form ¬φ ∧ φ be a proof-theoretically acceptable conclusion. If it is, fails to be a solution to the paradoxes. In for whether RND be legitimate solution to the the three questions in this chapter still need to be resolved. 4.5 Summary In this chapter, under assumption that reduction sequence the feature the paradoxes, we with three questions: (i) 'Which paradoxes (ii) 'Why should we accept only a normalizable and (iii) we consider only ⊥ as an conclusion?' In order for RND to be a solution to the paradoxes, at least three questions must be explicated. Section 4.1 deals with question and argues that has no clear for paradoxes. Tennant that the paradox is a genuine paradox but Russell's paradox is not, the argument of him suggests the opposite results that Russell's paradox genuine but the Liar not. Since he ignores fact a non-terminating reduction sequence to choice we shown Proposition 4.1.2 that his of Russell's paradox generates a reduction sequence by applying are two points of Tennant's argument. First, his of Russell's paradox does not satisfy TCPL TCPE). Second, as Proposition 4.1.1 shows, there is a closed full normal of ¬∃y(y ∈ x}) which states the rejection of the that there
adox overlooks the fact that tHe non-termiNatinG reDucTiOn seQuenCes are the key feATureS of the structure. Tennant (2016) Has alLeGEdly DEsCribeD the feaTUrE THat 'ThEsE arE tHE pRoof-tHeoRist's exPlication oF thE vIcious circulARiTy involved In pAradoxes.' The vIciOus cirCuLarITy is oFteN consIdered TO be the Primary chArACterisTIc 160 of the SELf-RefeRential paradoxes. IT ShOUld not be ignoreD when iNvEStIGAtiNg tHe prooftheOrEtic sTRucture OF tHE PAraDOxes. Petrolo anD Pistone's crITerIon of N-PaRadOX loses This pOiNT. NeVertheless, aS PetRolo and PiStone (2018) cLAim, any fORmula wiTh the fOrm ¬Φ ∧ φ cAn be A PrOoF-thEoREtiCAlLy aCCepTable conClUsIon. If It is, rnd FAils To bE a prOofthEoretic solutiOn tO the PAraDoxes. in ordEr foR eValuaTing whEther rNd can be a legitimaTe prOof-theoreTic SoLutIoN to thE ParadoXes, The Three quEstions IN thIs CHAPtEr still need to be resOlVED. 4.5 SUmmary In This chAPtEr, UNder the aSsUmpTion THAt the Non-tERmInating rEductiON sEqUence is ThE proof-ThEorEtiC featURe of The parAdoxes, we Have dEAlt with three quEStions: (i) 'Which pARaDOXeS Are gEnuIne paradoxeS', (ii) 'WHY shoUld wE AcCepT Only a NormaLiZAbLE derivation?', and (iii) 'ShOuLd we coNsideR only ⊥ as an unacCeptable coNCLUsion?' In oRder FOr rnD to be a proof-thEoretIc solution TO the paraDoxes, At least tHree questIONs must be ExpLicAteD. SeCTIoN 4.1 deals with the FIRst qUeStion anD arGues thaT TeNnaNt hAs nO cLear grounD for genuInE pArAdOxeS. AlthOUgh TennaNt BelIeVes That tHE Liar pAradoX is a GeNuINe pAradox bUT RUSSell'S pArAdox Is nOt, The siMilaR ArgUment of Him suggesTs tHE oppOsItE resultS that Russell's PaRadox is a geNuIne One but THE Liar is nOt. Since he ignores the fact THat a non-TerMinatIng rEduction sEquEnce is RelATive to Our choIce of ReDucTIOn proCEDuRe, wE hAve shown in pROpoSitioN 4.1.2 tHat hIs derivAtion of Russell's parADox Generates a looPinG redUCTiOn sEQuENce By APplYINg the Ekman-type rEduction ⊵GEf. THErE are two poiNTs oF TEnnant's ArgumenT. FirsT, His deriVation of RUssell's paRaDox dOES noT satisfy TCpL (or TCPE). second, as PROposiTIoN 4.1.1 showS, thEre is a ClOseD full Normal DEriVatioN of ¬∃y(y = {x|¬X ∈ x}) Which sTates ThE rejectiOn of the formula that there
adox overlooks the fact th at the non -term ina tin gredu ctio n sequences ar e  the  key features of the s truct ur e . Te n na nt (2 016) ha s  a l l ege dl ydes cr i be d the  fe ature t hat 'these  ar ethe proof-th e or ist's expl ica tion of thevic ious c ir cul a rityinv olved  in pa r adoxes .' The vi ci o us cir c ularity i softe n considered to b e  t h e primary char acteri st i c1 6 0 o f t he self-re fe renti a l parad o xe s . Its hould not beignored whe n  in vestig at ing the pr oofth eo r eti c structure  ofthe parad oxes.P etroloa nd Pist one'scri ter iono fN- par ad o x l o se s t h ispoint. N ev er thele ss,a s P etro loandPisto ne (2018) cla im,  any for mulawiththefo rm ¬φ  ∧ φ c an be  a  proof-theoreti call y accepta ble  c onc lu sion. If itis,  RN D fails  to bea  pr oo f t h eo retic solution toth e pa radoxes.  In or d er  f o r evalua ti ngwhet h e r RND  can be  a legit imatep ro of -theore ti c solu ti ontothe p a rado xes, t he three  ques t ions in this c h apter still n e ed t ob e re sol ved. 4.5 Su mmar y  Inthis ch apt e r, un der t he as s umption that the no n- termin ating  reduction se quence ist h e  proof-t heor e ti c  feature of th e par adoxes, we have dea lt wi th three  question s :  (i) 'Wh ich  pa rad oxe s ar e genuine par a d oxes ',  (ii) ' Why  should  we  ac cep t o nl y a norma lizablede ri va ti on? ', an d  (iii) ' Sh oul dweconsi d er onl y ⊥ a s an  u na c cep table c o nc l u sion ?'  I n or der  f or RN D to bea proof -theoreti c s o luti on  t o the p aradoxes, atle ast threequ est ions m u s t be exp licated. Section 4.1 de a ls with  th e fir st q uestion a ndargues  th a t Tenn ant ha s nocl ear g round f or  ge nu ine parado x e s.Altho ug h Te nnant b elieves that the L i arparadox is agen uine p ar ado x  b u t R us s ell ' s  paradox is not , the simi la r  a rgument of him  s uggests  the op posit e  result s that Ru ssell's p ar adox i s a  genuine o ne but t he Liar i s  not. Si nce h e i gnores  t hefactthat a non -term inatin greduct ion s eq uence is  relative to our choice  of re ducti onprocedure , w e  ha ve shownin P roposition  4. 1.2  that  hi s  deri vati o nofR ussel l'sp aradox ge n er ate s alooping red u c t ion  sequ enc e  by ap plyi ng the Ekman-type reduction ⊵GEF . Th e r e a ret wo p oi nts of Tennant 'sar g u ment. Fi rs t, his deri vation o fR ussel l's pa radoxdoes no t sa t isfy T CPL(or  TCPE). S eco nd ,  as Pro po si t ion 4. 1.1sh ows, t here i s  a c l o sed full normalderiv a t ion o f  ¬∃ y(y =  { x|¬x ∈x }) w hich state s the rejec tion o f th e for mula th at  there 
adox overlooks_the fact_that the non-terminating reduction_sequences are_the_key features_of_the structure. Tennant_(2016) has allegedly_described the feature that_'these are the_proof-theorist's_explication of the vicious circularity involved in paradoxes.' The vicious circularity is often considered_to_be the_primary_characteristic_160 of the self-referential paradoxes._It should not be ignored_when investigating_the prooftheoretic structure of the paradoxes. Petrolo and_Pistone's_criterion of N-paradox_loses this point. Nevertheless, as Petrolo and Pistone (2018)_claim, any formula with the form_¬φ ∧ φ_can_be_a proof-theoretically acceptable conclusion._If it is, RND fails to_be a prooftheoretic solution to the_paradoxes. In order for evaluating whether RND_can be a legitimate proof-theoretic solution_to the paradoxes, the three_questions in_this chapter still need to_be resolved. 4.5_Summary In_this chapter, under_the assumption that the non-terminating reduction_sequence is the_proof-theoretic feature of the paradoxes, we_have_dealt with three_questions:_(i)_'Which paradoxes_are genuine paradoxes',_(ii)_'Why should_we_accept only a normalizable derivation?', and_(iii)_'Should we consider only ⊥ as an_unacceptable conclusion?' In order_for_RND to be a_proof-theoretic solution to the paradoxes,_at least three questions must be_explicated. Section_4.1 deals_with the first question and argues that Tennant has no clear_ground for genuine paradoxes. Although Tennant_believes that the Liar_paradox is_a_genuine paradox but_Russell's_paradox is_not, the similar argument of him suggests_the opposite_results that Russell's paradox is a_genuine one but the_Liar_is not. Since he ignores the_fact that a non-terminating reduction sequence_is relative to our choice_of_reduction_procedure, we have shown in_Proposition 4.1.2 that his derivation of_Russell's paradox generates_a looping reduction sequence by applying the_Ekman-type_reduction ⊵GEF. There are two points_of_Tennant's argument. First, his derivation of_Russell's_paradox_does not satisfy TCPL (or_TCPE). Second, as Proposition 4.1.1 shows,_there is a closed full normal derivation of ¬∃y(y_= {x|¬x ∈_x}) which states the rejection_of_the_formula that there