 been a significant issue to the foundations of logic and mathematics. 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 Sainsbury (2009, p. 1) says that it is an unacceptable conclusion elicited from the acceptable premises via acceptable reasoning. A diagnosis of the paradoxes through the characterization can be that it is a trouble that acceptability leads to unacceptability. Thus, from the diagnosis with the characterization, three responses to the paradoxes can be proposed such that either the premises or the reasoning is not in fact acceptable, or else the conclusion is acceptable. As noted in Section 1.1, we shall call the first response the premise-rejection, the second the reasoning-rejection, and the last the 131 conclusion-acceptance. It is not to say that every solution to the paradoxes is understood as one of three solutions, but it is the case that Sainsbury's characterization is the simplest way to grasp the informal notion of a 'paradox.' Sainsbury regards a sentence (or a formula) with special characteristics a paradox. However, Tennant's criterion for paradoxicality TCP understands a paradox as a derivation (or an argument) which might have an unacceptable conclusion from the acceptable premises by the acceptable inference rules. As Sainsbury's characterization of a paradox is connected to three solutions, we may have a proof-theoretic solution to the paradoxes on the perspectives of TCP. Tennant (1982) proposes TCP as a criterion for genuine paradoxes and regards it as a conjecture namely that for any derivation D, D formalizes a genuine paradox iff D is a T-paradox. Focussing on normalizability, Tennant (2017, pp. 286–287) suggests a similar conjecture which is linked to his criterion for paradoxicality. How... are we to solve the paradoxes? It is not from this study to venture any new suggestions beyond those of Tennant (1982) and Tennant (1995). Those works provided... proofs, formalized as natural deductions, for all the major paradoxes.... They showed that all these... proofs... cannot be converted into normal form. The original proof-theoretic thesis stands: Genuine paradoxes are those whose associated proofs of absurdity, when formalized as natural deductions, cannot be converted into normal form. This conjecture provides a proof-theoretic criterion for the identification of genuine paradoxes.... The conject
been a significant issue to the foundations of logic and mathematics. It is often believe that solution to the paradoxes are closely tied with the portrayal of the paradoxes. For instance, an cozy characterization of a paradox proposed by Sainsbury (2009, p. 1) says that it is an impossible ending elicited from the satisfactory premises via acceptable reasoning. A diagnosis of the paradoxes through the characterization can be that it is a fuss that acceptability leads to unacceptability. Thus, from the diagnosis with the characterization, three responses to the paradoxes can be propose such that either the premises or the reasoning is not in fact acceptable, or else the conclusion is satisfactory. As noted in Section 1.1, we shall bid the first reaction the premise - rejection, the second the reasoning - rejection, and the last the 131 conclusion - acceptance. It is not to say that every solution to the paradoxes is understand as one of three solutions, but it is the case that Sainsbury's characterization is the simplest way to grok the cozy impression of a' paradox.' Sainsbury regards a sentence (or a formula) with special characteristics a paradox. However, Tennant's criterion for paradoxicality TCP sympathize a paradox as a deriving (or an argument) which might have an unacceptable ending from the acceptable premises by the acceptable inference dominion. As Sainsbury's characterization of a paradox is connected to three solutions, we may have a proof - theoretical solution to the paradoxes on the perspectives of TCP. Tennant (1982) proposes TCP as a criterion for genuine paradoxes and regards it as a conjecture namely that for any derivation D, D formalizes a genuine paradox iff D is a T - paradox. focus on normalizability, Tennant (2017, pp. 286–287) suggest a similar conjecture which is linked to his criterion for paradoxicality. How... are we to solve the paradox? It is not from this study to venture any new suggestions beyond those of Tennant (1982) and Tennant (1995). Those works provided... proofs, formalized as natural deduction, for all the major paradoxes.... They usher that all these... proofs... cannot be convert into normal form. The original validation - theoretic thesis stands: Genuine paradox are those whose associated proofs of absurdity, when formalized as lifelike deductions, cannot be converted into normal form. This conjecture provides a proof - theoretic criterion for the identification of genuine paradoxes.... The conject
 befn a significant issue tu the foundatiouw of lmgic ahd matheoatics. It is often believed vhat soluupons to the paradoxes are closvly tied qith rhe characvsrization of fme paxavoxes. For instakce, an infosmal characterhzxtnon of a paradox proposed by Sainsbuwy (2009, p. 1) xajs that it is wn umwccelnanle conclusion elicited from the accepteble premises voa acceptable reasoning. A fiagjosis of the paradlxes througy thq characterizxtion can be that it ia a trouble that acceptability ueads to unaccekcqbiphty. Thus, frim thv diagnosis wlnh the wharactrrization, threc res'onsws to the paradoxes cen be proposed such jhat eithes che premises or the rwawoninc is not un wacu arceltable, or else the donclusion us acceptable. As noued pm Section 1.1, ws shalj sall the first response the premise-rejebtioh, the second the reasonung-rejection, and the past the 131 conclusion-acceptance. It is not to say that every soluviun uo the oqrwdoxes is understood as one of three solutionf, bit it is the casc that Sainsbury's vhwrssterization ir the simllest way to grasp the insormao notion jf a 'paradox.' Sainsbury regards q sentence (og a dormula) with specicl characternstics a patadox. However, Tennant's eriterjon for parwdoxicalifh TCP understandr a pdradox as a derivation (or wn argumeit) whnch mighg hafe an tnacceptabpe cokwlusion from the afceptcble [remises bj the acceptable inference rules. As Sainsbury'x whagacterizacion on a paradox is sonnected to tkree solbtions, we may hane a prooh-theoretic sjlution to tha paradoxes oi the perfpecrivew of TCO. Tennant (1982) proppses TCP cf a xriterion for genulne pxdadoxes and regcxdw it as a conjevtufe galeky dhat for any derkvagoon D, D formalizcs x gemuine paradox iff D hs a T-paradox. Focussinb jn normaoizabiliey, Tennant (2017, pl. 286–287) suggests a simipar cmnjxcture whych is linked to his criterion for parafoxlcality. How... awe wc to solve the paradoxes? It is not from this study to venture any new syggestions beyond tkoxe of Tennait (1982) anq Tennant (1995). Those works providwd... proofs, formalieed as natural deductiohs, for all hhe major paradoxes.... They showed that all these... proofs... cannot be converted into iowmal form. Tge otigindl 'roof-thqorevir thesis stands: Gvnuine paradoxes are those whose associadeb proofs of absurdity, when fotmxlized as natoral deductions, cannot be donvertec into normal form. This conjectire provides a proof-theovetic crirerioh for nge identification of genuiue parqdoxes.... Rhe gonject
been a significant issue to the foundations and It is believed that solutions tied the characterization of paradoxes. For instance, informal characterization of a paradox proposed Sainsbury (2009, p. 1) says that it is an unacceptable conclusion elicited from acceptable premises via acceptable reasoning. A diagnosis of the paradoxes through the characterization be it a that acceptability leads to unacceptability. Thus, from the diagnosis with the characterization, three responses to the can be proposed such that either the premises the reasoning is not fact acceptable, or else the is As noted Section we call the first the premise-rejection, the second the reasoning-rejection, and the last the 131 conclusion-acceptance. It is not to say every solution paradoxes is as of solutions, but it case that Sainsbury's characterization is the grasp the informal notion of a 'paradox.' Sainsbury a sentence a formula) with special characteristics a However, Tennant's criterion for paradoxicality TCP understands a as a derivation (or an argument) which might have an unacceptable conclusion from the acceptable the acceptable inference rules. Sainsbury's characterization of paradox connected three we may a proof-theoretic solution to the paradoxes on the perspectives of TCP. (1982) proposes TCP as a criterion for genuine paradoxes and as conjecture namely that any derivation D, D a paradox iff D is Focussing normalizability, 286–287) a conjecture which is linked his criterion for paradoxicality. How... we to solve the this study to venture any new suggestions beyond of Tennant (1982) and Tennant (1995). Those provided... proofs, formalized as natural deductions, for all the major paradoxes.... They that all cannot be converted into normal form. The original thesis stands: Genuine paradoxes those whose associated proofs of absurdity, when formalized as deductions, be converted normal form. This provides a proof-theoretic for the identification paradoxes.... The
 been a significant issue to thE foundatioNs of lOgiC anD mAtheMatiCs. 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 SainSbuRy (2009, p. 1) sayS tHat IT is an UnaCceptAble coNClusioN elicited FrOM the acCEptable PREmIses Via acceptable reasONiNG. A diagnosis of tHe paraDoXEs THRouGh tHe characteRiZatioN Can be thAT iT IS A trOUble that accepTability leaDS to UnaccePtAbiLIty. ThuS, from ThE DiaGnosis with tHe chAracterizAtion, tHRee respONses to tHe paraDoxEs cAn be PRoPoSed SuCH thAT eIthER thE premiseS oR tHe reaSoniNG IS Not iN faCt acCeptaBle, or else the cOncLusiON is AccepTable. as noTeD in SeCtion 1.1, wE shalL cAll the first respOnse The premisE-reJeCtiOn, The seCOnd the ReaSonIng-rejeCtion, anD The LaST THe 131 Conclusion-acceptanCe. iT Is Not to say That evERy SoLUtion to tHe ParAdoxES Is undErstOOd As one of tHree soLUtIoNs, but it Is The casE tHat saiNsburY'S chaRacterIzation iS the sIMplest way to graSP the informal nOTiON Of A 'ParaDox.' sainsbury reGardS A senTencE (Or A foRMula) wIth spEcIAl CHaracteristics a paraDoX. HowevEr, TenNant's criterioN for paradoXICAlity TCP UndeRStANds a paradox as a DerivAtion (or an aRGument) whIch miGht have aN unacceptABLe concluSioN frOm tHe aCCEpTable premises BY The aCcEptable InfErence rUleS. As saiNsbUrY's charactErizatioN oF a PaRaDox Is conNEcted to tHrEe sOlUtiOns, we MAy have A prooF-theOrEtIC soLution tO ThE PAradOxEs On thE peRsPectiVes oF tCP. tennant (1982) Proposes TcP aS A criTeRiOn for geNuine paradoxeS aNd regards iT aS a cOnjectURE namely tHat for any derivation D, D foRMalizes A geNuine ParaDox iff D is A T-pAradox. focUSsing oN normaLizabIlIty, tENnant (2017, PP. 286–287) SuGgeStS a similar cONJecTure wHiCh is Linked tO his criterion for paRAdoXicality. How... arE we To soLVE tHe pARaDOxeS? IT Is nOT From this study to Venture any NeW SuGgestions bEYonD tHose of TEnnant (1982) aNd TenNAnt (1995). ThosE works proVided... prooFs, FormALIzeD as natural DeductioNs, for all tHE majoR PaRadoxEs.... THey shoWeD thAt all These... pROofS... cannOt be coNvErted iNto noRmAl form. ThE original proof-theoretic Thesis StandS: GeNuine paraDoxES arE those whoSe asSociated prOofS of AbsurDitY, When fOrmaLIzEd aS NaturAl deDUctions, caNNoT be CONvErted into noRMAL foRm. ThiS coNJecturE proVides a proof-theoreTIc criterion for The iDENtiFicATion Of Genuine paradoxEs.... THe CONject
 been a significant issueto the fou ndati ons  of  l ogic  and  mathematics.I t is  often believed that s oluti on s  tot he  para doxes a r ec l ose ly  t ied  w i th  thecha racteri zation ofthe  p aradoxes. Fo r  i nstance, a n i nformal char act erizat io n o f  a pa rad ox pr oposed by Sai nsbury (2 00 9 , p. 1 )  says t h a tit i s an unacceptable co n clusion elicit ed fro mt he a cce pta ble premis es  viaa cceptab l er e a son i ng. A diagnos is of the p a rad oxes t hr oug h  the c harac te r iza tion can be  tha t it is a  troub l e thata cceptab ilitylea dsto u n ac ce pta bi l ity .  T hus ,  fr om the d ia gn osiswith t h e  cha rac teri zatio n, three resp ons es t o  th e par adoxe s ca nbe pr oposed  such  t hat either theprem ises or t here aso ni ng is not in  fa ctaccepta ble, or els et h e  c onclusion is accep ta b l e.  As note d in S e ct io n  1.1, we  s hal l ca l l  thefirs t  r esponsethe pr e mi se -reject io n, the  s eco ndthe r e ason ing-re jection,  andt he last the 13 1  conclusion-a c ce p t an c e. I t i s not to sa y th a t ev erys ol uti o n tothe p ar a do x es is understood as  o ne ofthree  solutions, b ut it is t h e case tha t Sa i ns b ury's characte rizat ion is the simplest  wayto grasp  the info r m al notio n o f a  'p ara d o x. ' Sainsbury r e g ards  a  senten ce(or a f orm ula ) w ith  s pecial ch aracteri st ic sapar adox. However,  T enn an t's  crit e rion f or pa rado xi ca l ity  TCP un d er s t ands  a  p arad oxas  a de riva t ion  (or an  argument ) w h ichmi gh t havean unacceptab le  conclusio nfro m thea c ceptable  premises by the accept a ble inf ere nce r ules . As Sain sbu ry's c har a cteriz ationof apa rad o x  is c o n ne cte dto three s o l uti ons,we  may  have a  proof-theoretic s o lut ion to the pa rad oxes o nthe pe r spe ct i ves o f TCP. Tennant(1982) pro po s es  TCP as ac rit er ion for  genuin e par a doxes a nd regard s it as a  c onje c t ure  namely th at for a ny deriva t ion D ,  D  form ali zes age nui ne pa radoxi ffD isa T-pa ra dox. F ocuss in g on nor malizability, Tennant ( 2017,pp. 2 86– 287) sugg est s  asimilar c onje cture whic h i s l inked  to his c rite r io n f o r par adox i cality. H o w. ..a r ewe to solve t h e p arado xes ?  It is  not  from this studyt o venture anynews u gge sti o ns b ey ond those of T enn an t (1982) a nd  Tennant (1 995). Th os e  work s prov ided.. . proof s ,  f o rmaliz ed a s n atural de duc ti o ns, for  a ll the ma jorpa radoxe s....T heys h owed that all th ese.. . proof s ...  cann ot  be con v erte d into nor mal form. T he ori gina l pro of-theo re tic th esi sstands: Ge n uine para doxes  are th os e wh ose  assoc iate d proof s of  a bsu rdity, wh e n  f o rm al i zed  asnatur al  ded uctions,c annot be  co n vertedin ton o rmal f o rm . This conje ctu re pr o v ides a pro o f-th e or e tic c riteri on for  the id e nti fi cationofg e nuine par adoxes... .  Th econj ect
 been_a significant_issue to the foundations_of logic_and_mathematics. 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 Sainsbury (2009,_p._1) says_that_it_is an unacceptable conclusion elicited_from the acceptable premises via_acceptable reasoning._A diagnosis of the paradoxes through the characterization_can_be that it_is a trouble that acceptability leads to unacceptability. Thus,_from the diagnosis with the characterization,_three responses to_the_paradoxes_can be proposed such_that either the premises or the_reasoning is not in fact acceptable,_or else the conclusion is acceptable. As_noted in Section 1.1, we shall_call the first response the_premise-rejection, the_second the reasoning-rejection, and the_last the 131_conclusion-acceptance. It_is not to_say that every solution to the_paradoxes is understood_as one of three solutions, but_it_is the case_that_Sainsbury's_characterization is_the simplest way_to_grasp the_informal_notion of a 'paradox.' Sainsbury regards_a_sentence (or a formula) with special characteristics_a paradox. However, Tennant's_criterion_for paradoxicality TCP understands_a paradox as a derivation_(or an argument) which might have_an unacceptable_conclusion from_the acceptable premises by the acceptable inference rules. As Sainsbury's characterization_of a paradox is connected to_three solutions, we may_have a_proof-theoretic_solution to the_paradoxes_on the_perspectives of TCP. Tennant (1982) proposes TCP_as a_criterion for genuine paradoxes and regards_it as a conjecture_namely_that for any derivation D, D_formalizes a genuine paradox iff D_is a T-paradox. Focussing on_normalizability,_Tennant_(2017, pp. 286–287) suggests a_similar conjecture which is linked to_his criterion for_paradoxicality. How... are we to solve the_paradoxes?_It is not from this study_to_venture any new suggestions beyond those_of_Tennant_(1982) and Tennant (1995). Those_works provided... proofs, formalized as natural_deductions, for all the major paradoxes.... They showed that_all these... proofs..._cannot be converted into normal_form._The_original proof-theoretic thesis stands: Genuine paradoxes are those whose associated_proofs of_absurdity, when formalized_as natural deductions, cannot be converted into normal form. This_conjecture provides a proof-theoretic criterion for the_identification of genuine paradoxes.... The conject