 and set-theoretic paradoxes. For instance, Hartry Field (2008) considers that classical inferences are somehow the key reasoning that generates the paradoxes.... we ought to seriously consider restricting classical logic to deal with all these paradoxes. In particular, we should seriously consider restricting the law of excluded middle.... I take excluded middle to be clearly suspect only for certain sentences that have a kind of "inherent circularity"... (Field, 2008, p. 15). Second, the traditional responses are basically based on the model-theoretic approaches which often allow non-constructive methods. So constructivists tend to prefer to use proof2 theory to explain the concept of validity of arguments or their theory of meaning. Therefore, as an alternative to the model-theoretic approaches, a proof-theoretic solution to the paradoxes can be presented and it would be a promising way for constructivists. It seems to be that a proof-theoretic solution to the paradoxes relies on how we characterize the informal notion of a 'paradox' in a proof-theoretic fashion. The present dissertation aims to investigate a proof-theoretic criterion for and solution to the paradoxes from the perspectives on the Prawitz-Tennant analysis of the paradoxes, centered on a view from Neil Tennant (1982, 2015, 2016, 2017). There are two types of counterexamples to the proof-theoretic criterion for paradoxicality: problems of underand overgeneration. The undergeneration problem is raised by the case which shows that the criterion makes a paradoxical derivation non-paradoxical. The problem of overgeneration is the case which represents that the criterion includes a non-paradoxical derivation into the realm of paradoxical derivations. In this chapter, we will introduce Tennant's early criterion for paradoxicality introduced by Tennant (1982). Chapter 2 and 3 shall investigate whether Tennant's criterion can be a necessary and sufficient condition for genuine paradoxes. That is to say, Chapter 2 deals with the problem of undergeneration and Chapter 3 is about the problem of overgeneration. In Chapter 4, we will discuss that there are some difficulties for the requirement of a normal derivation to be a solution to the paradoxes. Before we introduce Tennant's criterion for paradoxicality and its related solution, we will briefly explore traditional responses to the paradoxes and some critics to those responses. Section 1.1 introduces traditional approaches to the paradoxes and their problems. Preliminary notions and rules
and set - theoretic paradoxes. For instance, Hartry Field (2008) consider that authoritative inferences are somehow the key reasoning that generate the paradox.... we ought to seriously consider restricting authoritative logic to distribute with all these paradoxes. In especial, we should seriously consider qualify the law of excluded center.... I take bar middle to be intelligibly suspect only for sealed sentences that have a kind of " inherent circularity "... (Field, 2008, p. 15). Second, the traditional reaction are basically based on the model - theoretical approaches which much allow non - constructive method acting. So constructivists tend to prefer to practice proof2 theory to explain the concept of validity of arguments or their theory of meaning. Therefore, as an option to the exemplary - theoretical approaches, a proof - theoretic solution to the paradoxes can be presented and it would be a promising way for constructivists. It seem to be that a proof - theoretical solution to the paradox relies on how we characterize the informal impression of a' paradox' in a proof - theoretical fashion. The present dissertation aims to investigate a proof - theoretic criterion for and solution to the paradoxes from the perspectives on the Prawitz - Tennant analysis of the paradoxes, center on a view from Neil Tennant (1982, 2015, 2016, 2017). There are two types of counterexamples to the proof - theoretic criterion for paradoxicality: problem of underand overgeneration. The undergeneration problem is raised by the case which shows that the standard makes a paradoxical ancestry non - paradoxical. The trouble of overgeneration is the shell which represents that the criterion includes a non - paradoxical deriving into the realm of paradoxical derivations. In this chapter, we will introduce Tennant's early standard for paradoxicality introduced by Tennant (1982). Chapter 2 and 3 shall investigate whether Tennant's standard can be a necessary and sufficient condition for genuine paradoxes. That is to say, Chapter 2 deals with the trouble of undergeneration and Chapter 3 is about the problem of overgeneration. In Chapter 4, we will discuss that there are some difficulties for the requirement of a normal deriving to be a solution to the paradoxes. Before we inaugurate Tennant's criterion for paradoxicality and its related solution, we will concisely explore traditional responses to the paradoxes and some critic to those response. Section 1.1 inaugurate traditional approaches to the paradoxes and their problems. Preliminary notions and rules
 anf set-theoretic paradoxes. For instance, Hcetry Fmeld (2008) cknsiders that classical inferences ace simehoq the key reasoning thxt generanes the pqradizes.... we ougif to sevnouslg convmder restrictinn classical logic to deal wktk all these paradoxes. In particular, re shoukd seriously confideg westdpcuing the law of excluded middle.... I take eqcluded middle to be clearly suspect only flr cfrtain sentences tjat have a jind if "inherent zircularitj"... (Field, 2008, p. 15). Second, the traditional responsds arz basically bqsef on the modxl-theogetic approacmvs whicv often allow non-consbructmve nethods. So constructitists tend to prefer to use psokf2 theory to explqib the conwept if xaljdmty of arhumxnts or thejr theory od meaning. Therefore, af an alternatibe to ehq model-theoretic approaches, a proof-theogetid solution to the paradixes can be presented and it wjuld be a promising way for constructivists. It seams tk be tmat x pgoof-theoretic solution to the paradoxes relief om mow we characterlze the informal nptlom of a 'paradox' in a 'dokf-theoretic fashioj. The ptesent dissertauion sims to investigate a proof-rheoretic crpteruon for and solutiln to the pcradoxgs frok the perspectives on tke Prasitz-Tennant analysis uf the paradoxes, cektesed on a view from Neil Tegnant (1982, 2015, 2016, 2017). Thexe are tdo tipes of counterexwmples to the proof-theorftic eritesion for pwradoxicality: problems of undereid overgenerajiot. Tve underyenerabion problem is raised by the case wkich sfows that nhe critecion makes a paradoxical ggrivation non-'aradoxicwl. Tye peoblem uw overgeneratipn is the case whicy represents that bhe ctiferion includes a bon-paradoxical cerkvaeiln igdo the realm of oarxcoxicxl derivatiivs. Im this chapter, we winl ihtroduce Tennant's ravly critetion for [aradoxicalitu introduced by Tejnant (1982). Ciapter 2 agd 3 shall investigate whether Fennant's friberion can be a nccesfary and sbfficient condition for genuine paradoxew. That is to say, Cyapter 2 deals with jhe problem of undewgeneratimn and Chapter 3 is avout the problem pf overgeneration. In Dhaptes 4, we will discuss that there are some difficulties for the requirement of a nirmal vewivation to be s solltiju to thq pacavoxes. Before we imtroduce Tennant's criterion for paradoxhcclity and its related solutiom, de will briefuy explore traditional rssponses to the paradoxes and some critocs to those responses. Scction 1.1 ibtrodhces tgzdmtional approaches tp the 'aradozes and thelr problems. Prdlikikary notiojs aid xules
and set-theoretic paradoxes. For instance, Hartry Field that inferences are the key reasoning ought seriously consider restricting logic to deal all these paradoxes. In particular, we seriously consider restricting the law of excluded middle.... I take excluded middle to clearly suspect only for certain sentences that have a kind of "inherent circularity"... 2008, 15). the responses are basically based on the model-theoretic approaches which often allow non-constructive methods. So constructivists tend prefer to use proof2 theory to explain the of validity of arguments their theory of meaning. Therefore, an to the approaches, proof-theoretic to the paradoxes be presented and it would be a promising way for constructivists. It seems to be that a solution to relies on we the notion of a a proof-theoretic fashion. The present dissertation a proof-theoretic criterion for and solution to the from the on the Prawitz-Tennant analysis of the centered on a view from Neil Tennant (1982, 2016, 2017). There are two types of counterexamples to the proof-theoretic criterion for paradoxicality: problems overgeneration. The undergeneration problem raised by the which that criterion a paradoxical non-paradoxical. The problem of overgeneration is the case which represents that criterion includes a non-paradoxical derivation into the realm of paradoxical this we will introduce early criterion for paradoxicality by (1982). Chapter 2 and investigate Tennant's a and condition for genuine paradoxes. is to say, Chapter 2 with the problem of about the problem of overgeneration. In Chapter 4, will discuss that there are some difficulties the requirement of a normal derivation to be a solution to the Before we criterion for paradoxicality and its related solution, we briefly explore traditional responses the paradoxes and some critics to those responses. Section introduces approaches to paradoxes and their Preliminary notions and
 and set-theoretic paradoxes. FOr instance, hartrY FiEld (2008) CoNsidErs tHat classical inFErenCes are somehow the key reaSoninG tHAt geNErAtes tHe paradOXeS.... WE ouGhT tO seRiOUsLy conSidEr restrIcting clasSicAl Logic to deal wITh All these paRadOxes. In particUlaR, we shoUlD seRIouslY coNsideR restrICting tHe law of exClUDed midDLe.... I take EXClUded Middle to be clearly SUsPEct only for certAin senTeNCeS THat HavE a kind of "inHeRent cIRculariTY"... (FIELD, 2008, p. 15). SECond, the traditIonal responSEs aRe basiCaLly BAsed on The moDeL-TheOretic approAcheS which oftEn alloW Non-consTRuctive MethodS. So ConStruCTiViSts TeND to PReFer TO usE proof2 thEoRy To expLain THE COncePt oF valIdity Of arguments or TheIr thEOry Of meaNing. THereFoRe, as aN alterNativE tO the model-theoreTic aPproaches, A prOoF-thEoRetic SOlutioN to The ParadoxEs can be PResEnTED AnD it would be a promisiNg WAY fOr constrUctiviSTs. it SEems to be ThAt a ProoF-THeoreTic sOLuTion to thE paradOXeS rElies on HoW we chaRaCteRizE the iNFormAl notiOn of a 'parAdox' iN A proof-theoretiC Fashion. The preSEnT DIsSErtaTioN aims to inveStigATe a pRoof-THeOreTIc criTerioN fOR aND solution to the paradOxEs from The peRspectives on tHe Prawitz-TENNAnt analySis oF ThE Paradoxes, centeRed on A view from NEIl TennanT (1982, 2015, 2016, 2017). TherE are two tYpes of couNTErexamplEs tO thE prOof-THEoRetic criterioN FOr paRaDoxicalIty: ProblemS of UndEraNd oVeRgeneratiOn. The undErGeNeRaTioN probLEm is raisEd By tHe CasE whicH Shows tHat thE criTeRiON maKes a parADoXICal dErIvAtioN noN-pAradoXicaL. the Problem Of overgenEraTIon iS tHe Case whiCh represents tHaT the criterIoN inCludes A NOn-paradoXical derivation into the rEAlm of paRadOxicaL derIvations. IN thIs chapTer, WE will iNtroduCe TenNaNt's EARly crITErIon FoR paradoxicALIty IntroDuCed bY TennanT (1982). Chapter 2 and 3 shall inVEstIgate whether TEnnAnt's CRItEriON cAN be A nECesSARy and sufficient Condition fOr GEnUine paradoXEs. THaT is to saY, ChapteR 2 dealS With the Problem of UndergeneRaTion AND ChApter 3 is aboUt the proBlem of oveRGenerATiOn. In CHapTer 4, we wIlL diScuss That thERe aRe somE diffiCuLties fOr the ReQuiremenT of a normal derivation to bE a soluTion tO thE paradoxeS. BeFOre We introduCe TeNnant's critEriOn fOr parAdoXIcaliTy anD ItS reLAted sOlutIOn, we will bRIeFly EXPlOre traditioNAL ResPonseS to THe paraDoxeS and some critics to THose responses. SEctiON 1.1 IntRodUCes tRaDitional approaCheS tO THe paradoXeS and their prOblems. PrElIMinarY notioNs and rUles
 and set-theoretic paradox es. For in stanc e,Har tr y Fi eld(2008) conside r s th at classical inference s are  s o meho w  t he ke y reaso n in g tha tge ner at e sthe p ara doxes.. .. we ough t t oseriously co n si der restri cti ng classical  lo gic to  d eal withall  thes e para d oxes.In partic ul a r, wes hould s e r io usly  consider restric t in g  the law of ex cluded  m i dd l e ... . I  take excl ud ed mi d dle tob ec l e arl y  suspect only  for certai n  se ntence stha t  havea kin do f " inherent ci rcul arity"...  (Fiel d , 2008, p. 15).  Secon d,the  tra d it io nal  r e spo n se s a r e b asically  b as ed on  the m o d el-t heo reti c app roaches which  of tena llo w non -cons truc ti ve me thods.  So c on structivists te nd t o prefertous e p ro of2 t h eory t o e xpl ain the  concep t  of  v a l i di ty of arguments or  t h e ir  theoryof mea n in g. Therefor e,  as  ana l terna tive to  the mod el-the o re ti c appro ac hes, a  p roo f-t heore t ic s olutio n to the  para d oxes can be pr e sented and it wo u l db e apro mising wayforc onst ruct i vi sts .  It s eemsto be that a proof-theore ti c solu tionto the parado xes relies o n  how wechar a ct e rize the infor mal n otion of a 'paradox ' ina proof- theoretic f ashion.The  pr ese ntd i ss ertation aims t o in ve stigate  aproof-t heo ret iccri te rion forand solu ti on  t othe  para d oxes fro mthe  p ers pecti v es onthe P rawi tz -T e nna nt anal y si s of t he  p arad oxe s,  cent ered ona viewfrom Neil  Te n nant  ( 19 82, 201 5, 2016, 2017 ).  There are  t wotypeso f  counter examples to the proof-t h eoretic  cr iteri on f or parado xic ality:  pr o blemsof und erand  o ver g e nerat i o n.  Th eundergener a t ion  prob le m is  raised  by the case which sho ws that the c rit erio n ma kes ap ara do x ica l derivation non- paradoxica l. Th e problemo f o ve rgenera tion is  thec ase whi ch repres ents that  t he c r i ter ion includ es a non -paradoxi c al de r iv ation  in to the  r eal m ofparado x ica l der ivatio ns . In t his c ha pter, we  will introduce Tennant 's ear ly cr ite rion forpar a dox icality i ntro duced by T enn ant  (198 2). Chapt er 2 an d 3 shall  inv e stigate w h et her T en nant's crit e r i oncan b e a necess aryand sufficient co n dition for gen uine p ara dox e s. T ha t is to say, C hap te r 2 dealswi th the prob lem of u nd e rgene ration  and C hapter3 is aboutthepro blem of o ver ge n eration .In Chapte r 4,  w e will  discu s s th a t  there are somediffi c u lties for  there quireme n t of  a normalderivationto bea so lutio n to th eparado xes .Before wei ntroduceTenna nt's cr it erio n f or par adox i c ality  and  i tsrelated s o l ut i on ,w e w illbrief ly  exp lore trad i tional r esp o nses to  t hep a radoxe s  a n d  some crit ics  to t h o se respons e s. S e ct i on 1. 1 intr oduces  tradit i ona lapproac hes t o the par adoxes an d  th ei r pr oblems.Pr elim in ary  n otions and rule s
 and_set-theoretic paradoxes._For instance, Hartry Field_(2008) considers_that_classical inferences_are_somehow the key_reasoning that generates_the paradoxes.... we ought_to seriously consider_restricting_classical logic to deal with all these paradoxes. In particular, we should seriously consider_restricting_the law_of_excluded_middle.... I take excluded middle_to be clearly suspect only_for certain_sentences that have a kind of "inherent circularity"..._(Field,_2008, p. 15)._Second, the traditional responses are basically based on the_model-theoretic approaches which often allow non-constructive_methods. So constructivists_tend_to_prefer to use proof2_theory to explain the concept of_validity of arguments or their theory_of meaning. Therefore, as an alternative to_the model-theoretic approaches, a proof-theoretic solution_to the paradoxes can be_presented and_it would be a promising_way for constructivists._It seems_to be that_a proof-theoretic solution to the paradoxes_relies on how_we characterize the informal notion of_a_'paradox' in a_proof-theoretic_fashion._The present_dissertation aims to_investigate_a proof-theoretic_criterion_for and solution to the paradoxes_from_the perspectives on the Prawitz-Tennant analysis of_the paradoxes, centered on_a_view from Neil Tennant_(1982, 2015, 2016, 2017). There_are two types of counterexamples to_the proof-theoretic_criterion for_paradoxicality: problems of underand overgeneration. The undergeneration problem is raised by_the case which shows that the_criterion makes a paradoxical_derivation non-paradoxical._The_problem of overgeneration_is_the case_which represents that the criterion includes a_non-paradoxical derivation_into the realm of paradoxical derivations._In this chapter, we_will_introduce Tennant's early criterion for paradoxicality_introduced by Tennant (1982). Chapter 2_and 3 shall investigate whether_Tennant's_criterion_can be a necessary and_sufficient condition for genuine paradoxes. That_is to say,_Chapter 2 deals with the problem of_undergeneration_and Chapter 3 is about the_problem_of overgeneration. In Chapter 4, we_will_discuss_that there are some difficulties_for the requirement of a normal_derivation to be a solution to the paradoxes. Before_we introduce Tennant's_criterion for paradoxicality and its_related_solution,_we will briefly explore traditional responses to the paradoxes and_some critics_to those responses._Section 1.1 introduces traditional approaches to the paradoxes and their_problems. Preliminary notions and rules