 he may believe that every T-paradox is a genuine paradox. Unlike the Liar paradox, Tennant (2016, pp. 8–12) asserts that Russell's paradox is not a genuine one. The derivation of ⊥ from Russell's paradox begins with the assumption that there is a set of all sets not members of themselves. He proves in his natural deduction system for the free logic of sets that there is a closed full normal derivation of the rejection of the assumption, i.e. there is no such set which contains all sets not members of themselves. He uses the result in order to support his view that Russell's paradox is not a genuine paradox. We first see the derivation in his free logic of sets which supports the view that Russell's paradox is not a genuine paradox. We show that, by the adoption of Ekman-type reductions introduced in Chapter 3, the derivation enters into loops and satisfies TCPE and TCPL. If every T-paradox is a genuine paradox, Russell's paradox becomes a genuine paradox. Furthermore, similar to his argument that Russell's paradox is not genuine, we use his early formalization of the Liar paradox in Tennant (1982, p. 271) with generalized elimination rules and put forward a derivation which represents that the Liar paradox is not a genuine paradox. Tennant (2016, Sec. 3) and Tennant (2017, pp. 294-298) show in his natural deduction system SF for the free logic of sets that there is a full normal closed derivation of ¬∃y(y = {x|¬x ∈ x}). His result will be introduced in Proposition 4.1.1. The derivation Σ3 of ¬∃y(y = {x|¬x ∈ x}) in Proposition 4.1.1 does not generate a looping reduction, and so is not a T-paradox. Since he believes that every genuine paradox is a T-paradox, he claims that Russell's paradox is not a genuine paradox. Tennant's natural deduction system SF differs from the system SN for the naive set theory 137 discussed in Section 1.3 of Chapter 1.1 Free logic is one whose quantifiers are interpreted in the usual way, but whose singular terms may denote objects outside of a domain or fail to denote at all. Therefore, unlike other logics, it has the rule of denotation. We abbreviate ∃x(x = t) as ∃!t which means
he may believe that every T - paradox is a genuine paradox. Unlike the Liar paradox, Tennant (2016, pp. 8–12) asserts that Russell's paradox is not a genuine one. The deriving of ⊥ from Russell's paradox begin with the assumption that there be a bent of all sets not members of themselves. He test in his natural deduction arrangement for the free logic of sets that there be a closed full normal ancestry of the rejection of the premise, i.e. there is no such set which contains all sets not members of themselves. He use the result in order to support his scene that Russell's paradox is not a genuine paradox. We first see the derivation in his complimentary logic of sets which supports the view that Russell's paradox is not a genuine paradox. We show that, by the borrowing of Ekman - type reductions introduced in Chapter 3, the derivation enters into loops and meet TCPE and TCPL. If every T - paradox is a genuine paradox, Russell's paradox becomes a genuine paradox. Furthermore, similar to his argument that Russell's paradox is not genuine, we use his early formalization of the Liar paradox in Tennant (1982, p. 271) with generalized elimination rules and put forward a deriving which represents that the Liar paradox is not a genuine paradox. Tennant (2016, Sec. 3) and Tennant (2017, pp. 294 - 298) appearance in his lifelike discount system SF for the free logic of sets that there is a full normal closed derivation of ¬∃y(y = { x|¬x ∈ x }). His resultant role will be introduced in Proposition 4.1.1. The derivation Σ3 of ¬∃y(y = { x|¬x ∈ x }) in Proposition 4.1.1 does not generate a looping reduction, and so is not a metric ton - paradox. Since he believes that every genuine paradox is a T - paradox, he claims that Russell's paradox is not a genuine paradox. Tennant's natural deduction arrangement SF differs from the system SN for the naive set theory 137 discourse in Section 1.3 of Chapter 1.1 Free logic is one whose quantifiers are interpret in the usual way, but whose singular terms may announce objects outside of a domain or fail to denote at all. Therefore, unlike early logics, it has the rule of denotation. We abridge ∃x(x = t) as ∃!t which means
 he may believe that every U-paradox is a genoibe paredox. Unmike the Liar paradox, Tennant (2016, pp. 8–12) adswrts ukat Russell's paradox is not a genuine one. Rhe derivavjon of ⊥ from Dmsseln'w paradox begiks with the assumption thdt tkere is a set of all sets not memberf of thrmdelves. He provgs in ris hatural deduction system for the fdee logpc of sets that tnere is a closed full normwl dfrivation of the rfjection of the qssumption, i.d. there is no such set which contains all sets not meobers of themseovws. Jg uses the rxsult pn order to smiport hhs view that Russell's pacadoz is not a genuine pacadox. We first see tre derivadikn in his free lotix of vets whizy sjppkrvs fhe vifw vhat Russelm's paradox us not a genuine pataqix. We show thzt, by ehq adoption of Ekman-type reductions intrmduded in Chapter 3, the deruvation enters into llops and fatisfies TCPE and TCPL. If every T-paradox is a getuine oarceox, Rjwsfll's paradox becomes a genuine paradox. Furthewjote, similar to hif argument yhwt Tussell's paradux is ukt genuine, we use hid early formqlization of yhe Liar paradox in Tennant (1982, p. 271) with geuerqlized elimination rules and 'ut fotward s derivation which reprzsents that the Llar paradkb is not a genuive iaragox. Tennant (2016, Sec. 3) and Tennwnt (2017, pp. 294-298) whow in his natoral dequction sydtem SF for the free loglc of sats that tjere is a full normal closed decmvation of ¬∃y(y = {f|¬x ∈ x}). His rzsult eill be introquced in Propoxition 4.1.1. The aerivation Σ3 of ¬∃y(y = {x|¬x ∈ x}) in [roposition 4.1.1 gles not genecate a lojpint reeuction, xnd so is not s T-paradoq. Fincw he believes that evgrg genuine paradix us a T-paradox, hr cuaiis tiat Rtvsell's paradmx ir nuy a gdnuine kcrabob. Temnant's natural deducdion system SF differs fvom the sistem SN sor the naive set theory 137 discudsed mn Secvion 1.3 pf Shapter 1.1 Free logic is one whoae quantivievs are interpwetee in the usucl way, but whose singular terms may denove objects outside of a domain or fail to bekote at all. Vherefjre, unlika other logics, it haw the rule of denptation. We abbreviate ∃x(x = t) as ∃!t which means
he may believe that every T-paradox is paradox. the Liar Tennant (2016, pp. is a genuine one. derivation of ⊥ Russell's paradox begins with the assumption there is a set of all sets not members of themselves. He proves his natural deduction system for the free logic of sets that there is closed normal of rejection of the assumption, i.e. there is no such set which contains all sets not members themselves. He uses the result in order to his view that Russell's is not a genuine paradox. first the derivation his logic sets which supports view that Russell's paradox is not a genuine paradox. We show that, by the adoption of Ekman-type introduced in the derivation into and TCPE and TCPL. T-paradox is a genuine paradox, Russell's genuine paradox. Furthermore, similar to his argument that paradox is genuine, we use his early formalization the Liar paradox in Tennant (1982, p. 271) generalized elimination rules and put forward a derivation which represents that the Liar paradox is genuine paradox. Tennant (2016, 3) and Tennant pp. show his deduction system for the free logic of sets that there is a full closed derivation of ¬∃y(y = {x|¬x ∈ x}). His result introduced Proposition 4.1.1. The Σ3 of ¬∃y(y = ∈ in Proposition 4.1.1 does a reduction, not T-paradox. he believes that every paradox is a T-paradox, he that Russell's paradox is natural deduction system SF differs from the system for the naive set theory 137 discussed Section 1.3 of Chapter 1.1 Free logic is one whose quantifiers are in the but whose singular terms may denote objects outside a domain or fail denote at all. Therefore, unlike other logics, it has rule denotation. We ∃x(x = t) ∃!t which means
 he may believe that every T-parAdox is a genUine pAraDox. unLike The LIar paradox, TennANt (2016, pp. 8–12) Asserts that Russell's parAdox iS nOT a geNUiNe one. the deriVAtION of ⊥ FrOm rusSeLL's ParadOx bEgins wiTh the assumPtiOn That there is a SEt Of all sets nOt mEmbers of themSelVes. He pRoVes IN his nAtuRal deDuctioN System For the freE lOGic of sETs that tHERe Is a cLosed full normal deRIvATion of the rejecTion of ThE AsSUMptIon, I.e. there is nO sUch seT Which coNTaINS All SEts not members Of themselveS. he uSes the ReSulT In ordeR to suPpORt hIs view that RUsseLl's paradoX is not A Genuine PAradox. WE first See The DeriVAtIoN in HiS FreE LoGic OF seTs which sUpPoRts thE vieW THAT RusSelL's paRadox Is not a genuine ParAdox. wE shOw thaT, by thE adoPtIon of ekman-tYpe reDuCtions introduceD in CHapter 3, the DerIvAtiOn EnterS Into loOps And SatisfiEs TCPE aND TCpL. iF EVeRy T-paradox is a genuiNe PARaDox, RusseLl's parADoX bEComes a geNuIne ParaDOX. FurtHermORe, Similar tO his arGUmEnT that RuSsEll's paRaDox Is nOt genUIne, wE use hiS early foRmaliZAtion of the Liar PAradox in TennaNT (1982, p. 271) WITh GEnerAliZed eliminatIon rULes aNd puT FoRwaRD a derIvatiOn WHiCH represents that the LIaR paradOx is nOt a genuine parAdox. TennanT (2016, sEC. 3) and TennAnt (2017, pP. 294-298) ShOW in his natural dEductIon system Sf For the frEe logIc of sets That there IS A full norMal CloSed DerIVAtIon of ¬∃y(y = {x|¬x ∈ x}). His RESult WiLl be intRodUced in PRopOsiTioN 4.1.1. ThE dErivation σ3 of ¬∃y(y = {x|¬x ∈ x}) In prOpOsItiOn 4.1.1 doeS Not generAtE a lOoPinG reduCTion, anD so is Not a t-pArADox. since he BElIEVes tHaT eVery GenUiNe parAdox IS a T-Paradox, He claims tHat rUsseLl'S pAradox iS not a genuine pArAdox. TennanT's NatUral deDUCtion sysTem SF differs from the systEM SN for tHe nAive sEt thEory 137 discuSseD in SecTioN 1.3 Of ChapTer 1.1 FreE logiC iS onE WHose qUANtIfiErS are interpRETed In the UsUal wAy, but whOse singular terms maY DenOte objects outSidE of a DOMaIn oR FaIL to DeNOte AT All. Therefore, unlIke other loGiCS, iT has the rulE Of dEnOtation. we abbreViate ∃X(X = t) as ∃!t whIch means
 he may believe that every  T-paradox  is a  ge nui ne  par adox . Unlike the L i ar p aradox, Tennant (2016,  pp.8– 1 2) a s se rts t hat Rus s el l ' s p ar ad oxis no t a g enu ine one . The deri vat io n of ⊥ fromR us sell's par ado x begins wit h t he ass um pti o n tha t t hereis a s e t of a ll sets n ot member s  of the m s el ves.  He proves in his na t ural deduction  syste mf or t hefre e logic of  s ets t h at ther e  i s a  cl o sed full norm al derivati o n o f there jec t ion of  theas s ump tion, i.e.ther e is no s uch se t  whichc ontains  all s ets  no t me m be rs  of  t h ems e lv es. Heuses the  r es ult i n or d e r to s upp orthis v iew that Russ ell 's p a rad ox is  nota ge nu ine p aradox . Wefi rst see the der ivat ion in hi s f re e l og ic of sets w hic h s upports  the vi e w t ha t R us sell's paradox isno t agenuineparado x .We show tha t,  by  the a dopti on o f  E kman-typ e redu c ti on s intro du ced in  C hap ter  3, t h e de rivati on enter s int o  loops and sat i sfies TCPE an d  T C P L. If e ver y T-paradox  isa  gen uine pa rad o x, Ru ssell 's pa r adox becomes a genu in e para dox.Furthermore,similar to h i s argume nt t h at Russell's para dox i s not genu i ne, we u se hi s earlyformaliza t i on of th e L iar  pa rad o x  i n Tennant (19 8 2 , p.  2 71) wit h g enerali zed  el imi nat io n rules a nd put f or wa rd  a  de rivat i on which  r epr es ent s tha t  the L iar p arad ox  i s  no t a gen u in e para do x.  Ten nan t(2016 , Se c . 3 ) and T ennant (2 017 ,  pp.  2 94 -298) s how in his na tu ral deduct io n s ystemS F  for the  free logic of sets tha t  thereisa ful l no rmal clos edderiva tio n  of ¬∃ y(y ={x|¬x  ∈  x} ) .  Hisr e su ltwi ll be intr o d uce d inPr opos ition 4 .1.1. The derivati o n Σ 3 of ¬∃y(y ={x| ¬x ∈ x })  in Pr o pos it i on4 . 1.1 does not ge nerate a l oo p in g reductio n , a nd  so isnot a T -para d ox. Sin ce he bel ieves tha tever y gen uine parad ox is aT-paradox ,  he c l ai ms th atRussel l' s p arado x is n o t a  genu ine pa ra dox. T ennan t' s natura l deduction system SF d iffers  from  th e systemSNf orthe naive  set  theory 13 7 d isc ussed  in Secti on 1 . 3ofC hapte r 1. 1  Free log i ciso n ewhose quant i f i ers  areint e rprete d in  the usual way, b u t whose singul ar t e r msmay deno te  objects outsi deof a  domainor  fail to de note atal l . The refore , unli ke othe r lo g ics, i t ha s t he rule o f d en o tation.  W ea bbrevi ate∃x (x = t ) as ∃ ! t wh i c h means
 he_may believe_that every T-paradox is_a genuine_paradox._Unlike the_Liar_paradox, Tennant (2016,_pp. 8–12) asserts_that Russell's paradox is_not a genuine_one._The derivation of ⊥ from Russell's paradox begins with the assumption that there is_a_set of_all_sets_not members of themselves. He_proves in his natural deduction_system for_the free logic of sets that there is_a_closed full normal_derivation of the rejection of the assumption, i.e. there_is no such set which contains_all sets not_members_of_themselves. He uses the_result in order to support his_view that Russell's paradox is not_a genuine paradox. We first see the_derivation in his free logic of_sets which supports the view_that Russell's_paradox is not a genuine_paradox. We show_that, by_the adoption of_Ekman-type reductions introduced in Chapter 3,_the derivation enters_into loops and satisfies TCPE and_TCPL._If every T-paradox_is_a_genuine paradox,_Russell's paradox becomes_a_genuine paradox._Furthermore,_similar to his argument that Russell's_paradox_is not genuine, we use his early_formalization of the Liar_paradox_in Tennant (1982, p._271) with generalized elimination rules_and put forward a derivation which_represents that_the Liar_paradox is not a genuine paradox. Tennant (2016, Sec. 3) and_Tennant (2017, pp. 294-298) show in_his natural deduction system_SF for_the_free logic of_sets_that there_is a full normal closed derivation of_¬∃y(y =_{x|¬x ∈ x}). His result will_be introduced in Proposition_4.1.1._The derivation Σ3 of ¬∃y(y =_{x|¬x ∈ x}) in Proposition 4.1.1_does not generate a looping_reduction,_and_so is not a T-paradox._Since he believes that every genuine_paradox is a_T-paradox, he claims that Russell's paradox is_not_a genuine paradox. Tennant's natural deduction_system_SF differs from the system SN_for_the_naive set theory 137 discussed_in Section 1.3 of Chapter 1.1_Free logic is one whose quantifiers are interpreted in_the usual way,_but whose singular terms may_denote_objects_outside of a domain or fail to denote at all._Therefore, unlike_other logics, it_has the rule of denotation. We abbreviate ∃x(x = t)_as ∃!t which means