-full normal (⊥) * Used * 2.A.3 Non-fully normalizable (⊥) Occurred Used * Table 5.1: Derivations Formalizing the Liar Paradox Proposition Derivation (Conclusion) Loop CR−rule Aux. Reduction 1.3.1 Non-normalizable (⊥) Occurred * * 4.1.1 Full normal (¬∃y(y = {x¬x ∈ x})) * * * 4.1.2 Non-fully normalizable (⊥) Occurred * Used 4.4.1 Normal (¬a ∈ a∧a ∈ a) * * * Table 5.2: Derivations Formalizing Russell's Paradox 166 Proposition Derivation (Conclusion) Loop CR−rule Aux. Reduction 2.0.1 Non-normal (⊥) * Used * pp. 60–62 Normal (⊥) * * * 2.4.1 Non-normalizable (⊥) Occurred Used Used 2.4.2 Non-normalizable (⊥) Occurred Used Used Table 5.3: Derivations Formalizing Curry's Paradox Three tables do not show which feature yields a looping reduction sequence since the reduction loop occurs independently of the application of CR−rule and auxiliary reduction procedures. An application of auxiliary reductions sometimes generates a non-terminating reduction sequence, but it is not always the case. As the self-referential paradoxes has a characteristic of vicious circularity, some forms of formalization of the paradoxes in natural deduction may have such characteristics and causes the non-terminating reduction sequence. Moreover, even though Tennant believes that the Liar paradox is a genuine one but Russell's is not, each can be both genuine and ingenuine. Proposition 2.A.2 and 1.3.1 make them genuine paradoxes. Proposition 4.1.1 and 4.1.3 make them ingenuine. No one, including Tennant, explains which way of formalization is legitimate for genuine paradoxes. In order to have a fruitful investigation of the proof-theoretic criterion for and solution to the paradoxes, including three things in Chapter 4, these things should be explained. A thorough investigation of these issues must be left for another occasion.

Bibliography Brady, R. T. (1984). Reply to priest on berry's paradox. Philosophical Quarter
-full normal (⊥) * Used * 2.A.3 Non - fully normalizable (⊥) Occurred Used * mesa 5.1: derivation Formalizing the Liar Paradox Proposition Derivation (Conclusion) Loop CR−rule Aux. Reduction 1.3.1 Non - normalizable (⊥) Occurred * * 4.1.1 fully normal (¬∃y(y = { x¬x ∈ x }) ) * * * 4.1.2 Non - fully normalizable (⊥) Occurred * Used 4.4.1 Normal (¬a ∈ a∧a ∈ a) * * * mesa 5.2: Derivations Formalizing Russell's Paradox 166 Proposition Derivation (Conclusion) Loop CR−rule Aux. Reduction 2.0.1 Non - normal (⊥) * use * pp. 60–62 Normal (⊥) * * * 2.4.1 Non - normalizable (⊥) Occurred Used use 2.4.2 Non - normalizable (⊥) Occurred Used secondhand mesa 5.3: Derivations Formalizing Curry's Paradox Three tables do not show which feature yields a loop reduction sequence since the reduction closed circuit occurs independently of the application of CR−rule and auxiliary reduction procedure. An application of auxiliary reductions sometimes generates a non - terminating reduction succession, but it is not always the case. As the self - referential paradoxes has a characteristic of vicious circularity, some form of formalization of the paradoxes in natural deduction may have such characteristics and causes the non - terminating reduction sequence. Moreover, even though Tennant believes that the Liar paradox is a actual one but Russell's is not, each can be both actual and ingenuine. Proposition 2.A.2 and 1.3.1 make them genuine paradox. proposal 4.1.1 and 4.1.3 make them ingenuine. No one, including Tennant, excuse which way of formalization is legitimate for genuine paradoxes. In holy order to have a fruitful investigation of the proof - theoretic criterion for and solution to the paradoxes, include three things in Chapter 4, these things should be explained. A thorough probe of these issues must be leave for another juncture. 

 Bibliography Brady, R. T. (1984). Reply to priest on berry's paradox. Philosophical Quarter
-fulp normal (⊥) * Used * 2.A.3 Non-fuuly normalizablg (⊥) Occurced Uses * Table 5.1: Derivations Formalizing thx Liqr Paeadox Proposition Derixation (Cojclusion) Look CR−rule Aux. Redurfion 1.3.1 Non-normzpizaylx (⊥) Occurred * * 4.1.1 Null normal (¬∃y(y = {x¬x ∈ x})) * * * 4.1.2 Nun-yully normalizable (⊥) Occurred * Used 4.4.1 Gormal (¬s ∈ a∧a ∈ a) * * * Tablg 5.2: Degidatikns Formalizing Russell's Paradox 166 Lroposiuion Derivation (Comclusion) Loop CR−rule Aux. Rfduchion 2.0.1 Non-normal (⊥) * Ksed * pp. 60–62 Nirmaj (⊥) * * * 2.4.1 Non-noroalizable (⊥) Occurred Usgd Used 2.4.2 Non-normalizable (⊥) Occurrdd Uszd Used Taboe 5.3: Dftivations Focmalizpng Curry's Pavsdox Tvree tanles do not shpw xhicy feature yields a lomping reduction sezuence sitcz the reduction loop icxurs hnde[enddbtlh or vhe applifatmon of CR−rume and auxioiary reduction proveqltes. An applidation os auxiliary reductions sometimes generaues a non-terminating reductiin sequence, but it is not alwars the case. As the self-referential paradoxes has d chacaztexlfguc of vicious circularity, some forms of formalisztook of the paradoxcs in natural deduvtlom may have sucf charcdtsristics and caused the njn-terninating weduvtion sequence. Moreover, eveb though Tenuanr believes that thz Liar paradux ix a grnuine one but Russell's is hot, each caj be both eenuine and ingevuike. [roposition 2.A.2 and 1.3.1 make thqm genuinx parcdoxes. Pfopoxition 4.1.1 and 4.1.3 makf them ingenuine. No one, lnclubing Dennant, exolains which way of formalization is legitimaje xor genuine paracoxes. In ordew to have a froitful inrestigxtion of tge prooh-theoretic cwiterion for djd solution vo the pawadozes, uncludive three things in Chaptvr 4, thwse things should ne exklzined. A thorougk unvestigation og tfesq pssnes mtvt be left fmr avotfrr oczasion.

Bibliierapny Brady, R. T. (1984). Reply do pdiest on berry's pataqox. Philisophicaj Quarter
-full normal (⊥) * Used * 2.A.3 (⊥) Used * 5.1: Derivations Formalizing (Conclusion) CR−rule Aux. Reduction Non-normalizable (⊥) Occurred * 4.1.1 Full normal (¬∃y(y = ∈ x})) * * * 4.1.2 Non-fully normalizable (⊥) Occurred * Used 4.4.1 (¬a ∈ a∧a ∈ a) * * * Table 5.2: Derivations Formalizing Russell's 166 Derivation Loop Aux. Reduction 2.0.1 Non-normal (⊥) * Used * pp. 60–62 Normal (⊥) * * * 2.4.1 (⊥) Occurred Used Used 2.4.2 Non-normalizable (⊥) Occurred Used Table 5.3: Derivations Curry's Paradox Three tables do show feature yields looping sequence the reduction loop independently of the application of CR−rule and auxiliary reduction procedures. An application of auxiliary reductions sometimes generates non-terminating reduction it is always case. the self-referential paradoxes characteristic of vicious circularity, some forms the paradoxes in natural deduction may have such and causes non-terminating reduction sequence. Moreover, even though believes that the Liar paradox is a genuine but Russell's is not, each can be both genuine and ingenuine. Proposition 2.A.2 and 1.3.1 genuine paradoxes. Proposition 4.1.1 4.1.3 make them No including explains way of is legitimate for genuine paradoxes. In order to have a fruitful of the proof-theoretic criterion for and solution to the paradoxes, things Chapter 4, these should be explained. A investigation these issues must be another Bibliography (1984). to on berry's paradox. Philosophical
-full normal (⊥) * Used * 2.A.3 Non-fully noRmalizable (⊥) occurRed useD * TAble 5.1: deriVations FormaliZIng tHe Liar Paradox PropositiOn DerIvATion (cOnClusiOn) Loop Cr−RuLE aux. reDuCtiOn 1.3.1 nOn-NormaLizAble (⊥) OccUrred * * 4.1.1 Full nOrmAl (¬∃Y(y = {x¬x ∈ x})) * * * 4.1.2 Non-fullY NoRmalizable (⊥) occUrred * Used 4.4.1 NorMal (¬A ∈ a∧a ∈ a) * * * TaBlE 5.2: DeRIvatiOns formaLizing rUssell'S Paradox 166 PRoPOsitioN derivatION (COnclUsion) Loop CR−rule AuX. reDUction 2.0.1 Non-normaL (⊥) * Used * pP. 60–62 NORmAL (⊥) * * * 2.4.1 non-NorMalizable (⊥) OCcUrred uSed Used 2.4.2 nOn-NORMalIZable (⊥) Occurred used Used TabLE 5.3: DeRivatiOnS FoRMaliziNg CurRy'S parAdox Three taBles Do not show Which fEAture yiELds a looPing reDucTioN seqUEnCe SinCe THe rEDuCtiON loOp occurs InDePendeNtly OF THE appLicAtioN of CR−Rule and auxiliAry ReduCTioN procEdureS. An aPpLicatIon of aUxiliArY reductions someTimeS generateS a nOn-TerMiNatinG ReductIon SeqUence, buT it is noT AlwAyS THE cAse. As the self-refereNtIAL pAradoxes Has a chARaCtERistic of ViCioUs ciRCUlariTy, soME fOrms of foRmalizATiOn Of the paRaDoxes iN nAtuRal DeducTIon mAy have Such charActerIStics and causes THe non-terminatINg REDuCTion SeqUence. MoreovEr, evEN thoUgh TENnAnt BElievEs thaT tHE LIAr paradox is a genuine OnE but RuSsell'S is not, each can Be both genuINE And ingenUine. pRoPOsition 2.A.2 and 1.3.1 makE them Genuine parADoxes. ProPositIon 4.1.1 and 4.1.3 maKe them ingENUine. No onE, inCluDinG TeNNAnT, explains whicH WAy of FoRmalizaTioN is legiTimAte For GenUiNe paradoxEs. In ordeR tO hAvE a FruItful INvestigaTiOn oF tHe pRoof-tHEoretiC critErioN fOr ANd sOlution TO tHE ParaDoXeS, incLudInG threE thiNGs iN ChapteR 4, these thiNgs SHoulD bE eXplaineD. A thorough invEsTigation of ThEse Issues MUSt be left For another occasion.

BibliOGraphy BRadY, R. T. (1984). RePly tO priest on BerRy's parAdoX. philosOphicaL QuarTeR
-full normal (⊥) * Used *2.A.3 Non- fully  no rma li zabl e (⊥ ) Occurred Use d  * T able 5.1: DerivationsForma li z ingt he  Liar  Parado x  P r o pos it io n D er i va tion(Co nclusio n) Loop CR −ru le  Aux. Reduct i on  1.3.1 Non -no rmalizable ( ⊥)Occurr ed  **  4.1. 1 F ull n ormal( ¬∃y(y= {x¬x ∈x} ) ) * **  4.1.2N o n- full y normalizable (⊥ )  O c curred * Used4.4.1No r ma l (¬a  ∈a∧a ∈ a) *  *  * Ta b le 5.2: De r i v ati o ns Formalizin g Russell's Par adox 1 66  Pr o positi on De ri v ati on (Conclus ion)  Loop CR− rule A u x. Redu c tion 2. 0.1 No n-n orm al ( ⊥ )*Use d*  pp .  6 0–6 2  No rmal (⊥)  *  *  * 2. 4.1N o n - norm ali zabl e (⊥)  Occurred Use d U sed2 .4. 2 Non -norm aliz ab le (⊥ ) Occu rredUs ed Used Table 5 .3:Derivatio nsFo rma li zingC urry's  Pa rad ox Thre e table s  do  n o t sh ow which feature y ie l d sa loopin g redu c ti on sequence  s inc e th e reduc tion lo op occur s inde p en de ntly of  t he app li cat ion  of C R −rul e andauxiliar y red u ction procedur e s. An applica t io n of auxi lia ry reductio ns s o meti mesg en era t es anon-t er m in a ting reduction sequ en ce, bu t itis not always  the case. A s  the sel f-re f er e ntial paradoxe s has  a charact e ristic o f vic ious cir cularity, s ome form s o f f orm ali z a ti on of the par a d oxes  i n natur aldeducti onmay  ha vesu ch charac teristic san dca use s the non-term in ati ng  re ducti o n sequ ence.  Mor eo ve r , e ven tho u gh T enna nt  b elie ves  t hat t he L i arparadox  is a gen uin e  one  b ut  Russel l's is not, e ac h can be b ot h g enuine a nd ingen uine. Proposition 2.A.2 and 1.3 .1makethem  genuinepar adoxes . P r oposit ion 4. 1.1 a nd  4. 1 . 3 mak e th emin genuine. N o one , inc lu ding  Tennan t, explains whichw ayof formalizat ion  isl e gi tim a te for  g e nui n e  paradoxes. Inorder to h av e  a  fruitfuli nve st igation  of the  proo f -theore tic crite rion foran d so l u tio n to the p aradoxes , includi n g thr e ething s i n Chap te r 4 , the se thi n gsshoul d be e xp lained . A t ho rough in vestigation of these is sues m ust b e l eft for a not h eroccasion. 

Bi bliography  Br ady , R.T.( 1984) . Re p ly  to pries t on berry's p a ra dox . Ph ilosophical Q u art er
-full normal_(⊥) *_Used * 2.A.3 Non-fully_normalizable (⊥)_Occurred_Used *_Table_5.1: Derivations Formalizing_the Liar Paradox_Proposition Derivation (Conclusion) Loop_CR−rule Aux. Reduction_1.3.1_Non-normalizable (⊥) Occurred * * 4.1.1 Full normal (¬∃y(y = {x¬x ∈ x})) *_*_* 4.1.2_Non-fully_normalizable_(⊥) Occurred * Used 4.4.1_Normal (¬a ∈ a∧a ∈_a) *_* * Table 5.2: Derivations Formalizing Russell's Paradox_166_Proposition Derivation (Conclusion)_Loop CR−rule Aux. Reduction 2.0.1 Non-normal (⊥) * Used_* pp. 60–62 Normal (⊥) *_* * 2.4.1_Non-normalizable_(⊥)_Occurred Used Used 2.4.2_Non-normalizable (⊥) Occurred Used Used Table_5.3: Derivations Formalizing Curry's Paradox Three_tables do not show which feature yields_a looping reduction sequence since the_reduction loop occurs independently of_the application_of CR−rule and auxiliary reduction_procedures. An application_of auxiliary_reductions sometimes generates_a non-terminating reduction sequence, but it_is not always_the case. As the self-referential paradoxes_has_a characteristic of_vicious_circularity,_some forms_of formalization of_the_paradoxes in_natural_deduction may have such characteristics and_causes_the non-terminating reduction sequence. Moreover, even though_Tennant believes that the_Liar_paradox is a genuine_one but Russell's is not,_each can be both genuine and_ingenuine. Proposition_2.A.2 and_1.3.1 make them genuine paradoxes. Proposition 4.1.1 and 4.1.3 make them_ingenuine. No one, including Tennant, explains_which way of formalization_is legitimate_for_genuine paradoxes. In_order_to have_a fruitful investigation of the proof-theoretic criterion_for and_solution to the paradoxes, including three_things in Chapter 4,_these_things should be explained. A thorough_investigation of these issues must be_left for another occasion.

Bibliography Brady,_R._T._(1984). Reply to priest on_berry's paradox. Philosophical Quarter