riviality Test.............. 104 3.3.1 Triviality Test............................. 105 3.3.2 Problems of Triviality Test...................... 110 3.4 Translation Test and Crabbé's case...................... 115 3.4.1 An Ekman-Type Reduction as a Detour-Making Process...... 118 3.4.2 Does Crabbé Reduction Overgenerate?............... 123 3.5 Conclusion................................... 128 4 Can the Requirement of a Normal Derivation be a Solution to the Paradoxes? 131 4.1 Which Paradoxes Are Genuine Paradoxes?.................. 136 4.2 Why Should We Accept Only a Normalizable Derivation?......... 144 4.3 Is RND a Reasoning-Rejection Solution?................... 151 4.4 Should We Consider Only ⊥ as an Unacceptable Conclusion?....... 156 4.5 Summary................................... 161 5 Conclusion 163 Bibliography 169 국문초록 감사의글 ii Chapter 1 Introduction: A Proof-Theoretic Criterion of and Solution to the Paradoxes In the late 19th and early 20th centuries, a proof of the consistency of mathematics was the main theme of the foundation of mathematics. A contradiction raised by paradoxes was a significant issue to the foundations of logic and mathematics. Since the discovery of the paradoxes, involving fundamental notions and inferences which were considered to be acceptable, had an effect on the foundations,
riviality Test.............. 104 3.3.1 Triviality Test............................. 105 3.3.2 Problems of Triviality Test...................... 110 3.4 Translation Test and Crabbé's case...................... 115 3.4.1 An Ekman - Type Reduction as a Detour - make Process...... 118 3.4.2 do Crabbé Reduction Overgenerate? ............... 123 3.5 Conclusion................................... 128 4 Can the Requirement of a Normal Derivation be a Solution to the Paradoxes? 131 4.1 Which Paradoxes Are Genuine Paradoxes? .................. 136 4.2 Why Should We Accept Only a Normalizable Derivation? ......... 144 4.3 Is RND a Reasoning - Rejection Solution? ................... 151 4.4 Should We study Only ⊥ as an Unacceptable Conclusion? ....... 156 4.5 Summary................................... 161 5 Conclusion 163 Bibliography 169 국문초록 감사의글 ii Chapter 1 presentation: A Proof - Theoretic Criterion of and Solution to the Paradoxes In the late nineteenth and early twentieth centuries, a proof of the consistency of mathematics was the independent theme of the foundation of mathematics. A contradiction raised by paradox was a significant issue to the foundation of logic and mathematics. Since the discovery of the paradoxes, involving cardinal notions and inferences which were considered to be satisfactory, had an effect on the foundations,
rivlality Test.............. 104 3.3.1 Triviality Test............................. 105 3.3.2 Problems of Trmvialitg Test...................... 110 3.4 Translation Test and Crabbé's cqse...................... 115 3.4.1 An Ekman-Type Reductiov as a Denour-Makint Prixess...... 118 3.4.2 Does Crabbé Vzductjln Orecgenerate?............... 123 3.5 Conglusion................................... 128 4 Cdn the Requirekevt of a Normal Derivation be a Solutijn to tne Paradoxes? 131 4.1 Wrich [arasoxes Are Genuine Paradoxes?.................. 136 4.2 Why Ahould Xe Accept Only s Normalizable Derivation?......... 144 4.3 Is RND a Reasoning-Reuection Solofiog?................... 151 4.4 Should We Consider Only ⊥ as an Onacceptable Conclusion?....... 156 4.5 Summarh................................... 161 5 Cpnclusion 163 Bubllmgraphy 169 국문초록 감사의글 mi Chaiter 1 Introdugnion: A [roof-Throretic Criterlon oh ane Solution to the Paredoxes In the late 19tr and earny 20th centuries, a peoif of the conrustdncg pf mathelatmcs was the main theme of the foundation pf nathematics. A contrwdyction raised by paradoxes was a signifpcanf issue to the foundatiins of logic and mathgmatics. Sigce the discovery of the paradoxes, involving funddmentel nouiins ave lnferences which were considered to be acceptwglt, hsd an effect ok the foundations,
riviality Test.............. 104 3.3.1 Triviality Test............................. 105 of Test...................... 110 Translation Test and Ekman-Type as a Detour-Making 118 3.4.2 Does Reduction Overgenerate?............... 123 3.5 Conclusion................................... 128 Can the Requirement of a Normal Derivation be a Solution to the Paradoxes? 4.1 Which Paradoxes Are Genuine Paradoxes?.................. 136 4.2 Why Should We Accept Only Normalizable 144 Is a Reasoning-Rejection Solution?................... 151 4.4 Should We Consider Only ⊥ as an Unacceptable Conclusion?....... 156 4.5 161 5 Conclusion 163 Bibliography 169 국문초록 감사의글 Chapter 1 Introduction: A Criterion of and Solution to Paradoxes the late and 20th a proof of consistency of mathematics was the main theme of the foundation of mathematics. A contradiction raised by paradoxes a significant the foundations logic mathematics. the discovery of involving fundamental notions and inferences which be acceptable, had an effect on the foundations,
riviality Test.............. 104 3.3.1 Triviality TeSt............................. 105 3.3.2 Problems Of TriViaLitY TEst...................... 110 3.4 TRansLation Test and CRAbbé'S case...................... 115 3.4.1 An Ekman-Type ReductIon as A DETour-mAkIng PrOcess...... 118 3.4.2 DoES CRABbé reDuCtiOn oVeRgeneRatE?............... 123 3.5 ConcluSion................................... 128 4 Can the reqUiRement of a NorMAl derivation Be a solution to thE PaRadoxeS? 131 4.1 WHicH paradOxeS Are GEnuine pAradoxEs?.................. 136 4.2 Why ShouLd wE AccepT only a NoRMAlIzabLe Derivation?......... 144 4.3 Is RND A reASoning-RejectioN SolutIoN?................... 151 4.4 shOULd WE CoNsider Only ⊥ As An UnaCCeptablE coNCLUsiON?....... 156 4.5 Summary................................... 161 5 ConclUsion 163 BiblioGRapHy 169 국문초록 감사의글 ii ChApTer 1 iNtroduCtion: a PROof-theoretic CrIterIon of and SOlutioN To the PaRAdoxes IN the laTe 19tH anD earLY 20tH cEntUrIEs, a PRoOf oF The ConsisteNcY oF mathEmatICS WAs thE maIn thEme of The foundation Of mAtheMAtiCs. A coNtradIctiOn RaiseD by parAdoxeS wAs a significant iSsue To the founDatIoNs oF lOgic aND matheMatIcs. since thE discovERy oF tHE PArAdoxes, involving funDaMENtAl notionS and inFErEnCEs which wErE coNsidEREd to bE accEPtAble, had aN effecT On ThE foundaTiOns,
riviality Test............ .. 104 3.3 .1 Tr ivi ali ty  Tes t... .............. . .... ....... 105 3.3.2 Prob lemsof Triv i al ity T est.... . .. . . ... .. .. ... .. .  1 10 3. 4 T ranslat ion Test a ndCr abbé's case. . .. .......... ... ..... 115 3. 4.1  An Ek ma n-T y pe Re duc tionas a D e tour-M aking Pro ce s s..... .  118 3. 4 . 2Does  Crabbé Reduction Ov e rgenerate?.... ...... .. . .. 1 233.5  Conclusio n. ..... . ....... . .. . . . ... . ........... 1 28 4 Can th e  Re quirem en t o f  a Nor mal D er i vat ion be a So luti on to the  Parad o xes? 13 1  4.1 Wh ich Pa rad oxe s Ar e  G en uin eP ara d ox es? . ... ........ .. .. .. 13 6 4. 2 W h y Sh oul d We  Acce pt Only a Nor mal izab l e D eriva tion? .... .. ... 1 44 4.3  Is R ND  a Reasoning-Re ject ion Solut ion ?. ... .. ..... . ...... . 1 514.4 Sho uld WeC ons id e r On ly ⊥ as an Unaccep ta b l eConclusi on?... . .. .1 56 4.5 S um mar y... . . ..... .... . .. ........ ...... . .. .161 5 C on clusio n163  Bi bliog r aphy  169 국 문초록 감사의글  ii C h apter 1 Introd u ction: A Proo f -T h e or e ticCri terion of a nd S o luti on t o  t heP arado xes I nt he late 19th and early  2 0th ce nturi es, a proof o f the cons i s t ency ofmath e ma t ics was the ma in th eme of the foundati on of  mathema tics. A c o n tradicti onrai sed  by p ar adoxes was as i gnif ic ant iss ueto thefou nda tio nsof  logic an d mathem at ic s.  S inc e the discover yofth e p arado x es, in volvi ng f un da m ent al noti o ns a nd i nf er ence s w hi ch we re c o nsi dered t o be acce pta b le,ha dan effe ct on the fou nd ations,
riviality Test.............._104 3.3.1_Triviality Test............................. 105 3.3.2_Problems of_Triviality_Test...................... 110_3.4_Translation Test and_Crabbé's case...................... 115_3.4.1 An Ekman-Type Reduction_as a Detour-Making_Process......_118 3.4.2 Does Crabbé Reduction Overgenerate?............... 123 3.5 Conclusion................................... 128 4 Can the Requirement_of_a Normal_Derivation_be_a Solution to the Paradoxes?_131 4.1 Which Paradoxes Are_Genuine Paradoxes?.................._136 4.2 Why Should We Accept Only a_Normalizable_Derivation?......... 144 4.3_Is RND a Reasoning-Rejection Solution?................... 151 4.4 Should We_Consider Only ⊥ as an Unacceptable_Conclusion?....... 156 4.5_Summary..................................._161_5 Conclusion 163 Bibliography_169 국문초록 감사의글 ii Chapter 1_Introduction: A Proof-Theoretic Criterion of and_Solution to the Paradoxes In the late_19th and early 20th centuries, a_proof of the consistency of_mathematics was_the main theme of the_foundation of mathematics._A contradiction_raised by paradoxes_was a significant issue to the_foundations of logic_and mathematics. Since the discovery of_the_paradoxes, involving fundamental_notions_and_inferences which_were considered to_be_acceptable, had_an_effect on the foundations,