-terminating reduction sequence. Hence, we may summarize our diagnosis in the following way. A Possible Diagnosis: A derivation formalizing a genuine paradox generates a non-terminating reduction sequence only if (i) every major premise in the derivation has a reduction procedure to eliminate it, or (ii) every formula including a principal constant (or operator) has a reduction procedure to eliminate it. The derivations in Proposition 2.0.1, 2.2.2, and 2.3.1 include a major premise which has no reduction procedure to get rid of the major premise. The derivation ∆12 of Curry's paradox using the axiom ¬a ∈ a ↔ a ∈ {x|x ∈ x → ⊥} includes a formula containing a principal constant ∈ which has no reduction process to remove the formula. In order to have a clearer diagnosis beyond our possible diagnosis, the general condition for a nonterminating reduction sequence should be given. From our diagnosis, we suggest a plausible solution to the problem of undergeneration. Provided that Proposition 2.0.1, 2.2.2, and 2.3.1 which raise the undergeneration problem are inappropriate counterexamples to TCPE, we have an additional condition that a derivation only uses harmonious rules. CR−rule in Proposition 2.0.1 and 2.2.2 has no its corresponding harmonious rule. Although the use of the axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is regarded 76 as an application of = I−rule, we cannot say that the use of the axiom ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ and = E−rule has a harmonious relation. An Additional Condition: A derivation formalizing a genuine paradox only uses harmonious rules. Harmonious I− and E−rules automatically have a standard reduction procedure which eliminates a maximum formula. If the suggested condition is acceptable, the condition can solve the problem of undergeneration. 2.5 Conclusion In this chapter, we have investigated derivations of Curry's and the Liar paradox which generate the problem of undergeneration. Under the assumption that Curry's and the Liar paradox are genuine paradoxes, at the beginning of this chapter, we introduce the Rogersontype derivation of Curry's paradox employing CR−rule which does not generate a
-terminating reduction sequence. Hence, we may summarize our diagnosis in the follow direction. A Possible Diagnosis: A derivation formalizing a actual paradox generates a non - terminating reduction sequence merely if (i) every major premise in the derivation have a reduction procedure to excrete it, or (ii) every formula including a principal changeless (or operator) has a reduction procedure to eliminate it. The derivation in Proposition 2.0.1, 2.2.2, and 2.3.1 include a major premise which has no reduction routine to get rid of the major premise. The derivation ∆12 of Curry's paradox using the axiom ¬a ∈ a ↔ a ∈ { x|x ∈ x → ⊥ } includes a formula incorporate a principal constant ∈ which has no reduction process to absent the formula. In order to have a clearer diagnosis beyond our possible diagnosis, the general condition for a nonterminating reduction sequence should be given. From our diagnosis, we suggest a plausible solution to the problem of undergeneration. Provided that Proposition 2.0.1, 2.2.2, and 2.3.1 which raise the undergeneration problem are inappropriate counterexamples to TCPE, we have an additional circumstance that a deriving only use harmonious rule. CR−rule in Proposition 2.0.1 and 2.2.2 has no its corresponding harmonious dominion. Although the use of the axiom ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ is regarded 76 as an application of = I−rule, we cannot say that the habit of the axiom ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ and = E−rule has a harmonious relation. An Additional Condition: A derivation formalizing a actual paradox only uses harmonious rules. Harmonious I− and E−rules automatically have a standard reduction procedure which excrete a maximum formula. If the suggested condition is acceptable, the circumstance can solve the problem of undergeneration. 2.5 Conclusion In this chapter, we have investigated derivations of Curry's and the Liar paradox which generate the problem of undergeneration. Under the assumption that Curry's and the Liar paradox are genuine paradox, at the beginning of this chapter, we introduce the Rogersontype derivation of Curry's paradox employing CR−rule which does not generate a
-terlinating reduction sequekce. Hence, we may summarmze our diagnosks in the following way. A Podsuble Eiagnosis: A derivation formalizpng a genyine paradox geisrates a non-tsvminaciig reduction seauence onlf if (i) every mdjur premise in the derivation has a requction pgocedure to elyminsee if, or (ii) every formula including a lrincipel constant (or pperator) has a reduction pgocefure to eliminate lt. The dericatijbs in Proposktion 2.0.1, 2.2.2, and 2.3.1 include a major premise which has no redjctiou procedure ti gfj rid of the majog premise. The derivathon ∆12 of Curry's paradow usiig tye axiom ¬a ∈ a ↔ a ∈ {x|x ∈ x → ⊥} includes a fotmula contdiuing a principal consrabt ∈ wvich has bo fedhcvioh procfss to remove the formulq. In order to have s soearer diagnoais berogd our possible diagnosis, the general cmndjtion for a nonterminatung reduction sequencg should bq given. From our diagnosis, we suggest a plausible soluviun uo the oeohlem of undergeneration. Provided that Proposiejom 2.0.1, 2.2.2, and 2.3.1 which rcise the undergemegayyon problem ate inap'dolriate counterexamoles to TCPE, we have wn acditional condition that a eerivation ouly uses harmonious rbles. CR−rule nn Prokositipn 2.0.1 and 2.2.2 has no its corxesponsing harmonlous rule. Xlthough the use of tve axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is regarded 76 as an ap'licacion of = I−ruke, we sannot say that the use of the axilm ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ dnd = E−rule has a harmonious relation. An Avvitional Condothon: A derivction normalizing a gqnuine paradox only ufes hxrmonious gules. Harkonious I− agd E−rules autmlatically hate a stanqard redyction ofocedure which eliminatvs a mqximum formula. If bhe sogfested conditiou us acceptable, tne zonqinioi can volve the prmbleo ow undefgeneraunon. 2.5 Convlusion In this chapder, se have investigatrd derivatuons of Surry's and thr Liar paradox whifh geierate the lroflem of undergeneration. Under fhe assumotijn that Currr's akd tre Liar paxadox are genuine paradoxes, at the beginiing of this chapter, wg introduce the Rogerxpntype deritation of Curry'v paradox employing XR−rule which does not generate a
-terminating reduction sequence. Hence, we may summarize in following way. Possible Diagnosis: A generates non-terminating reduction sequence if (i) every premise in the derivation has a procedure to eliminate it, or (ii) every formula including a principal constant (or has a reduction procedure to eliminate it. The derivations in Proposition 2.0.1, 2.2.2, 2.3.1 a premise has no reduction procedure to get rid of the major premise. The derivation ∆12 of Curry's using the axiom ¬a ∈ a ↔ a {x|x ∈ x → includes a formula containing a constant which has reduction to the formula. In to have a clearer diagnosis beyond our possible diagnosis, the general condition for a nonterminating reduction sequence be given. diagnosis, we a solution the problem of that Proposition 2.0.1, 2.2.2, and 2.3.1 undergeneration problem are inappropriate counterexamples to TCPE, we an additional that a derivation only uses harmonious CR−rule in Proposition 2.0.1 and 2.2.2 has no corresponding harmonious rule. Although the use of the axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is regarded an application of = we cannot say the of axiom (⌜Φ⌝)⌝= ⌜Φ⌝ = E−rule has a harmonious relation. An Additional Condition: A derivation a genuine paradox only uses harmonious rules. Harmonious I− and have standard reduction procedure eliminates a maximum formula. the condition is acceptable, the solve problem Conclusion this we have investigated derivations Curry's and the Liar paradox generate the problem of Curry's and the Liar paradox are genuine paradoxes, the beginning of this chapter, we introduce Rogersontype derivation of Curry's paradox employing CR−rule which does not generate a
-terminating reduction sequeNce. Hence, we May suMmaRizE oUr diAgnoSis in the followINg waY. A Possible Diagnosis: A deRivatIoN FormALiZing a Genuine PArADOx gEnErAteS a NOn-TermiNatIng reduCtion sequeNce OnLy if (i) every maJOr Premise in tHe dErivation has A reDuctioN pRocEDure tO elIminaTe it, or (II) every Formula inClUDing a pRIncipal CONsTant (Or operator) has a redUCtIOn procedure to eLiminaTe IT. THE DerIvaTions in ProPoSitioN 2.0.1, 2.2.2, And 2.3.1 inclUDe A MAJor PRemise which haS no reductioN ProCedure To Get RId of thE majoR pREmiSe. The derivaTion ∆12 Of Curry's pAradox USing the AXiom ¬a ∈ a ↔ a ∈ {X|x ∈ x → ⊥} incLudEs a FormULa CoNtaInINg a PRiNciPAl cOnstant ∈ wHiCh Has no ReduCTION proCesS to rEmove The formula. In oRdeR to hAVe a ClearEr diaGnosIs BeyonD our poSsiblE dIagnosis, the geneRal cOndition fOr a NoNteRmInatiNG reducTioN seQuence sHould be GIveN. FROM OuR diagnosis, we suggesT a PLAuSible solUtion tO ThE pROblem of uNdErgEnerATIon. PrOvidED tHat PropoSition 2.0.1, 2.2.2, ANd 2.3.1 WhIch raisE tHe undeRgEneRatIon prOBlem Are inaPpropriaTe couNTerexamples to TcpE, we have an addITiONAl COndiTioN that a derivAtioN Only Uses HArMonIOus ruLes. CR−RuLE iN proposition 2.0.1 and 2.2.2 has no ItS correSpondIng harmonious Rule. AlthouGH THe use of tHe axIOm ⌜¬t (⌜φ⌝)⌝ = ⌜Φ⌝ is regarded 76 as An appLication of = i−Rule, we caNnot sAy that thE use of the AXIom ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ anD = E−rUle Has A haRMOnIous relation. AN aDditIoNal CondItiOn: A deriVatIon ForMalIzIng a genuiNe paradoX oNlY uSeS haRmoniOUs rules. HArMonIoUs I− And E−rULes autOmatiCallY hAvE A stAndard rEDuCTIon pRoCeDure WhiCh ElimiNateS A maXimum foRmula. If thE suGGestEd CoNdition Is acceptable, tHe Condition cAn SolVe the pROBlem of unDergeneration. 2.5 Conclusion iN this chAptEr, we hAve iNvestigatEd dErivatIonS Of CurrY's and tHe LiaR pAraDOX whicH GEnEraTe The problem OF UndErgenErAtioN. Under tHe assumption that CuRRy's And the Liar parAdoX are GENuIne PArADoxEs, AT thE BEginning of this cHapter, we inTrODuCe the RogerSOntYpE derivaTion of CUrry's PAradox eMploying Cr−rule whicH dOes nOT GenErate a
-terminating reduction seq uence. Hen ce, w e m aysu mmar izeour diagnosisi n th e following way. A Pos sible  D i agno s is : A d erivati o nf o rma li zi ngag en uinepar adox ge nerates anon -t erminating r e du ction sequ enc e only if (i ) e very m aj orp remis e i n the  deriv a tion h as a redu ct i on pro c edure t o el imin ate it, or (ii) e v er y  formula inclu ding a  p r in c i pal  co nstant (or  o perat o r) hasa  r e d u cti o n procedure t o eliminate it.  The d er iva t ions i n Pro po s iti on 2.0.1, 2 .2.2 , and 2.3 .1 inc l ude a m a jor pre mise w hic h h as n o  r ed uct io n  pr o ce dur e  to  get rid  o fthe m ajor p r e mise . T he d eriva tion ∆12 of C urr y'sp ara dox u singtheax iom ¬ a ∈ a↔ a ∈  { x|x ∈ x → ⊥} in clud es a form ula  c ont ai ninga  princ ipa l c onstant  ∈ whic h  ha sn o re duction process to  r e m ov e the fo rmula. In  o r der to h av e a  cle a r er di agno s is  beyondour po s si bl e diagn os is, th egen era l con d itio n fora nonter minat i ng reduction s e quence should be g iv e n. F rom  our diagno sis, we s ugge s ta p l ausib le so lu t io n  to the problem ofun dergen erati on. Providedthat Propo s i t ion 2.0. 1, 2 . 2. 2 , and 2.3.1 wh ich r aise the u n dergener ation  problem  are inap p r opriatecou nte rex amp l e sto TCPE, we h a v e an  a ddition alconditi ontha t a  de ri vation on ly usesha rm on io usrules .  CR−rule  i n P ro pos ition 2.0.1and 2 .2.2  h as noits cor r es p o ndin gha rmon iou srule.  Alt h oug h the u se of the  ax i om ⌜ ¬T  ( ⌜Φ⌝)⌝ =  ⌜Φ⌝ is regar de d 76 as an  a ppl icatio n of = I−r ule, we cannot say that the use  of  theaxio m ⌜¬T (⌜Φ ⌝)⌝ = ⌜Φ⌝and = E−ru le has  a ha rm oni o u s rel a t io n.An  Additiona l Con ditio n:  A d erivati on formalizing a g e nui ne paradox on lyuses h ar mon i ou s  ru le s . H a r monious I− andE−rules au to m at ically hav e  ast andardreducti on pr o cedurewhich eli minates a  m axim u m  fo rmula. Ifthe sugg ested con d ition is  acce pta ble, t he  co nditi on can sol ve th e prob le m of u nderg en eration.  2.5 Conclusion In this  chapt er, w e h ave inves tig a ted  derivati onsof Curry's  an d t he Li arp arado x wh i ch  ge n erate  the problem o f  u nde r g en eration. Un d e r  th e ass ump t ion th at C urry's and the Li a r paradox aregenu i n e p ara d oxes ,at the beginni ngof t his chap te r, we intro duce the  R o gerso ntypederiva tion of C ur r y's pa rado x e mployingCR− ru l e which  d oe s  not g ener at e a
-terminating reduction_sequence. Hence,_we may summarize our_diagnosis in_the_following way._A_Possible Diagnosis: A_derivation formalizing a_genuine paradox generates a_non-terminating reduction sequence_only_if (i) every major premise in the derivation has a reduction procedure to eliminate_it,_or (ii)_every_formula_including a principal constant (or_operator) has a reduction procedure_to eliminate_it. The derivations in Proposition 2.0.1, 2.2.2, and_2.3.1_include a major_premise which has no reduction procedure to get rid_of the major premise. The derivation_∆12 of Curry's_paradox_using_the axiom ¬a ∈_a ↔ a ∈ {x|x ∈_x → ⊥} includes a formula_containing a principal constant ∈ which has_no reduction process to remove the_formula. In order to have_a clearer_diagnosis beyond our possible diagnosis,_the general condition_for a_nonterminating reduction sequence_should be given. From our diagnosis,_we suggest a_plausible solution to the problem of_undergeneration._Provided that Proposition_2.0.1,_2.2.2,_and 2.3.1_which raise the_undergeneration_problem are_inappropriate_counterexamples to TCPE, we have an_additional_condition that a derivation only uses harmonious_rules. CR−rule in Proposition_2.0.1_and 2.2.2 has no_its corresponding harmonious rule. Although_the use of the axiom ⌜¬T_(⌜Φ⌝)⌝ =_⌜Φ⌝ is_regarded 76 as an application of = I−rule, we cannot say_that the use of the axiom_⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ and_= E−rule_has_a harmonious relation._An_Additional Condition:_A derivation formalizing a genuine paradox only_uses harmonious_rules. Harmonious I− and E−rules automatically_have a standard reduction_procedure_which eliminates a maximum formula. If_the suggested condition is acceptable, the_condition can solve the problem_of_undergeneration._2.5 Conclusion In this chapter,_we have investigated derivations of Curry's_and the Liar_paradox which generate the problem of undergeneration._Under_the assumption that Curry's and the_Liar_paradox are genuine paradoxes, at the_beginning_of_this chapter, we introduce the_Rogersontype derivation of Curry's paradox employing_CR−rule which does not generate a