 pair of rules but RND is not. If the intrinsic harmony is a requirement for a legitimate pair of rules, RND cannot be a requirement for a prooftheoreically valid inference rule. Moreover, though RND can block the derivation Σ3 of ⊥, it does not single out a rule that is invalid in Σ3. If a reasoning-rejection solution to the paradoxes is a constraint on a specific rule in a paradoxical derivation, RND is not a reasoning-rejection but a structural restriction of all derivations in an intended system. Our question in this section is how RND can single out an inference rule which makes a derivation non-normalizable. If it cannot, then it will be regarded as a different kind of solution 152 than a reasoning-rejection in a usual sense. Interestingly, when Prawitz (1965, p. 95) mentions that the set-theoretical paradoxes are ruled out by the requirement that the derivations should be in normal form, he seems to request that the application of → E−rule, in our case ¬E−rule, has to be restricted. While defining ¬φ as φ →⊥, the derivation Σ12 is restated as the following derivation Σ′12. Σ ′ 10 ¬a ∈ a Σ ′ 11 a ∈ a → E⊥ With the derivation Σ′12 in his natural deduction system for the naive set theory, he remarked on the application of the last → E−rule in Σ′12.... the system [for the naive set theory] has serious disadvantages. Thus, although [→ E−rule] is a rule of the system, one cannot in general infer that [ψ] is provable given that [φ] and [φ → ψ] are provable, since there may be only a [derivation] of ψ [but not a normal derivation of ψ]. This renders investigations of the system rather difficult as it is not sufficient to derive the axioms of an ordinary mathematical theory in the system in order to conclude that also its theorems are provable in the system. (Prawitz, 1965, p. 95) Immediate subderivations Σ′10 and Σ ′ 11 are in normal form, but Σ ′ 12 is not. Since Σ ′ 10 and Σ′11 are canonical derivations, by Definition 4.2.1 of P−validity, they are P−valid but Σ′12 is
pair of rules but RND is not. If the intrinsic harmony is a requirement for a legitimate couple of rule, RND cannot be a requirement for a prooftheoreically valid inference rule. furthermore, though RND can block the ancestry Σ3 of ⊥, it does not single out a rule that is invalid in Σ3. If a reasoning - rejection solution to the paradoxes is a restraint on a specific principle in a paradoxical derivation, RND is not a reasoning - rejection but a geomorphologic restriction of all derivations in an mean system. Our question in this department is how RND can single out an inference dominion which makes a derivation non - normalizable. If it cannot, then it will be regarded as a different kind of solution 152 than a reasoning - rejection in a usual sense. Interestingly, when Prawitz (1965, p. 95) mention that the set - theoretical paradoxes are ruled out by the prerequisite that the derivations should be in normal form, he seems to request that the application of → E−rule, in our event ¬E−rule, has to be restricted. While defining ¬φ as φ →⊥, the derivation Σ12 is restated as the follow derivation Σ′12. Σ ′ 10 ¬a ∈ a Σ ′ 11 a ∈ a → E⊥ With the derivation Σ′12 in his natural deduction system for the uninitiate set theory, he remarked on the application of the last → E−rule in Σ′12.... the system [ for the naive set theory ] has dangerous disadvantages. Thus, although [ → E−rule ] is a principle of the arrangement, one cannot in general infer that [ ψ ] is provable given that [ φ ] and [ φ → ψ ] are demonstrable, since there may be only a [ deriving ] of ψ [ but not a normal derivation of ψ ]. This renders probe of the system rather difficult as it is not sufficient to derive the axioms of an ordinary numerical theory in the system in order to conclude that also its theorems are provable in the arrangement. (Prawitz, 1965, p. 95) Immediate subderivations Σ′10 and Σ ′ 11 are in normal form, but Σ ′ 12 is not. Since Σ ′ 10 and Σ′11 are canonical derivations, by Definition 4.2.1 of P−validity, they are P−valid but Σ′12 is
 palr of rules but RND is nut. If the intriuwic hacmony ia a requkrement for a legitimate paic of rulew, RND cannot be a requkrement flr a proifthtoreically valid mhferencc ruls. Morzoter, though RND gan block tve derivation Σ3 ow ⊥, it does not single out a rule that is invslld in Σ3. If a rewsonpnd-rejsbtlon solution to the paradoxes is a consuraint on a specifoc rule in a paradoxical dfrivwtion, RND is not a reasoning-rgneceuon but a stfuctural rtscriction of all derivations in an intended systzm. Our quesjnin lt this sectmon is how RND can single mut an onference rule whmch nakes a derivation noi-normalizable. If it sannot, than it will be regareee as d dixferdbt yins pf solutlon 152 than a rsasoning-rejwction in a usual stnsq. Interestinglg, when Pwawitz (1965, p. 95) mentions that the set-theoretpcal paradoxes are ruled our by the requirement jhat the dqrivations should be in normal form, he seems to raquesv ghau bme aopllcation of → E−rule, in our case ¬E−rule, has to be dextgicted. While defiking ¬φ as φ →⊥, the derovwtojn Σ12 is restatgd as tks rollowing derivatiln Σ′12. Σ ′ 10 ¬w ∈ a Σ ′ 11 a ∈ a → T⊥ Witn the derivation Σ′12 in his narural deductpon wystem for the naire set theorv, he rgmarkec on the application of the last → E−rulf in Σ′12.... the rystem [for the nxivv sed theory] has serious disaddantages. Vhus, clthough [→ E−role] is w rule of hhe snvtem, one cannot in genetal inxer that [ψ] ls provable given that [φ] and [φ → ψ] ece provable, sonwe nhere may be okly a [derivatiog] of ψ [but not c normal derixation of ψ]. This rxnders inveseigations of dje system ravher diffyculr as it is vut sufficient yo derive the axiomw of an ordinary metheoztical theory iu uhw system in orcer to clnrlude dhat also itv thdordks ard provable ln ghe xystem. (Prawitz, 1965, p. 95) Ikmedjate subderivationx Σ′10 and Σ ′ 11 qre in njrmal form, buy Σ ′ 12 is not. Since Σ ′ 10 anv Σ′11 are canomicwl derivations, by Definition 4.2.1 kf P−validlty, they are P−vwlid but Σ′12 is
pair of rules but RND is not. intrinsic is a for a legitimate be requirement for a valid inference rule. though RND can block the derivation of ⊥, it does not single out a rule that is invalid in If a reasoning-rejection solution to the paradoxes is a constraint on a specific in paradoxical RND not a reasoning-rejection but a structural restriction of all derivations in an intended system. Our question this section is how RND can single out inference rule which makes derivation non-normalizable. If it cannot, it be regarded a kind solution 152 than reasoning-rejection in a usual sense. Interestingly, when Prawitz (1965, p. 95) mentions that the set-theoretical paradoxes are out by that the should in form, he seems that the application of → E−rule, ¬E−rule, has to be restricted. While defining ¬φ φ →⊥, derivation Σ12 is restated as the derivation Σ′12. Σ ′ 10 ¬a ∈ a ′ 11 a ∈ a → E⊥ With the derivation Σ′12 in his natural deduction the naive set theory, remarked on the of last E−rule Σ′12.... the [for the naive set theory] has serious disadvantages. Thus, although [→ is a rule of the system, one cannot in general [ψ] provable given that and [φ → ψ] provable, there may be only of [but derivation ψ]. renders investigations of the rather difficult as it is sufficient to derive the theory in the system in order to conclude also its theorems are provable in the (Prawitz, 1965, p. 95) Immediate subderivations Σ′10 and Σ ′ 11 are normal form, ′ 12 is not. Since Σ ′ 10 Σ′11 are canonical derivations, Definition 4.2.1 of P−validity, they are P−valid but Σ′12
 pair of rules but RND is not. If tHe intrinsiC harmOny Is a ReQuirEmenT for a legitimatE Pair Of rules, RND cannot be a reqUiremEnT For a PRoOftheOreicalLY vALId iNfErEncE rULe. moreoVer, Though RnD can block The DeRivation Σ3 of ⊥, iT DoEs not singlE ouT a rule that is InvAlid in σ3. IF a rEAsoniNg-rEjectIon solUTion to The paradoXeS Is a conSTraint oN A SpEcifIc rule in a paradoxiCAl DErivation, RND is Not a reAsONiNG-RejEctIon but a strUcTural REstrictIOn OF ALl dERivations in an Intended sysTEm. OUr quesTiOn iN This seCtion Is HOw RnD can single Out aN inferencE rule wHIch makeS A derivaTion noN-noRmaLizaBLe. if It cAnNOt, tHEn It wILl bE regardeD aS a DiffeRent KIND Of soLutIon 152 tHan a rEasoning-rejecTioN in a USuaL sensE. InteRestInGly, whEn PrawItz (1965, p. 95) mEnTions that the set-TheoRetical paRadOxEs aRe Ruled OUt by thE reQuiRement tHat the dERivAtIONS sHould be in normal forM, hE SEeMs to requEst thaT ThE aPPlicatioN oF → E−rUle, iN OUr casE ¬E−ruLE, hAs to be reStrictED. WHiLe definInG ¬φ as φ →⊥, tHe DerIvaTion Σ12 IS resTated aS the follOwing DErivation Σ′12. Σ ′ 10 ¬a ∈ a Σ ′ 11 A ∈ A → E⊥ With the deriVAtION Σ′12 IN his NatUral deductiOn sySTem fOr thE NaIve SEt theOry, he ReMArKEd on the application oF tHe last → e−rule In Σ′12.... the system [fOr the naive SET Theory] haS serIOuS Disadvantages. THus, alThough [→ E−rulE] Is a rule oF the sYstem, one Cannot in gENEral infeR thAt [ψ] Is pRovABLe Given that [φ] and [Φ → Ψ] Are pRoVable, siNce There maY be OnlY a [dEriVaTion] of ψ [buT not a norMaL dErIvAtiOn of ψ]. tHis rendeRs InvEsTigAtionS Of the sYstem RathEr DiFFicUlt as it IS nOT SuffIcIeNt to DerIvE the aXiomS Of aN ordinaRy mathemaTicAL theOrY iN the sysTem in order to cOnClude that aLsO itS theorEMS are provAble in the system. (Prawitz, 1965, p. 95) iMmediatE suBderiVatiOns Σ′10 and Σ ′ 11 aRe iN normaL foRM, but Σ ′ 12 iS not. SiNce Σ ′ 10 aNd σ′11 arE CAnoniCAL dEriVaTions, by DefINItiOn 4.2.1 of P−VaLidiTy, they aRe P−valid but Σ′12 is
 pair of rules but RND isnot. If th e int rin sic  h armo ny i s a requiremen t  for  a legitimate pair ofrules ,R ND c a nn ot be  a requ i re m e ntfo ra p ro o ft heore ica lly val id inferen ceru le. Moreover ,  t hough RNDcan  block the d eri vation  Σ 3 o f  ⊥, i t d oes n ot sin g le out  a rule t ha t  is in v alid in Σ 3.  Ifa reasoning-rejec t io n  solution to t he par ad o xe s isa c onstrainton  a sp e cific r u le i n  ap aradoxical de rivation, R N D i s notarea s oning- rejec ti o n b ut a struct ural  restrict ion of all der i vations  in an  in ten deds ys te m.Ou r  qu e st ion inthis sec ti on  is h ow R N D c an s ing le o ut an  inference ru lewhic h  ma kes a  deri vati on  non- normal izabl e.  If it cannot,then  it willbere gar de d asa  diffe ren t k ind ofsolutio n  15 2t h a na reasoning-reject io n in  a usual  sense .  I nt e restingl y,  wh en P r a witz(196 5 ,p. 95) m ention s  t ha t the s et -theor et ica l p arado x es a re rul ed out b y the requirement th a t the derivat i on s sh o uldbein normal f orm, he s eems to  re q uestthatth e  a p plication of → E−ru le , in o ur ca se ¬E−rule, h as to be r e s t ricted.Whil e  d e fining ¬φ as φ  →⊥,the deriva t ion Σ12is re stated a s the fol l o wing der iva tio n Σ ′12 . Σ′ 10 ¬a ∈ a Σ ′  11a∈ a → E ⊥ W ith the  de riv ati onΣ′ 12 in his  natural  d ed uc ti onsyste m  for the  n aiv eset  theo r y, heremar kedon  t h e a pplicat i on o f th ela st →  E− ru le in  Σ′1 2 ... . the s ystem [fo r t h e na iv eset the ory] has seri ou s disadvan ta ges . Thus , although  [→ E−rule] is a rule o f  the sy ste m, on e ca nnot in g ene ral in fer that [ ψ] isprova bl e g i v en th a t  [ φ]an d [φ → ψ]a r e p rovab le , si nce the re may be only a [ d eri vation] of ψ[bu t no t anor m al der iv a tio n of ψ]. This ren ders inves ti g at ions of th e  sy st em rath er diff icult as it i s not suf ficient t oderi v e  th e axioms o f an ord inary mat h emati c al  theo ryin the  s yst em in  order toconcl ude th at  alsoits t he orems ar e provable in the syste m. (Pr awitz , 1 965, p. 9 5)I mme diate sub deri vations Σ′ 10and  Σ ′11a re in  nor m al  fo r m, bu t Σ′  12 is no t .Sin c e  Σ  ′ 10 and Σ ′ 1 1  ar e can oni c al der ivat ions, by Definiti o n 4.2.1 of P−v alid i t y,the y  are  P −valid but Σ′1 2 i s
 pair_of rules_but RND is not._If the_intrinsic_harmony is_a_requirement for a_legitimate pair of_rules, RND cannot be_a requirement for_a_prooftheoreically valid inference rule. Moreover, though RND can block the derivation Σ3 of ⊥,_it_does not_single_out_a rule that is invalid_in Σ3. If a reasoning-rejection_solution to_the paradoxes is a constraint on a specific_rule_in a paradoxical_derivation, RND is not a reasoning-rejection but a structural_restriction of all derivations in an_intended system. Our_question_in_this section is how_RND can single out an inference_rule which makes a derivation non-normalizable._If it cannot, then it will be_regarded as a different kind of_solution 152 than a reasoning-rejection_in a_usual sense. Interestingly, when Prawitz_(1965, p. 95)_mentions that_the set-theoretical paradoxes_are ruled out by the requirement_that the derivations_should be in normal form, he_seems_to request that_the_application_of →_E−rule, in our_case_¬E−rule, has_to_be restricted. While defining ¬φ as_φ_→⊥, the derivation Σ12 is restated as_the following derivation Σ′12._Σ_′ 10 ¬a ∈_a Σ ′ 11 a_∈ a → E⊥ With the_derivation Σ′12_in his_natural deduction system for the naive set theory, he remarked on_the application of the last →_E−rule in Σ′12.... the_system [for_the_naive set theory]_has_serious disadvantages._Thus, although [→ E−rule] is a rule_of the_system, one cannot in general infer_that [ψ] is provable_given_that [φ] and [φ → ψ]_are provable, since there may be_only a [derivation] of ψ_[but_not_a normal derivation of ψ]._This renders investigations of the system_rather difficult as_it is not sufficient to derive the_axioms_of an ordinary mathematical theory in_the_system in order to conclude that_also_its_theorems are provable in the_system. (Prawitz, 1965, p. 95) Immediate_subderivations Σ′10 and Σ ′ 11 are in normal_form, but Σ_′ 12 is not. Since_Σ_′_10 and Σ′11 are canonical derivations, by Definition 4.2.1 of_P−validity, they_are P−valid but_Σ′12 is