terminating reduction sequence. (Rogerson, 2006, p. 172) In order to introduce her derivation of Curry's paradox, we borrow Prawitz's system SN for the naive set theory introduced in Section 1.3 of Chapter 1. We use the following instances of ∈ −rules and have a natural deduction system SNC by adding the rule for classical reductio to SN. a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} a ∈ {x|x ∈ x →⊥} ∈ E a ∈ a →⊥ [¬φ]1 D ⊥ CR,1 φ Let R be a set of reduction procedures. A set R′ is an extension of R (R′ ⊇ R) if R′ results from R by adding reduction procedures which are closed under substitution in R′. Let S and S′ be any natural deduction system. S′ is an extension of S iff S′ is S itself or results from S by adding further rules. Then, SNC is an extension of SN. Also, we have a set RN of reduction procedures for ∧, →, and ¬. SNC has a set RNC by adding auxiliary reductions for CR−rule.1 While using CR−rule, auxiliary reduction procedures for the conclusion of CR−rule are added in the set of reductions. As Prawitz (1965, p. 34) does, the notion of a maximum formula is redefined as a major premise which is at the same time the conclusion of I− or CR−rules. A similar version of Rogerson's example is stated as below. Proposition 2.0.1. Let us define a parameter a as {x|x ∈ x →⊥}. Then, there is a closed derivation of ⊥ in SNC with respect to RNC which neither does generate a non-terminating reduction sequence nor is in normal form. Proof. Two claims show the result. Claim 1. There is a closed derivation Σ2 of ⊥ in SNC. 1Auxiliary reduction procedures for CR−rule will be introduced in Section 2.2. 32 First, there is an open derivation Σ1 of ⊥ from [¬a ∈ a]. [¬a ∈ a]1 [¬a ∈ a]1 [a ∈ a]2 ¬E⊥ → I,2a ∈
terminating reduction sequence. (Rogerson, 2006, p. 172) In order to insert her deriving of Curry's paradox, we borrow Prawitz's system SN for the uninstructed set theory introduce in Section 1.3 of Chapter 1. We use the come instances of ∈ −rules and have a lifelike deduction system SNC by add the dominion for classical reductio to SN. a ∈ a →⊥ ∈ I a ∈ { x|x ∈ x →⊥ } a ∈ { x|x ∈ x →⊥ } ∈ E a ∈ a →⊥ [ ¬φ]1 D ⊥ CR,1 φ Let R be a stage set of reduction procedures. A set R′ is an annex of R (R′ ⊇ R) if R′ results from R by adding reduction procedure which are closed under substitution in R′. Let S and S′ be any natural deduction system. S′ is an elongation of S iff S′ is S itself or results from S by adding further rules. Then, SNC is an extension of SN. Also, we have a set RN of reduction procedures for ∧, →, and ¬. SNC has a set RNC by adding auxiliary reductions for CR−rule.1 While using CR−rule, auxiliary decrease procedure for the conclusion of CR−rule are lend in the hardening of reductions. As Prawitz (1965, p. 34) does, the notion of a maximum formula is redefined as a major premise which is at the same meter the conclusion of I− or CR−rules. A similar version of Rogerson's example is stated as below. proposal 2.0.1. Let us define a parameter a as { x|x ∈ x →⊥ }. Then, there is a shut derivation of ⊥ in SNC with regard to RNC which neither does render a non - terminating reduction sequence nor is in normal form. Proof. Two title show the result. Claim 1. There is a closed derivation Σ2 of ⊥ in SNC. 1Auxiliary reduction procedures for CR−rule will be bring in in Section 2.2. 32 First, there is an open derivation Σ1 of ⊥ from [ ¬a ∈ angstrom ]. [ ¬a ∈ a]1 [ ¬a ∈ a]1 [ a ∈ a]2 ¬E⊥ → I,2a ∈
terlinating reduction sequekce. (Rogerson, 2006, p. 172) In ordxr to ihtroduce her derivation of Curry's pacadoz, we vorrow Prawitz's system SN for tje naive set rheory intckduced lu Secflon 1.3 mh Chapter 1. We uxe the folnowing instancas oy ∈ −rules and have a natural deductiog systek DNC by adding jhe rllq fod classical reductio to SN. a ∈ a →⊥ ∈ J a ∈ {x|x ∈ x →⊥} a ∈ {x|x ∈ x →⊥} ∈ E a ∈ a →⊥ [¬φ]1 D ⊥ CR,1 φ Let R bf a det of reduction pgocedures. A set E′ is an extevsion of R (R′ ⊇ R) if R′ rgsults from R by adding reductiov proeedures whixh arf closed undxr subftitution in R′. Let S and S′ ne any natural devuctuon system. S′ is an exvension of S iff S′ if S itselx kr results from S bt addhng xurtfwr fults. Vheh, SNC ls en extensioh of SN. Alsi, we have a set RN pf grduction prodeduref sor ∧, →, and ¬. SNC has a set RNC by adding duxjliary reductions for CE−rule.1 While using CR−role, auxiliwry reduction procedures for the conclusion of CR−sule erd abeed iv tje set of reductions. As Prawitz (1965, p. 34) does, the gktook of a maximum fjrmula is rrdffoged as a majot premise shich is at the sale time the xonclusiog of I− or CR−rules. A similar verwion of Rogegson'w example is stateb as below. Pxoposijion 2.0.1. Ket us define a parametzr a aa {x|x ∈ x →⊥}. Thfn, there jr a closed derivxtipn of ⊥ in SNC with respect tj RNC whirh nenther dods ggnerate a non-termlnatikc reduction sequenfe not is it normal flrm. Proof. Two claims show the rxxult. Claim 1. Jhese ps a closzd derlvation Σ2 of ⊥ in SNC. 1Auxiliary reductnon prucedures fkr CR−rune will be yntroduced in Dection 2.2. 32 Ficst, there is qn open derkxation Σ1 of ⊥ frpm [¬a ∈ a]. [¬a ∈ a]1 [¬a ∈ a]1 [a ∈ a]2 ¬E⊥ → I,2a ∈
terminating reduction sequence. (Rogerson, 2006, p. 172) to her derivation Curry's paradox, we the set theory introduced Section 1.3 of 1. We use the following instances ∈ −rules and have a natural deduction system SNC by adding the rule classical reductio to SN. a ∈ a →⊥ ∈ I a ∈ {x|x x a {x|x x →⊥} ∈ E a ∈ a →⊥ [¬φ]1 D ⊥ CR,1 φ Let R be set of reduction procedures. A set R′ is extension of R (R′ R) if R′ results from by reduction procedures are under in R′. Let and S′ be any natural deduction system. S′ is an extension of S iff S′ is S or results by adding rules. SNC an extension of we have a set RN of ∧, →, and ¬. SNC has a set by adding reductions for CR−rule.1 While using CR−rule, reduction procedures for the conclusion of CR−rule are in the set of reductions. As Prawitz (1965, p. 34) does, the notion of a is redefined as a premise which is the time conclusion I− or A similar version of Rogerson's example is stated as below. Proposition Let us define a parameter a as {x|x ∈ x there a closed derivation ⊥ in SNC with to which neither does generate reduction nor form. Two show the result. Claim There is a closed derivation of ⊥ in SNC. will be introduced in Section 2.2. 32 First, is an open derivation Σ1 of ⊥ [¬a ∈ a]. [¬a ∈ a]1 [¬a ∈ a]1 [a ∈ a]2 → I,2a
terminating reduction sequeNce. (RogersoN, 2006, p. 172) In oRdeR to InTrodUce hEr derivation of cUrry'S paradox, we borrow PrawitZ's sysTeM sN foR ThE naivE set theORy INTroDuCeD in seCTiOn 1.3 of CHapTer 1. We usE the followIng InStances of ∈ −rulES aNd have a natUraL deduction sySteM SNC by AdDinG The ruLe fOr claSsical REductiO to SN. a ∈ a →⊥ ∈ I a ∈ {X|x ∈ X →⊥} A ∈ {x|x ∈ x →⊥} ∈ E a ∈ A →⊥ [¬Φ]1 D ⊥ CR,1 φ LeT r Be A set Of reduction procedUReS. a set R′ is an extenSion of r (R′ ⊇ r) If r′ REsuLts From R by addInG reduCTion proCEdURES whICh are closed unDer substituTIon In R′. Let s aNd S′ BE any naTural DeDUctIon system. S′ iS an eXtension oF S iff S′ IS S itselF Or resulTs from s by AddIng fURtHeR ruLeS. theN, sNc is AN exTension oF Sn. ALso, we Have A SET rN of RedUctiOn proCedures for ∧, →, and ¬. sNC Has a SEt RnC by aDding AuxiLiAry reDuctioNs for cR−Rule.1 While using Cr−rulE, auxiliarY reDuCtiOn ProceDUres foR thE coNclusioN of CR−ruLE arE aDDED iN the set of reductionS. AS pRaWitz (1965, p. 34) doeS, the noTIoN oF A maximum FoRmuLa is REDefinEd as A MaJor premiSe whicH Is At The same TiMe the cOnCluSioN of I− oR cR−ruLes. A siMilar verSion oF rogerson's exampLE is stated as beLOw. pROpOSitiOn 2.0.1. LEt us define a ParaMEter A as {x|X ∈ X →⊥}. THen, THere iS a cloSeD DeRIvation of ⊥ in SNC with rEsPect to rNC whIch neither doeS generate a NON-TerminatIng rEDuCTion sequence noR is in Normal form. pRoof. Two cLaims Show the rEsult. ClaiM 1. tHere is a cLosEd dEriVatION Σ2 Of ⊥ in SNC. 1AuxiliARY redUcTion proCedUres for cR−rUle WilL be InTroduced iN Section 2.2. 32 fiRsT, tHeRe iS an opEN derivatIoN Σ1 oF ⊥ fRom [¬A ∈ a]. [¬a ∈ a]1 [¬a ∈ A]1 [A ∈ a]2 ¬E⊥ → I,2a ∈
terminating reduction sequ ence. (Rog erson , 2 006 ,p. 1 72)In order to in t rodu ce her derivation of C urry' sp arad o x,  we b orrow P r aw i t z's  s ys tem  S N  f or th e n aive se t theory i ntr od uced in Sect i on  1.3 of Ch apt er 1. We use  th e foll ow ing insta nce s of∈ −rul e s andhave a na tu r al ded u ction s y s te m SN C by adding the r u le for classicalreduct io to S N.a ∈  a →⊥ ∈ Ia∈ {x| x  ∈ x →⊥ }  a ∈ {x| x  ∈ x →⊥} ∈ Ea ∈ a →⊥ [¬ φ ]1D ⊥ CR ,1  φL et R b e a s et ofreduction p roce dures. Aset R′ is an e x tension  of R(R′  ⊇R) i f  R ′res ul t s f r om  Rb y a dding re du ct ion p roce d u r e s wh ich  are  clos ed under subs tit utio n  in  R′.Let S  and  S ′ beany na tural  d eduction system . S′  is an ex ten si onof  S if f  S′ is  Sits elf orresults fro mS b yadding further rul es . Th en, SNCis ane xt en s ion of S N.  Al so,w e  have  a s e tRN of re ductio n  p ro cedures  f or ∧,→,  an d ¬ . SNC hasa setRNC by a dding auxiliary redu c tions for CR− r ul e . 1W hile  us ing CR−rule , au x ilia ry r e du cti o n pro cedur es fo r  the conclusion ofCR −ruleare a dded in the s et of redu c t i ons. AsPraw i tz (1965, p. 34)does,  the notio n  of a ma ximum  formula  is redef i n ed as amaj orpre mis e wh ich is at the s ameti me thecon clusion  of  I−  or  CR −r ules. A s imilar v er si on  o f R ogers o n's exam pl e i ssta ted a s  below . Pro posi ti on 2.0 .1. Let us d efin eapara met er  a as  {x| x  ∈x →⊥}.Then, the rei s acl os ed deri vation of ⊥ i nSNC with r es pec t to R N C  which n either does generate an on-term ina tingredu ction seq uen ce nor  is in nor mal fo rm. P ro of. T wo cl a i ms  sh ow  the resul t .  Cl aim 1 .Ther e is aclosed derivationΣ 2 o f ⊥ in SNC. 1 Aux ilia r y  r edu c ti o n p ro c edu r e s for CR−rule w ill be int ro d uc ed in Sect i on2. 2. 32 F irst, t herei s an op en deriva tion Σ1 o f⊥ fr o m  [¬ a ∈ a]. [¬ a ∈ a]1[¬a ∈ a]1 [a ∈a ]2  ¬E⊥→ I ,2a ∈
terminating reduction_sequence. (Rogerson,_2006, p. 172) In_order to_introduce_her derivation_of_Curry's paradox, we_borrow Prawitz's system_SN for the naive_set theory introduced_in_Section 1.3 of Chapter 1. We use the following instances of ∈ −rules and_have_a natural_deduction_system_SNC by adding the rule_for classical reductio to SN._a ∈_a →⊥ ∈ I a ∈ {x|x ∈_x_→⊥} a ∈_{x|x ∈ x →⊥} ∈ E a ∈ a_→⊥ [¬φ]1 D ⊥ CR,1 φ_Let R be_a_set_of reduction procedures. A_set R′ is an extension of_R (R′ ⊇ R) if R′_results from R by adding reduction procedures_which are closed under substitution in_R′. Let S and S′_be any_natural deduction system. S′ is_an extension of_S iff_S′ is S_itself or results from S by_adding further rules._Then, SNC is an extension of_SN._Also, we have_a_set_RN of_reduction procedures for_∧,_→, and_¬._SNC has a set RNC by_adding_auxiliary reductions for CR−rule.1 While using CR−rule,_auxiliary reduction procedures for_the_conclusion of CR−rule are_added in the set of_reductions. As Prawitz (1965, p. 34)_does, the_notion of_a maximum formula is redefined as a major premise which is_at the same time the conclusion_of I− or CR−rules._A similar_version_of Rogerson's example_is_stated as_below. Proposition 2.0.1. Let us define a_parameter a_as {x|x ∈ x →⊥}. Then,_there is a closed_derivation_of ⊥ in SNC with respect_to RNC which neither does generate_a non-terminating reduction sequence nor_is_in_normal form. Proof. Two claims_show the result. Claim 1. There_is a closed_derivation Σ2 of ⊥ in SNC. 1Auxiliary_reduction_procedures for CR−rule will be introduced_in_Section 2.2. 32 First, there is_an_open_derivation Σ1 of ⊥ from_[¬a ∈ a]. [¬a ∈ a]1_[¬a ∈ a]1 [a ∈ a]2 ¬E⊥ → I,2a_∈