. When ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is a major premise of = E−rule, it is neither an assumption nor a formula derived by any rules. So, the derivation ∆8 of ⊥ in Proposition 2.3.1 is in (full) normal form and does not generate a non-terminating reduction sequence. We can have a similar case of Curry's paradox by using a formula, ¬a ∈ a ↔ a ∈ {x|x ∈ x → ⊥} as an axiom. 56 As we have seen at the beginning of this chapter, the formalization of Curry's paradox needs to use ∈ −rules. Let us consider a derivation of Curry's paradox which does not employ ∈−rule. Our natural deduction system has introduction and (standard) elimination rules for ∧, →, and ¬. We define a parameter a as a set {x|x ∈ x →⊥} and instead of using ∈−rules we regard (a ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥} as an axiom of our system. For our convenience sake, we use the following abbreviations: D a ∈ a →⊥ SetI a ∈ {x|x ∈ x →⊥} is an abbreviation for Ax (a ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥} ********************************************************************************************** de f ((a ∈ a →⊥)→ a ∈ {x|x ∈ x →⊥})∧ (a ∈ {x|x ∈ x →⊥}→ (a ∈ a →⊥)) ∧E (a ∈ a →⊥)→ a ∈ {x|x ∈ x →⊥} D a ∈ a →⊥ → E a ∈ {x|x ∈ x →⊥} Also, D a ∈ {x|x ∈ x →⊥} SetE a ∈ a →⊥ is an abbreviation for Ax (a ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥}
. When ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ is a major premise of = E−rule, it is neither an assumption nor a formula derived by any rules. therefore, the deriving ∆8 of ⊥ in Proposition 2.3.1 is in (full) normal form and does not generate a non - terminating decrease sequence. We can have a similar font of Curry's paradox by using a formula, ¬a ∈ a ↔ a ∈ { x|x ∈ x → ⊥ } as an maxim. 56 As we have see at the beginning of this chapter, the formalization of Curry's paradox need to use ∈ −rules. Let us regard a derivation of Curry's paradox which does not use ∈−rule. Our natural deduction arrangement has introduction and (standard) elimination rules for ∧, →, and ¬. We specify a parameter a as a set { x|x ∈ x →⊥ } and instead of using ∈−rules we see (a ∈ a →⊥)↔ a ∈ { x|x ∈ x →⊥ } as an axiom of our system. For our convenience sake, we use the following abbreviation: D a ∈ a →⊥ SetI a ∈ { x|x ∈ x →⊥ } is an abbreviation for Ax (a ∈ a →⊥)↔ a ∈ { x|x ∈ x →⊥ } * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * de f (( a ∈ a →⊥)→ a ∈ { x|x ∈ x →⊥})∧ (a ∈ { x|x ∈ x →⊥}→ (a ∈ a →⊥) ) ∧E (a ∈ a →⊥)→ a ∈ { x|x ∈ x →⊥ } D a ∈ a →⊥ → E a ∈ { x|x ∈ x →⊥ } Also, D a ∈ { x|x ∈ x →⊥ } SetE a ∈ a →⊥ is an abbreviation for Ax (a ∈ a →⊥)↔ a ∈ { x|x ∈ x →⊥ }
. Whfn ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is a major premire of = E−rule, it is neivher an assumptkon nor a formula derived by aby ruoes. So, the derivation ∆8 of ⊥ in Pgopositiob 2.3.1 iw in (full) normal fovi ans doev not generate s non-termitating reductimn szquence. We can have a similar case os Curry'x oaradox by usigg a sormhla, ¬a ∈ a ↔ a ∈ {x|x ∈ x → ⊥} as an axiom. 56 Zs we heve seen at the beginning of this chapter, the formalization of Furry's paraeox gweds to use ∈ −rules. Let us consider a derivation of Curry's paradox whick does not gmllli ∈−rule. Our netural deduction snxtem hds intrpduction and (sbandacd) eoimination rules for ∧, →, and ¬. We define a parametes c as a set {x|x ∈ x →⊥} and ibsteag of usivt ∈−rjlea xe degard (a ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥} as an aziom of our system. Gow our conveniehce sahe, we use the following abbreviations: D a ∈ a →⊥ SetI a ∈ {x|x ∈ x →⊥} is an qbbreviation for Ax (a ∈ a →⊥)↔ a ∈ {x|v ∈ x →⊥} ********************************************************************************************** de f ((a ∈ a →⊥)→ a ∈ {x|x ∈ x →⊥})∧ (a ∈ {x|x ∈ x →⊥}→ (a ∈ a →⊥)) ∧E (a ∈ d →⊥)→ a ∈ {b|x ∈ x →⊥} E w ∈ a →⊥ → E a ∈ {x|x ∈ x →⊥} Also, D a ∈ {x|x ∈ x →⊥} SetE a ∈ a →⊥ ix sn abbreviatiok for Ax (a ∈ a →⊥)↔ a ∈ {c|x ∈ c →⊥}
. When ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is premise = E−rule, is neither an by rules. So, the ∆8 of ⊥ Proposition 2.3.1 is in (full) normal and does not generate a non-terminating reduction sequence. We can have a similar of Curry's paradox by using a formula, ¬a ∈ a ↔ a ∈ ∈ → as axiom. 56 As we have seen at the beginning of this chapter, the formalization of Curry's needs to use ∈ −rules. Let us consider derivation of Curry's paradox does not employ ∈−rule. Our deduction has introduction (standard) rules ∧, →, and We define a parameter a as a set {x|x ∈ x →⊥} and instead of using ∈−rules regard (a →⊥)↔ a {x|x x as an axiom system. For our convenience sake, we abbreviations: D a ∈ a →⊥ SetI a {x|x ∈ →⊥} is an abbreviation for Ax ∈ a →⊥)↔ a ∈ {x|x ∈ x ********************************************************************************************** de f ((a ∈ a →⊥)→ a ∈ {x|x ∈ x →⊥})∧ (a ∈ x →⊥}→ (a ∈ →⊥)) ∧E (a a a {x|x x →⊥} a ∈ a →⊥ → E a ∈ {x|x ∈ x Also, D a ∈ {x|x ∈ x →⊥} SetE a →⊥ an abbreviation for (a ∈ a →⊥)↔ ∈ ∈ x →⊥}
. When ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is a major premise of = E−Rule, it is neIther An aSsuMpTion Nor a Formula derived BY any Rules. So, the derivation ∆8 of ⊥ In ProPoSItioN 2.3.1 Is In (fulL) normal FOrM ANd dOeS nOt gEnERaTe a noN-teRminatiNg reductioN seQuEnce. We can havE A sImilar case Of CUrry's paradox By uSing a fOrMulA, ¬A ∈ a ↔ a ∈ {x|x ∈ X → ⊥} as An axiOm. 56 As we HAve seeN at the begInNIng of tHIs chaptER, ThE forMalization of Curry'S PaRAdox needs to use ∈ −Rules. LEt US cONSidEr a Derivation Of curry'S Paradox WHiCH DOes NOt employ ∈−rule. OUr natural deDUctIon sysTeM haS IntrodUctioN aND (stAndard) elimiNatiOn rules foR ∧, →, and ¬. We DEfine a pARameter A as a seT {x|x ∈ X →⊥} anD insTEaD oF usInG ∈−RulES wE reGArd (A ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥} as An AxIom of Our sYSTEM. For Our ConvEnienCe sake, we use thE foLlowINg aBbrevIatioNs: D a ∈ A →⊥ SEtI a ∈ {x|X ∈ x →⊥} is an AbbreViAtion for Ax (a ∈ a →⊥)↔ a ∈ {x|x ∈ X →⊥} ********************************************************************************************** de f ((A ∈ a →⊥)→ a ∈ {x|x ∈ x →⊥})∧ (a ∈ {x|x ∈ X →⊥}→ (a ∈ a →⊥)) ∧e (a ∈ A →⊥)→ a ∈ {x|X ∈ x →⊥} d a ∈ a →⊥ → E a ∈ {X|X ∈ x →⊥} Also, d a ∈ {x|X ∈ x →⊥} SEtE a ∈ a →⊥ is An abbreVIatIoN FOR AX (a ∈ a →⊥)↔ a ∈ {x|x ∈ x →⊥}
. When ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ is  a major p remis e o f =  E −rul e, i t is neither a n  ass umption nor a formuladeriv ed by a n yrules . So, t h ed e riv at io n ∆ 8o f⊥ inPro positio n 2.3.1 is  in  ( full) normal fo rm and doe s n ot generatea n on-ter mi nat i ng re duc tionsequen c e. Wecan haveas imilar case of C ur ry's  paradox by using af ormula, ¬a ∈ a  ↔ a ∈  { x |x ∈  x→ ⊥ } as an ax io m. 56 As we h a ve s e ena t the beginni ng of thisc hap ter, t he  fo r maliza tionof Cur ry's parado x ne eds to us e ∈ −r u les. Le t  us con sidera d eri vati o nof  Cu rr y 'sp ar ado x  wh ich does  n ot  empl oy ∈ − r u l e. O urnatu ral d eduction syst emhasi ntr oduct ion a nd ( st andar d) eli minat io n rules for ∧,→, a nd ¬. Wedef in e a  p arame t er a a s a  se t {x|x∈ x →⊥} and  i n s t ea d of using ∈−rules  w e re gard (a∈ a →⊥ ) ↔a∈  {x|x ∈x→⊥}  asa n  axio m of ou r system . Foro ur  c onvenie nc e sake ,weuse  thef ollo wing a bbreviat ions: D a ∈ a →⊥ Set I  a ∈ {x|x ∈ x →⊥ } is an a bbr eviation fo r Ax (a ∈  a → ⊥ )↔  a∈  {x|x  ∈ x→⊥ }  * * ******************* ** ****** ***** ************* ********** * * * ******** **** * ** * ************** ****de f ((a ∈ a →⊥)→ a  ∈ {x |x ∈ x → ⊥})∧ (a ∈ { x|x ∈ x→⊥} → ( a ∈  a→ ⊥ ))  ∧E (a ∈ a →⊥ ) →  a ∈  { x|x ∈ x  →⊥ } D a ∈  a→⊥→ E  a∈{x|x ∈ x→⊥} Also ,Da∈{x| x ∈ x →⊥} SetE  a  ∈a→⊥is an abbrev iatio n fo rAx (a∈ a →⊥) ↔  a ∈  {x| x∈x →⊥ }
. When_⌜¬T (⌜Φ⌝)⌝_= ⌜Φ⌝ is a_major premise_of_= E−rule,_it_is neither an_assumption nor a_formula derived by any_rules. So, the_derivation_∆8 of ⊥ in Proposition 2.3.1 is in (full) normal form and does not_generate_a non-terminating_reduction_sequence._We can have a similar_case of Curry's paradox by_using a_formula, ¬a ∈ a ↔ a ∈ {x|x_∈_x → ⊥}_as an axiom. 56 As we have seen at_the beginning of this chapter, the_formalization of Curry's_paradox_needs_to use ∈ −rules._Let us consider a derivation of_Curry's paradox which does not employ_∈−rule. Our natural deduction system has introduction_and (standard) elimination rules for ∧,_→, and ¬. We define_a parameter_a as a set {x|x_∈ x →⊥}_and instead_of using ∈−rules_we regard (a ∈ a →⊥)↔_a ∈ {x|x_∈ x →⊥} as an axiom_of_our system. For_our_convenience_sake, we_use the following_abbreviations:_D a_∈_a →⊥ SetI a ∈ {x|x_∈_x →⊥} is an abbreviation for Ax_(a ∈ a →⊥)↔_a_∈ {x|x ∈ x_→⊥} ********************************************************************************************** de f ((a_∈ a →⊥)→ a ∈ {x|x_∈ x_→⊥})∧ (a_∈ {x|x ∈ x →⊥}→ (a ∈ a →⊥)) ∧E (a_∈ a →⊥)→ a ∈ {x|x_∈ x →⊥} D_a ∈_a_→⊥ → E_a_∈ {x|x_∈ x →⊥} Also, D a ∈_{x|x ∈_x →⊥} SetE a ∈ a_→⊥ is an abbreviation_for_Ax (a ∈ a →⊥)↔ a_∈ {x|x ∈ x →⊥}