ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D3 φ ⌜⌝E1 ψ ▷⌜⌝1 D3 φ D1 ψ [φ]1 D1 ψ [ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D4 ψ ⌜⌝E2 φ ▷⌜⌝2 D4 ψ D2 φ When we presume that a liar sentence Φ which is defined by ¬T (⌜Φ⌝) is expressible in our language, there is a derivation of the Liar paradox using the formula ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ which raises a non-terminating reduction sequence. Proposition 2.4.4. Let SL be a system containing T−, ¬−, and ⌜⌝−rules. SL has a set RL of standard reduction procedures for T (x), ¬, and ⌜⌝. Let us define a formula Φ as ¬T (⌜Φ⌝). Then, there exists a closed derivation Π11 of ⊥ in SL with respect to RL which generates a non-terminating reduction sequence, and so is not normalizable. Proof. Two claims establish the result. Claim 1. There is a closed derivation Π11 of ⊥ in SL with respect to RL. First, there is an open derivation Π9 of ⊥ from T (⌜Φ⌝). [Φ]1 ************* de f ¬T (⌜Φ⌝) [¬T (⌜Φ⌝)]2 **************** de f Φ ⌜⌝I,1,2⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]3 T E Φ ⌜⌝E1¬T (⌜Φ⌝) [T (⌜Φ⌝)]3 ¬
ψ]2 D2 φ ⌜ ⌝ I,1,2 ⌜ φ ⌝ = ⌜ ψ ⌝ D3 φ ⌜ ⌝ E1 ψ ▷ ⌜ ⌝ 1 D3 φ D1 ψ [ φ]1 D1 ψ [ ψ]2 D2 φ ⌜ ⌝ I,1,2 ⌜ φ ⌝ = ⌜ ψ ⌝ D4 ψ ⌜ ⌝ E2 φ ▷ ⌜ ⌝ 2 D4 ψ D2 φ When we presume that a liar sentence Φ which is defined by ¬T (⌜ Φ ⌝) is expressible in our language, there be a deriving of the Liar paradox using the formula ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ which raises a non - terminating decrease sequence. Proposition 2.4.4. Let SL be a arrangement containing T−, ¬−, and ⌜ ⌝ −rules. SL has a fit RL of standard reduction procedures for T (x), ¬, and ⌜ ⌝. lease us define a formula Φ as ¬T (⌜ Φ ⌝). Then, there exist a closed derivation Π11 of ⊥ in SL with respect to RL which generates a non - terminating decrease sequence, and so is not normalizable. Proof. Two claim establish the result. title 1. There is a shut derivation Π11 of ⊥ in shining path with respect to RL. First, there is an open ancestry Π9 of ⊥ from T (⌜ Φ ⌝). [ Φ]1 * * * * * * * * * * * * * de f ¬T (⌜ Φ ⌝) [ ¬T (⌜ Φ ⌝ )]2 * * * * * * * * * * * * * * * * de f Φ ⌜ ⌝ I,1,2 ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ [ T (⌜ Φ ⌝ )]3 T E Φ ⌜ ⌝ E1¬T (⌜ Φ ⌝) [ T (⌜ Φ ⌝ )]3 ¬
ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D3 φ ⌜⌝E1 ψ ▷⌜⌝1 D3 φ D1 ψ [φ]1 D1 ψ [ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D4 ψ ⌜⌝E2 φ ▷⌜⌝2 D4 ψ D2 φ When xe preshme that a liar sentence Φ which is dxfinwd by ¬T (⌜Φ⌝) is expressible in uur languwge, therw is q derivation of thc Liad parcdix using the fprmula ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ which raises a nln-terminating reduction sequence. Prjpositipn 2.4.4. Let SL be a fysttm sontzpnlng T−, ¬−, and ⌜⌝−rules. SL has a set RL of staidard reduction procedures for T (x), ¬, and ⌜⌝. Pet ks define a formulw Φ as ¬T (⌜Φ⌝). Thgh, trwre exists a closed degnvation Π11 of ⊥ in SL with respect to RL whicf genzrates a nob-twrmltating redurtion fequence, and so is nmt normslizable. Proof. Twm coaims establish the rxsult. Claim 1. There if a closeg berivation Π11 of ⊥ in SL wuth rgspecd to EL. Wirat, tgere id ai open deribation Π9 of ⊥ from T (⌜Φ⌝). [Φ]1 ************* de f ¬T (⌜Φ⌝) [¬U (⌜Φ⌝)]2 **************** ee f Φ ⌜⌝I,1,2⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]3 T E Φ ⌜⌝Q1¬T (⌜Φ⌝) [T (⌜Φ⌝)]3 ¬
ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D3 φ ▷⌜⌝1 φ D1 [φ]1 D1 ψ D4 ⌜⌝E2 φ ▷⌜⌝2 ψ D2 φ we presume that a liar sentence which is defined by ¬T (⌜Φ⌝) is expressible in our language, there is derivation of the Liar paradox using the formula ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ which a reduction Proposition Let SL be a system containing T−, ¬−, and ⌜⌝−rules. SL has a set RL of reduction procedures for T (x), ¬, and ⌜⌝. us define a formula as ¬T (⌜Φ⌝). Then, there a derivation Π11 ⊥ SL respect to RL generates a non-terminating reduction sequence, and so is not normalizable. Proof. Two claims establish the result. Claim There is derivation Π11 ⊥ SL respect to RL. is an open derivation Π9 of (⌜Φ⌝). [Φ]1 ************* de f ¬T (⌜Φ⌝) [¬T **************** de Φ ⌜⌝I,1,2⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]3 E Φ ⌜⌝E1¬T (⌜Φ⌝) [T (⌜Φ⌝)]3 ¬
ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D3 φ ⌜⌝E1 ψ ▷⌜⌝1 D3 φ D1 ψ [φ]1 D1 ψ [ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D4 ψ ⌜⌝e2 φ ▷⌜⌝2 D4 ψ D2 φ When We preSumE thAt A liaR senTence Φ which is dEFineD by ¬T (⌜Φ⌝) is expressible in ouR langUaGE, theRE iS a derIvation OF tHE liaR pArAdoX uSInG the fOrmUla ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ wHich raises A noN-tErminating reDUcTion sequenCe. PRoposition 2.4.4. LeT SL Be a sysTeM coNTainiNg T−, ¬−, And ⌜⌝−ruLes. SL hAS a set Rl of standaRd REductiON procedUREs For T (X), ¬, and ⌜⌝. Let us define a fORmULa Φ as ¬T (⌜Φ⌝). Then, theRe exisTs A ClOSEd dEriVation Π11 of ⊥ iN Sl with REspect tO rL WHICh gENerates a non-teRminating reDUctIon seqUeNce, ANd so is Not noRmALizAble. Proof. TwO claIms establIsh the REsult. ClAIm 1. There Is a cloSed DerIvatIOn π11 oF ⊥ in sL WIth REsPecT To Rl. First, thErE iS an opEn deRIVATion π9 of ⊥ From t (⌜Φ⌝). [Φ]1 ************* de F ¬T (⌜Φ⌝) [¬T (⌜Φ⌝)]2 **************** de f Φ ⌜⌝I,1,2⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [t (⌜Φ⌝)]3 T e Φ ⌜⌝E1¬T (⌜φ⌝) [t (⌜Φ⌝)]3 ¬
ψ]2 D2 φ ⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D 3 φ ⌜⌝E1 ψ  ▷⌜⌝1  D3  φD1  ψ [ φ]1D1 ψ [ψ]2 D2 φ ⌜⌝I, 1,2⌜φ⌝= ⌜ψ⌝ D4 ψ ⌜⌝E2φ ▷⌜⌝ 2D 4 ψD 2φ Whe n we pr e su m e  th at  a  li ar se ntenc e Φ  whichis defined  by  ¬ T (⌜Φ⌝) is e x pr essible in  ou r language,the re isader i vatio n o f the  Liarp aradox  using th ef ormula ⌜¬T (⌜Φ ⌝ ) ⌝= ⌜Φ ⌝ which raises an on - terminating re ductio ns eq u e nce . P roposition  2 .4.4. Let SLb ea s yst e m containingT−, ¬−, and ⌜⌝− rules.  S L h a s a se t RLof sta ndard reduc tion  procedur es for T (x),¬ , and ⌜ ⌝. Let  us  de fine afo rmu la Φ a s  ¬ T ( ⌜ Φ⌝) . Then,th er e exi stsa c l osed  de riva tionΠ11 of ⊥ in S L w ithr esp ect t o RLwhic hgener ates a  non- te rminating reduc tion  sequence , a nd  so  i s not normal iza ble . Proof . Two c l aim se s t ab lish the result. C la i m  1 . Thereis a c l os ed derivati on  Π1 1 of ⊥  in S L wi t hrespectto RL. Fi rs t, ther eis anop ender ivati o n Π9  of ⊥from T ( ⌜Φ⌝). [Φ]1 ********* * *** de f ¬T ( ⌜ Φ⌝ ) [¬ T  (⌜Φ ⌝)] 2 ********* **** * ** d e fΦ  ⌜ ⌝I, 1 ,2⌜Φ⌝ = ⌜¬T  ( ⌜ Φ⌝ ) ⌝ [T (⌜Φ⌝)]3 T E Φ⌜⌝ E1¬T ( ⌜Φ⌝)[T (⌜Φ⌝)]3 ¬
ψ]2 D2_φ ⌜⌝I,1,2⌜φ⌝=_⌜ψ⌝ D3 φ ⌜⌝E1_ψ ▷⌜⌝1_D3_φ D1_ψ_[φ]1 D1 ψ_[ψ]2 D2 φ_⌜⌝I,1,2⌜φ⌝= ⌜ψ⌝ D4 ψ_⌜⌝E2 φ ▷⌜⌝2_D4_ψ D2 φ When we presume that a liar sentence Φ which is defined_by_¬T (⌜Φ⌝)_is_expressible_in our language, there is_a derivation of the Liar_paradox using_the formula ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ which raises_a_non-terminating reduction sequence._Proposition 2.4.4. Let SL be a system containing T−,_¬−, and ⌜⌝−rules. SL has a_set RL of_standard_reduction_procedures for T (x),_¬, and ⌜⌝. Let us define_a formula Φ as ¬T (⌜Φ⌝)._Then, there exists a closed derivation Π11_of ⊥ in SL with respect_to RL which generates a_non-terminating reduction_sequence, and so is not_normalizable. Proof. Two_claims establish_the result. Claim_1. There is a closed derivation_Π11 of ⊥_in SL with respect to RL._First,_there is an_open_derivation_Π9 of_⊥ from T_(⌜Φ⌝)._[Φ]1 *************_de_f ¬T (⌜Φ⌝) [¬T (⌜Φ⌝)]2 ****************_de_f Φ ⌜⌝I,1,2⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]3_T E Φ ⌜⌝E1¬T_(⌜Φ⌝)_[T (⌜Φ⌝)]3 ¬