 to T−rules. [¬T (⌜φ⌝)]1 D ⊥ CR,1T (⌜φ⌝) ⊵CR(T (x)) [¬φ]2 [T (⌜φ⌝)]1 T E φ ¬E⊥ ¬I,1¬T (⌜φ⌝) D ⊥ CR,2 φ T I T (⌜φ⌝) ⊵CR(¬)− and ⊵CR(T (x))−reductions neither do eliminate a maximum formula nor lower the length of a derivation, but they lower the degree of the conclusion of CR−rule. Tennant (2015, pp. 585–588) proposes a derivation of the Liar paradox which does not employ CR−rule and does generate a looping reduction sequence. We will use a natural 47 deduction system SE which has ¬−, T−, =−rules. A set RE of reductions contains ▷¬−, ▷T (x)−, and ⊵Sub. Then, we have the result in SE with respect to RE. Proposition 2.2.1. Suppose that, for some formula Φ, ⌜Φ⌝ = ⌜¬T (⌜Φ⌝)⌝ is an axiom of SE. SE relative to RE has a closed derivation of ⊥ which generates a non-terminating reduction sequence, so is not normalizable. Proof. The result consists of two claims Claim 1. there is a closed derivation ∆3 of ⊥ in SE We begin with an open derivation ∆1 of ⊥ from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 = E T (⌜¬T (⌜Φ⌝)⌝) T E¬T (⌜Φ⌝) [T (⌜Φ⌝)]1 ¬E⊥ With the open derivation ∆1, we have the closed derivation ∆2 of T (⌜φ⌝). Ax2⌜¬T (�
to T−rules. [ ¬T (⌜ φ ⌝ )]1 D ⊥ CR,1 T (⌜ φ ⌝) ⊵CR(T (x) ) [ ¬φ]2 [ T (⌜ φ ⌝ )]1 T E φ ¬E⊥ ¬I,1¬T (⌜ φ ⌝) D ⊥ CR,2 φ T I T (⌜ φ ⌝) ⊵CR(¬)− and ⊵CR(T (x))−reductions neither do rule out a maximal formula nor lower the duration of a ancestry, but they lower the degree of the termination of CR−rule. Tennant (2015, pp. 585–588) nominate a derivation of the Liar paradox which does not use CR−rule and does generate a looping decrease sequence. We will use a natural 47 deduction arrangement SE which has ¬−, T−, = −rules. A set RE of reductions contains ▷¬−, ▷T (x)−, and ⊵Sub. Then, we suffer the result in SE with respect to RE. proposal 2.2.1. Suppose that, for some formula Φ, ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ is an axiom of SE. SE relative to RE have a closed derivation of ⊥ which generates a non - terminating reduction sequence, therefore is not normalizable. Proof. The result consists of two claims Claim 1. there is a closed derivation ∆3 of ⊥ in SE We begin with an open derivation ∆1 of ⊥ from [ T (⌜ Φ ⌝) ]. Ax1 ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ [ T (⌜ Φ ⌝ )]1 = E T (⌜ ¬T (⌜ Φ ⌝) ⌝) T E¬T (⌜ Φ ⌝) [ T (⌜ Φ ⌝ )]1 ¬E⊥ With the open derivation ∆1, we suffer the closed derivation ∆2 of T (⌜ φ ⌝). Ax2 ⌜ ¬T (�
 to T−rules. [¬T (⌜φ⌝)]1 D ⊥ CR,1T (⌜φ⌝) ⊵CR(T (x)) [¬φ]2 [T (⌜φ⌝)]1 T E φ ¬E⊥ ¬I,1¬T (⌜φ⌝) D ⊥ CR,2 φ T I T (⌜φ⌝) ⊵CR(¬)− and ⊵CR(T (x))−reductions neither do xlimunate a maximum formula nor lower thv length if a eerivation, but then lowsv the vegree of the cpnclusion mf CR−rule. Tenndng (2015, pp. 585–588) proposes a derivation of the Lyar parsdlx which does got tmpjoy DG−rmle and does generate a looping deductimn sequence. Wr will use a natural 47 deduftioj system SE which jas ¬−, T−, =−rulew. A fwt RE of redjctions contains ▷¬−, ▷T (x)−, znd ⊵Sub. Then, we have the result in SZ with respger tl RE. Proposivion 2.2.1. Fuppose that, for soma formuka Φ, ⌜Φ⌝ = ⌜¬T (⌜Φ⌝)⌝ is an axmom if SE. SE relative to CE has a closed deridation of ⊥ chich generates a non-reeminajing seduzriov stquxncs, so id nkt normaliaable. Proof. The result consistx jd two claims Dlaim 1. trere is a closed derivation ∆3 of ⊥ in SE Fe gegin with an open derication ∆1 of ⊥ from [T (⌜Φ⌝)]. Ad1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 = Q T (⌜¬T (⌜Φ⌝)⌝) T E¬T (⌜Φ⌝) [T (⌜Φ⌝)]1 ¬E⊥ With the open derivation ∆1, we haee thx zloweq eegivation ∆2 of T (⌜φ⌝). Ax2⌜¬T (�
to T−rules. [¬T (⌜φ⌝)]1 D ⊥ CR,1T (x)) [T (⌜φ⌝)]1 E φ ¬E⊥ φ I T (⌜φ⌝) and ⊵CR(T (x))−reductions do eliminate a maximum formula nor the length of a derivation, but they lower the degree of the conclusion CR−rule. Tennant (2015, pp. 585–588) proposes a derivation of the Liar paradox which not CR−rule does a looping reduction sequence. We will use a natural 47 deduction system SE which has ¬−, =−rules. A set RE of reductions contains ▷¬−, (x)−, and ⊵Sub. Then, have the result in SE respect RE. Proposition Suppose for formula Φ, ⌜Φ⌝ ⌜¬T (⌜Φ⌝)⌝ is an axiom of SE. SE relative to RE has a closed derivation of ⊥ generates a sequence, so not Proof. result consists of Claim 1. there is a closed ⊥ in SE We begin with an open ∆1 of from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ (⌜Φ⌝)]1 = E T (⌜¬T (⌜Φ⌝)⌝) T E¬T [T (⌜Φ⌝)]1 ¬E⊥ With the open derivation ∆1, we have the closed derivation ∆2 of Ax2⌜¬T (�
 to T−rules. [¬T (⌜φ⌝)]1 D ⊥ CR,1T (⌜φ⌝) ⊵CR(T (x)) [¬φ]2 [T (⌜φ⌝)]1 T E φ ¬e⊥ ¬I,1¬T (⌜φ⌝) D ⊥ CR,2 φ T I t (⌜φ⌝) ⊵CR(¬)− aNd ⊵Cr(T (x))−ReDuctIons Neither do elimiNAte a Maximum formula nor lower The leNgTH of a DErIvatiOn, but thEY lOWEr tHe DeGreE oF ThE concLusIon of CR−Rule. TennanT (2015, pp. 585–588) PrOposes a derivATiOn of the LiaR paRadox which doEs nOt emplOy cR−rULe and DoeS geneRate a lOOping rEduction sEqUEnce. We WIll use a NATuRal 47 dEduction system SE wHIcH Has ¬−, T−, =−rules. A set Re of redUcTIoNS ConTaiNs ▷¬−, ▷T (x)−, and ⊵Sub. thEn, we hAVe the reSUlT IN sE wITh respect to RE. proposition 2.2.1. sUppOse thaT, fOr sOMe formUla Φ, ⌜Φ⌝ = ⌜¬t (⌜Φ⌝)⌝ IS an Axiom of SE. SE RelaTive to RE hAs a cloSEd derivATion of ⊥ wHich geNerAteS a noN-TeRmInaTiNG reDUcTioN SeqUence, so iS nOt NormaLizaBLE. pRoof. the ResuLt conSists of two claIms claiM 1. TheRe is a CloseD derIvAtion ∆3 Of ⊥ in SE we begIn With an open derivAtioN ∆1 of ⊥ from [T (⌜Φ⌝)]. ax1⌜Φ⌝= ⌜¬t (⌜Φ⌝)⌝ [t (⌜Φ⌝)]1 = E t (⌜¬T (⌜φ⌝)⌝) T E¬T (⌜Φ⌝) [t (⌜φ⌝)]1 ¬E⊥ With The OpeN derivaTion ∆1, we hAVe tHe CLOSeD derivation ∆2 of T (⌜φ⌝). Ax2⌜¬T (�
 to T−rules. [¬T (⌜φ⌝)]1 D  ⊥ CR,1T ( ⌜φ⌝)⊵CR (T(x )) [ ¬φ]2  [T (⌜φ⌝)]1 TE  φ ¬ E⊥ ¬I,1¬T (⌜φ⌝) D ⊥ CR ,2 φTI  T ( ⌜ φ⌝ ) ⊵CR (¬)− an d  ⊵ C R (T(x )) −re du c ti ons n eit her doeliminatea m ax imum formula no r lower th e l ength of a d eri vation ,but theylow er th e degr e e of t he conclu si o n of C R −rule.T e nn ant(2015, pp. 585–58 8 )p roposes a deri vation  o f  t h e  Li arparadox wh ic h doe s  not em p lo y C R−r u le and does g enerate a l o opi ng red uc tio n  seque nce.We wil l use a nat ural  47 deduc tion s y stem SE which h as ¬−,  T− , = −rul e s.  A  se tR E o f  r edu c tio ns conta in s▷¬−,▷T ( x ) − ,  and  ⊵S ub.Then,  we have theres ulti n S E wit h res pect  t o RE.  Propo sitio n2.2.1. Supposethat , for som e f or mul aΦ, ⌜Φ ⌝  = ⌜¬T  (⌜ Φ⌝) ⌝ is an  axiomo f S E. S E  r elative to RE hasac l os ed deriv ationo f⊥w hich gen er ate s an o n-ter mina t in g reduct ion se q ue nc e, so i snot no rm ali zab le. P r oof.  The r esult co nsist s  of two claims Claim 1. ther e  i s ac lose d d erivation ∆ 3 of ⊥ in  SEW ebeg i n wit h anop e nd erivation ∆1 of ⊥ f ro m [T ( ⌜Φ⌝)] . Ax1⌜Φ⌝= ⌜¬T  (⌜Φ⌝)⌝ [T ( ⌜ Φ⌝)]1 =E T( ⌜¬ T  (⌜Φ⌝)⌝) T E¬T  (⌜Φ⌝ ) [T (⌜Φ⌝) ] 1 ¬E⊥ Wi th th e open d erivation ∆ 1, we ha vethe  cl ose d de rivation ∆2 o f T (⌜ φ⌝ ). Ax2⌜ ¬T(�
 to_T−rules. [¬T_(⌜φ⌝)]1 D ⊥ CR,1T_(⌜φ⌝) ⊵CR(T_(x))_[¬φ]2 [T_(⌜φ⌝)]1_T E φ_¬E⊥ ¬I,1¬T (⌜φ⌝)_D ⊥ CR,2 φ_T I T_(⌜φ⌝)_⊵CR(¬)− and ⊵CR(T (x))−reductions neither do eliminate a maximum formula nor lower the length_of_a derivation,_but_they_lower the degree of the_conclusion of CR−rule. Tennant (2015,_pp. 585–588)_proposes a derivation of the Liar paradox which_does_not employ CR−rule_and does generate a looping reduction sequence. We will_use a natural 47 deduction system_SE which has_¬−,_T−,_=−rules. A set RE_of reductions contains ▷¬−, ▷T (x)−,_and ⊵Sub. Then, we have the_result in SE with respect to RE._Proposition 2.2.1. Suppose that, for some_formula Φ, ⌜Φ⌝ = ⌜¬T_(⌜Φ⌝)⌝ is_an axiom of SE. SE_relative to RE_has a_closed derivation of_⊥ which generates a non-terminating reduction_sequence, so is_not normalizable. Proof. The result consists_of_two claims Claim_1._there_is a_closed derivation ∆3_of_⊥ in_SE_We begin with an open derivation_∆1_of ⊥ from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= ⌜¬T_(⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 =_E_T (⌜¬T (⌜Φ⌝)⌝) T_E¬T (⌜Φ⌝) [T (⌜Φ⌝)]1 ¬E⊥_With the open derivation ∆1, we_have the_closed derivation_∆2 of T (⌜φ⌝). Ax2⌜¬T (�