 atomic sentence Ψ and the atomic sentence Ψ becomes a major premise in ΨE−rule. Since most of introduction rules introduce a constant to formulas and make a complex formula as its conclusion, a major premise is often regarded as a complex formula. Second, ΨI−rule already has Ψ in the square quote of the premise which is introduced by the rule. Lastly, as Tennant (2016, p. 13) suggests, the reduction procedure for Ψ uses ¬I−rule.15 [T (⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ [¬T (⌜Ψ⌝)]2 D2 φ ΨE2 φ ▷Ψ [T (⌜Ψ⌝)]1 D1 ⊥ ¬I,1¬T (⌜Ψ⌝) D2 φ Though his Ψ−rules have special features, because of the specificity of paradoxical derivations, he may accept Ψ−rules as id est rules for the liar sentence Ψ. 15He actually applies a particular graphical form of the reduction procedure for Ψ which fits to the proof of cut-elimination for his Core Logic suggested in Tennant (2012, 2015). However, our discussion of the paradoxes formalized in natural deduction is not confined to his Core Logic. We do not use his graphical forms of reduction processes. 80 Let SΨ be a natural deduction system containing ¬−, T−, and Ψ−rules. The set RΨ of reduction procedures have reductions for ¬, T (x), and Ψ. We have a system SΨC by adding CR−rule to SΨ and have SΨCE by adding CRE−rule to SΨC. Also, RΨC is an extension of RΨ having ▷CRT. RΨCE is given by supplementing ▷CRE with RΨC. Tennant (2016, pp. 14–16) suggests the following result and claims that Liar paradox is a genuine paradox. Proposition 2.A.1. There is a closed derivation of ⊥ in SΨ relative to RΨ, which initiates a non-terminating reduction sequence and so is not fully normalizable. Proof. We begin with the proof of ⊥ and show that it fails to reduce a full normal derivation. Claim
atomic sentence Ψ and the atomic sentence Ψ become a major premise in ΨE−rule. Since most of initiation rules introduce a constant to formulas and cause a complex formula as its conclusion, a major premise is much regard as a complex formula. Second, ΨI−rule already has Ψ in the square quote of the premise which is insert by the rule. Lastly, as Tennant (2016, p. 13) suggests, the reduction procedure for Ψ uses ¬I−rule.15 [ thymine (⌜ Ψ ⌝ )]1 D1 ⊥ ΨI,1 Ψ [ ¬T (⌜ Ψ ⌝ )]2 D2 φ ΨE2 φ ▷Ψ [ T (⌜ Ψ ⌝ )]1 D1 ⊥ ¬I,1¬T (⌜ Ψ ⌝) D2 φ Though his Ψ−rules have special features, because of the specificity of paradoxical derivation, he may accept Ψ−rules as i d est rules for the liar sentence Ψ. 15He actually applies a especial graphical form of the reduction procedure for Ψ which fits to the validation of cut - elimination for his Core Logic suggested in Tennant (2012, 2015). However, our discussion of the paradoxes formalized in natural deduction is not enclose to his Core Logic. We do not practice his graphical forms of reduction processes. 80 Let SΨ be a natural deduction arrangement incorporate ¬−, T−, and Ψ−rules. The set RΨ of reduction procedure get reductions for ¬, T (x), and Ψ. We have a system SΨC by add CR−rule to SΨ and accept SΨCE by adding CRE−rule to SΨC. Also, RΨC is an elongation of RΨ having ▷CRT. RΨCE is given by supplementing ▷CRE with RΨC. Tennant (2016, pp. 14–16) suggests the following result and claim that Liar paradox is a genuine paradox. Proposition 2.A.1. There is a closed derivation of ⊥ in SΨ relative to RΨ, which initiates a non - terminating decrease succession and therefore is not fully normalizable. Proof. We begin with the validation of ⊥ and show that it fails to reduce a full normal derivation. Claim
 atlmic sentence Ψ and the auomic sentence Ψ bgcimes a major premise in ΨE−rule. Since most of intrldyctiob rules introduce a covstant to formulaw anv make a complex formula as ita conelnsion, a major ptemise is ofden regarded av x eomplex formula. Second, ΨI−rule already has Ψ im hhe square quoje of ehe lgenise which is introduced by tge rule. Lastly, as Temnant (2016, p. 13) suggests, the redkctiln procedure for Ψ kses ¬I−rule.15 [J (⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ [¬T (⌜Ψ⌝)]2 D2 φ ΨE2 φ ▷Ψ [T (⌜Ψ⌝)]1 D1 ⊥ ¬I,1¬T (⌜Ψ⌝) D2 φ Though gis Ψ−rules have special features, becabse of the wpwcivhcity of pacadoxibal derivatiokx, he mdy accelt Ψ−rules as id esv ruoes for the liar sentxnce Ψ. 15He actually apklies a pastncular graphical form od the redgctiub pfoctduce ror Ψ wjici fits to tge proof of cut-elimination for hyw Core Logic auggeseeq in Tennant (2012, 2015). However, our discussion ox tge paradoxes formalized in natural deduction is not cjnfined to his Core Logic. We do not use his graphhcal hofms on reaychion processes. 80 Let SΨ be a natural deduction fgsuem containing ¬−, T−, and Ψ−rules. Yhf xgt RΨ of reductkon procesures have reductilns for ¬, T (x), and Ψ. We rave a system SΨC by adding CR−ruoe to SΨ and kavw SΨCE by adding CRZ−rule to SΨC. Clso, RΨV is sn extension of RΨ haviny ▷CRT. DΨCE is givej by suppmdmenting ▷CRE witf RΨB. Tetnant (2016, pk. 14–16) suggests the fojlowing rxsult and clxims that Jiar paradlx is a genuine paradox. Oropodidion 2.A.1. Thege is a closed derivation of ⊥ ii SΨ relative jo SΨ, wvich iniciates a non-terminaeing reduction sequenee and so is not fully iormalizable. Proof. We beghj with the pcoof of ⊥ wnd whow that ig fails to reduve a full normal deeivation. Claim
atomic sentence Ψ and the atomic sentence a premise in Since most of to and make a formula as its a major premise is often regarded a complex formula. Second, ΨI−rule already has Ψ in the square quote of premise which is introduced by the rule. Lastly, as Tennant (2016, p. 13) the procedure Ψ ¬I−rule.15 [T (⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ [¬T (⌜Ψ⌝)]2 D2 φ ΨE2 φ ▷Ψ [T (⌜Ψ⌝)]1 ⊥ ¬I,1¬T (⌜Ψ⌝) D2 φ Though his Ψ−rules special features, because of specificity of paradoxical derivations, he accept as id rules the sentence Ψ. 15He applies a particular graphical form of the reduction procedure for Ψ which fits to the proof of for his suggested in (2012, However, discussion of the in natural deduction is not confined Logic. We do not use his graphical forms reduction processes. Let SΨ be a natural deduction containing ¬−, T−, and Ψ−rules. The set RΨ reduction procedures have reductions for ¬, T (x), and Ψ. We have a system SΨC CR−rule to SΨ and SΨCE by adding to Also, is extension of having ▷CRT. RΨCE is given by supplementing ▷CRE with RΨC. Tennant pp. 14–16) suggests the following result and claims that Liar a paradox. Proposition 2.A.1. is a closed derivation ⊥ SΨ relative to RΨ, a reduction is fully Proof. We begin with proof of ⊥ and show it fails to reduce
 atomic sentence Ψ and the atomIc sentence ψ becoMes A maJoR preMise In ΨE−rule. Since mOSt of Introduction rules introDuce a CoNStanT To FormuLas and mAKe A COmpLeX fOrmUlA As Its coNclUsion, a mAjor premisE is OfTen regarded aS A cOmplex formUla. second, ΨI−rule AlrEady haS Ψ In tHE squaRe qUote oF the prEMise whIch is intrOdUCed by tHE rule. LaSTLy, As TeNnant (2016, p. 13) suggests, the REdUCtion procedure For Ψ usEs ¬i−RuLE.15 [t (⌜Ψ⌝)]1 D1 ⊥ ψI,1 Ψ [¬t (⌜Ψ⌝)]2 D2 φ ΨE2 φ ▷Ψ [T (⌜Ψ⌝)]1 d1 ⊥ ¬I,1¬t (⌜Ψ⌝) D2 φ THOugh his ψ−RuLES HavE Special featurEs, because of THe sPecifiCiTy oF ParadoXical DeRIvaTions, he may aCcepT Ψ−rules as Id est rULes for tHE liar seNtence ψ. 15He ActUallY ApPlIes A pARtiCUlAr gRAphIcal form Of ThE reduCtioN PROCeduRe fOr Ψ wHich fIts to the proof Of cUt-elIMinAtion For hiS CorE LOgic sUggestEd in TEnNant (2012, 2015). However, our dIscuSsion of thE paRaDoxEs FormaLIzed in NatUraL deductIon is noT ConFiNED To His Core Logic. We do noT uSE HiS graphicAl formS Of ReDUction prOcEssEs. 80 LeT sψ be a nAturAL dEduction System COnTaIning ¬−, T−, aNd ψ−rules. thE seT RΨ Of redUCtioN proceDures havE reduCTions for ¬, T (x), and Ψ. wE have a system Sψc bY ADdINg CR−RulE to SΨ and havE SΨCe By adDing crE−RulE To SΨC. also, RψC IS aN Extension of RΨ having ▷cRt. RΨCE iS giveN by supplementIng ▷CRE with rψc. tennant (2016, pP. 14–16) sugGEsTS the following rEsult And claims tHAt Liar paRadox Is a genuiNe paradox. pROpositioN 2.A.1. THerE is A clOSEd Derivation of ⊥ iN sψ relAtIve to RΨ, WhiCh initiAteS a nOn-tErmInAting reduCtion seqUeNcE aNd So iS not fULly normaLiZabLe. proOf. We bEGin witH the pRoof Of ⊥ AnD ShoW that it FAiLS To reDuCe A fulL noRmAl derIvatIOn. CLaim
 atomic sentence Ψ and the  atomic se ntenc e Ψ  be co mesa ma jor premise in ΨE−r ule. Since most of int roduc ti o n ru l es  intr oduce a co n s tan tto  fo rm u la s and  ma ke a co mplex form ula  a s its conclu s io n, a major  pr emise is oft enregard ed  as a com ple x for mula.S econd,  ΨI−ruleal r eady h a s Ψ int h esqua re quote of the p r em i se which is in troduc ed by t herul e. Lastly,  a s Ten n ant (20 1 6, p .  13 )  suggests, th e reduction pro cedure  f orΨ  uses¬I−ru le . 15[T (⌜Ψ⌝)]1D1 ⊥  ΨI,1 Ψ [ ¬T (⌜Ψ ⌝ )]2 D2φ  ΨE2 φ▷Ψ [T(⌜Ψ ⌝)] 1 D1 ⊥¬I ,1¬ T( ⌜Ψ⌝ )  D 2 φ Tho ugh hisΨ− ru les h aves p e c ialfea ture s, be cause of thespe cifi c ity  of p arado xica lderiv ations , hema y accept Ψ−rule s as  id est r ule sfor  t he li a r sent enc e Ψ . 15Heactuall y  ap pl i e s  a  particular graphi ca l fo rm of th e redu c ti on procedur efor  Ψ w h i ch fi ts t o  t he proof  of cu t -e li minatio nfor hi sCor e L ogics ugge sted i n Tennan t (20 1 2, 2015). Howe v er, our discu s si o n  o f  the  pa radoxes for mali z ed i n na t ur ald educt ion i sn ot confined to his Cor eLogic.  We d o not use his  graphical f o rms of r educ t io n  processes. 80  LetSΨ be a na t ural ded uctio n system  containi n g  ¬−, T−,  an d Ψ −ru les . Th e set RΨ of r e d ucti on  proced ure s havered uct ion s f or  ¬, T (x) , and Ψ.  W eha ve  asyste m  SΨC byad din gCR− rulet o SΨ a nd ha ve S ΨC Eb y a dding C R E− r u le t oSΨ C. A lso ,RΨC i s an ext ensionof RΨ hav ing ▷CRT .RΨ CE is g iven by suppl em enting ▷CR Ewit h RΨC. T ennant ( 2016, pp. 14–16) sugges t s the f oll owing  res ult and c lai ms tha t L i ar par adox i s a g en uin e parad o x .Pro po sition 2.A . 1 . T hereis  a c losed d erivation of ⊥ inS Ψ r elative to RΨ , w hich i ni tia t es a n on - ter m i nating reductio n sequence  a n dso is notf ull ynormali zable.Proof .  We beg in with t he proofof  ⊥ a n d  sh ow that it  fails t o reducea  full no rmalder ivatio n.  Cl aim
 atomic_sentence Ψ_and the atomic sentence_Ψ becomes_a_major premise_in_ΨE−rule. Since most_of introduction rules_introduce a constant to_formulas and make_a_complex formula as its conclusion, a major premise is often regarded as a complex_formula._Second, ΨI−rule_already_has_Ψ in the square quote_of the premise which is_introduced by_the rule. Lastly, as Tennant (2016, p. 13)_suggests,_the reduction procedure_for Ψ uses ¬I−rule.15 [T (⌜Ψ⌝)]1 D1 ⊥ ΨI,1_Ψ [¬T (⌜Ψ⌝)]2 D2 φ ΨE2_φ ▷Ψ [T_(⌜Ψ⌝)]1_D1_⊥ ¬I,1¬T (⌜Ψ⌝) D2_φ Though his Ψ−rules have special_features, because of the specificity of_paradoxical derivations, he may accept Ψ−rules as_id est rules for the liar_sentence Ψ. 15He actually applies_a particular_graphical form of the reduction_procedure for Ψ_which fits_to the proof_of cut-elimination for his Core Logic_suggested in Tennant_(2012, 2015). However, our discussion of_the_paradoxes formalized in_natural_deduction_is not_confined to his_Core_Logic. We_do_not use his graphical forms of_reduction_processes. 80 Let SΨ be a natural_deduction system containing ¬−,_T−,_and Ψ−rules. The set_RΨ of reduction procedures have_reductions for ¬, T (x), and_Ψ. We_have a_system SΨC by adding CR−rule to SΨ and have SΨCE by_adding CRE−rule to SΨC. Also, RΨC_is an extension of_RΨ having_▷CRT._RΨCE is given_by_supplementing ▷CRE_with RΨC. Tennant (2016, pp. 14–16) suggests_the following_result and claims that Liar paradox_is a genuine paradox._Proposition_2.A.1. There is a closed derivation_of ⊥ in SΨ relative to_RΨ, which initiates a non-terminating_reduction_sequence_and so is not fully_normalizable. Proof. We begin with the_proof of ⊥_and show that it fails to reduce_a_full normal derivation. Claim