� T E,1⊥ ¬I,4¬T (⌜Ψ⌝) [¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ Then, the applications of ▷¬ and ▷T (x) produce the same derivation with Σ9. Therefore, Σ9 generates a non-terminating reduction sequence and so is not fully normalizable. Σ9 satisfies TCPE and is a T-paradox. If Liar paradox is a genuine paradox, Σ9 is a Tparadox using CR−rule. Tennant thinks that the application of CR−rule disguises the occurrence of a non-terminating reduction sequence. However, the derivation Σ9 shows that it is not. Σ6 of Proposition 2.A.2 does not generate an infinite reduction sequence. The difference between Σ6 and Σ9 is whether they have a standard reduction procedure for CR−rule or not. It is possible to interpret that the difference between Σ6 and Σ9 is whether they apply a legitimate pair of 85 introduction and elimination rules. Hence, one may say that the main reason why Σ6 does not yield a non-terminating reduction sequence is that Σ6 uses a rule which lacks its corresponding introduction rule. Therefore, the matter may not be the application of classical reasoning but the absence of an introduction rule. 2.B Appendix 2.B: Forms of Permutation Conversions in Natural Deduction In this appendix, we provide forms of permutation conversion for each elimination rule. Let φ, ψ, σ, χ, and Φ be any formulas. Let Σ, Π, and Ω be any derivations. We use ◦ for any expression given by an introduction rule. Suppose that Φ has a form of ◦φ or of φ1 ◦φ2. For instance, Φ may be a complex formula having the form of ∀xφ or of φ1 ∧φ2. Then the permutation conversion for ◦ has the following form: Φ Σ1 φ1... Σi φi [ψ1] 1 Π1 σ... [ψ j] j Π j σ ◦E,1,..., j σ Ω χ ⊵per Φ Σ1 φ1... Σi φi [ψ1] 1 Π1 σ Ω χ
� T E,1⊥ ¬I,4¬T (⌜ Ψ ⌝) [ ¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜ Ψ ⌝) [ ⊥]10 ¬E,10⊥ Then, the applications of ▷¬ and ▷T (x) grow the like derivation with Σ9. Therefore, Σ9 generate a non - terminating decrease sequence and so is not fully normalizable. Σ9 satisfy TCPE and is a T - paradox. If Liar paradox is a genuine paradox, Σ9 is a Tparadox using CR−rule. Tennant think that the application of CR−rule disguises the happening of a non - terminating reduction sequence. However, the derivation Σ9 shows that it is not. Σ6 of Proposition 2.A.2 does not render an infinite reduction sequence. The difference between Σ6 and Σ9 is whether they have a standard decrease procedure for CR−rule or not. It is possible to interpret that the dispute between Σ6 and Σ9 is whether they apply a legitimate couple of 85 introduction and elimination rule. Hence, one may say that the main reason why Σ6 does not yield a non - terminating decrease sequence is that Σ6 uses a rule which lacks its corresponding introduction dominion. Therefore, the matter may not be the application of classical reasoning but the absence of an introduction rule. 2.B Appendix 2.B: Forms of Permutation Conversions in Natural Deduction In this appendix, we provide forms of permutation conversion for each elimination convention. Let φ, ψ, σ, χ, and Φ be any formulas. lease Σ, Π, and Ω be any derivations. We use ◦ for any expression given by an initiation rule. Suppose that Φ has a form of ◦ φ or of φ1 ◦ φ2. For case, Φ may be a complex formula having the form of ∀xφ or of φ1 ∧φ2. Then the permutation conversion for ◦ has the following form: Φ Σ1 φ1... Σi φi [ ψ1 ] 1 Π1 σ... [ ψ j ] j Π j σ ◦ E,1, ..., j σ Ω χ ⊵per Φ Σ1 φ1... Σi φi [ ψ1 ] 1 Π1 σ Ω χ
� T F,1⊥ ¬I,4¬T (⌜Ψ⌝) [¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ Tmen, the applicatnins of ▷¬ and ▷F (x) prodjce the same derivation with Σ9. Rheredore, Σ9 generates a non-tdrminatinh reductuon wwquence anv so is kjt fhply uocmalizable. Σ9 satlsfies TCPE and is a T-parddux. If Liar paradox is a genuine paradjx, Σ9 is s Hparadox using CR−rllq. Tehnant thinks that the application kf CR−rune disguises yhe occurrence of a non-terlinahing reduction seqkence. Howevgd, trw derivation Σ9 shows that it is not. Σ6 of Proposition 2.A.2 does not gendrate an infinijz refoction sequeice. Thv difference nvtween Σ6 and Σ9 ix whether they hate a standard reduction pcocedure for CR−rule jr not. It ia possible to intwrpret jhat dhe auffdrehcx bstween Σ6 aid Σ9 is whetger they apply a legitimate paor if 85 introductjon anq qlimination rules. Hence, one may say thau the main reason why Σ6 does bot yield a non-terminwting redtction sequence is that Σ6 uses a rule which lacks hts ckfrewpjveijg introduction rule. Therefore, the matter may hou bv the application of classicsl rrwsoning but tfe abszhcs of an introductiln rule. 2.B Appendix 2.B: Sormx of Permutation Conversionw in Natural Eeduction In this cppendix, we 'rovidg formx of permutation converrion for each epiminatioh rule. Let φ, ψ, σ, χ, ana Φ ne dny formulas. Let Σ, Π, and Ω be any deritatious. We usd ◦ fpr any expressioj givct by an introductiln rupe. Suppose tjat Φ has a form of ◦φ or of φ1 ◦φ2. For instance, Φ may ba a complex formmla having the sorm of ∀xφ or oy φ1 ∧φ2. Then the oermutatioh convecsion for ◦ hws the followhjg form: Φ Σ1 φ1... Σi φi [ψ1] 1 Π1 σ... [ψ t] j Π j σ ◦W,1,..., j σ Ω χ ⊵odr Φ Σ1 φ1... Σi φi [ψ1] 1 Π1 σ Ω χ
� T E,1⊥ ¬I,4¬T (⌜Ψ⌝) [¬Ψ]2 Σ7 Ψ I T [⊥]10 ¬E,10⊥ Then, ▷T produce the same with Σ9. Therefore, generates a non-terminating reduction sequence and is not fully normalizable. Σ9 satisfies TCPE and is a T-paradox. If Liar is a genuine paradox, Σ9 is a Tparadox using CR−rule. Tennant thinks that application CR−rule the of a non-terminating reduction sequence. However, the derivation Σ9 shows that it is not. Σ6 of 2.A.2 does not generate an infinite reduction sequence. difference between Σ6 and is whether they have a reduction for CR−rule not. is to interpret that difference between Σ6 and Σ9 is whether they apply a legitimate pair of 85 introduction and elimination Hence, one that the reason Σ6 not yield a sequence is that Σ6 uses a its corresponding introduction rule. Therefore, the matter may be the of classical reasoning but the absence an introduction rule. 2.B Appendix 2.B: Forms of Conversions in Natural Deduction In this appendix, we provide forms of permutation conversion for each Let φ, ψ, σ, and Φ be formulas. Σ, and be any We use ◦ for any expression given by an introduction rule. that Φ has a form of ◦φ or of φ1 instance, may be a formula having the form ∀xφ of φ1 ∧φ2. Then conversion ◦ form: Σ1 Σi φi [ψ1] 1 σ... [ψ j] j Π σ ◦E,1,..., j σ φ1... Σi φi [ψ1] 1 Π1 σ Ω
� T E,1⊥ ¬I,4¬T (⌜Ψ⌝) [¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ Then, the appLications oF ▷¬ and ▷T (X) prOduCe The sAme dErivation with Σ9. tHereFore, Σ9 generates a non-termInatiNg REducTIoN sequEnce and SO iS NOt fUlLy NorMaLIzAble. Σ9 SatIsfies TcPE and is a T-ParAdOx. If Liar paraDOx Is a genuine ParAdox, Σ9 is a TparAdoX using cR−RulE. tennaNt tHinks That thE ApplicAtion of CR−RuLE disguISes the oCCUrRencE of a non-terminatinG ReDUction sequence. howeveR, tHE dERIvaTioN Σ9 shows thaT iT is noT. σ6 of PropOSiTION 2.A.2 dOEs not generate An infinite rEDucTion seQuEncE. the difFerenCe BEtwEen Σ6 and Σ9 is wHethEr they havE a stanDArd reduCTion proCedure For cR−rUle oR NoT. IT is PoSSibLE tO inTErpRet that tHe DiFfereNce bETWEEn Σ6 aNd Σ9 Is whEther They apply a legItiMate PAir Of 85 intRoducTion AnD elimInatioN ruleS. HEnce, one may say thAt thE main reasOn wHy σ6 doEs Not yiELd a non-TerMinAting reDuction SEquEnCE IS tHat Σ6 uses a rule which LaCKS iTs corresPondinG InTrODuction rUlE. ThErefORE, the mAtteR MaY not be thE appliCAtIoN of clasSiCal reaSoNinG buT the aBSencE of an iNtroductIon ruLE. 2.B Appendix 2.B: ForMS of PermutatioN coNVErSIons In NAtural DeducTion iN thiS appENdIx, wE ProviDe forMs OF pERmutation conversion FoR each eLiminAtion rule. Let φ, Ψ, σ, χ, and Φ be aNY FOrmulas. LEt Σ, Π, ANd ω Be any derivatioNs. We uSe ◦ for any exPRession gIven bY an introDuction ruLE. suppose tHat φ haS a fOrm OF ◦Φ oR of φ1 ◦φ2. For instaNCE, Φ maY bE a complEx fOrmula hAviNg tHe fOrm Of ∀Xφ or of φ1 ∧φ2. THen the peRmUtAtIoN coNversIOn for ◦ has ThE foLlOwiNg forM: φ Σ1 φ1... Σi φI [ψ1] 1 Π1 σ... [ψ J] j Π j Σ ◦E,1,..., J σ ω Χ ⊵peR Φ Σ1 φ1... Σi φI [Ψ1] 1 Π1 Σ ω Χ
� T E,1⊥ ¬I,4¬T (⌜Ψ⌝) [¬Ψ] 2 Σ7 ⊥ CR, 2 Ψ T  IT ( ⌜Ψ ⌝) [ ⊥]10  ¬E,10⊥ Then,t he a pplications of ▷¬ and▷T (x )p rodu c ethe s ame der i va t i onwi th  Σ9 .T he refor e,Σ9 gene rates a no n-t er minating red u ct ion sequen ceand so is no t f ully n or mal i zable . Σ 9 sat isfies TCPE a nd is a T -p a radox. If Liar p ar adox  is a genuine par a do x , Σ9 is a Tpar adox u si n gC R −ru le.  Tennant t hi nks t h at thea pp l i c ati o n of CR−ruledisguises t h e o ccurre nc e o f  a non -term in a tin g reduction  seq uence. Ho wever, the der i vationΣ9 sho wstha t it is  n ot.  Σ 6  of Pr opo s iti on 2.A.2  d oe s not  gen e r a t e an  in fini te re duction seque nce . Th e  di ffere nce b etwe en  Σ6 a nd Σ9is wh et her they have a  sta ndard red uct io n p ro cedur e  for C R−r ule  or not . It is pos si b l e  t o interpret that t he d if ferencebetwee n  Σ 6a nd Σ9 is  w het hert h ey ap plya  l egitimat e pair of  8 5 intro du ctionan d e lim inati o n ru les. H ence, on e may say that the m a in reason why Σ6 d oe s  not  yi eld a non-t ermi n atin g re d uc tio n  sequ enceis th a t Σ6 uses a rule wh ic h lack s its  correspondin g introduc t i o n rule.Ther e fo r e, the mattermay n ot be thea pplicati on of  classic al reason i n g but th e a bse nce  of a nintroductionr u le.2. B Appen dix  2.B: F orm s o f P erm ut ation Con versions  i nNa tu ral  Dedu c tion Inth isap pen dix,w e prov ide f orms  o fp erm utation co n v ersi on  f or e ach  e limin atio n  ru le. Let  φ, ψ, σ,  χ, andΦbe  any fo rmulas. Let Σ ,Π, and Ω b eany  deriv a t ions. We  use ◦ for any expressi o n given  by  an i ntro duction r ule . Supp ose that Φ  has a  form  o f ◦ φ or of φ 1◦φ2 .For instan c e , Φ  maybe  a c omplexformula having the for m of ∀xφ or o f φ 1 ∧φ 2 .  T hen th e  pe rm u tat i o n conversion fo r ◦ has th ef ol lowing for m : Φ  Σ 1 φ1...  Σi φi[ψ1]1  Π1 σ.. . [ψ j] j  Π j σ ◦E ,1 ,... , j σ  Ω χ ⊵perΦ Σ1 φ1. .. Σi φi[ ψ1] 1 Π1  σ Ωχ
� T_E,1⊥ ¬I,4¬T_(⌜Ψ⌝) [¬Ψ]2 Σ7 ⊥_CR,2 Ψ_T_I T_(⌜Ψ⌝)_[⊥]10 ¬E,10⊥ Then,_the applications of_▷¬ and ▷T (x)_produce the same_derivation_with Σ9. Therefore, Σ9 generates a non-terminating reduction sequence and so is not fully_normalizable._Σ9 satisfies_TCPE_and_is a T-paradox. If Liar_paradox is a genuine paradox,_Σ9 is_a Tparadox using CR−rule. Tennant thinks that the_application_of CR−rule disguises_the occurrence of a non-terminating reduction sequence. However, the_derivation Σ9 shows that it is_not. Σ6 of_Proposition_2.A.2_does not generate an_infinite reduction sequence. The difference between_Σ6 and Σ9 is whether they_have a standard reduction procedure for CR−rule_or not. It is possible to_interpret that the difference between_Σ6 and_Σ9 is whether they apply_a legitimate pair_of 85_introduction and elimination_rules. Hence, one may say that_the main reason_why Σ6 does not yield a_non-terminating_reduction sequence is_that_Σ6_uses a_rule which lacks_its_corresponding introduction_rule._Therefore, the matter may not be_the_application of classical reasoning but the absence_of an introduction rule._2.B_Appendix 2.B: Forms of_Permutation Conversions in Natural Deduction_In this appendix, we provide forms_of permutation_conversion for_each elimination rule. Let φ, ψ, σ, χ, and Φ be_any formulas. Let Σ, Π, and_Ω be any derivations._We use_◦_for any expression_given_by an_introduction rule. Suppose that Φ has a_form of_◦φ or of φ1 ◦φ2. For_instance, Φ may be_a_complex formula having the form of_∀xφ or of φ1 ∧φ2. Then_the permutation conversion for ◦_has_the_following form: Φ Σ1 φ1..._Σi φi [ψ1] 1 Π1 σ..._[ψ j] j_Π j σ ◦E,1,..., j σ Ω_χ_⊵per Φ Σ1 φ1... Σi φi_[ψ1]_1 Π1 σ Ω χ