 (iii) a derivation of ⊥ which only uses CR− rule and is not a T-paradox. The purpose of this appendix is to show that there is a T-paradox which the CR−rule is applied. Although the use of CR−rule is not necessary to formulate Liar paradox in natural deduction, it does not mean that the application of it makes a trouble that disguises the main feature of paradoxical derivations, i.e. the non-terminating reduction sequence. Our examples will show that our use of CR−rule does not always hide the occurrence of a non-terminating reduction sequence. Rather, we conclude that the occurrence of a non-terminating reduction sequence is related to a set of reduction procedures. We already have Proposition 2.2.1, 2.2.2, and 2.4.3. However, those results are based on the formalizations of Tennant (2015) which are not stated in generalized form. TCPE does not restrict the form of elimination rules. For our convenience, we will use in this appendix, TCPE rather than Tennant's later criterion. As we will see in Chapter 3, the latest criterion for paradoxicality introduced by Tennant (2016) has an additional condition that all elimination rules are stated in generalized form. Moreover, while Tennant (2016, pp. 12–16) asserts that Liar paradox is a genuine paradox, he proposes the id est rules for the liar sentence Ψ. 79 [T (⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ Ψ [¬T (⌜Ψ⌝)]1 D2 φ ΨE,1 φ Tennant (2016, p. 12) calls Ψ−rules the id est rules and said, [ΨI− and ΨE−rules] are the 'id est' rules for the Liar (so-called because of the familiar transitions '[Ψ], i.e. [¬T (⌜Ψ⌝)]'). The rules [ΨI− and ΨE−rules] ensure that the sentence called [Ψ] is interdeducible with [¬T (⌜Ψ⌝)] certainly a necessary (even if not sufficient) condition for the former to be the latter. Ψ−rules have several peculiar points. First, ΨI−rule introduces a seeming
(iii) a derivation of ⊥ which only uses CR− rule and is not a metric ton - paradox. The function of this appendix is to show that there is a T - paradox which the CR−rule is applied. Although the consumption of CR−rule is not necessary to formulate Liar paradox in natural tax write-off, it does not mean that the application of it make a trouble that disguises the independent feature of paradoxical derivations, i.e. the non - terminating decrease sequence. Our examples will show that our use of CR−rule does not always hide the occurrence of a non - terminating decrease sequence. Rather, we conclude that the happening of a non - terminating reduction sequence is related to a set of reduction procedure. We already have Proposition 2.2.1, 2.2.2, and 2.4.3. However, those results are based on the formalizations of Tennant (2015) which are not stated in generalized mannequin. TCPE does not restrict the form of elimination rules. For our convenience, we will use in this appendix, TCPE rather than Tennant's later criterion. As we will see in Chapter 3, the latest criterion for paradoxicality introduced by Tennant (2016) has an additional condition that all elimination rules are stated in generalized kind. Moreover, while Tennant (2016, pp. 12–16) asserts that Liar paradox is a actual paradox, he aim the i d est rule for the liar sentence Ψ. 79 [ T (⌜ Ψ ⌝ )]1 D1 ⊥ ΨI,1 Ψ Ψ [ ¬T (⌜ Ψ ⌝ )]1 D2 φ ΨE,1 φ Tennant (2016, p. 12) call Ψ−rules the i d est rules and said, [ ΨI− and ΨE−rules ] are the' i d est' dominion for the Liar (so - called because of the familiar transitions' [ Ψ ], i.e. [ ¬T (⌜ Ψ ⌝) ]'). The rules [ ΨI− and ΨE−rules ] ensure that the prison term called [ Ψ ] is interdeducible with [ ¬T (⌜ Ψ ⌝) ] certainly a necessary (even if not sufficient) condition for the former to be the latter. Ψ−rules give birth respective curious points. First, ΨI−rule introduces a seeming
 (iil) a derivation of ⊥ which only uses CR− rolw and ms not z T-paradux. The purpose of this appenvix us to show that there is a G-paradox ahich thw CR−cule is applied. Emthough the uac of ER−cule is not necgssary to fosmulate Liar pdrxdlx in natural deduction, it does not mean tnah the applicatyon ps it makes a trouble that disguises ths main heature of paracoxical derivations, i.e. the non-herminating reductlon sequencg. Ouw examples wiul show that our use or CR−rule does not always hide tfe oceurrence of a noj-jerminating ceductpon sequence. Vsther, fe conckude that the pccnrrebce of a non-terminatiig reduction sequencg is relatad to a set of reduxtuon ptocedgres. Qe xlrtadb hzve Prlpoaition 2.2.1, 2.2.2, ahd 2.4.3. However, those results are nafvc on the forjalizaeijns of Tennant (2015) which are not stated in geheralized form. TCPE doew not restrict the fotm of elimynation rules. For our convenience, we will use in dhis epoenblw, TCOW gather than Tennant's later criterion. As we wijm xev in Chapter 3, the latest criyegipg for paradoxkcalitv ihtroduced by Tennajt (2016) has an aeditional concition that all elimination rules are snatee in generalized flrm. Moreovex, whilg Tennsnt (2016, pp. 12–16) asserts that Lnar padadox is a henuine pzfadox, he proposer tme hd est rules for the liar fentence Ψ. 79 [T (⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ Ψ [¬T (⌜Ψ⌝)]1 C2 φ ΨE,1 φ Eennant (2016, p. 12) calls Ψ−rules the id est ruled dnd said, [ΨI− and ΨE−rules] are the 'id est' rules for the Liar (sm-canled beccuse on the familiar eransitions '[Ψ], i.g. [¬T (⌜Ψ⌝)]'). The xules [ΨK− and ΨE−rulvs] ensure that the sqntence calleg [Ψ] is interdevucible wyth [¬R (⌜Ψ⌝)] cwrtainlh a necessary (efen if non fuffucient) condition fpr jhs former to be cke latter. Ψ−rules hsve sedegal pqwuliar pointv. Fifst, ΨO−rule introduces x serming
(iii) a derivation of ⊥ which only rule is not T-paradox. The purpose show there is a which the CR−rule applied. Although the use of CR−rule not necessary to formulate Liar paradox in natural deduction, it does not mean the application of it makes a trouble that disguises the main feature of derivations, the reduction Our examples will show that our use of CR−rule does not always hide the occurrence of non-terminating reduction sequence. Rather, we conclude that the of a non-terminating reduction is related to a set reduction We already Proposition 2.2.2, 2.4.3. However, those are based on the formalizations of Tennant (2015) which are not stated in generalized form. TCPE does restrict the elimination rules. our we use in this rather than Tennant's later criterion. As in Chapter 3, the latest criterion for paradoxicality by Tennant has an additional condition that all rules are stated in generalized form. Moreover, while (2016, pp. 12–16) asserts that Liar paradox is a genuine paradox, he proposes the id for the liar sentence 79 [T (⌜Ψ⌝)]1 ⊥ Ψ [¬T D2 φ φ Tennant (2016, p. 12) calls Ψ−rules the id est rules said, [ΨI− and ΨE−rules] are the 'id est' rules for (so-called of the familiar '[Ψ], i.e. [¬T (⌜Ψ⌝)]'). rules and ΨE−rules] ensure that called is (⌜Ψ⌝)] a (even if not sufficient) for the former to be latter. Ψ−rules have several a seeming
 (iii) a derivation of ⊥ which only Uses CR− rule And is Not A T-pArAdox. the pUrpose of this apPEndiX is to show that there is a T-ParadOx WHich THe cR−rulE is applIEd. aLThoUgH tHe uSe OF Cr−rule Is nOt necesSary to formUlaTe liar paradox iN NaTural deducTioN, it does not meAn tHat the ApPliCAtion Of iT makeS a trouBLe that Disguises ThE Main feATure of pARAdOxicAl derivations, i.e. thE NoN-Terminating redUction SeQUeNCE. OuR exAmples will ShOw thaT Our use oF cR−RULE doES not always hidE the occurreNCe oF a non-tErMinATing reDuctiOn SEquEnce. Rather, wE conClude that The occURrence oF A non-terMinatiNg rEduCtioN SeQuEncE iS RelATeD to A Set Of reductIoN pRocedUres. wE ALReadY haVe PrOposiTion 2.2.1, 2.2.2, and 2.4.3. HoweveR, thOse rESulTs are Based On thE fOrmalIzatioNs of TEnNant (2015) which are not StatEd in generAliZeD foRm. tCPE dOEs not rEstRicT the forM of elimINatIoN RULeS. For our convenience, We WILl Use in thiS appenDIx, tCpe rather tHaN TeNnanT'S Later CritERiOn. As we wiLl see iN chApTer 3, the lAtEst criTeRioN foR paraDOxicAlity iNtroduceD by TeNNant (2016) has an additIOnal condition THaT ALl ELimiNatIon rules are StatED in gEnerALiZed FOrm. MoReoveR, wHIlE tennant (2016, pp. 12–16) asserts thaT LIar parAdox iS a genuine paraDox, he propoSES The id est RuleS FoR The liar sentencE Ψ. 79 [T (⌜Ψ⌝)]1 D1 ⊥ ψI,1 Ψ Ψ [¬T (⌜Ψ⌝)]1 D2 φ ΨE,1 Φ tennant (2016, p. 12) Calls ψ−rules thE id est rulES And said, [Ψi− anD ΨE−RulEs] aRE ThE 'id est' rules foR THe LiAr (So-calleD beCause of The FamIliAr tRaNsitions '[Ψ], I.e. [¬T (⌜Ψ⌝)]'). The rUlEs [ψI− AnD ΨE−Rules] ENsure thaT tHe sEnTenCe calLEd [Ψ] is iNterdEducIbLe WIth [¬t (⌜Ψ⌝)] certaINlY A NeceSsArY (eveN if NoT suffIcieNT) coNdition For the forMer TO be tHe LaTter. Ψ−ruLes have severaL pEculiar poiNtS. FiRst, ΨI−rULE introduCes a seeming
 (iii) a derivation of ⊥ w hich onlyusesCR−  ru le  and  isnot a T-parado x . Th e purpose of this appe ndixis to s h ow  that  therei sa T-p ar ad oxwh i ch  theCR− rule is  applied.Alt ho ugh the useo fCR−rule is  no t necessarytoformul at e L i ar pa rad ox in  natur a l dedu ction, it  d o es not mean th a t  t he a pplication of itm ak e s a trouble th at dis gu i se s the  ma in feature  o f par a doxical de r i v ati o ns, i.e. thenon-termina t ing  reduc ti ons equenc e. Ou re xam ples will s howthat ouruse of CR−rule does no t alwa yshid e th e  o cc urr en c e o f  a  no n -te rminatin gre ducti on s e q u e nce.  Ra ther , weconclude that  th e oc c urr enceof anon- te rmina ting r educt io n sequence is r elat ed to a s etof  re du ction proced ure s.We alre ady hav e  Pr op o s i ti on 2.2.1, 2.2.2, a nd 2 .4 .3. Howe ver, t h os er esults a re  ba sedo n  theform a li zationsof Ten n an t(2015)wh ich ar enot  st atedi n ge nerali zed form . TCP E  does not rest r ict the formo fe l im i nati onrules. Forourc onve nien c e,  we willuse i nt hi s  appendix, TCPE rat he r than  Tenn ant's later c riterion.A s we willseei nC hapter 3, thelates t criterio n  for par adoxi cality i ntroduced b y Tennan t ( 201 6)has a nadditional co n d itio nthat al l e liminat ion  ru les  ar estated in  general iz ed  f or m.Moreo v er, whil eTen na nt(2016 ,  pp. 1 2–16)  ass er ts tha t Liarp ar a d ox i sagenu ine  p arado x, h e  pr oposesthe id es t r u lesfo rthe lia r sentence Ψ.  7 9 [T (⌜Ψ⌝) ]1  D1  ⊥ ΨI, 1 Ψ Ψ [¬T(⌜Ψ⌝)]1 D2 φ ΨE,1 φ Ten n ant (20 16,  p. 1 2) c alls Ψ−ru les  the i d e s t rule s andsaid,  [ ΨI− a nd ΨE − r ul es]  a re the 'id e st'  rule sforthe Lia r (so-called becau s e o f the familia r t rans i t io ns' [Ψ ] , i .e .  [¬ T (⌜Ψ⌝)]'). The r ules [ΨI−an d  Ψ E−rules] e n sur ethat th e sente nce c a lled [Ψ ] is inte rdeducibl ewith [ ¬T(⌜Ψ⌝)] cer tainly a  necessar y  (eve n  i f not  su fficie nt ) c ondit ion fo r  th e for mer to  b e thelatte r.  Ψ−rules  have several peculiarpoints . Fir st,  ΨI−ruleint r odu ces a see ming 
 (iii)_a derivation_of ⊥ which only_uses CR−_rule_and is_not_a T-paradox. The_purpose of this_appendix is to show_that there is_a_T-paradox which the CR−rule is applied. Although the use of CR−rule is not necessary_to_formulate Liar_paradox_in_natural deduction, it does not_mean that the application of_it makes_a trouble that disguises the main feature of_paradoxical_derivations, i.e. the_non-terminating reduction sequence. Our examples will show that our_use of CR−rule does not always_hide the occurrence_of_a_non-terminating reduction sequence. Rather,_we conclude that the occurrence of_a non-terminating reduction sequence is related_to a set of reduction procedures. We_already have Proposition 2.2.1, 2.2.2, and_2.4.3. However, those results are_based on_the formalizations of Tennant (2015)_which are not_stated in_generalized form. TCPE_does not restrict the form of_elimination rules. For_our convenience, we will use in_this_appendix, TCPE rather_than_Tennant's_later criterion._As we will_see_in Chapter_3,_the latest criterion for paradoxicality introduced_by_Tennant (2016) has an additional condition that_all elimination rules are_stated_in generalized form. Moreover,_while Tennant (2016, pp. 12–16)_asserts that Liar paradox is a_genuine paradox,_he proposes_the id est rules for the liar sentence Ψ. 79 [T_(⌜Ψ⌝)]1 D1 ⊥ ΨI,1 Ψ Ψ_[¬T (⌜Ψ⌝)]1 D2 φ_ΨE,1 φ_Tennant_(2016, p. 12)_calls_Ψ−rules the_id est rules and said, [ΨI− and_ΨE−rules] are_the 'id est' rules for the_Liar (so-called because of_the_familiar transitions '[Ψ], i.e. [¬T (⌜Ψ⌝)]')._The rules [ΨI− and ΨE−rules] ensure_that the sentence called [Ψ]_is_interdeducible_with [¬T (⌜Ψ⌝)] certainly a_necessary (even if not sufficient) condition_for the former_to be the latter. Ψ−rules have several_peculiar_points. First, ΨI−rule introduces a seeming