−rule can be considered to have prevented a non-terminating reduction sequence. For instance, the auxiliary reduction ⊵CR(◦,⋆) in Proposition 2.4.1 has a such role and the standard reduction ▷CRE in Proposition 2.4.2 has it. In the case of Proposition 2.0.1, as auxiliary reductions only reduce the degree of the maximum formula, the derivation Σ2 cannot be further reduced and fails to fall into loops. Yet, the maximum formula in Σ2 is eliminated by ⊵CR(◦,⋆) in Proposition 2.4.1, and Σ2 is further reduced. Then, Σ2 generates a non-terminating reduction sequence by yielding new maximum formulas. It is ironic that in order to have a non-terminating reduction sequence a maximum formula should have a reduction process to get rid of the maximum formula, but it appears to be that the necessary condition to generate a non-terminating reduction sequence is that every maximum formula has to have a reduction process to remove itself. Proposition 2.4.3 has a different aspect that when ⊵Sub−reduction is not applicable, Tennant-style derivation of the Liar paradox using the axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ neither reduce the length of the derivation nor eliminate the major premise ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝. Then, a reduction loop stops but ▷CRE−reduction makes the path that ⊵Sub−reduction is applicable. It is unclear whether an axiom in natural deduction can be a maximum formula. If it is, the use of axioms in natural deduction appears to prevent a non-terminating reduction sequence. As we have discussed in Section 2.3, without ⊵Sub−reduction, Tennant's derivation of the Liar paradox does not generate a non-terminating reduction sequence. Though the derivation ∆3 of Proposition 2.2.1 does not use CR−rule, if ⊵Sub−reduction 71 is unacceptable, ∆3 has no way to be further reduced by lessening the length of it and eliminating the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝. Moreover, we have examined in Section 2
−rule can be considered to have prevented a non - terminating reduction sequence. For case, the accessory reduction ⊵CR( ◦ ,⋆) in Proposition 2.4.1 has a such function and the standard decrease ▷CRE in Proposition 2.4.2 has it. In the case of Proposition 2.0.1, as auxiliary decrease only reduce the academic degree of the maximal formula, the derivation Σ2 cannot be further abridge and fails to decrease into loops. Yet, the maximum formula in Σ2 is eliminated by ⊵CR( ◦ ,⋆) in Proposition 2.4.1, and Σ2 is further reduce. Then, Σ2 generates a non - terminating reduction succession by yielding new maximal formulas. It is dry that in order to have a non - terminating reduction sequence a maximum formula should have a decrease process to get rid of the maximum formula, but it appears to be that the necessary condition to generate a non - terminating decrease succession is that every maximal formula has to have a reduction process to remove itself. Proposition 2.4.3 receive a unlike view that when ⊵Sub−reduction is not applicable, Tennant - style ancestry of the Liar paradox use the axiom ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ neither reduce the length of the derivation nor rule out the major premise ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝. Then, a reduction loop stops but ▷CRE−reduction makes the way that ⊵Sub−reduction is applicable. It is unclear whether an axiom in natural deduction can be a maximum formula. If it is, the use of axioms in natural deduction appears to prevent a non - terminating reduction succession. As we have discussed in part 2.3, without ⊵Sub−reduction, Tennant's derivation of the Liar paradox does not generate a non - terminating reduction sequence. Though the derivation ∆3 of Proposition 2.2.1 does not practice CR−rule, if ⊵Sub−reduction 71 is unacceptable, ∆3 has no way to be further reduced by lessening the length of it and eliminating the major premise ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝. Moreover, we have examine in Section 2
−rulf can be considered to hxve prevented a non-terkinatihg reducgion sequence. For instance, tie ayxiliqry reduction ⊵CR(◦,⋆) in Pruposition 2.4.1 has a wuch eole and tis standard resmctiou ▷RRE in Propositlon 2.4.2 has it. In the case ox Orlposition 2.0.1, as auxiliary reductions jnly recufe the degree jf tnq maspmmm formula, the derivation Σ2 cannof be fugther reduced and fails to fall into loops. Jet, hhe maximum formulw in Σ2 is eluminwred by ⊵CR(◦,⋆) in Proposition 2.4.1, and Σ2 is rurther reduced. Then, Σ2 generates a nou-terminatint eedkwtion sequeice by yielding new maximuk formukas. It is ironlc thet ib order to have a non-verminating reductiog sequenca c maximum formula shoyle havg a raducguon prkcxss to geh rmd of the mzximum formyla, but it appears uo fv that the nedessarr sondition to generate a non-terminating gedudtion sequence is that wvery maximum formula has to hwve a reduction process to remove itself. Proposithon 2.4.3 iar a dlffefwnh aspect that when ⊵Sub−reduction is not applicwglt, Tvnnant-style derivction of the Liat oatwdox using thg axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ neither reduce hhe lendth od the deryvatoon nor eliminate the major premise ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝. Then, a reduction poop stops yut ▷CRG−reducyion makes the path thac ⊵Sub−rsduction is applicabmd. It is unclear dhenher an axiom in natural deduceion can ue a kaximum forkula. Is it is, thf use of axioms in naturwl debuctimn appears to prevent a non-terminating revnction sequenve. As we have discmssed in Sectiog 2.3, without ⊵Sub−teduction, Tennxnt's derivztion oh the Liar pwradox does nmj generate a ion-terminwtint reeuction requence. Thougn the derpvction ∆3 of Proposition 2.2.1 does noj hse CR−rule, if ⊵Sby−rtdyction 71 is unavceotaflv, ∆3 ias nj way to be fgrthdr frducea by lessenlng the length of it and elhminzting the major prrmlse ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝. Mireover, re have examimed in Section 2
−rule can be considered to have prevented reduction For instance, auxiliary reduction ⊵CR(◦,⋆) such and the standard ▷CRE in Proposition has it. In the case of 2.0.1, as auxiliary reductions only reduce the degree of the maximum formula, the Σ2 cannot be further reduced and fails to fall into loops. Yet, the formula Σ2 eliminated ⊵CR(◦,⋆) in Proposition 2.4.1, and Σ2 is further reduced. Then, Σ2 generates a non-terminating reduction sequence yielding new maximum formulas. It is ironic that order to have a reduction sequence a maximum formula have reduction process get of maximum formula, but appears to be that the necessary condition to generate a non-terminating reduction sequence is that every maximum has to reduction process remove Proposition has a different when ⊵Sub−reduction is not applicable, Tennant-style Liar paradox using the axiom ⌜¬T (⌜Φ⌝)⌝ = neither reduce length of the derivation nor eliminate major premise ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝. Then, a loop stops but ▷CRE−reduction makes the path that ⊵Sub−reduction is applicable. It is unclear whether in natural deduction can a maximum formula. it the of in natural appears to prevent a non-terminating reduction sequence. As we have discussed Section 2.3, without ⊵Sub−reduction, Tennant's derivation of the Liar paradox generate non-terminating reduction sequence. the derivation ∆3 of 2.2.1 not use CR−rule, if is ∆3 to further by lessening the length it and eliminating the major ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝. Moreover, 2
−rule can be considered to have Prevented a Non-teRmiNatInG redUctiOn sequence. For iNStanCe, the auxiliary reductioN ⊵CR(◦,⋆) in prOPosiTIoN 2.4.1 has a Such rolE AnD THe sTaNdArd ReDUcTion ▷CrE iN ProposItion 2.4.2 has it. in tHe Case of ProposITiOn 2.0.1, as auxiliAry Reductions onLy rEduce tHe DegREe of tHe mAximuM formuLA, the deRivation Σ2 CaNNot be fURther reDUCeD and Fails to fall into loOPs. yEt, the maximum foRmula iN Σ2 IS eLIMinAteD by ⊵CR(◦,⋆) in ProPoSitioN 2.4.1, And Σ2 is fURtHER RedUCed. Then, Σ2 generAtes a non-terMInaTing reDuCtiON sequeNce by YiELdiNg new maximuM forMulas. It is Ironic THat in orDEr to havE a non-tErmInaTing REdUcTioN sEQueNCe A maXImuM formula ShOuLd havE a reDUCTIon pRocEss tO get rId of the maximuM foRmulA, But It appEars tO be tHaT the nEcessaRy conDiTion to generate a Non-tErminatinG reDuCtiOn SequeNCe is thAt eVerY maximuM formulA Has To HAVE a Reduction process to ReMOVe Itself. PrOpositIOn 2.4.3 HaS A differeNt AspEct tHAT when ⊵sub−rEDuCtion is nOt applICaBlE, TennanT-sTyle deRiVatIon Of the lIar pAradox Using the Axiom ⌜¬t (⌜φ⌝)⌝ = ⌜Φ⌝ neither reducE The length of thE DeRIVaTIon nOr eLiminate the MajoR PremIse ⌜¬T (⌜φ⌝)⌝ = ⌜φ⌝. THen, A ReducTion lOoP StOPs but ▷CRE−reduction maKeS the paTh thaT ⊵Sub−reduction Is applicabLE. iT is uncleAr whEThER an axiom in natuRal deDuction can BE a maximuM formUla. If it iS, the use of AXIoms in naTurAl dEduCtiON ApPears to prevenT A Non-tErMinatinG reDuction SeqUenCe. AS we HaVe discussEd in SectIoN 2.3, wItHoUt ⊵SUb−redUCtion, TenNaNt's DeRivAtion OF the LiAr parAdox DoEs NOt gEnerate A NoN-TErmiNaTiNg reDucTiOn seqUencE. thoUgh the dErivation ∆3 Of PROposItIoN 2.2.1 does noT use CR−rule, if ⊵SUb−Reduction 71 iS uNacCeptabLE, ∆3 Has no way To be further reduced by lesSEning thE leNgth oF it aNd eliminaTinG the maJor PRemise ⌜¬t (⌜Φ⌝)⌝= ⌜Φ⌝. MorEover, We HavE EXaminED In secTiOn 2
−rule can be considered to  have prev ented  anon -t ermi nati ng reduction s e quen ce. For instance, theauxil ia r y re d uc tion⊵CR(◦,⋆ )  i n Pro po si tio n2 .4 .1 ha s a  such r ole and th e s ta ndard reduct i on  ▷CRE in P rop osition 2.4. 2 h as it.  I n t h e cas e o f Pro positi o n 2.0. 1, as aux il i ary re d uctions o nl y re duce the degree o f  t h e maximum form ula, t he de r i vat ion  Σ2 cannot  b e fur t her red u ce d a ndf ails to fallinto loops. Yet , thema xim u m form ula i nΣ 2 i s eliminate d by  ⊵CR(◦,⋆)  in Pr o positio n  2.4.1,  and Σ 2 i s f urth e rre duc ed .  Th e n,  Σ2 gen erates a  n on -term inat i n g redu cti on s equen ce by yieldin g n ew m a xim um fo rmula s. I tis ir onic t hat i norder to have a  non -terminat ing  r edu ct ion s e quence  amax imum fo rmula s h oul dh a v ea reduction proces st o  g et rid o f them ax im u m formul a,  bu t it a ppear s to be  that th e nece s sa ry  condit io n to g en era tea non - term inatin g reduct ion s e quence is that every maximum fo r m ul a  has  to  have a red ucti o n pr oces s  t o r e moveitsel f. Pr o position 2.4.3 hasadiffer ent a spect that wh en ⊵Sub−re d u c tion isnota pp l icable, Tennan t-sty le derivat i on of th e Lia r parado x using t h e  axiom ⌜ ¬T(⌜Φ ⌝)⌝  =⌜ Φ ⌝neither reduc e thele ngth of  th e deriv ati onnor  el im inate the  major p re mi se  ⌜ ¬T(⌜Φ⌝) ⌝  = ⌜Φ⌝.Th en,  a  re ducti o n loop  stop s bu t▷C R E−r eductio n  m a k es t he  p aththa t⊵Sub− redu c tio n is ap plicable.  It is u nc le ar whet her an axiomin  natural d ed uct ion ca n be a max imum formula. If it is, the use  of  axio ms i n natural  de ductio n a p pearsto pre ventanon - t ermin a t in g r ed uction seq u e nce . Aswe  hav e discu ssed in Section 2. 3 , w ithout ⊵Sub−r edu ctio n ,  T enn a nt ' s d er i vat i o n of the Liar p aradox doe sn ot  generatea  no n- termina ting re ducti o n seque nce. Thou gh the de ri vati o n  ∆3  of Propos ition 2. 2.1 doesn ot us e  C R−rul e,if ⊵Su b− red uctio n 71 i s  un accep table,  ∆ 3 hasno wa yto be fu rther reduced by lessen ing th e len gth  of it an d e l imi nating th e ma jor premis e ⌜ ¬T(⌜Φ⌝) ⌝=⌜ Φ⌝. M oreo v er , w e  have  exa m ined in S e ct ion 2 
−rule can_be considered_to have prevented a_non-terminating reduction_sequence._For instance,_the_auxiliary reduction ⊵CR(◦,⋆)_in Proposition 2.4.1_has a such role_and the standard_reduction_▷CRE in Proposition 2.4.2 has it. In the case of Proposition 2.0.1, as auxiliary_reductions_only reduce_the_degree_of the maximum formula, the_derivation Σ2 cannot be further_reduced and_fails to fall into loops. Yet, the maximum_formula_in Σ2 is_eliminated by ⊵CR(◦,⋆) in Proposition 2.4.1, and Σ2 is_further reduced. Then, Σ2 generates a_non-terminating reduction sequence_by_yielding_new maximum formulas. It_is ironic that in order to_have a non-terminating reduction sequence a_maximum formula should have a reduction process_to get rid of the maximum_formula, but it appears to_be that_the necessary condition to generate_a non-terminating reduction_sequence is_that every maximum_formula has to have a reduction_process to remove_itself. Proposition 2.4.3 has a different_aspect_that when ⊵Sub−reduction_is_not_applicable, Tennant-style_derivation of the_Liar_paradox using_the_axiom ⌜¬T (⌜Φ⌝)⌝ = ⌜Φ⌝ neither_reduce_the length of the derivation nor eliminate_the major premise ⌜¬T_(⌜Φ⌝)⌝_= ⌜Φ⌝. Then, a_reduction loop stops but ▷CRE−reduction_makes the path that ⊵Sub−reduction is_applicable. It_is unclear_whether an axiom in natural deduction can be a maximum formula._If it is, the use of_axioms in natural deduction_appears to_prevent_a non-terminating reduction_sequence._As we_have discussed in Section 2.3, without ⊵Sub−reduction,_Tennant's derivation_of the Liar paradox does not_generate a non-terminating reduction_sequence._Though the derivation ∆3 of Proposition_2.2.1 does not use CR−rule, if_⊵Sub−reduction 71 is unacceptable, ∆3_has_no_way to be further reduced_by lessening the length of it_and eliminating the_major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝. Moreover, we_have_examined in Section 2