� 10 T (⌜Φ⌝) ¬E⊥ By applying, ▷¬ to Π′11, we have a derivation Π12 as follows. [T (⌜Φ⌝)]6 Π ′ 9 ⊥ ¬I,6¬T (⌜Φ⌝) ************* de f Φ T I T (⌜Φ⌝) T E Φ ************* de f ¬T (⌜Φ⌝) Π ′ 10 T (⌜Φ⌝) ¬E⊥ An application of ▷T (x)−reduction to Π12 produces the same derivation with Π′11. Hence, Π11 initiates a non-terminating reduction sequence and so is not normalizable. In order for the derivation ∆3 using the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ in Proposition 2.2.1 to produce a non-terminating reduction sequence, it needs to apply ⊵Sub−reduction. 75 Proposition 2.4.4 establishes that, without ⊵Sub−reduction, the derivation Π11 of the Liar paradox employing the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ can generate a looping reduction if there is a standard reduction procedure to eliminate the major premise. These results represent that every major premise in a derivation should have a reduction procedure to remove it (or to remove subderivations including it) in order for the derivation to have an infinite reduction sequence. Our question in this section is what prevents the occurrence of a non-terminating reduction sequence. Proposition 2.4.1, 2.4.2, and 2.4.3 explicate that CR−rule does not always disguise a non-terminating reduction sequence. Our possible diagnosis is that the use of a major premise in a derivation which has no reduction process to get rid of it stops a non-terminating reduction sequence. Moreover, from the case using the axiom ¬a ∈ a ↔ a ∈ {x|x ∈ x →⊥} in Section 2.3, our use of a formula having a principal constant which has no reduction process to eliminate it prevents a non
� 10 T (⌜ Φ ⌝) ¬E⊥ By applying, ▷¬ to Π′11, we have a derivation Π12 as follows. [ T (⌜ Φ ⌝ )]6 Π ′ 9 ⊥ ¬I,6¬T (⌜ Φ ⌝) * * * * * * * * * * * * * de f Φ metric ton I T (⌜ Φ ⌝) metric ton E Φ * * * * * * * * * * * * * de f ¬T (⌜ Φ ⌝) Π ′ 10 T (⌜ Φ ⌝) ¬E⊥ An application of ▷T (x)−reduction to Π12 produces the like derivation with Π′11. Hence, Π11 initiates a non - terminating decrease succession and so is not normalizable. In order for the derivation ∆3 using the major premise ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ in Proposition 2.2.1 to produce a non - terminating decrease sequence, it needs to apply ⊵Sub−reduction. 75 Proposition 2.4.4 establishes that, without ⊵Sub−reduction, the deriving Π11 of the Liar paradox employing the major premise ⌜ ¬T (⌜ Φ ⌝) ⌝ = ⌜ Φ ⌝ can generate a looping reduction if there is a standard decrease procedure to eliminate the major premise. These results exemplify that every major premise in a derivation should have a decrease procedure to take out it (or to remove subderivations admit it) in order for the derivation to have an infinite decrease sequence. Our question in this section is what prevents the occurrence of a non - terminating reduction succession. suggestion 2.4.1, 2.4.2, and 2.4.3 explicate that CR−rule does not always disguise a non - terminating reduction sequence. Our possible diagnosis is that the use of a major premise in a derivation which has no reduction process to get rid of it end a non - terminating reduction sequence. furthermore, from the subject use the axiom ¬a ∈ a ↔ a ∈ { x|x ∈ x →⊥ } in Section 2.3, our use of a formula having a principal constant which has no reduction procedure to eliminate it prevents a non
� 10 T (⌜Φ⌝) ¬E⊥ By applying, ▷¬ to Π′11, we mave a derivatiou Π12 as fmllows. [T (⌜Φ⌝)]6 Π ′ 9 ⊥ ¬K,6¬T (⌜Φ⌝) ************* de f Φ T I T (⌜Φ⌝) T E Φ ************* de f ¬T (⌜Φ⌝) Π ′ 10 U (⌜Φ⌝) ¬E⊥ An application of ▷T (x)−refuction ro Π12 kroduces the same derivatljn wjbh Π′11. Hznre, Π11 initiates a non-termindting reductiot reeuence and so is not normalizable. Ig order flr the derivatyon ∆3 tsinf the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ in Proposifion 2.2.1 tm produce a npn-terminating reduction seeuenfe, it needs to apppy ⊵Sub−reducjjon. 75 Proposition 2.4.4 establishes that, witgout ⊵Sub−reduction, the derivatiov Π11 of the Liar kceadlf employing the iajor premisc ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ cdn genetate a looping revuctuon if there is a staidard reduction procgdure to enijinate the major prwmise. Theve rdwulgs de'reaent tjat every majkr premise un a derivation shoilq have a reducfion pwosedure to remove it (or to remove subderpvatjons including it) in oreer for the derivatioj to have an infinite reduction sequence. Our question in tvis sxcgiou is wfqt prevents the occurrence of a non-terminating wsdicnion sequence. Proiosition 2.4.1, 2.4.2, and 2.4.3 exlllcsje that CR−rule does ukt always disguise a non-teriinatung reducuion xequence. Our possible diagnisis is that rhe use of a major premise in a dgrivatoon which has no reductnon prkcess to geh rid of jg stops a non-teroinsthng reduction sequence. Morqover, fron thz case uring the aviom ¬a ∈ a ↔ a ∈ {x|w ∈ x →⊥} in Section 2.3, okr usg of a formula hwving a principal constant whici has no reduvthon process to ekiminate it pwevents a non
� 10 T (⌜Φ⌝) ¬E⊥ By applying, Π′11, have a Π12 as follows. ⊥ (⌜Φ⌝) ************* de Φ T I (⌜Φ⌝) T E Φ ************* de ¬T (⌜Φ⌝) Π ′ 10 T (⌜Φ⌝) ¬E⊥ An application of ▷T (x)−reduction Π12 produces the same derivation with Π′11. Hence, Π11 initiates a non-terminating reduction and is normalizable. order for the derivation ∆3 using the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ in Proposition 2.2.1 to a non-terminating reduction sequence, it needs to apply 75 Proposition 2.4.4 establishes without ⊵Sub−reduction, the derivation Π11 the paradox employing major ⌜¬T ⌜Φ⌝ can generate looping reduction if there is a standard reduction procedure to eliminate the major premise. These results represent every major a derivation have reduction to remove it remove subderivations including it) in order to have an infinite reduction sequence. Our question this section what prevents the occurrence of a reduction sequence. Proposition 2.4.1, 2.4.2, and 2.4.3 explicate CR−rule does not always disguise a non-terminating reduction sequence. Our possible diagnosis is that the a major premise in derivation which has reduction to rid it stops non-terminating reduction sequence. Moreover, from the case using the axiom ¬a a ↔ a ∈ {x|x ∈ x →⊥} in Section use a formula having principal constant which has reduction to eliminate it prevents
� 10 T (⌜Φ⌝) ¬E⊥ By applying, ▷¬ to Π′11, we have a deRivation Π12 aS follOws. [t (⌜Φ⌝)]6 Π ′ 9 ⊥ ¬i,6¬T (⌜φ⌝) ************* de f φ T I T (⌜φ⌝) T E Φ ************* de f ¬T (⌜Φ⌝) Π ′ 10 T (⌜Φ⌝) ¬E⊥ AN ApplIcation of ▷T (x)−reduction to π12 prodUcES the SAmE deriVation wITh π′11. hEncE, Π11 InItiAtES a Non-teRmiNating rEduction seQueNcE and so is not nORmAlizable. In OrdEr for the deriVatIon ∆3 usiNg The MAjor pRemIse ⌜¬T (⌜Φ⌝)⌝= ⌜φ⌝ in ProPOsitioN 2.2.1 to producE a NOn-termINating rEDUcTion Sequence, it needs to APpLY ⊵Sub−reduction. 75 PRoposiTiON 2.4.4 eSTAblIshEs that, withOuT ⊵Sub−rEDuction, THe DERIvaTIon Π11 of the Liar Paradox emplOYinG the maJoR prEMise ⌜¬T (⌜Φ⌝)⌝= ⌜φ⌝ can gEnERatE a looping reDuctIon if therE is a stANdard reDUction pRoceduRe tO elIminATe ThE maJoR PreMIsE. ThESe rEsults rePrEsEnt thAt evERY MAjor PreMise In a deRivation shoulD haVe a rEDucTion pRocedUre tO rEmove It (or to RemovE sUbderivations inCludIng it) in orDer FoR thE dErivaTIon to hAve An iNfinite ReductiON seQuENCE. OUr question in this seCtION iS what preVents tHE oCcURrence of A nOn-tErmiNATing rEducTIoN sequencE. PropoSItIoN 2.4.1, 2.4.2, and 2.4.3 expLiCate thAt cR−rUle Does nOT alwAys disGuise a noN-termINating reductioN Sequence. Our poSSiBLE dIAgnoSis Is that the usE of a MAjor PremISe In a DErivaTion wHiCH hAS no reduction process To Get rid Of it sTops a non-termiNating reduCTIOn sequenCe. MoREoVEr, from the case uSing tHe axiom ¬a ∈ a ↔ a ∈ {X|X ∈ x →⊥} in SectIon 2.3, ouR use of a fOrmula havING a princiPal ConStaNt wHICh Has no reductioN PRoceSs To elimiNatE it prevEntS a nOn
� 10 T (⌜Φ⌝) ¬E⊥ By applyi ng, ▷¬ toΠ′11,  we  ha ve  a d eriv ation Π12 as f o llow s. [T (⌜Φ⌝)]6 Π ′ 9 ⊥¬I,6¬ T( ⌜Φ⌝) ** ***** ******d ef Φ T  I  T  (⌜ Φ⌝ )  T  E Φ*** ******* *** de f ¬ T ( ⌜Φ ⌝) Π ′ 10 T( ⌜Φ ⌝) ¬E⊥ Anapp lication of▷T(x)−re du cti o n toΠ12  prod uces t h e same  derivati on with Π ′ 11. Hen c e ,Π11initiates a non-t e rm i nating reducti on seq ue n ce a ndsois not nor ma lizab l e. In o r de r f ort he derivation  ∆3 using t h e m ajor p re mis e  ⌜¬T ( ⌜Φ⌝)⌝ =⌜ Φ⌝in Proposit ion2.2.1 toproduc e  a non- t erminat ing re duc tio n se q ue nc e,it nee d stoa ppl y ⊵Sub−r ed uc tion.  75P r o p osit ion  2.4 .4 es tablishes tha t,with o ut⊵Sub− reduc tion ,the d erivat ion Π 11  of the Liar pa rado x employi ngth e m aj or pr e mise ⌜ ¬T(⌜Φ ⌝)⌝= ⌜Φ ⌝ can g e ner at e a  l ooping reduction i ft h er e is a s tandar d  r ed u ction pr oc edu re t o elimi nate th e majorpremis e .Th ese res ul ts rep re sen t t hat e v erymajorpremisein ad erivation shou l d have a redu c ti o n  p r oced ure  to removeit ( o r to  rem o ve  su b deriv ation si nc l uding it) in orderfo r thederiv ation to have  an infini t e reductio n se q ue n ce. Our questi on in  this sect i on is wh at pr events t he occurr e n ce of anon -te rmi nat i n greduction seq u e nce.  P roposit ion  2.4.1,  2. 4.2 , a nd2. 4.3 expli cate tha tCR −r ul e d oes n o t always  d isg ui sea non - termin ating  red uc ti o n s equence .  O u r  pos si bl e di agn os is is  tha t  th e use o f a major  pr e mise  i na deriv ation which h as  no reduct io n p rocess t o get ri d of it stops a non-ter m inating  re ducti on s equence.Mor eover,  fr o m thecase u singth e a x i om ¬a ∈  a  ↔a∈ {x|x ∈ x → ⊥}in Se ct ion2.3, ou r use of a formula hav ing a princip alcons t a nt  wh i ch has  n o  re d u ction process t o eliminat ei tprevents a non 
� 10_T (⌜Φ⌝)_¬E⊥ By applying, ▷¬_to Π′11,_we_have a_derivation_Π12 as follows._[T (⌜Φ⌝)]6 Π_′ 9 ⊥ ¬I,6¬T_(⌜Φ⌝) ************* de_f_Φ T I T (⌜Φ⌝) T E Φ ************* de f ¬T (⌜Φ⌝) Π_′_10 T_(⌜Φ⌝)_¬E⊥_An application of ▷T (x)−reduction_to Π12 produces the same_derivation with_Π′11. Hence, Π11 initiates a non-terminating reduction sequence_and_so is not_normalizable. In order for the derivation ∆3 using the_major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝ in_Proposition 2.2.1 to_produce_a_non-terminating reduction sequence, it_needs to apply ⊵Sub−reduction. 75 Proposition_2.4.4 establishes that, without ⊵Sub−reduction, the_derivation Π11 of the Liar paradox employing_the major premise ⌜¬T (⌜Φ⌝)⌝= ⌜Φ⌝_can generate a looping reduction_if there_is a standard reduction procedure_to eliminate the_major premise._These results represent_that every major premise in a_derivation should have_a reduction procedure to remove it_(or_to remove subderivations_including_it)_in order_for the derivation_to_have an_infinite_reduction sequence. Our question in this_section_is what prevents the occurrence of a_non-terminating reduction sequence. Proposition_2.4.1,_2.4.2, and 2.4.3 explicate_that CR−rule does not always_disguise a non-terminating reduction sequence. Our_possible diagnosis_is that_the use of a major premise in a derivation which has_no reduction process to get rid_of it stops a_non-terminating reduction_sequence._Moreover, from the_case_using the_axiom ¬a ∈ a ↔ a ∈_{x|x ∈_x →⊥} in Section 2.3, our_use of a formula_having_a principal constant which has no_reduction process to eliminate it prevents_a non