 problem. We now consider the second case that raises the problem of undergeneration without classical inferences. When we use generalized form of elimination rules with permutation conversions, there is a closed full normal derivation of ⊥ which formalizes the Liar paradox. For the example, we have the generalized elimination rules for the truth-predicate T (x) and its standard reduction process. D1 φ T I T (⌜φ⌝) T (⌜φ⌝) [φ]1 D2 ψ T E,1 ψ 9With respect to the problem of overgeneration, the issue with tests of a proper reduction will be discussed in Chapter 3. 10The assessment of his solution will be discussed in the next chapter. 54 The reduction procedure for T (x) D1 φ T I T (⌜φ⌝) [φ]1 D2 ψ T E,1 ψ ▷T (x) D1 φ D2 ψ Also, we take = E−rule having the form of the generalized elimination rule. t1 = t2 D1 φ(t1) [φ(t2)] 1 D2 ψ = E,1 ψ Then, Proposition 2.3.1 shows that there exists a full normal derivation of ⊥ from the Liar paradox. Proposition 2.3.1. Let SR be a system containing T−, ¬−, =−rules with their generalized form of elimination rules. SR has a set RR of reduction procedures for T (x) and ¬, and ⊵Sub with permutation conversion. Suppose that, for some formula Φ, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ is an axiom of SR. There exists a closed full normal derivation ∆9 of ⊥ in SR relative to RR. Proof. Two claims prove the result. Claim 1. there is a closed derivation ∆9 of ⊥. To begin with, we have an open derivation ∆7 of ⊥ from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 [T (⌜¬T (⌜Φ⌝)�
problem. We now consider the second case that raise the trouble of undergeneration without classical inferences. When we practice generalized form of elimination rule with permutation conversions, there be a closed full normal deriving of ⊥ which formalizes the Liar paradox. For the example, we get the generalized elimination rules for the truth - predicate T (x) and its standard reduction process. D1 φ T I T (⌜ φ ⌝) T (⌜ φ ⌝) [ φ]1 D2 ψ T E,1 ψ 9With esteem to the problem of overgeneration, the issue with tests of a proper decrease will be discussed in Chapter 3. 10The assessment of his solution will be discourse in the next chapter. 54 The decrease procedure for T (x) D1 φ triiodothyronine I T (⌜ φ ⌝) [ φ]1 D2 ψ T E,1 ψ ▷T (x) D1 φ D2 ψ Also, we take = E−rule having the mannequin of the generalized elimination rule. t1 = t2 D1 φ(t1) [ φ(t2) ] 1 D2 ψ = E,1 ψ Then, Proposition 2.3.1 shows that there exists a full normal derivation of ⊥ from the Liar paradox. Proposition 2.3.1. Let SR be a system containing T−, ¬−, = −rules with their generalized form of elimination rules. SR has a set RR of reduction procedures for T (x) and ¬, and ⊵Sub with permutation conversion. Suppose that, for some recipe Φ, ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ is an axiom of SR. There exists a shut wide normal ancestry ∆9 of ⊥ in SR relative to RR. Proof. Two claim rise the result. Claim 1. there constitute a closed derivation ∆9 of ⊥. To begin with, we have an open derivation ∆7 of ⊥ from [ metric ton (⌜ Φ ⌝) ]. Ax1 ⌜ Φ ⌝ = ⌜ ¬T (⌜ Φ ⌝) ⌝ [ T (⌜ Φ ⌝ )]1 [ T (⌜ ¬T (⌜ Φ ⌝) �
 prlblem. We now consider tht second case thaj eaises the pdoblem ow undergeneration without clessixal ibferences. When we use eeneralizvd form od elmmination rules xjth permutatikk conrecsions, there is a closed xull normal desixacion of ⊥ which formalizes the Liar pwradox. Gog the example, re hsde tgv neneralized elimination rules fod the tguth-predicate T (x) and its standard reductioj prlcess. D1 φ T I T (⌜φ⌝) T (⌜φ⌝) [φ]1 D2 ψ T E,1 ψ 9Wuth wwspect to thd problem of overgenerztion, the issue with tests of a propzr reductiob qilp be discussxd in Bhapter 3. 10The assessmett of hos solution wikl ue duscussed in the next rhapter. 54 The reductijn procedgrz for T (x) D1 φ T I T (⌜φ⌝) [φ]1 E2 ψ T E,1 ψ ▷T (x) D1 φ A2 ψ Auso, wx tzke = E−gulx having ths form of tye generalized elimonwnoon rule. t1 = f2 D1 φ(t1) [φ(e2)] 1 D2 ψ = E,1 ψ Then, Proposition 2.3.1 shows that thtre esists a full normal deruvation of ⊥ from the Piar paraqox. Proposition 2.3.1. Let SR be a system containing T−, ¬−, =−rulea wiuh theif gfneralized form of elimination rules. SR has a aeu RG of reduction prjcedures fot H (c) and ¬, and ⊵Sub with 'srjutation conversioj. Suppofe thqt, for soie fprmula Φ, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ is an axiom of WR. There exifrs a closed full nlrmal derivction ∆9 of ⊥ on SR relative to RR. Pruof. Fwo claims orove the fesult. Claim 1. thdre iv a closta derivation ∆9 of ⊥. To begin witk, we havd an open qerivation ∆7 of ⊥ from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 [T (⌜¬T (⌜Φ⌝)�
problem. We now consider the second case the of undergeneration classical inferences. When elimination with permutation conversions, is a closed normal derivation of ⊥ which formalizes Liar paradox. For the example, we have the generalized elimination rules for the T (x) and its standard reduction process. D1 φ T I T (⌜φ⌝) (⌜φ⌝) D2 T ψ 9With respect to the problem of overgeneration, the issue with tests of a proper reduction be discussed in Chapter 3. 10The assessment of solution will be discussed the next chapter. 54 The procedure T (x) φ I (⌜φ⌝) [φ]1 D2 T E,1 ψ ▷T (x) D1 φ D2 ψ Also, we take = E−rule having the form the generalized t1 = D1 [φ(t2)] D2 ψ = Then, Proposition 2.3.1 shows that there normal derivation of ⊥ from the Liar paradox. 2.3.1. Let be a system containing T−, ¬−, with their generalized form of elimination rules. SR a set RR of reduction procedures for T (x) and ¬, and ⊵Sub with permutation that, for some formula ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ an of There a closed normal derivation ∆9 of ⊥ in SR relative to RR. Proof. claims prove the result. Claim 1. there is a closed of To begin with, have an open derivation of from [T (⌜Φ⌝)]. Ax1⌜Φ⌝= [T [T
 problem. We now consider the seCond case thAt raiSes The PrObleM of uNdergeneration WIthoUt classical inferences. WHen we UsE GeneRAlIzed fOrm of elIMiNATioN rUlEs wItH PeRmutaTioN converSions, there Is a ClOsed full normAL dErivation oF ⊥ whIch formalizeS thE Liar pArAdoX. for thE exAmple, We have THe geneRalized elImINation RUles for THE tRuth-Predicate T (x) and its STaNDard reduction pRocess. d1 φ t i T (⌜Φ⌝) t (⌜Φ⌝) [φ]1 D2 Ψ T E,1 Ψ 9With respeCt To the PRoblem oF OvERGEneRAtion, the issue With tests of A ProPer redUcTioN Will be DiscuSsED in chapter 3. 10The aSsesSment of hiS solutIOn will bE DiscussEd in thE neXt cHaptER. 54 THe RedUcTIon PRoCedURe fOr T (x) D1 φ T I t (⌜φ⌝) [Φ]1 D2 Ψ T E,1 ψ ▷T (X) D1 φ D2 Ψ aLSO, we tAke = e−rulE haviNg the form of thE geNeraLIzeD elimInatiOn ruLe. T1 = t2 D1 φ(t1) [Φ(t2)] 1 D2 ψ = E,1 ψ then, PRoPosition 2.3.1 shows thAt thEre exists A fuLl NorMaL deriVAtion oF ⊥ frOm tHe Liar pAradox. PROpoSiTION 2.3.1. LEt SR be a system contaInING T−, ¬−, =−Rules witH their GEnErALized forM oF elIminATIon ruLes. Sr HaS a set RR oF reducTIoN pRocedurEs For T (x) aNd ¬, And ⊵sub With pERmutAtion cOnversioN. SuppOSe that, for some fORmula Φ, ⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ is an AXiOM Of sr. TheRe eXists a closeD fulL NormAl deRIvAtiON ∆9 of ⊥ in sR relAtIVe TO RR. Proof. Two claims prOvE the reSult. CLaim 1. there is a cLosed derivATIOn ∆9 of ⊥. To beGin wITh, WE have an open derIvatiOn ∆7 of ⊥ from [T (⌜Φ⌝)]. aX1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1 [T (⌜¬T (⌜φ⌝)�
 problem. We now considerthe second  case  th atra ises  the  problem of un d erge neration without class icalin f eren c es . Whe n we us e  g e n era li ze d f or m  o f eli min ation r ules withper mu tation conve r si ons, there  is  a closed fu llnormal  d eri v ation  of  ⊥ wh ich fo r malize s the Lia rp aradox .  For th e ex ampl e, we have the ge n er a lized eliminat ion ru le s  f o r  th e t ruth-predi ca te T( x) andi ts s t and a rd reductionprocess. D1 φ T  I T ( ⌜φ ⌝)T  (⌜φ⌝)  [φ]1  D 2  ψT E,1 ψ 9Wi th r espect to  the p r oblem o f  overge nerati on,  th e is s ue  w ith  t e sts of  ap rop er reduc ti on  will  bed i s c usse d i n Ch apter  3. 10The ass ess ment ofhis s oluti on w il l bediscus sed i nthe next chapte r. 5 4 The red uct io n p ro cedur e  for T  (x ) D 1 φ T I  T (⌜φ⌝ )  [φ ]1 D 2  ψ  T E,1 ψ ▷T (x) D1  φ D 2ψ Also,we tak e  =  E − rule hav in g t he f o r m oftheg en eralized  elimi n at io n rule.  t 1 = t2  D 1 φ (t1 ) [φ( t 2)]1 D2 ψ  = E,1 ψ  Then ,  Proposition 2 . 3.1 shows tha t  t h e re exis tsa full norm al d e riva tion of  ⊥f rom t he Li ar pa r adox. Proposition 2 .3 .1. Le t SRbe a system c ontainingT − ,  ¬−, =−r ules wi t h their genera lized  form of e l iminatio n rul es. SR h as a setR R  of redu cti onpro ced u r es  for T (x) an d ¬, a nd  ⊵Sub w ith  permut ati oncon ver si on. Suppo se that,  f or  s om e f ormul a  Φ, ⌜Φ⌝=  ⌜ ¬T(⌜ Φ⌝) ⌝ isa n axio m ofSR.Th er e  ex ists ac lo s e d fu ll  n orma l d er ivati on ∆ 9  of  ⊥ in S R relativ e t o  RR.  P ro of. Two  claims prove  t he result.  C lai m 1. t h e re is aclosed derivation ∆9 of ⊥. To b egi n wit h, w e have an  op en der iva t ion ∆7  of ⊥from[T  (⌜ Φ ⌝ )]. A x 1 ⌜Φ ⌝=⌜¬ T (⌜Φ⌝)⌝ [ T (⌜Φ ⌝)]1[T  (⌜¬ T (⌜Φ⌝) �
 problem._We now_consider the second case_that raises_the_problem of_undergeneration_without classical inferences._When we use_generalized form of elimination_rules with permutation_conversions,_there is a closed full normal derivation of ⊥ which formalizes the Liar paradox._For_the example,_we_have_the generalized elimination rules for_the truth-predicate T (x) and_its standard_reduction process. D1 φ T I T (⌜φ⌝)_T_(⌜φ⌝) [φ]1 D2_ψ T E,1 ψ 9With respect to the problem_of overgeneration, the issue with tests_of a proper_reduction_will_be discussed in Chapter_3. 10The assessment of his solution_will be discussed in the next_chapter. 54 The reduction procedure for T_(x) D1 φ T I T_(⌜φ⌝) [φ]1 D2 ψ T_E,1 ψ_▷T (x) D1 φ D2_ψ Also, we_take =_E−rule having the_form of the generalized elimination rule._t1 = t2_D1 φ(t1) [φ(t2)] 1 D2 ψ_=_E,1 ψ Then,_Proposition_2.3.1_shows that_there exists a_full_normal derivation_of_⊥ from the Liar paradox. Proposition_2.3.1._Let SR be a system containing T−,_¬−, =−rules with their_generalized_form of elimination rules._SR has a set RR_of reduction procedures for T (x)_and ¬,_and ⊵Sub_with permutation conversion. Suppose that, for some formula Φ, ⌜Φ⌝= ⌜¬T_(⌜Φ⌝)⌝ is an axiom of SR._There exists a closed_full normal_derivation_∆9 of ⊥_in_SR relative_to RR. Proof. Two claims prove the_result. Claim_1. there is a closed derivation_∆9 of ⊥. To_begin_with, we have an open derivation_∆7 of ⊥ from [T (⌜Φ⌝)]._Ax1⌜Φ⌝= ⌜¬T (⌜Φ⌝)⌝ [T (⌜Φ⌝)]1_[T_(⌜¬T_(⌜Φ⌝)�