 into loops. t1 = t2 D1 φ(t1) t2 = t1 [φ(t2)] 1 [φ(t2)] 1, [φ(t1)] 2 D2 ρ = E,2 ρ = E,1 ρ ⊵GEg′ t1 = t2 D1 φ(t1) [φ(t2)] 1, D1 φ(t1) D2 ρ = E,1 ρ Our choice of generalized elimination rules is independent of the occurrence of a looping reduction sequence but is dependent on our choice of reduction procedures. Thus, we only consider the first and the second cases in this section. To begin with, in order to find a solution to the problem of undergeneration, it is necessary to diagnose what the culprit of the problem is. As Tennant (2015, p. 589) said, 'The use of classical reductio has masked the real defect that lies at the heart of paradoxical reasoning.' He believes that CR−rule disguises the non-terminating reduction sequence. He does not seem to consider seriously the undergeneration problem. The use of CR−rule does not always block the occurrence of the non-terminating reduction sequence. So to speak, CR−rule is not the only culprit of the problem of undergeneration. For instance, in the beginning of this chapter, we see the derivation Σ2 of Curry's paradox in Proposition 2.0.1 suggested by Rogerson (2006). Σ2 uses CR−rule and does not initiate a non-terminating reduction sequence. Proposition 2.0.1 seems to establish that CR−rule masks the nonterminating reduction sequence, but it is not. Rogerson (2006) only considered standard reduction processes suggested by Prawitz (1965). However, Schroeder-Heister and Tranchini (2017, pp. 572–573) borrow the reduction proposed by Gunnar Stålmarck (1991, pp. 131–132) and claim that her example can be further reduced. With regard to our example, Σ2 in Proposition 2.0.1, their reduction 60 denoted by ⊵CR(◦,⋆) may be depicted schematically as follows: [¬φ]1 D1 ⊥ CR,1 φ ◦E ψ D2 σ ⋆E ρ ⊵CR(◦,⋆) [¬ρ]2 [φ
into loops. t1 = t2 D1 φ(t1) t2 = t1 [ φ(t2) ] 1 [ φ(t2) ] 1, [ φ(t1) ] 2 D2 ρ = E,2 ρ = E,1 ρ ⊵GEg′ t1 = t2 D1 φ(t1) [ φ(t2) ] 1, D1 φ(t1) D2 ρ = E,1 ρ Our choice of generalized elimination rules is independent of the happening of a loop reduction sequence but is pendent on our option of reduction procedures. Thus, we merely consider the first and the second casing in this section. To begin with, in club to find a solution to the problem of undergeneration, it is necessary to diagnose what the culprit of the trouble is. As Tennant (2015, p. 589) said,' The use of authoritative reductio has mask the real defect that lies at the heart of paradoxical reasoning.' He believe that CR−rule disguises the non - terminating reduction sequence. He does not seem to see seriously the undergeneration problem. The use of CR−rule does not always block the happening of the non - terminating reduction sequence. So to speak, CR−rule is not the only culprit of the trouble of undergeneration. For case, in the beginning of this chapter, we see the derivation Σ2 of Curry's paradox in Proposition 2.0.1 suggested by Rogerson (2006). Σ2 uses CR−rule and does not initiate a non - terminating reduction sequence. Proposition 2.0.1 seems to establish that CR−rule dissemble the nonterminating reduction sequence, but it is not. Rogerson (2006) entirely consider standard reduction processes suggested by Prawitz (1965). However, Schroeder - Heister and Tranchini (2017, pp. 572–573) adopt the reduction propose by Gunnar Stålmarck (1991, pp. 131–132) and claim that her example can be further reduce. With regard to our example, Σ2 in Proposition 2.0.1, their reduction 60 denoted by ⊵CR( ◦ ,⋆) may be describe schematically as follows: [ ¬φ]1 D1 ⊥ CR,1 φ ◦ E ψ D2 σ ⋆E ρ ⊵CR( ◦ ,⋆) [ ¬ρ]2 [ φ
 inho loops. t1 = t2 D1 φ(t1) t2 = t1 [φ(t2)] 1 [φ(t2)] 1, [φ(t1)] 2 D2 ρ = E,2 ρ = E,1 ρ ⊵GEg′ t1 = t2 S1 φ(t1) [φ(t2)] 1, D1 φ(t1) D2 ρ = E,1 ρ Our choice of genecalized eoimination rules is inaependent of the iccucrence of a loopmhg redugcion acquenee but is dependgnt on our cvoice of reducdiun procedures. Thus, we only consider tre firsy wnd the second casts yn tgps section. To begin with, in ordsr to fpnd a solution to the problem of undergenerwtioj, it is necessary ho diagnose whae the culprit of the problem is. As Jennant (2015, p. 589) said, 'The use of clasrical reductio yaw mwvked the reel defvct that lies at the veart og paradoxical veasoiing.' He believes that CR−rnle disguises the nog-terminathny reduction sequence. Ye does not seeo to cohsmded serilusmy the undsrgeneratiob problem. The use og SG−tule does nof alwars block the occurrence of the non-terminauing deduction sequence. So ti speak, CR−rule is not the only culprit of the problem of undergeneration. For invtancx, kn uhc cwglnning of this chapter, we see the derivation Σ2 kf Clrry's paradox in Iroposition 2.0.1 suggextfd fy Rogerson (2006). Σ2 uses ED−rhle and does not ijitiate a nob-terminatyng teduction sequence. Proposituon 2.0.1 seems tj establish that CR−xule masks tke nonjerminsting reduction sequencz, but jt is not. Rlgerson (2006) kvly considered sgancasd reducukon processes sugdested by Pracitz (1965). Hodevet, Schroqder-Heisteg and Tranchini (2017, pp. 572–573) borgow tke reguction prlposed by Gunnar Stålmarck (1991, pp. 131–132) eid claim that har vxample ccn be nurther reduced. With regard tp our erample, Σ2 in Propoaition 2.0.1, their redustion 60 denoteg by ⊵CR(◦,⋆) may bx depicteq scyemarically xs follows: [¬φ]1 D1 ⊥ CR,1 φ ◦E ψ D2 σ ⋆E ρ ⊵CR(◦,⋆) [¬ρ]2 [φ
into loops. t1 = t2 D1 φ(t1) t1 1 [φ(t2)] [φ(t1)] 2 D2 E,1 ⊵GEg′ t1 = D1 φ(t1) [φ(t2)] D1 φ(t1) D2 ρ = E,1 Our choice of generalized elimination rules is independent of the occurrence of a reduction sequence but is dependent on our choice of reduction procedures. Thus, we consider first the cases in this section. To begin with, in order to find a solution to the problem undergeneration, it is necessary to diagnose what the of the problem is. Tennant (2015, p. 589) said, use classical reductio masked real that lies at heart of paradoxical reasoning.' He believes that CR−rule disguises the non-terminating reduction sequence. He does not seem consider seriously problem. The of does always block the the non-terminating reduction sequence. So to not the only culprit of the problem of For instance, the beginning of this chapter, we the derivation Σ2 of Curry's paradox in Proposition suggested by Rogerson (2006). Σ2 uses CR−rule and does not initiate a non-terminating reduction sequence. seems to establish that masks the nonterminating sequence, it not. (2006) only standard reduction processes suggested by Prawitz (1965). However, Schroeder-Heister and Tranchini pp. 572–573) borrow the reduction proposed by Gunnar Stålmarck (1991, and that her example be further reduced. With to example, Σ2 in Proposition reduction denoted be schematically follows: [¬φ]1 D1 ⊥ φ ◦E ψ D2 σ ρ ⊵CR(◦,⋆) [¬ρ]2 [φ
 into loops. t1 = t2 D1 φ(t1) t2 = t1 [φ(t2)] 1 [φ(t2)] 1, [φ(t1)] 2 D2 ρ = E,2 Ρ = E,1 ρ ⊵GEg′ t1 = t2 D1 φ(T1) [φ(t2)] 1, D1 φ(T1) D2 ρ = e,1 ρ OUr ChoiCe of Generalized eliMInatIon rules is independent oF the oCcURrenCE oF a looPing redUCtION seQuEnCe bUt IS dEpendEnt On our chOice of reduCtiOn Procedures. ThUS, wE only consiDer The first and tHe sEcond cAsEs iN This sEctIon. To Begin wITh, in orDer to find A sOLution TO the proBLEm Of unDergeneration, it is NEcESsary to diagnosE what tHe CUlPRIt oF thE problem is. as tennaNT (2015, p. 589) said, 'THE uSE OF clASsical reductiO has masked tHE reAl defeCt ThaT Lies at The heArT Of pAradoxical rEasoNing.' He belIeves tHAt CR−rulE DisguisEs the nOn-tErmInatINg ReDucTiON seQUeNce. hE doEs not seeM tO cOnsidEr seRIOUSly tHe uNderGenerAtion problem. THe uSe of cr−ruLe doeS not aLwayS bLock tHe occuRrencE oF the non-terminatIng rEduction sEquEnCe. SO tO speaK, cR−rule Is nOt tHe only cUlprit oF The PrOBLEm Of undergeneration. FOr INStAnce, in thE beginNInG oF This chapTeR, we See tHE DerivAtioN σ2 oF Curry's pAradox IN PRoPositioN 2.0.1 sUggestEd By ROgeRson (2006). Σ2 USes Cr−rule aNd does noT initIAte a non-terminaTIng reduction sEQuENCe. pRopoSitIon 2.0.1 seems to eStabLIsh tHat Cr−RuLe mASks thE nontErMInATing reduction sequenCe, But it iS not. ROgerson (2006) only coNsidered stANDArd reducTion PRoCEsses suggested By PraWitz (1965). HoweveR, schroedeR-HeisTer and TrAnchini (2017, pp. 572–573) BORrow the rEduCtiOn pRopOSEd By Gunnar StålmARCk (1991, pp. 131–132) AnD claim tHat Her examPle Can Be fUrtHeR reduced. WIth regarD tO oUr ExAmpLe, Σ2 in pRopositiOn 2.0.1, TheIr RedUctioN 60 DenoteD by ⊵CR(◦,⋆) May bE dEpICteD schemaTIcALLy as FoLlOws: [¬φ]1 d1 ⊥ CR,1 Φ ◦E Ψ D2 σ ⋆E ρ ⊵cR(◦,⋆) [¬ρ]2 [φ
 into loops. t1 = t2 D1 φ( t1) t2 = t 1 [φ( t2) ] 1  [ φ(t2 )] 1 , [φ(t1)] 2 D2 ρ =E,2 ρ = E,1 ρ ⊵GEg′ t1  = t2  D 1  φ(t 1 )[φ(t2 )] 1, D 1  φ ( t 1)D2  ρ  =E, 1  ρ  Ourcho ice ofgeneralize d e li mination rul e sis indepen den t of the occ urr ence o fa l o oping  re ducti on seq u ence b ut is dep en d ent on our cho i c eof r eduction procedur e s. Thus, we onlyconsid er th e fir stand the se co nd ca s es in t h is s e cti o n. To begin w ith, in ord e r t o find  a  so l utionto th ep rob lem of unde rgen eration,it isn ecessar y  to dia gnosewha t t he c u lp ri t o ft hep ro ble m  is . As Ten na nt  (201 5, p . 5 8 9) s aid , 'T he us e of classica l r educ t iohas m asked  the  r eal d efectthatli es at the heart  ofparadoxic alre aso ni ng.'H e beli eve s t hat CR− rule di s gui se s t he  non-terminating r ed u c ti on seque nce. H e  d oe s  not see mtocons i d er se riou s ly  the und ergene r at io n probl em . Theus e o f C R−rul e  doe s notalways b lockt he occurrenceo f the non-ter m in a t in g  red uct ion sequenc e. S o  tospea k ,CR− r ule i s not  t h eo nly culprit of thepr oblemof un dergeneration . For inst a n c e, in th e be g in n ing of this ch apter , we see t h e deriva tionΣ2 of Cu rry's par a d ox in Pr opo sit ion  2. 0 . 1suggested byR o gers on  (2006) . Σ 2 usesCR− rul e a nddo es not in itiate a  n on -t er min ating reductio nseq ue nce . Pro p ositio n 2.0 .1 s ee ms toestabli s ht h at C R− ru le m ask sthe n onte r min ating r eductionseq u ence ,bu t it is  not. Rogerso n(2006) onl ycon sidere d standard  reduction processes su g gestedbyPrawi tz ( 1965). Ho wev er, Sc hro e der-He isterand T ra nch i n i (20 1 7 ,pp.  5 72–573) bo r r owthe r ed ucti on prop osed by Gunnar Stå l mar ck (1991, pp.  13 1–13 2 )  a ndc la i m t ha t  he r example can befurther re du c ed . With reg a rdto  our ex ample,Σ2 in Proposi tion 2.0. 1, theirre duct i o n 6 0 denotedby ⊵CR(◦ ,⋆) may b e  depi c te d sch ema ticall yasfollo ws: [¬ φ ]1D1 ⊥CR,1 φ  ◦ E ψ D2  σ ⋆E  ρ  ⊵CR(◦,⋆ ) [¬ρ]2 [φ
 into_loops. t1_= t2 D1 φ(t1)_t2 =_t1_[φ(t2)] 1_[φ(t2)]_1, [φ(t1)] 2_D2 ρ =_E,2 ρ = E,1_ρ ⊵GEg′ t1_=_t2 D1 φ(t1) [φ(t2)] 1, D1 φ(t1) D2 ρ = E,1 ρ Our choice_of_generalized elimination_rules_is_independent of the occurrence of_a looping reduction sequence but_is dependent_on our choice of reduction procedures. Thus, we_only_consider the first_and the second cases in this section. To begin_with, in order to find a_solution to the_problem_of_undergeneration, it is necessary_to diagnose what the culprit of_the problem is. As Tennant (2015,_p. 589) said, 'The use of classical_reductio has masked the real defect_that lies at the heart_of paradoxical_reasoning.' He believes that CR−rule_disguises the non-terminating_reduction sequence._He does not_seem to consider seriously the undergeneration_problem. The use_of CR−rule does not always block_the_occurrence of the_non-terminating_reduction_sequence. So_to speak, CR−rule_is_not the_only_culprit of the problem of undergeneration._For_instance, in the beginning of this chapter,_we see the derivation_Σ2_of Curry's paradox in_Proposition 2.0.1 suggested by Rogerson_(2006). Σ2 uses CR−rule and does_not initiate_a non-terminating_reduction sequence. Proposition 2.0.1 seems to establish that CR−rule masks the_nonterminating reduction sequence, but it is_not. Rogerson (2006) only_considered standard_reduction_processes suggested by_Prawitz_(1965). However,_Schroeder-Heister and Tranchini (2017, pp. 572–573) borrow_the reduction_proposed by Gunnar Stålmarck (1991, pp._131–132) and claim that_her_example can be further reduced. With_regard to our example, Σ2 in_Proposition 2.0.1, their reduction 60_denoted_by_⊵CR(◦,⋆) may be depicted schematically_as follows: [¬φ]1 D1 ⊥ CR,1_φ ◦E ψ_D2 σ ⋆E ρ ⊵CR(◦,⋆) [¬ρ]2 [φ