Φ⌝) = E T (⌜¬T (⌜Φ⌝)⌝) T E¬T (⌜Φ⌝) [¬Φ]3 ∆5 ⊥ CR,3 Φ T I T (⌜Φ⌝) ¬E⊥ There is no reduction process in RCE that we can apply to ∆6. ∆6 does not generate a non-terminating reduction sequence. If the Liar paradox is a genuine paradox, ∆6 is a counterexample to TCPE which raises the problem of undergeneration. It is the derivation of the genuine paradox but does not satisfy TCPE. Hence, TCPE fails to be a necessary condition to be the test of the genuine paradoxes. Tennant considers that the classical inference makes a trouble in the derivation of the Liar paradox. In his derivation of the Liar paradox employing CR−rule which does not apply ⊵CR(T (x))reduction, Tennant (2015, p. 289) describes a similar phenomenon as the classical rub. Now here's the classical rub: this proof appears to be in normal form. The 50 use of classical reductio has masked the real defect that lies at the heart of paradoxical reasoning (according to my account) – the abnormality that makes itself evident only when one hews to a constructivist line.... He thinks that a classical inference causes the trouble that a constructive reasoning does not make. Of course, there is a significant issue of whether the constructivist can accept the classical reductio and many constructivists do not agree that the reductio is a constructive reasoning. However, here the trouble that Tennant points out is not the issue on whether the classical reductio is constructive or not. His issue is that CR−rule makes a trouble when we investigate the proof-theoretic structure of the paradoxes. He considers that at least two things are troublesome. First, the derivation of the Liar paradox using CR−rule, such as ∆6 does not produce the heart of paradoxical reasoning, i.e. the non-terminating reduction sequence. Second, it appears to be a normal derivation.8 He proposes the methodological conjecture as the answer to the trouble, instead of arguing that the trouble only happens when using classical inferences. Paradoxes are never strictly classical. The kind of conceptual trouble that a paradox reveals will afflict the intuitionist
Φ ⌝) = E T (⌜ ¬T (⌜ Φ ⌝) ⌝) T E¬T (⌜ Φ ⌝) [ ¬Φ]3 ∆5 ⊥ CR,3 Φ T I T (⌜ Φ ⌝) ¬E⊥ There is no decrease procedure in RCE that we can apply to ∆6. ∆6 does not generate a non - terminating decrease sequence. If the Liar paradox is a actual paradox, ∆6 is a counterexample to TCPE which raises the problem of undergeneration. It is the ancestry of the actual paradox but does not satisfy TCPE. Hence, TCPE fails to be a necessary condition to be the test of the actual paradoxes. Tennant considers that the classical inference makes a fuss in the derivation of the Liar paradox. In his derivation of the Liar paradox employing CR−rule which does not apply ⊵CR(T (x))reduction, Tennant (2015, p. 289) describes a alike phenomenon as the classical rub. Now here's the authoritative rub: this proof appear to be in normal form. The 50 function of classical reductio has masked the real defect that lie down at the heart of paradoxical reasoning (according to my account) – the abnormality that makes itself evident only when one hews to a constructivist line.... He thinks that a classical inference causes the trouble that a constructive reasoning does not make. Of course, there is a significant issue of whether the constructivist can bear the classical reductio and many constructivist do not match that the reductio ad absurdum is a constructive reasoning. However, here the trouble that Tennant points out is not the issue on whether the classical reductio is constructive or not. His issue is that CR−rule makes a trouble when we investigate the proof - theoretical structure of the paradoxes. He considers that at least two things are troublesome. First, the derivation of the Liar paradox using CR−rule, such as ∆6 does not produce the heart of paradoxical reasoning, i.e. the non - terminating reduction sequence. Second, it appears to be a normal derivation.8 He project the methodological speculation as the answer to the trouble, instead of arguing that the trouble merely happens when using classical inference. Paradoxes are never rigorously classical. The kind of conceptual trouble that a paradox reveals will afflict the intuitionist
Φ⌝) = E T (⌜¬T (⌜Φ⌝)⌝) T E¬T (⌜Φ⌝) [¬Φ]3 ∆5 ⊥ CR,3 Φ T I T (⌜Φ⌝) ¬E⊥ There is no reductmon prodess in FCE that we can apply to ∆6. ∆6 dlew not generate a non-terminaging redubtion seqyenct. If the Liar paresox is a genujke paxavox, ∆6 is a countgrexample to TCPE which rahsds the problem of undergeneration. It ys the cegivation of thg genlige pzgabox but does not satisfy TCPE. Hsnce, TCKE fails to be a nrcessary condition to be tje tfst of the genuine paradoxes. Jsnnwbt considers that the bnassical ihference makes a trouble in the deriration of tye Liwt paradox. In his qerivation on the Lhar parsdox employing CR−cule which does not apply ⊵CR(T (x))reduction, Tegnant (2015, p. 289) dzscribes a similar phwnimenot as the xlarsidak dub. Noa hxre's the clzssical rub: this proof appears tj be in normal form. Ehq 50 use of classical reductio has masked ths real defect that lies at the heart of parafoxical rqasoning (according to my account) – the abnormality that oakts itseud fvident only when one hews to a constructivise lonv.... He thinks that c classical infetejcr causes the ttouble cgaf a constructive rfasonind doew not makt. Of vourse, there is a significabt issue of cherher the constructnvist can aceept tne clsssical reductio and mauy conatructivistd do not zeree that the reaucnio hs a constructive reasonind. However, herz the truublg that Eennant polnts out is not the issuf on chethar the cladsical reductio is constructive or not. His isxua iv that CX−rule kakes a troubje when we invgstigate che pruof-theoretpc structnre of the pwradoxes. He cmjsiders that at lease twi thungs ard troublesome. Forst, the btrivation if the Liar paradow usivf CR−rule, such aw ∆6 eoes not producr tfe rewrv of [dradoxical raasovine, i.e. tfe non-ttxmikatkng teduction sequence. Sacons, it appears to be a normal eerivatijn.8 He proposex the methodologicwl coijectuce as yhe answer to the trouble, instead of arguijg bhat the troufle inly happens when using classical inferences. Paradoees are never strictly xlassical. The kind pn conceptual troufle that d paradox reveals wiol afflict the inuuitionist
Φ⌝) = E T (⌜¬T (⌜Φ⌝)⌝) T [¬Φ]3 ⊥ CR,3 T I T reduction in RCE that can apply to ∆6 does not generate a non-terminating sequence. If the Liar paradox is a genuine paradox, ∆6 is a counterexample TCPE which raises the problem of undergeneration. It is the derivation of the paradox does satisfy Hence, TCPE fails to be a necessary condition to be the test of the genuine paradoxes. considers that the classical inference makes a trouble the derivation of the paradox. In his derivation of Liar employing CR−rule does apply (x))reduction, Tennant (2015, 289) describes a similar phenomenon as the classical rub. Now here's the classical rub: this proof appears be in The 50 of reductio masked the real lies at the heart of paradoxical my account) – the abnormality that makes itself only when hews to a constructivist line.... He that a classical inference causes the trouble that constructive reasoning does not make. Of course, there is a significant issue of whether the accept the classical reductio many constructivists do agree the is constructive reasoning. here the trouble that Tennant points out is not the issue whether the classical reductio is constructive or not. His issue CR−rule a trouble when investigate the proof-theoretic structure the He considers that at things troublesome. of Liar using CR−rule, such as does not produce the heart paradoxical reasoning, i.e. the appears to be a normal derivation.8 He proposes methodological conjecture as the answer to the instead of arguing that the trouble only happens when using classical inferences. are never The kind of conceptual trouble that a paradox will afflict the intuitionist
Φ⌝) = E T (⌜¬T (⌜Φ⌝)⌝) T E¬T (⌜Φ⌝) [¬Φ]3 ∆5 ⊥ CR,3 Φ T I T (⌜Φ⌝) ¬E⊥ There is No reductioN procEss In RcE That We caN apply to ∆6. ∆6 does noT GeneRate a non-terminating redUctioN sEQuenCE. IF the LIar paraDOx IS A geNuInE paRaDOx, ∆6 Is a coUntErexampLe to TCPE whIch RaIses the problEM oF undergeneRatIon. It is the deRivAtion oF tHe gENuine ParAdox bUt does NOt satiSfy TCPE. HeNcE, tCPE faILs to be a NECeSsarY condition to be the TEsT Of the genuine paRadoxeS. TENnANT coNsiDers that thE cLassiCAl inferENcE MAKes A Trouble in the dErivation of THe LIar parAdOx. IN His derIvatiOn OF thE Liar paradoX empLoying CR−rUle whiCH does noT Apply ⊵CR(t (x))reduCtiOn, TEnnaNT (2015, p. 289) DeScrIbES a sIMiLar PHenOmenon as ThE cLassiCal rUB. nOW herE's tHe clAssicAl rub: this prooF apPearS To bE in noRmal fOrm. THe 50 Use of ClassiCal reDuCtio has masked thE reaL defect thAt lIeS at ThE hearT Of paraDoxIcaL reasonIng (accoRDinG tO MY AcCount) – the abnormalitY tHAT mAkes itseLf evidENt OnLY when one HeWs tO a coNSTructIvisT LiNe.... He thinKs that A ClAsSical inFeRence cAuSes The TroubLE thaT a consTructive ReasoNIng does not make. oF course, there iS A sIGNiFIcanT isSue of whetheR the COnstRuctIViSt cAN accePt the ClASsICal reductio and many cOnStructIvistS do not agree thAt the reducTIO Is a constRuctIVe REasoning. HoweveR, here The trouble THat TennaNt poiNts out is Not the issUE On whetheR thE clAssIcaL REdUctio is constrUCTive Or Not. His iSsuE is that cR−rUle MakEs a TrOuble when We investIgAtE tHe ProOf-theORetic strUcTurE oF thE paraDOxes. He ConsiDers ThAt AT leAst two tHInGS Are tRoUbLesoMe. FIrSt, the DeriVAtiOn of the liar paradOx uSIng Cr−rUlE, such as ∆6 Does not producE tHe heart of pArAdoXical rEASoning, i.e. The non-terminating reductIOn sequeNce. seconD, it aPpears to bE a nOrmal dEriVAtion.8 HE propoSes thE mEthODOlogiCAL cOnjEcTure as the aNSWer To the TrOublE, insteaD of arguing that the tROubLe only happens WheN usiNG ClAssICaL InfErENceS. pAradoxes are neveR strictly cLaSSiCal. The kind OF coNcEptual tRouble tHat a pARadox reVeals will Afflict thE iNtuiTIOniSt
Φ⌝) = E T (⌜¬T (⌜Φ⌝)⌝) T E ¬T (⌜Φ⌝) [ ¬Φ]3∆5⊥ C R, 3 ΦT IT (⌜Φ⌝) ¬E⊥ Th e re i s no reduction process  in R CE that we  canapply t o  ∆ 6 .  ∆6  d oe s n ot ge nerat e a  non-te rminatingred uc tion sequenc e .If the Lia r p aradox is agen uine p ar ado x , ∆6isa cou nterex a mple t o TCPE wh ic h  raise s  the pr o b le m of  undergeneration. It is the derivat ion of  t h eg e nui neparadox bu tdoesn ot sati s fy T C PE. Hence, TCPE f ails to bea  ne cessar ycon d itionto be  t h e t est of thegenu ine parad oxes.T ennantc onsider s that  th e c lass i ca linf er e nce ma kes a t rouble i nth e der ivat i o n of t heLiar  para dox. In his d eri vati o n o f the  Liar  par ad ox em ployin g CR− ru le which does n ot a pply ⊵CR( T ( x) )re du ction ,  Tenna nt(20 15, p.289) de s cri be s a  s imilar phenomenonas t he  classic al rub .  N ow here's t he  cl assi c a l rub : th i sproof ap pearst obe  in nor ma l form .The  50  useo f cl assica l reduct io ha s  masked the re a l defect that li e s  a t  the  he art of para doxi c al r easo n in g ( a ccord ing t om ya ccount) – the abnor ma lity t hat m akes itself e vident onl y w hen onehews to a constructivi st li ne.... Het hinks th at aclassica l inferen c e  causesthe  tr oub let h at  a constructi v e  rea so ning do esnot mak e.Ofcou rse ,there isa signif ic an tis sue  of w h ether th econ st ruc tivis t  can a ccept  the  c la s sic al redu c ti o andma ny  con str uc tivis ts d o  no t agree  that the  re d ucti ois  a cons tructive reas on ing. Howev er , h ere th e troublethat Tennant points out is notthe  issu e on  whetherthe  class ica l  reduc tio is  cons tr uct i v e orn o t.  Hi sissue is t h a t C R−rul emake s a tro uble when we inves t iga te the proof- the oret i c  s tru c tu r e o ft hep a radoxes. He con siders tha ta tleast twot hin gs  are tr oubleso me. F i rst, th e derivat ion of th eLiar p ara dox usingCR−rule,  such as∆ 6 doe s  n ot pr odu ce the  h ear t ofparado x ica l rea soning ,i.e. t he no n- terminat ing reduction sequence.  Secon d, it  ap pears tobea  no rmal deri vati on.8 He pr opo ses  themet h odolo gica l  c onj e cture  ast he answer to  th e tr ouble, inst e a d  of  argu ing that t he t rouble only happe n s when using c lass i c alinf e renc es . Paradoxes ar e n ev e r  strictl yclassical.The kind  o f  conc eptual  troub le that a  p a radoxreve als  will aff lic tt he intu it io n ist
Φ⌝) =_E T_(⌜¬T (⌜Φ⌝)⌝) T E¬T_(⌜Φ⌝) [¬Φ]3_∆5_⊥ CR,3_Φ_T I T_(⌜Φ⌝) ¬E⊥ There_is no reduction process_in RCE that_we_can apply to ∆6. ∆6 does not generate a non-terminating reduction sequence. If the_Liar_paradox is_a_genuine_paradox, ∆6 is a counterexample_to TCPE which raises the_problem of_undergeneration. It is the derivation of the genuine_paradox_but does not_satisfy TCPE. Hence, TCPE fails to be a necessary_condition to be the test of_the genuine paradoxes._Tennant_considers_that the classical inference_makes a trouble in the derivation_of the Liar paradox. In his_derivation of the Liar paradox employing CR−rule_which does not apply ⊵CR(T (x))reduction,_Tennant (2015, p. 289) describes_a similar_phenomenon as the classical rub._Now here's the_classical rub:_this proof appears_to be in normal form. The_50 use of_classical reductio has masked the real_defect_that lies at_the_heart_of paradoxical_reasoning (according to_my_account) –_the_abnormality that makes itself evident only_when_one hews to a constructivist line.... He_thinks that a classical_inference_causes the trouble that_a constructive reasoning does not_make. Of course, there is a_significant issue_of whether_the constructivist can accept the classical reductio and many constructivists do_not agree that the reductio is_a constructive reasoning. However,_here the_trouble_that Tennant points_out_is not_the issue on whether the classical reductio_is constructive_or not. His issue is that_CR−rule makes a trouble_when_we investigate the proof-theoretic structure of_the paradoxes. He considers that at_least two things are troublesome._First,_the_derivation of the Liar paradox_using CR−rule, such as ∆6 does_not produce the_heart of paradoxical reasoning, i.e. the non-terminating_reduction_sequence. Second, it appears to be_a_normal derivation.8 He proposes the methodological_conjecture_as_the answer to the trouble,_instead of arguing that the trouble_only happens when using classical inferences. Paradoxes are never_strictly classical. The_kind of conceptual trouble that_a_paradox_reveals will afflict the intuitionist