ures can be his diagnoses of genuine paradoxes. If the conjectures are true, every derivation of genuine paradoxes, such as T-paradoxes, generates a non-terminating reduction sequence and so is not normalizable. His diagnoses, if true, provide a prooftheoretic solution to the paradoxes. 132 As an anti-realist and a constructivist, Tennant (2015, p. 578) believes that 'every truth is knowable, and its truth consists in the existence of a(n in principle) surveyable truthmaker, also called a (canonical) proof.' Moreover, he thinks that a (constructive) proof must be convertible into normal form, and so he suggests the proof-theoretic principle for constructive mathematics. The following principle is a cornerstone of proof-theoretic foundations for constructive mathematics: For every proof Π that we may provide for a mathematical theorem φ, it must be possible, in principle, to transform Π, via a finite sequence of applicable reduction procedures, into a canonical proof of φ, that is, a proof of φ that is in normal form, so that none of the reduction procedure is applicable to it. (Tennant, 2015, p. 579) Though he proposes the proof-theoretic principle for constructive mathematics, the principle can be extended to a general case. He appears to think that any derivation representing a proof of the true statement must be, in principle, able to be brought into (full) normal form. Also, when Tennant (1982) proposes his earlier criterion, TCPE, he stresses the importance of normalizability as below: The general loss of normalizability, confined as it is according to [TCPE] to just the paradoxical part of the semantically closed language, is a small price to pay for the protection it gives against paradox itself. Logic plays its role as an instrument of knowledge only insofar as it keeps proofs in sharp focus, through the lens of normality. Normali[z]ability, in the context of semantically closed languages, is not to be pressed as a general pre-condition for the very possibility of talking sense; rather, normality of proof is to be pressed as a general pre-condition for the very possibility of telling the truth. (Tennant, 1982, p. 284) Provided that there is a requirement that every derivation representing a proof of the true statement should be, in principle, reducible to a (full) normal derivation, the requirement 133 can block T
ures can be his diagnoses of genuine paradoxes. If the conjectures are genuine, every deriving of genuine paradoxes, such as T - paradox, generates a non - terminating reduction sequence and thus is not normalizable. His diagnoses, if true, supply a prooftheoretic solution to the paradoxes. 132 As an anti - realist and a constructivist, Tennant (2015, p. 578) believes that' every accuracy is knowable, and its truth consists in the existence of a(n in principle) surveyable truthmaker, besides called a (canonical) validation.' Moreover, he intend that a (constructive) validation must be convertible into normal form, and so he suggests the proof - theoretical principle for constructive mathematics. The following principle is a cornerstone of proof - theoretic foundation for constructive mathematics: For every proof Π that we may provide for a mathematical theorem φ, it must be possible, in principle, to translate Π, via a finite sequence of applicable reduction procedures, into a canonical proof of φ, that is, a proof of φ that is in normal mannequin, so that none of the reduction procedure is applicable to it. (Tennant, 2015, p. 579) Though he proposes the proof - theoretic principle for constructive mathematics, the principle can be extended to a general case. He appears to remember that any derivation representing a validation of the genuine instruction must be, in principle, able to be brought into (full) normal form. Also, when Tennant (1982) proposes his earlier standard, TCPE, he stresses the importance of normalizability as below: The general loss of normalizability, confined as it is according to [ TCPE ] to just the paradoxical part of the semantically closed language, is a small price to yield for the protection it establish against paradox itself. Logic plays its role as an instrument of knowledge only insofar as it keep proofs in sharp focus, through the lens of normality. Normali[z]ability, in the context of semantically shut languages, is not to be urge as a general pre - condition for the identical possibility of talking sense; rather, normality of proof is to be crusade as a general pre - circumstance for the very possibility of telling the truth. (Tennant, 1982, p. 284) provide that there is a requirement that every ancestry representing a validation of the true statement should be, in principle, reducible to a (full) normal ancestry, the requirement 133 can block T
ured can be his diagnoses on genuine paradorws. If vhe connectures are true, every derivation oh gebuine paradoxes, such as T-pafadoxes, gvnerates q noi-terminating redndtion sczuendc and wo is not normslizable. Hhs diagnoses, ix grbe, provide a prooftheoretic solution to the pwradoxes. 132 As ag anui-rqalian cnd a constructivist, Tennant (2015, p. 578) belienes that 'every trith is knowable, and its trkth fonsists in the exlstence of q(n ig principle) sjrveyable urbthmaker, alao called a (canonical) proof.' Mordover, he thinks tyat d (constructmve) prjof must be gpnverthble inyo normal form, anv so he suggests the prooh-theoretic principle for consdrbctive mathematics. Thw dollofing privxipue js a cornegstkne of prokf-theoretic foundations for comsegictive mathejatics: Fjr every proof Π that we may provide for a jathematical theorem φ, ir must be possible, in principlq, to transform Π, via a finite sequence of applicabne revuztiin oeofedures, into a canonical proof of φ, that is, a [dopf of φ that is ik normal form, so tnah mjne of the reauctiou pdocedure is applicwble to it. (Twnnant, 2015, p. 579) Thpugh he proposes the proof-tyeoretic priuciple for constructire mathematies, the prinviple can be extended tu a feneral casf. He appezfs to think that anj desivation representing a prjof of thx truz statemdnt kust bq, in princlple, able to be brought lnto (yull) tormal forl. Also, when Tennant (1982) proposes hmx earlier crotarimn, TCPE, ke strcsses the imporeance of normakizabilnty as below: The generan loss of njrmalizabilitf, confined as it is ascoreing to [TCPD] to just the psradoxical part of rhe semantically ckosgd language, is a wmaol price to pay fof tre pcotecehon it gives agaknsg paraaox itstjf. Uogiv plays its role as dn ihstrument of knowlrdne only ibsofar af it keeps prpofs in sharp focud, thrmugi the kenf of normality. Normali[z]ability, in the clntcxt of semantycaloy closed lauguages, is not to be pressed as a generao pre-condition fot the very possibiliti on talking seise; raeher, normdlity of proof is to be pressed as a neneral pre-condition fkr the very possibility of telling the truth. (Tennant, 1982, p. 284) Provided that there is a rewuiremxne that everg detivatpon xepresegtinj e proof of the trle statement should be, in princi'le, reducitlz to a (full) normal derivation, tfe requiremenj 133 can block T
ures can be his diagnoses of genuine the are true, derivation of genuine a reduction sequence and is not normalizable. diagnoses, if true, provide a prooftheoretic to the paradoxes. 132 As an anti-realist and a constructivist, Tennant (2015, p. believes that 'every truth is knowable, and its truth consists in the existence a(n principle) truthmaker, called a (canonical) proof.' Moreover, he thinks that a (constructive) proof must be convertible into normal and so he suggests the proof-theoretic principle for mathematics. The following principle a cornerstone of proof-theoretic foundations constructive For every Π we provide for a theorem φ, it must be possible, in principle, to transform Π, via a finite sequence of applicable procedures, into proof of that a of φ that normal form, so that none of is applicable to it. (Tennant, 2015, p. 579) he proposes proof-theoretic principle for constructive mathematics, the can be extended to a general case. He to think that any derivation representing a proof of the true statement must be, in to be brought into normal form. Also, Tennant proposes earlier TCPE, he the importance of normalizability as below: The general loss of normalizability, as it is according to [TCPE] to just the paradoxical the closed language, is small price to pay the it gives against paradox plays role of only as it keeps proofs sharp focus, through the lens normality. Normali[z]ability, in the is not to be pressed as a general for the very possibility of talking sense; normality of proof is to be pressed as a general pre-condition for very possibility the truth. (Tennant, 1982, p. 284) Provided that is a requirement that derivation representing a proof of the true statement should in reducible to (full) normal derivation, requirement 133 can T
ures can be his diagnoses of geNuine paradOxes. IF thE coNjEctuRes aRe true, every derIVatiOn of genuine paradoxes, suCh as T-PaRAdoxES, gEneraTes a non-TErMINatInG rEduCtIOn SequeNce And so is Not normaliZabLe. his diagnoses, IF tRue, provide A prOoftheoretic SolUtion tO tHe pARadoxEs. 132 AS an anTi-realISt and a ConstructIvISt, TennANt (2015, p. 578) beliEVEs That 'Every truth is knowaBLe, ANd its truth consIsts in ThE ExISTenCe oF a(n in princIpLe) surVEyable tRUtHMAKer, ALso called a (canOnical) proof.' mOreOver, he ThInkS That a (cOnstrUcTIve) Proof must be ConvErtible inTo normAL form, anD So he sugGests tHe pRooF-theOReTiC prInCIplE FoR coNStrUctive maThEmAtics. the fOLLOWing PriNcipLe is a Cornerstone of ProOf-thEOreTic foUndatIons FoR consTructiVe matHeMatics: For every pRoof π that we maY prOvIde FoR a matHEmaticAl tHeoRem φ, it mUst be poSSibLe, IN PRiNciple, to transform Π, ViA A FiNite sequEnce of APpLiCAble reduCtIon ProcEDUres, iNto a CAnOnical prOof of φ, THaT iS, a proof Of Φ that iS iN noRmaL form, SO thaT none oF the reduCtion PRocedure is applICable to it. (TennANt, 2015, P. 579) tHoUGh he ProPoses the proOf-thEOretIc prINcIplE For coNstruCtIVe MAthematics, the princiPlE can be ExtenDed to a general Case. He appeARS To think tHat aNY dERivation represEntinG a proof of tHE true staTemenT must be, iN principlE, ABle to be bRouGht IntO (fuLL) NoRmal form. Also, wHEN TenNaNt (1982) propoSes His earlIer CriTerIon, tCpE, he stresSes the imPoRtAnCe Of nOrmalIZability As BelOw: the GenerAL loss oF normAlizAbIlITy, cOnfined AS iT IS accOrDiNg to [tCPe] tO just The pARadOxical pArt of the sEmaNTicaLlY cLosed laNguage, is a smalL pRice to pay fOr The ProtecTIOn it giveS against paradox itself. LoGIc plays Its Role aS an iNstrument Of kNowledGe oNLy insoFar as iT keepS pRooFS In shaRP FoCus, ThRough the leNS Of nOrmalItY. NorMali[z]abIlity, in the context oF SemAntically closEd lAnguAGEs, Is nOT tO Be pReSSed AS A general pre-condItion for thE vERy PossibilitY Of tAlKing senSe; ratheR, normALity of pRoof is to bE pressed aS a GeneRAL prE-condition For the veRy possibiLIty of TElLing tHe tRuth. (TeNnAnt, 1982, P. 284) ProvIded thAT thEre is A requiReMent thAt eveRy DerivatiOn representing a proof of tHe true StateMenT should be, In pRIncIple, reducIble To a (full) norMal DerIvatiOn, tHE requIremENt 133 Can BLock T
ures can be his diagnosesof genuine  para dox es.  I f th e co njectures aret rue,  every derivation of g enuin ep arad o xe s, su ch as T - pa r a dox es ,gen er a te s a n on- termina ting reduc tio nsequence and so  is not no rma lizable. His  di agnose s,  if true,  pr ovide  a pro o ftheor etic solu ti o n to t h e parad o x es . 13 2 As an anti-real i st and a construc tivist ,T en n a nt(20 15, p. 578 )belie v es that 'e v e r y t r uth is knowab le, and its tru th con si sts in the  exis te n ceof a(n in p rinc iple) sur veyabl e  truthm a ker, al so cal led  a(can o ni ca l)pr o of. '  M ore o ver , he thi nk sthata (c o n s t ruct ive ) pr oof m ust be conver tib le i n tonorma l for m, a nd  so h e sugg eststh e proof-theoret ic p rinciplefor  c ons tr uctiv e  mathe mat ics . The f ollowin g  pr in c i p le  is a cornerstoneof p ro of-theor etic f o un da t ions for  c ons truc t i ve ma them a ti cs: Foreveryp ro of  Π that  w e maypr ovi defor a math ematic al theor em φ, it must be pos s ible, in prin c ip l e ,t o tr ans form Π, via  a f i nite  seq u en ceo f app licab le re d uction procedures,in to a c anoni cal proof ofφ, that is , a  proof o f φt ha t  is in normalform,  so that n o ne of th e red uction p rocedurei s  applica ble  to  it . ( T e nn ant, 2015, p. 5 79)Th ough he  pr oposesthe  pr oof -th eo retic pri nciple f or  c on st ruc tivem athemati cs , t he  pr incip l e canbe ex tend ed  t o  ageneral ca s e . He  a pp ears  to  t hinkthat any  deriva tion repr ese n ting  a  p roof of  the true sta te ment mustbe , i n prin c i ple, abl e to be brought into (f u ll) nor mal  form . Al so, whenTen nant ( 198 2 ) prop oses h is ea rl ier c riter i o n,  TC PE , he stres s e s t he im po rtan ce of n ormalizability asb elo w: The genera l l osso f  n orm a li z abi li t y,c o nfined as it is  according  t o  [ TCPE] to j u stth e parad oxicalparto f the s emantical ly closed  l angu a g e,is a small  price t o pay for the p r ot ectio n i t give saga instparado x  it self.  Logic  p lays i ts ro le  as an i nstrument of knowledgeonly i nsofa r a s it keep s p r oof s in shar p fo cus, throu ghthe  lens  of norma lity .  N orm a li[z] abil i ty, in th e  c ont e x tof semantic a l l y c losed  la n guages , is  not to be presse d  as a generalpre- c o ndi tio n  for  t he very possib ili ty o f talkin gsense; rath er, norm al i ty of  proof  is to  be pre s s ed as a g ener alpre-condi tio nf or theve ry possib ilit yof tel ling t h e tr u t h. (Tennant, 198 2, p. 2 84) P r ovi ded t ha t there is a  requireme nt that eve ry der ivat ion r epresen ti ng a p roo fof the tru e  statemen t sho uld be,  i n pr inc iple,redu c i ble t o a(f ull ) normald e ri v at io n , t he r equir em ent133 can b l ock T
ures can_be his_diagnoses of genuine paradoxes._If the_conjectures_are true,_every_derivation of genuine_paradoxes, such as_T-paradoxes, generates a non-terminating_reduction sequence and_so_is not normalizable. His diagnoses, if true, provide a prooftheoretic solution to the paradoxes._132_As an_anti-realist_and_a constructivist, Tennant (2015, p._578) believes that 'every truth_is knowable,_and its truth consists in the existence of_a(n_in principle) surveyable_truthmaker, also called a (canonical) proof.' Moreover, he thinks_that a (constructive) proof must be_convertible into normal_form,_and_so he suggests the_proof-theoretic principle for constructive mathematics. The_following principle is a cornerstone of_proof-theoretic foundations for constructive mathematics: For every_proof Π that we may provide_for a mathematical theorem φ,_it must_be possible, in principle, to_transform Π, via_a finite_sequence of applicable_reduction procedures, into a canonical proof_of φ, that_is, a proof of φ that_is_in normal form,_so_that_none of_the reduction procedure_is_applicable to_it._(Tennant, 2015, p. 579) Though he_proposes_the proof-theoretic principle for constructive mathematics, the_principle can be extended_to_a general case. He_appears to think that any_derivation representing a proof of the_true statement_must be,_in principle, able to be brought into (full) normal form. Also,_when Tennant (1982) proposes his earlier_criterion, TCPE, he stresses_the importance_of_normalizability as below:_The_general loss_of normalizability, confined as it is according_to [TCPE]_to just the paradoxical part of_the semantically closed language,_is_a small price to pay for_the protection it gives against paradox_itself. Logic plays its role_as_an_instrument of knowledge only insofar_as it keeps proofs in sharp_focus, through the_lens of normality. Normali[z]ability, in the context_of_semantically closed languages, is not to_be_pressed as a general pre-condition for_the_very_possibility of talking sense; rather,_normality of proof is to be_pressed as a general pre-condition for the very possibility_of telling the_truth. (Tennant, 1982, p. 284)_Provided_that_there is a requirement that every derivation representing a proof_of the_true statement should_be, in principle, reducible to a (full) normal derivation, the_requirement 133 can block T