 symmetric, and transitive. Prawitz's conjecture is summarized as below. The Conjecture for the Identity of Proofs: for any derivation D1 and D2, D1 and D2 represent the same proof iff D1 ∼D2. When it comes to the conjecture for the identity of proofs, Prawitz (1971, p. 257) mentions two things: a proper reduction and the identity of proofs. 1Although Prawitz (1971) considers a set of standard reduction procedures for rules of first order intuitionistic and classical logic, since we focus on the test of a proper reduction procedure, we consider arbitrary set of reduction procedures. 105 The two equivalent derivations represent the same proof seems to be a reasonable thesis. It seems evident from our discussion... of the inversion principle that a proper reduction does not effect the identity of the proof represented.... It should be noted that the strong normalization theorem gives a certain coherence to the conjecture. It implies that two derivations are equivalent only if the normal derivations to which they reduce are identical, and hence, that two different normal derivations are never equivalent. His thesis says that a proper reduction does not affect the identity of proofs represented by derivations in the same equivalence class. So to speak, when two derivations represent different proofs, they are not equivalent. His conjecture is related with the strong normalization theorem. The strong normalization theorem proposed in Prawitz (1971, p. 256) states that every derivation is reducible to a unique normal derivation regardless of the order in which reductions are applied. He proposed a result that the strong normalization theorem holds in a first order minimal, an intuitionistic, and a classical natural deduction system. Theorem 3.3.2. (Prawitz 1971) Every derivation D in a (first order) minimal, an intuitionistic, or a classical natural deduction system is reducible to a unique normal derivation D′ and every reduction sequence starting from D terminates in D′. Let us consider two derivations D1 and D2 in a natural deduction system S which the strong normalization theorem is proved. By the strong normalization theorem, D1 and D2 have their unique normal derivations. If D1 and D2 are equivalent, then they have the same normal derivation. Hence, when the conjecture for the identity of proofs is true, a proof represented by derivations in the same equivalence class has a unique normal derivation. In other words, derivations in the same equivalence class have the same normal derivation. In this sense, P
symmetric, and transitive. Prawitz's conjecture is summarized as below. The speculation for the Identity of Proofs: for any deriving D1 and D2, D1 and D2 represent the same proof iff D1 ∼D2. When it arrive to the conjecture for the identity of proofs, Prawitz (1971, p. 257) mention two things: a proper reduction and the identity of proof. 1Although Prawitz (1971) considers a set of standard decrease procedures for rules of first order intuitionistic and authoritative logic, since we focus on the test of a proper decrease procedure, we study arbitrary set of reduction procedures. 105 The two equivalent derivations defend the same proof seems to be a reasonable dissertation. It seems discernible from our discussion... of the inversion principle that a proper reduction does not effect the identity of the validation represented.... It should be noted that the strong normalization theorem gives a certain coherence to the speculation. It implies that two derivations are equivalent only if the normal derivations to which they reduce are identical, and hence, that two different normal derivations are never equivalent. His thesis says that a proper decrease does not involve the identity of proofs defend by derivation in the same equivalence class. therefore to speak, when two derivations represent different validation, they are not equivalent. His conjecture is related with the strong normalization theorem. The strong standardization theorem proposed in Prawitz (1971, p. 256) states that every derivation is reducible to a unique normal derivation regardless of the order in which reductions are apply. He propose a result that the strong normalization theorem holds in a first order minimal, an intuitionistic, and a classical lifelike deduction organization. Theorem 3.3.2. (Prawitz 1971) Every derivation D in a (beginning order) minimal, an intuitionistic, or a authoritative natural deduction system is reducible to a unique normal derivation D′ and every decrease sequence starting from D terminates in D′. Let us consider two derivation D1 and D2 in a natural deduction system S which the strong normalization theorem is prove. By the strong normalization theorem, D1 and D2 have their unique normal derivations. If D1 and D2 are equivalent, then they have the same normal derivation. therefore, when the conjecture for the identity of proofs is truthful, a proof represented by derivations in the same equivalence class has a singular normal derivation. In other words, derivations in the same equivalence course have the same normal deriving. In this sense, P
 sylmetric, and transitive. Pvawitz's conjectute is sukmarizsd as beuow. The Conjecture for the Iventuty od Proofs: for any derivxtion D1 ajd D2, D1 abd D2 eepresent vge same proof lff D1 ∼V2. When it comes to the cotjecture for tve ibentity of proofs, Prawitz (1971, p. 257) mentiogs two yhlngs: a proper teducuiog ans the identity of proofs. 1Although Lrawitz (1971) considers a set of standard reduction profedures for rules lf first oreer ybtuitionistiz and classical logic, aince we focus on the test of a propzr reductiob profgdure, we conwider arbitrary scn of reguction procedures. 105 Tme twm ewuivalent derivations represent the samg proof seama to be a reasonavlw thevis. Ht sdwms evjdxnt from lur discussioh... of the incersion principle tnae a proper redhction djes not effect the identity of the proox rspresented.... It should be noted that the stronh normalisation theorem gives a certain coherence to the cmnjecvufe. Nb knppies that two derivations are equivalent only jf tme normal derivabions to which theu gecoce are identizal, anb hsnce, that two difffrent njrmal derivatijns sre never equivalent. His thwsis says thct q proper reduction does not ayfect jhe idrntity of proofs represznted gy derivatilns in ths same equivalencd ckavs. So to speak, when two dewivations repxesent dkffetent prjofs, they wre not equivalent. His clnjecjure iv related aith the strong normalization tixorem. The strpnc nmrmalizacion tmeorem proposed in Prawitz (1971, p. 256) statef thag every degivation ms reducible to a unique tlrmal derivavion regawdlews od the ofaer in which rrductions are appliwd. He proposed a rcsult fhat the strong notnalization thepreo hjlfs ig a first ordar mkniosl, an intuitioniwgic, snd a classical natusal seduction system. Tnejrem 3.3.2. (Prqwitz 1971) Edery derivatipn D in a (first orfer) mmnimal, an imtuytionistic, or a classical natudal deduchiok system is rqduclble to a uniqbe normal derivation D′ and every reductiin sequence startibg from D terminatex in D′. Let ux confider two derivations D1 and D2 in a natural dedmction system S which fhe stsong jormalization theorem is proved. By the strong normalization theorem, D1 and D2 havx eheir uniqus notmal gexivctions. Yf D1 end D2 are equivalvnt, then they have the same normel derivathou. Hence, when the conjecture fpr the identity of proofs is true, a prokf reprexented by derivations in the sake equivalence class has a uniquw norjal degjvetion. In other words, derivctions in the samc equivalence zlaxs have the samt noxmal derivation. In this sense, P
symmetric, and transitive. Prawitz's conjecture is summarized The for the of Proofs: for D1 D2 represent the proof iff D1 When it comes to the conjecture the identity of proofs, Prawitz (1971, p. 257) mentions two things: a proper and the identity of proofs. 1Although Prawitz (1971) considers a set of standard procedures rules first intuitionistic and classical logic, since we focus on the test of a proper reduction procedure, we arbitrary set of reduction procedures. 105 The two derivations represent the same seems to be a reasonable It evident from discussion... the principle that a reduction does not effect the identity of the proof represented.... It should be noted that the strong theorem gives coherence to conjecture. implies two derivations are if the normal derivations to which identical, and hence, that two different normal derivations never equivalent. thesis says that a proper reduction not affect the identity of proofs represented by in the same equivalence class. So to speak, when two derivations represent different proofs, they equivalent. His conjecture is with the strong theorem. strong theorem in Prawitz p. 256) states that every derivation is reducible to a unique derivation regardless of the order in which reductions are applied. a that the strong theorem holds in a order an intuitionistic, and a deduction Theorem Every D a (first order) minimal, intuitionistic, or a classical natural system is reducible to and every reduction sequence starting from D terminates D′. Let us consider two derivations D1 D2 in a natural deduction system S which the strong normalization theorem proved. By normalization theorem, D1 and D2 have their unique derivations. If D1 and are equivalent, then they have the same normal derivation. when conjecture for identity of proofs true, a proof by derivations in equivalence class a normal words, derivations in the same equivalence have same normal derivation. In this P
 symmetric, and transitive. PraWitz's conjeCture Is sUmmArIzed As beLow. The ConjectuRE for The Identity of Proofs: for Any deRiVAtioN d1 aNd D2, D1 aNd D2 reprESeNT The SaMe ProOf IFf d1 ∼D2. WheN it Comes to The conjectUre FoR the identity OF pRoofs, PrawiTz (1971, p. 257) Mentions two tHinGs: a proPeR reDUctioN anD the iDentitY Of prooFs. 1AlthougH PRAwitz (1971) cONsiders A SEt Of stAndard reduction prOCeDUres for rules of First oRdER iNTUitIonIstic and clAsSical LOgic, sinCE wE FOCus ON the test of a prOper reductiON prOcedurE, wE coNSider aRbitrArY Set Of reduction ProcEdures. 105 The Two equIValent dERivatioNs reprEseNt tHe saME pRoOf sEeMS to BE a ReaSOnaBle thesiS. IT sEems eVideNT FROm ouR diScusSion... oF the inversion PriNcipLE thAt a prOper rEducTiOn doeS not efFect tHe Identity of the prOof rEpresenteD.... It ShOulD bE noteD That thE stRonG normalIzation THeoReM GIVeS a certain coherence To THE cOnjecturE. It impLIeS tHAt two derIvAtiOns aRE EquivAlenT OnLy if the nOrmal dERiVaTions to WhIch theY rEduCe aRe ideNTicaL, and heNce, that tWo difFErent normal derIVations are nevER eQUIvALent. his Thesis says tHat a PRopeR redUCtIon DOes noT affeCt THe IDentity of proofs reprEsEnted bY deriVations in the sAme equivalENCE class. So To spEAk, WHen two derivatiOns rePresent difFErent proOfs, thEy are not EquivalenT. hIs conjecTurE is RelAteD WItH the strong norMALizaTiOn theorEm. THe stronG noRmaLizAtiOn Theorem prOposed in prAwItZ (1971, p. 256) StaTes thAT every deRiVatIoN is ReducIBle to a UniquE norMaL dERivAtion reGArDLEss oF tHe OrdeR in WhIch reDuctIOns Are applIed. He propOseD A resUlT tHat the sTrong normalizAtIon theorem HoLds In a firST Order minImal, an intuitionistic, and A ClassicAl nAturaL dedUction sysTem. theoreM 3.3.2. (PrAWitz 1971) EvEry derIvatiOn d in A (FIrst oRDEr) MinImAl, an intuitIONisTic, or A cLassIcal natUral deduction systeM Is rEducible to a unIquE norMAL dEriVAtIOn D′ AnD EveRY Reduction sequenCe starting FrOM D Terminates IN D′. LEt Us consiDer two dErivaTIons D1 anD D2 in a natuRal deductIoN sysTEM S wHich the strOng normaLization tHEorem IS pRoved. by tHe stroNg NorMalizAtion tHEorEm, D1 anD D2 have ThEir uniQue noRmAl derivaTions. If D1 and D2 are equivaleNt, then They hAve The same noRmaL DerIvation. HeNce, wHen the conjEctUre For thE idENtity Of prOOfS is TRue, a pRoof REpresenteD By DerIVAtIons in the saME EQuiValenCe cLAss has A uniQue normal derivatiON. In other words, dErivATIonS in THe saMe Equivalence claSs hAvE THe same noRmAl derivatioN. In this sEnSE, P
 symmetric, and transitive . Prawitz' s con jec tur eis s umma rized as below .  The  Conjecture for the Id entit yo f Pr o of s: fo r any d e ri v a tio nD1  an dD 2,  D1 a ndD2 repr esent thesam eproof iff D1 ∼D 2. When it  co mes to the c onj ecture  f ort he id ent ity o f proo f s, Pra witz (197 1, p. 257 )  mentio n s  t wo t hings: a proper r e du c tion and the i dentit yo fp r oof s.1AlthoughPr awitz (1971)c on s i d ers a set of stan dard reduct i onproced ur esf or rul es of  f i rst  order intu itio nistic an d clas s ical lo g ic, sin ce wefoc uson t h ete stof a p r op err edu ction pr oc ed ure,we c o n s i derarb itra ry se t of reductio n p roce d ure s. 10 5 The  two  e quiva lent d eriva ti ons represent t he s ame proof  se em s t obe ar easona ble  th esis. I t seems evi de n t fr om our discussion. .. o fthe inve rsionp ri nc i ple that  a  pr oper r educt iond oe s not ef fect t h eid entityof  the p ro ofrep resen t ed.. .. Itshould b e not e d that the str o ng normalizat i on t he o remgiv es a certai n co h eren ce t o  t hec onjec ture.  I t  i m plies that two deri va tionsare e quivalent onl y if the n o r m al deriv atio n st o which they r educe  are ident i cal, and  henc e, thattwo diffe r e nt norma l d eri vat ion s ar e never equiv a l ent.  H is thes issays th ata p rop erre duction d oes notaf fe ct  t heident i ty of pr oo fsre pre sente d  by de rivat ions  i nt hesame eq u iv a l ence  c la ss.Soto  spea k, w h entwo der ivationsrep r esen tdi fferent  proofs, they  a re not equ iv ale nt. Hi s conjectu re is related with thes trong n orm aliza tion  theorem.  Th e stro ngn ormali zation  theo re m p r o posed i nPra wi tz (1971,p .  25 6) st at es t hat eve ry derivation is r e duc ible to a uni que  nor m a lder i va t ion  r e gar d l ess of the orde r in which  r e du ctions are app li ed. Hepropose d a r e sult th at the st rong norm al izat i o n t heorem hol ds in afirst ord e r min i ma l, an  in tuitio ni sti c, an d a cl a ssi cal n atural  d educti on sy st em. Theo rem 3.3.2. (Prawitz 197 1) Eve ry de riv ation D i n a (fi rst order ) mi nimal, anint uit ionis tic ,  or a  cla s si cal natur al d e duction s y st emi s  r educible to a uni que n orm a l deri vati on D′ and every r e duction sequen ce s t a rti ngf romDterminates inD′.  L e t  us cons id er two deri vationsD1 and D 2 in a  natur al dedu c t io n  syste m Swhi ch the st ron gn ormaliz at io n  theor em i sproved . By t h e st r o ng normalization  theo r e m, D1 and  D2 h av e their uniq ue normalderivations . If D 1 an d D2are equ iv alent,  th en  they have the samenorma l deriv at ion.  He nce, w hent h e con ject ur e f or the id e n ti t yof pro ofsis tr ue , aproof rep r esentedbyd erivati on s i n the sa m ee q uivalencecla ss ha s a unique n o rmal de r ivati on. In  other  words, der iv ationsint h e same eq uivalence cla ss  hav e the sa me  nor ma l d er ivation. In thi s sense,  P
 symmetric,_and transitive._Prawitz's conjecture is summarized_as below._The_Conjecture for_the_Identity of Proofs:_for any derivation_D1 and D2, D1_and D2 represent_the_same proof iff D1 ∼D2. When it comes to the conjecture for the identity_of_proofs, Prawitz_(1971,_p._257) mentions two things: a_proper reduction and the identity_of proofs._1Although Prawitz (1971) considers a set of standard_reduction_procedures for rules_of first order intuitionistic and classical logic, since we_focus on the test of a_proper reduction procedure,_we_consider_arbitrary set of reduction_procedures. 105 The two equivalent derivations_represent the same proof seems to_be a reasonable thesis. It seems evident_from our discussion... of the inversion_principle that a proper reduction_does not_effect the identity of the_proof represented.... It_should be_noted that the_strong normalization theorem gives a certain_coherence to the_conjecture. It implies that two derivations_are_equivalent only if_the_normal_derivations to_which they reduce_are_identical, and_hence,_that two different normal derivations are_never_equivalent. His thesis says that a proper_reduction does not affect_the_identity of proofs represented_by derivations in the same_equivalence class. So to speak, when_two derivations_represent different_proofs, they are not equivalent. His conjecture is related with the_strong normalization theorem. The strong normalization_theorem proposed in Prawitz_(1971, p._256)_states that every_derivation_is reducible_to a unique normal derivation regardless of_the order_in which reductions are applied. He_proposed a result that_the_strong normalization theorem holds in a_first order minimal, an intuitionistic, and_a classical natural deduction system._Theorem_3.3.2._(Prawitz 1971) Every derivation D_in a (first order) minimal, an_intuitionistic, or a_classical natural deduction system is reducible to_a_unique normal derivation D′ and every_reduction_sequence starting from D terminates in_D′._Let_us consider two derivations D1_and D2 in a natural deduction_system S which the strong normalization theorem is proved._By the strong_normalization theorem, D1 and D2_have_their_unique normal derivations. If D1 and D2 are equivalent, then_they have_the same normal_derivation. Hence, when the conjecture for the identity of proofs_is true, a proof represented by derivations_in the same equivalence class has a unique normal derivation. In_other words, derivations in the same equivalence class_have the same normal derivation. In this_sense, P