 a natural deduction system developed by Prawitz (1965, 1971) but first introduced by Gentzen (1935, 2008). When Gentzen (1936) attempted to show the consistency of arithmetic, he believed that if there was a fault in the paradoxes, it must be sought in the logical reasoning employed. One of the main purposes of the proof-theoretic analysis of the paradoxes is to find any errors in the reasoning. It appears to be convincing that a suitable proof-theoretic solution to the paradoxes can be the reasoning-rejection solution. RND may be the reasoning-rejection solution in a broad sense. However, as our second question asks, it should be explained why a nonnormalizable derivation is unacceptable. The Second Question (ii): Why should we accept only a normalizable derivation? A promising answer is that a non-normalizable derivation would not be proof-theoretically valid. In this section, we will briefly investigate Prawitz's idea of proof-theoretic validity and the relation between proof-theoretic validity and normalizability. If proof-theoretic validity implies (full) normalizability, then paradoxical derivations, i.e. T-paradoxes which are not (fully) normalizable, are not proof-theoretically valid. Then, since there is no nonnormalizable derivation which is proof-theoretically valid, it can be explained why any non-normalizable derivations are unacceptable. The name 'proof theory' was originally coined by David Hilbert. The aim of his proof theory is to obtain a reduction of mathematics to some more elementary part of it, such 144 as finitistic or constructive mathematics, by analyzing the proofs of mathematical theories. Prawitz (1971, 1973) thinks that Hilbert's proof theory is only a tool to obtain the reduction because it does not aim at studying the very proofs. He calls it reductive proof theory. He suggests general proof theory as the study of the notion of proof. The subject matter of general proof theory is thus proofs considered as a process by which we get to know the theorems of a theory or the validity of an argument, and this process is studied here in its own right. (Prawitz, 1971, p. 237) In general proof theory, we are... interested in understanding the very proofs themselves, i.e., in understanding not only what deductive connections hold but also how they are established,... (Prawitz, 1973, p. 225) One of the main
a natural deduction system developed by Prawitz (1965, 1971) but foremost introduce by Gentzen (1935, 2008). When Gentzen (1936) attempted to picture the consistency of arithmetical, he believed that if there was a fault in the paradox, it must be seek in the logical reasoning use. One of the main purposes of the proof - theoretical analysis of the paradoxes is to find any errors in the reasoning. It appear to be convincing that a suitable proof - theoretic solution to the paradoxes can be the reasoning - rejection solution. RND may be the reasoning - rejection solution in a broad common sense. However, as our second question asks, it should be explained why a nonnormalizable ancestry is unacceptable. The Second Question (ii ): Why should we accept merely a normalizable derivation? A bright answer is that a non - normalizable ancestry would not be proof - theoretically valid. In this section, we will briefly investigate Prawitz's idea of proof - theoretical validity and the relation between proof - theoretic validity and normalizability. If proof - theoretic validity incriminate (entire) normalizability, then paradoxical derivations, i.e. T - paradoxes which are not (fully) normalizable, are not proof - theoretically valid. Then, since there is no nonnormalizable derivation which is proof - theoretically valid, it can be excuse why any non - normalizable derivation are unacceptable. The name' proof theory' was in the first place coin by David Hilbert. The aim of his proof theory is to obtain a decrease of mathematics to some more elementary part of it, such 144 as finitistic or constructive mathematics, by analyze the proofs of mathematical theory. Prawitz (1971, 1973) thinks that Hilbert's proof theory is only a tool to obtain the reduction because it does not calculate at studying the very proofs. He calls it reductive validation theory. He indicate cosmopolitan proof hypothesis as the study of the notion of proof. The subject matter of general proof theory is therefore proofs study as a process by which we get to sleep together the theorems of a hypothesis or the validity of an argument, and this process is studied here in its own right. (Prawitz, 1971, p. 237) In cosmopolitan proof theory, we are... interested in understanding the identical proofs themselves, i.e., in understanding not only what deductive connection hold but also how they are established, ... (Prawitz, 1973, p. 225) One of the main
 a jatural deduction system developed by Ptaqitz (1965, 1971) but fjrst intfoduced by Gentzen (1935, 2008). When Geitzeb (1936) atuvmpted to show the covsistency of aritymetmc, he believed tizt if tmzre wzd a yanlt in the paracoxes, it mgst be sought hn tke logical reasoning employed. One of the maon purposes of tre pgojf-thsoretic analysis of the paradoxes js to fpnd any errors in the reasoning. It appears ho bf convincing that w suitable kdoos-rheoretic souution to uhz paradoxes can be the reasoning-rejection rolutnon. RND may bw tjg reasoning-rxjectijn solution lm a brmad senxe. However, as pur sexond question asks, it should be explaingd why a nmnuormalizable derivatiin is utaccaptacoe. Ghe Sxcohd Quedtikn (ii): Why ahould we axcept only a normalozwvle derivatioh? A prjmysing answer is that a non-normalizable gerjvation would not be priof-theoretically valif. In this section, we will briefly investigate Prawitz's ided of 'ruof-uhcjfwtlc validity and the relation between proof-thejdeuic validity and kormalizability. If pgops-theoretic vauidity imllies (full) normalixabiliti, then paradoxisal cerivations, i.e. T-paradoxes wyich are not (dully) normalizable, are not pruof-tneoreyically valid. Then, sincz thers is no nonjormalizague derivation whkch iv proof-theoretically valid, it can bx expkained dhy sny nog-normalizahle dcsivations are unacfeptayle. Tve name 'prlof theory' was originally coinev by David Hikbart. The aim of hls proof theory is to obtain c reductnon of mathematibs to somx more elemegtary part of lt, such 144 as hinitistis or conwtructixd mathematics, ny analyzpny the proifs of mathematicak tfsories. Prawitz (1971, 1973) tyinks that Hilbrrt'r pwolf traory is only a tuol yo obgain tht rcduztiom because it does nod aij at studying the fevy proofs. He callf it reductivr proof theory. He duggevts genetal proof theory as the study of fhe notioj on proof. The stbjegt mwtter of gzneral proof theory is thus proofs consivered as a process by wyich we get to know uhe theorems pf a eheory or the validity of an qrgument, and this process is studied hsre in its lwn right. (Prawitz, 1971, p. 237) In general proof theory, we are... interested in undersrandinj ehe very prkofs themvejres, i.e., yn uidxrstanding not onky what deductive connections huld but dlfo how they are established,... (Lrxwitz, 1973, p. 225) One of the main
a natural deduction system developed by Prawitz but introduced by (1935, 2008). When the of arithmetic, he that if there a fault in the paradoxes, it be sought in the logical reasoning employed. One of the main purposes of proof-theoretic analysis of the paradoxes is to find any errors in the reasoning. appears be that suitable proof-theoretic solution to the paradoxes can be the reasoning-rejection solution. RND may be the reasoning-rejection in a broad sense. However, as our second asks, it should be why a nonnormalizable derivation is The Question (ii): should accept a normalizable derivation? promising answer is that a non-normalizable derivation would not be proof-theoretically valid. In this section, we will investigate Prawitz's proof-theoretic validity the between validity and normalizability. validity implies (full) normalizability, then paradoxical which are not (fully) normalizable, are not proof-theoretically Then, since is no nonnormalizable derivation which is valid, it can be explained why any non-normalizable are unacceptable. The name 'proof theory' was originally coined by David Hilbert. The aim of theory is to obtain reduction of mathematics some elementary of such 144 finitistic or constructive mathematics, by analyzing the proofs of mathematical theories. (1971, 1973) thinks that Hilbert's proof theory is only a obtain reduction because it not aim at studying very He calls it reductive He general the of notion of proof. The matter of general proof theory thus proofs considered as get to know the theorems of a theory the validity of an argument, and this is studied here in its own right. (Prawitz, 1971, p. 237) In proof theory, interested in understanding the very proofs themselves, i.e., understanding not only what connections hold but also how they are established,... (Prawitz, p. One of main
 a natural deduction system deVeloped by PRawitZ (1965, 1971) buT fiRsT intRoduCed by Gentzen (1935, 2008). WhEN GenTzen (1936) attempted to show the ConsiStENcy oF ArIthmeTic, he beLIeVED thAt If TheRe WAs A faulT in The paraDoxes, it musT be SoUght in the logICaL reasoning EmpLoyed. One of thE maIn purpOsEs oF The prOof-TheorEtic anALysis oF the paradOxES is to fINd any erRORs In thE reasoning. It appeaRS tO Be convincing thAt a suiTaBLe PROof-TheOretic soluTiOn to tHE paradoXEs CAN Be tHE reasoning-rejEction solutIOn. RnD may bE tHe rEAsoninG-rejeCtIOn sOlution in a bRoad Sense. HoweVer, as oUR second QUestion Asks, it ShoUld Be exPLaInEd wHy A NonNOrMalIZabLe derivaTiOn Is unaCcepTABLE. The secOnd QUestiOn (ii): Why should We aCcepT OnlY a norMalizAble DeRivatIon? A prOmisiNg Answer is that a noN-norMalizable DerIvAtiOn Would NOt be prOof-TheOreticaLly valiD. in tHiS SECtIon, we will briefly inVeSTIgAte PrawiTz's ideA Of PrOOf-theoreTiC vaLidiTY And thE relATiOn betweeN proof-THeOrEtic valIdIty and NoRmaLizAbiliTY. If pRoof-thEoretic vAlidiTY implies (full) noRMalizability, tHEn PARaDOxicAl dErivations, i.E. T-paRAdoxEs whICh Are NOt (fulLy) norMaLIzABle, are not proof-theorEtIcally Valid. then, since therE is no nonnoRMALizable dErivATiON which is proof-tHeoreTically valID, it can be ExplaIned why aNy non-normALIzable deRivAtiOns Are UNAcCeptable. The naME 'ProoF tHeory' waS orIginallY coIneD by davId hilbert. ThE aim of hiS pRoOf ThEorY is to OBtain a reDuCtiOn Of mAthemATics to Some mOre eLeMeNTarY part of IT, sUCH 144 as fInItIstiC or CoNstruCtivE MatHematicS, by analyzIng THe prOoFs Of matheMatical theoriEs. prawitz (1971, 1973) thiNkS thAt HilbERT's proof tHeory is only a tool to obtaiN The reduCtiOn becAuse It does not Aim At studYinG The verY proofS. He caLlS it REDuctiVE PrOof ThEory. He suggESTs gEneraL pRoof Theory aS the study of the notiON of Proof. The subjeCt mAtteR OF gEneRAl PRooF tHEorY IS thus proofs consIdered as a pRoCEsS by which we GEt tO kNow the tHeorems Of a thEOry or thE validity Of an argumEnT, and THIs pRocess is stUdied herE in its own RIght. (PRAwItz, 1971, p. 237) IN geNeral pRoOf tHeory, We are... iNTerEsted In undeRsTandinG the vErY proofs tHemselves, i.e., in understandIng not Only wHat Deductive ConNEctIons hold bUt alSo how they aRe eStaBlishEd,... (PRAwitz, 1973, P. 225) One OF tHe mAIn
 a natural deduction syste m develope d byPra wit z(196 5, 1 971) but first intr oduced by Gentzen (193 5, 20 08 ) . Wh e nGentz en (193 6 )a t tem pt ed  to  s h ow  thecon sistenc y of arith met ic , he believe d  t hat if the rewas a faultinthe pa ra dox e s, it  mu st be  sough t  in th e logical  r e asonin g  employ e d .Oneof the main purpo s es of the proof-t heoret ic an a l ysi s o f the para do xes i s  to fin d  a n y err o rs in the rea soning. Ita ppe ars to  b e c o nvinci ng th at a s uitable pro of-t heoreticsoluti o n to th e  parado xes ca n b e t he r e as on ing -r e jec t io n s o lut ion. RND  m ay  be t he r e a s o ning -re ject ion s olution in abro ad s e nse . How ever,  asou r sec ond qu estio nasks, it should  beexplained  wh ya n on norma l izable  de riv ation i s unacc e pta bl e . Th e Second Question(i i ) :Why shou ld wea cc ep t  only ano rma liza b l e der ivat i on ? A prom isinga ns we r is th at  a non -n orm ali zable deri vation  would n ot be proof-theoreti c ally valid. I n  t h i ss ecti on,  we will br iefl y  inv esti g at e P r awitz 's id ea of proof-theoretic val id ity an d the  relation bet ween proof - t h eoreticvali d it y  and normaliza bilit y. If proo f -theoret ic va lidity i mplies (f u l l) norma liz abi lit y,t h en  paradoxicald e riva ti ons, i. e.T-parad oxe s w hic h a re  not (ful ly) norm al iz ab le , a re no t  proof-t he ore ti cal ly va l id. Th en, s ince  t he r e i s no no n no r m aliz ab le  der iva ti on wh ichi s p roof-th eoretical lyv alid ,it  can be  explained wh yany non-no rm ali zabled e rivation s are unacceptable. The name 'p roo f the ory'  was orig ina lly co ine d  by Da vid Hi lbert .The a im of h is  pr oo f theory i s toobtai na re duction  of mathematics to som e more elemen tar y pa r t  o f i t ,s uch  1 4 4 a s finitistic or c onstructiv em at hematics,b y a na lyzingthe pro ofs o f  mathem atical th eories. P ra witz ( 197 1, 1973) t hinks th at Hilber t 's pr o of  theo ryis onl ya t ool t o obta i n t he re ductio nbecaus e itdo es not a im at studying the very  proof s. He  ca lls it re duc t ive  proof th eory . He sugge sts  ge neral  pr o of th eory as  th e  stud y of the notio n  o f p r o of . The subje c t mat ter o f g e neralproo f theory is thusp roofs consider ed a s a p roc e ss b ywhich we get t o k no w the theo re ms of a the ory or t he valid ity of  an ar gument, a nd this p roce ssis studie d h er e  in its  o wn right.  (Pr aw itz, 1 971, p .  237 ) In general proof  theo r y , wea re. .. in te restedi n un derstandin g the veryproofs  the mselv es, i.e .,  in un der st anding not only what  dedu ctive c on nect ion s hold  but a lso h ow t he y a re establ i s he d ,. .. (Pr awit z, 19 73 , p.  225) One of the m ain 
 a_natural deduction_system developed by Prawitz_(1965, 1971)_but_first introduced_by_Gentzen (1935, 2008)._When Gentzen (1936)_attempted to show the_consistency of arithmetic,_he_believed that if there was a fault in the paradoxes, it must be sought_in_the logical_reasoning_employed._One of the main purposes_of the proof-theoretic analysis of_the paradoxes_is to find any errors in the reasoning._It_appears to be_convincing that a suitable proof-theoretic solution to the paradoxes_can be the reasoning-rejection solution. RND_may be the_reasoning-rejection_solution_in a broad sense._However, as our second question asks,_it should be explained why a_nonnormalizable derivation is unacceptable. The Second Question_(ii): Why should we accept only_a normalizable derivation? A promising_answer is_that a non-normalizable derivation would_not be proof-theoretically_valid. In_this section, we_will briefly investigate Prawitz's idea of_proof-theoretic validity and_the relation between proof-theoretic validity and_normalizability._If proof-theoretic validity_implies_(full)_normalizability, then_paradoxical derivations, i.e._T-paradoxes_which are_not_(fully) normalizable, are not proof-theoretically valid._Then,_since there is no nonnormalizable derivation which_is proof-theoretically valid, it_can_be explained why any_non-normalizable derivations are unacceptable. The_name 'proof theory' was originally coined_by David_Hilbert. The_aim of his proof theory is to obtain a reduction of_mathematics to some more elementary part_of it, such 144_as finitistic_or_constructive mathematics, by_analyzing_the proofs_of mathematical theories. Prawitz (1971, 1973) thinks_that Hilbert's_proof theory is only a tool_to obtain the reduction_because_it does not aim at studying_the very proofs. He calls it_reductive proof theory. He suggests_general_proof_theory as the study of_the notion of proof. The subject_matter of general_proof theory is thus proofs considered as_a_process by which we get to_know_the theorems of a theory or_the_validity_of an argument, and this_process is studied here in its_own right. (Prawitz, 1971, p. 237) In general proof_theory, we are..._interested in understanding the very_proofs_themselves,_i.e., in understanding not only what deductive connections hold but_also how_they are established,..._(Prawitz, 1973, p. 225) One of the main