 a natural deduction system, we restrict our concern on the proof-theoretic validity as a property of formal derivations in the system. We will borrow Prawitz's notion of proof-theoretic validity introduced in Prawitz (1971, 1973, 1974, 2006), and it will be applied to formal derivations relative to a set of reduction procedures but not to his notion of an argument. Prawitz (1965) shows that by iterated application of reduction processes, every derivation in an intuitionistic natural deduction system can be converted into a normal derivation. It has a corollary that every closed derivation in the system can be restated to one using an introduction rule in the last step. Prawitz (1971, 1973, 1974, 2006) interprets the collorary as the requirement of a valid inference that a valid closed derivation is able to be reduced to one using an introduction rule in the last step. The results are connected to Prawitz's proof-theoretic validity through his inversion principle that whatever follows from a formula must follow from the direct ground for deriving that formula. As we have seen in Section 1.2, standard reduction procedures for ∧, ∨, →, ¬, ∀, and ∃ show that a pair of introduction and elimination rules of each constant satisfy the inversion principle. The inversion principle reflects Gentzen's idea that the meaning of an operator (or a constant) is determined by an introduction rule and determines an elimination rule. The idea gives a semantic interpretation of an introduction rule 146 that nothing is gained by an application of an elimination rule when its major premise has been derived by means of an introduction rule which confers a meaning of a constant. With the semantic role of an introduction rule, Prawitz (1971, 1973, 1974, 2006) introduces his definitions of proof-theoretic validity based on introduction rules.3 The main idea is that introduction rules preserve proof-theoretic validity and elimination rules are justified by (standard) reduction procedures. Simply put, a derivation is proof-theoretically valid if either it reduces to a derivation of an atomic formula, or it reduces to a derivation whose last step is an introduction rule and whose immediate subderivations are proof-theoretically valid. To introduce Prawitz's definition of proof-theoretic validity more precisely, we borrow some terminologies from Schroeder-Heister (2006). Let D1,...,Dn be derivations where n is a natural number, P be a set of production rules which derives
a natural deduction system, we restrict our business on the proof - theoretical robustness as a property of conventional deriving in the system. We will borrow Prawitz's impression of proof - theoretic validity introduced in Prawitz (1971, 1973, 1974, 2006), and it will be apply to formal derivations proportional to a set of reduction procedures but not to his notion of an controversy. Prawitz (1965) shows that by iterated application of reduction processes, every derivation in an intuitionistic lifelike deduction system can be converted into a normal ancestry. It has a corollary that every closed deriving in the system can be repeat to one using an insertion rule in the last step. Prawitz (1971, 1973, 1974, 2006) rede the collorary as the requirement of a valid inference that a valid closed derivation is able to be reduced to one using an initiation rule in the last step. The results are connected to Prawitz's proof - theoretic validity through his inversion principle that whatever follows from a recipe must follow from the direct flat coat for deriving that formula. As we have seen in Section 1.2, standard decrease procedures for ∧, ∨, →, ¬, ∀, and ∃ usher that a pair of introduction and elimination dominion of each constant satisfy the inversion principle. The inversion principle reflects Gentzen's estimate that the meaning of an operator (or a constant) is determined by an introduction dominion and determines an elimination convention. The idea gives a semantic interpretation of an introduction rule 146 that nothing is gain by an application of an elimination rule when its major premise has been derived by mean of an introduction principle which confers a meaning of a constant. With the semantic role of an insertion rule, Prawitz (1971, 1973, 1974, 2006) introduces his definition of proof - theoretic cogency based on presentation rules.3 The main theme is that introduction rules keep proof - theoretic robustness and elimination rules are justified by (standard) reduction procedures. Simply put, a derivation is proof - theoretically valid if either it reduces to a deriving of an atomic formula, or it reduce to a derivation whose last step is an introduction rule and whose immediate subderivations are proof - theoretically valid. To insert Prawitz's definition of proof - theoretic validity more precisely, we borrow some terminologies from Schroeder - Heister (2006). Let D1,... ,Dn be derivations where n is a natural numeral, phosphorus embody a set of production rules which derives
 a jatural deduction system, we restrict out xonceri on ths proof-tfeoretic validity as a propecty if foemal derivations in thd system. Ae will vorriq Prawitz's notion of prokn-theoxevic validity injroduced in [rawitz (1971, 1973, 1974, 2006), ang kt will be applied to formal derivatijns relstlve to a set os rectctikn procedures but not to his notioh of an argument. Praeitz (1965) shows that by iteratfd aoplication of reduftion procewses, wvery derivagion in an intuitionisjic natural deduction system can be cpnverted ibti a tormal deritation. It has a covpllary that efery closed devivatmon un the system can be cestated to one usind an intrmdbction rule in the lawt step. Prafitz (1971, 1973, 1974, 2006) inuer'refs the comlorary as the requirwment of a valid ingewvmce that a vzlid cjofed derivation is able to be reduced to ons using an introduction rule in the last stek. The resujts are connected to Prawitz's proof-theoretic valigity vhfouym fus inversion principle that whatever follows frjj s normula must foljow from thr fitgct ground for derivnhg that formula. As wf have feen un Sectiog 1.2, syandard reduction procedurew for ∧, ∨, →, ¬, ∀, aud ∃ show that a pair lf introduccion amd elomination rules of each conatant satisvy the inbdrsion principle. Thv ineersion kfinciple reflects Gentzen's idec that tfe mganing jf an operwtor (or a constant) is dehermiued bf an introfuction rule and determines an xkimination role. Thv idea gires a xemantic intewpretation of cn introbuctiov rule 146 thzt nothmng is gaineq by an appliwwtion of an xliminatijn ryle qhen itr major premise has been derived bt means of an intrpduzfion rule which cobfers a meaning of a sojsvant. Rhth the semattic rour of xn introducbiov ruke, Prawitz (1971, 1973, 1974, 2006) introgucea his definitions pf proof-thgoretic vwlidity based on introduction rkles.3 Vhe mamn ides if that introduction rules pressrve proov-thcoretic validyty qnd eliminatnon rules are justified by (standard) redurtion procedures. Simpli put, a derivation is kroof-theoretirally dalid if aither it reduces to a derivation of sn atomic formula, or jt redgces ho a derivation whose last step is an introduction rule and whose immediare subvewivations ade ptoof-tvejxeticaljy velmd. To introduce Pgawitz's definition of proof-theorxtic validhtv more precisely, we borrow soke terminologier from Schroeder-Heister (2006). Let D1,...,Dn be derivations where n is a nayural number, P be a set pf produxtion rules whirh derives
a natural deduction system, we restrict our the validity as property of formal will Prawitz's notion of validity introduced in (1971, 1973, 1974, 2006), and it be applied to formal derivations relative to a set of reduction procedures but to his notion of an argument. Prawitz (1965) shows that by iterated application reduction every in intuitionistic natural deduction system can be converted into a normal derivation. It has a corollary that closed derivation in the system can be restated one using an introduction in the last step. Prawitz 1973, 2006) interprets collorary the of a valid that a valid closed derivation is able to be reduced to one using an introduction rule in last step. are connected Prawitz's validity his inversion principle follows from a formula must follow ground for deriving that formula. As we have in Section standard reduction procedures for ∧, ∨, ¬, ∀, and ∃ show that a pair introduction and elimination rules of each constant satisfy the inversion principle. The inversion principle reflects that the meaning of operator (or a is by introduction and determines elimination rule. The idea gives a semantic interpretation of an introduction 146 that nothing is gained by an application of an when major premise has derived by means of introduction which confers a meaning constant. the an rule, (1971, 1973, 1974, 2006) his definitions of proof-theoretic validity on introduction rules.3 The rules preserve proof-theoretic validity and elimination rules are by (standard) reduction procedures. Simply put, a is proof-theoretically valid if either it reduces to a derivation of an formula, or to a derivation whose last step is an rule and whose immediate are proof-theoretically valid. To introduce Prawitz's definition of proof-theoretic more we borrow terminologies from Schroeder-Heister Let D1,...,Dn be where n is number, P a of derives
 a natural deduction system, we Restrict ouR concErn On tHe ProoF-theOretic validity AS a prOperty of formal derivatiOns in ThE SystEM. WE will Borrow PRAwITZ's nOtIoN of PrOOf-TheorEtiC validiTy introducEd iN PRawitz (1971, 1973, 1974, 2006), and it wILl Be applied tO foRmal derivatiOns RelatiVe To a SEt of rEduCtion ProcedURes but Not to his nOtIOn of an ARgument. pRAwItz (1965) sHows that by iterateD ApPLication of reduCtion pRoCEsSES, evEry Derivation In An intUItionisTIc NATUraL Deduction systEm can be convERteD into a NoRmaL DerivaTion. IT hAS a cOrollary thaT eveRy closed dErivatIOn in the SYstem caN be resTatEd tO one USiNg An iNtROduCTiOn rULe iN the last StEp. prawiTz (1971, 1973, 1974, 2006) inTERPRets The CollOrary As the requiremEnt Of a vALid InferEnce tHat a VaLid clOsed deRivatIoN is able to be reduCed tO one using An iNtRodUcTion rULe in thE laSt sTep. The rEsults aRE coNnECTEd To Prawitz's proof-theOrETIc Validity ThrougH HiS iNVersion pRiNciPle tHAT whatEver FOlLows from A formuLA mUsT follow FrOm the dIrEct GroUnd foR DeriVing thAt formulA. As we HAve seen in SectiON 1.2, standard reduCTiON PrOCeduRes For ∧, ∨, →, ¬, ∀, and ∃ show tHat a PAir oF intROdUctIOn and ElimiNaTIoN Rules of each constant SaTisfy tHe invErsion principLe. The inverSION principLe reFLeCTs Gentzen's idea That tHe meaning oF An operatOr (or a Constant) Is determiNED by an intRodUctIon RulE ANd Determines an eLIMinaTiOn rule. THe iDea giveS a sEmaNtiC inTeRpretatioN of an intRoDuCtIoN ruLe 146 thaT Nothing iS gAinEd By aN applICation Of an eLimiNaTiON ruLe when iTS mAJOr prEmIsE has BeeN dEriveD by mEAns Of an intRoduction RulE WhicH cOnFers a meAning of a constAnT. With the seMaNtiC role oF AN introduCtion rule, Prawitz (1971, 1973, 1974, 2006) introduCEs his deFinItionS of pRoof-theorEtiC validIty BAsed on IntrodUctioN rUleS.3 tHe maiN IDeA is ThAt introducTIOn rUles pReServE proof-tHeoretic validity anD EliMination rules Are JustIFIeD by (STaNDarD) rEDucTIOn procedures. SimPly put, a derIvATiOn is proof-tHEorEtIcally vAlid if eIther IT reduceS to a derivAtion of an AtOmic FORmuLa, or it reduCes to a deRivation wHOse laST sTep is An iNtroduCtIon Rule aNd whosE ImmEdiatE subdeRiVationS are pRoOf-theoreTically valid. To introduce prawitZ's defIniTion of proOf-tHEorEtic validIty mOre preciseLy, wE boRrow sOme TErminOlogIEs FroM schroEder-hEister (2006). Let d1,...,dn Be dERIvAtions where N IS A naTural NumBEr, P be a Set oF production rules wHIch derives
 a natural deduction syste m, we rest rictour  co nc ernon t he proof-theor e ticvalidity as a property  of f or m al d e ri vatio ns in t h es y ste m.  W e w il l  b orrow  Pr awitz's  notion of  pr oo f-theoreticv al idity intr odu ced in Prawi tz(1971,  1 973 ,  1974 , 2 006),  and i t  willbe applie dt o form a l deriv a t io ns r elative to a seto fr eduction proce duresbu t  n o t  to  hi s notion o fan ar g ument.P ra w i t z ( 1 965) shows th at by itera t edapplic at ion of red uctio np roc esses, ever y de rivationin ani ntuitio n istic n atural  de duc tion sy st emca n  be co nve r ted  into ano rm al de riva t i o n . It  ha s acorol lary that eve ryclos e d d eriva tionin t he  syst em can  be r es tated to one us ingan introd uct io n r ul e int he las t s tep . Prawi tz (197 1 , 1 97 3 , 19 74, 2006) interpre ts t he  collora ry ast he  r e quiremen tofa va l i d inf eren c ethat a v alid c l os ed  deriva ti on isab letobe re d uced  to on e usingan in t roduction rule in the last s t ep . Th e  res ult s are conne cted to P rawi t z' s p r oof-t heore ti c  v a lidity through hisin versio n pri nciple that w hatever fo l l o ws froma fo r mu l a must followfromthe direct ground f or de riving t hat formu l a . As wehav e s een  in S ec tion 1.2, sta n d ardre duction  pr ocedure s f or∧,∨,→,  ¬, ∀, an d ∃ show  t ha tapai r ofi ntroduct io n a nd  el imina t ion ru les o f ea ch  c o nst ant sat i sf y thein ve rsio n p ri ncipl e. T h e i nversio n princip ler efle ct sGentzen 's idea thatth e meaningof  an  opera t o r (or aconstant) is determined by an i ntr oduct ionrule anddet ermine s a n  elimi nation  rule .The i dea g i v es  ase mantic int e r pre tatio nof a n intro duction rule 146 t h atnothing is ga ine d by a napp l ic a tio no f a n elimination rul e when its  m a jo r premiseh asbe en deri ved bymeans of an i ntroducti on rule w hi ch c o n fer s a meanin g of a c onstant.W ith t h eseman tic  roleof  an  intr oducti o n r ule,Prawit z(1971,  1973 ,1974, 20 06) introduces his defi nition s ofpro of-theore tic val idity bas ed o n introduc tio n r ules. 3 T h e mai n id e aist hat i ntro d uction ru l es  pr e s er ve proof-th e o r eti c val idi t y andelim ination rules are justified by ( stan d a rd)  re d ucti on  procedures. S imp ly p ut, a de ri vation is p roof-the or e tical ly val id ifeitheri t  r e ducesto a  de rivationofan atomicfo rm u la, or  itre ducesto a d e riva t i on whose last st ep is a n int r odu ction  r ule and whos e immediat e subderiva tionsareproof -theore ti callyval id . To intro d uce Prawi tz'sdefinit io n of  pr oof-th eore t i c val idit ymor e precise l y ,w ebo r row  som e ter mi nolo gies from Schroede r-H e ister ( 20 06) . Let D1 , .. . , Dn be deri vat ionsw h ere n is a natu r al numbe r, P b e a se t of pr o duc ti on rule s w h i ch derive s
 a_natural deduction_system, we restrict our_concern on_the_proof-theoretic validity_as_a property of_formal derivations in_the system. We will_borrow Prawitz's notion_of_proof-theoretic validity introduced in Prawitz (1971, 1973, 1974, 2006), and it will be applied_to_formal derivations_relative_to_a set of reduction procedures_but not to his notion_of an_argument. Prawitz (1965) shows that by iterated application_of_reduction processes, every_derivation in an intuitionistic natural deduction system can be_converted into a normal derivation. It_has a corollary_that_every_closed derivation in the_system can be restated to one_using an introduction rule in the_last step. Prawitz (1971, 1973, 1974, 2006)_interprets the collorary as the requirement_of a valid inference that_a valid_closed derivation is able to_be reduced to_one using_an introduction rule_in the last step. The results_are connected to_Prawitz's proof-theoretic validity through his inversion_principle_that whatever follows_from_a_formula must_follow from the_direct_ground for_deriving_that formula. As we have seen_in_Section 1.2, standard reduction procedures for ∧,_∨, →, ¬, ∀,_and_∃ show that a_pair of introduction and elimination_rules of each constant satisfy the_inversion principle._The inversion_principle reflects Gentzen's idea that the meaning of an operator (or_a constant) is determined by an_introduction rule and determines_an elimination_rule._The idea gives_a_semantic interpretation_of an introduction rule 146 that nothing_is gained_by an application of an elimination_rule when its major_premise_has been derived by means of_an introduction rule which confers a_meaning of a constant. With_the_semantic_role of an introduction rule,_Prawitz (1971, 1973, 1974, 2006) introduces_his definitions of_proof-theoretic validity based on introduction rules.3 The_main_idea is that introduction rules preserve_proof-theoretic_validity and elimination rules are justified_by_(standard)_reduction procedures. Simply put, a_derivation is proof-theoretically valid if either_it reduces to a derivation of an atomic formula,_or it reduces_to a derivation whose last_step_is_an introduction rule and whose immediate subderivations are proof-theoretically valid._To introduce_Prawitz's definition of_proof-theoretic validity more precisely, we borrow some terminologies from Schroeder-Heister_(2006). Let D1,...,Dn be derivations where n_is a natural number, P be a set of production rules_which derives