 elimination harmony (GE-harmony for short). Unlike intrinsic harmony, GE-harmony delivers a method for generating GE-harmonious rules. Recently, the application of generalized elimination rules and the notion of GEharmony has been developed by Dyckhoff and Francez (2012) and Negri and Von Plato (2001). Especially, Negri and Von Plato (2001, p. 6) suggest the generalized version of 7Let S be a system of rules. A rule with the premises φ1,...,φn and the conclusion ψ is admissible in S if, whenever the premises φ1,...,φn are derivable in S, the corresponding conclusion ψ is derivable in S. 43 inversion principle that whatever follows from the direct grounds for deriving a formula must follow from that formula. We call it the Generalized Inversion Principle: The Generalized Inversion Principle: Let Di be any immediate subderivation of an introduction rule for deriving a major premise of an elimination rule, Dj be any derivation of minor premises of the elimination rule. Whatever is derivable by Di together with Dj is a consequence of the major premise. The principle is standardly taken to be formally represented by Dyckhoff and Francez (2012). They have proposed the general form of introduction and generalized elimination rules which satisfy the generalized inversion principle. The general forms of introduction and generalized elimination rules with their standard reduction procedures are called the GE-schema. All rules and their standard reductions introduced in Section 2.1.1 are instances of the GE-schema. GE-harmony relation is defined in terms of the GE-schema such that a pair of introduction and elimination rules is GE-harmonious iff an elimination rule has been induced from the introduction rule by means of the GE-schema. The intrinsic harmony and the GE-harmony have been considered to be the main contemporary accounts of harmony. It is encouraged to use intrinsically harmonious rules (or GE-harmonious rules if generalized elimination rules are employed). It is still an open question of whether we should only use harmonious rules or not. Although it is desirable to use harmonious rules, it seems to be a too strong requirement that only harmonious rules are acceptable. For our purpose of investigating the problem of undergeneration, the next three sections will argue that applications of axioms having a principal operator (or a principal constant) that no I-rule introduces or of rules without having its corresponding harmonious rules can block the occurrence of a non-terminating reduction sequence
elimination harmony (GE - harmony for short). Unlike intrinsic harmony, GE - harmony delivers a method for generating GE - harmonious rule. Recently, the lotion of generalized elimination rules and the notion of GEharmony has been develop by Dyckhoff and Francez (2012) and Negri and Von Plato (2001). specially, Negri and Von Plato (2001, p. 6) suggest the generalized version of 7Let S be a organization of rule. A rule with the premise φ1,... ,φn and the ending ψ is admissible in S if, whenever the premises φ1,... ,φn are derivable in S, the comparable conclusion ψ is derivable in S. 43 inversion principle that whatever follows from the lineal grounds for deriving a rule must follow from that formula. We name it the Generalized Inversion Principle: The Generalized Inversion Principle: Let Di be any immediate subderivation of an introduction principle for deriving a major premise of an elimination rule, Dj be any derivation of minor premises of the elimination convention. Whatever is derivable by Di together with Dj is a consequence of the major premise. The principle is standardly taken to be formally represented by Dyckhoff and Francez (2012). They have proposed the cosmopolitan form of introduction and generalized elimination rules which satisfy the generalized inversion principle. The cosmopolitan form of introduction and generalized elimination rules with their standard reduction procedures are called the GE - schema. All rules and their standard reductions introduced in Section 2.1.1 are instances of the GE - schema. GE - harmony sexual intercourse is defined in terms of the GE - schema such that a couple of initiation and elimination rule is GE - harmonious iff an elimination rule has been induced from the initiation rule by means of the GE - schema. The intrinsic harmony and the GE - harmony have been considered to be the main contemporaneous accounts of harmony. It is encouraged to use intrinsically harmonious rules (or GE - harmonious rules if generalized elimination rules are employed). It is still an open question of whether we should only use harmonious principle or not. Although it is desirable to practice harmonious dominion, it seems to be a too strong necessity that only harmonious rules are acceptable. For our function of investigating the problem of undergeneration, the next three sections will argue that application of axioms have a principal hustler (or a principal constant) that no I - rule introduces or of rules without induce its corresponding harmonious rule can block the occurrence of a non - terminating reduction sequence
 ellmination harmony (GE-harmuny for short). Uuoike iitrinsid harmonh, GE-harmony delivers a methov foe gentgating GE-harmonious rjles. Recejtly, the appoucation of generallved emlminaciin rules and tme notion ox GEharmony hav cezn developed by Dyckhoff and Francez (2012) and Nrggi and Von Plajo (2001). Ex[ecizlly, Negri and Von Plato (2001, p. 6) suggeat the jeneralized verxion of 7Let S be a system lf rkles. A rule with tje premises φ1,...,φn age the conclurion ψ is admissible in S if, whenever the premises φ1,...,φn afe dexivable in W, rhe worrespondiig conblusion ψ is dcgivable in S. 43 onversion pringiple thqt whatever follows fcom the direct grounqs for desiring a formula must filoow ftom tvat wirmjla. Wx czll it thx Generalizsd Inversiob Principle: The Gentrajpaed Inversioh Prinsi[le: Let Di be any immediate subderivatimn kf an introduction rule for deriving a major premise jf an elimination rule, Dj be any derivation of mitor pceoists of tfw flimination rule. Whatever is derivable by Di ekgtthvr with Dj is a cjnsequence pf tng major premisg. The pxjndiple is standardlj taken to bw formallr relresented by Dyckhoff and Feancez (2012). They yave proposed the yeneral form of ontrocuction and generalized elijination rupes which ratisfy the genefalpzed inversion principle. The gqneral focms oy introdjctipn and generalizfd ellkination rules witj thenr stdndard redkction procedures are called thx GE-schema. Alk sulvs and thzir stsndard reductyons introduceb in Seccion 2.1.1 xre instanbes of thx GE-schema. GQ-harmony reladlon is definxd in teris od thw GE-schdoa such that a pair of pnuroduction and elimination rmles ka GE-harmonious nyf an elimination ruue rad ueen ytduced from dhe kntfpductkon rult bn mdans of the GE-schema. The intdinsic harmony and tme GE-harminy have been consideted to be the main contxmporacy accpunjs of harmony. It is encouraged fo use inhriksically harmjnioms roles (or GE-hcrmonious rules if generalized eliminatiin rules are emploted). It is still an pken question pf whqther we vhould only use harminious rules or npt. Although it is desjrable to ude harmonious rules, it seems to be a too strong requirement that only harnonious rules are zcceltablv. Fjx our ptrpodx of investigatinb the problem of undergeneratioi, the next tkree sections will argue that aoplications ow axioms having a princilal operstor (or a principal constant) thst no I-rule introduces ov of rulew witgout havinj its corresponding narmonnous ryles cab bllck the occurfenve of a non-hermmnacing reduction sequence
elimination harmony (GE-harmony for short). Unlike intrinsic delivers method for GE-harmonious rules. Recently, rules the notion of has been developed Dyckhoff and Francez (2012) and Negri Von Plato (2001). Especially, Negri and Von Plato (2001, p. 6) suggest the version of 7Let S be a system of rules. A rule with the φ1,...,φn the ψ admissible in S if, whenever the premises φ1,...,φn are derivable in S, the corresponding conclusion ψ derivable in S. 43 inversion principle that whatever from the direct grounds deriving a formula must follow that We call the Inversion The Generalized Inversion Let Di be any immediate subderivation of an introduction rule for deriving a major premise of an rule, Dj derivation of premises the rule. Whatever is Di together with Dj is a major premise. The principle is standardly taken to formally represented Dyckhoff and Francez (2012). They have the general form of introduction and generalized elimination which satisfy the generalized inversion principle. The general forms of introduction and generalized elimination rules standard reduction procedures are the GE-schema. All and standard introduced Section 2.1.1 instances of the GE-schema. GE-harmony relation is defined in terms of GE-schema such that a pair of introduction and elimination rules iff elimination rule has induced from the introduction by of the GE-schema. The and GE-harmony to the contemporary accounts of harmony. is encouraged to use intrinsically rules (or GE-harmonious rules employed). It is still an open question of we should only use harmonious rules or Although it is desirable to use harmonious rules, it seems to be too strong only harmonious rules are acceptable. For our purpose investigating the problem of the next three sections will argue that applications of having principal operator a principal constant) no I-rule introduces of rules without corresponding harmonious can the non-terminating reduction sequence
 elimination harmony (GE-harmoNy for short). unlikE inTriNsIc haRmonY, GE-harmony deliVErs a Method for generating GE-hArmonIoUS rulES. REcentLy, the apPLiCATioN oF gEneRaLIzEd eliMinAtion ruLes and the nOtiOn Of GEharmony hAS bEen developEd bY Dyckhoff and fraNcez (2012) anD NEgrI And VoN PlAto (2001). EsPecialLY, Negri And Von PlaTo (2001, P. 6) SuggesT The geneRALiZed vErsion of 7Let S be a sySTeM Of rules. A rule wiTh the pReMIsES Φ1,...,φn And The conclusIoN ψ is aDMissiblE In s IF, WheNEver the premisEs φ1,...,φn are derIVabLe in S, tHe CorREspondIng coNcLUsiOn ψ is derivaBle iN S. 43 inversiOn prinCIple thaT WhateveR folloWs fRom The dIReCt GroUnDS foR DeRivINg a Formula mUsT fOllow From THAT FormUla. we caLl it tHe Generalized invErsiON PrIncipLe: The geneRaLized inversIon PrInCiple: Let Di be any ImmeDiate subdEriVaTioN oF an inTRoductIon RulE for derIving a mAJor PrEMISe Of an elimination rulE, DJ BE aNy derivaTion of MInOr PRemises oF tHe eLimiNATion rUle. WHAtEver is deRivablE By di TogetheR wIth Dj iS a ConSeqUence OF the Major pRemise. ThE prinCIple is standardLY taken to be forMAlLY RePReseNteD by Dyckhoff And FRAnceZ (2012). TheY HaVe pROposeD the gEnERaL Form of introduction aNd GeneraLized Elimination ruLes which saTISFy the genEralIZeD Inversion princIple. THe general fORms of intRoducTion and gEneralizeD ELiminatiOn rUleS wiTh tHEIr Standard reducTIOn prOcEdures aRe cAlled thE GE-SchEma. all RuLes and theIr standaRd ReDuCtIonS intrODuced in SEcTioN 2.1.1 aRe iNstanCEs of thE GE-scHema. gE-HaRMonY relatiON iS DEfinEd In TermS of ThE GE-scHema SUch That a paIr of introDucTIon aNd ElIminatiOn rules is GE-haRmOnious iff aN eLimInatioN RUle has beEn induced from the introduCTion rulE by Means Of thE GE-schema. the IntrinSic HArmony And the gE-harMoNy hAVE been CONsIdeReD to be the maIN ConTempoRaRy acCounts oF harmony. It is encourAGed To use intrinsiCalLy haRMOnIouS RuLEs (oR Ge-HarMONious rules if genEralized elImINaTion rules aRE emPlOyed). It iS still aN open QUestion Of whether We should oNlY use HARmoNious rules Or not. AltHough it is DEsiraBLe To use HarMoniouS rUleS, it seEms to bE A toO stroNg requIrEment tHat onLy HarmonioUs rules are acceptable. For Our purPose oF inVestigatiNg tHE prOblem of unDergEneration, tHe nExt Three SecTIons wIll aRGuE thAT applIcatIOns of axioMS hAviNG A pRincipal opeRATOr (oR a priNciPAl consTant) That no I-rule introdUCes or of rules wiThouT HAviNg iTS corReSponding harmonIouS rULEs can bloCk The occurrenCe of a non-TeRMinatIng redUction SequencE
 elimination harmony (GE-h armony for  shor t).  Un li ke i ntri nsic harmony,G E-ha rmony delivers a metho d for  g e nera t in g GE- harmoni o us r ule s.  R ece nt l y,  theapp licatio n of gener ali ze d eliminatio n  r ules and t henotion of GE har mony h as  be e n dev elo ped b y Dyck h off an d Francez  ( 2 012) a n d Negri a nd  Von  Plato (2001). Es p ec i ally, Negri an d VonPl a to ( 200 1,p. 6) sugg es t the general i ze d v ers i on of 7Let Sbe a system ofrules.  A  ru l e with  thepr e mis es φ1,...,φ n an d the con clusio n  ψ is a d missibl e in S  if , w hene v er  t hepr e mis e sφ1, . .., φn are d er iv ablein S , t h e co rre spon dingconclusion ψisderi v abl e inS. 43  inv er sionprinci ple t ha t whatever foll owsfrom thedir ec t g ro undsf or der ivi nga formu la must fol lo w f ro m that formula. We  c a l lit the G eneral i ze dI nversion  P rin cipl e :  TheGene r al ized Inv ersion Pr in ciple:Le t Di b eany  im media t e su bderiv ation of  an i n troduction rul e  for deriving am a jo r  pre mis e of an eli mina t ionrule ,  D j b e  anyderiv at i on of minor premises o fthe el imina tion rule. Wh atever isd e r ivable b y Di to g ether with Djis aconsequenc e  of themajor  premise . The pri n c iple issta nda rdl y t a k en  to be formal l y  rep re sentedbyDyckhof f a ndFra nce z(2012). T hey have  p ro po se d t he ge n eral for mofin tro ducti o n andgener aliz ed  e l imi nationr ul e s  whi ch  s atis fyth e gen eral i zed  invers ion princ ipl e . Th ege neral f orms of intro du ction andge ner alized e liminati on rules with their sta n dard re duc tionproc edures ar e c alledthe GE-sch ema. A ll ru le s a n d  thei r st and ar d reductio n s  in trodu ce d in  Sectio n 2.1.1 are instan c esof the GE-sch ema . GE - h ar mon y  r e lat io n  is d efined in terms  of the GE -s c he ma such th a t a  p air ofintrodu ction and eli minationrules isGE -har m o nio us iff aneliminat ion ruleh as be e ninduc edfrom t he  in trodu ctionr ule  by m eans o fthe GE -sche ma . The in trinsic harmony and the  GE-ha rmony  ha ve been c ons i der ed to bethemain conte mpo rar y acc oun t s ofharm o ny . I t  is e ncou r aged to u s eint r i ns ically harm o n i ous  rule s ( o r GE-h armo nious rules if ge n eralized elimi nati o n  ru les areem ployed). It is  st il l an openqu estion of w hether w es hould  onlyuse ha rmoniou s ru l es ornot.  Al though it  is  d e sirable  t ou se har moni ou s rule s, its eems t o be a too stron g req u i remen t  th at on ly  harmon i ousrules areacceptable.  For o ur p urpos e of in ve stigat ing  t he problem of underg enera tion, t he  nex t t hree s ecti o n s wil l ar gu e t hat appli c a ti o ns  o f  ax ioms  havi ng  a p rincipalo perator(or a princ ip alc o nstant )  t h a t no I-rul e i ntrod u c es or of r u lesw it h out h avingits co rrespon d ing  h armonio usr u les can b lock theo ccu rr ence  of a no n- term in ati ng  reduction sequ ence
 elimination_harmony (GE-harmony_for short). Unlike intrinsic_harmony, GE-harmony_delivers_a method_for_generating GE-harmonious rules._Recently, the application_of generalized elimination rules_and the notion_of_GEharmony has been developed by Dyckhoff and Francez (2012) and Negri and Von Plato_(2001)._Especially, Negri_and_Von_Plato (2001, p. 6) suggest_the generalized version of 7Let_S be_a system of rules. A rule with the_premises_φ1,...,φn and the_conclusion ψ is admissible in S if, whenever the_premises φ1,...,φn are derivable in S,_the corresponding conclusion_ψ_is_derivable in S. 43_inversion principle that whatever follows from_the direct grounds for deriving a_formula must follow from that formula. We_call it the Generalized Inversion Principle:_The Generalized Inversion Principle: Let_Di be_any immediate subderivation of an_introduction rule for_deriving a_major premise of_an elimination rule, Dj be any_derivation of minor_premises of the elimination rule. Whatever_is_derivable by Di_together_with_Dj is_a consequence of_the_major premise._The_principle is standardly taken to be_formally_represented by Dyckhoff and Francez (2012). They_have proposed the general_form_of introduction and generalized_elimination rules which satisfy the_generalized inversion principle. The general forms_of introduction_and generalized_elimination rules with their standard reduction procedures are called the GE-schema._All rules and their standard reductions_introduced in Section 2.1.1_are instances_of_the GE-schema. GE-harmony_relation_is defined_in terms of the GE-schema such that_a pair_of introduction and elimination rules is_GE-harmonious iff an elimination_rule_has been induced from the introduction_rule by means of the GE-schema._The intrinsic harmony and the_GE-harmony_have_been considered to be the_main contemporary accounts of harmony. It_is encouraged to_use intrinsically harmonious rules (or GE-harmonious rules_if_generalized elimination rules are employed). It_is_still an open question of whether_we_should_only use harmonious rules or_not. Although it is desirable to_use harmonious rules, it seems to be a too_strong requirement that_only harmonious rules are acceptable._For_our_purpose of investigating the problem of undergeneration, the next three_sections will_argue that applications_of axioms having a principal operator (or a principal constant)_that no I-rule introduces or of rules_without having its corresponding harmonious rules can block the occurrence of_a non-terminating reduction sequence