 a reasoning-rejection. The related issues will be discussed in Section 4.2. Regarding the non-terminating reduction sequence as the main feature of the paradoxes, Tennant (1982) proposes the criterion for genuine paradoxes. He admits the lesson of Kripke (1975) that some paradoxes are relative to the empirical facts, and he put forward to the criterion with respect to a given model which can contain the empirical facts. Let M be any model and θ(M) be a set of sentences relative to M. Tennant (1982, p. 283) said, A set of sentences is paradoxical relative to M iff there is some proof of [⊥] 27 from θ(M), involving those sentences in id est inferences, that has a looping reduction sequence. His first stipulation of the criterion for paradoxicality is not purely described in prooftheoretic fashion because he uses the notion of'model.' Since Tennant (1982) admits the lesson of Saul Kripke (1975) that some paradoxes are relative to the empirical facts, he proposes the criterion for paradoxicality with respect to a given model of the empirical facts. Instead of using the notion of'model' and'set of sentences,' we will only use a 'derivation.' Tennant (1982) appears to think that some id est inferences are legitimated by empirical facts and some paradoxical derivations have open assumptions. We will consider that both open and closed derivations of ⊥ can be paradoxical. Let us summarize the early version of Tennant's criterion for paradoxicality, TCPE, as follows: The Early Version of Tennant's Criterion for Paradoxicality(TCPE): Let D be any derivation of a given natural deduction system S. D is a T-paradox if and only if (i) D is a (closed or open) derivation of ⊥ 8, (ii) id est inferences (or rules) are used in D, (iii) a reduction procedure of D generates a non-terminating reduction sequence, such as a reduction loop.9 While he confines his attention to intuitionistic proofs, Tennant (1982, p. 285) conjectures that every derivation which formulates a genuine paradox is a T-paradox, by saying, I undertook at the beginning to confine my attention as much as possible to intuitionistic proofs.... It appears to me an open question whether every para8⊥ is not the
a reasoning - rejection. The related issues will be discussed in Section 4.2. Regarding the non - terminating decrease succession as the independent feature of the paradox, Tennant (1982) proposes the standard for genuine paradoxes. He accept the moral of Kripke (1975) that some paradoxes are proportional to the empiric facts, and he put fore to the criterion with respect to a given exemplar which can contain the empirical facts. Let M be any model and θ(M) be a stage set of sentences relative to M. Tennant (1982, p. 283) said, A hardening of sentences is paradoxical relative to M iff there is some proof of [ ⊥ ] 27 from θ(M), involve those sentences in i d est inferences, that has a looping decrease sequence. His first stipulation of the criterion for paradoxicality is not purely described in prooftheoretic manner because he use the notion of'model.' Since Tennant (1982) admits the lesson of Saul Kripke (1975) that some paradoxes are relative to the empirical facts, he proposes the criterion for paradoxicality with esteem to a given model of the empiric fact. rather of using the notion of'model' and'set of prison term,' we will only use a' derivation.' Tennant (1982) appears to remember that some i d est inferences are legitimated by empirical facts and some paradoxical derivations have capable assumptions. We will consider that both open and closed derivation of ⊥ can be paradoxical. Let us sum up the early interpretation of Tennant's criterion for paradoxicality, TCPE, as follows: The Early Version of Tennant's Criterion for Paradoxicality(TCPE ): Let D be any derivation of a give natural deduction system S. D is a T - paradox if and merely if (i) D is a (closed or open) derivation of ⊥ 8, (ii) i five hundred est inferences (or rule) are used in D, (iii) a decrease procedure of D generates a non - terminating reduction sequence, such as a decrease loop.9 While he confines his attention to intuitionistic proof, Tennant (1982, p. 285) conjectures that every derivation which formulate a genuine paradox is a T - paradox, by say, I undertook at the beginning to confine my attention as much as possible to intuitionistic proofs.... It appears to me an exposed question whether every para8⊥ is not the
 a geasoning-rejection. The rtlated issues will be divcusses in Secgion 4.2. Regarding the non-termiiatibg reeuction sequence as thd main fewture of the paradoxes, Vsnnant (1982) propoacs thz rriterion for ggnuine paradmxes. He admits tfe lesson of Kripke (1975) that some paradoves are rflative to the empprycal facts, and he put forward to the cditerioi with respect yo a given model which can conhain the empirical facts. Let N be qny model ana θ(M) be a stt of sentendes relative to M. Tennant (1982, p. 283) sxid, A set of sebtwncfv is paradoeical gelative to M iff these is spme proof of [⊥] 27 frmm θ(N), involving those senvences in id est infgrences, thdt has a looping reeuxtion seqgencd. Hir fjrxt stipupatmon of the driterion fir paradoxicality ix git purely desdribed ig prooftheoretic fashion because he usev tge notion of'model.' Since Tennant (1982) admits the pesson of Saul Kripke (1975) that some paradoxes are relative to the xmoirngwu fwcts, he proposes the criterion for paradoxicajjtu eith respect tj a given mpdfl jf the empirizal faefs. Instead of using hhe notyon od'model' anq'set of sentences,' we will only yse a 'derivanion.' Tennant (1982) appears co think thac some id ext inferences are legitnmated by empiricwl facts zvd some paradoxizal darivations have open assum[tions. We wilk consiaer jhat boeh open anf closed derivations of ⊥ can ye pasadoxical. Pet us summarize the early versmpn of Tennanj's wrinerion fox paracoxicality, TC[E, as follows: Jhe Early Verskon of Tenhant's Cciterion for Paradoxicalidi(TCPE): Let D bx any deryvatuon if a gixdn natural dediction system S. D iw a T-paradox if anv onug if (i) D is a (coostd or open) derivstiun jf ⊥ 8, (iy) id est infesencds (ut rulds) are used kn D, (iii) a reduction prowedude of D generates s kon-terminqting requction sequemce, such as a reduftion loup.9 Whole he confines his attention to jntuitionlstlc proofs, Tengant (1982, p. 285) conjectbres that every derivation which formulaves a genuine paradox iw a T-paradox, by saynnn, I undertool at ehe beginting to confine my artention as much ss possible to intuitjonisthc prlofs.... It appears to me an open question whether every para8⊥ is not the
a reasoning-rejection. The related issues will be Section Regarding the reduction sequence as paradoxes, (1982) proposes the for genuine paradoxes. admits the lesson of Kripke (1975) some paradoxes are relative to the empirical facts, and he put forward to criterion with respect to a given model which can contain the empirical facts. M any and be a set of sentences relative to M. Tennant (1982, p. 283) said, A set of is paradoxical relative to M iff there is proof of [⊥] 27 θ(M), involving those sentences in est that has looping sequence. first stipulation of criterion for paradoxicality is not purely described in prooftheoretic fashion because he uses the notion of'model.' Since (1982) admits of Saul (1975) some are relative to facts, he proposes the criterion for to a given model of the empirical facts. of using notion of'model' and'set of sentences,' we only use a 'derivation.' Tennant (1982) appears to that some id est inferences are legitimated by empirical facts and some paradoxical derivations have We will consider that open and closed of can paradoxical. us summarize early version of Tennant's criterion for paradoxicality, TCPE, as follows: The Version of Tennant's Criterion for Paradoxicality(TCPE): Let D be any a natural deduction system D is a T-paradox and if (i) D is or derivation (ii) est (or rules) are used D, (iii) a reduction procedure D generates a non-terminating reduction loop.9 While he confines his attention to proofs, Tennant (1982, p. 285) conjectures that derivation which formulates a genuine paradox is a T-paradox, by saying, I at the confine my attention as much as possible to proofs.... It appears to an open question whether every para8⊥ is not the
 a reasoning-rejection. The relAted issues Will bE diScuSsEd in sectIon 4.2. Regarding thE Non-tErminating reduction seqUence As THe maIN fEaturE of the pARaDOXes, teNnAnt (1982) PrOPoSes thE crIterion For genuine ParAdOxes. He admits THe Lesson of KrIpkE (1975) that some parAdoXes are ReLatIVe to tHe eMpiriCal facTS, and he Put forwarD tO The criTErion wiTH ReSpecT to a given model whiCH cAN contain the empIrical FaCTs. lET M bE anY model and θ(m) bE a set OF sentenCEs RELAtiVE to M. Tennant (1982, p. 283) sAid, A set of seNTenCes is pArAdoXIcal reLativE tO m ifF there is somE proOf of [⊥] 27 from θ(m), involVIng thosE SentencEs in id Est InfErenCEs, ThAt hAs A LooPInG reDUctIon sequeNcE. HIs firSt stIPULAtioN of The cRiterIon for paradoxIcaLity IS noT pureLy desCribEd In proOftheoRetic FaShion because he uSes tHe notion oF'moDeL.' SiNcE TennANt (1982) admiTs tHe lEsson of saul KriPKe (1975) tHaT SOMe Paradoxes are relatiVe TO ThE empiricAl factS, He PrOPoses the CrIteRion FOR paraDoxiCAlIty with rEspect TO a GiVen modeL oF the emPiRicAl fActs. INSteaD of usiNg the notIon of'MOdel' and'set of seNTences,' we will oNLy USE a 'DErivAtiOn.' Tennant (1982) apPearS To thInk tHAt SomE Id est InferEnCEs ARe legitimated by empiRiCal facTs and Some paradoxicAl derivatiONS Have open AssuMPtIOns. We will consiDer thAt both open ANd closed DerivAtions of ⊥ Can be paraDOXical. Let Us sUmmAriZe tHE EaRly version of TENNant'S cRiterioN foR paradoXicAliTy, TcPE, As Follows: ThE Early VeRsIoN oF TEnnAnt's CRIterion fOr parAdOxiCalitY(tCPE): LeT D be aNy deRiVaTIon Of a giveN NaTURal dEdUcTion SysTeM S. D is A T-paRAdoX if and oNly if (i) D is A (clOSed oR oPeN) derivaTion of ⊥ 8, (ii) id est InFerences (or RuLes) Are useD IN D, (iii) a reDuction procedure of D geneRAtes a noN-teRminaTing Reduction SeqUence, sUch AS a reduCtion lOop.9 WhIlE he CONfineS HIs AttEnTion to intuITIonIstic PrOofs, tennant (1982, P. 285) conjectures that evERy dErivation whicH foRmulATEs A geNUiNE paRaDOx iS A t-paradox, by sayinG, I undertooK aT ThE beginning TO coNfIne my atTention As mucH As possiBle to intuItionistiC pRoofS.... iT apPears to me aN open queStion whetHEr eveRY pAra8⊥ is Not The
 a reasoning-rejection. Th e relatedissue s w ill  b e di scus sed in Section 4.2.  Regarding the non-ter minat in g  red u ct ion s equence as t hema in  fe at u re  of t heparadox es, Tennan t ( 19 82) proposes th e criterio n f or genuine p ara doxes.  H e a d mitsthe  less on ofK ripke(1975) th at some p a radoxes a re  rel ative to the empi r ic a l facts, and h e putfo r wa r d  to  th e criterio nwithr espectt oa g ive n  model whichcan contain the  empir ic alf acts.Let M  b e  an y model and  θ(M ) be a se t of s e ntences relativ e to M . T enn ant( 19 82 , p .2 83) sa id, A s et of se nt en ces i s pa r a d o xica l r elat ive t o M iff there  is  som e  pr oof o f [⊥]  27fr om θ( M), in volvi ng  those sentence s in  id est i nfe re nce s,  that has aloo pin g reduc tion se q uen ce . H is  first stipulation  o f th e criter ion fo r  p ar a doxicali ty  is  not p urely  des c ri bed in p roofth e or et ic fash io n beca us e h e u ses t h e no tion o f'model. ' Sin c e Tennant (198 2 ) admits thel es s o no f Sa ulKripke (197 5) t h at s omep ar ado x es ar e rel at i ve to the empirical fa ct s, hepropo ses the crite rion for p a r a doxicali ty w i th respect to a g ivenmodel of t h e empiri cal f acts. In stead ofu s ing thenot ion  of 'mo d e l'  and'set of s e n tenc es ,' we w ill  only u sea ' der iva ti on.' Tenn ant (198 2)  a pp ea rsto th i nk thatso meid  es t inf e rences  arelegi ti ma t edby empi r ic a l  fac ts  a nd s ome  p arado xica l  de rivatio ns have o pen assu mp ti ons. We  will conside rthat bothop enand cl o s ed deriv ations of ⊥ can be para d oxical.  Le t ussumm arize the  ea rly ve rsi o n of T ennant 's cr it eri o n  forp a ra dox ic ality, TCP E ,  as  foll ow s: T he Earl y Version of Tenna n t's  Criterion fo r P arad o x ic ali t y( T CPE ): Let D  be any derivat ion of a g iv e nnatural de d uct io n syste m S. Dis aT -parado x if andonly if ( i)  D i s a ( closed oropen) de rivationo f ⊥ 8 ,  ( ii) i d e st inf er enc es (o r rule s ) a re us ed inD,  (iii)  a re du ction pr ocedure of D generatesa non- termi nat ing reduc tio n  se quence, s uchas a reduc tio n l oop.9  Wh i le he  con f in esh is at tent i on to int u it ion i s ti c proofs, T e n n ant  (198 2,p . 285)  con jectures that eve r y derivation w hich f orm ula t es a  g enuine paradox  is  a T -paradox ,by saying,I undert oo k  at t he beg inning  to con f i ne my att enti onas much a s p os s ible to  i nt u itioni stic  p roofs. ... It appe a r s to me an openquest i o n whe t her  ever ypara8⊥i s no t the
 a_reasoning-rejection. The_related issues will be_discussed in_Section_4.2. Regarding_the_non-terminating reduction sequence_as the main_feature of the paradoxes,_Tennant (1982) proposes_the_criterion for genuine paradoxes. He admits the lesson of Kripke (1975) that some paradoxes_are_relative to_the_empirical_facts, and he put forward_to the criterion with respect_to a_given model which can contain the empirical facts._Let_M be any_model and θ(M) be a set of sentences relative_to M. Tennant (1982, p. 283)_said, A set_of_sentences_is paradoxical relative to_M iff there is some proof_of [⊥] 27 from θ(M), involving_those sentences in id est inferences, that_has a looping reduction sequence. His_first stipulation of the criterion_for paradoxicality_is not purely described in_prooftheoretic fashion because_he uses_the notion of'model.'_Since Tennant (1982) admits the lesson_of Saul Kripke_(1975) that some paradoxes are relative_to_the empirical facts,_he_proposes_the criterion_for paradoxicality with_respect_to a_given_model of the empirical facts. Instead_of_using the notion of'model' and'set of sentences,'_we will only use_a_'derivation.' Tennant (1982) appears_to think that some id_est inferences are legitimated by empirical_facts and_some paradoxical_derivations have open assumptions. We will consider that both open and_closed derivations of ⊥ can be_paradoxical. Let us summarize_the early_version_of Tennant's criterion_for_paradoxicality, TCPE,_as follows: The Early Version of Tennant's_Criterion for_Paradoxicality(TCPE): Let D be any derivation_of a given natural_deduction_system S. D is a T-paradox_if and only if (i) D_is a (closed or open)_derivation_of_⊥ 8, (ii) id est_inferences (or rules) are used in_D, (iii) a_reduction procedure of D generates a non-terminating_reduction_sequence, such as a reduction loop.9_While_he confines his attention to intuitionistic_proofs,_Tennant_(1982, p. 285) conjectures that_every derivation which formulates a genuine_paradox is a T-paradox, by saying, I undertook at_the beginning to_confine my attention as much_as_possible_to intuitionistic proofs.... It appears to me an open question_whether every_para8⊥ is not_the