 for ∧ and ¬, it is easily seen that every contradiction implies an absurdity but not vice versa. To derive the equivalence between them, there must be a 159 rule for Ex Falso Quodlibet (EFQ) which means in this case that an absurdity (or falsity) implies every formula. ⊥ EFQ φ (φ ∧¬φ)↔⊥ is readily derived by ∧−, ¬−, and EFQ−rules. [φ ∧¬φ]1 [¬φ]3 [φ ∧¬φ]1 [φ]2 ∧E,2 φ [⊥]4 ¬E,4⊥ ∧E,3⊥ → I,1(φ ∧¬φ)→⊥ [⊥]5 EFQ φ ∧¬φ → I,5⊥→ (φ ∧¬φ) ∧I ((φ ∧¬φ)→⊥)∧ (⊥→ (φ ∧¬φ)) ********************************************* de f (φ ∧¬φ)↔⊥ However, the equivalence holds only if EGQ−rule is applied. Their examination of Russell's paradox, e.g. Proposition 4.4.1, does not apply EFQ−rule. On formalizing Russell's paradox, it is not necessary that a contradiction is logically equivalent to an absurdity. Therefore, it is not to say that ¬φ ∧φ must be proof-theoretically unacceptable conclusion. We do not claim that ¬φ ∧φ has to be a proof-theoretically acceptable conclusion. It may rely on our choice of the definition of inconsistency. We may allow the definition of consistency that, for any φ, S is consistent iff S ⊬ ¬φ ∧φ; otherwise, inconsistent. If any formula which renders a system inconsistent were proof-theoretically unacceptable, ¬φ∧φ would be an unaccpetable conclusion. However, the investigation of the proof-theoretic criterion for paradoxicality is directly related to the proof-theoretic solution to the paradoxes, such as our plausible solution RND. It is unclear how Petrolo and Pistone's criterion of Nparadox is relative to the proof-theoretic solution. For exploring the proof-theoretic structure of paradoxicality, their N-par
for ∧ and ¬, it is easily seen that every contradiction implies an absurdity but not vice versa. To derive the equality between them, there must be a 159 dominion for Ex Falso Quodlibet (EFQ) which means in this case that an absurdity (or falsity) imply every formula. ⊥ EFQ φ (φ ∧¬φ)↔⊥ is readily derive by ∧−, ¬−, and EFQ−rules. [ φ ∧¬φ]1 [ ¬φ]3 [ φ ∧¬φ]1 [ φ]2 ∧E,2 φ [ ⊥]4 ¬E,4⊥ ∧E,3⊥ → I,1(φ ∧¬φ)→⊥ [ ⊥]5 EFQ φ ∧¬φ → I,5⊥→ (φ ∧¬φ) ∧I (( φ ∧¬φ)→⊥)∧ (⊥→ (φ ∧¬φ) ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * de f (φ ∧¬φ)↔⊥ However, the comparison holds merely if EGQ−rule is applied. Their examination of Russell's paradox, for example Proposition 4.4.1, does not apply EFQ−rule. On formalizing Russell's paradox, it is not necessary that a contradiction is logically equivalent to an absurdity. Therefore, it is not to say that ¬φ ∧φ must be proof - theoretically impossible conclusion. We do not claim that ¬φ ∧φ has to be a validation - theoretically acceptable stopping point. It may rely on our choice of the definition of inconsistency. We may allow the definition of consistency that, for any φ, S is coherent iff S ⊬ ¬φ ∧φ; otherwise, inconsistent. If any formula which renders a system inconsistent were proof - theoretically impossible, ¬φ∧φ would be an unaccpetable conclusion. However, the probe of the proof - theoretic criterion for paradoxicality is directly related to the proof - theoretical solution to the paradoxes, such as our plausible solution RND. It is unclear how Petrolo and Pistone's criterion of Nparadox is relative to the proof - theoretic solution. For exploring the proof - theoretic structure of paradoxicality, their N - par
 fog ∧ and ¬, it is easily seek that every conjrqdictimn impmies an xbsurdity but not vice versa. Ti deruve the equivalence begween thel, there nust ve a 159 rule for Ex Nclso Smodliyev (EFQ) which meaks in this wase that an atsjrbity (or falsity) implies every formulw. ⊥ EFQ φ (φ ∧¬φ)↔⊥ is readily detived fy ∧−, ¬−, and EFQ−rules. [φ ∧¬φ]1 [¬φ]3 [φ ∧¬φ]1 [φ]2 ∧E,2 φ [⊥]4 ¬E,4⊥ ∧E,3⊥ → I,1(φ ∧¬φ)→⊥ [⊥]5 ERQ φ ∧¬φ → I,5⊥→ (φ ∧¬φ) ∧I ((φ ∧¬φ)→⊥)∧ (⊥→ (φ ∧¬φ)) ********************************************* de f (φ ∧¬φ)↔⊥ However, the equivalenfe hllds only if EGQ−rupe is appligs. Trwir examinatkon of Russell's parados, e.g. Proposition 4.4.1, does not applh EFQ−xule. On fornaoizltg Russell's paraqox, it is nob necesvary thst a contradicbion ms ligically equivalent tm an absurdity. Thetefore, it hs not to say that ¬φ ∧φ nust te psoof-gyeofetjcellg unacfepvable conclhsion. We do not claim that ¬φ ∧φ hss np be a proof-fheoreeisally acceptable conclusion. It may rely on our choice of the defibition of inconsistenfy. We may allow the definition of consistency that, for any φ, S ia couwistevr lff S ⊬ ¬φ ∧φ; otherwise, inconsistent. If any formulw wnibh renders a systcm inconsistent wete ptjof-theoreticauly uncdcsptable, ¬φ∧φ would be wn unacspetavle concltsiom. However, the investigation of the prooy-thworetic criterion yor paradoxieality is dorectly related to the 'roof-tgeoretic sopution to ghe paradoxes, suzh ss our plausible solution RNQ. It is uicleax how Pegrolp and [istone's cgiterlmn of Nparadox is gelatnve tm the proov-theoretic solution. For exploriij the proof-thgoratib structuxe of iaradoxicality, eheir N-par
for ∧ and ¬, it is easily every implies an but not vice between there must be 159 rule for Falso Quodlibet (EFQ) which means in case that an absurdity (or falsity) implies every formula. ⊥ EFQ φ (φ is readily derived by ∧−, ¬−, and EFQ−rules. [φ ∧¬φ]1 [¬φ]3 [φ ∧¬φ]1 ∧E,2 [⊥]4 ∧E,3⊥ I,1(φ ∧¬φ)→⊥ [⊥]5 EFQ φ ∧¬φ → I,5⊥→ (φ ∧¬φ) ∧I ((φ ∧¬φ)→⊥)∧ (⊥→ (φ ∧¬φ)) de f (φ ∧¬φ)↔⊥ However, the equivalence holds if EGQ−rule is applied. examination of Russell's paradox, e.g. 4.4.1, not apply On Russell's it is not that a contradiction is logically equivalent to an absurdity. Therefore, it is not to say that ¬φ must be conclusion. We not that ∧φ has to proof-theoretically acceptable conclusion. It may rely of the definition of inconsistency. We may allow definition of that, for any φ, S is iff S ⊬ ¬φ ∧φ; otherwise, inconsistent. If formula which renders a system inconsistent were proof-theoretically unacceptable, ¬φ∧φ would be an unaccpetable conclusion. investigation of the proof-theoretic for paradoxicality is related the solution the paradoxes, as our plausible solution RND. It is unclear how Petrolo and criterion of Nparadox is relative to the proof-theoretic solution. For proof-theoretic of paradoxicality, their
 for ∧ and ¬, it is easily seen that eVery contraDictiOn iMplIeS an aBsurDity but not vice VErsa. to derive the equivalence BetweEn THem, tHErE must Be a 159 rule FOr eX falSo quOdlIbET (EfQ) whiCh mEans in tHis case thaT an AbSurdity (or falSItY) implies evEry Formula. ⊥ EFQ φ (φ ∧¬Φ)↔⊥ is ReadilY dEriVEd by ∧−, ¬−, aNd EfQ−rulEs. [φ ∧¬φ]1 [¬φ]3 [φ ∧¬Φ]1 [Φ]2 ∧E,2 φ [⊥]4 ¬E,4⊥ ∧E,3⊥ → I,1(Φ ∧¬φ)→⊥ [⊥]5 EFQ φ ∧¬φ → I,5⊥→ (φ ∧¬Φ) ∧I ((Φ ∧¬Φ)→⊥)∧ (⊥→ (φ ∧¬φ)) ********************************************* de f (Φ ∧¬Φ)↔⊥ HoweveR, THe EquiValence holds only iF eGq−Rule is applied. THeir exAmINaTIOn oF RuSsell's paraDoX, e.g. PrOPositioN 4.4.1, DoES NOt aPPly EFQ−rule. On fOrmalizing RUSseLl's parAdOx, iT Is not nEcessArY ThaT a contradicTion Is logicalLy equiVAlent to AN absurdIty. TheRefOre, It is NOt To Say ThAT ¬φ ∧φ MUsT be PRooF-theoretIcAlLy unaCcepTABLE conCluSion. we do nOt claim that ¬φ ∧φ Has To be A ProOf-theOretiCallY aCceptAble coNclusIoN. It may rely on our ChoiCe of the deFinItIon Of InconSIstencY. We May Allow thE definiTIon Of CONSiStency that, for any φ, S Is CONsIstent ifF S ⊬ ¬φ ∧φ; otHErWiSE, inconsiStEnt. if anY FOrmulA whiCH rEnders a sYstem iNCoNsIstent wErE proof-ThEorEtiCally UNaccEptablE, ¬φ∧φ would Be an uNAccpetable concLUsion. However, tHE iNVEsTIgatIon Of the proof-tHeorETic cRiteRIoN foR ParadOxicaLiTY iS Directly related to thE pRoof-thEoretIc solution to tHe paradoxeS, SUCh as our pLausIBlE Solution RND. It iS unclEar how PetrOLo and PisTone's CriterioN of NparadOX Is relatiVe tO thE prOof-THEoRetic solution. fOR expLoRing the ProOf-theorEtiC stRucTurE oF paradoxiCality, thEiR N-PaR
 for ∧ and ¬, it is easily  seen that  ever y c ont ra dict ionimplies an abs u rdit y but not vice versa.To de ri v e th e  e quiva lence b e tw e e n t he m,  th er e  m ust b e a  159 ru le for ExFal so  Quodlibet ( E FQ ) which me ans  in this cas e t hat an  a bsu r dity(or  fals ity) i m pliesevery for mu l a. ⊥ E F Q φ (φ∧ ¬ φ) ↔⊥ i s readily derived by ∧−, ¬−, and EF Q−rule s. [φ ∧ ¬φ] 1 [ ¬φ]3 [φ ∧¬ φ] 1 [φ] 2  ∧E,2 φ [⊥ ] 4 ¬E, 4 ⊥ ∧E,3⊥ → I,1 (φ ∧¬φ)→⊥ [ ⊥ ]5EFQ φ∧¬ φ → I,5⊥→(φ ∧¬ φ) ∧I((φ ∧¬φ)→⊥) ∧ (⊥ → (φ ∧¬φ) ) **** * ******* * ******* ****** *** *** **** * ** ** *** *d e f (φ  ∧¬ φ )↔⊥  However ,th e equ ival e n c e  hol dsonly  if E GQ−rule is ap pli ed.T hei r exa minat ionof  Russ ell'sparad ox , e.g. Proposit ion4.4.1, do esno t a pp ly EF Q −rule.  On  fo rmalizi ng Russ e ll' sp a r ad ox, it is not nece ss a r ythat a c ontrad i ct io n  is logi ca lly  equ i v alent  toa nabsurdit y. The r ef or e, it i snot to  s aytha t ¬φ∧ φ mu st beproof-th eoret i cally unaccept a ble conclusio n .W e  d o  not  cl aim that ¬φ  ∧φh as t o be apro o f-the oreti ca l ly acceptable conclusi on . It m ay re ly on our cho ice of the d e finition  ofi nc o nsistency. Wemay a llow the d e finition  of c onsisten cy that,f o r any φ,  Siscon sis t e nt  iff S ⊬ ¬φ ∧ φ ;  oth er wise, i nco nsisten t.Ifany  fo rm ula which  renders  a  s ys te m i ncons i stent we re  pr oo f-t heore t ically  unac cept ab le ,  ¬φ ∧φ woul d  b e an u na cc peta ble  c onclu sion .  Ho wever,the inves tig a tion  o fthe pro of-theoreticcr iterion fo rpar adoxic a l ity is d irectly related to thep roof-th eor eticsolu tion to t heparado xes ,  suchas our  plau si ble s oluti o n  R ND.  I t is uncle a r  ho w Pet ro lo a nd Pist one's criterion of Npa radox is rela tiv e to t he  pr o of - the or e tic s olution. For ex ploring th ep ro of-theoret i c s tr uctureof para doxic a lity, t heir N-pa r
 for_∧ and_¬, it is easily_seen that_every_contradiction implies_an_absurdity but not_vice versa. To_derive the equivalence between_them, there must_be_a 159 rule for Ex Falso Quodlibet (EFQ) which means in this case that_an_absurdity (or_falsity)_implies_every formula. ⊥ EFQ φ_(φ ∧¬φ)↔⊥ is readily derived_by ∧−,_¬−, and EFQ−rules. [φ ∧¬φ]1 [¬φ]3 [φ ∧¬φ]1_[φ]2_∧E,2 φ [⊥]4_¬E,4⊥ ∧E,3⊥ → I,1(φ ∧¬φ)→⊥ [⊥]5 EFQ φ ∧¬φ_→ I,5⊥→ (φ ∧¬φ) ∧I ((φ_∧¬φ)→⊥)∧ (⊥→ (φ_∧¬φ))_*********************************************_de f (φ ∧¬φ)↔⊥_However, the equivalence holds only if_EGQ−rule is applied. Their examination of_Russell's paradox, e.g. Proposition 4.4.1, does not_apply EFQ−rule. On formalizing Russell's paradox,_it is not necessary that_a contradiction_is logically equivalent to an_absurdity. Therefore, it_is not_to say that_¬φ ∧φ must be proof-theoretically unacceptable_conclusion. We do_not claim that ¬φ ∧φ has_to_be a proof-theoretically_acceptable_conclusion._It may_rely on our_choice_of the_definition_of inconsistency. We may allow the_definition_of consistency that, for any φ, S_is consistent iff S_⊬_¬φ ∧φ; otherwise, inconsistent._If any formula which renders_a system inconsistent were proof-theoretically unacceptable,_¬φ∧φ would_be an_unaccpetable conclusion. However, the investigation of the proof-theoretic criterion for paradoxicality_is directly related to the proof-theoretic_solution to the paradoxes,_such as_our_plausible solution RND._It_is unclear_how Petrolo and Pistone's criterion of Nparadox_is relative_to the proof-theoretic solution. For exploring_the proof-theoretic structure of_paradoxicality,_their N-par