 and D is P−valid relative to R. F is logically valid iff, for every P′ ⊇ P and R′ ⊇ R, F is P′−valid relative to R′. Let SN be a natural deduction system for the naive set theory which only contains the rules for ∧, →, ¬, and ∈. SN has a set R of standard reductions for ∧, →, ¬, and ∈, and its set P of production rules. Then, SN has the derivation Σ12 (or Σ′12) of Russell's paradox. With respect to Definition 4.3.1, since Σ10 and Σ11 are P−valid relative to R but Σ12 is not, the last inference rule, i.e. ¬E−rule, is not P−valid relative to R. Tranchini (2016) accepts Prawitz's definition of P−validity of an inference rule as the correctness of the rule and argues that the correctness of the rule is different from the validity of it. He examines a similar phenomenon and says, 154 It is not [¬E−rule] to be blamed for not preserving validity. The source of the problem should rather be identified with the presence of [the parameter a defined as a ∈ {x|¬x ∈ x}]. How can this intuition be spelled out?... the availability of reduction procedures usually suffices to warrant the correctness of the elimination rule to which they are associated. It should now be clear that, when the language contains [the parameter a which raises paradoxical derivations], this is no more the case. To repeat, while in standard cases the existence of reduction procedures associated to the rule is enough to show that the rule preserves validity, this is not so in general. (Tranchini, 2016, pp. 505–506) Though he thinks that ¬E−rule is correct, when the parameter a as a ∈ {x|¬x ∈ x} is associated, he agrees that ¬E−rule is not P−valid relative to R. Prawitz (1965, p. 95) already has considered the similar phenomenon. So, we conclude that in the case of Russell's paradox, ¬E−rule fails to preserve validity of derivations. If the application of ¬E−rule is restricted when the parameter a as a ∈ {x|¬x ∈ x} is involved in the language of the system SN for the naive
and D is P−valid relative to R. F is logically valid iff, for every P′ ⊇ P and R′ ⊇ R, F is P′−valid relative to R′. Let SN be a lifelike discount system for the naive hardening hypothesis which only contains the rules for ∧, →, ¬, and ∈. SN have a set R of standard reductions for ∧, →, ¬, and ∈, and its set P of output rules. Then, SN has the ancestry Σ12 (or Σ′12) of Russell's paradox. With regard to Definition 4.3.1, since Σ10 and Σ11 are P−valid relative to R but Σ12 is not, the final inference rule, i.e. ¬E−rule, is not P−valid relative to R. Tranchini (2016) accept Prawitz's definition of P−validity of an inference rule as the correctness of the rule and argues that the correctness of the principle is different from the validity of it. He examines a similar phenomenon and state, 154 It is not [ ¬E−rule ] to be blamed for not preserving validity. The source of the problem should preferably be identified with the presence of [ the parameter a defined as a ∈ { x|¬x ∈ x } ]. How can this intuition be spelled out? ... the availability of reduction procedures usually suffices to warrant the correctness of the elimination rule to which they are associated. It should now be clear that, when the language contains [ the argument a which raise paradoxical derivations ], this is no more the casing. To reprise, while in standard cases the existence of reduction routine associated to the rule is enough to show that the dominion preserves validity, this is not so in general. (Tranchini, 2016, pp. 505–506) Though he thinks that ¬E−rule is correct, when the argument a as a ∈ { x|¬x ∈ x } is associated, he agrees that ¬E−rule is not P−valid relative to R. Prawitz (1965, p. 95) already has considered the similar phenomenon. So, we conclude that in the font of Russell's paradox, ¬E−rule fails to preserve validity of derivations. If the application of ¬E−rule is restrict when the parameter a as a ∈ { x|¬x ∈ x } is involved in the language of the arrangement SN for the naive
 anf D is P−valid relative tu R. F is logically valmd iff, ror everh P′ ⊇ P and R′ ⊇ R, F is P′−valid celarive uj R′. Let SN be a nagural dedlction sywtem dor the nambe set bkeory ahick inly contains jhe rules fos ∧, →, ¬, and ∈. SN hds a set R of standard reductions for ∧, →, ¬, and ∈, snf its set P of proctctikn rules. Then, SN has the derivatioh Σ12 (or Σ′12) mf Russell's psradox. With respect to Deflnitlon 4.3.1, since Σ10 and Σ11 age P−valid rgmatyce to R but Σ12 is not, tht nast infergnce rule, i.e. ¬E−rule, is not P−valid relacive to R. Ttcbchlti (2016) accepts Prawptz's definition of P−vdlidity of an inferenge rune qs the correctness of the rule and argugs that tha eorrectness of the ruoe is dhffesent droo tge vzliditj oh it. He exajines a simular phenomenon and swjx, 154 It is not [¬E−rule] tj be blamed for not preserving validity. Ths source of the problem should rather be idejtified wyth the presence of [the parameter a defined as a ∈ {x|¬x ∈ e}]. Fow cwv tjis intuition be spelled out?... the availability kf rvduction procedurcs usually sufficex ho rarrant the currectussa of the eliminatiln rule to wyich they are associated. It should now bw clear that, qhen the language eontains [the parsmetet a which raises paradorical serivations], this is hu more the case. Go gepedt, while in standard cases the exisvence of redjctipn prosedures asdociabad to the rule is fnougk to vhow that hhe rule preserves validity, this is not so in ganegal. (Tranckini, 2016, ip. 505–506) Though he trinks that ¬E−ruke is cjrrecg, when the paramever a as a ∈ {v|¬x ∈ x} is assowlated, he agrxes that ¬Q−rulw is not P−vxuid relative tp R. Prawinz (1965, p. 95) already has consiverea the similar phzuonenon. So, we convluae ehwt ig the case of Rusrelu'x parxdox, ¬E−rule naius tp preserve validity mf dsrivations. If the spilication of ¬E−rulq is restrictrd when the parameuer a es a ∈ {e|¬x ∈ x} os ynvolved in the language of ths system DN nor the naive
and D is P−valid relative to R. logically iff, for P′ ⊇ P is relative to R′. SN be a deduction system for the naive set which only contains the rules for ∧, →, ¬, and ∈. SN has set R of standard reductions for ∧, →, ¬, and ∈, and its P production Then, has the derivation Σ12 (or Σ′12) of Russell's paradox. With respect to Definition 4.3.1, since Σ10 Σ11 are P−valid relative to R but Σ12 not, the last inference i.e. ¬E−rule, is not P−valid to Tranchini (2016) Prawitz's of of an inference as the correctness of the rule and argues that the correctness of the rule is different from validity of examines a phenomenon says, It is not be blamed for not preserving validity. the problem should rather be identified with the of [the a defined as a ∈ {x|¬x x}]. How can this intuition be spelled out?... availability of reduction procedures usually suffices to warrant the correctness of the elimination rule to are associated. It should be clear that, the contains parameter which raises derivations], this is no more the case. To repeat, while in cases the existence of reduction procedures associated to the rule to that the rule validity, this is not in (Tranchini, 2016, pp. 505–506) thinks ¬E−rule the a a ∈ {x|¬x ∈ is associated, he agrees that is not P−valid relative 95) already has considered the similar phenomenon. So, conclude that in the case of Russell's ¬E−rule fails to preserve validity of derivations. If the application of ¬E−rule restricted when a as a ∈ {x|¬x ∈ x} is in the language of system SN for the naive
 and D is P−valid relative to R. F iS logically Valid Iff, For EvEry P′ ⊇ p and r′ ⊇ R, F is P′−valid relATive To R′. Let SN be a natural deduCtion SySTem fOR tHe naiVe set thEOrY WHicH oNlY coNtAInS the rUleS for ∧, →, ¬, and ∈. sN has a set R Of sTaNdard reductiONs For ∧, →, ¬, and ∈, and iTs sEt P of productIon Rules. THeN, SN HAs the DerIvatiOn Σ12 (or Σ′12) OF RusseLl's paradoX. WITh respECt to DefINItIon 4.3.1, sInce Σ10 and Σ11 are P−valiD ReLAtive to R but Σ12 is Not, the LaST iNFEreNce Rule, i.e. ¬E−rulE, iS not P−VAlid relATiVE TO R. TRAnchini (2016) acceptS Prawitz's deFIniTion of p−vAliDIty of aN infeReNCe rUle as the corRectNess of the Rule anD Argues tHAt the coRrectnEss Of tHe ruLE iS dIffErENt fROm The VAliDity of it. he ExAmineS a siMILAR pheNomEnon And saYs, 154 It is not [¬E−rulE] to Be blAMed For noT presErviNg ValidIty. The SourcE oF the problem shouLd raTher be ideNtiFiEd wItH the pREsence Of [tHe pArameteR a definED as A ∈ {x|¬X ∈ X}]. hOw Can this intuition be SpELLeD out?... the aVailabILiTy OF reductiOn ProCeduRES usuaLly sUFfIces to waRrant tHE cOrRectnesS oF the elImInaTioN rule TO whiCh they Are assocIated. iT should now be clEAr that, when the LAnGUAgE ContAinS [the parametEr a wHIch rAiseS PaRadOXical DerivAtIOnS], This is no more the case. to Repeat, While In standard casEs the existENCE of reducTion PRoCEdures associatEd to tHe rule is enOUgh to shoW that The rule pReserves vALIdity, thiS is Not So iN geNERaL. (Tranchini, 2016, pp. 505–506) THOUgh hE tHinks thAt ¬E−Rule is cOrrEct, WheN thE pArameter a As a ∈ {x|¬x ∈ x} is AsSoCiAtEd, hE agreES that ¬E−ruLe Is nOt p−vaLid reLAtive tO R. PraWitz (1965, P. 95) aLrEAdy Has consIDeRED the SiMiLar pHenOmEnon. SO, we cONclUde that In the case Of RUSselL's PaRadox, ¬E−rUle fails to preSeRve validitY oF deRivatiONS. If the apPlication of ¬E−rule is restrICted wheN thE paraMeteR a as a ∈ {x|¬x ∈ x} iS inVolved In tHE languAge of tHe sysTeM SN FOR the nAIVe
 and D is P−valid relative  to R. F i s log ica lly  v alid  iff , for every P′ ⊇ Pand R′ ⊇ R, F is P′−va lid r el a tive to  R′.Let SNb ea nat ur al  de du c ti on sy ste m for t he naive s etth eory which o n ly  containsthe  rules for ∧ , → , ¬, a nd  ∈. SN ha s a  setR of s t andard  reductio ns for ∧, →, ¬, a n d  ∈ , an d its set P of pr o du c tion rules. Th en, SN  h a st h e d eri vation Σ12  ( or Σ′ 1 2) of R u ss e l l 'sp aradox. Withrespect toD efi nition  4 .3. 1 , sinc e Σ10  a n d Σ 11 are P−va lidrelativeto R b u t Σ12 i s  not, t he las t i nfe renc e  r ul e,i. e . ¬ E −r ule ,  is  not P−v al id  rela tive t o R. T ran chin i (20 16) accepts P raw itz' s  de finit ion o f P− va lidit y of a n inf er ence rule as th e co rrectness  of  t heru le an d  argue s t hat  the co rrectne s s o ft h e  r ule is different f ro m th e validi ty ofi t.  H e  examine sa s imil a r  phen omen o nand says , 154I tis  not [¬ E− rule]to  be  bl amedf or n ot pre servingvalid i ty. The source of the proble m  s h o ul d  rat her  be identif iedw iththep re sen c e of[thepa r am e ter a defined as a∈{x|¬x∈ x}] . How can thi s intuitio n b e spelle d ou t ?. . . the availabi lityof reducti o n proced uresusuallysufficest o  warrant  th e c orr ect n e ss  of the elimi n a tion  r ule towhi ch they  ar e a sso cia te d. It sho uld nowbe  c le ar  th at, w h en the l an gua ge  co ntain s  [theparam eter  a  w h ich  raises pa r a doxi ca lderi vat io ns],this isno more  the case . T o  rep ea t,  whilein standard c as es the exi st enc e of r e d uction p rocedures associated to the rul e i s eno ughto show t hat  the r ule preser ves va lidit y,  th i s  is n o t  s o i ngeneral. ( T r anc hini,  2 016,  pp. 50 5–506) Though he t h ink s that ¬E−rul e i s co r r ec t,w he n  th ep ara m e ter a as a ∈ {x |¬x ∈ x} i sa ss ociated, h e  ag re es that  ¬E−rul e isn ot P−va lid relat ive to R.  P rawi t z  (1 965, p. 95 ) alread y has con s idere d  t he si mil ar phe no men on. S o, wec onc ludethat i nthe ca se of  R ussell's  paradox, ¬E−rule fails  to pr eserv e v alidity o f d e riv ations. I f th e applicat ion  of  ¬E−r ule is re stri c te d w h en th e pa r ameter aa sa ∈ { x| ¬x ∈ x} isi n v olv ed in  th e  langu ageof the system SNf or the naive
 and_D is_P−valid relative to R._F is_logically_valid iff,_for_every P′ ⊇_P and R′_⊇ R, F is_P′−valid relative to_R′._Let SN be a natural deduction system for the naive set theory which only_contains_the rules_for_∧,_→, ¬, and ∈. SN_has a set R of_standard reductions_for ∧, →, ¬, and ∈, and its_set_P of production_rules. Then, SN has the derivation Σ12 (or Σ′12)_of Russell's paradox. With respect to_Definition 4.3.1, since_Σ10_and_Σ11 are P−valid relative_to R but Σ12 is not,_the last inference rule, i.e. ¬E−rule,_is not P−valid relative to R. Tranchini_(2016) accepts Prawitz's definition of P−validity_of an inference rule as_the correctness_of the rule and argues_that the correctness_of the_rule is different_from the validity of it. He_examines a similar_phenomenon and says, 154 It is_not_[¬E−rule] to be_blamed_for_not preserving_validity. The source_of_the problem_should_rather be identified with the presence_of_[the parameter a defined as a ∈_{x|¬x ∈ x}]. How_can_this intuition be spelled_out?... the availability of reduction_procedures usually suffices to warrant the_correctness of_the elimination_rule to which they are associated. It should now be clear_that, when the language contains [the_parameter a which raises_paradoxical derivations],_this_is no more_the_case. To_repeat, while in standard cases the existence_of reduction_procedures associated to the rule is_enough to show that_the_rule preserves validity, this is not_so in general. (Tranchini, 2016, pp._505–506) Though he thinks that_¬E−rule_is_correct, when the parameter a_as a ∈ {x|¬x ∈ x}_is associated, he_agrees that ¬E−rule is not P−valid relative_to_R. Prawitz (1965, p. 95) already_has_considered the similar phenomenon. So, we_conclude_that_in the case of Russell's_paradox, ¬E−rule fails to preserve validity_of derivations. If the application of ¬E−rule is restricted_when the parameter_a as a ∈ {x|¬x_∈_x}_is involved in the language of the system SN for_the naive