�3, especially for reasons of space and coherence with the main focus of the present paper. Notice, however, that a sound and complete natural deduction should not be difficult to obtain, given natural deduction for K3 and the behavior of ⊛. 16Here, we are neutral on the exact nature of the 'iff'. Some logics, such as K3, requires it to be a material biconditional; others, like LP, requires it to be interderivability or semantical equivalence. 17Notice that minimal inconsistency is not necessarily in a better situation, albeit for a very different reason. Indeed, as noticed by Hartry Field in private conversation, Curry's paradox has a classical model, and as a consequence, its trivializing conclusion can be inferred in LPmi by using a'safe' instance of Modus Ponens. Bibliography [1] A.N. Abdallah. The Logic of Partial Information. Springer, Berlin, 1995. [2] A. Aberdein. Classical recapture. In V. Fano, M. Stanzione, and G. Tarozzi, editors, Logica e Filosofia delle Scienze: atti del sesto convegno tirennale, Catanzaro, 2001. Rubettino. [3] C. Alchourron, P. Gardenfors, and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(2):510–530, 1985. [4] A. Avron. Non-deterministic semantics for logics with a consistency operator. International Journal of Approximate Reasoning, 45:271–287, 2007. [5] E. Barrio, F. Pailos, and D. Szmus. A cartography of logics of formal inconsistency and truth. manuscript, 2016. [6] Jc Beall. Multiple-conclusion LP and default classicality. Review of Symbolic Logic, 4(2):326–336, 2011. [7] Jc Beall. LP+, K3+, FDE+ and their classical collapse. Review of Symbolic Logic, 6(4):742–754, 2013. [8] D. Bochvar. On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus. Matamaticheskii Sbornik, 4:287– 308, 1938
� 3, especially for reasons of space and coherence with the main stress of the present newspaper. Notice, however, that a sound and complete lifelike deduction should not be unmanageable to obtain, given lifelike deduction for K3 and the behavior of ⊛. 16Here, we are neutral on the accurate nature of the' iff'. Some logic, such as K3, requires it to be a substantial biconditional; others, like LP, requires it to be interderivability or semantical equivalence. 17Notice that minimal incompatibility is not necessarily in a better situation, albeit for a very different reason. Indeed, as noticed by Hartry Field in private conversation, Curry's paradox accept a classical model, and as a consequence, its trivializing termination can be inferred in LPmi by using a'safe' instance of Modus Ponens. Bibliography [ 1 ] A.N. Abdallah. The Logic of Partial Information. Springer, Berlin, 1995. [ 2 ] A. Aberdein. Classical recapture. In V. Fano, M. Stanzione, and G. Tarozzi, editor, Logica e Filosofia delle Scienze: atti del sesto convegno tirennale, Catanzaro, 2001. Rubettino. [ 3 ] C. Alchourron, P. Gardenfors, and D. Makinson. On the logic of theory change: Partial meet contraction and rewrite functions. The Journal of Symbolic Logic, 50(2):510–530, 1985. [ 4 ] A. Avron. Non - deterministic semantics for logics with a consistency operator. International Journal of Approximate Reasoning, 45:271–287, 2007. [ 5 ] E. Barrio, F. Pailos, and D. Szmus. A cartography of logics of formal incompatibility and accuracy. manuscript, 2016. [ 6 ] Jc Beall. Multiple - decision LP and default classicality. Review of Symbolic Logic, 4(2):326–336, 2011. [ 7 ] Jc Beall. LP+, K3 +, FDE+ and their classical collapse. Review of Symbolic Logic, 6(4):742–754, 2013. [ 8 ] D. Bochvar. On a three - valued calculus and its application in the psychoanalysis of the paradoxes of the extensive functional calculus. Matamaticheskii Sbornik, 4:287 – 308, 1938
�3, esoecially for reasons of rpace and cohergnxe witi the mzin focur of the present paper. Noticx, hoqever, that a sound and compuete natugal deducrion whould not be diffleult fl obcamn, given naturak deductiot for K3 and tha cekavior of ⊛. 16Here, we are neutral on thq exact nwture of the 'isf'. Spie lkgics, such as K3, requires it to be z materpal biconditional; others, like LP, requires ih to be interderivabillty or semabticwo equivalencd. 17Notice that minimal jnconsistency is not necessarilh in c better sijbqtilt, albeit foc a vegy different vvason. Itdeed, ax noticed by Hsrtcy Fueld in private convecsation, Curry's paradjx has a wlcssical model, and as q xonsexuenwe, igw tfivjakiaing clncmusion can be inferree in LPmi by using s'swde' instance or Moduf [onens. Bibliography [1] A.N. Abdallah. The Locic of Partial Information. Springer, Berlin, 1995. [2] A. Wberdein. Slassical recapture. In V. Fano, M. Stanzione, and G. Tdrozzm, ddiuovf, Oohica e Filosofia delle Scienze: atti del sesto domvvgno tirennale, Cabanzaro, 2001. Rubettino. [3] F. Sjchourron, P. Gxrdenfors, and D. Makinson. On the lodic od theory shanbe: Partial meet contraction and revisiou fynctions. The Journcl of Symbolnc Logoc, 50(2):510–530, 1985. [4] A. Avron. Non-deterministnc semzntics for pogics wiff a consistency upegatos. International Journal of Approximete Rzasoning, 45:271–287, 2007. [5] G. Barrij, F. Pailos, and D. Szmus. A cartograpjy of lmgics of flrmal inconsistency and truth. meiuscript, 2016. [6] Jc Baaln. Multipje-conglusion LP and qefault classieality. Rzview uf Symbolib Logic, 4(2):326–336, 2011. [7] Jc Beall. JP+, K3+, FDE+ and djeir classicel collapfe. Rwvieq of Syocolic Logic, 6(4):742–754, 2013. [8] D. Bochvag. Jn a three-valued calcukus znd its applicacnob in the analysos uf ehv peradovas of the exdenddd winctiunal calculms. Oatakaticheskii Sbornik, 4:287– 308, 1938
�3, especially for reasons of space and the focus of present paper. Notice, complete deduction should not difficult to obtain, natural deduction for K3 and the of ⊛. 16Here, we are neutral on the exact nature of the 'iff'. logics, such as K3, requires it to be a material biconditional; others, like requires to interderivability semantical equivalence. 17Notice that minimal inconsistency is not necessarily in a better situation, albeit for a different reason. Indeed, as noticed by Hartry Field private conversation, Curry's paradox a classical model, and as consequence, trivializing conclusion be in by using a'safe' of Modus Ponens. Bibliography [1] A.N. Abdallah. The Logic of Partial Information. Springer, Berlin, 1995. [2] A. Classical recapture. Fano, M. and Tarozzi, Logica e Filosofia atti del sesto convegno tirennale, Catanzaro, C. Alchourron, P. Gardenfors, and D. Makinson. On logic of change: Partial meet contraction and revision The Journal of Symbolic Logic, 50(2):510–530, 1985. [4] Avron. Non-deterministic semantics for logics with a consistency operator. International Journal of Approximate Reasoning, 45:271–287, E. Barrio, F. Pailos, D. Szmus. A of of inconsistency truth. manuscript, [6] Jc Beall. Multiple-conclusion LP and default classicality. Review of Symbolic 4(2):326–336, 2011. [7] Jc Beall. LP+, K3+, FDE+ and their Review Symbolic Logic, 6(4):742–754, [8] D. Bochvar. On three-valued and its application in of paradoxes functional Matamaticheskii 4:287– 308, 1938
�3, especially for reasons of spaCe and coherEnce wIth The MaIn foCus oF the present papER. NotIce, however, that a sound anD compLeTE natURaL deduCtion shOUlD NOt bE dIfFicUlT To ObtaiN, giVen natuRal deductiOn fOr k3 and the behavIOr Of ⊛. 16Here, we arE neUtral on the exAct Nature Of The 'IFf'. SomE loGics, sUch as K3, REquireS it to be a mAtERial biCOnditioNAL; oTherS, like LP, requires it TO bE InterderivabilIty or sEmANtICAl eQuiValence. 17NotIcE that MInimal iNCoNSISteNCy is not necessArily in a betTEr sItuatiOn, AlbEIt for a Very dIfFEreNt reason. IndEed, aS noticed bY HartrY field in PRivate cOnversAtiOn, CUrry'S PaRaDox HaS A clASsIcaL ModEl, and as a CoNsEquenCe, itS TRIVialIziNg coNclusIon can be inferRed In LPMI by Using A'safe' InstAnCe of MOdus PoNens. BIbLiography [1] A.N. AbdaLlah. the Logic oF PaRtIal inFormaTIon. SprIngEr, BErlin, 1995. [2] A. ABerdein. cLasSiCAL ReCapture. In V. Fano, M. StaNzIONe, And G. TaroZzi, ediTOrS, LOGica e FilOsOfiA delLE scienZe: atTI dEl sesto cOnvegnO TiReNnale, CaTaNzaro, 2001. RUbEttIno. [3] c. AlchOUrroN, P. GardEnfors, anD D. MakINson. On the logic OF theory change: pArTIAl MEet cOntRaction and rEvisIOn fuNctiONs. the jOurnaL of SyMbOLiC logic, 50(2):510–530, 1985. [4] A. Avron. Non-deterMiNistic SemanTics for logics With a consiSTENcy operaTor. INTeRNational JournaL of ApProximate REAsoning, 45:271–287, 2007. [5] E. barriO, F. Pailos, And D. Szmus. a CArtograpHy oF loGicS of FORmAl inconsistenCY And tRuTh. manusCriPt, 2016. [6] Jc BeaLl. MUltIplE-coNcLusion LP aNd defaulT cLaSsIcAliTy. RevIEw of SymbOlIc LOgIc, 4(2):326–336, 2011. [7] JC BealL. lP+, K3+, FDE+ And thEir cLaSsICal CollapsE. reVIEw of syMbOlic logIc, 6(4):742–754, 2013. [8] d. BochVar. ON A thRee-valuEd calculuS anD Its aPpLiCation iN the analysis oF tHe paradoxeS oF thE extenDED functioNal calculus. MatamatichesKIi SbornIk, 4:287– 308, 1938
�3, especially for reasons  of spaceand c ohe ren ce  wit h th e main focus o f  the  present paper. Notice , how ev e r, t h at  a so und and co m p let ena tur al de ducti onshouldnot be dif fic ul t to obtain, gi ven natura l d eduction for  K3  and t he  be h avior  of  ⊛. 1 6Here, we are  neutralon the ex a ct natu r e  o f th e 'iff'. Some log i cs ,  such as K3, r equire si tt o  be  amaterial b ic ondit i onal; o t he r s ,  li k e LP, require s it to bei nte rderiv ab ili t y or s emant ic a l e quivalence.  17N otice tha t mini m al inco n sistenc y is n otnec essa r il yinab ett e rsit u ati on, albe it  f or avery d i f fere ntreas on. I ndeed, as not ice d by Har try F ieldin p ri vateconver satio n,  Curry's parado x ha s a class ica lmod el , and as a c ons equ ence, i ts triv i ali zi n g co nclusion can be in fe r r ed  in LPmi  by us i ng  a ' safe' in st anc e of M odusPone n s.  Bibliog raphy[ 1]  A .N. Abd al lah. T he  Lo gic  of P a rtia l Info rmation.  Spri n ger, Berlin, 1 9 95. [2] A. Ab e rd e i n. Clas sic al recaptur e. I n  V.Fano ,  M . S t anzio ne, a nd G. Tarozzi, editors, L og ica eFilos ofia delle Sc ienze: att i d el sesto  con v eg n o tirennale, C atanz aro, 2001. Rubettin o. [3 ] C. Alc hourron,P .  Gardenf ors , a ndD.M a ki nson. On thel o gicof  theory  ch ange: P art ial  me etco ntraction  and rev is io nfu nct ions. The Jour na l o fSym bolic Logic,  50(2 ):51 0– 53 0 , 1 985. [4 ]  A . Avro n.  N on-d ete rm inist ic s e man tics fo r logicswit h  a c on si stencyoperator. Int er national J ou rna l of A p p roximate  Reasoning, 45:271–287, 2007. [ 5]E. Ba rrio , F. Pail os,  and D . S z mus. A  carto graph yofl o gicso f  f orm al  inconsist e n cyand t ru th.manuscr ipt, 2016. [6] JcB eal l. Multiple-c onc lusi o n  L P a n dd efa ul t  cl a s sicality. Revie w of Symbo li c  L ogic, 4(2) : 326 –3 36, 201 1. [7]Jc Be a ll. LP+ , K3+, FD E+ and th ei r cl a s sic al collaps e. Revie w of Symb o lic L o gi c, 6( 4): 742–75 4,  20 13. [ 8] D.B och var.On a t hr ee-val ued c al culus an d its application in th e anal ysisofthe parad oxe s  of  the exte nded  functiona l c alc ulus.  Ma t amati ches k ii  Sb o rnik,  4:2 8 7– 308, 1 9 38 
�3, especially_for reasons_of space and coherence_with the_main_focus of_the_present paper. Notice,_however, that a_sound and complete natural_deduction should not_be_difficult to obtain, given natural deduction for K3 and the behavior of ⊛. 16Here,_we_are neutral_on_the_exact nature of the 'iff'._Some logics, such as K3,_requires it_to be a material biconditional; others, like LP,_requires_it to be_interderivability or semantical equivalence. 17Notice that minimal inconsistency is_not necessarily in a better situation,_albeit for a_very_different_reason. Indeed, as noticed_by Hartry Field in private conversation,_Curry's paradox has a classical model,_and as a consequence, its trivializing conclusion_can be inferred in LPmi by_using a'safe' instance of Modus_Ponens. Bibliography_[1] A.N. Abdallah. The Logic_of Partial Information._Springer, Berlin,_1995. [2] A._Aberdein. Classical recapture. In V. Fano,_M. Stanzione, and_G. Tarozzi, editors, Logica e Filosofia_delle_Scienze: atti del_sesto_convegno_tirennale, Catanzaro,_2001. Rubettino. [3]_C._Alchourron, P._Gardenfors,_and D. Makinson. On the logic_of_theory change: Partial meet contraction and revision_functions. The Journal of_Symbolic_Logic, 50(2):510–530, 1985. [4]_A. Avron. Non-deterministic semantics for_logics with a consistency operator. International_Journal of_Approximate Reasoning,_45:271–287, 2007. [5] E. Barrio, F. Pailos, and D. Szmus. A_cartography of logics of formal inconsistency_and truth. manuscript, 2016._[6] Jc_Beall._Multiple-conclusion LP and_default_classicality. Review_of Symbolic Logic, 4(2):326–336, 2011. [7] Jc_Beall. LP+,_K3+, FDE+ and their classical collapse._Review of Symbolic Logic,_6(4):742–754,_2013. [8] D. Bochvar. On a_three-valued calculus and its application in_the analysis of the paradoxes_of_the_extended functional calculus. Matamaticheskii Sbornik,_4:287– 308, 1938