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J. Aus1n & Lorenzo Pe9a +x  Paraconsistent Deontic Logic with Enforceable Rights  !   e   Frontiers of Paraconsistent Logic   %ed. by D. Batens, Ch. Mortensen, G. Priest & J.P. van Bendegem kBaldford (England): Research Studies Press Ltd. (RSP)  [Logic and Computation Series], 2000 'pp. 2947 !  $ISBN 086302532m=n/n/n/ +x   o ۟ESES ,.,. 6&6&StandardII.PS)LAURENTI.PRSx  6&finitif@p@@FF MMx6&Standardwfte simple interl.DINA4 sans N   #XpiP;rEXP#      fG   Paraconsistent Deontic Logic with Enforceable Rights `r"#Lă   y]dddy #NxzPC$'P#X01Í ÍhhX01ÍÍ. $#$-t\  PC%qP#, ,  h   Paraconsistent Deontic Logic with Enforceable Rights  Z F. J. Aus1n & Lorenzo Pe9a  R "> (Contents: , 1." Preliminary considerations 2." The Underlying Quantificational Calculus 3." Deontic Postulates 4." Strengthening the Transitive System of Deontic Logic 5." Some Outstanding Results  R 6." A Remark about Gaps and Gluts in Juridical Contexts 7." Classical Deontic Principles Rejected 8." Concluding Remarks  R. 9." Bibliography "  Z E v  1." Preliminary considerations  ZX , kkThis paper is devoted to proposing a system of gradualistic (fuzzy) paraconsistent deontic logic, a logic which implements the idea not just of degrees of truth (and falseness)  Z< but also that of degrees of obligatoriness and licitness. The system we propose (transitive  Z/ deontic logic) tries to implement a correct treatment of quantified deontic and juridical propositions, chiefly those concerning positive rights, which are couched in terms of existential quantifiers (the right to have a dwelling, a job, to enjoy medical care and so on). Even negative rights are expressed through universal quantifiers. Thus, without an adequate treatment of quantifiers, no system of deontic logic is satisfactory. However, standard systems of deontic logic make it extremely difficult, if not downright impossible, to implement quantifiers in any reasonable way. In order to achieve such a treatment, new postulates are put forward, but at the same time most of the usual axioms of standard systems of deontic logic are dropped. In fact, our approach departs from the common assimilation between deontic and modal notions. We start with three preliminary considerations. kk1." Our underlying conception of obligatoriness (mandatoriness, bindingness) is as  Z follows. First, obligatoriness is a property of states of affairs.z ~Jt ԍMany obligations, and indeed also many rights, exist because certain facts exist. While in usual modal logics we do not have  ~J& comparativos , Lenguajes naturales y lenguajes formales II , ed. by Carlos Mart1n Vide, Barcelona: University of Barcelona Press, 1987, pp. 33550 [ISBN 8476651414.] After writing the present essay, we have come across a line of research we hitherto were completely ignorant of, namely comparative logic developed in Italy by E. Casari, Pierluigi Minari and Francesco  ~J( Paoli; see of the latter author: On Strong Comparative Logic , Logique et Analyse N$ 155156 (1996), pp. 27183. An examination of the deep divergences between our respective approaches goes far beyond the scope of this paper. What is perhaps more interesting is the existence of a number of coincidences or partial agreements between the two lines of research. The main difference lies is the fact that, unlike those Italian logicians, we regard degrees of truth as (all of them, except one) also degrees of falseness, and conversely; thus we look upon degrees and comparisons as resulting from a blend of truth and+o.,, falseness " in a vein which owes much to the influence of Heracleitus, Anaxagoras, and other Presocratics, as well as to Plato's  ~JX Parmenides and Sophist dialogues.e! o.,,55Ԍ Z kkTransitive logic (system P10 ) is built up by strenthening the logic of entailment E .  Z (Our notational conventions are ! la Church: no hierarchy among connectives; a dot stands for a left parenthesis with its mate as far to the right as possible; remaining ambiguities are dispelled by associating leftwards.) kkTransitive logic introduces a distinction between strong () and simple (N) negation. Simple contradictions are nothing to be afraid of, whereas contradictions involving strong negation are completely to be rejected. Alternatively, transitive logic has a primitive functor  Z of strong assertion, `H', such that  p  abbreviates  HNp  ( Hp  can be read as It is completely (entirely, wholly, totally) the case that p ). Within transitive logic disjunctive  Z syllogism holds for strong negation; thus we define a conditional `D', such that  pDq   Z abbreviates  pVq . Notice that conjunction `&' is such that  p&q  is as true as  q ,  Z provided  p  is not utterly false; we read  p&q  as `It being the case that p, q'. `U' is simple  Z conjunction, `and' ( p&q  is defined as  pUq ). We also define  p\q  as  pqU(qp)   Z ( p\q  means that it is less true that p than that q). Our reading of `' is as follows:  pq   Z is read: To the extent [at least] that p, q . We add a further definition: let  p  abbreviate  Z  N(pNp)  "  p  meaning that  p  is true enough. h ~J ԍ p  could be read `It is sufficiently (or amply or abundantly, or the like) true that p', or something like that. Last let `' be a primitive sentential constant meaning the conjunction of all truths.  Z% kkHere is an axiomatization of system P10 . Primitives: U, , N, H, . definitions of  Z `\', `', `' are as above.  pVq  abbreviates  N(NpUNq) .  h  AXIOMS: X` hp x (#%'0*,.8135@8: