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Lorenzo Pe9a 0x  v !L  Quantificational Entitlements  vG !" and Relevantoid Deontic Logic   e * Lorenzo Pe9a &   v  Logique et Analyse , N 150151152 (1995)  v/ +pp. 209238 0x  =m/m/m/ ,x   o ۟ UKES ,.,. 6&A46&StandardII.PS)LAURENTI.PRSXN\ 6&finitif@p@@FF MMx6&Standardwfte simple interl.DINA4 sans N   #XpiP;rEXP#    ESUK1UKUK#NxzPC 'P#X01Í ÍX01ÍÍ.  X1Í.X*Í  # -t\  PC'qP#% iI   Quantificational Entitlements and Relevantoid Deontic Logic  by Lorenzo Pe9a`l"#Uă   ybdddy%= 1  h Quantificational Entitlements and Relevantoid Deontic Logic  Z +| Lorenzo Pe9a *UKUKThis paper appeared under the joint authorship of Lorenzo Pe9a and F.J. D1ez Aus1n. What the reader finds here is Lorenzo Pe9a's own contribution, which was wholly and solely written by him. '  h #  Logique et Analyse , N 150151152 (1995)  h ,pp. 209238 '  R$ -Contents . 0." Preliminaries  R 1." Introducing the Deontic System DAP 2." Objections and Apparent Counterexamples 3." A Plausible Extension of DAP: Two Principles of Mildness  R@ 4." Principles Not Included in Our System DAP 5." A Discussion about the Principle of Permission  R 6." References '  Z ) v  0." Preliminaries  . kkSystems of fuzzy deontic logic have been developed which are based on a nonconser Z vative extension of Anderson & Belnap's system of entailment E. Unlike some of such  Z systems, the present one is based, not on P5, but on P10. The chain of systems P58P10 is  Z constituted by nonconservative extensions of entailmental logic E proposed by Anderson &  Z Belnap;eH ~J[ ԍFor a detailed presentation of those systems, see Pe9a (1993).e unlike P5, P10 contains a classical negation and is a conservative extension of  Z classical logic. The present paper offers a new system of deontic logic ( DAP ), based on a  Z sentential calculus which we call relevantoid logic. kkBut first we briefly present the (Hilbertstyle) axiomatic system:  Z  Primitive symbols : `U', `N', `', `H'. `p', `q' etc are used as schematic letters. Notational conventions are ! la Church: all twoplace functors have the same weight and associate leftwards; a dot after a twoplace functor stands for a left parenthesis with its mate as far to the right as possible.  Z  Definitions :  Z5  pVq  abbr  N(NpUNq)   Z  pq  abbr  pqU.qp   Z"  Lp  abbr  NHNp   Zv#  p&q  abbr  LpUq   Z$  pDq  abbr  HNpVq   ZP&  p  abbr  HNp  `' is strong negation, meaning `not at all'. `H' is read: `It is altogether true that'. `D' is  Z( a [mere] conditional, read `only if'. Implication, `', is a connective such that P10pr  Z) only if ddnn CLpDr, nndd but not conversely. ( CL is of course classical logic.))X=o.o.o.Ԍ Z ' The chain of systems: ă  Zg P0 = E  Z P1 = P0 + Factor (i.e.  pq.p.pUq )  Z= P2 = P1 + Linearization ( pqV.qp )  Z P3 = P2 + the SelfSelf Principle ( N(ppN(pp)).pp )  Z P4 = P3 + IF [Implicational Funnel] ( pqV.pqr )  Z Xkk P5 = P4 + one of these: Aristotle ( N(pNp) ), Boethius ( pqN(pNq) ) or  Z{ Contradiction (namely, for some particular sentential constant j:  jUNj )#k  Z P3.5 = P3 + both these principles:  pq.NHNHpHq  and  HpqV.Npr   ZU Xkk P8 = P3.5 + these two:    and  pV.pq #k kk(`' is a sentential constant meaning the conjunction of all truths)  Z/ P10 = P8 +  HN(N)   Z  Rule of inference:  Z  DMP (i.e. disjunctive modus ponens): for n1:  Zp p1qV(p2q)V8Vpnq, p1, 8, pn  q  Z kk MP [Modus Ponens] is a particular case of the rule " the one wherein n=1.  Z Adjunction is a derived inference rule.  Z9  Quantification:  Z kkOur quantificational extension of system P10 is obtained by adding further axiomatic schemata plus three inference rules. We introduce universal quantifier as primitive and  Z define  yxp  as  NzxNp . We also define  p\q  as  pqU(qp)  ( p\q  means that it  Z{ is less true that p than that q ). By  r[(x)]  we mean a formula  r  with no free occurrence of variable `x'. Additional axiomatic schemata: yx(zxqUp)zx(yxpUq) zx(pUq)(zxpUq) zxs\r[(x)]Dyx(s\r) zxpUyxqyx(pUq) zxpyxp  Z$  Additional Inference Rules :  Z[& Xkk rinfq01 (universal generalization): p q#k  Z' Xkk(provided  q  results from prefixing  p  with a finite number of universal quantifiers)#k  Z)) Xkk rinfq02 (Free-variables change rule): p q#k  Z* Xkk(provided  q  results from  p  by uniformly replacing every free ocurrence (in  p ) of a certain variable with an also free ocurrence of another variable)#k+o.,,::Ԍ Z Xkk rinfq03 (alphabetic variation rule): p q#k  Zg Xkk(provided  q  is an alphabetic variant of  p ).#k '  Z5   1." Introducing the Deontic System DAP  . kkOne of the motivations of our current proposal is that we want to secure what seems to us a correct treatment of quantified deontic and juridical propositions. Most deonticlogic  Z accounts and systems are deficient in this respect.  ~J ԍAlthough in the first important modern collection of essays on deontic logic (Hilpinen 1971) some authors, such as Kanger and Hintikka, pointed out the relevance and indispensability of quantifiers in deontic logic  (Hintikka 1971, p. 61), only a few approaches to the logical analysis of deontic notions have dealt with quantificational issues " among them Casta9eda's (1975), Makinson's (1981), van Eck's (1982), and Puga's (1985). That feature of our current proposal is all the more important as positive rights are couched in terms of quantifiers: the right to have a dwelling, a job, to enjoy medical care and so on. (Cf. the Universal Declaration of Human Rights, arts. 2226, and especially the International Compact for Economic, Social and Cultural Rights, signed in New York on 16121966, whose art. 11 states `the right of every person to an adequate welfare, 8, and to