This article proposes a general strategy to overcome normative conflicts, namely, paradoxes represented in Standard Deontic Logic. This solution is based on preference relations between norms that circumvent situations of conflict. Pragmatic justifications of the proposed method are also given

Testa, Rafael

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Solving Normative Conflicts using PreferenceRelationsRafael R. Testa and Marcelo E. ConiglioCentre for Logic, Epistemology and the History of Science (CLE)and Department of Philosophy (IFCH)State University of Campinas - UNICAMPC.P. 6133 - 13083-970 Campinas, SP, Brazil{rafaeltesta, coniglio}@cle.unicamp.brFebruary 14, 2008AbstractThis article proposes a general strategy to overcome normative con-flicts, namely, paradoxes represented in Standard Deontic Logic. Thissolution is based on preference relations between norms that circum-vent situations of conflict. Pragmatic justifications of the proposedmethod are also given.1 The Standard Deontic LogicFrom now on, the following notation will be used:• L is the propositional language over the connectives ¬, →, ∨, ∧;• LD is the propositional language extending L by adding the unaryconnectives O and P ;• PCL is the propositional classical logic defined over the language L.Consider the following list of axioms (schemes) and rules of inference:1(A0) All the instances (in the DL language) of PCL tautologies(A1) Pα↔ ¬O¬α(A2) O(α→ β)→ (Oα→ Oβ)(A3) Oα→ Pα(A4) Oα→ OOα(A5) POα→ Oα(A6) O(Oα→ α)(A7) O(POα→ α)(R1) modus ponens : if α and α→ β then β(R2) O-necessitation: if α is a theorem then Oα is a theoremAs usual in the literature in the area, we will call SDL the logic consistingof A0-A3, R1 and R2.1 One can note that the above axioms form a basis todefine the well known logics OK, OM, OS4, OB, OS5, OK+, OM+, OS4+,OB+ and OS5+, defined as follows (Assuming R1 and R2 for every system):OK = A0-A2OM = A0-A2, A6OS4 = A0-A2, A4, A6OB = A0-A2, A6, A7OS5 = A0-A2, A4, A5 (hence A6 and A7 are theorems of OS5)Let L be any logic above; so L+ is the logic L plus the axiom A3.With such settings in view, we can see that SDL is nothing else thatthe system OK+, or the system D (also called KD) of modal logic (withthe obvious differences in the interpretation of modalities). The observantreader may notice that the paradoxes presented below may be formalizedusing any of the ten systems above. Moreover, our proposed strategy is alsoapplicable to them. The choice of SDL obeys to two reasons: firstly, SDL isthe most notorious and studied system of deontic logic. Secondly, we believe1Frequently P is not taken as a primitive connective and so A1 is presented as alinguistic definition.2that the intuitive notion of obligation is better, but not fully, represented bythis logic.22 Some Normative Conflicts2.1 The Chisholm ParadoxOne could say that paradoxes and dilemmas related to deontic logic, as wellas the efforts to overcome them, emerged right after the publication of vonWright’s seminal paper of 1951. The Paradox of Chisholm [2] is consideredby several authors as the most notable example of deontic paradoxes – so thatsome authors consider that any system of deontic logic unable of dealing withit is an inadequate tool to express deontic reasoning. Others (cf. [3]) suggestthat this paradox was responsible for solidifying the status of deontic logicas a distinct logic from the normal modal logics. The fact is that Chisholmwas right – and this is one of the few deontic logic areas in which there isa certain consensus: the type of conditional obligation expressed in [2] andother related paradoxes can not be satisfactorily expressed in SDL.Several versions of the puzzle are known in the literature. We will considerthe following version:(1) It ought to be that John does not impregnate Suzy Mae.(2) Not-impregnating Suzy Mae commits John to not marrying her.(3) Impregnating Suzy Mae commits John to marry her.(4) John impregnates Suzy Mae.The set of sentences above can be formalized in SDL as follows:(1’) O¬i(2’) O(¬i→ ¬m)(3’) i→ Om2This discussion is beyond the scope of this article, but it is of crucial importance tounderstand the motivations behind the present proposal – the reader only concerned withthe formal part of this article may, without any problem, substitute the references to SDLto any of the ten systems presented here.3(4’) iUndesirable consequence: We obtain O¬m from 1’ and 2’ (by A2 in 2’ andMP), and Oi from 3’ and 4’ (by MP). Thus, we have Oi ∧ O¬i and, by A3(and A1), it follows ⊥, so we lost the consistency of (1)-(4).32.2 The minor aged murderer paradoxThis paradox was originally presented by Alchourrón in [1]. Consider thefollowing normative system:(i) The judges must punish the murderers.(ii) The judges should not punish minors.(i’) h→ Op(ii’) m→ O¬pAt first glance such a system does not seem to lead to a conflict of obligations.Any lawyer and even any reasonable person non expert in law would say thatthe normative system above is quite clear in saying that all murderers, unlessthe minor aged ones, should be punished. Moreover, the norm (ii) explicitlyprohibits the punishment of minors. On the other hand, by SDL, we havethat (h ∧m) → Op and (h ∧m) → O¬p, thus, in the case of a minor agedmurderer, it is simultaneously obligatory (for the judges) to punish and notto punish.3 The preference relations approach3.1 Pragmatic JustificationsIt is usual to believe that the legislator, that is, the agent that creates therules of a particular normative system,4 issues rules in order to make their3One may note that there are two other ways of formalizing (1)-(4):(1”) O¬i, (2”) O(¬i→ ¬m), (3”) O(i→ m), (4”) iUndesirable consequence: 3” follows from 1” in SDL.(1’’’) O¬i, (2’’’) ¬i→ O¬m, (3’’’) i→ Om, (4’’’) iUndesirable consequence: 2’’’ follows from 4’’’ in SDL.That is, in both cases we lost the logical independence of (1)-(4).4Notice that the legislator has, usually, no “real existence”, that is, such entity onlydesignates the legislative authority that set the rules.4commands achieve certain objectives. In other words, the norms encourage(require or allow) the actions taken as means to achieve the objectives ofthe legislator or, otherwise, restrict those actions that might bring resultsqualified as “undesirable”.What underlies this assumption is the expectation, or even the require-ment, that the legislator does not dictate arbitrary norms, without any pur-pose, nor dictate norms impossible – logically or physically – to be performed.For this reason, we call this legislator a rational one.The identification of the reason (the purpose) of the legislator may involvea creative process of the interpreter (usually, the judge). This identification ismade, for instance, by taking into account the possible results of the actionsprescribed, as well as the (moral, ethics, etc) values accepted at the timeof the interpretation or at the time of promulgation of the norm. In otherwords, there is no method free of subjectivity for the identification of thelegislator purposes. However, what interest us here is not whether or not itcan be identified such purposes, but how, in view of them, the interpreter canadapt the normative system to tailor it to specific requirements ofconsistency, namely, by introducing a preference relation between norms.Let us see now how such a relation can be defined.3.2 Main conceptsDefinition 3.1. A norm is a sentence in the language LD of the form Oβ,α → Oβ or O(α → β), such that α and β are sentences in the language L.The sentence α is called the case of the norm, and the sentence Oβ is calledthe normative solution of the norm. The case of the norms of the form Oβis the sentence >. We write case(N) and ns(N) to denote the case and the normative solu-tion of a norm N , respectively. Notice that ns(Oβ) = Oβ.Definition 3.2. Let N = {N1, . . . , Nk} be a set with k norms such thatcase(Ni) = Ai, for 1 ≤ i ≤ k. A relevance order of cases on N is a strictpartial order  ⊆ {A1, . . . , Ak}2, that is,  is antisymmetric and transitive(and, therefore, irreflexive). Definition 3.3. Let N be a set of norms and let  be a relevance orderof cases on N . The preference order of norms on N induced by  is the5relation  ⊆ N 2 given by:N  N ′ iff case(N) case(N ′).Definition 3.4. A situation is a set Σ of sentences in LD that contains anon empty set N 6= ∅ of norms. A situation Σ is a conflicting situation, ora deontic dilemma, if there exist situations Σ1 and Σ2 contained in Σ and asentence A in L such that:• Σ1 `SDL OA but Σ1 − {B} 0SDL OA, for any B ∈ Σ1;• Σ2 `SDL O¬A but Σ2 − {B} 0SDL O¬A, for any B ∈ Σ2;• Σ1 6`SDL O¬A and Σ2 6`SDL OA. Definition 3.5. Let Σ1 and Σ2 be situations contained in a situation Σ, letN be a norm in Σ1 and let  be the preference order of norms induced by apreference order of cases  defined on the set of cases of Σ. The relevanceweight of the norm N compared to Σ2 is the natural numberrwn(N,Σ2) = card({N ′ ∈ Σ2 : N  N ′}(where card(X) denotes the cardinal of a set X). In the same way, letA ∈ Σ1 be a formula (not a norm) which is the case of some norm in Σ (notnecessarily in Σ1); the relevance weight of a case A compared to Σ2 is thenatural numberrwc(A,Σ2) = card({A′ ∈ Σ2 : A A′}).Definition 3.6. Let Σ1, Σ2 and Σ be situations as in the previous definition.The argumentative relevance weight of Σ1 compared to Σ2 is the naturalnumberarw(Σ1,Σ2) =∑N∈Σ1rwn(N,Σ2) +∑A∈Σ1rwc(A,Σ2).63.3 Analyzing some paradoxes3.3.1 The minor aged murderer paradox revisitedThis paradox is simpler than the others, so let us examine it first. Recallsubsection 2.2. One can note that the intention here is to punish all mur-derers, unless they are minors, that is, in the case of a minor aged murdererthe second norm normative solution O¬p is preferred, since the fact of beingminor aged is more relevant than being murderer. Formally, let N1 and N2be the norms of (i’) and (ii’), respectively. That is, N1 is h→ Op and N2 ism→ O¬p. So, according to the definitions of Section 3.2:• case(N1) = h;• case(N2) = m;• ns(N1) = Op;• ns(N2) = O¬p.Let N = {N1, N2}. We define a relevance order of cases on N asm  h.Induced by it, we built a preference order of normsN2  N1.Moreover, it is easy to prove that Σ = {N1, N2, h ∧ m} is a conflictingsituation, for Σ1 = {N1, h ∧m} and Σ2 = {N2, h ∧m}. Note that:• rwn(N1,Σ2) = 0;• rwn(N2,Σ1) = 1;• rwc(h ∧m,Σ2) = 0;• rwc(h ∧m,Σ1) = 0.Therefore• arw(Σ1,Σ2) = 0;• arw(Σ2,Σ1) = 1.We conclude that the logical consequences of Σ2 are more relevant thanthose of Σ1 and so the normative solution O¬p prevails over Op.73.4 The Chisholm paradox revisitedRecall subsection 2.1. Let N1, N2 and N3 be the norms (1’), (2’) and (3’),respectively. That is: N1 is O¬i, N2 is O(¬i→ ¬m) and N3 is i→ Om. LetN = {N1, N2, N3}. We define the relevance order of cases on N asi  ¬i.The induced preference order of norms isN3  N2.Note that Σ = {N1, N2, N3, i} is a conflicting situation for Σ1 = {N1, N2}and Σ2 = {N3, i}. It is easy to see that• rwn(N1,Σ2) = 0;• rwn(N2,Σ2) = 0;• rwn(N3,Σ1) = 1;• rwc(i,Σ1) = 0.Therefore• arw(Σ1,Σ2) = 0;• arw(Σ2,Σ1) = 1.We conclude that the logical consequences of Σ2 are preferable to thoseof Σ1 and so the normative solution Om prevails over O¬m.3.5 Preference and subjectivityOne can note that the definitions above require objective information (prefer-ence order of the cases, norms, etc). In this section we discuss the situationin which the information above is subjective, that is, provided by severalagents. For this, given a group G of agents, for each agent a ∈ G a relevanceorder of cases a on a given set of norms N will be considered.8Definition 3.7. Let N be a set of norms and G a group of agents. For eachagent a ∈ G leta be a relevance order of cases on N . The preference orderof norms a on N of the agent a induced by a is given as the definition3.3. The functions rwna, rwca and arwa about the agent a are built over aas the definitions 3.5 and 3.6, respectively. As we’ve said, there is no method free of subjectivity to identify the pur-poses of the rational legislator. Thus, by changing the normative systemto tailor it to specific requirements of consistency, that is, by introducinga relationship of preference between the norms, each agent can identify dif-ferent purposes and, inevitably, introduce different orders of relevance andpreference: so, different agents may get different normative solutions in aconflicting case.4 Concluding RemarksThe use of relevance orders of cases as well as the induced preference relationsbetween norms allows to define the notion of argumentative relevance weightarw(Σ1,Σ2) of a set Σ1 compared to another set Σ2 (cf. Definition 3.6). Thisparameter can be seen as a measure of the relevance of the premises of Σ1with respect to Σ2 and vice-versa. Thus, if arw(Σ1,Σ2) > arw(Σ2,Σ1) thenthe normative solutions inferred from Σ1 are preferable to those derived fromΣ2. The case arw(Σ2,Σ1) > arw(Σ1,Σ2) is symmetric. On the other hand,if arw(Σ1,Σ2) = arw(Σ2,Σ1) then the inferences from Σ1 and Σ2 are equallyrelevant and so the normative conflict between both sets persist in this case.We believe that argumentative relevance weight can be analyzed behindthe scope of normative conflicts, as a general tool in theory of argumentation.Finally, the relationship between preference relations and Belief Revision(more specifically, revision of belief bases) deserves future research.9Bibliographic References[1] ALCHOURRÓN, Carlos E., “Condicionalidad y la representación de lasnormas juŕıdicas”. In Alchourrón, C. y Bulygin, E. (eds.), Análisis lógico yDerecho, C.E.I., 1991[2] CHISHOLM, R.M., “Contrary-to-duty Imperatives and Deontic Logic”,Analisis 23 (1963), pp.33-36[3] MC NAMARA, Paul, “Making Room for Going Beyond the Call.” Mind105(419):415-450, 1996.10

Solving Normative Conflicts Using Preferences Relations

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Solving Normative Conflicts Using Preferences Relations

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