Abstract

ABSTRACT Deontic logic is standardly conceived as the logic of true statements about the existence of obligations and permissions. In his last writings on the subject, G. H. von Wright criticized this view of deontic logic, stressing the rationality of norm imposition as the proper foundation of deontic logic. The present paper is an attempt to advance such an account of deontic logic using the formal apparatus of update semantics and dynamic logic. That is, we first define norm systems and a semantics of norm performatives as transformations of the norm system. Then a static modal logic for norm propositions is defined on that basis. In the course of this exposition we stress the performative nature of (i) free choice permission, (ii) the sealing legal principle and (iii) the social nature of permission. That is, (i) granting a disjunctive permission means granting permission for both disjuncts; (ii) non-prohibition does not entail permission, but the authority can declare that whatever he does not forbid is thereby permitted; and (iii) granting permission to one person means that all others are committed to not prevent the invocation of that permission. Keywords: deontic logic, dynamic semantics, update semantics, imperatives, performatives. It is a fundamental feature of norms that they are imposed and adopted (and promulgated, reaffirmed, and so on). 1 To say this is not to claim that norms are arbitrary: as human beings, we are prone to impose and adopt only certain norms and not others. Rather, it indicates that norms are not existing in and of themselves. We may feel compelled to condemn killing human beings, but until we actually do so there is no norm against it, only our revulsion (and inclination to condemn). Because norms are not natural entities, the 'logic of norms' cannot be grounded in the logical structure of reality. To explain why p∨q and ¬ p∧¬q cannot both be true, perhaps we appeal to a correspondence theory of truth. To explain why an object necessarily is not both green all over and red all over, we may appeal to its intrinsic properties and to the metaphysics of (secondary) qualities. Yet, to explain why 1 This relation between the norm and its introduction is constitutive and not necessarily temporal: a person may adopt a norm in accepting blame, but the blame itself, as well as its acceptance, can only be understood as such by presupposing that the norm is (thereby) adopted. 83 EuJAP | VOL. 7 | No. 2 | 2011 two norms are conflicting, we cannot do so by means of an appeal to the impossibility of them co-existing. Despite our inclinations to accept only certain norms and not others, we may still decide to accept norms that turn out to be conflicting in rare or unforseen cases. Legal experts and judges commonly have to deal with the co-existence of conflicting norms. The tension between privacy and security is a familiar source of examples. Security issues may instigate us to endorse a more stringent security policy (e.g., one involving CCTV cameras), overlooking the potential conflicts of such a norm with the privacy norms to which we have committed ourselves already in legally binding international agreements. No doubt the ten commandments in the New Testament did not reckon with the possibility of IVF, cloning, HIV, ultrasound, peer-to-peer file sharing, and more-leading to a potential conflict, either among them, or with certain norms concerning those new phenomena. So, potential and actual quandaries in our systems of norms do not make it impossible that such norm systems be put forward and become real. Consequently, we cannot explain the 'conflicting' of two norms in terms of the impossibility of an actual norm system encompassing both norms. That is, the proposition stating that the one norm is binding and the proposition stating that the other norm is binding may both be true, and yet those norms need not be consistent. What then, if anything, does it mean that norms are 'consistent'? Note that this question is not answered by the statement that two norms are consistent just in case the proposition that one complies with both is contingent. That statement, if true, would still only concern the question which norms are consistent. Moreover, it does not clarify what, if any, constraint permissive norms impose on consistency. In some of his last works on deontic logic, von Wright proposed that the logic of norms is to be understood in terms of the practice of norm imposition. Although conflicts may occur in actual norm systems, an authority who imposes these norms on you is not being rational. Such an authority puts you in a potential quandary, making it impossible for you to comply with the norms it has put forward. Similarly, we may add, a person who accepts or endorses such norms (or the authority of the norm-giver) is not being rational either. In this way, von Wright proposes to explain the logic of norms by an appeal to the rationality of the practice of introducing (imposing and endorsing) norms. p and ¬ p are mutually contradictory. But why should O p and O¬ p be deemed so? Answer: A norm-giver who demands that one and the same state of affairs both be and not be the case cannot have his demand satisfied. He is "crying for the moon." His issuing the norms is irrational. (von Wright 1996, 40) 84 R. Mastop | Norm Performatives and Deontic Logic The concept of consistency to be invoked in explaining norm consistency is therefore not that the norms can jointly exist, but that they can be subsequently imposed without thereby manifesting irrationality. 2 Importantly, this account of norm consistency does not only offer an alternative grounding of the logic of norms: it also leads us to ask which logical validities obtain for normative language. Which acts of norm imposition can be subsequently performed without manifesting irrationality? The simple answer would be that these are all series of acts leading to a body of norms such that the proposition that those norms are jointly fulfilled is consistent. Von Wright rejects this answer, on the basis of an account of permission. Standard deontic logic defines permission as the absence of prohibition. That would mean that permission imposes no constraints on the possibility to comply with a body of norms. Intuitively, that seems clearly to get the facts wrong. But why? According to von Wright, in order to explain this we need to consider the act of giving permission. Giving permission is a kind of "binding one's hands." It is somewhat like giving a promise or like saying "you are free to do this, I am not going to interfere." One could also say that the permission-giver imposes a prohibition on himself not to prevent the permission-holder from availing himself of the permission. (von Wright 1999, 37) Various authors have stressed the importance of the act of permission giving for our understanding of permission-and its logic. Von Wright argues that the relation between obligation and permission is thus misconstrued by standard deontic logic as a logical connection between certain statements about (existing) norms. Permission is not the absence of prohibition, but it is a distinct category of norms. As a consequence, the principle that "whatever is not prohibited is thereby permitted" has been misread as a statement of logic, whereas it is in fact a meta-norm-comparable to the legal formula nullum crimen sine praevia lege poenali. Its imposition is an act whereby a norm system is "closed off", i.e., determining the normative status of any action not yet covered by any of the preceding norm performatives. In what follows, I present an attempt to characterise the logic of norm performatives. It purports to do for the logic of norms, roughly, what prescriptivism (e.g. Hare 1952) does for their content. As will be clear from the above, this involves at least separate acts of obligation and permission, a multi-actor approach to represent the positive element of permission, and an act of closure whereby permission is granted for anything that has not yet been prohibited. First, I introduce the concept of an effectivity relation and define some properties and operations in terms of it. Second, I state an update semantics for (conditional) norm performatives, with an aside concerning free choice permission. Third, I formulate a dynamic semantics for norm propositions (i.e., statements about the existing norms) in which we can study the 'static' consequences of a consistent 'dynamics' of imposing norms. Fourth and last, I define the 'sealing legal act' of permitting whatever has not been forbidden. Before embarking on all of this, I first make some preliminary clarifying remarks. Preliminaries: actions, coalitions, performatives What is the object of a norm? If you get permission to open the window, and opening the window in the present circumstances will cool down the room, does this mean that you have been given permission to cool down the room? In the framework presented in this paper, the permission does not contradict a prohibition against cooling down the room. 4 In von Wright's terminology, we may distinguish the 'result' of an action from its 'consequences'. The result of opening the window is that the window is open, whereas the cooling down of the room is a consequence. The consequences of an action are of course largely dependent on the contingent circumstances. Because of this, we had best define norms in terms of only the result of the permitted action. 4 That is, unless we add a static principle that the 'coalition with zero members' has permission to do all that is causally necessary to happen. In that case, the agglomeration principle of 'additive closure' implies that whenever you get permission to open the window you also get permission to open the window plus everything that is causally necessary to happen as a consequence. See also the discussion on playable and closed world effectivity functions in Broersen et.al. 2009. R. Mastop | Norm Performatives and Deontic Logic For characterising the positive aspect of permission, we need to logically relate the permissions and obligations of different actors. The framework presented here imports various elements from coalition logic (Pauly 2001). Norms will be attributed to coalitions, or groups, which are represented as arbitrary sets of actors. This allows for an elegant characterisation of intersubjective consistency of norm systems: if one coalition C gets permission to act in such a way that A will necessarily become true, then the complement of C (in the society of norm subjects) cannot subsequently be given permission to act in such a way as to prevent A from becoming true-at least not without the norm-giver thereby "crying for the moon". Von Effectivity relations Pauly Given a domain of states and a finite set N of agents, the effectivity relation E attributes sets of states (propositions) to sets of agents (called 'coalitions'), for any given states. Intuitively, if 〈s, C , X 〉 ∈ E, then coalition C 'can' (in some further to be specified sense) see to it that X (at the subsequent moment), at state s. Definition 2.1 (Effectivity relation). An effectivity relation is a set E ⊆ × ow(N ) × ow( ). The E-alternatives for C at s are s E C = def {X | 〈s, C , X 〉 ∈ E}. Given a coalition C and sets of states X and Y , we define for the pairwise intersection of members of and . The complement of any set A in its obvious domain is written A. The effectivity function f E can be reconstructed from the relation E by mapping each state s onto the function mapping each coalition C onto s E C . The set E XC Y will be used later to represent the content of particular norm performatives, to the extent that C must, or may, see to it that Y under circumstances X . This will be made more clear later on. In the formal analysis of effectivity functions, two important formal properties are outcome monotonicity and superadditivity (see Pauly 2001). Outcome monotonicity says that if a coalition is effective for some set X it is also effective for any larger set: if the coalition can ensure that the next state will be some state in X , then eo ipso it can ensure that the next state will be some state in X ∪ Y . For instance, if the majority of U.S. voters can ensure that Obama is re-elected, then (trivially) they can also ensure that either Obama re-elected or Canada leaves the Commonwealth of Nations. Superadditivity relates the powers of coalitions to the powers of its members. If one party can ensure that the window is open and another party can ensure that the door is open, then together they can ensure that both the door and the window are open. This can only apply as a logical principle if the two coalitions are disjoint: e.g., if Sally is needed both for opening the window and for opening the door, then she will have to chose with which party to collaborate. The reason for calling this property superadditive is that it leaves open the possibility that some larger coalition has powers that extend beyond the powers of its members. As Gärdenfors points out: . . . the rights of a group G is, normally, not just the union of the rights of the individuals in G, but the group may agree on contracts and have other forms of collectivistic rights which essentially extend the power of the group beyond the individual rights. (Gärdenfors 1981, 344) In keeping with the more process-oriented perspective on deontic logic, we define two closure operations: the outcome monotonic closure and the additive closure. The latter leaves open the abovementioned possibility of a group having abilities extending beyond the sum of powers of its members, but it is not a 'superadditive closure' since no such powers are included in the additive operation. Definition 2.2 (Closure). Given an effectivity relation E, its outcome monotonic closure is E ↑ : Its additive closure is E + , which is defined by induction on the size of C ∪ D: , provided C and D are disjoint. To construct the additive closure we begin with the empty set, which partitions into disjoint sets and -therefore it intersects all the things for which it is effective. Then we move to all singletons, partitioning into itself and the empty set; all pairs of agents; and so on until we reach the entire domain of agents in the last step. Although perhaps formally complex, or involved, the meaning of additive closure is straightforward: when one coalition C can independently force the next state to be in the set X and another, disjoint coalition D can independently force the next state to be in the set Y , then by additive closure C ∪ D can force the next state to be in (X ∩ Y ). If we close permission additively, then we make permissions independent of allegiance: if you are permitted to open the door and I am permitted to open the window, then by additive closure we are jointly permitted to vent the room. Definition 2.3 (Properties) . In terms of the closures defined above, we define the following properties. • E is outcome monotonic iff E ↑ = E; • E is regular iff 〈s, C , 〉 ∈ E + , for any s and C . Intersubjective compatibility of coalitional power is called 'regularity' in coalition logic Regularity is the formal representative of the idea that, in giving permission, the norm giver is "binding one's hands". A norm giver who makes the effectivity function irregular is neglecting the commitments undertaken earlier through his other acts of norm imposition. Maintaining regularity will therefore be the way we will understand the practice of consistent norm imposition, in the next section. The first proposition connects regularity to the definition given by Pauly (2001). Proposition 2.1. If there is some X and some C such that X ∈ s E ↑ C and X ∈ s E ↑ C , then E is not regular. This proposition can easily be seen to be true, given the fact that if E meets the stated condition, then E + will contain 〈s, N , 〉. The next proposition states that we can characterise additivity equivalently in a different way. This also makes the connection with the definition of superadditivity in Coalition Logic (Pauly 2001) more evident. Proposition 2.2. E is additive iff ( * ) for all s, disjoint C 1 and C 2 , and X 1 and X 2 , if X 1 ∈ s E C 1 and X 2 ∈ s E C 2 , then Y ∈ s E C 1 ∪C 2 for some Y ⊆ (X 1 ∩ X 2 ). Proof. ⇐: Suppose that E satisfies condition ( * ) but E is not additive. Then + , since additive closure is an additive operation. Given the inductive definition of additive closure, there must be some coalitions A, B, C , with A and B disjoint and C = A∪ B, such that 〈s, A, X 〉 and 〈s, B, Y 〉 are members of E ↑ , but 〈s, C , (X ∩ Y )〉 is not. By outcome monotonic closure, there must be X ⊇ X and Y ⊇ Y such that 〈s, A, X 〉 and 〈s, B, Y 〉 are members of E. On the basis of ( * ) we can then conclude that there is some set , which contradicts our earlier assumption. ⇒: Suppose that E is additive but does not satisfy condition ( * ). Then there are s, X 1 , X 2 , and disjoint A and B such that X 1 ∈ s E A and X 2 ∈ s E B but for no Y ⊆ (X 1 ∩ X 2 ), Y ∈ s E A∪B . If we now take the outcome monotonic closure of E, then this set will include 〈s, A, X 1 〉 and 〈s, B, X 2 〉 but not 〈s, A∪B, (X 1 ∩X 2 )〉. Then additive closure of E ↑ will add that element, which contradicts our assumption that To come to a characterisation of updating with norm performatives, we need to introduce operations of adding particular norms to a given norm system. This is done by defining two operations on effectivity relations: joining two of them together and merging the powers incorporated in two of them. The first of these operations is, under special circumstances a form of adding possibilities to an effectivity relation while preserving additivity, as will be shown below. Definition 2.4 (Operations). Let E 1 and E 2 be two effectivity relations. The join operation and the merge operation are defined in the following manner. • E 1 E 2 = def E 1 ∪E 2 ∪{〈s, C , (X ∩Y )〉 | ∃D ⊆ C : X ∈ s E 1D and Y ∈ s E 2(C \D) }; R. Mastop | Norm Performatives and Deontic Logic Some light can be shed on these definitions in the form of some propositions. The next proposition states that intersection of two effectivity functions preserves outcome monotonicity. Merging effectivity relations preserves additivity, as is stated and proved next. Proposition 2.4. If E 1 and E 2 are additive, then so is E 1 E 2 . Proof. Using Proposition 2.2. Suppose that there are s, X , Y and disjoint C and D such that X ∈ s(E 1 E 2 ) C and Then the definition of tells us that there must be X 1 , X 2 , Y 1 and Y 2 such that X = (X 1 ∩ X 2 ) and Y = (Y 1 ∩ Y 2 ) and X 1 ∈ s E 1C , X 2 ∈ s E 2C , Y 1 ∈ s E 1D and Y 2 ∈ s E 2D . But the assumption is that E 1 and E 2 are additive, so (X 1 ∩ Y 1 ) ∈ s E 1(C ∪D) and similarly for E 2 . Then, applying the definition of once more, we conclude that Next, we show that additivity is preserved by the join operation under certain circumstances. This is relevant, because the update semantics for permission performatives will be defined as being of this form. Proposition 2.5. If E 1 is additive and E 2 = E XC Y for some X , C and Y , then E 1 E 2 is additive. Proof. For brevity we define Suppose that there are some s, X 1 , X 2 and disjoint C 1 and C 2 such that X 1 ∈ s(E 1 E 2 ) C 1 and X 2 ∈ s(E 1 E 2 ) C 2 whereas for no Y ⊆ (X 1 ∩ X 2 ), Y ∈ s(E 1 E 2 ) C 1 ∪C 2 . X 1 and X 2 must be members of either E 1 , E 2 or E * . We proceed by cases: • X 1 ∈ s E 1C 1 and X 2 ∈ s E 1C 2 : by additivity of E 1 , (X 1 ∩ X 2 ) ∈ s E 1(C 1 ∪C 2 and so • X 1 ∈ s E 2C 1 and X 2 ∈ s E 2C 2 : impossible since E 2 is specific to only one coalition. • X 1 ∈ s E * C 1 and X 2 ∈ s E * C 2 : impossible because all coalitions in E * overlap; • X 1 ∈ s E 1C 1 and X 2 ∈ s E * C 2 : by definition of , there must be some X 3 and X 4 and disjoint C 3 and C 4 , such that (X 3 ∩ X 4 ) = X 2 and (C 3 ∪ C 4 ) = C 2 , and such that X 3 ∈ s E 1C 3 and X 4 ∈ s E 2C 4 . But then it is also true that C 1 and C 3 are disjoint. So by additivity of E 1 , we conclude that there is some Z ⊆ (X 1 ∩ X 3 ) such that Z ∈ s E 1(C 1 ∪C 3 ) . And since C 4 is also disjoint from . We observe that (Z ∩ X 4 ) ⊆ (X 1 ∩ X 2 ), and . So for some subset of (X 1 ∩ X 2 ), it is the case that this set is an element of s E * C 1 ∪C 2 and hence an element of s(E 1 E 2 ) C 1 ∪C 2 , which contradicts our assumption; • X 1 ∈ s E 2C 1 and X 2 ∈ s E * C 2 : impossible because any member of E * concerns a coalition superset of the one coalition occurring in E 2 ; • X 1 ∈ s E 1C 1 and X 2 ∈ s E 2C 2 : by definition of , , which contradicts our assumption, given E * ⊆ E 1 E 2 . Lastly, if E 2 is already contained in the outcome monotonic closure of some additive effectivity relation E 1 , then joining E 2 to E 1 will not yield anything new in that outcome monotonic closure. Proposition 2.6. If E 1 is additive and Proof. Let us abbreviate the set {〈s, It is enough to prove that . Suppose that this is not the case, so 〈s, C , (A∩B)〉 ∈ E * \E ↑ 1 . Then there is some D ⊆ C such that 〈s, D, A〉 ∈ E 1 and 〈s, (C \D), B〉 ∈ E 2 . But then both tuples are also members of E ↑ 2 . By additivity of E 1 , this means that 〈s, C , (A ∩ B)〉 ∈ E ↑ 1 , which contradicts our assumption. 3. Update semantics for norm performatives 3.1. General introduction into update semantics Philosophers of logic have often thought that the logical concepts-consistency, entailment, contradiction, and so on-must be explicated in terms of truth. Von R. Mastop | Norm Performatives and Deontic Logic Wright claimed that this idea wrongly limits the applicability of such concepts. Deontic logic gets part of its philosophical significance from the fact that norms and valuations, though removed from the realm of truth, yet are subject to logical law. This shows that logic, so to speak, has a wider reach than truth. (von Wright 1957, vii) The insistence that this is not a genuine application of logical concepts is, according to von Wright (1996, 45), "simply stubbornness". As indicated earlier, von Wright's reference to "norms and valuations" will be understood here as applying to norm performatives. So, I contend that logical concepts of consistency and entailment apply to norm performatives despite the fact that truth evaluation is not applicable to such performatives-or any other performatives for that matter. Update semantics In this section an update system for norm performatives is presented. 5 This system characterises the meanings of the norm performatives as such: its object language is a language of performatives. In the following section this update semantics is used to define a dynamic deontic logic. There the object language is 'constative' or descriptive, allowing us to reason about the updates effected by norm performatives, and its consequences for the norm system. 5 Formal analyses of permission change have been provided by An update system The goal in update semantics is to define an update system, consisting of a language (a space of message-conveying expressions), a space of possible norm systems, and an update function that assigns, in a systematic way, to every message-conveying expression in the language, an update function determining what norm system results from accepting the message conveyed by that expression in a given initial norm system. We first define a norm system, then a language of norm performatives and third an update semantics. Definition 3.1 (Norm system). Given a set P of simple proposition letters, their valuation is an assignment V : P → ow( ). A norm system S is a tuple 〈F S , D S , R S 〉, such that F S ⊆ and D S and R S are effectivity relations. The empty norm system 0 is 〈 , D 0 , R 0 〉, where D 0 = ( × ow(N ) × { }) and R 0 = . S is called regular iff D S , R S and (D S R S ) are regular. S is called possible iff F S = . A norm system has three parameters: a representation of the factual information available; an effectivity relation D representing the duties of coalitions at any given state; and an effectivity relation R for the rights of coalitions at any given state. The empty norm system is one in which there is no information (every state is possible), there are no rights, and only the trivial outcome is obligatory. 6 A norm system is regular if the combination of rights and duties is intersubjectively consistent. Every coalition must be able to combine one way to fulfil all of its duties with one right, without thereby making it logically impossible for the others to do the same thing. Admittedly, this requirement is somewhat arbitrary. A libertarian might insist that every combination of individual choices for each agent of a way to fulfil its duties plus one right should be consistent. Moreover, depending on what rights we consider it might be argued that rights agglomerate. The definition of regularity follows a definition given by von Wright, cf. below. We write S ↑ for 〈F S , D ↑ S , R ↑ S 〉. When it comes to assessing whether a norm system supports some sentence, new duties the fulfilment of which entails fulfilling already existing duties should be considered as already supported. This way derived obligations and derived permissions are supported as well. 6 This last feature can be compared to the deontic logic axiom that O( p ∨ ¬ p). Here it is not an axiom but merely a feature of the 'null' state in which there are no substantial norms. 94 R. Mastop | Norm Performatives and Deontic Logic Definition 3.2 (Performative language). Given a set P of propositional atoms, L P is the usual propositional language based on P . The language L 1 is the smallest set containing L P and O C (φ|ψ) and P C (φ|ψ) for all φ and ψ in L P . The language L 1 is interpreted below as a language of performatives. The propositional sentences are interpreted as informative of the facts concerning which state we are in. In the terminology of Dynamic Epistemic Logic Definition 3.3 (Update semantics). Given a norm system 〈F S , D S , R S 〉, the update of the norm system with a L 1 expression is defined in the following way. = 〈F