AppendixAPPENDIX 413 ELEMENTS FOR A FORMALISATION OF THE THEORY OF NORMS DEVELOPED IN THIS VOLUME by Alberto Artosi, Antonino Rotolo, Giovanni Sartor, and Silvia Vida* 1. Preliminaries The aim of this appendix is to sketch a formalisation of some aspects of the theory of norms developed in this volume. In this regard, the appendix does not provide any new logical contribution: The purpose is simply to use some logical tools, standard in the literature, to reconstruct some concepts and defi- nitions proposed, in particular, in Chapters 6 through 10. Let us first define our formal language. Here it suffices that it consist of the language of classical first-order logic with variables x, y, z … and constants a, b, d … to denote agents belonging to a finite set G of agents, the identity symbol, and the following intensional operators and supplementary connec- tives: • the obligation operator ‘O’; this operator, as we shall see, will be in- dexed by agent variables and constants to express directed obligations (cf. Herrestad and Krogh 1995; 1996); • the action operator ‘Doesx’, such that an expression like ‘Doesx A’ means that the agent x brings about that A (cf. Santos and Carmo 1996); • the operators ‘Uttersx’ and ‘Procx’ to express agents’ generic actions of utterance and the speech act of institutional proclamation respectively (see Gelati, Governatori, Rotolo, and Sartor 2002a and 2002b; Sartor, Volume 5 of this Treatise, Chapter 23); • the modal operators ‘Knowx’ and ‘Belx’ for the agent’s knowledge and beliefs respectively (cf. Meyer and van der Hoek 1995); • the binary connective ‘⇒’ to represent normative conditionality (cf. Prakken 1997; Nute 1997); • the binary connective ‘≡>’ to capture the notion of typicality or normal- ity (cf. Delgrande 1987); • the unary temporal operators ‘A’ and ‘E’, which read “for all possible courses of future events” and “for at least one possible course of future events,” respectively, and the binary temporal operator “until” ‘U’, such that * Thanks are due to Guido Governatori for his comments on an earlier version of this appendix. 414 TREATISE, 1 - THE LAW AND THE RIGHT the expression ‘A U B’ means that A will be true until B is true (cf. Emerson and Halpern 1986; Schild 2000). It is worth noting that we will not commit ourselves to adopting any specific logical characterisations of such notions: A number, and sometimes a plethora, of different axiomatisations are in fact available. The reader is thus recommended to consult at least the literature cited above for further details. It will suffice here to focus on some well-known properties that seem indis- putable and, above all, required for developing the formalisation of the theory of norms proposed in this volume. As is well known, a very minimal characterisation of the obligation opera- tor is that it does not enjoy axiom T O A → A (1) Of course, if A is obligatory, this should not imply that A is true. On the contrary, the logic for the action operator ‘Does’ has to be charac- terised at least by the schema → Doesx A A (2) since such an operator is meant to represent successful actions. In addition, the logic for ‘Does’ as well as that for ‘O’ is normally closed under logical equivalence: ⏐= A ≡ B entails ⏐= X A ≡ X B (3) where X stands for either O or Doesx. As understood in this volume, obliga- tions, permissions, and prohibitions are interdefinable: ¬ ¬ P A =def O A (4) ¬ F A =def O A (5) Like ‘Does’, the operators ‘Utters’ and ‘Proc’ are action operators. An action of utterance is not in general successful. On the other hand, the notion of pro- claiming is used to cover all those acts by which a subject makes a statement expressing a certain proposition, and this statement has the function of mak- ing this proposition true. However, even ‘Proc’ is not necessarily successful: Procx A is only an attempt to achieve A. Whether it will be successful or not, within a certain institution, linguistic context, or system, depends on whether the institution, the linguistic context, or the system contains (implicitly or ex- plicitly) a rule that guarantees its effectiveness. In addition, the logics for ‘Ut- ters’ and ‘Proc’ are closed under logical equivalence. APPENDIX 415 The main difference between knowledge and belief is often as follows: If the agent x knows that A, then A is true, whereas this does not apply for be- liefs. In other words, only ‘Knowx’ enjoys the schema T. Intuitively, it may be said that knowledge implies belief: → Knowx A Belx A (6) As regards normative conditionality, we have different options depending on the approach we want to adopt. For example, we can view ‘⇒’ as either a monotonic or a non-monotonic link. A great part of the recent literature fo- cuses in particular on the second alternative, since it is widely acknowledged that normative reasoning is basically defeasible (cf., e.g., Nute 1997; Prakken 1997). In general, conditional logics for modelling normative reasoning are characterised by different axiomatisations. Most of them do not adopt any ex- plicit or unrestricted form of detachment of the consequent. Here a (re- stricted) form of detachment will be adopted. However, since no explicit ref- erence is made in this volume to the notion of defeasibility (but see Treatise Volume 5)—and there is no requirement here for the essential use of other formal properties of ⇒—the logical behaviour of normative conditionality looks more like that of material implication. Now to the binary connective ‘≡>’. An expression like ‘A ≡> B’ means that, normally, B is true whenever A is true. This exemplifies a typicality- based notion of plausible consequence which is often characterised within the framework of default reasoning. Here, we are not required to provide any axiomatisation for this connective. The same applies to the temporal opera- tors we have introduced above, for which it is sufficient to assume that ¬ ¬ E A =def A A (7) In addition to that, in both cases it is sufficient to bear in mind the intuitive meaning of these operators. The interested reader may refer to the relevant literature we have previously cited. 2. The Definition of “Norm” According to the view expounded in this volume, the content of a norm can be expressed propositionally. In this regard, it may correspond to the descrip- tion of one conditioning type of circumstance and one conditioned type of ac- tion; this last is qualified deontically as obligatory, permitted, or prohibited. In very abstract terms, a norm can thus be conceived of as a conditional struc- ture stating that a given type of action ought to be performed if a certain type of circumstance occurs. In the present context, we shall confine our analysis 416 TREATISE, 1 - THE LAW AND THE RIGHT to the case in which the “ought” of the norm corresponds to the deontic op- erator ‘O’. Thus, a norm n could be trivially represented as follows: A ⇒ OB (8) which is simply one of the usual representations of conditional obligation. However, a crucial point of this volume’s approach is that a norm exists qua norm if and only if at least one person believes it to be a norm (doxia): No type of action that no one believes to be objectively binding is a norm, and no norm can exist without belief; therefore, if something is to be a norm it must have at least one believer. Accordingly, the existence (definition) of a norm n as binding is strictly connected with the existence of a norm-believer b: ⇒ Belb (A OB) (9) Trivially, the notion of adoxia (the nonexistence of a norm n in a subject b who is a nonbeliever) is rendered as follows: ¬ ⇒ Belb (A OB) (10) and, if applied to all agents in the group G, it corresponds to ∧ ¬ ⇒ b ∈G Belb (A OB) (11) 3. Duty-Holder (Deontia) and Right-Holder (Exousia). The Being-in-Force and Not-Being-in-Force of a Norm. Efficaciousness and Inefficaciousness of a Norm As a second step, we will show how to give an account of the concepts of deontia and exousia: The former corresponds to the fact that one or more subjects are actual duty-holders under a norm n; the latter is related to the fact that one or more subjects are actual right-holders under n (Section 6.4). While the notion of deontia can be easily defined via the usual obligation operator (indexed by the agents who bear such an obligation), the same does not apply to the case of exousia. As is well known, the logical represen- tation of normative positions such as rights, duties, etc., is one of the most discussed issues in contemporary logic of norms.1 We will not enter here into a discussion of this problem. The interested reader is referred to Krogh 1997. 1 Since the seminal contributions of Hohfeld are some of the authors who have worked on this issue S. Kanger, L. Lindahl, D. Makinson, I. Pörn, A. Jones and M. Sergot. APPENDIX 417 Following Herrestad and Krogh’s (1995; 1996) approach (developed in particular to deal with the so-called directed obligations in contracts), the rep- resentation of normative relations between two subjects corresponding to the right/duty bilaterality requires 1. that the obligation operator be indexed by the subjects/agents who are the holders of such an obligation; 2. a distinction between two kinds of obligation: (a) a bearer-relativised ought-to-do obligation Od on the agent d (duty-holder) to bring about a cer- tain state of affairs B with respect to another agent r (right-holder); (b) a counterparty-relativised ought-to-be statement relative to r that d makes it so that B holds with respect to r.