Synthese (2012) 185:1–17Knowledge, Rationality & Action 331–347DOI 10.1007/s11229-011-9953-1

D�L: a dynamic deontic logic

Krister Segerberg

Received: 24 December 2003 / Accepted: 20 April 2011 / Published online: 6 September 2011© Springer Science+Business Media B.V. 2011

Abstract This paper suggests that it should be possible to develop dynamic deonticlogic as a counterpart to the very successful development of dynamic doxastic logic (ordynamic epistemic logic, as it is more often called). The ambition, arrived at towardsthe end of the paper, is to give formal representations of agentive concepts such as“the agent is about to do (has just done) α” as well as of deontic concepts such as “itis obligatory (permissible, forbidden) for the agent to do α”, where α stands for anaction (event).

Keywords Actions · Norms · The logic of “obligatory”, “permitted” and “forbidden”

1 Introduction

It seems fair to say that the formal representation of action raises a challenge that stillhas not been met by philosophers. Computer scientists have modellings of action,suited to their interests, but those have not attracted the interest of philosophers.Significantly, today’s leading treatise on the philosophical logic of action mentionsactions only indirectly; in the formal semantics they have no counterpart (Belnap et al.2001).

In memory of Georg Henrik von Wright (1916–2003).

This paper was completed during the author’s stay, as part of a Humboldt Prize, in the philosophydepartment at the Goethe University of Frankfurt-am-Main 2010–2011. For the history of this paper, seethe postscript.

K. Segerberg (B)Uppsala University, Uppsala, Swedene-mail: krister.segerberg@filosofi.uu.se

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In the present paper is described a formal modelling, inspired by Georg Henrik vonWright’s philosophy of action, which was an effort to take seriously the existence ofactions and the way we think about them. Sections 2, 3, 4, 5, 6 and 7 below describea modelling of action. In Sects. 8, 9 and 10 a logic of action, based on that modelling,is briefly outlined. In Sects. 11 and 12 the discussion is extended to norms and thelogic of certain normative concepts, respectively. The main novelty of the paper overthe author’s previous attempts in this area (Segerberg 2002, 2003, 2005) is perhapsthe deployment of hypertheories in the normative setting of Sect. 11. (One reason fordeveloping the model theory described in the early part of the paper was actually thehope of finding an appropriate place for that extremely useful concept.)

This paper is part of a project in dynamic deontic logic which we call D�L, not tobe confused with its cousin DDL, dynamic doxastic logic (Segerberg 1999, 2001).

2 Excerpts from von Wright

von Wright’s unpretentious and open-minded approach to all things philosophical,his careful way of expressing his thoughts, and his concentration on what he saw asimportant, combine to make his writings a store-house of fruitful ideas. In this sectionare listed some quotations, suggestive even out of context, that have been with thepresent author for years:

The notion of a human act is related to the notion of an event, i.e., a change inthe world. What is the nature of this relationship? It would not be right, I think,to call acts a kind or species of events. An act is not a change in the world. Butmany acts may quite appropriately be described as bringing about or effecting(‘at will’) of a change. To act is, in a sense, to interfere with ‘the course of nature’.( . . . ) To every act . . . there corresponds a change or an event in the world. (VonWright 1963a, pp. 35f., 39)

An act is the bringing about or production at will of a change in the world, e.g., theunlocking of a door. The change brought about we call the result of the act, e.g.,the fact that a certain door, which was locked, is now open. The ‘way of doing’again is some act or activity which ‘leads up to’ the result of an act, e.g., theturning of a key and pulling of a handle, which opens the door. The tie between‘way of doing’ and ‘thing done’ is an intrinsic connexion between a kind of actor activity and some generic state of affairs. (Von Wright 1963b, p. 115)

It is useful to distinguish between the result and the consequences of an act. Theresult of an act is that state of affairs which must obtain, if we are to say truly thatthe act has been done. For example, the result of the act of opening a window isthat a certain window is open (at least for a short time). The consequences of anact are states of affairs which, by virtue of causal necessity, come about whenthe act has been done. … For example: A consequence of the act of openinga window may be that the temperature goes down. The relation between an actand its result is intrinsic; the relation between an act and its consequences againextrinsic. … There is an act of cooling the room. Its result is that the temperatureis now lower than it was before. It is a different act from the act of opening the

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window, to which it has a causal, not an intrinsic, relation. … They are differentbecause the result of one is a consequence of the result of the other. But theactivity which I display in performing the two acts, i.e., the manipulations of thewindow, is the same in both. (In this sense the two acts could be said to ‘look’the same.) (Von Wright 1963b, pp. 116, 123f.)

Action, one could say, normally presents two aspects: an “inner” and an “outer.”… When the outer aspect of an action consists of several causally related phases,it is normally correct to single out one of them as the object of the agent’s inten-tion. It is the thing the agent intends to do. This is the result of the action. …An intention is an intention to do something. (Von Wright 1971, pp. 86, 89; VonWright (1963b), p. 123).

The model theory delineated in this paper represents the author’s attempt to translatevon Wright’s philosophy of action into the language of set theory. But philosophy, likepoetry, does not translate easily. There is no way of knowing how much of our theoryvon Wright himself would have recognised—let alone accepted—as a re-statement orexplication of his ideas. No doubt he would have found much to disagree with. Nodoubt he would have balked at the more ideosyncratic speculations.

Nevertheless, the basic ideas are his.

3 Informal background

To begin with, let us adopt the following perspective: in a setting where it makes senseto talk about actions being performed, it must be meaningful to postulate an underlyingsystem involving two elements: agents and the world.1 The system works according torule (some rule, some set of laws). What happens, happens in the world. In fact, whathappens is determined by the system in a certain way: by agents and by the currentstate of the world.2

The agents are outside the world, and the effects of what they “do” is inside (whetherthey intend it or not, whether the know it or not). The world is always in some state orother, a state that changes (or may change) over time. This change, we shall assume,we can witness as analysts, but what the agents “really do” we cannot witness. Thusif we watch a certain system, we can see a course of history—call it the world-line—develop before our eyes: world state following world state. We may feel a need tomake intelligible what we see, to understand what is going on (in humans this seemsto be an innate need) and to do so we may well speculate about what goes on amongthe agents and what they are “really doing”. In this way we come up with all kindsof descriptions, explanations and interpretations. But all we can actually see is theemerging world-line—the course of nature, to use von Wright’s phrase.

1 It would perhaps have been better to use ‘will’ in stead of ‘agent’. In some modellings, the agents’ bodiesmay well be “in the world”. But if their wills were also part of the world, then the agents would not be ableto play the rôles that have been assigned to them in our modelling.2 In this paper, by ‘state of the world’ we shall always mean ‘total state of the world’. Philosophers, includ-ing von Wright, sometimes talk about possible states-of-affairs. The latter can perhaps be represented inour modelling as sets of total states.

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That we don’t have access to the agent’s “will” will not prevent us from talkingabout it, but we shall prefer to call it the contribution (from outside the world) ofthe agents. We assume that there are moments of action at which agents provide acontribution, and moments of non-action at which they don’t. At moments of thelatter sort, the system works on its own, as it were.

It is helpful to think of a system along the lines of something like a Turing machine.There are inner states about which we may not know much or indeed anything. Thereare inputs: the current total world state and the current total contribution from theagents. There are outputs, consisting of a sequence of total states.

To go over this in more detail, assume that T is a system with two input tapes (tape1 and tape 2), and that there are n agents 1, 2, . . . , n, for some n � 0. On tape 1 isprinted (a representation of) a world state. On tape 2 is printed (a representation of)the agents’ contribution, which may be thought of as an n-tuple c = (c1, c2, . . . , cn),where each ci is the individual contribution of agent i .3 We assume that T works insuch a way that the world state on tape 1 is always the current world state, and thatthe n-tuple on tape 2 always represents the current contribution from the agents. Call(s, u) an occasion if the system is in inner state s and the current world state is u(and, consequently, u is printed on tape 1). If the total contribution of the agents isc = (c1, c2, . . . , cn) (the contents on tape 2), then T will print on the output tape asequence 〈v0, v1, . . . 〉 of (representations of) total world states which is either finite(the system halts) or infinite (the system never halts). This sequence we call the con-tinuation of the world-line (relative to that occasion), and we denote it by T (s, u, c)or T (s, u, c1, . . . , cn). (If there are no agents—that is, if n = 0, then (the continuationof) the world-line is T (s, u).)

We may say that the effect of a contribution c = (c1, c2, . . . , cn) on a certainoccasion (s, u) is the world-line T (s, u, c). Thus, given an occasion, the effect of c isa completely determined continuation of the world-line. However, often our interest isfocussed on one particular agent, say i . In that case, the world-line is not determinedby i’s individual contribution z alone, since the contributions of other agents may alsoinfluence the resulting effect. In this case, rather than a particular continuation of theworld-line, it is a set {T (s, u, c) : ci = z} of possible continuations of the world-linethat we are dealing with. Thus, even though T is a deterministic system, here is asource of epistemic indeterminacy.

On any occasion, the system works by realizing the world states on the output tape,one after the other: first v0, then v1, and so on. Moreover, at the same time as the sys-tem realises the next world state written on the output tape, say w, it also replaces theworld state on tape 1 byw and deletes the old contents on tape 2. However, at any timeit is possible for a new contribution from the agents to appear on tape 2; the systemdeletes it as soon as it has been taken notice of it, but the agents may make a newcontribution before the entire continuation of the world-line has been printed out. This

3 An interesting special case in when an agent fails to make a contribution. There are actually (at least) twocases in which that might happen. In one case the agent leaves a contribution not amounting to anything,namely, the null contribution; in the other, there is no contribution whatsoever. If we were to extend ourmodelling to include, for each agent i and world state u, a set Xi,u of possible contributions of which one isto represent i’s contribution at u, then Xi,u must include two special elements: one for the null contributionand the other for the absence of a contribution.

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is an important complication, and one which illustrates how actions can be interrupted.Suppose that on one occasion a contribution c is made that defines a certain (finiteor infinite) continuation T (s, u, c) = 〈v0, v1, . . .〉. Suppose also that the system hasarrived at (and is in) world state vi , which is not the last world state in the continuation,and that the system is currently in inner state s′. Then it is not precluded that the agentsmake a new contribution c′; if they do, a new continuation T (s′, vi , c′) = 〈v′

0, v′1, . . .〉

of the world-line will be determined, which in general will be quite different from〈vi+1, vi+2, . . .〉. Let us say that the action defined by (s, u) and c is interrupted at(s′, vi ). Interruptions are extremely common in everyday contexts, even when only oneagent is involved. Clearly it is an important phenomenon worth studying. However,because of the complexity involved, we will not do so in this paper. On the contrary,one assumption implicit in what follows is that actions are not interrupted.

Another assumption, made with the same ambition of trying to reduce complica-tion, is that agents neither collaborate nor compete with one another: in the modellingpresented below, agents lack means of coordinating their contributions. This limitationis of course equally notable.

In spite of the metaphysical ring of terms like “will” and “world”, the informalbackground sketched in this section is not meant to be either profound or controver-sial but only to give an intuitive background for the abstract and tortuous modellingthat will be described in the following sections. For example, there is no suggestionthat our modelling has a bearing on the problem of free will. As long as the agentcontributions c1, . . . , cn are treated as independent parameters, employing a model-ling of the kind outlined commits one to the assumption that the will of the agents isfree. But this assumption is relative to the modelling, call in M. It is conceivable thatsomeone might wish to consider (at the same time) another modelling, call it M′, thatincorporates M as well as a mechanism for generating, in some way, what we havecalled the agents’ contribution: then the will of the agents is not free relative to M′.If the two modellings are consistent—and why would they not be?—then M′ offersa deterministic and M a nondeterministic perspective.

4 Events

If U is a nonempty set of points—in this paper often referred to as a universe—thenby a path in U we mean a sequence of points. The set of finite paths in U is denotedby U ◦. We use letters p, q, r , etc., for paths, writing pq for the concatenation of pand q (in that order) and pqr for the concatenation of p, q, and r (in that order); thelast means that we identify (pq)r and p(qr). The length of a path is the number ofpoints of which it consists. A sequence of length 1 is identified with its sole element;thus, for any point u, u is both an element of U and a sequence of length 1. There isone sequence of length 0, namely, the empty path. For any nonempty path p, we writep(∗) for the first and p(†) for the last element of p. We say that p is an initial subpathof q, in symbols p � q, if there is some path r such that pr = q; we say that p is aproper initial subpath of q, in symbols p < q, if there is some nonempty path r suchthat pr = q. Thus p � q if and only if p < q or p = q.

A (real) event type (or often just event) in U is a set of finite, nonempty pathsin U . If e is an event type and p ∈ e, then p is a (possible) realisation of e. An

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individual event is a pair (e, p)where p ∈ e. There are three logically constant events:U ◦(the universal event), U (the trivial event), and ∅ (the impossible event). The uni-versal event is unavoidable: it is realised by every finite path. At the other extreme isthe impossible event, which is not realised by any path. In between is the trivial event,which is realised by any path of length 1. We use the letter e for events in U . If eand e′ are events we write ee′ for the succession of e and e′ (in that order), defined asee′ = {pq : p ∈ e & q ∈ e′ & p(†) = q(∗)}. Note that (ee′)e′′ = e(e′e′′), and thateU = e = Ue and U ◦U ◦ = U ◦ and ∅e = ∅ = e∅.

5 Action frames

We say that (U, E, A, R) is an action frame if (i) U is a nonempty set of points, (ii) E isa set of events in U , (iii) A is a set of natural numbers, and (iv) R is a subset of A×E×U ◦such that (a, e, p) ∈ R only if p ∈ e. Informally, the points of the universe U arethought of as representing the possible total states of the world, The elements of A arecalled agents, while the elements of R are called (possible) individual actions. Noticethat if (a, e, p) ∈ R, it is not excluded that (a, e, p′) ∈ R, where p′ is a path other thanp but p(∗) = p′(∗); nor is it excluded that (a, e′, p) ∈ R, where e′ is an event typeother than e; nor is it excluded that (a′, e, p) ∈ R, where a′ is an agent other than a.

Informally, if (a, e, p) ∈ R and the world is in total state u = p(∗), then it is inprinciple possible that agent a should bring about a realisation of the event (e, p). (Inprinciple, meaning that if he sets himself to doing it, he will, provided that he is notprevented, his action is not interfered with, etc.) Even more informally, refer to thepicture outlined in the introduction. Suppose, assuming a system T and an occasion(s, u), that there is a total agent contribution c, the individual contribution of a beingca . The path defined by c and the set of paths defined by ca correspond to the path pand the event e, respectively:

p = T (s, u, c) e = {T (s, u, c′) : c′a = ca}

As long as there is a “real” contribution ca of a (cf. footnote 3 above) the agentis deemed to be one of the “authors” of the event (e, p). With this background weshall feel free to use locutions such as the following: “a does e”, “e is the event cor-responding to a’s action”, “a’s action was realised by p”, “a is (partially) responsiblefor bringing off p”.

Among the topics not properly dealt with in this paper is the analysis of ‘ability’and ’opportunity’. Here is one brief suggestion. Let us introduce yet another primitiveconcept, that of “repertoire”, a function from the set of agents to the set of events. Inother words, let there be a function rep from A to the power set PE of E such that, foreach agent a, rep(a) is a subset of E , intuitively, the set of events that a is (sometimes)able to set off.4 We could then adopt the following definitions:

4 We might add as a postulate that the repertoire of an agent a contain an event e only if (a, e, p) is apossible individual action, for some p; in symbols,

e ∈ rep ⇒ ∃p((a, e, p) ∈ R)

But not the converse.

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a is in principle able to do e iff e ∈ rep(a),

a is able to do e at u iff e ∈ rep(a) and there is some path p such that u = p(∗) and(a, e, p) ∈ R,

there is an opportunity at u to do e iff there is some path p such that u = p(∗) andp ∈ e.

These definitions may not completely match our informal understanding of ‘abil-ity’ and ‘opportunity’.5 At the same time it has to be remembered that that informalunderstanding is not all that exact.6

6 Chronicles, records, histories

We need a distinction between two conceptions of history and also terms to denotethem; therefore we somewhat arbitrarily commandeer two words: “chronicle” and“record”.

A chronicle is a sequence of individual actions subject to two conditions: (i) thesequence is either finite or of the order type of the set of natural numbers;7 (ii)if (a, e, p) and (a′, e′, p′) are consecutive individual actions in the sequence, thenp(†) = p′(∗). We shall use letters h, g, f for chronicles.

A chronicle consisting of only one individual action is identified with that action.The unique chronicle made up of the empty sequence of individual actions is called theempty chronicle and is denoted by ø. If h is a chronicle whose first individual action is(a, e, p), then we write h(∗) = p(∗). If h also has a last individual action (a′, e′, p′),then we write h(†) = p(†). If h and g are chronicles such that h(†) = g(∗), then wewrite hg for the sequence consisting of h immediately followed by g. We considerthat (hg) f = h(g f ). By convention we also accept that hø = h = øh.

We say that h is an initial segment, or just an initial, of h′ if there is a chronicleg such that hg = h′; a proper initial if hg = h′, for some nonempty chronicle g.Similarly, we say that h is a final segment of h′ if there is some chronicle g suchthat gh = h′; a proper final segment if gh = h′, for some nonempty chronicle g. Achronicle h is maximal if there is no chronicle h′ such that h is a proper initial of h′;that is, if, for all chronicles g, if hg is a chronicle, then g is empty.

5 But some agreement there is, and here is an interesting example. Since we do not assume the converse ofthe formula in footnote 3, it is possible that there could be a, e and p such that (a, e, p) ∈ R but e /∈ rep(a).This means that our modelling allows for the possibility that an agent might on occasion produce an individ-ual action without having the ability to do so (such as when by “luck” an incompetent golf player managesan “impossible” shot).6 For example, imagine an inebriated virtuoso, famous for his brilliant interpretations of Liszt, being ledto a piano. He finds, as does his audience, that he cannot play. What is it he lacks, ability or opportunity?It appears that one may choose between two alternatives. One: he has the ability for which he is famous,but on this occasion his regrettable intoxication prevents him from exercising it—he lacks the opportunity.The other: he has the opportunity, but he lacks the ability; for the ability that he does have is just to be ableto play under normal conditions, the latter not including his being under the influence.7 Even though we don’t do so here, one might wish also to consider chronicles of other order types, forexample, of the order type of the set of all integers.

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Every chronicle h implicitly defines a certain path which we will call the trace ofthe history and denote by tr(h). If h is a finite chronicle (a0, e0, p0) . . . (an, en, pn)

or a denumerable chronicle (a0, e0, p0) . . . (an, en, pn) . . ., then tr(h) = p0 . . . pn ortr(h) = p0 . . . pn . . ., respectively. In a similar way, h implicitly defines a certainfunction that we will call the signature of h, denoted by sign(h). Let � be any fixedobject other than a pair consisting of an agent and an event.

If h = (a, e, p), then sign(h) is defined as the function s such that, for all initialsq < p,

s(q) ={(a, e) if q = ø,� if ø < q < p.

Suppose that sign(h) has been defined and that the trace of h is p (that is, thattr(h) = p). Suppose also that h(a, e, q) is a chronicle (and so h(†) = q(∗)). Thenthe signature of h(a, e, q) is the function s′ defined as follows: for all proper initialsr < pq,

s′(r) =⎧⎨⎩

s(r) if r < p,(a, e) if r = p,� if p < r < pq.

Thus if h is a chronicle, then the signature sign(h) will be a function defined forall proper initials of h and taking as values either ordered pairs (a, e), where a is anagent and e is an event, or �; in the former case there is a path p such that the action(a, e, p) appears in h after the proper initial in question. Evidently, the ordered pair(tr(h), sign(h)) will carry exactly as much information as h itself; we will refer to itas a record of h.

An example will clarify these definitions. Suppose that each of three agents per-forms one action, one immediately after the other: agent a1 performing e1, agent a2performing e2, and agent a3 performing e3. Furthermore, suppose that the outcomesof these actions are three paths u1u2u3, v1v2v3v4v5 andw1w2w3, respectively, whereu3 = v1 and v5 = w1. Representing this history as a chronicle results in a sequence

(a1, e1, u1u2u3), (a2, e2, v1v2v3v4v5), (a3, e3, w1w2w3)

whereas representing it as a record results in an ordered pair consisting of a trace

u1u2u3v2v3v4v5w2w3

and a signature

((a1, e1), ∗, (a2, e2), ∗, ∗, ∗, (a3, e3), ∗, ∗)

The ends the example.The notion of a record is thus more general than that of a chronicle; for while two

historians who disagree over a record would also disagree over the corresponding

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chronicle, the converse is not always true—they might agree on the trace but not onthe signature.

If p is a path and s is a function defined on the set of proper initials of p, then letus say that the pair (p, s) is a record without loose ends if there are finite nonemptypaths p, called the defining paths, such that (i) either (the finite case) p = p0 . . . pn ,for some n � 0, or (the denumerable case) p = p0 . . . pn . . ., and (ii) for all properinitials r < p,

s(r) =⎧⎨⎩(ai , ei ) for some agent ai and event type ei such that (ai , ei , pi ) ∈ R,

if r = p0 . . . pi−1(for any i � n or for any i, respectively),� otherwise.

We say that h is a record with a loose beginning if there is a finite nonempty path qand a (finite or denumerable) record (p, s) without loose ends such that h = (p′, s′),where p′ = qp and, for all proper initials r < p′,

s(r) ={

s(r ′) if r = qr ′,� if r < q.

We say that h is a finite record with a loose tail if there is a finite nonempty pathq and a finite record (p, s) without loose ends such that h = (p′, s′), where p′ = pqand, for all proper initials r < p′,

s(r) =

⎧⎪⎪⎨⎪⎪⎩

s(r) if r < p,(a, e) if r = p, for some a and e such that, for some q ′,

q < q ′ and (a, e, q ′) ∈ R,� if p < r < pq.

The concepts ’initial segment’ and ’final segment’ are readily extended to records.There is a one-one correspondence between chronicles and records without loose

ends. To elaborate this claim we introduce the following terminology. If h is a chron-icle, then we write R(h) for the record (tr(h), sign(h)). If h = (p, s) is a recordwithout loose ends, then we write C(h) for the chronicle

(a0, e0, p0) . . . (an, en, pn),

if h is finite with defining paths p0, . . . , pn , or

(a0, e0, p0) . . . (an, en, pn) . . . ,

if h is denumerable with defining paths p0, . . . , pn, . . ., where (ai , ei , pi ) =s(p0 . . . pi−1), for all i � n or for all i , respectively.

Observation If h is a chronicle, then R(h) is a record without loose ends. If h is arecord without loose ends, then C(h) is a chronicle. Moreover, in the former case,CR(h), in the latter RC(h) = h.

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From now on we shall use the term “history” for records with or without looseends, except that we shall sometimes feel free to use the same term to refer to thecorresponding chronicle of a record without loose ends. If a distinction is necessary,we may speak of histories-as-records versus histories-as-chronicles.

Every action frame determines a set of histories-as-chronicles and so, indirectly, aset of histories-as-records, namely, the smallest set H that contains, for every chron-icle h in that frame, the corresponding record R(h) and, furthermore, contains everyinitial and final segment of any of its members. In a similar fashion several other setsof histories are uniquely determined: the set H f in or all finite histories, the set Hmax

of all maximal histories, the set H f ut and H past of all future and all past histories,respectively:

H f in = {h ∈ H : h is finite},Hmax = {h ∈ H : h is maximal},H f ut = {h ∈ H : ∃g(g ∈ H & g �= ø & gh ∈ Hmax )},

H past = {h ∈ H : ∃g(g ∈ H & g �= ø & hg ∈ Hmax }.

If h is a past history (and so ends at some point), then let us write cont (h) andcont◦(h) for the set of all maximal continuations of h and the set of finite continua-tions of h, respectively:

cont (h) = {g : g ∈ H & hg ∈ Hmax } = {g : g ∈ H f ut & g(∗) = h(†)},cont◦(h) = {g : g ∈ cont (h) & g ∈ H f in} = {g : g ∈ H f in & g(∗) = h(†)}.

Let us say that two chronicles are past-compatible if one is an initial of the other.It is clear that past-compatibility is an equivalence relation. A maximal equivalenceclass under under this equivalence relation bears some (some!) relationship to whatsome (some!) philosophers might call a possible world.

Before leaving this section we define yet another technical notion of equiva-lence between histories: let ∼= be the least equivalence relation such that, for allh, h′, a, a′, e, e′, p, p′,

h(a, e, p)(a, e′, p′)h′ ∼= h(a, ee′, pp′)h′.

We say that h and h′ are agent-equivalent if h ∼= h′. Notice that agent-equivalenthistories have identical traces: if h ∼= h′ then tr(h) = tr(h′).

7 Limitations of our formalism

Given the background of Sect. 3, the formal modelling presented in the precedingsections should be almost self-explanatory. A history, unfolding from the past towardsthe future, is just that: a chronicle or record of what might take place when agents actin certain ways. If, at a certain stage of an unfolding history, the world is in total stateu, then that history can (in a weak sense of ‘can’) be continued by an agent a who is

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in a position to do something at u; that is, if there is some event e and path p suchthat p(∗) = u and (a, e, p) ∈ R. The modelling itself does not regulate or predictwho will be acting next; our assumption is only that the history may grow, and thatif it does we can see what happens. (Note that time is not a primitive concept in ourmodelling: time is, you may say, action driven.)

An agent may have it in his power to select a certain event to be realised, but he hasno control over which among perhaps many possible paths will be the one actually torealise the event. In other words, the set {p : p ∈ e & p(∗) = u & (a, e, p) ∈ R},which may be a proper subset of the set {p : p ∈ e & p(∗) = u}, contains exactly thepaths that are possible after a at u has selected e, and a himself is (usually) not ableto choose between them.

The present modelling, like any modelling, has its limitations.8 It may be worthlisting some of them.

1. Agents act one at the time. This feature suits certain formal or regulated applica-tions, but it rules out collaboration, cooperative action, and plural action generally.In the informal background we allowed several agents to act at the same time. Butthe present modelling does not permit this.

2. Actions are never interrupted. This assumption is unrealistic in many applications.It also blocks an analysis of at least one important kind of “trying to do some-thing”.9 Again, our modelling fails to provide for something that was allowed inthe informal background.

3. There is no consideration of knowledge or belief. To add such an analysis wouldbe as complicated as it would be important.

4. There is no analysis of agent contribution. Clearly a complete theory of actionmust provide an analysis of such concepts as intention, goal, purpose, deliberation,following a plan, executing an action, monitoring a performance of an action, etc.

5. There is no analysis of causality. This is perhaps the limitation that would be themost difficult to overcome.

These limitations of our modelling notwithstanding, many concepts can be given akind of analysis within it.

8 As one referee points out, there are modellings that do not suffer from one or several of the limitationsmentioned here.9 In general there seem to be at least three different kinds of trying to something. (1) The agent selects anaction, the realisation of which is thwarted. (2) The agent has a certain goal in mind and selects an actioneven though only some of the defining paths realise that goal. That is, the agent wishes to achieve a goal(state-of-affairs) P , and he selects an event e such that the set {p(†) : p ∈ e} only partially overlaps withP . If a path q in e is realised such that q(†) /∈ P , the agent is likely to comment, afterwards, that at leasthe tried to achieve P by doing e. (3) The agent has a certain goal P in mind and believes that runningevent e will lead to that result, even though in fact {p(†) : p ∈ e} is disjoint from P . Again, the agent maycomment, after the event, that at least he tried to achieve P by doing e.

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8 Models and truth in models

We assume a denumerable set of propositional letters and a truth-functionally completeset of boolean connectives; in addition there will be further operators as explained inlater sections. We use letters like φ,ψ and θ for formulæ.

A valuation in a set U is a function assigning to each propositional letter a subset ofU . A model is an action frame together with a valuation in the universe of the frame;the model is sometimes said to be on the frame. By an articulated history we meana pair (h, g) of histories (in the action frame in question) such that hg is a maximalhistory.

We define truth of a formula in a model at10 an articulated history as follows; wewrite (h, g) �M φ for “φ is true in M at (h, g)” and (h, g) �

M φ for “φ is false (thatis, not true) in M at (h, g)”, where φ is a formula, M is a model, and (h, g) is anarticulated history in M.

(h, g) �MP iff either h(†) ∈ V (P) and h �= ø, or g(∗) ∈ V (P) and g �= ø (or

both), where P is a propositional letter,

(h, g) �M φ ∧ ψ iff (h, g) �M φ and (h, g) �M ψ ,

(h, g) �M ¬φ iff not (h, g) �M φ,

etc.

(From now on we omit the reference to M when this can be done without causingconfusion.)

A formula is valid in an action frame if it is true in all models on the frame at allarticulated histories.

9 Temporal and historical operators

It is easy to see that we may take over, without any difficulty, all the usual operatorsof tense logic. In this paper we only mention temporal operators formalising ‘alwaysin the future’, ‘some time in the future’ and ‘unless’:11

(h, g) � [f]φ iff ∀h′, g′ ((hg ∼= h′g′ & h � h′) ⇒ (h′, g′) � φ),(h, g) � 〈f〉φ iff ∃h′, g′ (hg ∼= h′g′ & h � h′ & (h′, g′) � φ),(h, g) � (until φ)θ iff ∀g0, g1((g ∼= g0g1 & (hg0g1) � φ) ⇒. ∀g′, g′′((g ∼= g′g′′ & g′ < g0) ⇒ (hg′, g′′) � φ) ⇒. ∀g′, g′′((g ∼= g′g′′ & g′ < g0) ⇒ (hg′, g′′) � θ ))).

Our operators [f] and 〈f〉 are in effect the same as Prior’s G and F , while (until)comes from Kamp. These operators may be termed operators for future time. It wouldbe routine to introduce corresponding operators for past time: [p] and 〈p〉 and (since).

10 At? In? For? Prepositions are difficult.11 In these definitions the informal quantifiers ∀ and ∃ range over histories-as-records. They make it clearwhy officially we view histories as records rather than as chronicles.

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The term “historical” in the section heading means “defined in terms of histories”.We give only one operator here— [h]. For [h]φ, where φ is a formula, read “it ishistorically necessary that φ” or (to use a term recommended by Belnap) “it is settledtrue that φ”. The corresponding truth condition is

(h, g) � [h]φ iff ∀g′(g′ ∈ cont (h) ⇒ (h, g′) � φ).

10 Agentive operators

At this stage let us assume a denumerable set of event letters, disjoint from the setof propositional letters. The event letters will be called terms; in this paper they arethe only terms. We introduce proposition-forming operators doesa,donea, realisesa ,realiseda, occursa, occurreda , on terms, where a is supposed to range over the set ofpositive integers. The following informal readings may be helpful:

doesaα a is just about to do α,doneaα a has just done α,realisesaα a is just about to do α,realisedaα a has just realised α,occursα α is just about to occur,occurredα α has just occurred.12

Let M be a model with valuation V . Assume that the domain of V is extendedto include the set of event letters, and that V assigns to each event letter an eventin the universe of the frame of M. We use letters α and β for terms. For V (α) weusually prefer the notation [[α]] if it is clear which valuation is understood. There arethe following meaning conditions (where α is restricted to range over the set of agentsof the frame M):

(h, g) � doesaα iff ∃g0, g1, e, p (g ∼= g0g1 & s(ø) = (a, e) & p = tr(g0)

& e = [[α]]), where s = sign(g0),

(h, g) � realisesaα iff ∃g0, g1, e, p (g ∼= g0g1 & s(ø) = (a, e) & p = tr(g0)

& p ∈ [[α]]), where s = sign(g0),

(h, g) � occursα iff ∃p, p′(pp′ = tr(g) & p ∈ [[α]]) ;

(h, g) � doneaα iff ∃h0, h1, e, p (h ∼= h0h1 & s(ø) = (a, e) & p = tr(h1)

& e = [[α]]), where s = sign(h1),

(h, g) � realisedaα iff ∃h0, h1, e, p (h ∼= h0h1 & s(ø) = (a, e)& p = tr(h1) & p ∈ [[α]]), where s = sign(h),

(h, g) � occurredα iff ∃p, p′(pp′ = tr(h) & p′ ∈ [[α]]).

12 Strictly speaking, we should write “a is about to do [[α]]”, etc. However, for greater ease of reading andwriting we omit the brackets, both here and in what follows.

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11 Norms

By a hypertheory in a set S we mean a nonempty set of subsets of S that is wellorderedby inclusion. Note that if S is a hypertheory, then

⋂S ∈ S. If S is a set, we write

hth(S) for the set of hypertheories in S.13

We adopt the following technical definitions. If h is any past history, g any finitecontinuation of h, and X any set of maximal continuations of h, then we write X g forthe set of continuations of hg that are final segments of elements of X . Schematically,for all X ∈ cont (h),

X g =df { f : f ∈ cont (hg) & g f ∈ X}.

Similarly, if H is a set of subsets of cont (h), then we write Hg for the set of nonemptysubsets Y of cont (hg) such that Y = X g , for some X ∈ H. Schematically,

Hg = {X g : X ∈ H & X g �= ∅}.

Observation (i) If X and Y are elements of H, then X ⊆ Y only if X g ⊆ Y g. (ii) If His a hypertheory in cont(h), then Hg is a hypertheory in cont(hg).

Proof (i) Suppose that X and Y are elements of H such that X ⊆ Y . If f ∈ X g

then g f ∈ X , whence g f ∈ Y by assumption. Hence f ∈ Y g . (ii) Suppose thatH ∈ hth(cont (h)). That Hg is ordered by inclusion follows by (i). We must show thatthe ordering is a well-ordering. Let I be an indexing set such that C = {Yi : i ∈ I }is a nonempty collection of elements of Hg . It will be enough to show that

⋂C is an

element of C . For each i ∈ I there is some some element Xi of H such that X gi = Yi .

Since H is well-ordered, the set {Xi : i ∈ I } contains a smallest element, say X j , forsome particular index j . Evidently, X g

j = Y j is an element of C . Suppose that Yk is anelement of C . Suppose that Yk is any element of C such that Yk ⊆ Y j . Then there issome element Xk in H, for some index k, such that X g

k = Yk . Since X j is the smallestelement in {Xi : i ∈ I }, and since Xk is of course also an element of {Xi : i ∈ I },it follows that X j ⊆ Xk . Hence by (i), X g

j ⊆ X gk . In other words, Y j ⊆ Yk , whence

Y j = Yk . This shows that Y j is the smallest element of C . ��We can now explain what a norm function for an action frame (U, E, A, R) is: a

function assigning to each past history in the frame a normative position (intuitively,the normatively position holding after the unfolding of that particular past history).The difficulty for the formal analyst is to find a counterpart of ‘normative position’.In this paper the following definition is natural, if technically complicated: normative

13 The concept of a hypertheory goes back to Lewis’s so-called sphere systems in Lewis (1973). (Thewell-ordering condition used here is stronger than his Limit Assumption.) Hypertheories (under variousnames) play an important rôle in the theory of belief revision and therefore in DDL. (See Alchourrón et al.(1985) for the classic paper on belief revision. For DDL, see Grove (1988), and Segerberg (1999, 2001)and the references listed there.) It is worth recalling that the original concern of Alchourrón—a professorof jurisprudence and one of the authors of Alchourrón et al. (1985)—was not belief change but change inlegal contexts.

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positions are hypertheories. One difficulty in spelling out this position is the preform-al intuition that successful normative positions “hang together” or cohere in a certainway: the norm stays the same even though the circumstances change. As will be seen,that problem is solved by the use of hypertheories.

We say that N is a norm function, or just a norm, if N is a function defined on theset of past histories, such that, for every h ∈ H past , N (h) is a hypertheory in the setof continuations of h; schematically,

(1) N (h) ∈ hth(cont (h)).

Furthermore, N is coherent in the sense that, for all g ∈ cont◦(h),(2) N (hg) = {X g : X ∈ N (h) & X g �= ∅}.

N (h) is called the normative position at h. (This definition is an improvement on thetentative discussion in Segerberg (2005).)

The intuitive idea here is that certain future histories are normal in the sense ofbeing in accordance with the norm, while others are not—this seems to be the pointof any norm system: to separate the sheep from the goats. In this spirit, let us defineany continuation of a past history h as normal (after h) if it falls within

⋂N (h), that

is, the innermost layer of the hypertheory assigned to h by the norm. In other words,if g ∈ cont (h), then g is normal if and only if g ∈ ⋂

N (h). Something similar holdsfor finite continuations: if g ∈ cont◦(h), then we say that g is normal (after h) if andonly if g � g′, for some g′ ∈ ⋂

N (h). (Clearly, the two uses of the term “normal” areconsistent.)

Observation Let N be a norm function. Suppose that h is a past history and g a finitecontinuation of h. Then

(i) (⋃

N (h))g = ⋃N (hg).

Furthermore, if there exists a smallest set X in N(h) such that X g �= ∅, then

(ii)⋂

N (hg) = X g.

In particular, if g is normal after h, then

(iii)⋂

N (hg) = (⋂

N (h))g.

12 Deontic operators

If a norm function is available, we can define a number of deontic notions. For exam-ple, we might say that a proposition is “deontically necessary” if it is true (in a modelwith an associated norm N ) with respect to all continuations within the intersectionof the hypertheory determined by the past history. More precisely, let [d] be a newoperator with the reading “it is deontically necessary that”, and define

(h, g) � [d]φ iff ∀g(g′ ∈ ⋃N (h) ⇒ (h, g′) � φ).

However, the concept ‘deontically necessary’ is not meant to be an explication of‘ought’—even senses of ‘ought’ that are deontic rather than evaluative must be ana-lysed with more care than so. The easiest, within the present modelling, is probably

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to begin with term operators; that is, starting with ‘ought-to-do’ rather than ‘ought-to-be’. Thus let us introduce formula-producing term-operators oba and pma with theinformal reading “α is obligatory for a” and “α is forbidden for a”, respectively. Theintended semantic conditions have a forbidding look to them, so here we are contentto offer syntactic characterisations instead:

(1) obaα =df [h](until doneaα)[d]〈f〉doneaα,(2) fba =df [h][f][d][f]¬doneaα.

The sense of obaα in (1) is that as long as the agent has not done α, then he shoulddo so—his obligation is discharged only when he has done α. To define obligationsimply as

(3) obaα =df [d]〈f〉doneaα,

would not do since the obligation to do α would disappear in any normal continuationof the current past history. A more interesting variation is

(4) obaα =df [h](until occurred α)[d]〈f〉doneaα,

with the sense that the agent should do α as long as α has not occurred; in this casehis obligation vanishes if α is brought about in some other way than by his action. Asan explication of “a ought to do α”, (4) is sometimes more reasonable than (1).

The preferred definitions of obligation and prohibition in (1) and (2) lack the sim-ple symmetrical relationship deontic logicians have come to expect. But many otherdefinitions would be possible. A definition of prohibition that better balances (1) is

(5) fbaα =df [h](until doneaα)[d][f]¬doneaα.

According to (5), it is the first realisation that is prohibited; once α has been done, theprohibition disappears (which is not to say that it is replaced by permission). Perhapsone can call this sense of prohibition “one-time prohibition” and the sense of obli-gation in (1) “one-time obligation”. By contrast, (2) defines a concept of “standingprohibition”: α must never be done. A corresponding “standing obligation” defined as

(6) obaα =df [h][f][d][f]doneaα

seems excessively demanding.There are certainly many concepts of standing obligation: some involving con-

ditional action (“Close the door behind you whenever you enter the room!”), someinvolving processes rather than action (“In the reading room, silence is to be main-tained at all times!”), some involving ways of acting rather than action (“Always behonest!”). There seems to be no way of giving a reasonably faithful formalisation ofany of these concepts within our present modelling.

It is worth remarking that, within the present modelling, ‘permissible’ seems to bemore difficult to formalise than ‘obligatory’ and ‘forbidden’.

Acknowledgements postscript 2011: The work on this paper was essentially finished in 2003; the dateon what seems to be the final print-out of the draft of the paper is December 24, 2003. It was submitted tothis journal, reports were written by two referees, but for reasons unknown this writer became aware of themonly in February 2011. I am grateful that the editor is still willing to publish the paper. I am also grateful tothe referees who wrote detailed and helpful reports. I have tried to correct the errors they pointed out and

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have made some minor revisions, but their suggestions for enlargement I have declined on the principlethat new patches on old cloth are to be avoided (Mark 2:21). One criticism made by the referees was thatreferences to similar work in computer science are missing. Yes, here I must plead guilty of ignorance. It isin general to be regretted that there is so little contact between the several communities that are interested inaction: the philosophical logicians, the computer scientists, the linguists and—last but not least—the “real”(informal) philosophers.

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