joint action and group action made precise

GABRIEL SANDU AND RAIMO TUOMELA

JOINT ACTION AND GROUP ACTION MADE PRECISE

ABSTRACT. The paper argues that there are two main kinds of joint action, direct joint bringing about (or performing) something (expressed in ten'as of a DO-operator) and jointly seeing to it that something is the case (expressed in terms of a Stit-operator). The former kind of joint action contains conjunctive, disjunctive and sequential action and its central subkinds. While joint seeing to it that something is the case is argued to be necessarily intentional, direct joint performance can also be nonintentional. Actions performed by social groups are analyzed in terms of the notions of joint action (basically DO and Silt).

A precise semantical analysis of the aforementioned kinds of joint action is given in terms of "time-trees". With each participant a tree is connected, and the trees are joined defining joint possible worlds in terms of state-expressing nodes from the trees, Sentences containing DO and Stit are semantically evaluated with respect to such joint possible worlds. Intentional joint actions are characterized in terms of the notion of "we-intention" (joint intention), characterized formally by means of a special operator.

1, INTRODUCTION

The main concern of this paper is the logical features of joint action - such as jointly can3,ing a table upstairs, singing a duet, or jointly building a house. A joint social action in its broadest sense is an action performable by several agents who share a "we-attitude" (involving a joint goal, belief, or the like) and act on this we-attitude. In this paper we shall require - in analogy with the single-agent case - of a joint action that it, furthermore, be based on joint intention (basically a shared "we-intention" about which there is a mutual belief). This is fulI-blown joint action. ~ In contrast, there is "coaction", collective action in which agents - without a joint intention - have the same goal, perhaps mutually believing so and possibly interacting in various ways. On the other hand, joint action is to be distinguished from action performed by a group. A group's performing an action obviously presupposes that some of the members (or representatives) perform an action - or bring about a state of affairs - in virtue of which a relevant action can be attributed to the group. Groups can also act in a somewhat more general sense, other than by affecting a change in the world. For instance, a battalion of soldiers carl see to it that nobody offends the border. In most cases they can fulfil the task by "doing nothing at all." Obviously,

Synthese 105: 319-345, 1996. @ 1996 Kluwer Academic Publishers. Prh~ted in the Netherlands.

320 GABRIEL SANDU AND RAIMO TUOMELA

also in the case of the notion of seeing to it that something is the case the group's activity must be based on its members' (or representatives') relevant activity (or control-involving agentive "passivity").

In this paper we shall investigate joint action and group action from a logical point of view. We have elsewhere presented an account of single agent action and defined two action notions, represented by the operators called DO and Stit, respectively, in order to account for ordinary action performances and intentional seeings to it that something is the case. 2 In that paper we also defined an operator, DO*, corresponding to DO, to account for intentional action.

We will start by reviewing some of the ideas of our earlier work, partly in order to make this paper self-contained.

2. SINGLE-AGENT qREES

In this section we present the technical framework for analyzing action concepts in the case of single agents. 3 This is necessary for a better under- standing of the joint agent case, which will be an extension of the single- agent case. We use the symbols t0, tl , . . . . to denote "states" or"occasions". (Possible worlds in the sense to be introduced later will be collections of states to, tl, .... ) One way to individuate a state t would be to take it to consist of states of affairs P, Q, etc., which, formally speaking, would be the referents of the propositional symbols (sentences) P , Q, etc., of a formal language L. From a strictly formal point of view, this is, however, not the route we are going to take in this paper. Instead, we shall proceed in the usual way and drop states of affairs altogether from our model. We will identify a state t straightforwardly with a truth-assignment, i.e. a function which assigns one of the values True or False to the propositional symbols (sentences) of a formal language (cf. below). However, informally, we will go on speaking about states of affairs (e.g., the window is open), meaning in that case what a sentence refers to. Also, for convenience, we will often not distinguish formally between a sentence and the state of affairs the sentence refers to. The context should make it clear which is meant.

Our idea of single agency is represented in the following way: we compare two possible states, to and tl , such that to occurs or obtains earlier than tl . In our model, nothing happens without a cause, and when causes are nonhuman, they are ascribed to a particular agent called Nature. Nature is allowed to have decision points with choice alternatives. The basic idea is that if a sentence P is false at an earlier state to, but is true at tl , then it is inferred that the change is due to some agent's action at to. In this case we will say that P is the result brought about by some agent's

JOINT ACTION AND GROUP ACTION MADE PRECISE 321

action at to, or that the agent in question is the direct agent of P, or, simply, that the agent directly effected P. We also say that to is a choice point or decision point for that agent with respect to P. It then becomes clear that, for arbitrary P, if P is true at the state t and we want to find out who "did" P, viz. performed the action with P as its conceptually inbuilt "result" (in the sense of the action of opening the window having the state of affairs of the window's being open as its result), we have to look backwards for an earlier state *0 at which P is not true and see to whom to was assigned as a decision point with respect to P. Recalling that we metaphorically regard Nature as an agent, it can be said that in our model aI1 changes are agentive.

We start our formal analysis with an exact description of the language L. L consists of:

(a) a set of primitive propositional symbols (atomic sentences);

(b) a set X = {~1, c~2, ... . Nature . . . . } of individual constants denoting individual agents;

(c) a binary operator DO acting on individual constants and sentences (we write DO(a, P) as DOg, P);

(d) a set consisting of temporal sentential connectives: ; ('and then') and II ('in the same time');

(e) a set of standard propositional logical constants.

The set of well-formed sentences of L is defined in the usual way as the smallest set including (a) and closed under the standard Boolean connectives and the connectives in (c), (d), and (e). We read 'DO~P ' as 'The agent t~ directly effected (is the direct agent of) P ' ; ' (P ; Q)' as ' P and then Q'; ' (P II Q)' as ' P and Q at the same time' ('in parallel').

A model for L is a tupte M = ( T , X 5r, Dec r , HT, StrT...) whose elements are described below.

A tree T is a tuple T = (T, <, Rime, - . - ) , where T is a set of states or occasions and, for each t E T, the set {t t C T : t ~ < t} is linearly ordered. (We assume that the ordering _< is discrete). We will call such a set a branch of T and a finite segment of it a segment. Intuitively, each branch of T represents an infinite history made up of (possible) states.

Ttime , i.e., the time of T, is a set of instants of time linearly ordered by the "earlier" (<) relation. Each state t in T will be assigned an instant of time it belonging to rtime. All the states in T which are assigned the same instant of time are said to be parallel. Figure 1 provides an example of parallel states. Here the instant it is associated with the set of parallel states {to, tl, t2, t3}.


to t l t2 t3

l l I ",,,,,,,,,I ....

Fig. 1.

As pointed out above, each state t E 7' will be taken to be a function which assigns a truth value t ( P ) to each primitive propositional symbol P (t(P) E {True, False}).

The set X T is a set of individuals X T = {c~i, c~j, Nature .... }, called the agents of T serving as the values of the constants in X. In the sequel we will use the same symbol (,~,i for the individual constant ~i in X and for its value c~i.

The function Dec T, called the decision function of T, assigns to every agent in X T a subset of T. The idea is that DecT(c~) represents the nodes of the tree which are decision points for ~, We shall impose the following constraint on the function Dec T:

(C1) For any i ¢ j : DecXr'(oz~:) n Dec'r(c~j) = ~.

In other words, c~i and o~j cannot have the same decision points. This is the reason why we call the trees T single-agent trees.

The function H T assigns to each agent c~ and states t, t I and t" (t < t f < t rt) a set Hr (c t , t, t ~, t") of sentences of L. Informally, we interpret P E H:r(cr, t, t t, t") as "oz has an intention between t and t ~ to bring about P at t"." (We leave the development of a full-fledged logic of intentions for another occasion.)

Let t be a state in T and t ~ a later state, i.e., t < t t. T~' will denote the subtree of T whose root is t and whose leaves are the states corresponding to the instant it, determined by t ~. A strategy in the tree T~' is any function f such that dom(f ) = { t l : it <_ it1 < it ,}, and range(f) = {t" t E T~'} such that for any t C dora(f) we have t < f ( t ) .

The function Str r assigns to each agent cr and states t and t r (t < t ~) a set StrZ(c~, t, t r) whose members are pairs (P, f ) consisting of sentences

• , • , t ' /

P of L and strategms f in the subtree T~ When (P, f ) C Str~'(c~, t, t ), we say that c~ has a strategy in T between the states t and t t with respect


to P. The set StrT(oe, t, t') obeys several constraints that will be listed in Section 5 below where the notion of strategy will be used for the analysis of Stit.

Still a notational point: When T' C dom(f) , we let f ]" T' denote the restriction of the function f to the set T'.

The truth of a propositional symbol P at a state t (i.e. t e P) is defined inductively in the following way:

DEFINITION 1.

(i) t V P iff t(P) = True, for a primitive P.

(ii) t ~ -~P iff not t ~ P

(iii) t ~ ( P A Q) i f f t ~ P a n d t ~ Q

(iv) t ~ (P; Q) iff ~tl < t(tl > (-~Q A P) A gt2(ti < t2 <_ t -+ t2 (P A 0)))

(v) t ~ ( P I] Q) iff 3tl < t(tl ~ (~P A -~O) A V~2(tl < Z2 ~ ~

(t2 (P a 0))) (vi) t ~ D O ~ P iff 3tl < t(t~ > -,P k t~ E Dec(a) A gt2(t, < t2 <

* --+ t 2 ~ P ) ) .

Our analysandum is the perfomaance of an action, a doing, as we might say. According to Definition 1 (vi), that an agent directly effected (is the direct agent of) P means that he has the last decision point t x in the situation at hand where -~P holds true. This last decision point need not, however, be located at the instant of time which immediately precedes t. When 'DO~P ' is true, we can say that a performed an action having (the tokening of) P as its conceptually inbuilt "result". While P must occur after t × and no later than t, it is still open for us to adopt the position that our agent's action occurs during the segment (t*, t), if we so want.

Aithough we are not directly dealing with the specific contentual features of o~'s action, viz., the "mode" with which he effects the result P , our present account seems to be as close as one can come in formal terms in capturing direct agency, viz. the direct, non-mediated performance of an achievement-related action. We would like to emphasize that the end states P can be agentive. Thus in the case of an agent o,'s opening the window we can take P = 'the window is open due to c~'s opening it', or simply P = 'ee opened the window'. This guards against some well-known criti- cisms against the action-operator approach but makes the account perhaps conceptually less illuminating, because it means regarding the notion of action as a primitive concept. Illuminating or not, we take this to be right and do not believe in reductionist approaches to action.


Direct agency in our sense includes the possibility of using tools as long as the tools are not the agent's other actions. If an agent effects P in the sense of our operator DO, he can be said to bring about the propositional state of affairs P. However, not all instances of bringing about a state of affairs are direct performances in our present sense.

A technical remark is in order here. There is an alternative way of interpreting sentences of the form D O , P , which has its motivation in dynamic logic. In this other framework, DOs will denote a program, i.e., a binary relation R on the tree T. Informally, R(t, t*) will be in the interpretation of DOs whenever t and t* will be as in Definition 1 (vi), for some P. However, we did not choose this alternative here, but the two frameworks are intertranslatable.

D O ~ P is not required to be intentional with respect to P. However, intentionality can be made explicit in the tree T in different ways, one of which - perhaps the simplest one - will be adopted below. In a finer analysis we would need also the notions of want and belief, especially the latter. The present account actually makes the simplifying assumption that the participating agents' beliefs are true and that hence we can speak in objective terms of what the action-trees in question contain. 4

Intentions are formally implemented in our model by the function H T which assigns to each agent c~ and states t, t t, and t" (it < t f <_ t") a set HT(c~, t, t ~, t") consisting of propositional symbols P. Intuitively, the set HT(c~, t, t ~, t") specifies the states of affairs that a intends between t and t r to hold at t". In a more recent terminology we could say that, if P belongs to the set H~C(c~, t, t ~, t"), then a has an intention that P holds at t" that persists between t and t r. We can say somewhat loosely that c~ intends (to bring about by his action) that a proposition expressed by P be true if and only if he either has the (future-directed or present-directed) intention to "do P " or has the "intention-in-action" (or if he "endeavors" or "wills") to effect or bring about P .

We can now define a notion of intentional action which is obtained by adding to Definition 1 (vi) a clause for intentions, and a clause for the existence of a "counter":

DEFINITION 2. t ~ DO~P ("a intentionally directly effected P") if and only if

(i) t ~ P.

(ii) ~ P is true at some state in the tree earlier than t, and the last such state, say t*, is a decision point for c~.


(iii) ~ P is true at some state t' belonging to the instant it.

(iv) P c It~'(c~, t*, t, t).

Clause (iii) ensures that the decision point t ~ is a real one with respect to P, that is, ~ had a choice between P and - ,P at t ~. It is a mark of an agent's intentional action that he could have done othm~ise.

That o: directly effected P entails that he did not do P via some other agent or with the help of some other agent.

It can easily be seen that, at each state t in T : t ~ DO~P ~ DO~P. That is, if an agent acts intentionally, that agent also acts simpliciter.

THEOREM t.

(i) t ~ -~(P; P )

(ii) t>(P; Q)--+ -~(Q; P) (iii) t ~ ( P ; Q) A (O; R) -+ ( P ; R)

(iv) t ~ DOa,(P * Q) -+ (P • Q) (* stands for any sentential connective)

(v) t ~ D O s ( P ; Q) ~ DOaQ

(vi) t ~ DO~(P A Q) --+ (DOo, P V DO~xQ)

(vii) t ~ D Q ( P II Q) --+ D O ~ f I1 DOoQ) (viii) DO~(P V Q) -+ (DO~P V DO~O),

Proof straightforward from the definitions. For illustration, we shall prove (v).

Assume, for contradiction, that: (*) t ~ DO~(P ; Q) A -~DO,:~Q. Then, from (iv), t > ( P ; Q). Let t~ be the last moment before t such that tl V -~(P; Q) (note that such moment exists). Clearly tl is a decision point for a. Since t > ( P ; Q), there is t2 < t such that t2 ~ (PA =Q), and for all moments above t2, P and Q hold. Now, if t2 = tl, then t > DO, Q, which contradicts (*). If ~2 ~( tl then tl ~ (Q; P) , which again is a contradiction.

(vi) cannot be strengthened to DO~(P A Q) -+" (DO~P A DO, Q). A counterexample is provided by Figure 2. In this picture we assume that t2 occurs immediately earlier than tl, which occurs immediately earlier than t, tt is the last moment before t at which -~Q holds, and t2 is the last moment before * at which -~P holds, and also that tl E Dec(c~), and t2 ~ Dec(J) .


P,,,O,, DOoP, ~ DOeO ,, [ It

t - ° ...... 1

Fig. 2.

Ttirne

3. JOINT ACTION

Let us now analyze joint social actions in our model. We recall from the introduction that joint actions in the full-blown sense are actions performed together by several agents on the basis of their shared we-intention.

We must distinguish between the notions of joint action and group action, the latter notion being understood to be an action performed by a social group. The former notion is the more basic notion, which can be used to analyze the latter, although it should be emphasized that the notion of an intentional joint action makes use of the notion of "us" or "our" group. In this section we will only be concerned with joint actions. Group action will be discussed in Section 7.

Joint actions may be classified in a number of ways. One way is to divide them into causally generated and conventionally or, more broadly, "conceptually" generated joint actions. 5 For instance, the agents' jointly carrying a table is causally brought about by their component actions of carrying the table. But when the agents perform a toast by lifting their glasses, then the joint action (an action token with the result state of the kind P) of the toast being performed is conventionally generated by the result events of the kinds P1 and P2 of the individual glass liftings, in the sense that P1 and P2 are redescribed as P. Another criterion of classification is whether the original agents themselves carry out the whole joint action or whether at some point they employ some representative or "operative" agent in the terminology of Section 7 (cf. the prime minister representing a state in a negotiation). Representation can perhaps also be regarded as a species of action generation.

Let us think of a case of causally generated action like jointly carrying a table (whose result will be denoted by P) performed by some agents cx 1, .... c~. The result-state of this action - the state of affairs of the table's


having been carried - is a (proximate) joint end of the agents. In order for this action to be individuated as an intentional joint action, two elements must be present:

(1) Constitutive Condition: There are PI,. . . ,P~, such that DO~IP1, . . . , DO~,~P~ hold true and P1 , . . . , P~ "make u p " P (here "DO" is the operator introduced at the beginning of this paper); and

(2) Intentionality Condition: c~,. . . , c~ jointly intend (share the "we-intention") to bring about the joint end-state P.

These two conditions are conceptual requirements of intentionally performed joint action in a given situation with n participants. According to (1), each agent c~i will have a "part" or "share" Pi of the joint action to perform (behaviorally "empty" parts can be allowed), and the performances of the factually or conceptually parts generate the joint action. Here we speak of the parts P1 , . . . , P~ "making up" the joint end P.

Condition (2) presents the central requirement for the intentionality of the joint action: the participants must jointly intend the joint end P, an end meant to be realized by the agent's joint action. (The notion of a joint end is regarded as primitive in our logical formalism.)

An agent's we-intention to bring about a state P jointly with the others amounts to his intention to do his part of the joint action in question in accordance with and because of the agents' "plan" or "agreement" to perform that action jointly, and the relevant belief of his that the "joint action opportunities" will obtain (see Tuomela 1995). ActualIy, an intentional performance of the joint action with P as its inbuilt result requires a we-intention only in an indirect way. That is, when participating in the joint action, a we-intention to bring about P jointly can be expressed by the sentence: "We will bring about P." Each participant can be assumed to accept the following simple inference:

(i) We will bring about P.

(ii) Therefore, I will do my share of our bringing about P.

Furthermore, when a part-division exists, (ii) entails for each o~i the state- ment:

(ii*) I will do Pi,

where doing Pi is c~i's part of the joint action (to bring about P jointly). According to (ii*), a we-intending agent in the last analysis will intend to perform his part of the joint action. This joint action to bring about P - when it indeed is a "full-blown" joint action - can be argued to be


i i P, -~Q

I ,oi l-P, ol I P.o

t l -~P, -10

T•m8

ii

io

Fig. 3.

based on an explicit or implicit agreement by the participants to perform it. Indeed, any full-blown or "proper" joint action can be argued to be based on agreement-making, an agreement (or plan, if you prefer) concerning what the agents will do together and how they will do it.

A joint intention, viz., here a shared, mutually known we-intention, need not be preformed and can therefore express a joint action-intention, viz., what the agents jointly aim at by their performances of their parts of a joint action. We would like to note, however, that our tree-semantics allows for joint (and collective) actions in a wider sense, although we cannot discuss all the possibilities here. To mention just one way of relaxing it, instead of operating with proper joint intentions, we can get along with mere "shared" intentions, viz., the agents' personal intentions concerning a jointly realizable state. 6

Our formal treatment of joint action will be a generalization of the single-action case. However, we shall have to be more careful here, for the following reason.

In Section 2 when introducing single-agent trees, we assumed that each node in the tree T can be a decision point only for at most one agent. We will relax this assumption now, as exemplified by Figure 3. In this figure we assume that tl is a decision point both for ~ and/3. Thus at the state tl at which -~P and -~Q hold true, there are four possible courses of action resulting in four possible states: t2, t3, t4 and I5. However, it is not possible to say on the basis of the formal model developed so far, which one of the agent brings about, e.g., P, and which one brings about Q at t4. Eor this reason, we need to introduce more structure on the single-agent trees

J O I N T A C T I O N A N D G R O U P A C T I O N M A D E P R E C I S E 329

in order to be able to differentiate between different agents. Formally this will be done by changing the definition of the function Dec.

Another remark is in order. Here we shall not offer a complete for- malization of the relation existing between the parts P1 . . . . , Pa and the whole P. Instead we will deal only with very simple cases, that is, those cases in which n = 2 and in which P has the form t P , Q), (P II Q), or simply the Boolean ( P A Q) and ( P V Q). The case with r~ arbitrary is a straightforward generalisation from this particular case.

4. J O I N T - A G E N T T R E E S

The formal model for joint action is roughly the same as that for single action.

The formal language L contains the same logical and nonlogical constants as the one introduced in Section 2, plus the operators Int~ lnt! °int ~ Og i ~Oe 2 , and we Int~, . The well formed sentences of L are defined by induction in the following way:

(i) all the primitive propositional symbols are sentences; (ii) if P~ and/)2 are sentences, so are --7 P~, ( P1 A P2), (P~ V P2), (P1 ; P2 ),

and (PI tt P2); (iii) if P is a sentence, and a an agent constant, then DOs P is a sentence; (iv) if P and Q are sentences, and ~1 and ct 2 are agent constants, then

Inta 1 P, lnT tJointcgl,~2 ~U'lr~ and Intwe(p, Q) are sentences ("P is intended by

oq ", " P is jointly intended by cq and a2", and °'Q is weqntended by c~ as his "part" of (the agents' jointly effecting) P".

Notice that here we restrict our attention to the case in which the joint intention involves only two agents. In the general case, we would need also arbitrary operators T ejoint lnlEa 1 ,...,~, •

We now describe the models M of L. They are identical with the ones in Section 2, except for the definition of the function Dec and the functions H T and Str T assigning intentions and joint-strategies to individual agents and respectively pairs of individual agents from X T. In addition, we shall have a function G T ascribing we-intentions to the individual agents in X T. More precisely, a model M is a tuple M = ( T, X 7, Dec T , / / T G T, Str z ), such that

(a) T is a joint-action tree T = (T, <, Ttime), where T is a set of states partially ordered by <, and rtime is the time of T, linearly ordered by < as before. Each state t in T is a function which ascribes a truth-value for each propositional symbol P.


(b) X T is a set of individual agents. (c) Dec T is a function which assigns to every agent c~ in X T and state

t a set Decr(c~, t) of sentences satisfying the following constraints:

(C2) Decr(o~i, t) N DecT(c~j, t) = ~, for any i 7 ~ j .

(The parts of any two agents in a joint action are disjoint.) (d) The function H r is extended so that its arguments are, in addition

to agents and states t, t', and t", also pairs of agents and states t, t', and t" (t < t ' _< t"). As in the single-agent case, the values of H T are sets of sentences of L. When P E H T ( ( a l , a2), t, t ', t"), we say that between t and t ' ~1 and ee2 have a joint intention that P holds at t".

(e) The function G T ascribes to every agent in X T and states t, t', and t" in T a set GT(a, t, t ', t") = {(P, Q) : P , Q are sentences of L and Q is a subsentence of P} . The meaning of (P, Q) E GT(a, t~ t', t") is that between t and t' Q is we-intended by a as a "part" of P at *".

We impose the following closure conditions on the sets H T and GT:

(C3) (Q, P) C GT(oz, t, t ', fit) :=~ r E [/T(c~, t, t t, fit)

(C4) (/'1#/'2) c ttT((al,a2),t,t',t ") ((&#P2),Pd C GT(ai, t, t', t")

for each i -= 1,2. Here # stands for ]1, ;, and A.

(C5) (P1 V P2) C H Y ( ( a l , a 2 ) , t , t ' , t '') =~ [((P1 V P2),P1) E GT(o~I, t, t ', fit)) V ((P1 V P2), P2) C GT(o~2, t, if, fit)].

Notice that if e.g., (P1 II P2) is jointly intended by al and a2 between t and t', to hold at l", i.e., (P1 It P2) E gT( (cq , o~2), t, l t, fit), then by (C4), P1 is we-intended by cq as a "part" of (P1 N P2), and/:'2 is we-intended by oz 2 as a part of (Pt II Pz), i.e., ((P1 11 P2), e l ) E GT(al , t, if, t"), and ((Pt II t"2), P2) ar( Z, t, t', ¢'). Then by (C3), PI is intended by a l and P2 is intended by a2. Thus the agents al and c~2 have a joint-intention with respect to (PI II/:'2), and each of them has a we-intention with respect to his "part". In other words, the constraints on the functions H y and G r ensure that the practical syllogism (i)-(ii*) from Section 3 holds.

(f) The function Str r will be described in Section 6. Our main goal in this section is to define the notion of the agents a l and

a2 jointly bringing about or being the direct agents of P, where P is a joint end the agents have (possibly only inadvertently - although the intentional case is our main concem). As in the single-agent case, we will use here the linguistically more neutral expression "c~ and a2 jointly directly effected P". As we said before, we will be concerned here only with the cases in


which P has the form (P1 * P2), with * being either ;, ,It, or the standard Boolean connectives (see our comments below).

The truth of a sentence P in the model M is defined analogously with the single-agent case. The only new clauses are:

DEFINITION 3.

we p t((P,O) E GT(ctl tt, t : t ) ) ; (i) t ~ Int~ ( , Q) iff 3t' <

(ii) t > Int~ 1P iff 3I' < t(P ¢ HT'(al , t', t, t,));

(iii) t ~ IntJ01nt2p iff Et' < t ( e e ItT((cq, c~;), *', t, t)).

We can now define the notion of the agents czl and o'2 jointly directly effecting P. There are several notions involved here.

Think of several agents jointly carrying a table upstairs. In this case the agents bring about the state of affairs of the table having been upstairs at the same time (in parallel). Or think of 4 agents running 400 meters, in relay so that each of them runs 100 meters. In this case the joint action is done sequentially. There are cases of joint actions in which the Boolean conjunction of the separate results suffice (think of a safety lock which can only be opened by your and my keys used conjunctively). Finally, think of a group of mathematicians who are given the task of solving a problem. Even if only one member of the group is successful, there is still a definite sense in which the agents may be said to have solved the problem jointly.

Formally speaking, the notion of joint action wilt be defined with the help of other notions, namely DO~,~2 ( Pt A 1:' 2 ) (0~t and ~2 directly effected P~ and P2 conjunctively (with Pi as the part of ~)),7 DO~<,~2(pl ; P2) (al and O/2 directly effected P1 and/='2 sequentially), DO~I,~2(P I [] P2) (oq and a2 directly effected P1 and P2 in parallel), and DO~l,a,2(P I V P2) (o,1 and O~ 2 directly effected P1 and P2 disjunctively), s

DEFINITION 4.

(i) DOc~,c~2(P1 [I P2) {=1" (,DOc~IP1 [[ DOa2P2);

(ii) DOo,~,~,2(P1 ; P2 ) ~ (DOg,P1 ; DO~eP2);

(iii) DOc~>~2(P1 A 1)2) <g> (DO~.,PI k DOQ,2P2);

(iv) DOo.~,~.2(P1 V P2) ¢:> (DOo,~P1 V DO~2P2);

Several comments are needed at this point. (i) It should be clear by now how the concepts introduced here can be

extended to cover the general case in which n is arbitrary. For instance, (iii) of Definition 4 would be replaced by


tl

DOel,a2(P 1 A P2) -1 DOaI,o2(Pll I P=) -~ D O a l , e 2 ( P G P 2 )

Pl P2

-~ D O e l P 1 -~ D O o 2 P 1

"~ P1 P2

t2 I -1 Pl "~ P2

Fig. 4.

T,im.

i2

(1) DO~l,~2,...,~,(P1 A/:'2 A- . . A Pr~) ¢ > Ai~=1 DO~Pi .

(ii) DO~I,~2(P 1 ]l P2), DO~l,~2(P1 ;P2), DO~,~z(P1 A P2), and DOcq,~2(D1 V P2) are the simplest type of actions involving two agents in which P1 and P2 are effected in parallel, sequentially, conjunctively, and disjunctively, respectively. They are not yet instances of intentional joint actions, because the "we-intention" element is missing (cf. below).

(iii) DOcq,c~2(P1 N P2), DOcq,c,2(Pl;-P2), DO~,c~2(PI A /°2), and DOcq,c~2(rl V /192) are all defined notions. If we spell out the definitions, we get, e.g.,

(2) t ~ DOcq,cc2(P1 H 1)2) iffqt l( t > tl A t 1 N (-~P1 A -~P2) A Vt2(tl < t2 <_ t --+ t2~ (P1 AP2)) A Pi C Dec(~i , t l ) ) .

(iv) DO~1,~,2(P1 II P2) and DO~I,~2(P1 ;P2) are stronger notions than DO~,~2(P~ A P2). In other words, it always holds that

(3) DOc¢1,c~2( p l [I P2) --+ DO~l,~2(P1 A 1°2)

(4) DO~,~2(P1 ;-P2) --+ DO~,,~2(Pt A P2).

However, it is easy to see that the converse is not true. A counterexample to both of them is given in Figure 4 below.

Let P1,/:'2 be atomic sentences, and consider three possible states t:, tl, and ~, such that t: is immediately earlier than tl which is immediately earlier than t. In addition, let t (Pl) = t(P2) = tl(P2) =


(6)

(7)

(8)

are valid.

True, tl(1,1) • t2(P1) : t2(F2) --- False, P1 C Oec(c~i,t2), 1)2 E Dec(o<2,t2), P1 E Dec(oq, t l ) , P2 C Dec(ct2, tl). Then it is easy to check that t ~ DO~,~2(P 1 A 1,2), but t ~ -~DO~I,~2(& [[ P2) and t > -~DO~,,c~2(P1 ;P2).

(v) Our remarks following Theorem 1 show that

(5) DOo.~(1,1 ,', 1,2) ~ DO~.t,~2(P1 A Pz)

is not valid. However, Theorem 1 (vi), (vii), and (viii) show that

D O m ( P 1 A P2) -+ DOcq,c<(1"1 V !)2)

DQi(1,1 II 1,2)+ II P2) DO~,(P1 V P2) --+ DO~,I,~q(P1 V P2)

(vi) One of the basic elements in the analysis of the concept of joint action is the Constitutive Condition described in the previous section. In connection with this condition we spoke about the "parts" P~ . . . . , P,, "making up" P. In the formal model given in this section, we limited ourselves only to the case in which P has one of the forms (1,1 tl P2), (P1 ;P2), (1'1 A P2), and (P~ V P2), the parts being obviously P1 and P2.

We may now consider whether the part-whole relation in question is exhausted by these four cases. That is, we are considering the following

Conjecture: All joint actions (viz. joint actions in our achievement- sense with inbuilt results) can be characterized by means of the elementary notions of parallelity (1[), sequentiality (;), conjunction (A), or disjunction (v).

It is not an outlandish claim that our conjecture is tenable, given some qualifications. First, there is the problem of describing the actions in the right way. We may here speak of the problem of conceptual generation. Thus, for instance, in the joint action of asking and answering obviously the questioner's question and the answerer's answer must be described in the right way if we are to speak of an intentionally performed sequential joint action. We are accordingly assuming that our language for describing joint and other states is conceptually adequate in this sense; and we will also assume that cases of representation in the sense of an agent representing the participants in the context of a joint action such as jointly building a house can be treated as cases of conceptual generation. The second qualification that is needed for making our conjecture defensible is that we must in some cases explicitly make Nature involved. Nature is involved in our tree account in two ways. First, we have assumed that Nature cooperates in


upholding the (deterministic) relationships that our whole setup assumes (basically that there are such and such choice possibilities leading to such and such outcomes). Secondly, in cases like building a house jointly or hunting a deer jointly, it may be necessary to give Nature the role of an agent with choice points in the tree to account for jointly causally generated states. But given these two qualifications, it seems that our conjecture is tenable or is acceptable relative to our present knowledge.

The formal analysis of joint action given above is incomplete in the sense that the intentional element is missing. If one is willing to assume that joint action in its full-blown sense requires explicit or implicit agreement- making (roughly, intentional acceptance of a joint plan), we can say that every joint action is intentional at least to some extent because agreement- making is necessarily intentional. 9 Accordingly, the participants cannot be mistaken about their performing something together, although they may be mistaken about some details of their joint actions.

We can now define the notion of intentional joint action by adding a clause for intentions to each of the clauses of Definition 4. What we get is the definition of parallel, sequential, conjunctive, and disjunctive intentional joint action:

DEFINITION 5.

ncg°i" (1"1 (i) --~1,~2 (ii) ~ i o i n t I p

I--/L~,~ 1,~2 k t (iii) oint ' DC0~,,~2(P1 (iv) r~c~ioint r D

IJ~-Y~ 1 ~o~2 ~ i I

Recall that the agents

joint 1[ P2 ) ~ (DOcq ,c~2(P l l] /02.) A Int~ 1 c~2(P1 1[ f )2 ) ) ,

^ ,_,joint r p " P 2 ) ) ; ;P2) ¢ > (DO~1,~2(P1 ;P2) / , mt~,,~=~, 1,

A P2) © (DO~,~2(P1 A P2) A Int~.<j°int, c~a(,P1 A P2)); ^ i_tjoint r ~ v *'2) ¢ > (DOo , (P1 v P2),, 1 v P2)).

al and Og 2 having a joint intention that (Pl * 1°2) (i.e., "-tj°int f o * m~1,~ak~ 1 /:'2)) implies that each of them has a we-intention with respect to his own part (i.e., IntW~((P1 * i°2) , P1) and IntW ((el * p~), P2)). Thus e.g., the agents c~1 and era directly effecting jointly and in parallel P1 and P2 implies that each of them had a we-intention with respect to his part.

The clauses of Definition 5 are simplified in the sense that they do not really require that the agents act jointly because of their joint action. This is basically the well-known problem of the "wayward" causal chains, which we will not attempt to solve in this paper. In a more refined account a relevant account of "purposive causation" (as called in Tuomela 1977, 1984) should be employed.

Actually the above analysis of the notion of intentional joint action is still incomplete in one respect, viz. the belief element is missing. In a complete analysis of joint action, we should add on the right side of each


of the clauses of Definition 5 a conjunct expressing the fact that the agents or1 and c~2 have a mutual belief that each of them has a we-intention to bring about his part. (Mutual belief can be analyzed, roughly, as an iterated shared be l i e f - see Tuomela, 1995, Chapter 1.)

We have argued in this section that our approach applies to all joint achievement-involving actions. In particular, we have shown in detail how it applies to purely conjunctive, purely disjunctive, purely parallel and sequential actions.l°

5. INTRODUCTION TO STIT

The activity of somebody's seeing to it that P is the case is, at least in our view, necessarily intentional. Namely, in all cases, if you see to it that P , then you intentionally see to it that P. Accordingly, if I see to it that the door is closed, then this action is necessarily intentional. Seeing to it that the door is closed expresses intentional control over the state of affairs of the door's being closed and requires success (viz., the agent has not seen to it that the door is closed unless it is closed). This, in turn, involves the idea of the agent having and carrying out a strategy through which he, if necessary, not only brings about P but also blocks (prevents) all the possible interferences by other agents endeavoring to bring about -~P. In our example, this means that the agent c~ would close the door if it is open, or he would omit to open the door if it is closed, or, in case some other agent would open the door, the agent in question would interfere and close the door. Let us once again emphasize that in all the above cases the agent can be roughly said to exercise intentional control over the situation.

If we say that c~ sees to it that P, then we attribute intentional activity to o~ in the general sense of his exercising intentional control over P. If c~ sees to it that P he must be taken to intend that P , and c~ intends that P only if c~ is committed by his actions to the obtaining of P. Thus c~ is committed to do whatever he must (in his view) do in order to obtain P. More precisely, if c~ intends that P there will be actions of his that are required, in his view, of him to be performed which are conducive to the obtaining of P (and which he must be capable, in his view, of performing with some nonnegligible probability).

We consider that the notion of seeing to it (Stit) that a state of affairs, say P, obtains concerns achievement-related intentional activity in a highly general sense] 1 Broadly speaking, we think that the notion of Stir is analyzable in terms of effective intention involving commitment and the actions carrying out that intention according to an intention-containing plan. From aformal point of view, these ideas will be implemented in our


model by requiring that the agent in question has a "strategy over P" and intentionally "uses" it. If an agent intends P, then, he ought to believe he can bring about P, at least with some probability. In standard cases he ought to have the belief that he has a successful strategy - or at least a probable successful strategy - over P. In these cases - which we connect with Stit - a normally rational agent has no problems with satisfying this doxastic requirement. There are, however, nonstandard cases where an agent intends something which is not likely to come about due to his acting. For instance, an athlete may intend to run a four-minute mile, believing that he probably will not succeed (while also believing that it is not completely impossible for him to do it). In this latter kind of case, there need be no successful strategy, nor does the agent believe there is one. The action - if it is performed - is not a Stit but only a doing (in the sense of DO*).

What was said above may explain in part why there has been so much discussion about the doxastic conditions related to intention: In the case of Stit, a clear-cut doxastic condition is available, viz., that the agent must (and can be taken to) believe that he can or probably can bring about P. But when a doing or a bringing about which falls short of Stir is at stake, the only satisfiable doxastic condition is that achieving P not be impossible for the agent in the situation at hand.

6. JOINT STIT

Armed with the above philosophical and conceptual points, we are ready to discuss Stit from a technical point of view. We shall start our analysis with joint Stit; that is, we shall clarify what the agents' Oel,..., c~,~ jointly seeing to it that R amounts to. As before, we shall restrict our analysis to the case in which n = 2.

To arrive at an adequate definition of joint StitR, we shall require that a t and c~2 jointly intend R, and believe that they have a strategy regarding /~, and successfully use this strategy. However, as in the previous cases, we shall not be concerned here with the belief element. Instead, we shall concentrate on implementing the notion of strategy in our agent-trees.

Let us fix a tree T = (t47, <, Ttime ). We aim at defining the truth in a state t of the sentence 'the agents cq and oe 2 jointly saw to it that R' .

We enrich the formal language L with two operator symbols Stit and Stit j°int. The former acts on agent constants and sentences, and the latter on pairs of agent constants and sentences. As before, we write Stit~P instead of Stit(o~, P), and ,Rtit j°int P instead of StitJ°int(c~, a2, P).

. . . . O~ 1 ~O~ 2


The models of L will be the same as those defined for joint DO. The crucial element in this case is the function Str r which assigns to every individual (or pair of individuals) and states t, t~(t < t') a set consisting of pairs of strategies and sentences of L. The idea here is that, whenever (P, f ) C StrT(ai, t, t '), the agent oz~ has a strategy f with respect to P between ~ and t ~. This interpretation justifies the following constraints on Strr:

(C6) (P, f ) ~ StrT(ai,t,t ') ~ Vtl(tl E dom(f ) ~ P E DecZ(a~i, f( t l)))

(C7) (P, f ) E s t r r ( a i , t, t ') ~ Vtl(~ 1 ~ val(f) (? it, =~ tl > P)

(C8) (P, f ) E StrT(al,~2, t , t ') ~ Vii(t1 E dom(f ) ~ ( P E DecT(al , f ( t l)) V P E Decr(a2, f( t l)))

(C9) (P, f ) E StrT(al , o~2, t, t') ~ gtl (tl C val(f) • it, ~ tl ~ P) ( e l0 ) (P, f ) E StrT(al , a2, t, t') ~ (P, f ) E St r r (o i , t, t')

(C11) (P, f ) E StrT(ai , a l , t,*') and t and t" belong to the same instant ~ (P, f ) C StrT(al , t, t").

Constraint (C6) says that the strategy f of o~i with respect to P yields always a later state in the relevant subtree which is a decision point of o~ i with respect to P. This is a natural thing to require, given the fact that a strategy for ai associated with P is supposed to be a method or rule telling o~i what to do next in order to "protect" p.12 Constraint (C7) expresses the fact that a strategy ~ has between t and t r with respect to P leads him always to a state belonging to the instant it, at which P holds. Constraint (C9) expresses the same condition for pairs of agents. Constraint (C8) expresses the interaction of the agents o~l and c~2 in a joint strategy. What the joint strategy of a l and a2 does in this case is to lead to a next state in the relevant subtree which is a decision point (with respect to P) for either one. Notice that by (C3), such a state cannot be a decision point for both of them. Figure 5 below exhibits an example of a joint strategy of al and e2 which does not reduce to an individual strategy for either one of them.

Assume that (P, f ) E StrT(al,a2, tl,tS), where f is defined as: f ( t l ) = t3, f( t2) = t4, f ( t 3 ) = t6, and in addition P C Decr(o~2, tl), P E Dec r ( a l , t 2 ) , P E DecT(a2, t3). Then (P, f ) ~ St rT(~ l , t l , t s ) , and (P, f ) ¢~ StrT(c~2, tl, ts), because in both cases (C6) is violated. Con- straints (C10) and (C11) are self-explanatory.

We are now ready to define Stit~P and StitJ°in~2 P:


t 4

P

I t21

I t.

I t~

Fig. 5.

t3J

t,J ~ P

DEFINITION 6.

(i) t ~ Sti t~P iff ~t 1 <( t3f[(P, f) ~ St r ; (~ , t l , t) A t ¢ va l ( f ) A Int~P];

(ii) t ~ Stir j°int P iff ~tl < t3f[(P, f) E StrT(oq, ~2, tl t) A t - - - -Cel ~O~ 2 -

I joint p . v a l ( / ) A ntc~l,C~ 2 ]

In other words, 'S t i t~P ' is true at t, if there is a state tl earlier than t such that (i) c~ has a strategy between '1 and t with respect to P, (ii) this strategy leads to t, and (iii) ct intends P .

In the general case, clause (ii) of Definition 6 would be replaced by

(ii)* t ~ StitJ°{n~.,~ P iff ~t 1 < t3f[(P, f) ¢ strr(a, , . . . ,a~,t l , t ) A t ¢ va l ( f ) A IntJ°{n.t..,~, f ] .

The notion of St i t~P as defined here is a strong one: the agents c~ jointly saw to it that P because of their joint intention to achieve P. As noted earlier, in order for the agents rationally to have that joint intention they must believe that they jointly can achieve P (at least with some nonnegligible probability).

If we return to Figure 5, and make the additional assumption that IntJ°{nt 2 P holds at both t4 and t6, we get ta > __Stir j°int_oe 1 ,c~2 P and t6 ~ Stlt~l- joint, c¢2 p .

• joint Notice that the truth of Stlt~l ,a2P at t4 and t6 does not depend on whether P holds or not at t l , t2 and t3.

Consider now Figure 6 below, which is exactly like Figure 5, except that P is true at the states t l , t2, and t3. Let all the other assumptions be

• joint the same. Then in this case it holds that: t4 > ~.~t aRt:dointl,o~2/-,D t6 ~ Stlt~l,~op '

JOINT ACTION AND GROUP ACT[ON MADE PRECISE 339

it, tsl t,l i P i Pi L PI

] 1

J t l

Fig. 6.

i3

i2

but t4 > -,DO< ,a2 D and t 6 ~ -nDOa. l , a2P . Thus Figure 6 provides us with an example of a joint Stit which is not reducible to a joint DO.

7~ GROUP ACTION

' jointly directly effecting P, The notions of the agents Ctl,...,o~r~ (DOJ°int.,~ P, in its different forms), and jointly seeing to it that P

(Stit~in.t..,~ P) have been regarded here as the most basic types of social actions. Other related notions, for instance the notion of group action, viz. an action performed by a social group can be analyzed in terms of them. (A social group obviously amounts to more than the collection of the agents it contains - cf. Tuomela, 1995, Chapter 4.) Consider some examples. If one nation declares war against another nation, this may take place through appropriate actions by the members of its government, its parliament, or by its president. Or consider a hockey team scoring. Some player, or perhaps players, did the scoring. Let us say that it was the "operative" members of the team who did it. The team's scoring was constituted by their actions. The point to be made here is that the notion of joint action involved in these examples can be further explicated in terms of the notions of DoJ°i,nt..,~,~ P

and StitJ°in.t..,,~ P, technically elucidated in this paper. We shall below for- mulate a necessary condition for intentional group action. What we are interested in here is actions performed by a social group as a whole in a


sense normatively binding the group members to the action. This notion is to be distinguished from group (or collective) action in which the persons act not as group members but rather as separate persons having the same goal or following the same social norm, or something like that. This latter kind of collective action may be termed "coaction". In the case of group- binding action, the action must be based on a joint plan, produced by some kind of "group-decision-making" system or mechanism for creating shared group-intentions. This system has been called an authority systemJ 3

Given this, we can present the following thesis giving a necessary condition for group action (here P is taken to be the result event of a group action): I4

(GAD A group G performed an action P intentionally in certain "fight" social and normative circumstances C only if in C there were operative agents ~1, . . . , O~r~ of G, such that cq , . . . , an, when performing their social tasks in their respective social positions in G and due to their exercising the relevant authority system of G, intentionally jointly brought about P.

Analogously, we get for a group action P, which need not be fully intentional:

(GA) A group G performed the action P in the "fight" social and normative circumstances C only if in C, there were operative agents a l , - .., o~n of G, such that a 1,. . . , as , when performing their social tasks in G and due to their exercising the relevant authority system of G, jointly brought about P.

The "right" social and normative conditions can be characterized by the set of rules constitutively and regulatively G and its activities. It is not, however, necessary here to discuss them.

While (GAI) concerns only intentional group action, (GA) covers all kinds of group actions proper, viz., both intentional and unintentional actions. We can see that the most central element in them is that a group action P is brought about by some (if not all) group members' jointly bringing about P. We will here reformulate the above analyses in terms of the notion of Stit j°int and thus make use of all the technical rigor that our approach involves. Let us see what we get. We replace (GAI) by a new analysis where the analysandum now is Sti taP, viz., the group G saw to it that P, recalling that Stit is necessarily intentional and that P can be taken to be the result event (type) of a group action P:


(GAI) 'S t i t cP ' is true, viz., G saw to it that P, in the social and normative circumstances C, only if in C there were operative agents Ctl,..., ce~ of G, such that ~1, . . . , c~.~, when performing their social tasks in their respective social positions in G and due to their exercising the relevant authority system of G, jointly saw to it that P in the following sense: them was a state Q such that the operative agents saw to it that Q (in the technical sense of making true Stit!°in.t.,c~,, Q), and their seeing to it that Q generated, and was believed and purported by the operative members to generate, the state P.

In the case of (GA) we face the problem of whether one can unintentionally or mistakenly see to it that something is the case. As argued, one cannot mistakenly see to it that P (not to speak of more strongly unintentional activity). But there seems to be a closely related notion which can be used here: one can have a strategy regarding P and occasionally use it unintentionally. So we might call this Stir# and give it the following analysis, which accordingly allows for mistaken and other unintentional group actions:

(GA) 'S t i t#cP ' is true, viz., G saw# to it that P, in the social and normative circumstances C only if in C, there were operative agents ~1, . . . , ~,r~ of G, such that o~1,..., c~,~, when engaged in performing their social tasks in G and due to their exercising the relevant authority system of G, jointly saw# to it that P (made true Stit~ 1 ..... ~,~.P, where Stit~l ..... ~,~ is the counterpart of Stir j°int obtained from Definition 6 by omitting the conjunct

Int~l ..... ~=P).

It is central in the theory developed by Tuomela that over and above "standard" actions such as the group's building a house, also acceptances are regarded as group actions.15 Thus the group can accept a proposition as its view or its goal. Suppose that a group G accepts that P (e.g., P = 'the earth is flat'). Can we analyze acceptance in terms of Stit? In the case under discussion we are dealing with acceptance in the sense that the operative members of G have mutually agreed, and formed the joint intention, to accept P. Thus we can speak of the operative members' we-intention to accept P - and indeed of G's intention to accept/:,.t6 So, when G accepts P, G can be said to see to it that it puts P in its "belief-box" or "view- acceptance box", metaphorically speaking. There is C's intention (viz., its operative members' shared we-intention) to accept P and this we-intention

342 GABRIEL SANDU AND RAIMO T U O M E L A

is realized, resulting in G's acceptance of P (or its state of having accepted P or having put P in its belief-box).

While having a strategy is a strong notion in general, it surely is appropriate, in the case of acceptance, for acceptance involves control and success in the required sense. Thus considering intentional acceptance (the only relevant case when speaking of beliefs), if a single or collective agent intentionally accepts as his belief that the earth is flat, he must be in full control of his acceptance and must be successful; and that makes the agent see to it that he puts P in his belief-box. Thus our conclusion is that if a group action P is the action of accepting a view (or goal or a proposition of any kind), v~e are dealing with action in the sense of Stit (rather than a weaker senseJsuch as DO or DO*).

Above we have spoken about group action proper, viz. group action involving (at least normatively involving) the whole group. Group action in a weaker sense, or parallel action by the group members toward the same goat seems never to be characterizable in terms of the notion of S - ' d o i n t 17

[l[oe 1 ,...,o~ n •

As Stit and DO do not always go together, there being cases of one holding true without the other holding true, we might want to use an even more general notion of action to be able to cover both group action proper and weaker kinds of collective action. What could it be? Considering a single or collective agent A, we may technically speak of activity (ACT) in something like the following sense: ACTAR =dr StitAR V USAR V DOAR V (~P)DOA(P, R) V BAAR. Here US means successfully using one's strategy. This covers cases where the (single or collective) agent intentionally or unintentionally does something which he can do in the sense of having a successful strategy. The operator DOA(P, R) means bringing about R by means of P. BA means bringing about - note that there can be bringings about which are not even indirect doings and which do not involve the agent's having a successful strategy. Also recall that every doing - be it direct or indirect - is a bringing about. In the case of group action based on an exercise of the group's authority system for action - viz., the case dealt with by (GA), we can in fact use the notion of ACT in the analysandum, viz. ACTGR. In the analysans of ACT we would use the aforementioned disjunction in its definition, except that the corresponding jointness-notions appear there. Thus, for instance, ' StitA R' wilt be replaced by Sti0 °int and similarly for the other disjuncts. The

disjunction can be called ACTJ°intR. Analogously, in the case of intentional group action ACTOr R =df Stita/~

V D O ~ R V (~P)DO~(P , R) V BA*AR, where BA* represents intentional bringing about (and the other symbols are used with their previ-


ous meanings). Analogously, we can also derive the notion to be called ACy*joint/~.

The above general notions can be used on some occasions when analyzing group concepts in terms of the joint activities of the group's operative members. One such case is the very notion of collective activity related to a group G, meaning all of its activities and not only its group-binding actions (analyzed above). Whether or not such a broadest possible notion of group activity will be useful in itself, obviously it is important to have available analytic concepts (appearing as disjuncts above) to deal with all the cases one may encounter when analyzing and discussing collective activities.

8. CONCLUDING REMARKS

Our main task in this paper has been to make precise some central concepts related to joint action and group action, action here understood to involve achievement. To accomplish this, we have employed a time-tree semantics to analyze two central kinds of joint action concepts. These were termed direct joint performance (or doing) and joint seeing to it that something is the case. The former was logically characterized by an operator D O j°int and the latter - and essentially intentional notion - by Stir j°int. In terms of these action concepts we analyzed actions performed by social groups, making more precise a part of the theory of group action developed by Tuomela. I s

Our approach employs the mentioned action concepts and the notion of "we-intention ' , shared we-intentions - when there is a mutual belief about them - amounting to joint intentions. However, we did not attempt to give a satisfactory logic of we-intentions in this paper, partly because we could accomplish our main task satisfactorily without doing that. Furthermore, we did not explicitly use a notion of belief in our logical treatment. This makes our present approach somewhat simplified, but does not seriously diminish its value.

NOTES

* We wish to thank Professor Risto Hilpinen for pointing out some problems due to a constraint adopted in a previous version of our paper. 1 For a discussion, see Tuomela (1984) and (1995). This paper concentrates on the logic of joint action. The conceptual and philosophical features related to joint action are treated in the mentioned works. With the exception of Levesque et al. (1990) (see footnote 8 below) and Belnap and Perloff (forthcoming), we are not aware of other logical works analyzing the central elements of joint action.


2 See Tuomela and Sandu (1994). 3 With some slight changes, we follow here the article of Tuomela and Sandu cited in note 2. 4 Following Tuomela (1977), and many others, we take the notion of intention as primitive. A similar treatment may be found in Rao and Georgeff (1991) and also in Singh (manuscript). For a different approach, see Cohen and Levesque (1990). As far as the concerns of this paper go, it does not matter much, if at all, which particular account of the notion of intention is accepted. 5 See chapter 6 of Tuomela (1984) for action generation. 6 For a more detailed analysis of these matters, see Tuomela (1991), and Tuomela (1993a). 7 With the exception of Singh cited in note 4 above, we are not aware of similar attempts in the literature to analyze parallel actions. In the standard approaches, it is usually pointed out from the start that the case of several actions being performed in parallel is not considered. Such is the case, for instance, with Levesque, Cohen and Nunes (1990).

The model introduced here differs from Singh (manuscript) in at least one respect: He takes a possible world to be a history" of times branching in the future with basic actions being assigned intervals over histories. In our approach the notion of possible world is more basic and a history is taken to be a sequence of possibte worlds linearly ordered in time. 8 See Tuomela (1984) cited in note 5. 9 See Tuomela (1995) cited in note 1. lo Given our approach, one can define utilities concerning the joint outcomes, viz., leaves, in the trees of our framework for each participating individual. (In principle one can also define probabilities for the branches to obtain structures reminiscent of the extensive forms of games in the sense of game theory.) With utilities so defined one can investigate features of cooperative and noncooperative joint actions in terms of correlations of preferences (utilities) as sketched in Tuomela (1993b) (note 6), and Tuomela, (manuscript). 11 In this respect, our notion of Stit differs essentially from that of Belnap and Perloff (1988). 12 The notion of strategy employed here is to a large extent game-theoretically motivated (i.e., a future-oriented rule telling the agents what to do "come what may") and is not derived from regular programs in dynamic logic, as in Singh (manuscript) (note 4). 13 See Tuomela (1995) cited in note 1. 14 The full analyses (GAI) and (GA), giving also sufficient conditions, may be found in Tuomela (1995), Chapter 5. In Chapter 2 of the book various kinds of joint action are discussed. Both our D(Y~ ~t and Sti t ; int represent joint action in a full sense, viz., joint action performed because of a joint intention to perform that action. According to the agreement-view of joint action advocated in the aforementioned book, all planned and jointly intended actions involving joint obligations for the participants to carry out the joint action in question can be said to be based on explicit or implicit agreement to perform it. 15 The theory is presented in Tuomela (1995). 16 See Tuomela (1995), Chapter 6. 17 This is also the case with collective action called "coaction" in Tuomela (1995). t8 See note 15 above.

REFERENCES

Belnap, N. and M. Perloff: 1988, 'Seeing to It That: a Canonical Form for Agentives', Theoria 54, 175-99.


Cohen, R R. and H. J. Levesque: 1990, 'Intention is Choice with Commitment', Artificial Intelligence 42, 213-61.

Levesque, H., R Cohen, and J. Nunes: 1990, 'On Acting Together', in Proceedings of Eight National Conference on Artificial Intelligence. vol L The MIT Press, MA, pp. 94-99.

Rao, A. and M. Georgeff: 1991, 'Modeling Rational Agents within a BDI-Architecture', in Allen, J. et al. (eds.), Proceedings of the Secondlnternational Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufmann Publishers, Sann Mateo, CA, pp. 473-484.

Sadek, M. D.: t 992, 'A Study in the Logic of Intention', in Nebe!, B. et al. (eds.), Proceed- ings of the Third International Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufmann Publishers, San Mateo, ppo 462-469.

Singh, M. E: (manuscript), 'Intentions for Muttiagent Systems'. Tuomela0 R.: 1977, Human Action andlts Explanation, Reidel, Dordrecht and Boston. Tuomela, R.: I984, A Theory ofSocialAction, Synthese Library, Reidel Publishing Com-

pany, Dordrecht and Boston. Tuometa, R.: 1991, 'We Will Do It: An Analysis of Group-Intentions', Philosophy and

PhenomenologicalResearch LI, 249-77. Tuomela, R.: 1993a, 'What Are Joint Intentions?', in Casati, R. and White, G. (eds.), Phi-

losophy and the Cognitive Sciences, Austrian Ludwig Wittgenstein Society, Kirchberg, pp. 543-547.

Tuomela, R.: 1993b, 'What is Cooperation?', Erkenntnis 38, 87-101. Tuomela, R.: 1995, The Importance of Us: A Philosophical Study of Basic Social Notions,

Stanford University Press, Stanford. Tuomela, R.: (manuscript), Cooperation: A Philosophical Study. Tuomela, R. and G. Sandu: 1994, 'Action as Seeing to It that Something Is the Case', in R

Humphreys (ed.), Patrick Suppes, Scientific Philosopher, Kluwer Academic Publishers, Dordrecht and Boston, pp. 193-221.

Department of Philosophy University of Helsinki Helsinki Finland

joint action and group action made precise

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