Some Forms of Collectively Bringing About or ‘Seeing to it that’

One of the best known approaches to the logic of agency are the ‘stit’ (‘seeing to it that’) logics. Often, it is not the actions of an individual agent that bring about a certain outcome but the joint actions of a set of agents, collectively. Collective agency has received comparatively little attention in ‘stit’. The paper maps out several different forms, several different senses in which a particular set of agents, collectively, can be said to bring about a certain outcome, and examines how these forms can be expressed in ‘stit’ and stit-like logics. The outcome that is brought about may be unintentional, and perhaps even accidental; the account deliberately ignores aspects such as joint intention, communication between agents, awareness of other agents’ intentions and capabilities, even the awareness of another agent’s existence. The aim is to investigate what can be said about collective agency when all such considerations are ignored, besides mere consequences of joint actions. The account will be related to the ‘strictly stit’ of Belnap and Perloff (Annals of Mathematics and Artificial Intelligence 9(1–2), 25–48 1993) and their suggestions concerning ‘inessential members’ and ‘mere bystanders’. We will adjust some of those conjectures and distinguish further between ‘potentially contributing bystanders’ and ‘impotent bystanders’.

branching-time structure of some kind. The best known examples are the 'stit' ('seeing to it that') logics associated with Belnap and colleagues (see e.g. [2,3,5,16,17]). There are many variants, differing primarily in their treatment of temporal features. There are also connections to formalisms in game theory and theoretical computer science: the relationship between stit and Coalition Logic [19] for example is well established [9,10], and there is a range of stit and stit-like logics interpreted on semantical structures such as transition systems [21,22], concurrent game structures [6], models of distributed processes [8], and others. A recent paper by Ciuni and Lorini [12] compares various semantics for a temporal stit logic and also provides a brief classification and references to other forms of temporal stit.
Often, it is not the actions of an individual agent that bring about a certain outcome but the joint actions of a group of agents. Although stit logics typically provide operators for group as well as individual agency, to allow expressions of the form 'group (set of agents) G sees to it that ϕ', group agency in stit has received comparatively little attention, beyond the observation that group stit has the property of superadditivity: if a group (set of agents) G sees to it that ϕ then every superset of G also sees to it that ϕ. There are exceptions. 1 Herzig and Schwarzentruber [15] investigate properties of group agency in an atemporal stit logic (the 'deliberative stit') including satisfiability and axiomatisability. Belnap and Perloff [4] discuss a form of stit they call 'strictly stit'and make a number of suggestions for further investigation. Belnap and Perloff's suggestions will be examined in the second half of the paper.
The aim of this paper it to map out several different forms, several different senses in which the joint actions of a particular set of agents can be said to bring about a certain outcome, and to examine how these forms can be expressed in stit logics. The outcome that is brought about may be unintentional, and perhaps even accidental; the account deliberately ignores aspects of joint action such as joint intention, communication between agents, awareness of other agents' intentions and capabilities, even the awareness of another agent's existence. The aim is to investigate what can be said about collective agency when all such considerations are ignored, besides mere consequences of joint action.
Suppose, for example, that two agents a and b are positioned at either end of a table. Each can lift or lower its end. On the table stands a vase. If one agent lifts and the other does not, or if one agent lowers its end and the other does not, then the table tilts, and if it tilts, then the vase falls and breaks. Suppose that one agent lifts and the other lowers its end and the vase falls and breaks. Which of the two agents, if either, 'brings about', or is responsible for, the breaking of the vase? It seems wrong to pick on one or other of them; their actions collectively brought it about. If we add another agent into the picture, an agent c whose actions cannot affect the table or the vase, or interfere in any way with the lifting and lowering by a and b, then it also seems right to say that when a lifts and b lowers, it is the set {a, b} of agents that brings about the breaking of the vase and not the set of agents {a, b, c}. c had nothing to do with it. In this example the set {a, b} of contributing agents is unique; that need not always be so.
In examples like this it is possible that the agents are coordinating their actions. Perhaps it is their intention to dislodge the vase and break it. But it is also possible that they are acting quite independently; one lifts, the other lowers, for whatever reason (or perhaps for no reason at all), and the tilting of the table and the falling of the vase are incidental. Even then it is meaningful to say that agents a and b collectively 'brought it about' that the table tilted and the vase fell. The phrase 'saw to it that' is less appropriate, inasmuch as it hints at some kind of joint intention or purpose. It is important to distinguish between collective action (deliberative or not) and acting collectively with joint/shared intention. This paper considers only the first.
The account will be developed in a general formal framework that is common to many forms of stit and stit-like logics. Differences in specific kinds of stit are often due to their treatment of temporal features, and those features play no role in this paper. The development will then be related to the 'strictly stit' of Belnap and Perloff [4] and their further suggestions concerning the distinction between what they call 'inessential members' and 'mere bystanders'. We will adjust some of their conjectures and distinguish further between what we call 'potentially contributing bystanders' and 'impotent bystanders'.
Of course the stit framework is not the only way to talk about actions or the responsibility of an agent or group of agents for a certain outcome. This paper is limited to investigating how forms of group agency can be expressed within stit.
One final caveat: as in [4] we will not discuss in this paper the treatment of cases where the outcome of collective action is determined by the sequential actions of the members. We discuss only simultaneous joint actions, or actions that can be thought of as simultaneous. Section 2 of the paper introduces the formal framework and some of the terminology encountered in the stit literature. Section 3 investigates how one can express in stit that G is a minimal set of agents whose actions jointly bring about a certain outcome and some of the properties of that construction. Section 4 introduces another sense of group agency-the set of agents who are the 'contributors' to the bringing about of the outcome in the sense that they are the members of those minimal sets. The second part of Section 4 relates that notion to Belnap and Perloff's distinction between 'inessential members' and 'mere bystanders'. Section 5 distinguishes further between 'potentially contributing bystanders' and 'impotent bystanders'.

Syntax and Semantics
Ag is a non-empty, finite set of agents.
The language extends classical propositional logic with operators , [x] and [G] for every agent x in Ag and every non-empty subset G of Ag.
We have the usual truth-functional abbreviations. ♦, x and G are the respective duals.
Semantics In abstract terms, models are relational structures of the form W is a non-empty set of possible worlds, ∼ and every ∼ x are equivalence relations on W , and V is an evaluation function mapping propositional atoms to subsets of W . We further require that, for every x ∈ Ag: These structures are similar to the abstract models of the deliberative stit (dstit) discussed in [1] except that the models presented there have a simpler form (the relation ∼ is omitted) because they incorporate an extra, very strong 'independence of agents' assumption characteristic of stit. They can be presented equivalently as models of the form Eq. 1 with additional (rather strong) restrictions on the ∼ x relations over and above Eq. 2.
For a less abstract reading, W can be seen as the set of transitions in a transition system. For any transitions τ, τ in W , τ ∼ τ represents that τ and τ are transitions from the same initial state, and τ ∼ x τ that τ and τ are transitions from the same initial state (∼ x ⊆ ∼) in which agent x performs the same action in τ as it does in τ . This is the reading adopted in [21,22] for example.
For readers familiar with stit logics, and the deliberative stit (dstit) in particular, W can be thought of as the set of moment-history pairs m/ h. τ ∼ τ then represents two moment-history pairs τ = m/ h and τ = m/ h through the same moment m. The equivalence relations ∼ x determine each agent x's choice function.
The formal exposition that follows does not depend on any particular concrete reading however. We will tend to refer to the elements of W simply as 'situations'.
The truth conditions are We write |= ϕ to denote that ϕ is valid in all models. It is also convenient to employ a functional notation. Let: alt is for 'alternative'. (alt(τ ) and alt x (τ ) are thus the ∼ and ∼ x equivalence classes containing τ , respectively.) Condition Eq. 2 says that, for every x ∈ Ag and every τ ∈ W , The truth conditions can be expressed as: where ϕ M is the 'truth set' of formula ϕ in the model M, i.e., the set of worlds in M in which ϕ is true. In terms of transitions, alt(τ ) is the set of transitions from the same initial state as τ , and alt x (τ ) is the set of transitions from the same initial state as τ in which x performs the same action as it does in τ . alt x (τ ) is the ∼ x -equivalence class to which τ belongs: it can be thought of as the particular action performed by x (deliberatively, intentionally, or possibly unwittingly) in the transition τ . The set of all possible actions that could be performed by x at τ is the partition In terms of dstit-models and moment-history pairs, when τ = m/ h, alt(τ ) can be thought of as the set of histories passing through m, and alt x (τ ) is Choice m x (h), i.e., the action performed by x at moment m in history h, or equivalently, the subset of histories passing through m in which x performs the same action at moment m as it does at moment m in history h. and each [x] are normal modal operators of type S5. The schema is valid for all agents x in Ag. [x] is what some authors (e.g. Horty [16]) call the 'Chellas stit'.

Group Actions
This account generalises naturally to dealing with the joint actions of groups (sets) of agents. Let G be a non-empty subset of Ag. alt x (τ ) represents the action performed by x in τ , which is the subset of alt(τ ) in which x performs the same action as it does in τ . x∈G alt x (τ ) is the subset of alt(τ ) in which every agent in G performs the same action as it does in τ , and is thus a representation of the joint action performed by the group G in τ . (Again, readers familiar with stit logics will recognise the construction.) The truth conditions are: where ϕ could also be defined as shorthand for [∅]ϕ, which simplifies the statement of some formal properties. For example, 'necessity' Eq. 14 would then just be a special case of 'superadditivity' Eq. 15.

Two 'Brings it About' Modalities
Let ∂ x ϕ represent that agent x brings it about, perhaps deliberatively, perhaps unwittingly, that ϕ. ∂ x ϕ is defined as follows: ∂ x ϕ is satisfied at τ in a model M when: (1) (necessity) M, τ |= [x]ϕ, that is, ϕ is true in all alternatives alt x (τ ) of τ in which x acts in the same way as it does in τ , or as we also say, ϕ is necessary for how x acts in τ , or x guarantees that ϕ at τ ; and (2) (counteraction) had x acted differently than it did in τ then ϕ might have been false: there is an alternative τ of τ in which ϕ is false (and hence in which x acts differently than it does in τ ). This is exactly the construction used in the definition of 'deliberative stit' [16,17] The notation ∂ x ϕ is chosen in preference to dstit partly because it is more concise, but more importantly, because dstit, in common with other forms of stit, incorporates a very strong 'independence of agents' assumption. 2 Nothing in this paper depends on those additional 'agent independence' properties. Since dstit models are a subclass of models of form Eq. 1 all logical properties of ∂ x are also properties of [x dstit: · ]. [x] is of type S5, and so, amongst other things: Notice that ∂ x ϕ ∧ ∂ y ϕ is satisfiable even when x = y. Indeed Suppose for example that a and b are both pushing against a spring-loaded door (from the same side) and thereby keeping it shut, though either one of them is strong enough by itself to keep it shut. If k represents that the door is shut, then ∂ a k ∧ ∂ b k is true: it is necessary for what a does that the door remains shut (a is strong enough by itself), and there is an alternative counterfactual situation in which the door fails to remain shut (namely, where neither a nor b pushes). So ∂ a k is true. And likewise for b. The same example (an example of what is often referred to as 'overdetermination') works for dstit: [x dstit: ϕ] ∧ [y dstit: ϕ] for x = y is satisfiable in dstit models. It is possible to define a stronger kind of 'brings it about' modality. Let: ∂ + x ϕ is satisfied in τ in a model M when: (1) (necessity) ϕ is necessary for how x acts in τ , M, τ |= [x]ϕ; and (2) (counteraction) had x acted differently than it did in τ then ϕ might have been false even if all other agents, besides x, had acted in the same way as they did in τ .
We have the following properties: ∂ + x ϕ represents a sense in which it is x and x alone who is responsible for bringing about that ϕ.
Note that: Eq. 14), ∂ + G ϕ can be defined equivalently, for all subsets G of Ag: Operators corresponding to ∂ G where G is a set of agents are found in the literature on stit. The definition takes the form [G stit: ϕ] = def [G]ϕ ∧ ¬ ϕ. Operators corresponding to ∂ + G are not encountered in the literature on stit but they could be defined in analogous fashion, as It is straightforward to re-express the axiomatisation of [G] in terms of ∂ G . We note only the following properties for future reference, for all subsets G of Ag: The following is standard terminology in the stit literature.
¬∂ G ϕ on the other hand is not agentive in G: |= ¬∂ G ϕ → ∂ G ¬∂ G ϕ; the property Eq. 28 is weaker. These are also all features of dstit: they are properties of models of the form Eq. 1 and do not depend on the 'independence of agents' constraint incorporated in dstit models.
For more general forms of stit, ψ is said to be agentive in G when ψ is equivalent to [G stit: ψ]. Belnap and Perloff [4, p39] say that [G stit: [G stit: ϕ]] is equivalent in stit theory to [G stit: ϕ] and that ¬[G stit: ϕ] is not in general agentive in G. From the proof of their Theorem 17 it seems that the analogue of property Eq. 28 also holds for the version of stit employed in that paper. 3 Note that ∂ + 4 For future reference it is not difficult to derive: from which follows 3 In presenting the features of stit in that paper, Belnap and Perloff refer to earlier works, including [2,3] amongst others, but say [4, p26] that for convenience of exposition the postulates are summarised in a 'conceptually different but mathematically equivalent' variant and limited to ideas 'that we think work equally well for either stit or dstit'. It is likely that stit refers to the 'achievement stit' [5] but some details are omitted. It is difficult to check every detail of the formal development without reference to these earlier works. 4 This may seem slightly surprising at first sight but not if we read the [G]ϕ modalities as expressing 'distributed knowledge' in group G. ∂ + x ϕ would then be read as saying that x is the only one who knows that ϕ. If x is the only one who knows that ϕ, we would not infer that x is the only one who knows that x is the only one who knows that ϕ. Or even that x knows that x is the only one who knows that ϕ.
To see this, note first that |= ¬ Herzig and Schwarzentruber [15] identify a number of further properties of group agency operators in dstit (the 'deliberative stit') as well as investigating satisfiability and axiomatisability. Those properties however depend crucially on the 'independence of agents' assumption in dstit and will not be employed in this paper.

Collective agency: Inessential members
and therefore also Belnap and Perloff [4, p40] observe that this is also a feature of the version of stit they employ. Their 'Fact 14' states that, given Clearly neither ∂ G ϕ nor ∂ + G ϕ expresses that it is the set G of agents that is responsible for bringing it about that ϕ, except in a very weak sense indeed.
We can show: So although G and H need not be unique they must have some members in common. Notice the special case:

Minimal sets of agents
∂ G and ∂ + G express a very weak kind of collective agency. If the set G of agents collectively brings it about that ϕ then so, in a very weak sense, does every superset of G; indeed the set Ag also collectively brings it about that ϕ. But if our aim is to ascribe notions of responsibility for ϕ, this is not what we are aiming at. In the tablevase example when a and b collectively tilt the table and break the vase, c has nothing to do with it.
A natural way of looking at it is that the 'necessity' condition [G]ϕ is too weak: G can be 'too big'-it might contain x who contributes nothing to the bringing about of ϕ: [G−{x}]ϕ might also be true for some x ∈ G. Horty [16, footnote 15, p33] refers to such an x as a 'free-rider'. We will follow Belnap and Perloff [4] and refer to such an x as an 'inessential member' of G. In later sections we will distinguish further between 'inessential members' and 'mere bystanders'.
It seems inescapable to look at the subsets of G in an expression [G]ϕ, and insist that for the necessity condition, G should be minimal in some sense. There are several possible ways of expressing this requirement. Let us consider the obvious one first.

Definition 3
For every subset G of Ag: (2) there is an alternative (in which G, collectively, acted differently than it did in τ ) in which ϕ is false. The minimality condition (1 ) could be regarded as part of the 'necessity' condition or as part of the counteraction condition. It makes no difference.
Notice that, as defined above, |= [∅] min ϕ ↔ ϕ, and |= Δ min ∅ ϕ ↔ ⊥. For the case G = ∅, by 'necessity' Eq. 14, we have |= ¬[H ]ϕ → ¬ ϕ for every H ⊆ Ag and so: Although we are primarily interested here in Δ min G ϕ it is still useful to refer to [G] min ϕ from time to time and so we will retain it. For the case G = {x}: |= Δ min {x} ϕ ↔ ∂ x ϕ. We can easily derive: Equation 37 corresponds exactly to Belnap and Perloff's definition of 'strictly stit' in [4]. What we are calling the 'minimality condition' they call the 'no inessential members' condition. We will discuss that definition in more detail in Section 3.2 below. It is very useful to have an alternative characterisation of [G] min ϕ and hence of Δ min G ϕ. We have: and also This alternative formulation explains Belnap and Perloff's choice of the term 'no inessential members'. As they say [4, p27], 'strictly stit', in our notation Δ min G ϕ, expresses that 'the bearers of G, without any outside help, and with the essential input of each of them, guarantee that ϕ'.
What about some of the properties of Δ min G ? -Obviously we have: Indeed, we have for all non-empty subsets G and H of Ag: In other words, if ϕ is brought about by any set of agents, then there exists at least one minimal set H such that Δ min H ϕ.
Belnap and Perloff say that G 'strictly sees' to it that ϕ (at τ ) when every member of G is essential for ∂ G ϕ at τ . Let us confirm. When G is non-empty: Following Belnap and Perloff, let us define nim(G, ϕ) (for 'no inessential members').

Definition 5
For every subset G of Ag: Belnap and Perloff's definition of nim(G, ϕ) is not expressed as a formula. In our notation their definition [4, p41, Theorem 17] is: That is equivalent since |= ¬∂ ∅ ϕ ↔ . nim(G, ϕ) could also be defined as: It is perhaps worth noting that |= nim({x}, ϕ) ↔ . (The empty conjunction is true. The empty disjunction is false.) Re-stated in terms of nim(G, ϕ), the 'strictly stit' Δ min G ϕ is: Belnap and Perloff now prove (for [G stit: ϕ] in place of ∂ G ϕ) that the following three expressions are equivalent (their Theorem 17 in our notation): They show that the first implies the second, the second implies the third, and the third (trivially) implies the first. The essential point is that the first implies the third, i.e.
This says that Δ min G ϕ is (by Definition 2) 'agentive in G'. Further (their Lemma 20, our notation): Taking these together we have and hence In other words (by Belnap and Perloff's definition), Δ min G ϕ is strictly agentive in G.

Definition 6 An expression ψ is strictly agentive in
It is quite easy to derive these results in the logic by following, more or less, the semantic arguments given by Belnap and Perloff. Here is a slightly more direct derivation of the main result.
is not really necessary as it is already implied by (ii) if we allow the special case is type S5) and |= ¬ ϕ → [G]¬ ϕ ( is type S5 and 'necessity' Eq. 14) so: For part (ii) we can prove the stronger:

A stronger form of collective agency
Can we find a stronger form of collective agency, denoted Δ sole G say, such that to ∂ x ϕ. One possibility, for example, is to try: which is also (as it turns out) G has some reasonable formal properties but does not seem to correspond to any particularly meaningful notion of collective agency.
The counteraction condition employed in the definition of Δ min G ϕ is: The strongest counteraction condition that can be defined of similar form is: This suggests the following construction as a plausible candidate.

Definition 7
For every subset G of Ag: ϕ, that is, the joint actions of G guarantee that ϕ; and (2) (counteraction) for every x ∈ G, there is an alternative in which ϕ is false and in which all other agents Ag−{x}, not only those in G, act in the same way as they do in τ .
Since |= ¬[Ag−{x}]ϕ → ¬ ϕ for every x ∈ Ag ('necessity' Eq. 14), the definition can be expressed equivalently as: and hence (and also |= Δ sole G ϕ → G ϕ). -For any subsets G and H of Ag: In the pushing-the-door example where any two agents pushing together are strong enough to keep the door shut, we have Δ min G k true for any pair G of distinct agents from Ag (at least two agents are required to keep the door shut, and if Δ min G k is true then Δ min H k is not true for any H ⊃ G). Δ sole G k on the other hand is not true for any particular pair G ⊆ Ag. This seems quite natural and satisfactory.
The definition of Δ sole G ϕ may seem contrived: the only motivation offered was an appeal to the strongest counteraction condition of a particular form. Here is an alternative characterisation.

Lemma 1 Let G and H be (non-empty) subsets of Ag.
Proof From Eq. 54 and the previous lemma.
In other words, Δ sole G ϕ holds when G is the only set (if it exists) such that Δ min G ϕ. By analogy with ∂ + G ϕ, properties Eqs. 29 and 30, we can also derive the following.

Proposition 3
Proof As in the derivation of the corresponding properties Eqs. 29 and 30 of ∂ + G ϕ Finally, we noted earlier that yields something interesting? It does not. We have: but this is not an equivalence: in the pushing-the-door-shut example we have Δ min {a,b} k and ¬Δ min {c} k but not Δ sole {a,b} k. The construction Eq. 57 seems to have no particular significance. Similarly, we have But this is not an equivalence. It says only that |= Δ sole G ϕ → G ϕ.

Summary
We have identified a range of possible forms of collective agency with implications between them as summarised in the following diagram: Of these, it is Δ min G and Δ sole G that are deserving of attention. They are the analogues of ∂ x and ∂ + x , respectively, in that Δ min G ϕ allows for the possibility that several distinct sets G of agents bring about that ϕ, while Δ sole G ϕ expresses that any such set G, if it exists, is unique. Both imply that the set G of agents is minimal: |= Δ min G ϕ → [G] min ϕ, and |= Δ sole G ϕ → Δ min G ϕ. Δ min G corresponds to what Belnap and Perloff call 'strictly stit'. Δ min G ϕ ('strictly stit') implies ∂ G ϕ ('stit') but, obviously, not the converse. Δ min G ϕ can also be defined in terms of 'no inessential members' as ∂ G ϕ ∧ nim(G, ϕ). G has a natural technical definition as ∂ + G ϕ ∧ nim(G, ϕ) and some reasonable formal properties but does not appear to express any meaningful notion of collective agency. For the special case of singleton sets, |= Δ min {x} ϕ ↔ ∂ x ϕ and |= Δ sole {x} ϕ ↔ ∂ + x ϕ.

Contributors: Δ max
G is the set of contributors to ϕ at τ .
The condition τ |= ∂ Ag ϕ is to guard against the possibility that there are no sets H such that τ |= Δ min H ϕ. τ |= ∂ Ag ϕ is equivalent to saying that τ |= Δ min H ϕ for some H ⊆ Ag. In the definition it could be replaced equivalently by G = ∅.
The following properties follow more or less immediately.

vi) From the two previous observations
For the case of singleton sets, |= Δ max {x} ϕ ↔ ∂ + x ϕ.

Bystanders
We will now relate the definition of Δ max G and 'contributors' to some suggestions by Belnap and Perloff [4]. The following are their definitions [4,p40,Definition 16] (but with ∂ G in place of their stit).
Definition 9 x is essential [inessential, a mere bystander, not a mere bystander] for ϕ at τ iff ∃G τ |= ∂ G ϕ and for every [not all, no, some] G such that τ |= ∂ G ϕ, x is essential for ∂ G ϕ.
(There is a typographical error in Belnap and Perloff's statement of the definition but it is clear from context what was intended.) As in Belnap and Perloff, our main interest is in (mere) bystanders, and in particular the case where ∂ G ϕ is true and G contains no mere bystanders for ϕ. But first, notice that 'essential for ϕ' as defined above is not the same as, and is much stronger than, the notion of 'essential for ∂ G ϕ' in terms of which nim(G, ϕ) was earlier defined. 'essential for ∂ G ϕ' (Definition 4) is ∂ G ϕ ∧ ¬∂ G−{x} ϕ. The terminology is unfortunate and potentially confusing. One concept (Definition 4) is group dependent. The other, stronger (Definition 9) is group independent.
Definition 9 partitions the set Ag of agents, when τ |= ∂ Ag ϕ, in two different ways: 'essential for ϕ' and 'inessential for ϕ' on the one hand, and 'mere bystanders for ϕ' and 'not mere bystanders for ϕ' on the other. 'essential for ϕ' implies 'not mere bystander for ϕ'; 'inessential for ϕ' does not imply 'mere bystander for ϕ'.
The group independent concept 'essential for ϕ' is extremely strong indeed. Here is an equivalent formulation which is easier to work with.
Proposition 6 x is essential for ϕ at τ iff: Proof From Definition 9: x is essential for We see that x is essential for ϕ at τ when x is a member of every set H such that τ |= Δ min H ϕ. That is extremely strong. We will comment briefly in Section 4.5 on what it might signify.
Let us turn to mere bystanders. In discussion of directions for further work, Belnap and Perloff [4, p46] say that G contains no mere bystander for ϕ when every x in G is such that [H stit: ϕ] for some x ∈ H . There seems to be a mistake here, though it might be just a typographical slip: from the definitions it seems that strictly stit [H sstit: ϕ] was intended rather than plain stit, i.e., in our notation Δ min H ϕ rather than ∂ H ϕ. Let us confirm that.
Proposition 7 x is not a mere bystander for ϕ at τ iff: Proof From Definition 9: x is not a mere bystander for ϕ at τ Belnap and Perloff refer to nmb(G, ϕ) 'contains no mere bystander'. Let us define it as follows.

Definition 10
For every subset G of Ag: Expressed as a formula: nmb(G, ϕ) = def x∈G x∈H Δ min H ϕ.
'Evidently', according to Belnap and Perloff [4,p46], nim(G, ϕ) implies nmb (G, ϕ). Informally, that is true: from Definition 9 'x is essential for ϕ' implies 'x is not a mere bystander for ϕ'. But that is not the validity of nim(G, ϕ) → nmb(G, ϕ) because nim(G, ϕ) is defined in terms of the group dependent 'x is essential for ∂ G ϕ' as in Definition 5 not in terms of the group independent 'x is essential for ϕ'. However, the basic intuition is correct, with a minor adjustment. From Eq. 46 we have: ϕ). So together we have: Now Belnap and Perloff [4, p46]: 'There is the statement omb(G, ϕ) that outside of G there are only mere bystanders for ϕ'. They go on to suggest that Δ min G ϕ ∧ omb(G, ϕ) might then express that 'G is the one and only joint agent for ϕ'. Δ min G ϕ∧ omb(G, ϕ) is equivalent to: And indeed that does turn out to express one sense in which G is the one and only joint agent for ϕ. We will look at it presently. First, the more obvious construction to look at is the following: That says ∂ G ϕ is true, there are no mere bystanders for ϕ in G, and outside G there are only mere bystanders for G: in other words, that G is precisely the set of agents who are not mere bystanders for ϕ. Belnap and Perloff's suggestion Eq. 60 is different. It has nim(G, ϕ) (no inessential members, in the group dependent sense) in place of nmb(G, ϕ) (no mere bystanders, group independent). It remains to define omb(G, ϕ), 'outside G there are only mere bystanders for ϕ': That is or equivalently We thus take the following definition.
Proof The first is Definition 8 expressed in terms of nmb and omb. The second follows by Proposition 8.
Corollary 1 Δ max G ϕ can be expressed as a formula, as follows: What we call the contributors to ϕ at τ , that is, the set G such that τ |= Δ max G ϕ, is exactly what Belnap and Perloff call the set of not mere bystanders for ϕ at τ .
Finally, what of Belnap and Perloff's conjecture Eq. 60 that Δ min G ϕ ∧ omb(G, ϕ) expresses that 'G is the one and only joint agent for ϕ'? That turns out to be Δ sole G ϕ.

Summary
For any τ |= ∂ Ag ϕ the set of agents Ag is partitioned into 'essential for ϕ'/'inessential for ϕ' and 'contributors to ϕ'/'mere bystanders for ϕ' as depicted in the following diagram.
The notation in the diagram is intended to be mnemonic. More exactly: Given ϕ and τ , the set G such that τ |= Δ max G ϕ is unique. G is the set of 'contributors' to ϕ: they are the agents who are not mere bystanders for ϕ. If τ |= ∂ Ag ϕ, i.e., ∃ G τ |= ∂ G ϕ, then Δ max G ϕ is always true at τ for some non-empty set G. Δ sole G ϕ is stronger. Δ sole G ϕ implies Δ max G ϕ but not the other way round. It may be that there is no G such that τ |= Δ sole G ϕ. Consider a version of the pushing-the-door-shut example where a, b and c are all pushing and any two of them are strong enough to keep the door shut (k). Then Δ min {a,b} k, Δ min {b,c} k, Δ min {a,c} k and Δ max {a,b,c} k are true, but Δ sole G k is not true for any G.
If τ |= Δ sole G ϕ for some G then τ |= Δ max G ϕ and τ |= Δ min G ϕ: in that case the contributors to ϕ are all essential for ϕ, and the agents who are not essential for ϕ are the mere bystanders for ϕ. The essential/inessential and contributors/mere bystanders partitions then coincide.
Example (Door) Here is another variant of the pushing-the-door example, for illustration. Suppose there are agents a, b, c, and d,  pushers In the first row, a, b, and c all push and so the door remains shut even though d also pushes from its side. d is not a contributor to k and is a 'mere bystander'. In the second row, pushing by {a, b, c} is sufficient to guarantee the door is shut, irrespective of whether d pushes or not. Pushing by {a, b} or by {c} is also sufficient to keep the door shut-as long as d does not push: d is a contributor to k (not a mere bystander). In the last two rows d does not push. There {a, b} and {c} by themselves guarantee the door remains shut, given that d does not push. d is a contributor (not a mere bystander). The table also shows the agents 'essential for k' in each case, in Belnap and Perloff's strong group-independent sense.

Is Δ max G ϕ agentive?
A standard question in writings on stit is whether an expression of interest is 'agentive'. Is Δ max G ϕ agentive in any of the following senses?
: even the first of these is false. And that is surely to be expected, for whether Δ max G ϕ is true will depend not only on the actions of the members of G but also potentially on the actions of agents outside G, who might perhaps have contributed to bringing about that ϕ had they acted otherwise. (They could not have contributed to bringing about that ¬ϕ by acting otherwise, because then [G]ϕ would be false.)

A comment about 'essential for ϕ'
As defined by Belnap and Perloff, the group-independent notion x is essential for ϕ at τ is equivalent to saying (Proposition 6) that x belongs to all sets H such τ |= Δ min H ϕ. That is very much stronger than merely being a 'contributor' (not mere bystander). Having introduced it, nothing much is made of it thereafter by Belnap and Perloff. What might it represent? Intuitively, 5 it can be seen as saying that, given the actions of the other agents, the agent x had the power of unilaterally avoiding that ϕ was brought about. x is not just involved in the bringing about of ϕ; with the actions of the other agents fixed, x's action is necessary to guarantee that ϕ.
How might this reading be expressed? It turns out to be simple. antidote while a, b and d acted as they did then Δ min {a,b,c} k and Δ min {a,b,d} k would be true and c would not be a bystander for k. e on the other hand is always a bystander for k: there is no alternative situation in which e is not a bystander for k.
This suggests distinguishing between two kinds of bystanders: those whose actions did not contribute to the bringing about of ϕ on this occasion but who might have contributed had they acted otherwise-let us call them 'potentially contributing bystanders'-and those others whose actions would make no difference to the bringing about of ϕ, by any G. These bystanders are not potentially contributing: let us call them 'impotent bystanders'.
To take another example: in the Brexit referendum of 2016 in which the majority of those voting indicated a wish for the United Kingdom to leave the European Union, it was the 'leave' voters and the abstainers together who brought about the outcome. They were the 'contributors'. Those who voted 'remain' were 'bystanders'. An abstainer is someone who is entitled to vote but does not. Abstainers contributed to the outcome because had they voted 'remain' the outcome might have been different. The 'remain' voters were 'bystanders'-but not 'impotent bystanders'. Had they voted 'leave' or abstained they would have been 'contributors'. The 'impotent bystanders' were those who were not entitled to vote.

Potentially contributing and impotent bystanders
Definition 12 x is a potentially contributing bystander for ϕ at τ iff x is a mere bystander for ϕ at τ and x is not a mere bystander for ϕ at some alternative τ ∼ τ .
x is an impotent bystander for ϕ at τ iff x is a mere bystander for ϕ at τ but not a potentially contributing bystander for ϕ at τ . Now instead of nmb(G, ϕ) and omb(G, ϕ) we will have nib(G, ϕ) and oib(G, ϕ) (for 'no impotent bystanders' and 'outside only impotent bystanders').
Proposition 12 x is an impotent bystander for ϕ at τ iff: Proof x is a potentially contributing bystander for ϕ at τ x is an impotent bystander for ϕ at τ whose consequent is, by definition Γ G Γ G ϕ. Γ G ϕ expresses a meaningful sense in which it is the set G of agents that collectively brings it about that ϕ. In the Brexit referendum all those entitled to vote, including the 'remain' voters, brought about the outcome in this sense. It must be noted however that Γ G ϕ does not represent any sense of responsibility for ϕ: a potentially contributing bystander for ϕ is still a bystander for ϕ and could not reasonably be counted as belonging to the set of agents whose actions are responsible for ϕ. Those who voted 'remain' in the Brexit referendum could not reasonably be counted among those responsible for bringing about that the vote was to leave.

Necessarily 'essential for ϕ'
Belnap and Perloff's group-independent notion 'x is essential for ϕ' (Definition 9) is already very strong. It says that x belongs to all sets H such that Δ min H ϕ (Proposition 6) or equivalently, assuming ∂ Ag ϕ is true, that given the actions of the other agents, without x the outcome ϕ is not guaranteed: [Ag−{x}]ϕ is false (Proposition 11). One might wonder 6 whether there is a further distinction that can be made analogous to that between 'impotent bystanders' and 'potentially contributing bystanders', in particular, whether one can distinguish between agents that are merely essential for ϕ, and agents that are necessarily essential for ϕ, in the sense that they are essential for ϕ whenever ϕ is true, or rather, whenever ∂ Ag ϕ is true.
Let us say that x is necessarily essential for ϕ at τ if x is essential for ϕ at τ and x is essential for ϕ at every alternative τ ∼ τ where τ |= ∂ Ag ϕ.
That can be expressed (c.f. the treatment of impotent bystanders in Section 5 and the observation that |= Δ min H ϕ → ∂ Ag ϕ) as: or equivalently (Proposition 11) as a formula thus: In the first three rows, where a pushes the button and b does not, b's action is required for the detonation of the bomb because if b acts otherwise and pushes, the bomb does not detonate. Likewise in the last two rows, where b pushes and a does not: if both were to push the bomb would not detonate. One can see from the table that a and b are 'necessarily essential' for the detonation of the bomb. In no combination however are the actions of {a, b} alone sufficient to detonate the bomb: a technician is always required as well.

Conclusion
In the 2016 Brexit referendum it was the 'leave' voters and the abstainers together who brought about the outcome. They were the 'contributors'. Those who voted 'remain' were not contributors; they were 'mere bystanders' in the terminology of Belnap and Perloff [4]. There is a sense, a very weak sense, in which all of us-those who voted 'leave', those who voted 'remain', abstainers, those who were not even entitled to vote, all of us together-brought about the referendum result. This is the sense of agency expressed by plain group stit, which has the property of 'superadditivity'. In principle, with access to individual voting records, one could identify the minimal subsets of 'leave' voters and abstainers whose voting actions brought about the outcome. Each of those minimal subsets would 'strictly stit' the outcome in the terminology of Belnap and Perloff [4]. It is also meaningful to say, in yet another sense, that it was those who were entitled to vote who brought about the referendum result. Those who were not entitled to vote were 'impotent bystanders'.
There are various different senses in which a set G of agents, collectively, can be said to bring about a certain outcome. Besides stit itself, we identified three forms for special attention. Δ min G ϕ, which corresponds exactly to what Belnap and Perloff [4] call 'strictly stit', expresses that the set G of agents is a minimal (not necessarily unique) set af agents whose joint actions collectively bring about ϕ. A variant Δ sole G ϕ holds if there is exactly one such minimal set. It turns out that Δ sole G ϕ is Belnap and Perloff's suggestion for 'the one and only joint agent for ϕ'. It is noteworthy that Δ min G ϕ is 'strictly agentive' in G, in the sense that Δ min G ϕ implies (and is equivalent to) Δ min G Δ min G ϕ. One can also derive the conditions under which Δ sole G ϕ is equivalent to Δ sole G Δ sole G ϕ. A second general form, Δ max G ϕ, expresses that G is the union of all minimal sets H such that Δ min H ϕ. G is the set of 'contributors' to ϕ. It turns out that the contributors G are what Belnap and Perloff called 'not mere bystanders' for ϕ. If ϕ is brought about by any set of agents then such a G always exists and is unique. It is another natural candidate for 'the one and only joint agent for ϕ'. Δ max G ϕ is not 'strictly agentive' nor even 'agentive' in G. The third form, Γ G ϕ, distinguishes further between 'potentially contributing bystanders' (who happen to be 'mere bystanders' but might not have been) and 'impotent bystanders', who are necessarily bystanders. Γ G ϕ says that G brings about that ϕ but excludes exactly the 'impotent bystanders' for ϕ. Although Γ G ϕ is not 'strictly agentive' in G it is 'strictly agentive' in any subset H of G for which Δ min H ϕ holds. Γ G ϕ is also 'collectively agentive' in G, in the sense that Γ G ϕ implies (and is equivalent to) Γ G Γ G ϕ.
All of these constructions are definable in stit logics. The formal results of the paper hold for the 'deliberative stit' dstit, since models for dstit are (or can be seen as) a special case of the model structures used in the paper. The formal results also hold for all forms of temporal stit in which an atemporal stit operator is combined with a separate temporal logic, as in the transition-based stit-like formalism in [21,22] for example and the temporal stit logics of e.g. [12,18]. It is likely that the formal results hold also for those forms of stit where the temporal aspects are incorporated in the stit operator itself, such as Broersen's xstit [7] and, in particular, the general form of stit (the 'achievement stit') used by Belnap and Perloff in [4]. Details for these other forms of stit remain to be checked.
The logic of possibly unwitting, mere behavioural collective agency is surprisingly rich. There are other accounts of agency and responsibility, besides stit, that could also be explored in similar fashion. We might also think about adding more features, such as communication between agents, mutual awareness, and joint intention, which are essential ingredients of genuine collective action and have been the focus of other studies.
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