No Need for Infinite Iteration

3.1 Dispositional Knowledge, Ordinary Beliefs

Even if one thinks – for whatever reason – that would-be members of a group need to form beliefs about the beliefs of other would-be members, and not simply about their intentions, this still would not lead to an infinite iteration of beliefs, at least not of any kind that renders the individualistic approach psychologically implausible. Not even requiring that agents iterate their belief to some arbitrarily high degree would have this result.

To see this, consider that any kind of belief implies a potentially infinite number of other beliefs, without this stopping us from ascribing beliefs to ordinary agents in other settings. A person holding the belief that Oslo is in Norway need not simultaneously hold the belief that Oslo is in Norway or on Mars, even if that is implied by the first belief. The usual epistemological response to the problem of infinite iteration of beliefs, which Schmid somewhat unsatisfactorily treats in a footnote to his discussion (2005, p. 125, footnote 19), is to say that any belief implied by those we hold explicitly can be seen as both dispositional and implicit, and hence do not encumber the person to whom we ascribe the explicit belief (Schwitzgebel 2014).

Only if the beliefs in question are explicit beliefs, i.e. beliefs actually entertained by Anna at a given point in time, can there be any psychological problem. It is not sufficient for Schmid’s argument that Anna could form these beliefs if she were to pursue a chain of reasoning; the beliefs cannot merely be implicit in her other beliefs, be derivable from them. The simple fact that Anna, reflecting on her belief about Bertha’s beliefs about Anna’s intentions, is disposed to form additional beliefs of the “I believe that Bertha believes that I believe that Bertha believes” – kind does not in itself pose any problem for the psychological plausibility of the individualistic approach (see Lewis 1969).

3.2 Objection: Not Ordinary Beliefs, Rational Grounds for Action

To this, a defender of Schmid’s position might respond that even if it is true that implied beliefs may safely rest undisturbed in the implicit dispositional domain in the case of most beliefs, this is not the case for beliefs about the intentions of others to contribute to collective acts. Borrowing a point from Jane Heal (1978), Schmid points out that an adequate conception of mutual belief or common knowledge must not only give us something that is psychologically plausible (i.e. not entailing an infinite iteration of explicit beliefs), but also something that (together with the appropriate desires) provides rational grounds for action. And, continues Schmid, rational grounds for action cannot contain contradictory beliefs (Schmid 2005, p. 126–127).

Given the right circumstances, whether or not an agent has rational grounds for intending her behaviour as a contribution to a collective act may depend on a high order of iterated belief about the beliefs of other potential contributors. Schmid gives the following example: Anna has stopped some ways in front of Bertha on the trail, and Bertha sees this. But when Anna looks back toward Bertha behind her on the trail, Anna is almost blinded by the setting sun. Anna can still make out Bertha against the strong backlight, but she is not sure that Bertha knows that she can see her. Anna then remembers telling Bertha earlier about the way her glasses makes it almost impossible for her to see in backlit situations, and so Anna concludes that Bertha, being intelligent enough to take this information into account, has concluded that Anna cannot see her, and thus cannot see that she sees that Anna has stopped to wait for her. Anna then concludes that Bertha (falsely) believes that Anna does not know that Bertha knows that Anna has stopped to wait for Bertha (Schmid 2005, p. 128–129). In this case, Anna’s explicit, third-order iterated belief about the beliefs of Bertha contradicts the non-iterated belief that Bertha intends to contribute, and so Anna does not have rational grounds for contributing.^[12]

The problem is that, given less “intelligent” agents, a situation in which there are grounds for forming beliefs that are contradictory to the ones currently held, but where neither agent follows the iteration up to the level of contradiction, the success conditions of each agent’s contribution would be fulfilled. In this case, there would be mutual belief or common knowledge, and the collective act would occur, even if there (supposedly) were no rational grounds for it. I am not entirely sure why this is a problem for the individualistic approach. Schmid himself here asks, rhetorically: what should we say about a situation in which there is common knowledge, even if, following closer examination, rational agents would not form common knowledge (Schmid 2005, p. 130)? What indeed? Saying that this should not count as common knowledge is simply affirming the consequent, and even if we think there is something less than fully rational in each of the two agents forming the requisite beliefs, this in itself does not pose a conceptual problem for the individualistic approach. Such an approach need not exclusively account for the collective actions of ideally rational agents. As long as it can explain when and how it would be rational for an individual to contribute to a collective act, it might well be said to be a strength that a theory is also able to explain collective action by less than fully rational agents.

As for where to set the bar for “rational,” we would certainly think it strange if an agent, after having iterated her beliefs to some level and having found a contradiction, then proceeded to act on this contradictory set of beliefs. But it is far less obvious that there is something wrong with an agent who does not iterate her belief, and acts on what she takes to be true, even if she would not take this to be true had she iterated her beliefs. Neglecting to iterate belief in itself does not give rise to any contradictions, and if the charge is that an agent would be irrational to act on a contradictory set of explicit beliefs, then the lazy iterator is not being irrational. If the charge is that it is irrational to act on any set of beliefs that contain contradictions, whether the beliefs in question are explicitly entertained or merely implicit and dispositional, the criterion of rationality applied would condemn as irrational much of what we normally call rational behaviour – anyone might discover that some of their beliefs contradict each other. Philosophical inquiry is, since Socrates, to a large extent an exploration such beliefs. To require of us that we be certain that we always act on a non-contradictory set of beliefs for our actions to count as rational is clearly too strict.

There might of course be reasons for which we would require that agents iterate their beliefs about other would-be contributors to some degree above zero, at least for some kinds of collective action, such as when we are planning a collective endeavour.^[13] For other, simpler cases, I think we should be careful in calling the lazy iterator irrational. We could perhaps say that it is epistemically irresponsible not to iterate belief to some level before acting, but I am sceptical of discounting, a priori, the rationality of acting on non-iterated beliefs. Even if there are grounds for forming a contradicting belief at a low level of iteration, if neither agent iterates their belief, but both simply assume that the other will contribute and proceed to contribute themselves, they will in fact be acting on correct mutual belief. In such cases, rather than facilitating collective action, iteration of beliefs gets in the way of collective action.^[14] One might think that rational agents could recognise this possibility, and adapt a strategy of not overthinking their contributions to collective acts as a way of maximising their successes in such endeavours. If so, non-iteration could sometimes be the rational thing to do.

4.1 An Intentional Somersault

After having discussed the problem of infinite iteration, Schmid undertakes an outing into “early phenomenology” in order to provide alternate ways of approaching the problem of collectivity (Schmid 2005, p. 132–141). The gist of this section of Schmid’s book is the claim that a collective precedes rather than is created by individually held beliefs in the collective experience of something, a move that Schmid, borrowing from Walther, calls an “intentional somersault” (intentionalen Purzelbaum). As interesting a read as this is in itself, it depends for its pertinence to the current argument on a problematic assumption, namely that there is a defensible equality to the prima facie different phenomena of collective experience and collective action. If Schmid is to derive support from these sources, he needs to show that their insights are relevant to collective action.

Schmid attempts this transference using a pair of complementary examples to show (a) that common knowledge is what renders the (infinite) iteration of individual belief possible and (b) that (infinite) iteration of individual belief cannot produce common knowledge (Schmid 2005, p. 141–144). Together this sets the stage for an “intentional somersault” like that of Walther, whereby commonality of knowledge is put prior to the beliefs individuals might have about the subject of that knowledge. This in turn is the first part of Schmid’s proposed Copernican revolution, where commonality of knowledge is explained by the ontological priority of collectivity over individuality. The first example is a version of “The Barbecue Problem” (Vanderschraaf and Sillari 2014), and will not be treated here. I will simply grant Schmid that individual knowledge can be derived from collective knowledge. The crucial point Schmid needs to prove is that the common knowledge or mutual belief necessary for collective action cannot be constructed from (iterations of) individual belief, and for this, he needs the second example.

The second example is a version of the Coordinated Attack problem, first described in the distributed systems literature (Chant and Ernst 2008). In Schmid’s telling of it, two infantry detachments have separated as part of a tactical manoeuvre. They are trying to fix the time of the upcoming attack to six o’clock the next morning, and the first detachment sends a messenger with that message to the second. Given the possibility that the first messenger could have been intercepted, and seeing as how the first detachment cannot know whether the second has received the order or not, the second detachment decides to send a messenger in return confirming the time of attack, thereby assuring the first detachment that they know the order and will commence the attack at the time set. The second messenger also arrives, but given the possibility that this messenger could also have been intercepted, the first detachment reasons that the second detachment cannot know whether the first detachment knows that the second detachment has received the order and will commence the attack at the time set. If they do not know this, the whole structure collapses, making any previous communication worthless in the process. The first detachment therefore sends out a third messenger, confirming that they have received the second detachment’s confirmation – et ceteraad infinitum. This shows that no amount of back-and-forth can produce the knowledge the two detachments need as rational grounds for commencing the attack at the time set. The exchange of messengers – like the iteration of beliefs – fails to establish the common knowledge necessary for collective action (Schmid 2005, p. 144).

Although it relies on some unrealistic assumptions, the Coordinated Attack problem does prove that there are instances where full-fledged common knowledge is required for each would-be participant to have rational grounds to contribute to the collective act, and where this is unachievable (Chant and Ernst 2008, p. 552–555).^[15] It does not show, however, that common knowledge can never be established by individual agents forming beliefs about each other (nor that communicating intention is never effective in so doing). Using the term “interactive knowledge”^[16] for the knowledge one agent has about the knowledge of another (with mutual interactive knowledge constituting common knowledge), Chant and Ernst give an example from the other end of the scale:

• Two friends staying in the same hotel in a strange city get separated. It is obvious that they may be perfectly capable of meeting up together again at the hotel rather than, say, at a local landmark. The reason is that this choice is highly salient to both people. For our purposes here, we note that only a minimal level of interactive knowledge is necessary for them to achieve this coordination. Perhaps each needs only believe that the other person is capable of reaching the hotel, for example. No other peculiar features of the situation force them to acquire higher levels of interactive knowledge. (Chant and Ernst 2008, p. 555).

Schmid’s use of the Coordinated Attack problem to set the stage for an intentional somersault is, as it stands, partly a straw-man argument. The proponent of an individualistic approach can accept that the two infantry detachments would be unable to fix a time for the upcoming attack in this way, but deny that this is of any great consequence for any but the most artificially constrained examples. In particular, for most if not all real-world situations, there will be other factors influencing belief formation, providing, at least in some cases, rational grounds for contributive action. The real question is not, or rather should not be, whether would-be contributors to a collective act can come to have mutual belief or common knowledge purely through the iteration of beliefs about the intentions or beliefs of the other would-be contributors. As I have argued, the third criterion does not necessitate any iteration of beliefs – it suffices that each would-be member of a group forms the belief that the other would-be member(s) intend to contribute. The real question is when or whether individual agents have grounds for forming the required beliefs. I finish by indicating one of the ways the proponent for an individualistic approach might answer Schmid’s challenge to explain how individual agents can come to have rational grounds for their contributions to collective acts.

4.2 Grounds for Belief

To recapitulate: At a minimum, the individualistic approach can be seen to posit three criteria for collective acts to occur:

1. Each would-be member of a group acts in an appropriately coordinated manner

2. Each would-be member intends their behaviour as their contribution to the collective act

3. Each would-be member believes that the other would-be members have the intention to contribute to the collective act

Schmid has argued that C3 implies that would-be members must infinitely iterate their beliefs about the intentions of others to contribute, prompting the revised version of C3:

1. Each would-be member has infinitely iterated beliefs to the effect that the other would-be members have the intention to contribute to the collective act

So far, I have argued 1) that C3 does not imply C3^*, and 2) that C3 only gives rise to an infinite iteration of beliefs in the most artificially constrained examples, where iteration also proves to be insufficient for establishing rational grounds for contributing to the collective act. The proponent of an individualistic approach should simply accept that there are such situations, and deny that this is of any great importance. In most cases, there is no need for infinite iterations, nor for psychological mechanisms to compensate for such iterations. However, both the mental parts and the actual behaviour are required, and both (or all) parties must fulfil all criteria, if a collective act is to occur. If we are looking purely at the would-be contribution of a single agent, this can occur even if no other agent contributes: the only requirement is that the agent in question actually has the required belief and intention, and shows the requisite behaviour. Take Tuomela’s example of two people carrying a table (Tuomela 2005, p. 327–328) and focus on the first part of the collective action, the lifting of the table: If I think that you will lift your end of it, and I begin to lift my own, my act would truly be a contribution to the would-be collective act of lifting the table – there is no sense in trying to say that I did not do my part. The collective act fails because you did not contribute, not because I was wrong in something. Of course, if this happened, it would quickly become obvious that I was mistaken in thinking that you would do your bit, but that does not alter the fact that my action fulfilled the criteria for my contribution to the collective act. If, ceteris paribus, you had contributed, the collective act of lifting the table would have occurred, with zero change in my part of that act. Unless there was some obvious or immediate reason for me to think that you would not contribute, I neither could be sure, nor needed to be sure about your intentions; all that was required was something like a reasonable confidence on my part about you doing yours.

When it is no longer required of each agent that they be certain that the other(s) will do or are intending to do their part – or that they be certain that they have no contradictory beliefs about this – the problem of mutual belief is reduced to a question of whether or when agents have grounds for forming such beliefs. But here a whole host of different kinds of evidence can come into play, and we are no longer locked inside the head of the individual contributor (see, e.g. Bratman 2014). The ongoing behaviour of the other (potential) contributors, any communication of intention or belief that they might make, their responses to your behaviour and communication – even solipsistic mind-games of iterated beliefs, as well as background knowledge or information – can all influence belief formation. While the possibilities are endless, this should not be seen as a weakness of the approach. The simple fact that people tend to be able to act collectively guarantees that there are practicable limits to the kinds of things that go into belief formation in such cases, and this way of approaching the question brings into view the complex sets of factors that might influence an individual in an instance of collective action.

This suggestion is also in line with what, e.g. Robert Sugden has already argued about the ways in which individual agents can come to have “reason to believe” that other agents will use “team reasoning” in their interactions (Sugden 2003, p. 176–179), thus making it logically consistent and/or rational for each individual to apply team reasoning. Sugden’s ‘assurance condition’ and to some extent also Michael Bacharach’s ‘interdependence hypothesis’ (2006) can be seen as formalised versions of the kind of belief-forming procedure suggested here. Both have been integrated into theories of collective action demonstrating that common knowledge or mutual belief, while reliant on belief about the beliefs of others, at least does not depend on infinite iterations of belief (e.g. Gold and Sugden 2007 and Tuomela 2013).^[17]

4.3 Objection: Social Environment

Here it might be objected that we are covertly importing collectivity into the individual analysis, since the grounds for belief stem from something like a pre-existing social environment, e.g. in form of the shared norms and knowledge of the individuals involved in collective action.^[18] According to this objection, if we think, in Schmid’s example, that the army detachments in most normal cases would have rational grounds for launching their collective attack, we do so because we (implicitly) appeal to things such as their belonging to the same army, or the existence of a pre-established chain of command. But these elements may be seen to carry in them the very collectivity or common knowledge that Schmid argues that individual belief-formation presupposes. And so the individualistic analysis ends up depending on collectivity after all.

This objection will not work as it stands, for it presupposes that whatever pre-established collectivity we might appeal to in explaining individual belief formation in particular cases itself necessarily cannot be the result of individual contributions that are not dependent on such collectivity. It trades on a highly constrained and contrived example to draw a general conclusion about all forms of collective action and thus assumes the consequent; that no forms of collective action can be given an individualistic explanation. Whether in the end all forms of collective action presuppose some form of pre-established collectivity remains an open question, but it cannot be assumed against the individualistic approach that this is the case.