WINNER OF THE PHILOSOPHICAL ESSAY PRIZE 2010: The simplicity of mutual knowledge

Abstract

Mutual perceptual knowledge is a prevalent feature of our everyday lives, yet appears to be exceptionally difficult to characterise in an acceptable way. This paper argues for a renewed understanding of Stephen Schiffer's iterative approach to mutual knowledge, according to which mutual knowledge requires an infinite number of overlapping, embedded mental states. It is argued that the charge of ‘psychological implausibility’ that normally accompanies discussion of this approach can be offset by identifying mutual knowledge, not with the infinite iterations themselves, but with the finite base which Schiffer proves is capable of generating those iterations. An understanding of this finite base as a primitive, relational property holding between two or more people, allows us to understand the iterations as an implicit and ‘harmless’ intrapersonal feature of what is an interpersonal phenomenon. The paper concludes by relating the account to joint attention in infant interaction.

Keywords:

• intersubjectivity
• social psychology
• cognitive science
• folk psychology
• developmental psychology

Not so long ago, debates about the nature of social-cognitive understanding were almost exclusively dominated by the turf wars that surrounded the competing merits and demerits of theory-theory and simulation theory (Davies and Stone 1995a, 1995b; Carruthers and Smith 1996; Nichols and Stich 2003; Goldman 2006). More recently, however, there has been a wealth of interdisciplinary work (encompassing, amongst other areas, philosophy of mind, phenomenology, developmental psychology and the burgeoning field of social neuroscience) which has loosely converged on an alternative approach to social cognition that we might call the primary intersubjective theory (PIT) (Gallagher 2001; Hobson 2002; Hutto 2004; Carpendale and Lewis 2006; Eilan 2007; Ratcliffe 2007; Reddy 2008; Striano and Reid 2009).

In this paper I shall attempt to contribute to one strand of this newly evolving approach to social cognition by articulating an understanding of the notion of mutual knowledge that draws on seminal work by Schiffer (1972), and which might help to strengthen the case for PIT, while weakening the case for its more established competitors. Although mutual knowledge itself has not been at the forefront of PIT work thus far, its closely related counterpart of joint attention in infant interaction has been, and the difficulties that have been observed in providing a satisfactory characterisation of that phenomena (Eilan 2005, 25; Heal 2005, 39; Griffin and Dennett 2009, 257) are clearly of a type with those which have, for quite some time, bedevilled researchers attempting to characterise mutual knowledge (Schiffer 1972; Smith 1982; Barwise 1989). A psychologically plausible, and ontologically respectable, understanding of mutual knowledge might, therefore, be of some help to those researchers who are attempting to ground the PIT approach to social cognition in the well-documented evidence of infantile joint attention.

The structure of this paper, then, is as follows. In section 1, I shall provide a rough characterisation of the phenomenology of mutual knowledge, which will stress its pervasive and basic presence in our everyday (adult) social lives, and then point out the initial difficulties that arise with attempting to articulate the nature of the phenomena. In section 2, I shall outline Schiffer's (1972) iterative approach to mutual knowledge, but argue that it suffers from certain internal pressures that require a modification of the account so as to avoid the internal difficulties that inflict it, with the result that the core of the approach – and the theoretical benefits of that core – can be preserved by means of construing mutual knowledge as a primitive, relational mental state that is necessarily and irreducibly shared by two (or more) subjects, rather than as an infinite series of overlapping individualistic mental states held by two (or more) people. In section 3, I shall discuss the virtues of relational accounts of mutual knowledge and argue that the modified Schifferian account outlined in the previous section is well placed to provide a satisfactory account of mutual knowledge that could serve a role within both intrapersonal and interpersonal psychology. In section 4, I shall argue that the reformulated Schifferian analysis of mutual knowledge has the psychological simplicity required of it to be capable of serving as a plausible model for characterising salient elements of joint attention in infant exchanges. Finally, I shall argue that the fact that it is so well-suited to modelling the kind of joint attentional infant social exchanges that form the developmental basis from which our full-blown adult social abilities arise, helps to off-set at least some of the initial worries one might have about the plausibility of a dual-subject psychological state that cannot be reduced to two individuals’ single-subject psychological states.

1. What is the paradox of mutual knowledge?

Imagine the following situation. You are standing with a friend at a fireworks display, watching the Catherine Wheel in front of you. There is nothing mysterious or untoward about the situation: you are both symmetrically and fully aware that you are both standing and admiring the Catherine Wheel in front of you. Mutual knowledge of this kind – where knowledge of a situation, event or object is out-in-the-open between two people – should surely be recognised as a prevalent and basic feature of our everyday lives. That is to say, there are innumerable situations in our social lives where the knowledge that one possesses of a certain fact or situation does not appear to operate at a higher level of perspicuity or certainty than the accompanying knowledge that one's awareness of the fact or situation is shared by another.

The question that initially confronts us is: how should we characterise mutual knowledge of this kind? And a quick answer might simply be this: where there is mutual knowledge, the knowledge possessed by each of the protagonists is transparent, overt and open to each. And it is this transparency and openness that is missing in an uncertain situation, where there is some doubt or concealment about what is going on.

Nevertheless, while such a characterisation as this might well have a certain phenomenological familiarity, it is clearly inadequate if the aim is to offer a characterisation of mutual knowledge that might serve a role within a psychological theory; failing, as it does, to move beyond the metaphorical concepts of ‘transparency’ and ‘openness’. What we want to uncover, then, is a way of cashing out these metaphors in a manner that would allow the concept to play an explanatory role within a comprehensive psychological theory, while preserving the intuition that what is distinctive about situations such as that described above is exactly the transparency and openness that is lacking in uncertain situations.

The central problem of mutual knowledge can now move into focus. Knowledge that occurs in social situations (be it a case of mutual knowledge or more uncertain situations where there is some doubt about what each other knows) differs from knowledge that is abstracted from social situations (i.e. knowledge simpliciter, where we are considering an isolated individual's knowledge of a fact about the environment) to the apparent extent that the former advert to higher-order, or metarepresentational, knowledge of another's mental states, while the latter need not. So, whereas knowledge simpliciter can be represented schematically as:

•	O1 knows that p

in social situations, where the participants have some knowledge of each other's awareness, we require not just shared knowledge:

•	O1 knows that p
•	O2 knows that p

but also a schematic representation of the fact that the knowledge one has of p can be known to be shared by the other:

•	O1 knows that O2 knows that p

and perhaps:

•	O2 knows that O1 knows that p

Having set things up in this way, however, the problem of distinguishing situations of mutual knowledge from uncertain situations proliferates. For, we can certainly imagine cases where the above schema is fulfilled, but where there is none of the transparency and openness that is required of mutual knowledge. We could, for instance, imagine two spies covertly observing each other at a fireworks display, each standing before the same Catherine Wheel, but with each oblivious to the fact that he is being watched by the other. Furthermore, even supposing that it comes to dawn on both the spies that they are watching each other looking at the Catherine Wheel, thus adding an extra level of knowledge such that:

•	O1 knows that O2 knows that O1 knows that p
•	O2 knows that O1 knows that O2 knows that p

this would still be consistent with the supposition that neither spy is yet aware that it has dawned on the other that they are watching each other – and thus that the situation is not transparent or open in the requisite sense.

The obvious worry that arises from this is that we could go on in this manner indefinitely, adding an extra level of embedded knowledge at each step, without ever reaching a point that would serve to distinguish an open situation from an uncertain one. This, then, is the problem of mutual knowledge: how can one characterise mutual knowledge without being committed to an infinite regression of mental states.

It is important to note the nature of this problem. There is a temptation to suppose that the problem of mutual knowledge is something of a philosopher's abstraction; a quest for a definition of a pure, idealised state of shared knowledge, but not, ultimately, a problem that needs to be addressed when it comes to cognitively limited, ‘real-time’ human agents. Michael Tomasello, for instance, claims that:

Overall, a reasonable way of dealing with all of this … is simply to say that the recursive spiral is not infinite but only indefinite; we compute it as far as we need to or are able to, which is typically only a few levels up. (Tomasello 2008, 96)

This temptation, shared by many,1 simply supposes that – abstract philosophical worries aside – there is no problem of mutual knowledge, since the threatened regress can be cut short ‘a few levels up’ on the grounds that real-time human agents either ‘don't need to’ or ‘are not able to’ entertain embedded thoughts of a more complex nature.

This, however, would be to misunderstand the nature of the problem. The problem of mutual knowledge does not arise from a supposed misfit between psychological limitations on human agents on the one hand, and an idealised notion of what mutual knowledge should consist in on the other but rather from a misfit between psychological limitations on human agents on the one hand, and the need for a psychologically expedient notion of what mutual knowledge consists in on the other. And what the problem of mutual knowledge throws up is the apparent incompatibility of these two latter factors. Let me explain.

The truncated approach, as adopted by Tomasello in the quote above, does not provide a psychologically expedient notion of what mutual knowledge consists in because it is, for want of a better phrase, inherently unstable. The difficulty involved with characterising mutual knowledge is not merely the vague intuition that, as Naomi Eilan puts it, ‘we think that … [a] recognition by each of us that the other is aware of our perception must be included in the total account of what we are each aware of – that this is part of what complete transparency requires’ (Eilan 2005, 25; italics added), but rather that, for any given level on the hierarchy, that level either reveals itself to be useless for the purposes of mutual knowledge, or else requires a further step up the hierarchy. As a result, any attempt to cut the regress short ends up undermining the utility of the iteration itself.

To get this point in clearer view, we can briefly shift focus and consider cases of coordinated attack. A typical example of a problem of coordinated rational attack can be outlined as follows:

Two divisions of an army, each commanded by a general, are camped on two hilltops overlooking a common valley. In the valley awaits the enemy. It is clear that if both divisions attack the enemy simultaneously they will win the battle, while if only one division attacks it will be defeated. As a result, neither general will attack unless he is absolutely sure that the other will attack with him. In particular, a general will not attack if he receives no messages. The commanding general of the first division wishes to coordinate a simultaneous attack (at some time the next day). The generals can communicate only by means of messengers. Normally it takes the messenger one hour to get from one encampment to the other. However, it is possible that he will get lost in the dark or, worse yet, be captured by the enemy. Fortunately, on this particular night, everything goes smoothly. How long will it take them to coordinate the attack? (Fagin et al. 1995, 190–1)

The standard answer is that the generals will never get round to coordinating their attack, even supposing that every message sent reaches its target. Suppose General A sends a message to General B to the effect of ‘attack at dawn’. Although B will now know of A's plan, A cannot be sure that his message has reached B, and so it will not be rational to attack. Knowing A's predicament, B could then send a message of receipt. But, from B's perspective, it will not be rational to attack on this basis, because he will be unaware if A has received the message of receipt, and he will know that without receiving such a message, A would not be in a position to attack. Knowing B's predicament – that is, that B now needs to know whether A has received his message – A could then send a further message of receipt back to B. But, knowing that B requires this further message in order to attack, A requires yet a further message to confirm B has received this last message… and so on, back and forth, indefinitely; always one step behind the point where it is rational for the armies to co-ordinate their attacks.

One might sum-up the viciousness of the coordinated attack paradox as so2: clearly if no message is sent at all, this will not suffice for coordinated attack. Furthermore, for the iteration to begin there must be at least one message k which is sent and received but which does not suffice for co-ordinated attack. If we were then to suppose (as the truncated approach suggests) that, for some k, k + 1 messages do suffice for attack, then the sender of that (k + 1)st message attacks without knowing whether that message has been received by the other. Since the generals only attack in unison, we must suppose that the intended recipient of the (k + 1)st message will attack regardless of whether he receives the message or not. But then this goes to show that the (k + 1)st message was actually useless, and that k messages suffice after all, which in turn undermines the purpose of the iteration; for that k may as well be the first message sent.

The problem that this paradox uncovers is not just structurally similar to that of mutual knowledge, but simply is the problem of mutual knowledge illustrated within the context of coordinated action: it outlines a situation in which nothing less than mutual knowledge would suffice for appropriate action, and then raises a problem of how such mutual knowledge could be achieved. That is to say, the paradox does not raise a problem about how mutual knowledge can be translated into action (it is assumed that if the appropriate state of knowledge is achieved, then the generals will act as they should). Nor is there anything quixotic about the kind of shared knowledge that would be required to dissolve the paradox: what is required is the kind of transparency that is evident in many ordinary communicative situations (it is assumed that if the Generals had access to flares or flags which would be mutually observable, then they will act as they should). Rather, what the paradox highlights is that once one gets embroiled in supposing that an act of transparent communication or shared knowledge requires a set of hierarchical to-ing and fro-ing about who knows what, then there will be no end to the matter. If one attempts to cut short the hierarchy at a logically arbitrary level – even on grounds of psychological limitations – then one will not arrive at the kind of shared knowledge that is required for transparency, but will rather undermine the purpose of the iteration itself.

To sum up: we appear to require a notion of mutual knowledge that is both psychologically plausible and psychologically expedient. But the only psychologically expedient notion of mutual knowledge that we have to hand is not psychologically plausible; while the only psychologically plausible account we have to hand is not psychologically expedient. That is to say, mutual knowledge appears to require an infinite regression of overlapping, embedded states, but, at least prima facie, it is not clear how this could be so much as possible for cognitively limited human agents.

2. Stephen Schiffer's iterative approach to mutual knowledge

One of the most influential, and widely accepted accounts of mutual knowledge is that offered by Schiffer (1972). Schiffer's account is interesting because, although it argues for the problematic claim that mutual knowledge requires an infinite string of mental states, it does so by means of provably showing how it is possible for the whole infinite hierarchy to be generated from a finitely specified situation.

By itself, showing that the infinite hierarchy can be generated from a finite situation does not answer our dilemma, for one shall still lack an account of how that infinite, embedded hierarchy of thoughts can be entertained by finite human beings.

What I shall be arguing over the next two sections, however, is that something very similar to Schiffer's approach is capable of solving our dilemma. Rather than reductively identifying mutual knowledge with the infinite string of thoughts which follow from the finite base, I shall argue that mutual knowledge can be identified with the finite base itself. Such a move allows us to preserve the requirement that mutual knowledge have the equivalent epistemic power of an infinite string of mental states, while maintaining that it is ontologically grounded in a finite situation. Mutual knowledge, I shall argue, is – contrary to initial analytic appearances – a very simple thing.

In this section I shall outline Schiffer's iterative approach to mutual knowledge. The attention shall be on the aforementioned finite base, which, as we shall see, requires some modification. In the following sections, I shall argue that the suggested modification shows how Schiffer's iterative analysis could provide an account of mutual knowledge that is both psychologically plausible and psychologically expedient.

Schiffer's theory, then, embraces the infinite regress, such that:

A speaker (S) and an audience (A) mutually know that p provided that:

1.	S knows that p A knows that p
2.	S knows that A knows that p A knows that S knows that p
3.	S knows that A knows that S knows that p A knows that S knows that A knows that p … etc … etc … etc ad infinitum

Our interest, however, is not so much in this regressive result (the utility of which we shall discuss in the next section), but rather the finite means by which this result is achieved.

We are to suppose that S is F, and that A is G, where S = the speaker; A = the audience3; F = the property of being a ‘normal’ (regarding sensory faculties, reasoning capacities, etc), conscious individual who is identical with S; G = the property of being a ‘normal’, conscious individual who is identical with A. The following, according to Schiffer, will then provide necessary and sufficient conditions for the iteration:

S and A mutually know that p iff there are properties F and G such that:

1.	S is F
2.	A is G
3.	Both being F and being G are sufficient for knowing that p, that S is F, and that A is G; i.e. (x) (Fx v Gx → Kxp & KxFS & KxGA)
4.	For any proposition q, if both being F and being G are sufficient for knowing that q, then both being F and being G are sufficient for knowing that both being F and being G are sufficient for knowing that q; i.e. (q) ((x) (Fx v Gx → Kxq) → (y) (Fy v Gy → Ky(z) (Fz → Kzq) & Ky(w) (Gw → Kwq)) (Schiffer 1972, 34–5)

Given this finite base, it would seem that one would have the infinite number of iterations that we have claimed are necessary to characterise the openness of a situation of mutual knowledge. For, the propositional knowledge that the generating properties4 generated in (3) can be fed into (4), and the result of that function can then be reapplied to (4), and the result of that function can then be reapplied to (4) again, and so on ad infinitum. As Christopher Peacocke puts it, in his instructive commentary on Schiffer's approach:

The hierarchy of iterations of knowledge on the part of each mutual knower is attained by the knower by inference from his knowledge that he and the other have the relevant generating properties. These iterations really do follow from this simple theory. The elegance of the approach is undeniable. (Peacocke 2005, 310)

That Schiffer has shown how mutual knowledge can, theoretically, be seen to be generated from a finite situation, should not be underestimated; even if, in and of itself, it takes us no further to solving our dilemma.

What I shall shortly argue is that something like Schiffer's analysis has the potential to be of benefit for those theorists who have recognised the need for a robust understanding of mutual knowledge in psychological theory, but who have balked at the seemingly intolerable consequences such an understanding would wreck. A renewed understanding of Schiffer's approach, I shall argue, has the resources to underpin a workable understanding of the role that mutual perceptual knowledge has to play within a person's mental economy, such that (a) it respects the requirement that mutual knowledge have the epistemic power of an infinite string of embedded knowledge states; (b) it respects the requirement that mutual knowledge be achievable by cognitively-limited human agents; (c) it respects (but does not demand) a realist construal of propositional attitude psychology in general; and (d) it has the potential to shed light on the foundational role that mutual knowledge has to play in our understanding of other minds, and in the development of that understanding. The key to this understanding of the utility of Schiffer's account is in the generating base itself. However, there are problems with Schiffer's account of this base as it stands, and I shall ask for the reader's patience as we unpick these problems.

The renewed understanding of Schiffer's approach that I am urging begins to emerge if we take a closer look at the generating properties that are meant to hold of the individuals in question. The initial thing to focus on here is the definition of the generating properties. Schiffer tells us that the generating properties are ones that specify that the person of whom that property is true is ‘normal’ – ‘a visibly “normal” open-eyed, conscious person’ (Schiffer 1972, 35). As a definition of a concept that is to play a central role in his account of mutual knowledge, this is remarkably underspecified. However, on closer inspection, it soon becomes clear that this appeal to ‘normality’ is not doing any work in the analysis, or, if it is, it is doing far more than could reasonably be asked of it.

Obviously, the mere fact that two people are ‘normal’ does not guarantee mutual knowledge about any specific facts. Take myself and Bill Clinton. Let us suppose that we are both, as far as these things go, pretty normal. Well, clearly, this does not guarantee that we have mutual knowledge about anything, for Bill Clinton is in all probability completely unaware of my existence. Furthermore, even supposing he were aware of my existence, our visible normality would not suffice to guarantee that, under all and every conditions, we would know that each other is normal, and so would not even get us as far as (3) on Schiffer's analysis. ‘Visible normality’ does not ipso facto entail mutual knowledge on any subject.

Perhaps, however, one might defend the proposal by trying to outline a more specific situation in which mutual perceptual knowledge can be expected to occur. Let us imagine, then, like Schiffer does, that we are sitting in a restaurant with a candle placed between us: this surely is the archetype of a situation of mutual knowledge, and one might expect that the mere fact of visible normality will here suffice to establish mutual knowledge of the candle.

In fact, however, Schiffer's analysis produces a problem of grain that a sweeping appeal to normality is ill-equipped to handle. The difficulty comes with the universal claim of the fourth clause, essential to generating the regression of embedded knowledge states, but which, once combined with the appeal to normality, is far too strong to generate a case of mutual knowledge even when a candle is placed between the two protagonists. Suppose, for instance, that in addition to the candle placed between us there is also a large mirror behind your back which, unbeknownst to you, I can see into, allowing me to see by reflection all that you can see directly. This addition to the scenario will immediately undermine the claim that we have mutual knowledge of the candle on Schiffer's analysis because clause (4) will now be unfulfilled were we to suppose that there were an event (say, the waiter on his way to our table with the bill) that you observe directly, and I observe via the mirror. That is to say, in such a situation, there is a proposition q (that the waiter is coming our way), such that being F and being G (i.e. being ‘visibly normal’) are both sufficient for knowing, but for which it is not the case that being F and being G are both sufficient for knowing that they are both sufficient for knowing that q. Thus, even though we are both facing each other with a candle between us, we do not, on the analysis as it stands, have mutual knowledge that there is a candle between us, because the universal claim in (4) is not fulfilled.

It would be too hasty to reject Schiffer's analysis on these grounds, because the problem does not arise from the analysis per se, but from the combination of the analysis with the supposition that the generating properties in question can be filled out by appeal to the ‘normality’ of the protagonists.

What the analysis requires instead, I think, is a way of filling out the generating properties, such that they apply to the individuals in question as relativised to the situation in which they are: it is not the fact that we are both ‘normal’ that suffices for our mutual knowledge of the candle, but rather the fact that, inter alia, we are both sitting in the restaurant, have a candle in front of us, are facing each other, with well-functioning sensory capacities and are suitably responsive to the relevant aspects of the situation in some respect. That is to say, we might suppose that there is a unique generating property in play to generate a situation of mutual knowledge about the candle, but that it is of a rather complex sort, such that it cannot be generalised to cover all situations, is specific to the situation in which the individuals are, and suffices to fulfil Schiffer's analysis on that occasion about that fact.

Specified to a fine enough grain, then we might suppose that the generating properties that suffice for the individuals’ respective knowledge of the candle, and of the overlapping iterations that follow from that situation, do not also thereby suffice for knowledge of propositions which are not related to that very scenario; whatever generates our respective knowledge of the waiter's movements are generated by other, distinct generating properties with their own distinct logic.

Now, if this is correct, then it turns out that we have cause to change the analysis in a further, critical respect. On Schiffer's analysis (in combination with the widely held assumption that knowledge is factive), the two generating properties are necessary and sufficient conditions for each other. This is provable by means of the following argument:

1.	(x) (Fx v Gx → Kxp & KxFS & KxGA) (assumption from Schiffer)
2.	(x) (q) (Kxq → q) (factivity assumption)
3.	FS → KS(GA) (from 1)
4.	GA → KA(FS) (from 1)
5.	FS → GA (from 2 and 3)
6.	GA → FS (from 2 and 4)
7.	GA ↔ FS (from 5 and 6)

Having dispensed with the appeal to normality, however, and having subsequently supposed that the generating properties are such that they are to be relativised to the particular situation at hand, we are left with the following puzzle.

Just so long as the generating properties were understood to be filled out by appeal to a sweeping notion of ‘normality’, then the bi-conditional that holds between S being F and A being G could be explicable by means of the supposition that what it is for someone to have ‘normal’ sensory and reasoning capacities would be at least partly a matter of having the same kind of sensory and reasoning capacities of his conspecifics. But, without the appeal to normality, we do not have this explanation of the logical connection between the two generating properties; rather we are left with the bare claim that, if there is to be a situation of mutual knowledge, then a property true of one individual is necessary and sufficient for a distinct property to be true of another individual.

This should, at the very least, strike one as a rather puzzling state of affairs. To say of two metaphysically distinct properties that they are necessary conditions for each other is hard to countenance. Take an analogy: suppose that Sally and Andrew are so grossly in love with each other that they are happy if and only if the other is. Well, if so, then poor Sally and Andrew: In contrast to the property of having normal capacities, the property of being happy is a transient thing – it comes and goes depending on the situation. And given that Sally is happy if and only if Andrew is, then neither has the wherewithal to be happy without the other being happy. Consequently, neither can be happy as a temporal consequence of the other being happy. Rather, we must suppose that if they are ever to be happy, happiness must strike them simultaneously or not at all. Short of a cosmic coincidence (or, for the incorrigibly romantic, a Leibnizian pre-established harmony), Sally and Andrew seem destined for a miserable life.

To return to cases of mutual knowledge, a similar, spontaneous and simultaneous ‘outbreak’ of the requisite generating properties must occur for mutual knowledge to take place. Given that S and A are not already respectively F and G, then neither S nor A can reason themselves into the desired state necessary for mutual knowledge, without the other participant necessarily and simultaneously reasoning themselves into their respective state (because of the biconditional relation that holds between F and G), and the idea of co-reasoning, would, in this context, appear to be as obscure and unhelpful as the idea of mutual knowledge that we are hoping to illuminate. Correspondingly, if we were to suppose that the situation of mutual knowledge was not so much a matter of reasoning, but rather a matter of being an effect of an external cause, then this would require a common external cause that was such that it was sufficient to cause both S to be F and A to be G, but not sufficient to cause either individually. But why might this be? We are meant to suppose that F and G are metaphysically distinct properties, true of distinct individuals. So why should they be inextricably tied to each other in this logically heavy-handed way? We seem to be left with a puzzle.

One response to this, of course, would be just to say so much the worse for Schiffer's analysis. Nevertheless, given the effective way in which the analysis demonstrates how mutual knowledge can be based on a finite situation, a preferable way out of this difficulty, I submit, would be to construe the bi-conditional not as registering a necessary connection between metaphysically distinct properties, but as registering a metaphysical identity between the properties said to be true of S and A. That is to say, rather than supposing that there are two generating properties in play – one true of A and the other true of S – we should suppose that there is a single generating property that holds of the two individuals in question. This proposal does not do damage to the outcome of Schiffer's analysis, for the analysis can be re-formulated as so:

S and A mutually know that p iff there is a relational property H true of S and A, such that:

1.	S and A are H. (i.e. a three-place relation between A, S and the property H: HAS)
2.	Being H is sufficient for knowing that p, and that S and A are H; i.e. (x) (y) (Hxy → Kxp & Kyp & KxHAS & KyHAS)
3.	For any proposition q, if being H is sufficient for knowing that q, then being H is sufficient for knowing that being H is sufficient for knowing that q; i.e. (q) ((x) (y) (Hxy → Kxq & Kyq) → (z) (w) (Hzw → Kz(x*)(y*) (Hx*y* → Kx*q & Ky*q) & Kw(x*)(y*) (Hx*y* → Kx*q & Ky*q)))

Understood in this way, mutual knowledge is irreducible to the individual cognitive states of the individuals in question; it is something that holds of S and A as a pair. This reformulation does not do damage to the outcome of Schiffer's analysis, since it yields the same iterations as Schiffer's original analysis did. Further, it removes some of the puzzlement regarding how the individuals in question could become sufficiently calibrated to yield the iterations, because we can suppose that the calibration is not the result of two separate properties managing to conveniently mirror each other, but of a single, relational property holding between the two individuals in question. There is no coincidence to account for in the fact that Sally and Andrew are sitting opposite each other. And the same, we might suppose, goes for the fact that they have mutual knowledge of the candle that is placed between them.

3. The simplicity of mutual knowledge

In an article on joint attention and mutual knowledge, John Campbell makes the following distinction:

… we can distinguish between relational and reductive analyses [of joint attention]. I will say that an analysis is ‘reductive’ if it is possible to say which individualistic states of x matter here, without this already implying that there is joint attention involving x and another. By contrast, an analysis is ‘relational’ if ascribing the relevant psychological states to x already implies that there is someone with whom x is jointly attending. (Campbell 2005, 288)

Our reformulation of the Schifferian analysis gives us an account of mutual knowledge that is not reducible to what is known by either of the parties taken individually. The analysis it provides is, in Campbell's terms, a relational one, rather than a reductive one. In this section I shall outline the appeal of a relational analysis of mutual knowledge, and show how the reformulated Schifferian analysis fits into this picture.

There is a sense in which relational analyses of mutual knowledge constitute a break from the more traditional attempts to characterise mutual knowledge.5 John Barwise, for instance, takes it as a given that the aim is to provide ‘characterisations of [mutual] knowledge in terms of ordinary knowledge’ (Barwise 1988, 366). By claiming that mutual knowledge is irreducible to ordinary knowledge, however, the relational approach clearly eschews this order of explanation.

There are both dangers and virtues in doing so. On the one hand, by construing mutual knowledge in a basic relational manner, then the use to which mutual knowledge can be put in practical situations, such as the General's Paradox, becomes all but conspicuous. For, on a relational analysis, mutual knowledge is a matter of the participants standing in a relation to each other as joint perceivers – as co-subjects – rather than as objects of each other's perceptual knowledge. Although it is far from obligatory to put the matter in these terms, one might be tempted to say that on a relational analysis we have a dual-subject psychological state, to the effect that S-and-A knows6 that p, where such a psychological state is not reducible or equivalent to S knows that p and A knows that p.7 The idea being (to put it as baldly as I dare) that, as co-subjects, the participants are in direct, unmediated cognitive contact with each other to the extent that they literally share the mental state of mutual knowledge.8 Understood in this way, the participants have automatically reached the very stage towards which the original iterative approach could only strive in an infinite number of steps. As such, to the extent that mutual knowledge leads to rationally coordinated action, then, on a relational analysis, the two participants are fully equipped to launch the kind of coordinated attack required of the General's Paradox: together they know that it is time to attack, so together they can attack.9

On the other hand, however, one cannot help but feel glared at by the attendant worries that come with such a seemingly straightforward solution to the General's Paradox. Firstly, the idea of a literally shared mental state might seem unintuitive and ontologically suspect, to say the least. I shall postpone discussion of this matter to the following section. Secondly, one might object that a mere appeal to something like a dual-subject psychological state ends up isolating that joint psychological state from psychological theory in general, and thus, for our purposes, is next to useless. That is to say, although perhaps an unanalysed notion of mutual knowledge might, and indeed does, have a place-holding role to play in accounts of joint action and rational coordination,10 one might object that the kind of reductive analysis that Barwise deemed to be constitutive of an account of mutual knowledge is required after all if our aim is to understand the role that mutual knowledge has to play within ordinary psychological theory.

There is substance to this second worry, but falling back on a reductive analysis is not the only way of assuaging it. Rather, what the relational analysis suggests is an inversion of the more traditional reductive picture, whereby, rather than characterising mutual knowledge in terms of ordinary knowledge, one might take mutual knowledge as primitive and try to characterise the ordinary knowledge that follows from it in terms of that primitive.11

It is here that the reformulated Schifferian analysis can come into play as a distinctive version of a relational approach to mutual knowledge. For the iterations that follow from the analysis can be seen as a point of contact between, on the one hand, a primitive notion of mutual knowledge, fit to take an unadulterated role in coordinated reasoning, and, on the other, ordinary, intrapersonal psychological explanation.

The idea is as follows. By taking mutual knowledge as a primitive relational property holding of two (or potentially more) individuals, then we allow it to take the same free role in accounts of coordinated action and reasoning as an unanalysed notion of mutual knowledge would take: the point being that there is nothing further to analyse – it is simply a semantically primitive psychological state.12 Nevertheless, according to the reformulated Schifferian analysis, this primitive psychological state does not stand in isolation from ordinary psychology, since it suffices to yield the catalogue of iterations that Schiffer's original analysis assumed to be necessary for mutual knowledge. The difference now, however, being that these iterations are not necessary for mutual knowledge itself (the primitive relational property suffices for that inter-personal notion) but, rather, suffice to characterise the individuals’ intra-personal understanding of the situation.

Now, this move has the following boon. Just so long as the iterations were seen as necessary for securing the openness of mutual knowledge, then any attempt to offset the threat of psychological implausibility by appeal to tacit or implicit knowledge would have been inadequate. Schiffer, for instance, defends the iterations as ‘harmless’ on the following grounds:

First, it is no objection to the claim that S knows that p that the thought that p never once entered S's head. For example, I trust that it is true of each philosophy don in Oxford that he knows that his maternal grandmother was never married to Benito Mussolini. Second, it is no objection to the claim that S knows A knows S knows A knows S knows that p that S may have to be ‘convinced’ or ‘brought to see’ that he is entitled to this claim. (Schiffer 1972, 36)

What Schiffer appears to be arguing here is that the infinite iterations can be implicitly known by the participants even though neither participant explicitly entertains them (he may have to be ‘convinced’ or ‘brought to see’ that he is entitled to the claim). By taking this line, Schiffer can effectively offset the worry about the ‘psychological implausibility’ of agents entertaining an infinite number of mental states.

However, given that what we are trying to characterise is the kind of openness that would, for example, enable two generals to coordinate an attack in a life-or-death situation, then this response does not fit the bill. Implicit knowledge – understood as that which is logically implied by what is explicitly known (Dennett 1982/1983, 216) – would not suffice for the openness required for joint action because, not only would each agent have to recognise these implications him/herself, but he/she would also have to be assured that the other did so too. This is because ‘implicit’ knowledge (as opposed to ‘explicit’ or even ‘tacit’ knowledge) is exactly that which needs to be drawn upon, and made explicit, in order to have any effect on the agent's choices and actions. That is to say, one can explicitly know that p, and explicitly know that p⊃q, and thus implicitly know that q. But, if one is not in some sense aware that one is entitled to draw that final claim q, then that claim will remain idle in the agent's mental economy. In effect, then, the further step of being ‘convinced’ that one is entitled to the claim of mutual knowledge might seem to be exactly what would be required for the ‘openness’ needed for joint action: without it, mutual knowledge would be idle. Yet, in fact, the further step could not thereby provide an effective solution to the problem of mutual knowledge insofar as it arises in the General's Paradox, because the paradox simply reappears at another level: each would now need to be convinced that the other was equally convinced that they were entitled to the claim of mutual knowledge. Thus, Schiffer's suggestion that mutual knowledge can be reductively analysed away in terms of an infinite string of embedded knowledge, is stuck between a rock and a hard place: in order for a mutually known fact to constitute an element fit to take a role in practical reasoning it would need to be more than implicit; yet, if it is more than implicit, it either falls foul of the problem of ‘psychological implausibility’, or it simply pushes the paradox one step back. So, it seems that we cannot identify mutual knowledge itself with the infinite string of iterations, as Schiffer suggests we can.

The matter, however, is somewhat different once we suppose that the sole role that the iterations play is in drawing out consequences in intra-personal reasoning. That is to say, by allowing that the primitive relational property suffices, on its own, to enable coordinated action, the iterations can take on just the kind of implicit, ‘harmless’ role that Schiffer envisaged for them. Phenomenologically, at least, this would seem to be the right way of looking at things: in a case of genuinely open, mutual knowledge, both Generals should feel unreflectively assured that they are in a position to coordinate an attack, without needing to undergo unfulfillable mental gymnastics. Of course, the particularly jittery – or philosophically minded – General might, on reflection, start to wonder whether things really are as out-in-the-open as they seem. But, for such a General, there is an epistemically empowering procedure, implicit within his relational situation, which can be brought to the fore if he so wishes.13 That is, there are internal reasons there for the generals to exploit in order to reflectively justify their coordinated attack (an infinite number of them, indeed), but there is no need for those reasons to be exploited for the generals to rationally coordinate their attack. If this is the correct way of looking at things, then there is no need to suppose that the iterations are explicitly represented. The iterations, rather, could just be seen as being an implicit procedure which could be brought to use if need be, much like, for instance, once one has the ability to count by adding 1, then one can tacitly know that 10,546 follows on in the sequence from 10,545, without needing to represent this fact explicitly.

4. Demystifying mutual knowledge: mutual knowledge and joint attention

I return now to the lingering worries about the legitimacy of a positing an irreducible, semantically primitive state.

At least part of the reason why a nonreductive notion of mutual knowledge initially appears unexplanatory or mysterious might be owing to the fact that it does not fit easily into what Jane Heal calls the ‘mainstream analytic understanding’ (Heal 2005, 34) of social cognition as a matter of employing a ‘theory of mind’. Such a view of social understanding construes understanding another as essentially a matter of a third-person, ‘spectatorial’ engagement – the person with whom one is engaging is, on this view, understood not from the position of a second-person interaction, but as essentially just another denizen of the physical universe, albeit an ‘object’ towards which one will make use of a dedicated and complex set of concepts and strategies. Viewed from this paradigm, an unreduced notion of mutual knowledge would appear to be something of an anomaly: it would be a conceptually distinct species of understanding that would have no counterpart with the way in which we generally understand matters; neither matters regarding the physical world (‘x knows that p’), nor matters regarding the psychological states of others (‘x knows that [y knows that [p]]’). Our proposal would, rather, have introduced an apparently anomalous type of psychological state, or propositional attitude, of the form ‘x and y mutually know that p’ that is not to be understood as a simple conjunction of what two people know, but as a basic attitude that necessarily relates two subjects to a proposition or fact.

It would not be understating things to say that, viewed from the paradigm of a traditional, functionalist approach to social understanding (and, indeed, to cognition in general), the proposal would appear to be at best highly anomalous and mysterious, and at worst, simply mistaken. And it seems likely that this is a central reason for why an unreduced notion of mutual knowledge was not even considered an option for the traditional analytic analyses of mutual knowledge mentioned by Jon Barwise.

Nevertheless, the traditional functionalist picture is not quite the unchallenged monolith it once was, and recent work in cognitive science, especially developmental psychology, might well serve not only to validate a relational, non-reductive notion of mutual knowledge, but also allow us to develop a picture in which that concept lies at the centre of both our cognitive and socio-cognitive lives. For instance, Jane Heal hypothesises that:

… when we delve back to the conceptualisations or proto-conceptualisations from which fully-fledged psychological competence arises, what we find is not the infant with a sense of itself as contrasted with an external world, which then turns out to contain a variety of different things, animate and inanimate, with which the infant must learn to deal. Rather, the normal infant from the earliest moments has a sense of him or herself as part of an ‘us’, an ‘us’ whose shared life is already moving forward in the context of an external, spatial world. What the infant learns with growing conceptual sophistication is to play his or her part in that life. Playing that part requires mastering both the nature of the material world and the intricacies of the social world. (Heal 2005, 41)

Variants on this suggestion have informed much recent discussion regarding folk psychology and infant's psychological development,14 and what it would be useful to reflect on, for our purposes, is this. First, by situating the notion of mutual knowledge within a developmental perspective, the apparent air of mystery which hangs over the notion of mutual knowledge can be seen to dissipate somewhat. Mutual knowledge – regardless of whether taken as relational, reductive or unanalysed – has been recognised to be both an indispensable concept throughout a number of intellectual disciplines,15 yet also, as we have seen, notoriously difficult to capture in a satisfactory way. By placing an unreduced notion of mutual knowledge at the fulcrum of a child's communicative and psychological development, in the way hinted at by Heal, however, the anomalous nature of the concept becomes, to a degree, more explicable: mutual knowledge is not ‘just’ another cognitive state, or folk psychological concept, but is something that lies at the constitutive heart of our cognitive and socio-cognitive lives, one that serves to drive and condition our skills at dealing with others and with the material world; the child is seen as amassing more and more skills to enable her to exchange and receive information, and achieve shared understanding on matters distal and proximal, hypothetical and real, psychological and material.

Further, by taking this dynamical developmental view – where mutual knowledge is conceived as having not just a role to play in particular communicative and proto-communicative exchanges, but also as a driving principle behind a child's psychological development – then there is the possibility of tackling two extant problems that have come to the fore in recent work on infant social understanding. The first problem has been outlined by Richard Breheny:

Prominent accounts of language use and human communication face something of a dilemma. The dilemma arises because it is assumed (a) that basic communicative situations essentially involve propositional attitude-like states of the participating agents and (b) that competent language users have the conceptual abilities to represent agents as being in such states and make folk-psychological inferences about agents so represented. These assumptions conflict with one or more robust findings in developmental psychology: that children below the age of four years do not yet possess these abilities. The conflict arises because it is widely agreed in research on language development that children below the age of three years are competent language users and communicators in the basic sense. (Breheny 2006, 74)

The second was outlined sometime ago by Jerry Fodor16:

Much of the recent discussion of [children's acquisition of a folk-psychological theory] has centred on data that appear to disclose a stage-like discontinuity in the child's progress: children of about three years regularly fail the ‘false belief’ task, whereas children of about four years regularly succeed. These data have been widely viewed as implying that the younger child's theory of mind differs in quite radical ways from the one that the older children share with adults; for example, that the young child either lacks the notion of belief entirely, or fails to realise that beliefs can misrepresent the world, and thus lacks a notion of false belief. If any of this is correct, then we are badly in need of an account of how, somewhere between the ages of three and four, maturational processes, or learning processes, or a combination of the two, could eventuate in what amounts to a conceptual revolution in the child's theory of mind. It would be putting it mildly to say that, as things now stand, there is no such account on offer. So there appears to be something like a crisis in the theory of cognitive development. (Fodor 1992, 109)17

Clearly, both these problems arise out of attempts to apply standard functional, ‘theory-theory’ accounts of social understanding to children and infants. Although the matter is far from closed, it is becoming increasingly clear that standard accounts have more than a few problems in this area. But, if we are willing to countenance the suggestion that a central, constitutive element of understanding others is the ability to mutually know a particular fact or proposition, and that mutual knowledge is itself a brutely simple semantic state (even if, in most adult exchanges, the means by which it is often achieved are sophisticated), then both of these problems can be seen to lack the force that they might otherwise have had. That is to say, the young child's ability to exhibit a genuine, if limited, understanding of others might, if the relational account is right, require nothing more complex than an ability to stand in an appropriately responsive relation to another; while the older child's increasingly sophisticated ability to understand others – including their false-beliefs – might well require not so much a ‘conceptual revolution’ in his understanding, but rather a honing of a skill that he has had since early infancy.

Of course, these final sketchy remarks require a lot of beefing up. But, to all appearances, there is something brewing at the foot of cognitive science, and it is my hunch that the role of joint attention in infants is going to have a central role in what is to come. If so, then perhaps the idea of a primitive, relational notion of mutual knowledge will not appear so mysterious after all; and Schiffer's analysis suggests a means of keeping that notion just about within the purview of ordinary, cognitive psychology.

Notes on contributor

Michael Wilby is a Lecturer in Philosophy at Anglia Ruskin University. He recently completed his PhD, on the relation between social cognition and intentionality, at the University of York, UK.

Acknowledgements

Thanks to Tom Stoneham, Mike Beaney, Matthew Ratcliffe, Chris Dowling, Anna Armstrong and the members of the Philosophical Explorations Essay Prize jury for helpful comments and suggestions.

Notes

See also Bach and Harnish (1979).

This is adapted, with a few changes, from Halpern and Moses (1990, 556; quoted in Syverson (2003, 46)).

I have followed Schiffer's convention of referring to a speaker and audience, although the account which follows can be thought to apply just as much to a mutually perceived event as to a mutually perceived utterance.

The term ‘generating property’ comes from Peacocke (2005). I am taking the term to refer to a finite, physically reducible property, true of an individual, that is sufficient for that individual to have knowledge of a particular fact. For instance, an agent facing a tree in broad daylight, has knowledge that there is a tree in front of him due to certain physically reducible properties that hold true of him: e.g. that he has a well-functioning perceptual system, is not cognitively damaged in any significant way, is standing at such-and-such a distance and angle from the tree, etc. I take it that it is a plausible working assumption that these properties can, theoretically, be cashed-out in purely physical terms.

Traditionally, the three major players have been the ‘iterative approach’ (Schiffer 1972); the ‘fixed point’ approach (Harman 1977); and the ‘shared situations’ approach (Barwise 1989).

At the beckoning of Microsoft word-check, I have used the third-person form of the verb, rather than the first person plural, so as to emphasise the notion of a genuinely plural subject. As J. David Velleman puts it ‘a truly “plural subject” ought to be a single subject that isn't singular – or, if you like, a plural subject that isn't a plurality of subjects’ (Velleman 1997, 30).

See, for instance, Gilbert (1989) and Eilan (2007). Campbell's (2005) approach is slightly different, because on his account the co-attenders are ‘present as constituent[s of each other's] experience’ (p. 292). What allows Campbell to make this claim, without lapsing back into the problems that arise when each participant is considered to be the object of the other's perception, is his rejection of a representational understanding of perception in general.

In slightly more (speculative) detail: I would suggest that the idea of a literally shared state of mutual knowledge, should be primarily understood as applying to mutual perceptual knowledge (including mutually perceived utterances), where the shared perceptual knowledge in question is understood as being conceptual, as opposed to non-conceptual (Gunter 2003). The further question of whether that perceptual knowledge should be understood as experiential (rather than purely propositional, a la Davidson) is dependent on the contentious issue of whether the experiential aspects of perception should themselves be understood as conceptual in any sense, a question which is beyond the scope of the present paper.

The claim here is that the appeal to a relational account suffices to give a straightforward solution to the problem of joint action, insofar as that problem arises in the General's Paradox. However, it should be noted that the General's Paradox does not appear to be primarily a problem regarding the conditions required for joint action as such, but rather a problem regarding the necessity of mutual knowledge for joint action (section 1). The proposal here, then, attempts to dissolve the problem of mutual knowledge that drives the General's Paradox, but is silent on what, if any, further matters of co-ordination might be required for an actual launch of a coordinated attack, beyond the mutual knowledge of a mutually understood signal.

See, for instance, Bratman (1993).

As we shall see in section 4, the idea that social-cognitive understanding has its basis in externalised, bodily/behavioural-based relations has a rich history in psychology, especially within developmental psychology. See especially the work in the Vygotskian tradition.

Again, for those who are getting impatient with this seemingly implausible claim, I urge you to postpone judgement until the discussion in section 4.

It is worth pointing out that the possibility that one of the Generals might get jittery about his knowledge does not thereby throw us back into the paradox, such that each must now be convinced that the other is not feeling jittery. The reason for this, is that the worries that such a General might entertain are negative, and do not encompass the state of mutual knowledge per se, but rather express an idiosyncratic need for a complete absence of doubt about that knowledge, and thus involve the prior assumption that there is a state of mutual knowledge towards which one might entertain doubts. To understand the problem of mutual knowledge as a problem regarding the absence of doubt about an item of knowledge (as Davies (1987) does), is to make the mistake of supposing that the problem of mutual knowledge is just a special case of the Cartesian search for philosophical certainty. However, as I argued in section 1, the problem of mutual knowledge does not arise from this traditional form of philosophical scepticism, but rather arises from the need to acknowledge a psychologically expedient state of mutual knowledge, with a role to play in psychological theory and in action. It is no objection to an account of mutual knowledge that the agents in question might have doubts about that mutual knowledge, or make mistakes with regards to it (perhaps one of the Generals just perversely cannot be bothered to attack, despite knowing the conditions obtain), just as it is no objection to the idea that one might have knowledge simpliciter about a particular fact, even though one might reflectively entertain doubts about it, or make a mistake with regards to it.

See especially Bruner (1990), Gallagher (2001), Hutto (2004), Carpendale and Lewis (2006), Ratcliffe (2007), Eilan (2007) and Striano and Reid (2009).

Barwise (1989) mentions game theory, law, and communication theory (p. 198), although he might also have mentioned economics and sociology.

For a more recent version of the worry, see Hutto (2008, 170–3).

Fodor (1987, 132), for his part, does not actually think this problem spells doom for the theory of cognitive development, because he embraces the idea that folk psychological understanding is innate, and thus not in need of development.