Unawareness and Implicit Belief Edward Elliott∗ School of Philosophy, Religion and History of Science University of Leeds December 27, 2018 Abstract Possible worlds models of belief have difficulties accounting for unawareness, the inability to entertain (and hence believe) certain propositions. Accommodating unawareness is important for adequately modelling epistemic states, and representing the informational content to which agents have in principle access given their explicit beliefs. In this paper, I develop a model of explicit belief, awareness, and informational content, along with an sound and complete axiomatisation. I furthermore defend the model against the seminal impossibility result of Dekel, Lipman and Rustichini, according to which three intuitive conditions preclude nontrivial unawareness on any possible worlds model of belief. 1. Introduction Iggy is an agent, who may or may not be ideally rational. In his capacity as an agent, Iggy must have at least some of what we'll refer to broadly as epistemic attitudes regarding some range of propositions. There will be some propositions that Iggy knows, some he believes, some he disbelieves, and some that he's contemplated but he's not sure about. Question: for any proposition φ, must Iggy have some epistemic attitude or other regarding φ? Most will want to say that he need not. To know, believe, be uncertain about, or even suspend judgement regarding a proposition requires at least the capacity to entertain that proposition, to represent it some way or another in thought-and there may be some propositions that Iggy has never entertained, will never entertain, and perhaps cannot even entertain. For example, suppose that Iggy belongs to an isolated tribe located deep in a far off jungle, which has had no contact with any other cultural group for hundreds of years. Most people wouldn't want to say that Iggy knows, believes, or is uncertain about the proposition that we would ordinarily express were we to utter the sentence 'The habitat of the eastern grey kangaroo extends as far north as the Cape York Peninsula'. It would furthermore be unnatural to say that Iggy is agnostic, or has suspended judgement, on the matter of eastern grey kangaroo habitats. Suspension of judgement is typically understood as a kind of ∗Email: e.j.r.elliott@leeds.ac.uk. Acknowledgements removed. mental act that one might perform upon entertaining some proposition, when one decides that there's insufficient reason to believe one way or the other. But Iggy might not even know that there's a proposition here for him to suspend judgement about. Indeed, he may not have the basic mental resources necessary to even think the thought, let alone decide whether he has sufficient evidence to believe it. So let us admit, at least in principle, that there may be some propositions of which Iggy is unaware. I use this phrase semi-stipulatively: to be unaware of a proposition φ is to lack the basic representational resources required to entertain φ-thoughts or otherwise have what were above called epistemic attitudes with φ as their content. To be aware of φ is to not be unaware of it.1 If this kind of unawareness exists, and is widespread, then it seems to be the kind of thing to which any developed formal theory of the epistemic attitudes ought to pay attention. Consequently, after saying more in the next section about what I mean by 'awareness' and why we should care about it, in Section 3 I will characterise a class of models-Possible Worlds Awareness (PWA) models-aimed at representing differences in states of awareness and unawareness, and how these interact with the epistemic attitudes and questions of information access. I focus on modelling the interaction of unawareness with the attitudes of belief and knowledge, though much of my discussion will also be of direct relevance to Bayesian models of uncertainty under unawareness (see, e.g., Bradley (2017), Cozic (2016)). As the name suggests, PWA models adopt the kind of coarse-grained approach to propositions that's generally associated with possible worlds semantics. The models are therefore subject to the influential triviality result of Dekel, Lipman and Rustichini Dekel et al. (1998). In Section 4, I describe Dekel et al.'s result and some natural extensions thereof, while in Sections 5–8, I critically discuss the intuitions and implicit philosophical assumptions that motivate it. As we will see, the force of their result depends much on the intended interpretation of the model, and (moreover) on questions regarding the nature and granularity of mental content. 2. Background Before anything else, let me first say more about how 'awareness' is understood in the wider literature and how I will be using the term (Section 2.1), and why we should care about awareness so-understood (Section 2.2). 2.1 What is awareness? In their seminal work on the topic, Fagin and Halpern Fagin and Halpern (1988) were intentionally ambiguous with their use of 'awareness'. For some of their discussion, the term is used in connection with a notion of entertainability that's 1 It is worth emphasising the distinction in English between being aware that something is the case, and being aware of some thing. The former is synonymous with knowing a true proposition: Iggy is aware that φ iff φ is true and Iggy knows it; Iggy is unaware that φ iff φ is true and Iggy doesn't know it. On the other hand, Iggy would be aware of a proposition in something like the way he might be aware of an idea, an argument, or an oncoming train: by consciously attending to (a representation of) the object of one's awareness, or at least being in a position to do so. 2 at least similar to that which I characterised above. However, they also use it to mark a distinction between the beliefs that a subject is consciously attending to versus her background beliefs; and between those possibilities that the agent has considered and incorporated into her deliberations versus those she's not yet considered or perhaps considered but forgotten about. And in still other cases, Fagin and Halpern use 'awareness' and 'unawareness' to distinguish between those conclusions which can and cannot be derived from a 'database' of stored beliefs within a given amount of time. In other words, we have had from the start four distinct (though by no means unrelated) notions all being discussed under the same heading. Roughly, we have awareness of φ as: Entertainability: Being able to entertain the possibility that φ. Attentional: Consciously attending to the possibility that φ. Deliberative: Having considered the possibility that φ in one's reasoning. Derivability: Being able to derive that φ within a given time. All four admit of differences in degree and further precisifications. For example, some possibilities might be more or less at the forefront of Iggy's conscious attention, and some things might be more or less easy to derive, or derivable only given certain conditions. Unfortunately, matters have not in general become clearer since Fagin and Halpern's essay. Awareness is often characterised in the first instance as a "lack of ability to conceive", or a "lack of concept" (e.g., in Heifetz et al. (2008), Schipper (2014), and Walker (2014)), which perhaps suggests that something like the entertainability sense is intended. Moreover, many of the formal conditions that awareness is standardly taken to satisfy seem to fit best with something like the entertainability sense, whereas they fit relatively poorly with the attentional, deliberative, and derivability senses unless strong assumptions are made about the capacities of the agent in question. (See, e.g., SYM, CONA, and especially the AGPP principle; these will be discussed in Section 3.) However, characterisations of 'awareness' in the literature tend to be perfunctory at best, and detailed exposition on the matter is rare and sometimes conflicting, making confident interpretation difficult.2 I note all this in order to draw attention to the fact that I will be understanding 'awareness' in this paper in a rather specific way-viz., as a particular instance of the entertainability notion. This will be important throughout my discussion, and I certainly do not want to claim that all of my conclusions apply with equal force to every understanding of 'awareness' present in the literature. Hence, it will be useful to get clearer on just what 'awareness' means in the present context. The following is a good rough-and-ready way to picture the kind of thing that I have in mind. Suppose we adopt a very näıve version of the Language of Thought theory combined with the Representational Theory of Mind, at least as applied to the epistemic attitudes (see Pitt (2018) for details). We assume that for Iggy, and any agent like him, mental representation is underwritten by 2 For exceptions to this rule, see Fritz and Lederman (Fritz and Lederman) and Sillari (2008a). An orthogonal distinction in kinds of awareness, between awareness that attaches to propositions versus awareness that attaches to sentences or theorems, is also discussed in Fritz and Lederman (Fritz and Lederman). 3 an essentially linguistic system of meaningful symbols. Moreover, this symbolic 'language' looks a lot like a simplified version of English. Somewhere inside his head Iggy has, or is capable of 'uttering', various word-like representations which might denote individuals, properties, and relations, as well as perhaps operators, quantifiers, connectives, and so on. Call these his concepts. Iggy's concepts can be composed in a systematic way into sentence-like structures, which have the propositional contents that they do as a consequence of the meanings of their constituent concepts and the ways that those concepts are arranged. For Iggy to have any epistemic attitudes whatsoever regarding a proposition φ, he must be psychologically related to some sentence composed out of concepts he possesses which means that φ (cf. Field (1978), Fodor (1975)). Presumably, different agents will tend to have different concepts-different mental vocabularies, as it were. Iggy lacks any concepts that pick out kangaroos, for example, whereas the average Australian would typically possess such a concept. Call the concepts that Iggy does have his basic representational resources. If we were to accept all this, then we could say that Iggy is aware of φ, in the sense that I have in mind, just in case he possesses concepts sufficient to compose a sentence that means that φ. Thus, he need not be presently and consciously attending to φ in order to be aware of it, and the entertaining itself might require significant mental effort-in fact it may even be physically impossible. So, for example, Iggy may be aware of the proposition expressed by the near-infinitely long English sentence 'Iggy believes that Jiggy believes that Iggy believes that. . . that Jiggy believes that φ', merely by being able to represent himself, his colleague Jiggy, φ and the belief relation-even if actually 'uttering' the relevant sentence in his Language of Thought would require many more neurons than could fit in any ordinary-sized human brain. Now, I don't myself attach much credence to that picture of how the mind works. Even if mental representation is typically language-like in the way that the picture supposes, it's highly doubtful that the content of every epistemic attitude that Iggy has will itself be the meaning of some sentence-like symbol stored somewhere in his head (see esp. Cummins (1986), Cummins (1987)). (Not that advocates of the Language of Thought and/or the Representational Theory of Mind have ever thought otherwise-I called it "näıve" for a reason.) Furthermore, the picture gives the potentially misleading impression that the possession of certain very basic logical concepts is a mere contingent matter, a question of whether the agent has a particular word in their mental lexicon. Had Iggy merely lacked the concepts for conjunction and negation, say, then even if he could entertain φ and ψ individually, he might still have been unable to entertain ¬φ and φ ∧ ψ. Compare, for example, an account where the format of our mental representations is more map-like in nature (cf. Camp (2007), Lewis (1994)). A map that represents that Sydney is northeast of Melbourne and that Sydney is northeast of Canberra automatically represents that Sydney is northeast of Melbourne and Canberra, and it doesn't need a special symbol denoting conjunction to do so-the propositional contents of maps are closed under conjunctions just as a consequence of the way that maps represent the world in general. Similarly, a map-like representational system that can represent that Sydney is on the coast is, by that very fact, able to represent that Sydney is not on the coast (e.g., simply move whatever represents the city inland); and if 'X' marks the spot where Blackbeard's treasure is, then the absence of an 'X' marks the spots 4 where it isn't. Maps don't usually have symbols on them that represent negation or absence, but they still manage to tell us a lot about what is and isn't the case without them. Hence, Iggy's ability to entertain ¬φ and φ ∧ ψ might be an immediate consequence of his ability to entertain φ and ψ, due to the format that his mental representations take and the operation of the systems that use them, rather than being dependent on the presence of some special mental symbols that specifically denote logical operators and connectives. (I'll say more in relation to this in Section 3.1) So the näıve picture isn't perfect, but it does manage to capture the core idea of what I mean by 'awareness' well enough, and it does so in a relatively intuitive way. In general: Iggy can represent different ways the world might be. Presumably, his capacity to represent those different ways depends on his having some stock of basic representational resources that can be systematically restructured or recombined to represent new possibilities (as argued in Fodor (1975), Fodor (1987)). This seems highly plausible regardless of the exact format that our mental representations take, map-like or sentence-like or otherwise. Some of those possibilities that Iggy has the basic resources to represent he never will in fact represent, and some may be too complex or time-consuming to actually represent given his limited cognitive resources-but there will still be a good sense in which he possesses the basic representational resources required to do so. On the other hand, some propositions Iggy will be unable to represent by virtue of lacking the requisite representational resources; some distinctions between ways the world might be won't even be on Iggy's mental radar. Of these, he is unaware. 2.2 Why should we care about awareness? I am inclined to think that all four senses of 'awareness' are important, each in their own way and to varying extents. So why should we care about the particular kind of awareness qua entertainability that I've specified? Here's one reason: it is the weakest of the four senses. One gets to be aware of φ in the attentional, deliberative, or derivability senses only if one is aware of φ in the entertainability sense. Moreover, awareness in any of the other three senses will tend to vary dependent on the rational capacities of the agent in question. To put the point roughly: amongst those propositions that Iggy can in principle entertain, he will be able to consciously attend to more, incorporate more into his reasoning, and derive more faster, the bigger and better his brain is. What an agent can entertain in principle thus sets bounds on their awareness in the other senses, bounds which depend only on that agent's basic representational resources (and not on their capacity for reasoning, attentional resources, etc.). Furthermore, if unawareness in this sense is widespread,3 then it is an important phenomenon that ought to be represented in our best models of the epistemic attitudes-including (but not limited to) our models of ideally rational agents. Consider, for instance, the standard possible worlds models of belief in the style of Hintikka Hintikka (1962). We begin with a rich space of possible worlds, Ω, subsets of which represent the contents of our attitudes.4 Our agent 3 See Section 6 for discussion on how widespread we should expect 'awareness' to be. 4 By my use of 'possible worlds,' I mean to exclude specifically worlds which are either not maximally specific, or inconsistent with classical propositional logic. Much of what I say will not hang on any specific account of what 'worlds' are; though see Section 6 for discussion on some issues that depend on exactly which worlds constitute Ω. 5 Iggy (i) can then be associated with a binary relation Ri on Ω which for each world ω (effectively) picks out a proposition Ri(ω) containing all and only those worlds that Iggy considers possible at ω. We then say that Iggy believes that φ at ω iff φ is true at every world in Ri(ω). Problem: if Ri(ω) is even the least bit specific, then a vast number of things will be true at every world in Ri(ω), and we can't expect Iggy to believe all of them-and not just because he may not be a very good deductive reasoner. Make Iggy as logically gifted as you like, give him unlimited working memory, and all the time in the world; still, if he can't even entertain certain propositions then it's wrong to say that he believes them. A simple example of what I mean. Suppose that there are only two atomic propositions, φ and ψ, and exactly four possible worlds ω1, ω2, ω3, ω4 in Ω corresponding to the different combinations of φ and ψ: φ ∧ ψ φ ∧ ¬ψ ¬φ ∧ ψ ¬φ ∧ ¬ψ ω1 ω2 ω3 ω4 Suppose that Ri(ω1) = {ω1, ω2}. This is just the set of worlds where φ is true, so we can say that Iggy believes φ. Now presumably, if Iggy believes φ then he can entertain φ; and if he can entertain φ then he can distinguish between φ and ¬φ. So it looks fair to say that Iggy is able to entertain ¬φ at ω1. Likewise, since he can entertain φ and ¬φ, he can entertain φ ∧ ¬φ and φ ∨ ¬φ. The latter is true at every world in Ri(ω1), and it's reasonable enough to expect that Iggy believes φ ∨ ¬φ. At the very least, he can certainly reason his way to that conclusion given his other beliefs without much difficulty. However, imagine that Iggy is wholly unaware of the ψ/¬ψ distinction. Perhaps specifying the distinction requires some concept that Iggy lacks. Would we be happy to say that Iggy believes φ ∨ ψ and φ ∨ ¬ψ at ω1? Both are true at every world in Ri(ω1). But for Iggy to believe φ ∨ ψ or φ ∨ ¬ψ would require him to make distinctions between possibilities which, ex hypothesi, he is unable to entertain. For the sake of clarification, it will be helpful to discuss two very common responses to this kind of example before we move on. 1. First, you may have the thought that it would be equally wrong to say that Iggy is able to entertain ψ∨¬ψ, even though ψ∨¬ψ picks out just the same set of possible worlds that φ ∨ ¬φ does; thus the possible worlds model of content is inadequate for understanding states of unawareness in any case. If you do have that thought, then you will probably also think that possible worlds theories of content suffer from problems relating to hyperintensionality more generally. And you may well might be right. (I'll consider this matter again in Section 8.) But the phenomenon that I want to discuss in this paper is very general, and arises regardless of how granular you think mental content is. 6 Even if you accept a very coarse-grained conception of propositions-i.e., one such that φ ∨ ¬φ and ψ ∨ ¬ψ are by virtue of their classical logical equivalence the very same proposition just described or picked out in two different ways, or under different modes of presentation-then you will still need to deal with the possibility of unawareness. It's not plausible to say that Iggy believes φ∨ψ under any mode of presentation. (Henceforth, we'll refer to advocates of coarse-grained theories of content as possible worlds theorists.) 2. Second, you may think that the relational model has no real problems with unawareness at all-at least, not once it's interpreted in the right way. After all, it is frequently said that the relational model is best viewed as a way of representing an agent's implicit rather than explicit beliefs, where the former are understood to capture something like the informational content contained in the beliefs that the agent actually has 'stored' somewhere in her head. Thus, if we interpret the model this way, there need not be any issues whatsoever with claiming that Iggy implicitly believes both φ ∨ ψ and φ ∨ ¬ψ, merely on the basis of the assumption that he explicitly believes φ. After all, the former are in a perfectly well-defined sense built into the informational content of the latter. Now I want to be clear that I have no objections to interpreting the relational model like this; if that's the kind of thing you want to represent, then so be it. Nevertheless, in response to the point I do want to introduce a distinction that I think is very important when it comes to thinking about the informational content of our epistemic attitudes, and one which underscores the need for incorporating some representation of unawareness into our formal models. Say that in a broad sense, the informational content of a set of (explicitly represented, or 'stored') epistemic attitudes consists exactly in anything and everything that's entailed by the contents of those attitudes, whether separately or in conjunction. It's in this sense that it would be unproblematic to say that Iggy's explicit belief that φ "contains the information" that φ ∨ ψ and φ ∨ ¬ψ. After all, the information is there, just waiting for someone with the appropriate representational resources to draw it out. But Iggy is not one of those with the required representational resources. For him, some of that broad informational content is in a very strong sense inaccessible: it is not available for him to use in reasoning, inference and decision making (cf. Yalcin (2018)). For all rational purposes, that information is for him completely invisible. Iggy may not even know that there's a distinction that he's failed to consider, and no amount of reasoning with the distinctions that he does have will lead him to recognise what he's been missing. And information which is wholly invisible to Iggy is, from his own perspective, not really information at all. It seems clear then that there's an important theoretical role for a second, narrow notion of informational content-one that's specifically dependent upon awareness qua entertainability. This point is, I think, especially straightforward if mental representation does turn out to be sentence-like: obviously, and regardless of how rational he is, Iggy cannot possibly derive from that which he explicitly believes any sentences the content of which can only be expressed using concepts that he lacks. But the distinction between broad and narrow informational content is still useful even if mental representations are map-like. Imagine, for example, that Iggy and Jiggy 7 each has their own 'mental map', Mi and Mj respectively, which have slightly different keys but are in all other respects similar: Mi Mj KEY: \ = coastline S = Sydney k = kang KEY: \ = coastline S = Sydney k = kangaroos S S Though both maps explicitly represent that Australia is shaped like so, and that Sydney is located on its southeast coast, Mj represents a somewhat more specific possibility than Mi does: Mj explicitly represents that there are no kangaroos to be found anywhere within or around the country; whereas Mi is neutral on the matter of whether there are any kangaroos, and if so, where they might be located. But, more importantly, given the richer key associated with Mj , Jiggy could represent the presence of kangaroos at any number of different locations, by simply adding a 'k' on the appropriate part of the map. Iggy's map, by contrast, lacks the resources to represent the presence or absence of kangaroos-draw a 'k' on any part of Mi and you're left with a meaningless squiggle, or perhaps a very odd coastline, and in Iggy's case the absence of a 'k' doesn't represent anything at all. So now consider: φ = Sydney is on the southeast coast of Australia. ψ = There are kangaroos as far north as the Cape York Peninsula. The 'explicit' contents of both Iggy's and Jiggy's mental maps entail that φ∨ψ and φ ∨ ¬ψ, since they both clearly represent at least φ. We can all agree that φ∨ψ and φ∨¬ψ belong to the broad informational contents of both maps. But Jiggy knows what it would take for φ ∨ ψ and φ ∨ ¬ψ to be true-he has the capacity to represent all the different ways those propositions might be true, he can imagine and entertain those possibilities and play around with them in his hypothetical reasoning. Iggy, by contrast, cannot: his map entails φ∨ψ and φ∨¬ψ, but he's not aware of these entailments. The fact that Iggy's mental map includes φ ∨ ψ and φ ∨ ¬ψ in its broad informational content is of no rational use to him even in principle, given his relatively limited conceptual resources. So I take it that the distinction between broad and narrow informational content is theoretically well-motivated, and independent of questions concerning representational format. Furthermore, if we draw the natural corresponding distinction between broad and narrow implicit beliefs, then it's arguably the latter which fit better with the central function that the notion of 'implicit beliefs' were originally introduced to play-viz., as a way of marking a distinction between that information which is explicitly represented by a thinker versus that which is not explicitly represented but is still accessible for use on the basis of that which is.5 5 See Braddon-Mitchell and Jackson (1996), pp. 185ff, for an illuminating discussion of the notion of implicit belief in the context of sentence-like and map-like theories of mental representation. 8 But it doesn't really matter what words we use. What's important is whether the distinctions are useful-and the fact that an agent's narrow implicit beliefs have much closer ties to matters concerning the agent's rationality suggests that they are. If Iggy fails to act appropriately given his narrow implicit beliefs then he is rationally criticisable, at least to some (possibly very small) extent. On the other hand, if Iggy fails to act appropriately given his non-narrow broad implicit beliefs, then any such criticism seems wrong-headed. One isn't the least bit epistemically blameworthy for lacking certain representational resources, and not having a way to mentally represent kangaroos doesn't make Iggy irrational under any remotely reasonable conception of rationality. It's worth noting also that given a narrow understanding of informational content, there is still plenty of room for Stalnaker's Stalnaker (1991) useful distinction between content which is more or less accessible for different kinds of applications. It's plausible to think that the way we represent information about the world makes a difference to how easily we might access it and reason with it on a given occasion, and consequently that at any time we probably only attend to a fragment of the total information we have represented across the full range of our epistemic attitudes. As Stalnaker notes, recognising the limits of attention and information access goes some of the way towards solving the problems of logical omniscience that all possible worlds theories of content have to deal with (cf. also Egan (2008), Lewis (1982), Stalnaker (1984), Yalcin (2018)). But any such a solution cannot be complete without also recognising the role that awareness qua entertainability has as a precondition for genuine informational access simpliciter. In summary: awareness qua entertainability is theoretically important, as are the notions of narrow informational content and narrow implicit belief which directly depend upon it. Hence, in the next section I aim to outline a class of models that will distinguish appropriately between arbitrary agents' (i) states of awareness qua entertainability, (ii) their explicit knowledge and belief, and (iii) the narrow informational content that's in principle accessible given those explicit attitudes, all within the context of a possible worlds theory of content. Ultimately, it would be nice to have a model that incorporates all of the different kinds of unawareness-variabilities in attentional awareness, for example, with perhaps a sliding scale for different degrees of informational accessibility and deliberative capacity, and multiple belief states at once to represent fragmentation. It would in any case be useful to have some representation of (i) what information an agent has access to in principle-what they could derive from their explicit beliefs if they were ideally rational-from (ii) that information that the agent does have access as a matter of fact, and (iii) that information that's readily available (or ready-to-degree-x available) to the agent given their bounded capacities. Each notion has a role to play in explaining action, reasoning, and ideal versus bounded rationality. But that is much more than I can hope for here, and an initial model for just one small part of that much larger picture will have to suffice here. 3. Possible Worlds Awareness To construct our model, we will first need a formal language. Given a finite set of agents Ag = {1, . . . , n} and a countable set of atomic propositions Φ (with 9 typical element p), we characterise LAXI(Φ) by the following grammar: φ := p | ¬φ | φ ∧ ψ | Aiφ | Xiφ | Iiφ, where p ∈ Φ and i ∈ Ag. We abbreviate with '∨,' '→,' and '↔' in the usual ways, and we will let '>' stand for p ∨ ¬p, and '⊥' for p ∧ ¬p. 'Aiφ' should be read as saying that i is aware of φ. (Henceforth I will simply say 'awareness' rather than 'awareness qua entertainability'.) I will leave the interpretation of the X operators ambiguous for now, between an explicit knowledge and an explicit belief reading. The I operators are used to stand for either narrow implicit knowledge or narrow implicit belief, depending on how we choose to read the Xs. To make things easier, I'll focus just on the belief interpretation for this section and the next. I will have more to say on the different interpretations of X and I as we progress through later sections. As noted, the usual way of generating a semantics for epistemic modals involves assigning a relation Ri to each agent i which picks out those worlds i considers possible at any given world ω. However, I will not assume that agents' beliefs satisfy any closure conditions except closure under necessary equivalence (which is essential to any possible worlds theory of content). Nor will I assume that any given agent is irrational. I want the model to be flexible enough to allow for both rationality and irrationality of explicit belief. Thus, in characterising agents' explicit beliefs, I will use the strictly more general neighbourhood structures of Montague (1970) and Scott (1970). A standard neighbourhood model M consists in (i) a set of worlds Ω, (ii) a set of what we will call belief functions Xi (one for each i ∈ Ag) which map each world ω to a (potentially empty) set of (potentially empty) subsets of Ω, and (iii) a propositional valuation function π : Φ 7→ 2Ω. The satisfaction conditions for φ in the sub-language LX(Φ) can then be given as follows: M,ω |= p iff ω ∈ π(p), for p ∈ Φ M,ω |= ¬φ iff it's not the case that M,ω |= φ M,ω |= φ ∧ ψ iff M,ω |= φ and M,ω |= ψ M,ω |= Xiφ iff {ω′ : M,ω′ |= φ} ∈ Xi(ω) If we let '‖φ‖M ' (the truth set of φ in M) refer to the set of worlds ω such that M,ω |= φ, then the final clause can be stated a little more perspicuously as: M,ω |= Xiφ iff ‖φ‖M ∈ Xi(ω) Thus, we can use belief functions to characterise directly those propositions that we want to say i believes at a given world ω. Note that there is no inbuilt assumption, for any particular φ and any ω, that ‖φ‖M must belong to Xi(ω). Likewise, we don't assume that if ‖φ‖M ∈ Xi and ψ is true at every world in ‖φ‖M , then ‖ψ‖M ∈ Xi. Indeed, the logic associated with neighbourhood models is very weak, consisting of just: PROP. All classical propositional tautologies MP. From φ and φ→ ψ, infer ψ REP. From φ↔ φ′, infer ψ ↔ ψ[φ/φ′] Where 'ψ[φ/φ′]' denotes any sentence that results from the replacement of zero or more instances of φ in ψ with φ′. (For example, (Xi(p∨¬p))[p/q] can refer to 10 Xi(p∨¬p), Xi(q∨¬p), Xi(p∨¬q), or Xi(q∨¬q).) Following Chellas (1980), we will refer to any axiom system which contains PROP, MP, and REP as classical. The flexibility afforded by neighbourhood structures will also be useful in characterising states of awareness, which we will represent by an additional set of awareness functions, Ai. Like the belief functions, every awareness function is a mapping from worlds to sets of sets of worlds. However, in keeping with the intended interpretation of the model, we will want to place a four additional conditions upon each agent's awareness function and its relationship with belief functions. I'll discuss these in turn. 3.1 Basic conditions on belief and awareness functions pla. The most obvious condition concerns the relation between awareness and the explicit attitudes. An agent's awareness cannot float free of her explicit beliefs: if she isn't aware of a proposition, then she cannot explicitly believe it. I take it that this is analytic of 'awareness' if anything is, given how I've characterised the notion. For reasons that will become apparent below, in the axiom system we'll refer to this as PLAa: PLAa. Xiφ→ Aiφ We capture PLAa with the condition pla, that Xi(ω) ⊆ Ai(ω), for all ω. sym. Next, and a little less obviously, I will assume that awareness is closed under negations. So, if an agent is aware of φ, then she is aware of ¬φ. Hence, SYM. Aiφ→ Ai¬φ We ensure that this axiom holds by assuming that Ai is closed under complements (sym). In any classical logic SYM implies Aiφ↔ Ai¬φ. I expect that SYM will be generally uncontroversial, and I have already mentioned some reasons to think that our actual states of awareness might be automatically closed under negations in Section 2.1. Nevertheless, allow me to here add some considerations in favour of SYM. First: on any account of concepts, one possesses the concept for some kind F only if one is able to tell the Fs from the not-Fs. Concepts as they are widely understood either literally are, or otherwise form the representational bases for, our recognitional and sortal capacities; so, if you can think that a is an F you can think that a is a not-F , which to the possible worlds theorist just is to think that a is not an F . The same is plausibly true for propositions more generally. One can't really understand what it is for some possibility to be the case if one can't understand what it would take for it to not be the case. There is also a long and widespread tradition going back to Wittgenstein (1922)-one that's particularly common amongst possible worlds theorists-according to which one isn't able to represent the possibility that φ without being able to represent ¬φ. According to this tradition, to believe φ is just to carve the space of possibilities Ω down the line that divides φ and ¬φ, and toss the ¬φ half away. So it's at least arguable, and quite widely assumed, that the capacity to believe anything seems to presuppose 'possession of the concept of negation', in at least the weak sense characterised by SYM.6 6 Something similar holds on almost every account of uncertainty within the Bayesian tradition: having any uncertainty regarding φ implies having some uncertainty regarding ¬φ, while learning φ requires setting one's uncertainty regarding ¬φ to zero. 11 con. We will also assume that awareness is closed under conjunctions: if an agent is aware of φ and ψ, then she is aware of φ ∧ ψ. CONA. (Aiφ ∧Aiψ)→ Ai(φ ∧ ψ) Thus we will assume that the Ai are closed under binary intersections (con). In conjunction with SYM, this means that for all worlds ω and agents i, Ai(ω) is either empty or a finitely additive Boolean algebra on Ω. The reasons for accepting CONA are not dissimilar from those in favour of SYM. Suppose that Iggy believes that φ, and believes that ψ. He may or may not therefore believe that φ∧ψ, but there can be hardly any doubt that he is aware of that possibility-in the sense that he has at least the basic representational resources required to represent that proposition. Even if those beliefs are stored in different 'fragments', there will be a good sense in which he could bring those beliefs together, and guide his behaviour on the basis of the information contained in the pair of them. As we've seen already, Iggy doesn't have to have a sentence in his head that specifically means φ ∧ ψ in order to represent that proposition. It would be enough if he merely represents at once that the world is as φ says it is and that the world is as ψ says it is, and is able to draw conclusions as appropriate from that basis. This, I assume, is something that anyone with the capacity to represent φ and ψ individually has the requisite representational resources to do. A clarificatory note: we will not be assuming that an awareness of φ ∧ ψ implies an awareness of φ and awareness of ψ. In other words, we don't assume: DISA. Ai(φ ∧ ψ)→ (Aiφ ∧Aiψ) In any classical logic, having DISA alongside SYM and CONA would quickly lead to triviality. As pointed out in Fritz and Lederman (Fritz and Lederman), if an agent i is aware of any φ, then she's aware of φ∧¬φ, and hence (by REP) aware of ψ∧¬ψ for arbitrary ψ. Awareness of any φ would then imply awareness of every ψ, which is clearly unacceptable. But the arguments just given in favour of CONA don't also support DISA, and more generally it should be apparent that DISA is only plausible under the presupposition of a fine-grained theory of content. (Consider again the example from Section 2.2: Iggy is aware of φ, which just is (φ∨ψ)∧ (φ∨¬ψ), yet Iggy isn't aware of either φ∨ψ or φ∨¬ψ.) Obviously, this is only a reason to reject DISA if we have good reasons to accept a coarse-grained conception of mental content-but that discussion will have to wait until Section 8. nax. Finally, and mostly for the sake of simplicity, I will make one further assumption about the relationship between awareness and explicit belief: NAX . Ai> → Xi> In the context of the other axioms, this implies that if i is aware of anything at all, then she is aware of >. In context, then, NAX states that i has awareness only if she explicitly believes >. Where REP has already been accepted, this is quite weak. It is hard to imagine an agent worthy of the title who does not accept even the simplest of tautologies, and the most popular theories of coarse-grained mental content 12 imply that agents don't even have a choice about whether to believe >. We capture this in the model by use of the condition nax, that if Ω belongs to Ai, then it belongs to Xi. Building nax into the definition of a PWA model simplifies the relationship between explicit and implicit attitudes by ruling out the possibility of awareness without any explicit attitudes, without going so far as to commit us to the more common (and stronger) axiom N (discussed in Section 4). If the reader is unwilling to accept NAX , we can do without it by (i) removing condition nax from Definition 3.1, and (ii) replacing the axiom NAX in Σ with the weaker axiom Ai> → Ii>. The soundness and completeness proofs will be left mostly unchanged. 3.2 PWA models I will discuss further potential conditions on awareness and belief functions below, but for now let us move on to the basic PWA models: Definition 3.1 A model M = (Ω, {Xi}i∈Ag, {Ai}i∈Ag, π) belongs to the class of PWA models M iff: 1. Ω is a non-empty set 2. Xi and Ai are functions from Ω to 22 Ω satisfying (for all ω ∈ Ω): pla. P ∈ Xi(ω) only if P ∈ Ai(ω) sym. P ∈ Ai(ω) only if Ω \ P ∈ Ai(ω) con. P1 ∈ Ai(ω) and P2 ∈ Ai(ω) only if P1 ∩ P2 ∈ Ai(ω) nax. Ω ∈ Ai(ω) only if Ω ∈ Xi(ω) 3. π is a propositional valuation function Definition 3.2 then characterises what it is for an element of LAXI(Φ) to be true at a given world ω in a PWA model M (i.e., M,ω |= φ): Definition 3.2 For all φ ∈ LAXI(Φ), M,ω |= p iff ω ∈ π(p), for p ∈ Φ M,ω |= ¬φ iff it's not the case that M,ω |= φ M,ω |= φ ∧ ψ iff M,ω |= φ and M,ω |= ψ M,ω |= Xiφ iff ‖φ‖M ∈ Xi(ω) M,ω |= Aiφ iff ‖φ‖M ∈ Ai(ω) M,ω |= Iiφ iff ‖φ‖M ∈ Ai(ω) and ⋂ Xi(ω) ⊆ ‖φ‖M Definition 3.2 should make it clear how the awareness functions work in relation to the belief functions. If i believes each of a collection of propositions φ1, . . . , φn which jointly imply ψ, then we will say that i (narrowly) implicitly believes ψ whenever she's aware of ψ. This is modelled by first characterising (for each world ω) the largest set of worlds P1 ⊆ Ω consistent with all of i's explicit attitudes at ω as the intersection of all the sets of worlds in Xi(ω). We then say that i implicitly believes any P2 such that P1 ⊆ P2 just in case i is aware of P2. So, Iggy has the narrow implicit belief that φ just in case φ is something Iggy can entertain that's entailed by the conjunction of his explicit beliefs. Thus, Ai acts as a filter or sieve over the propositions that i implicitly 13 believes in the broad sense, letting through only those that i implicitly believes in the narrow sense. We say that φ is valid in M iff M,ω |= φ for every ω ∈ M ; and valid in M (i.e., M |= φ) iff φ is valid in every M ∈M. The classical logic associated with the class of PWA models can then be axiomatised by the following system, which we'll label Σ: PROP. All classical propositional tautologies SYM. Aiφ→ Ai¬φ CONA. (Aiφ ∧Aiψ)→ Ai(φ ∧ ψ) XI. Xiφ→ Iiφ IA. Iiφ→ Aiφ KAI . (Iiφ ∧ Ii(φ→ ψ))→ (Aiψ → Iiψ) NAX . Ai> → Xi> MP. From φ and φ→ ψ, infer ψ REP. From φ↔ φ′, infer ψ ↔ ψ[φ/φ′] Say that φ is a theorem of Σ (i.e., `Σ φ) just in case φ is either an axiom of Σ or can be derived from the axioms by finite applications of the inference rules. We can then show that φ is a theorem of Σ if and only if it is valid in the class of PWA models: Theorem 3.1 Σ is sound and complete with respect to M and LAXI(Φ); i.e., for all φ ∈ LAXI(Φ), M |= φ if and only if `Σ φ. Proof. See Appendix A. Σ is best understood as a logic for belief rather than knowledge, as it imposes no requirement of veridicality on X (and I). Where a minimal conception of knowledge takes it to be a species of veridical (or 'non-delusional,' 'factive') belief, we would want our logic to include: TI . Iiφ→ φ In combination with XI, this will get us: TX . Xiφ→ φ We can ensure TI and TX by adding a reflexivity condition of the form: tx. If P ∈ Xi(ω), then ω ∈ P Since, under condition tx, Xiφ is true only at worlds where φ is true, and Iiφ is only true at worlds ω where Aiφ and ⋂ Xi(ω) ⊆ ‖φ‖M , the addition of tx will also get us that Iiφ is true at ω only if φ is too (since ω ∈ ⋂ Xi(ω)). Finally, it's worth noting that if we wanted to capture instead the other senses of 'awareness' outlined in Section 2.1, we might be able to do this by removing pla, sym and/or con from Definition 3.1 and modifying the axiom system appropriately. For example, I see no strong reasons to think that PLAa and CONA are appropriate for either the attentional or deliberative senses of 'awareness', unless we make very strong idealising assumptions about the agent's capacities. 14 3.3 Relationship to other models Before I discuss Dekel et al.'s triviality result and my response to it, in this subsection I'll briefly discuss the relationship between PWA models and other models in the literature. It should be emphasised that the notions of 'awareness' and 'implicit belief' referred to in the other works discussed herein need not be the specific kinds of awareness and implicit belief that I take to be represented by PWA models, though they are of course clearly related. Considered primarily as a way of representing the relationship between implicit belief and explicit belief (i.e., ignoring the awareness operator), PWA models resemble the local reasoning structures of Fagin and Halpern (Fagin and Halpern 1988, pp. 58ff), which are intended to represent a 'fragmented' belief system. Indeed, PWA models are essentially generalised local reasoning structures with the addition of awareness functions acting as sieves over the supersets of⋂ Xi(ω). In the same vein, PWA models are also quite similar to the more recent model of implicit and explicit belief developed by Velazquez-Quesada VelázquezQuesada (Velázquez-Quesada). Velazquez-Quesada's construction works by taking a neighbourhood function Xi defined for a finite space Ω (interpreted as specifying i's explicit beliefs), and using it to systematically construct another function X ?i which contains Ω and is closed under supersets and binary intersections, interpreted as specifying i's implicit beliefs. In outline, this construction of X ?i is quite similar to my construction of ‖Iiφ‖M as a subset of {ω : ⋂ Xi(ω) ⊆ ‖φ‖M}, though there are some important differences. One of these concerns the axiom XI, which is not valid on Velazquez-Quesada's class of models. Velazquez-Quesada provides an interesting defence of this characteristic of his model. For present purposes, however, XI seems essential inasmuch as I is understood to represent the informational content contained in an agent's explicit beliefs-every explicit belief that φ contains at least the information that φ, after all. In terms of the relationship between implicit belief and awareness, the use of the Ai functions in PWA models is conceptually similar to the way that awareness functions are used in the Kripke structures for general awareness of Fagin and Halpern Fagin and Halpern (1988). Fagin and Halpern's models use awareness functions that take sets of sentences as values rather than sets of (possible-worlds) propositions, where those sets of sentences need not be closed under logical equivalence; by virtue of this, they are able to generate extremely fine-grained distinctions between states of awareness. No particular restrictions are placed on Fagin and Halpern's awareness functions, and they define agents' explicit beliefs as a subset of their implicit beliefs-specifically, those the content of which they are aware. PWA models also share many similarities with the partitional models of Fritz and Lederman Fritz and Lederman (Fritz and Lederman). (Especially given the presence of sym and con and with pla dropped and stronger rationality conditions placed on the Xi functions.) Their models are intended to capture a somewhat stronger logic of belief (and knowledge) and its interaction with awareness also within the context of a coarse-grained approach to modelling content. Like Fagin and Halpern, Fritz and Lederman take a notion of implicit belief as a primitive and define explicit beliefs as those states of implicit belief the content of which the agent is aware. PWA models have much in common with Yalcin's resolution-sensitive models of belief Yalcin (2018). Yalcin uses a partition over logical space to represent 15 different 'resolutions' at which a non-ideal agent might conceive of a space of possibilities. That partition acts to filter between accessible and inaccessible possible-worlds propositions in essentially the same way that the Ai functions do, or in much the same way that a Boolean sub-algebra of 2Ω might be used on a probabilistic model of uncertainty to represent the idea that an agent need not have degrees of belief regarding propositions at all levels of specificity. Yalcin is primarily concerned with representing limited informational accessibility under a holistic model of belief for non-ideal agents, where the inaccessibility might result from multiple sources including, for example, attentional or computational limitations, rather than merely conceptual limitations. Many of the motivations for Yalcin's resolution-sensitive models clearly overlap with those for PWA models, and the former are in effect what PWA models would be if nax were removed and the Xi functions could only ever take singleton sets of propositions as values. So much for similarities; now let us consider perhaps the main respect in which PWA models differ from a great many alternative models of awareness in the literature. In particular, it is not only common for models of awareness to satisfy at least the axiom DISA mentioned earlier, but also more generally the property of Awareness Generated by Primitive Propositions (AGPP); i.e., that i is aware of every primitive proposition in Ψ ⊆ Φ if and only if she is aware of every φ ∈ LAXI(Ψ). (See, for example, Cozic (2016), Halpern (2001), Heifetz et al. (2006), Li (2009), and Schipper (2015).) AGPP immediately implies DISA, and just like DISA it is quite obviously not going to play nicely with any remotely coarse-grained theory of content. Given what I'm trying to model, however, AGPP seems highly implausible. According to AGPP, for Iggy to be aware of, say, Ajp, he need only be aware of the primitive proposition p. But that cannot be right: he also needs to be aware of the other agent j, and the relation of awareness. (Proviso: I am assuming that Ajp is contingent. See Section 6 for discussion.) This suggests two independent problems that we might raise against the principle. First, there are no compelling reasons to think that Iggy must able to represent any agent in Ag as an automatic consequence of being able to represent anything at all. Iggy may not even be able to represent himself, let alone another agent who he may never have been in any kind of contact with-or who may not even actually exist. Second, Iggy's capacity to entertain p-thoughts doesn't automatically come with the capacity to represent that some agent or other is awareness-related to p, in the specific way that I defined 'awareness' in Section 2.1. (A lot of concepts went into my characterisation of that notion, so an inability to represent the relation of awareness wouldn't be at all surprising!) The same, of course, goes for narrow implicit belief relation, and even belief and knowledge. It's not a precondition of agenthood that agents in general should necessarily have the resources to represent those specific kinds of propositional attitude relations.7 One can readily imagine a member of an alien species, say, or an animal with a less developed theory of mind than our own, who lacks any notion of our spe7 Introspection principles-e.g., Xiφ→ XiXiφ and ¬Xiφ→ Xi¬Xiφ-would imply that knowledge/belief in any proposition requires higher-order knowledge/belief, and hence awareness, in one's own first-order knowledge/belief states. But these are dubious axioms even for normative logics of knowledge/belief under the assumption of full awareness Williamson (2002); they're even less plausible for a descriptively-oriented logic of knowledge/belief that incorporates unawareness such as the one I'm aiming to construct. 16 cific folk-psychological attitudes and yet can still be said to have such attitudes themselves. Consequently, AGPP is too strong as a general principle for modelling awareness as I'm understanding it. Likewise, and for the reasons just noted, we will also want to reject each of the following consequences of AGPP: AIA. Aiφ→ AiAjφ AIX . Aiφ→ AiXjφ AII . Aiφ→ AiIjφ By contrast, SYM and CONA (in conjunction with PLAa) are only strong enough to get us the result that if an agent i is aware of every φ in a set X ⊆ LAXI(Φ), then i will also be aware of everything in the closure of X under ¬ and ∧. 4. The Impossibility of Unawareness In this section, I outline a lightly modified version of Dekel et al.'s well-known impossibility result. It centres on three straightforward assumptions about the relationship between awareness and the epistemic attitudes: PLA. ¬Aiφ→ (¬Xiφ ∧ ¬Xi¬Xiφ) AUR. Ai¬Aiφ→ Aiφ XUI. ¬Xi¬Aiφ Dekel et al. themselves describe PLA (for 'Plausibility') as the most plausible of their three axioms (hence the name), and many in the literature have followed them in treating it as fundamental-or even definitional-for any adequate understanding of awareness. See, for instance, Chen et al. (2012), Cozic (2016), Heifetz et al. (2006), Modica and Rustichini (1994), Modica and Rustichini (1999), Walker (2014), though cf. Halpern (2001), where PLA is derivable only under certain assumptions. Given a classical logic, PLA is of course just the conjunction of PLAa with: PLAb. ¬Aiφ→ ¬Xi¬Xiφ PLAb says that if an agent i is unaware of φ, then she cannot believe that she doesn't believe φ. AUR (for 'AU Reflection') says that if i is aware of the proposition ¬Aiφ, then she is ipso facto aware of φ itself. The intuitions motivating PLAb and AUR are the same, and is easiest to grasp if we view awareness and belief as relations that hold between a thinking subject and those propositions she entertains-the intended idea being that anyone who can think that a is not R-related to b has the conceptual resources to think in terms of a, R, and b. Thus, for PLAb: if i can believe that she's not belief-related to φ, the she can think in terms of φ directly. Likewise, for AUR: if i can entertain the idea that she is not awareness-related to φ, then she can entertain φ directly. Finally, XUI (for 'XU Introspection') says it's impossible for i to believe that she is unaware of some specific proposition φ. To be clear: i might be aware that there exists some propositions she's not aware of, and which some other agent may be aware of. All of the axioms we discuss in this paper are consistent with saying this much. But i cannot be aware of a specific state of unawareness 17 that she has, towards a particular proposition φ. To express the existentially quantified thoughts, we would need more than the non-quantificational language that I'm employing here.8 In Sections 5–7, we will examine the plausibility of these axioms in more detail; for now, let us focus on the impossibility result: Theorem 4.1 Suppose Σ∗ is a classical logic which includes PLA, AUR, and XUI. Then, if Σ∗ includes N, then for every φ, `Σ∗ Aiφ; and if Σ∗ includes MON, then for all φ and ψ, `Σ∗ ¬Aiφ→ ¬Xiψ. Where: N. Xi> MON. From φ→ ψ, infer Xiφ→ Xiψ Proof. From AUR, PLAb: ¬Aiφ → ¬Ai¬Aiφ and ¬Ai¬Aiφ → ¬Xi¬Xi¬Aiφ. With XUI and REP, this gets us ¬Aiφ→ ¬Xi>. So, ¬Aiφ is inconsistent with N. Furthermore, MON implies that for any ψ, Xiψ → Xi>; or equivalently, ¬Xi> → ¬Xiψ. So, ¬Aiφ→ ¬Xiψ. Thus, for any classical logic Σ∗, the very presence of a non-trivial Ai operator is incompatible with PLA, AUR, XUI, and at least one of N or MON. Moreover, neither N nor MON (nor PLAa, for that matter) are essential to generating a serious problem, as the following corollary indicates: Corollary 1 Suppose Σ∗ is a classical logic which includes PLAb, AUR, and XUI. Then, for all φ, `Σ∗ ¬Aiφ→ ¬Xi>. And this is already a very troubling result-unawareness shouldn't preclude belief in simple propositional tautologies! We can generalise the badness a little further, by noting that XUI can be broken down into two separate axioms: XUIa. ¬Aiφ→ ¬Xi¬Aiφ XUIb. Aiφ→ ¬Xi¬Aiφ XUIa is a simple consequence of AUR and PLAa, so what XUI actually brings to the table is better understood through XUIb. Hence, Corollary 2 Suppose Σ∗ is a classical logic which includes PLAa, PLAb, AUR, and XUIb. Then, for all φ, `Σ∗ ¬Aiφ→ ¬Xi>. What's worse, if we add SYM, CONA, and NAX back into the mix, then we again get that if i is unaware of anything, then she's unaware of everything: Corollary 3 Suppose Σ∗ is a classical logic which includes PLAa, PLAb, AUR, XUIb, SYM, CONA, and NAX . Then, for all φ and ψ, `Σ∗ ¬Aiφ→ ¬Aiψ. 8 Given the straightforward nature of PWA models, I would not anticipate any difficulties in extending them to allow for quantification over propositional variables. I have not done this since the issues relevant to the present paper arise already with non-quantificational languages. For relevant work, see esp. Fritz and Lederman (Fritz and Lederman), as well as Halpern and Rêgo (2009), Sillari (2008b), and Walker (2014). 18 Proof. From Corollary 2, ¬Aiφ → ¬Xi>, and from NAX , ¬Xi> → ¬Ai>. We've seen that in any classical logic with PLAa, SYM, and CONA, Aiφ implies Ai(φ ∨ ¬φ), and hence Ai>. So, ¬Ai> → ¬Aiψ; and for all φ and ψ, ¬Aiφ→ ¬Aiψ. Since PLAa is secure if anything is, if we want non-trivial awareness then we have a choice: 1. Reject classical logic, and/or 2. Reject at least one of XUIb, AUR, or PLAb. In the following three sections, I will discuss (in turn) XUIb, AUR, and PLAb. Overall, I will argue that the possible worlds theorist is entitled to reject at least one of XUIb, AUR, or PLAb, and perhaps all three. However, matters are complicated: which of those three axioms we ought to reject in particular will depend partly on (i) how we want to interpret X, (ii) the specific coarsegrained theory of content that we want to adopt, and (iii) whether there exist any necessarily unentertainable propositions. 5. Critical Discussion: XUIb Assume first that we want X to specifically represent knowledge, or indeed any other veridical epistemic attitude. (For example, it's plausible that rational certainty is veridical: correctly updating on one's evidence should never lead one to become immutably convinced of a falsehood.) Since XUIb follows immediately from TX , and TX just is the veridicality axiom, there is no plausible way out of the triviality result via XUIb under such an interpretation. On the other hand, assume that we want X to represent belief. Under this interpretation, XUIb says that if i is aware of φ, then she does not explicitly believe that she's not aware of φ. And this does not seem especially plausible.9 It would be odd, perhaps, for Iggy to have false beliefs about his own state of awareness. After all, if Iggy is aware of φ, then you might think that he has access evidence justifying the belief that he's aware of φ-at least where φ is a proposition that Iggy can actually entertain. (Recall from Section 2.1 that entertainability in principle doesn't mean entertainability in practice, and Iggy can of course only believe what he can in fact entertain.) But XUIb says that Iggy cannot falsely believe that he is in some state of unawareness. And even if he does have evidence that he's aware of φ, he might have even more evidence that he's not aware of φ. Or perhaps he has evidence that he's aware of φ, but he doesn't weigh up that evidence appropriately. As a non-ideal agent, then, Iggy can presumably have all sorts of absurd and unjustified beliefs that fly in the face of his evidence. Why should his beliefs about awareness be any different? Consider, for instance, the following case: 9 It's worth noting that, although Dekel et al. express their result only in terms of 'knowledge,' they don't presuppose that TX is valid in general, and (more importantly) the result has been widely taken to apply to all epistemic models which use Aumann structures or something closely analogous, including those intended to represent non-veridical states. See, e.g., (Cozic 2016, p. 3), Schipper (2013), (Sillari 2008b, p. 516) and the models outlined in Schipper (2015). 19 Example 1. After long discussions on the matter with leading philosophers and psychologists, Iggy ends up committed to an error theory about folk psychology. According to this theory, all talk of being 'aware of,' 'entertaining,' or even 'believing' abstract entities like propositions is bunk, a manner of thinking associated with an outdated and generally inaccurate theory of how our minds work.10 Iggy has thought a lot about folk psychology and about his stance on it, so he's clearly aware of the proposition Iggy does not believe that folk psychology is an accurate theory. However, he doesn't believe that he's aware of that proposition-in fact, he strongly (and wrongly) believes that he isn't. Let φ be the proposition that Iggy does not believe that folk psychology is an accurate theory. We might view this as a case where Iggy has overwhelming evidence that ¬Aiφ, or one where he has insufficient evidence that ¬Aiφ but believes it anyway. Either way, we have a plausible situation where Aiφ∧Xi¬Aiφ is true, and thus a counterexample to XUIb. So, here is our first lesson: while XUIb is clearly unavoidable if our goal is to model a veridical attitude like knowledge, it does not look like we should want to keep it around if our goal is to model non-veridical belief. Note that the point being made here is entirely independent of how fine-grained or coarse-grained we take mental contents to be. XUIb is dubious for non-veridical belief given any position on the granularity of mental content, and should therefore be dropped regardless of whether we want to stick with a classical logic or not. 6. Critical Discussion: AUR Since AUR doesn't interact in any interesting way with TX , we can consider its plausibility independently of how we interpret the X and I operators. Nevertheless, AUR turns out to be much trickier than XUIb to evaluate, and doing so will involve opening more than one can of worms. The large number of issues here necessitates some brevity, and I will not pretend to have stated the last word on any of them. It's unsurprising that AUR is plausible given a fine-grained theory of content, so the point of the present section is only to consider whether and to what extent AUR can also be considered plausible from the perspective of a coarse-grained theory of content. That is, the question I'll be considering is: supposing we've already accepted a coarse-grained theory of content, should we also accept AUR? By the end of the section I will conclude that (i) counterexamples to AUR seem to exist on a 'metaphysical' conception of coarse-grained content, while (ii) on an alternative 'epistemic' conception of coarse-grained content, AUR is defensible. 6.1 Metaphysical conceptions of coarse-grained content Let's begin with what we can call the metaphysical conception of coarse-grained content, according to which if φ↔ ψ holds as a matter of metaphysical necessity, 10 An error theorist about a domain of discourse holds that at least all positive, first-order, atomic and non-trivial or non-analytic sentences in the domain are meaningful (truth-apt), yet systematically false. An error theorist about folk psychology would deny the truth of any sentence of the form Aiφ, Xiφ, or Iiφ, but may accept the truth of, e.g., ¬Aiφ, Aiφ→ Aiφ, or Aiφ∨ψ (i.e., if ψ is true). In LAXI(Φ), Aiφ, Xiφ, or Iiφ are not 'atomic,' but each corresponds to an atomic sentence in the ordinary languages where folk psychological discourse usually occurs. 20 then φ and ψ denote one and the same content. Ω on this picture can be thought of as the space of metaphysical possibilities, such that to be aware of (or believe that) Water is H2O is just to be aware of (or believe that) H2O is H2O. I suspect that most readers, when they think of a coarse-grained theory of content, will have this kind of conception in mind; it is the kind of account most typically (but not necessarily) associated with externalism about content.11 On the metaphysical conception, counterexamples to AUR will take the form of a proposition ψ such that ¬Aiφ↔ ψ is metaphysically necessary, but Aiψ → Aiφ is not. Such counterexamples can exist only where the ψ in question is either metaphysically contingent or impossible: if ψ and ψ ↔ Aiφ are both metaphysically necessary, then so is Ai¬Aiφ → Aiφ. I will discuss whether there are any necessarily unentertainable propositions separately (Section 6.2), so here we will focus only on those cases where the ψ in question is contingent. Prima facie, counterexamples of this form should be easy to find: for any given proposition ψ, there will usually be many ways-different modes of presentation- under which a different agents might entertain something metaphysically equivalent to ψ despite having very different conceptual resources. I think that the relevant counterexamples probably do exist. But their existence is not quite as easy to establish as first appearances suggest. Consider the following case: Example 2. Iggy has just made first contact with an alien from the planet Gzorp, with whom he's attempting to communicate. Gzorp is 65.24 light years away from Earth, bearing 92.45° at an elevation of 30.13°. In its own language, the Gzorpian asserts that Iggy is not aware of the proposition that skirnobs are poisonous, where a skirnob is a kind of Gzorpian fruit found all over the planet. Iggy does not understand, of course, but he knows that the Gzorpian just asserted something, and that whatever it is, it's true. Let φ denote the proposition skirnobs are poisonous, and let ψ pick out that set of worlds where the proposition the Gzorpian actually just expressed is true. Then it's presumably the case that Aiψ, and ψ ↔ ¬Aiφ. If it's also the case that ¬Aiφ, then we have a counterexample to AUR. However, for the case to work, it needs to be true that Iggy has no way of entertaining the thought that skirnobs are poisonous under any mode of presentation whatsoever. So, for instance, we must suppose that he doesn't have the resources to think anything from the following (non-exhaustive) list: 1. The kind of fruit the Gzorpian just referred to is poisonous. 2. The kind of fruit with purple stripes and pink feathers is poisonous. 3. The kind of fruit, an instance of which is 65.24 light years away from me bearing 92.45° at an elevation of 30.13°, is poisonous. 4. The kinds of things referred to by the sound /sk@:n6b/ in the language of those things are poisonous. 5. The kinds of things in the category my community refers to when they make the sound /fru:t/, an instance of which is 6.17218×1017 metres away from me bearing 92.45° at an elevation of 30.13°, are poisonous. 11 Thanks to [anonymised] for highlighting to me the importance of discussing AUR under the metaphysical conception, and for discussion on these issues more generally. I owe Examples 3 and 4 below to him. 21 Generally: give Iggy some basic sortal concepts and a way to represent arbitrary relative distances, directions, and times, and there won't be many objects that actually exist/existed/will exist, or properties that are/were/will be instantiated, that Iggy won't in principle have the resources to think about under some mode of presentation or other (cf. Hofweber (2016)). He almost certainly never will have all these thoughts, but he could-and that's all that's required for awareness. So, perhaps Iggy could be aware of the proposition Iggy is not aware of the proposition that skirnobs are poisonous without being aware of skirnobs are poisonous, but this would require that he's quite representationally impoverished indeed. I suspect that it is possible for agents to be impoverished in this way, but we couldn't expect it of any (or many) actual human beings. Thus, while we have here a potential counterexample to AUR, the case also points to a kind of practical triviality for the notion of awareness under this conception of coarsegrained content. That is, awareness can come very cheap on the metaphysical conception, and this cheapness serves to undermine the theoretical importance of the notion. Even deep in the jungle, Iggy could still think a thought with the same content as that which we would ordinarily express upon uttering 'The habitat of the eastern grey kangaroo extends as far north as the Cape York Peninsula'; in fact there's dozens of obvious and easy ways by which he could do this, though Iggy himself wouldn't recognise them as such. The problem, of course, is that the flexibility in constructing alternative modes of presentation by which one might entertain Aiφ will tend to go handin-hand with the same degree of flexibility in constructing alternative modes of presentation for entertaining φ itself. It's easy to imagine various ways of describing the set of metaphysically possible worlds where Aiφ is true that don't specifically mention φ; what's a lot less easy is to imagine an agent with the capacity to entertain the former without also having any capacity to entertain something metaphysically equivalent to the latter. One more example to drive the point home: Example 3. As it turns out, the state of being aware of the proposition that skirnobs are poisonous can be necessarily identified with a particular psychophysical state, S. Iggy is a brilliant neuroscientist, and can represent himself as being in state S. This time, letting ψ denote that set of worlds where Iggy is in state S, ψ ↔ Aiφ is necessary. And since Aiψ, so we also have Ai¬ψ. But do we have Aiφ? This would be easier to argue for if we were internalists about content. In that case S could be identified with some internal (presumably neurological) state, and it's quite plausible that Iggy could represent an arbitrary neurological state without being able to represent that skirnobs are poisonous. (Merely looking at a brain in that state and thinking inside my head is like that would suffice.) But internalism sits poorly with the metaphysical conception, and few adherents to the latter would want to commit themselves to the former. Externalists will have a much tougher time of showing that Aiφ holds in this example. After all, on an externalist's theory, S would presumably be a highly disjunctive psychophysical state, where at least some of the disjuncts would involve causal connections between the agent and skirnobs. And being able to represent that kind of state without being able to represent skirnobs in one way or another is not obviously possible. 22 There are additional examples we could consider, involving for example Burgean social externalism and the apparent capacity to entertain a thought about arthritis without really knowing what that is merely by being situated in the relevant linguistic community. I will not try to consider them all. What is clear is that there is an argument-or rather, several arguments-to be made for the existence of counterexamples to AUR under the metaphysical conception of coarse-grained content. With that said, there is also a genuine concern that awareness is a little too cheap on the metaphysical conception, at least for ordinary agents whom we can expect to have reasonably rich basic representational resources. 6.2 Necessarily unentertainable propositions Perhaps there is an easier route to finding counterexamples to AUR: if, for some agent i, there are any propositions φ of which i is necessarily unaware, then ¬Aiφ is necessary, and there will be (vastly many) ways that i might entertain ¬Aiφ without entertaining φ. (For example, i could entertain ¬Aiφ by entertaining ψ ∨ ¬ψ, for any ψ that i can in fact entertain.) But are there any propositions φ and agents i such that it's metaphysically impossible that i lacks the basic representational resources to entertain φ? It's not obvious that there are.12 Let me consider two general kinds of argument for necessarily unentertainable propositions. First, you might think that some propositions cannot be entertained by beings us as a result of our cognitive limitations. For instance, due to limited resources of memory and processing, there may be some φ which are simply too complex or specific for an ordinary agent like Iggy to entertain. Of course, limitations of memory or processing are insufficient to make the point: these limitations are contingent, and there are no good reasons to suspect that there aren't any possible worlds where they have been overcome (including, potentially, in our own future). And as we've already discussed, the notion of awareness that we're playing with here isn't bound by the contingent limitations of the ordinary human brain (Section 2.1). But there are other examples in the vicinity that might deserve some more weight. In particular, you might think that there are hard-wired limitations on the kinds of phenomenological experiences we might undergo (Nagel 1986, pp. 90ff). Given the kind of being that Iggy is, perhaps he cannot-as a matter of metaphysical necessity-know what it is like to be a bat. Now, if (i) it is metaphysically necessary of Iggy that he is a human, (ii) it is metaphysically essential to being human that one's brain is hard-wired in this kind of way (i.e., such as to preclude the possibility of undergoing certain kinds of experiences), and (iii) 'what it is like to be a bat' can be cashed out in terms of some propositional content (or contents), then we might have a counterexample to AUR here. I will leave this matter hanging, though it should go without saying that not one of (i)–(iii) is unassailable. 12 For a recent and thorough treatment of several arguments relating to the present discussion, see Hofweber (2016). It's important to note, however, that Hofweber's discussion is centred on the issue of whether there are any aspects of reality which cannot be expressed in language or thought by beings like us-i.e., whether we might express or entertain every proposition given the way we actually are. His concern is not directly on whether there exists any propositions that are necessarily unentertainable tout court, or necessarily unentertainable for some agent i. 23 The second kind of argument purports to establish that there aren't enough entertainable propositions to go around. Perhaps the best known of these can be found in Lewis (Lewis 1986, pp. 104–7), as part of his response to the RussellKaplan paradox (cf. Davies (1981), Kaplan (1995)). Lewis gives three reasons for thinking that there are unentertainable propositions. The first is that it provides a way out of the paradox, though few commentators have been convinced by this-at most it helps with a very specific version of the paradox, but doesn't address the deeper problem (e.g., Bueno et al. (2014), Chalmers (2011), Uzquiano (2015)). The second reason rests heavily on the specifics of Lewis' views on mental content and the role that relative naturalness plays within it. I'll describe this briefly below, but a full discussion would take us too far afield. The third and final reason arises from Lewis' functionalism more generally, and it's this reason that we'll focus on here. As Lewis points out, if one accepts a certain kind of functionalism about what it is to believe (or desire, be uncertain about, etc.) a proposition-I'll say more about the functionalist assumptions in a moment-then there can be no more states of believing (or desiring, being uncertain about, etc.) a proposition φ than there are functional roles with which to define those states. However, while there are probably at least i3 sets of possible worlds, Lewis also asserts that there are probably no more than countably many functional roles. So, there are more possible worlds propositions than there are distinct states of believing that φ. Most propositions are in fact unbelievable. Likewise, most are undesirable, and most we cannot have any uncertainty about, and so on. If we add now the reasonable premise that if it's possible for i to be aware of φ, then it's possible for i to believe (or desire, be uncertain about, etc.) that φ, then we get the conclusion that there are propositions of which i cannot possibly be aware.13 There are two obvious ways to resist this conclusion. One would be to deny that there are fewer functional roles than there are propositions. Lewis does not say much to support this, only that he 'cannot see the slightest prima facie reason to think that there are even uncountably many of the definitive roles' (pp. 106–7). The thought, perhaps, was this: for the typical functionalist of the reductive physicalist variety, a definitive functional role for a given mental state S is characterised in terms of causal relations between that state and some range of possible sensory inputs, behavioural outputs, and other (physical, e.g., neurological) states. Suppose therefore that there are only finitely many relevantly distinct possible sensory inputs, behavioural outputs, and other neurological states. This is plausible enough for beings like us; we have limited capacities for making meaningful distinctions between sensory inputs, for example, and our behavioural dispositions can only get so fine-grained. If so, then for each finite n there would be no more than finitely many n-ary relevantly distinct relations that hold between those inputs, outputs, and other states; and hence, no more than countably many functional roles could be characterised in terms of those relations (or continuum many if we allow also relations which take a countably infinite number of arguments). The problem, again, is that it's not clear whether these finite limitations are metaphysically essential to thinkers in general. As noted in (Chalmers (2011), pp. 91–2) if there can be arbitrarily complex infinitary beings, then there's no 13 Lewis conducts his discussion in terms of 'thoughts' and 'thinking' rather than 'beliefs', 'desires' and 'uncertainties', as I've done here. This won't make any difference to my response, but it will help to avoid some ambiguities. 24 reason why there cannot be arbitrarily many functional roles. If there are k relevantly distinct input, output, or neurological states, then there are (at least) 2k n-ary relations between those states.14 And to reiterate the point made earlier, it's at least arguable that Iggy, or any other agent i, could have been such an infinitary being, capable of making arbitrarily fine-grained distinctions in perception or behaviour, or with an infinitely complex neurological state. Alternatively, given a non-reductive functionalism-such as that found in Schwitzgebel's dispositionalism for instance; see Schwitzgebel (2002)-we might allow definitive functional relations to hold between a richer variety of phenomena. Schwitzgebel himself characterises 'beliefs' partly in terms of their relationships with nonreduced phenomenological properties, and (for all we know) there may well be an arbitrarily large number of these. A second way to resist Lewis' conclusion would be to deny the implicit assumption that for each φ such that an agent can believe (or desire, be uncertain about, etc.) φ, there must be a separate functional role by which believing (or desiring, or being uncertain about, etc.) φ is to be defined. There are multiple ways you might go about this, but I'll just discuss the one that I find the most interesting. In particular, you might think that instead of each individual attitude receiving a separate definition, it's entire or 'holistic' belief-plus-desire states that are functionally defined. Lewis himself seems to have held a view like this. Roughly, given a sequence of perceptual or evidential inputs E and a suite of appropriately characterised behavioural dispositions D, we are to assign to an agent (or an appropriate state of the agent, most likely a brain state; see Lewis (1983), p. 373) the maximally 'eligible' set of credences and utilities that rationalises D given E; see Lewis (1974), Lewis (1983). Lewis thought that, given any D and E pair, there will always be multiple sets of credences and utilities that rationalise D given E. Hence, he thought that there are more holistic credence-utility states than there are definitive functional roles, and since only one or at most a relative few of those will be maximally 'eligible', there will thus be some credence-utility states which are never assigned at any possible world. I have argued elsewhere that this was a mistake (see [anonymised]; cf. also Schwarz (2014)), but that's neither here nor there. Even supposing it's true that there are, let's imagine, only countably many definitive functional roles with which to define the possible assignments of holistic credence-utility states, it simply does not follow from this that there are only countably many propositions to which an agent might have credences and/ or utilities. Similarly, imagine that there are only countably many pairs of sets of beliefs and desires that can be functionally characterised-this is entirely consistent with saying that there are an arbitrarily large number of distinct propositions which might belong to one of those sets. Again, there is more that could be said on each of the above points. Perhaps there are brute necessities, for instance, and one of those brute necessities is that i never happens to be aware of the proposition φ. I don't see any way that we could rule that out, short of settling once and for all the difficult matter of whether there are any brute necessities tout court. Or perhaps there are facts about transworld identity which preclude any human agent from possibly being like a bat, or from having an infinitely complex brain. I am not convinced. 14 Assume, e.g., that there is at least one distinct binary relation on any set S for every nonempty set of ordered pairs of members of S; then there are at least as many binary relations on S as there are non-empty subsets of S × S. 25 More importantly, it's not obvious that there are any necessarily unentertainable propositions. The easy route to finding counterexamples to AUR doesn't seem so easy after all. 6.3 Epistemic conceptions of coarse-grained content So much for the metaphysical conception. Let's now consider an alternative, epistemic conception of coarse-grained content.15 Say that φ is epistemically possible just in case it cannot be ruled out a priori. The thought that φ can then be said to correspond to an epistemic possibility, a way the world might be for all one might know a priori. Given this, let Ω designate the space of maximally specific epistemic possibilities; i.e., for each ω and every φ, ω either a priori entails φ or ¬φ, and ω entails both φ and ψ only if φ∧ψ is epistemically possible. Then, φ and ψ are a priori equivalent just in case φ and ψ are entailed by all the same ω. All logical truths and falsehoods will be a priori equivalent, as will be, e.g., bachelors are bachelors and bachelors are unmarried available men. And, most importantly, if two thoughts φ and ψ under two different modes of presentation are metaphysically equivalent, yet the equivalence of those modes of presentation is not a priori, then φ and ψ will correspond to distinct subsets of Ω. Most theorists who make use of the epistemic conception are pluralists about content: they accept that the metaphysical conception is explanatorily useful for many purposes, but also that it cannot play all of the explanatory roles for which we might want a notion of content to play. (Though this is not true of everyone; cf. Sandgren (2018).) The epistemic conception, in particular, seems better equipped to account for the phenomenon of cognitive significance. That H2O is H2O is trivial and easily discovered upon a priori reflection, but no amount of reasoning absent empirical evidence will get us to water is H2O. Where our task is to model the kind of information an agent has available to her via reasoning from her explicit beliefs, the epistemic conception thus seems particularly apt. But lest it be said that the epistemic conception does not "really" give us a coarse-grained theory to content, let me note some points before moving on. First, one should not assume that the epistemic conception involves merely supplementing the original space of metaphysically possible worlds with a number of metaphysically impossible worlds so as to let us distinguish between, e.g., those worlds where Hesperus is Hesperus and those where Hesperus is Phosphorus. For one thing, at least some metaphysical possibilities seem to be a priori false. For instance, where 'watery' designates the property of being the clear, potable liquid around here that fills the lakes and oceans and falls from the sky as rain, then the thought that water is watery is arguably a priori, but it's certainly not metaphysically necessary that H2O is watery. Or a less controversial example: if something is watery, then the stuff that is actually watery is watery is clearly a priori, but it is not metaphysically necessary. The epistemic conception is not merely the metaphysical conception with a few extra impossible worlds. 15 I include here only a bare-bones development of the epistemic conception. More detailed developments can be found in Chalmers (2002), Chalmers (2006), Chalmers (2011), Jackson (1998), Jackson (2009). I have defended the epistemic conception elsewhere; see [anonymised]. 26 Indeed, on so-called 'one-spaceist' views (see e.g., Jackson in Jackson (2009) and Chalmers in (Chalmers 2006, p. 82)) the set of (centred) metaphysically possible worlds and the set of maximally specific epistemic possibilities are one and the same: every (centred) world can be thought of in one of two ways: as an hypothesis about how the world might be for all one might know a priori, or as an hypothesis about how it could have been given the way things actually are. According to one-spaceism, then, there's no sense in which the epistemic conception is more (or less) fine-grained than the metaphysical conception- contents on either conception are simply subsets of a single space of worlds, Ω, so each posits a notion of content which is exactly as coarse-grained as that which is posited by the other. Supposing then that classical logic is a priori, if we adopt the epistemic conception then we will end up with a model on which PROP, MP, and REP are valid, but which also lets us draw some distinctions between contents under different modes of presentation that are unavailable on the metaphysical conception. For instance, on the epistemic conception, no pair from the earlier list of modes of presentation for thinking that skirnobs are poisonous are a priori equivalent, so each will hold relative to a different subset of Ω. Moreover, cases like Examples 2 and 3 fail to generate counterexamples to AUR, for the proposition ψ that Iggy is supposed to be aware of in those cases is not a priori equivalent to Aiφ. To consider just the latter example, it is certainly not a priori that Iggy is in brain state S if and only if Iggy is aware of the proposition that Iggy is aware of the proposition that skirnobs are poisonous. Awareness is not as cheap on the epistemic conception as it is on the metaphysical conception. It should be clear that other purported counterexamples which rely on a posteriori metaphysical identities or rigidified definite descriptions will fail on the epistemic conception for similar reasons.16 There is a general reason for thinking that counterexamples won't arise once we've got a notion of content that cuts as fine as cognitive significance-for how could Iggy entertain some content ψ that has the very same cognitive significance as Iggy is awareness-related to φ, without having the conceptual resources to represent Iggy, awareness, and φ? Supposing that ψ is epistemically contingent, then AUR looks essentially right. Unlike >, there are only so many ways to entertain the thought that Iggy is aware of φ under a mode of presentation that's a priori equivalent to Iggy is aware of φ, and it's reasonable to expect that they all go hand-in-hand with the capacity to entertain φ itself. Furthermore, there's a stronger case to be made on the epistemic conception that there are no φ such that it's a priori that Iggy is not aware of φ. Arguments from cognitive limitations don't seem to get any grip: for any limitations that are supposedly essential Iggy qua human being, it is not a priori for Iggy that he is subject to those limitations. For all he knows a priori, he could have been a bat. (See also Chalmers (2011), §9, for a detailed discussion of whether there are any a priori unentertainable propositions.) 16 The same points apply to cases that involve linguistic deference and social externalism, though I have not discussed these. When a non-expert thinks to themselves, I have arthritis in my thigh, their grasp of arthritis is distinct from the understanding of an expert. Roughly, it is something like the disease the experts refer to when they say 'arthritis'. What the nonexpert knows a priori when they know something "merely by being situated in a linguistic community" is quite different than what the experts know. Cf. (Chalmers 2002, §9). 27 Before we move on, let me consider one final case for the existence of a proposition that's a priori unentertainable: Example 4. Iggy decides to let 'Silly' name that proposition φ such that, from amongst those propositions of which he is unaware, is expressible by the shortest English sentence. He thinks to himself: I am not aware of Silly. Assuming that Iggy's designation succeeds-i.e., there is indeed a unique φ that satisfies the description-then Iggy's thought is certainly a priori. But we have to be careful here. The case does not establish the existence of a proposition φ such that it's a priori for Iggy that he is not aware of φ, because it is not a priori what proposition 'Silly' picks out (if in fact it picks out any proposition at all). What Iggy knows a priori is that if 'Silly' picks out something, then whatever proposition it picks out, he is not aware of that proposition. At one epistemic possibility ω1, 'Silly' might designate φ1; at ω2, φ2. It's not a priori what proposition 'Silly' picks out, so there's no specific φ that Iggy knows he's not aware of, and AUR is safe. 7. Critical Discussion: PLAb Finally, let us consider PLAb. As with Section 6, I will discuss the plausibility of this axiom from the perspective of one who has already accepted a coarsegrained theory of content. There are two main cases: those where there are counterexamples to AUR (e.g., if we've adopted a metaphysical conception), and those where there are no counterexamples to AUR (e.g., if we've adopted an epistemic conception). 7.1 Counterexamples from AUR to PLAb I will here argue that any counterexamples to AUR can be expected to generate a counterexamples to PLAb, at least inasmuch as X represents knowledge. This should be unsurprising: the potential counterexamples to AUR that were discussed in the previous section can quite clearly be tweaked to provide counterexamples to PLAb. Consider Example 2, and substitute 'Iggy is not aware of the proposition that. . . ' for 'Iggy does not know that. . . '; everything else about can be left unchanged. The case would be one in which Xiψ is true and ψ ↔ ¬Xiφ is necessarily true, which combined with ¬Aiφ would constitute a counterexample to PLAb. (The substitution of 'aware of' for 'knows that' should make no difference to the plausibility of ¬Aiφ in that example.) Similarly, if it turns out that there is some φ of which a given agent i is necessarily unaware, then φ is ipso facto also something that i necessarily cannot know or believe. In that case, > ↔ ¬Xiφ is necessary, leading to a counterexample against PLAb. However, there are more general reasons to think that there are no counterexamples to AUR that do not also generate counterexamples to PLAb, in the presence of TX . Thus, at least where X represents knowledge, we should be willing to reject AUR only if we're willing to reject PLAb. (This point will not be especially important for the rest of my discussion, so if you're satisfied with what's been said already then you may wish to skip directly on to Section 7.2.) First of all, I assume that if Xi¬Aiφ∧¬Aiφ can be true, then Xi¬Xiφ∧¬Aiφ can also be true. If Iggy can know that Iggy isn't aware of φ (despite not being 28 aware of φ), then Iggy can know that Iggy doesn't know that φ (despite not being aware of φ). After all, the latter is an obvious consequence of the former, and Iggy should be able to know any obvious consequences of what he knows if he's also aware of all the relevant propositions. And if being unaware of φ doesn't prevent him from being aware of ¬Aiφ, then there's no reason to think that it should prevent him from being aware of ¬Xiφ. Second, note that we have a counterexample to AUR only if there is some possible situation where Ai¬Aiφ ∧ ¬Aiφ; and a counterexample to PLAb only if there is a situation where Xi¬Xiφ ∧ ¬Aiφ. So let ψ = ¬Aiφ, and assume that there does exist some counterexample to AUR while there are also no counterexamples to PLAb. Obviously this requires that ψ be contingent, given the points made above; likewise φ must also be contingent, else ¬Aiφ would be incompatible with Ai¬Aiφ. In light of the assumption of the previous paragraph and the fact that Xiψ already implies ψ, this would have to mean that Aiψ ∧ψ somehow precludes the possibility of Xiψ-which would be very strange, since if ψ is true and i is aware of ψ, then i should be able to know ψ. It would be one thing if the truth of ψ for some reason precluded awareness of ψ, which would in turn naturally rule out any knowledge of ψ; but if the truth of ψ is compatible with awareness of ψ, then it should also be compatible with knowledge that ψ. (Simply imagine a case where Aiψ ∧ ψ, and i has, let's say, plenty of reliable testimonial evidence that ψ.) Furthermore, since ¬Aiψ already implies ¬Xiψ, the assumption that AUR faces counterexamples while PLAb doesn't would mean that ψ alone implies ¬Xiψ; so, Xiψ would have to be impossible tout court. By PLAb, ¬Aiψ implies ¬Xi¬Xiψ, so ¬Aiψ implies ¬Xi>; and by NAX , ¬Xi> implies ¬Ai>. Putting these two points together, ¬Aiψ would imply ¬Ai>; and we already know that Aiψ implies Ai>. Hence, Ai> ↔ Aiψ would be necessary. Yet ψ cannot be impossible, by hypothesis; likewise, ψ cannot be necessary, since then Xi> would be impossible and so would Aiψ, which again it isn't by hypothesis. Thus, for PLAb to avoid falling foul of the whatever counterexamples exist for AUR, we would need that there are no propositions φ of which i is necessarily unaware, and for each and every φ such that Ai¬Aiφ ∧ ¬Aiφ is possible, AiAiφ↔ Ai> is necessary. In sum: for there to be counterexamples to AUR yet no counterexamples to PLAb, there would need to be some pair of contingent propositions, φ and ψ, such that (i) ψ is true just in case i is not aware of φ, (ii) i is aware of ψ whenever i is aware of anything whatsoever, and (iii) i cannot know ψ even when ψ is true and i is aware of it. I think we can safely assume that no such pair φ and ψ exists, even setting aside the already very strange property (iii). For what reason do we have to think that i's being aware of anything at all would require being aware of ¬Aiφ, for some contingent φ? The only somewhat plausible suggestion for what φ could be in this case is Ai>, or (what amounts to the same thing given SYM and CONA) 'i is aware of something'. But I have argued already in Section 3.3 that i's being aware of something or other doesn't presuppose the capacity to represent either i or awareness itself, so we don't get Ai¬AiAi> as a consequence for i's being aware of anything at all out of some simple iterative principles like AIA. And if I am right about that, then more generally we shouldn't expect awareness of any contingent proposition ¬Aiφ to be a consequence of awareness simpliciter-after all, any other way of entertaining that same proposition would require some other suite of representational resources which i may well also lack. 29 7.2 The implausibility of 'Plausibility' So the possible worlds theorist should reject PLAb, if they think that there are in fact some counterexamples to AUR. However, I have also argued that, at least on an epistemic conception of coarse-grained content, AUR is plausible. Consequently, I will here argue that where X represents knowledge, if the possible worlds theorist is willing to accept AUR, then they should reject PLAb-indeed, in part because of the acceptance of AUR. This might seem a little surprising. AUR and PLAb rest on more or less the same intuition. So, just as you might expect that if there are any problems for the former then there will be analogous problems for the latter, likewise you might expect that if there are any problems for the latter then there will be analogous problems for the former. And if we've decided that AUR is plausible after all, then shouldn't we do the same for PLAb? The problem with this thought is that AUR and PLAb are not entirely symmetrical in all relevant respects, once TX is factored into the equation. In particular, AUR implies the existence, for each agent i, of a specific class of propositions that i necessarily cannot know. Assume that there are no counterexamples to AUR. Then, for any φ, suppose that i knows that i is not aware of φ. Knowledge presupposes awareness, so i must be aware of i's being unaware of φ, and therefore he must be aware of φ. However, by veridicality, i cannot be aware of φ. Contradiction. Consequently, if AUR is accepted, then there are certain propositions that i cannot know-i.e., anything necessarily equivalent to ¬Aiφ-simply because the truth of those propositions is incompatible with i's knowledge thereof. On the other hand, PLAb does not imply the existence of any propositions of which i is necessarily unaware. And this is an important asymmetry. For we have seen that AUR is plausible from the perspective of a coarse-grained theory of content only if there are no propositions φ of which i is necessarily unaware-simply because, if ¬Aiφ is impossible, then i might come to be aware of ¬Aiφ in a myriad of ways without being aware of φ (or himself, or the awareness relation). By the same token, PLAb is plausible from the perspective of a coarse-grained theory of content only if there are no propositions that i cannot possibly know-simply because, if Xiφ is impossible, then i might come to know ¬Xiφ in a myriad of ways without being aware of φ (or himself, or the awareness relation). Thus, on a coarse-grained theory of content, the acceptance of AUR generates problems for PLAb, but not vice versa. But we don't need to go via AUR to find problems with PLAb. Given TX , there are plenty of other kinds of impossible knowledge states. Suppose we introduced into the language ei, interpreted as 'i exists', and stipulate that it's true at a world ω just in case Ai(ω) is non-empty. (An agent who's not aware of anything whatsoever is not an agent at all, so the agent i exists only if i is aware of something.) Now consider Xi¬Xi¬ei: does Ai¬ei follow? No: the agent i might come to know > in any number of ways, and ipso facto come to know ¬Xi¬ei, without necessarily being able to represent himself or his own (in)existence. Or suppose that b is true just in case there are beliefs; then Xi¬b is impossible, T ↔ ¬Xi¬b necessary, yet knowing > doesn't imply being aware of b: having beliefs does not require being able to think thoughts about the existence of beliefs. 30 8. In Defence of Classical Logic The points made in Section 6 and Section 7 obviously hang on the prior acceptance of a coarse-grained theory of content in some form or another. I don't expect to have convinced anyone who thinks that the contents of thought cut finer than necessary equivalence that they ought to reject AUR and/or PLAb. To such a reader it will likely appear that I am seeing counterexamples to AUR and PLAb where I really ought to be seeing counterexamples to the coarse-grained theory of content. I've argued, for example, that since ¬Xi¬Aiφ is necessary and contents are coarse-grained, therefore i can know ¬Xi¬Aiφ without being aware of Aiφ. But perhaps I should have argued instead that since i can't know and be aware of ¬Xi¬Aiφ without being aware of Aiφ, therefore contents are not coarse-grained. (This was the essential point of Dekel et al.'s argument, after all.) But therein lies I think the most important lesson of the foregoing discussion: to the extent that incorporating some representation of unawareness into classical logics of belief and knowledge presents possible worlds theorists with any problems at all, it doesn't seem to add any specific issues over and above the more general concerns about hyperintensionality of which we are all already aware. For the possible worlds theorist, who has considered the arguments and intuitions in favour of fine-grained contents and found them wanting, it's not at all troubling to be told that there's something counterintuitive to saying that i can know and be aware of ¬Xi¬Aiφ even while unaware of Aiφ, given that Xi¬Aiφ is impossible. This is precisely on a par with being told that one can know p ∨ ¬p even while one is unaware of p-and no possible worlds theorist is going to give up on their position because of that! Or consider again the axiom DISA: DISA. Ai(φ ∧ ψ)→ (Aiφ ∧Aiψ) There is no denying that DISA has prima facie plausibility. But that plausibility stems entirely from the intuition that mental contents cut finer than necessary equivalence-that to be aware of φ∧ψ is to be aware of some structured entity which has φ and ψ as its essential parts. Any triviality argument against a coarse-grained account of unawareness that's based on DISA (like the one we saw in Section 3.1) is by virtue of this no more compelling a reason to reject the coarse-grained view than the already apparent strangeness of the claim that by knowing p ∨ ¬p you therefore also automatically know ¬(q ∧ ¬q). If there were something special about the phenomenon of unawareness itself which rendered it incompatible with classical logic-something separable from presuppositions about granularity-then that would be a major blow for the possible worlds theorist. But there is nothing new here, just an old problem dressed up in new clothes. And, importantly, possible worlds theorists have a suite of tools to help address the problems that arise from the apparent hyperintensionality of thought. Given a Two-Dimensionalist approach to mental content and an appropriately characterised space of epistemically possible worlds, for example, many of the classic problem cases for possible worlds semantics-e.g., the need to distinguish between metaphysically equivalent but epistemically non-equivalent water-beliefs and H2O-beliefs-can be dealt with very naturally without deviating from the basic idea that propositional contents are sets of possible worlds. And conversational pragmatics can help deal with linguistic intuitions about 31 the substitutability of that-clauses within the context of propositional attitude verbs even in cases of logical or a priori equivalence (see esp. Stalnaker (1978), Stalnaker (1984)).17 The real concern for coarse-grained theories of content arises when they are taken in conjunction with a plausible theory of action; viz., that agents will typically act so as to maximise their desire or preference satisfaction given the way they believe the world to be (cf. Stalnaker (1991)). If a theory of content has empirically false behavioural implications then it must be rejected-and, prima facie, the typical subject doesn't seem to act as we might expect given beliefs in every necessary truth unless we posit that they have very strange desires. Thus the coarse-grained theorist has to tell us a story about why, for example, a mathematician might spend her days trying to work out whether or not the Riemann hypothesis is true. (She already knows the answer to that question; she just doesn't know whether the sentence 'The Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2 ' expresses the necessary truth-the task is to attribute some plausible set of beliefs and desires that rationalises the time spent discovering yet another complicated way to say >.) For AUR and/or PLAb to generate empirical problems for the coarse-grained theory along similar lines, we would need to have a case of an ordinary agent i with presumably ordinary desires who, with respect to some φ such that it's impossible for i to be aware of φ (or such that i cannot know that she is unaware of φ), tends to behave as if she doesn't believe that she's unaware of φ (or as if she doesn't believe that she doesn't know she's unaware of φ). But how does the typical agent generally fail to act so as to suggest that she doesn't have the relevant beliefs? Examples are not easy to find, since (under REP) i will also believe that > → ¬Aiφ (or > → ¬Xi¬Aiφ), which will tend to mute any behavioural consequences that we might otherwise have expected would be generated by those beliefs. For instance, perhaps i doesn't like being unaware of anything; hence, for i, ¬Aiφ represents an undesirable state of affairs. But if she knows that > → ¬Aiφ, then she won't try to do anything to improve her state of awareness in this respect. More generally, according to the view of rational action that is supposed to generate the problem, agents' actions are a response to those things they believe they can change so as improve their situation. So considered in combination with a coarse-grained account of content, we shouldn't expect a belief in a necessary proposition to have interesting behavioural consequences: agents who believe a necessary proposition also believe that it's necessary, that it will be a fact of the world regardless of what they choose. So it's certainly not obvious that there would be any behavioural consequences of saying that i believes ¬Aiφ, or ¬Xi¬Aiφ, which aren't in fact borne out by i's behaviour. It may turn out that there is really is no way to assign plausible coarsegrained beliefs and desires so as to rationalise the actions of ordinary agents. If so, we'll need to either adopt a more fine-grained approach to content, or revise our theory of action. I take this is still an open empirical question. The success of standard models of decision-making-which generally make use of 17 'Fragmentation' also helps deal with additional concerns relating to information access and logical closure properties. However, these are much less pressing issues for classical logics in general, which are only committed to closure under necessary equivalence. Similarly for PWA models, where explicit beliefs are not closed under implication and implicit beliefs are only closed under implication relative to awareness. 32 coarse-grained content-provides a limited reason to think that the behavioural data can be accommodated within a classical logic. Or, at the least, they provide reasons to continue modelling contents using sets of possible worlds. And whatever issues such models may or may not have, they are independent of considerations arising from unawareness. 9. Rational Belief with Non-Trivial Unawareness Let's take stock. Where X represents knowledge, XUIb is plausible but PLAb isn't. WhereX represents belief, XUIb is implausible, while PLAb seems about as plausible as AUR is. Whether AUR is plausible is independent of how we interpret X, but does seem to depend on (a) the particular theory of coarse-grained content, and (b) whether there are any necessarily or a priori unentertainable propositions. Thus, where X represents belief, the status of AUR and PLAb is uncertain. In this final section, I will prove that supplementing Σ with AUR and PLAb is consistent with non-trivial awareness in the absence of XUIb. In fact, I'll prove something stronger than that-suppose we add the following three axioms to Σ: ARA. AiAjφ→ Aiφ ARX . AiXjφ→ Aiφ ARI . AiIjφ→ Aiφ Given SYM, ARA implies AUR; and given SYM and PLAa, ARX gets us to PLAb. The three in combination say that for all agents i and j, if i is aware of j's having some attitude (A, X, or I) regarding φ, then i must herself be aware of φ. I take it that this is a natural generalisation of the shared intuition that motivates both PLAb and AUR-so, if we can show that the combination of ARA, ARX , and ARI is compatible with non-trivial awareness in the absence of XUIb, then we'll also have given strong reasons to believe that there are no other triviality results in the vicinity of Dekel et al.'s that rest on the same intuitions but don't go through XUIb. Let Σ† refer to the system Σ∪{ARA,ARX ,ARI}, and let M† refer to that class of PWA models which satisfies the following additional constraints: ara. If {ω′ : P ∈ Aj(ω′)} ∈ Ai(ω), then P ∈ Ai(ω) arx. If {ω′ : P ∈ Xj(ω′)} ∈ Ai(ω), then P ∈ Ai(ω) ari. If {ω′ : P ∈ Aj(ω′) and ⋂ Xi(ω) ⊆ P} ∈ Ai(ω), then P ∈ Ai(ω) The following is then easy to prove given the proof of Theorem 3.1: Theorem 9.1 Σ† is sound and complete with respect to M† and LAXI(Φ). Proof. See Appendix B. We can now show that there are PWA models of (rational) believers belonging to M† which allow for non-trivial awareness. For the model Me = (Ω,Xi,Ai, π), let Ω = {ω1, ω2, ω3}, and Φ = {p, q, r}. We will suppose that 33 π(p) = {ω1}, π(q) = {ω2}, and π(r) = {ω3}, and we let Iggy's awareness and belief functions be defined as follows: Ai(ωn) =  {∅, {ω1}, {ω2, ω3},Ω}, if n = 1 {∅,Ω}, if n = 2 2Ω, if n = 3 Xi(ωn) =  {{ω2, ω3},Ω}, if n = 1 {Ω}, if n = 2 {Ω}, if n = 3 The world of interest is ω1, where Iggy's awareness is non-trivial: he is able to draw a distinction between p-worlds and ¬p-worlds, but he is unable to draw a distinction between q-worlds and r-worlds. From the model Me, Theorem 9.2 follows: Theorem 9.2 For all φ,ψ, M† 6|= ¬Aiφ→ ¬Xi> and M† 6|= ¬Aiψ → ¬Aiφ. Proof. The model obviously satisfies pla, sym, con, and nax. That Me satisfies ara can be seen by noting first of all that: {ω : P ∈ Ai(ω)} =  Ω, if P ∈ {∅,Ω} {ω1, ω3}, if P ∈ {{ω1}, {ω2, ω3}} {ω3}, otherwise So there are exactly three propositions P (i.e., Ω, {ω1, ω3}, {ω3}) such that for some φ, P = ‖Aiφ‖M e . In all three cases it's easy to check that if Ω, {ω1, ω3} or {ω3} belongs to Ai(ω), then the required proposition P also belongs to Ai(ω). Similarly for arx: {ω′ : P ∈ Xi(ω)} = { Ω, if P = Ω {ω1}, if P = {ω2, ω3} So there are exactly two propositions P (i.e., Ω, {ω1}) such that for some φ, P = ‖Xiφ‖M e . {ω1} is in Ai(ω) whenever {ω2, ω3} is, and Ω always belongs to Ai(ω). Finally, ari is equivalent in this case to arx, since Xi(ω) is already closed under supersets of ⋂ Xi(ω). Furthermore, at every world in the model, Iggy is rational in at least the sense that his beliefs satisfy: KX . (Xi(φ→ ψ) ∧Xiφ)→ Xiψ D. Xiφ→ ¬Xi¬φ N. Xi> KX is stronger than we need to call Iggy rational, of course, but its satisfaction straightforwardly implies the more generally plausible rationality axioms: KAX . (Xi(φ→ ψ) ∧Xiφ)→ (Aiψ → Xiψ) CONX . (Xiφ ∧Xiψ)→ Xi(φ ∧ ψ) 34 Together these mean that there's no difference between Iggy's implicit and explicit beliefs-Iggy has drawn every classical logical consequence that can be drawn from what he believes given everything of which he is aware. Thus, let M†† designate the restriction of M† models to those where the Xi satisfy the following constraints (for all i, ω, P1, P2): kax. If P1 ∈ Xi(ω) and P1 ⊆ P2, then P2 ∈ Xi(ω) if P2 ∈ Ai(ω) conx. ⋂ Xi(ω) ∈ Xi(ω) d. If P ∈ Xi(ω), then Ω \ P /∈ Xi(ω) n. Ω ∈ Xi(ω) Then the model also obviously establishes: Corollary 4 For all φ,ψ, M†† 6|= ¬Aiφ→ ¬Xi> and M†† 6|= ¬Aiψ → ¬Aiφ. That is, even under the strengthened awareness conditions (arx, ara and ari) which generalise PLAb and AUR, plus basic rationality conditions (kax, conx, d and n), there are going to PWA models which can accommodate the possibility of non-trivial unawareness, in the sense that: 1. An agent can be unaware of φ and still believe >. 2. An agent can be unaware of φ without being unaware of everything. Consequently, where the goal is just to develop a logic of non-veridical belief with unawareness, the possible worlds theorist has at least one sure way out of Dekel et al.'s triviality result-and they can rest easy that there will be no further triviality results in the nearby vicinity, either. 10. Conclusion The very large majority of epistemic or doxastic logics that have been developed over the past two decades which feature an awareness operator in some form or another have been non-classical. All of these logics are incompatible with a coarse-grained approach to content. This sets much the work on unawareness somewhat at odds with research elsewhere in formal epistemology, where coarsegrained models of propositional content are still very much standard-and for good reason. Moreover, the present state of affairs leaves the possible worlds theorist without an appropriate way to understand the impact that unawareness has on belief and informational content. I have shown that it's possible to retain PLA and AUR-widely considered to be of special importance to any model of unawareness-within a coarse-grained content model of belief with unawareness, and indeed we can include stronger axioms that generalise the intuitions behind PLA and AUR without introducing triviality. We can do this as long as we give up XUI, which I've argued we should do regardless of how fine-grained we take mental content to be. There are no formal reasons arising from PLA and AUR for not adopting a model that makes use of coarse-grained, sets-of-possible-worlds contents. Whether we ought to keep both of those axioms, on the other hand, is a trickier matter. I have also argued that when it comes to modelling knowledge, the arguments in favour of PLA add nothing over and above already existing arguments against coarse-grained contents. Theorem 4.1 and its corollaries give us no reason to think that a classical logic with unawareness is any worse off than the 35 classical logic of belief was already with respect to the puzzles associated with hyperintensionality. There is a rich body of philosophical work dealing with exactly these puzzles within the framework of traditional possible worlds semantics, and all the reason in the world to think that the very same work can be marshalled in support of a possible worlds model of awareness. Appendix A The soundness part of Theorem 3.1 is straightforward by induction and left to the reader. To prove completeness we will construct a canonical PWA model. For any φ ∈ LAXI(Φ), say that φ is consistent (relative to the system Σ) iff it's not the case that `Σ ¬φ; a finite set Γ = {φ1, . . . , φn} is consistent iff φ1∧* * *∧φn is consistent; and an arbitrary set Γ = {φ1, φ2, . . . } is consistent iff every finite subset of Γ is consistent. Finally, say that Γ is a maximal consistent set (i.e., MaxΓ) just in case Γ is consistent and maximal, in the sense that any strict superset of Γ is inconsistent. On the ordinary way of constructing canonical models, the set of 'worlds' is just the set of all maximal consistent sets of formulas. For the present proof, however, it will be easier to include two 'worlds' for every maximally consistent set Γ. In this, I am applying a modified version of a strategy from Fagin and Halpern Fagin and Halpern (1988). For the sequel, then, let Ω0 = {Γn : MaxΓ, n = 0}, and Ω1 = {Γn : MaxΓ, n = 1}, with Ωc = Ω0 ∪Ω1. We will also make frequent use of the following abbreviations. We use '|φ|' to refer to the proof set of φ in Ωc; i.e., the set of all Γn ∈ Ωc such that φ ∈ Γn. We use 'B(X)' to refer to the closure of X under complements and binary intersections. And finally, for all i and Γn, Γn\Ai = {φ : Aiφ ∈ Γn} P ?α,Γn = {∆1 ∈ Ωc : Γn\Ii ⊆ ∆1} 'Γn\Xi' and 'Γn\Ii' are defined in a similar fashion. We can now define the canonical PWA model, M c: Definition 10.1 M c = (Ωc, {X ci }i∈Ag, {Aci}i∈Ag, πc), where: 1. Ωc = Ω0 ∪ Ω1 2. X ci (Γn) =  ∅, if Γn\Ai = ∅ {|φ| : Xiφ ∈ Γn} ∪ {Ω1,Ω0}, if Γn\Ai 6= P ?α,Γn = ∅ {|φ| : Xiφ ∈ Γn} ∪ {P ?α,Γn}, if P ?α,Γn 6= ∅ 3. Aci (Γn) = { ∅, if Γn\Ai = ∅ B({|φ| : Aiφ ∈ Γn} ∪ 2Ω 1), if Γn\Ai 6= ∅ 4. πc(p) = |p| We will also need a few basic lemmas. Lemma 10.1 For all φ,ψ ∈ LAXI(Φ), 1. |¬φ| = Ωc \ |φ| 36 2. |φ ∧ ψ| = |φ| ∩ |ψ| 3. |>| = Ωc Proof. The proof is no different than for standard canonical models where Ωc = {Γ : MaxΓ}, and therefore omitted. Lemma 10.2 B({|φ| : Aiφ ∈ Γn}) = {|φ| : Aiφ ∈ Γn} Proof. That {|φ| : Aiφ ∈ Γn} is closed under complementation: Suppose that P ∈ {|φ| : Aiφ ∈ Γn}. So, P = |φ|, for some formula φ such that Aiφ ∈ Γn. By SYM, it follows that Ai¬φ ∈ Γn. Hence, |¬φ| ∈ {|φ| : Aiφ ∈ Γn}, and by Lemma 10.1, |¬φ| = Ωc \ |φ|. That {|φ| : Aiφ ∈ Γn} is closed under binary intersections follows a basically similar structure, using CONA and Lemma 10.1. Lemma 10.3 |φ| ∈ B({|φ| : Aiφ ∈ Γn} ∪ 2Ω 1) just in case |φ| ∈ {|φ| : Aiφ ∈ Γn}, whenever Γn\Ai 6= ∅ Proof. The right-to-left direction is trivial. To establish the left-to-right direction, we suppose throughout that Γn\Ai 6= ∅, and hence {|φ| : Aφ ∈ Γn} 6= ∅. We first show that if a proposition P belongs to B({|φ| : Aiφ ∈ Γn} ∪ 2Ω 1), then P satisfies at least one of the following three conditions: c1 P ∈ {|φ| : Aiφ ∈ Γn} c2 P ⊆ Ω1 c3 P ∩ Ω0 = P ′ ∩ Ω0, for some P ′ ∈ {|φ| : Aiφ ∈ Γn} It's trivial that if P ∈ {|φ| : Aiφ ∈ Γn} ∪ 2Ω 1 , then P satisfies c1 or c2. So we only need to show that closing {|φ| : Aiφ ∈ Γn} ∪ 2Ω 1 under complements and binary intersections will not leave us with any P which don't satisfy (at least) one of c1, c2, or c3. We'll proceed in two steps. For the first step, note that: (a) If P satisfies c1, then Ωc \ P satisfies c1 (b) If P satisfies c2 or c3, then Ωc \ P satisfies c3 (a) follows immediately from Lemma 10.2. For (b), note first that if P satisfies c2, then P ∩ Ω0 = ∅, which belongs to {|φ| : Aiφ ∈ Γn} and therefore satisfies c3 trivially. Furthermore, supposing that P satisfies c3, P includes exactly those states of Ω0 which belong to some |φ|. So, (Ωc \ P ) ∩ Ω0 includes just those states of Ω0 which aren't included in P , which are exactly those in |¬φ| ∩ Ω0. Next, (c) If {ω1} satisfies c2, then {ω1} ∩ P2 satisfies c2 (for any P2) (d) If {ω1} and P2 both satisfy c1, then {ω1} ∩ P2 satisfies c1 (e) If {ω1} and c2 both satisfy c3, then {ω1} ∩ P2 satisfies c3 (f) If {ω1} satisfies c1, and P2 satisfies c3, then {ω1} ∩ P2 satisfies c3 (c) is obvious, and (d) is an immediate consequence of Lemma 10.2. Similarly, (e) and (f) follow more or less directly from Lemma 10.2, which entails that if P , P ′ ∈ {|φ| : Aiφ ∈ Γn}, then P ∩ P ′ ∩ Ω0 = P ′′ ∩ Ω0, for some P ′′ ∈ {|φ| : Aiφ ∈ Γn}. 37 Given facts (a) through (f) plus the fact that every P ∈ {|φ| : Aiφ ∈ Γn}∪2Ω 1 satisfies either c1 or c2, we know that for any P ∈ B({|φ| : Aiφ ∈ Γn} ∪ 2Ω 1), P satisfies c1, c2, or c3. So now let P refer to the set of all proof sets. We now prove that if P ∈ B({|φ| : Aiφ ∈ Γn} ∪ 2Ω 1) then P ∈ P only if P ∈ {|φ| : Aiφ ∈ Γn}, which gets us to the left-to-right direction of the present lemma. By definition, if P satisfies c1, then it belongs to {|φ| : Aiφ ∈ Γn}. Likewise, if P satisfies c2, then P ∈ P only if P = ∅, in which case it also belongs to {|φ| : Aiφ ∈ Γn}. And finally, if P satisfies c3, then P ∈ P only if P ∈ {|φ| : Aiφ ∈ Γn}. For c3 implies that P , whatever it is, contains exactly those states of Ω0 which are also contained in some P ′ ∈ {|φ| : Aiφ ∈ Γn}. Suppose then that P ∈ P . Then, Γ1 ∈ P iff Γ0 ∈ P , from which it follows that P = P ′ for some P ′ ∈ {|φ| : Aiφ ∈ Γn}. With that out of the way, we first want to show that the canonical model so-characterised actually belongs to the class of PWA models, M: Lemma 10.4 M c ∈M Proof. That Aci and X ci satisfy pla: For the case when Aci = ∅, X ci = ∅ by definition. For the cases where Aci is non-empty, note that when P ?α,Γn = ∅, then Ω1 and Ω0 are in both X ci (Γn) andAci (Γn); and when P ?α,Γn 6= ∅, P ?α,Γn ∈ X ci (Γn) and (trivially) P ?α,Γn ∈ Aci (Γn). So, it suffices to show that {|φ| : Xiφ ∈ Γn} ⊆ {|φ| : Aiφ ∈ Γn} This follows from XI and IA. For suppose that Bφ ∈ Γn. Then `Σ Xiφ → (Iiφ → Aiφ). So Aiφ is derivable from Xiφ in Σ, hence Aiφ ∈ Γn. Hence, |φ| ∈ {|φ| : Xiφ ∈ Γn} only if |φ| ∈ {|φ| : Aiφ ∈ Γn}. That Aci satisfies sym and con is true by construction; that Aci and X ci satisfy nax is straightforward and left to the reader. The next step is to establish a truth lemma: Lemma 10.5 M c,Γn |= φ iff φ ∈ Γn Proof. The proof proceeds by induction on the complexity of the formula φ. For the cases where φ = p, φ = ¬ψ, and φ = ψ ∧ γ, the argument is standard. Hence we will only focus on the following three cases: (a) φ = Xiψ (b) φ = Aiψ (c) φ = Iiψ For case (a): Under the inductive hypothesis, ‖ψ‖Mc = |ψ|, so ‖ψ‖Mc ∈ X ci (Γn) iff |ψ| ∈ X ci (Γn). Suppose that M c,Γn |= Xiψ. By Definition 3.2, it follows that ‖ψ‖Mc = |ψ| ∈ X ci (Γn). As before, let P refer to the set of all proof sets. Ω0 /∈ P and P ?α,Γn ∈ P only if P ?α,Γn = ∅ (since ∅ ⊆ P ?α,Γn ⊆ Ω1). So by Definition 10.1, if |ψ| ∈ X ci (Γn), then |ψ| ∈ {|φ| : Xiφ ∈ Γn}, and so Xiψ ∈ Γn. In the other direction, if Xiψ ∈ Γn, then |ψ| = ‖ψ‖M c ∈ X ci (Γn), so M c,Γn |= Xiψ. For case (b): Given Lemma 10.3, the argument that M c,Γn |= Aiψ iff Aiψ ∈ Γn is exactly parallel to case (a). 38 For case (c): Note first that P ?α,Γn ⊆ |φ| for any |φ| ∈ {|φ| : Xiφ ∈ Γn}. (Recall that P ?α,Γn = {∆1 : Γn\Ii ⊆ ∆1}, and by XI, Γn \Xi ⊆ Γn\Ii.) Given this and Definition 10.1, ⋂ X ci (Γn) = P ?α,Γn Given this and Definition 3.2, M c,Γn |= Iiψ iff Γn ∈ ‖Aiψ‖M c and P ?α,Γn ⊆ ‖ψ‖Mc . So suppose that Iiψ ∈ Γn. Given IA, we know then that Aiψ ∈ Γn; so Γn ∈ |Aiψ|, and by the inductive hypothesis, Γn ∈ ‖Aiψ‖M c . Furthermore, we know that ψ ∈ ∆1 for every ∆1 ∈ P ?α,Γn , so P ?α,Γn ⊆ |ψ| = ‖ψ‖M c , and P ?α,Γn ⊆ ‖ψ‖M c . Hence, if Iiψ ∈ Γn, then M c,Γn |= Iiψ. For the other direction, suppose that M c,Γn |= Iiψ. By the points just established, it follows that P ?α,Γn ⊆ ‖ψ‖M c , and given the inductive hypothesis, P ?α,Γn ⊆ |ψ|. From this it follows that the set Γn\Ii ∪ ¬ψ is inconsistent. For, suppose that Γn\Ii∪¬ψ ⊆ Λ, for some maximal consistent set Λ. It would follow that M c,Λ1 |= ¬ψ, so by Lemma 10.1, Λ1 /∈ |ψ|. However this cannot be, since Λ1 ∈ P+α,Γn ⊆ |ψ|. Since Γn\Ii ∪¬ψ is inconsistent, some finite set {φ1, . . . , φn,¬ψ} ⊆ Γn\Ii ∪ ¬ψ is inconsistent. Thus: `Σ φ1 → (. . . (φn → ψ) . . . ) Given NAX and XI, we have Ii(φ1 → (. . . (φn → ψ) . . . )) ∈ Γn whenever Ai(φ1 → (. . . (φn → ψ) . . . )) ∈ Γn. Next, note that since φ1, . . . , φn ∈ Γn\Ii, we also get that Iiφ1, . . . , Iiφn are in Γn. Under IA, it follows that Aiφ1, . . . , Aiφn are also in Γn. Furthermore, note that by Definition 3.2, Γn ∈ ‖Aiψ‖M c , which given the earlier points means that Aiψ ∈ Γn. Finally, for any pair of formulas φ and γ, if Aiφ ∈ Γn and Aiγ ∈ Γn, then by applications of SYM and CONA, Ai¬(φ ∧ ¬γ) ∈ Γn. Since we can also show that awareness is closed under logical equivalence, so Ai(φ → γ) ∈ Γn. Putting these points together, in Γn we have: Aiψ, Ai(φn → ψ), Ai ( φn−1 → (φn → ψ) ) ... Ai ( φ1 → ( . . . (φn → ψ) . . . )) Now, by finitely many applications of KAI , we can derive that Iiψ ∈ Γn. Hence, if M c,Γn |= Iiψ, then Iiψ ∈ Γn. We can then apply a standard argument to get from Definition 10.1, Lemma 10.4 and Lemma 10.5 to establish Theorem 3.1. See (Chellas 1980, pp. 59ff) for details. Appendix B For Theorem 9.1, I'll just sketch a proof that Σ ∪ {ARX} is complete for the class of PWA models which satisfy arx; the full proof proceeds along the same lines for the ara and ari. We keep the characterisation of the canonical models M c as given in Definition 10.1, with the obvious modification that Ωc is now 39 composed of maximal consistent sets relative to Σ ∪ {ARX}. The proof is then the same as for Theorem 3.1, with the following addition to Lemma 10.4: That M c satisfies arx: Let Q = {∆i : P ∈ X cj (∆i)}. We need to show that if Q ∈ Aci (Γn), then P ∈ Aci (Γn). Assume that Q ∈ Aci (Γn). Now, either P /∈ P , or P ∈ P . If the former, then by Definition 10.1, either P ⊆ Ω1 or P = Ω0; in either case, P ∈ Aci (Γn), so arx is straightforwardly satisfied whenever P /∈ P . Suppose then that P ∈ P . Since Definition 10.1 implies in general that X cj (∆1) = X cj (∆0), so ∆1 ∈ Q iff ∆0 ∈ Q. By points already established (Lemma 10.3), then, Q ∈ Aci (Γn) only if Q ∈ {|φ| : Aiφ ∈ Γn}; so Q = |φ| for some φ such that Aiφ ∈ Γn. Furthermore, since by hypothesis P /∈ (2Ω 1 \ {∅}) ∪ {Ω0}, P will be in X cj (∆i) if and only if P = |ψ| for some ψ such that Xjψ ∈ ∆i. So Q = |Xjψ|, for some ψ. Putting these two facts together, we know that AiXjψ ∈ Γn; and by ARX , Aiψ ∈ Γn; so |ψ| ∈ {|φ| : Aiφ ∈ Γn}, which is of course a subset of Aci (Γn). References Braddon-Mitchell, D. and F. Jackson (1996). Philosophy of Mind and Cognition. Malden: Blackwell. Bradley, R. (2017). Decision Theory with a Human Face. Cambridge: Cambridge University Press. Bueno, O., C. 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