Our senses provide us with information about the world, but what exactly do they tell us? I argue that in order to optimally respond to sensory stimulations, an agent’s doxastic space may have an extra, “imaginary” dimension of possibility; perceptual experiences confer certainty on propositions in this dimension. To some extent, the resulting picture vindicates the old-fashioned empiricist idea that all empirical knowledge is based on a solid foundation of sense-datum propositions, but it avoids most of the problems traditionally associated with that idea. The proposal might also explain why experiences appear to have a non-physical phenomenal character, even if the world is entirely physical.

## 1. Learning from Experience

Through the window I can see that it is still raining. A stream of water is running down the street into the gutters. But can I tell, just be looking, that it is water? Couldn’t it be a stream of vodka? To be sure, that is an outlandish possibility. But if for whatever reason I had taken the vodka hypothesis seriously before looking outside, my visual experience wouldn’t put me in a position to rule it out. So if we define the information provided by my visual experience in terms of the possibilities the experience allows me to rule out, then the information I receive from my senses does not entail that there is a stream of water on the road. Nor does it entail that it is raining. What looks like rain could be a setup for a movie scene. My windows could have been replaced with sophisticated LCD screens. Again, my visual experience by itself does not put me in a position to rule out these possibilities.

This line of thought naturally leads to the old empiricist idea that the information we receive from our senses is in the first place information not about the external world, but about a special, luminous, internal realm of appearances or sense data: the possibilities I can rule out are all and only the possibilities in which things do not appear as they actually do. Yet this view also faces problems. Aren’t we often ignorant or mistaken about how things appear? How could everything we know about the world be inferred from facts about appearance? How are appearance facts supposed to fit into a naturalistic account of the mind?

There are other ways of defining the information provided by an experience. For example, if my experience is in fact caused by rain, and experiences of the same type are caused by rain across a variety of nearby worlds, then there is a good (causal) sense in which my experience carries the information that it is raining. But it is not clear how this sense of information bears on how the experience should change my beliefs. After all, I should not become absolutely certain that it is raining. With suitable background beliefs, my credence in the rain hypothesis should even decrease.

So perhaps we should drop the assumption that perceptual experiences put us in a position to conclusively exclude possibilities. Instead, my experience, perhaps together with my background beliefs, merely allows me to conclude tentatively and defeasibly that it is raining. In general, on this view, experiences combine with background beliefs to confer degrees of plausibility or probability to various claims about the world, without making anything certain.

But things are not so easy. To bring out why, let’s try to model the present idea in the framework of Bayesian epistemology. Here we assume that beliefs come in degrees that satisfy the mathematical conditions on a probability measure. How should these probabilities change under the impact of perceptual experience? Classical Bayesianism suggests the following answer. For each type of perceptual experience there is a proposition $$E$$ such that, whenever a rational agent has the experience, her new probability equals her previous probability conditional on $$E$$; that is, for all $$A$$,

$$P_{{new}}(A) = P_{{old}}(A/E) = P_{{old}}(A\land E)/P_{{old}}(E), \text{ provided P_{old}(E) > 0}.$$

Here, $$P_{{new}}$$ is said to come from $$P_{{old}}$$ by conditionalizing on E.[1] Since $$P_{{old}}(E/E) = 1$$, the new probability of $$E$$ is 1. So $$E$$ can hardly be an ordinary proposition about the world. Again, we seem forced to postulate a mysterious realm of sense-datum propositions.

To avoid commitment to such propositions, Richard Jeffrey developed what he called radical probabilism as an alternative to the classical Bayesian picture (see Jeffrey 1965: Chapter 11; 1992). Radical probabilism rejects the idea that subjective probabilities require a bedrock of certainty. To use a well-known example from Jeffrey (1965), imagine you catch a glimpse of a tablecloth in a poorly lit room. According to Jeffrey, the direct effect of this experience on your beliefs may be that you come to assign credence 0.6 to the hypothesis that the cloth is green and 0.4 to the hypothesis that it is blue; these probabilistic judgments need not be inferred from anything that has become certain.

In general, Jeffrey’s model assumes that an experience is directly relevant to some propositions and not to others. Suppose $$E_1,\ldots,E_n$$ is a list of pairwise exclusive and jointly exhaustive propositions whose probabilities change in response to an experience so that their new probabilities are $$x_1,\ldots,x_n$$ respectively. If the experience is directly relevant only to $$E_1,\ldots,E_n$$, then probabilities conditional on these propositions should be preserved. It follows that the new probability of any proposition A is given by

$$P_{{new}}(A) = \sum_i P_{{old}}(A / E_i) \cdot x_i.$$

This transformation from $$P_{{old}}$$ to $$P_{{new}}$$ is known as Jeffrey conditionalization.[2]

At first glance, Jeffrey’s model seems to deliver just what we were looking for. Instead of assuming that each type of perceptual experience is associated with a sense-datum proposition E rendered certain by the experience, we only need to assume that there is some assignment of probabilities $$x_1,\ldots,x_n$$ to the elements of some partition $$E_1,\ldots,E_n$$ of ordinary propositions such that, when a rational agent has the experience then her degrees of belief evolve by the corresponding instance of Jeffrey conditionalization.

More concretely, we might assume that for every perceptual experience there is a proposition E that captures how the experience intuitively represents the world as being. We do not require agents to become absolutely certain of $$E$$ when they have the experience. Instead, we might say that they should assign some intermediate credence x (maybe 0.95) to $$E$$, and consequently $$1\!-\!x$$ to $$\neg E$$. The complete update is then determined by the following special case of Jeffrey’s rule:

$$P_{{new}}(A) = P_{{old}}(A / E) \cdot x + P_{{old}}(A / \neg E) \cdot (1-x).$$

We would still need to explain why this response is justified: why it is OK to tentatively assume that the world is as it appears to be. But at least we seem to have a structurally sound model of belief change that frees us from the implausible commitments of the classical model.

Unfortunately, the present model won’t do either—not if experiences are individuated by their physiology or phenomenology.[3] For then the rational response to a given experience should depend on the agent’s background information. Your new beliefs about the colour of the tablecloth, for instance, should be sensitive to background beliefs about the colour of other tablecloths in the house. My belief about the weather should be sensitive to background beliefs about whether or not people are filming a rain scene outside my window.

So we cannot associate experience types with fixed posterior probabilities $$x_1,\ldots,x_n$$ over fixed propositions $$E_1,\ldots,E_n$$. We must also take into account the agent’s previous probabilities $$P_{{old}}$$ . But how does a given type of experience, together with an agent’s previous probabilities $$P_{{old}}$$ , determine the “inputs” to a Jeffrey update: the evidence partition $$E_1,\ldots,E_n$$ and the associated probabilities $$x_1,\ldots,x_n$$?

This question is sometimes called the input problem for Jeffrey conditionalization. It was first raised by Carnap in his 1957 correspondence with Jeffrey (published in Jeffrey 1975). Carnap reports that he had himself toyed with the idea of relaxing the classical Bayesian account along Jeffrey’s lines but had given up because he couldn’t find an answer to the input problem. Since then, nobody else has found a plausible answer either. It is widely thought that the problem simply can’t be solved.[4]

Jeffrey, in any case, never gave an answer to the input problem. His radical probabilism is silent on how perceptual experiences together with previous beliefs and possibly other factors yield new probabilities $$x_1,\ldots,x_n$$ over a partition $$E_1,\ldots,E_n$$ such that probabilities conditional on the partition cells are preserved. All Jeffrey says is that if somehow or other this happens, then the new probabilities ought to result from the old ones by the relevant instance of Jeffrey conditioning. But that much is a simple consequence of the probability calculus. Jeffrey’s account therefore doesn’t provide a substitute for conditionalization as the second norm of Bayesian epistemology. His alternative threatens to collapse into the first norm, probabilistic coherence.

This leaves a serious gap in Bayesian epistemology (as noted by Carnap and reiterated, e.g., in Field 1978 and Christensen 1992). The demands of epistemic rationality go well beyond probabilistic coherence. There are substantive norms on how one’s beliefs may change through perception.[6] For example, when a chemist uses a litmus strip to test whether a solution is basic or acidic, they are not free to change their beliefs in any way they please in response to the outcome. Likewise, my visual experience of the rain supports the hypothesis that it is raining, but not that it is snowing or that Tycho Brahe was poisoned by Johannes Kepler. (“How do you know?”—“I looked through the window”.)

Even if there were no such norms, we would have a gap in Bayesian psychology. A psychological model of rational agents should have something to say on how belief states change under the impact of perceptual experience. If this could not be done within the Bayesian framework, we should conclude that something is wrong with the framework. But the problem isn’t internal to Bayesianism. The general problem, illustrated by examples like the repeated tablecloth experience, is that if beliefs only pertain to ordinary external-world propositions, then the rational impact of a perceptual experience on an agent’s beliefs is not determined by the nature of the experience, her previous beliefs, and the environment. Something else plays a role. We need to know what it is and how it works.

## 2. Armchair Robotics

To make progress on the problem raised in the previous section, it may help to change perspectives and think about how we would design an ideal agent. Imagine we are to build a robot whose task is to find certain objects—mushrooms perhaps, or tennis balls, or landmines. To this end, the robot has a database in which it can store probabilistic information about the environment. It also has sense organs to receive new information. How should the probabilities in the database change in response to activities in the sense organs?

A sense organ is a physical device whose internal state systematically and reliably varies with certain features of the environment. Let’s assume our robot has a visual sense organ consisting of a two-dimensional array of photoreceptors, like in the human eye. When hit by light of suitable wavelengths, each photoreceptor produces an electrical signal. Different colours, shapes and arrangements of objects in the environment give rise to different patterns of light waves activating the photoreceptors, which in turn lead to different signals produced by the sense organ—that is, to different patterns of electrochemical activity in the “output” wires of the device.

It would be convenient if one could read off the exact colours, shapes, and spatial arrangement of objects in the environment from the signal produced by the robot’s sense organ. In practice, this is not possible, because different configurations of the environment lead to the very same activation of photoreceptors and thus to the very same sensory signal: a small cube nearby can cause the very same signal as a larger cube further away; a convex shape with light from above can cause the same signal as a concave shape with light from below; a red cube under white light can cause the same signal as a white cube under red light; and so on.

So the functional architecture of a sense organ only determines, for each sensory signal $$S$$, a range of alternative hypotheses about the environment $$E_1,\ldots, E_n$$ that could be responsible for $$S$$. Typically, some of these environmental conditions will be much more common than others. If our robot traverses the surface of the Earth, it will mostly find itself in situations where roughly white light is coming roughly from above. Nevertheless, the robot arguably shouldn’t become certain that a particular one of $$E_1,\ldots, E_n$$ obtains, giving zero probability to all the others. A better idea is to implement a form of Jeffrey conditionalization, where the new probabilities $$x_1,\ldots,x_n$$ over $$E_1,\ldots, E_n$$ might reflect something like the ecological relative frequency or objective chance with which the conditions obtain when the signal is produced.

But that is still not an optimal solution. The new probability assigned to the $$E_i$$ ’s should be sensitive not only to the sensory signal (equivalently, to the upstream activation of photoreceptors) but also to the old probabilities. For example, suppose signal $$S$$ occurs just as often under condition $$E_1$$ as under $$E_2$$, so that the ecological frequencies $$x_1$$ and $$x_2$$ are the same. Suppose further that before the arrival of $$S$$ the robot has received information that supports $$E_1$$ over $$E_2$$. On the present account, the new signal will wipe out this information, setting $$P_{{new}}(E_1) = P_{{new}}(E_2)$$ This is clearly not ideal. Relatedly, our robot should be able to learn whether it is in an environment where $$S$$ generally goes with $$E_1$$ or $$E_2$$; in the present model, the $$x_i$$ values are fixed once and for all.

A better idea, which gets us closer to actual approaches in Artificial Intelligence (see, e.g., Russell & Norvig 2010: Chapters 15 and 17), is to fix not the probability of $$E_i$$ given $$S$$, but the inverse probability of $$S$$ given $$E_i$$ . That is, let’s endow our robot with a “sensor model” that defines a probability measure π over possible signals $$S$$ conditional on possible world states $$E_i$$ . The new probabilities over the world states can then be computed by a variant of Bayes’ Theorem:

$$P_{new}(E_i) = \frac{\pi(S/E_i)P_{old}(E_i)}{\sum_j \pi(S/E_j)P_{old}(E_j)}$$

Note that $$P_{new}(E_i)$$ is sensitive to $$P_{old}(E_i)$$, as desired.

Ideally, the sensor model should not be fixed once and for all either. Ignoring matters of computational tractability, this problem is easily patched by merging the sensor model π into the robot’s main probability function $$P$$. That is, we extend the domain of $$P$$ by the set of possible sensory signals, and define $$\pi(S/E_i)$$ as $$P_{old}(S/E_i)$$. Intuitively, we assume that our robot has opinions about what kinds of signals it is likely to receive in what kinds of environments. These opinions can themselves change through sensory experience.

If we replace $$\pi(S/E_i)$$ in the above variant of Bayes’ Theorem by $$P_{{old}}(S/E_i)$$, then $$P_{{new}}(E_i)$$ is simply $$P_{{old}}(E_i/S)$$. Moreover, Jeffrey conditionalizing on a partition $$E_1,\ldots, E_n$$ whose new probabilities $$x_1,\ldots,x_n$$ are given by $$P_{old}(E_1/S),\ldots,$$ $$P_{old}(E_n/S)$$, respectively, is equivalent to strict conditionalization on $$S$$. So we might as well bypass the evidence partition $$E_1,\ldots, E_n$$ and simply say that for all $$A$$,

$$P_{{new}}(A) = P_{{old}}(A/S).$$

The new probability of any proposition $$A$$ is the old probability of $$A$$ conditional on the current sensory signal. Our robot has become a classical (“strict”) conditionalizer. What it conditionalizes on when it receives a sensory signal is not any of the relevant propositions $$E_1,\ldots, E_n$$ about the environment, but rather the signal itself.

There’s something odd about this approach. It looks like our robot must now have well-defined subjective probabilities over sensory signals—over the occurrence of complicated electrochemical events at the interface of its sense organs. One might have thought that a robot in search of mushrooms wouldn’t need to be trained in electrochemistry, and that it wouldn’t need to have perfect knowledge about the internal workings of its perceptual system. Indeed, a little reflection makes clear that this is not required.

Suppose the robot’s probabilities are originally defined only for certain propositions $$\mathbf{R}$$ about the macroscopic environment. The above considerations suggest that we need to extend the domain of the probability function by further elements $$\mathbf{I}$$ such that when a signal $$S$$ arrives, the robot conditionalizes on a corresponding proposition $$\rho(S) \in \mathbf{I}$$, where ρ is some function mapping distinct signals to distinct elements of $$\mathbf{I}$$. The elements of $$\mathbf{I}$$ thereby “represent” or “denote” electrochemical events in some causal sense, but this need not in any way be transparent to the robot or reflected in its probability space. For example, if at some point we wanted to replace the robot’s photoreceptors with ones that produce different electrochemical outputs, we would need to adjust the mapping ρ, but we might not have to change the content of the robot’s database. Similarly, if we eventually wanted to train our robot in electrochemistry, the elements of $$\mathbf{I}$$ would still not need to stand in interesting logical relationships to the electrochemical propositions in the robot’s belief space.

It may help to imagine that our robot stores information in the form of English sentences, so that its database associates sentences like ‘there is a mushroom to the left’ with numbers between 0 and 1. If we restrict the database language to ordinary sentences about the macroscopic environment, we will run into problems when we want to specify how the database should be updated in response to activity in the robot’s sense organs. To optimally deal with sensory input, I suggest, we need to extend the robot’s probability space by new sentences such that whenever a sensory signal arrives, the robot becomes certain of one of these sentences. But there is no good reason why these sentences must be correct and detailed descriptions of the relevant electrochemical signal. In principle, the update works just as well if the new sentences are bare tags, ‘$$A$$’, ‘$$B$$’, ‘$$C$$’, etc.

I will have more to say on what, if anything, the elements of $$\mathbf{I}$$ represent, and whether the robot’s probabilities over $$\mathbf{I}$$ should be understood as degrees of belief. But these are matters of interpretation; they don’t affect the coherence of the architecture I have outlined.

Formally, the required extension of a probability measure is a straightforward product construction. Take the simplest case where everything is finite. Let $$\textbf{R}$$ be the set of propositions about the environment on which we want the robot to have an opinion. Probability theory requires that $$\textbf{R}$$ is a Boolean algebra; so we can identify each proposition in $$\textbf{R}$$ with a set of “possible worlds”: the atoms of the algebra. Now let $$\mathbf{I}$$ be an arbitrary set disjoint from $$\textbf{R}$$ such that there is a one-one correspondence between $$\mathbf{I}$$ and the signals the robot can receive.[7] Each pair $$\langle{w,i}\rangle$$ of a possible world $$w$$ and an element $$i$$ of $$\mathbf{I}$$ is then an atom in the extended doxastic space; each set of such atoms is a bearer of probability.

Some terminological stipulations will be useful. I will call anything to which the extended probability measure assigns a value a (complex) proposition. The members of $$\mathbf{R}$$ are real propositions; subsets of $$\mathbf{I}$$ are imaginary propositions—in analogy to complex, real, and imaginary numbers, and to highlight the fact that imaginary propositions do not have to be understood as genuine propositions about a special subject matter. Individual members of $$\mathbf{I}$$ I will call sense data, since their role in the present model in some ways resembles the role of sense data in the classical empiricist model of perception (see Section 4 below). In the robot’s complex doxastic space, a real proposition $$A$$ can be re-identified with the set of atoms $$\langle{w,i}\rangle$$ whose possible world coordinate $$w$$ lies in $$A$$; similarly for imaginary propositions and sense data.

In some respects, the construction of complex propositions is analogous to a popular construction of centred propositions in the modelling of self-locating beliefs. Arguably, our doxastic space contains not only propositions about the universe as a whole, but also propositions about our own current place in the world: an agent might know every truth about the world from a God’s eye perspective and still be ignorant about who they are or what time is now (see Lewis 1979). Thus the atoms in an agent’s doxastic space are often modelled as pairs $$\langle{w,c}\rangle$$ of a possible world $$w$$ and a “centre” $$c$$ that fixes an individual and a time in $$w$$. In the resulting doxastic space, an objective proposition $$A$$ about the world is then re-identified with the set of “centred worlds” $$\langle{w,c}\rangle$$ whose possible-world coordinate $$w$$ lies in $$A$$.

In fact, there are good reasons to make the propositions in our robot’s doxastic space centred as well. Imagine our robot is moving towards a wall. At time $$t_1$$ it receives a signal $$S$$ which (by the robot’s lights) indicates that the wall is about 5 metres away. A little later, at $$t_2$$, the robot receives another signal $$S'$$ indicating that the wall is about 4 metres away. At this point, we don’t want the robot to conclude that the wall is most likely both 5 and 4 metres away. Nor should it conclude that the previous signal was faulty. Rather, it should realize that the first signal indicated that the wall was 5 metres away at the time, which is perfectly compatible with the distance now being 4 metres.

Here is how the model I have outlined could be adjusted to accommodate the passage of time. (The details will not be important for what follows, so feel free to skip.) First, we make the objects of probabilities centred. So probabilities are now defined over a three-fold product $$\mathbf{R}\times\mathbf{I}\times\mathbf{T}$$ (or a sub-algebra of that product), where $$\mathbf{T}$$ is a suitable set of relative time indices. Assuming for simplicity that time is linear and discrete, we might identify $$\mathbf{T}$$ with the set of integers, interpreting 0 as now, 1 as the next point in the future, and so on. When signal $$S$$ arrives, the robot conditionalizes not on $$\mathbf{R} \times \{ \rho(S) \}$$, as above, but on $$\mathbf{R} \times \{ \rho(S) \} \times \{ 0 \}$$ intuitively, on the indexical proposition that $$\rho(S)$$ is true now. Such indexical beliefs must be updated constantly to keep track of the passing time. Thus at the next point in time, the robot’s certainty of $$\mathbf{R} \times \{ \rho(S) \} \times \{ 0 \}$$ should have evolved into certainty of $$\mathbf{R} \times \{ \rho(S) \} \times \{ -1 \}$$. (Intuitively, the robot should now be certain that $$\rho(S)$$ was true one moment ago.) These updated probabilities are then conditionalized on the new evidence $$\mathbf{R} \times \{\rho(S')\} \times \{ 0 \}$$. See Schwarz (2017) for further details and motivation.

## 3. From the Armchair to Cognitive Science

I have described a model of how subjective probabilities change under the impact of sensory stimulation. The model requires an agent’s doxastic space to be extended by an “imaginary” dimension whose points are associated with sensory signals in such a way that when a given signal arrives, the agent assigns probability 1 to the corresponding imaginary proposition; the probability of any real proposition is then set to its prior probability conditional on that imaginary proposition.

As I mentioned in passing, this general approach is hardly new: it closely resembles standard treatments in artificial intelligence. It is also well-known in the neuroscience of perception, where similar models have proved a useful paradigm (see Yuille & Kersten 2006). In these areas, the propositions on which an agent or her perceptual system is assumed to conditionalize are called ‘percepts’, ‘sense data’, or ‘input strings’, and people rarely pause to reflect on their representational features or on what the postulated models imply for the epistemology of perception.

But it’s worth pausing and reflecting. Suppose, in line with evidence from cognitive science, that our own cognitive system approximates something like the model I have described. What would that mean? Would it vindicate classical empiricist foundationalism? Would it provide an answer to the problems from Section 1?

To begin, we need to clarify how the extended probability function that figures in the model should be understood. Does it represent the agent’s degrees of belief? Unsurprisingly, the answer depends on what we mean by ‘degrees of belief’. Philosophers often use terms like ‘belief’ or ‘credence’ in a demanding intellectualist sense tied to conceptual structure, conscious thought, and linguistic assertion. In that sense, it is doubtful whether cats, dogs, or robots have beliefs. The model I have described does not assume that the relevant agents have a language, or that they store information in the form of “conceptually structured propositions” (whatever that might mean). So the extended probability function in the model may not fit the job description for intellectualist degrees of belief.

An alternative to the intellectualist conception of belief is a family of functionalist conceptions on which belief and other intentional states are defined by their causal-functional role. On a crude version of this approach (still popular in some parts of economics), beliefs and desires are defined by an agent’s behavioural dispositions: to have such-and-such beliefs and desires means to be disposed to make such-and-such choices. Less crude versions of functionalism identify beliefs and desires with causally efficacious internal states whose defining functional role links them not just to behavioural output but also to one another and to sensory input: to have such-and-such beliefs is to be in some state or other that bears the right connection to sensory input, to behavioural output, and to other internal states similarly individuated by their functional role.[8]

On the functionalist approach, ‘belief’ is implicitly defined by a certain theory, or model. Different models define somewhat different notions of belief. Let’s revisit the classical Bayesian model from this perspective.

The Bayesian model assumes that agents have a credence and a utility function. What does it take for a lump of flesh and blood or silicon to have a particular credence and utility function? On the functionalist conception, an agent has a particular credence and utility function just in case she is in some state that plays the role the model attributes to these functions. One aspect of that role links the state to the agent’s actual and counterfactual choices: when facing a decision, standard Bayesianism assumes that an agent chooses an option that maximizes expected utility in light of her credence and utility function. Another aspect of the role describes how probabilities and utilities change over time. This is where the input problem arises. As we saw, it is hard to specify how an agent’s probabilities should change in response to sensory stimulations (or perceptual experiences, if you want) if we don’t assume that these provide infallible access to non-trivial facts about the world.

To get around the input problem, I have suggested that we should extend the domain of the model’s probability function by extra, “imaginary” elements associated with sensory signals in such a way that sensory stimulation leads to conditionalization on the associated imaginary element. So what does it take for a real agent to have such an extended probability function? As before, the agent must be in some state or other that plays the functional role the model assigns to the extended probabilities—to a sufficient degree of approximation.[9]

Even on the functionalist conception, however, there are limits to what one can sensibly call ‘belief’. If a functional role deviates too far from the role we ordinarily associate with the word ‘belief’, it would be better to use a different name. On these grounds, one might argue that an agent’s probabilities over non-real propositions should not be called ‘credences’ or ‘degrees of belief’. After all, to have a belief is to represent the world as being a certain way. Beliefs can be true or false; partial beliefs can be accurate or inaccurate to various degrees. It is not obvious whether probabilities assigned to imaginary propositions satisfy these conditions.

One might argue that imaginary propositions represent the corresponding sensory signals, so that $$\rho(S)$$ is true iff the relevant signal $$S$$ is received. From that perspective, probabilities over imaginary propositions might look like ordinary degrees of belief; our robot will be interpreted as having sophisticated beliefs about electrochemistry, albeit under an opaque ‘mode of presentation’.

I prefer to understand degrees of belief as pertaining directly to ways a world might be, not to intermediary entities which in turn represent ways a world might be, perhaps relative to a mode of presentation. More importantly, there is really no need to assign truth-values to imaginary and complex propositions. In the model I have described, the purpose of having probabilities assigned to these propositions is not to represent special information about the world. Rather, the extended probabilities encode the agent's dispositions to change her (genuine) beliefs about the world in response to sensory input. For example, what is encoded by our robot's assigning greater probability to $$A$$ & $$\rho(S)$$ than to $$\neg A$$, &, $$\rho(S)$$ is (to a large part) that receiving $$S$$ would make the robot assign greater probability to $$A$$ than to $$\neg A$$.

So there are reasons to not call an agent's probabilities over imaginary propositions `degrees of belief'. An agent's degrees of belief, on that usage, are the real part of her subjective probability function. These degrees of belief do not evolve by strict conditionalization, since the relevant sense data are not in the domain of the belief function. Instead, they typically evolve by Jeffrey conditionalization. This is because the redistribution of an agent’s probabilities over real propositions brought about by conditionalizing the extended probability function on a sense datum $$I_S$$ can usually be modelled as an instance of Jeffrey conditionalization. Concretely, assume that $$\{ E_1, \ldots, E_n \}$$ is some partition of real propositions that “screens off” the sense datum $$I_S$$ from any other real proposition—meaning that $$P_{{old}}(A/E_i \land I_S) = P_{{old}}(A/E_i)$$ for any real $$A$$ and any $$E_i\in \{E_1,\ldots,E_n\}$$. The agent’s degrees of belief then change by Jeffrey conditionalization on $$\{ E_1,\ldots,E_n \}$$, with the new credences $$x_1,\ldots,x_n$$ set by $$P_{{old}}(E_1/S_I), \ldots, P_{{old}}(E_n/S_I)$$, respectively. If the agent’s real probability space is finite, there will always be some such partition (in the worst case, the partition of individual worlds).[10]

The model I have outlined therefore provides an answer to the input problem, insofar as we can explain how a given experience together with an agent’s prior cognitive state determines the input parameters to a Jeffrey update—provided that the “prior cognitive state” includes an extended probability function of the kind I described. If we only look at the agent’s prior probabilities over real propositions, the problem can’t be solved: there is no fixed way in which perceptual experiences should affect an agent’s beliefs about the world.

To illustrate how the present model gets around the problems from Section 1, imagine our robot finds itself in the repeated tablecloth scenario. Let $$S$$ be the perceptual signal the robot receives both times when it looks at the cloth in the dimly lit room, and let $$I_S$$ be the sense datum associated with $$S$$. Let’s say the robot initially assigned probability ⅓ to each of the possibilities $$\{ {Red}, {Green}, {Blue} \}$$. Moreover, let’s assume it assigned greater probability to $$I_S$$ conditional on Green than to $$I_S$$ conditional on Red or Blue. Concretely, let’s assume that $$P_{{old}}(I_S/{Green}) =$$ ¼ and $$P_{{old}}(I_S/{Red}) = P_{{old}}(I_S/{Blue}) =$$ ⅛. As you can check, the robot’s new probabilities over $$\{ {Red}, {Green}, {Blue} \}$$ will then be ¼, ½, ¼, respectively. At the same time, the robot’s probability for $$I_S$$ increases to 1. When the robot takes a second look at the cloth from the exact same perspective, we can assume that its extended probability measure assigns high probability to the hypothesis that it is going to learn $$I_S$$ again. As a consequence, the second look at the tablecloth will barely affect the robot’s beliefs about the cloth’s colour.

Here I have made various assumptions about the robot’s prior probabilities. The formal model alone does not guarantee sensible results. For example, if the robot assigned high prior probability to $$I_S$$-now conditional on Red and to $$I_S$$-in-a-moment conditional on Blue, the first look at the tablecloth might make it confident that the cloth is red and the second that the cloth is blue. In this respect, the present model is on a par with the classical Bayesian model in which beliefs evolve by strict conditionalization. That, too, leads to sensible posterior beliefs only if the agent starts out with sensible prior beliefs. I will return to the “problem of the priors” in the next section. First I want to discuss another important caveat.

Bayesian models of rationality are often highly idealized. The model I have described certainly is. Such idealized models can still be useful, and not just as normative ideals. Even if they don’t fit all the phenomena, they can capture important patterns in the phenomena—central aspects of our psychology, ignoring friction and air resistance, as it were. On the other hand, it is also useful to study how reality deviates from the ideal. Here too one can often find interesting patterns. In fact, many peculiarities of our cognitive system can arguably be explained as consequences of the short-cuts evolution has taken to approximate the model I have described.

If we tried to build our robot, with its central database of probabilities updated by conditionalizing on sense data, we would quickly hit insurmountable problems. Conditionalizing a high-dimensional probability measure is a non-trivial, often intractable computational task. Computer science has come up with several tricks to make it more tractable. For example, instead of computing exact conditional probabilities we could employ Monte Carlo sampling (Griffiths, Kemp, & Tenenbaum 2008) or variational approximations (Seeger & Wipf 2010). Restricting the mathematical form of prior probabilities also proves useful in this context. Ideas from predictive coding could be used to exploit regularities in sensory signals (Clark 2013). It also helps to decompose an agent's (‘joint’) probability measure into probabilistically independent components, perhaps in the structure of a Bayes net (Pearl & Russell 2001).[11] There is evidence that our nervous system employs these and other tricks to approximate the simple Bayesian ideal -- see Weiss, Simonvelli, and Adelson (2002), Vul, Alvarez, Tenenbaum, and Black (2009), Sanborn, Griffiths, Navarro, and To (2010), Gershman and Daw (2012), Gershman, Vul, and Tenenbaum (2012), Howhy (2014), Griffiths, Lieder, and Goodman (2015).

Another respect in which my model is idealized is its assumption of precise probabilities. This, too, could easily be relaxed. Instead of storing a precise probability function, we might only store certain constraints on probabilities: that $$P$$ is more probable than $$Q$$, that $$R$$ is probabilistically independent from $$S$$, and so on. The agent’s doxastic state would then be represented by a whole set of probability functions: all those that meet the constraints (Jeffrey 1984). On closer inspection, however, it is not clear whether imprecise probabilities improve computational tractability, since standard methods for updating a probability function (such as Monte Carlo sampling) do not straightforwardly generalize to sets of probability functions (Zhang, Dai, Beer, & Wang 2013).[12] But there are other reasons to allow for imprecise probabilities. For example, it has been argued that only imprecise credences adequately reflect a certain kind of inconclusive evidence (Joyce 2005, Sturgeon 2008). These considerations would carry over to an agent’s extended probability function, suggesting that for many sense data $$S$$ and real propositions $$A$$, rational agents should not assign a strict probability to $$S$$ given $$A$$.

Such a two-tiered implementation might help to explain why perceptual experiences generally seem to present the world as being a particular way. When I look at the Müller-Lyer illusion, there is a sense in which my visual experience suggests to me that one line is longer than the other. This is a kind of “perceptual content” (indeed, it is what philosophers mostly have in mind when they talk about perceptual content), but it is clearly not what I conditionalize on, since I do not become certain that one line is longer. I know that the lines are the same length, but the mechanism that produces the categorical interpretation is not sensitive to that information.

It is not my aim in the present paper to speculate about how our nervous system approximates the Bayesian ideal. This is a task for cognitive science. The above remarks are only meant to illustrate what a more refined model that takes into account our cognitive limitations might look like, and how the required compromises might account for salient features of our psychology that are not predicted by the simple model from the previous section.

Here it is important not to conflate different levels of modelling. Hypotheses about the “Bayesian brain” (Doya, Ishii, Pouget, & Rao 2007) are often understood as conjectures about the internal processes involved in perception and action. The model I want to defend is largely neutral on these issues. For example, it does not settle whether perceptual input is processed in classical bottom-up style or in the more top-down fashion postulated by recent accounts of predictive coding.

The only suggestion I have for lower-level Bayesian models in cognitive science concerns the interpretation of such models. Cognitive scientists commonly describe perception as a process of inferring facts about the environment from sensory stimuli; they talk about the surprisingness of incoming signals, or about the construction of internal models that predict those signals. Taken literally, this suggests that our cognitive system is given direct and infallible information about complicated electrochemical events in its periphery and then faces the task of explaining or predicting the occurrence of these events (see, e.g., Rieke 1999). Yet most of us are fairly ignorant of the electrochemical processes in our nervous system: the infallible basis of our empirical knowledge appears to get lost through cognitive processing. What I want to suggest is that from the perspective of a cognitive system, sensory inputs are not represented as electrochemical events, even though that is what they are. We can distinguish between the inputs themselves and the corresponding elements in the domain of a system’s probability measure. What is “given” in perception are not sophisticated facts about neurophysiology, but imaginary sense-datum propositions that don’t directly settle any substantive question about the world.

## 4. Softcore Empiricism

Let me now explore some epistemological consequences of the model I have outlined. As I mentioned in Section 1, Jeffrey developed his alternative to classical conditionalization because he rejected the view he called ‘hardcore empiricism’: that our knowledge of the world rests on a foundation of infallible and indubitable beliefs about present experience. On the most familiar version of the empiricist picture, all our knowledge is derived from an infallible perceptual basis, drawing on a priori connections between the verdicts of experience and statements about the external world. If the connections are logical, we get the striking phenomenalist view that the world (or at least all we can ever know about it) is a logical construction out of sense data.

The model I have defended bears a superficial similarity to this empiricist picture: experiences confer absolute certainty on a special class of (“imaginary”) propositions; beliefs about the external world are then adjusted according to their prior connections with these propositions. But there are also important differences.

First of all, imaginary propositions do not represent true features of the relevant experiences—at least not in a cognitively transparent way. They do not distinguish real ways the world could be at all. (No wonder they can never turn out to be false.) As a corollary, there are no intrinsic logical or analytical connections between imaginary propositions and real propositions. To get from sense data to claims about the world, we need external bridge principles, encoded in our prior (conditional) probabilities.

In this way, the model is tailored to accommodate the holism of confirmation. Whether sensory information $$E$$ supports a genuine hypothesis $$H$$ about the world always depends on the agent’s background beliefs. Depending on the prior probabilities, the very same experience can rationally lead to very different beliefs. There is no once-and-for-all right or wrong interpretation of sensory signals. The only propositions that are directly and unrevisably supported by sensory experience are imaginary propositions without real empirical content.

Second, the model I have put forward is not committed to an ontology of sense data or irreducible phenomenal properties. Relatedly, it makes no claims about what we see (or hear or taste), or about what we are directly aware of in experience. Surely what we see are in general such things as trees and tables and tigers. Nothing I have said suggests that that we also, or primarily, see non-physical ideas, impressions, or sense-data.

Third, the model I have outlined does not assume that perceivers have special introspective access to imaginary propositions, nor does it assume that imaginary propositions are objects of “belief” in the intellectualist sense that dominates discussions in epistemology. The claim is not that whenever we have a perceptual experience, we become certain of a special “observation sentence” from which we then deduce other, perhaps probabilistic, statements about the world. Conditionalization is not an inference, with premises and conclusion, and it is not supposed to be a conscious, deliberate activity. Perceivers don’t need words or concepts that capture the imaginary content of their perceptions, and they don’t need to conceptualize their experiences as reasons for their beliefs. As Sellars (1956) and others have pointed out, these commitments render the foundationalist picture highly unappealing.

If we focus on the intellectualist sense of ‘belief’ it is hard to explain the epistemic impact of perceptual experience. Since perceptions don’t seem to have the required sentence-like, “conceptual” content, how can they support or justify beliefs with that kind of content? Is the link between perception and belief merely causal, outside the domain of epistemology? That also seems wrong, for there clearly are rational constraints on how one’s beliefs may change through perceptual experience. Again, scientists are not free to change their beliefs in any way they please when observing a litmus paper that has turned red.

To make progress on these issues, we should accept that epistemology is not confined to intellectualist belief. The functionalist notion of belief (or credence) popular in the Bayesian tradition brings us one step further, but it still doesn’t reach far enough, at least if we restrict it to real propositions about the world. As I’ve argued in Sections 1 and 2, how an agent’s representation of the world should change under the impact of perceptual experience is not a function of the experience and the agent’s representation of the world before the experience. Something else plays a role. In the model I have described, that something else is represented by the agent’s extended probabilities.

So what I’m advocating is not a rebranding of classical empiricist foundationalism. In many respects, the model I want to advertise looks more like Jeffrey’s “softcore empiricism”: it offers a systematic account of how perceptions affect rational attitudes about the world, without making the agent certain of any substantive propositions, and without assuming any fixed probabilistic connection between experiences and propositions about the world.

This brings me back to the problem of priors. Suppose you’re a scientist and you’ve just observed a litmus strip turning red. Absent unusual background assumptions, you should become confident that the strip is red and the tested substance acidic. Why is that? The model I have put forward does not give an answer. It only says that your new probability in the strip being red should equal your previous probability conditional on the imaginary proposition associated with your experience. But why should that conditional probability be high?

Now, one advantage of the model I have proposed is that the relevant conditional probabilities can themselves be adjusted through learning. So we can explain why the effect of your visual experience is sensitive to background information about the workings of litmus stripts, the present lighting conditions, or your eye sight. But that doesn’t fully answer the question. Suppose (unrealistically, of course) that your probabilities evolved from an ultimate prior probability function by successive conditionalization on sensory evidence. Would that automatically make your probabilities epistemically rational? Arguably not. With sufficiently deviant ultimate priors, your entire history of perception could lead to a state in which you treat the experience of the red litmus paper as strong evidence that the paper is blue (or that Tycho Brahe was poisoned by Johannes Kepler). So there must be substantive, non-formal constraints even on ultimate priors—equivalently, on what one may believe in light of an entire history of sensory input.

How tight are these constraints? Some have argued that there is a unique rational prior, so that rational agents with the same history of sensory input would always arrive at the same credences (e.g., White 2005). Others disagree (e.g., Meacham 2014). The approach I want to advertize is compatible with either position. I do assume, however, that there are some non-formal constraints on ultimate priors.

Where do these constraints come from? It is doubtful that they could be defended by non-circular a priori reasoning. Perhaps they reflect irreducible epistemic norms. Or perhaps they can be explained as (in some sense) constitutive of the relevant intentional states: perhaps what makes it the case that a given brain state is a belief that we’re looking at something red is in part that the state is normally caused by perceptions of red things. Alternatively, we might try to vindicate some constraints by objective, external correlations. Suppose sensory stimulus $$S$$ is triggered mostly under external circumstances $$C$$, and robustly so. Then we might say that $$S$$ objectively supports $$C$$. More generally, if the objective chance of $$C$$ given $$S$$ is $$x$$, we might say that $$S$$ objectively supports $$C$$ to degree $$x$$. And so we might say that a subject is justified in assigning conditional credence $$x$$ to $$C$$ given $$\rho(S)$$, absent relevant evidence, iff $$x$$ matches the degree to which $$S$$ objectively supports $$C$$ (see Dunn 2015, Tang 2016, and Pettigrew in press).

Most of these issues and options are familiar from contemporary discussions in epistemology. The model I have suggested does little to resolve them. What it does is provide a credible background story. The empiricism I’m trying to sell you does not presuppose an outdated, 18th century view of perception and the mind. On the contrary, it naturally goes with 21st century cognitive science.

## 5. Puzzles of Consciousness

Before concluding, I want to explore one further application of my model, to the puzzle of conscious experience.

I have suggested that terms like ‘credence’ or ‘belief’ might be reserved for an agent’s attitudes towards real propositions, since the non-real part of an agent’s probability function does not serve a straightforwardly representational function. But this difference between real and imaginary propositions need not be transparent to the agent. An agent’s cognitive system need not draw a sharp line between the two kinds of attitudes. Our robot, for example, does not need a special database for real propositions in addition to its database for imaginary (and complex) propositions. From the robot’s perspective, it might simply appear as if reality had an extra dimension, an extra respect of similarity and difference. Perceptual experience will then appear to convey direct and certain information about this aspect of reality, and only uncertain information about everything else. Conversely, ordinary information about the world will never suffice to fix the apparent further dimension of reality: there may be no conjunction of real propositions conditional on which any sense datum proposition has probability 1. The robot may then be tempted to conclude that physics is incomplete, that there are special phenomenal facts revealed through experience that are not implied by or reducible to physical facts about the arrangement and dynamics of matter. Yet our robot could well exist in a completely physical world.

These intuitions put pressure on physicalist accounts of experience. If experiences are brain states, and we can always rule out situations in which we have different experiences, it would seem to follow that merely in virtue of being in a given brain state we can rule out situations where we are in different brain states. That seems wrong. As Lewis put it, “making discoveries in neurophysiology is not so easy!” (1995: 329). Lewis concludes that physicalists should reject the folk psychological Identification Thesis, that when we have an experience of a certain type, we can rule out possibilities in which we have experiences of a different type.

The model I have put forward suggests a different response. Both the reading experience and the skiing experience are associated with imaginary propositions in your extended doxastic space—the propositions of which the relevant experience would make you certain. When you entertain the hypothesis that you are in the skiing scenario, what you entertain includes certain imaginary propositions. And these propositions are incompatible with the imaginary propositions associated with the reading experience: their conjunction is the empty proposition. Hence the reading experience allows you to conclusively rule out the skiing scenario—not by its physical features, but by its “imaginary features”, so to speak.

Along the same lines we can explain other phenomena that seem to put pressure on physicalism. Consider Mary (from Jackson 1982), who has learned all physical facts about colours and colour vision without having seen colours. If Mary’s probability space has an imaginary dimension, her physical knowledge may still leave open many possibilities along the imaginary dimension. For example, let $$I_R$$ be an imaginary proposition associated in Mary’s cognitive system with experiences of whatever physical type $$X$$ is typically caused by looking at red things. In Mary’s extended probability space, this association will plausibly be contingent: $$P(I_R/X) \ll 1$$. So when she is eventually presented with some coloured chips, without being told their colour, Mary will become certain of $$I_R$$, but she won’t be able to tell that she is in physical state $$X$$. Conversely, if she learns that she will be in state $$X$$ tomorrow, this will leave her uncertain about $$I_R$$-tomorrow. All this will be so even if Mary lives in a completely physical world.

Similarly, if $$I_R$$ is an imaginary proposition associated with red experiences, and P is the totality of all physical truths, we can explain why both $$P$$ & $$I_R$$ and $$P$$ & $$\neg I_R$$ are a priori conceivable (see Chalmers 2009), even if the world is completely physical.

In short, the phenomena that appear to support dualism about consciousness might be artefacts of the way we process sensory information.

To be clear, the model I have outlined makes no direct claims about consciousness. I never mentioned consciousness when I introduced the model. Indeed, empirical evidence (e.g., about binocular rivalry, see Blake & Logothetis 2002) makes clear that our conscious experience does not simply track the stimulation patterns in our sense organs. Consciousness rather seems to play a role in something like the two-stage processing about which I speculated in Section 3. There I suggested that our sensory systems might compute a single (coarse-grained) hypothesis about the environment which is then turned into an input signal for personal-level probabilities (as well as driving immediate behavioural reactions). Clearly, this is mere speculation. I have no qualified views on the functional role of consciousness in our cognitive architecture, and I don’t claim that my model makes much progress on this issue—on what Chalmers (1995) calls the “easy problem” of consciousness. But it might help with the “hard problem”, the problem of explaining how physical processes in the brain seem to create a phenomenon that can’t be understood in physical or functional terms at all. My suggestion is that, for reasons to do with the efficient processing of sensory signals, our subjective picture of the world has an added dimension which makes it appear as if perceptual experiences carry a special kind of information that goes beyond physical and functional information.

I want to close with one more puzzle about consciousness that has not received much attention.[14] The puzzle is the apparent fit between the phenomenal character of mental states and their functional role. To see what I mean, compare again the skiing experience with your present reading experience. Both experiences have a distinctive phenomenal character. For the skiing experience, this involves the phenomenology of feeling the wind, seeing the slopes, moving your legs, and so on. My claim is that this phenomenal character goes well with the external circumstances that cause the experience and with the behaviour it causes. Imagine a world where the phenomenal characters are swapped, where ordinary skiing events are associated with the actual phenomenology of reading a paper, and vice versa. That would be a world where phenomenal character doesn’t fit functional role.

Are “inverted qualia” worlds like this conceivable? Not if the phenomenal truths are a priori entailed by broadly physical truths. But many philosophers— physicalists and dualists alike—deny the thesis of a priori entailment. They hold that there is an epistemic gap between the physical and the phenomenal. This suggests that worlds with thoroughly inverted qualia should be epistemically possible.[15] Epistemically speaking, it is then just a coincidence that in our world phenomenology nicely fits functional role. For all we know a priori, it could have been that skiing experiences are associated with the phenomenology of reading philosophy papers. Or it could have been that everyone’s phenomenology is running two hours late, so that, when people eat breakfast and listen to the news in the morning, they have the experience of still sleeping; when they have started working, they have the experience of eating breakfast and listening to the news, and so on. How convenient that we don’t live in a world like that! (If indeed we don’t. For can we really be sure?)

This is the puzzle. Here is the solution. In our extended doxastic space, imaginary propositions are compatible with many, perhaps all, real propositions. There are points in our doxastic space where the imaginary proposition actually associated with skiing stimulations is conjoined with a reading scenario. On the other hand, in order for perception to provide us with information about the world, there must be strong a priori constraints on the interpretation of sensory signals and thus on the probabilities of real propositions conditional on imaginary propositions. (Recall the discussion of non-formal constraints in Section 4.) Absent unusual background information, a specific sense datum must be regarded as strong evidence for a narrow range of hypotheses about the world. These connections can change through experience, but the functioning of our perceptual system demands that we give low a priori probability to possibilities where a given type of experience—as represented by the associated imaginary propositions—is caused by an unusual environment. The above inverted qualia scenarios are extreme cases of this type. The model I have outlined suggests that they must have negligible prior probability. They are almost a priori ruled out.

## 6. Conclusion

Much of what we know about the world we know through perception. But how does perception provide us with that knowledge? Perceptual experiences do not seem to deliver direct and certain information about the external world. The classical empiricists held that perceptual experience instead delivers information about an internal world of sense data, from which we infer hypotheses about the external world. But the idea of a luminous internal world is hard to square with a naturalistic picture of cognition and our general fallibility. A more sober response rejects the assumption that rational belief requires a bedrock of certainty: perceptions may increase or decrease the credibility of various external-world hypotheses without rendering anything certain. Unfortunately, once we try to fill in the details, this response runs into serious problems. It seems that an agent’s belief state prior to a given experience does not contain enough information to settle how the beliefs should change through the experience.

I have proposed a way out that takes a step back towards the old empiricist account. In order to adequately respond to sensory stimulation, I have argued, it may be useful to extend the domain of an agent’s subjective probability function by an “imaginary” dimension whose points are associated with sensory signals in such a way that when a signal $$S$$ arrives, a corresponding “imaginary proposition” $$\rho(S)$$ becomes certain; the probability of real propositions is then adjusted in accordance with their prior probability conditional on $$\rho(S)$$. The space of imaginary propositions plays an epistemological role somewhat analogous to the empiricist’s internal world. In Section 5, I have suggested that the similarities might go further insofar as imaginary propositions might correspond to the phenomenal properties that appear to present themselves to us in perception.

The model I have presented is abstract and formal. It does not settle that— let alone explain why—a given type of red experience should make an agent confident that she confronts something red. Nor does it imply any particular algorithm or mechanism for computing the new probabilities. As such, it remains neutral on many contentious and difficult questions in epistemology and cognitive science. Nonetheless, it may provide a useful background for tackling some of the more substantive questions in these areas.

## Acknowledgments

Ancestors of this paper were presented in Bielefeld (2006), Canberra (2006), Cologne (2010), Melbourne (2013), Leeds (2014), and Bochum (2017). I thank the audiences at these events for helpful feedback. Special thanks to David Chalmers, Colin Klein, Christian Nimtz, Mark Sprevak, Daniel Stoljar, and two anonymous referees for comments on earlier versions.

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## Notes

1. The conditional probabilities $$P_{{old}}(A / E)$$ are often computed via Bayes’ Theorem, which is why conditionalization is also known as Bayes’ Rule.

2. For ease of exposition, I have assumed that the experience is directly relevant only to a finite partition $$E_1,\ldots,E_n$$; the model is easily extended to infinite cases; see Diaconis and Zabell (1982: Section 6).

3. Experiences can of course be typed in other ways. For example, we might say that two experiences are of the same type iff they lead to the same rational posterior beliefs. The difficulties I am going to discuss then resurface as the problem of determining when two experiences are of the same type.

4. The input problem also arises for standard (“strict”) conditionalization if we don’t assume that the proposition on which agents conditionalize is directly given by the relevant experience. For example, Skyrms (1980) points out that the effect of Jeffrey conditionalization can be mimicked by strict conditionalization on propositions about (posterior) degrees of belief, so that what you learn for certain in the tablecloth scenario is (say) that you have become 60 percent confident that the tablecloth is green. But where do these 60 percent come from? How does your experience together with your prior beliefs determine that this is the appropriate new credence in the tablecloth being green?

5. The present point is inspired by Garber (1980), where it is used to argue against a particular answer to the input problem suggested in Field (1978). Hawthorne (2004) presents a model that gets around the problem by making the input parameters to Jeffrey conditionalization depend not only on present experience and old probabilities, but also on earlier experiences; the dynamics of rational credence is thereby rendered unattractively non-Markovian. Further (though related) challenges to solving the input problem arise from the holistic character of evidential support; see Christensen (1992), Weisberg (2009), Wagner (2013), and the discussion of parochialism in Jeffrey (1988). The basic worry here is that if probabilities are only defined over ordinary external-world propositions, then it may be impossible to find a non-trivial evidence partition $$E_1,\ldots,E_n$$ that screens off the experience from all other propositions in the sense that $$P_{{new}}(A/E_i) = P_{{old}}(A/E_i)$$. Weisberg (2009) also points out that a result in Wagner (2002) seems to entail that the failed proposal of Field (1978) is the only systematic answer to the input problem that satisfies a desirable commutativity condition (roughly, that it makes no difference to the final probabilities which of two experiences arrives first).

6. Jeffrey agreed; see especially Jeffrey (1970).

7. $$\mathbf{I}$$ is an “arbitrary” set because the identity of its members is irrelevant to the functional specification of our robot. In this respect, the framework of probability theory is a little artificial, since it forces us to make a choice.

8. See, e.g., Lewis (1974), Stalnaker (1984), or Braddon-Mitchell and Jackson (1996) for classical expositions of this kind of functionalism in the philosophy of mind.

9. A full specification of the relevant functional role would need spell out other controversial details in the Bayesian model—for example, whether choices should maximize causal or evidential expected utility. The complete model should also include non-formal constraints on utilities and probabilities, for the reasons discussed in Lewis (1974) and Lewis (1983): without such constraints, the model will plausibly allow for too many assignments of probabilities and utilities, many of which are far removed from what we would intuitively take to be the agent’s beliefs and desires. I will turn to some such non-formal constraints on extended probabilities in a moment.

10. In the infinite case, we may need to invoke some generalization of Jeffrey’s formula like the one mentioned in Footnote 2, but there may still be cases where Jeffrey’s model is inapplicable. Never mind: we have an answer to the problem Jeffrey’s model was meant to solve.

11. In a Bayes net, assumptions of conditional independence are directly reflected in the structure of the network, which might allow for a more perspicuous identification of the input partition in a Jeffrey update. Along these lines, Schwan and Stern (2017), drawing on Pearl (1988), suggest that an agent should Jeffrey conditionalize on a partition $$\{ E_i,\ldots E_n \}$$ iff every node $$A$$ in the network is d-separated from the input node $$i$$ by the elements $$E_i$$ of the partition. Schwan and Stern and Pearl, however, do not treat $$i$$ as an imaginary element of the agent’s doxastic space. Rather, they take $$i$$ to be an ordinary fact about the world of which the agent could become certain if only it were represented in her doxastic space. In reality, there often won’t be any such fact $$i$$: there is no suitable fact about the world of which my experience of the rain could rationally make me certain.

12. Thanks here to an anonymous referee.

13. The intermediate hypothesis produced by the sensory modules need not be a pure, objective proposition about the environment; it might also involve “imaginary” elements, locating the present state of the world in a high-dimensional phenomenal space, as (effectively) suggested, e.g., in Shepard (2001).

14. Latham (2000) discusses a version of the puzzle.

15. Strictly speaking, one could deny that the phenomenal is entailed by the physical but also deny the coherence of the described scenario. The idea would be that there is partial entailment from the physical to the phenomenal: given a state’s physical and functional properties, one can a priori rule out many candidate phenomenal properties; the entailment is partial because more than one candidate is left standing. However, most philosophers who believe in an explanatory gap believe that the gap is fairly wide, so that physical information entails very little about phenomenal character. As long as the gap is sufficiently wide, we can construct strange inversion scenarios, even if not the exact scenario from above.