Fragmentation and Representation Processing: Two Distinct Approaches to Information Access, Deduction, and Logical Omniscience

Abstract

Ordinary thinkers don’t know every necessary truth - they are not “logically omniscient.” Furthermore, they don’t know everything that follows from their beliefs - they are not “deductively omniscient.” Stalnaker has famously argued that this is partly due to the “fragmentation” of our beliefs, which are divided into different belief-states. Elga and Rayo have developed this idea, arguing that an agent has access to different information under different conditions, such that her doxastic state is divided (“fragmented”) into different informational states. In this article, I distinguish fragmentation from an alternative approach to limited information access. According to this second approach, our limited access to information is explained by our representation-processing mechanisms - the ways in which mental representations are processed by our cognitive systems. I argue that this approach must be sharply distinguished from fragmentation: The limitations that result from our representation-processing mechanisms need not amount to a fragmented doxastic state. The distinction is especially significant given recent objections against fragmentation: Stalnaker’s and Elga and Rayo’s views still seem to attribute an implausible amount of knowledge to ordinary thinkers. I show that an account based on representation processing avoids these objections, thereby opening up a viable alternative to fragmentation.

Keywords:
fragmentation; mental representation; information access; problem of logical omniscience; problem of deduction

1 Introduction

Stalnaker’s theory of propositional attitudes has been highly influential in the literature, but it has also faced well-known challenges. In particular, his theory seems to have the implausible consequence that ordinary thinkers are logically omniscient - they know every necessary truth - and deductively omniscient - they know everything that follows from their beliefs. Stalnaker (1984) proposes a solution, which he refines and develops in later works (Stalnaker, 1986, 1991, 1999, 2006, 2021). A central component of Stalnaker’s solution is the idea that our beliefs are fragmented: They are divided into distinct belief-states, and if a certain necessary truth follows from beliefs in different belief-states, the subject may not know the truth in question. Other authors have endorsed fragmentation,¹ and Elga and Rayo have applied the idea to a number of philosophical problems in their recent work (Rayo, 2013; Elga and Rayo, 2021, 2022). However, some argue that fragmentation doesn’t fully solve the problem (Field, 1986; Richard, 1990; Speaks, 2006; Soysal, 2022; Bjerring and Tang, 2023), since Stalnaker’s and Elga and Rayo’s views still attribute an implausible amount of knowledge to ordinary thinkers.

In this article, I contrast fragmentation with an alternative approach. According to Stalnaker and Elga and Rayo, fragmentation can account for “limitations on information access” (Elga and Rayo, 2022, 37),² which in turn explain why we are neither deductively nor logically omniscient. In the first part of the article, I distinguish fragmentation from a different explanation of limited information access. This alternative explanation appeals to our representation-processing mechanisms - the ways in which mental representations are processed by our cognitive systems - so I refer to it as a “representation-processing explanation” (RPE). I argue that a RPE can account for various cases of limited information access while avoiding some of the objections that have been raised against fragmentation. In the second part of the paper, I spell out the implications of RPEs for the problems of deduction and logical omniscience. I will not provide a solution to these two problems, but I will explain how RPEs could be developed to provide such a solution, and I will identify the goals that a solution based on RPEs would have to accomplish.

I begin by presenting the problems of deduction and logical omniscience. I then discuss Elga and Rayo’s proposal and their endorsement of fragmentation, followed by objections that have been raised against it (Section 2). In Section 3, I contrast fragmentation with RPEs: I explain why they provide distinct explanations of limited information access, and argue that RPEs overcome some of fragmentation’s problems. In Section 4, I explain why RPEs of limited access can contribute to solving the problems of deduction and logical omniscience, and I identify the steps that should be taken to develop such a solution.³

2 The Possible Worlds Theory and Fragmentation

2.1 The Possible Worlds Theory and its Problems

I will begin by briefly presenting a well-known theory of belief: the possible worlds theory. According to this theory, the beliefs held by a subject at a time form a belief state, which is represented as a set of possible worlds (Stalnaker, 1984, 82). Individual beliefs are “properties of such a belief state: To believe that P is for the proposition that P to be true in all the possible worlds in the belief state. [...] to believe that P is simply to be in a belief state which lacks any possible world in which P is false” (ibid., p. 69).

The possible worlds theory entails a number of problematic closure principles. One of them is:⁴

Closure under Necessary Equivalence: If A believes that p, and p and q are true in the same possible worlds, then A believes that q. (Yalcin, 2018, 26)

This closure principle has implausible consequences concerning our attitudes towards necessarily equivalent propositions. One widely discussed instance of the problem arises due to necessarily true propositions, which are true in exactly the same set of worlds - namely, the set of all worlds. If the possible worlds theory were true, any subject who believed some necessary truth would thereby believe all necessary truths. If p and q are necessarily true propositions, then they are true in the same worlds, namely in all possible worlds. That is, the set of p-worlds (worlds where p is true) and the set of q -worlds (worlds where q is true) are the same set. Now consider a subject A who believes that p. Since p is true in all possible worlds in the subject’s belief state, q is also true in all those worlds, so A believes that q. More generally, the possible worlds theory entails that, if a subject believes some necessary proposition, then she believes all necessary propositions. But this is clearly false. Someone who is competent in first-order logic may believe a certain logical truth without thereby believing all logical truths. Someone who believes a truth of arithmetic after making a simple calculation doesn’t ipso facto believe all mathematical truths. Generally speaking, an ordinary subject believes some, but not all necessarily true propositions. Thus the possible worlds theory ascribes to ordinary subjects a number of necessarily true beliefs that they clearly don’t have.⁵ Following Greco (2021, 56), I’ll refer to the problem I’ve just presented as the problem of logical omniscience.

Closure under Necessary Equivalence follows from a more general closure principle:

Closure under Entailment: If A believes that p, and p is true in a world w only if q is true in w, then A believes that q. (Yalcin, 2018, 27)

Closure under Entailment also gives rise to problematic consequences, which go beyond the specific case of necessary truth, which was examined above. Suppose that q is true in all the worlds where p is true, i.e., that p entails q. Now consider a subject A who believes that p. Since p is true in all possible worlds in A’s belief state, q is also true in all possible worlds in A’s belief state. So A believes that q according to the possible worlds theory. More generally, the possible worlds theory entails that, if a subject believes some proposition, then she believes all the propositions which are entailed by it. But this seems clearly false, since the subject might fail to make the inference which consists in deducing q from p (Greco, 2021, 57). Following Greco (ibid.), I’ll refer to this as the problem of deduction.

2.2 Fragmentation and its Problems

Elga and Rayo have recently proposed a solution to the problems of deduction and logical omniscience, as well as other related puzzles (Rayo, 2013; Elga and Rayo, 2021, 2022). One of their main observations is that information can be accessible to an agent for some purposes but not others. For instance:

Consider [...] a pair of crossword-puzzle solvers trying to fill in the blanks below to complete a word of English:

________ MT

The first puzzlist fills in just the right letters. The second scratches his head and leaves the puzzle blank. Suppose further that each puzzlist knows that dreamt is a word of English, and knows how to spell it. (Elga and Rayo, 2022, 718)

According to Elga and Rayo, both subjects have the relevant information, but only one of them has been able to access that information for the purpose of solving the puzzle. Yet the other subject will have access to that same information for other purposes - for instance, she will be able to answer the question: “Is ‘dreamt’ a word of English ending in MT?” To account for a subject’s behavior, then, it’s not enough to specify what information she possesses: We must also specify whether she has access to that information relative to a specific behavioral purpose (Elga and Rayo, 2021, 40). We can specify this through an “access table” like the following (Elga and Rayo, 2022, 718-19):

The table represents the information that Elga and Rayo’s puzzled puzzlist has access to, relative to different choice conditions. Each choice condition is associated with a probability function, which assigns probabilities to sets of possible worlds (ibid., p. 719). In the table above, for instance, “P ₁ is a probability function that assigns high credence to the set of worlds in which dreamt is a word of English spelled D-R-E-A-M-T, and P ₂ is a credence function that assigns low credence to that set” (ibid.). The same set of worlds can thus be assigned different degrees of credence in different choice conditions. On the other hand, the credence function associated with a specific choice condition is “probabilistically coherent” (Bjerring and Tang, 2023, 2144), and it assigns a single degree of credence to a set of worlds.

In this example, the information about the spelling of “dreamt” is accessible in the first choice condition, but not the second. This explains why the puzzlist can answer the question “Is ‘dreamt’ a word of English ending in MT?”, while she’s unable to solve the fill-in-the-blanks puzzle. As the table also indicates, salience can determine whether certain information is accessible in a given condition. For instance, when we ask “Is ‘dreamt’ a word of English ending in MT?”, we make the word “dreamt” salient, and this allows the puzzlist to access information that’s unavailable under other conditions.

Elga and Rayo provide a similar account for failures of deduction and logical omniscience. We can expect a “logically competent” agent to detect “obvious entailments” between “salient sentences,” where a sentence is salient in a choice condition if it is a sentence “that an agent is attending to - that she has in the forefront of her mind” (Elga and Rayo, 2022, 720), and an entailment is obvious “if it is guaranteed by the meanings of the Boolean connectives as they apply to sentences that are salient in that condition” (ibid., p. 721). At the same time, the agent may well fail to detect non-obvious entailments between non-salient sentences. If so, then logical information about the entailment is inaccessible to the agent, and she doesn’t know that the entailment holds. In a similar fashion, Rayo argues that limited access to information can account for mathematical ignorance (Rayo, 2013, ch. 4). An agent might know that the Dedekind Axioms are true yet be unable to answer the question: “Is it the case that $\sqrt{81}=9$?” (ibid., p. 102). According to Rayo, the agent possesses all the information required for the purpose of answering the question, because of her knowledge of the axioms, yet she cannot access that information for the purpose of answering the question (ibid., pp. 101-04). Summing up: On Elga and Rayo’s view, limited access to information accounts for failures of deduction and logical omniscience.

Elga and Rayo draw explicitly on Stalnaker’s views. More specifically, the “core motivation” for their proposal was Stalnaker’s idea that “logical omniscience failures can be understood in terms of fragmented belief states”⁶ (Elga and Rayo, 2021, 37). What is fragmentation? According to Stalnaker, a subject’s overall doxastic state may be fragmented, consisting of different belief states (“fragments”) that are not integrated with each other:

A person may be disposed, in one kind of context, or with respect to one kind of action, to behave in ways that are correctly explained by one belief state, and at the same time be disposed in another kind of context or with respect to another kind of action to behave in ways that would be explained by a different belief state. This need not be a matter of shifting from one state to another or vacillating between states; the agent might, at the same time, be in two stable belief states, be in two different dispositional states which are displayed in different kinds of situations. (Stalnaker, 1984, 83)

As previously noted (Section 2), according to Stalnaker, the content of a belief state is a set of worlds. Given fragmentation, then, a subject’s doxastic state is represented by multiple sets of worlds, where each set is the content of a different belief state or “fragment.” Similarly, on Elga and Rayo’s view, a subject’s doxastic state is represented by associating sets of worlds with different choice conditions and specifying the degree of credence assigned to each set of worlds under the choice condition in question. Different information will then be accessible under different conditions, much like different belief states are displayed in different situations on Stalnaker’s view.

Note that, on Stalnaker’s view, each belief state is still subject to Closure under Deduction and Closure under Necessary Equivalence, since the content of a belief state is still a set of worlds. Mutatis mutandis, the same is true of Elga and Rayo’s view, where the credence function for each choice condition assigns a single degree of credence to a set of worlds in a probabilistically coherent way. The two views also have a further element in common: Elga and Rayo’s accessible information guides action in the same way that Stalnaker’s fragments do. On Elga and Rayo’s view, a subject can behave differently under different choice conditions because different information is accessible in those conditions; on Stalnaker’s view, different fragments (belief states) lead to different behaviors in different contexts.

The idea of fragmentation is a key component of Stalnaker’s and Elga and Rayo’s views, but it has recently been criticized by Soysal (2022) and Bjerring and Tang (2023). Their arguments against fragmentation are similar: Generally speaking, the problem is that fragmentation doesn’t fully solve the problems of deduction and logical omniscience, for the information that is accessible relative to a fragment is still rich enough to entail several truths that the subject doesn’t seem to believe.⁷ As Soysal says:

[...] it seems that for the one task of solving mathematical problems-or, to take another example, for the one task of winning a chess game-thinkers have, or at least can have, all the rules and axioms or board positions in one belief state. Just as playing chess doesn’t make one cease to know the rules of the game and the board positions, proving theorems doesn’t make one cease to know the axioms and rules of inference. Solving the closure problem by diagnosing every closure-problem case as a case of fragmentation is ad hoc [...]. (Soysal, 2022, 462)

Consider, for instance, the following objection, which is adapted from Bjerring and Tang (2023, 2141-47). Suppose formula s ₁ entails formula s ₂ by some inference rule r and the entailment is obvious to a logically competent agent. Furthermore, suppose formula s ₂ entails formula s ₃ by r and this entailment is also obvious to the same agent. Our agent could nevertheless fail to detect that s ₁ entails s ₃; the problem is that this is incompatible with Elga and Rayo’s account. The first two entailments (s ₁ s ₂, and s ₂ s ₃) are both obvious to the agent, so there should be some choice condition c where the information about the entailments is accessible to the agent. For instance, the agent can correctly answer questions like “Does s ₁ entail s ₂?” and “Does s ₂ entail s ₃?” in c. On Elga and Rayo’s view, the agent’s credence function in c must therefore “respect” (Bjerring and Tang, 2023, 2142) those entailments: That is, for any world w, the degree of credence that the function assigns to s ₁’s being true at w is less than or equal to the degree of credence it assigns to s ₂’s being true at w ; the same goes for s ₂ and s ₃, mutatis mutandis. But then, since the function in c is probabilistically coherent, the degree of credence that it assigns to s ₁’s being true at w is less than or equal to the degree of credence it assigns to s ₃’s being true at w. Less formally: All the worlds compatible with the agent’s credences in c are worlds where s ₁ s ₃. By Elga and Rayo’s view, then, the information about this non-obvious entailment is accessible to the agent in c. Therefore, the view predicts (incorrectly) that the agent can detect the entailment when she is in c.

One possible response is that even if the agent has access to the information that s ₁ entails s ₃ in condition c, she might not have access to this information in other choice conditions, much like the puzzlist in the “dreamt” example. If so, then the agent will fail to detect the entailment s ₁ ⊢ s ₃ whenever she is in one of these other choice conditions. But this reply misses the point. As we’ve seen, fragmented states are supposed to explain the subject’s actions in the relevant contexts or conditions. If there is a condition c where the agent assigns high credence to the proposition that s ₁ entails s ₃, then she is disposed to behave in accordance with this credence state when she is in c. But the agent may not have any such disposition if the entailment (s ₁ s ₃) is sufficiently hard to detect. For instance, the agent may be unable to answer the question “Does s ₁ entail s ₃?” correctly, even though she can answer questions about the “easier” entailments (s ₁ s ₂, s ₂ s ₃). In other words: If fragmented doxastic states explain behavior, there shouldn’t be any choice condition where the agent has access to the information that s ₁ entails s ₃.

In this section, I have presented the possible worlds theory of belief, and explained why it must face the problems of deduction and logical omniscience. I have then presented Elga and Rayo’s view, which develops Stalnaker’s idea of belief fragmentation. Finally, I have discussed some recent objections against Elga and Rayo’s proposal: Elga and Rayo offer a a fragmentation-based account of information access, but the account still seems to attribute an implausible amount of knowledge to ordinary thinkers. In the next section, I discuss an alternative account of information access. This account will not postulate fragmentation, and it will avoid the problem that affects Elga and Rayo’s view.

3 Representation-Processing Explanations (RPEs) of Limited Access

3.1 The AB- and ABC-Machines

Interestingly, Stalnaker himself provides examples that serve to illustrate a “fragmentation-free” approach to information access. Stalnaker takes these examples to support fragmentation, but I’ll show that they can be interpreted in a different way.

Stalnaker imagines a simple machine with the following features (Stalnaker, 2021, 1986):

Two integers are inputs to the machine, and it has two components, A and B, that each compute whether one of the two numbers is odd or even. Since the machine has the information whether each of the input integers is odd or even, it implicitly has the information whether the sum of the two integers is odd or even, but it will not have access to this information if the two components are not connected. If we add a third component C, an exclusive or gate, with the binary outputs of A and B as inputs, then the output of C will carry the binary information (odd or even) about the sum.

[...] before the third component was added, the information was only implicit since the machine did not then have access to it; that is, the machine was not in a position to make a decision (in the sense in which a machine can make a decision) based on that information. (Stalnaker, 2021, 190-91)

Stalnaker’s example will be important in the discussion to follow, so it’s worth spelling it out in some detail. At the outset, Stalnaker describes a machine that only has two components, A and B - call this the AB-machine. A third component, C, is then added, which results in a more complex system - call this the ABC-machine. The two machines can receive one of four pairs of inputs:

1. ⟨even, even⟩

2. ⟨even, odd⟩

3. ⟨odd, even⟩

4. ⟨odd, odd⟩

Input pairs (1) and (4) both have an even sum. The ABC-machine produces the same output for the two input pairs, while the AB-machine produces different outputs. So the ABC-machine can produce an output that correlates with the inputs’ property of having an even sum, while the AB-machine cannot. Mutatis mutandis for input pairs (2) and (3), which have an odd sum. According to Stalnaker (ibid.), this is what makes the difference in the machines’ ability to access information: Both machines carry the information about the sum of the inputs’ being even or odd, but only the ABC-machine can access that information, because “its output will covary systematically” (ibid.) with the sum of the inputs’ being even or odd.⁸ We can thus give a first-pass definition of Stalnaker’s notion of access:⁹ Having access to a certain piece of information requires the ability to produce a behavioral output that correlates with the information in question.

3.2 Bjerring and Tang’s Case: The ENT-Machine

Let’s now return to the case from the previous section, which poses a problem for Elga and Rayo’s account. In that case, an agent knows that, by rule r, s ₁ entails s ₂ and s ₂ entails s ₃. So she has access to the information about these entailments, relative to some choice condition c. By Elga and Rayo’s view, however, the agent must then have access to the information that s ₁ entails s ₃, relative to c. But if this prediction were correct, the agent would be able to detect that s ₁ entails s ₃ when she is in c - something that an ordinary, logically competent thinker might well be unable to do.

I’ll now describe a system that behaves in a more realistic way: It can detect that s ₁ entails s ₂ and s ₂ entails s ₃, while failing to detect that s ₁ entails s ₃. Consider a system that’s capable of detecting entailment relations between some formulas of propositional logic. A formula s _a is input into a first component A, and a formula s _b is input into a second component B. The joint state of components A and B is then input into a third component C - that is, C receives a signal carrying information about A’s and B ’s respective states. Based on A and B ’s joint state, C computes whether s _a entails s _b . C then produces an output, which determines the final behavioral response of the system.

This abstract description can be implemented in different ways. It will help to have an example before us, so I’ll now describe one system that meets the above description, constructed along the lines of Stalnaker’s AB- and ABC-machines. I’ll call it the ENT-machine. In the ENT-machine, component C processes s _a through mechanical manipulations that implement a certain inference rule. If C transforms s _a into s _b in a given time window, the system’s response is to state that s _a entails s _b , then halt. If C doesn’t transform s _a into s _b in the available time, the system’s only response is to halt. Note that the ENT-machine displays the familiar features of a mechanical computational system, such as a Turing machine. This may seem worrying in the present dialectical context, in light of Stalnaker’s criticisms of the “linguistic picture” of mental representation (Stalnaker, 1984) and the “sentence-storage model” of belief (Stalnaker, 1991). Later on, I’ll explain how my argument can also be applied to systems that don’t operate on linguistic and symbolic representations; for now, I’ll stick to the present example.

In Bjerring and Tang’s case, the ENT-machine could detect that s ₁ entails s ₂ and s ₂ entails s ₃, without detecting that s ₁ entails s ₃. For instance, suppose the inputs are s ₁ and s ₂: Component A receives the input formula s ₁, while component B receives s ₂. Given these inputs, component C is capable of transforming s ₁ into s ₂ in the time available, so the system outputs that s ₁ entails s ₂. A similar process occurs when the inputs are s ₂ and s ₃. But if s ₁ and s ₃ are input into the system, C may not complete all the necessary manipulations in the available time, in which case it will fail to transform s ₁ into s ₃, and the system will fail to detect that s ₁ entails s ₃.

Now, the ENT-machine carries information that’s inaccessible to it, just like the AB- and the ABC-machines. When it receives s ₁ and s ₃ as inputs, it does carry the information that there is an entailment among the current inputs, because s ₁ does entail s ₃ and the system carries the information that the current inputs are s ₁ and s ₃. However, the information is inaccessible, because the system is incapable of making its behavior depend on that information. On the other hand, when the input pair is s ₁ , s ₂ or s ₂ , s ₃ , information about the entailment is both present and accessible.

Why is the machine’s access to information limited in this way? Because of its representation-processing mechanisms; fragmentation is not part of the explanation. The machine’s ability to access information is explained by specific features of its representation-processing mechanisms, which determine its computational power. These limitations need not amount to a fragmented doxastic state; for instance, they need not involve a collection of probabilistically coherent credence functions, as on Elga and Rayo’s view. Therefore, we’re not forced to make implausible predictions about the machine’s behavior in Bjerring and Tang’s case. We are not postulating a probabilistically coherent credence function that drives the machine’s behavior. Its behavior is instead determined by its representation-processing mechanisms, and given those mechanisms, the machine might well detect the entailments s ₁ s ₂ and s ₂ s ₃ while failing to detect the entailment s ₁ s ₃. This is the correct result; unlike Elga and Rayo’s fragmentation account, an account based on representation-processing yields the right predictions about the system’s behavior in Bjerring and Tang’s case.

3.3 The Difference between RPEs and Fragmentation

We can now explain in more general terms the difference between fragmentation and the approach I am proposing. The above cases illustrate a certain kind of explanation of limited information access. In Stalnaker’s AB-machine, certain information is inaccessible because the system’s components cannot produce outputs that correlate with the information in question. Once component C is added, this structural limitation is overcome, the system’s representation-processing mechanisms change, and the information becomes accessible. A similar explanation was provided in Bjerring and Tang’s case. There, I described a mechanical system (the ENT-machine) which has limited access to information about some entailments because it cannot transform certain representations in the available time. These explanations respectively make reference to the internal architecture and computational power of the systems. In the AB- and ABC-machines, the explanation is based on the features of the processing units and the relations between those units; in the ENT-machine, the explanation is based on the computational power available to the system for the purpose of representation processing. In more general terms, these explanations make reference to the way in which the system processes representations. Call this a Representation-Processing Explanation (RPE): an explanation of limited access that makes reference to the way in which a system processes (manipulates, transforms) representations.¹⁰

Elga and Rayo also aim at explaining limited access, but their explanation is different: They identify a state of limited access with a fragmented doxastic state, while RPEs do not. Again, consider entailment relations between logical formulas. According to Elga and Rayo, an agent may assign high credence to the worlds where s _a entails s _b relative to some choice conditions while assigning low credence to the same worlds relative to other choice conditions (Elga and Rayo, 2022, 719-21). On the contrary, RPEs require no such assumption. In the ENT-machine, for instance, knowledge of inference rules consists in the machine’s disposition to transform inputs in accordance with those rules. Because of the system’s processing mechanisms, some transformations are completed in the available time, while others are not. These features and limitations explain the system’s behavior without postulating fragmentation: The explanation is compatible with the system’s being in a single doxastic state.

This also marks a difference from Stalnaker, who is one of the most influential proponents of fragmentation.¹¹ It’s worth noting that Stalnaker doesn’t reject what I’ve called RPEs of limited access. Indeed, he notes: “The manifest fact that we are not logically omniscient is a fact about our computational limitations - the fact that some of the information that is implicit in what we know or believe is, because of computational limitations, not accessible to us”(Stalnaker, 1991, 251). My point is that these computational limitations need not amount to a fragmented belief state. And in fact, Stalnaker himself acknowledges that fragmentation doesn’t account for all cases of limited access:

The fact that information is fragmented or distributed is one way, but not the only way, that knowledge may be merely implicit in the sense of unavailable. In our minimal example [the AB-machine], the machine also has the information about whether each of the input integers is prime or composite, and those items of information are each implicit in single components of the machine. The items of information concerning whether the integer is prime are also inaccessible, even though each is carried by a single component that represents the input integer. (Stalnaker, 2021, 191)

Stalnaker then notes (ibid.) that the machine would need “further computational components” in order to access the information about the input’s being prime or composite. This is what I’ve called an RPE: Stalnaker is claiming that the architecture of the system explains why certain information is inaccessible to it. So, in this case, Stalnaker himself favors an RPE over a fragmentation-based explanation. But Stalnaker also thinks that limited access does amount to a fragmented doxastic state in other cases, and this is where we part ways. I propose a uniform explanatory strategy: We can explain all cases of limited access through the representation-processing mechanisms of the relevant system, without appealing to fragmentation.

Let’s now provide a more rigorous argument to show that an RPE need not presuppose fragmentation. We begin by defining the notion of “access.” My definition will follow Stalnaker’s usage of the term: On this usage, as we’ve seen (Section 3.1), a system has access to certain input information if it can produce a behavioral output which correlates with that information. For instance, the AB-machine doesn’t have access to the information that the sum of the inputs is even, because it is incapable of making its output correlate with that property of the inputs (having an even sum).

Our definition of access must also take other parameters into account. As Stalnaker and Elga and Rayo point out, the condition (context, situation) where information is used and the behavioral purpose for which it is used are both relevant to information access. For instance, the ENT-machine (Section 3) doesn’t have access to the information that there is an entailment among certain formulas, because it doesn’t have enough time and computational power to do the necessary processing. If we appropriately modify these conditions (time, computational power), the machine will be able to bring about the desired input-output correlation, i.e. it will have access to the information about the entailment among the input formulas.

As for behavioral purposes, a non-idealized system can only bring about a correlation between specific behaviors and input information. Consider the ABC-machine, for instance. There is a specific output that the machine produces just in case the sum of the inputs is even. This could be an expression in natural language, such as the sentence: “The sum of the inputs is even.” Now, as Stalnaker (1999, 267-68) notes, it follows from Goldbach’s conjecture that the sum of the inputs is even just in case it is the sum of two odd primes (for any sum greater than two). Yet the machine may have no disposition to output the sentence “The sum of the inputs is the sum of two odd primes” when the sum of the inputs is even. This is a case where the system has access to the information that the sum is even for the purpose of producing one output sentence but not the other.

In light of these considerations, I define access as follows:¹²

System a has access to information i in condition c for the purpose of producing behavior b =_df a is disposed to make b depend on i in c.

Dependence can in turn be defined as counterfactually robust correlation:¹³

x depends on y =_df x would occur if y occurred; and x would not occur if y did not occur.¹⁴

Therefore, if a system a is disposed to make behavior b depend on information i, then a is disposed to produce b just in case it receives i.

With these definitions in hand, we can more clearly explain why RPEs need not presuppose fragmentation. Consider the ABC-machine again. Suppose an even numeral is input into component A. By hypothesis, there is an output that A produces just in case it receives an even input, so the following information is accessible to A:

(I ) The current input to component A is even.

Yet I is inaccessible to the system (the ABC-machine). By design, system component C delivers the same output for the input pairs even, even and odd, odd , so C ’s output doesn’t correlate with I. Therefore, the behavior of the system doesn’t correlate with I either, because (by design) C ’s output determines the system’s behavior. In other words, our conditions for information access are not met: There is no output b such that the machine is capable of producing b just in case component A receives an even number as input. Thus information I is inaccessible to the system, although it is accessible to one of its components.

What I have just provided is an RPE, which explains why the ABC-machine does not have access to I. However, the ABC-machine is not in a fragmented belief state. If it were, then it would be able to access information I and use it to guide its behavior in at least some contexts or choice conditions. This is a key claim of the fragmentation-based accounts that were presented in Section 2: If a subject is in a fragmented belief state, then the beliefs in a certain fragment guide the subject’s behavior in some (actual or possible) context. This is not what happens in our example. Information I is inaccessible to the ABC -machine in all contexts and choice conditions: The system is simply incapable of behaving in a way that correlates with I, no matter the current context or task.¹⁵

I have provided an example of an RPE-explanation which does not presuppose fragmentation. In more general terms, there is a fundamental difference between the two types of explanation. To accept fragmentation is to say something about the doxastic state of a subject, claiming that it is divided into multiple belief states or credence states, which guide the subject’s behavior in certain situations. But when we describe the representation-processing mechanisms of a system, we need not be making any claims about the system’s beliefs and their connection to the system’s behavior. The two descriptions are located at different levels.

3.4 Lewis’s Case

We now turn to a different type of case, which does not involve logical or mathematical truth:¹⁶

I used to think that Nassau Street ran roughly east-west; that the railroad nearby ran roughly north-south; and that the two were roughly parallel. (By ‘roughly’ I mean ‘to within 20◦.’) So each sentence in an inconsistent triple was true according to my beliefs, but not everything was true according to my beliefs. Now, what about the blatantly inconsistent conjunction of the three sentences? I say that it was not true according to my beliefs. My system of beliefs was broken into (overlapping) fragments. Different fragments came into action in different situations, and the whole system of beliefs never manifested itself all at once. The first and second sentences in the inconsistent triple belonged to - were true according to - different fragments; the third belonged to both. The inconsistent conjunction of all three did not belong to, was in no way implied by, and was not true according to, any one fragment. That is why it was not true according to my system of beliefs taken as a whole. Once the fragmentation was healed, straightway my beliefs changed: now I think that Nassau Street and the railroad both run roughly northeast-southwest. (Lewis, 1982, 436)

Lewis treats this as a case of belief fragmentation, but if fragmentation is construed along the lines of Elga and Rayo or Stalnaker’s views, problems arise. Lewis attributes to himself three beliefs, which are organized in two fragments:

• Lewis’s beliefs

• 1. Nassau Street runs east-west.

• 2. The railroad runs north-south.

• 3. Nassau Street and the railroad are parallel.

• First fragment

• 1. Nassau Street runs east-west.

• 3. Nassau Street and the railroad are parallel.

• Second fragment

• 2. The railroad runs north-south.

• 3. Nassau Street and the railroad are parallel.

As we have seen (Section 2), a fragment is subject to closure principles. By Closure under Entailment, the two fragments will thus include a number of further beliefs, such as (4) and (5):

• First fragment

• 1. Nassau Street runs east-west.

• 3. Nassau Street and the railroad are parallel.

• 4. The railroad does not run north-south.

• Second fragment

• 2. The railroad runs north-south.

• 3. Nassau Street and the railroad are parallel.

• 5. Nassau Street does not run east-west.

Fragmentation views also maintain that the beliefs in a fragment determine the subject’s behavior in some (actual or possible) situations. If so, there must be some (actual or possible) situations where Lewis behaves in accordance with (4) - situations where we can explain and predict Lewis’s behavior by ascribing (4) to him. But this need not be so - Lewis may have no disposition to behave in a way that is explained and predicted by (4). For instance, he may have no disposition to assent to “The railroad does not run north-south” - given his description of the case, this is a sentence that he may well fail to deduce from the other sentences that he does accept. Belief (4) might also provide a poor explanation of Lewis’s non-verbal behavior. If Lewis finds himself on the railroad, and desires to see the rest of the railroad, he might be disposed to move north or south, with no disposition to move in any other direction. This behavior is not predicted or explained by belief (4). A parallel argument can be provided for belief (5), mutatis mutandis.¹⁷

In sum, if Lewis respectively believed (4) and (5) relative to the two fragments, he would not behave as he does. We are thus forced to conclude that he does not hold those beliefs relative to the two fragments. More generally, Lewis might hold beliefs (1), (2), and (3) without believing (4) and (5) relative to any fragment. Again, if he did believe (4) and (5) relative to some fragment or other, then there would have to be some context or other where his behavior was correctly explained and predicted by those beliefs. And this need not be so, for the reasons I have just given.

My diagnosis is that we are faced with another instance of Bjerring and Tang’s problem: If we assume fragmentation, the individual fragments will still be rich enough to include a number of beliefs that the subject does not appear to hold. Here, too, an RPE would make more plausible predictions. As our previous examples show, a system might have to perform various operations on internal representations in order to gain access to a certain piece of information. For instance, Lewis’s cognitive system carried the following information:

(4) The railroad does not run north-south.

For Lewis believes (1) and (3), which jointly entail (4). But while the system carries information (4), it may not have access to it. Lewis’s cognitive system might have to perform a series of operations on (1) and (3) before gaining access to (4), as we already saw in our previous examples involving simple mechanical systems. And Lewis’s cognitive system might not have the necessary resources to perform these operations, at least up to a certain point in Lewis’s story. I will say (a bit) more about the relevant operations later on; for now, I will only point out that these operations may be defined on symbolic representations, as in the ENT-machine, but they may also be defined on mental representations in a different, non-symbolic format. Once the relevant operations have been performed, the system would be able to make its behavior depend on information (4) - it would have access to this information. It would then be able to make its behavior depend on (4): assenting to “The railroad does not run north-south;” not heading north or south to see the rest of the railroad; and so on. Until then, however, the system would have no access to information (4), even though the information was present all along in the system itself. As in the ENT-case, this is presumably due to performance limitations, since Lewis was logically competent enough to draw the necessary inferences, had he devoted enough cognitive resources to it. What the system was unable to do, then, was to make the necessary operations given the available resources (time, attention, and so on). Until these operations were performed, information (4) was present, but inaccessible.

In this section, I have introduced the notion of an RPE and I have distinguished it from fragmentation. I have argued that the former approach yields better predictions about the behavior of the relevant systems in a number of cases, including the case that Bjerring and Tang use against Elga and Rayo’s account. In the next section, I discuss the implications of RPEs for the issues we began with - the problems of deduction and logical omniscience.

4 A Framework for RPEs

4.1 From Access to Belief

The problems of deduction and logical omniscience have to do with belief and other doxastic states: A certain theory of belief - the possible worlds theory - was shown to have incorrect consequences concerning the beliefs of ordinary thinkers (Section 2). What are the implications of RPEs for these two problems?

As I will now explain, information access is connected with belief via the notion of a system’s behavior. Recall our definition of access:

System a has access to information i in condition c for the purpose of producing behavior b =_df a is disposed to make b depend on i in c.

By this definition, a system’s access to information consists in a certain type of behavioral disposition: the disposition to make a specific behavior depend on specific information. For instance, if the AB- and the ABC-machines both receive “1571” and “1591” as inputs, each of them carries the information that the sum of the current inputs is even, but only the ABC-machine has access to this information. The ABC-machine has access to this information because it can produce a specific output that correlates with the sum of the current inputs’ being even: There is an output that the machine produces just in case the sum of the current inputs is even. By contrast, there is no output that the AB-machine produces just in case the sum of the current inputs is even, because it lacks the processing component that would be required to produce such an output. This is the first observation we need to link information access with belief: A system’s access to information consists in its disposition to make its behavior depend on that information.

The second premise will link the system’s behavior with the system’s beliefs. It is a platitude that our belief ascriptions are partly based on people’s behavior. This is a platitude that most philosophers will accept, even if they disagree on other, more controversial issues concerning the relationship between action, belief, and belief ascription. The platitude can be illustrated through countless examples from the philosophical literature:

Consider a favorite belief of philosophers: the belief that there is beer in the fridge. Some of the dispositions associated with this belief include: the disposition to say, in appropriate circumstances, sentences like ‘There’s beer in my fridge’; the disposition to look in the fridge if one wants a beer; a readiness to offer beer to a thirsty guest; the disposition to utter silently to oneself, in appropriate contexts, ‘There’s beer in my fridge’; an aptness to feel surprise should one go to the fridge and find no beer; the disposition to draw conclusions entailed by the proposition that there is beer in the fridge (e.g., that there is something in the fridge, that there is beer in the house); and so forth. (Schwitzgebel, 2002, 251)

Schwitzgebel is describing the “dispositional stereotype” (ibid., p. 250) that we generally associate with the belief that there is beer in the fridge: a set of dispositions that we take to be typically possessed by those who believe there is beer in the fridge. Many of the dispositions Schwitzgebel cites are behavioral dispositions, including both verbal and non-verbal behavior, just as our platitude would have it.

In light of our platitude and our first observation, we can put forward a hypothesis: In cases of deductive failure and logical ignorance, we refuse to attribute certain beliefs to the relevant subject because of the subject’s behavior, which is in turn determined by the subject’s limited access to information. If the hypothesis is right, then RPEs would indeed be relevant to the problems of deduction and logical omniscience: RPEs explain why a certain system does not have access to certain information, which in turn explains why the system fails to believe certain necessary truths or certain deductive consequences of its beliefs.¹⁸

The hypothesis is supported by our cases. Consider Lewis’s case first. Here Lewis fails to draw certain deductive inferences, which involve three contingent propositions:

• 1. Nassau Street runs east-west.

• 2. The railroad runs north-south.

• 3. Nassau Street and the railroad are parallel.

For instance, he fails to deduce from (1) and (3) that (2) is false, and the railroad does not run north-south. Given an adequate RPE for Lewis’s cognitive system, we could explain why Lewis fails to infer that (2) is false. First, the RPE would describe how the system processes the relevant representations, which is what determines the system’s behavior. For instance, the RPE would explain why Lewis isn’t disposed to assent to “The railroad doesn’t run north-south;” why he is disposed to head north or south to see the rest of the railroad; and so on. Then these facts about Lewis’s behavior would explain why he does not believe that (2) is false: Borrowing Schwitzgebel’s phrase, Lewis lacks the “dispositional stereotype” associated with this belief, so he does not believe that (2) is false. An RPE could thus contribute to explaining Lewis’s deductive failure, thereby addressing the problem of deduction in this case.

4.2 The Role of Meta-Representational Information

Our last case involved contingent propositions (1), (2), and (3). When we turn from contingent to necessary propositions, the RPE approach faces a problem. To explain what the problem is, we must get say more about the nature of information - a notion that is obviously central to our discussion, but has so far been left undefined. An RPE explains limited access to information, but what is information? Various notions of information have been employed in the philosophical literature. One notion that is relevant to the present debate is that of Dretskean information. Dretske (1981) defined information as the elimination of possibilities:¹⁹

Information theory identifies the amount of information associated with, or generated by, the occurrence of an event [...] with the reduction in uncertainty, the elimination of possibilities, represented by that event or state of affairs. (Dretske, 1981, 4)

For instance, suppose there’s a group of eight employees, one of whom must be assigned some unpleasant task (ibid.). They decide by drawing straws, and Herman is selected; in this case, Herman’s drawing the shortest straw carries information about which employee was chosen to do the job. The event (Herman’s drawing the shortest straw) will then reduce eight possibilities to one, and that determines the amount of information associated with the event.

On the Dretskean definition of information, then, the information carried by a signal²⁰ can be represented as a set of possible worlds - the set of possibilities that are compatible with that signal.²¹ This raises a problem. The information in a signal is determined by the possible alternatives that the signal eliminates, but a necessary state has no possible alternatives. Therefore, a signal cannot carry any information about a necessary state, since it cannot eliminate any possible alternatives to that state.²² So it’s impossible for any system to carry information about a necessary state, no matter how the system processes representations. But then RPEs would seem irrelevant to the problem of logical omniscience: The reason why a system doesn’t have access to information about necessities is not that its representation-processing mechanisms are so-and-so, but rather that there is no such information in the first place!

Following Stalnaker and Dretske, however, I note that a system also carries meta-representational information, and this information is present in cases involving necessary truths.²³ More precisely, the output of a processing unit can carry meta-representational information about the state of that unit, and that state is not a necessary one. For instance, suppose we input the numerals “1571” and “1591” into either the AB- or the ABC-machine. Each of system components A and B will then produce an output. Their joint output cannot carry any information about the (necessary) fact that the sum of 1571 and 1591 is even, but it can carry information about the (contingent) fact that the sum of the current inputs is even. The latter fact is contingent because the sum of the inputs could have been odd; for instance, the sum of the inputs would have been odd if the machine had received “1572” and “1591” as inputs. And the information about this fact is meta-representational because it is information about the representational properties of a state of the system - in this example, the state of A and B.²⁴

We can now explain how an RPE would address failures of logical omniscience: A system may be unable to access certain meta-representational information, which means that its behavior doesn’t correlate with the information in question; in turn, lack of correlation between the system’s behavior and the meta-representational information explains why the system does not believe certain necessary truths. Consider the AB- and ABC-machines, and suppose that they are cognitive systems which are capable of belief. As we have just seen, if they both receive “1571” and “1591” as inputs, then they both carry the (meta-representational) information that the sum of the current inputs is even. However, only the ABC-system has access to this information, and we have seen how to explain this difference in access by providing RPEs for the two systems (Section 3.1). Now, the difference in access is a difference in behavior: The ABC-system can produce a behavior that correlates with the sum of the inputs being even, while the AB-system cannot. And since the two systems’ respective behaviors guide our belief attributions, we will attribute different beliefs to the two systems. More specifically, we will not attribute to the AB-system the belief that the sum of 1571 and 1591 is even, since the system’s behavior doesn’t correlate with the sum of the inputs being even. In sum: Through an RPE, we explain the behavioral dispositions of the AB-system, which in turn explain why the system does not believe that the sum of 1571 and 1591 is even - a necessary truth. RPEs can thus contribute to explaining failures of logical omniscience.

A parallel explanation could be provided in Bjerring and Tang’s example (Section 2), which involves entailment. Why is it that our subject knows that s ₁ entails s ₂ and that s ₂ entails s ₃, without knowing that s ₁ entails s ₃? The subject considers two pairs of formulas: s ₁ , s ₂ and s ₂ , s ₃ . Her cognitive system carries the (meta-representational) information that the first element of each pair entails the second element, so it also carries the (meta-representational) information that an element of the first pair entails an element of the second. However, the subject might be unable to make her behavior depend on the latter piece of information, and an RPE could explain why, as we have seen in the case of the ENT-machine (Section 3). For instance, the subject might be unable to produce a certain verbal behavior (“The first formula entails the second”) when considering s ₁ , s ₃ , while she is able to produce that behavior when considering s ₁ , s ₂ and s ₂ , s ₃ respectively. An RPE would thus explain the subject’s behavioral dispositions, which in turn explain why the subject does not believe that s ₁ entails s ₃. Here, too, an RPE contributes to explaining a failure of logical omniscience.

Our last two examples involved linguistic behavior, but the same point applies to other cases, where the relevant behavioral output is not linguistic (Stalnaker, 1991, 1999; Rayo, 2013). Borrowing an example from Rayo (2013, ch. 4), consider a farmer who has to buy fencing for her square piece of land, knowing how much fencing is required for one side. She picks up the material at the store, and once the amount of fencing in her cart is four times the amount required for one side, she stops and proceeds to check out. Suppose she buys the right amount of fencing on several other occasions for square patches of land of different sizes. The farmer arguably knows that the perimeter of a square piece of land is four times the length of one of its sides. Now, the farmer may be incapable of articulating her knowledge linguistically, but the basic structure of the account does not change: The farmer’s knowledge consists in her ability to access certain meta-representational information. The farmer’s memory carries the information that a certain amount of material is required to fence one side; the farmer’s perception at the store carries the information that a certain amount of material is in the cart. These two representations jointly carry the information that the amount of material in the cart is four times the amount of material required for one side. Once the farmer enters this representational state, her behavior changes and she proceeds to check out. Thus the farmer’s behavior correlates with meta-representational information about her current representational state; by our definition, she has access to this information. This is what grounds the farmer’s knowledge of a necessary geometrical fact, even though she does not articulate that knowledge linguistically.

Relatedly, note that RPEs do not presuppose that the mental representations of the system have a linguistic or symbolic format. This is important, because one might worry that RPEs presuppose the linguistic picture of mental representation or the sentence-storage model of belief that Stalnaker criticized (Stalnaker, 1984, 1991). If RPEs carried such a commitment with them, that would indeed be problematic in the present dialectical context: RPEs would not be a viable option for those who reject linguistic and sentence-storage pictures, such as Stalnaker himself. But RPEs carry no such commitment. While some of our examples have involved symbolic representations - numerals and logical formulas - one could also provide RPEs for systems that manipulate non-symbolic representations. Consider representations in map format, for instance. In Lewis’s example, the subject’s cognitive system might be processing two representations of Princeton’s geography that have the format of a visual map. One representation depicts Nassau Street as running east-west, while the second representation depicts Nassau Street and the railroad as parallel. If the two representations are correct, then the railroad must also run east-west, yet the system may not have access to the information that the rail-road runs east-west. Here, too, the system’s lack of access can be explained by its representation-processing mechanisms, without presupposing fragmentation; yet the representations being manipulated in this case are neither linguistic nor symbolic. More generally, there is no reason to think that RPEs cannot be applied to map-like or iconic representations, which might of course be involved in important cognitive functions like perception and memory.²⁵

In the literature on the problem of logical omniscience, the appeal to meta-representation faces a well-known challenge.²⁶ The objection is that one might not know a certain necessary truth yet have access to the relevant meta-representational information. Going back to one of our examples, suppose a certain logically competent subject doesn’t know whether formula s ₁ entails formula s ₃, while knowing all the relevant semantic facts: The subject knows the meaning of the formulas, she knows that certain inference rules are truth-preserving, and so on. Since she knows all this, she also has access to the relevant meta-representational information - that the formulas have a certain meaning, that certain inference rules are truth-preserving, and so on. However, s ₁ entails s ₃ in all the worlds where these semantic facts hold, so it seems that the subject must also have access to the information that s ₁ entails s ₃. But how could she have access to this information yet not know that s ₁ entails s ₃?

Following Rayo (2013, 110-12) and Soysal (2022), my reply is that the subject has access to the information that s ₁ entails s ₃ for some behavioral purposes and conditions; for other purposes and conditions, she doesn’t have access to that information, and that’s why she doesn’t know that s ₁ entails s ₃. Let’s see why that is. First, recall that our definition (Section 3.3) relativizes access to specific behaviors and conditions. When a system has access to certain information - including meta-representational information about semantic facts - it has access to that information under certain conditions, for the purpose of producing certain behaviors; relative to other conditions and behaviors, the system may not have access to that same information. Now, one premise of the objection is that the subject has access to information concerning certain semantic facts - that the formulas have a certain meaning, that certain inference rules are truth-preserving, and so on. We would then need to specify: Relative to what conditions and behavioral purposes does she have access to this semantic information? Given the description of the case, it’s reasonable to assume that our subject will be able to answer questions about the meaning of the expressions in the formulas, and indicate which inference rules are truth-preserving. She can thus access the meta-representational information for these purposes - answering questions about meaning, and indicating which inference rules are truth-preserving. In light of these behavioral dispositions, we acknowledge that the subject knows the relevant semantic facts: She knows that the formulas have a certain meaning, and that certain inference rules are truth-preserving.

At the same time, however, the description of the case makes clear that there is a different type of question which the subject cannot answer: If someone asks “Does s ₁ entail s ₃?”, our subject is not able to provide the right answer. The subject does possess the information needed to answer the question, since she possess the semantic information about the formulas and inference rules, and it follows from this semantic information that s ₁ entails s ₃. However, she cannot access this semantic information for the purpose of answering the question: “Does s ₁ entail s ₃?” Yet this is the verbal behavior that is required of someone who knows that s ₁ entails s ₃; since our subject does not behave in this way, she does not know that the entailment holds. In sum, the subject’s access to information warrants the following two claims. On the one hand, the subject knows that the formulas have a certain meaning and that certain inference rules are valid; on the other hand, she does not know that one formula entails the other. As in previous cases, then, the subject’s epistemic position is determined by her ability to access information for certain specific purposes and conditions; in this case, however, the information in question is semantic and meta-representational.²⁷

4.3 Filling the Framework: An Agenda for RPEs

My discussion so far has focused on specific cases, where I described a system’s limitations in accessing information and then explained how those limitations affect the system’s beliefs. However, it would be desirable to formulate a general principle along the following lines:

A subject A believes that p only if A has access to information i ₁ , i ₂ , . . . , i _n in conditions c ₁ , c ₂ , . . . , c _n for the purpose of producing behaviors b ₁ , b ₂ , . . . , b _n .

Providing this kind of principle requires overcoming at least two challenges. First, the principle must not be too demanding. If we require the subject to have access to the relevant information in all conditions, for all behavioral purposes, then we may make it too difficult for the subject to count as having the relevant belief in the first place (Stalnaker, 1999, 265). Second, we must specify what information is relevant for the belief in question. There will be cases where this information cannot simply be identified with the content of the belief, at least given a Dretskean notion of information. For instance, consider a subject who does not know that the sum of 1571 and 1591 is even. On my approach, this is explained by the fact that certain information is inaccessible to the subject. However, this information is not that the sum of 1571 and 1591 is even - as we have seen, there is no such information, at least on a Dretskean conception. Instead, I have identified the inaccessible information with the meta-representational information that the sum of the current inputs is even, in a condition where the inputs are “1571” and “1591.”

It would also seem desirable to provide a biconditional, rather than a mere conditional:

A subject A believes that p iff A has access to information i ₁ , i ₂ , . . . , i _n in conditions c ₁ , c ₂ , . . . , c _n for the purpose of producing behaviors b ₁ , b ₂ , . . . , b _n .

A mere conditional would only provide a necessary condition for belief: Given a description of the information that the subject cannot access, this principle would tell us what beliefs she does not have. However, it would not tell us what beliefs the subject does have, given the information that she can access.²⁸ For instance: if the subject has access to the information that P Q, relative to certain behavioral purposes and conditions, does she thereby believe that P, and does she thereby believe that Q ?

A third limitation of the RPEs I have sketched has to do with the kind of system that those RPEs target. Some examples from previous sections involved simple artificial systems, which process representations in the way we stipulate - this was the case with the AB-and ABC-machines, or the ENT-machine. Other examples, like Lewis’s Princeton case, involved complex cognitive systems, which do not obey our stipulations. There is an obvious difference between the two types of systems. With the former type, it is guaranteed that our RPEs provide an accurate description of their representation-processing mechanisms (at least if there are possible systems which satisfy all our stipulations). Providing an RPE for an actual cognitive system is a different matter.²⁹ We cannot stipulate that a human cognitive system will process representations in a certain way. To provide an RPE for such a system, one must provide an empirically backed description of its representation-processing mechanisms.

In this paper, I do not provide RPEs for actual cognitive systems - that is a much larger task than I can undertake here. There have been attempts in this direction, however. Solaki et al. (2019) propose a formal model for the operations of the two systems (System 1 and System 2) postulated by dual-process theories of reasoning. Their approach aims to avoid logical omniscience (ibid., pp. 19-20), while modeling the inferences that a logically competent agent would make, given their limited resources (“time, memory, computational power, etc.”; ibid., p. 26). As they observe, this approach requires empirical evidence concerning the cognitive system that we are modeling. For example, empirical evidence is needed to determine how “expensive” it is to apply a given inference rule (“not all inference rules require equal cognitive effort, as indicated by experimental evidence”; ibid., p. 9). This is precisely the kind of empirical evidence that an RPE would require when the system in question is an actual cognitive system: To know whether the system would perform a given operation (e.g. applying an inference rule), we must know what resources it has available and how resource-consuming it would be for the system to complete the operation in question.³⁰

Cherniak (1986, ch. 3) represents another example of the approach I am suggesting. He proposes a theory of representation processing to account for failures of deduction and logical omniscience, making substantial empirical claims about the structure of human memory. For instance, Cherniak appeals to influential psychological theories concerning long-term and short-term memory; having explained how memory structures affect our access to information, he then takes limited information access to explain certain cognitive limitations, e.g. the failure to draw valid deductive inferences from our beliefs (Cherniak, 1986, 56-57). This is precisely the general approach I’ve described: Cherniak proposes what I’ve called an RPE of limited information access, and he then takes limited information access to explain failures of deduction and logical omniscience. Note that I am not endorsing the specific RPE that Cherniak proposes. Cherniak argues that human memory is “compartmentalized” (Cherniak, 1986, 66-70), and his notion of compartmentalization is at least very close to fragmentation; as such, it may well fall prey to the same criticisms. If it does, then Cherniak’s RPE will be incorrect, but his general explanatory strategy will still represent a good example of the approach I’ve been defending.

Relatedly, it’s worth noting that the fragmentation views which were discussed in the first part of the paper are not backed by an empirical description of the relevant systems. I suggest that this is the root cause of their problems. As we have seen, fragmentation views uphold closure principles for single fragments; at the same time, they claim that the beliefs in each fragment will guide behavior in specific contexts. These two ideas fit poorly: Given the action-guiding role of belief, the closure principles end up making the wrong predictions about the relevant systems’ behavior, as we have seen in a number of cases. I submit that the problem is the following: The closure principles are not constrained by an empirical description of the representation-processing mechanisms of the relevant system. It is the representation-processing mechanisms that determine the system’s behavior, which in turn determines the beliefs that we can plausibly attribute to the system. Therefore, closure principles for belief must be constrained by a description of representation-processing mechanisms, particularly their internal organization and the resources that are available to them. And this description will inevitably be an empirical one.

Summarizing, I have proposed an agenda for an RPE approach to deduction and logical omniscience. To offer a complete solution, this approach must accomplish at least two goals. The first goal is to link information access with belief in a general way. The second goal is to provide an empirical description of the representation-processing systems which are involved in our deductive reasoning and in our knowledge of necessary truth.

4.4 The Pragmatic Picture

Before ending the paper, it will be useful to discuss a possible reaction to the problems of deduction and logical omniscience, which were presented in Section 2. Since I presented those problems as challenges for the possible worlds theory of content, it might be thought that there is a simple solution - abandon the possible worlds theory and endorse instead one of the alternative theories on the market. For instance, we might opt for a structured theory of content³¹ or appeal to the question-sensitivity and topic-sensitivity of doxastic attitudes.³²

I won’t be able to compare RPEs to these alternative approaches here, but I will argue that we cannot solve the problem by simply rejecting the possible worlds theory. As Greco (2021) shows, our problems also arise for a view which is more widely accepted than the possible worlds theory. This is the view which Stalnaker (1984) dubs the pragmatic picture. According to this picture:

Belief and desire [...] are correlative dispositional states of a potentially rational agent. To desire that P is to be disposed to act in ways that would tend to bring it about that P in a world in which one’s beliefs, whatever they are, were true. To believe that P is to be disposed to act in ways that would tend to satisfy one’s desires, whatever they are, in a world in which P (together with one’s other beliefs) were true. (Stalnaker, 1984, 15)

Stalnaker notes that the pragmatic picture motivates the possible worlds theory (Stalnaker, 1984, 23). Nevertheless, they are separate views, and many theorists who reject the possible worlds theory are nevertheless attracted to the pragmatic picture (Greco, 2021, 61-62). Therefore, it is important to note that the pragmatic picture alone leads to the problems of deduction and logical omniscience, no matter what theory of content we combine it with. This shows the true extent of the problem: As Greco would put it, the problems of deduction and logical omniscience are “(almost) everyone’s” problem (ibid., p. 61).

Following Greco (2021, 56-58, 61-65), let’s see why the pragmatic picture faces the problem of deduction. Suppose that p entails q, and A believes p. On the pragmatic picture, A is disposed to act in ways that satisfy her desires in worlds in which p is true (call them p-worlds). But q is also true in all these worlds, since p entails q - the set of p-worlds is a subset of the set of q -worlds. Therefore, A also acts in ways that satisfy her desires in worlds where q - together with A’s other beliefs - is true. On the pragmatic picture, then, A also believes that q. In other words, the pragmatic picture entails Closure under Entailment (Section 2), which in turn leads to the problem of deduction.³³

The previous argument presupposes that the space of worlds only includes possible worlds, but some views also include impossible worlds in the space of worlds.³⁴ Given this enriched space of worlds, what becomes of the argument? That is, does the pragmatic picture still lead to the problems of deduction and logical omniscience once we bring impossible worlds into the picture?³⁵ I argue that it does, by adapting an argument from Greco (2021, 62-66). Consider Lewis’s case again. Lewis believes (3):

Nassau Street and the railroad are parallel.

but he has failed to deduce from this that ¬(1 ∧ 2):

It is not the case that: Nassau Street runs east-west and the railroad runs north-south.

Once impossible worlds are included in the space of worlds, there are (impossible) worlds where (3) is true and (1 2) is not true, as well as worlds where (3) is true and (1 2) is true. Note that, since there are (impossible) worlds where (1 2) is not true, we are no longer forced to conclude that (1 2) is true in all worlds where (3) is true. So perhaps the pragmatic picture can indeed avoid Closure under Entailment by appealing to impossible worlds?

The problem with this strategy is the following. Since Lewis believes (3), then according to the pragmatic picture, he is disposed to act in a way that satisfies his desires in all worlds - both possible and impossible - where (3) is true. However, it is not at all clear that Lewis’s actions would satisfy his desires in the impossible worlds where (3) is true but ¬(1 ∧ 2) is not true. Suppose Lewis is on Nassau Street and desires to get to the railroad, and suppose his actual behavioral disposition is to turn perpendicular to Nassau Street (Greco, 2021, 63-64). Would Lewis’s behavior satisfy his desire in the impossible worlds where (3) is true but (1 2) is not true? An affirmative answer is plausible, since (3) is true in these worlds, so Nassau Street and the railroad are parallel. However, a negative answer is just as plausible: Since (1 2) is not true, in these worlds Nassau Street runs east-west and the railroad runs north-south, in which case turning perpendicular to Nassau Street will not satisfy Lewis’s desire to get to the railroad. And if Lewis’s behavior doesn’t satisfy his desires in all worlds where (3) is true, then the pragmatic picture entails that Lewis doesn’t believe (3), contra hypothesis.

It does not help to note that Lewis has different behavioral dispositions in different situations, as Lewis (1982, 436) himself suggests in describing the case. Suppose that Lewis is disposed to turn perpendicular to Nassau Street in some situations, while he is disposed to continue walking on Nassau Street in other situations. Then, in the impossible worlds I have described, his behavioral disposition in the first class of situations does and does not satisfy his desire to reach the railroad, and the same is true for his behavioral disposition in the second class of situations. In sum: Impossible worlds would only avoid the problem of deduction for the pragmatic picture if the pragmatic picture’s conditions for belief attribution were satisfied in such worlds, and it is unclear whether they are.³⁶

5 Conclusion

In this article, I have contrasted fragmentation with an alternative approach, which involves representation-processing explanations (RPEs) of limited information access. I have argued (Section 3) that RPEs provide a better account of limited information access than fragmentation, since they avoid some of the objections faced by the latter approach. I have then explained (Section 4) how RPEs could be developed into a full solution to the problems of deduction and logical omniscience. This is not a solution in itself, but I hope it serves as a useful framework for future research applying the RPE approach to the problems of deduction and logical omniscience.

Research Data Availability:

All research data are available within the main text of the article.

Acknowledgments:

The author would like to thank Derek Ball, Carolyn Benson, Michael Murez, Dilip Ninan, two anonymous reviewers, and the audience at the Mind and Language seminar of the Arché Philosophical Research Centre (University of St. Andrews) for their very helpful feedback.

5 References

• Aronowitz, S. (2023). Review: The fragmented mind. Philosophical Review, 132(2).
• Berto, F. (2024). Hyperintensionality and overfitting. Synthese, 1(4):1-21.
• Berto, F. and Jago, M. (2019). Impossible Worlds. Oxford University Press.
• Bjerring, J. C. and Tang, W. H. (2023). Fragmentation, metalinguistic ignorance, and logical omniscience. Philosophical Studies, 180(7):2129-2151.
• Borgoni, C., Kindermann, D., and Onofri, A., editors (2021). The Fragmented Mind. Oxford University Press.
• Cherniak, C. (1986). Minimal Rationality. MIT Press.
• Dretske, F. (1981). Knowledge and the Flow of Information. MIT Press.
• Egan, A. (2008). Seeing and believing: Perception, belief formation and the divided mind. Philosophical Studies , 140(1):47-63.
• Egan, A. (2021). Fragmented models of belief. In Borgoni, C., Kindermann, D., and Onofri, A., editors, The Fragmented Mind, pages 108-133. Oxford University Press.
• Elga, A. and Rayo, A. (2021). Fragmentation and information access. In Borgoni, C., Kindermann, D., and Onofri, A., editors, The Fragmented Mind, pages 37-53. Oxford University Press.
• Elga, A. and Rayo, A. (2022). Fragmentation and logical omniscience. Noûs, 56(3):716-741.
• Field, H. (1978). Mental representation. Erkenntnis, 13(July):9-61.
• Field, H. (1986). Critical notice: Robert Stalnaker, Inquiry. Philosophy of Science, 53(3):425-448.
• Greco, D. (2021). Fragmentation and coarse-grained content. In Borgoni, C., Kindermann, D., and Onofri, A., editors, The Fragmented Mind, pages 54-77. Oxford University Press.
• Hawke, P., Özgün, A., and Berto, F. (2020). The fundamental problem of logical omniscience. Journal of Philosophical Logic, 49(4):727-766.
• Hoek, D. (2022). Questions in action. Journal of Philosophy, 119(3):113-143.
• Hoek, D. (2025). Minimal rationality and the web of questions. In van Elswyk et al., editor, Unstructured Content. Oxford University Press.
• Kindermann, D. and Onofri, A. (2021). The fragmented mind: An introduction. In Borgoni, C., Kindermann, D., and Onofri, A., editors, The Fragmented Mind, pages 1-33. Oxford University Press.
• Kripke, S. (1980). Naming and Necessity. Harvard University Press.
• Lewis, D. (1982). Logic for equivocators. Noûs , 16(3):431-441.
• Loewer, B. M., editor (1990). Meaning in Mind: Fodor and His Critics. Blackwell, Cambridge.
• Millikan, R. G. (1984). Language, Thought, and Other Biological Categories. MIT Press.
• Neander, K. (2017). A Mark of the Mental: A Defence of Informational Teleosemantics. MIT Press, Cambridge, USA.
• Norby, A. (2014). Against fragmentation. Thought: A Journal of Philosophy , 3(1):30-38.
• Özgün, A. and Berto, F. (2021). Dynamic hyperintensional belief revision. Review of Symbolic Logic, (3):766-811.
• Rayo, A. (2013). The Construction of Logical Space. Oxford University Press.
• Rescorla, M. (2020). The Computational Theory of Mind. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2020 edition.
• Richard, M. (1990). Propositional Attitudes: An Essay on Thoughts and How We Ascribe Them. Cambridge University Press, Cambridge; New York.
• Schwitzgebel, E. (2002). A phenomenal, dispositional account of belief. Noûs , 36(2):249-275.
• Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, (27):379--423, 623--656.
• Shea, N. (2018). Representation in Cognitive Science. Oxford University Press.
• Shea, N. (2023). Organized representations forming a computationally useful processing structure. Synthese, 202(6):1-20.
• Silva, F. (2024). Question-relative knowledge for minimally rational agents. Inquiry: An Interdisciplinary Journal of Philosophy , pages 1-31.
• Soames, S. (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press.
• Soames, S. (2006). Understanding assertion. In Thomson, J. J. and Byrne, A., editors, Content and Modality: Themes From the Philosophy of Robert Stalnaker, pages 222-250. Oxford University Press.
• Solaki, A., Berto, F., and Smets, S. (2019). The logic of fast and slow thinking. Erkenntnis , 86(3):733-762.
• Soysal, Z. (2022). A metalinguistic and computational approach to the problem of mathematical omniscience. Philosophy and Phenomenological Research, 106:455-474.
• Speaks, J. (2006). Is mental content prior to linguistic meaning?: Stalnaker on intentionality. Noûs , 40(3):428-467.
• Stalnaker, R. C. (1978). Assertion. Syntax and Semantics, 9:315-332.
• Stalnaker, R. C. (1981). Indexical belief. Synthese, 49(1):129-151.
• Stalnaker, R. C. (1984). Inquiry. Cambridge University Press.
• Stalnaker, R. C. (1986). Replies to Schiffer and Field. Pacific Philosophical Quarterly, 67(2):113-123.
• Stalnaker, R. C. (1991). The problem of logical omniscience, I. Synthese, 89(3):425-440.
• Stalnaker, R. C. (1999). The problem of logical omniscience, II. In Context and Content: Essays on Intentionality in Speech and Thought, pages 255-273. Oxford University Press.
• Stalnaker, R. C. (2006). Responses. In Thomson, J. J. and Byrne, A., editors, Content and Modality: Themes From the Philosophy of Robert Stalnaker, pages 251-295. Oxford University Press.
• Stalnaker, R. C. (2021). Fragmentation and singular propositions. In Borgoni, C., Kindermann, D., and Onofri, A., editors, The Fragmented Mind, pages 183-98. Oxford University Press.
• Williamson, T. (2018). Counterpossibles. Topoi, 37(3):357-368.
• Yablo, S. (2014). Aboutness. Oxford: Princeton University Press.
• Yalcin, S. (2018). Belief as question-sensitive. Philosophy and Phenomenological Research , 97(1):23-47.

• 1
  See Lewis (1982) and Cherniak (1986) for two influential formulations of the idea. For a more recent defense of fragmentation, see Egan (2008); Yalcin (2018), as well as several papers in Borgoni et al. (2021). See Kindermann and Onofri (2021) for an overview of the literature.
• 2
  More precisely, Stalnaker believes that fragmentation is applicable in some cases, while other cases require a different explanatory strategy; I’ll argue that we can provide the same type of explanation across the board. More on this in Section 3.
• 3
  While I argue that RPEs explain limited information access better than fragmentation, I remain neutral on the merits and limitations of fragmentation as an explanation of other cognitive phenomena. For all I say in the paper, fragmentation may or may not provide a good account of other aspects of our cognitive life, such as implicit bias - see the articles in Borgoni et al. (2021, part V) for a recent discussion. This is a separate issue, which I can remain neutral about for the purposes of this paper.
• 4
  For a comprehensive overview of the relevant closure principles, see for instance Özgün and Berto (2021); Hawke et al. (2020); Silva (2024).
• 5
  The same argument can be applied to necessary a posteriori truths (Kripke, 1980); see Soames (2006) and Stalnaker (2006, 2021) for discussion. I’ll focus on logical and mathematical necessities in what follows.
• 6
  Concerning Stalnaker’s solution to the problems of deduction and logical omniscience, see especially Stalnaker (1984, 2006, 2021). See Greco (2021) and Kindermann and Onofri (2021) for an overview of Stalnaker’s solution, and see Stalnaker (1991, 1999) for further discussion of the two problems.
• 7
  See Field (1986, 445), Richard (1990, 9-16), and Speaks (2006, 449-50) for similar objections raised against Stalnaker’s appeal to fragmentation. For other types of objections to fragmentation, see Hoek (2022) and Norby (2014).
• 8
  In the same spirit, Stalnaker (1999, 265) notes that, intuitively, an agent has access to certain information when she can make her actions depend on that information. He then provides a refined version of this definition (ibid., pp. 265-66), noting its limitations and implications for the problem of logical omniscience. This notion of access is at least very close to Dretske’s notion of “selective sensitivity” (Dretske, 1981, 179-89). But see Stalnaker (1999, fns. 5-6) for a criticism of Dretske in connection with the problem of logical omniscience.
• 9
  I’ll provide a more rigorous definition in Section 3.
• 10
  One could also talk of “information processing,” but this terminology is controversial (Rescorla, 2020, section 6.1).
• 11
  One difference between Stalnaker and Elga and Rayo is that Stalnaker relativizes belief states to contexts, while Elga and Rayo relativize credences to contexts (what they call “choice conditions”). This doesn’t affect my argument, though, for RPEs don’t presuppose either kind of fragmentation: They don’t relativize belief states or credences to contexts. Bjerring and Tang (2023, 2135-41) argue against Stalnaker’s solution to the problem of logical omniscience, noting that it still attributes an implausible amount of knowledge to ordinary thinkers. Their objection against Stalnaker’s view is similar to their objection against Elga and Rayo’s view, which isn’t surprising since the latter is inspired by the former. I won’t discuss Bjerring and Tang’s objection against Stalnaker, but I believe it should be handled in a similar way as the objection against Elga and Rayo.
• 12
  I’m adapting a definition discussed by Stalnaker (1999, 265).
• 13
  Thanks to an anonymous reviewer for asking me to clarify this.
• 14
  I believe this is Dretske’s notion of dependence, and it is closely related to his notion of information: See Dretske (1981, 23, 37-39). It’s worth noting that, according to Dretske, informational dependence is distinct from causal dependence. Furthermore, informational dependence does not require a physical channel linking the two events (ibid., pp. 37-39).
• 15
  We could modify the system’s internal organization, so that it was capable of producing an output that correlates with I. For instance, this could be achieved if A was capable of controlling the system’s behavior without going through C. A further component could also be introduced which served as a switch, “deciding” whether A or C will control the next behavioral output. These modifications would result in a new system, though; unless we modify it in one of these ways, the ABC-machine does not have access to I. Also note that information I became inaccessible to the system once component C was added; at the same time, adding C made certain other information accessible. As Dretske (1981, 140-41, 182-83) points out, this is a characteristic feature of information-processing systems in general, and of cognitive systems in particular.
• 16
  Thanks to an anonymous reviewer for urging me to discuss this case in more detail.
• 17
  We don’t know enough about Lewis’s actual predicament to know whether he had this dispositional profile, but it doesn’t matter. It is enough for my purposes to show that someone in Lewis’s doxastic position might be disposed to behave in the way I have described.
• 18
  This approach to the problems of deduction and logical omniscience is based on Stalnaker’s insights: One of Stalnaker’s main ideas was that “[t]he problem of logical omniscience [...] is the problem of accessibility”(Stalnaker, 1991, 254). My disagreement with Stalnaker resides elsewhere: We disagree on whether fragmentation accounts for limited accessibility, as I have explained in the first part of the paper.
• 19
  Dretske’s influential account was based on the mathematical theory of information (Shannon, 1948). Picking an alternative notion of information might have significant implications for the present discussion - see for instance Solaki et al. (2019, fn. 4) for a weaker, non-factive notion. It’s worth noting that “information” is a technical term here. Therefore, the definition under discussion isn’t meant to capture our ordinary usage, and tension with ordinary usage shouldn’t lead us to reject the definition. That said, I agree that there is a tension. For instance, the discussion of Egan (2021) in Aronowitz (2023, 335) seems to bring out a clash between the ordinary usage of “information” and the technical notion of information.
• 20
  What counts as a signal? Dretske’s notion of a signal is very broad: If r is a signal carrying the information that s is F, then r could be any “event, condition, or state of affairs” which depends on whether s is F (Dretske, 1981, 65).
• 21
  I’m setting aside some aspects of Dretske’s account here. In particular, Dretske distinguishes the quantity of information in a signal from the informational content of the signal. The distinction can be informally stated as follows: The quantity of information in a signal depends on how many possibilities it eliminates, while its informational content depends on what possibilities it eliminates (Dretske, 1981, chs. 1-3). The distinction between quantity of information and informational content isn’t essential for our purposes, so I’ll just speak of the “information” that’s carried by a signal.
• 22
  This can also be proved through Dretske’s formulas (ibid., pp. 4-12). Note that Dretskean information and possible worlds content are subject to the same closure principles.
• 23
  This is a version of what is often called Stalnaker’s “meta-linguistic strategy” (Stalnaker, 1986, 120). As Stalnaker notes, the label is misleading because the strategy can also involve non-linguistic representations (ibid., p. 121). This is why I talk about “meta-representational” rather than “meta-linguistic” information in the main text. Stalnaker’s strategy is also known as the “diagonalization” strategy (Stalnaker, 2021, 185), making reference to the formal treatment of the idea in Stalnaker (1978). For Stalnaker’s attempt to combine meta-representation and fragmentation to address the problem of logical omniscience and other related problems, see (Stalnaker, 1984, 1986, 2021); see especially Greco (2021, 60) and Stalnaker (2021, 190-93, 196-97) for an explanation of how the two strategies complement each other. Meta-representational information is also central to Dretske’s discussion of necessary truth (Dretske, 1981, ch. 9). For reasons of space, I won’t be able to discuss Dretske’s argument here.
• 24
  This type of RPE explains why a system has limited access to information about the semantic properties of its own representations. If so, the RPE presupposes that the system’s representations have a certain content; it doesn’t explain why those representations have that content. An RPE is not a theory of intentionality. The goal of an RPE is to explain why a system has limited access to information about the content of its own representations; its goal is not to explain why those representations have that content in the first place. It’s important to keep this in mind if we adopt a Dretskean notion of information, since it’s notoriously difficult to reduce mental content to Dretskean information alone - see for instance the discussion in (Loewer, 1990). I acknowledge that a theory of intentionality might well have to appeal to other notions in addition to Dretskean information; for instance, it may be necessary to complement information with function, following a teleosemantic approach to intentionality (Millikan, 1984; Neander, 2017; Shea, 2018).
• 25
  It’s also worth keeping in mind that, as we have seen, Stalnaker himself proposes RPEs (as I’ve called them) in some of our cases. So if RPEs did presuppose the linguistic picture or the sentence-storage model, Stalnaker’s own explanation of limited access in these cases would be inconsistent with his general view of intentionality. See Shea (2023) for a recent discussion and examples of representation processing with different formats, both symbolic and non-symbolic.
• 26
  A similar objection was raised against Stalnaker’s solution to the problems of deduction and logical omniscience: Field (1986) raises the objection against Stalnaker’s account of logical and mathematical ignorance, while Soames (2006) presents a related objection against Stalnaker’s approach to necessary a posteriori truth. Stalnaker (1984, 76) himself notes that his account faces this challenge. To address the problem, Greco (2021, 60) and Stalnaker (2021, 190-93, 196-97) claim that meta-representational knowledge is itself fragmented. Field (1986, 445) anticipates this response and briefly argues against the idea that meta-representational knowledge is fragmented in the relevant cases. In what follows, I argue that limited access to meta-representational information is explained by the system’s representation-processing mechanisms, without presupposing fragmentation.
• 27
  What about cases of inconsistent belief, like Lewis’s case? In that case, Lewis attributes to himself three jointly inconsistent beliefs, but he also claims that the “blatantly inconsistent conjunction” of his beliefs was not true “according to [his] system of beliefs taken as a whole.” Can an RPE explain how this is possible? Here, too, it is the subject’s behavior that drives our belief attributions. We describe Lewis as not believing the “blatantly inconsistent conjunction” because he does not assent to the blatantly inconsistent sentence that expresses the contradiction. At the same time, Lewis does assent to each conjunct of the inconsistent sentence, so we attribute to him three corresponding beliefs, which respectively have the content of each conjunct. Note that access limitations are still relevant here, because they explain why Lewis does not acknowledge the inconsistency. Lewis’s cognitive system does carry the information that, if his current beliefs were all true, then the inconsistent sentence would be true. However, he does not have access to this information. To have access to it, his cognitive system would have to perform a number of operations on the relevant representations, but his cognitive system does not have the resources to complete these operations up to a certain point in the story, when he identifies the inconsistent consequences of his beliefs and reforms them accordingly.
• 28
  Stalnaker (1981) raised a similar issue for the “sentence-storage” model of belief. The issue is closely related to the requirement of “minimal rationality,” which has been widely discussed in the literature on deduction and logical omniscience — see for instance Cherniak (1986); Hoek (2025).
• 29
  Stalnaker (1999, 266) raises a related concern.
• 30
  Not all failures of deduction and logical omniscience are explained by resource limitations. There are different kinds of limitations in information access, and these must be explained by different kinds of RPEs. Our previous examples illustrate this. Stalnaker’s AB-machine seems unable to detect whether the input sum is odd or even for any pair of inputs. On the contrary, the ENT-machine was capable of detecting some entailments but not others. The ENT-machine is similar to an ordinary, logically competent subject: A system with this kind of logical competence is able to detect at least some entailments, and when it fails to do so this is due not to lack of “competence” but to “performance” limitations - there isn’t enough time, the proof is too long given the system’s computational power, and so on. On the contrary, the AB-machine suffers from a competence limitation of sorts. The two systems thus have different kinds of limitations in their access to information, and these limitations are explained by different RPEs. The article by Solaki, Berto, and Smets also illustrates how a formal approach and an empirical approach are not mutually exclusive, but rather complement each other: They present an epistemic logic framework enriched with non-normal worlds, then apply it to model the reasoning processes described by empirical research in cognitive psychology. I thank an anonymous reviewer for bringing this paper to my attention.
• 31
  See for instance Field (1978); Richard (1990); Soames (2002).
• 32
  See, for instance, Yablo (2014); Yalcin (2018); Hoek (2022); Berto (2024). See Silva (2024) for an approach that combines question-sensitivity with impossible worlds.
• 33
  A parallel argument shows that the pragmatic picture entails Closure under Necessary Equivalence, which leads to the problem of logical omniscience.
• 34
  See for instance Berto and Jago (2019); Silva (2024).
• 35
  Thanks to an anonymous reviewer for urging me to discuss this.
• 36
  As Greco notes, the discussion of impossible worlds is connected with the discussion of counterpossibles: “subjunctive conditionals with impossible antecedents.” (Greco, 2021, 62) The received view holds that all such conditionals are vacuously true (Williamson, 2018), but the received view can be rejected by those who include impossible worlds in the space of worlds (Berto and Jago, 2019, ch. 12). As Greco (2021, 65-66) notes, however, rejecting the received view on counterpossibles is not enough to save the pragmatic picture. Even if we grant that some counterpossibles are non-vacuously true, while others are false, the pragmatic picture would need something more than that - it would need some specific counterpossibles to come out non-vacuously true, including for instance: If Nassau Street and the railroad were parallel, and Nassau Street ran east-west while the railroad ran north-south, then turning perpendicular to Nassau Street would satisfy Lewis’s desires. But as the argument in the main text shows, turning perpendicular to Nassau Street would both satisfy and fail to satisfy Lewis’s desires in those (impossible) worlds where the antecedent is true. Therefore, it’s not at all clear whether this counterpossible is (non-vacuously) true.
• Article Info:
  CDD: 121
• Funding:
  No funding was received to assist with the preparation of this manuscript.

Edited by

• Editor-in-Chief
  MARCO RUFFINO ORCID: https://orcid.org/0000-0001-5619-0016

Publication Dates

• Publication in this collection
  19 Jan 2026
• Date of issue
  2025

History

• Received
  11 Jan 2025
• Reviewed
  21 Sept 2025
• Accepted
  23 Oct 2025