Precedent and rest stop convergence in reflective equilibrium

Abstract

The method of reflective equilibrium is typically characterized as a process of two kinds of adjustments: hold fixed one’s current set of commitments/intuitions and adjust rules/principles to account for them, then hold fixed those rules while making adjustments to one’s set of commitments. Repeat until no further adjustments are required. Such characterizations ignore the role of precedent, i.e., information about the commitments and rules of others and how those might serve as guides in one’s own process of deliberation. In this paper we develop a model that narrows in on a local part of a reflective equilibrium process in which a group of peer agents start with identical lists of commitments and work towards finding the rule that captures that set. If successful, we say they reach rest stop interpersonal convergence. The trouble is that, without precedent, reaching this convergence rests on implausible deterministic assumptions. We analyze how far the inclusion of a kind of soft precedent goes towards aiding the group in reaching said convergence when those assumptions are relaxed.

1 Introduction

Consider the dolphin. It seems like you’d want to count it as a kind of fish. It lives in the water; it has fins. Whatever your prototype is of a fish, a dolphin comes pretty close to it visually and behaviorally. But of course dolphins are mammals: they get their oxygen from the air and give live birth. We make some updates to our ideas about what lives out its days in the sea. Not everything that swims like a fish is a fish.

In a similar way, not everything that looks like knowledge is knowledge. When Jones believes that the man with the ten coins in his pocket will get the job, it seems that he knows but he doesn’t. Jones is our dolphin. The old idea of knowledge as justified, true belief includes too much. Not everything in the ocean is a fish; not all instances of justified, true belief are knowledge.¹

For both Jones and the dolphin, we have some settled intuitions about what is what: ocean-dwelling flippered things are fish; knowledge is JTB; organisms that breathe air aren’t fish; Jones got lucky in his belief. When agents engage in the process of reflective equilibrium (RE), they make mutual adjustments to judgments of cases and the systematic principles that account for them.² Biologists might have shifted their ideas about fish to include dolphins instead of thinking about dolphins as non-fish. Philosophers could throw their hands up and say, “maybe Jones really did know and the JTB account is just fine!" but this isn’t how history played out. These “dolphin dilemmas," as we’ll call them, emphasize an important aspect of the method of RE as employed in groups: their results are often historically contingent. It turned out that biologists, for some reasons over others, chose to classify dolphins as mammals, but we can imagine an alternate history in which swimming through the sea is key for classifying animals.³

A common goal of RE, as exemplified by Rawls (1971), Goodman (1983), and Rechnitzer (2022), is justification of some system. In these cases, either justification belongs to the subject that is engaging in the process, or the justification is “free-floating” in some abstract sense. Either way, such applications underestimate or even overlook the social environment and historical mechanisms involved in the development of our judgments and related adjustments in them. Recent work in social epistemology has begun addressing this kind of oversight for the process of knowledge production.⁴ Despite the acknowledgment of collective decision-making and other interpersonal goals as aims of the RE process (Rechnitzer, 2022), the role of a social historical landscape in that process has not been analyzed or modeled.

One exception might be the work of Kelly and McGrath (2010). They distinguish between intrapersonal and interpersonal convergence. Questions about intrapersonal convergence are concerned with whether, for a particular person, the process of RE will yield a unique equilibrium. Questions about interpersonal convergence are about whether different individuals each deploying the process of RE would arrive at the same equilibrium. Kelly and McGrath argue that while intrapersonal convergence is necessary for the interpersonal kind, it is not sufficient (individuals might have different starting points, for example). But all that still leaves open a host of questions about ways in which intrapersonal convergence might help facilitate some kind of progress towards the interpersonal version, and whether and how this is further aided by the kind of considerations of interest to social epistemologists.

One plausible way forward is to consider the phenomenon of precedent. As an institutionalized example, judges in many legal systems are obliged to make their rulings as consistent as reasonably possible with previous judicial decisions on the same subject. Effectively, this means that each case that is decided by a court of law becomes a precedent or guideline for subsequent decisions. Future judges stand on the shoulders of past judges.

There are also plenty of non-institutionalized examples, including the practice of doing philosophy. For instance, when presenting an example, an instructor may report to the students which analysis is most widely accepted, thus nudging students to shape their intuitions or principles towards making similar classifications of that case.⁵ A related example is the role of a publication, which invites intellectual peers to consider similar reasons for accepting some conclusion. Previous publications can frame a debate, allowing authors some clearance to forego the considerations of some intuitions over others.⁶

Our rough-and-ready characterization of precedent is: a guide for how to approach a case without delivering a judgment on that case. Stealing a phrase from Chisholm (2018) in another context, precedent “inclines without necessitating." Consequently, the challenge of theorizing about the role of precedent in the context of RE involves assumptions about how to navigate three different categories of considerations: commitments, principles, and external precedents set by others. This is a significant complication to the standard conceptualization of the process of adjustments at the heart of RE, which consists of two iterative steps: (i) hold constant the current set of commitments and adjust the existing principles in order to best account for the set of commitments, then (ii) hold constant the newly adjusted principles and adjust the set of commitments to maximize agreement with the principles.⁷

In this paper we develop a model for how a process of adjustments might navigate between the three categories. We opt to narrow in on a local part of the global RE process. We imagine that we have a group of peer agents, each with their own but identical set of cases on which commitments have been made in a previous iteration.⁸ We prefer to call such commitments intuitions on account of the history of the term in epistemology and because of how we conceive of their role in making adjustments (more below).⁹ The task of each agent is to find a rule or principle that best accounts for their intuitions. But because agents are not all-seeing, they consider cases one at a time: What about this case? Now what about that one? And so on. All the while, they are making adjustments to whatever their current rule is, making announcements to their peers about how they’ve classified cases, and taking into considerations the announcements of others (i.e. precedent).¹⁰ By the end of this process, the hope is that all the agents identify the same rule to account for the set intuitions they all initially shared - what we call rest stop interpersonal convergence.¹¹ But previous work shows that this hope is easily dashed (Baumgaertner & Lassiter, 2023). Agents do not consider cases in the exact same order; nor are they equipped with a deterministic procedure for making adjustments to their rules. Insisting otherwise is far too demanding in our eyes. But these two degrees of freedom are individually sufficient for agents to fail at reaching interpersonal convergence (even though each agent reaches intrapersonal convergence).

We suspect that precedent can help constrain the diffusion of rules brought about by the process of RE. Compared to making the RE process deterministic or precisely ordering the presentation of cases, precedent is not nearly as demanding as a modeling assumption. It is also independently plausible. In this paper, we present a very weak conception of it that we call soft precedent: its primary function is to break ties. It will turn out that the role of soft precedent is a social mechanism by which the hope of rest stop interpersonal convergence is restored. However, it does so at the price of being historically contingent, which some may see as too high of a cost. In any case, we see this as a new constraint on the reflective equilibrium literature.

2 Guiding assumptions and model design

We mentioned previously that, on our view of the process of RE, agents are balancing commitments, principles, and precedents. Due to space constraints, we leave a more in-depth defense of these to the side.¹² Even so, we expect these are at the very least prima facie plausible and broadly consistent with much of the literature on reflective equilibrium. Some might suggest our model is better thought of as a characterization of narrow reflective equilibrium, but even that might not be quite right.¹³ Our primary and modest aim is to make some advances in our understanding of the process of RE when balancing three different sorts of constraints, particularly when “holding fixed” as much as possible intuitions about cases. To that end we make decisions that are charitable and favor success of the method (given computational constraints).

2.1 Representing commitments and principles

We’ll get to incorporating precedents later, after we’ve introduced our implementation of the process of RE. Since balancing precedents depends on others’ judgments about cases, it’s easier on exposition to describe the elements and process of RE in our model first.

2.1.1 Commitments: cases and intuitions

How do we represent commitments? First, there are two things to consider: cases and intuitions about them.

Case representation Thought experiments can be specified by a finite series of relevant propositions. Formally then, a case is represented as a finite binary string of yes/no responses to questions about the truth of the relevant propositions. Each bit is said to be a feature of a case.

Intuitions We have something in mind akin to “considered judgments” about a case: similar to Rawls, we think of them as stable over time, issued when one is able to concentrate without distraction, but it can be reasonable to exclude some over others, or even suppress them over the course of reflecting. Sometimes we will also refer to intuitions as commitments, acknowledging Brun’s work Brun (2014) that intuitions are non-inferential. What ultimately matters for us is that we functionally represent intuitions as labels on cases. “IA" means the intuition is to accept the case as an instance of the prototype. Our dolphin case is an example: initially, there is an intuition to accept dolphins as a kind of fish. “IR” means to reject the case as an instance of the prototype, like when philosophers do not want to credit Jones with knowledge. “NI” means the agent has no intuition about it. We will say more on the role of intuitions when we describe the reflective process.

2.1.2 Rules/principles

Now, how we represent principles:

Rules (Principles) A rule consists of a core case and a tolerance threshold that says how similar cases must be to the core case. This pair defines the extension of the rule. We think about the extension of a rule as representing instances of a prototypical case. The extension of a rule is said to be incoherent when it contains a pair of cases that are pairwise inconsistent. From here it can be more or less coherent depending on its width.¹⁴

Coherence We operationalize the idea of the coherence of a set of cases in terms of the width of the set: the distance (as measured by the similarity score) between the minimally similar pair of cases.

Similarity Cases are more or less similar as measured by the Hamming distance between them, i.e., the number of positions in the string on which the cases differ. Two cases that disagree on each position are said to be pair-wise inconsistent. The Hamming distance is normalized by the length of the strings so that we have a similarity score between 0 and 1, where 0 means is minimally similar (pairwise inconsistent) and 1 is maximally similar (which is just the case compared to itself).

2.2 Representing the RE process

We follow the constraints set by some “verbal models” of the RE process in which one moves “back and forth between [initial principles and initial beliefs] and eliminating, adding to, or re-vising the members of either set until one ends up with a final set of beliefs and principles which cohere with each other.” Cath (2016)¹⁵

When an agent “considers" a case C, it checks the normalized Hamming distance between C and its core case. If the distance between C and the core case is less than the tolerance threshold, then the case goes onto the ACCEPT list. Otherwise, it goes onto the REJECT list. Going back to our fish example, if the proposed creature is similar enough to our prototype, then it is accepted as a fish. Otherwise, it counts as something else.

But recall that every case is tagged with an intuition label: intuitive accept (IA), intuitive reject (IR), and no intuition (NI). In the case of NI, agents put the case on the ACCEPT list if it is sufficiently similar to the center case, i.e., if the normalized Hamming distance is less than the agent’s tolerance threshold. Otherwise, it goes on the REJECT list. In the case of IA, an agent’s disposition is to put it on the ACCEPT list. But what happens when the case isn’t close enough to the center case?¹⁶ The reflective process is triggered. The agent considers what changes it can make to its center rule in order to accept IA cases or reject IR cases. Changes to the center case are constrained by cases already on the ACCEPT and REJECT lists. If the agent can change the center case and all classified cases can stay where they are, then the agent makes the change. So our agents will make small changes to their idea of a fish provided that they don’t have to reclassify any animals.

If a case is tagged IA (or IR) but is too far from (or too close to) the agent’s center case, then the IA tag is replaced with an NI tag and put on the POSTPONE list. POSTPONE is motivated by how things seem to go in our own experiences. People might sometimes have an intuition about a case but it would require reclassifying what’s already settled. In these cases, people might put off consideration of that case until a later time, when principles and cases are more firmly settled.¹⁷

As we have seen, an agent keeps track of four lists: UNCLASSIFIED (where all the cases start out), ACCEPT, REJECT, and POSTPONE. To reach intra-personal convergence there are three conditions to be satisfied: (i) every member of ACCEPT falls under the extension of the agent’s principle, (ii) every member of REJECT falls under the complement of the principle’s extension, and (iii) UNCLASSIFIED and POSTPONE are empty. We say a pseudo-convergence is obtained when conditions (i) and (ii) are satisfied, but cases remain in UNCLASSIFIED or POSTPONE.

2.3 Precedent

In Sects. 2.1 and 2.2, we described our operationalization of the reflective equilibrium process for individual agents. Now we can describe the last piece: precedent.

We conceptualize the role of precedent in the process of reflective equilibrium as a tie-breaker in conflicts of between cases and principles. In our fish case, suppose you have the intuition that a dolphin is a kind of fish but your current principle for classifying animals as fish doesn’t admit dolphins into its extension. The precedent set by other agents helps you figure out what to do. If they judge that a dolphin isn’t a fish, then perhaps you revise your initial intuition about a dolphin being a fish, leaving your principle untouched.

Whenever an agent classifies a case, it “announces" it to its neighbors. If the neighbors have already classified the case, then they ignore the announcement. If they have not already classified the case, then there are twelve possibilities to consider.¹⁸ These are mapped out in Table 1 and motivated by assumptions we discuss shortly.

Table 1 The outcomes of how an agent balances a precedent against their own intuition and rule about a case

In each case, the neighbor does not actually classify the case. Once the precedent case has been considered, the case is put back into its original spot in UNCLASSIFIED or POSTPONE. In this way, neighbors hear how someone else has classified the case and makes the corresponding changes to the center case or intuition label but handle the case when they get to it in their own time.¹⁹

2.4 Background assumptions for our representation of RE

Any model has to make assumptions about how to represent the target system. Here are ours. We classify these as armchair assumptions.

Armchair convergence When agents reflect in their armchairs, the method of reflective equilibrium converges in a finite time so that (i) every case in the domain is classified as accept or as reject, (ii) the extension of the rule includes all the accepted cases and the complement of the rule includes all the rejected cases.

Armchair conservativity The rule is updated conservatively. Agents make as few changes as required for the rule to be congruent with the intuition on the case.

Armchair conservation The rule is updated to be congruent with the current intuition only if that change maintains congruency with cases it has classified in the past.

It is worth noting that the model so far also satisfies armchair contingency. It is an open question whether a unique equilibrium is inevitable as the method of reflective equilibrium is practiced. The armchair convergence principle guarantees that a reflective equilibrium is reached (and in this sense is inevitable), but nothing about the domain of cases nor the reflective procedure guarantees that the equilibrium is unique. That is, if an agent were to run the reflective procedure again from the exact same starting point, it is possible they land at a different equilibrium (as measured by their rule at the end of the reflective procedure).

2.5 Precedent assumptions

Here are our assumptions concerning interactions between agents. Agents need to balance three considerations: their own intuitions and rule (as above) as well as a precedent given by an interlocutor. We call these precedent assumptions.

Indirectness precedent never classifies a case directly for an agent. At most, a precedent will invite an agent to suppress its intuition or change its rule (“invite” because the against is constrained by the armchair assumptions above).

Externalism A precedent is not a stand-in for an agent’s own intuitions. As such, a precedent can never create or eliminate an intuition, though it can suppress them (e.g. replacing the IA tag with an NI tag).

Tie-breaking When an agent reflects on precedent and its own commitments, it uses precedent to break ties in deliberation. When an agent’s intuition about a case and its rule are incongruent, then the agent is invited to “change” whichever of the two is also incongruent with the precedent.

Precedent nudging In the absence of an agent’s own intuition, a precedent that is incongruent with an agent’s rule will invite the agent to change its rule.

These assumptions together form what we call soft precedent, as they place constraints on how much precedent influences the reflective process. These assumptions about precedent yield the outcomes listed in Table 1 for how an agent balances the different contributions of their intuition, rule, and precedent of a case.

Note that the precedent procedure is guaranteed to terminate. To see this intuitively, notice that at the heart of it lies the armchair reflective procedure, which will terminate for each agent eventually. When we consider the outcomes in Table 1 we notice that the precedent procedure respects armchair conservativity and armchair precedence. Hence each agent continues to make progress towards their own intrapersonal convergence - precedent never undoes any armchair work (potential interactions here are considered in the discussion section). Rather, precedent “pushes and pulls” agents towards certain rules that are still accessible to them given how far along they already are in their own armchair process.

2.6 Interpersonal networks and model flow

In addition to how an individual agent balances precedent, intuitions, and rules, we also need to consider the reach of precedent by specifying who interacts with whom. There are two extremes. One is where everyone is directly connected with everyone else. In effect, this is like a workshop in which everyone in a group will have multiple turns to present to the group. The other extreme is where no one interacts with anyone, effectively eliminating the idea of precedent entirely. There are many kinds of social networks between these extremes. We limit ourselves in terms of “degrees of separation” between agents—the longest path of connections between two agents. For example, in a network of seven agents, moving from a complete network to a 4-regular network increases the degrees of separation from one to two, and moving to the ring network increases this to three (see Fig. 1). The effect of precedent will be null in the empty network and increase as the degrees of separation decrease in considering the ring network, the 4-regular network (aka “ring2”), and then the complete network.²⁰

There are several ways of thinking about how agents could take turns in the reflective process. Given that we are thinking of interpersonal convergence as a kind of “intermittent” or “rest stop to rest stop” collaborative process while agents are also doing their own armchair work, we opt for the following.

Stated first informally: An agent is selected at random, picks a case from its UNCLASSIFIED list, and goes through its reflective procedure (see Sect. 2.2 above). If the results of its reflective procedure on that case ended in the case being accepted or rejected, that classification becomes a “precedent” and the agent announces it to its neighbors. Each of the neighbors then uses the conditions specified in Table 1 to decide how to update, if at all, based on the precedent just announced for that case. After all the neighbors have been invited to update based on the announced precedent, another agent is selected at random and the procedure is repeated.

Stated more formally: A round begins from the set of agents who have not yet reached intrapersonal convergence. From this set an agent $A_m$ is randomly selected. $A_m$ then picks a case $C_n$ from its UNCLASSIFIED list. It assigns $C_n$ to one of the ACCEPT, REJECT, or POSTPONE lists using the method described in Sect. 2.2. If $A_m$ accepts or rejects $C_n$, it “announces" this result to all its neighbors, $N_{A_m}$, where $N_{A_m}$ is determined by the network structure (e.g. in the complete network $N_{A_m}$ is everyone (minus $A_m$ itself) and in the ring network $N_{A_m}$ is the agents to the left and right of $A_m$). Each neighbor $A_n \in N_{A_m}$ who gets the announcement then searches its UNCLASSIFIED and POSTPONE lists. If $C_n$ has already been classified by $A_n$, then $A_n$ is done for the round. Otherwise, $A_n$ is faced with one of the twelve possibilities in Table 1. After each $A_n$ has done their classification the round comes to an end and a new one begins (unless all agents have reached intrapersonal convergence).

Fig. 1

The four types of network structures used to analyze the impact of precedent, selected on the basis of their diameters (the longest shortest path between two agents). From left to right: In the empty network there are no links and thus there is no precedent, which serves as a control. In the ring network two agents are connected by at most three “degrees of separation”. In the 4-regular network (“ring2”) there are two degrees of separation. In the complete network there is one degree of separation (everyone is connected to everyone)

3 The search after “truth”

Malebrancheans don’t get many shout-outs these days. This one goes out to them

In a standard model of the RE process there are two types of adjustments: one for commitments (type 1) and one for principles (type 2). These adjustments are made iteratively: (i) hold constant the principles and adjust the commitments, and (ii) hold constant the set of commitments and adjust the principles. We now have a model in which a third category of considerations, precedent, is involved. In this paper we are primarily interested in adjustments of type 2. Accordingly, our model has largely taken into consideration how precedent might be involved when using (“holding fixed”) commitments in an attempt to adjust principles. In many ways our approach resembles the use of intuitions and thought experiments in the analysis or engineering of concepts, and for this reason we’ve adopted ‘intuition’ as our term of choice.²²

Here our simplification of thinking of principles akin to prototypes will be of benefit. We will exclude the possibility of what we call dolphin dilemmas. A dolphin dilemma occurs when there is a case that “looks and swims like a fish, but ain’t a fish”. Resolving such tensions requires more sophisticated developments of concepts or rules than our model currently allows. We see it as an open question how it is that, e.g. biologists, manage to navigate a logic of discovery.²³ So we opt to make that logic of discovery transparent and in principle possible within the confines of our model. To do that we initialize our agents with a set of intuitions about cases such that it is a coherent set for which there is a “true” rule. We do not tell our agents, however, what that true rule is.

Avoiding dolphin-dilemmas still leaves us with important choices for how to guide our agents. Their task is to try to converge on a single, pre-determined string as the center case, but each agent begins with a center case selected at random. For ease of exposition, our agents are looking for the string [11111]. In addition, we will always assume that the true rule and any rule agents consider have a pre-set tolerance level of 0.2, i.e., the rule extension includes the core case and all cases that differ from it by one digit. Without any further guidance our agents would bumble through the dark looking for the right answer. Even by seeding cases with a coherent set of intuitions we still have to make some important choices for how we do this. We choose two ways to seed agents.

The first is a bare seeding. For strings that are five bits long, there are 32 candidate cases. It is a matter of fact that only three intuitions are needed to uniquely determine our true rule. With a tolerance of 0.2 — meaning that they accept strings that differ from the chosen string in one place, which again we assume throughout — we seed with [11110] and [11101] as IA and [11100] as IR. We assign all other cases NI.

The second is a maximum seeding. For all 32 strings, we assign IA to all strings that differ from the target string by one bit. All other strings get IR.

Now we have another decision: how are these seeded cases distributed among the agents? The seed cases could be distributed over the entire population or we could assign them to some agents and not others. We choose to do this at random. But again a choice: how many copies of these sets of seeded cases should be distributed? Suppose we had one copy in the bare seeding scenario. That means three cases are distributed among agents. Even if we have only seven agents, that’s very little signal. But we could have two, three, or four copies of those seeded cases distributed among agents and increase the signal to noise ratio (where “noise” includes an absence of evidence, i.e. “no intuition”). The upper bound of copies is the size of the population. We call this signal-noise ratio token saturation. We will set our population size to be 7 agents. Thus we can consider token saturation levels ranging from 7 (every agent has a copy of the seeding) to 1 (only 1 copy of the seeding is dispersed across the population, with possibly one agent having all copies).²⁴

We thereby recast type 2 processes of adjustment (in which commitments/intuitions are “held fixed”) as a kind of constrained search that a small group of agents engage in. There are lots of ways to make this process more complex, but we now have sufficient detail on a sufficiently simple and transparent model for us to consider how the assumptions fit together. It is time to look at some results.²⁵

4 Results

We begin with the bare seeding case and the two extremes of our network structures. The outcome we’re most focused on is what proportion of the largest clusters converge on the truth, as if the ultimate decision of the group would be made by plurality voting. But note that the largest cluster is largest relative to a run of the model: it’s entirely possible that the largest cluster arriving at the true rule has only 1 member. This, in fact, is what we find: the mean size of the largest cluster finding the truth is 1.66 with a standard deviation of 0.84. So when there’s relatively little signal, it tends to be individuals and pairs who find the true case.

Fig. 2

For the bare seeding condition, we see that connectivity makes a big impact when saturation levels are high

Notice in Fig. 2 that as token saturation decreases, fewer of the largest clusters converge on the true case. This makes intuitive sense: the signal is harder to track as it gets weaker. We can likewise see that connectivity makes a difference, at least as the ratio of signal to noise grows. Connectivity makes less of a difference as the signal grows weaker. We find, then, an interaction effect with respect to token saturation and network structure: the effect of decreasing the signal to noise ratio is magnified by a lack of connectivity, though even the interactions differ. Notice that the slope for the empty network is roughly straight while the curve for the complete network is convex. We find drops in the proportion of largest clusters converging on the truth, but the drop-off is more dramatic when connectivity is maximized.

Fig. 3

For the max seeding condition, we see that connectivity makes little difference when saturation levels are at the extremes. For both networks, the proportion landing on the true string drops quickly once saturation levels hit five

There’s a similar story to be told for the max seeding condition, pictured in Fig. 3. The average size of the largest cluster converging on the truth is 4.16, with a standard deviation of 2.07. So when agents’ intuitions are entirely truth-tracking, then the majority tends to land on the truth.

As token saturation decreases, so does the proportion of largest clusters converging on the truth. Both curves in this case are concave and decrease at similar rates as token saturation decreases. When there is complete token saturation, the presence of a network makes practically no difference.²⁶ At the other end of the saturation scale, the network makes some difference. For every largest cluster that converged on the truth in the empty network, there is 1.2 clusters that so converged in the complete network. The tipping point is at a token saturation of six (i.e. six complete sets of intuitions were distributed among seven agents). Once saturation goes below that value, then, we see network structure making a difference.

Fig. 4

This plot, featuring all networks, shows the absolute proportions of all converged simulations as connectivity and saturation levels decrease

In Fig. 4 we see the differences in network structure as we move from unconnected to completely connected. Consider first the bare seeding facet. If we look at a token saturation level of seven, then we find marginal increases in largest clusters arriving at the true string as connectivity goes up. As saturation levels decrease, the marginal contribution of network structure likewise decreases.

In the case of max seeding at a token saturation level of seven, network structure makes no marginal contribution: the largest cluster always converged on the truth. The largest marginal increase is always from the empty network to the ring. The marginal increase peaks at saturation levels three and four and then shrink as the signal to noise ratio decreases. See Appendix 1 for the relevant numerical values.

A word of caution in interpreting this final plot (i.e., Fig. 4). If you, like one of us,²⁷ are tempted to read across seeding conditions: don’t. You might notice, for instance, that the area for the complete network under the bare determination at saturation level seven is the same as the area for the complete network under the max seeding condition with a saturation level of two. (Go to the Appendix for details.) What can we infer from this? It’s hard to say because the information available to the agents in the different seeding conditions function differently. In terms of the sheer volume of information—if we were to count up the number of strings with the IA and IR tags—max two has 64 strings to distribute among seven agents and bare seven has 21. But it’s not just a matter of counting strings. There’s redundancy in both cases: in the bare case, it’s three strings copied again and again. In the max case, it’s 32 strings copied twice. In the bare case, the “essential" strings for determining the truth are copied and distributed. In the max case, it’s those essential strings plus 29 others that are redundant. The remaining 29 cases confirm what is already implicit in the initial three: that the true string that fits with the assigned intuitions is [11111]. There’s redundancy in both cases but different kinds of redundancy.²⁸ We hope to explore this in later work. For now, it’s less wrong²⁹ to treat the bare and max seeding variables as categorical rather than continuous.

5 Discussion

Let’s take stock. We looked at how intuitions could be used by a group of agents to make adjustments to agents’ rules at an intermittent stage of a reflective equilibrium process. We called this intermittent stage a rest stop - it is where all agents use a starting set of intuitions to step-wise classify all cases into (proposed) accept or reject lists, i.e. each agent reaches intrapersonal convergence. We supposed that the intuitions would uniquely determine the correct rule. We also supposed that the rule was sufficiently simple (center case plus a similarity threshold) so that the logic of discovering the rule is transparent and possible given our setup (and traceable by us as the modelers). We then explored the effects of assumptions regarding: (i) different ways that intuitions might be distributed in the group, (ii) a relatively weak characterization of precedent, and (iii) different levels of how wide-spread precedent could be (network types). Our primary measurement was how similar the largest sub-group’s rule was to the truth, i.e. how close did the group come to interpersonal rest stop convergence?

We saw that network structure makes a difference in reaching rest stop interpersonal convergence. This seems intuitive from the armchair. Less intuitive is that it’s one factor among many: even a complete network doesn’t guarantee it, even when all agents achieve intrapersonal convergence. What in the model might explain this?

Here’s one possibility. The precedent-consideration process, as we’ve implemented it, is a push-and-pull process. When agents consider precedents, they are given opportunities to adjust their rules. Each opportunity is a chance to increase or decrease interpersonal convergence. When changing its center case, the agent has to choose what site(s) to change to accommodate the precedent. This can have a diverging effect. At the same time, increased connectivity spreads information, so more agents know that another agent accepted or rejected a case and use that to inform their own classification. So increased connectivity has two effects: there can be a diverging effect from an increased number of choice points (since every agent can consider cases multiple times), and there is also the converging effect of sharing information about how a case is classified.

This push-pull tension is resolved, however, when we flood the group with intuitions. Specifically, we need to both include an intuition about every possible case, and every agent needs to have these intuitions. This is the max seeding condition with complete saturation: every agent has a complete set of intuitions which unambiguously point to the truth. To our eyes this is an implausible assumption: not only do many of us have different intuitions than others, but many of us don’t have an intuition about a case when others do (and hence one reason why we might have rules to classify cases where intuition is lacking). In addition, flooding the group with intuitions removes any need for interactions between agents, including precedent. It’s when we relax this “flooding” assumption - which we showed can be done in two different but related ways, i.e. bare seeding and saturation - that we begin to see the role that precedent can play, namely increasing the chances of reaching rest stop interpersonal convergence.

There are numerous directions to pursue from here. We deliberately picked a group size of seven because the three non-empty regular networks had the convenient property of increasing degrees of separation by one (not to mention smaller groups are computationally tractable when having to compare their different lists). Real groups are not only bigger, but are less likely to have regular structure and more likely to be scale-free, clustered, or small-world. We predict that relaxing our modeling choices here would make rest-stop interpersonal convergence less likely rather than more likely. Nevertheless, exploring models with larger group sizes and different network structures would be one interesting direction.

Another direction would build on the possibility that our representation of real world precedent is too soft. Here are two fruitful options to “harden” precedent. Consider again Table 1. At lines five and 12, agents do nothing when there is no intuition. We might strengthen the role of precedent by having our agents develop an intuition in these cases where there was no intuition before. Or they might even classify the case based on the precedent. Early work suggests that these small differences in how to think about precedent have significant outcomes in the model.

Another fruitful option connects to our previous point about network structures. One could, for example, enable hierarchies of agents in the group. We might designate one or two agents as “experts” with others in the group updating their rules in line with the experts. The expert-novice dynamic in this model would end up being one in which the experts’ reasons effectively take the place of one’s own reasoning (cf. Zagzebski, 2012). Less drastically, there may be ways to think of the role of deference or outsourcing in our reasoning (Rabb et al., 2019; Levy, 2021).

Putting such possibilities to the side, the main takeaway is this. To the extent that it is possible to reach interpersonal convergence in the reflective equilibrium process, there are a host of details that need to be fleshed out. Some of these details can be gleaned from actual applications of the process (see Rechnitzer, 2022 for a recent example). Some are to be explored using formal models that highlight tradeoffs we’re likely to face (see Beisbart et al., 2021). We see our contribution as a first step towards understanding how a social mechanism like precedent would contribute to the reflective equilibrium process, and it’s not nearly as simple as one might think, even with the numerous simplifying assumptions that allow us to model the process.

In particular, we see there is a tradeoff with respect to historical contingency and convergence. As simulated agents are more highly connected, they are more likely to converge on a single rule—and possibly irrespective of whether that rule is the “truth." But how simulated agents arrive at that rule will be historically contingent—things could have gone otherwise to varying degrees, even if starting places were the same. Since the model is highly constrained towards favoring interpersonal convergence, this makes it a live possibility that collectively reasoned conclusions in actual practice could have turned out otherwise.³⁰ To the extent that precedent plays a role in actual practice then, we see it as placing a constraint on the reflective equilibrium literature that ought not to be ignored.³¹

Appendix A: Detailed view of interactions

Here we provide numerical values for the plot in Fig. 4. The first is the bare seeding condition (Table 2).

Table 2 The proportion of largest clusters that arrived at the truth under bare seeding conditions

Table 3 lists marginal increases for the bare seeding conditions. Notice that as token saturation decreases, the marginal increases across successive pairs of network structures consistently decreases.

Table 3 The marginal increase in largest clusters that arrived at the truth under bare seeding conditions by increased connectivity

Next up, we provide values for the max seeding conditions, again looking at the proportion of largest clusters that converged on the truth (Table 4).

Table 4 The proportion of largest clusters that arrived at the truth under max seeding conditions

Finally, Table 5 lists marginal changes as connectivity changes. Notice that marginal changes are smaller or nonexistent at the boundaries of saturation. Marginal changes swell at levels three and four, only to shrink as the saturation level decreases to one.

Table 5 The marginal increase in largest clusters that arrived at the truth under max seeding conditions by increased connectivity