This competition entry has been selected for publication by the Forum team.
Summary
For expected value maximization, indeterminate beliefs allow for a situation where multiple undominated objects of choice have overlapping expected value intervals. The key difference from this to having a set of objects of choice which the agent is indifferent between is that the agent cannot think of those objects of choice as being equally good. This makes them clueless, instead of just indifferent, between which object of choice to take from the undominated set. The situation is particularly relevant to impartial consequentialists because the undominated set is often considerable. This leaves the agent with very little, if any, action guidance and invites considering further ways to gain action guidance.
One possible avenue to resolve cluelessness is to narrow down the consequences that the agent considers to what their epistemic access allows for to find a single or just a smaller set of undominated objects of choice. This sentiment can be made precise as maximizing welfare in relation to the maximal sets of beneficiaries with determinate sign of impact. This is called Top Down Bracketing (TDB) (Kollin et al. 2025)[1]. The objects of choice are still overlapping for the original EV so this is not merely an extension of being impartially consequentialist but is instead best thought of as a distinct theory as is pointed out by Clifton. This theory uses consequentialism as the base and asks the consequentialist to also bring a way to index locations of value.
Another suggestion is to find the largest set of normative views that still give action guidance in aggregate. This sentiment is made precise by metanormative bracketing (MNB). It uses a set of weighted normative views for aggregate voting. This theory asks the agent to bring a set of normative views, a weighting and an aggregation rule. It does not rely on consequentialism as the base.
A third suggested avenue is to simply break the ties left by consequentialism with other normative reasons. For example, the EV maximizing agent could consider some other consequentialist or non-consequentialist values or by the virtuousness of choosing each object of choice. This text aims to make this sentiment precise via lexically ordered normative views and calls it lexical filtering. This theory is applicable to agents with a totally ordered set of utility functions (or merely ranking rules in the abstract version).
This text looks at the scenario at two levels of abstraction:
Lexical filtering might offer an overall less arbitrary framework than existing approaches because it does not rely on choosing among different ways to represent something whereas both TDB (locations of value) and MNB (topology of normative views) do. This is because lexicality is defined by the possible discontinuity (for example read as categorically unacceptable tradeoffs between values) present in the space of normative views. Discontinuity is a fact about the values and not a modelling choice. Of course, it can be that the space is fully continuous in which case there would only be one lexical rank. Another attractive property of LF is that it does not need to discount or ignore anything whereas TDB works by discounting some part of the locations of value and MNB works by discounting some normative views. -filtering as explored in Appendix C does discount.
Chapter 1 discusses lexicality in general and justifies why endorsing lexicality is not irrational outright. Chapter 2 introduces the matrix representation that falls out of combining common ways to model lexicality and imprecise beliefs. Chapter 3 explores which way of working with the representor is more justified. Chapter 4 specifies the intuition behind LF, introduces the tie-breaking principle and formal LF. Chapter 5 explores conflicts with three intuitive principles, and the dynamic inconsistencies two of them allow. Chapter 6 considers the applicability of the reasoning in this text to bounded and unaware agents like humans. Finally, chapter 7 concludes the text. Additionally, Appendix A discusses why most of this text assumes an ideal agent and the used notation. Appendix B explores how the matrix representation expands to accommodate for within-rank normative incompleteness and Appendix C lays out a tentative way to use lexical filtering for something similar to TDB, together with my reason for thinking it fails at what it is trying to do.
Take a classical utilitarian who is not lexical (i.e. satisfies continuity/Archimedeanity). If there exists no acceptable trade-off between some other value besides welfare, the agent cannot have a preference for that value. For example, to prefer a more beautiful world over a less beautiful one all else (crucially welfare) being equal, the agent would need to be willing to sacrifice at least an arbitrarily small amount of welfare for some amount of beauty.
Also, a non-lexical negative utilitarian cannot have a preference for more positive welfare if they wouldn’t sacrifice some amount of suffering for it. They for example cannot prefer an unbothered existence to non-existence. A non-lexical agent who prefers to live longer cannot prefer red socks to green ones if they wouldn’t sacrifice some probability of immortality to make that choice, as is pointed out by Academian.
For ideal agents lexicality is essentially “free” in that incorporating a secondary value function to the agent’s current value structure has no ex ante impact by the agent’s current lights and the only impact is to the added value function. For an example of this, consider a classical utilitarian who finds themselves sympathetic to some form of virtue ethics. They never want to violate utilitarianism but also do not want to be unnecessarily lacking in virtue, or to put it another way, they think utilitarianism is infinitely more important than virtue, but that virtue is still important. A natural way to represent this would be that the agent considers virtue lexically after utilitarianism. This allows the agent to choose an object of choice that is ex ante optimal on utilitarianism without compromise but also choose the object of choice that is more virtuous among those that utilitarianism deems equally optimal. This notion of “free” lexicality is challenged for LF with dynamic choice (more).
At least to me, these examples highlight the intuitiveness of lexicality. I might not be willing to sacrifice any amount of positive welfare for more neutral welfare, but I still prefer worlds with more of neutral welfare ceteris paribus. Nor would I trade any welfare for beauty, but I still prefer worlds with more of it ceteris paribus.
Lexicality finds a purpose in minimizing arbitrariness in choosing among options to the extent that the agent has access to the lexical structure of their own values. Personally, I am able to produce a set of nine rankings before I don’t have strong intuitions as to whether there exist acceptable trade-offs and which direction those go. A more comprehensive discussion on ethics and metaethics is outside the scope of this text (see Bergman 2025 for a recent discussion on both the ethics and decision theory of unacceptable tradeoffs in longtermism). Nevertheless, the formalized tie-breaking offered by lexicality seems like a uniquely reasonable property in the option space of morality to reflect on, especially when one often faces multiple options with equal or comparatively indeterminate EV in regard to the value one primarily cares about.
With lexicographic utility we have a totally ordered set of utility functions (Hausner 1954; notation from Fishburn 1971):
where and each is a standard complete and Archimedean utility function. The criterion for having many different utility functions is if there exists no acceptable trade-off between what they map value to, which is also called non-Archimedeanity.
This is normally combined with a precise probability distribution to calculate a utility vector:
For choosing between two objects of choice, the calculation can be stopped at the first differing dimension which rules the preference. This gives a simple determinate answer of preference or indifference due to having a totally ordered set.
For incomplete beliefs we have a set of probability distributions called the representor (Bradley 2017, §11)[2]:
where each is a precise complete probability distribution and is not ordered.
Combining the utility vector with the representor, the expected utility forms a matrix:
For illustration, here is a comparison of how much better A is over B:
It is at least not immediately obvious what to do with this matrix. We know that for choice worthiness the upper rows are strictly more important than lower ones within each column but there is no rule to compare the columns. The key question is what to do if different beliefs give different preferences under the same utility function (i.e. different signs on the same row). Let’s first think of this through the comparison of just two objects of choice. I've thought of two possible rule candidates:
We see that the decision guidance of this extension of lexical utility is not self-evident. In lexical utilities, the agent starts to consider the next utility function when multiple objects of choice don't differ in terms of expected value for the more primary utility function . This is not the same as having an indeterminate sign of difference, so we need a new assumption for interpreting indeterminacy.
Toy Example: With two probability distributions and two utility functions comparing EV of object of choice A to B we could have .
An agent using SLD would find the incoherent preference under and declare the preference indeterminate and take either of A or B. An agent using LF would look to and find that A dominates B there and then choose A.
Other remarks:
Let’s think about this in terms of the whole set of objects of choice (S) instead of just A and B. Say that an agent has more than one undominated object of choice (let be this permissible set) under their primary utility function . This includes capacity building such as doing more research and there truly is no single best or set of equally good objects of choice the agent could take to maximize the EV of over all other objects of choice. The agent can suspend judgement on those objects of choice or consider their secondary utility function to possibly narrow down the current permissible set. Which option is the agent more justified in choosing by?
If we think of any object of choice in as being as good as any other object of choice in the answer is clearly to consider (this coincides with the ceteris paribus intuitions). Again, the objects of choice are not as good as each other and are instead indeterminate compared to each other. This also means that by it is indeterminate whether narrowing down with is worse.
Then again, why wouldn't the agent consider ? They still care about their lexically secondary utility function and have no reasoning to not consider it. Considering cannot give an ex ante determinately worse object of choice than would be chosen with just . It would be strange to go with an object of choice that would not be permissible under when we have no reason from to do that. To me this is enough justification to consider narrowing permissible objects of choice further on to under indeterminacy as a superior option. (The conflicts section introduces some challenges for “why not?”)
Here is my attempt at specifying the intuition used in the previous chapter and defining it into a principle that can stand on its own merits:
Tie-breaking intuition: “When more important reasons do not determine which option is better, less important ones can be used.”
Tie-breaking principle: For a set of objects of choice that are permissible at rank i, the objects of choice are compared at rank i+1, if such exists, and the permissible set narrows to those not determinately worse than another at that rank.[3]
This can be formally coded as iterated maximality over a well-ordered sequence of relations:
Let when the comparison between and comes out determinate at rank in favour of . Assume has no infinite ascending chains on the menu. Let the ranks be indexed by well-ordered . The iteration is:
where is the full menu.
For successor ranks : .
For limit ranks : let and . For infinite menus, assume is non-empty.
The verdict is at the greatest rank if has one, and otherwise.
This is a more abstract and general principle than what is used in the rest of the text. The representor with lexical utilities is only one instance of the rule: holds exactly when for all , with strict inequality for at least one. This makes the principle applicable for agents who cannot or do not compute expectations at all as long as their values induce determinate comparisons rank by rank.
SLD needs an additional relation for indifference because just a relation that does not ascend infinitely need not distinguish a tie from incomparability, since both show up as neither nor . In this sense LF asks less of each rank.
I am aware of three principles that seem to stand on their own merits but are incompatible with LF. These are the sure-thing principle, Sen’s alpha and acyclicity. Alpha and acyclicity violations arise because the rank at which a comparison resolves is not fixed by the options being compared. Any rule that tie-breaks on indeterminacy has this because indeterminacy can be intransitive.[4] The ranks can disagree arbitrarily, so any shift in the resolving rank can flip the verdict. The conflicts require both lexicality and the representor[5]: lexicality with a single probability distribution gives a total order, and representor dominance with a single utility function gives an acyclic relation. The violations are therefore direct costs of adopting LF over SLD[6]. STP has a different source, which is discussed below.
Thanks to Anthony DiGiovanni for finding that LF, and forms of bracketing, can violate this. See the original example and more discussion on violating STP here.
“If an option is most preferred given a proposition and given its negation, it should be most preferred even without knowing if the proposition is true or not”. This intuition is formalized as the sure-thing principle (STP). A violation of STP means that an agent is guaranteed to regret their ex ante decision no matter which side of the partition they are on. They would therefore trade some value to not learn which partition they are in. This then violates an intuition that value of information is never negative.
Consider the following example. Let both members of the representor assign probability 1/2 to G. Let . Each entry contains in three information states: (ex ante, given G, given not-G)
| (4, 10, -2) | (4, -2, 10) | |
| (-1, -1, -1) | (-1, -1, -1) |
Ex ante, resolves and gives A. Both given G and not-G, is indeterminate. LF considers which resolves and gives B and therefore according to STP, LF should also give B ex ante.
The comparison being determinate ex ante and indeterminate in every cell of the partition is the general condition for violating STP. Here the only way this can happen is if the representor has larger credal intervals after conditioning. This is called dilation which is a standard phenomenon for imprecise credence. It is also the usual reason why imprecise models can have negative value of information (Bradley and Steele 2016). Allowing for incomplete u, there is at least one other way LF can meet the general condition (see Appendix B for incompleteness). DiGiovanni gives an example of this with a version of opaque sweetening.
To illustrate value trading in an environment with sequential choice and learning (these are not assumptions the model makes in general), suppose the agent has a choice of taking information about whether G is true, and not taking it costs some utility. Suppose information gathering is an object of choice like any other; they evaluate it under first: without taking it, the future agent assumes probability x of G and in the choice immediately following would take A. Taking it makes the future agent take B regardless of which way the information goes. So and , and not taking is determinately better whenever the cost is less than the infimum over the representor of , which is 4 in this example. This payment is a dynamic cost similar to the pump in the acyclicity section and can be prevented in the same way (see the next subsection).[7]
SLD does not consider under indeterminacy so just gives , which here is equivalent to just with a representor. It vacuously satisfies STP and has no guaranteed regret or negative value of information. It does violate a weaker intuition that incomparability given a proposition and given its negation should be preserved ex ante, though as DiGiovanni notes, this feels less plausible than STP itself. For LF, the violation is always from the perspective of a lower ranked value function against a higher ranked one.
“The option that is most preferred from a set of options should remain the same when other options from the set are removed”. This is also called the independence of irrelevant options and the intuition is formalized as Sen’s alpha: if is permissible in and , then is permissible in .
Consider the following example. Let . Entries are .
| A | B | C | |
|---|---|---|---|
| (2, 0) | (0, 1) | (1, 2) | |
| (0, 0) | (5, 5) | (-1, -1) |
Under , C beats B on both distributions, so B is eliminated. A is indeterminate against both and survives, leaving . A dominates C under so the verdict is A.
Restriction to : under the two are indeterminate, so both survive. Under , B beats A. The verdict is B.
So A is chosen from the larger set and not chosen from a subset of it because C removes B before is consulted. Removing C makes B survive to the second rank, where it wins.
I do not think this is much of a problem on its own. To me, the intuition behind Sen’s alpha is trying to say that removing options that do not matter should not affect the choice. Crucially, not mattering is different from not being the most preferred (being in the final permissibility set) in the unrestricted set. In the example above, C does matter as it is the reason for to not permit B. In general, LF does satisfy the motivating intuition behind alpha if we accept that eliminating other options qualifies an option as mattering.
If we also have the stronger version of the intuition that not being in the final permissibility set means an option should not matter, the example above represents a real conflict. The options that reach are by definition what did not consider dominated and dominance is relative to the menu. As stated in the preamble to this section, a rule that avoids this does not tie-break indeterminacy.
SLD satisfies alpha. It considers {A, C} permissible from {A, B, C} and {A, B} permissible from {A, B} so alpha only applies to A which is in both. I consider this to be essentially a vacuous satisfaction. SLD can still choose an option from that is not permitted for , but it is just not forced to.
Unlike the STP violation, a single alpha failure cannot cost . Any two options permissible in and in are indeterminate or equal under . This does not extend to multiple choices in aggregate, as the next section shows.
“If a is preferred to b and b to c, then a is not dispreferred to c”
Again, consider the following example. Let . Entries are .
| A | B | C | |
|---|---|---|---|
| (0, 0) | (2, -1) | (1, 1) | |
| (10, 10) | (5, 5) | (0, 0) |
In pairwise comparisons:
| S= | LF verdict | ||
|---|---|---|---|
| {A, B} | indeterminate | A | A |
| {B, C} | indeterminate | B | B |
| {A, C} | C | A | C |
So, C is chosen against A, A against B, and B against C which is a cycle. This also contains an alpha failure: B is permissible in , but B is not permissible in .
The cycle can be pumped. If each swap costs determinate value of such that subtracting from the comparisons doesn’t flip the choice, the cycle returns the agent to the start with determinate of . For this example, . None of the three swaps gives up ex ante value for value, but their aggregate does.
The cycle is inherent to LF's verdicts across menus such as in the example, but the pump requires the additional assumption that choices arrive as sequences. In such a decision situation, the pump can be blocked in the same way that Kollin et al. do for bracketing: (1) the agent can take complete plans as objects of choice and (2) adopt wise choice.[8] Fixing cyclicity itself would have to reject one of the pairwise verdicts, and each is just the tie-breaking principle applied to its own menu. LF cannot give one up without giving up the principle. Again, SLD vacuously satisfies the intuition. It leaves {A, B} and {B, C} indeterminate and so has no verdicts forcing it to cycle but it is still permitted to cycle.
Let’s assume a more explicit idea of an agent for this section. Let D be a set that includes everything the agent individuates. This can be thought of as the things the agent’s (internal) world consists of including abstract things like language or concepts of self. Assume the agent has some procedure for gathering the elements of D that are considered as objects of choice for a decision. Call that set S. Assume well-ordered I and the ranks indexed by I are elements of D. Let the information the ranks have access to be limited to D. Assume the agent has a procedure for LF using the ranks and the verdict of LF is the choice. D can also contain anything else.
Now we see unawareness as missing something in D that should be there. This can cause S to be lacking and the ranks to not have information that they should have. This can also cause missing ranks. For example, for a rank comparing with EV, the missing information could be some propositions or having coarse propositions. “Missing” and “coarseness” are relative to something. Having access to the reference point would mean not being unaware relative to the reference point. More precisely, having access to the witnesses of being unaware means not being unaware. The existential statement of any reference point for unawareness can therefore not be evaluated (as in finding a witness or not finding any witness for it).
Boundedness can be interpreted as a limitation on the extent to which procedures can be conducted. It can also be a limitation on how much information D can contain. If the agent has a procedure for generating awareness growth or refinement, boundedness also limits that. Knowing that one has stopped a procedure artificially does not need an additional reference point beyond the procedure itself.
Running procedures wrong in other ways than stopping short or having elements in D that should not be there is not considered since these are not boundedness or unawareness. Awareness refinement is not considered since that requires having propositions so is not statable at this level of abstraction. Especially refinement is still worth considering but outside the scope of this text.
Below is a list exploring the effects of boundedness and unawareness on S, the ranks themselves, the rank family and LF itself on the verdict of LF. Not having boundedness failures is detectable by running the procedure to completion but not having unawareness failures is not detectable.
Boundedness and unawareness do not affect the ability to use LF given that the agent can compare options with the ranks independently[9] and that there are only finitely many ranks[10]. That is to say that it is nearly trivially true that LF is applicable to unaware and bounded agents. There are two more interesting questions:
(1) Can a bounded and unaware agent be thought to be satisfying the intuition behind LF by using LF? To put it another way, does using LF as a bounded and unaware agent still amount to what LF was for?
(2) How does applying LF to bounded and unaware agents affect the conflicts with other intuitions?
(1) To me, true LF is for (i) comparing with rank i+1 exactly when rank i is indeterminate between the options and (ii) using all the normative information one has so exhausting all the ranks. This is verified if all the options are actually compared properly meaning that failing to run any rank comparison procedure invalidates serving the original intuition. However, for bounded agents, indeterminacy seems better interpreted as not being able to find determinacy rather than just the negation of determinacy. Now the (i) purpose of LF would be “Comparing with rank i+1 when I have not been able to find determinacy with rank i.” This would justify ignoring parts of S for meta reasons. Stopping LF before all ranks are exhausted seems fine too. An agent is perfectly justified in using some meta reasons, like running out of compute time, to not bother consulting some parts of their lexically less important normative reasons and just using the action guidance they got from their more important normative reasons. The prerequisite for running LF such that it satisfies the original intuition is that S, the number of ranks and the rank comparisons themselves are such that the agent is not bounded with regard to the task. By interpreting indeterminacy and the use of normative reasons in a way that is more suitable to bounded agents, that or any of its parts are not required.
As said above, unawareness is relative to something. The agent can consider themselves unaware due to a track record of awareness expansion (e.g. for themselves or for similar agents) and they can consider unawareness to cause failures relative to a reference point due to witnessing failures relative to having current D as a reference point. As opposed to the failures caused by boundedness, it is not clear how the failures caused by unawareness relative to the reference point affect serving the intuition since discussing the reference point without having access to it is conceptually weird. This remains an open question.
One possible way for the agent to make unawareness concrete is to consider deliberate awareness growth as one object of choice and take previous unawareness as evidence for indeterminate comparisons for other objects of choice. Then, they could grow awareness if it is one of the undominated options.
(2) As mentioned in the failure list above, the alpha violations of LF can force unaware agents to choose an option the reference point does not consider permissible. Again, SLD does not violate alpha, so it does not force to choose against the reference point (but it is still permitted to do so). The pump and STP require propositions and sequential choices so assume the decision situation is now such. The patch suggested for avoiding pumps and paying for ignorance is quite demanding (requiring complete plans instead of just actions) and might not be available to a bounded agent even if they pass the prerequisite for the strict reading of the intuition from the previous question. Due to unawareness, the agent might not conceive of the pumping or paying for ignorance plans and fail to rule them out before getting into these. An acyclic and STP satisfying decision rule does not need to be able to conceive of a pump or paying for ignorance beforehand to not fall for them, so at least some part of these failures is attributable to LF.
Similar to the negative VOI example for STP violation, an agent can find that awareness growth (or refinement) was determinately worse by their previously less aware lights. They can therefore come to have evidence of awareness growth (or refinement) having negative value. This is not similarly statable as just one instance like it is with the negative VOI example since modelling what awareness growth (or refinement) would lead to requires already having that awareness.
To summarize:
I think these reflect the weaker standing of the bounded and unaware agent in general rather than LF amplifying it. Also, I still do not find LF to violate the things I value about the intuitions given the situation LF is trying to provide action guidance for. Therefore, even though LF for unaware and bounded agents is more inclined to violate the other principles discussed, I still do not consider LF to be disqualified for the violations.
I find the model of combining lexical utilities with imprecise beliefs satisfying and the tie-breaking principle seems very intuitive. Compared to MNB or TDB lexical filtering is a very conservative option that could also be described as the lame and intuitive solution to cluelessness.
LF violates STP and Sen’s alpha and can cycle but the most serious instances of these failures, the ways these could lead to sure value loss in dynamic choice scenarios, can be avoided in the same way that is done for TDB. LF still violates the intuitions behind these principles, but the intuitions do not seem to apply cleanly to the situations LF is trying to give guidance for. More precisely, these intuitions are most salient when one wants to be consistent when choosing for the same reasons, but lexicality is about choosing for many different (ordered) reasons.
The abstract formal version of LF is applicable to bounded at least when interpreting “not determinately better” in a way that is more suitable to bounded agents.
As explored in Appendix B, the model allows for normative incompleteness or more complex substructures within ranks. Appendix C formalizes a tentative way to make decisions in the spirit of impartial consequentialism when that itself gives more than one permissible object of choice. Crucially, however, the method faces an individuation problem like bracketing and might not actually serve the intuition behind it well.
Open questions:
Thanks to Jim Buhler, Jesse Clifton and Anthony DiGiovanni for comments on an earlier draft of this text. Thanks to Claude Opus 5 for editing suggestions, feedback and discussion on the ideas. Some sections started from drafts by Claude that I then rewrote. The text in this version is my own.
What do you mean by ideal agent?
By ideal agent I refer to a decision maker that has a complete awareness set and Boolean algebra over that set and is able to compute EV for each of those. The discussion in this paper is not sensitive to the expected utility theory the agent uses. For clarity, this paper still uses a classic way of representing expected value with separate and . To for example read this directly as JB EUT, read as .
Why would ideal agents just not be precise Bayesian EV maximizers?
It might be that ideal agents actually can find a uniquely justified single probability distribution such as is required for precise EV maximization. Still, my best guess is that even agents that are for example able to set Occam priors via Solomonoff induction might not arrive at such a privileged epistemic situation and instead have multiple possible distributions. The possible arbitrariness of Solomonoff resulting in a representor is not the only hurdle. DiGiovanni’s Clarifying “wisdom” highlights for example meta-philosophy, and Evidential Cooperation in Large Worlds as issues that I take to challenge ideal agents being precise Bayesians.
Why discuss lexical filtering from the perspective of ideal agents?
The matrix representation makes discussing LF more concrete and much of the discussion this text is aiming to contribute to happens with credences and utilities. It also seems at least somewhat compelling to be continuous with my idea of an ideal agent so I wanted to see if LF is applicable to it.
While utility functions most intuitively represent utilitarianism or other forms of consequentialism, other values can also be thought through lexical utilities as they allow for fundamentally unacceptable trade-offs present in e.g. some deontological views. The tie-breaking principle itself is also compatible with normative incompleteness.
Normative incompleteness within some utility function can be allowed for by giving a third dimension to the matrix at each utility function such that is a set of functions that form a set similar to the representor. Here, for some and , gives a determinate + / - / 0 iff every difference gives that, and otherwise gives an indeterminate answer. Completeness among the is still needed for the lexical ordering.
Should a preference between A and B under some be declared indeterminate if even a single difference is indeterminate or is that indeterminacy taken out of the count for a dominant object of choice? Consider an example:
Let's say I have found that under object of choice A (instead of B) improves my friendships but degrades my creativity but under both are improved and I consider these values incommensurable. Now I have a reason to think that the value of A over B is under some possible belief indeterminate and under another positive. As we are trying to find a determinately better object of choice, this combination leads to me considering the whole of A over B under this indeterminate.
This seems like a good way to consider incomplete utilities, allowing for incorporation of deontological rules and virtue ethics etc. without issue. To summarize this extended LF with a 3D matrix:
Expanding to a 3D matrix is a simple example of modelling a more complex preference structure but the model could accommodate any structure of lexical or incomplete utility functions by using the decision rules of
(1) resolving lexical sets at the first utility function that gives coherent preference, and
(2) resolving incomplete sets as indeterminate when they give incoherent preference.
For example this allows for a model where there is an incomplete set of five utility functions that all share the second lexical ranking and within those have a new lexically ordered set.
Caveat: There are probably better ways to do filtering based on arbitrariness of beliefs or at least this is not a very good one. Consider this a tentative exploration of a possible use of LF. Thanks to Jesse Clifton for motivating a redraft of this appendix and this note, and for the example below.
Summary: -filtering works by creating a series of secondary utility functions, , each of which is identical to but zeroes out the value of any proposition where the agent's credal interval conditional on an object of choice is wider than a threshold . Using this requires an atomic breakdown of the utility relevant propositions which I don’t have a solution for.
This appendix explores a way to formalize consequentialists considering only propositions they have some level of confidence in (interpreted as credal width) while ignoring the ones that they don’t with LF. Take an epistemically accessible subset from the awareness set which only includes the (maximally precise with respect to ) propositions where the interval is at most some in length and set other and [11]. As usual these propositions need to be maximally precise w.r.t. , but this introduces the problem of proposition individuation because the way the agent defines propositions affects the filtering.[12] This discounting of unsure propositions might seem simple in principle but (i) modelling this behavior is tricky because each only has an input from that and cannot consider the other outright to find these intervals and (ii) I do not have a good way to individuate propositions. Proposition individuation is a problem for other situations as well. For example, Bradley notes that the Principle of Indifference and MaxEnt are sensitive to how finely the outcomes are described (2017, §13.2). I provide a solution for the first problem, but the second one is left open.
Problem (i) is solved by ordering the computation such that computing the comparison first (that does not need interval widths) makes the interval lengths available to . This requires a constraint on one of the utility functions before [13]: the maximally precise prospects it considers need to be maximally precise for as well. Now it is also clear that this is not normatively purely consequentialist as it puts weight on the agent’s information in calculating utility.
Lexicality also allows us to consider a utility function with a less strict, larger to be infinitely better than a more narrowed down one, so that the agent decides on the least aggressive filter to give determinacy. The most intuitive option would be to index the filters by the continuum but that cannot be used since the continuum is not well-ordered as is required by the tie-breaking principle. The filters can be indexed by a countable descending sequence fixed in advance. The agent could still have utility functions to consider if the filtered ranks do not give a determinate object of choice even at the smallest .
To be clear, this formalization requires the agent to reason over maximally precise hypotheses which is not possible for bounded agents who can only consider vague hypotheses. For the purpose of a considering welfare, the propositions would be all the snapshots of the process of valenced experiences in space-time, which is much more precise than can be expected of a bounded agent.
Jesse Clifton proposed the following example: Let the utility relevant propositions be , and with , and , and let the representor give and . The unfiltered expected utility of A is . For any the second proposition is filtered out and the filtered expected utility is , which is the supremum of the unfiltered interval.
Another example: Take , of width . The unfiltered expected utility is while the filtered value, for any , is . A is assigned value that no endorses. Note that this is a fundamental contrast to TDB, which only ignores locations with indeterminate sign.
To me, these examples illustrate that the intuition -filtering is trying to serve is not about credal width and that it cannot be thought of as serving the real intuition. The real intuition seems to be: “Less arbitrary parts of beliefs could be lexically more important than more arbitrary parts and that could be used for filtering”. The arbitrariness of beliefs seems to have more to do with what those beliefs are based on rather than their numerical representations but I have not found a way to make this sentiment precise or action guiding.[14]
Kollin, Sylvester, Jesse Clifton, Anthony DiGiovanni, and Nicolas Macé. 2025. "Bracketing Cluelessness." Working paper, September 8, 2025. https://longtermrisk.org/media/Bracketing_Cluelessness.pdf.
Hausner, Melvin. 1954. "Multidimensional Utilities." In Decision Processes, edited by R. M. Thrall, C. H. Coombs, and R. L. Davis, 167–80. New York: Wiley.
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A note on use inside LF: Kollin et al. present Top Down Bracketing as a ranking rule, not a function, so it cannot be entered into the matrix. On the abstract rule, ranking rules in general can be used but due to possible cyclicity TDB specifically cannot.
(i) See DiGiovanni (2025) for a concise justification even for ideal agents. (ii) Bradley (2017) uses a set of states of mind (each S=(P, V)) but since here the normative lexicality is independent of the epistemic incompleteness we’ll use a different approach. The same could however be modelled as just having V be the constant set of lexical utility functions.
(i) An alternative principle that also chooses LF over SLD (put forward by Claude Opus 5): No Residual Dominance: “If a and b are both permissible, there is no rank at which a is determinately worse than b.” (ii) Levi (1986) uses a somewhat related structure, with maximin breaking ties left by E-admissibility. Compared to his method, LF's nonprimary ranks carry normative content, filter by maximality, iterate over an arbitrarily many well-ordered rankings, and do not discard information.
(i) This is also why top-down bracketing can cycle. (ii) Write for rank i when neither nor . If is transitive it is an equivalence relation whose classes are ordered by , and LF collapses to maximality under . That satisfies alpha and is acyclic.
For STP, lexicality and any way to have incomparability within a rank is enough and a representor is one way.
More generally, they do not occur for SLD because it descends only on exact indifference, which is transitive.
u1 sees that it should not let u2 make the decision at the later node but at that node it has no say and it is willing to pay to avoid this. Plans with wise choice is a way for u1 to override u2 at the node without having to pay.
Wise choice (Rabinowicz 2020) considers a plan permissible if (1) it is feasible, meaning that at no node that the plan reaches does another feasible plan that also reaches that node give a determinately better prospect, and (2) it results in a prospect that is not dominated by any other feasible plan. The pumping plan is infeasible since the null plan reaches the root node too and dominates. LF is applied to feasibility rank by rank (as opposed to applying feasibility to LF's verdict since it can cycle). At each rank i, the feasibility constraint only requires acting at nodes against ranks below i. Similar to what Kollin et al. concede for TDB, this is a weak form of resoluteness. (Resolute in that it requires commitment to the plan, weak in that the commitment comes from more important reasons constraining less important ones.)
This is a debatable assumption. On one hand, requiring LF to be usable with ranks that are unusable in themselves would require LF to change (i.e. misrepresent) the content of those ranks. On the other hand, one of the main motivations for LF is to be able to try to consult something like consequentialism. Trying to use unusable ranks requires some way to avoid getting stuck on them. See question 1 below for more on this.
This is a reasonable assumption for bounded agents.
There are also other intuitive ways to define epistemic access. For example, maximum relative difference (e.g. 1% and 0.01% would give a delta of 100 instead of 0.99%).
Notice that this directly violates the non-atomicity of classical Jeffrey-Bolker.
Most simply u1 but that isn’t required as the presented u2 could be at any point in the lexical set.
Hill's confidence rankings (2013) do something related by asking the agent to bring an ordering of probability distributions by confidence. A way to produce such an ordering in accordance with the intuition is unclear to me since serving the intuition requires separating the more and less arbitrary parts of a belief whereas each distribution contains all the parts.