Agents often face a choice of what to do. And it seems that, in most of these choice situations, the relevant reasons do not require performing some particular act, but instead permit performing any of numerous act alternatives. This is known as the basic belief. Below, I argue that the best explanation for the basic belief is not that the relevant reasons are incommensurable (Raz) or that their justifying strength exceeds the requiring strength of opposing reasons (Gert), but that they (...) are imperfect reasons—reasons that do not support performing any particular act, but instead support choosing any of the numerous alternatives that would each achieve the same worthy end. In the process, I develop and defend a novel theory of objective rationality, arguing that it is superior to its two most notable rivals. (shrink)

John DePoe has criticized the self-defeat argument for Phenomenal Conservatism. He argues that acquaintance, rather than appearance, may form the basis for non-inferentially justified beliefs, and that Phenomenal Conservatism conflicts with a central motivation for internalism. I explain how Phenomenal Conservatism and the self-defeat argument may survive these challenges.

This paper is the introduction to the volume. It gives an argumentative view of the philosophical landscape concerning incommensurability and incomparability. It argues that incomparability, not incommensurability, is the important phenomenon on which philosophers should be focusing and that the arguments for the existence of incomparability are so far not compelling.

The first part of the chapter surveys some of the main ways in which the Sorites Paradox has figured in arguments in practical philosophy in recent decades, with special attention to arguments where the paradox is used as a basis for criticism. Not coincidentally, the relevant arguments all involve the transitivity of value in some way. The second part of the chapter is more probative, focusing on two main themes. First, I further address the relationship between the Sorites Paradox and (...) the main arguments discussed in the first part, by elucidating in what sense they rely on (something like) tolerance principles. Second, I briefly discuss the prospect of rejecting the respective principles, aiming to show that we can do so for some of the arguments but not for others. The reason is that in the latter cases the principles do not function as independent premises in the reasoning but, rather, follow from certain fundamental features of the relevant scenarios. I also argue that not even adopting what is arguably the most radical way to block the Sorites Paradox – that of weakening the consequence relation – suffices to invalidate these arguments. (shrink)

Suppose that you prefer A to B, B to C, and C to A. Your preferences violate Expected Utility Theory by being cyclic. Money-pump arguments offer a way to show that such violations are irrational. Suppose that you start with A. Then you should be willing to trade A for C and then C for B. But then, once you have B, you are offered a trade back to A for a small cost. Since you prefer A to B, you (...) pay the small sum to trade from B to A. But now you have been turned into a money pump. You are back to the alternative you started with but with less money. This Element shows how each of the axioms of Expected Utility Theory can be defended by money-pump arguments of this kind. The Element also defends money-pump arguments from the standard objections to this kind of approach. This title is also available as Open Access on Cambridge Core. (shrink)

Derek Parfit was one of the most important and influential moral philosophers of the late 20th and early 21st centuries. This Element offers a critical introduction to his wide-ranging ethical thought, focusing especially on his two most significant works, Reasons and Persons and On What Matters, and their contribution to the consequentialist moral tradition. Topics covered include: rationality and objectivity, distributive justice, self-defeating moral theories, Parfit's Triple Theory, personal identity, and population ethics.

Joe Horton argues that partial aggregation yields unacceptable verdicts in cases with risk and multiple decisions. I begin by showing that Horton’s challenge does not depend on risk, since exactly similar arguments apply to riskless cases. The underlying conflict Horton exposes is between partial aggregation and certain principles of diachronic choice. I then provide two arguments against these diachronic principles: they conflict with intuitions about parity, prerogatives, and cyclical preferences, and they rely on an odd assumption about diachronic choice. Finally, (...) I offer an explanation, on behalf of partial aggregation, for why these diachronic principles fail. (shrink)

Philosophers have spent a great deal of time debating whether intransitive preferences can be rational. I present a risky decision that poses a challenge for the defender of intransitivity. The defender of intransitivity faces a trilemma and must either: (i) reject the rationality of intransitive preferences, (ii) deny State-wise Dominance, or (iii) accept the bizarre verdict that you can be required to pay to relabel the tickets of a fair lottery. If we take the first horn, then we have a (...) synchronic refutation of intransitivity, an improvement on widely criticized diachronic arguments. I sketch possible responses that may rescue intransitivity and argue that each response is possible but generates an explanatory debt. I conclude by showing how the challenge here clarifies the foundations of decision theory without transitivity, conditional on some explanatory debt being payable. (shrink)

A persistent argument against the transitivity assumption of rational choice theory postulates a repeatable action that generates a significant benefit at the expense of a negligible cost. No matter how many times the action has been taken, it therefore seems reasonable for a decision-maker to take the action one more time. However, matters are so fixed that the costs of taking the action some large number of times outweigh the benefits. In taking the action some large number of times on (...) the grounds that the benefits outweigh the costs every time, the decision-maker therefore reveals intransitive preferences, since once she has taken it this large number of times, she would prefer to return to the situation in which she had never taken the action at all. We defend transitivity against two versions of this argument: one in which it is assumed that taking the action one more time never has any perceptible cost, and one in which it is assumed that the cost of taking the action, though (sometimes) perceptible, is so small as to be outweighed at every step by the significant benefit. We argue that the description of the choice situation in the first version involves a contradiction. We also argue that the reasoning used in the second version is a form of similarity-based decision-making. We argue that when the consequences of using similarity-based decision-making are brought to light, rational decision-makers revise their preferences. We also discuss one method that might be used in performing this revision. (shrink)

This paper uses the idea of a proxy, which figures in discussions of bounded rationality, to construct an argument for a revisionary conclusion about ideal instrumental rationality. I consider how subjective responses can figure as proxies in heuristics and develop the following argument: (1) Proxies, even if relatively easy to recognize, can sometimes be messy, prompting incomplete or cyclic preferences. (2) From the point of view of ideal instrumental rationality, it is permissible for an agent to be concerned with a (...) proxy rather than with what it is a proxy for. And so, (3) although it is standardly assumed, as part of the prevailing conception of ideal instrumental rationality, that rational preferences are complete and acyclic, neither incomplete preferences nor cyclic preferences can be dismissed as invariably irrational from the point of view of ideal instrumental rationality. (shrink)

Is listening to music like looking through a kaleidoscope? Formalists contend that music is meaningless. Most music theorists concede that this austere thesis is surprisingly close to the truth. Nevertheless, they refute formalism with a little band of diffusely referential phenomena, such as musical quotation, onomatopoeia, exemplification, and leitmotifs. These curiosities ought to be pressed into a new campaign against assumptions that vagueness can only arise in the semantically lush setting of language. Just as the discovery of extremophilic bacteria led (...) biologists to revise their opinions about the scope and preconditions of life, the marginal forms of reference that survive in the semantic desert of absolute music should lead philosophers to revise their assumptions much about scope and preconditions of vagueness. (shrink)

It is commonly accepted that what we ought to do collectively does not imply anything about what each of us ought to do individually. According to this line of reasoning, if cooperating will make no difference to an outcome, then you are not morally required to do it. And if cooperating will be personally costly to you as well, this is an even stronger reason to not do it. However, this reasoning results in a self-defeating, yet entirely predictable outcome. If (...) everyone is rational, no one will cooperate, resulting in an aggregate outcome that is devastating for everyone. This dismal analysis explains why climate change and other collective action problems are so difficult to ameliorate. The goal of this paper is to provide a different, exploratory framework for thinking about individual reasons for action in collective action problems. I argue that the concept of commitment gives us a new perspective on collective action problems. Once we take the structure of commitment into account, this activates requirements of diachronic rationality that give individuals instrumental reasons to cooperate in collective action problems. (shrink)

An example shows that 'pairwise preferences' (certain hypothetical choices) can cycle even when rational. General considerations entail that preferences tout court (certain relations of actual valuations) cannot cycle. A world-bayesian theory is explained that accommodates these two kinds of preference, and a theory for rational actions that would have them maximize and be objects of ratifiable choices. It is observed that choices can be unratifiable either because of troublesome credences or because of troublesome preferences. An appendix comments on a third (...) way in which efforts to maximize can be frustrated. (shrink)

ABSTRACT Is uncertainty about preferences rationally possible? And if so, does it matter for rational decision? It is argued that uncertainty about preferences is possible and should play the same role in rational decision-making as uncertainty about worldly facts. The paper develops this hypothesis and defends it against various objections.

The philosophy of mind encompasses a familiar set of topics: consciousness, intentionality, mental causation, emotion, whatever topics in psychology happen to capture our interest (concepts, mindreading . . .), and so on. There is a topic deserving of addition to this list, a topic that should be receiving regular attention from philosophers of mind but is not: practical rationality. The philosophy of mind bears directly upon what can be called the ‘meta-theory’ of practical rationality, and meta-theories of rationality likewise impose (...) constraints on the philosophy of mind. Philosophers of mind who do not engage with work on practical rationality risk developing theories that are blind to important theoretical demands. They also risk allowing ethicists—who have been at work on the meta-theory of practical rationality—to unilaterally set their theoretical agendas. It is therefore time to get to work. (shrink)

The much-discussed ‘Argument from Small Improvement’ has been advanced both as a reason to reject (tripartite) Completeness, one of the standard axioms of decision theory, and to accept the possibility of rationally incomparable choices. But this form of argument cannot be an effective basis for either of these conclusions, unless one already has some prior, independent reason to prefer Transitivity to Completeness as a constraint on rational preferences (or rational values). In particular, I show how a reverse argument from small (...) improvement can be constructed, starting from the assumption of tripartite Completeness, to the conclusion that Transitivity fails. I conclude that this form of argument as it has been standardly presented in the literature is a kind of ‘red herring’. We can only make progress here by evaluating the reasons independent of either such argument to prefer one or other of Transitivity or Completeness. (shrink)

Larry Temkin famously argues that what he calls spectrum arguments yield strong reason to reject Transitivity, according to which the ‘all-things-considered better than’ relation is transitive. Spectrum arguments do reveal that the conjunctions of independently plausible claims are inconsistent with Transitivity. But I argue that there is very strong independent reason to reject such conjunctions of claims, and thus that the fact that they are inconsistent with Transitivity does not yield strong reason to reject Transitivity.

One of the currently popular dogmata of anti-consequentialism is that consequentialism doesn't respect, recognize, or in some important way account for what is referred to as the The charge is often made, but rarely explained in any detail, much less argued for. In this paper I explain what I take to be the most plausible interpretation of the separateness of persons charge. I argue that the charge itself can be deconstructed into at least two further objections to consequentialist theories. These (...) objections amount to (i) the rejection of axiological aggregation, and (ii) the rejection of deontic aggregation. Of these two objections, I argue that the first one, though often made, is untenable. I also argue that the second objection, in its various forms, relies on distinctions whose moral significance is vigorously denied by almost all consequentialist theorists. I thus argue that the separateness of persons objection poses no special threat to consequentialism. (shrink)

This paper discusses the inefficacy problem that arises in contexts of “collective harm.‘ These are contexts in which by acting in a certain sort of way, people collectively cause harm, or fail to prevent it, but no individual act of the relevant sort seems to itself make a difference. The inefficacy problem is that if acting in the relevant way won’t make a difference, it’s unclear why it would be wrong. Each individual can argue, “things will be just as bad (...) whether or not I act in this way, so there’s no point in doing otherwise.‘ The goal of this paper is to give an overview of some of the main responses available to the problem and to highlight central issues that arise for each type of response. In the final section, I explain what I take to be the most promising strategy and discuss the form that this strategy should take. (shrink)

There are many cases in which people collectively cause some morally significant outcome (such as a harmful or beneficial outcome) but no individual act seems to make a difference. The problem in such cases is that it seems each person can argue, ‘it makes no difference whether or not I do X, so I have no reason to do it.’ The challenge is to say where this argument goes wrong. My approach begins from the observation that underlying the problem and (...) motivating the typical responses to it is a standard, intuitive assumption. The assumption is that if an act will not make a difference with respect to an outcome, then it cannot play a sig- nificant, non-superfluous role in bringing that outcome about. In other words, helping to bring about an outcome requires making a difference. I argue that the key to solving the problem is to reject this assumption. I develop an account of what it is to help to bring about an outcome, where this does not require making a difference, and I use this explain our reasons for action in the problem cases. This account also yields an error theory that explains why the standard assumption is so tempting, even though it is mistaken. (shrink)

There are several powerful motivations for neutral value‐based deontic theories such as Act Consequentialism. Traditionally, such theories have had great difficulty accounting for partiality towards one's personal relationships and projects. This paper presents a neutral value‐based theory that preserves the motivations for Act Consequentialism while vindicating some crucial intuitions about reasons to be partial. There are two central ideas. The first is that when it comes to working out what you ought to do, your friends’ interests, the needs of your (...) family, the significance of your own projects and ideals, etc. have more weight than the interests and needs of strangers. Your friends’ interests are not (thereby) more neutrally valuable than the interests of others. So there is a difference between the value of an outcome and its deontic significance. The second familiar idea is that reasons are modifiable. Reasons of partiality are reasons the weights of which are a function of the value of the relevant outcome modified by facts about the value of caring about the outcome in question. The resulting principle has various further explanatory advantages; in particular, it accounts for project‐ and relationship‐specific permissions and requirements, both at a time and across time. (shrink)

Climate change has made pressing the question of why we do little to reduce greenhouse gas emissions. By analogy to the puzzle of the self-torturer, I argue that even if interpersonal and intergenerational conflicts of interest were resolved, we may still end up in a regretful environmental state when we aim to maximize our net benefit derived from polluting activities. This is because a rational agent with transitive preferences making climate change decisions faces incentives to over-pollute. This is caused by (...) the presence of marginal costs that are uninformative of well-being in an uncertain and intertemporal decision problem. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

I defend the 'Repugnant' Conclusion that for any possible population of happy people, a population containing a sufficient number of people with lives barely worth living would be better. Four lines of argument converge on this conclusion, and the conclusion has a simple, natural theoretical explanation. The opposition to the Repugnant Conclusion rests on a bare appeal to intuition. This intuition is open to charges of being influenced by multiple distorting factors. Several theories of population ethics have been devised to (...) avoid the Repugnant Conclusion, but each generates even more counterintuitive consequences. The intuition opposing the Repugnant Conclusion is thus among the best candidates for an intuition that should be revised. (shrink)

Two objects of valuation are said to be incommensurable if neither is better than the other, nor are they equally good. This negative, coarse-grained characterization fails to capture the nuanced structure of incommensurability. We argue that our evaluative resources are far richer than orthodoxy recognizes. We model value comparisons with the corresponding class of permissible preference orderings. Then, making use of our model, we introduce a potentially infinite set of degrees of approximation to better, worse, and equally good, which we (...) interpret as degrees of commensurability.One payoff is the solution our approach provides to a paradox in population ethics, generated by Parfit's “Continuum Argument”. Parfit imagines a sequence of populations, starting with one consisting of excellent lives and, by a sequence of apparent improvements, reaching a much larger population of lives barely worth living. What he dubs “the Repugnant Conclusion” is that the final population is better than the first. Developing Parfit's response, we argue that some of the populations in the sequence are merely almost better than their immediate predecessors. Almost better is not transitive (unlike better). We offer analogies to other ‘spectrum arguments’, Condorcet's paradox, and to developments in formal epistemology. (shrink)

Utility cascades occur when a utilitarian’s reduction of support for an intervention reduces the effectiveness of that intervention, leading the utilitarian to further reduce support, thereby further undermining effectiveness, and so on, in a negative spiral. This paper illustrates the mechanisms by which utility cascades occur, and then draws out the theoretical and practical implications. Theoretically, utility cascades provide an argument that the utilitarian agent should sometimes either ignore evidence about effectiveness or fail to apportion support to effectiveness. Practically, utility (...) cascades call upon utilitarians to rethink their relationship with the social movement known as Effective Altruism, which insists on the importance of seeking and being guided by evidence concerning effectiveness. This has particular implications for the ‘Institutional Critique’ of Effective Altruism, which holds that Effective Altruists undervalue political and systemic reforms. The problem of utility cascades undermines the Effective Altruist response to the Institutional Critique. (shrink)

If A is better than B and B is better than C, then A is better than C, right? Larry Temkin and Stuart Rachels say: No! Betterness is nontransitive, they claim. In this paper, I discuss the central type of argument advanced by Temkin and Rachels for this radical idea, and argue that, given this view very likely has sceptical implications for practical reason, we would do well to identify alternative responses. I propose one such response, which employs the idea (...) that rational agents might regard some options as incommensurate in value, and will reasonably employ a heuristic of status quo maintenance to avoid suboptimal choices from incommensurate goods. (shrink)

Some normative theories are self-defeating. They tell us to respond to our situations in ways that bring about outcomes that are bad, given the aims of the theories, and which could have been avoided. Across a wide range of debates in ethics, decision theory, political philosophy, and formal epistemology, many philosophers treat the fact that a normative theory is self-defeating as sufficient grounds for rejecting it. I argue that this widespread and consequential assumption is false. In particular, I argue that (...) a theory can be self-defeating and still internally consistent, action-guiding, and suitable as a standard for criticism. (shrink)

There are at least two threads in our thought and talk about rationality, both practical and theoretical. In one sense, to be rational is to respond correctly to the reasons one has. Call this substantive rationality. In another sense, to be rational is to be coherent, or to have the right structural relations hold between one’s mental states, independently of whether those attitudes are justified. Call this structural rationality. According to the standard view, structural rationality is associated with a distinctive (...) set of requirements that mandate or prohibit certain combinations of attitudes, and it’s in virtue of violating these requirements that incoherent agents are irrational. I think the standard view is mistaken. The goal of this paper is to explain why, and to motivate an alternative account: rather than corresponding to a set of law-like requirements, structural rationality should be seen as corresponding to a distinctive kind of pro tanto rational pressure—i.e. something that comes in degrees, having both magnitude and direction. Something similar is standardly assumed to be true of substantive rationality. On the resulting picture, each dimension of rational evaluation is associated with a distinct kind of rational pressure—substantive rationality with (what I call) justificatory pressure and structural rationality with attitudinal pressure. The former is generated by one’s reasons while the latter is generated by one’s attitudes. Requirements turn out to be at best a footnote in the theory of rationality. (shrink)

Sometimes, a group of people can produce a morally bad outcome despite each person’s individual act making no difference to whether the outcome is produced. Since each person’s act makes no difference, it seems the effects of the act cannot provide a reason not to perform it. This is problematic, because if each person acts in accordance with their reasons, each will presumably perform the act—and thus, the bad outcome will be brought about. I suggest that the key to solving (...) this problem is to make it true of each person that their act would in fact make a difference to the relevant outcome. Fortunately, I contend, this can be accomplished by each person simply forming a particular type of attitude. I argue that each person has an obligation to form the relevant attitude in collective action cases, on pain of being immoral or irrational. (shrink)

Sergio Tenenbaum and Diana Raffman contend that ‘vague projects’ motivate radical revisions to orthodox, utility-maximising rational choice theory. Their argument cannot succeed if such projects merely ground instances of the paradox of the sorites, or heap. Tenenbaum and Raffman are not blind to this, and argue that Warren Quinn’s Puzzle of the Self-Torturer does not rest on the sorites. I argue that their argument both fails to generalise to most vague projects, and is ineffective in the case of the Self-Torturer (...) itself. (shrink)

Two options are ‘incommensurate’ when neither is better than the other, but they are not equally good. Typically, we will say that one option is better in some ways, and the other in others, but neither is better ‘all things considered’. It is tempting to think that incommensurability is vagueness—that it is (perhaps) indeterminate which is better—but this ‘vagueness view’ of incommensurability has not proven popular. I set out the vagueness view and its implications in more detail, and argue that (...) it can explain most of the puzzling features of incommensurability. This argument proceeds without appeal to John Broome’s ‘collapsing principle’. (shrink)

We develop a mathematical theory for comparative sorites, considered in terms of a system mapping pairs of stimuli into a binary response characteristic whose values supervene on stimulus pairs and are interpretable as the complementary relations 'are the same' and 'are not the same' (overall or in some respect). Comparative sorites is about hypothetical sequences of stimuli in which every two successive elements are mapped into the relation 'are the same', while the pair comprised of the first and the last (...) elements of the sequence is mapped into 'are not the same'. Although soritical sequences of this kind are logically possible, we argue that their existence is grounded in no empirical evidence and show that it is excluded by a certain psychophysical principle proposed for human comparative judgements in a context unrelated to soritical issues. We generalize this principle to encompass all conceivable situations for which comparative sorites can be formulated. (shrink)

This paper argues for the existence of a fourth positive generic value relation that can hold between two items beyond ‘better than’, ‘worse than’, and ‘equally good’: namely ‘on a par’.

This short paper states a new objection against “spectrum arguments” for nontransitive betterness. It is shown that defenders of such arguments must reject one of two very plausible principles.

Suppose you resolve now to resist an expected temptation later while knowing that once the temptation arrives your preference or evaluative assessment will shift in favor of that temptation. Are there defensible norms of rational planning agency that support sticking with your prior intention in the face of such a shift at the time of temptation and in the absence of relevant new information? This article defends the idea that it might be rational to stick with your prior intention in (...) part because of your anticipation of regret in the further future if you were to give in to the temptation. It does this by appeal to the significance of anticipated future regret to the agent ’s present standpoint, and to the role of the agent ’s standpoint in her self-governance. (shrink)

According to a widely held view, rationality demands that the preferences of a person be transitive. The transitivity assumption is an axiom in standard theories of rational choice. It is also prima facie very plausible. I argue here that transitivity is not a necessary condition of rationality; it is a constraint only in some cases. The argument presented here is based on the non-linearity of differential utility functions. This paper has four parts. First, I present an argument against the transitivity (...) assumption (I, II). Second, I discuss some objections; this will also bring out some features of the view defended here, like the essentially comparative nature of choice (III). Then, I discuss certain proposals that promise to avoid my conclusion (IV). Finally, I go into some implications concerning the nature of rational decision making (V). To accept some intransitivities as rational does not make our life easier; it can even lead to hard choices. However, it does not lead to skepticism about rational choice. (shrink)

There is a well-known moral quandary concerning how to account for the rightness or wrongness of acts that clearly contribute to some morally significant outcome – but which each seem too small, individually, to make any meaningful difference. One consequentialist-friendly response to this problem is to deny that there could ever be a case of this type. This paper pursues this general strategy, but in an unusual way. Existing arguments for the consequentialist-friendly position are sorites-style arguments. Such arguments imagine varying (...) a subject’s predicament bit by bit until it is clear that a relevant difference has been achieved. The arguments offered in this paper are structurally different, and do not rely on any sorites series. For this reason, they are not vulnerable to objections that have been leveled against the sorites-style arguments. (shrink)

Morally supererogatory acts are those that go above and beyond the call of duty. More specifically: they are acts that, on any individual occasion, are good to do and also both permissible to do and permissible to refrain from doing. We challenge the way in which discussions of supererogation typically consider our choices and actions in isolation. Instead we consider sequences of supererogatory acts and omissions and show that some such sequences are themselves problematic. This gives rise to the following (...) puzzle: what problem can we have with a sequences of actions if each individual act or omission is itself permissible? In this paper, we develop a response to this question, by exploring whether solutions analogous to those proposed in the rational choice literature are available in the case of supererogatory sequences. Our investigation leads us to the view that making sense of the supererogatory requires accepting that there are global moral norms that apply to sequences of acts alongside the local moral norms that apply to individual acts. (shrink)

Why should we typically act in accordance with our resolutions when faced with the temptation to do otherwise? A much-maligned view suggests that we should do so because resolutions themselves provide us with reasons for action. We defend a version of this view, on which resolutions provide second-order reasons. This account avoids the objections typically taken to be fatal for the view that resolutions are reasons, including the prominent bootstrapping objections.

Moral puzzles about actions which bring about very small or what are said to be imperceptible harms or benefits for each of a large number of people are well known. Less well known is an argument by Warren Quinn that standard theories of rationality can lead an agent to end up torturing himself or herself in a completely foreseeable way, and that this shows that standard theories of rationality need to be revised. We show where Quinn's argument goes wrong, and (...) apply this to the moral puzzles. (shrink)