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# A Broomean model of rationality and reasoning

Franz Dietrich, Antonios Staras, Robert Sugden

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A Broomean model of rationality and reasoning

Franz DIETRICH, Antonios STARAS, Robert SUGDEN

2018.30

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# A Broomean model of rationality and reasoning

July 2018

Franz Dietrich

Antonios Staras

Robert Sugden

Paris School of Economics & CNRS

U. of East Anglia

U. of East Anglia

## Abstract

John Broome has developed an account of rationality and reasoning which gives philosophical foundations for choice theory and the psychology of rational agents. We formalize his account into a model that differs from ordinary choice-theoretic models through focusing on psychology and the reasoning process. Within that model, we ask Broome's central question of whether reasoning can make us more rational: whether it allows us to acquire transitive preferences, consistent beliefs, non-akratic intentions, and so on. We identify three structural types of rationality requirements: consistency requirements, completeness requirements, and closedness requirements. Many standard rationality requirements fall under this typology. Based on three theorems, we argue that reasoning is successful in achieving closedness requirements, but not in achieving consistency or completeness requirements. We assess how far our negative results reveal gaps in Broome's theory, or deficiencies in choice theory and behavioural economics.

## 1 Introduction

John Broome's (2007, 2013) widely discussed philosophical theory of rationality and reasoning is strikingly different from ordinary rational choice theory. Firstly, it focuses on psychology and mental states – against the behaviourist revealed-preference tradition, but in line with the recent trend towards 'behavioural economics' which takes psychology more seriously.<fnref n="1" /> Secondly, it explicitly addresses the process of reasoning by which mental states change, the blind spot of choice theory. Thirdly, it is largely informal, which explains why choice theory has so far managed to ignore it, despite its strong interest to philosophers (e.g., Parfit and Broome 1997, Kolodny 2005, 2007, Wright 2014, Boghossian 2016, Cullity 2016, Pettit 2016, Southwood 2016).

In this paper, we formalize Broome's framework, in the hope to open it towards choice theorists, behavioural economists, and logicians; and we present sharp results about Broome's central question of whether we can become more rational through reasoning. First though we should give some context.

Broome understands rationality as a static concept, and reasoning as a dynamic concept. The rationality of a human agent ('you' in his language) is taken to consist

<footnote n="1">¹The debate between behaviourism and mentalism is ongoing; e.g., Hausman (2007, 2012), Dietrich and List (2016a) and Okasha (2016).</footnote>
in ‘order’ or ‘coherence’ in your mind (2013: 152). You are rational if your current
mental states or attitudes – your beliefs, intentions, etc. – cohere with one another.
Reasoning, by contrast, is regarded as a rule-governed mental process of forming
new attitudes from existing ones, through consciously operating on the contents of
your attitudes. This distinguishes reasoning from other causal psychological processes
which are automatic and subpersonal.

To what extent can you become rational through active reasoning? This is Broome's
central question. The question poses a problem for those 'who write on rationality
[and] seem to think that they have finished their job when they have described re-
quirements of rationality' (2013: 208). Why do many think describing rationality is
enough? Broome suggests - and we agree - that they take for granted that rationality
is achievable through reasoning: they 'must believe that, starting from knowledge of
a particular requirement, you can reason your way actively to satisfying that require-
ment' (2013: 208-209). So they simply presuppose a positive answer to Broome's
open question.

Broome’s question is of existential importance for choice theory and behavioural
social sciences. These disciplines would get into trouble by a negative answer to
Broome’s question, although they rarely ask how you come to satisfy rationality
axioms.<fnref n="2" /> Indeed, whenever choice theorists interpret their axioms normatively, the
principle of ‘ought implies can’ seems to commit them to assuming that rationality is
achievable, presumably through reasoning.<fnref n="3" /> As for behavioural social science, a major
strand attributes observed violations of rationality to ‘reasoning errors’ or ‘absence
of reasoning’ – an interpretation that once again presupposes that reasoning can in
principle achieve rationality.<fnref n="4" />

Despite all this, Broome’s theory has so far had little impact on social science. This comes not only from its informal nature, but perhaps also from Broome’s parallel pursuit of two orthogonal lines of investigation: a ‘structural’ investigation into the concepts of requirements and reasoning, and an (often only tentative) ‘substantive’ investigation into what precisely is rationally required and when reasoning is intuitively ‘correct’. Both dimensions are important, but should be disentangled and addressed separately, we believe.

In this paper, we address the *structural* dimension of Broome’s programme. We formalize the central features of Broome’s approach, including ‘requirements of rationality’, ‘theories of rationality’, and ‘reasoning rules’, while setting aside substantive questions of which theory and reasoning rules are ‘correct’. Our model differs from ordinary rational-choice models through focusing on psychology and through modeling the reasoning process, including theoretical reasoning (reasoning with beliefs) and

<footnote n="2">²Lewis’s (1969) theory of common knowledge is an important exception. In contrast to what is now standard practice in game theory, his analysis treats players’ common knowledge as a product of specific modes of reasoning (see Cubitt and Sugden 2003).</footnote>

<footnote n="3">³In a canonical statement of rational-choice theory, Savage (1954: 20) says that the main use he would make of its rationality axioms ‘is normative, to police my own decision for consistency and, where possible, to make complicated decisions to depend on simpler ones’. This concept of ‘policing’ seems to presuppose that reasoning can in principle achieve rationality.</footnote>

<footnote n="4">⁴Infante et al. (2016) document the widespread use of the concept of ‘reasoning error’ in behavioural economics.</footnote>
practical reasoning (reasoning towards intentions).

Psychologists and thereafter behavioural social scientists have often classified men-
tal processes into two systems: the fast and automatic ‘System 1’ which generates im-
pressions, intuitions, feelings, and impulses, and the slow, conscious, and deliberative
‘System 2’ which, operating on the outputs of System 1, constructs explicit thoughts
in an orderly way (Wason and Evans, 1975; Kahneman, 2011: 19–30). Behavioural
economists often presuppose without question that System 2 is able to create fully
rational preferences and beliefs, and surprisingly do not explicitly describe the reas-
oning process by which this is supposed to happen.<fnref n="5" /> Our Broomean model can be
regarded as an explicit description of the reasoning underlying System 2.

Within our model, we derive sharp results about whether specific types of ration-
ality requirements can or cannot be achieved through reasoning. As it turns out, some
common types of requirement can be achieved by reasoning, but others can not. Our
negative results cast doubts on the mentioned idea that System 2 is able to achieve
full rationality. Using a simple version of conventional choice theory as our leading
example, we discuss how far our negative results reveal deficiencies in received ideas
about rationality, and or gaps in Broome’s theory.

2 Key features of Broome’s structural theory

Let us give some background that will guide our model. Broome fundamentally dis-
tinguishes between ‘rationality’ and ‘reasoning’. Rationality is understood as one of
many possible sources of requirements on your mental states. Other sources of require-
ments might be morality, prudence, fashion, or Catholicism (2013: 26, 116). Broome
leaves open whether rationality is normative, i.e., generates normative ‘oughts’ (2013:
146). A distinctive feature of rationality is that it requires coherence between mental
states: ‘your mental states [are] properly related to each other’ (2013: 152). Broome
is primarily concerned with synchronic requirements: requirements on mental states
held simultaneously. He takes ordinary rationality requirements to have wide scope:
for example, for Broome an enkrasia requirement takes the form ‘Rationality requires
of you that if you believe you ought to do F then you intend to do F’, rather than
‘If you believe you ought to do F, rationality requires of you that you intend to do
F’ in which the requirement has a narrow scope (2013: 31–32). Our model is thus
designed to represent wide-scope synchronic requirements. Broome’s contention that
ordinary rationality requirements have wide scope has been questioned, in particular
by Kolodny (2005, 2007); we need not take a position in this debate. It is however
uncontroversial that the principal rationality requirements found in choice theory are
both synchronic and wide-scope, which establishes (at least) the relevance of Broome’s
notion of rationality and our model thereof.

Unconventionally, Broomean reasoning is reasoning with multiple attitudes. This
goes beyond reasoning with beliefs (‘theoretical reasoning’) and reasoning towards in-
tentions (‘practical reasoning’), although Broome gives special importance to these
two classic types of reasoning (2013: 250). Broome describes reasoning as ‘a rule-
governed operation on the contents of your conscious attitudes’ (2013: 234); as a

<footnote n="5">⁵See Infante et al. (2016) for a documentation of this claim.</footnote>
causal psychological process through which ‘some of your attitudes cause you to acquire a new attitude’ (2013: 225); and as a conscious activity under your control, to be distinguished from automatic psychological processes which also affect attitudes. Structurally, a mental state or attitude is given by (i) its content, a proposition, and (ii) the type of attitude held towards that content, for example belief or intention (2013: 251–252). Your mental experience of reasoning consists in bringing to mind a set of premise-attitudes and then finding (in a way that can be expressed as ‘So…’) that some conclusion-attitude follows from them. The sense of ‘following from’ is that of (implicitly or explicitly) being guided by some rule, in a way that ‘seems right to you’; whether that rule is correct in some external sense is beside the point (2013: 237–238). Crucially, reasoning is an operation on the *contents* of your attitudes: you bring to your consciousness the contents of attitudes (e.g., that it rains, or that you take an umbrella), not the attitudes towards these contents (e.g., that you believe that it rains, or intend that you take an umbrella). Nonetheless, the attitudes held towards contents do not get lost in reasoning: they determine ‘how’ you consciously entertain the contents (as beliefs? as intentions? etc.), the exact meaning of which belongs to the philosophically trickiest elements of Broome’s theory.

Broome distinguishes higher-order from first-order attitudes; the former are attitudes towards propositions about your own attitudes, such as your intention that you believe something (2013: 236). He treats reasoning with first-order attitudes as the fundamental kind of reasoning, insisting that ‘higher-order beliefs are not necessary for reasoning’ (2013: 236). Thus, for example, a child can carry out modus ponens reasoning on the contents of his beliefs without any awareness of having beliefs, indeed without any concept of ‘belief’ (2013: 229). Although our model does not rule out higher-order attitudes, our applications concentrate on first-order attitudes.

Against this background, we now present our Broomean model.

## 3 Mental states formalized

An agent has many mental states, like beliefs, intentions, or preferences. A mental state is an attitude towards something (the ‘object’ or ‘content’ of that attitude). An example is the belief that it rains: the attitude is belief, the content is that it rains. Another example is the intention that I swim: the attitude is intention, the content is that I swim.

Each attitude has (i) a number of places and (ii) a domain. We explain both in turn. Belief and intention are one-place attitudes: their object is a single thing. Preference is a two-place attitude, since something is being preferred to something else. A mental state with a one-place attitude is represented by an object-attitude pair $(p, a)$, for example (*I swim, intention*). In general, a state with an *n*-place attitude is represented by a tuple $(p_1, ..., p_n, a)$ where *a* is the attitude and $(p_1, ..., p_n)$ its *n*-ary object. For example, (*I swim, I eat, preference*) represents preference of swimming over eating. We use the term ‘object’ of attitudes in two ways: the state $(p_1, ..., p_n, a)$ is said to have *n* objects $p_1, ..., p_n$, or to have just one object $(p_1, ..., p_n)$. Both usages are equivalent for one-place attitudes.

The domain of an attitude is its set of possible objects: the belief domain contains
whatever can be believed, the intention domain contains whatever can be intended, the preference domain contains the things between which preferences can be held, etc. Some might claim that any attitude can be held towards anything: you can believe anything, intend anything, etc. Others might insist that you can intend only things you can bring about.

Formally, our model has two primitives:

* a fixed non-empty set *L* of *objects*. Philosophers might think of them as propositions, choice-theorists as options, events, or other things. More on this soon.

* a fixed non-empty set *A* of *attitudes* or more exactly *types of attitudes*, each coming with a domain *D* ⊆ *L* of possible objects and a number of places *n* in {1, 2, ...}. *A* might contain one-place attitudes of belief *bel* and intention *int* and a two-place preference attitude ≻, each with some domain of possible objects. Some or all attitudes might have universal domain of applicability *D* = *L*. Choice-theoretic models (when recast in our framework) typically involve attitudes with restricted domains: in game theory, players hold preferences w.r.t. final outcomes (or outcome lotteries), intentions w.r.t. own actions, and beliefs w.r.t. actions of opponents.

**Definition 1** *A (mental) state or attitude is a tuple (p<sub>1</sub>, ..., p<sub>n</sub>, a) – called the 'attitude of type a towards p<sub>1</sub>, ..., p<sub>n</sub>' – where a is an attitude type in A, n is its number of places, and p<sub>1</sub>, ..., p<sub>n</sub> belong to its domain. Let M be the set of all mental states.*

Note that we say ‘attitude’ both for ‘mental states’ in *M* and ‘attitude types’ in *A*. No ambiguity will arise. Mental states whose attitude type is belief (intention, preference, ...) are called belief states (intention states, preference states, ...), or simply beliefs (intentions, preferences, ...).

We call the totality of an agent’s mental states at a given time his current ‘constitution’:

**Definition 2** *A (mental) constitution is a set $C \subseteq M$ of mental states; it contains the mental states held by the agent.*

We shall often interpret objects in *L* as propositions. One might indeed argue that all attitudes are held fundamentally towards propositions: that desiring apples ultimately means desiring that one eats apples, that intending to swim ultimately means intending that one swims, etc. But one may alternatively take *L* to consist of sentences, properties, actions, events, states of affairs, or even a mixture of things such as actions (the domain of intention) and events (the domain of belief).

Choice theorists, by contrast, might interpret objects in *L* as choice options, goods, Savage acts, nature events, strategies, or a mixture thereof.

We now give a philosophical or logical application, followed by a choice-theoretic application.

**Application 1: agent-internal language for practical reasoning.** Following Broome, think of *L* as containing all propositions relevant to practical reasoning, such as *it rains* or *I take an umbrella*. Let *A* contain at least the one-place attitudes of
belief *bel* and intention *int*, the two central attitudes in Broome's analysis of practical reasoning. Your constitution might contain mental states such as (*it rains*, *bel*), (*I stay dry*, *int*), (*carrying an umbrella is a means implied by staying dry*, *bel*), and (*I carry an umbrella*, *int*).

Following Broome (2013: 251, 260–261), one might not formalize *L*: one might let *L* be an abstract set of primitive objects called ‘propositions’. But if one needs a model of propositions – to capture their logical relations and combine them through operations like ‘and’ – then there are two routes, defined precisely in the companion paper Dietrich et al. (2018). A set-theoretic or extensional model lets propositions be sets of possible worlds; so *L* contains subsets of a given set of possible worlds. A syntactic or intensional model lets propositions be sentences; so *L* contains the sentences of a suitable formal language. This is not to claim that sets of worlds or sentences metaphysically ‘are’ the contents of attitudes, but that they can formally ‘represent’ these contents. The intensional model is more Broomean: Broome needs an intensional notion of proposition, as agents might believe (intend, ...) some proposition without believing (intending, ...) an equivalent one.

**Application 2: choice under certainty.** We now apply the Broomean approach to choice theory, in the absence of uncertainty or risk. Consider a fixed non-empty set *X* of mutually exclusive choice options, e.g., goods or political candidates. The agent has attitudes of preference and indifference towards options. In each choice context, certain options from *X* are feasible; they form the feasible set, formally a non-empty subset *Y* ⊆ *X* from which the agent chooses one element. Choice theory implicitly assumes the feasible set to be known. Our mentalistic model rather takes the agent to have beliefs about what the feasibility set is; the model is silent about whether those beliefs are correct. Following Broome, acts of choice are not mental states, but non-mental occurrences caused by particular mental states, namely intentions. So our model contains not acts, but intentions as their mental counterparts. Overall, as objects of attitudes we need:

• options *x* in *X*, the objects of intention, preference, and indifference,

• feasible sets *Y* in 2*X*\{Ø}, the objects of feasibility beliefs.

Formally, $L = X \cup (2^X \setminus \emptyset)$. Whoever wants attitudes to have propositional content can re-interpret an $x$ as the proposition that $x$ is chosen, and a $Y$ as the proposition that $Y$ is the feasible set.

Next, let $A = \{int, bel, \succ, \sim\}$, where:

• int is a one-place attitude with domain X; (x, int) represents intention to choose x;

• ≻ and ∼ are two-place attitudes with domain X; (x, y, ≻) represents (strict) preference of x to y, (x, y, ∼) represents indifference between x and y;

• bel is a one-place attitude with domain 2^X \ {Ø}; (Y, bel) represents belief that the feasible set is Y.

One could replace $\succ$ and $\sim$ by a single weak-preference attitude $\gtrsim$; then $A = \{int, bel, \gtrsim\}$. This would come closer to choice-theoretic models, which usually start from a single weak-preference relation, not two relations of strict preference and indif-ference. Within our mentalist framework, we prefer working with $\succ$ and $\sim$, because a
weak-preference attitude ≳ seems artificial. Mentalistically, the weak preference relation in choice theory is best interpreted as a mathematically convenient representation of the disjunction of two attitudes. Technically, the difference between working with > and ~ and working with ≳ is small: using > and ~ one can express weak preference (as preference-or-indifference), and conversely, using ≳ one can express both strict preference (as weak preference without indifference) and indifference (as weak preference in both directions). Nonetheless, working with > and ~ is more general, because it allows violation of three very basic rationality assumptions, namely asymmetry of preference, symmetry of indifference, and mutual exclusiveness between preference and indifference, defined in R2-R4 below.

# 4 Requirements on mental states

We now formalize the notion of requirement, in its most standard sense aligned with choice theory and endorsed by Broome (i.e., the ‘synchronic’ and ‘wide-scope’ sense mentioned in Section 2). In our terminology, such requirements are restrictions on your constitution. In this section, we define ‘requirements’ generically, without yet caring about whether they are imposed by rationality.

Since a requirement classifies constitutions into those satisfying and those violating it, we identify each requirement with the set of constitutions satisfying it:

**Definition 3** A *requirement* is a set $R$ of constitutions; constitutions in $R$ ‘satisfy’ the requirement, others ‘violate’ it.

Among the numerous potential requirements (sets of constitutions), the tautological requirement allows all constitutions ($R = 2^M$) and the contradictory one allows no constitution ($R = \emptyset$).

We now give examples of requirements, more precisely requirement schemas since the requirements involve parameters:

**Examples of requirements within Application 1:** Broome treats the following properties, or versions thereof, as requirements of rationality:<fnref n="6" />

* **Non-contradiction:** you do not believe both *p* and *not p*, formally $R = \{C : (p, bel) \in C) \Rightarrow (not p, bel) \notin C\}$. Parameter: $p \in L$.

* **Modus ponens:** believing *p* and *if p then q* implies believing *q*, formally $R = \{C : (p, bel), (if p then q, bel) \in C) \Rightarrow (q, bel) \in C\}$. Parameters: $p, q \in L$.

* **Enkrasia (non-akrasia):** believing *obligatorily p* implies intending *p*, formally $R = \{C : (obligatorily p, bel) \in C \Rightarrow (p, int) \in C\}$. Parameter: $p \in L$.

* **Necessary means:** intending *p* and believing *q* is a means implied by *p* implies intending *q*, formally $R = \{C : (p, int), (q is a means implied by p, bel) \in C \Rightarrow (q, int) \in C\}$. Parameters: $p, q \in L$.

If propositions have no formal structure, these definitions of requirements are informal, since then we cannot give formal meaning to composite propositions such as

<footnote n="6">⁶Broome’s non-contradiction requirement is exactly ours (2013: 155). His modus ponens requirement includes as a third premise that *I care whether q* (2013: 157). His necessary means and enkrasia requirements include the additional premise that *I believe that q is up to me* (2013: 169, 171).</footnote>
not p, if p then q and obligatorily p (where p, q ∈ L). But these definitions can, if
wished, be turned into formal definitions by adopting the syntactic (intensional) or
set-theoretic (extensional) model of propositions sketched above (see Dietrich et al.
2018 for details).

**Examples of requirements within Application 2:**

**R1: Transitivity.** The transitivity schema consists of four subschemas, namely transitivity of strict preference, transitivity of indifference, and two mixed transitivities:

**R1**: Preferring *x* to *y* and *y* to *z* implies preferring *x* to *z*, formally *R* = {*C* : if (*x*, *y*, ≻), (*y*, *z*, ≻) ∈ *C* then (*x*, *z*, ≻) ∈ *C*}. Parameters: *x*, *y*, *z* ∈ *X*.

**R1~:** Indifference between *x* and *y* and between *y* and *z* implies indifference between *x* and *z*, formally *R* = {*C* : if (*x*, *y*, ~), (*y*, *z*, ~) ∈ *C* then (*x*, *z*, ~) ∈ *C*}. Parameters: *x*, *y*, *z* ∈ *X*.

**R1~,**~: Preference of *x* to *y* and indifference between *y* and *z* imply preference of *x* to *z*, formally *R* = {*C* : if (*x*, *y*, ≻), (*y*, *z*, ~) ∈ *C* then (*x*, *z*, ≻) ∈ *C*}. Parameters: *x*, *y*, *z* ∈ *X*.

**R1~,**~: Indifference between *x* and *y* and preference of *y* to *z* imply preference of *x* to *z*, formally *R* = {*C* : if (*x*, *y*, ~), (*y*, *z*, ≻) ∈ *C* then (*x*, *z*, ≻) ∈ *C*}. Parameters: *x*, *y*, *z* ∈ *X*.

**R2:** Asymmetry of ≻. Preferring *x* to *y* excludes preferring *y* to *x*, formally *R* = {*C* : (*x*, *y*, ≻) ∈ *C* ⇒ (*y*, *x*, ≻) ∉ *C*}. Parameters: *x*, *y* ∈ *X*.

**R3:** Symmetry of ~. Indifference between *x* and *y* implies indifference between *y* and *x*, formally *R* = {*C* : (*x*, *y*, ~) ∈ *C* ⇒ (*y*, *x*, ~) ∈ *C*}. Parameters: *x*, *y* ∈ *X*.

**R4:** Exclusiveness between ≻ and ~. Preference of *x* to *y* excludes indifference between *x* and *y*, formally *R* = {*C* : (*x*, *y*, ≻) ∈ *C* ⇒ (*x*, *y*, ~) ∉ *C*}. Parameters: *x*, *y* ∈ *X*.

**R5:** Preference completeness. The options *x* and *y* are compared, formally *R* = {*C* : (*x*, *y*, ≻) ∈ *C* or (*y*, *x*, ≻) ∈ *C* or (*x*, *y*, ~) ∈ *C*}. Parameters: *x*, *y* ∈ *X*.

**R6:** No conflicting feasibility beliefs. You do not believe both *Y* and *Y'* to be the feasible set, formally *R* = {*C* : (*Y*, *bel*) ∈ *C* ⇒ (*Y'*, *bel*) ∉ *C*}. Parameters: distinct feasible sets *Y*, *Y'* ∈ 2*^X*\{Ø}.

**R7:** No conflicting intentions. You do not intend both *x* and *y*, formally *R* = {*C* : (*x*, *int*) ∈ *C* ⇒ (*y*, *int*) ∉ *C*}. Parameters: distinct options *x*, *y* ∈ *X*.

**R8:** Economic enkrasia. If you believe that *Y* is the feasible set and rank the options in its subset *Z* top within *Y* then you intend some option in *Z*, formally *R* = {*C* : if (*Y*, *bel*) ∈ *C*, (*x*, *y*, ≻) ∈ *C* for all *x* ∈ *Z*, *y* ∈ *Y*\* *Z*, and (*x*, *z*, ~) ∈ *C* for all distinct *x*, *z* ∈ *Z*, then (*x*, *int*) ∈ *C* for some *x* ∈ *Z*}. Parameters: *Y* ⊆ *X* and *Z* ⊆ *Y* (*Z* ≠ Ø).

In conjunction, R1-R8 are interpretable as representing the conventional theory of
rational choice under certainty, re-expressed mentalistically. To see this, suppose your
constitution satisfies the requirements R1-R8. Conditions R1-R5 ensure fully classical
preferences; our non-standard axiomatization R1-R5 comes from using two attitudes,
strict preference and indifference, rather than a single weak-preference attitude.<fnref n="7" />

<footnote n="7">⁷R1 is our counterpart of classic choice-theoretic transitivity, which requires for all x, y, z ∈ X</footnote>
Conditions R6-R7 exclude contradictory intentions or feasibility beliefs. Condition R8 reflects the classical preference-maximization hypothesis: you intend something that you most prefer among what you believe to be feasible. R8 also constitutes a choice-theoretic counterpart of ordinary enkrasia (Application 1), which regulates the formation of intentions. R8 differs from ordinary enkrasia in that intentions respond to preferences, not ought-beliefs.

Notice a crucial consequence of working with intentions rather than (as choice theory does) with choice acts: even if you rank more than one option top in your feasible set (so that $|Z| \ge 2$), you intend a specific option (from Z). Conventional choice theory evades the question of how you manage to choose between top-ranked options by representing behaviour through ‘choice correspondences’ that can generate possibly non-singleton sets of chosen options. We return to this issue in Section 8.

Whether reasoning can achieve a requirement depends on the type of requirement. Before we can show this, we first introduce our typology of requirements.

**Definition 4** *A consistency requirement is a requirement R that forbids holding certain mental states simultaneously; formally, $R = \{C : \text{not } F \subseteq C\}$ for some non-empty set F of mental states, the 'forbidden set'.*

In Application 1, the non-contradiction requirement is a consistency requirement ($F = \{(p, bel), (not p, bel)\}$). Application 2 contains consistency requirements in R2 ($F = \{(x, y, \succ), (y, x, \succ)\}$), R4 ($F = \{(x, y, \succ), (x, y, \sim)\}$), R6 ($F = \{(Y, bel), (Y', bel)\}$), and R7 ($F = \{(x, int), (y, int)\}$). Our requirements R1-R4 imply preference acyclicity, another schema of consistency requirements.<fnref n="8" />

**Definition 5** *A completeness requirement is a requirement R that forbids holding none of certain mental states; formally, $R = \{C : C \cap U \neq \emptyset\}$ for some non-empty set U, the 'unavoidable set'.*

Completeness requirements are non-abstention requirements. Examples are the preference-completeness requirements in R5 ($U = \{(x, y, \succ), (y, x, \succ), (x, y, \sim)\}$). In Application 1 one might require ‘belief completeness’ relative to certain propositions p; here $U = \{(p, bel), (not p, bel)\}$.

Finally, a closedness requirement demands that holding certain (premise) states implies holding a certain (conclusion) state:

**Definition 6** *A closedness requirement is a requirement R demanding that if certain mental states are held then a certain mental state is held; formally, $R = \{C : P \subseteq C \Rightarrow c \in C\}$ for some set of ('premise') states P and some ('conclusion') state c.*

<footnote>that weak preference of x to y and of y to z implies weak preference of x to z (formally $R = \{C : \text{if } [(x, y, \succ) \in C \text{ or } (x, y, \sim) \in C] \& [(y, z, \succ) \in C \text{ or } (y, z, \sim) \in C] \text{ then } [(x, z, \succ) \in C \text{ or } (x, z, \sim) \in C]\}$). R5 is our counterpart of classic choice-theoretic completeness, which requires for all $x, y \in X$ that x be weakly preferred to y or y to x (formally, $R = \{C : [(x, y, \succ) \in C \text{ or } (x, y, \sim) \in C] \text{ or } [(y, x, \succ) \in C \text{ or } (y, x, \sim) \in C]\}$). R2-R4 have no classic counterparts, as they correspond to properties which the classic approach treats as true by definition (through the way it defines strict preference and indifference from weak preferences). In fact, each of R1 and R5 implies its classic counterpart, and is equivalent to it given R2-R4.</footnote>

<footnote n="8">⁸Preference acyclicity forbids a preference cycle over options $x_1, \dots, x_n$; so $F = \{(x_1, x_2, \succ), (x_2, x_3, \succ), \dots, (x_{n-1}, x_n, \succ), (x_n, x_1, \succ)\}$. Parameters: $n \ge 2$ and $x_1, \dots, x_n \in X$.</footnote>
Modus ponens, enkrasia and necessary means in Application 1 are schemas of closedness requirements, as are transitivity R1 and symmetry of indifference R3 in Application 2.

Our three types of requirement are mutually exclusive, except for one special case: requirements of the form $R = \{C : m \in C\}$ ('you must hold state m', for given $m \in M$) are expressible both as degenerate completeness requirements $R = \{C : C \cap \{m\} \neq \emptyset\}$ and as degenerate closedness requirements $R = \{C : \emptyset \subseteq C \Rightarrow m \in C\}$. All requirements listed for Applications 1 and 2 fall under our typology, with the important exception of economic enkrasia, which we address and 'categorize' in Section 8.

# 5 Theories of rationality and their requirements

A theory of rationality expresses a specific conception of what rationality requires of you. We now formalize the notion of ‘theory of rationality’, without attempting to adjudicate between alternative theories. A brute-force approach to specifying a theory would be to list all its requirements. But formally defining a ‘theory of rationality’ as a set of requirements would be both unparsimonious and too permissive.<fnref n="9" /> We thus identify a theory, not with its set of requirements, but with the set of constitutions it deems rational:

**Definition 7** A notion or theory of rationality is a set *T* of constitutions; constitutions in *T* are 'rational' under *T*, others are 'irrational' under *T*.

A theory of rationality is formally the same object as a requirement: it is a set of constitutions. But that set is interpreted differently: it contains only the fully rational constitutions, not all constitutions satisfying one particular requirement. A theory of rationality implies certain requirements, representing the requirements of rationality and defined as follows:

**Definition 8** Given a theory of rationality *T*, the requirements of *T* are those requirements *R* which follow from *T*, i.e., for which $T \subseteq R$.

The strongest requirement of a theory *T* is the theory itself *R* = *T*; the weakest is the tautological requirement *R* = *M*. Incidentally, a theory *T* implies not just requirements, but also permissions. It might permit rather than require transitive preferences.<fnref n="10" />

Our definitional setup has proceeded in a top-down direction, by starting with a notion of rationality and deriving individual requirements. In practice, one proceeds in a bottom-up direction: one first comes up with requirements of rationality, whose conjunction then defines one’s theory of rationality. We have already argued that a mentalistic version of the standard theory of rational choice under certainty can be

<footnote n="9">⁹ *Unparsimonious*: theories usually have enormously many requirements, most of which are artificial (e.g., whenever *R* and *R'* are requirements of the theory, so are their conjunction *R*∩*R'* and disjunction *R* ∪ *R'*). *Too permissive*: viewing any set of requirements as a theory allows for awkward theories, since something can then be required without something logically weaker being required.</footnote>

<footnote n="10">¹⁰ *T* 'permits' the constitution to be of a given kind (formally, to fall into a given set *P* of constitutions) if some constitution in *T* is of that kind (formally, if $P \cap T \neq \emptyset$).</footnote>
axiomatized by R1-R8. Formally, the intersection of those requirements defines a
theory of rationality T. Notice that exactly the same theory could be axiomatized
by a different set of requirements. For example, we might replace transitivity R1 by
Savage's (1954: 17-21) 'negative transitivity'.<fnref n="11" />

**6 Reasoning and becoming rational**

Can you become rational through reasoning? This is Broome's key question. We now formalize the notions of 'reasoning' and 'becoming rational'. As explained in Section 2, reasoning for Broome is a conscious, rule-governed mental process by which you form new attitudes based on existing ones. A 'reasoning rule' encodes this idea. For example, if you have the intention to swim and the belief that swimming requires undressing, you might then reason that 'So I shall undress' (where 'shall' is a linguistic marker of intention and 'So' expresses the internal sense of rule-following). In terms of our model, you apply a rule to (certain attitudes in) your constitution, which adds a new attitude to your constitution. Formally:

**Definition 9** A reasoning rule is a pair $r = (P, c)$ of a set of (premise) states $P \subseteq M$ and a (conclusion) state $c \in M$. The revision of a constitution $C$ through a rule $r = (P, c)$ is the constitution $C|r$ obtained by adding the conclusion state provided all premise states are held, formally

$$C|r = \begin{cases} C \cup \{c\} & \text{if } P \subseteq C \text{ (the rule 'applies' to } C\text{)} \\ C & \text{if } P \not\subseteq C \text{ (the rule 'does not apply' to } C\text{).} \end{cases}$$

Your way to reason is defined by the set of rules you follow, to be called your
'reasoning system'.

**Definition 10** A reasoning system is a set $S$ of reasoning rules. A constitution $C$ is closed under $S$ if for each rule $r = (P, c)$ in $S$, possession of the premise states implies possession of the conclusion state, i.e., $P \subseteq C \Rightarrow c \in C$ (equivalently, $C|r = C$).

If your constitution is not yet closed under your reasoning system $S$, then reasoning
leads to the addition ('formation') of new attitudes, until the constitution is finally
closed under $S$, i.e., until each rule in $S$ which applies has been applied. We call the
so-reached new constitution the 'revision of $C$ through $S$':

**Definition 11** The revision (or closure) of $C$ through a reasoning system $S$ is the constitution $C|S$ obtained from $C$ by adding mental states until the constitution is closed under $S$. Formally, $C|S$ is the minimal extension of $C$ closed under $S$.<fnref n="12" />

<footnote n="11">¹¹Negative transitivity consists in the requirements $R = \{C : [(x, y, \succ) \notin C \& (y, z, \succ) \notin C] \Rightarrow (x, z, \succ) \notin C\}$ for all options $x, y, z \in X$. It is equivalent to transitivity R1, given R2-R5. Negative transitivity requirements are requirements of a fourth type: they are 'negative closedness' requirements, given by $R = \{C : C \cap P = \emptyset \Rightarrow c \notin C\}$ for certain $P \subseteq M$ and $c \in M$.</footnote>

<footnote n="12">¹²This minimal extension exists and is unique, because the intersection of all extensions of $C$ closed under $S$ is itself an extension of $C$ closed under $S$, and it is a minimal one, indeed the only minimal one.</footnote>
Revising a constitution through a reasoning system resembles the familiar opera-
tion of deductively closing your belief set, but it applies to your constitution or full
'psychology' rather than your beliefs only.

What is a 'good' reasoning system, given a theory of rationality? We postulate
two criteria or desiderata:

* Desideratum 1: Reasoning should achieve many of the theory's requirements of rationality.

* Desideratum 2: Reasoning should never destroy the consistency of a constitution, a minimal demand defined shortly.

So, loosely speaking, reasoning should improve rationality by Desideratum 1, without elsewhere compromising rationality by Desideratum 2. The following definition makes Desideratum 1 precise:

**Definition 12** *A reasoning system S achieves a requirement R if for each constitution C its revision $C|S$ satisfies R.*

The next two definitions clarify Desideratum 2:

**Definition 13** *Given a theory of rationality *T*, a constitution *C* is *consistent* if its states can be rationally held together, i.e., if some expanded constitution *C'* ⊇ *C* is rational under *T*.*

How does consistency of a constitution relate to our notion of consistency require-
ments? As one can show, a constitution is consistent if and only if it satisfies all
consistency requirements of the theory.<fnref n="13" />

**Definition 14** *Given a theory of rationality, a reasoning system *S* preserves consistency if for each consistent constitution *C* its revision *C*|*S* is still consistent.*

**7 Achieving (or not achieving) rationality through reasoning**

We are now in a position to provide answers to Broome’s question. Specifically, we
address three subquestions: can reasoning help you satisfy consistency requirements,
completeness requirements, and closedness requirements, respectively?

**7.1 Can you achieve closedness requirements?**

Fortunately, a general possibility result holds about closedness requirements:

**Theorem 1** *Given any theory of rationality *T*, there exists a reasoning system which achieves each closedness requirement of *T* and preserves consistency.*

<footnote n="13">¹³First, assume *C* is inconsistent. Then no rational constitution includes *C*. So *C* is the forbidden set of a consistency requirement of *T*. *C* violates this requirement. Conversely, if *C* violates some consistency requirement of *T*, every extension *C'* ⊇ *C* also violates it and is thus irrational, implying inconsistency of *C*.</footnote>
So, whatever theory of rationality $T$ we adopt, all its closedness requirements are achievable through reasoning. For instance, you can achieve the transitivity requirements R1 in Application 2. The achievability of closedness requirements would be trivial if we did not require consistency-preservation, since then you could use the reasoning system $S$ containing all rules: this would transform each constitution $C$ into the maximal constitution $C|S = M$, which trivially satisfies all closedness requirements, but is inconsistent under any non-degenerate theory.

Typically, many alternative reasoning systems $S$ achieve all closedness requirements of a given theory $T$ (and preserve consistency). The theorem’s proof introduces a tailor-made rule for each closedness requirement of $T$: $S$ contains a rule $(P, c)$ whenever $T$ makes the corresponding requirement $R = \{C : P \subseteq C \Rightarrow c \in C\}$. This reasoning system is unnecessarily rich in rules and unrealistic for agents with limited cognitive skills. It is also peculiar, as it achieves each closedness requirement of $T$ in a single reasoning step, by applying a tailor-made rule. Slimmer and psychologically more natural reasoning systems need more reasoning steps, but still achieve all closedness requirements of $T$ (and preserve consistency). This suggests a fundamental trade-off: the richer the reasoning system is, the more rules the agent must internalize or ‘store’, but the faster he can form some attitudes.

Theorem 1 formalizes the fundamental truth that reasoning, as understood by Broome, is well adapted to achieving closedness requirements. Intuitively, this comes from the structural analogy between reasoning rules and closedness requirements: closedness requirements have an if-then structure whose antecedent and consequent are that certain attitudes are present, and reasoning is a process by which the presence of certain attitudes causes the presence of another attitude.

## 7.2 Can you achieve consistency requirements?

The picture reverses as we consider consistency requirements:

**Theorem 2** *No reasoning system achieves any consistency requirement.*

So reasoning can for instance not achieve requirements of non-contradictory beliefs (Application 1), or of asymmetry of preference R2, no conflicting intentions R7, and preference acyclicity (Application 2).

The proof is simple. Informally, if a constitution violates a consistency requirement, then that requirement is achievable only through removing certain attitudes. Yet Broomean reasoning can only add, not remove attitudes.

Theorem 2’s negative result can be traced back to the impossibility to reason towards absences of attitudes, a central Broomean thesis. To see why, notice that a consistency requirement forbids simultaneously holding all attitudes from some set $F$, which can be expressed as the if-then requirement that if certain $F$-states (all $F$-states except a fixed one) are present then the remaining $F$-state is absent. One could have achieved this requirement if one could reason from the presence of the former states towards the absence of the latter state. Yet reasoning ‘cannot conclude in an absence’, following Broome.

Broome (2013: 278–280) offers a cautious defence of an apparently similar claim to Theorem 2: through ‘correct reasoning’, you cannot achieve a requirement which
forces you to give up some attitude (a ‘consistency requirement’, in our terminology). Specifically, he argues that if you hold beliefs in *p* and also in *not p*, then ‘correct reasoning’ cannot conclude in non-belief in *p*. You might derive from your belief in *p* a positive belief in *I do not believe p*, or in *I ought not to believe p*, but these beliefs crucially differ from non-belief in *p*, and they come from ‘incorrect reasoning’, following Broome. Theorem 2 says something simpler and more general: reasoning cannot achieve any consistency requirements simpliciter, regardless of correctness considerations.

Broome argues that where reasoning fails ‘automatic processes will normally prevent you from having contradictory beliefs’ (2013: 279–280). We believe that reasoning can nonetheless play a crucial role in achieving consistency requirements, i.e., in removing some attitude from a given set *F* of mutually inconsistent attitudes within your constitution. We see three routes:

1. Reasoning might remove the grounds on which you hold or had formed some attitude in *F*. In a second step, the ‘ground-less’ attitude disappears automatically. There might be many ways to remove the grounds of an attitude. One might be to form the belief that you ought not to hold that attitude. Another might be to form some attitude on whose absence that attitude is premised.

2. Plausibly, some attitudes in *F* are not hard-wired in your constitution. They disappear after a while – they ‘expire’ or fail to be ‘refreshed’ – should the conditions for their survival no longer be met. In particular, some attitudes (beliefs, intentions, preferences etc.) might ultimately disappear should you develop certain other attitudes which conflict with them. Reasoning can thus block the survival of attitudes through generating attitudes in conflict with them. The belief that it rains might not be refreshed if you meanwhile form a belief in sunshine.

3. Psychologists often assume that your attitudes are of two sorts, ‘explicit’ and ‘implicit’ ones (e.g., Wilson et al. 2000). Implicit attitudes might be described as unconscious, never actively formed, or the result of automatic processes or ‘System 1’. Explicit attitudes might be described as conscious, actively formed, or the result of explicit reasoning or ‘System 2’. Explicit attitudes can sometimes crowd out implicit attitudes if there is some direct conflict between the two (Wilson et al. 2000: 102). By producing explicit attitudes, reasoning can thus make some implicit attitudes disappear, thereby restoring consistency. For example, suppose you hold inconsistent beliefs. By reasoning you come to believe a proposition *p*, which directly conflicts with your existing belief of *not p*. Your explicit belief of *p* then makes your (supposedly implicit) belief of *not p* disappear.

## 7.3 Can you achieve completeness requirements?

Consider a completeness requirement: you should hold at least one mental state from *U*, formally $R = \{C : C \cap U \neq \emptyset\}$. There is an easy (but, as will turn out, unsatisfactory) way to achieve *R*: fix an attitude $c \in U$, and adopt the rule to ‘always form attitude c’, formally the rule $r = (\emptyset, c)$ with an empty premise set. For example, let *T* be the theory of rationality defined by R1–R8 in Applica-
tion 2, and consider the preference-completeness requirement in R5 for options x, y,
formally $R^* = \{C : (x, y, \succ) \in C \text{ or } (y, x, \succ) \in C \text{ or } (x, y, \sim) \in C\}$. Any of the
rules $r_1 = (\emptyset, (x, y, \succ))$ ('always come to prefer $x$ to $y$'), $r_2 = (\emptyset, (y, x, \succ))$ ('always
come to prefer $y$ to $x$') or $r_3 = (\emptyset, (x, y, \sim))$ ('always become indifferent between $x$
and $y$') would achieve that requirement $R^*$. But this solution is unsatisfactory for
two reasons. Firstly, in many contexts, such arbitrary rules seem unjustified: why
for example should one systematically come to prefer $x$ to $y$ in the name of prefer-
ence completeness? But there is a second problem: while achieving a completeness
requirement, such a rule often causes violation of a consistency requirement. Sup-
pose you initially have no preference or indifference between $x$ and $y$, violating $R^*$,
but you prefer $y$ to another option $z$, and $z$ to $x$. So your initial constitution is
$C = \{(y, z, \succ), (z, x, \succ), \dots\}$, where ‘...’ stands for other states. As in the putative
solution, let your reasoning system $S$ contain the rule $r_1$, so that after reasoning the
revised constitution is $C|S = \{(x, y, \succ), (y, z, \succ), (z, x, \succ), \dots\}$. While $C|S$ satisfies
the completeness requirement $R^*$, it violates a preference-acyclicity requirement of
the theory (defined in footnote 8). A completeness requirement has been achieved at
the cost of a consistency requirement. The rules $r_2$ and $r_3$ lead to analogous problems.

In general, when can a rule of type $(\emptyset, m)$ cause a consistency violation? To answer
this question, we need the concept of falsifiability:

**Definition 15** *Given any theory of rationality T, a mental state m is falsifiable if some consistent constitution becomes inconsistent through adding m.*

In plausible theories of rationality, almost all states are falsifiable. For example,
$(p, bel)$ in Application 1 is falsifiable because it rules out (not $p, bel$); and $(x, y, \succ)$
in Application 2 is falsifiable because it rules out $(y, x, \succ)$, and $(x, y, \sim)$, and the
combination of $(y, z, \succ)$ and $(z, x, \succ)$, etc.

Our general result about completeness requirements states as follows:

**Theorem 3** *Given any theory of rationality T,*

(a) *some reasoning system achieves all completeness requirements of T, but*

(b) *no consistency-preserving reasoning system achieves any completeness require-*
*ment of T whose *U*-states are all falsifiable.*

The negative result (b) can be traced back to Broome's thesis that we cannot reason
from absences of attitudes towards attitudes – a thesis also endorsed by Kolodny (2005: 527–528). Reasoning from absences would have helped us achieve completeness re-
quirements because such requirements are expressible as if-then requirements where
the antecedent is the absence of certain states (all but one of the *U*-states) and the
consequent is the presence of an attitude (the remaining *U*-state). In sum, the diffi-
culty to achieve consistency requirements (Theorem 2) and completeness requirements
(Theorem 3(b)) stems from the Broomean impossibility to reason towards absences
and from absences, respectively.

Why does our negative finding in (b) not contradict our positive finding in Theorem
1 about closedness requirements, although certain degenerate completeness require-
ments (those with a single *U*-state) are also special closedness requirements? Part (b)
rules out such degenerate completeness requirements because their single *U*-state is
non-falsifiable, as one easily checks.

Although completeness requirements share with consistency requirements the un-
achievability through reasoning, completeness requirements seem even less under your
conscious control. In terms of the dual-system model, the conscious reasoning of Sys-
tem 2 operates on inputs generated by the unconscious processes of System 1. If these
inputs are insufficient to allow conscious reasoning to arrive at particular types of
conclusion, this does not seem to be a fault of reasoning – nor, one might think, a
contravention of rationality. Take the case of the preference completeness requirement
of choice theory. If, given the preferences and beliefs you actually hold, there is no
way of reasoning towards any specific preference or indifference between options x and
y, one might doubt that that rationality requires you to hold some such preference or
indifference, while being silent about which (Hausman 2012: 19).<sup>14</sup> However, banish-
ing completeness requirements altogether from theories of rationality may go too far.
As we now argue, certain completeness-like requirements have a strong claim to count
as rationality requirements.

8 Economic enkrasia and Buridan’s ass

The economic enkrasia requirement in R8 is neither a closedness requirement, nor a consistency requirement, nor a completeness requirement. Whether it is achievable through reasoning is thus not settled by Theorems 1-3. This question is however crucial for forming intentions and making choices. To address it, consider the famous problem of Buridan’s ass.

In the story, the ass is exactly equidistant from two identical bales of hay, one to
its right and one to its left. Unable to decide between the two, it starves to death.
Representing this situation in Application 2 of our model, let the ass face the feasible
set $Y = \{\mathit{left}, \mathit{right}, \mathit{starve}\}$, and have the initial constitution $C = \{(\mathit{left}, \mathit{right}, \sim), (\mathit{left}, \mathit{starve}, \succ), (\mathit{right}, \mathit{starve}, \succ), (Y, \mathit{bel})\}$. In our interpretation, the ass fails to
form an intention for *left* or *right*, which leads to the outcome *starve*, the only feasible
outcome requiring no intention.

Could the ass have solved its problem by reasoning? Broome (2013: 189–190, 263–
4) argues that the answer is ‘Yes’. His solution is strikingly simple: in our terminology,
the ass’s reasoning system can contain both of the rules $r_1 = (C, (\mathit{left}, \mathit{int}))$ and
$r_2 = (C, (\mathit{right}, \mathit{int}))$; and (something we do not engage with) both rules are correct.
According to Broome, ‘there is room for choice in reasoning’ (2013: 264), and so the
ass is free to choose which of the two rules to apply. Having chosen one, it gets to a
bale of hay and survives.

Viewed through our model, this solution is problematic, because there is nothing
in the domain of reasoning to stop the ass from forming both intentions and, if this
happens, reasoning cannot ex-post eliminate one of them. Psychologically, conflicting
intentions might have much the same unfortunate effects as the absence of intentions.

In having no intention, the ass effectively violates economic enkrasia R8. Economic

<footnote n="14"><sup>14</sup>It is sometimes claimed that, if your preferences are not complete and transitive, you are neces-
sarily vulnerable to ‘money pumps’. This is incorrect (e.g., Cubitt and Sugden 2001).</footnote>
enkrasia requirements are completeness requirements of the following generalized type:

**Definition 16** A conditional-completeness requirement is a requirement $R$ whereby possession of certain (premise) states implies possession of at least one of certain possible (conclusion) states; formally, $R = \{C : P \subseteq C \Rightarrow C \cap U \neq \emptyset\}$ for some sets of states $P$ and $U \neq \emptyset$.

Conditional-completeness requirements indeed generalize completeness requirements, obtained when $P = \emptyset$. Are conditional-completeness requirements achievable through reasoning? Theorem 3 generalizes as follows.

**Definition 17** Given any theory of rationality $T$ and any set $P \subseteq M$ of mental states, a mental state $m \in M \setminus P$ is **falsifiable given** $P$ if some consistent constitution that includes $P$ becomes inconsistent through adding $m$.

**Theorem 4** Given any theory of rationality $T$,

(a) some reasoning system achieves all conditional-completeness requirements of $T$, but

(b) no consistency-preserving reasoning system achieves any conditional-completeness requirement whose $U$-states are all falsifiable given $P$.

Illustrating part (a), if the ass's reasoning system contains either (or both) of the rules $r_1$ and $r_2$, the ass can form an intention to go to one bale of hay, thereby achieving the economic enkrasia requirement. But this leads to conflicting intentions whenever (against the story) another intention was already present – which illustrates part (b).

Could different reasoning rules solve this problem? Generalizing the notion of ‘rule’, might one replace $r_1$ and $r_2$ with a single ‘rule’ that derives (right, int) from the presence of the states in $C$ and the absence of (left, int)? But that would be reasoning from an absence, which Broome explicitly rejects. Or (making the strong assumption of negative introspection), one might assume that whenever the ass lacks an intention it believes that it lacks it, and then replace $r_1$ and $r_2$ with a single rule that derives (right, int) from the union of $C$ and {(no option is currently intended, bel)}. But that would require second-order reasoning, which seems contrary to the spirit of Broome’s approach, although compatible with our Broomean notion of a ‘rule’.

Perhaps the most Broomean response to the problem would be to keep $r_1$ and $r_2$ and to argue that automatic processes will induce consistency by eliminating ‘surplus’ intentions. That at least would have the virtue of offering a unified solution to the difficulty to achieve requirements of consistency, completeness, and conditional completeness: automatic causal processes jump in where reasoning fails.

## 9 Do behavioural anomalies come from reasoning errors?

When behavioural social scientists observe choice patterns that violate rationality axioms of choice theory, they often claim that these patterns stem from imperfect reasoning. Such claims are sometimes used to justify public policies such as ‘nudging’ that are said to help individuals avoid reasoning errors (e.g. Thaler and Sunstein,
2008). We now assess these claims in the light of our Broomean analysis. We focus on *context-dependent choice reversals*, one of the most common types of ‘anomaly’ investigated in behavioural economics. Two choice contexts *K* and *K'* both offer the options *x* and *y*, and possibly others. In terms of all properties which might plausibly matter, choosing *x* in context *K* is equivalent to choosing *x* in *K*', and similarly for *y*. Yet the agent persistently chooses *x* in *K* and *y* in *K'* (and in each case is even willing to pay a small premium for the chosen option). Contextual features that have been found to influence choices include which option is framed as the status quo, whether the ranking of two options is elicited through a binary choice or through separate money valuations, the degree to which the choosers attention is drawn to the each option, and the presence or absence of ‘irrelevant’ options in the feasible set. Are such violations of choice-theoretic rationality really evidence of reasoning failure?

On one account of reasoning failure in behavioural economics (e.g. Thaler and Sunstein 2008: 40-41), the agent has a context-independent and fully conscious preference for one of the options over the other, say for *x* over *y*. But in context *K'* psychological ‘cues’ activate a temptation to take *y* instead. This is a failure of self-control: in terms of the dual-system model, System 1 is cutting in to obstruct System 2 reasoning. In context *K'* the agent fails to reason towards the right choice intention. In terms of our analysis, reasoning fails to achieve Economic Enkrasia, by not properly deriving intentions from preferences.

But not all context-dependent choice reversals can credibly be explained by lack of self-control or ‘economic akrasia’. Another explanatory strategy still assumes classical and context-independent preferences, but assumes that, because optimizing with respect to them is cognitively costly, the agent uses rules of thumb to simplify decision problems. For example, Manzini and Mariotti (2012) explain violations of the weak axiom of revealed preference by assuming that the agent first selects a shortlist of feasible options through a context-dependent heuristic, and then chooses the most-preferred option from that shortlist.

Yet another explanation assumes that the agent mentally represents each option as a bundle of ‘motivationally salient properties’, and maximizes preferences over such property bundles. Although these preferences are context-independent, choice reversals happen because the context influences which properties are motivationally salient and hence how options are perceived (Dietrich and List 2016; also Dietrich and List 2017).

All three explanations assume context-independent preferences of some kind: be-
havioural anomalies occur because of context-dependence in the agent's exposure to
temptation, in the choice heuristic used, or in the agent's perception of options. This
background assumption is common to most behavioural economists' accounts of 'im-
perfect reasoning'. But, as noted in Section 1, such accounts presuppose a positive
answer to Broome's fundamental question of whether reasoning can achieve rational-
ity. Given the mixed results of our paper, that presupposition seems questionable.

Our findings about the unachievability of completeness requirements through reas-
oning suggest a different explanation of choice reversals: preference incompleteness.
While making no reasoning errors, the agent lacks any preference or indifference
between the options in question, a possibility defended by Infante et al. (2016). There-
fore, enkratic reasoning rules do not apply, so that no choice intention is formed. The automatic processes of System 1 then step in and lead to context-dependent choice.

## A A general type of requirement

All types of requirements considered above are special cases of a single type, namely a generalized version of conditional-completeness requirements which no longer imposes that $U \neq \emptyset$:

**Definition 18** A *unified requirement* is a requirement *R* whereby possession of certain (premise) states implies possession of at least one of certain possible (conclusion) states; formally, $R = \{C : P \subseteq C \Rightarrow C \cap U \neq \emptyset\}$ for some sets of states *P* and *U*, not both empty.

Unified requirements are unificatory in two senses. Firstly, they simultaneously generalize our three earlier requirement classes:

**Remark 1** *Unified requirements* $R = \{C : P \subseteq C \Rightarrow C \cap U \neq \emptyset\}$ reduce to

* completeness requirements if $P = \emptyset$, as then $R = \{C : C \cap U \neq \emptyset\}$,

* consistency requirements if $U = \emptyset$, as then $R = \{C : \text{not } P \subseteq C\}$,

* closedness requirements if $U$ is a singleton $\{c\}$, as then $R = \{C : P \subseteq C \Rightarrow c \in C\}$.<fnref n="15" />

Secondly, every theory can be axiomatized in terms of unified requirements only:

**Theorem 5** Every theory of rationality *T* is an intersection (conjunction) of unified requirements.

Given the generality of unified requirements, it is natural to ask how far they are achievable through reasoning. Our earlier theorems provide the answer. If $U = \emptyset$, our negative conclusion about consistency requirements applies (Theorem 2). If $U$ is singleton, our positive conclusion about closedness requirements applies (Theorem 1). Otherwise our essentially negative conclusion about completeness or conditional-completeness requirements applies (Theorems 3 and 4).

## B Proofs

**Proof of Theorem 1.** Let *T* be a theory. Define the reasoning system *S* as containing all rules corresponding to closedness requirements of *T*. So *S* = {*(*P*, *c*)* : the closedness requirement given by (*P*, *c*) is a requirement of *T*}. Now consider any initial constitution *C* and any closedness requirement *R* of *T*, given by a pair (*P*, *c*).

*Claim 1:* $C|S$ satisfies the requirement *R*. This is true because if $P \subseteq C|S$, then $c \in C|S$ as $C|S$ is closed under *S* which contains (*P*, *c*).

<footnote n="15">¹⁵Furthermore, unified requirements reduce to negative-closedness requirements (defined in footnote 11), which are obtained if *P* is a singleton {*c*} as then $R = \{C : c \in C \Rightarrow C \cap U \neq \emptyset\}$ or equivalently $R = \{C : C \cap U = \emptyset \Rightarrow c \notin C\}$.</footnote>
*Claim 2:* *S* preserves consistency. Assume *C* is consistent, hence a subset of a rational constitution *C*\* ∈ *T*. We show that *C*|*S* ⊆ *C*\* (which establishes consistency). This follows from two facts. The first is that *C*\* is closed under *S*, because it is rational and hence in particular satisfies all closedness requirements of *T*. The second is that *C*|*S* is by definition the smallest extension of *C* closed under *S*. ■

**Proof of Theorem 2.** Consider a reasoning system *S*, and a consistency requirement *R*, say *R* = {*C* : not *F* ⊆ *C*}. It suffices to specify a constitution *C* such that *C*|*S* violates *R*. Simply let *C* be any constitution such that *F* ⊆ *C*. Since *F* ⊆ *C* and *C* ⊆ *C*|*S*, we have *F* ⊆ *C*|*S*. So the revised constitution *C*|*S* violates *R*. ■

**Proof of Theorems 3 and 4.** Since Theorem 3 is the special case of Theorem 4 in which *P* = *Ø*, it suffices to prove Theorem 4. Consider any theory *T*.

(a) By definition, each conditional-completeness requirement of *T* is conditional on some set *P* of states, and has at least one *U*-state. For each such requirement, fix an arbitrary member *m*<sub>*U*</sub> of *U*, and define the rule *r*<sub>*U*</sub> = (*P*, *m*<sub>*U*</sub>). Let *S* be any reasoning system containing one such rule for each conditional-completeness requirement. Clearly, *S* achieves all conditional-completeness requirements of *T*.

(b) Consider any conditional-completeness requirement of *T*, defined by some *P* and *U* (where *U* ≠ *Ø*). Suppose some consistency-preserving reasoning system *S* achieves this requirement. Since the requirement is achieved, there must be some *m'* ∈ *U* such that *m'* ∈ *P*|*S*. To complete the proof, it suffices to show that *m'* is not falsifiable given *P*. To that end, we consider a consistent *C'* ⊇ *P*, and must show consistency of *C'* ∪ {*m'*}. Because *S* is consistency-preserving, *C'*|*S* is consistent. But *m'* ∈ *P*|*S* and *P* ⊆ *C'* imply *m'* ∈ *C'*|*S*. As *C'*|*S* (= (*C'*|*S*) ∪ {*m'*}) is consistent, so is its subset *C'* ∪ {*m'*}. ■

**Proof of Theorem 5.** Consider a theory *T*. For any constitution *C* ∉ *T*, *T* makes the requirement not to have that constitution, i.e., the ‘unique-exclusion’ requirement *R*<sub>*C*</sub> = {*C' : C' ≠ C*}. The theory is expressible as the conjunction (intersection) of its unique-exclusion requirements: *T* = ∩<sub>*C*⊆*M*:*C*∉*T*</sub>*R*<sub>*C*</sub>. Each *R*<sub>*C*</sub> is a unified requirement, as it equivalently demands that possession of all states in *C* implies possession of at least one state not in *C*, formally *R*<sub>*C*</sub> = {*C' : C ⊆ *C'* ⇒ *C'* ∩ (*M* ∩ *C*) ≠ *Ø*}. ■

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