Franz Dietrich

Robert Sugden

A Broomean model of rationality and reasoning

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 di¤ers 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 de…ciencies in choice theory and behavioural economics.

Introduction

John [START_REF] Broome | Wide or narrow scope?[END_REF][START_REF] Broome | Rationality through reasoning[END_REF] widely discussed philosophical theory of rationality and reasoning is strikingly di¤erent 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. 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., [START_REF] Par…t | Reasons and Motivation[END_REF][START_REF] Kolodny | Why be rational?[END_REF], 2007[START_REF] Wright | Comment on Paul Boghossian[END_REF][START_REF] Boghossian | Rationality, reasoning and rules: re ‡ections on Broome's rationality through reasoning[END_REF][START_REF] Cullity | Describing rationality[END_REF][START_REF] Pettit | Broome on reasoning and rule-following[END_REF][START_REF] Southwood | The motivation question[END_REF].

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 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 …nished their job when they have described requirements 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 requirement ' (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.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. 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. 4Despite 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 di¤ers from ordinary rational-choice models through focusing on psychology and through modelling the reasoning process, including theoretical reasoning (reasoning with beliefs) and practical reasoning (reasoning towards intentions).

Psychologists and thereafter behavioural social scientists have often classi…ed mental processes into two systems: the fast and automatic 'System 1'which generates impressions, 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 reasoning process by which this is supposed to happen. 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 speci…c types of rationality 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 de…ciencies in received ideas about rationality, and or gaps in Broome's theory.

Key features of Broome' s structural theory

Let us give some background that will guide our model. Broome fundamentally distinguishes between 'rationality'and 'reasoning'. Rationality is understood as one of many possible sources of requirements on your mental states. Other sources of requirements 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 [START_REF] Kolodny | Why be rational?[END_REF][START_REF] Kolodny | State or process requirements?[END_REF]; 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 intentions ('practical reasoning'), although Broome gives special importance to these two classic types of reasoning (2013: 250). Broome describes reasoning as 'a rulegoverned operation on the contents of your conscious attitudes ' (2013: 234); as a 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 a¤ect 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 …nding (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 …rst-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 …rst-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 …rst-order attitudes.

Against this background, we now present our Broomean model.

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; pref erence) 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 …xed 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 …xed 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 f1; 2; : : :g. 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. …nal outcomes (or outcome lotteries), intentions w.r.t. own actions, and beliefs w.r.t. actions of opponents.

De…nition 1 A (mental) state or attitude is a tuple (p 1 ; :::; p n ; a) -called the 'attitude of type a towards p 1 ; :::; p n ' -where a is an attitude type in A, n is its number of places, and p 1 ; :::; p n 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': De…nition 2 A (mental) constitution is a set C 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 a¤airs, 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, de…ned precisely in the companion paper [START_REF] Dietrich | Psychology analysed logically: when are our attitudes consistent, complete, and deductively closed? working paper[END_REF]. 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 …xed non-empty set X of mutually exclusive choice options, e.g., goods or political candidates. The agent has attitudes of preference and indi¤erence 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 indi¤erence, feasible sets Y in 2 X nf?g, the objects of feasibility beliefs. Formally, L = X [ (2 X nf?g). 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 = fint; bel; ; g, 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 indi¤erence between x and y; bel is a one-place attitude with domain 2 X nf?g; (Y; bel) represents belief that the feasible set is Y . One could replace and by a single weak-preference attitude %; then A = fint; bel; %g. This would come closer to choice-theoretic models, which usually start from a single weak-preference relation, not two relations of strict preference and indifference. Within our mentalist framework, we prefer working with and , because a weak-preference attitude % seems arti…cial. Mentalistically, the weak preference relation in choice theory is best interpreted as a mathematically convenient representation of the disjunction of two attitudes. Technically, the di¤erence between working with and and working with % is small: using and one can express weak preference (as preference-or-indi¤erence), and conversely, using % one can express both strict preference (as weak preference without indi¤erence) and indi¤erence (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 indi¤erence, and mutual exclusiveness between preference and indi¤erence, de…ned in R2-R4 below.

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 de…ne 'requirements' generically, without yet caring about whether they are imposed by rationality.

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

De…nition 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 = ?).

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: 6Non-contradiction: you do not believe both p and not p, formally R = fC : (p; bel) 2 C) ) (not p; bel) 6 2 Cg. Parameter: p 2 L. Modus ponens: believing p and if p then q implies believing q, formally R = fC : (p; bel); (if p then q; bel) 2 C) ) (q; bel) 2 Cg. Parameters: p; q 2 L. Enkrasia (non-akrasia): believing obligatorily p implies intending p, formally R = fC : (obligatorily p; bel) 2 C ) (p; int) 2 Cg. Parameter: p 2 L. Necessary means: intending p and believing q is a means implied by p implies intending q, formally R = fC : (p; int); (q is a means implied by p; bel) 2 C ) (q; int) 2 Cg. Parameters: p; q 2 L. If propositions have no formal structure, these de…nitions of requirements are informal, since then we cannot give formal meaning to composite propositions such as not p, if p then q and obligatorily p (where p; q 2 L). But these de…nitions can, if wished, be turned into formal de…nitions by adopting the syntactic (intensional) or set-theoretic (extensional) model of propositions sketched above (see [START_REF] Dietrich | Psychology analysed logically: when are our attitudes consistent, complete, and deductively closed? working paper[END_REF] 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 satis…es 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 indi¤erence, rather than a single weak-preference attitude. 7Conditions R6-R7 exclude contradictory intentions or feasibility beliefs. Condition R8 re ‡ects 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 di¤ers 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 jZj 2), you intend a speci…c 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 …rst introduce our typology of requirements.

De…nition 4 A consistency requirement is a requirement R that forbids holding certain mental states simultaneously; formally, R = fC : not F Cg for some nonempty set F of mental states, the 'forbidden set'.

In Application 1, the non-contradiction requirement is a consistency requirement (F = f(p; bel); (not p; bel)g). Application 2 contains consistency requirements in R2 (F = f(x; y; ); (y; x; )g), R4 (F = f(x; y; ); (x; y; )g), R6 (F = f(Y; bel); (Y 0 ; bel)g), and R7 (F = f(x; int); (y; int)g). Our requirements R1-R4 imply preference acyclicity, another schema of consistency requirements. 8

De…nition 5 A completeness requirement is a requirement R that forbids holding none of certain mental states; formally, R = fC : C \ U 6 = ?g for some non-empty set U , the 'unavoidable set'.

Completeness requirements are non-abstention requirements. Examples are the preference-completeness requirements in R5 (U = f(x; y; ); (y; x; ); (x; y; )g). In Application 1 one might require 'belief completeness'relative to certain propositions p; here U = f(p; bel); (not p; bel)g.

Finally, a closedness requirement demands that holding certain (premise) states implies holding a certain (conclusion) state: De…nition 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 = fC : P C ) c 2 Cg for some set of ('premise') states P and some ('conclusion') state c.

that weak preference of x to y and of y to z implies weak preference of x to z (formally

R = fC : if [(x; y; ) 2 C or (x; y; ) 2 C] & [(y; z; ) 2 C or (y; z; ) 2 C] then [(x; z; ) 2 C or (x; z; ) 2 C]g).
R5 is our counterpart of classic choice-theoretic completeness, which requires for all x; y 2 X that x be weakly preferred to y or y to x (formally, R = fC : [(x; y; ) 2 C or (x; y; ) 2 C] or [(y; x; ) 2 C or (y; x; ) 2 C]g). R2-R4 have no classic counterparts, as they correspond to properties which the classic approach treats as true by de…nition (through the way it de…nes strict preference and indi¤erence from weak preferences). In fact, each of R1 and R5 implies its classic counterpart, and is equivalent to it given R2-R4. Modus ponens, enkrasia and necessary means in Application 1 are schemas of closedness requirements, as are transitivity R1 and symmetry of indi¤erence R3 in Application 2.

Our three types of requirement are mutually exclusive, except for one special case: requirements of the form R = fC : m 2 Cg ('you must hold state m', for given m 2 M ) are expressible both as degenerate completeness requirements R = fC : C \ fmg 6 = ?g and as degenerate closedness requirements R = fC : ? C ) m 2 Cg. 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.

Theories of rationality and their requirements

A theory of rationality expresses a speci…c 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 de…ning a 'theory of rationality'as a set of requirements would be both unparsimonious and too permissive. 9 We thus identify a theory, not with its set of requirements, but with the set of constitutions it deems rational: De…nition 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 di¤erently: 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 de…ned as follows:

De…nition 8 Given a theory of rationality T , the requirements of T are those requirements R which follow from T , i.e., for which T 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. 10 Our de…nitional 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 …rst comes up with requirements of rationality, whose conjunction then de…nes one's theory of rationality. We have already argued that a mentalistic version of the standard theory of rational choice under certainty can be 9 Unparsimonious: theories usually have enormously many requirements, most of which are arti…cial (e.g., whenever R and R 0 are requirements of the theory, so are their conjunction R\R 0 and disjunction R [ R 0 ). 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.

1 0 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 \ T 6 = ?).

axiomatized by R1-R8. Formally, the intersection of those requirements de…nes a theory of rationality T . Notice that exactly the same theory could be axiomatized by a di¤erent set of requirements. For example, we might replace transitivity R1 by Savage's (1954: 17- Your way to reason is de…ned by the set of rules you follow, to be called your 'reasoning system'.

De…nition 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 C ) c 2 C (equivalently, Cjr = 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 …nally 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':

De…nition 11 The revision (or closure) of C through a reasoning system S is the constitution CjS obtained from C by adding mental states until the constitution is closed under S. Formally, CjS is the minimal extension of C closed under S. 12 Revising a constitution through a reasoning system resembles the familiar operation 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 de…ned shortly. So, loosely speaking, reasoning should improve rationality by Desideratum 1, without elsewhere compromising rationality by Desideratum 2. The following de…nition makes Desideratum 1 precise: De…nition 12 A reasoning system S achieves a requirement R if for each constitution C its revision CjS satis…es R.

The next two de…nitions clarify Desideratum 2:

De…nition 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 0 C is rational under T .

How does consistency of a constitution relate to our notion of consistency requirements? As one can show, a constitution is consistent if and only if it satis…es all consistency requirements of the theory. 13 De…nition 14 Given a theory of rationality, a reasoning system S preserves consistency if for each consistent constitution C its revision CjS is still consistent.

Achieving (or not achieving) rationality through reasoning

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

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.

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 CjS = M , which trivially satis…es 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 = fC : P C ) c 2 Cg. 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-o¤: 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.

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 con ‡icting 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 …xed 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) o¤ers a cautious defence of an apparently similar claim to Theorem 2: through 'correct reasoning', you cannot achieve a requirement which 13 Documents de travail du Centre d'Economie de la Sorbonne -2018.30 forces you to give up some attitude (a 'consistency requirement', in our terminology). Speci…cally, 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 di¤er 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 con ‡ict with them. Reasoning can thus block the survival of attitudes through generating attitudes in con ‡ict 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., [START_REF] Wilson | A model of dual attitudes[END_REF]. 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 con ‡ict 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 con ‡icts with your existing belief of not p. Your explicit belief of p then makes your (supposedly implicit) belief of not p disappear.

Can you achieve completeness requirements?

Consider a completeness requirement: you should hold at least one mental state from U , formally R = fC : C \ U 6 = ?g. There is an easy (but, as will turn out, unsatisfactory) way to achieve R: …x an attitude c 2 U , and adopt the rule to 'always form attitude c', formally the rule r = (?; c) with an empty premise set. For example, let T be the theory of rationality de…ned by R1-R8 in Applica-tion 2, and consider the preference-completeness requirement in R5 for options x; y, formally R = fC : (x; y; ) 2 C or (y; x; ) 2 C or (x; y; ) 2 Cg. Any of the rules r 1 = (?; (x; y; )) ('always come to prefer x to y'), r 2 = (?; (y; x; )) ('always come to prefer y to x') or r 3 = (?; (x; y; )) ('always become indi¤erent 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 unjusti…ed: why for example should one systematically come to prefer x to y in the name of preference completeness? But there is a second problem: while achieving a completeness requirement, such a rule often causes violation of a consistency requirement. Suppose you initially have no preference or indi¤erence between x and y, violating R , but you prefer y to another option z, and z to x. So your initial constitution is C = f(y; z; ); (z; x; ); : : :g, 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 CjS = f(x; y; ); (y; z; ); (z; x; ); : : :g. While CjS satis…es the completeness requirement R , it violates a preference-acyclicity requirement of the theory (de…ned 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 (?; m) cause a consistency violation? To answer this question, we need the concept of falsi…ability:

De…nition 15 Given any theory of rationality T , a mental state m is falsi…able if some consistent constitution becomes inconsistent through adding m.

In plausible theories of rationality, almost all states are falsi…able. For example, (p; bel) in Application 1 is falsi…able because it rules out (not p; bel); and (x; y; ) in Application 2 is falsi…able because it rules out (y; x; ), and (x; y; ), and the combination of (y; z; ) and (z; x; ), 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 requirement of T whose U -states are all falsi…able.

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 requirements 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 di¢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 …nding in (b) not contradict our positive …nding in Theorem 1 about closedness requirements, although certain degenerate completeness requirements (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-falsi…able, as one easily checks.

Although completeness requirements share with consistency requirements the unachievability through reasoning, completeness requirements seem even less under your conscious control. In terms of the dual-system model, the conscious reasoning of System 2 operates on inputs generated by the unconscious processes of System 1. If these inputs are insu¢ cient 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 speci…c preference or indi¤erence between options x and y, one might doubt that that rationality requires you to hold some such preference or indi¤erence, while being silent about which (Hausman 2012: 19). 14 However, banishing 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.

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 = flef t; right; starveg, and have the initial constitution C = f(lef t; right; ); (lef t; starve; ); (right; starve; ); (Y; bel)g. In our interpretation, the ass fails to form an intention for lef t 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; (lef t; int)) and r 2 = (C; (right; 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, con ‡icting intentions might have much the same unfortunate e¤ects as the absence of intentions.

In having no intention, the ass e¤ectively violates economic enkrasia R8. Economic 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 0 both o¤er 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 0 , and similarly for y.

Yet the agent persistently chooses x in K and y in K 0 (and in each case is even willing to pay a small premium for the chosen option). Contextual features that have been found to in ‡uence 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 chooser's 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 0 psychological 'cues'activate a temptation to take y instead. This is a failure of selfcontrol: in terms of the dual-system model, System 1 is cutting in to obstruct System 2 reasoning. In context K 0 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, [START_REF] Manzini | Categorize then choose: Boundedly rational choice and welfare[END_REF] explain violations of the weak axiom of revealed preference by assuming that the agent …rst 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 in ‡uences 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: behavioural 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 'imperfect reasoning'. But, as noted in Section 1, such accounts presuppose a positive answer to Broome's fundamental question of whether reasoning can achieve rationality. Given the mixed results of our paper, that presupposition seems questionable.

Our …ndings about the unachievability of completeness requirements through reasoning suggest a di¤erent explanation of choice reversals: preference incompleteness. While making no reasoning errors, the agent lacks any preference or indi¤erence between the options in question, a possibility defended by [START_REF] Infante | Preference puri…cation and the inner rational agent: a critique of the conventional wisdom of behavioural welfare economics[END_REF]. 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 6 = ?: De…nition 18 A uni…ed requirement is a requirement R whereby possession of certain (premise) states implies possession of at least one of certain possible (conclusion) states; formally, R = fC : P C ) C \ U 6 = ?g for some sets of states P and U , not both empty.

Uni…ed requirements are uni…catory in two senses. Firstly, they simultaneously generalize our three earlier requirement classes:

Remark 1 Uni…ed requirements R = fC : P C ) C \ U 6 = ?g reduce to completeness requirements if P = ?, as then R = fC : C \ U 6 = ?g, consistency requirements if U = ?, as then R = fC : not P Cg, closedness requirements if U is a singleton fcg, as then R = fC : P C ) c 2 Cg. 15 Secondly, every theory can be axiomatized in terms of uni…ed requirements only:

Theorem 5 Every theory of rationality T is an intersection (conjunction) of uni…ed requirements.

Given the generality of uni…ed requirements, it is natural to ask how far they are achievable through reasoning. Our earlier theorems provide the answer. If U = ?, 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 conditionalcompleteness requirements applies (Theorems 3 and 4).

B Proofs

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

Claim 1: CjS satis…es the requirement R. This is true because if P CjS, then c 2 CjS as CjS is closed under S which contains (P; c).

Claim 2 : S preserves consistency. Assume C is consistent, hence a subset of a rational constitution C 2 T . We show that CjS C (which establishes consistency). This follows from two facts. The …rst is that C is closed under S, because it is rational and hence in particular satis…es all closedness requirements of T . The second is that CjS is by de…nition the smallest extension of C closed under S. (a) By de…nition, 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, …x an arbitrary member m U of U , and de…ne the rule r U = (P; m U ). 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 , de…ned by some P and U (where U 6 = ?). Suppose some consistency-preserving reasoning system S achieves this requirement. Since the requirement is achieved, there must be some m 0 2 U such that m 0 2 P jS. To complete the proof, it su¢ ces to show that m 0 is not falsi…able given P . To that end, we consider a consistent C 0 P , and must show consistency of C 0 [ fm 0 g. Because S is consistency-preserving, C 0 jS is consistent. But m 0 2 P jS and P C 0 imply m 0 2 C 0 jS. As C 0 jS (= (C 0 jS) [ fm 0 g) is consistent, so is its subset C 0 [ fm 0 g. Proof of Theorem 5. Consider a theory T . For any constitution C 6 2 T , T makes the requirement not to have that constitution, i.e., the 'unique-exclusion'requirement R C = fC 0 : C 0 6 = Cg: The theory is expressible as the conjunction (intersection) of its unique-exclusion requirements: T = \ C M :C6 2T R C . Each R C is a uni…ed requirement, as it equivalently demands that possession of all states in C implies possession of at least one state not in C, formally R C = fC 0 : C C 0 ) C 0 \ (M \ C) 6 = ?g.

C References

8

  Preference acyclicity forbids a preference cycle over options x1; : : : ; xn; so F = f(x1; x2; ); (x2; x3; ); : : : ; (xn 1; xn; ); (xn; x1; )g. Parameters: n 2 and x1; :::; xn 2 X.

  Proof of Theorem 2. Consider a reasoning system S, and a consistency requirement R, say R = fC : not F Cg. It su¢ ces to specify a constitution C such that CjS violates R. Simply let C be any constitution such that F C. Since F C and C CjS, we have F CjS. So the revised constitution CjS violates R.Proof of Theorems 3 and 4. Since Theorem 3 is the special case of Theorem 4 in which P = ?, it su¢ ces to prove Theorem 4. Consider any theory T .

  for details).

	Examples of requirements within Application 2:
	R1: Transitivity. The transitivity schema consists of four subschemas, namely
	transitivity of strict preference, transitivity of indi¤erence, and two mixed trans-
	itivities:

R1 : Preferring x to y and y to z implies preferring x to z, formally R = fC : if (x; y; ); (y; z; ) 2 C then (x; z; ) 2 Cg. Parameters: x; y; z 2 X. R1 : Indi¤erence between x and y and between y and z implies indi¤erence between x and z, formally R = fC : if (x; y; ); (y; z; ) 2 C then (x; z; ) 2 Cg. Parameters: x; y; z 2 X. R1 ; : Preference of x to y and indi¤erence between y and z imply preference of x to z, formally R = fC : if (x; y; ); (y; z; ) 2 C then (x; z; ) 2 Cg. Parameters: x; y; z 2 X. R1 ; : Indi¤erence between x and y and preference of y to z imply preference of x to z, formally R = fC : if (x; y; ); (y; z; ) 2 C then (x; z; ) 2 Cg. Parameters: x; y; z 2 X. R2: Asymmetry of . Preferring x to y excludes preferring y to x, formally R = fC : (x; y; ) 2 C ) (y; x; ) 6 2 Cg. Parameters: x; y 2 X. R3: Symmetry of . Indi¤erence between x and y implies indi¤erence between y and x, formally R = fC : (x; y; ) 2 C ) (y; x; ) 2 Cg. Parameters: x; y 2 X. R4: Exclusiveness between and . Preference of x to y excludes indi¤erence between x and y, formally R = fC : (x; y; ) 2 C ) (x; y; ) 6 2 Cg. Parameters: x; y 2 X. R5: Preference completeness. The options x and y are compared, formally R = fC : (x; y; ) 2 C or (y; x; ) 2 C or (x; y; ) 2 Cg. Parameters: x; y 2 X. R6: No con ‡icting feasibility beliefs. You do not believe both Y and Y 0 to be the feasible set, formally R = fC : (Y; bel) 2 C ) (Y 0 ; bel) 6 2 Cg. Parameters: distinct feasible sets Y; Y 0 2 2 X nf?g. R7: No con ‡icting intentions. You do not intend both x and y, formally R = fC : (x; int) 2 C ) (y; int) 6 2 Cg. Parameters: distinct options x; y 2 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 = fC : if (Y; bel) 2 C, (x; y; ) 2 C for all x 2 Z; y 2 Y nZ, and (x; z; ) 2 C for all distinct x; z 2 Z, then (x; int) 2 C for some x 2 Zg. Parameters: Y X and Z Y (Z 6 = ?).

The debate between behaviourism and mentalism is ongoing; e.g.,[START_REF] Hausman | The Philosophy of Economics: An Anthology[END_REF]

2012), Dietrich and List (2016a) and Okasha (2016). 1 Documents de travail du Centre d'Economie de la Sorbonne -2018.30

[START_REF] Lewis | Convention: A Philosophical Study[END_REF] 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 speci…c modes of reasoning (see[START_REF] Cubitt | Common knowledge, salience and convention: a reconstruction of David Lewis's game theory[END_REF].

In 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.

[START_REF] Infante | Preference puri…cation and the inner rational agent: a critique of the conventional wisdom of behavioural welfare economics[END_REF] document the widespread use of the concept of 'reasoning error'in behavioural economics.2 Documents de travail du Centre d'Economie de laSorbonne -2018.30

See Infante et al. (2016) for a documentation of this claim.3 Documents de travail du Centre d'Economie de laSorbonne -2018.30

Documents de travail du Centre d'Economie de laSorbonne -2018.30

Broome'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).

7Documents de travail du Centre d'Economie de laSorbonne -2018.30

R1 is our counterpart of classic choice-theoretic transitivity, which requires for all x; y; z 2 X

Documents de travail du Centre d'Economie de laSorbonne -2018.30

1 Negative transitivity consists in the requirements R = fC : [(x; y; ) 6

C&(y; z; ) 6 2 C] ) (x; z; ) 6 2 C)g for all options x; y; z 2 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 = fC : C \ P = ? ) c 6 2 Cg for certain P M and c 2 M . 1 2 This 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.

3 First, 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 0 C also violates it and is thus irrational, implying inconsistency of C.12 Documents de travail du Centre d'Economie de laSorbonne -2018.30

4 It is sometimes claimed that, if your preferences are not complete and transitive, you are necessarily vulnerable to 'money pumps'. This is incorrect (e.g.,Cubitt and

[START_REF] Cubitt | On money pumps[END_REF].

5 Furthermore, uni…ed requirements reduce to negative-closedness requirements (de…ned in footnote 11), which are obtained if P is a singleton fcg as then R = fC : c

C ) C \ U 6 = ?g or equivalently R = fC : C \ U = ? ) c 6 2 Cg.19 Documents de travail du Centre d'Economie de laSorbonne -2018.30

enkrasia requirements are completeness requirements of the following generalized type:

De…nition 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 = fC : P C ) C \ U 6 = ?g for some sets of states P and U 6 = ?.

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

De…nition 17 Given any theory of rationality T and any set P M of mental states, a mental state m 2 M nP is falsi…able 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 falsi…able 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 con ‡icting intentions whenever (against the story) another intention was already present -which illustrates part (b).

Could di¤erent 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 (lef t; 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 f(no option is currently intended; bel)g. 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 o¤ering a uni…ed solution to the di¢ culty 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,