1 Advancing the debate

The debate on the content and nature of rational requirements that govern attitudinal coherence faces a serious obstacle.Footnote 1 On the one hand, most parties to the debate agree about numerous combinations of attitudesFootnote 2 that violate a requirement of rationality. For example, if you hold a pair of contradictory beliefs or intentions, fail to intend a means you acknowledge as necessary to your ends, have no intention of doing what you believe you ought to do, hold intransitive preferences, or assign inconsistent probabilities to states of affairs, you violate a requirement of rationality.

On the other hand, there is no unified agreement on how we should formulate the requirements that are violated in the above examples. Philosophers disagree about the logical form of rational requirements and, in particular, about whether those requirements are best formulated as requirements with conditional contents or as conditionals whose consequents express a rational requirement.

Consider an illustration of this problem. Suppose you are akratic: you have no intention to X even though you believe that you ought to X. I assume that you violate a requirement of rationality.Footnote 3 But which precise requirement do you violate? How should we formulate it?

The answer to this question is not clear. There are at least two plausible formulations, both of which do an equally good job of generating the violation. According to the first formulation, rationality simply prohibits the combination of the two attitudes. What is required is that you not [believe that you ought to X and not intend to X]. Or, put in its equivalent material conditional form:

(W):

Necessarily: rationality requires of you that [(you believe you ought to X) → (you intend to X)],

where → represents a material conditional. In (W) ‘rationality requires’ has a wide conditional scope: it governs the entire material conditional ‘(you believe you ought to X) → (you intend to X)’. This guarantees that by believing that you ought to X without intending to X, you violate a rational requirement. (The conditional content of the requirement then turns out to be false.) It also guarantees that you can satisfy this requirement by either intending to X or not believing that you ought to X. (The content of the requirement then turns out to be true.) Accordingly, (W) represents a first prima facie plausible requirement formulation that would ensure the irrationality of akrasia.

According to the second formulation, ‘rationality requires’ does not govern a material conditional. Instead, it says that either it is not the case that you believe that you ought to X or rationality requires you to intend to X. Or, again, expressed with the help of a material conditional:

(N):

Necessarily: (you believe you ought to X) → [rationality requires of you that (you intend to X)].

In (N) ‘rationality requires’ has a narrow conditional scope: it governs only the consequent of the material conditional ‘(you believe you ought to X) → (you intend to X)’. This also ensures that akrasia is irrational. Your belief that you ought to X entails that you are rationally required to intend to X. But, being akratic, you do not intend to X. Thus, you violate a requirement. You can satisfy this requirement in only one way, i.e. by intending to X. (N) represents a second prima facie plausible formulation that would ensure the irrationality of akrasia.Footnote 4

Which formulation should we prefer: (W) or (N)? Also, which formulation should we employ as a model for stating other conditional requirements of rationality? In the current debate, these are open questions. Good arguments are presented for and against both (types of) formulations.Footnote 5 But are these arguments really worth the effort? Is the question whether we use (W)- or (N)-type formulations to represent a rational requirement important?

A close reading of the current debate suggests that it might well be. Indeed, there are good reasons to view the difference between (W)- and (N)-type requirements as in some sense fundamental to the nature of rational requirements. For example, the choice between either (W)- or (N)-type requirements is thought to influence whether the requirements of rationality …

(i) …:

give us genuinely symmetric options regarding how we can satisfy them (Schroeder 2004)Footnote 6;

(ii) …:

are prone to issue contradictory requirements (Broome 2007a, b, 2013a; Kolodny 2007b)Footnote 7;

(iii) …:

pick out necessary conditions for being fully rational (Broome 2007b)Footnote 8;

(iv) …:

can guide the formation of attitudes (Reisner 2009)Footnote 9;

(v) …:

are apt for constituting standards for correct reasoning (Kolodny 2005, 2007b; Broome 2013a);Footnote 10

(vi) …:

are attitude sensitive in their application (Broome 2007b; Lord 2014b).Footnote 11

The most important difference, however, lies elsewhere. Many philosophers suppose that rational requirements are in some way normative (Kolodny 2007a, p. 230; see Broome 2005, 2013a). Interpreted in a minimalFootnote 12 yet non-trivial way,Footnote 13 this is to say that rational requirements entail normative reasons.Footnote 14 Or, more precisely:

Reasons entailment. Necessarily: if rationality requires you to X, there is a normative reason for you to X.

Should we endorse this entailment? The difference between (W)- and (N)-type requirements is fundamental to answering this question. Consider a logical consequence of conjoining (N) with Reasons entailment. (N) says that, necessarily, if you believe you ought to X, then rationality requires you to intend to X. Reasons entailment implies that, necessarily, if rationality requires you to intend to X, then there is a normative reason for you to intend to X. By virtue of transitive implication, this entails:

Implication. Necessarily: if you believe you ought to X, then there is a normative reason for you to intend to X.

That is: by adopting a belief that you ought to X, you guarantee the existence of a normative reason to intend to X. Should we accept Implication?

According to a central finding in the philosophy of reasons and rationality, the answer is clearly ‘no’. Implication is subject to the so-called ‘bootstrapping objection’ (Bratman 1987; Broome 2001; Piller 2013). It entails that for any X (e.g. ‘driving drunk’, ‘becoming a terrorist’, etc.) you can create a reason to intend to X simply in virtue of adopting a belief that you ought to X. Put succinctly, Implication licences the creation of normative reasons where there aren’t any.

Let me illustrate this point. Suppose you have no normative reason to intend to kill yourself. Nothing speaks in favour of such an intention. You also do not believe that you ought to kill yourself. Suppose now that, as a result of someone’s spiking your coffee with a dangerous pill, you come to believe that you ought to kill yourself. Does this suffice to create a normative reason to intend to kill yourself? That is, can we create a reason to intend to X by merely adopting the view that we ought to? Evidently, the answer is ‘no’. A mere belief that you ought to kill yourself cannot ‘bootstrap’ such a reason into existence. It cannot create reasons where there are none. In general, reasons for intending to X depend on whether (intending to) X is good or choiceworthy (in the relevant sense). They do not depend on whether one judges X to be good, choiceworthy, or obligatory.

The consequence of this is plain: Implication states an untenable entailment relation between what you believe you ought to do and what reasons there are. Since Implication is a logical consequence of conjoining (N) with Reasons Entailment, at least one of these two propositions must be incorrect too.

Unlike (N), (W) does not give rise to incredible bootstrapping when conjoined with Reasons entailment. Suppose, necessarily, there is a normative reason for you to satisfy (W). This does not imply that your belief that you ought to kill yourself entails a reason to intend to kill yourself. Instead, it implies that there must be a reason to satisfy the following relation: either it is not the case that you believe that you ought to kill yourself or you intend to kill yourself. This is a reason to be ought-belief/intention coherent. It is not a reason to have a particular intention. Hence, accepting (W) and Reasons entailment does not force one to embrace a kind of bootstrapping of normative reasons that goes beyond a normative reason for being coherent.

In sum: conjoining (N) and Reasons entailment entails Implication. By contrast, conjoining (W) and Reasons entailment does not entail Implication. Since Implication represents an unacceptable form of bootstrapping, Reasons entailment does not seem tenable in conjunction with a regime of (N)-type requirements. Thus, in order to advance the debate on the normativity of rational requirements, we first need to settle the choice between (W)- and (N)-type requirements. We need to devise a method that tells us whether ‘rationality requires’ takes a wide or a narrow logical scope in formulating conditional requirements of rationality.

How can we choose between (W) and (N)? In general: how can we identify what rationality requires and when it requires something of us? Not many systematic approaches have been put forward. Broome, and others, find themselves ‘… forced to appeal largely to our intuitions’ (Broome 2013a, p. 150). However, in his recent work, Broome (2007a, b, 2013a) argues against the very possibility of a seemingly promising strategy. I will call it the ‘property strategy’. Roughly, the property strategy says that we can employ the property of full rationality in deciding what rationality requires of us and when it does so.

This strategy has three steps: first, we develop a theory about which combinations of attitudes and mental processes are in/consistent with having the property of full rationality. Second, we determine whether (or which) (W)- or (N)-type requirements are (more) conducive to our theory of fully rational attitudes. And finally, we choose between (W)- and (N)-type requirements on the basis of which logical form fits best with our theory of fully rational attitudinal combinations and mental processes.

Broome views this strategy as futile. He argues this as a technical point (2007a, pp. 363–364, 2013a, p. 134): as far as the property of full rationality is concerned, (W) and (N) turn out to be equivalent. I shall call this ‘property equivalence’. That is, the combinations of attitudes under which you are fully rational under a regime of (N)-type requirements are identical to the combinations of attitudes under which you are fully rational under a regime of corresponding (W)-type requirements. Thus, replacing an (N)-type requirement with a (W)-type requirement (or vice versa) does not affect the combinations of attitudes that are consistent with full rationality. Any attempt to decide between (N) and (W) by considering the property of full rationality is thus destined to fail.

However, Broome’s rejection of the property strategy is too quick. In this paper, I will argue that we should not dispense with the property of rationality as a guide in settling the debate between (W)- and (N)-type requirements. Replacing an (N)-type with a (W)-type requirement can indeed alter the combinations of attitudes that are consistent with having the property of full rationality. This does not, of course, undermine Broome’s own defence of the wide-scope form of conditional requirements of rationality. Yet it shows that we should make use of the property of rationality in settling the debate on the scope of rational requirements.

Part of what makes Broome’s argument authoritative is that he tries to establish property equivalence with a formal theorem and a corresponding proof (2007a, pp. 369–70, 2007b, pp. 39–40, 2013a, p. 148). Although Broome’s theorem is correct (as I show in Sect. 6), it fails to establish what he intends to establish, i.e. property equivalence. In particular, the theorem is premised on the implicit but implausible exclusion of a particular entailment relation that may hold between rational requirements. The theorem excludes, for instance, the existence of requirements that prohibit intending that X if one knows that X will lead one to have an attitude one is rationally required not to have. So, while Broome manages to prove property equivalence for a system of rational requirements that excludes such requirements, I will argue that there is no justification for excluding them. Broome is thus unable to sustain his methodological point. There are situations in which we can decide between (W)- and (N)-type requirements by considering attitudinal combinations that are fully rational. Thus the property of rationality might help us to make progress on some of the questions that have traditionally been linked to the scope of rational requirements.

This paper proceeds as follows. Section 2 introduces the semantic framework for rational requirements that Broome employs in making his equivalence claim. Section 3 defines Broome’s equivalence claim, i.e. the claim that the property of full rationality remains unaffected by replacing a (W)-type with an (N)-type requirement (or vice versa). Sections 4 and 5 advance an argument against property equivalence. I will put forward a counterexample according to which one is fully rational under a (W)-type requirement, but not so under a corresponding (N)-type requirement. Section 6 turns to Broome’s formal theorem, which purports to vindicate property equivalence. I shall demonstrate that Broome’s theorem fails to justify property equivalence, with the upshot that the property of rationality is still available to us as a possible guide in constructing the logical form of rational requirements. There are situations in which the property strategy can determine whether (N)- or (W)-type requirements represent rational requirements correctly, thereby helping us, for instance, to form a correct view on whether the bootstrapping objection poses a threat to the normativity of rational requirements.

2 The code of rationality

How can we decide whether (W)- or (N)-type formulations correctly represent a conditional requirement of rationality? Two things need to be taken into account: (i) ‘rationality requires’ is by and large ‘… a philosopher’s phrase of art’ (Kolodny 2005, p. 515); and (ii) (W)- and (N)-type requirements seem to differ significantly with regard to their logical properties. Consequently, as a first step towards deciding on the correctness of (W)- and (N)-type requirements, we need to work towards a semantics of ‘rationality requires’ (Broome 2007a, p. 361).

Broome (2007a) proposes a general semantic framework for rational requirements. His framework is in fact so general that it could be adopted for other systems of requirements, such as morality or prudence (Broome 2007b, 2013a). Nonetheless, Broome employs this framework to render precise key notions concerning rationality and its requirements and to expose the exact difference between (W)- and (N)-type requirements. He also uses the semantic framework in his proof that (W)- and (N)-type requirements are equivalent with respect to the property of rationality.

In essence, Broome’s (2007a, pp. 361–363) framework rests on two general stipulations. First, for every possible world w, there is a set of propositions that rationality requires of you. That is, at w, p is a member of the set of required propositions if and only if, at w, rationality requires of you that p. Second, there is a function that relates worlds to sets of required propositions.

This function is called the ‘code of rationality’ (or just ‘code’ hereafter). A code is a mapping from worlds to sets of required propositions. A complete code thus determines for all worlds w and all propositions p whether or not p is a member of the set of required propositions at w. Less technically, a code specifies what rationality requires of you and when it does so.

Broome’s general code semantics also allows us to define when a person possesses the property of full rationality. That is, at w, you are fully rational if and only if, at w, every proposition in w’s set of required propositions—call it RP(w)—is true at w (Broome 2007a, p. 362).

In assigning required propositions to worlds, a code will satisfy a variety of constraints. Some constraints will be purely formal. Since a code is a function, it cannot assign two different sets of required propositions to the same world. Other constraints will stem from the logic of rational requirements. A correct code may, for example, satisfy a ‘no-conflict’ constraint, as it exists in standard deontic logic (Broome 2007a, p. 365; 2013a, p. 122). That is, necessarily, if, at w, p is an element of RP(w), then it is not the case that not-p is an element of RP(w). Other constraints will be more substantive. I assume, for instance, that if, at w, you possess no capacity for rationality (i.e. logical and reasoning abilities), then RP(w) will be the empty set.Footnote 15

Some constraints will represent individual requirements. I will thus call them ‘individual constraints’. Put roughly, an individual constraint signifies how a code injects a particular proposition into RP(w). Individual constraints allow us to express precisely when a code instantiates a wide- and/or a narrow-scope requirement. In principle, a code instantiates a narrow-scope requirement if and only if (NC) describes how a code injects a proposition into the set of required propositions. This description reads as follows:

(NC):

For all w: (p ∈ w) → [q ∈ RP(w)],

where w represents a possible world, p and q stand for individual propositional attitudes of yours, ∈ reads as ‘is an element of’, and RP(w) represents the set of rationally required propositions at w. (NC) says that for all possible worlds in which p holds true, it also holds true that q is a proposition that is required by rationality.

Analogously, a code instantiates a wide-scope requirement if and only if (WC) describes how a code injects a proposition into the set of required propositions. This description reads as follows:

(WC):

For all w: (p → q) ∈ RP(w).

(WC) says that it holds for all possible worlds that the truth of the material conditional (p → q) is rationally required of you.

In sum, the code semantics helps us to expose the precise difference between wide- and narrow-scope requirements. (WC)-type requirements require the truth of the material conditional (p → q) [i.e. either (p and q), or (p and not-q), or (not-p and not-q)]. They do so at all possible worlds. (NC)-type requirements require the truth of q. They do so at all worlds in which p holds true.

3 Property equivalence

I now turn to the claim that (WC)- and (NC)-type requirements are equivalent when it comes to the property of full rationality. Consider two individual propositional attitudes, p and q. Suppose that the conjunction of [p and not-q] (but neither p nor not-q individually) necessarily results in your violating a rational requirement. (As indicated in Sect. 1, you could, for example, think of p as standing for ‘You believe that you ought to X’ and q as standing for ‘You intend that you X’.) Which individual constraint should we assign to a code in order to guarantee this?

As far as this point is concerned, we have a choice: we could assign either (WC) or (NC). Any code that satisfies at least one of these constraints will ensure that whenever [p and not-q] is an element of w, the set of required propositions at w—RP(w)—will contain at least one false proposition. But how can we decide between (WC) and (NC)? That is, which constraint—(WC) or (NC)—correctly represents a conditional requirement?

In answering this question, Broome rejects the strategy of choosing between (WC) and (NC) by considering when one is fully rational. In Rationality Through Reasoning, he emphasises this point as follows:

One putative approach to answering the question about scope will definitely not work. It will do no good to think about the property that corresponds to the source of requirements we are investigating. […] You might think we could start by working out implications wide-scope and narrow-scope requirements have for the property of [full] rationality. It might turn out that one gives a better account of the property than the other. But actually this is not so. (Broome 2013a, pp. 133–134)

That is to say: we cannot first define the attitudinal combinations that are in/consistent with having the property of full rationality and then choose between (WC) or (NC) on the basis of which formulation best matches our account of fully rational attitudes. This strategy is untenable, Broome argues, because (WC) and (NC) are too similar. (WC) and (NC) are ‘property equivalent’ when it comes to the property of full rationality:

There is less difference between [(WC) and (NC)] than one might think. Perhaps the most important question a system of rational requirements needs to settle is whether you are [fully] rational—have the property of [full] rationality. It turns out that the answer to this question is unaffected by the choice between narrow and wide scope. The proposition that you are rational is the same whichever way a conditional requirement is formulated. Either way, you are rational at exactly the same worlds. (Broome 2007a, p. 363; emphasis added)

In sum, the logical form of a conditional requirement is immaterial to the question of when a person is fully rational. By manipulating a code only to the effect that it satisfies (NC) instead of (WC) (or vice versa), one does not change the set of combinations of attitudes that are consistent with full rationality.

For the sake of precision, let me make this point more formally. Take a code—R 1—that satisfies (WC) for a pair of propositions p and q. That is to say: R 1 distributes (p → q) to the set of required propositions at all worlds. Perform the following operation on R 1. First, only remove (WC) from R 1. Thus R 1 no longer distributes (p → q) to the set of required propositions at all worlds. Second, apply (NC) to R 1. Call the resulting code R 2. So, R 2 distributes q to the set of required propositions at all worlds where p holds true. Then R 1 and R 2 are ‘property equivalent’ in the following sense:

Property equivalence. Necessarily: at w, you are fully rational under R 1 if and only if, at w, you are fully rational under R 2.

4 A counterexample

As noted above, Property equivalence has not received much critical attention.Footnote 16 This is surprising because its correctness could reduce the significance of the ongoing wide/narrow scope debate.Footnote 17 To the degree that ‘… the most important question a system of rational requirements needs to settle is whether you are [fully] rational—have the property of [full] rationality’ (Broome 2007a, p. 363), Property equivalence threatens the importance of the scope distinction.Footnote 18

However, this lack of critical attention is also a mistake, since Property equivalence is incorrect. By changing a code merely such that it now satisfies (WC) rather than (NC), one can alter the combinations of attitudes that are consistent with being fully rational. So, even when we are only concerned with the property of rationality, the question of scope is highly significant after all.

Consider first an entirely schematic counterexample to Property equivalence.Footnote 19 Compare two codes: R W and R N. Both satisfy two individual constraints: R W satisfies (WC) and (LR). (LR) reads as follows:

(LR):

For all w: [q ∈ RP(w)] → [r ∈ RP(w)].Footnote 20

(LR) constrains a code as follows: suppose that, at w, q is among the rationally required propositions. Then, at w, r is also among the rationally required propositions. That is, you cannot be rationally required to q without being required to r.

Correspondingly, R N satisfies (NC) and (LR). R N is thus the result of only one operation performed on R W, namely the replacement of (WC) with (NC). Accordingly, R W and R N fall within the range of Property equivalence. Schematically, the situation is as follows:

Code (RW)

Code (RN)

(WC) For all w: (p → q) ∈ RP(w).

(NC) For all w: (p ∈ w) → [q ∈ RP(w)].

(LR) For all w: [q ∈ RP(w)] →

(LR) For all w: [q ∈ RP(w)] →

[r ∈ RP(w)].

[r ∈ RP(w)].

Compare R W with R N. According to Property equivalence, the following proposition should hold true: necessarily, at w, you are fully rational under R W if and only if, at w, you are fully rational under R N. But this is not the case. There is a possible situation in which you are fully rational under R W and are less than fully rational under R N.

Suppose that, at w’, (p, q, and not-r) signifies a conjunction of your attitudes. Under R N, you cannot be fully rational at w’. To show this, conjoin (NC) with p. This entails that, at w’, q is a required proposition. Next, conjoin the fact that q is a required proposition with (LR). This entails that, at w’, r is a required proposition. However, at w’, not-r holds true. So, you cannot be fully rational at w’ under R N.

This is not so under R W, however. You can be fully rational under R W at w’. First, (WC) assigns (p → q) as a required proposition to w’. The fact that, at w’, both p and q hold true ensures the truth of (p → q). (LR) says that r is a required proposition at w’ whenever q is a required proposition at w’. However, given what we know of R W and w’, under R W, there is no need to assume that q is a required propositions at w’. Also, there is no need to assume that, under R W, r is a required proposition at w’. So, under R W, you can be fully rational at w’.

Of course, this result requires two things. First, conjoining p and (WC) does not entail that q is a required proposition via ‘factual detachment’. Second, it is not the case that p is a necessary attitude of yours. I take both conditions to be unproblematic.

Put schematically, ‘factual detachment’ licenses an inference from

 

(i) at w, ‘p → q’ is a required proposition

and

 
 

(ii) at w, p

to

 
 

(iii) at w, q is a required proposition.

If this were correct, [p, q, and not-r] would not be consistent with being fully rational under R W. At w’, by conjoining p and (WC), q would turn out to be a required proposition. (LR) would then imply that, at w’, r is a required proposition. Ad hypothesis, r is not the case at w’. You would not be fully rational under R W.

Likewise, we need to assume that p is not necessarily true. Otherwise, q (and therefore r) would again turn out to be required under R W—this time via not ‘factual’ but rather ‘necessary’ detachment (Broome 2013a, p. 123), i.e. a plausible inference to (iii) from (i) and

(ii’):

at w, necessarily p

As above, [p, q, and not-r] would not be consistent with being fully rational under R W.

It is easy to avoid ‘necessary detachment’. We are, of course, entitled to stipulate that p stands for a non-necessary attitude. That is, p represents an attitude of yours that you have at some but not all possible worlds. I assume this holds true for most, if not all, attitudes.

Likewise, ‘factual detachment’ does not pose a problem either. I assume that it is invalid. First, it resembles an invalid modal inference:

 

Necessarily: [(I am unmarried) → (I am a bachelor)]

and

 
 

I am unmarried.

So

 
 

Necessarily: I am a bachelor.

No doubt, this is incorrect (cf. Rippon 2011, pp. 4–5). ‘Factual detachment’ is also philosophically unattractive. If we were to allow it, (WC)-codes would be as open to the bootstrapping objection as (NC) ones. I therefore reject ‘factual detachment’.

The upshot, then, is this: (p, q, and not-r) are jointly consistent with being fully rational at w’ under R W. However, this is not so under R N. There is a combination of attitudes that permits full rationality under R W but does not do so under R N. R W and R N are not equivalent with respect to the property of full rationality. Property equivalence is incorrect.

5 Beyond the conceptual result

So far, I have shown that, conceptually, the choice between wide- and narrow-scope requirements is not negligible when it comes to the property of full rationality. Replacing a wide- with a narrow-scope requirement can logically alter the circumstances in which you are fully rational.

This is, I believe, an interesting theoretical result. However, it does not yet imply that there is a correct code for which replacing a wide- with a narrow-scope requirement will actually change the circumstances in which you are fully rational under that code. This depends on whether we can find a code for which either (WC) or (NC), taken together with (LR), represents a correct individual constraint. Only then can we guarantee that the property of full rationality can actually help us to determine whether (WC) or (NC) represents a rational requirement correctly.

Recall (LR). It says that one required proposition entails another required proposition. However, you might think that required propositions are not related in this way. Hence, a code that satisfies (LR) cannot represent a correct code of rationality.

This reaction would clearly be ad hoc. It is not unnatural to constrain a code such that a required proposition implies another required proposition. Take a code that injects a conjunction of beliefs—‘You believe that a and you believe that b’—into the set of required propositions at all worlds. It seems natural to think that this code will also inject each conjunct—‘You believe that a’ and ‘You believe that b’—into the code of required propositions at all worlds. A required proposition may thus entail another required proposition.

Next, consider a concrete example, where replacing a wide- with a narrow-scope requirement changes the circumstances in which you are fully rational. Suppose, at world w*, (i) you believe that you ought give up smoking and (ii) you do not believe that it is not the case that you ought to give up smoking. However, (iii) you believe that if you have a conversation with SimonFootnote 21 (a passionate smoker and a master of persuasion), you will (instantly) believe that it is not the case that you ought to give up smoking. Nevertheless, (iv) you intend to have a conversation with Simon.

Let us construe a code—R W*—that permits your being fully rational at w*. R W* may, for instance, satisfy the following (WC)-type constraint:

Wide ought-belief consistency. For all w: [B(O) → ¬B(¬O)] ∈ RP(w),

where B stands for ‘You believe that’, O for ‘You ought to give up smoking’, and ¬ for negation. Expressed informally, Wide ought-belief consistency says that rationality requires you not to have contradictory ought-beliefs. More formally, it says that, at all possible worlds, the material conditional ‘(you believe you ought to give up smoking → it is not the case that you believe that it is not the case that you ought to give up smoking)’ is a rationally required proposition.

Wide ought-belief consistency may constrain R W*. R W* is a code under which you can be fully rational at w*. At w*, (i) you believe you ought to give up smoking, and (ii) you have no belief that it is not the case that you ought to give up smoking. So, at w*, you satisfy the requirement that wide ought-belief consistency represents.

Consider another possible constraint on R W*:

Safety. For all w: {[¬B(¬O) ∈ RP(w)] & B[X → B(¬O)]} → [¬I(X) ∈ RP(w)],

where I stands for ‘You intend that’, and X stands for ‘You have a conversation with Simon’. The general idea behind Safety is that rationality requires you not to intend anything that you believe will bring about a situation that violates a requirement of rationality. That is, suppose ‘You do not believe that it is not the case that you ought to give up smoking’ is a rationally required proposition. Suppose also that you believe that if you have a conversation with Simon, then you will believe that it is not the case that you ought to give up smoking. Then, as Safety expresses, ‘You do not intend to have a conversation with Simon’ is also a rationally required proposition.Footnote 22

As with Wide ought-belief consistency, Safety may constrain R W*. It permits you to be fully rational at w*. This requires, of course, that (ii) ‘You do not believe that it is not the case that you ought to give up smoking’ is not a required proposition at w*. For if (ii) were a required proposition at w*, Safety would, in conjunction with (iii) and (iv), entail that RP(w*) contains at least one false proposition at w*.

However, we need not assume that (ii) is a member of RP(w*). First, as I deny factual detachment, conjoining (i) with Wide ought-belief consistency does not imply that (ii) is required at w*. Second, I assume that, at w*, it is not the case that (i) is a necessary attitude of yours. This prevents (ii)’s being injected into RP(w*) via necessary detachment.Footnote 23 Safety can hence constrain R(w*) too.

Moreover, I think that Safety is a plausible constraint on a code of rationality. Consider an analogy with moral requirements. Suppose that racist beliefs are gravely immoral (cf. Appiah 1990; Lengbeyer 2004). Racist beliefs are not only false but also dispositions to act immorally. Thus, if r expresses racist content, then morality requires you not to believe r. Suppose now that you believe that having a conversation with William—a persuasive racist—will lead you to form a cluster of racist beliefs. It seems very plausible that in this situation morality requires you not to intend to have a conversation with William. Put generally, morality requires you, among other things, not to intend to engage in behaviour that will make you gravely immoral.

A similar argument suggests itself for Safety and rationality. Rationality, I have already assumed, aims at attitudinal coherence. Roughly, this is to say (as I have argued in detail in Fink 2014, ms) that rationality aims at ensuring that your attitudes can simultaneously fulfil their ‘constitutive aims’ (or ‘success conditions’). For example, I assume that beliefs aim constitutively at truth, intentions at implementation, and ought-beliefs at both truth and implementation.

Suppose now that rationality requires you to abstain from believing that it is not the case that you ought to give up smoking. Given my account of coherence, this is the case only if abstaining from having this belief is pivotal to preserving your attitudes’ ability simultaneously to fulfil their constitutive aims. Suppose you believe, however, that having a conversation with Simon will cause you to believe that it is not the case that you ought to give up smoking. Intending to have a conversation with Simon therefore jeopardizes your attitudes’ ability to fulfil their constitutive aims. Rationality thus requires you not to have this intention. In general, as in the moral example, rationality requires you not to intend anything you believe will cause you to have an attitude for which it happens to be true that rationally requires you not to have it.

Here is another brief way to consolidate this point. Having a pair of contradictory intentions is one clear way to be incoherent. You cannot implement both intentions. Thus, rationality requires you not to have a pair of contradictory intentions. However, intending to have a pair of contradictory intentions is also a distinctive way of being incoherent. It is to aim to implement something that cannot itself be implemented. This is why, I assume, rationality requires you not to intend to have a pair of contradictory intentions. This lends support to viewing Safety as a plausible requirement of rationality.

I now return to Property equivalence. Recall that R W* is a code under which you are fully rational at w*. Apply the instructions from Property equivalence: remove Wide ought-belief consistency from R W* and inject its narrow-scope equivalent (i.e. ‘narrow ought-belief consistency’) into R W*. Call the resulting code R N*. ‘Narrow ought-belief consistency’ reads as follows:

Narrow ought-belief consistency. For all w: [B(O) ∈ w] → [¬B(¬O) ∈ RP(w)].

Narrow ought-belief consistency says that whenever you believe you ought to give up smoking, ‘You do not believe that it is not the case that you ought to give up smoking’ is a rationally required proposition. In general, as long as you believe something, rationality requires you not to believe its negation.

In sum, R W* and R N* compare as follows:

Code (RW*)

Code (RN*)

(Wide ought-belief consistency)

(Narrow ought-belief consistency)

For all w: [B(O) → ¬B(¬O)] ∈ RP(w).

For all w: [B(O) ∈ w] → [¬B(¬O) ∈ RP(w)].

(Safety)

(Safety)

For all w: {[¬B(¬O) ∈ RP(w)] & B[X → B(¬O)]} → [¬I(X) ∈ RP(w)].

For all w: {[¬B(¬O) ∈ RP(w)] & B[X → B(¬O)]} → [¬I(X) ∈ RP(w)].

Property equivalence predicts that you are fully rational under R W* if and only if you are fully rational under R N*. However, w* shows this to be incorrect.

I have already shown that you are fully rational under R W*. However, this is not so under R N*. Recall w*: (i) you believe that you ought to give up smoking, and (ii) you do not believe that it is not the case that you ought to give up smoking. Furthermore, (iii) you believe that if you have a conversation with Simon, you will believe that it is not the case that you ought to give up smoking. Also, (iv) you intend to have a conversation with Simon. Conjoining (i) with Narrow ought-belief consistency of R N* implies that ‘It is not the case that you believe that it is not the case that you ought to give up smoking’ is a required proposition at w*. Conjoining this with the fact that (iii) you believe that [if you have a conversation with Simon, then you believe that it is not the case that you ought to give up smoking], implies, via Safety, that, at w* ‘It is not the case that you intend to have a conversation with Simon’ is a required proposition. However, ad hypothesis, at w*, (iv) you intend to have a conversation with Simon. So while, at w*, you are fully rational under R W*, this is not so under R N*.

Again, Property equivalence proves incorrect.Footnote 24 This time, however, it does so for what I take to be a plausible set of codes. This shows that there are situations in which the property of full rationality can actually help us to determine the scope of a conditional requirement of rationality. Wide- and narrow-scope requirements differ practically in more respects than Broome suggests.

6 Broome’s theorem

This result is surprising. Broome does not defend Property equivalence in passing. Rather, he attempts to prove Property equivalence by formulating a general theorem and a corresponding proof.

This final section turns to Broome’s theorem. I argue that it fails to vindicate Property equivalence. Though formally correct, the theorem neither represents nor entails Property equivalence. In fact, the theorem is too weak to support the following two main points: (i) you can replace a wide- with a narrow-scope requirement (and vice versa) without changing the conditions under which you are fully rational; and (ii) ‘[…] we cannot decide between the wide-scope and narrow-scope formulations by considering when you have the property of rationality’ (Broome 2007a, p. 364).

To see this, first consider Broome’s theoremFootnote 25:

Theorem. Let R 1 and R 2 be two codes that are the same except that, for one pair of propositions p and q, q ∈ R 1(w) for all w at which p is true (and this may not be so for R 2) whereas (p → q) ∈ R 2(w) for all w (and this may not be so for R 1). Then ‘You are [fully] rational’ is true under R 1 at exactly those worlds where it is true under R 2. (Broome 2007a, p. 369)

To avoid confusion, note that R 1 and R 2 represent entire codes. R 1(w) and R 2(w) stand for corresponding sets of rationally required propositions. Theorem claims the following. Suppose two codes, R 1 and R 2, differ only in the following way: for a pair of propositions p and q, R 1 satisfies (NC) and does not satisfy (WC). Analogously, R 2 satisfies (WC) and does not satisfy (NC). In all other respects, R 1 and R 2 are identical. Then the two codes pick out the same set of worlds at which you are fully rational.

Unlike Property equivalence, Theorem is correct.Footnote 26 It is relatively easy to show this. Let R 1 and R 2 differ in terms of precisely the two properties that Theorem assigns to R 1 and R 2. That is: R 1 satisfies (WC) and not (NC); R 2 satisfies (NC) and not (WC). In all other respects, R 1 and R 2 are identical. This effectively turns the comparison of R 1 and R 2 into a comparison between R 1* and R 2*, where both codes only satisfy one constraint: R 1* only satisfies (NC) and R 2* only satisfies (WC).

Let us compare R 1* and R 2*. Under R 1*, q is rationally required of you at all worlds where p holds true, whereas under R 2* (p → q) is rationally required of you at all worlds. Thus, under both R 1* and R 2*, you violate a rational requirement (and are not fully rational) at w if and only if, at w, p and not-q hold true of you. In all other situations, you are fully rational. Consequently, Broome’s Theorem is correct.

But why does this fail to vindicate Property equivalence? Recall that Property equivalence implies that by replacing (NC) with (WC) (or vice versa) in a code R 1, one creates another code R 2, which picks out the same set of worlds (and combinations of attitudes) at which you are fully rational as that picked out by R 1. However, in contrast, Theorem implies that if R 1 and R 2 differ only in the described way, then R 1 and R 2 pick out an identical set of worlds (or combination of attitudes) at which you are fully rational. The difference is subtle, yet significant.

Here is when it is significant. Suppose you perform only the following operation on a code R: you remove (WC) and inject (NC). As a consequence, this may entail that the resulting code, call it R*, differs in ways that go beyond satisfying (NC) rather than (WC). It may differ in another significant respect. By replacing (WC) with (NC), one may indirectly apply to R* another constraint that does not apply to R. This further constraint may manipulate the set of required propositions such that it becomes possible for you to be fully rational under the original but not under the resulting code.

The counterexamples in Sects. 4 and 5 represent cases in point. Recall my entirely schematic counterexample. In creating R N, I followed the instructions of Property equivalence. I removed (WC) from R W and applied (NC) instead. In doing so, however, I did not create a code that differs from the original code R W with respect to at most two properties. Instead, I created a code that differs from R W with respect to more than two properties. By replacing (NC) with (WC), I indirectly injected another constraint—call it (CC)—into R N. (CC) reads as follows: ‘For all w: (p ∈ w) → [q ∈ RP(w)]’. But I injected (CC) into R N only indirectly, in virtue of the fact that the constraints on R N are closed under inference by modus ponens. The following table represents this situation:

Code (RW)

Code (RN)

(WC) For all w: (p → q) ∈ RP(w).

(NC) For all w: (p ∈ w) → [q ∈ RP(w)].

(LR) For all w: [q ∈ RP(w)] → [r ∈ RP(w)].

(LR) For all w: [q ∈ RP(w)] → [r ∈ RP(w)].

 

(CC) For all w: (p ∈ w) → [r ∈ RP(w)].

I showed that under R W, (p, q, and not-r) is consistent with being fully rational, while this is not so under R N. Replacing a (WC)-type constraint with an (NC)-type constraint can thus make a difference as to which attitudinal combinations count as fully rational.

However, Theorem evades this result. It does so by stipulating that the two codes can differ only with respect to two particular constraints. Yet R W and R N differ with respect to more than two constraints. As a consequence, Theorem disallows the comparison of a (WC)-code and an (NC)-code if these codes satisfy (LR), i.e. one required proposition implying another required proposition. (LR)-codes are excluded from Theorem, and this ensures that R W and R N thereby fall outside the theorem’s scope. Hence, Theorem remains immune to my counterexamples.

But this immunity comes at a heavy cost. As I have argued in Sect. 5, there is no good reason to exclude the existence of (LR)-type constraints on a code. In Sect. 5, I presented (LR)-type constraints that I find plausible. Also, Broome has not offered any reasons to doubt their existence. This, however, undermines the aim in terms of which Broome advances his theorem.

Broome advances his theorem in order to demonstrate that ‘[…] we cannot decide between the wide-scope and narrow-scope formulations by considering when you have the property of [full] rationality’ (Broome 2007a, p. 364). Under Property equivalence, this would hold true. By excluding (LR)-type requirements, Theorem fails to support it.

Again, the counterexample in Sect. 5 shows this. Here is a schematic version of it. Suppose you believe you ought to A and you do not believe that it is not the case that you ought to A. Suppose, further, that you believe that if you do not believe that it is not the case that you ought to A, then you do not X. Yet you intend to X. Are you fully rational?

Suppose the answer is ‘yes’. Then, as long as Safety holds, this excludes the correctness of Narrow ought-belief consistency (i.e. if you believe you ought to A, then rationality requires of you that you do not believe that it is not the case that you ought to A). Narrow ought-belief consistency, in conjunction with Safety, entails that a required proposition fails to hold true of you. Under Narrow ought-belief consistency, you would not be fully rational.

By contrast, suppose that the answer is ‘no’. Then, as long as Safety holds, Wide ought-belief consistency (i.e. rationality requires you not to [believe that you ought to A] and [believe that it is not the case that you ought to A] simultaneously) cannot be correct, because Wide ought-belief consistency, in conjunction with Safety, does not entail that any required proposition fails to hold true of you. Under Wide ought-belief consistency, you would be fully rational. In sum—pace Broome—the property of full rationality can help to determine whether a conditional requirement takes a wide or a narrow scope. The property of rationality can therefore help us to determine what rationality requires.

7 Conclusion

In the debate over the logical form of rational requirements, it has been argued (most prominently by Broome) that the difference between wide- and narrow-scope requirements is immaterial to the issue of when a person possesses the property of rationality. Suppose we propose a wide-scope requirement of rationality and replace it with its narrow-scope equivalent. This replacement will not make any difference when it comes to determining when a person is fully rational under the proposed requirement. There is no combination of attitudes that turns out to be rational under a regime of wide-scope requirements that would not turn out to be rational under a regime of equivalent narrow-scope requirements.

In this paper, I show that this is incorrect. First, the equivalence claim cannot be established by Broome’s theorem and proof (see Sect. 6). Second, replacing a wide- with a narrow-scope requirement (or vice versa) can make a difference as to when a person is fully rational. As I have demonstrated in Sects. 4 and 5, there are combinations of attitudes that are rational under a wide-scope requirement but fail to be so under the same narrow-scope requirement. This is the case when the introduction of a narrow-scope requirement triggers the detachment of a further requirement that was not detachable under a wide-scope requirement (as Sect. 4 shows for a schematic code of requirements, and Sect. 5 with an actual code).

This result presents an opportunity for the debate on rational requirements. By following the equivalence claim, we were misled into accepting that we cannot overcome the scope debate by examining whether wide- or narrow-scope requirements give a better account of the property of rationality. But this position is too sceptical. In fact, it deprives us of an important opportunity to make progress in the scope debate. There are situations in which deciding whether you are fully rational will also determine whether a particular requirement has a wide or a narrow logical scope. So, by working out when a person is fully rational, we can also make progress on the question of which logical form represents conditional requirements of rationality.

This does not only mean progress for the scope debate. It will also prove helpful for advancing answers to some of the most important questions linked to the requirements of rationality. Scope and logical form decide, among other things, whether rational requirements can explain the correctness of reasoning, can guide our attitude formation, and are apt to serve as sources of normative reasons.

Perhaps most importantly, the scope of rational requirements decides whether the so-called bootstrapping objection entails a forceful argument against the normativity of rational requirements. As explained in Sect. 1, to assume that narrow-scope requirements are normative leads to incredible bootstrapping. It would licence the spurious fabrication of normative reasons where clearly no such reasons exist. In order to maintain a normative interpretation of rational requirements, these requirements must have a wide logical scope. Thus, until we settle the scope question, the consequences of the bootstrapping objection for the normativity of rational requirements will remain undecided. This underlines why the result of the presented argument is important: if the property of rationality can help us to resolve the scope debate, it will also prove useful for deciding whether or not rational requirements are normative.

Thus, research on rationality should no longer ignore the property of rationality. Indeed, we should try to give an account of the property of rationality that is independent of its requirements—an account on the basis of which we can then determine precisely what rationality requires.Footnote 27