3.1. Top-down: from rationality to logical notions

The first way to model the three logical notions starts with a theory of rationality (given for instance by axioms such as R1–R7) and then constructs the logical notions. This can be done as follows:

What is the intuition behind these definitions?

• Consistency means that your attitudes cohere with one another, i.e. do not rule out one another. You are permitted to have your attitudes simultaneously. You might be forbidden to hold only them; but you can hold at least them. For instance, suppose you intend p, believe q is a means implied by p, but do not intend q. Your constitution is then not rational, assuming rationality requires Enkrasia R5; but your constitution is consistent, as long as it could be made rational by adding suitable attitudes including the intention of q.

• Completeness means that you have ‘enough’ attitudes. Your attitudes do not require additional attitudes. You are permitted to have no more than your attitudes. You might be forbidden to hold all your attitudes; but you can hold no more than them. For instance, assume you prefer p to q and also prefer q to p. Then your constitution is not rational, assuming rationality requires Preference Acyclicity R6; but your constitution is complete, as long as it could be made rational by removing suitable attitudes, including one of the two mentioned preferences.

• Closedness means that you have each attitude that rationally follows from your attitudes. For instance, supposing rationality requires Instrumental Rationality R4, then believing obligatorily p rationally entails intending p; so, closed constitutions containing the mentioned belief also contain the intention. Closed constitutions need not be consistent or complete, let alone rational. For instance, the maximal constitution $C = M$ – where you believe everything, intend everything, etc. – is trivially closed, but it is irrational (in fact, inconsistent) under plausible theories of rationality. At the other extreme, the empty constitution $C = \emptyset $ – where you have no attitude whatsoever – is, under some theories of rationality, closed but irrational (in fact, incomplete).

3.2. Bottom-up: from logical notions to rationality

Under the previous approach, the three logical notions are children of rationality. We now take the opposite approach. We start from logical notions and derive a theory of rationality – like when logicians use logical notions to define which belief sets are rational. But what do we mean by logical notions, in all abstract generality?

(b) A completeness notion is a set $COM \subseteq {2^M}$ of (‘complete’) constitutions such that whenever $C \in COM$ and $C \subseteq {C'}$ ( $ \subseteq M$ ) then ${C'} \in COM$ (‘gaining attitudes preserves completeness’).

(c) A closedness notion is a set $CLO \subseteq {2^M}$ of (‘closed’) constitutions that consists of all constitutions which are closed under some classical consequence operator, i.e. that equal $\{ C \subseteq M:C = Cn(C)\} $ for some classical consequence operator Cn over M. ^{Footnote 14}

A consistency notion CON captures the absence of tensions between attitudes by some standard (and is accordingly closed under taking subsets). A completeness notion COM captures the presence of enough attitudes by some standard (and is accordingly closed under taking supersets). A closedness notion CLO captures the presence of all attitudes that follow from present attitudes by some standard (and is accordingly closed under some classical consequence operator).

In section 3.1 we had defined special logical notions based on a theory of rationality T. We henceforth denote them by

$$\hskip-68ptCO{N_T} = \{ C:C \subseteq {C'}\;\;{\rm{for}}\,\,{\rm{some}}\,{C'} \in T\} $$

$$\hskip-72ptCO{M_T} = \{ C:C \supseteq {C'}\;\;{\rm{for}}\,\,{\rm{some}}\,{C'} \in T\} $$

$$CL{O_T} = \{ C:C\;\;{\rm{contains}}\,\,{\rm{all}}\,m \in M\,{\rm{s}}{\rm{.t}}{\rm{.}}\,C\;T {\rm{\hbox{-}entails}}\,\,m\} .$$

These are indeed logical notions in the general sense of Definition 4, because $CO{N_T}$ is closed under taking subsets, $CO{M_T}$ is closed under taking supersets, and $CL{O_T}$ consists of the closed constitutions under the consequence operator $C{n_T}$ that maps any $C \subseteq M$ to

$$\hskip-118ptC{n_T}(C) = {\rm{set}}\,\,{\rm{of}}\,\,{\rm{attitudes}}\,\,T\!\hbox- {\rm{entailed}}\,\,{\rm{by}}\,\,C$$

$$ \hskip 14pt= \{ m \in M:m\,\,{\rm{is}}\,\,{\rm{in}}\,\,{\rm{all}}\,\,{C'} \in T\,{\rm{s}}{\rm{.t}}{\rm{.}}\,C \subseteq {C'}\} = { \cap _{{C'} \in T:C \subseteq {C'}}}{C'}.$$

Deriving logical notions from a full-fledged theory of rationality T is a ‘top-down’ approach to logical notions. But under a ‘bottom-up’ approach, where could the logical notions CON, COM and CLO come from? They could emerge from individual axioms about attitudes. For instance, the axiom schemas R1–R7 in our ‘illustration’ in section 2 can serve to define logical notions, where we must carefully select the right axioms for each logical notion:

• A consistency notion CON can be defined by the consistency-type ^{Footnote 15} schemas R2 (Non-Contradictory Desires) and R6 (Preference Acyclicity). Formally, $CON = \{ C:C$ satisfies R2 & R6 $\} $ .

• A completeness notion COM can be defined by the completeness-type ^{Footnote 16} schema R7 (Preference Completeness).

• A closedness notion CLO can be defined by the closedness-type ^{Footnote 17} schemas R1 (Modus Ponens), R3 (Enkrasia), R4 (Instrumental Rationality), and R5 (Preference Transitivity).

Any logical notions can be used to define a theory of rationality:

One might wonder about the appropriateness of requiring completeness for rationality, given that preferences and logical beliefs are often not required to be complete. We discuss this issue in section 3.4, but let us anticipate that this problem is only apparent since one can assume a vacuous completeness notion $COM{ = 2^M}$ , in which case rationality is effectively generated by consistency and closedness alone.

3.3. Comparing the top-down and bottom-up approaches to logical notions

We have considered two opposite approaches:

• Starting from a theory of rationality T and generating logical notions $CO{N_T}$ , $CO{M_T}$ , $CL{O_T}$ .

• Starting from logical notions CON, COM, CLO and generating a theory of rationality $T = CON \cap COM \cap CLO$ .

Interpretive differences aside, are both approaches formally equivalent? That is, are theories of rationality and logical notions interdefinable through some one-to-one correspondence? The answer is negative, for two reasons.

For one, some notions of rationality T are not reducible to any logical notions at all, because they are simply not structured along strict logical lines. So to say, rationality could go beyond logic. Rationality notions that are reducible to logical notions will be called ‘classical’, to highlight the parallel to classical notions of rational beliefs, which are indeed derived from the logical notions (cf. Appendix A). Formally:

For instance, the illustrative theory in section 2, $T = \{ C:C$ satisfies R1–R7 $\} $ , is classical, being generated by the consistency notion $CON = \{ C:C$ satisfies R2 & R6 $\} $ , the completeness notion $COM = \{ C:C$ satisfies R7 $\} $ , and the closedness notion $CLO = \{ C:C$ satisfies R1, R3, R4 & R5 $\} $ (cf. section 3.2).

For another, one and the same (classical) notion of rationality can usually be generated by two different triples of logical notions. ^{Footnote 18} So the ‘true’ logical notions are underdetermined by the notion of rationality. Yet, despite this underdetermination, one triple of logical notions stands out as canonical in that it consists in the logically strongest logical notions that generate the given (classical) theory T. The canonical logical notions are precisely the logical notions $CO{N_T}$ , $CO{M_T}$ and $CL{O_T}$ from the top-town approach in section 3.1. We now formally state this result, proved in Appendix B.

This result gives some salience to the logical notions from section 3.1, and provides some support for the top-down approach to modelling logical notions.

3.4. Completeness – really?

In logic just as in rational-choice theory, completeness assumptions are often regarded as convenient assumptions rather than a requirement of rationality. Arguably, rationality does not require holding beliefs about everything, or preferences between any two options. Does this idea clash with our analysis that makes completeness a requirement of rationality? No, because our concept of completeness is very flexible. We allow the vacuous completeness notion ${2^M}$ , which deems all constitutions complete – even the empty constitution $C = \emptyset $ . If one deems it permissible to hold no beliefs and no preferences, then one effectively endorses a completeness notion that does not require any beliefs or preferences. A slightly more restrictive completeness notion requires believing tautologies and being indifferent between options and themselves, without requiring anything else about beliefs or preferences.

This represents a departure of our abstract completeness notion from standard belief- or preference-theoretic completeness. ‘Our’ completeness is by definition rationally required but can be undemanding or even vacuous. ^{Footnote 19} ‘Standard’ completeness (of beliefs or preferences) is very demanding but might not be rationally required. In principle, something similar applies to both other logical conditions: ‘our’ consistency and closedness are by definition rationally required but can be vacuous, whereas ‘standard’ consistency and closedness (of beliefs or preferences) may or not be rationally required. But here the contrast is smaller than for completeness, since the rationality of ‘standard’ consistency and closedness is far less controversial.

The point we just made holds under both approaches, i.e. regardless of the priority between the rationality notion and the logical notions. Let us restate the point formally:

Convention When generating a theory of rationality from logical notions (bottom-up approach), vacuous logical notions need not be mentioned, as they drop out of the intersection of logical notions. For instance, we call a theory $T$ generated by $CON$ and $CLO$ if $T = CON \cap CLO$ , i.e. if $T$ is generated by $CON$ , ${2^M}$ and $CLO$ .

To honour the fact that standard theories of rationality often impose no (non-vacuous) completeness requirement, let us call a classical theory ‘fully classical’ if it is generatable without completeness notion, i.e. with a vacuous completeness notion:

Theorem 1 has a corollary for fully classical theories, as shown in Appendix B:

4.1. The conceptual difference between logical and standard requirements

Our logical requirements and standard rationality requirements such as those in R1–R7 share an obvious feature: both are rationality requirements. Let us spell this fact out formally.

Having made this trivial point, let us see how logical and standard requirements differ.

1. 1. Abstract versus concrete. Logical requirements are abstract and structural, since their definitions do not refer to the type or content of attitudes, but to structural relations between attitudes. Standard rationality requirements are concrete and attitude-specific, since they are defined in terms of particular attitudes, such as preferences (in R5–R7) or intentions and beliefs (in R3–R4).

2. 2. Global versus local. Logical requirements are global: they affect the constitution as a whole. Standard requirements are local: they concern only the (non-)possession of certain attitudes, regardless of the rest of the constitution. They are effectively constraints on a small subset of the constitution C. For instance, an instance of Preference Transitivity R5 concerns only C’s intersection with $\{ (p,q, \succ ),(q,r, \succ ),(p,r, \succ )\} $ , and an instance of Enkrasia R3 concerns only C’s intersection with $\{ (obligatorily$ $p,bel),(p,int)\} $ . Christensen (Reference Christensen2004) draws a similar global/local distinction, but for beliefs only.

3. 3. Rationality-determined versus rationality-determining. This difference arises only under our current top-down approach of modelling logical notions as derivative objects; it should therefore not be universalised. While logical requirements are (under the top-down approach) determined by rationality, standard rationality requirements typically determine rationality. For instance, the ‘illustration’ in section 2 invokes schemas R1–R7 of standard requirements that jointly determine or define a theory of rationality, which in turn determines or defines logical requirements. This striking difference in status or priority between standard and logical requirements could be given thinner or heavier meanings depending on what is read into ‘determining’. Possible interpretations range from a mere functional or supervenience relationship between both objects to an explanatory relationship or even a relationship of metaphysical grounding.

4.2. The formal correspondence between logical and standard requirements

Despite all differences, logical and standard requirements of rationality stand in a tight formal relationship: each logical requirement is equivalent to a particular class of rationality requirements of standard type. But first, what are rationality requirements of standard type? A simple inspection of the rationality requirements discussed in philosophy or choice theory reveals that most of them, including those in the schemas R1–R7, fall into a three-kind typology. This typology is implicit in the work of Broome and others and formally introduced in Dietrich et al. (Reference Dietrich, Staras and Sugden2019):

The conditions in R1–R7 fall into this typology:

• Non-Contradictory Desires R2 and Preference Acyclicity R6 are schemas of consistency conditions, with forbidden set $\{ (p,des),(not$ $p,des)\} $ or $\{ ({p_1},{p_2}, \succ ),({p_2},{p_3}, \succ ), \ldots ,({p_{k - 1}},{p_k}, \succ ),({p_k},{p_1}, \succ )\} $ , respectively.

• Preference Completeness R7 is a schema of completeness conditions, with unavoidable set $\{ (p,q, \succ ),(q,p, \succ ),(p,q, \sim )\} $ . Another schema of completeness conditions is the schema in footnote 15, an instance of which requires holding some probabilistic belief in a given proposition.

• Modus Ponens R1, Enkrasia R3, Instrumental Rationality R4, and Preference Transitivity R5 are schemas of closedness conditions. In R1, the set of premise-attitudes is $\{ (p,bel),(if$ p then $q,bel)\} $ and the conclusion-attitude is $(q,bel)$ .

Having formalized logical conditions as well as conditions of standard type, we are ready to state the formal relationship between both kinds of condition. A tight correspondence holds by the following theorem.

Parts (a)–(c) connect the logical world of abstract requirements to the choice-theoretic or philosophical world of rationality requirements of standard type. Part (d) is an addendum, of interest in its own right.

Figure 1 displays schematically the requirements of a typical theory of rationality T. As usual in choice theory, the theory has been constructed from some set ${\cal A}$ of basic principles or ‘axioms’, for example the instances of the schemas R1–R7. That is, the rational constitutions are the constitutions satisfying the conditions in ${\cal A}$ :

$$T = T({\cal A}) = \{ C:C \in R\,{\rm{for}}\,\,{\rm{all}}\,R \in {\cal A}\} = { \cap _{R \in {\cal A}}}R.$$

Figure 1. The rationality requirements of the theory $T({\cal A})$ .

Let all axioms be of a standard type: ${\cal A}$ consists of consistency conditions, completeness conditions, and closedness conditions. Of course, some other theories have no axiom of one of the three standard types (e.g. no completeness axiom) or have additional axioms of non-standard type; but in Figure 1 all three standard types, and only these types, occur among the axioms. The theory implies plenty of other requirements besides the axioms. As indicated by the different areas in Figure 1, some additional requirements still fall within the standard typology. ^{Footnote 22} The most salient requirements outside the typology are perhaps the three logical requirements; each of them is equivalent to a class of requirements of standard type by Theorem 2, as the arrows ‘ $ \leftrightarrow $ ’ in Figure 1 indicate. Other requirements outside the typology are often artificial, including conjunctions or disjunctions of axioms.

Since the set of axioms ${\cal A}$ can be partitioned into sets ${{\cal A}_{{\rm{con}}}}$ , ${{\cal A}_{{\rm{com}}}}$ and ${{\cal A}_{{\rm{clo}}}}$ of consistency, completeness or closedness conditions, respectively, the resulting theory of rationality $T = T({\cal A})$ is classical, being generated by (i.e. being the intersection of) the logical notions

$$CON = \{ C:C \in R\,{\rm{for}}\,\,{\rm{all}}\,R \in {{\cal A}_{{\rm{con}}}}\} $$

$$COM = \{ C:C \in R\,{\rm{for}}\,\,{\rm{all}}\,R \in {{\cal A}_{{\rm{com}}}}\} $$

$$CLO = \{ C:C \in R\,{\rm{for}}\,\,{\rm{all}}\,R \in {{\cal A}_{{\rm{clo}}}}\} .$$

The theory would be even fully classical if, unlike in Figure 1, we had an empty set of completeness axioms ${{\cal A}_{{\rm{com}}}} = \emptyset $ and hence a vacuous completeness notion $COM{ = 2^M}$ .

5.1. Reasoning in attitudes

Let us first introduce the Broomean notion of reasoning and formalize it following Dietrich et al. (Reference Dietrich, Staras and Sugden2019). For Broome (Reference Broome2013), reasoning is a process of forming attitudes from existing attitudes: forming beliefs from beliefs, or intentions from beliefs and intentions, or preferences from preferences, etc. The process is causal. Unlike other causal processes, it is conscious and constitutes a mental act. It is explicit: you bring the premise-attitudes to mind by ‘saying’ their contents to yourself, usually through internal speech, which causes you to ‘construct’ and thereby acquire some conclusion-attitude, again using (usually internal) speech.

Here is a stylized instance of reasoning with a single premise. You say this to yourself:

$$Doctors\,\,recommend\,\,resting.\,\,So,\,\,I\,shall\,\,rest.$$

This is reasoning from a belief into an intention. The ‘So’ is not part of the conclusion, but expresses the act of drawing the conclusion. In reasoning, you say to yourself, not contents of attitudes simpliciter, but marked contents, i.e. contents with a marker indicating how you entertain the content: as a belief, or an intention, etc. In reaching the intention with content ‘I rest’, you say ‘I shall rest’, here using ‘shall’ as a marker for intention. ^{Footnote 23} The English language provides markers for various attitudes, including desire and preference (Broome Reference Broome2013). Beliefs are special: they need no explicit marker (in English), as the same sentence expresses the content and the marked content. Note that you do not reason about your attitudes. Reasoning about attitudes is a meta-level process by which you discover that you have an attitude rather than forming that attitude; for details, see section 6.2.

Reasoning in attitudes is rule-governed: you draw the conclusion by following a rule. Rules can be individuated more or less broadly. In the example, the rule could be

• specific: from believing that doctors recommend resting, to intending to rest.

• broader: from believing that doctors recommend $\phi $ -ing to intending to $\phi $ . Parameter: any act $\phi $ .

• even broader: from believing that expert E recommends $\phi $ -ing towards intending to $\phi $ . Parameters: any expert E (such as a doctor) and act $\phi $ .

We will work with specific rules, to avoid dealing with schemas and parameters. Nothing hinges on this technical choice: our results could be re-stated (more clumsily) using a broader notion of rule. Given our choice, we identify a rule with a specific premises/conclusion combination. Technically, a reasoning rule is a pair $(P,c)$ of a set of (‘premise-’)attitudes $P \subseteq M$ and a (‘conclusion-’)attitude $c \in M$ , representing the formation of c from P. In the rule in the example above, P contains just believing that doctors recommend resting, and c is intending to rest.

In his concept of reasoning, Broome distinguishes between your ‘endorsement’ of a rule and its ‘correctness’. A rule is yours if it captures premise-to-conclusion processes that you endorse, i.e. that ‘seem right to you’ (Broome Reference Broome2013: 237–238). Whenever you follow a rule in a way that seems right to you, you are reasoning. You are reasoning correctly only if that rule is correct according to some universal or intersubjective standard. For example, the rule $(\{ (p,q, \succ ),(q,r, \succ )\} ,(p,r, \succ ))$ may seem right to most people and is arguably correct, but its putative rightness and correctness are outside the scope of our abstract model.

The totality of your rules is your ‘reasoning system’, representing your reasoning policy. Technically, a reasoning system is a set S of reasoning rules. Starting from your initial constitution, you can reason with your rules: whenever you have a rule’s premise-attitudes, you can form the rule’s conclusion-attitude, which is added to your constitution. You can do this until your constitution is stable. A constitution C is stable under S (‘under reasoning’) if reasoning makes no change, i.e. C already contains the conclusion-attitude of each rule in S whose premise-attitudes it contains. The stable constitution reached by reasoning from your initial constitution C using your reasoning system S is denoted $C|S$ and called the revision of C through S (‘through reasoning’). Technically, $C|S$ is defined as the minimal extension of C stable under S. ^{Footnote 24} Provided your reasoning system S is finite, you can reach $C|S$ in finitely many reasoning steps. You first apply a rule $(P,c)$ in S that is effective (‘difference-making’) on C, i.e. for which $P \subseteq C$ but $c \,\notin \,C$ ; your constitution becomes $C \cup \{ c\} $ . You then apply another rule $({P'},{c'})$ in S that is effective on $C \cup \{ c\} $ ; your constitution becomes $C \cup \{ c,{c'}\} $ . You continue until all your rules are ineffective. The order in which you reason, i.e. apply rules, is irrelevant: you inevitably converge to the same stable constitution $C|S$ . All this can be stated formally. ^{Footnote 25} For infinite reasoning systems S, one might argue that $C|S$ was defined too largely, as including even attitudes that are reachable ‘in infinitely many steps’ (so to say) without being ‘really’ reachable. For infinite S our definition of $C|S$ is therefore appropriate for ‘infinite reasoners’ rather than ‘real reasoners’ – a questionable but convenient idealization.

5.2. Comparison with broader accounts of reasoning

Our Broomean account of reasoning differs from other accounts. We now discuss some key differences; our formal result will hinge on them.

Broomean reasoning is broad in that it operates within general attitudes, not just beliefs. But it is narrow in that it (i) forms but never removes attitudes, and (ii) is based on the presence but never the absence of attitudes. In short, you cannot reason to, or from, absences; you for instance cannot reason from not believing something to no longer intending something. These two features make the Broomean reasoning operator inclusive and monotonic. ^{Footnote 26}

But our Broomean approach does not deny the existence of other mental processes that produce, or start from, absences of attitudes, such as processes of losing intentions based on lacking certain beliefs. On the contrary, our Broomean approach regards such processes as a central element of psychology: an automatic element distinct from reasoning. Such automatic processes help improve rationality where reasoning alone is unsuccessful (Broome Reference Broome2013; Dietrich et al. Reference Dietrich, Staras and Sugden2019). This idea will be confirmed in section 5.3.

But let us mention possible criticisms of our Broomean account of reasoning. For one, this account is restricted to deterministic reasoning: your current attitudes fully determine what conclusion-attitude you form. Under a generalized account, the reasoner can choose between possible conclusions. Choice in reasoning is studied in Dietrich and Staras (Reference Dietrich and Staras2022) using indeterministic rules. For another, by precluding reasoning to or from absences, the Broomean account seems to clash with belief elimination in AGM-type belief revision theory (Alchourrón et al. Reference Alchourrón, Gärdenfors and Makinson1985) and with non-monotonic logics (Horty Reference Horty and Goble2001). One might try to reconcile Broomean reasoning with these formal developments by interpreting AGM-type belief revision and non-monotonic logical consequence as capturing not reasoning alone but a combination of reasoning and automatic mental processes. We cannot explore here whether and how such a reconciliation works.

At a more philosophical level, Drucker (Reference Drucker2021) has recently challenged Broome’s concept of reasoning, defending a broader concept (based on Boghossian Reference Boghossian2018). He argues that reasoning can not just add, but also remove attitudes. Roughly, according to his central thesis called ‘Argumentalism’, you reason towards an arbitrary attitudinal change (e.g. an attitude loss) when you run an argument that convinces you and that ends with a conclusion whose utterance expresses this change. For instance, suppose you have the initial belief that it rains. You reason towards losing that belief if you run an argument that convinces you and that concludes that it does not rain. In convincing you, the argument has a causal effect on your attitudes: you gain the belief that it does not rain and lose the belief that it rains. By uttering the conclusion of the argument, you express both the belief acquisition and the belief loss.

Unlike Broomean reasoning, Druckerian reasoning is not explicit all the way. It is explicit in the sense that it follows an argument in language. But Drucker leaves the explicit paradigm in attributing to reasoning various implicit attitudinal changes that the argument induces. In the ‘rain’ example, the explicit reasoning by which you acquire the ‘no rain’ belief is both Druckerian and Broomean reasoning towards a belief, but the loss of the ‘rain’ belief is attributable to reasoning only in an implicit and non-Broomean sense.

Broome and Drucker have different notions of ‘expressing’. For Broome, your sentence expresses its (literal) content, in that it denotes or represents it. For Drucker, your utterance of a sentence expresses an attitude (change) of yours just if, according to the rules of the language, you could not utter that sentence sincerely while knowing that the utterance is not caused non-deviantly by the occurrence of the attitude (change). ^{Footnote 27} Thus, for Broome, the conclusion sentence ‘it does not rain’ expresses the marked content of the ‘no rain’ belief. For Drucker, uttering that sentence expresses both the acquisition of the ‘no rain’ belief and the loss of the ‘rain’ belief.

In sum, Druckerian reasoning goes beyond Broomean reasoning in including processes that would count as automatic under our Broomean account. We find the Broomean notion of reasoning useful – for philosophy, but also cognitive science, decision theory and behavioural science – because it aims at a clear conceptual separation between processes under a reasoner’s explicit control and automatic processes beyond such control. This mirrors the psychological distinction between System 2 and System 1 processes (Watson and Evans Reference Watson and Evans1974; Kahneman Reference Kahneman2011). However, we acknowledge that the two kinds of process can interact in ways that are not represented explicitly in the Broomean model. ^{Footnote 28}

5.3. Which logical requirements are achievable through reasoning?

What would it mean to achieve a logical requirement or even full rationality through reasoning? Given a theory of rationality, a reasoning system S achieves consistency, completeness, closedness, or (full) rationality if for each initial constitution $C \subseteq M$ the revision $C|S$ is, respectively, consistent, complete, closed or rational.

We shall want reasoning to not only achieve certain requirements, but also to preserve consistency. Formally, a reasoning system S preserves consistency if for each consistent constitution C its revision $C|S$ is still consistent. Preserving consistency matters because there would be little point in achieving some logical requirement if one thereby lost consistency, the arguably most basic and ‘least sacrificeable’ logical requirement.

By Theorem 2, achieving consistency, completeness or closedness is respectively equivalent to achieving certain rationality requirements of standard type. But whether these standard-type requirements are achievable is known; it is informally contained in Broome’s work, and formally presented in Dietrich et al. (Reference Dietrich, Staras and Sugden2019). Details aside, reasoning can successfully achieve closedness requirements, but not consistency or completeness requirements. Using this fact, Theorem 2 implies another theorem as a corollary, which (roughly) says that

• reasoning can achieve closedness while preserving consistency,

• reasoning cannot achieve consistency,

• reasoning can achieve completeness, but only while sacrificing consistency.

Formally:

In (b)–(d), ‘unless’ can be read not only in its weak sense (‘if it is not the case that’), but even in its strong sense (‘if and only if it is not the case that’). So Theorem 3 provides necessary and sufficient conditions for the possibility of successful reasoning, in four senses of ‘successful’. ^{Footnote 31}

The message of Theorem 3 is gloomy, though quite ‘Broomean’: you cannot reason towards two of three logical requirements, just as (following Broome) you cannot reason towards many ordinary rationality requirements. This result is independent of the attitude type: it even holds for ordinary ‘theoretic’ reasoning in beliefs.

A more nuanced picture emerges after cashing in that other mental processes than reasoning could jump in to make your attitudes inch closer to completeness (by creating attitudes) or consistency (by removing attitudes). For instance, some beliefs or intentions might crowd out other ones that are inconsistent with them, making you ‘more consistent’. We can become ‘more logical’, but not through reasoning alone. Furthermore, if the concept of reasoning were defined more broadly to include indeterministic reasoning (as in Dietrich and Staras Reference Dietrich and Staras2022), then part (c) of Theorem 3 would no longer hold, so that you could reason (indeterministically) to completeness.

We now discuss each part in turn.

Part (a): the achievability of closedness. By part (a), you can develop closed attitudes through reasoning – without losing consistency. Why? By Theorem 2, closedness is achieved once all the theory’s closedness requirements are achieved. A closedness requirement says: having a certain set of attitudes P implies having a certain attitude c. You achieve this requirement if you have the rule $r = (P,c)$ . You achieve all of the theory’s closedness requirements if you have all corresponding rules. If these are your only rules, reasoning provably preserves consistency. Although this reasoning system does the job, it is peculiar: it is so rich in rules that you can reason towards each closedness requirement of the theory in a single step. In practice, much slimmer (and cognitively more plausible) reasoning systems also achieve closedness and preserve consistency. You only need rules corresponding to some of the theory’s closedness requirements. Suppose rationality requires that believing p and if p then q implies believing q, and that believing q implies intending s. Then rationality also requires that believing p and if p then q implies intending s. If you have the rules corresponding to the first two closedness requirements,

$$r = (\{ (p,bel),(if\,p\,\,then\,\,q,bel)\} ,(q,bel))\,\,{\rm{and}}\,\,{r'} = (\{ (q,bel)\} ,(s,int)){\rm{,}}$$

then you need not have the rule corresponding to the third requirement, ${r{''}} = (\{ (p,bel),(if$ p then $q,bel)\} ,(s,int))$ , because the third requirement is achievable through applying first r and then ${r'}$ . Real people presumably reason with few and simple rules.

Part (b): the unachievability of consistency. Part (b) is mathematically trivial, but philosophically disturbing. It is trivial (without even consulting Theorem 2) because Broomean reasoning never removes attitudes, hence never makes inconsistent constitutions consistent. Broome acknowledges that inconsistencies often disappear, but insists that they disappear, not through reasoning, but through automatic mental processes, such as when you find yourself losing a belief after realizing a conflict with other beliefs. The impossibility to reason yourself out of inconsistency is disturbing because consistency is a more basic normative desideratum than completeness and closedness. One would have hoped that reasoning can at least make consistent. Instead reasoning can make closed, but not consistent. The problem is only avoided for trivial theories of rationality that deem all constitutions consistent.

Part (c): the unachievability of completeness. Why does part (c) hold? Given the theory of rationality, we call a set of attitudes avoidable if some rational constitution contains none of its states, and unavoidable otherwise. Two possible avoidable sets are $\{ (p,bel),(not$ $p,bel)\} $ and $\{ (p,q, \succ ),(q,p, \succ ),(p,q, \sim )\} $ , for propositions p and q – though these sets are unavoidable if the theory requires holding ‘beliefs about anything’ and ‘preferences between any options’. The theory’s unavoidable sets stand in one-to-one correspondence with the theory’s completeness requirements: a set $U \subseteq M$ is unavoidable if and only if the theory makes the completeness requirement of having some attitude from U. Now by Theorem 2, completeness is achieved once you satisfy the theory’s completeness requirements, or equivalently, once you have acquired some attitude from each unavoidable set. There is a trivial (but implausible) way to acquire such attitudes: for each unavoidable set U, you simply have a rule that always generates a given attitude in U (formally, a rule $r = (\emptyset ,m)$ which has no premise-attitudes and some conclusion-attitude m in U).

This trivial way to reason towards completeness is unconvincing. It seems ad hoc, if not stubborn and blind, to always acquire the same attitude from a given unavoidable set U, regardless of the web of existing attitudes. What matters is not just that you form an intention (from an unavoidable set of intentions U), but also which intention you form. Otherwise the new intention can be inconsistent with your beliefs, preferences or other existing attitudes. Formally, the trivial reasoning system achieves completeness by sacrificing consistency. Unfortunately, also all other reasoning systems that achieve completeness fail to preserve consistency.

This argument presupposes that completeness is not essentially trivial, as shown in the proof. Completeness is trivial if the theory deems all constitutions complete; or equivalently, the empty constitution is rational. Here there are no unavoidable sets U. More generally, completeness is essentially trivial if all constitutions containing at least the unfalsifiable attitudes (if any) are complete; or equivalently, some constitution containing at most unfalsifiable attitudes is rational. An attitude m is unfalsifiable if it never conflicts with other attitudes, i.e. if $\{ m\} \cup C$ is consistent whenever C is consistent. The main example are attitudes that are ‘tautological’, i.e. contained in all rational constitutions. Standard theories of rationality deem all attitudes aside from tautological ones falsifiable: desiring p is falsifiable by conflicting with desiring not p; preferring p to q is falsifiable by conflicting with being indifferent between p and q, or with preferring q to p, or with preferring q to r and also r to p; etc.

Part (d): the unachievability of full rationality. Since consistency is unachievable by part (b), so is full rationality. This again presupposes that not all constitutions count as consistent – otherwise you could trivially become rational by having all reasoning rules, making you form all attitudes.

6.1. The statics of multiple attitudes

The statics of multi-attitude psychology concern your attitudes at a given time. Our abstract logical requirements – consistency, completeness, closedness – are purely static. An alternative to our abstract approach would be to use some concrete logic of attitudes. Mono-modal logics involve just one attitude, for instance belief in ‘doxastic logics’ (e.g. Halpern Reference Halpern2017) or preferences in ‘preference logics’ (e.g. Liu Reference Liu2011). Multi-modal logics involve two or more attitudes, for instance beliefs, desires and intentions in ‘BDI logics’ (e.g. Van der Hoek and Wooldridge Reference Van der Hoek and Wooldridge2003). Attitudes are represented by modal operators, and rationality by axioms that constrain attitudes. This machinery provides concrete representations of attitude types (through attitude operators), but also of attitude contents (through logical sentences). Since these contents can themselves involve attitudes, one can explicitly form and study nested attitudes (meta-attitudes) such as intentions to desire to believe something. Like our abstract model, such a concrete logical model can of course be used to define notions of attitudinal consistency, completeness and closedness, though one would be limited to the (often few) attitudes present in the logic in question.

6.2. The dynamics of multiple attitudes

The dynamics of multi-attitude psychology concern attitude change. Modal logics of the sort just discussed can model (deductive ^{Footnote 32} ) reasoning about attitudes, through the entailment relation. But reasoning about attitudes is a process of attitude discovery, not attitude change; it differs from reasoning in attitudes (or ‘with’ attitudes, in Broome’s words), a process of attitude formation. Establishing that this difference is real and could not be easily overcome through some formal reduction requires a careful analysis, which we undertake in Dietrich and Staras (Reference Dietrich and Staras2022). Here, a few remarks should suffice.

If someone reasons about your attitudes, then what changes are not your attitudes, but the reasoner’s beliefs about them. Even when it is you yourself who reasons about some of your attitudes, then not those attitudes change, but your (meta-)beliefs about them. ^{Footnote 33} In our earlier example, you reason in your attitudes by saying:

$$Doctors\,\,recommend\,\,resting.\,\,So,\,\,I\,\,shall\,\,rest.$$

You thereby form an intention from a belief. An observer (possibly you) might reason about your attitudes by saying:

$$You\,\,believe\,\,doctors\,\,recommend\,\,resting.\,\,So,\,\,you\,\,intend\,\,to\,\,rest.$$

This and other reasoning about attitudes can be modelled modal-logically, using entailments between atomic attitude-sentences of type ‘you hold attitude such-and-such towards such-and-such’, formally $O(\phi )$ with an operator O representing the attitude type and a sentence $\phi $ representing the attitude content. Thanks to building appropriate rationality axioms into the logic, the right entailments hold between such ‘atomic’ attitude-sentences. The logic also provides entailments between plenty of ‘non-atomic’ attitude-sentences, such as: ‘you do not desire this’, ‘you either believe this or intend that’, etc. Reasoning about attitudes can thus start from, or conclude in absences of attitudes (or disjunctions of attitudes etc.) – meaning that the reasoner discovers that absence (or disjunction etc.). But Broomean reasoning in attitudes cannot start from, or conclude in, absences – meaning that reasoning starts from attitudes you have and generates rather than removes attitudes (cf. section 5.2).

Overall, analogies between our abstract approach to multi-attitude psychology and concrete attitude logics are easier to draw at the static level than at the dynamic level. At the static level, both approaches include notions of consistency, completeness and closedness. At the dynamic level, our abstract model of reasoning departs from logical entailment between attitudes, as it captures reasoning in attitudes rather than about attitudes.