2. Lewis’s account of common knowledge

In this section, I outline Lewis’s theory of common knowledge. His definition goes as follows:

According to Lewis, a basis for common knowledge generates an infinite chain of higher-order reasons to believe: everyone in P has reason to believe that ___, everyone in P has reason to believe that everyone in P has reason to believe ___, and so on ad infinitum. Let us call this ‘Lewis’s theorem’.

Note that Lewis’s theorem does not say or imply that a basis for common knowledge generates an infinite series of knowledge states. Hence, the term ‘common knowledge’ is a misnomer, as Lewis himself came to realise:

Therefore, it would be more appropriate to call this infinite sequence of reasons to believe ‘common reason to believe’ instead of ‘common knowledge’. Commentators on Lewis’s theory usually keep the name ‘common knowledge’ (Clark and Marshall Reference Clark, Marshall, Joshi, Webber and Sag1981; Vanderschraaf Reference Vanderschraaf1998; Cubitt and Sugden Reference Cubitt and Sugden2003), but some use the term ‘common reason to believe’ (Sillari Reference Sillari2005). In the following both terms will be used, conforming to the authors discussed.

Although ‘reason to believe’ and ‘indication’ are central to Lewis’s definition of common knowledge, he does not provide formal definitions of these notions. In the following, I will try to provide definitions that stay as close as possible to Lewis’s text. An obvious starting point for defining ‘reason to believe’ is to treat it as a relation between individuals and propositions. This is the approach taken by philosophers who have analysed Lewis’s argument (Vanderschraaf Reference Vanderschraaf1998; Cubitt and Sugden Reference Cubitt and Sugden2003; Sillari Reference Sillari2005). I will discuss this approach in section 4 and then proceed to offer a novel approach, based on the idea that reasons to believe are objects of a sort, at least in the sense that they can be quantified over. For now, I will remain neutral on this point and use ${R_i}\varphi $ as an abbreviation for ‘i has reason to believe $\varphi $ ’.^{Footnote 2}

‘Indication’ is defined by Lewis as follows:

Following Vanderschraaf (Reference Vanderschraaf1998), Cubitt and Sugden (Reference Cubitt and Sugden2003) and Sillari (Reference Sillari2005), I treat ‘indication’ as a relation between an individual, a state of affairs, and a proposition, using $In{d_i}{\rm{\,}}A\varphi $ as an abbreviation for ‘the state of affairs A indicates to agent i that $\varphi $ ’. The question now is how we should interpret this relation. The difficulty is that Lewis uses the subjunctive mood and ‘if … thereby’ instead of the standard logical operator ‘if … then’. This indicates that he does not mean a material implication. Moreover, the assumption that a material implication is meant leads to implausible conclusions. For, assume:

(A0) $$In{d_i}{\rm{\,}}A\varphi \leftrightarrow \left( {{R_i}\left( {A\;{\rm{holds}}} \right) \to {R_i}\varphi } \right).$$

The left-to-right part of (A0) states

$$In{d_i}{\rm{\,}}A\varphi \to \left( {{R_i}\left( {A\;{\rm{holds}}} \right) \to {R_i}\varphi } \right)$$

which is equivalent to

(A1) $$\left( {In{d_i}{\rm{\,}}A\varphi \wedge {R_i}\left( {A\;{\rm{holds}}} \right)} \right) \to {R_i}\varphi $$

This seems intuitively plausible, and most importantly, it captures Lewis’s statement: “Consider next that our definition of indication yields a principle of detachment: if A indicates to x that ___ and x has reason to believe that A holds, then x has reason to believe that ___.” (Lewis Reference Lewis1969: 53). Lewis uses this inference in every step of his argument.

However, on the face of it, at least, the right-to-left part of (A0)

$$\left( {{R_i}\left( {A\;{\rm{holds}}} \right) \to {R_i}\varphi } \right) \to In{d_i}{\rm{\,}}A\varphi $$

does not seem correct. For one thing, it makes $In{d_i}{\rm{\,}}A\varphi $ vacuously true if i does not have a reason to believe A. However, in Lewis’s analysis, the indication relation is only used in contexts in which an individual does have reason to believe A, and therefore, this objection does not carry much weight. A more serious objection is that, according to the right-to-left part of (A0), $In{d_i}{\rm{\,}}A\varphi $ is true for every proposition $\varphi $ such that i has reason to believe $\varphi $ , which makes the indication relation trivial.

Apparently, the indication relation must be stronger than material implication. In section 4, I propose a new account of indication that does better justice to Lewis’s formulation.

2.1. Lewis’s theorem

We can now begin to explicate Lewis’s definition of a basis for common knowledge as follows: a state of affairs A is a basis for common knowledge of a proposition $\varphi $ in a group P if and only if A holds and the following hold for all i and j in P:

(C1) $${R_i}\left( {A\;{\rm{holds}}} \right)$$

(C2) $$In{d_i}{\rm{\;}}A\left( {{R_j}\left( {A\;{\rm{holds}}} \right)} \right)$$

(C3) $$In{d_i}{\rm{\,}}A\varphi $$

According to Lewis, these conditions are necessary but not sufficient for generating an infinite chain of higher-order reasons to believe. In addition, “suitable ancillary premises regarding our rationality, inductive standards, and background information” (Lewis Reference Lewis1969: 53) are required. Lewis develops these premises as follows.

Consider that if A indicates something to i, and if j shares i’s inductive standards and background information, then A must indicate the same thing to j. (53)

because

What A indicates to i will depend, therefore, on i’s inductive standards and background information. (53)

Lewis then states:

Therefore, if A indicates to i that j has reason to believe that A holds, and if A indicates to i that ___, and if i has reason to believe that j shares i’s inductive standards and background information, then A indicates to i that j has reason to believe that ___ (this reason being j’s reason to believe that A holds). (53)^{Footnote 3}

Accordingly, Lewis assumes that the members of P have reason to believe they share the same inductive standards and background information, at least nearly enough so that A will indicate the same things to all of them and, on that basis, he assumes the following:

(S) $$In{d_i}{\rm{\,}}A\left( {{R_j}\left( {A{\rm{\;holds}}} \right)} \right) \wedge In{d_i}{\rm{\,}}A\varphi \to In{d_i}{\rm{\,}}A\left( {{R_j}\varphi } \right)$$

However, Lewis offers no formal proof that this follows from his definition of indication, or from his description of rationality, inductive standards and background information. Therefore, we have to treat this as an axiom. This axiom is needed to introduce higher-order indications that, using (A1), then give rise to higher-order reasons to believe.

Lewis now uses (S) to outline a proof that, if conditions (C1)–(C3) are met, there is common knowledge of $\varphi $ , i.e. that all finitely nested formulas ${R_i}{R_j} \ldots {R_k}\varphi $ hold for all $i,j, \ldots ,k$ in P (Lewis’s theorem). The first steps of his proof are these:

1. — first-order reasons to believe

2. (1) $ \forall i \in P:In{d_i}{\rm{\,}}A\varphi \hskip 9.55pc {\rm C}3$

3. (2) $\forall i \in P:{R_i}\varphi \;\hskip 10.65pc{\rm{C}}1,1,{\rm{A}}1$

4. — second-order reasons to believe

5. (3) $\forall i,j \in P:In{d_i}{\rm{\,}}A\left( {{R_j}\varphi } \right)\hskip 7.6pc{\rm{C}}2,1,{\rm{S}}$

6. (4) $\forall i,j \in P:{R_i}{R_j}\varphi \;\hskip 9.37pc{\rm{C}}1,3,{\rm{A}}1$

7. — third-order reasons to believe

8. (5) $\forall i,j,k \in P:In{d_i}{\rm{\,}}A\left( {{R_j}{R_k}\varphi } \right)\hskip 6.03pc{\rm{C}}2,3,{\rm{S}}$

9. (6) $\forall i,j,k \in P:{R_i}{R_j}{R_k}\varphi \;\hskip 7.8pc{\rm{C}}1,5,{\rm{A}}1$

10. — higher-order reasons to believe

11. (7) ${\rm{etc}}.$

In the next section, I will first explain how Cubitt and Sugden (Reference Cubitt and Sugden2003) formalise this account and thereby clarify the plausibility of (S) and how Sillari (Reference Sillari2005) proposes a very different approach to prove the theorem.

3. Previous proposals for developing the theory

3.1. Cubitt and Sugden

Cubitt and Sugden (Reference Cubitt and Sugden2003) start their reconstruction of Lewis’s theory by giving an interpretation of the key concepts of this theory. First, ‘having a reason to believe’ is interpreted as a two-place relation between agents and propositions, as follows:

To say that some individual i has reason to believe some proposition x is to say that x is true within some logic of reasoning that is endorsed by (that is, accepted as a normative standard by) person i. For x to be true within such a logic of reasoning, it must either be treated as self-evident or be derivable from propositions that are treated as self-evident using the inference rules of the logic. Self-evidence may be either a priori or obtained through observation; the rules of inference may be deductive or inductive. (Cubitt and Sugden Reference Cubitt and Sugden2003: 184)

Second, ‘indication’ is interpreted as a three-place relation between agents, states of affairs and propositions. Lewis’s formulation “if i had reason to believe that A held, i would thereby have reason to believe that x” is read as “i’s reason to believe that A holds provides i’s reason for believing that x is true”. When combined with the above definition of reason to believe, this leads to:

Our interpretation of the formula ‘A ind $\rm {_i}$ x’ is that, in the logic of reasoning that i endorses, there is an inference rule which legitimates inferring x from ‘A holds’. (187)

The central concept in this interpretation of Lewis’s account is thus the logic of reasoning that someone endorses. This idea has been further elaborated in Cubitt and Sugden (Reference Cubitt and Sugden2014). In this paper, the phrase ‘logic of reasoning’ is replaced by ‘reasoning scheme’. Each agent has his own reasoning scheme, which consists of a non-empty set of axioms and a set of inference rules that contains all rules of logically valid inferences. This reasoning scheme is taken to be consistent, i.e. no contradiction can be derived.

On this basis they assume that the indication relation has the following properties:^{Footnote 4}

(A1) $$\forall i \in P:\left( {In{d_i}{\rm{\,}}A\varphi \wedge {R_i}\left( {A \ {\rm{holds}}} \right)} \right) \to {R_i}\varphi $$

(C4) $$\forall i,j \in P:In{d_i}{\rm{\,}}A\,\psi \to {R_i}\left( {In{d_j}\,A\,\psi } \right)$$

(A6) $$\forall i,j \in P:\left( {In{d_i}{\rm{\,}}A\left( {{R_j}\left( {A\;{\rm{holds}}} \right)} \right) \wedge {R_i}\left( {In{d_j}\,A\varphi } \right)} \right) \to In{d_i}{\rm{\,}}A\left( {{R_j}\varphi } \right)$$

(A1) is implied by the above definition of indication. It is just the left-to-right part of Lewis’s definition of indication, discussed in section 2.

(C4) is based on Lewis’s assumption “Suppose you and I do have reason to believe we share the same inductive standards and background information, at least nearly enough so that A will indicate the same things to both of us …” (Lewis Reference Lewis1969: 53). This axiom is not required to hold for all possible propositions, but only for the proposition $\varphi $ in the definition of a basis for common knowledge and for propositions that express first-order or higher-order reasons to believe $\varphi $ .

(A6) is motivated as follows:

(A6) says that if i’s logic legitimates an inference from ‘A holds’ to the proposition that j has reason to believe ‘A ${\rm{'}}$ holds’, and if i has reason to believe that j’s logic legitimates an inference from ‘A ${\rm{'}}$ holds’ to x, then i’s logic legitimates an inference from ‘A holds’ to the proposition that j has reason to believe x.’ (Cubitt and Sugden Reference Cubitt and Sugden2003: 188)

As a further explanation they add

If i has reason to believe that j’s logic legitimates an inference from (A ${\rm{'}}$ holds) to x, then, because i’s logic obeys the rules of deductive inference, i has reason to believe $[{R_j}$ (A’ holds) $ \to {R_j}\left( x \right)]$ . Hence, given the antecedent of the previous sentence, if i’s logic allows an inference from (A holds) to ${R_j}$ (A ${\rm{'}}$ holds), it also allows an inference from (A holds) to ${R_j}$ (x). Notice that A6 attributes deductive inference only to i’s reasons to believe. It does not postulate anything about what i has reason to believe about the inference rules endorsed by j. (188, fn17)

Interestingly, Lewis’s axiom (S) deductively follows from (C4) and (A6), as is easy to see.

These three assumptions suffice to prove Lewis’s theorem. The first steps of the proof in this reconstruction are as follows:

1. — first-order reasons to believe

2. (1) $\forall i \in P:In{d_i}{\rm{\,}}A\varphi\hskip 10pc {\rm{C}}3$

3. (2) $\forall i \in P:{R_i}\varphi \hskip 11.35pc {\rm{C}}1,1,{\rm{A}}1$

4. — second-order reasons to believe

5. (3) $\forall i,j \in P:{R_i}\left( {In{d_j}\,A\varphi } \right)\hskip 8pc 1,{\rm{C}}4$

6. (4) $\forall i,j \in P:In{d_i}{\rm{\,}}A\left( {{R_j}\varphi } \right)\hskip 8.06pc {\rm{C}}2,1,{\rm{A}}6$

7. (5) $\forall i,j \in P:{R_i}{R_j}\varphi \hskip 10.06pc {\rm{C}}1,4,{\rm{A}}1$

8. — third-order reasons to believe

9. (6) $\forall i,j,k \in P:{R_i}\left( {In{d_j}\,A\left( {{R_k}\varphi } \right)} \right)\hskip 5.89pc 5,{\rm{C}}4$

10. (7) $\forall i,j,k \in P:In{d_i}{\rm{\,}}A\left( {{R_j}{R_k}\varphi } \right)\hskip 6.5pc {\rm{C}}2,5,{\rm{A}}6$

11. (8) $\forall i,j,k \in P:{R_i}{R_j}{R_k}\varphi \hskip 8.5pc {\rm{C}}1,5,{\rm{A}}1$

12. — higher-order reasons to believe

13. (9) ${\rm{etc}}.$

Cubitt and Sugden have clarified Lewis’s theory by offering an interpretation of the two basic concepts of ‘reason to believe’ and ‘indication’. These interpretations are based upon a model in which each person has a logic of reasoning with axioms and inference rules. In addition, they have replaced the crucial axiom (S) by two axioms, both of which, with the help of the aforementioned interpretations, they have argued to be plausible. These two axioms together imply Lewis’s axiom (S). However, Cubitt and Sugden’s interpretation of ‘has reason to believe’ has some drawbacks.

First, it offers no way of representing that someone may have reasons for believing two contradictory propositions. If an agent i has reason to believe a proposition $\varphi $ then $\varphi $ is a theorem in the reasoning scheme that i endorses, according to Cubitt and Sugden (Reference Cubitt and Sugden2014). But $\varphi $ and not- $\varphi $ cannot both be true in i’s reasoning scheme because a reasoning scheme is supposed to be consistent. It is, however, quite conceivable, that someone has reasons to believe two contradictory propositions. Take, for example, the Nixon diamond scenario (Horty Reference Horty2012: 27-28). The example is set in the time when Nixon was president of the US. The fact that Nixon is a Quaker and that Quakers usually are pacifists is a reason to believe that Nixon is a pacifist. At the same time, the fact that Nixon is a Republican and that Republicans tend not to be pacifists is a reason to believe that Nixon is not a pacifist.

Second, Cubitt and Sugden’s account implies that every agent has reason to believe every tautology because his reasoning scheme contains all rules of valid inferences. This consequence is not implausible in itself but it is unclear whether it is in the spirit of Lewis’s approach, according to which our actual beliefs are governed by our reasons to believe: “Anyone who has reason to believe something will come to believe it, provided he has a sufficient degree of rationality.” (Lewis Reference Lewis1969: 55) In other words, provided it is not too hard to see that we have reason to believe that $\varphi $ we will believe $\varphi $ . Hence, on Cubitt and Sugden’s account, we believe all tautologies that aren’t too difficult to work out, and it may be doubted whether this prediction is plausible.

3.2. Sillari

Whereas Cubitt and Sugden’s account is syntactic, Sillari (Reference Sillari2005) adopts a semantic approach to prove Lewis’s theorem. In this section, I will successively discuss how Sillari interprets the notions ‘reason to believe’ and ‘indication’, how he defines ‘common reason to believe’, and how he proves Lewis’s theorem.

For Sillari ‘i has reason to believe ___’ is a modal operator ${R_i}$ on propositions that has the following axiomatisation:^{Footnote 5}

(B1) $$\left( {{R_i}\varphi \wedge {R_i}\left( {\varphi \to \psi } \right)} \right) \to {R_i}\psi $$

(B6) $${\rm{From\;}}\varphi \;{\rm{infer}}\;{R_i}\varphi $$

(B1) is the K axiom of modal logic, which seems acceptable in this context. (B6) is the rule of necessitation that is standard in modal logic. It says that every agent has reason to believe all tautologies. Sillari argues that “it makes sense to require that an agent has reason to believe what logic dictates, although it is not the case that the agents in the model will come to actually believe all logical truths” (Sillari Reference Sillari2005: 289). For reasons discussed at the end of the last subsection, this is potentially problematic.

The axiom (B1) and the rule (B6) make ${R_i}$ a normal modal operator, for which Sillari gives a Kripke semantics. For each agent i there is a relation ${{\cal R}_i}$ on a set of possible worlds W. The semantics of ${R_i}\varphi $ is given in the usual way: ${R_i}\varphi $ is true in a world w in a model M if and only if $\varphi $ is true in each world that is accessible from w:

(S1) $$\left( {M,w} \right) \models {R_i}\varphi \;{\rm{iff\;}}\left( {M,v} \right) \models \varphi \;{\rm{for\;all\;}}v{\rm{\;such\;that\;}}\left( {w,v} \right) \in {{\cal R}_i}.$$

‘Indication’ is treated as a modal operator on pairs of propositions^{Footnote 6} with the following axiomatisation:^{Footnote 7}

(B2) $$\left( {{R_i}\varphi \wedge \left( {\varphi \to \psi } \right)} \right) \to In{d_i}\,\varphi \psi $$

(B3) $$\left( {{R_i}\varphi \wedge In{d_i}\,\varphi \psi } \right) \to {R_i}\psi $$

(B3) is unproblematic: it is just the left-to-right part of Lewis’s definition of indication. But (B2) may be too strong an assumption. It entails that agents have reasons to believe all logical consequences of everything they have reason to believe, which is not uncontroversial. Another objection to this axiom is that it does not fit with Lewis’s definition of indication because that definition contains no implication from $\varphi $ to $\psi $ .

The semantics of $In{d_i}\varphi \psi $ is given by^{Footnote 8}

(S2) $$\left( {M,w} \right) \models In{d_i}\,\varphi \,\psi \;{\rm{iff\;}}\left( {M,w} \right) \models {R_i}\varphi \;{\rm{and\;}}\left( {M,w} \right) \models \left( {\varphi \to \psi } \right).$$

This definition is inconsistent with (B3). To see this, note that while (B3) is equivalent to $In{d_i}\,\varphi \,\psi \to \left( {{R_{i}}\varphi \to {R_i}\psi } \right)$ , (S2) yields $In{d_i}\,\varphi \,\psi \to \left( {\varphi \to \psi } \right)$ . Now we construct a counterexample with two worlds, w and v:

According to (S2), we have $w \models In{d_i}\,\varphi \,\psi $ , because $w \models {R_i}\varphi $ and $w \models \left( {\varphi \to \psi } \right)$ . So, according to (B3), we have $w \models {R_i}\psi $ . On the other hand, according to (S1), $w \, \,{{\models}\hskip-8.7pt{\not}}\,\, {R_i}\psi $ , because it is not the case that $v \models \psi $ .^{Footnote 9}

This contradiction poses a problem because both (B3) and (S2) are essential to Sillari’s account. The former represents Lewis’s definition of indication. The latter plays an essential role in Sillari’s semantical proof of Lewis’s theorem. There is no obvious way, as far as I can see, to reformulate these principles in such a way that the problem is circumvented.

Clearly, this is a serious problem, but let us put it on one side and consider Sillari’s proof of Lewis’s theorem (Sillari Reference Sillari2005: 390-391). Sillari first introduces a new modal operator $C{R_G}$ for a group of agents G that stands for “members of G have common reason to believe that …”. In order to conveniently express the semantic clause for $C{R_G}$ , he uses the concept of G-reachability: “we say that a world v is G-reachable in k steps from world w iff there is a path of length k from v to w such that the edges between adjacent worlds are labelled by the accessibility relations of members of G”. It follows that: “ $\left( {M,w} \right) \models C{R_G}\,\varphi $ iff $\left( {M,v} \right) \models \varphi $ for all worlds v that are G-reachable from w in any numbers of steps.”

The proof of Lewis’s theorem proceeds by induction on the length of the path. Unfortunately, the inductive step of this proof is either incorrect or the formulation of the theorem is trivial, as is shown in detail in the appendix, and it is doubtful that this can be fixed.

In conclusion, Sillari’s proposal is flawed in several respects. Most importantly, it uses two principles, (B3) and (S2), that are mutually inconsistent. Moreover, it uses assumptions that are too strong to be plausible and it replaces Lewis’s argument with something entirely different, and therefore it hardly counts as an explication of Lewis’s account.

In the next section, I will present a new reconstruction of Lewis’s theory by developing the notion of ‘reason to believe’ a bit further than Lewis did.

4. Reasoning with reasons

Sillari and Cubitt and Sugden treat ‘reason to believe’ as an operator on propositions. This treatment misses an important aspect of Lewis’s account, as the following quotes illustrate:

In the first quote, Lewis identifies one reason with another, which suggests that reasons should be treated as objects, at least in the sense that two reasons can be the same or different. This suggestion is strengthened by the use of ‘thereby’ in the second quote, which seems to be saying that the reason to believe ___ is based upon the reason to believe that A holds.

A reason to believe something may be seen as a piece of evidence that supports that belief (Égré et al. Reference Égré, Marty and Renne2021: 689). For example, seeing a bright flash of light followed by a loud bang is a reason to believe that a thunderstorm is underway. Or, having seen many people driving on the right and never one on the left, is a reason to believe that almost everyone drives on the right (cf. Lewis Reference Lewis1969: 40).

Based on these observations, I propose to analyse Lewis’s argument using a logic that allows us to talk about reasons. My logic is a minimal extension of a subset of justification logic (Artemov and Fitting Reference Artemov and Fitting2019; Kuznets and Studer Reference Kuznets and Studer2019). Obviously, justifications and reasons are related concepts (Brandom Reference Brandom2009: 5), and therefore it shouldn’t come as a surprise that a version of justification logic can be used to reason about reasons to believe.

It bears emphasising that justification logic was originally developed for quite different purposes than the one pursued here. In mathematical logic, it provides a semantics for intuitionistic logic in which justification terms represent proofs. In this context, $t:X$ represents that t is a proof of proposition X. In formal epistemology, justifications have been used to complement standard possible-worlds analyses of knowledge and belief. So, unlike the proposals discussed in the foregoing, the present account of reasons to believe is motivated entirely on independent grounds.

The language of our logic is based on a countable set of atomic propositions, a finite set of (names for) agents, and countable sets of constants and variables that represent reasons. The formulas of the language are defined inductively as follows:

1. 1. each atomic proposition is a formula

2. 2. if $\alpha $ and $\beta $ are formulas then $\alpha \to \beta $ and $\alpha \wedge \beta $ are formulas

3. 3. if $\alpha $ is a formula, i is an agent and r is a reason, then $r\ {:_i}\ \alpha $ is a formula. This formula is to be read as ‘r is a reason for agent i to believe $\alpha $ ’

4. 4. if $\alpha $ is a formula and i is an agent, then $\exists r\left( {r\ {:_i}\ \alpha } \right)$ is a formula.

The logic features an operator, ‘ $ \cdot $ ’, for constructing composite reasons:^{Footnote 10} if r and s are reasons, then $r \cdot s$ is also a reason. Composite reasons are used to define a version of modus ponens:

$$s\ {:_i}\ \left( {\alpha \to \beta } \right) \to \left( {\left( {t\ {:_i}\ \alpha } \right) \to \left( {s \cdot t{\ :_i}\ \beta } \right)} \right)$$

If an agent i has reason s to believe the implication $\alpha \to \beta $ and reason t to believe $\alpha $ , then $s \cdot t$ represents his reason to believe $\beta $ . So two reasons are needed to infer $\beta $ . The operator is called ‘application’ because a reason to believe an implication may be seen as a device that applies to any reason to believe $\alpha $ so as to produce a reason to believe $\beta $ .

Besides a set of axioms for classical propositional logic and the rule of modus ponens, our logic has the following application rule:^{Footnote 11}

(AR) $$s\ {:_i}\ \left( {\alpha \to \beta } \right),{\rm{\;}}t\ {:_i}\ \alpha \;\;\vdash {\rm{\;\;}}s \cdot t\ {:_i}\ \beta $$

In contrast to the accounts of Cubitt and Sugden and Sillari, this logic does not require that agents have a reason to believe for every tautology. This premise is not needed to prove Lewis’s theorem, and as we will see, a relatively weak logic suffices for a proof of that theorem: we only have to assume that agents have a reason to believe that $\alpha \wedge \beta $ follows from $\alpha $ and $\beta $ , and a reason to believe that $\alpha \to \gamma $ follows from $\alpha \to \beta $ and $\beta \to \gamma $ . The appendix shows that these two assumptions are sufficient.

With this logic in place, we can give a new interpretation of the two basic concepts of Lewis’s account.

To begin with, I will interpret ‘i has reason to believe that $\varphi $ ’ as ‘i has a reason to believe that $\varphi $ ’. I will keep the notation ${R_i}\varphi $ , but with a new meaning, which will help to compare our account with those of Cubitt and Sugden (Reference Cubitt and Sugden2003) and Sillari (Reference Sillari2005). The new meaning of ${R_i}\varphi $ is: ‘there is a reason r such that $r\ {:_i}\ \varphi $ ’. So ${R_i}\varphi $ is an existential claim: that i has a reason to believe $\varphi $ . Hence, whereas from $r\ {:_i}\ \varphi $ we can infer ${R_i}\varphi $ , it is not possible to conclude $r\ {:_i}\ \varphi $ from ${R_i}\varphi $ .

Let us now turn to the notion of indication. According to Lewis, $In{d_i}\,A\varphi $ means that if i had a reason to believe that A holds, he thereby would have a reason to believe $\varphi $ . Now, suppose i has a reason, say r, to believe that A holds and thereby has reason to believe $\varphi $ . I interpret ‘thereby’ as meaning that the reason to believe $\varphi $ is, at least partly, provided by r. In our logic, we can formalise this by saying that there is a reason s that, together with r, is the reason to believe $\varphi $ . An obvious choice for the latter reason then is $s \cdot r$ . Consequently, s is a reason to believe $A \to \varphi $ .^{Footnote 12}

Therefore, we interpret ‘the state of affairs A indicates to i that $\varphi $ ’ as ‘i has a reason to believe that A implies $\varphi $ ’. Accordingly, $In{d_i}{\rm{\,}}A\varphi $ is an abbreviation of ‘there is a reason r such that $r\ {:_i}\ \left( {A \to \varphi } \right)$ ’. This interpretation closely follows what Lewis says about indication. In particular, this interpretation takes into account the word ‘thereby’ in Lewis’s definition of indication quoted on page 4.

To demonstrate this, let us assume that A indicates $\varphi $ to i. In our interpretation, this means that there exists a reason s such that $s\ {:_i}\ \left( {A \to \varphi } \right)$ . Then we have to show that if i has a reason to believe that A holds he thereby has a reason to believe $\varphi $ . Suppose i has a reason, say r, to believe that A holds. By rule (AR), we know that there indeed is a reason to believe $\varphi $ , namely $s \cdot r$ . This reason contains r as a constituent, which does justice to the suggestion of the word ‘thereby’ that the reason to believe $\varphi $ is partly provided by the reason to believe that A holds.

Interestingly, this interpretation of indication is consistent with “in the logic of reasoning that i endorses, there is an inference rule which legitimates inferring x from ‘A holds’ ” in Cubitt and Sugden (Reference Cubitt and Sugden2003: 187).

Our interpretations of the concepts of reason to believe and indication enable us to close the gaps in Lewis’s argument and to formally derive Lewis’s theorem. We begin with reformulating the conditions for a state of affairs A that holds to be a basis for common knowledge of a proposition $\varphi $ .

(C1) $${R_i}A \ \rightsquigarrow \ \exists r\left( {r\ {:_i}\ A} \right)$$

(C2) $$In{d_i}{\rm{\,}}A\left( {{R_j}{\rm{\,}}A} \right){\rm{\;}} \rightsquigarrow {\rm{\;}}{R_i}\left( {A \to {R_j}{\rm{\;}}A} \right){\rm{\;}} \rightsquigarrow {\rm{\;}}\exists r\left( {r\ {:_i}\ \left( {A \to \exists s\left( {s\ {:_j}\ A} \right)} \right)} \right)$$

(C3) $$In{d_i}{\rm{\,}}A\varphi {\rm{\;}} \rightsquigarrow {\rm{\;}}{R_i}\left( {A \to \varphi } \right){\rm{\;}} \rightsquigarrow {\rm{\;}}\exists r\left( {r\ {:_i}\ \left( {A \to \varphi } \right)} \right)$$

Lewis assumed that the agents have a reason to believe that they share their inductive standards and background information and therefore have a reason to believe that A indicates the same things to them. Cubitt and Sugden translated this assumption into their axiom (C4). By replacing all occurrences of $In{d_i}{\rm{\;}}A\psi $ in this axiom with ${R_i}\left( {A \to \psi } \right)$ we get:

(C4) $$\forall i,j \in P:{R_i}\left( {A \to \psi } \right) \to {R_i}{R_j}{\rm{\;}}\left( {A \to \psi } \right)$$

We will only need this assumption in so far as $\psi $ is one of the propositions $\varphi ,{R_i}\varphi $ , ${R_i}{R_j}\varphi $ , ${R_i}{R_j}{R_k}\varphi $ , and so on.

Before spelling out Lewis’s argument, we formulate three theorems of our logic that will simplify the proof:

(E1) $${R_i}\left( {\alpha \to \beta } \right),{\rm{\;}}{R_i}\alpha\;\vdash \;{R_i}\beta $$

(E2) $${R_i}\left( {\alpha \to \beta } \right),{\rm{\;}}{R_i}\left( {\beta \to \gamma } \right) \;\vdash \;{R_i}\left( {\alpha \to \gamma } \right)$$

(E3) $${R_i}{R_j}{\rm{\;}}\left( {\alpha \to \beta } \right) \;\vdash \;{R_i}\left( {{R_j}{\rm{\;}}\alpha \to {R_j}{\rm{\;}}\beta } \right)$$

The proofs of these theorems, which can be found in the appendix, require only a relatively weak logic. In particular, there is no need to assume that the agents have a reason to believe for every tautology, as in the accounts of Cubitt and Sugden and Sillari.

The axiom schemas (A1) and (A6) that Cubitt and Sugden needed to prove Lewis’s theorem can be easily deduced in our theory. The proofs can be found in the appendix.

Now, we can complete the proof of Lewis’s theorem:

1. — first-order reasons to believe

2. (1) $\forall i \in P:{R_i}{\rm{\;}}A\;\hskip 10.65pc{\rm{C}}1$

3. (2) $\forall i,j \in P:{R_i}\left( {A \to {R_j}A} \right)\hskip 7.3pc{\rm{C}}2$

4. (3) $\forall i \in P:{R_i}\left( {A \to \varphi } \right)\hskip 8.7pc{\rm{C}}3$

5. (4) $\forall i \in P:{R_i}\varphi \;\hskip 11pc 3,1,{\rm{E}}1$

6. — second-order reasons to believe

7. (5) $\forall i,j \in P:{R_i}{R_j}\left( {A \to \varphi } \right)\hskip 7.55pc 3,{\rm{C}}4$

8. (6) $\forall i,j \in P:{R_i}\left( {{R_j}A \to {R_j}\varphi } \right)\hskip 6.8pc 5,{\rm{E}}3$

9. (7) $\forall i,j \in P:{R_i}\left( {A \to {R_j}\varphi } \right)\hskip 7.5pc2,6,{\rm{E}}2$

10. (8) $\forall i,j \in P:{R_i}{R_j}\varphi \;\hskip 9.9pc 7,1,{\rm{E}}1$

11. — third-order reasons to believe

12. (9) $\forall i,j,k \in P:{R_i}\left( {{R_j}\left( {A \to {R_k}\varphi } \right)} \right)\hskip 5.4pc 7,{\rm{C}}4$

13. (10) $\forall i,j,k \in P:{R_i}\left( {{R_j}A \to {R_j}{R_k}\varphi } \right)\hskip 5.3pc 9,{\rm{E}}3$

14. (11) $\forall i,j,k \in P:{R_i}\left( {A \to {R_j}{R_k}\varphi } \right)\hskip 6pc 2,10,{\rm{E}}2$

15. (12) $\forall i,j,k \in P:{R_i}{R_j}{R_k}\varphi \;\hskip 8.4pc 11,1,{\rm{E}}1$

16. — higher-order reasons to believe

17. (13) ${\rm{etc}}.$

The last column of this proof shows the role of the different axioms and theorems. Axiom (C4) enables the step to the next higher-order proposition, while theorems (E2) and (E3) allow for this proposition to be transformed into a higher-order reason to believe. These theorems thus do the work that axiom (A6) did in Cubitt and Sugden’s formalisation. At first sight, explicit reasons do not seem to play a role in this proof. However, their role is visible in the proofs of theorems (E2) and (E3) which can be found in the appendix.

This proof, like Lewis’s own proof, provides just the first steps of an infinite series of steps. However, that is not sufficient for a proper proof. A rigorous proof by mathematical induction is given in the appendix.

In conclusion, I have given new interpretations of ‘i has reason to believe’ as ‘i has a reason to believe’ and of ‘A indicates $\varphi $ to i’ as ‘i has a reason to believe that A implies $\varphi $ ’. A major advantage of this approach is that the two basic concepts of ‘having reason to believe’ and ‘indication’ are reduced to a single concept ‘reason to believe’. But more importantly, a minimal logic for reasoning about these reasons with minimal assumptions about inferences enabled us to close the gaps in the proof of Lewis’s theorem.

5. From ‘reason to believe’ to ‘actual belief’

Lewis’s theory aims to describe how we coordinate our activities and, therefore, it requires an understanding of beliefs. Specifically, the theory must explain how individuals convert their reasons to believe into actual beliefs because it is only actual beliefs that can shape their behaviour. According to Lewis, only the first few orders of actual belief will be formed because common belief, which is an infinite series of ever-higher orders of belief, is unattainable. This is because the iteration of reasons to believe is “a chain of implications, not of steps in anyone’s actual reasoning” but actual reasoning is required to form actual beliefs, necessitating ancillary premises about rationality. In this context, rationality does not refer to decision-theoretic or game-theoretic concepts, such as maximizing expected utility or having unlimited deductive power. Instead, it signifies that someone who has reason to believe something will come to believe it. This kind of rationality comes in degrees, and the higher the order of a reason to believe, the higher the degree of rationality it takes to come to actual belief.

In this paper, three reconstructions of Lewis’s theory of reasons to believe have been discussed. In this section, I will consider whether these reconstructions can shed some light on the transition of a reason to believe to an actual belief.

Cubitt and Sugden (Reference Cubitt and Sugden2003) investigate what Lewis’s analysis would look like if they added some strong assumptions about the agents’ rationality, acknowledging that these assumptions are unrealistic and not in line with Lewis’s intentions. The analysis does, however, lead to an interesting conclusion, namely that a basis A for common knowledge of a proposition $\varphi $ may produce not only common reason to believe $\varphi $ but also common (actual) belief of this proposition. This requires assumptions about the rationality of the agents, the basis for common knowledge, and the indication relation. First, every agent has to be a faultless reasoner, that is, he must believe everything he has reason to believe. Furthermore, this basis A must also be a basis for common knowledge that everyone in P reasons faultlessly. Finally, there are two ancillary premises constraining indication:

(A3) $$In{d_i}{\rm{\;}}A\varphi \wedge In{d_i}{\rm{\;}}A\psi \to In{d_i}{\rm{\;}}A\left( {\varphi \wedge \psi } \right)$$

(A5) $$In{d_i}{\rm{\;}}A\varphi \wedge \left( {\varphi \;{\rm{entails\;}}\psi } \right) \to In{d_i}{\rm{\;}}A\psi $$

With these assumptions in place, the agents will have common belief of $\varphi $ .

Sillari (Reference Sillari2005) models the difference between reasons to believe and actual beliefs by introducing an ‘awareness structure’ on the set of possible worlds. For each agent i there is a set ${A_i}$ of formulas the agent is aware of. A formula ${A_i}\varphi $ , i.e. agent i is aware of $\varphi $ , means that agent i is sufficiently rational to believe $\varphi $ if he has reason to believe $\varphi $ . So an agent actually believes a formula if and only if two conditions are met: he has reason to believe it, and the formula belongs to his awareness set. This can be expressed as ${B_i}\varphi \equiv {R_i}\varphi \wedge {A_i}\varphi $ .

The concept of degrees of rationality, as proposed by Lewis, can be modeled using the awareness structure. First-order rationality of an agent i then means that $\varphi $ is in his awareness set. Second-order rationality means that all formulas of the form ${B_j}\varphi $ are in his awareness set. Now suppose that i also has reason to believe that j has first-order rationality. Then he will have the second-order beliefs ${B_i}{B_j}\varphi $ . Similarly, it is possible to model any level of belief. While this account ties in with Lewis’s explanation of the conversion of reasons to believe into actual beliefs, it lacks explanatory power because it reduces rationality to a list of formulas that are taken to be given (Paternotte Reference Paternotte2011: 264).

These two accounts show that it is difficult to formalise Lewis’s account of acquiring higher-order actual beliefs. Lewis’s explanation starts with the generation of an infinite series of ever-higher orders of reasons to believe. He then describes the conversion of reasons to believe into actual beliefs. This conversion follows the structure of the series of reasons to believe. For each step in the conversion, an agent needs a higher degree of rationality and reasons to ascribe a higher degree of rationality to the other agents. Lewis does not spell out what kind of rationality is involved in this process. Neither of the two accounts succeeds in clarifying this account.

The reconstruction proposed in section 4 provides insight into why each step in the hierarchy of higher-order reasons to believe is an implication that does not require actual reasoning, whereas each step in the transition from reason to believe into actual belief requires an ever higher degree of rationality. To proceed from n-order reasons to believe to $n + 1$ -order reasons to believe only some weak principles of logic are needed. So, logically speaking, this transition is not complex at all. However, the reasons themselves become increasingly complex at each step. As can be seen in the proof of Lewis’s theorem, each step requires the use of theorems (E2) and (E3), each of which uses the application rule (AR) that produces composite reasons. This might explain why the demands on the rationality required for the conversion from reasons to believe into actual beliefs increase rapidly. So, further research along this formalisation of Lewis’s theory, might be useful.