VARIETIES OF INTERPERSONAL COMPATIBILITY OF BELIEFS

Giacomo Bonanno and Klaus Nehring Department of Economics University of California Davis, CA 95616-8578, USA

gfbonanno@ucdavis.edu kdnehring@ucdavis.edu

Contents

1 Introduction 2 2 The basic system K_{n}^{∗} 2 3 Some properties of K_{n}^{∗} 3 4 Interpersonal compatibility of beliefs 4 5 Concluding remarks 9 6 Appendix 10

1 Introduction

Since Lewis’s (1969) and Aumann’s (1976) pioneering contributions, the concepts of common knowledge and common belief have been discussed extensively in the literature, both syntactically and semantically^{1}. At the individual level the difference between knowledge and belief is usually identified with the presence or absence of the Truth Axiom (□_{i}A → A), which is interpreted as ”if individual i believes that A, then A is true”. In such a case the individual is often said to know that A (thus it is possible for an individual to believe a false proposition but she cannot know a false proposition). Going to the interpersonal level, the literature then distinguishes between common knowledge and common belief on the basis of whether or not the Truth Axiom is postulated at the individual level. However, while at the individual level the Truth Axiom captures merely a relationship between the individuals’ beliefs and the external world, at the interpersonal level it has very strong implications. For example, the following is a consequence of the Truth Axiom: □_{i}□_{j}A → □_{i}A, that is, if individual i believes that individual j believes that A, then individual i herself believes that A. Thus, in contrast to other axioms, the Truth Axiom does not merely reflect individual agents’ “logic of belief”. (The reason why the Truth Axiom is much stronger in an interpersonal context than appears at first glance is that it amounts to assuming that agreement of any individual’s belief with the truth is common knowledge). Given its logical force, it is not surprising to find that it has strong implications for the logic of common knowledge. In particular, if each individual’s beliefs satisfy the strongest logic of knowledge (namely S5 or KT5), the associated common knowledge operator satisfies this logic too. Such is not the case for belief: bereft of the Truth Axiom, even the strongest logic for individual belief (KD45) is insufficient to ensure the satisfaction of the “Negative Introspection” axiom for common belief: ¬□_{∗}A → □_{∗}¬□_{∗}A (where □_{∗} denotes the common belief operator). That is to say, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. Indeed the Negative Introspection axiom for common belief implies restrictions on individual beliefs of an intersubjective nature. In this paper we consider a variety of intersubjective compatibility restrictions on the beliefs of the individuals and study their relationship. We also provide a characterization of Negative Introspection for common belief.

2 The basic system K_{n}^{∗}

We consider a multimodal system with n+1 operators □_{1}, □_{2}, ..., □_{n}, □_{∗} where, for i = 1, ..., n, the interpretation of □_{i}A is “individual i believes that A”, while □_{∗}A is interpreted as “it is common belief that A”. The basic system K_{n}^{∗} is given by a suitable axiomatization of Propositional Calculus together with the following axiom schemata and rules of inference:

1See, fo example, Fagin et al (1995) and references therein.

\begin{matrix}K & □(A → B) → (□A → □B) & (∀□ ∈ \{□_{1}, ..., □_{n}, □_{∗}\}) \\ CB1 & □_{∗}A → □_{i}A & (∀i = 1, ..., n) \\ CB2 & □_{∗}A → □_{i}□_{∗}A & (∀i = 1, ..., n) \end{matrix}

CB3 □_{∗}(A → □_{1}A ∧ ... ∧ □_{n}A) → (□_{1}A ∧ ... ∧ □_{n}A → □_{∗}A)
MP (modus ponens) \frac{A, A→B}{B}
RN (necessitation) \frac{A}{□A} (∀□ ∈ \{□_{1}, ..., □_{n}, □_{∗}\})

We now turn to the semantics. A standard model is a tuple

M = ⟨W, R_{1}, ..., R_{n}, R_{∗}, V ⟩

where W is a non-empty set of possible worlds, R_{1}, ..., R_{n}, R_{∗} are binary accessibility relations on W and V is a valuation, that is, a function that associates with every atomic proposition p the set of possible worlds where p is true. The valuation is extended to the set of formulas in the usual way; we denote the fact that formula A is true at world w in model M by M, w ⊧ A. Thus, in particular, for i = 1, ..., n, M, w ⊧ □_{i}A if and only if M, w′ ⊧ A for all w′ such that wR_{i}w′. Similarly, M, w ⊧ □_{∗}A if and only if M, w′ ⊧ A for all w′ such that wR_{∗}w′. The following result is well-known (cf., for example, Bonanno, 1996).

Theorem 1 The system K_{n}^{∗} is sound and complete with respect to the class of standard models where R_{∗} is the transitive closure of R_{1} ∪ ... ∪ R_{n}.^{2}

A standard model where R_{∗} is the transitive closure of R_{1} ∪ ... ∪ R_{n} will be called a CB-model.

We will investigate extensions of K_{n}^{∗} obtained by adding one or more of the following axioms for individual beliefs:

\begin{matrix}D & □_{i}A → ¬□_{i}¬A & (i = 1, ..., n) \\ T & □_{i}A → A & (i = 1, ..., n) \\ 4 & □_{i}A → □_{i}□_{i}A & (i = 1, ..., n) \\ 4c & □_{i}□_{i}A → □_{i}A & (i = 1, ..., n) \\ 5 & ¬□_{i}A → □_{i}¬□_{i}A & (i = 1, ..., n) \end{matrix}

In the next section we list some theorems and derived rules of inference for K_{n}^{∗} which will be used later.

3 Some properties of K_{n}^{∗}

For every modal operator □ ∈ {□_{1}, ..., □_{n}, □_{∗}} we write ♢ for ¬□¬ (thus, for example, ♢_{i}A stands for ¬□_{i}¬A). Furthermore, PL stands for ”Propositional Logic”.

2That is, aR_{∗}b if and only if there are sequences ⟨w_{1}, ..., w_{m}⟩ and ⟨i_{1}, ..., i_{m}⟩ such that (1)

w_{1} = a, (2) w_{m} = b and (3) for every k = 1, ..., m − 1, w_{k}R_{i_{k}}w_{k+1}.

It is well-known (see, for example, Chellas, 1984, Lismont and Mongin, 1994) that K_{n}^{∗} has the following theorems and rules of inference:

RK \frac{A→B}{□A→□B} for every □ ∈ \{□_{1}, ..., □_{n}, □_{∗}\}
RK♢ \frac{A→B}{♢A→♢B}  for every ♢ ∈ \{♢_{1}, ..., ♢_{n}, ♢_{∗}\}
M  □(A ∧ B) ↔ □A ∧ □B for every □ ∈ {□_{1}, ..., □_{n}, □_{∗}}
R1CB \frac{A → (□_{1}A ∧ ... ∧ □_{n}A)}{(□_{1}A ∧ ... ∧ □_{n}A) → □_{∗}A} (apply RN to the hypothesis,

then use CB3 and MP)

R2CB \frac{A → (□_{1}A ∧ ... ∧ □_{n}A)}{A → □_{∗}A}  (apply R1CB to the hypothesis,

then use PL)

Proofs of the following lemma and corollary are given in the Appendix.

Lemma 2 The following is provable in K_{n}^{∗}:

CB4  (□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A

Corollary 3 The following is provable in K_{n}^{∗} + 4c:

CB5  (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A

Remark 1 It is well-known (cf. Chellas, 1984) that axiom 4c is provable in K_{n}^{∗} + 5. Hence CB5 is provable in K_{n}^{∗} + 5.

4 Interpersonal compatibility of beliefs

In general, the common belief operator □_{∗} does not inherit all the properties of the individuals’ belief operators. Consider, for example, the counterpart for common belief of axiom 5 for individual beliefs:

5^{∗} ¬□_{∗}A → □_{∗}¬□_{∗}A.

Now, 5^{∗} is not provable in the system obtained by adding D, 4 and 5 to K_{n}^{∗}, which will be denoted by K_{n}^{∗} + D45 (a fortiori it is not provable in a weaker system such as K_{n}^{∗} + 5). This is shown in the following example.

Example 1 Consider the following CB-model: W = {a, b}, R_{1} = {(a, a), (b, b)}, R_{2} = {(a, b), (b, b)}. Thus R_{∗} = {(a, a), (a, b), (b, b)}. Let p be an atomic proposition which is true at b and false at a. Then the formula (¬□_{∗}p → □_{∗}¬□_{∗}p), which is an instance of 5^{∗}, is false at a. Since the system K_{n}^{∗} + D45 is sound with respect to the class of CB-models where R_{i} is serial, transitive and euclidean (i = 1, ..., n) and this model satisfies these properties, it follows that 5^{∗} is not provable in K_{n}^{∗} + D45.

It follows from Example 1 that axiom 5^{∗} must involve further restrictions on the beliefs of the individuals which presumably are intersubjective in nature. In this section we consider various requirements of intersubjective compatibility of beliefs and study their relationship.

The following axioms capture interpersonal restrictions of various strength on the beliefs of the individuals.

C  □_{i}A → ♢_{j}A
TN □_{i}□_{j}A → □_{j}A
TP □_{i}♢_{j}A → ♢_{j}A
IN □_{i}□_{j}A → □_{i}A
IP ♢_{i}□_{j}A → ♢_{i}A
SW □_{i}□_{∗}A → □_{j}□_{∗}A
T^{CB} □_{i}□_{∗}A → □_{∗}A

C rules out the possibility that one individual believes A and at the same some other individual believes ¬A. TN requires individuals to be correct in their beliefs about what others believe, while TP requires them to be correct in their beliefs about what others consider possible. By IN individuals must share the beliefs that they attribute to others, while IP is a weakening of this: if individual i considers it possible that individual j believes A then i himself must at least consider A possible. SW (for Shared Worlds) requires the individuals’ beliefs about what is commonly believed to agree. Finally T^{CB} requires individuals to be correct in their beliefs about common belief.

Remark 2 T, the Truth Axiom for individuals beliefs, plays a much stronger role in multi-agent contexts than in single-agent ones. For a single individual the Truth Axiom captures merely a relationship between her beliefs and the external world; at the interpersonal level, on the other hand, it implies strong intersubjective compatibility of beliefs. Indeed, all of the above axioms (C, TN, TP, IN, IP, SW and T^{CB}) are provable in K_{n}^{∗} +T. The reason why the Truth Axiom has strong implications in an interpersonal context is that it amounts to assuming that agreement of any individual’s beliefs with the truth is commonly believed.

The following theorem establishes the relationship between the first five axioms, namely C, TN, TP, IN and IP.

Theorem 4

(i) K_{n}^{∗} + IP + TP ⊢ C
(ii) K_{n}^{∗} + 4 + C ⊢ IP
(iii) K_{n}^{∗} + 4 + C ⊢ TP
(iv) K_{n}^{∗} + 5 + C ⊢ TN

Proof. The proof of (i) is as follows.

1. ♢_{i}□_{j}¬A → ¬□_{i}A (IP)
2. □_{i}A → □_{i}♢_{j}A  (1,PL)
3. □_{i}♢_{j}A → ♢_{j}A  (TP)
4. □_{i}A → ♢_{j}A  (2,3,PL)

(ii) Next we prove that K_{n}^{∗} + 4 + C ⊢ IP:

1. □_{i}¬A → ¬□_{j}A  (C)
2. □_{i}□_{i}¬A → □_{i}¬□_{j}A (1,RK)
3. □_{i}¬A → □_{i}□_{i}¬A  (axiom 4)
4. □_{i}¬A → □_{i}¬□_{j}A  (2,3,PL)
5. ♢_{i}□_{j}A → ♢_{i}A  (4,PL)

(iii) The proof of (iii) is as follows:

1. □_{j}¬A → □_{j}□_{j}¬A  (axiom 4)
2. □_{j}□_{j}¬A → ¬□_{i}¬□_{j}¬A (C)
3. □_{j}¬A → ¬□_{i}¬□_{j}¬A  (1,2,PL)
4. □_{i}♢_{j}A → ♢_{j}A  (3,PL)

(iv) The proof that K_{n}^{∗} + 5 + C ⊢ TN is as follows:

1. ¬□_{j}A → □_{j}¬□_{j}A  (axiom 5)
2. □_{j}¬□_{j}A → ¬□_{i}□_{j}A (C)
3. ¬□_{j}A → ¬□_{i}□_{j}A  (1,2,PL)
4. □_{i}□_{j}A → □_{j}A  (3,PL)

By Theorem 4, Compatibility (axiom C) has strong implications for the truth of intersubjective beliefs (TP and TN); moreover, it crucially involves some form of intersubjective agreement (IP). However, as the following example shows, intersubjective truth alone (TP and TN) fails to imply any intersubjective agreement (IP or IN) and therefore fails to imply C.

Example 2 Consider the following model: W = {a, b}, R_{1} = {(a, b), (b, b)}, R_{2} = {(a, a), (b, a)}. Thus R_{∗} (the transitive closure of R_{1}∪R_{2}) is the universal relation. Let p be an atomic proposition which is true at a and false at b. Then at a all of the following are true □_{1}□_{2}p, ♢_{1}□_{2}p, ¬□_{1}p, □_{1}¬p, □_{2}p, thus falsifying IN, IP, and C. On the other hand, both TP and TN are true at every world.^{3}

Similarly, the following example shows that intersubjective agreement (IP and IN) does not imply intersubjective truth (TP and TN) nor does it imply compatibility (C).

Example 3 Let

W = {a, b, c}, R_{1} = {(a, a), (b, a), (c, c)}, R_{2} = {(a, a), (b, c), (c, c)}.

3This is a consequence of the following fact, which is proved in the appendix. Axioms TN and TP are valid in the class of CB-models where, ∀i, j ∈ N, ∀w, w′, w′′ ∈ W, (1) R_{i} is serial and (2) if wR_{i}w′ and w′R_{j}w′′ then wR_{j}w′′. The model of the above example satifies this property.

Thus R_{∗} = {(a, a), (b, a), (b, c), (c, c)}. Let p be an atomic proposition which is true at a and false at b and c. Here at world b all the following are true: □_{1}□_{2}p, □_{1}♢_{2}p, ¬□_{2}p, □_{2}¬p and □_{1}p. Thus at b TP, TN and C are falsified. On the other hand both IP and IN are valid in this model.^{4}

We now turn to the relationship between SW and T^{CB}.

Theorem 5

(i) K_{n}^{∗} + T^{CB} ⊢ SW
(ii) K_{n}^{∗} + 4c + SW ⊢ T^{CB}

Proof. For (i)

\begin{matrix}1. & □_{i}□_{∗}A → □_{∗}A & (T^{CB}) \\ 2. & □_{∗}A → □_{j}□_{∗}A & (CB2) \\ 3. & □_{i}□_{∗}A → □_{j}□_{∗}A & (1,2,PL) \end{matrix}

For (ii)

1.  □_{i}□_{∗}A → □_{1}□_{∗}A  (SW)
...  ...  (SW)
n.  □_{i}□_{∗}A → □_{n}□_{∗}A  (SW)
n+1. □_{i}□_{∗}A → (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) (1,...,n,PL)
n+2. (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A  (CB5: cf. Corollary 3)
n+3. □_{i}□_{∗}A → □_{∗}A  (n+1, n+2, PL)

The next theorem relates T^{CB} to 5^{∗}.

Theorem 6

(i) K_{n}^{∗} + D + 5^{∗} ⊢ T^{CB}
(ii) K_{n}^{∗} + 5 + T^{CB} ⊢ 5^{∗}

Proof. The proof of (i) is as follows:

1. ¬□_{∗}A → □_{∗}¬□_{∗}A  (axiom 5^{∗})
2. □_{∗}¬□_{∗}A → □_{i}¬□_{∗}A (CB1)
3. □_{i}¬□_{∗}A → ¬□_{i}□_{∗}A (D)
4. ¬□_{∗}A → ¬□_{i}□_{∗}A  (1,2,3,PL)
5. □_{i}□_{∗}A → □_{∗}A  (4,PL)

Next we prove (ii).

4This is a consequence of the following fact, which is proved in the Appendix. Axioms IN and IP are valid in the class of CB-models where, ∀i, j ∈ N, ∀w ∈ W, ∃w′ ∈ W such that (1)

wR_{i}w′ and (2) ∀w′′ ∈ W, if wR_{i}w′′ then w′R_{j}w′′. The model of the above example satifies this property.

1. □_{i}□_{∗}A → □_{∗}A  (T^{CB})
2. ¬□_{∗}A → ¬□_{i}□_{∗}A  (1,PL)
3. ¬□_{i}□_{∗}A → □_{i}¬□_{i}□_{∗}A (axiom 5)
4. ¬□_{∗}A → □_{i}¬□_{i}□_{∗}A  (2,3,PL)
5. □_{∗}A → □_{i}□_{∗}A  (CB2)
6. ¬□_{i}□_{∗}A → ¬□_{∗}A  (5,PL)
7. □_{i}¬□_{i}□_{∗}A → □_{i}¬□_{∗}A (6,RK)
8. ¬□_{∗}A → □_{i}¬□_{∗}A  (4,7,PL)

Now, a repetition of steps 1-8 for every i = 1, ..., n leads to (by 8 and PL)

\begin{matrix}9. & ¬□_{∗}A → (□_{1}¬□_{∗}A ∧ ... ∧ □_{n}¬□_{∗}A) \\ 10. & ¬□_{∗}A → □_{∗}¬□_{∗}A & (9, R2CB) \end{matrix}

Remark 3 Since (cf. Remark 1) K_{n}^{∗} + 5 ⊢ 4c, it follows from (ii) of Theorem 5 that K_{n}^{∗} + 5 + SW ⊢ T^{CB}. Thus, by Theorem 6, K_{n}^{∗} + 5 + SW ⊢ 5^{∗}. Furthermore, by Theorems 5 and 6, K_{n}^{∗} + D + 5^{∗} ⊢ SW. It follows from Theorem 6 that K_{n}^{∗} + D5 + SW and K_{n}^{∗} + D5 + 5^{∗} are the same system.

We now turn to the relationship between axioms TN, TP, IN, IP, C and the Shared Worlds axiom (SW).

Proposition 7 None of TN, TP, IN, IP and C is provable in K_{n}^{∗}+D45 + SW (thus, a fortiori, in a weaker system such as K_{n}^{∗} + SW).

Proof. First of all, it is straightforward to show that the system K_{n}^{∗} + D45 + SW is sound with respect to the class of CB-models where (1) ∀i ∈ N, R_{i} is serial, transitive and euclidean, and (2) ∀i, j ∈ N, ∀x, y, z ∈ W, if xR_{j}y and yR_{∗}z then ∃w ∈ W such that xR_{i}w and wR_{∗}z. The following model belongs to this class: W = {a, b, c}, R_{1} = {(a, b), (b, b), (c, c)}, R_{2} = {(a, a), (b, c), (c, c)}, R_{3} = {(a, a), (b, b), (c, a)}; thus R_{∗} is the universal relation. Let p be an atomic proposition which is true at c and false at a and b. Then this model validates all the theorems of K_{n}^{∗} + D45 + SW (in particular SW itself)^{5}. It is easily checked that

□_{1}□_{2}p → □_{2}p (which is an instance of TN) is false at a
□_{1}♢_{2}p → ♢_{2}p (which is an instance of TP) is false at a
□_{1}□_{2}p → □_{1}p (which is an instance of IN) is false at a
♢_{1}□_{2}p → ♢_{1}p (which is an instance of IP) is false at a
□_{2}p → ¬□_{3}¬p (which is an instance of C) is false at b

Theorem 8

5Since R_{∗} is the universal relation, for every formula A and every world w, w ⊧ □_{∗}A if and only if A is valid (i.e. true at every world). It follows from seriality of R_{i} that if individual i believed that A is common belief then A is indeed commonly believed (thus T^{CB} is valid).

Hence everybody shares i’s belief that A is common belief (thus SW is valid).

(i) K_{n}^{∗} + TN ⊢ SW
(ii) K_{n}^{∗} + D5 + TP ⊢ SW

Proof. For (i)

1. □_{∗}A → □_{j}□_{∗}A  (CB2)
2. □_{i}□_{∗}A → □_{i}□_{j}□_{∗}A (1,RK)
3. □_{i}□_{j}□_{∗}A → □_{j}□_{∗}A (TN)
4. □_{i}□_{∗}A → □_{j}□_{∗}A  (2,3,PL)

For (ii)

1. □_{∗}A → □_{j}□_{∗}A  (CB2)
2. ♢_{j}□_{∗}A → ♢_{j}□_{j}□_{∗}A  (1,RK♢)
3. ¬□_{j}□_{∗}A → □_{j}¬□_{j}□_{∗}A (axiom 5)
4. ♢_{j}□_{j}□_{∗}A → □_{j}□_{∗}A  (3,PL)
5. ♢_{j}□_{∗}A → □_{j}□_{∗}A  (2,4,PL)
6. □_{j}□_{∗}A → ♢_{j}□_{∗}A  (axiom D)
7. □_{i}□_{∗}A → □_{i}□_{j}□_{∗}A  (1,RK)
8. □_{i}□_{j}□_{∗}A → □_{i}♢_{j}□_{∗}A (6,RK)
9. □_{i}□_{∗}A → □_{i}♢_{j}□_{∗}A  (7,8,PL)
10. □_{i}♢_{j}□_{∗}A → ♢_{j}□_{∗}A  (TP)
11. □_{i}□_{∗}A → ♢_{j}□_{∗}A  (9,10,PL)
12. □_{i}□_{∗}A → □_{j}□_{∗}A  (5,11,PL)

5 Concluding remarks

We have considered a variety of interpersonal compatibility restrictions on the beliefs of the individuals. The strongest restrictions are obtained by imposing the Truth Axiom on individual beliefs because of the implied common knowledge of the correctness of beliefs. Of the remaining conditions C is the strongest, since it implies all the others. The weakest of all is SW, which in the system K_{n}^{∗} + D5 turns out to be equivalent to 5^{∗} Negative Introspection of Common Belief. The relationship among all the axioms considered in this paper (T, C, TN, TP, IN, IP, SW, T^{CB}, 5^{∗}) is summarized in Figure 1.

Weaker intersubjective restrictions on beliefs can be obtained from C, TN, TP, IN, IP and SW by replacing □_{j} with □_{∗}. In this case TN becomes T^{CB}, while the remaining axioms become

C^{∗}  □_{i}A → ♢_{∗}A
TP^{∗} □_{i}♢_{∗}A → ♢_{∗}A
IN^{∗} □_{i}□_{∗}A → □_{i}A
IP^{∗} ♢_{i}□_{∗}A → ♢_{i}A
SW^{∗} □_{i}□_{∗}A → □_{∗}□_{∗}A

It is straightforward to show that all of the above are provable in K_{n}^{∗} + D45.^{6}

6Indeed, C^{∗}and TP^{∗} are provable in K_{n}^{∗} + D, IN^{∗} is provable in K_{n}^{∗} + 4c and IP^{∗} is

6 Appendix

Proof of Lemma 2.

1. (□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) →  (M,PL)
□_{1}(A ∧ □_{∗}A) ∧ ... ∧ □_{n}(A ∧ □_{∗}A)
2. □_{∗}A → □_{1}A ∧ ... ∧ □_{n}A  (CB1,PL)
3. □_{∗}A → □_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A  (CB2,PL)
4. □_{∗}A → □_{1}(A ∧ □_{∗}A) ∧ ... ∧ □_{n}(A ∧ □_{∗}A)  (1,2,3,PL)
5. A ∧ □_{∗}A → □_{∗}A  (PL)
6. A ∧ □_{∗}A → □_{1}(A ∧ □_{∗}A) ∧ ... ∧ □_{n}(A ∧ □_{∗}A)  (4,5,PL)
7. □_{1}(A ∧ □_{∗}A) ∧ ... ∧ □_{n}(A ∧ □_{∗}A) → □_{∗} (A ∧ □_{∗}A)  (6, R1CB)
8. □_{∗} (A ∧ □_{∗}A) → □_{∗}A ∧ □_{∗}□_{∗}A  (M)
9. (□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A ∧ □_{∗}□_{∗}A (1,7,8,PL)
10. □_{∗}A ∧ □_{∗}□_{∗}A → □_{∗}A  (PL)
11. (□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A  (9,10,PL)

Proof of Corollary 3.

1. □_{∗}A → □_{i}A  (CB1)
2. □_{i}□_{∗}A → □_{i}□_{i}A  (1,RK)
3. □_{i}□_{i}A → □_{i}A  (axiom 4c)
4. □_{i}□_{∗}A → □_{i}A  (2,3,PL)
5. (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → (□_{1}A ∧ ... ∧ □_{n}A)  (4,PL)
6. (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) (PL)
7. (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) →
(□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A)  (5,6,PL)
8. (□_{1}A ∧ ... ∧ □_{n}A) ∧ (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A (CB4)
9. (□_{1}□_{∗}A ∧ ... ∧ □_{n}□_{∗}A) → □_{∗}A  (7,8,PL)

Lemma 9 Axioms TN and TP are valid in the class of CB-models where, ∀i, j ∈ N, ∀w, w′, w′′ ∈ W, (1) R_{i} is serial and (2) if wR_{i}w′ and w′R_{j}w′′ then wR_{j}w′′.

Proof. First we prove validity for TN. Fix an arbitrary model that satisfies the above two properties. Fix arbitrary i, j, a and an arbitrary formula A. Suppose that a ⊧ □_{i}□_{j}A. By seriality of R_{i} there exists a b such that aR_{i}b. Fix an arbitrary such b. Then b ⊧ □_{j}A. By seriality of R_{j}, there exists a c such that bR_{j}c. Fix an arbitrary such c. Then c ⊧ A. By property (2) aR_{j}c. Hence a ⊧ □_{j}A.

(2) Next we prove validity of TP. Fix an arbitrary model that satisfies the above two properties. Fix arbitrary i, j, a and an arbitrary formula A. Suppose that a ⊧ □_{i}♢_{j}A. By seriality of R_{i} there exists a b such that aR_{i}b. Then b ⊧ ♢_{j}A. Hence there exists a c such that bR_{j}c and c ⊧ A. By property (2) aR_{j}c. Hence a ⊧ ♢_{j}A.

provable in K_{n}^{∗} + D4 or in K_{n}^{∗} + D5.

Lemma 10 Axioms IN and IP are valid in the class of CB-models where, ∀i, j ∈ N, ∀w ∈ W, ∃w′ ∈ W such that (1) wR_{i}w′ and (2) ∀w′′ ∈ W, if wR_{i}w′′ then w′R_{j}w′′.

Proof. First we prove validity for IN. Fix an arbitrary model that satisfies the above properties. Fix arbitrary i, j, a and an arbitrary formula A. Suppose that a ⊧ □_{i}□_{j}A. By the assumed properties, there exists a b such that aR_{i}b. Choose an arbitrary such b. Then b ⊧ □_{j}A. Choose an arbitrary c such that aR_{i}g. By the assumed properties, bR_{j}c. Hence c ⊧ A. Therefore a ⊧ □_{i}A.

(2) Next we prove validity for IP. Fix an arbitrary model that satisfies the above properties. Fix arbitrary i, j, a and an arbitrary formula A. Suppose that a ⊧ ♢_{i}□_{j}A. Then there exists a b such that aR_{i}b and b ⊧ □_{j}A. By the assumed properties (choosing c = b), bR_{j}b. Hence b ⊧ A. Therefore, a ⊧ ♢_{i}A.

References

[1] Aumann, Robert. 1976. Agreeing to disagree, Annals of Statistics, 4, 1236-1239.

[2] Bonanno, Giacomo. 1996. ‘On the logic of common belief’. Mathematical Logic Quarterly, 42: 305-311.

[3] Chellas, Brian. 1984. Modal logic: an introduction. Cambridge University Press.

[4] Fagin, Ronald, Joseph Halpern, Yoram Moses and Moshe Vardi. 1995, Reasoning about knowledge. MIT Press.

[5] Lewis, David. 1969. Convention: a philosophical study, Harvard University Press.

[6] Lismont, Luc and Philippe Mongin. 1994. ‘On the logic of common belief and common knowledge’, Theory and Decision, 37: 75-106.