To appear in: G. Meggle (Ed.), Social Facts & Collective Intentionality, Special Issue of Grazer Philosophical
Studies, 2002, S. 205-223.
Georg Meggle
∗
Mutual Knowledge and Belief
The logics of Interpersonal Relations have long been utterly neglected. This holds even for
Interpersonal Attitudes. As far as Common Belief and Common Knowledge (:= everybody believes
/ knows that everybody believes / knows that ...) are concerned, I have already tried to start
changing this situation. In this paper I try to do the same for their Mutual relatives (:= everybody
believes / knows that everybody else believes that ... .)
I
What we will do or not do very often depends on what we think the others will do; and
we all know that the very same applies to the others as well. Their actions and forbearances
too are often guided by what they think we will or will not do. And they may know as well
that what we will do depends on what we think they will do and on what we think they will
think we will do, etc. However, it is also often the case that we also think or even know that
the others’ convictions are incorrect – in which case, although we (believe we) know their
convictions, we do not share them. If so, we cannot (and should not) strictly speak of
Common Belief – but instead only of something weaker, namely Mutual Belief.
The logics of both kinds of Interpersonal Beliefs have been utterly neglected for a long time.
In Common Belief and Common Knowledge I started to take my part in changing this
situation, at least for these very concepts. In the following I’ll try to do the same for their
weaker Mutual relatives.1
II
Interpersonal Beliefs are webs of beliefs knotted out of related beliefs of at least two
different individuals. Hence the concepts of beliefs of individuals are the basis for any
explication of Interpersonal-Belief concepts. (This individualistic approach may turn out to be
too narrow as soon as explications of special Group-Belief Concepts are at stake – if there
really are such non-interpersonal Group-Beliefs.)
Again, let’s start by taking B(X, A) – as short for X believes (at time t) that A – to be the
strongest possible concept of Belief, i.e. as standing, first, for the so-called strong belief (firm
conviction in contrast to mere supposition) and, second, for a strictly rational belief to be
governed by the following principles:
∗
This paper is the twin of my Common Belief and Common Knowledge article, i.e. Meggle (2001), of which
several slightly differing German versions exist. Both papers are parts of my manuscript Interpersonal
Knowledge and Belief, in which both Common and Mutual Belief/Knowledge concepts are discussed in more
detail; and this manuscript again is an offshoot of my investigations made in connection with my Habilitation
“Handlungstheoretische Semantik” (ms. 1984). Those investigations were carried out under the guidance and
with the help of my philosophical teachers and friends Franz von Kutschera and Wolfgang Lenzen.
Common and Mutual Beliefs, though these terms are used in ordinary language, don’t have a clear ordinary
language sense. As used here, they are technical terms. In order to make this transparent, I have decided to write
these terms with capitals.
1
Again, I will work with very strong idealizations, accepting the axioms of the logics of strong rational belief as
developed in Kutschera (1976) and Lenzen (1980). I am not going to discuss the semantics of that logic. For that
alternative kind of discussion see, e.g., M. Colombetti (1993).
RB:
B1:
B2:
B3:
B4:
A ├ B(X, A)
B(X, A ⊃ B) ⊃ (B(X, A) ⊃ B(X, B))
B(X, A) ⊃ ¬B(X, ¬A)
B(X, A) ⊃ B(X, B(X, A))
¬B(X, A) ⊃ B(X, ¬B(X, A))
If we put
(K)
K(X, A) := B(X, A) ∧ A
X knows that A iff X is right in his conviction that A.
it can easily be proved that:
(K.1)
K(X, A) is governed by the same principles as B(X, A) – with B4
being the only exception.
For the corresponding K-substitute of principle RB or Bn, let’s write RK or Kn, respectively.
So much for the basics.2
III.1 Though this paper will focus on the Mutuality of Beliefs, let me in this § III just
summarize the respective essentials of their Commonality in order to be able to make in due
course some of the relevant distinctions.
If we write X ∈ P as short for “X belongs to population P” (or “X is a member of P”),
Common Belief can be explicated in this way:
D1.a:
CB1(P, A) := ΛX(X ∈ P ⊃ B(X, A))
It is Common Belief of the 1st level within P that A iff every member of P
believes that A.
D1.b:
CBn+1(P, A) := CB1(P, CBn(P, A))
It is Common Belief of the n+1-th level within P that A iff it is Common Belief
of the 1st level within P that it is Common Belief of the nth level within P that
A.
D1.c:
CB(P, A) := ΛnCBn(P, A)
It is a Common Belief within P that A iff within P it is Common Belief at all
levels that A.
II.2
Accordingly, Common Knowledge can be determined as follows:
D2.a:
CK1(P, A) := ΛX(X ∈ P ⊃ K(X, A))
D2.b:
CKn+1(P, A) := CK1(P, CKn(P, A))
D2.c:
CK(P, A) := ΛnCKn(P, A)
Thus, in direct analogy to definition (K), we get:
2
For more on this, cf. Kutschera (1976) and Lenzen (1980).
2
(CK)
CK(P, A) ↔ CB(P, A) ∧ A
It is Common Knowledge within P that A iff it is Common Belief within P that
A, and A really is the case.
III.3 Between these concepts of Common Belief and Knowledge on the one hand and the
basic concepts of simple Belief and Knowledge on the other, there exists a whole range of
parallels. The most important one is this:
(CB/CK.1)
Both Common Belief and Common Knowledge are governed by precisely the
analogous laws as the basic concepts of belief and knowledge themselves –
with (the parallels of) B4 being the only exception.
In particular it therefore holds that:
(CB/CK.2)
(a) If a B-Theorem can be proved merely by using rule RB and the laws B1 to
B3, then the analogous CB and CK sentences can also be proved accordingly.
(b) If a B-Theorem can be proved merely by using rule RB and the laws B1 and
B2, then the analogous CBn and CKn sentences can also be proved
accordingly – for any n ≥ 1.
Hence, as for the sentence-form ♣ [Ω] substituted by B(X, Ω), not only the theorems3
R:
1:
2:
3:
A ├ ♣ [A]
♣ [A ⊃ B] ⊃ (♣ [A] ⊃ ♣ [B] )
♣ [A] ⊃ ¬♣ [¬A]
♣ [A] ⊃ ♣ [♣ [A]]
but also, for instance, the following ones can be proved to meet these if-conditions of
(CB/CK.2),
T.1:
T.2:
T.3:
A ⊃ B ├ ♣ [A] ⊃ ♣ [B]
♣ [A] ∧ ♣ [B] ⊃ ♣ [A ∧ B]
♣ [A ≡ B ] ⊃ ( ♣ [A] ≡ ♣ [B] )
all these theorems hold not only for the substitution K(X, Ω), but also for CB(P, Ω) and CK(P,
Ω) as well! And all of them minus 3 and T.3 also hold for the corresponding CBn or CKn
theorems, respectively.
And this very fact not only makes working with the various CB/CK-concepts very easy; it also
gives us the deepest possible explanation for why it seems to be so natural to think of groups
(population P) as being Belief-and-Knowledge-Subjects in precisely the same way as we do of
individual persons. The explanation is: Both types of concepts (Group Beliefs and
Individualistic Beliefs) have the same logical deep structure.
So, if – where P continues to refer exclusively to the respective Common Belief population –
we choose to define4
3
This is the trivial case – these ‘theorems’ just being the B-axioms RB and B1-B3 themselves.
The following analogy would stop, if we would, as Kutschera (1976) and Lenzen (1980) do, extend our basics,
adding (for Belief)
4
3
(CB*)
(CK*)
B(P, A) := CB(P, A)
K(P, A) := CK(P, A)
B(P, A) and K(P, A) could be treated just as being special instances of the more general
B(X, A) and K(X, A). So, as far as Belief and Knowledge are concerned, P’s may be
substituted for X’s: Groups are special kinds of epistemic individuals!
But observe that, as proved already by means of the above definitions D1/D2, this linguistic
usage alone would not be a good argument for multiplying belief-subjects ontologically as
well.
III.4 Thus, (K.1) and (CB/CK.1) lead us, for instance, immediately to the following list of
valid principles. (Please insert, wherever appropriate, “for any n ≥ 1” yourself.)
RK:
RCB:
RCK:
RCB.1:
RCK.1:
A ├ K(X, A)
A ├ CB(P, A)
A ├ CK(P, A)
A ├ CBn(P A)
A ├ CKn(P, A)
K1:
K(X, A ⊃ B) ⊃ (K(X, A) ⊃ K(X, B))
CB1:
CB(P, A ⊃ B) ⊃ (CB(P, A) ⊃ CB(P, B))
CK1:
CK(P, A ⊃ B) ⊃ (CK(P, A) ⊃ CK(P, B))
etc. for the nth-level concepts.
K2:
K(X, A) ⊃ ¬K(X, ¬A)
CB2:
CB(P, A) ⊃ ¬CB(P, ¬A)
CK2:
CK(P, A) ⊃ ¬CK(P, ¬A)
th
etc. for the n -level concepts.
K3:
K(X, A) ⊃ K(X, K(X, A))
CB3:
CB(P, A) ⊃ CB(P, CB(P, A))
CK3:
CK(P, A) ⊃ CK(P, CK(P, A))
with no parallel for the nth-level concepts.
T.B1:
A ⊃ B ├ B(X, A) ⊃ B(X, B)
T.K1:
A ⊃ B ├ K(X, A) ⊃ K(X, B)
T.CB1:
A ⊃ B ├ CB(P, A) ⊃ CB(P, B)
T.CK1:
A ⊃ B ├ CK(P, A) ⊃ CK(P, B)
etc. for the nth-level concepts.
4:
¬♣ [A] ⊃ ♣ [¬♣ [A]]
as an additional axiom. The analogies to 4 would neither hold for Knowledge simpliciter, nor for B(P, A) or
K(P, A) respectively.
By the way: The following strong concepts of Group-Belief and Knowledge (i.e., B(P, A) and K(P, A)) should
not be mixed up with their extremely weaker parallels:
(CB0)
B0(P, A) := CB1(P, A)
0
(CK )
K0(P, A) := CK1(P, A)
4
T.B2:
B(X, A) ∧ B(X, B) ⊃ B(X, A ∧ B)
T.K2:
K(X, A) ∧ K(X, B) ⊃ K(X, A ∧ B)
T.CB2:
CB(P, A) ∧ CB(P, B) ⊃ CB(P, A ∧ B)
T.CK2:
CK(P, A) ∧ CK(P, B) ⊃ CK(P, A ∧ B)
th
etc. for the n -level concepts
T.B3:
B(X, A ≡ B) ⊃ (B(X, A) ≡ B(X, B))
T.K3:
K(X, A ≡ B) ⊃ (K(X, A) ≡ K(X, B))
T.CB3:
CB(P, A ≡ B) ⊃ (CB(P, A) ≡ CB(P, B))
T.CK3:
CK(P, A ≡ B) ⊃ (CK(P, A) ≡ CK(P, B))
etc. for the nth-level concepts.
III.5
And in direct analogy to the connections between simple Belief and Knowledge,
T.B/K1:
T.B/K2:
T.B/K3:
B(X, A) ⊃ K(X, B(X, A))
B(X, A) ⊃ B(X, K(X, A))
K(X, A) ⊃ B(X, K(X, A))
the following principles apply as well, for the same reasons:
T.CB/CK1:
T.CB/CK2:
T.CB/CK3:
CB(P, A) ⊃ CK(P, CB(X, A))
CB(P, A) ⊃ CB(P, CK(X, A))
CK(P, A) ⊃ CB(P, CK(X, A))
And in particular we have this parallel:
T.B/K0:
T.CB/CK0:
B(X, A) ↔ B(X, K(X, A))
CB(P, A) ↔ CB(P, CK(P, A))
As for a person to Strongly Believe that A just means for her to Believe to Know that A, for a
group to have the Common Belief that A just means that the group Commonly Believes to
Know that A.
So much for the Commonality of P-Belief and Knowledge.
IV
What about Mutual Beliefs? Whereas Common Belief that A entails that every member
of the respective Common Belief Group believes herself that A, Mutual Belief does not. It
only entails Beliefs about the other’s Beliefs (about the other’s Beliefs, etc.). Now, before
defining Mutual Beliefs generally, let’s look more closely at how they differ in their initial
stages from their big sisters, i.e. Common Beliefs. Let’s take the most simple (2 person) case
of P = {a, b}. For this extremely restricted case, let’s now spell out the first 3 levels for both
Common and Mutual Belief en bloc:5
5
To keep things simple, let’s (as in the Common Belief and Common Knowledge paper) again pretend that X, Y
etc. are so-called standard-names and that Knowledge about group-membership is itself part of Common
Knowledge of the groups considered. Without these assumptions, we would have to consider, even only in the
simple case of P = {a, b} and MB(a, P, F( Ŷ )), instead of
(O)
B(a, B(b, F(a))
the following different readings (equivalent with (*), given our assumptions):
(a)
VyVy(x=a ∧ y=b ∧ B(a,B(y,F(x))))
5
Common Belief (CB)
1st level:
2nd level:
3rd level:
B(a, A) ∧ B(b, A)
(1st level) ∧ B(a, B(b, A)) ∧ B(b, B(a, A))
(2nd level) ∧ B(a, B(b, B(a, A))) ∧ B(b, B(a, B(b, A)))
Mutual Belief (MB)
1st level:
2nd level:
3rd level:
B(a, B(b, A)) ∧ B(b, B(a, A))
B(a, B(b, B(a, A))) ∧ B(b, B(a, B(b, A)))
B(a, B(b, B(a, B(b, A)))) ∧ B(b, B(a, B(b, B(a, A))))
There are several striking differences. First, as already mentioned, from Mutual Belief no
conclusions can be drawn about whether what is mutually believed (by the members of P) is
also individually believed (believed by the P-members themselves) or not. Second, in Mutual
Belief there is no Common Ground (as is the 1st level for Common Belief), on which all
higher levels are built. Third, contrary to Common Belief, Mutual Belief is not even weakly
conservative: There is no level n (n≥1) which is part of (at least) one of the higher levels n+m
(m≥1). To put it generally, whereas the following principles
T.CB4:
T.CB5:
T.CB6:
CBn(P, A) ⊃ CB1(P, A)
CB(P, A) ⊃ CB1(P, A)
CBn+m(P, A) ⊃ CBn(P, A)
for any n≥1
for any n, m ≥ 1
do hold for Common Belief,6 none of their Mutual counterparts would.
V7
One consequence of the T.CB6’s Mutual Belief counterpart not being valid is that at
any Mutual-Belief level it may be the case that each member (of the respective Mutual Belief
Group) holds that she is the only one whose beliefs (regarding the relevant Belief-content in
question) are true – the beliefs of all the others being taken by her to be false.
Let’s look for example at the 3rd MB-level in the 3-person-case (with P = {a, b, c}) – and
concentrate just on those components which only present a’s perspective:
MB3-3-person case / a’s perspective:
(i)
(ii)
(iii)
(iv)
(v)
B(a, B(b, B(a, B(b, A)))) ∧
B(a, B(b, B(a, B(c, A)))) ∧
B(a, B(b, B(c, B(b, A)))) ∧
B(a, B(b, B(c, B(a, A)))) ∧
B(a, B(c, B(a, B(b, A)))) ∧
(b)
Vy(y=b ∧ B(a,Vy(x=a ∧ B(y,F(x)))))
(c)
Vy(y=b ∧ B(a,B(y, F(a))))
(d)
B(a,Vx(x=a ∧ B(b, F(x))))
In Interpersonal Knowledge and Belief these restrictions will be skipped.
6
Please, don’t forget the presuppositions referred to in fn. 4.
7
This § is also meant to provide some kind of exercise for those readers who are not yet very keen in working
with the doxastic logics machinery. If for some these exercises might be too heavy to start with, they can just
skip this § and go to the next one.
6
B(a, B(c, B(a, B(c, A)))) ∧
B(a, B(c, B(b, B(a, A)))) ∧
B(a, B(c, B(b, B(c, A))))
(vi)
(vii)
(viii)
That a believes herself to be the only person (out of the relevant MB3-group with respect to A)
means that she takes all the underlined propositions to be false, meaning that – if we restricted
our attention to the third line (iii) only – in addition to (iii) it holds also that:
(iv.1)
B(a, ¬B(c, B(b, A)))
And this again might be backed by
(iv.1*)
B(a, B(c, [¬B(b, A)]))
or even by8
(iv.1**)
B(a, B(c, [B(b, ¬A)]))
But none of these additional (n.1/1*/1.**) beliefs would be compatible with what person a
would have to believe in the corresponding (i.e., 3rd level) case of Common Belief. Thus, in
order to become clear about the logical structure of our Mutual Beliefs, it is of the utmost
importance to keep the individual perspectives involved distinct from the very beginning in
this structure.
For this reason I will split the explication of Mutual Belief Concepts into two steps,
introducing at first (in VI.1) some concepts which make the individual-perspective-relativity
explicit. In the second step (VI.2) we will then skip this restriction.
VI.1
Mutual Belief from one person’s perspective9
D3.a: MB1(X, P, F( Ŷ )) := B(X, ΛY (Y ≠ X ∧ Y ∈ P ⊃ F(Y)))
From X’s viewpoint, Mutual Belief of the 1st level exists in P that within P the characteristic
F(Ŷ ) exists iff X believes that everyone else in P has the property F.
D3.b: MBn+1(X, P, F( Ŷ )) := MB1(X, P, MBn( Ŷ , P, F( Ẑ )))
8
(iv.1) is entailed by (iv.1*) and that again by (iv.1**). Notice that with (iv.1*) and (iv.1**), too, the same game
may start again: a may think that c’s beliefs are wrong (i.e. their [] bracketed contents being false) , which would
lead us to B(a, B(b, A)) and B(a, ¬B(b, ¬A)), respectively. And then, maybe, again the same with B(a, B(b, A)),
leading us to B(a, ¬A).
9
Notice, that D3.a does not totally match with
(α)
MB / 1st level: G(a, G(b, A)) ∧ G(b, G(a, A))
as spelled out, for P = {a, b}, in IV above, as D3.a, instead, amounts to:
(β)
G(a, F(b)) ∧ G(b, F(a))
(α) is only a special case of (β), resulting if F(x) := G(x, A). Why this difference? My motivation behind it is to
get more of systematics of a kind into all this stuff. Mutual Belief is to be treated as only a special case of Mutual
Attitudes; and these include also Mutual Preferences as if, e.g., every member of P wants that (i) everyone else
believes that A, or that (ii) everyone else wants that A, or that (iii) everyone else sees to it that A. Now, it would
be fine, if Mutual Preference in P – as Mutual Attitude in P in general – would possess as much the same
structure as MB does. And this aim is reached better if we just start with D3.a (covering all the three contents (i)
to (iii)) instead of the special case MB1(X, P, B( Ŷ , A)), which is tantamount only to the attitude-content (i).
7
D3.c: MB(X, P, F( Ŷ )) := ΛnMBn(X, P, F( Ŷ ))
VI.2
Real Mutual Belief = from more than one person’s perspective
D3.d: MB(P, F( Ŷ )) := ΛX(X ∈ P ⊃ MB(X, P, F( Ŷ ))
It is Mutually Believed in P that within P it is the case that F( Ŷ ).
Now, the equivalence (*) being true already for logical reasons,
(*)
ΛX(X ∈ P ⊃ MB(X, P, F( Ŷ ))) ↔ ΛnΛX(X ∈ P ⊃ MBn(X, P, F( Ŷ )))
we are allowed to make use of the following abbreviation, too:
D3.e: WGn(P, F( Ŷ )) := ΛX(X ∈ P ⊃ WGn(X, P, F( Ŷ )))
for any n ≥ 1
th
It is n -level Mutual Belief in P that within P it is the case that F( Ŷ ).
VI.3 Mutual Knowledge can then again be determined as the corresponding Mutual Beliefs
(Convictions) being true.
D4.a: MK1(X, P, F( Ŷ )) := MB1(X, P, F( Ŷ )) ∧ ΛY (Y ≠ X ∧ Y ∈ P ⊃ F(Y))
i.e.: K(X, ΛY (Y ≠ X ∧ Y ∈ P ⊃ F(Y)))
From X’s viewpoint, it is 1st level Mutual Knowledge in P that F(Ŷ ) within P iff X believes
that everyone else in P has the property F and this belief of X is true.
D4.b: MKn+1(X, P, F( Ŷ )) := MK1(X, P, MKn( Ŷ , P, F( Ẑ )))
D4.c: MK(X, P, F( Ŷ )) := ΛnMKn(X, P, F( Ŷ ))
D4.d: MK(P, F( Ŷ )) := ΛX(X ∈ P ⊃ MK((X, P, F( Ŷ )))
VII.1 Now, although the structure of all these Mutual Belief / Knowledge concepts is much
more complicated than the structure of their Common relatives, applications of Mutuality
concepts are very much facilitated – as soon as you notice that they, too, follow the same
logical deep structure as the Commonality concepts. Their internal complications
notwithstanding, even for Mutual Belief and Knowledge the same holds as what already held
for the Common Belief and Knowledge and for Belief and Knowledge simpliciter:
(MB/MK.1)
Both Mutual Belief and Mutual Knowledge are governed by precisely the
analogous laws as the basic (individualistic) concepts of belief and knowledge
themselves – again with (the parallels of) B4 being the only exception.
And again we thus get the important corollary:
(MB/MK.2)
(a) If a B-Theorem can be proved merely by using rule RB and the laws B1 to
B3, then the analogous MB and MK sentences can also be proved accordingly.
(b) If a B-Theorem can be proved merely by using RB, B1 and B2, then the
analogous MBn and MKn sentences can also be proved accordingly – for any
n ≥ 1.
8
VII.2 Let’s visualize this happy result again via the joint Belief/Knowledge deep structure
already introduced in III.3 above:
R:
1:
2:
3:
A ├ ♣ [A]
♣ [A ⊃ B] ⊃ (♣ [A] ⊃ ♣ [B])
♣ [A] ⊃ ¬♣ [¬A]
♣ [A] ⊃ ♣ [♣ [A]]
T.1:
T.2:
T.3:
A ⊃ B ├ ♣ [A] ⊃ ♣ [B]
♣ [A] ∧ ♣ [B] ⊃ ♣ [A ∧ B]
♣ [A ≡ B] ⊃ ( ♣ [A] ≡ ♣ [B])
All these sentence-forms – in the nth-level cases again minus form 3 – remain valid if
substituted accordingly.
VII.3 Now, let’s enjoy the beauty of such substitutions for a moment. Here are some
instances of the myriad of possible applications of the central (MB/MK.2) meta-theorem.
Again, just insert “for any n ≥ 1” yourself wherever you please.
RMB:
A ├ MB(P, B( Ŷ , A))
RMK:
A ├ MK(P, B( Ŷ , A))
RMB.1:
A ├ MB(X, P, B( Ŷ , A))
A ├ MK(X, P, B( Ŷ , A))
RMK.1:
etc. for the nth level cases.
MB1:
MB(P, (F( Ŷ ) ⊃ F*( Ŷ ))) ⊃ (MB(P, F( Ŷ )) ⊃ MB(P, F*( Ŷ )))
MK1:
MK(P, (F( Ŷ ) ⊃ F*( Ŷ ))) ⊃ (MK(P, F( Ŷ )) ⊃ MK(P, F*( Ŷ )))
MB1.1:
MB(X, P, (F( Ŷ ) ⊃ F*( Ŷ ))) ⊃ (MB(X, P, F( Ŷ )) ⊃ MB(X, P, F*( Ŷ )))
MK1.1:
MK(X, P, (F( Ŷ ) ⊃ F*( Ŷ ))) ⊃ (MK(X, P, F( Ŷ )) ⊃ MK(X, P, F*( Ŷ )))
etc. for the nth level cases.
MB(P, F( Ŷ )) ⊃ ¬MB(P, ¬F( Ŷ ))
MB2:
MK2:
MK(P, F( Ŷ )) ⊃ ¬MK(P, ¬F( Ŷ ))
MB2.1:
MB(X, P, F( Ŷ )) ⊃ ¬MB(X, P, ¬F( Ŷ ))
MK,(X, P, F( Ŷ )) ⊃ ¬MK(X, P, ¬F( Ŷ ))
MK2.1:
etc. for the nth level cases.
MB3:
MB(P, F( Ŷ )) ⊃ MB(P, MB( X̂ , P, F( Ŷ )))
MK3:
MK(P, F( Ŷ )) ⊃ MK(P, MK( X̂ , P, F( Ŷ )))
MB3.1:
MB(X, P, F( Ŷ )) ⊃ MB(X, P, MB( Ŷ , P, F( Ẑ )))
MK3.1:
MK(X, P, F( Ŷ )) ⊃ MK(X, P, MK( Ŷ , P, F( Ẑ )))
with no parallels for the nth level cases.
T.MB1:
T.MK1:
T.MB1.1:
A ⊃ B ├ MB(P, B( Ŷ , A)) ⊃ MB(P, B( Ŷ , B))
A ⊃ B ├ MK(P, B( Ŷ , A)) ⊃ MK(P, B( Ŷ , B))
A ⊃ B ├ MB(X, P, B( Ŷ , A)) ⊃ MB(X, P, B( Ŷ , B))
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T.MK1.1:
A ⊃ B ├ MK(X, P, B( Ŷ , A)) ⊃ MK(X, P, B( Ŷ , B))
etc. for the nth level cases.
T.MB2:
MB(P, F( Ŷ )) ∧ MB(P, F*( Ŷ )) ≡ MB(P, F( Ŷ ) ∧ F*( Ŷ ))
MK(P, F( Ŷ )) ∧ MK(P, F*( Ŷ )) ≡ MK(P, F( Ŷ ) ∧ F*( Ŷ ))
T.MK2:
T.MB2.1:
MB(X, P, F( Ŷ )) ∧ MB(X, P, F*( Ŷ )) ≡ MB(X, P, F( Ŷ ) ∧ F*( Ŷ ))
T.MK2:
MK(X, P, F( Ŷ )) ∧ MK(X, P, F*( Ŷ )) ≡ MK(X, P, F( Ŷ ) ∧ F*( Ŷ ))
etc. for the nth level cases.
T.MB3:
MB(P, F( Ŷ ) ≡
MK(P, F( Ŷ ) ≡
T.MK3:
T.MB3.1:
MB(X, P, F( Ŷ )
T.MK3.1:
MB(X, P, F( Ŷ )
etc. for the nth level cases.
F*( Ŷ ))
F*( Ŷ ))
≡ F*( Ŷ
≡ F*( Ŷ
⊃
⊃
))
))
(MB(P, F( Ŷ )) ≡
(MK(P, F( Ŷ )) ≡
⊃ (MB(X, P, F( Ŷ
⊃ (MK(X, P, F( Ŷ
MB(P, F*( Ŷ )))
MK(P, F*( Ŷ )))
)) ≡ MB(X, P, F*( Ŷ )))
)) ≡ MK(X, P, F*( Ŷ )))
VIII In all Interpersonal Relations, the 2-person case is the most simple case. To simplify
our notation by getting it nearer to our usual B(a, B(b, B(a, ...)))-formulations, let’s introduce
some special explications.
If P = {a, b}, then MB1(a, P, F( Ŷ )) or MB3(a, P, F( Ŷ )), respectively, look like (γ) or (δ):
(γ)
(δ)
B(a, F(b))
B(a, B(b, B(a, F(b))))
and, correspondingly, for MB2(a, P, F( Ŷ )) or MB4(a, P, F( Ŷ )), we have:
(ε)
(ϕ)
B(a, B(b, F(a)))
B(a, B(b, B(a, B(b, F(a)))))
More generally: If P = {a, b}, then, for all uneven n’s, MBn(a, P, F( Ŷ )) is of the form
(η)
B(a, ... F(b) ... ))
or
B(b, ... F(a) ... ))
whereas for all even n’s it is of the form:
(ι)
B(a, ... F(a) ... ))
or
B(b, ... F(b) ... ))
To distinguish these cases, let’s define:
D4.1
D4.2
MB1(X, P, F( Ŷ ))
MB2(X, P, F( Ŷ ))
:=
:=
Λn(Vj(n = 2j-1) ⊃ MBn(a, P, F( Ŷ )))
Λn(Vj(n = 2j) ⊃ MBn(a, P, F( Ŷ )))
Thus, it trivially holds that:
T.MB4:
MB1(X, P, F( Ŷ )) ∧ MB2(X, P, F( Ŷ ))
≡
MB(X, P, F( Ŷ ))
Now, if X in MB1(X, P, F( Ŷ )) or MB1(X, P, F( Ŷ )) is fixed, Y will be fixed, too. Therefore,
we can simplify things and just write
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D4.1a
D4.2a
MB(a, {a, b}, F(b))
MB(a, {a, b}, F(a))
for: MB1(X, P, F( Ŷ )) ∧ X = a
for: MB2(X, P, F( Ŷ )) ∧ X = a
IX
A Mutual Belief is something weaker than a Common Belief. Therefore, my closing
question is: How (and under what conditions) do we get from a Mutual Belief to a Common
Belief and vice versa? In other words, how are Mutual and Common Belief related? The
answer is,
T.MB/CB:
(???)
Well, it’s tea-time – and so you are kindly asked to find out the answer by yourselves.10
References
Bach, K., “Analytic Social Philosophy – Basic Concepts”, in: J. Theory Soc. Behaviour 5,
1975, 182-214.
Colombetti, M., Formal Semantics for Mutual Belief, in: Artificial Intelligence 62, 1993, 341353.
Gilbert, M., On Social Facts, London, 1980.
Lewis, D., Convention: A Philosophical Study, Cambridge/Mass., 1969.
Kutschera, F. von, Einführung in die intensionale Semantik, Berlin, 1976.
Lenzen, W., Glauben, Wissen und Wahrscheinlichkeit, Wien / New York, 1980.
Meggle, G., Handlungstheoretische Semantik, 1984 (Habilitation-Ms.); Berlin / New York (de
Gruyter), forthcoming.
Meggle, G., Common Belief and Common Knowledge, in: Matti Sintonen, Petri Ylikoski,
Kaarlo Miller (Hg.), Realism in Action, Dordrecht (Kluwer), 2001, p. 244-251.
Schiffer, S., Meaning, Oxford, 1972.
Tuomela, R., The Importance of Us: A Philosophical Study of Basic Social Notions, Stanford,
1995.
10
But may I invite you to check your result with mine ! Cf. the very end of Meggle (2001).
11