A Fuzzy Dynamic Belief Logic
Xiaoxin Jing and Xudong Luo
∗
Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China
Keywords:
Belief Revision, Fuzzy Logic, Dynamic Logic.
Abstract:
This paper introduces a new logic approach to reason about the dynamic belief revision by extending well-
known Aucher’s dynamic belief revision approach to fuzzy environments. In our system, propositions take a
valuation in linguistic truth term set, and a belief revision is also in a qualitative way. Moreover, we reveal
some properties of our system in epistemic style and do a comparison between our fuzzy belief system with
famous AGM postulats.
1 INTRODUCTION
The problem of belief change is an active topic in
logic (Aucher, 2006; van Ditmarsch, 2005; Shapiro
et al., 2011) and artificial intelligence (Sardina and
Padgham, 2011; Casali,Godob and Sierra, 2011). Its
focus is to understand how an agent should change
his belief in the light of new information. Re-
searchers have developed various models for belief
revision. Epistemic plausibility model is one of main
approaches to model the dynamics of the belief. It
adds the plausibility ordering in the epistemic model
for each agent, i.e., a pre-order w ≤ v that says agent
i considers world w at least as plausible as v.
In order to reason about this structure, the epis-
temic language is extended with a conditional belief
operator. van Benthem (2007) developed a dynamic
belief revision frame based on the dynamic epistemic
logic and then adds the conditional belief operator in
it. Baltag (2006) uses the epistemic plausibility model
for conditional belief in a multi-agent epistemic envi-
ronment, and introduces plausibility pre-order on ac-
tions, notated as action plausibility models to display
the dynamic setting, which is somehow the extension
of well-known Aucher’s method in (Aucher, 2006).
Another typical approach for belief revision is epis-
temic probability model (Baltag and Smets, 2006),
which is closed to the epistemic plausibility model
with probability measures in the place of plausibility
ordering. In these models, various arithmetical for-
mulas are used to compute the probability of belief of
the output-states from the probabilities of the input-
states and the probabilities of actions.
In the logic above, the beliefs are all crisp. How-
ever in real-life, it is not always the case (Wu and
∗
Corresponding author.
Zhang, 2012). In fact, our belief is often fuzzy. For
example, “I think the temperature will be high tomor-
row”. Here “high” is a fuzzy conception because there
is not any crisp division between “high” and “low”.
And suppose that we want to book an air ticket, we
thought ticket A is good at first and then when we
know that the flight would be delayed, we need to re-
vise our previous beliefs about the ticket. On the one
hand, “the ticket is good” is a fuzzy proposition be-
cause there are no crisp standards for a good ticket.
It will depend on many factors such as the price, the
departure time and service, and so on. On the other
hand, we might not be able to decide whether or not
to change our previous belief completely. Rather, we
can only say the ticket is better or worse than before.
How can we handle this kind of fuzzy belief revision?
Fuzzy theory can provide a powerful tool to han-
dle this situation, which is studied in various areas,
such as mathematics, logic and computer science.
Fuzzy logic began with the 1965 proposal of fuzzy
set theory by Zadeh (1965) and it is a form of many-
valued logic (Zadeh, 1975). It deals with reason-
ing that is approximate rather than accurate. In tra-
ditional logic, a proposition is usually true or false,
while fuzzy logic proposition can have a truth value
in-between 0 and 1. In this sense, fuzzy logic is a bet-
ter way for us to handle uncertain reasoning for fuzzy
belief revision.
The structure of the paper is listed as follows. Sec-
tion 2 briefs the fuzzy theory, which we will use for
developing our logic. Section 3 defines our fuzzy sys-
tem based on the dynamic belief logic. Section 4
studies some properties of our logic. Section 5 gives
a comparison between our fuzzy logic system with
AGM postulats. Section 6 discusses the related work
to confirm that our work has advanced the state of art.
289
Jing X. and Luo X..
A Fuzzy Dynamic Belief Logic.
DOI: 10.5220/0004257302890294
In Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART-2013), pages 289-294
ISBN: 978-989-8565-39-6
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Linguistic truth.
Section 7 concludes the paper with future work.
2 PRELIMINARIES
This section recaps fuzzy set and logic theory. Let
X = {x} be a space of points with a generic element,
denoted as x, of X. A fuzzy set A in X is characterised
by a membership function of µ(x), which associates
with each point in X a real number in interval [0, 1],
with the values of µ(x) at x representing the member-
ship degree of x in A. In fuzzy logic, truth value comes
in continuous degrees. There are different forms for
truth value. By a linguistic variable we mean a vari-
able whose values are words or sentences in a natural
language (Zadeh, 1975; Luo et al., 2002).
Defininition 1 (Linguistic Truth). A proposition can
take truth on linguistic truth rather than classical
proposition truth {0,1}, i.e.,
LTT S = {absolutely-true,very-true,true,
f airly-true,undecided, f airly- f alse,
f alse,very- f alse,absolutely- f alse}.(1)
For convenience, we denote
LTT S
t
={absolutely-true,very-true,true, f airly-true},
LTT S
f
={absolutely- f alse,very- f alse,
f alse, f airly- f alse}.
Thus, we have
LTT S = LT T S
t
∪ LTT S
f
∪ {undecided}.
Semantics of the terms in this term set are defined
as shown in Table 1 (Luo et al., 2002).
The operation on linguistic variables are defined
according to the corresponding operation on numeri-
cal variables by using the extension principle as fol-
lows:
Defininition 2 (Extension Principle). Suppose f is a
function with n arguments x
1
,...,x
n
,denoted by~x. Let
µ
i
(x
i
) be the membership function of argument x
i
(1 ≤
i ≤ n ). Then
µ(y) = sup{µ
1
(x
1
) ∧ ... ∧ µ
n
(x
n
) | f (~x) = y}, (2)
where sup denote the supremum operation on a set, ∧
means the conjunction of the items. Let the fuzzy set
corresponding to µ be B, and let the fuzzy set corre-
sponding to µ
i
be A
i
. For convenience, we denote the
operation of the extension principle as ⊗. i.e.,
B = ⊗(A
1
,...,A
n
, f ). (3)
If an operation on some linguistic terms is not
closed in the predefined linguistic term set, a linguis-
tic approximation technique in necessary in order to
find a term in the term set, whose meaning (member-
ship function) is the closest to that of the result of the
operation.
Defininition 3 (Linguistic Approximation). The most
straight forward approach, the BEST FIT, uses the
Euclidean Distance (ED) as follows:
ED(A,B) =
q
∑
{(µ
A
(x) − µ
B
(x))
2
| x ∈ [0,1]} (4)
between fuzzy sets A and B defined on [0, 1], to eval-
uate which one in the term set is the closest to the set
being approximated. Namely, τ ∈ LT T S, being the
closest to τ
00
should satisfy
∀τ
0
∈ LT T S,ED(τ, τ
00
) ≤ ED(τ
0
,τ
00
). (5)
For convenience, we denote the above operation of
linguistic approximation as , i.e.
τ = (τ
00
) (6)
3 FUZZY DYNAMIC BELIEF
LOGIC
This section will present our logic, which will be de-
fined on the linguistic truth set and combines with the
belief dynamic system of (van Ditmarsch, 2005).
3.1 Language
We define the language of our fuzzy dynamic belief
logic as follows:
Defininition 4 (Language of belief-epistemic logic).
φ := p | ¬φ | φ ∧ ψ | K
j
φ | B
j
φ | [∗φ]ψ
where p is an atomic formula as usual, ¬φ and φ ∧ ψ
are the Boolean combination of the propositional for-
mulas, K
j
φ means agent j knows proposition φ, B
j
φ
means agent j believes proposition φ, and [∗φ]ψ ex-
presses that proposition ψ holds after revising an-
gent’s belief with formula φ.
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Table 1: Linguistic truth.
µ
absolutely−true
(x) =

1 if x = 1
0 otherwise
µ
absolutely− f alse
(x) =

1 if x = 0
0 otherwise
µ
very−true
(x) = µ
2
true
(x) µ
very− f alse
(x) = µ
2
f alse
(x)
µ
true
(x) = x,∀x ∈ [0,1] µ
f alse
(x) = 1 − x,∀x ∈ [0,1]
µ
f airly−true
(x) = µ
1/2
true
(x) µ
f airly− f alse
(x) = µ
1/2
f alse
(x)
µ
undecided
(x) =

1 if x ∈ (0,1)
0 otherwise
3.2 Semantics
The semantics of our system is completely different
from the one in the classical two-valued logic. The
truth value of the proposition is taken on the lan-
guage truth term set, LT T S, given by (1). The func-
tions of different linguistic truths’ membership de-
grees have been listed in Table 1 and their figure have
been showed in Fig.1 correspondingly.
Then we will give the semantics of the formulas
in three steps. First, we will give the semantics of
the simple formulas, (i.e., the no-epistemic formulas),
then the epistemic formulas with epistemic operator
K, B, and finally we will define the semantics of the
belief revision.
First in the simplest situation, the semantics of the
non-epistemic formulas (i.e., the ones in the classi-
cal predicate logic). We can obtain the value of the
formulas from estimates for the valuation of its sub-
formulas. For example, if P is a Boolean combina-
tion of P
1
and P
2
, Val(P
1
) = τ
1
∈ LT T S, Val(P
2
) =
τ
2
∈ LT T S, then Val(P) = τ ∈ LTT S, and we can ob-
tain τ by doing some operation on τ
1
and τ
2
. Specifi-
cally, we apply the extension principle ⊗ and the lin-
guistic approximation technique  on the valuation
of the sub-formulas, and at last obtain the result we
expected.
Defininition 5. Let Val(P) = τ ∈ LT T S. Let µ
τ
(x) de-
notes the membership degree of a proposition on the
linguistic value LT T S when given a value x between
[0,1]. Then
1. Val(¬P) = τ
¬P
, and µ
τ
¬P
(x) = µ
τ
P
(1 − x),∀x ∈
[0,1];
2. Val(P
1
∧ P
2
) = (⊗(τ
P
1
,τ
P
2
),4);
3. Val(P
1
∨ P
2
) = (⊗(τ
P
1
,τ
P
2
),5); and
4. Val(P
1
→ P
2
)
=

absolutely-true if Val(P
1
) ≤ Val(P
2
),
(1 − µ(x) + µ(y)) if Val(P
1
) > Val(P
2
).
According to (Bonissone and Decker, 1986), 4
denotes the operator given by one of 4
1
, 4
2
and 4
3
defined as follows:
4
1
(x,y) = max{0,x + y − 1},
4
2
(x,y) = x × y,
4
3
= min{x, y};
and 5 denotes the operator given by one of 5
1
,5
2
and 5
3
defined as follows:
5
1
(x,y) = min{1,x + y},
5
2
(x,y) = x + y − x × y,
5
3
(x,y) = max{x, y}.
As we can see in the literatures of dynamic
epistemic logic (van Ditmarsch, 2005; Sardina and
Padgham, 2011), the semantic of the epistemic formu-
las are interpreted in the style of modal logic. There-
fore, before we take the second step to define the truth
value of the epistemic formulas, we need to define the
belief-epistemic logic model in advance.
Defininition 6. A belief-epistemic model is a 4-tuple
of (W,R,V,κ
j
), where
1. W is the set of the possible worlds;
2. R is the accessibility relation between the possible
worlds;
3. V is an function from the propositions to LT T S
(i.e., we associate each proposition with a set of
possible worlds, meaning the valuation of the for-
mulas in these possible worlds); and
4. κ is a function from the possible worlds in W to
natural numbers.
Given an actual state, with which we associate a
factual description of the world, an agent may be un-
certain about which of a set of different states is actu-
ally the case. This is the set of plausible states. Any
of those plausible states may have its own associated
set of plausible states, relative to that state. In dy-
namic epistemic logic, term ‘plausible’ means ‘acces-
sible’. In classical modal logic, the accessibility rela-
tion is crisp, there are exactly two cases: R(w,w
0
) and
¬R(w,w
0
). R(w,w
0
) means that starting with world w,
world w
0
is accessible using R, while ¬R(w, w
0
) means
that starting with world w, world w
0
is not accessible
using R. Now we want to get more expressive power
AFuzzyDynamicBeliefLogic
291

and so we replace the crisp relation by a “soft” acces-
sibility relation of R : W ×W → LT T S, where LT T S
is given by (1).
Given two plausible states, the agent may think
that one is more likely to be the actual state than the
other. In other words, the agent has a preference
among states. The difference between plausible and
implausible states is: preferences among plausible
states are comparable, some plausible states are pre-
ferred over other plausible states, while the implausi-
ble states are all equally implausible. Preferences are
assumed to be partially ordered and we write < for
an agents preference relation on the set of plausible
states given an actual world. Here we take κ to repre-
sent a preference degree of the possible worlds for an
agent; the less κ the more an agent prefers that pos-
sible world. Then, we can give the semantics of the
epistemic formulas as follows:
Defininition 7. Given model M and any possible
world w in M,
1. Val(K
j
φ,M,w) = inf{Val(φ,M,w
0
) | w,w
0
∈ W ,
R(w,w
0
) = absolutely-true};
2. Val(B
j
φ,M,w) = sup{Val(φ,M,w
0
) | w,w
0
,w
00
∈
W , R(w,w
0
),R(w,w
00
) 6= absolutely- f alse, and for
all w
0
,w
00
∈ W,κ(w
0
) ≤ κ(w
00
)}.
In the above definition, item 1 expresses that the
valuation of K
j
φ in possible world w under model M is
the infimum of the valuations of φ in worlds w
0
, where
R(w,w
0
) is absolutely-true. Item 2 expresses the val-
uation of B
j
φ in possible world w in model M is the
supremum of the valuations of φ in world w
0
, where
R(w,w
0
) and R(w, w
00
) are not absolutely- f alse, and
the preference degree of w
0
is not less than that of any
other possible world w
00
.
Defininition 8. Given belief-epistemic logic model
M = (W, R,V, κ) and a belief-epistemic logic model
M
0
= (W
0
,R
0
,V
0
,κ
0
), which is an arbitrary belief-
epistemic model for a set of atoms P, a set of agents
N, if
Val([∗φ]ψ,M,w) = Val(ψ, M
0
,w
0
),
where (M
0
,w
0
) : (M,w) k [∗φ] k (M
0
,w
0
), then M
0
is the
new model by revising the original model M with [∗φ],
and after the revision we transfer from the possible
world w to a new possible world w
0
.
Formula [∗φ]ψ reads as “after revision with φ, ψ
holds”. The semantics that we propose is typical for
a dynamic modal operator: a state transformer [∗φ]
induces a binary relation between belief-epistemic
states. That is, if we revise belief-epistemic state w
in model M = (W, R,V, κ) with formula [∗φ], then
we can get a new belief-epistemic state, denoted as
w
0
, in a new model, denoted as M
0
= (W
0
,R
0
,V
0
,κ
0
).
Here we get W = W
0
and V = V
0
because the possible
worlds in these two models are the same and the facts
in possible worlds do not change, but for an agent the
preference for the possible worlds (the result of func-
tion κ) has been changed by revision with [∗φ]. In this
case, the definition becomes:
Val([∗φ]ψ,(W,R,V,κ),w) = Val(ψ,(W,R
0
,V,κ
0
),w
0
).
There are different methods for belief revision.
Here we proposed a revision method based on
Aucher’s (Aucher, 2006), which is called successful
minimal belief revision.
Defininition 9 (Revision Method). If we revise a
model M with a formula φ, then
κ
∗
(w)=







κ(w)−Min{κ(v)|Val(φ,M,v)∈LT T S
t
}
if Val(φ,M,w) ∈ LT T S
t
,
κ(w)+1−Min{κ(v)|Val(¬φ,M,v)∈LT T S
t
}
otherwise.
Defininition 10 (Validity).
1. Σ |=
τ
φ iff for any model M and world w in M, if
∀B ∈ Σ,Val(B) ≥ τ, then Val(φ) ≥ τ.
2. Σ |= φ iff ∀τ ∈ LT T S, Σ |=
τ
φ.
4 PROPERTIES
According to the validity condition we defined above,
we can prove some theorems for our fuzzy belief logic
system.
Theorem 1. K
j
φ |= φ.
This theorem is coincide with the intuitions of hu-
man beings: if we know a proposition φ, then φ is true
in our world.
Theorem 2. K
j
φ |= B
j
φ.
The theorem means if agent j knows proposition
φ, then he must believe it.
Theorem 3. If the frame is transitive then K
j
φ |=
K
j
K
j
φ.
5 COMPARISON WITH THE
AGM POSTULATES
The AGM postulates (named after the names of their
proponents, Alchourrn, G
¨
ardenfors, and Makinson)
are properties that an operator that performs revision
should satisfy in order for that operator to be consid-
ered rational. The belief set of an agent is denoted
by a theory set of K, which is deductively closed set
for formulas in the logical language, φ is the revision
formula. The postulates are listed as follows:
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1. K ∗ φ is a theory type
2. φ ∈ K ∗ φ success
3. K ∗ φ ⊆ K + φ upper bound
4. if ¬φ ∈ K, then K + φ ⊆ K ∗ φ lower bound
5. K ∗ φ = K⊥ iff φ is inconsistent triviality
6. if φ is equivalent to ψ then K ∗ φ = K ∗ ψ exten-
sionality
7. K ∗ (φ ∧ ψ) ⊆ (K ∗ φ) + ψ iteration upper bound
8. if¬ψ ∈ K ∗ φ, then (K ∗ φ) + ψ ⊆ K ∗ (φ ∧ ψ) iter-
ation lower bound
In this section we will check whether or not the
AGM postulates are fulfilled after an update in our
logic system. As a preliminary step, we need to define
belief sets, and decide the type of the revision formu-
las. In the reminder, let (M, s) be a belief state, where
M is defined in Definition 6; let Σ = {φ | M, s |= Bφ}
be a belief set, and ψ be a revision formula; and we re-
strict the revision formulas on propositional formulas.
Then formally, we have:
Defininition 11.
K = {χ | Val(Bχ,M, s) 6= absolutely- f alse};
K ∗ φ = {χ | Val([∗φ]Bχ, M,s) 6= absolutely- f alse};
K + φ = K ∪ {φ}.
Now we are going to check whether or not the 8
postulates can be verified. Formally, we have the fol-
lowing theorem:
Theorem 4. In our fuzzy belief logic system, AGM
postulates
• K
1
, K
5
, K
6
are verified;
• K
3
,K
4
,K
7
,K
8
are undecided; and
• K
2
is weakly satisfied, i.e., if we revise our previ-
ous belief with a proposition of φ and Val(φ) =
{τ | τ ∈ LTT S
t
} in the revision method pro-
posed in Definition 9, then we will believe propo-
sition φ to an extent of τ
0
such that τ
0
∈ LT T S
t
and τ
0
= Val(φ,M, v), where κ(v) ≤ κ(v
0
) and
Val(φ,M, v),Val(φ, M,v
0
) ∈ LT T S
t
.
6 RELATED WORK
Generally speaking, so far few people have study the
fuzzy form of dynamic belief logic although fuzzy
modal logic is the foundation of the fuzzy belief logic
and fuzzy dynamic logic because they are actually the
extension of modal logic.
Researchers started to study fuzzy modal logic
in about 1970 (Schotch, 1975; Zadeh, 1965; Zadeh,
1975). Moreover, some scholars have developed
a complete system of fuzzy modal logic (Mironov,
2005). In most of the literature on fuzzy modal logic,
they deal the fuzzy environment with the many-value
logic, while in this paper a new method is proposed
to depict the fuzziness, i.e. the linguistic variable
terms (LTT S). On one hand, LTT S can give the
propositions’ truth value in a continuous degree, like
Lukasiewicz many-value logic. On the other hand,
LTT S is more closer to our daily language and it may
play an more important part in application.
In the study of fuzzy belief logic (Zhang and Liu,
2012), researchers formalize reasoning about fuzzy
belief and fuzzy common belief, reduce the belief
degrees to truth degrees, and finally prove the com-
pleteness of the fuzzy belief and fuzzy common be-
lief logic. However, they have just studied the fuzzy
version of belief and knowledge in the state situation.
While in daily life, actually if our belief or knowl-
edge have changed, there must be a new actions or an
event happened. Like Public Announcement Logic
(van Benthem, 2002), we may know φ after someone
has declared proposition φ or some other proposition
ψ related to φ. We can say that the actions or events
actually cause to the belief or knowledge changing.
So, it is really important to consider the dynamics
when we reason about the beliefs. However, the ex-
isting work did this little.
In addition, the dynamic fuzzy logic can handle
the fuzzy environments (Hughes,Esterline and Kimi-
aghalam, 2012). However, belief revision has not be
handled like what we did in this paper. On the other
hand, although dynamic epistemic logic (van Ben-
them, 2002; Ditmarsch,Hoek and Kooi, 2007) and
dynamic belief logic (van Ditmarsch, 2005; van Ben-
them, 2007) are studied vastly, there are few literature
to expand the dynamic epistemic logic and dynamic
belief logic into the fuzzy realm. However in this pa-
per, we give a new approach to reason about dynamic
belief revision in fuzzy environments.
7 CONCLUSIONS
This paper is a fuzzy extension of dynamic belief re-
vision with quantitative method depicting belief revi-
sion. We give the syntax and semantics of the fuzzy
dynamic logic and use the Linguistic Truth Term Set
as the truth values for the fuzzy propositions. Then
we expose some properties for the logic. Our fuzzy
method to deal with dynamic belief logic is not only
of great importance for theory study like epistemic
logic and belief logic, but also of great application in
Artificial Intelligence and Multi-agent Systems.
In the future, we will give the complete axiomatic
system and prove the soundness and completeness of
AFuzzyDynamicBeliefLogic
293

our logic,
2
and we will also integrate the fuzzy logic
for other dynamic belief logic besides Aucher’s the-
ory.
ACKNOWLEDGEMENTS
This paper is supported by National Natural Science
Foundation of China (No. 61173019), Bairen plan
of Sun Yat-sen University, National Fund of Phi-
losophy and Social Science (No. 11BZX060), Of-
fice of Philosophy and Social Science of Guangdong
Province of China (No. 08YC-02), Fundamental Re-
search Funds for the Central Universities in China,
and major projects of the Ministry of Education (No.
10JZD0006). We also appreciate reviewers’ valuable
comments, which helped us to improve the paper sig-
nificantly.
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