The Dynamics of Narrow-minded Belief

Shoshin Nomura

, Norihiro Arai

and Satoshi Tojo

National Institute of Informatics, Japan

Japan Advanced Institute of Science and Technology, Japan

Keywords:

Dynamic Epistemic Logic, Public Announcement Logic, Logic of Passion, Narrow-minded Belief.

Abstract:

The purpose of this paper is to consider and formalize an important factor of human intelligence, belief affected

by passion, which we call narrow-minded belief. Based on Public Announcement Logic, we deﬁne our logic,

Logic Of Narrow-minded belief (LON), as that which includes such belief. Semantics for LON is provided

by the Kripke-style semantics, and a proof system for it is given by the Hilbert-style proof system. We then

provide a proof of the semantic completeness theorem for the proof system with the innermost strategy of

reducing a formula for LON. Using LON, we formally analyze the mental state of the hero of Shakespeare’s

tragedy Othello as an example of narrow-minded belief and its formalization.

1 INTRODUCTION

Love is blind, and hatred is also blind. To gener-

alize these phrases, we may say that passion causes

narrow-mindedness. It is not unusual that people can-

not emotionally stop believing what they do not want

to believe without any speciﬁc reason to believe so.

The hero of William Shakespeare’s play, Othello, is

involved in a pitiful but possible situation where he

wants to believe his wife’s chastity but he cannot since

he heard a bad rumor about her. It may be difﬁcult to

answer whether or not he believes that his wife is a

betrayer of their marriage given that he has heard this

rumor. In this situation, Othello has at least two dif-

ferent types of belief and/or knowledge. One is pas-

sionate or narrow-minded belief, which he is willing

to believe or cannot stop believing emotionally. The

other is belief, which is more rational (less passionate)

or, without considering any philosophical discussions

regarding the relationship between knowledge and be-

lief, it may even be said, is knowledge whereby he

judges something based on information attained via

rational inferences. The latter type of knowledge or

belief is treated by a standard epistemic (or doxastic)

logic and the current researchers would like to intro-

duce the former belief, passionate belief or narrow-

minded belief.

In fact, the notion of passion has a philosophically

and psychologically profound meaning in terms of be-

lief, and it is highly possible that such emotional be-

lief plays a signiﬁcant role in rationality. In A Treatise

of Human Nature, Hume famously (or even notori-

ously) wrote the following quotation.

[T]he principle, which opposes our passion,

cannot be the same with reason, and is only

called so in an improper sense. We speak not

strictly and philosophically when we talk of

the combat of passion and of reason. Reason

is, and ought only to be the slave of the pas-

sions [...]. (Hume, 1739, Book II, Sec. 3, Part

3).

Here, Hume says not only that passion has the same

signiﬁcance as rationality, but also that reason is a

subordinate of passion. We introduce one more quo-

tation from modern literature, Damasio’s Descartes’

error, to support the importance of consideration on

the relationship between passion and rationality.

[T]here may be a connecting trail, in anatom-

ical and functional terms, from reason to feel-

ings to body. It is as if we are possessed by a

passion for reason [...]. Reason, from the prac-

tical to the theoretical, is probably constructed

on this inherent drive by a process which re-

sembles the mastering of a skill or craft. Re-

move the drive, and you will not acquire the

mastery. But having the drive does not auto-

matically make you a master. (Damasio, 1994,

Part III, Chap. 11)

By referring neurological evidence, Damasio argues

that feeling (or passion) and rationality are strongly

connected with other, and they cannot be separated

Nomura, S., Arai, N. and Tojo, S.

The Dynamics of Narrow-minded Belief.

DOI: 10.5220/0007394502470255

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 247-255

ISBN: 978-989-758-350-6

247

as Descartes thought. The current researchers would

like to take a similar stance to that of Damasio, where

passion and rationality (in our term, narrow-minded

belief and knowledge) are related to one another in a

formal language of epistemic logic.

In this paper, we treat such a paradigm of agent

communication that each agent changes his/her belief,

after receiving messages from others, to strengthen/

weaken his/her tolerance. Towards this motivation,

we present a logic that adequately reﬂects human

minds which tends to be biased by certain kinds of

information.

The outline of the paper is as follows. In Section 2,

we introduce logic of narrow-minded belief (LON)

which is based on Public Announcement Logic by

Plaza (Plaza, 1989) and refers to the ideas of explicit

and implicit belief in dynamic epistemic awareness

logic by van Benthem & Vel

azquez-Quesada (van

Benthem and Vel

azquez-Quesada, 2010). Its seman-

tics are given by an expansion of the Kripke-style se-

mantics. In Section 3, we attempt to investigate and

formalize a person’s belief and emotion through fo-

cusing on a literary work, Othello since this is a story

of delicate transition of the hero’s narrow-minded be-

lief towards his wife. In Section 4, we introduce a

Hilbert-style proof system LON of LON, and some

proof theoretic properties. In Section 5, we give

a proof of the semantic completeness of our proof

system LON (Theorem 5.4) through the innermost

strategy for reducing a formula for LON into a for-

mula without announcement operators. In Section 6,

we introduce related epistemic/doxastic logics to the

present work.

2 LANGUAGE AND SEMANTICS

OF LON

2.1 Language

First of all, we address the syntax of LON. Let

Atom = {p,q,. ..} be a countable set of atomic propo-

sitions. Then, formula ϕ of the language L

(KN⊕!)

inductively deﬁned as follows (p ∈ Atom):

ϕ ::= p | ¬ϕ | (ϕ → ϕ) | Kϕ | Nϕ | [ϕ]ϕ | [⊕ϕ]ϕ | [.!.ϕ]ϕ.

We deﬁne other Boolean connectives such as ϕ ∧ χ,

ϕ ∨ χ, ϕ ↔ χ and ⊥ in a usual manner. We call oper-

ators [⊕ϕ], [ϕ] and [.!.ϕ] announcement operators.

Besides,

K is deﬁned by ¬K¬ and

N is deﬁned by

¬N¬. Note that K and N can be considered as the

box operator 2 in modal logic, and

K and

N can be

considered as the diamond operator 3.

• Kϕ reads ‘the agent knows that ϕ’,

• Nϕ reads ‘the agent narrow-mindedly believes

that ϕ’,

• [ϕ]χ reads ‘after obtaining information ϕ which

may strengthen the agent’s narrow-mindedness, χ

holds’,

• [⊕ϕ]χ reads ‘after obtaining information ϕ which

may weaken the agent’s narrow-mindedness, χ

holds,’ and

• [.!.ϕ]χ reads ‘after obtaining truthful information

(announcement) ϕ, χ holds’.

2.2 Semantics

Let us go on to the semantics of LON. We call the tu-

ple hS, R,V i an epistemic model (or a Kripke model)

if the domain S is a nonempty set of states, the ac-

cessibility relation R is an equivalence relation on S

and V : Atom → P (S) is a valuation function. The

set S is called domain of M and may be denoted

by D(M ). Subsequently, we deﬁne an epistemic

narrow-doxastic model (or simply en-model) M =

hS, R, Q,V i where the components of S, R and V are

the same as that of the epistemic model, and Q is a

binary relation on S such that Q ⊆ R.

Deﬁnition 2.1 (Satisfaction relation). Given an en-

model M , a state s ∈ D(M ), and a formula ϕ ∈

(KN⊕!)

, we deﬁne the satisfaction relation M , s |= ϕ

as follows:

M , s |= p iff s ∈ V (p),

M , s |= ¬ϕ iff M , s 6|= ϕ,

M , s |= ϕ → χ iff M , s |= ϕ implies M , s |= χ,

M , s |= Kϕ iff for all x ∈ S : sRx implies M , x |= ϕ,

M , s |= Nϕ iff for all x ∈ S : sQx implies M , x |= ϕ,

M , s |= [ϕ]χ iff M

ϕ

,s |= χ,

M , s |= [⊕ϕ]χ iff M

⊕ϕ

,s |= χ,

M , s |= [.!.ϕ]χ iff M , s |= ϕ implies M

!ϕ

,s |= χ,

where the notations M

ϕ

, M

⊕ϕ

and M

!ϕ

above re-

spectively indicate the en-models deﬁned by M

ϕ

hS, R, Q

ϕ

,V i, M

⊕ϕ

= hS, R, Q

⊕ϕ

,V i and M

!ϕ

h[[ϕ]]

!ϕ

i with

[[ϕ]]

:= {x ∈ S | M , x |= ϕ},

!ϕ

:= R ∩ [[ϕ]]

×[[ϕ]]

ϕ

:= Q ∪ {(s,t) ∈ R | s ∈ [[ϕ]]

or t ∈ [[ϕ]]

!ϕ

:= Q ∩ [[ϕ]]

×[[ϕ]]

⊕ϕ

:= Q ∩ S × [[ϕ]]

!ϕ

(p) := V ∩ [[ϕ]]

(where p ∈ Atom).

Then we deﬁne the validity of a formula in a usual

way.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

248

Deﬁnition 2.2 (Valid). A formula ϕ is valid at M if

M ,s |= ϕ for any s ∈ D(M ), and we write M |= ϕ.

A formula ϕ is valid if M |= ϕ, for any en-model M ,

and we write |= ϕ.

We conﬁrm that an en-model, which is modiﬁed

by announcement operators [⊕ϕ],[ϕ] and [.!.ϕ], pre-

serves frame properties, i.e., R is an equivalence rela-

tion and the subset relation Q ⊆ R.

Proposition 2.1 (Preserving frame properties). Let

ϕ ∈ L

(KN⊕!)

be any formula. If M = hS, R, Q,V i

is an en-model, then M

ϕ

= hS,R, Q

ϕ

,V i, M

⊕ϕ

hS, R, Q

⊕ϕ

,V i and M

!ϕ

= h[[ϕ]]

!ϕ

i are

also en-models.

Proof. What we wish to show is that (1) R

!ϕ

is an

equivalence relation (i.e., it satisﬁes reﬂexivity, Eu-

clidicity and transitivity), and (2) the subset relation

!ϕ

⊆ R

!ϕ

, (3) the subset relation Q

ϕ

⊆ R and (4) the

subset relation Q

⊕ϕ

⊆ R. We only treat one of (1) and

(3) in the following.

(1)-2 R

!ϕ

satisﬁes Euclidicity. Fix any x, y,z ∈

[[ϕ]]

. Suppose xR

!ϕ

y and xR

!ϕ

z, and show yR

!ϕ

Since R is Euclidean i.e., xRy and xRz jointly

imply yRz for all x,y, z ∈ S. By the assumption

x, y, z ∈ [[ϕ]]

⊆ S, we have x, y, z ∈ S and yRz.

so we get the goal R

!ϕ

is also Euclidean with

y, z ∈ X .

(3) Fix any (x, y) ∈ Q ∪ {(x, y) ∈ R | x ∈ [[ϕ]]

or y ∈

[[ϕ]]

}, and we show (x,y) ∈ R. Suppose (x, y) ∈

Q. Then we obtain (x, y) ∈ R with Q ⊆ R. Suppose

(x, y) ∈ {(x,y) ∈ R | x ∈ [[ϕ]]

or y ∈ [[ϕ]]

}. So,

we obtain (x, y) ∈ R.

3 EXAMPLES OF

FORMALIZATION OF

NARROW-MINDED BELIEF

3.1 Comments of Knowledge and

Narrow-minded Belief Operators

Before moving on the topic of narrow-minded belief,

we add some comments on the general features of

knowledge operator K and accessibility relation R in

epistemic logics. Let us look at the epistemic model

hS, R,V i = h{w, v}, S

,V i where V (p) = {v} (that can

be regarded as an en-model M = h{w, v}, S

,∅,V i),

and the graphic form of this model is as follows.

¬p

In this model, at world w, the agent is ignorant about

p’s truth-value. This is because the formula

Kp ∧

K¬p, which intuitively means that the agent does not

know whether p, is true at w. As it implies, in epis-

temic logic, an arrow between states has a negative

meaning in general. In other words, van Ditmarsch

et al. state that “the more worlds an agent considers

possible, the less he believes, and vice versa.” (van

Ditmarsch et al., 2008, p.55). The operator

K repre-

sents at least one arrow in an epistemic model. The

narrow-minded belief operator

N basically preserves

these features; nevertheless, we cannot say that ‘the

more worlds an agent considers possible, the less he

believes, and vice versa’ in case of the operator N

since the narrow-mind belief is affected by uncertain

information or even the agent’s imagination and may

be wrong. In other words, to express such capricious

belief, we introduce the operator N.

Additionally, we note on the frame property of

R and Q. The accessibility relation R represents the

accessibility relation for knowledge, and so we as-

sume that the agent is an introspective agent, i.e.,

R is an equivalence relation. Moreover, the for-

mulas of Kϕ → ϕ, Kϕ → KKϕ (positive introspec-

tion) and ¬Kϕ → K¬Kϕ (negative introspection) are

valid at M where its accessibility relation is equiva-

lence relation. However, since Q represents a narrow-

minded belief, we do not assume the agent is intro-

spective since introspectiveness is based on some kind

of rationality, which is the exact opposite of narrow-

mindedness. That is why Q does not have any frame

property. By distinguishing these two accessibility re-

lations, we formally express the distinction between

knowledge and narrow-minded belief.

3.2 Formalizing Othello’s

Narrow-minded Belief

As mentioned in the introduction, our target, which

we consider and formalize, is Shakespeare’s Othello

as it depicts a typical case of the change in a person’s

delicate mental state. Its story depicts how the lives of

the four main characters (Othello, Desdemona, Iago

and Cassio) are woven together and driven by passion.

The following is the short summary of the play:

General in the Venetian military Othello was

recently married to a rich senator’s daughter

Desdemona. Although there is a great dispar-

ity of age between the two, they build a good

relationship of trust, and Othello and Desde-

mona love and believe each other from their

The Dynamics of Narrow-minded Belief

249

hearts. However, Othello’s trusted subordi-

nate Iago who secretly holds a deep grudge

against Othello tells him a rumor that Desde-

mona is having an affair with a young hand-

some soldier named Cassio. This causes Oth-

ello to feel uncertain towards his wife’s inno-

cence. Deepening Othello’s doubt against his

wife, Iago steals Desdemona’s handkerchief,

a present from Othello, and leads Cassio up

to ﬁnd it. Using the handkerchief as proof,

Iago succeeds in convincing Othello that Des-

demona has engaged in an immoral relation-

ship with Cassio. Finally, Othello narrow-

mindedly believes what Iago has told him and

he feels great jealousy and anger towards his

wife. Even though Desdemona protests her in-

nocence, Othello, who is now mad with jeal-

ousy, kills his wife in a ﬁt of passion. Follow-

ing her death, Desdemona’s servant confesses

that her mistress was innocent and that Iago

fabricated the story, which resulted in such a

tragedy. Othello comes to his senses and real-

izes his mistake, at which point he loses hope

and takes his own life.

Of course, this summary is extremely simpliﬁed and

actual tale is more intricately woven. There are at

least four main scenes in the story, which highlight

Othello’s narrow-minded belief, and we would like to

focus on these in this paper. The four main points are

as follows:

1. Othello believes Desdemona from the heart.

2. Iago spreads a bad rumor about Desdemona,

which causes doubt about her innocence in Oth-

ello’s mind.

3. Iago uses fake evidence (a handkerchief) to con-

vince Othello of Desdemona’s immoral actions

and he narrow-mindedly believes it.

4. A servant truthfully informs Othello that Desde-

mona is innocent.

Othello’s mind, including narrow-minded belief in

each of the four scenes, may be semantically modeled

as follows. We note that, in the graphic form of en-

models, the double circle indicates the actual state. In

addition, arrows of the straight line represent the line

of R and arrows of the dotted line represent that of Q.

Moreover, let an atomic proposition p to read ‘Desde-

mona is having an affair,’ and Atom = {p}.

(1) Othello deeply believes his wife. In the initial

stage, Othello, who was recently married, believes his

wife from the depth of his heart and does not doubt

her immorality. However, Othello does not have any

speciﬁc evidence that Desdemona is having an af-

fair and he does not actually know if she is innocent

or not at this stage. Therefore, the initial stage al-

ready includes some contradiction in his mind, i.e.,

he does not explicitly know if she is innocent, but

he narrow-mindedly believes her. Thus, the mental

state of Othello at the opening of the play may for-

mally be expressed by en-model M = hS,R, Q,V i =

h{s,t}, S

,{(s,s)}, {p 7→ {t}}i. Therefore, we may

say that, at this stage, formulas

Kp ∧

K¬p and N¬p

are valid at M .

¬p

Figure 1: M .

(2) Iago spreads a bad rumor about Desdemona,

which leads to doubts in Othello’s mind. After

Iago tells Othello a bad rumor ([p]) about Desde-

mona, he begins to doubt his wife. In other words,

he is now unsure about her constancy and does not

know if she is innocent or not. Separately from Oth-

ello’s narrow-minded belief, his state of knowledge

remains unchanged since he has not obtained any new

truthful information and can only go by Iago’s story

in which his wife is accused of inﬁdelity. Then the

mental state of Othello at the second stage of the

play may formally be expressed by en-model M

p

hS, R, Q

p

,V i = h{s,t}, S

,{p 7→ {t}}i, where the

formula

Np ∧

N¬p is now valid at this en-model. This

formula represents a confusion in his mind about his

wife’s innocence.

¬p

Figure 2: M

p

(3) Iago uses fake evidence to convince Othello

of Desdemona’s immorality. At this stage of the

play, Iago attempts to deceive his superior, Oth-

ello, by using fake evidence (Desdemona’s hand-

kerchief) to pretend she spent her time with Cas-

sio, and Othello is completely taken in. Conse-

quently, Othello completely loses his self-control, and

strongly and narrow-mindedly believes that his wife

is having an affair with Cassio. This is also rep-

resented by en-model M

p⊕p

= hS, R, Q

p⊕p

,V i =

h{s,t}, S

,{(t,t),(s,t)}, {p 7→ {t}}i. Formally, in his

mind, the formula

Np is valid at this en-model, but

N¬p is not anymore. Let us remind the reader that in

¬p

Figure 3: M

p⊕p

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

250

the case of the operator

N, it does not mean that if the

number of arrows is reduced, then the agent’s igno-

rance is reduced.

(4) A servant truthfully informs that Desde-

mona is innocent. In the last scene of the play,

Desdemona’s faithful servant truthfully tells the

fact that Desdemona is innocent, implying that

Othello’s narrow-minded belief regarding his wife

is completely erroneous. Othello faces such a

surprising fact and he is heart-broken by the

confession. This is represented by en-model

p⊕p!¬p

= h[[¬p]]

p⊕p

!¬p

p⊕p!¬p

!¬p

i =

h{s},{(s, s)},

0}i. Formally, by the truthful in-

formation of ¬p, a state t where p holds is elim-

inated, and as a result, while the agent (Othello)

knows ¬p (his wife is innocent), the arrow of narrow-

minded belief is empty. This means that he narrow-

mindedly believes everything even if it is a contra-

diction M

p⊕p!¬p

|= ⊥, i.e., he is going crazy. As a

result, the tragedy ends with the suicide of Othello in

the ﬁnal scene.

¬p

Figure 4: M

p⊕p!¬p

4 HILBERT-SYSTEM FOR LON

We move on the topic of a proof theory for LON.

Hilbert-system for LON (LON), is deﬁned in Ta-

ble 1. Axioms (4) and (5) indicate what we call pos-

itive introspection and negative introspection, respec-

tively. Axiom (K&N) indicates a relation of knowl-

edge and narrow-mined belief, in which if the agent

knows something, he/she also narrow-mindedly be-

lieves. This implies that narrow-minded belief is one

of the bases of our knowledge, and this view of be-

lief and knowledge can be supported by philosophers

and/or psychologists like Hume and Damasio, as dis-

cussed in the introduction. Axioms (RA∗) are called

reduction axioms. Through the reduction axioms and

rules, each theorem of LON may be reduced into a

theorem of the language L

(KN)

which will be shown

in Section 5.

We provide some basic deﬁnitions and properties

for proofs in the next section.

Deﬁnition 4.1 (Derivable). A derivation in LON con-

sists of a sequence of formulas of L

(KN⊕!)

each of

which is an instance of an axiom or is the result of

applying an inference rule to formula(s) that occur

earlier. If ϕ is the last formula in a derivation in LON,

then ϕ is derivable in LON, and we write `

LON

ϕ (or

just ` ϕ).

Let ϕ be a formula of L

(KN⊕!)

and Γ be a subset of

languageKakko. Then, ϕ is derivable from Γ (Γ `

LON

ϕ) if there exists a ﬁnite subset Γ

of Γ such that `

LON

→ ϕ.

Proposition 4.1. Let ϕ, χ, ψ be arbitrary formulas of

(KN⊕!)

. Then the following holds.

1. `

LON

[∗ϕ](χ∧ψ) ↔ ([∗ϕ]χ∧[∗ϕ]ψ) (where ∗ ∈

{⊕,, .!.})

2. `

LON

ϕ ↔ χ implies `

LON

[∗ψ]ϕ ↔

[∗ψ]χ (where ∗ ∈ {⊕, , .!.})

3. `

LON

ϕ ↔ χ implies `

LON

2ϕ ↔

2χ (where 2 ∈ {K,N}

Deﬁnition 4.2 (Substitution). The substitution for

formula ϕ





means p appearing in a formula ϕ is

Table 1: Hilbert-system for LON : LON.

Axioms for knowledge andnarrow−minded belief

(taut) all instantiations of

propositional tautologies

) K(ϕ → χ) → (Kϕ → Kχ)

) N(ϕ → χ) → (Nϕ → Nχ)

(T) Kϕ → ϕ

(4) Kϕ → KKϕ

(5) ¬Kϕ → K¬Kϕ

(K&N) Kϕ → Nϕ

Inference Rules

(MP) From ϕ and ϕ → χ, infer χ

(NecK) From ϕ, infer Kϕ

(NecN) From ϕ, infer Nϕ

(Nec[∗]) From ϕ, infer [∗χ]ϕ

where ∗ ∈ {, ⊕,.!.}

Reduction Axioms for []

(RA1) [ψ]p ↔ p

(RA2) [ψ]¬ϕ ↔ ¬[ψ]ϕ

(RA3) [ψ](ϕ → χ) ↔ ([ψ]ϕ → [ψ]χ)

(RA4) [ψ]Kϕ ↔ K[ψ]ϕ

(RA5) [ψ]Nϕ ↔ N[ψ]ϕ∧

(ψ → K[ψ]ϕ) ∧ K(ψ → [ψ]ϕ)

Reduction Axioms for [⊕]

(RA⊕1) [⊕ψ]p ↔ p

(RA⊕2) [⊕ψ]¬ϕ ↔ ¬[⊕ψ]ϕ

(RA⊕3) [⊕ψ](ϕ → χ) ↔ ([⊕ψ]ϕ → [⊕ψ]χ)

(RA⊕4) [⊕ψ]Kϕ ↔ K[⊕ψ]ϕ

(RA⊕5) [⊕ψ]Nϕ ↔ N(ψ → [⊕ψ]ϕ)

Reduction Axioms for [.!.]

(RA!1) [.!.ψ]p ↔ (ψ → p)

(RA!2) [.!.ψ]¬ϕ ↔ (ψ → ¬[.!.ψ]ϕ)

(RA!3) [.!.ψ](ϕ → χ) ↔ ([.!.ψ]ϕ → [.!.ψ]χ)

(RA!4) [.!.ψ]Kϕ ↔ (ψ → K[.!.ψ]ϕ)

(RA!5) [.!.ψ]Nϕ ↔ (ψ → N(ψ → [.!.ψ]ϕ))

The Dynamics of Narrow-minded Belief

251

replaced by χ, and deﬁned as follows:





:= q (if q 6= p),





:= χ (if q = p),

(◦ψ)





:= ◦(ψ





(ψ

→ ψ

)





:= (ψ





) → (ψ





([∗ψ

]ψ

)





:= [∗ψ

](ψ





where ◦ ∈ {¬, K, N} and ∗ ∈ {,⊕, !}.

Proposition 4.2. Let ∗ ∈ {, ⊕, !}. If `

LON

ϕ ↔ χ,

then `

LON





↔ ψ





for any ϕ, χ, ψ ∈ L

(KN⊕!)

and p ∈ Atom.

Proof. Fix any ϕ, χ ∈ L

(KN⊕!)

and p ∈ Atom. Sup-

pose `

LON

ϕ ↔ χ. Then we show `

LON





↔ ψ





By induction on ψ ∈ L

(KN⊕!)

5 COMPLETENESS

Let us move onto a proof of the completeness theorem

of LON with a similar argument in (van Ditmarsch

et al., 2008, Section 5).

5.1 Semantic Completeness of LON

Let the language L

(KN)

be our formal lan-

guage L

(KN⊕!)

without announcement operators

([],[⊕] and [.!.]). For an en-model M and s ∈

D(M ) and ϕ ∈ L

(KN)

, the satisfaction relation

M ,s |= ϕ is naturally deﬁned by following the def-

inition in Section 2. Additionally, Hilbert-system

LON

is also generated by rejecting the reduction ax-

ioms and inference rules of (Nec[]), (Nec⊕) and

(Nec[.!.]) in Table 1. Note that deﬁnitions of the

derivation and derivability of LON

are given in the

same manner as that of LON in Deﬁnition 4.1.

Theorem 5.1 (Soundness of LON

). For any formula

ϕ ∈ L

(KN)

LON

ϕ implies |= ϕ.

Proof. Fix any ϕ ∈ L

(KN)

such that ϕ is derivable

in LON

. We show that ϕ is valid by induction on

the height of the derivation. In the base case, the

derivation height is 0 i.e., it consists of only an ax-

iom. Therefore, we show the validity of each axiom

of LON.

A direct proof of the completeness theorem of LON

can be shown in a usual manner with Lindenbaum’s

lemma.

Theorem 5.2 (Completeness for LON

w.r.t. the se-

mantics of L

(KN)

). For any ϕ ∈ L

(KN)

, the following

holds:

|= ϕ implies `

LON

ϕ.

5.2 Semantic Completeness of LON

Based on the completeness theorem of LON

, we ex-

pand the discussion to the completeness of LON. A

proof of the completeness theorem of LON is given

in this section by the reduction method whose ba-

sic idea was introduced in the previous work (Plaza,

1989, Theorem 2.7). The essential idea of this method

is based on the fact that every formula in L

(KN⊕!)

reducible into a formula in L

(KN)

which will be shown

in Lemma 5.3.

Remark 5.1. We note that reduction axioms for se-

quential announcement operators e.g.,

(RA!6) [.!.χ][.!.ψ]ϕ ↔ [.!.(χ ∧ [.!.χ]ψ)]ϕ

are not included since, without them, any formula

with announcement operators can be reducible. It is

known that there are at least two strategies to reduce

a formula with announcement operators into a for-

mula without any such operator. Let us consider the

formula [!p][!q]r − (i). One approach, we may call

it ‘outermost strategy’, focuses on the outermost oc-

currence of announcement operator, for example [!p]

of the above formula (i). Following this strategy, an

axiom like (RA!6) is required for reducing the for-

mula. By using (RA!6), we may obtain [!(p∧[!p]q)]q.

Then (RA!1) becomes applicable, and so we obtain

the formula which does not include any announce-

ment operator but is equivalent to the initial formula.

This approach is introduced by (van Ditmarsch et al.,

2008). The other strategy may be called ‘innermost

strategy’ and focuses on the innermost occurrence of

announcement operator, for example [!q] of (i). Thus,

by applying (RA!1) to the innermost occurrence i.e.,

[!q]r, we obtain [!p](q → r). After that, (RA!3) and

(RA!1) are subsequently applicable, and so we ob-

tain the formula without any announcement operator

but equivalent to the initial formula of (i). The latter

strategy does not require reduction axioms for reduc-

ing sequential announcement operators into a single.

Therefore, we employ this strategy to avoid introduc-

ing many and messy axioms.

The idea of this inner-

most strategy was introduced by (van Benthem et al.,

2006), and (van Benthem, 2011, p.54). Furthermore,

an attentive proof for reducibility of a formula of Dy-

namic logic into a formula of standard modal logic

If we follow the outermost strategy, six additional ax-

ioms (e.g, axioms for reducing combination of [⊕A][B]

and [.!.A][⊕B] etc.) are required.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

252

by using the innermost strategy is given in (Aucher,

2003, Proposition 3.3.5).

At ﬁrst, we treat the soundness theorem of it which

is straightforward.

Theorem 5.3 (Soundness of LON). For any formula

ϕ ∈ L

(KN⊕!)

LON

ϕ implies |= ϕ.

Proof. For the soundness theorem of LON, it sufﬁces

to show the validity of reduction axioms, additional

axioms to that of LON

and additional inference rules

(Nec[]),(Nec[⊕]) and (Nec[.!.]). The validity of ad-

ditional rules and axioms are also easily shown by fol-

lowing semantics of LON.

Next, we give some deﬁnitions and lemmas for

proof of the completeness.

Deﬁnition 5.1 (Length). The length function

` : L

(KN⊕!)

→ N is inductively deﬁned as follows:

`(p) := 1, `(Nϕ) := 1 + `(ϕ),

`(¬ϕ) := 1 + `(ϕ), `([⊕ϕ]χ) := (5 + `(ϕ))

`(χ)

`(ϕ → χ) := 1 + `(ϕ) + `(χ), `([ϕ]χ) := (5 + `(ϕ))

`(χ)

`(Kϕ) := 1 + `(ϕ), `([.!.ϕ]χ) := (5 + `(ϕ))

`(χ)

With these settings, we may show the following

lemma.

Lemma 5.1. Let ∗ be ⊕,  or !. Then for all reduction

axioms [∗ϕ]χ ↔ ψ, `([∗ϕ]χ) > `(ψ) holds.

Proof. We only conﬁrm the following case.

Case of (RA  5). The less-than relation

`(N[ψ]ϕ ∧(ψ → K[ψ]ϕ)∧ K(ψ → [ψ]ϕ)) <

`([χ]Nϕ) holds by the following equations:

`(N[ψ]ϕ ∧ (ψ → K[ψ]ϕ) ∧ K(ψ → [ψ]ϕ))

= `(N[ψ]ϕ → ¬(ψ → K[ψ]ϕ) ∧ K(ψ → [ψ]ϕ))

= 2 + `(N[ψ]ϕ) + `(ϕ → K[ψ]ϕ) + `(Kϕ → K[ψ]ϕ)

= 8 + `([ψ]ϕ) + `(ϕ) + `([ψ]ϕ) + `(ϕ) + `([ψ]ϕ)

= 8 + 2 · `(ϕ) + 3 · (5 + `(ψ))

`(ϕ)

Then we can prove that `([ψ]Nϕ) = k

1+`(ϕ)

8 + 2 · `(ϕ) + 3 · k

`(ϕ)

(where k = (5 + `(ψ)) ≥ 6)

holds.

We add one more comment for a technical difference

between the two strategies. In the outermost strategy of

public announcement logic, we need to include axiom like

(RA!6) to reduce sequential announcement operators into a

single, but the inference rule of (Nec[!]) is derivable. On

the other hand, the rule is indispensable in the case of the

innermost strategy, instead of economizing the number of

axioms.

Let `(ϕ) = n and f

(k) ≡ k

n+1

− 3k

− 2n − 8. Then

obviously f

(k) = k

(k − 3) − 2n − 8 > 0 for k ≥ 6 for any

ﬁxed n ≥ 1, as well as for ﬁxed n ≥ 1 for any k ≥ 6.

Lemma 5.2. For any p ∈ Atom, ϕ, χ, ψ ∈ L

(KN⊕!)

if `(ϕ) > `(χ) and ψ





6= ψ





, then `(ψ





) >

`(ψ





) holds.

Proof. Fix any p ∈ Atom and ϕ, χ ∈ L

(KN⊕!)

. Then

assume `(ϕ) > `(χ). We conduct the proof by induc-

tion on ψ ∈ L

(KN⊕!)

Case of ψ is of the form ψ

→ ψ

. Assume (ψ

→

)





6= (ψ

→ ψ

)





. We show `((ψ

→

)





) > `((ψ

→ ψ

)





). Therefore, it suf-

ﬁces to show `(ψ





) + `(ψ





) > `(ψ





) +

`(ψ





). From the assumption, we obtain that

at least one of C

and C

satisﬁes ψ









. Without loss of generality, assume that





6= ψ





. By induction hypothesis, we ob-

tain `(ψ





) > `(ψ





). Therefore, no mat-

ter whether ψ





6= ψ





or not, we obtain the

goal.

Other cases are similar to the above.

Deﬁnition 5.2. `

: L

(KN⊕!)

→ N is deﬁned as fol-

lows.

(ϕ) :=







0 if ϕ ∈ L

(KN)

`(ϕ) otherwise

Lemma 5.3 (Reduction lemma). For any ϕ ∈

(KN⊕!)

, there exists ψ ∈ L

(KN)

such that `

LON

ϕ ↔

ψ.

Proof. By induction on `

(ϕ). We only treat the fol-

lowing case.

Case: `

(ϕ) > 0. In this case, ϕ ∈ L

(KN⊕!)

in-

cludes at least one subformula which is of the

form [∗χ

]χ

(where ∗ ∈ {⊕, , !} and χ

∈

(KN⊕!)

,χ

∈ L

(KN)

). On the other hand, there

is a reduction axiom which has the form of

[∗χ

]χ

↔ χ

, and let this reduction axiom be

(RA∗). The formula ϕ is equal to D



[∗χ

]χ



and

is equal to D





for some D ∈ L

(KN⊕!)

, and

so ﬁx such D. Then we may obtain the following

derivation.

1. ` [∗χ

]χ

↔ χ

(RA*)

2. ` ϕ ↔ ϕ

1 and Proposition 4.2

3. ` ϕ

↔ ψ Induction hypothesis

4. ` (ϕ ↔ ϕ

) → ((ϕ

↔ ψ) → (ϕ ↔ ψ)) (taut)

5. ` ϕ ↔ ψ 2,3 and 4 with (MP)

The Dynamics of Narrow-minded Belief

253

where ϕ = D



[∗χ

]χ



and ϕ

= D





. Induc-

tion hypothesis in the above derivation is applica-

ble, since the less-than relation `



[∗χ

]χ



) >





) holds by Lemma 5.1 and Lemma 5.2.

Actually, Lemma 5.3 is the core of the proof of the

completeness theorem. Through this, we may easily

show the theorem as follows.

Theorem 5.4 (Completeness of LON w.r.t. the se-

mantics of L

(KN⊕!)

). For any formula ϕ ∈ L

(KN⊕!)

the following holds:

|= ϕ implies `

LON

ϕ.

Proof. Fix any ϕ ∈ L

(KN⊕!)

such that |= ϕ. By

Lemma 5.3, we obtain `

LON

ϕ ↔ χ for some χ ∈

(KN)

. Then consider such χ ∈ L

(KN)

. By Theo-

rem 5.3 (the soundness of LON), we obtain |= ϕ ↔ χ.

With the assumption |= ϕ, we have |= χ. Next, by

Theorem 5.2 (the completeness of LON

), we obtain

LON

χ, and so `

LON

χ trivially holds; therefore, we

obtain `

LON

ϕ with `

LON

ϕ ↔ χ again. That is what

we desired.

6 RELATED WORKS

In this section, we introduce some related epis-

temic/doxastic logics. An epistemic logic for implicit

and explicit belief by (Vel

azquez-Quesada, 2014) is

perhaps the closest concept we can ﬁnd to that of

LON. This logic is based on the logic of awareness

logic (van Benthem and Vel

azquez-Quesada, 2010),

and it distinguishes the sense of belief into two, im-

plicit and explicit belief, to avoid the logical omni-

science in epistemic logic. A traditional approach

to mix knowledge and belief operators, sometimes

called epistemic-doxastic logic (e.g., see (Voorbraak,

1993)), is another system similar to ours since K and

N of LON may be interpreted as a mixture of these

two different human tendencies.

One of differences between LON and the above

existing works may relate to the deﬁnition of the sat-

isfaction relation of LON:

M , s |= [ϕ]χ iff M

ϕ

,s |= χ,

where Q

ϕ

:= Q ∪ {(s,t) ∈ R | s ∈ [[ϕ]]

or t ∈ [[ϕ]]

Here, we include a mechanism of adding arrows i.e.,

a mechanism in which some of the information may

confuse the agent.

In addition, there are some other attempts to in-

troduce a distinction in our belief/knowledge from

a different point of view. Intuitionistic epistemic

logic (Artemov and Protopopescu, 2014; Williamson,

1992) is one of them; this epistemic logic is based

on intuitionistic logic, which distinguishes knowledge

into two: standard knowledge, which normal epis-

temic logics treat and knowledge in the strict sense.

In other words, this aims at introducing a distinction

in knowledge, more strict and rational knowledge and

not strict knowledge, which is an opposite perspec-

tive to our attempt, which introduced a distinction be-

tween belief with passion and knowledge.

There are also some logics which deal with human

emotion; for example (Lorini and Schwarzentruber,

2011) and (Dastani and Lorini, 2012). We may, for

the further development, need to consider relevance

to these existing logics about emotion.

7 CONCLUSION AND FURTHER

DIRECTIONS

We introduced logic of narrow-minded belief (LON),

a variant of dynamic epistemic logic. This aims to

formally express a human’s passionate and narrow-

minded belief, and as an example of the application

of LON, we formalized Shakespeare’s play Othello.

Philosophers and neuropsychologists believe that pas-

sion, or belief affected by passion, is an indispensable

factor and even a basis for our reason. Without pas-

sion or emotions, human intelligence may be never

realized. Therefore, we hope that our attempt in the

present work will contribute to formal expressions of

the human mind.

It may be possible to further develop our attempt

in various directions. For example, we did not re-

gard the problem of the logical omniscience; the logic

of awareness is one of the candidates to be added

to LON, as it is difﬁcult to interpret the meaning of

awareness in the context of passion. Another inter-

esting feature that should be considered and added

to LON is ‘a lie’ as it pertains to dynamic epistemic

logic by van Ditmarsch (van Ditmarsch, 2011). Actu-

ally, Iago’s rumor should be regarded as a lie, as our

passion or narrow-mindedness is easily affected by

such dubious information. Therefore, it might be in-

teresting to consider these aspects in future researches

regarding the logic of passion.

ACKNOWLEDGMENT

The authors thank the anonymous reviewers for their

careful reading of our manuscript and their many in-

sightful comments. This work was supported by JST

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

254

CREST Grant Number JPMJCR1513 and JSPS kaken

17H02258.

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