INTERMATHS, VOL. 4, NO. 2 (2023), 28–37
https://doi.org/10.22481/intermaths.v4i2.14133
Article
cb licença creative commons
A basic epistemic logic and its algebraic model
Hércules de Araujo Feitosa
a,⋆
, Mariana Matulovic
b
, and Ana Claudia de J. Golzio
a
a
UNESP, School of Sciences, Bauru, SP, Brazil;
b
UNESP, School of Sciences and Engineering, Tupã, SP, Brazil
* Correspondence: hercules.feitosa@unesp.br
Abstract: In this paper we propose an algebraic model for a modal epistemic logic. Although
it is known the existence of algebraic models for modal logics, considering that there are so
many different modal logics, so it is not usual to give an algebraic model for each such system.
The basic epistemic logic used in the paper is bimodal and we can show that the epistemic
algebra introduced in the paper is an adequate model for it.
Keywords: Epistemic logic; Knowledge and belief; Algebraic logic; Algebraic model.
Classification MSC: 03B45; 03G27; 03C60.
Resumo: Neste artigo propomos um modelo algébrico para uma bem simples lógica modal
epistêmica. Embora seja conhecida a existência de modelos algébricos para as lógicas modais,
devido ao fato de que existe uma quantidade enorme de lógicas modais, não é usual apresentar-
mos modelos algébricos para cada tal particular caso modal. O sistema de lógica epistêmica
usado nestas notas é bimodal e podemos demonstrar que a álgebra espistêmica introduzida no
artigo é modelo algébrico adequado à lógica considerada.
Palavras-chave: Lógica epistêmica; Conhecimento e crença; Lógica algébrica; Modelo al-
gébrico.
1 Introduction
Algebraic logic formalizes some aspects of logic and then places these aspects in a
general algebraic environment. Usually, the constructions with algebraic logic consider
and develop some other branches of mathematics, such as topology, filters and ideals,
and set theory, among others.
The contemporary logics have several types of models, and algebraic models are one
of these distinct contexts for logical interpretation.
The first steps of algebraic logic appeared in the XIX century with Boole and other
thinkers, but they were well-developed in the Polish tradition with Tarski, Rasiowa and
Sikorski, in the next century (Dunn and Hardgree [4]), (Rasiowa and Sikorski [10]).
In general, it is easier to recognize the properties of a logic in its models than in its
deductive systems. In particular, the algebraic models can express many logical laws in
a simple way.
Submitted 13 December 2023; Accepted 20 December 2023; Available online: 30 December 2023.
ISSN 2675-8318 Copyright ©2023 INTERMATHS. Published by Edições UESB. This is an Open Access article under the CC BY 4.0 license.
Of course, it is more direct for people with some mathematical experience, but in the
next pages we will observe that it is not difficult to see an algebraic model for a very
basic logic.
For the purposes of this article, we selected an Epistemic Logic (EL) among the formal
systems proposed to model this relationship between known facts and the facts about
knowing highlights. One can find more information about the relationship between
knowledge and Epistemic Logic in (Meyer and Van der Hoek [7]).
In 1999, Mortari [
9
] points out that the notion of knowledge is linked to the verb to
know, a verb that is said of propositional attitudes, i.e, knowledge refers to “attitudes
that an intelligent agent can have in relation to any proposition p”.
Thus, the formalization of epistemic logics can connect aspects of epistemology
with the most up-to-date technologies of information and I.A. devices (Halpern [
5
]),
(Rosenschein [12]).
In Section 2, we present the modal system of Basic Epistemic Logic BEL. In the
next section, we introduce, as an original contribution, the correspondent algebra
B
, for
short
BEL
-algebra, motivated by the algebraic logic tradition and interest in BEL. In
Sections 4 and 5 we show the theorems of soundness and completeness, developed for
the class of
BEL
-algebras, which are shown to be completely adequate to the logical
system BEL.
2 A basic epistemic logic
In this section we present the system of epistemic logic for which we will give an
algebraic model. We follow Mortari [9] and denote this logic by BEL.
Modal logics have been deeply studied in (Carnielli and Pizzi [
1
]), (Chagrov and
Zakharyachev [2]) and (Chellas [3]).
The propositional logic BEL is constructed over the propositional language
L
=
{¬, ∧, →, K
i
, B
i
, p
1
,
p
2
, p
3
, ...}
, where knowledge and belief are represented via the modal
operators
K
i
and
B
i
, for some
m,
1
≤ i ≤ m
. Also
⊤
(“top”) and
⊥
(“bottom”) are
used to denote the constantly true proposition and the constantly false proposition,
respectively. The calculus for the system is the Hilbert calculus for classical propositional
logic with the following additional axioms and rules:
(CPC) φ, if φ is a tautology
(K
K
) K(φ → ψ) → (Kφ → Kψ)
(K
B
) B(φ → ψ) → (Bφ → Bψ)
(T) Kφ → φ
(D) Bφ → ¬B¬φ
(M) Kφ → Bφ
H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio INTERMATHS, 4(2), 28–37, December 2023 | 29
(MP) φ → ψ, φ / ψ
(RN
K
) ⊢ φ / ⊢ Kφ.
Formulas
K
a
φ
and
B
a
φ
are then read like “agent
a
knows that
φ
” and “agent
a
believes that φ”. We will not detach the agents a, b, ....
As usual, we write ⊢ φ to indicate that φ is a theorem.
So it holds
K
(
φ ∨ ¬φ
) or
K ⊤
, considering that
⊤
denotes a random theorem of
BEL.
If Γ
∪ {φ}
is a set of formulas, then Γ deduces
φ
, what is denoted by Γ
⊢ φ
, if there
is a finite sequence of formulas
φ
1
, ..., φ
n
such that
φ
n
=
φ
and, for every
φ
i
, 1
≤ i ≤ n
:
• φ
i
is an axiom, or
• φ
i
∈ Γ, or
• φ
i
is obtained from previous formulas of the sequence by some of the deduction
rules.
Let
F or
BEL
be the set of all propositional formulas and
V ar
BEL
be the set of all
propositional variables of the system BEL.
We don’t need to add
RN
B
as an inference rule because it can be derived from (
RN
K
)
and (M), see below:
Proposition 2.1. (i) [RN
B
] If ⊢ φ, then ⊢ Bφ;
(ii) If ⊢ φ → ψ, then ⊢ Kφ → Kψ;
(iii) If ⊢ φ → ψ, then ⊢ Bφ → Bψ.
Proof: (i)
1. φ Hypothesis
2. Kφ → Bφ M
3. Kφ RN
K
in 1
4. Bφ MP in 2 and 3.
(ii)
1. φ → ψ Hypothesis
2. K(φ → ψ) RN
K
in 1
3. K(φ → ψ) → (Kφ → Kψ) K
K
4. Kφ → Kψ MP in 2 and 3.
(iii) As (ii) using RN
B
.
Proposition 2.2. The following formulas are theorems:
(i) (Bφ ∧ Bψ) ↔ B(φ ∧ ψ);
(ii) (Kφ ∧ Kψ) ↔ K(φ ∧ ψ);
(iii) ¬K⊥.
Proof:
(i) (⇒)
30 | https://doi.org/10.22481/intermaths.v4i2.14133 H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio
1. φ → (ψ → (φ ∧ ψ)) CPC
2. B(φ → (ψ → (φ ∧ ψ))) RN
B
in 1
3. B(φ → (ψ → (φ ∧ ψ))) → (Bφ → B(ψ → (φ ∧ ψ))) K
B
4. Bφ → B(ψ → (φ ∧ ψ)) MP in 2 and 3
5. B(ψ → (φ ∧ ψ)) → (Bψ → B(φ ∧ ψ)) K
B
6. Bφ → (Bψ → B(φ ∧ ψ)) CPC in 4 and 5
7. (Bφ ∧ Bψ) → B(φ ∧ ψ) CPC in 6.
(⇐)
1. (φ ∧ ψ) → φ CPC
2. B(φ ∧ ψ) → Bφ 2.1 (iii) in 1
3. (φ ∧ ψ) → ψ CPC
4. B(φ ∧ ψ) → Bψ 2.1 (iii) in 3
5. B(φ ∧ ψ) → (Bφ ∧ Bψ) CPC in 2 and 4.
(ii) The proof is similar to the previous item using the axiom
K
K
and the rule
RN
K
.
(iii)
1. K⊥ → ⊥ T
2. ¬⊥ → ¬K⊥ Contrapositive in 1.
3. ⊤ → ¬K⊥ Substitution in 3.
4. ⊤ Theorem
5. ¬K⊥ MP in 3 and 4.
3 The epistemic algebra
Considering the system BEL, we introduce the correspondent algebra
B
,
BEL
-algebra
for short.
More information about algebraic models for logical systems can be seen in (Dunn
and Hardgree [4]), (Miraglia [8]), (Rasiowa and Sikorski [10]) and (Rasiowa [11]).
Definition 3.1. An epistemic algebra is a tuple
B
= (
B,
0
,
1
, ∼, ⊓, ⊔, k, ♭
) such that
(
B,
0
,
1
, , ∼, ⊓, ⊔
) is a Boolean algebra,
k
:
B → B
is an operator for the notion of
knowledge, and ♭ : B → B is an operator for the notion of belief, such that:
(a) ♭0 = 0
(b) ♭(a ⊓ b) = ♭a ⊓ ♭b
(c) k(a ⊓ b) = ka ⊓ kb
(d) k1 = 1
(e) ka ≤ a
(f) ka ≤ ♭a.
From items (a), (d) and (f) it is immediate that
k
0 = 0 and
♭
1 = 1. We do not have
any data on the order for a and ♭a.
Considering that
B
is Boolean, then
a → b
=
∼ a ⊔ b
and naturally the De Morgan
laws are valid.
Proposition 3.2. For any Boolean algebra, it holds the following equivalence:
a ≤ b ⇔ a → b = 1.
Proof: If
a ≤ b
, then
a ⊔ b
=
b ⇒ ∼ a ⊔
(
a ⊔ b
) =
∼ a ⊔ b ⇒
(
∼ a ⊔ a
)
⊔ b
=
∼ a ⊔ b ⇒
1 ⊔ b = a → b ⇒ 1 = a → b.
If
a → b
= 1, then
∼ a ⊔ b
= 1
⇒ a ⊓
(
∼ a ⊔ b
) =
a ⊓
1
⇒
(
a⊓ ∼ a
)
⊔
(
a ⊓ b
) =
a ⇒
0 ⊔ (a ⊓ b) = a ⇒ a ⊓ b = a ⇒ a ≤ b.
H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio INTERMATHS, 4(2), 28–37, December 2023 | 31
Proposition 3.3. If B = (B, 0, 1, ∼, ⊓, ⊔, k, ♭) is an epistemic algebra, then:
(i) a ≤ b ⇒ ka ≤ kb;
(ii) a ≤ b ⇒ ♭a ≤ ♭b.
Proof: (i) If a ≤ b, then a ⊓ b = a ⇒ k(a ⊓ b) = ka ⇒ ka ⊓ kb = ka ⇒ ka ≤ kb.
(ii) The same justification.
Proposition 3.4. If B = (B, 0, 1, ∼, ⊓, ⊔, k, ♭) is an epistemic algebra, then:
(i) a ⊓ z ≤ b ⇒ z ≤ a → b;
(ii) k(a → b) ≤ (ka → kb).
Proof: (i) If
a ⊓ z ≤ b
, then
z →
(
a ⊓ z
)
≤ z → b ⇒ a →
(
z →
(
a ⊓ z
))
≤ a →
(
z → b
).
As
a →
(
z →
(
a ⊓ z
)) = 1, then
a →
(
z → b
) = 1 and
z →
(
a → b
) = 1. Thus
z ≤ a → b.
(ii) For this item we use the items (c) of Definition 3.1 and (i) of Propositions 3.3 and
3.4.
a ⊓ b ≤ b ⇒
0
⊔
(
a ⊓ b
)
≤ b ⇒
(
a⊓ ∼ a
)
⊔
(
a ⊓ b
)
≤ b ⇒ a ⊓
(
∼ a ⊔ b
)
≤ b ⇒ k
(
a ⊓
(
∼
a⊔b
))
≤ kb ⇒ k
(
a
)
⊓k
(
∼ a⊔b
))
≤ kb ⇒ k
(
a
)
⊓k
(
a → b
))
≤ kb ⇒ k
(
a → b
)
≤
(
ka → kb
).
4 Soundness
Now, we need to show that the class of
BEL
-algebras is an appropriate model for
BEL.
Definition 4.1. A restrict valuation is a function
v
:
V ar
BEL
→ B
, that maps each
variable of BEL in an element of B.
Definition 4.2. A valuation is a function
v
:
F or
BEL
→ B
, that extends natural and
uniquely v as follows:
(i) v(p) = v(p)
(ii) v(¬φ) = ∼ v(φ)
(iii) v(φ ∧ ψ) = v(φ) ⊓ v(ψ)
(iv) v(φ → ψ) = ∼ v(φ) ⊔ v(ψ)
(v) v(Kφ) = k v(φ)
(vi) v(Bφ) = ♭ v(φ).
As usual, operator symbols in the left sides represent logical operators and those in
right sides represent algebraic operators.
Of course, v(φ ∨ ψ) = v(φ) ⊔ v(ψ).
Definition 4.3. A valuation
v
:
F or
BEL
→ B
is a model for a set Γ
⊆ F or
BEL
if
v(φ) = 1, for each formula φ ∈ Γ.
In particular, a valuation
v
:
F or
BEL
→ B
is a model for a formula
φ ∈ F or
BEL
when
v(φ) = 1.
Definition 4.4. A formula
φ ∈ F or
BEL
is valid in a
BEL
-algebra
B
if each valuation
v : F or
BEL
→ B is a model for φ.
32 | https://doi.org/10.22481/intermaths.v4i2.14133 H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio
Definition 4.5. A formula
φ BEL
-valid, what is denoted by
⊨ φ
, when it is valid in
every BEL-algebra.
The soundness theorem must show that the
BEL
-algebras are correct models for the
logic
BEL
, that is, that every theorem of
BEL
is valid in any
BEL
-algebra and that
the BEL-rules preserve the validity.
Since each
BEL
-algebra is a Boolean algebra, then we will use the Proposition 3.2
several times.
Theorem 4.6. (Weak Soundness) If ⊢ γ then ⊨ γ .
Proof: If
B
= (
B,
0
,
1
, ∼, ⊓, ⊔, k, ♭
) is a generic
BEL
-algebra, then we must show that
the axioms (
K
K
)
,
(
K
B
)
,
(
T
)
,
(
D
) and (
M
) are valid in
B
, and that the rule (
RN
K
),
according Definition 3.1, preserves validity in B. The Boolean part works as usually.
(T): By (e),
k v
(
φ
)
≤ v
(
φ
), so considering Proposition 2.2,
k v
(
φ
)
→ v
(
φ
) = 1 and
then v(Kφ → φ) = 1.
(M): By (f), k v(φ) ≤ ♭ v(φ), then k v(φ) → ♭ v(φ) = 1 and v(Kφ → Bφ) = 1.
(D):
v
(
Bφ → ¬B¬φ
) =
∼
(
♭ v
(
φ
)
⊓ ♭ ∼ v
(
φ
)). But by (b) and (a),
♭ v
(
φ
)
⊓ ♭ ∼
v(φ) = ♭ (v(φ)⊓ ∼ v(φ)) = ♭0 = 0. So, v(Bφ → ¬B¬φ) = 1.
(
K
K
): By Proposition 3.4 (ii), we have
v
(
K
(
φ → ψ
))
≤ v
(
Kφ → Kψ
) and hence
v(K(φ → ψ) → (Kφ → Kψ)) = 1.
(K
B
): The proof is analogous to (K
K
) by using Proposition 3.4.
(RN
K
): If v(φ) = 1, using (d) we have v(Kφ) = k v(φ) = k1 = 1.
Corollary 4.7. (Strong Soundness) If Γ ⊢ φ, then Γ ⊨ φ.
Proof: Suppose Γ ⊢ φ, and let B be an algebraic model such that B ⊨ Γ.
The proof is by induction on the the length of the deduction Γ ⊢ φ.
• If n = 1, then φ is an axiom (theorem) or belongs to Γ.
If it is an axiom, the result is given by the preceding theorem. If
φ
belongs to Γ, then
naturally Γ ⊨ φ.
• Let now n > 1, then φ is obtained by (MP) or (RN
K
).
But these two rules preserve the validity, then B ⊨ φ.
The next corollary shows that if a set of formulas has a model, then it is consistent.
Corollary 4.8. The logic BEL is consistent.
Proof: Suppose that
BEL
is not consistent. Then there is
φ ∈ F or
BEL
such that
⊢ φ
and ⊢ ¬φ.
So, by the soundness theorem,
φ
and
¬φ
are valid. Let
v
be a valuation in a
BEL
-
algebra with exactly two elements 2 =
{
0
,
1
}
. Since
φ
is valid, then
v
(
φ
) = 1 and
v(¬φ) = ∼ v(φ) = 0. But this contradicts the fact of ¬φ is valid.
H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio INTERMATHS, 4(2), 28–37, December 2023 | 33
5 Completeness
For the completeness we will use the Lindenbaum algebras.
Let (
F or
BEL
, ⊥, ⊤, ¬, ∧, →, K, B
) be the algebra of formulas of
BEL
, such that
⊥
and
⊤
are constants,
¬
,
K
and
B
are unary operators,
∧
and
→
are binary operators,
and as usual φ ∨ ψ =
df
¬(¬φ ∧ ¬ψ).
So, we define the Lindenbaum Algebra of BEL.
Definition 5.1. For Γ ⊆ F or
BEL
, we define the relation ≡ by:
φ ≡ ψ ⇔ Γ ⊢ φ → ψ and Γ ⊢ ψ → φ.
The relation
≡
, more than an equivalence relation, is a congruence, for by rule
RN
K
:
φ ≡ ψ ⇔ Γ ⊢ φ ↔ ψ ⇒ Γ ⊢ Kφ ↔ Kψ ⇔ Kφ ≡ Kψ.
Also, as we have ⊢ φ / ⊢ Bφ, then:
φ ≡ ψ ⇔ Γ ⊢ φ ↔ ψ ⇒ Γ ⊢ Bφ ↔ Bψ ⇔ Bφ ≡ Bψ.
If Γ
∪ {ψ} ⊆ F or
BEL
, we denote the equivalence class of
ψ
modulo
≡
and Γ by:
[ψ]
Γ
= {σ ∈ F or
BEL
: σ ≡ ψ} .
Definition 5.2. The Lindenbaum algebra of
BEL
, denoted by
B
Γ
(
BEL
), is the quotient
algebra defined by:
B
Γ
(BEL) = (F or
BEL
|
≡
, 0, 1, ¬
≡
, ∧
≡
, →
≡
, K
≡
, B
≡
), such that:
(i) 0 = [φ ∧ ¬φ] = [⊥]
(ii) 1 = [φ ∨ ¬φ] = [⊤]
(iii) ¬
≡
[φ] = [¬φ]
(iv) [φ] ∧
≡
[ψ] = [φ ∧ ψ]
(v) [φ] →
≡
[ψ] = [φ → ψ]
(vi) K
≡
[φ] = [Kφ]
(vii) B
≡
[φ] = [Bφ].
In general, we will not indicate the index ≡ in the operations.
When Γ = ∅ we denote the Lindenbaum algebra of BEL by B(BEL).
Proposition 5.3. In B
Γ
(BEL) it holds: [φ] ≤ [ψ] ⇔ Γ ⊢ φ → ψ.
Proof: [
φ
]
≤
[
ψ
]
⇔
[
φ
]
∧
[
ψ
] = [
φ
]
⇔
[
φ ∧ ψ
] = [
φ
]
⇔
Γ
⊢ φ ∧ ψ ↔ φ ⇔
Γ
⊢ φ → ψ
.
Proposition 5.4. The algebra B
Γ
(BEL) is a BEL-algebra.
Proof: (a) From axiom D,
Bφ → ¬B¬φ
, we have
¬
(
Bφ ∧ B¬φ
) iff
¬
(
B
(
φ ∧ ¬φ
)). Then
[¬(B(φ ∧ ¬φ))] = 1 and [B(φ ∧ ¬φ)] = 0. Then, [B⊥] = 0 ⇒ B[⊥] = 0.
34 | https://doi.org/10.22481/intermaths.v4i2.14133 H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio
(b) From Proposition 2.2 (ii),
B
(
φ ∧ ψ
)
↔ Bφ ∧ Bψ
. Then [
B
(
φ ∧ ψ
)] = [
Bφ ∧ Bψ
]
and [B(φ ∧ ψ)] = [Bφ] ∧ [Bψ].
(c) From Proposition 2.2 (i),
K
(
φ ∧ ψ
)
↔ Kφ ∧ Kψ
. Then [
K
(
φ ∧ ψ
)] = [
Kφ ∧ Kψ
]
and [K(φ ∧ ψ)] = [Kφ] ∧ [Kψ].
(d) As ⊢ K(φ ∨ ¬φ), then [K⊤] = 1 ⇒ K[⊤] = 1.
(e) From (T) Kφ → φ, then [Kφ] ≤ [φ] ⇒ K[φ] ≤ [φ].
(f) From (M) Kφ → Bφ, we have [Kφ] ≤ [Bφ] ⇒ K[φ] ≤ B[φ].
Definition 5.5. The algebra B
Γ
(BEL) is the canonical model of Γ ⊆ F or
BEL
.
We denote a valuation on the canonical model by
v
0
:
F or
BEL
→ B
Γ
(
BEL
). When
Γ = ∅ we have v
0
: F or
BEL
→ B(BEL).
Corollary 5.6. Let
φ ∈ F or
BEL
and
B
(
BEL
) be the canonical model for
BEL
. If
φ
is
a theorem of BEL, then [φ] = 1, and if φ is irrefutable, then [φ] ̸= 0.
Proof: If
⊢ φ
, considering that
B
(
BEL
) is a BEL-algebra, by the soundness theorem we
have [φ] = 1.
Now, φ is irrefutable iff ⊬ ¬φ iff [¬φ] ̸= 1 iff ¬[φ] ̸= 1 iff [φ] ̸= 0.
From the preceding proposition and the definitions of
0
and
1
in the Lindenbaum
algebra it results that for each formula φ:
[φ] = 1 iff ⊢ φ and
[φ] = 0 iff ⊢ ¬φ.
Theorem 5.7. For φ ∈ F or
BEL
, the following assertions are equivalent:
(i) ⊢ φ;
(ii) ⊨ φ;
(iii) φ is valid in every BEL-algebra of sets B = (B, ∅,
C
, ∩, ∪, K, B);
(iv) v
0
(φ) = 1, for the canonical valuation in A(BEL).
Proof: (i) ⇒ (ii): from the Soundness Theorem.
(ii) ⇒ (iii): is immediate.
(
iii
)
⇒
(
iv
): as every BEL-algebra is isomorphic to a BEL-algebra of sets B =
(B, ∅,
C
, ∩, ∪, K, B) and A(BEL) is a BEL-algebra, the result follows.
(
iv
)
⇒
(
i
): if
φ ∈ F or
BEL
and it is not derivable in BEL, by Corollary 5.6, [
φ
]
̸
= 1
in A(BEL) and then v
0
(φ) ̸= 1. Therefore φ is not a valid formula.
Corollary 5.8. (Completeness) For each
φ ∈ F or
BEL
, if
φ
is valid, then
φ
is derivable
in BEL.
The next result shows the strong adequacy of the algebraic models given by BEL-
algebras for the logic system BEL.
As usual, Γ ⊨ φ denotes that every model of Γ is a model of φ too.
Definition 5.9. A model v : F or
BEL
→ A is strongly adequate for Γ when:
Γ ⊢ φ ⇔ Γ ⊨ φ.
H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio INTERMATHS, 4(2), 28–37, December 2023 | 35
Proposition 5.10. If Γ
⊆ F or
BEL
is consistent, then the canonical valuation is a
correct model for Γ.
Proof: Considering the canonical valuation
v
0
:
F or
BEL
→ A
(
BEL
), that maps
v
0
(
φ
) =
[
φ
], by Corollary 5.6 and Proposition 4.7,
v
0
(
φ
) = 1 iff Γ
⊢ φ
. Therefore we have that
v
0
is a correct model for Γ.
Theorem 5.11. For Γ ⊆ F or
BEL
, the following conditions are equivalent:
(i) Γ is consistent;
(ii) there is a correct model for Γ;
(iii) there is a correct model for Γ in a BEL-algebra of sets B = (
B, ∅,
C
, ∩, ∪, K, B
);
(iv) there is a model for Γ.
Proof: (i) ⇒ (ii) As in the previous proposition.
(ii) ⇒ (iii) As A(BEL) is a BEL-algebra and every BEL-algebra is isomorphic to a
BEL-algebra of sets B = (B, ∅,
C
, ∩, ∪, K, B), then the result follows.
(iii) ⇒ (iv) Immediate.
(vi) ⇒ (i) It results directly by Corollary 4.8.
Corollary 5.12. (Strong adequacy) Let Γ
∪ {φ} ⊆ F or
BEL
. If Γ is consistent, the
following conditions are equivalent:
(i) Γ ⊢ φ;
(ii) Γ ⊨ φ;
(iii) every model of Γ in a BEL-algebra of sets B = (
B, ∅,
C
, ∩, ∪, K, B
) is a model
for φ;
(iv) v
0
(φ) = 1 for the canonical valuation v
0
.
This way, we have shown that the BEL-algebras are adequate models for this basic
epistemic logic. If we need some more complex epistemic system to formalize some
situation, we think that we can use this algebra as an initial point and get other algebraic
models too.
6 Final considerations
Among the several reasons for justifying the aim of any algebraic model for a logic,
or an epistemic logic, is the simplicity in the use of algebraic structures, as well as the
intuitive use of algebraic notions for problems involved with the present-day technologies
(Meyer and Van der Hoek [7]), (Halpern and Moses [6]).
We chose a basic system, but we consider that we can extend the method to several
other epistemic systems. Maybe we can think about what more laws can be included
into BEL and preserve algebraic models.
In another direction, we could study more relations between the two epistemic
operators and other operators defined from these.
36 | https://doi.org/10.22481/intermaths.v4i2.14133 H. de Araujo Feitosa, M. Matulovic, A. C. de J. Golzio
Acknowledgments. This work was supported by the Brazilian National Council for
Scientific and Technological Development - CNPq [grant numbers: 421782/2016-1] and The
São Paulo Research Foundation - FAPESP [grant numbers: 2019/08442-9].
Disclosure statement. The authors declare no conflict of interest in the writing of the
manuscript, or in the decision to publish the results.
ORCID
Hércules de Araujo Feitosa https://orcid.org/0000-0003-0023-4192
Mariana Matulovic https://orcid.org/0000-0001-6626-4621
Ana Claudia de J. Golzio https://orcid.org/0000-0003-3185-7552
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