Technion - Computer Science Department - Tehnical Report CS-2007-04 - 2007
Non-monotonic modal logic of belief
Michael Kaminski
Michael Tiomkin
Department of Computer Science
Technion – Israel Institute of Technology
Haifa 32000
Israel
Abstract
We propose an alternative non-monotonic modal formalism called
non-monotonic modal logic of belief. It is based on replacing the classical fixpoint equation E = T hS (A ∪ {M ϕ : E 6`S ¬ϕ}) with the
belief fixpoint equation E = T hS (A ∪ {M ϕ : E 6`S ¬M ϕ}). The solutions of the belief fixpoint equation, called belief S-expansions, are
tightly related to the logic of belief KD45. We show interpretation
of default logic in non-monotonic modal logic of belief and present
minimal model semantics for belief expansions.
Keywords: Non-monotonic modal logic; Logic of belief; Default logic;
Minimal model semantics
1
Introduction
Conclusions by failure to prove the opposite are frequently used in reasoning
about an incompletely specified world. This naturally leads to logics for nonmonotonic reasoning where introducing new facts can invalidate previously
made conclusions. Non-monotonic modal logics known from the literature
were intended to simulate the process of human reasoning by providing a
formalism for deriving consistent conclusions from an incomplete description
of the world. The language of non-monotonic modal logics contains a modal
connective L (known or believed) and its dual M (consistent or compatible),
Technion - Computer Science Department - Tehnical Report CS-2007-04 - 2007
and the logics themselves are described by expansions which are fixpoints of
a monotonic provability operator. Loosely speaking, expansions are obtained
by augmenting the underlying modal logic S with the “inference rule” of the
form
6`S ¬ϕ
(1)
Mϕ
called possibilitation, see [9], or negative introspection, see [8, p. 224]. We
refer the reader to [8] for a comprehensive study of various formalisms of
non-monotonic reasoning.
The first definitions of non-monotonic and default logic (cf. [13, 10, 9])
allowed existence of inconsistent expansions. Later works considered only
consistent expansions (cf. [8, Definition 9.2]). In this paper we deal only
with consistent expansions. In particular, we prefer to say that there are no
expansions to saying that there exists only one inconsistent expansion.
One of the major features of non-monotonic expansions is that they are
theories of Kripke interpretations with the total accessibility relation.1 Consequently, if we think of the knowledge set of an agent as the set of formulas of
the form Lϕ which are true in every world of a Kripke interpretation with a
total accessibility relation, in non-monotonic expansions modal connective L
should naturally be interpreted as known, see [20]. However, in many works
(cf. [16, 11, 12, 7]) L is interpreted as believed, but non-monotonic expansions do not address the belief set of an agent that consists of the formulas of
the form Lϕ which are true in every world of a Kripke interpretation whose
accessibility relation is not necessarily total. This even happens in the case
when the underlying modal logic is contained in the logic of belief KD45. For
example, a natural belief theory {Lψ}, where both ψ and ¬ψ are satisfiable
modal-free formulas, has no KD45-expansion, see [7, Example 2.2].
This paper introduces a different non-monotonic modal formalism – nonmonotonic modal logic of belief, where belief theories of the form {Lψ} can
be consistently expanded to fixpoints of the appropriate operators. Nonmonotonic modal logic of belief replaces the negative introspection rule (1)
with
6`S ¬M ϕ
,
(2)
Mϕ
cf. [17, default rule (1)]. Accordingly, we call the latter fixpoints belief Sexpansions. Note that the form of (2) resembles Reiter’s normal default
1
In particular, they include modal logic S5.
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without prerequisites ([13]).
We show that non-monotonic modal logic of belief is an extension of the
classical non-monotonic modal logic. Namely, each ordinary expansion is
also a belief expansion. Moreover, if the underlying modal logic S contains
T then each belief S-expansion is also an ordinary expansion. However, if S
is the logic of belief KD45 or weaker, belief S-expansions are tightly related
to KD45, instead of S5 for the ordinary expansions, cf. [16, Note 5]. In particular, belief S-expansions are not necessarily the ordinary ones, e.g., there
exists a belief KD45-expansion for a KD45-consistent {Lψ}, see Corollary 68
in Section 3.3. Still, if a belief expansion includes T then it is also an ordinary
expansion, see Proposition 52 in Section 3.1.
In this paper we establish some basic properties of non-monotonic modal
logic of belief, show that Truszczyński’s translation of Reiter’s default logic
to non-monotonic modal logic ([18, 19]) also applies to non-monotonic modal
logic of belief, and present a semantical description of belief expansions that
is similar to the Schwarz semantics of non-monotonic modal logic ([15]).
The paper is organized as follows. In the next section we recall the definitions of propositional non-monotonic modal logic and propositional default
logic and list some of their basic properties. Section 3 is the “belief counterpart” of Section 2. We first give the formal definition of propositional
non-monotonic modal logic of belief and derive some of its immediate consequences. Then we translate Reiter’s default logic into non-monotonic modal
logic of belief using Truszczyński’s translation. We end Section 3 with the
minimal model semantics of non-monotonic modal logic of belief. Finally,
Section 4 contains some concluding remarks.
2
Propositional non-monotonic logics
In this section we recall the definitions of propositional non-monotonic modal
logic and propositional default logic and list some of their basic properties.
2.1
Propositional modal logic
We start with the classical propositional logic that contains propositional
variables and only two classical propositional connectives ⊥ (a logical constant falsity) and ⊃ (implication). Connectives > (truth), ¬ (negation), ∧
(conjunction), ∨ (disjunction), and ≡ (equivalence) are defined in the usual
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manner, e.g., ¬ϕ is ϕ ⊃ ⊥. We assume a standard set of axioms and the
modus ponens rule of inference.
The language of propositional modal logic is obtained from the language
of classical propositional logic by extending it with a unary modal connective
L (believed). As usual, the dual connective M (consistent) is defined by
¬L¬. Formulas not containing L are called ground or modal-free. The set of
all propositional modal formulas will be denoted by F m and the set of all
propositional ground formulas will be denoted by GF m. Also, for a set of
formulas X we denote the sets of formulas {M ϕ : ϕ ∈ X} and {Lϕ : ϕ ∈ X}
by M X and LX, respectively.
The weakest normal modal logic K results from the classical propositional
logic by adding the inference rule
NEC ϕ ` Lϕ
called necessitation and the axiom scheme
k L(ϕ ⊃ ψ) ⊃ (Lϕ ⊃ Lψ).
The “classical” modal logics are obtained by adding to K all instances of
some axiom schemes, e.g.,
t Lϕ ⊃ ϕ
d M>
4 Lϕ ⊃ LLϕ
f (ϕ ∧ M Lψ) ⊃ L(M ϕ ∨ ψ)
5 M ϕ ⊃ LM ϕ
Adding t to K results in T, adding 4 to T results in S4, and adding f
to S4 results in S4F, etc., see [8, page 197]. Modal logics including K are
called normal, see [3, Definition 1.2.8, p. 377]. In this paper we assume that
all modal logics under consideration include K.
For a modal logic S and a set of formulas A, called (proper) axioms, we
write A `S ϕ if there exists a sequence of formulas ψ1 , ψ2 , . . ., ψn = ϕ such
that each ψi is either an axiom from S, or belongs to A, or is obtained from
some of the formulas ψ1 , ψ2 , . . . , ψi−1 by modus ponens or necessitation. We
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define the (monotonic) theory of A, denoted T hS (A), as T hS (A) = {ϕ : A `S
ϕ}. The classical propositional consequence (i.e., without NEC and modal
axioms) is denoted by just `.
The Kripke semantics of propositional modal logics is defined as follows.
A Kripke interpretation is a triple M = hU, R, Ii, where U is a non-empty
set of possible worlds, R ⊆ U × U is an accessibility relation on U , and I is
an assignment to each world in U of a set of propositional variables. In what
follows we identify an ordinary propositional interpretation with the set of all
propositional variables of the underlying language it satisfies.
Definition 1 Let M = hU, R, Ii be a Kripke interpretation, u ∈ U , and let
ϕ ∈ F m. We say that the pair (M, u) satisfies ϕ, denoted (M, u) |= ϕ if the
following holds.
• If ϕ is a propositional variable p, then (M, u) |= ϕ if and only if p ∈
I(u);
• (M, u) 6|= ⊥;
• (M, u) |= ϕ ⊃ ψ if and only if (M, u) 6|= ϕ or (M, u) |= ψ;
• (M, u) |= Lϕ if and only if for every v such that (u, v) ∈ R, (M, v) |= ϕ.
We say that a formula ϕ is valid in a Kripke interpretation M, denoted
M |= ϕ if and only if for every u ∈ U , (M, u) |= ϕ, and we say that a set of
formulas X is valid in M, denoted M |= X, if and only if M |= ϕ for every
ϕ ∈ X. The set of all formulas valid in M is called the theory of M and is
denoted T h(M). That is, T h(M) = {ϕ : M |= ϕ}.
Remark 2 Let M = hU, R, Ii be a Kripke interpretation and let u ∈ U .
Then Mu = hU u , Ru , I u i denotes the Kripke interpretation whose set of
worlds consists of the worlds of U which are reachable from u by means
of R, and Ru and I u are the restrictions of R and I on U u , respectively. It
is well-known that for a formula ϕ and u ∈ U , (M, u) |= ϕ if and only if
(Mu , u) |= ϕ.
Remark 3 Let ϕ be a modal formula, M = hU, R, Ii be a Kripke interpretation, and let P contain all propositional variables of ϕ. Let M|P =
hU, R, I|P i, where I|P is the restriction of I to P. Then for every u ∈ U ,
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(M, u) |= ϕ if and only if (M|P , u) |= ϕ. That is, validity of ϕ in M depends
only on the assignment to the propositional variables which appear in ϕ.
The Kripke semantics is sound and complete for K. That is, X `K ϕ if
and only if ϕ is valid in all Kripke interpretations in which X is valid. Kripke
interpretations with a reflexive accessibility relation are sound and complete
for T, Kripke interpretations with a reflexive and transitive accessibility relation are sound and complete for S4, and Kripke interpretations where the
accessibility relation is an equivalence relation are sound and complete for
S5, see [8, Corollary 7.51, p. 214].
For the minimal model semantics of non-monotonic modal logic and nonmonotonic modal logic of belief we shall also need the following definition.
Definition 4 Let C be a class of Kripke interpretations and let S be a modal
logic. We say that S is characterized by C if the following holds.
• For every set of formulas X and every formula ϕ, X `S ϕ if and only
if for every Kripke interpretation M ∈ C, M |= X implies M |= ϕ.
Definition 5 Kripke interpretations of the form hU, U × U, Ii are called S5models (or clusters) and denoted hU, Ii.
It can be readily seen that S5 is characterized by the class of all S5-models
(e.g., see [8, Theorem 7.52, p. 216]).
Remark 6 We call an S5-model hU, Ii reduced if I(u1 ) 6= I(u2 ) for all different u1 , u2 ∈ U . Thus, we can identify a reduced S5-model hU, Ii with the set
of propositional interpretations I(U ) = {I(u) : u ∈ U }. It easily follows from
the definition that S5 is characterized by the class of all reduced S5-models.
Definition 7 Kripke interpretations hU, R, Ii such that R = U × Uc , where
Uc is a non-empty subset of U are called KD45-models and denoted hU, Uc , Ii.
For a KD45-model M we denote by Mc the corresponding S5-model hUc , Ic i,
where Ic is the restriction of I onto Uc .
It is well-known that KD45 is characterized by the class of all KD45models, e.g., see [8, Theorem 7.52, p. 216]. Note that for a KD45-model M,
a formula from M F m ∪ LF m is valid in M if and only if it is valid in Mc .
Finally, we call a KD45-model hU, Uc , Ii reduced if I(u1 ) 6= I(u2 ) for
all different u1 , u2 ∈ U . Like in the case of reduced S5-models, we can
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identify a reduced KD45-model hU, Uc , Ii with a pair of sets of propositional
interpretations hI(U ), I(Uc )i, cf. Remark 6.
Definition 8 Let M0 = hU 0 , Uc0 , I 0 i and M00 = hU 00 , Uc00 , I 00 i be KD45-models.
We say that the worlds u0 ∈ U 0 and u00 ∈ U 00 are ground equivalent, denoted
u0 ≈ u00 , if I 0 (u0 ) = I 00 (u00 ). We say that the models M0 and M00 are ground
equivalent, denoted M0 ≈ M00 , if for every world u0 ∈ Uc0 there exists a world
u00 ∈ Uc00 such that u0 ≈ u00 and vice versa.
Proposition 9 Let M0 = hU 0 , Uc0 , I 0 i and M00 = hU 00 , Uc00 , I 00 i be KD45models, u0 ∈ U 0 , and u00 ∈ U 00 . If M0 ≈ M00 and u0 ≈ u00 , then for every
formula ϕ, (M0 , u0 ) |= ϕ if and only if (M00 , u00 ) |= ϕ.
Proof The proof is by induction on the complexity of ϕ. The basis is trivial,
because satisfiability of a modal-free formula depends only on I 0 (u0 ) and
I 00 (u00 ) which are the same.
For the induction step, the case of the ordinary propositional connective ⊃
is immediate. Let ϕ be of the form Lψ. It suffices to show that (M0 , u0 ) |= Lψ
implies (M00 , u00 ) |= Lψ. So, let (M0 , u0 ) |= Lψ. Then, for every w0 ∈ Uc0 ,
(M0 , w) |= ψ. It follows from the ground equivalence of the models and
the induction hypothesis that for every w00 ∈ Uc00 , (M00 , w00 ) |= ψ. That is,
(M00 , u00 ) |= Lψ.
Corollary 10 Modal logic KD45 is characterized by the class of all reduced
KD45-models.
Proof Let M = hU, Uc , Ii be a KD45-model. Consider a KD45-model Mr =
hU r , Ucr , I r i, where U r = I(U ), Ucr = I(Uc ), and I r (u) = u, for all u ∈ U r . By
definition, Mr is reduced, and for all u ∈ U , u ≈ I(u). Therefore, M ≈ Mr ,
and by Proposition 9, M and Mr validate the same formulas.
The following two simple lemmas will be used in the sequel.
Lemma 11 Let M0 = hU 0 , I 0 i and M00 = hU 00 , I 00 i be S5-models such that
{M ϕ : M0 |= M ϕ} ⊆ {M ϕ : M00 |= M ϕ}. Then T h(M0 ) = T h(M00 ).
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Proof If ϕ ∈ T h(M00 ) then M ¬ϕ 6∈ T h(M00 ). Hence M ¬ϕ 6∈ T h(M0 ),
implying ϕ ∈ T h(M0 ).
For the converse inclusion, if ϕ ∈ T h(M0 ) then M Lϕ ∈ T h(M0 ). Hence
M Lϕ ∈ T h(M00 ), implying ϕ ∈ T h(M00 ).
Lemma 12 Let M0 = hU 0 , Uc0 , I 0 i and M00 = hU 00 , Uc00 , I 00 i be KD45-models
such that
1. T h(M0 ) ∩ GF m = T h(M00 ) ∩ GF m and
2. T h(M0 ) ∩ M F m = T h(M00 ) ∩ M F m.
Then T h(M0 ) = T h(M00 ).
Proof It suffices to show that M0 |= ϕ implies M00 |= ϕ.
Let M0 |= ϕ. By [8, Theorem 7.3, p. 191], each formula is equivalent to a
conjunction of formulas of the form
Lα1 ∨ · · · ∨ Lαm ∨ M β ∨ γ,
(3)
where α1 , . . . , αm , β ∈ F m and γ ∈ GF m. Therefore, we may assume that
ϕ itself is of the form (3).
If for some u ∈ U 0 and i = 1, 2, . . . , m, (M0 , u) |= Lαi or (M0 , u) |= M β,
then M00 |= ϕ follows from prerequisite 2 of the lemma, otherwise we have
M0 |= γ, and M00 |= ϕ follows from prerequisite 1.
2.2
Propositional non-monotonic modal logic
Here we recall the definition of propositional non-monotonic modal logics
based on the McDermott and Doyle fixpoint equation ([10]). Definition 13
below is a relativization to a modal logic S of McDermott’s original definition ([9]) that only dealt with the classical modal logics T, S4, or S5. Note
that we deal only with consistent expansions. A general form of McDermott’s
definition is as follows.
Definition 13 ([8, Definition 9.2, p. 252]) Let A be a set of modal formulas
(axioms) and let S be a modal logic. An S-consistent set of formulas E is
called an S-expansion for A if
E = T hS (A ∪ {M ϕ : E 6`S ¬ϕ}).
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That is, S-expansions for A can be considered as the “deductive closures”
of A in S extended with (1).
Some (of the most important) properties of expansions are listed below.
Lemma 14 (Cf. [17, Proposition 2].) An S-consistent set of formulas E is
an S-expansion for A if and only if there exists a set of formulas D such that
the following two conditions are satisfied.
(i) For each formula ψ 6∈ D, A ∪ {M ϕ : ϕ ∈ D} `S ¬ψ, and
(ii) E = T hS (A ∪ {M ϕ : ϕ ∈ D}).
Proposition 15 ([9, Theorem 1]) Let A be a set of formulas, S be a modal
logic, and let E be an S-expansion for A. Then S5 ⊆ E.
Actually, expansions possess a much stronger property than just including
S5. To state it we need the following definition.
Definition 16 ([16], [11]) A propositionally consistent set of formulas X is
called stable if it satisfies the following three conditions.
1. X is closed under classical propositional consequence `.
2. For every formula ϕ, if ϕ ∈ X then Lϕ ∈ X.
3. For every formula ϕ, if ϕ 6∈ X then ¬Lϕ ∈ X.
Proposition 17 ([12], see also [8, Theorem 8.10, p. 228, and Theorem 8.12,
p. 229].) A set of formulas is stable if and only if it is the theory of an
S5-model.
Proposition 18 ([8, Theorem 9.4, p. 253]) Let A be a set of formulas, S be
a modal logic, and let E be an S-expansion for A. Then E is stable.
Proposition 19 below immediately follows from Propositions 17 and 18.
Proposition 19 Let A be a set of formulas, S be a modal logic, and let E
be an S-expansion for A. Then E is the theory of an S5-model.2
2
Consequently, E is the theory of a reduced S5-model, see Remark 6.
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Proposition 20 ([8, Theorem 9.6, p. 256]) Let A be a set of formulas and
let S1 ⊆ S2 ⊆ S5. Then each S1 -expansion for A is also an S2 -expansion for
A.
Proposition 21 ([9, Theorem 12], see also [8, Corollary 9.11, p. 258].) Let ψ
and A be a modal formula and a set of axioms, respectively. Then ψ belongs
to all S5-expansions for A if and only if A `S5 ψ.
Example 22 ([7, Example 2.2]) Let ψ ∈ GF m be such that 6` ψ and 6` ¬ψ.
Then for S ⊆ KD45, {Lψ} has no S-expansion.
2.3
Non-monotonic default modal logic
Below we recall the definition of non-monotonic default modal logic ([17])
and its relationship to non-monotonic modal logic from the previous section.
Definition 23 ([17]) Let A and D be sets of formulas and let S be a modal
logic. An S-consistent set of formulas E is called a default D,S-expansion for
A if
E = T hS (A ∪ {ϕ ∈ D : E 6`S ¬ϕ}).
Proposition 24 ([17, Remark 1]) Let A be a set of formulas and let S be
a modal logic such that S ∪ A includes T. Then E is a default M F m,Sexpansion3 for A if and only if it is an S-expansion for A.
2.4
Propositional default logic and its interpretation
in non-monotonic modal logics
This section contains Truszczyński’s interpretation of default logic in nonmonotonic modal logics, see [18, 19].
Default logic deals with inference rules called defaults which are expressions of the form
α : β1 , . . . , βm
,
γ
3
That is, D = M F m.
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where m ≥ 0, and α, β1 , . . . , βm and γ are propositional formulas.4 The
formula α is called the prerequisite, the formulas β1 , . . . , βm are called the
justifications, and the formula γ is called the conclusion of the default rule.
Roughly speaking, the intuitive meaning of a default is as follows. If α is
believed and the βi s are consistent with one’s beliefs, then one is permitted
to deduce γ and add it to the “belief set.” A default theory is a pair (D, A),
where D is a set of defaults and A is a set of propositional formulas (axioms).
Note that similarly to Definition 13 and contrary to the definitions in
[13, 8], in Definition 25 below we restrict ourselves to consistent default extensions.
Definition 25 ([13]) Let (D, A) be a default theory. For any set of formulas
S let Γ(D,A) (S) be the smallest set of formulas B (beliefs) that satisfies the
following three properties.
D1. A ⊆ B.
D2. B = {ϕ : B ` ϕ}, i.e., B is deductively closed.
D3. If
α : β1 , . . . , βm
∈ D, α ∈ B, and ¬β1 , . . . , ¬βm 6∈ S, then γ ∈ B.
γ
A consistent set of formulas E is an extension for (D, A) if Γ(D,A) (E) = E,
i.e., if E is a fixpoint of the operator Γ(D,A) .
Truszczyński’s interpretation of default logic in non-monotonic modal
α : β1 , . . . , βm
logic is as follows. For a default d =
we denote by τ (d) the
γ
Vm
modal formula (Lα ∧ i=1 LM βi ) ⊃ γ.
Theorem 26 ([18, Theorem 3.1] and [19, Theorem 4.1], see also [8, Theorem 12.1, p. 353].) Let (D, A) be a default theory and let S ⊆ S4F. A
set of formulas E is an extension for (D, A) if and only if there exists an
S-expansion E τ for A ∪ {τ (d) : d ∈ D} such that E = E τ ∩ GF m.
α : M β1 , . . . , M βm
.
Actually, in [13], [8], and [18] the default rules appear in the form
γ
In this paper we employ a bit different notation to avoid a possible confusion when the
modal connective M appears in default rules.
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2.5
Minimal model semantics of non-monotonic modal
logic
This section contains Schwarz’s semantics of propositional non-monotonic
modal logics ([15]). It is based on Definitions 27–31 below. Note that the
corresponding definitions in [15] are different from those in [8], and in this
paper we adopt the definitions of Marek and Truszczyński from [8].
Definition 27 ([15]) Let M = hU, R, Ii and M0 = hU 0 , R0 , I 0 i be two Kripke
interpretations such that U ∩U 0 = ∅. The concatenation of M to M0 , denoted
M0 M, is a Kripke interpretation M00 = hU 00 , R00 , I 00 i, where
• U 00 = U 0 ∪ U ,
• R00 = R0 ∪ (U 0 × U ) ∪ R, and
I(u) if u ∈ U
00
00
• I is defined by I (u) =
.
I 0 (u) if u ∈ U 0
Definition 28 (Cf. [15]) Let M = hU, Ii and M0 = hU 0 , R, I 0 i be an S5model and a Kripke interpretation, respectively. We say that M0 M is
preferred over M, denoted M0 M < M, if there is a world u0 ∈ U 0 and a
(modal-free) formula θ ∈ GF m such that I 0 (u0 ) |= θ, but M |= ¬θ.
Remark 29 It immediately follows from the definition that M0 M < M
if and only if
T h(M0 M) ∩ GF m ⊂ T h(M) ∩ GF m.
Definition 30 ([15]) Let C be a class of Kripke interpretations and A be
a set of modal formulas. An S5-model M ∈ C is called C-minimal for A if
M |= A and for every Kripke interpretation M0 such that M0 M ∈ C and
M0 M |= A, M0 M 6< M.
Definition 31 ([8]) A class C of Kripke interpretations is called cluster
closed if it contains all S5-models and at least one of the two following conditions is satisfied.
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1. For every S5-model M and every Kripke interpretation M0 ∈ C, M0 M ∈ C.
2. Every Kripke interpretation in C is either an S5-model, or of the form
M0 M. Moreover, for every M0 M ∈ C, and every S5-model N,
M0 N ∈ C.5
At last, we have arrived to a connection between C-minimality and Sexpansions.
Theorem 32 [8, Corollary 9.22, p. 266], see also ([15, Theorem 3.1].) Let S
be a modal logic contained in S5 and characterized by a cluster closed class C
of Kripke interpretations, and let A be a set of formulas. A set of formulas
E is an S-expansion for A if and only if there exists an S5-model M such
that M is C-minimal for A and E = T h(M).
Remark 33 It is easy to see that we can remove the condition M ∈ C from
Definition 30 and the condition that the class C contains all S5-models from
Definition 31, without affecting Theorem 32.
Also note that the conditions that S is contained in S5 and M is an S5model are not needed in this theorem. Every modal logic that is characterized
by a cluster closed class is a subset of S5, because every S5-model is a righthand part of some concatenation that belongs to C. Also, by Definition 30,
every C-minimal model is an S5-model.
2.6
Notes on Schwarz’s semantics
As it was pointed out in [6], Schwarz’s semantics does not adequately extend
to first-order non-monotonic modal logic, where a workable solution can be
based on the notion of weak preference. In first-order logic the weak preference is indeed weaker than that introduced in Definition 28, but it is still
equivalent to Schwarz’s preference when restricted to propositional logic.
Definition 34 ([6]) Let M = hU, Ii and M0 = hU 0 , R0 , I 0 i be an S5-model
and a Kripke interpretation, respectively. We say that M0 M is weakly
preferred over M, denoted M0 M <w M, if there is a world u0 ∈ U 0 and a
5
In [2] classes of Kripke interpretation satisfying property 2 of the definition are called
cluster decomposable.
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formula θ ∈ F m (not necessarily modal-free) such that (M0 M, u0 ) |= θ,
but M |= ¬θ.
It follows that M0 M <w M if and only if T h(M0 M) ⊂ T h(M), cf.
Remark 29.
Proposition 35 ([6, Proposition 4.1]) Let M and M0 be an S5-model and
a Kripke interpretation, respectively. Then M0 M < M if and only if
M0 M <w M.
In Section 3.3 we introduce a similar notion of the preference and weak
preference relations with respect to KD45-models. Like in the case of S5models they are equivalent in the propositional case. We use the weak preference relation for the proof of the belief counterpart of Theorem 32.
3
Non-monotonic modal logic of belief
This section is the “belief counterpart” of Section 2. It consists of three
parts. The first part contains the definition of belief expansions and its
consequences, and the second and the third deal with interpretation of default
logic in non-monotonic modal logics of belief and semantics of non-monotonic
modal logic of belief, respectively.
3.1
Belief expansions and their properties
We start with the definition of belief expansions.
Definition 36 (Cf. Definitions 13 and 23.) Let S be a modal logic and A
be set of formulas (axioms). An S-consistent set of formulas E is called a
belief S-expansion for A if
E = T hS (A ∪ {M ϕ : E 6`S ¬M ϕ}).
That is, inference rule (1) is replaced with inference rule (2).
Remark 37 (Cf. Proposition 24.) It immediately follows from the definition that belief S-expansions are default M F m,S-expansions. Thus, by
Proposition 24, if S ∪ A includes T then each belief S-expansion for A is an
S-expansion for A and vice versa.
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Lemma 38 ([17, Proposition 2], cf. Lemma 14.) An S-consistent set of
formulas E is a belief S-expansion for A if and only if there exists a set of
formulas D such that the following two conditions are satisfied.
(i) For each formula ψ 6∈ D, A ∪ {M ϕ : ϕ ∈ D} `S ¬M ψ, and
(ii) E = T hS (A ∪ {M ϕ : ϕ ∈ D}).
Proof For the proof we just observe that the proof of [17, Proposition 2]
does not use the assumption that S contains T made throughout [17].
Corollary 39 (Cf. Proposition 21.) Let ψ and A be a modal formula and
a set of axioms, respectively. Then ψ belongs to all belief KD45-expansions
for A if and only if A `KD45 ψ.
Proof The “if” part of the corollary is immediate, and for the proof of the
“only if” part it suffices to show that for each formula ψ 6∈ T hKD45 (A) there
exists a belief KD45-expansion E of A such that ψ 6∈ E.
So, assume ψ 6∈ T hKD45 (A) and let M be a KD45-model in which A is
valid, but ψ is not. Let D = {ϕ : M |= M ϕ}. We shall prove that D
satisfies condition (i) of Lemma 38 for A and KD45. Let ϕ 6∈ D. Then
M |= ¬M ϕ, implying M |= M L¬ϕ. Therefore, L¬ϕ ∈ D. Hence, by 5,
A ∪ M D `KD45 L¬ϕ which, by definition, is ¬M ϕ.
Now, let E = T hKD45 (A ∪ M D). Since M |= A ∪ M D, E is KD45consistent. Therefore, by Lemma 38, E is a belief KD45-expansion for A.
Since M 6|= ψ, ψ 6∈ E, and the proof is complete.
Note that when we can derive a formula L⊥ in a theory, we have an
unnatural modal logic where every formula of the form Lϕ is true, and every
formula of the form M ϕ is false. These belief inconsistent logics follow the
words of Tertullian of Carthage credo quia absurdum, and the corresponding
modal system is called Ver, cf. [5, pp. 66, 108]. To avoid such theories, in some
cases we shall restrict ourselves to belief consistent theories, see Definition 40
below.
Definition 40 Let S be a modal logic. A set of formulas A is called belief
S-consistent, if A 6`S L⊥ (cf. “classical S-consistency” of A: A 6`S ⊥).
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Corollary 41 Let S be a modal logic and A be an S-consistent, but belief
S-inconsistent set of formulas. Then A has a unique belief S-expansion
T hS (A).
Proof To show that T hS (A) is a belief S-expansion for A we just put D = ∅
in Lemma 38. Now uniqueness follows from Remark 37 and [17, Corollary 1
to Proposition 2], by which no belief S-expansion can be a proper subset of
another belief S-expansion.
Proposition 42 (Cf. Proposition 15.) Let A be a set of formulas, S a modal
logic, and let E be a belief S-expansion for A. Then K45 ⊆ E. Moreover, if
A is belief S-consistent then d ⊂ E as well.
Proof The proof of inclusion K45 ⊆ E is similar to the proof of [9, Theorem 1] and is omitted. For the proof of the second part of the proposition, let
A be a belief S-consistent set of formulas and let E be a belief S-expansion
for A. Let D be the set of formulas satisfying conditions (i) and (ii) of
Lemma 38 for E, S and A. It suffices to show that D 6= ∅.
Were D = ∅, by condition (i) of the lemma with ϕ being >, we would
have A `S L⊥, which contradicts the proposition assumption.
The following proposition is the belief counterpart of Proposition 20.
Proposition 43 (Cf. Proposition 20.) Let A be a belief S2 -consistent set of
formulas and let
S1 ⊆ S2 ⊆ KD45.
(4)
Then each belief S1 -expansion for A is also a belief S2 -expansion for A.
Proof Let E be a belief S1 -expansion for A and let D be the set of formulas
satisfying conditions (i) and (ii) of Lemma 38 for E, S1 and A. Since S1 ⊆ S2 ,
D also satisfies condition (i) for E, S2 and A, and the proof will be complete
if we show that E = T hS2 (A ∪ M D). We have
E = T hS1 (A ∪ M D) ⊆ T hS2 (A ∪ M D) ⊆ T hKD45 (A ∪ M D) = E,
where the last equality follows from Proposition 42.
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Proposition 44 Let A be a set of formulas, S a modal logic, and let E be
a belief S-expansion for A. If M = hU, Uc , Ii is a KD45-model of E then
T h(Mc ) = {ϕ : Lϕ ∈ E}.
Proof Let Mc |= ϕ and assume to the contrary that Lϕ 6∈ E. Then M ¬ϕ ∈
E, implying M |= M ¬ϕ. That is, for some u ∈ Uc , (Mc , u) |= ¬ϕ, which
contradicts our assumption Mc |= ϕ.
Conversely, if Lϕ ∈ E then M |= Lϕ. That is, for all u ∈ Uc , (Mc , u) |= ϕ,
which completes the proof.
Corollary 45 (Cf. Proposition 18.) Let S be a modal logic, A a belief Sconsistent set of formulas, and let E be a belief S-expansion for A. Then the
set of formulas {ϕ : Lϕ ∈ E} is stable.6
Proof By Proposition 17, theories of S5-models are stable and the result
follows from Proposition 44.
Corollary 46 Let A be a set of formulas, S a modal logic, and let E be
a belief S-expansion for A. Let M and N be KD45-models of E. Then
T h(Mc ) = T h(Nc ).
Proof By Proposition 44, both T h(Mc ) and T h(Nc ) are equal to {ϕ : Lϕ ∈
E}.
Corollary 47 Let A be a set of formulas, S a modal logic, and let E be a
belief S-expansion for A. Let M = hU, Uc , Ii be a KD45-model of E and let
ϕ ∈ LF m ∪ M F m. Then M |= ϕ if and only if ϕ ∈ E.
Proof The “if” part of the corollary is immediate. Assume that ϕ is of
the form Lψ. Then M |= ϕ implies Mc |= ψ, and ϕ ∈ E follows from
Proposition 44.
Assume that ϕ is of the form M ψ. Then M |= ϕ implies Mc 6|= ¬M ψ.
Therefore, E 6`S ¬M ψ. Since E is a belief S-expansion, M ψ ∈ E follows.
6
In other words, the set of beliefs of a “rational agent” acting in a belief expansion is
stable.
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Corollary 48 Let A be a set of formulas, S a modal logic, E be a belief
S-expansion for A, and let M = hU, Uc , Ii be a KD45-model of E. Then
D = {ϕ : M |= M ϕ} satisfies the conditions of Lemma 38 for S, E, and A.
Proof Let D0 satisfy the conditions of Lemma 38 for E. Condition (ii)
of the lemma implies D0 ⊆ {ϕ : M |= M ϕ}. In addition, it follows from
condition (i) of the lemma that for each ϕ 6∈ D0 , M 6|= M ϕ. Consequently,
{ϕ : M |= M ϕ} = D0 .
Theorem 49 below is the belief counterpart of Proposition 19.
Theorem 49 (Cf. Proposition 19.) Let S be a modal logic, A a set of
formulas, and let E be a belief S-consistent belief S-expansion for A. Then
there exists a KD45-model N such that E = T h(N).7
For the proof of Theorem 49 we need the following technical lemma that
shows that if the S5 parts of KD45-models are equivalent then worlds can
“migrate” between the models without changing the validity of modal formulas.
(λ)
Lemma 50 Let M = hU, Uc , Ii and M(λ) = hU (λ) , Uc , I (λ) i, λ ∈ Λ, be
KD45-models with pairwise disjoint sets of worlds such that
T h(hUc(λ) , Ic(λ) i) = T h(hUc , Ic i).
(5)
S
Let U 0(λ) ⊆ U (λ) and U 0 = U ∪ λ∈Λ U 0(λ) . Let N = hU 0 , Uc , Ji, where J is an
extension of I onto U 0 defined by J|U 0(λ) = I (λ) |U 0(λ) , for every λ ∈ Λ. Then
for every formula ϕ, every u ∈ U , every λ ∈ Λ, and every v ∈ U 0(λ) , (N, u) |=
ϕ if and only if (M, u) |= ϕ, and (N, v) |= ϕ if and only if (M(λ) , v) |= ϕ.
Proof Let Pϕ be the (finite) set of all propositional variables which appear
in ϕ. First we observe that all M|Pϕ , M(λ) |Pϕ , λ ∈ Λ, and N|Pϕ are ground
equivalent (see Definition 8). Indeed, finite propositional interpretations are
definable by ground formulas. In addition, it follows from (5) that every
(λ)
ground formula valid in either of Mc , Mc , λ ∈ Λ, or Nc is also valid in all
the others.
7
Consequently, E is the theory of a reduced KD45-model, cf. Corollary 10.
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Hence, by Proposition 9, for every u ∈ U , and every λ ∈ Λ, v ∈ U 0(λ) ,
(N|Pϕ , u) |= ϕ if and only if (M|Pϕ , u) |= ϕ, and (N|Pϕ , v) |= ϕ if and only if
(M(λ) |Pϕ , v) |= ϕ. Therefore, the lemma follows from Remark 3.
Proof of Theorem 49 By Proposition 42, KD45 ⊆ E. Let Φ = {ϕ ∈
F m : E 6`KD45 ϕ}. Let M = hU, Uc , Ii be a KD45-model of E and let
(ϕ)
M(ϕ) = hU (ϕ) , Uc , I (ϕ) i be a KD45-model of E in which ϕ is not valid, for
every ϕ ∈ Φ. Renaming appropriate worlds in the above KD45-models if
necessary, we may assume that their sets of worlds are pairwise disjoint.
Let u(ϕ) ∈ U (ϕ) be such that (M(ϕ) , u(ϕ) ) 6|= ϕ; and let N = hU ∪ {u(ϕ) :
ϕ ∈ Φ}, Uc , Ji, where J is an extension of I onto U ∪ {u(ϕ) : ϕ ∈ Φ} defined
by J(uϕ ) = I (ϕ) (uϕ ).
(ϕ) (ϕ)
By Corollary 46, T h(hUc , Ic i) = T h(hUc , Ic i). Hence, by Lemma 50,
N |= E and for every ϕ ∈ Φ, N 6|= ϕ. Therefore, by the definition of Φ,
T h(N) = E.
We conclude this section with the relationship between the ordinary and
belief expansions.
Proposition 51 Let A be a set of modal formulas and let S be a modal logic.
Then each S-expansion for A is also a belief S-expansion for A.
Proof Let E be an S-expansion for A and let D = {ϕ : M ϕ ∈ E}). Because
E is S-consistent, by condition (ii) of Lemma 14, we have condition (ii) of
Lemma 38.
Let ϕ 6∈ D. Then M ϕ 6∈ E, implying E `S ¬ϕ. Hence, by NEC,
E `S L¬ϕ (≡ ¬M ϕ). That is, condition (i) of Lemma 38 is satisfied as well.
Next, we separate the ordinary expansions from the belief ones, which
“completes” Proposition 51.
Proposition 52 Let A be a set of modal formulas, S be a modal logic, and
let E be a belief S-expansion for A. Then E is an S-expansion for A if and
only if T ⊆ E.8
8
In other words, a belief expansion is an ordinary one if and only if it is the theory of
an S5-model, cf. Theorem 49 and Proposition 19.
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Proof The “only if” direction is immediate, because, by Proposition 15, all
ordinary expansions include S5.
Let T ⊆ E. Then for each formula ϕ, E 6`S ¬ϕ if and only if E 6`S ¬M ϕ.
Therefore, T hS (A ∪ {M ϕ : E 6`S ¬ϕ}) = T hS (A ∪ {M ϕ : E 6`S ¬M ϕ}), and
the proposition follows from Definitions 13 and 36.
Finally, Proposition 53 below is a “weak converse” of Proposition 51.
Proposition 53 Let A be a set of formulas, S be a modal logic, E be a belief
S-expansion for A, and let M be a KD45-model of E. Then T h(Mc ) is an
S ∪ T-expansion for A.
Proof First we shall prove that for each formula ϕ, E 6`S ¬M ϕ if and only
if E 6`S∪T ¬M ϕ. If E `S ¬M ϕ then, trivially, E `S∪T ¬M ϕ. If E 6`S ¬M ϕ
then E `S M ϕ, implying Mc |= M ϕ. Thus, E 6`S∪T ¬M ϕ.
Let E 0 = T hS∪T (E). Since Mc |= E 0 , E 0 is consistent. In addition,
E 0 = T hS∪T (A∪{M ϕ : E 6`S ¬M ϕ}) = T hS∪T (A∪{M ϕ : E 6`S∪T ¬M ϕ}) =
T hS∪T (A ∪ {M ϕ : E 0 6`S∪T ¬M ϕ}). Hence E 0 is an S ∪ T-expansion for A.
Finally, we observe that for each formula ϕ, E `S Lϕ if and only if
E 0 `S∪T ϕ. If E `S Lϕ then, trivially, E 0 `S∪T ϕ. If E 0 `S∪T ϕ then Mc |= ϕ,
and therefore, by Corollary 47, E `S Lϕ.
This completes the proof because, by Proposition 44, T h(Mc ) = {ϕ :
Lϕ ∈ E}, implying T h(Mc ) = E 0 .
3.2
Interpretation of default logic in non-monotonic
modal logics of belief
Theorem 54 below shows that Truszczyński’s translation τ works for nonmonotonic modal logic of belief as well. Recall that we consider only consistent default extensions, see Definition 25.
Theorem 54 (Cf. Theorem 26.) Let (D, A) be a default theory and let
S ⊆ KD4F. A set of formulas E is an extension for (D, A) if and only
if there exists a belief S-expansion E τ for A ∪ {τ (d) : d ∈ D} such that
E = {ϕ ∈ GF m : Lϕ ∈ E τ }.
Proof Let E be an extension for (D, A). Then, by Theorem 26, there exists
an S-expansion E τ for A ∪ {τ (d) : d ∈ D} such that E = E τ ∩ GF m.
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Since by Proposition 15, E τ `S S5, for each formula ϕ, ϕ ∈ E τ if and
only if Lϕ ∈ E τ . Therefore, E = {ϕ ∈ GF m : Lϕ ∈ E τ }. Finally, by
Proposition 51, E τ is also a belief S-expansion for A ∪ {τ (d) : d ∈ D}, which
proves the “only if” part of the theorem.
Conversely, let E τ be a belief S-expansion for A ∪ {τ (d) : d ∈ D}, and
let M = hU, Uc , Ii be a KD45-model such that E τ = T h(M). In addition,
let E = {ϕ ∈ GF m : Lϕ ∈ E τ }. Obviously, for a formula ϕ, Lϕ ∈ T h(M)
if and only if ϕ ∈ T h(Mc ). Therefore, E = T h(Mc ) ∩ GF m. Since, in
addition, S ⊆ KD4F implies S∪T ⊆ S4F, the result follows from Theorem 26
and Proposition 53.
3.3
Minimal model semantics of non-monotonic modal
logic of belief
This section is the belief counterpart of Section 2.5. Definition 55 below is
similar to Definition 27.
Definition 55 (Cf. Definition 27.) Let M = hU, Uc , Ii and M0 = hU 0 , R0 , I 0 i
be a KD45-model and a Kripke interpretation, respectively, such that U ∩
U 0 = ∅. The concatenation of M to M0 , also denoted M0 M, is a Kripke
interpretation M00 = hU 00 , R00 , I 00 i, where
• U 00 = U 0 ∪ U ,
• R00 = R0 ∪ (U 0 × Uc ) ∪ (U × Uc ), and
I(u) if u ∈ U
00
• I (u) =
.
I 0 (u) if u ∈ U 0
Note that if M is an S5-model then Definition 55 is equivalent to Definition 27.
Definition 56 (Cf. Definition 28.) Let M = hU, Uc , Ii and M0 = hU 0 , R, I 0 i
be a KD45-model and a Kripke interpretation, respectively. We say that
M0 M is belief preferred (respectively, weakly belief preferred) over M,
denoted M0 M <b M (respectively, M0 M <w,b M) if there are a formula
θ ∈ GF m (respectively, θ ∈ F m) and a world u0 ∈ U 0 such that (M0 M, u0 ) |= M θ, but M |= L¬θ.
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The intuition lying behind Definition 56 is as follows, cf. [15, Section 3].
If we interpret the belief set of an agent as the set of formulas of the form
Lϕ satisfied by each world of a Kripke interpretation, then M0 M <b M
implies that the belief set with respect to M0 M is included in the belief
set with respect to M. That is, we prefer models of beliefs where the belief
set is smaller.
Remark 57 Note that if θ is modal-free then in the above definition we may
replace hM0 M, u0 i |= M θ with hM0 , u0 i |= M θ.
Proposition 58 (Cf. [6, Proposition 4.1].) Let M = hU, Uc , Ii and M0 =
hU 0 , R0 , I 0 i be a KD45-model and a Kripke interpretation, respectively. Then
M0 M <b M if and only if M0 M <w,b M.
For the proof of Proposition 58 we shall need the following definition.
Definition 59 For a formula ϕ we define the modal depth of ϕ, denoted
dL (ϕ) by the following recursion.
• If ϕ ∈ GF m then dL (ϕ) = 0;
• dL (ϕ ⊃ ψ) = max{dL (ϕ), dL (ψ)}; and
• dL (Lϕ) = dL (ϕ) + 1.
Proof of Proposition 58 The “only if” part of the proposition is immediate.
For the proof of the “if” part assume that M0 M <w,b M. Let θ ∈ F m
and u0 ∈ U 0 be such that (M0 M, u0 ) |= M θ, but M |= L¬θ. We shall prove
that there is a ground formula θ0 and v ∈ U 0 such that (M0 M, v) |= M θ0 ,
but M |= L¬θ0 . The proof is by induction on dL (θ).
Basis: If dL (θ) = 0 then θ is a ground formula and we just put θ0 = θ and
v = u0 .
We precede the induction step with the observation that if θ is of the
form θ1 ∨ θ2 , we can replace it with θ1 if (M0 M, u0 ) |= M θ1 , and with θ2
otherwise.
Induction step: By [8, Theorem 7.3, p. 191], θ is equivalent to disjunction of
formulas of the form
Lα ∧ M β1 ∧ · · · ∧ M βn ∧ γ,
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where dL (α), dL (βj ) < dL (θ), j = 1, . . . , n, and γ ∈ GF m. By the observation preceding the proof of the induction step, we may assume that θ itself
is of the form (6).
Replacing θ with (6) we obtain
(M0 M, u0 ) |= M (Lα ∧ M β1 ∧ · · · ∧ M βn ∧ γ)
(7)
M |= L¬(Lα ∧ M β1 ∧ · · · ∧ M βn ∧ γ).
(8)
and
By the definition of |= and , it follows from (7) and (8) that for some
v ∈ U0
(M0 M, v) |= Lα,
(9)
(M0 M, v) |= M βj ,
j = 1, . . . , n,
(10)
and
(M0 M, v) |= γ.
(11)
Since M is a KD45-model, by the definition of , (9) implies
M |= Lα.
(12)
Therefore, it follows from (8) and (12) that
M |= L(L¬β1 ∨ · · · ∨ L¬βn ∨ ¬γ).
(13)
If M |= L¬γ then, by (11), we can put θ0 to be γ, because γ ∈ GF m.
Otherwise, for some u ∈ Uc , (M, u) 6|= ¬γ. Therefore, by (13), for some
i = 1, . . . , n, (M, u) |= L¬βi , implying M |= L¬βi , because M is a KD45model. Since, by (10), (M0 M, v) |= M βi , we can replace θ with βi , which,
together with dL (βi ) < dL (θ), completes the proof of the induction step.
Proposition 60 Let S be a modal logic, A be a set of formulas, E be a
belief S-expansion for A, and let M be a KD45-model of E. Then for every
KD45-model M0 such that M0 M |= A, M0 M 6<b M.
Proof Assume to the contrary that there exists a Kripke interpretation M0 =
hU 0 , R0 , I 0 i, a world u0 ∈ U 0 , and a formula θ such that
M0 M |= A,
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hM0 M, u0 i |= M θ,
(15)
M |= L¬θ.
(16)
but
Let D = {ϕ : M |= M ϕ}. Then, (16) implies θ 6∈ D. However, (14)
and (15) imply A ∪ M D 6`S ¬M θ, which, by Corollary 48, contradicts condition (i) of Lemma 38.
Definition 61 (Cf. [8, Definition 11.24, p. 343].) Let M = hU, Uc , Ii and
M0 = hU 0 , Uc , I 0 i be KD45-models such that U ⊂ U 0 and I 0 |U = I. We say
that M0 is preferred over M with respect to GF m, denoted M0 ≺ M, if
T h(I 0 (U 0 )) ⊂ T h(I(U )).9
Proposition 62 Let S be a modal logic, A be a set of formulas, E be a belief
S-expansion for A, and let M be the KD45-model such that E = T h(M).
Then for every KD45-model M0 of A, M0 6≺ M.
Proof Assume to the contrary that there exists a KD45-model M0 of A
such that M0 ≺ M. Then M0 |= {M ϕ : E 6`S ¬M ϕ}. Since M0 |= A, by
Corollary 48 and Lemma 38, M0 |= E. However, it follows from M0 ≺ M
that T h(M0 ) ⊂ T h(M) (= E), which contradicts M0 |= E.
Definition 63 (Cf. Definition 30 and Remark 33.) Let C be a class of Kripke
interpretations and A be a set of formulas. A KD45-model M is called belief
C-minimal (respectively, strongly belief C-minimal) for A if
1. M |= A,
2. for every KD45-model M0 such that M0 |= A, M0 6≺ M, and
3. for every Kripke interpretation M0 such that M0 M ∈ C and M0 M |=
A, M0 M 6<b M (respectively, M0 M 6<w,b M). In case that the
worlds of M and M0 are not disjoint, we can rename the worlds of M.
9
As usual, for a set of propositional interpretations I we denote by T h(I) the set of
all ground formulas satisfied by each interpretation from I.
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In continuation to the intuitive explanation following Definition 56, if A
is the initial knowledge of an agent, we look for a minimal model according
to our preference relations in which the agent knows A, cf. [15, Section 3].
Proposition 64 below is an immediate corollary to Proposition 58.
Proposition 64 (Cf. Proposition 35.) Let C be a class of Kripke interpretations and let A be a set of formulas. Then a KD45-model M is belief
C-minimal for A if and only if it is strongly belief C-minimal for A.
Definition 65 (Cf. Definition 31 and Remark 33.) A class C of Kripke
interpretations is called KD45-model closed if at least one of the two following
conditions is satisfied.
1. For every Kripke interpretation M0 ∈ C and every KD45-model M
whose set of worlds is disjoint from that of M0 , M0 M ∈ C.
2. Every Kripke interpretation in C is of the form M0 M. Moreover, for
every M0 M ∈ C and every KD45-model N whose set of worlds is
disjoint from that of M0 , M0 N ∈ C. (Note that M0 M ∈ C implies
that M is a KD45-model.)
Remark 66 Following Remark 33, we do not impose the conditions M ∈ C
and M0 ∈ C in Definition 63, and the condition that the class C contains
all KD45-models in Definition 65. Still, we have the belief counterpart of
Schwarz’s Theorem 32, see Theorem 67 below.
Theorem 67 (Cf. Theorem 32.) Let S be a modal logic characterized by a
KD45-model closed class C of Kripke interpretations and let A be a set of
formulas. A set of formulas E is a belief S-expansion for A if and only if
there exists a KD45-model M such that M is belief C-minimal for A and
E = T h(M).
The proof of the “only if” part of Theorem 67 is an easy corollary to the
results which we have established so far. It is presented below. The proof
of the “if” part of the theorem is rather long. It is presented in the next
section.
Proof of the “only if ” part of the Theorem 67 Let E be a belief Sexpansion for A and let N be the KD45-model of E provided by Theorem 49.
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Then E = T h(N), which trivially implies condition 1 of Definition 63. By
Proposition 62, N satisfies condition 2 of Definition 63, and, by Proposition 60, N also satisfies condition 3 of that definition. That is, N is belief
C-minimal for A.
Corollary 68 (Cf. Example 22.) Let ψ be a satisfiable modal-free formula
and let S ⊆ KD45. Then {Lψ} has a belief S-expansion.
Proof By Proposition 43, it suffices to show that {Lψ} has a belief Kexpansion. Consider a KD45-model M = hU, Uc , Ii, where U is the set of all
propositional interpretations of the underlying language, Uc is the set of all
propositional interpretations satisfying ψ, and I is the identity function. By
Theorem 67, it suffices to show that M is belief minimal for A with respect
to the class of all Kripke interpretations.
By definition, M |= Lψ and for every KD45-model M0 of Lψ, M0 6≺ M.
Thus, by Remark 57, the proof will be complete if we show that for every Kripke interpretation M0 = hU 0 , R0 , I 0 i in which Lψ is valid, for every
u0 ∈ U 0 , and for every modal-free formula θ such that (M0 , u0 ) |= M θ, there
exists a world u ∈ Uc such that I(u) |= θ. (Recall that we identify an ordinary propositional interpretation with the set of all propositional variables
it satisfies.)
Let u00 ∈ U 0 be such that (u0 , u00 ) ∈ R0 and (M0 , u00 ) |= θ. Since θ is modalfree and M0 |= Lψ, u00 |= ψ ∧ θ. Thus, by the definition of M, I(u00 ) ∈ Uc ,
and the proof is complete.
Remark 69 Similarly to Corollary 68, one can show that for a satisfiable
modal-free formula ψ such that 6` ψ, {M Lψ} has a belief K-expansion. However, {M Lψ} has no S4-expansion, see [17, Proposition 5]. Thus inclusion (4)
of Proposition 43 cannot be relaxed to S5.
3.4
Proof of the “if ” part of Theorem 67
We shall need the following two lemmas.
Lemma 70 (Cf. [8, Lemma 9.20, p 264].) Let M = hU, Uc , Ii be a KD45model, and let M0 = hU 0 , R0 , I 0 i be a Kripke interpretation such that M0 |=
{M ϕ : M |= M ϕ}. Then for each formula ϕ and for each u0 ∈ U 0 , (M0 M, u0 ) |= ϕ if and only if (M0 , u0 ) |= ϕ.
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Proof The proof is by induction on the complexity of ϕ.
The basis (in which ϕ is a modal-free formula) is immediate, because both
relations (M0 M, u0 ) |= ϕ and (M0 , u0 ) |= ϕ depend on I 0 (u0 ) only.
For the induction hypothesis, the case of implication is straightforward.
Let ϕ be of the form M ψ and assume that (M0 , u0 ) |= M ψ. That is, for
some u00 ∈ U 0 such that (u0 , u00 ) ∈ R0 , (M0 , u00 ) |= ψ. Then, by the induction
hypothesis, (M0 M, u00 ) |= ψ and, by the definition of the accessibility
relation in M0 M, (M0 M, u0 ) |= M ψ.
Assume now that (M0 M, u0 ) |= M ψ. Then, by the definition of the
accessibility relation in M0 M, either
• for some uc ∈ Uc , (M, uc ) |= ψ,
or
• for some u00 ∈ U 0 such that (u0 , u00 ) ∈ R0 , (M0 M, u00 ) |= ψ.
In the former case, M |= M ψ, implying M0 |= M ψ. Hence, (M0 , u0 ) |=
M ψ.
In the latter case, by the induction hypothesis, (M0 , u00 ) |= ψ, which
implies (M0 , u0 ) |= M ψ.
Lemma 71 (Cf. [8, Lemma 9.19, p. 263].) Let M0 = hU 0 , R0 , I 0 i be a Kripke
interpretation and let M = hU, Uc , Ii and M00 = hU 00 , Uc00 , I 00 i be KD45-models
such that T h(Mc ) = T h(M00c ). Then for each modal formula ϕ and each
world u ∈ U 0 , (M0 M, u) |= ϕ if and only if (M0 M00 , u) |= ϕ.
Proof The proof is by induction on the complexity of ϕ. The induction basis
is immediate, because satisfiability of a propositional variable depends only
on I 0 (u).
For the induction step, the case of ⊃ is trivial. We assume that ϕ is of the
form M ψ and shall prove that (M0 M, u) |= ϕ implies (M0 M00 , u) |= ϕ.
The proof of the converse implication will follow after interchanging M and
M00 .
Let (M0 M, u) |= M ψ. Then, by the definition of the accessibility
relation in M0 M, either
• for some uc ∈ Uc , (M, uc ) |= ψ,
or
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• for some u0 ∈ U 0 such that (u, u0 ) ∈ R0 , (M0 M, u0 ) |= ψ.
In the former case Mc |= M ψ and, since T h(M00c ) = T h(Mc ), for some
u00c ∈ Uc00 , (Mc , u00c ) |= ψ. Therefore, by the definition of the accessibility
relation in M0 M, (M0 M00 , u) |= M ψ.
In the latter case, by the induction hypothesis, (M0 M00 , u0 ) |= ψ, implying (M0 M00 , u) |= M ψ.
Proof of the “if ” part of the Theorem 67 Let M = hU, Uc , Ii be a
KD45-model that is belief C-minimal for A and let E = T h(M). Let D =
{ϕ : M |= M ϕ}. We shall prove that E and D satisfy conditions (i) and (ii)
of Lemma 38 for A and S. Note that whenever we use an expression M0 M
in the proof below, we rename the worlds of M, if necessary, to make them
disjoint from those of M0 .
To show that condition (i) of Lemma 38 is satisfied we shall distinguish
between cases 1 and 2 of Definition 65.
Case 1. Assume to the contrary that for some θ 6∈ D, A ∪ M D 6`S ¬M θ.
Then there exists a Kripke interpretation M0 = hU 0 , R0 , I 0 i ∈ C such that
M0 |= A ∪ M D, but for some u0 ∈ U 0 , (M0 , u0 ) |= M θ. Therefore, by
Lemma 70, (M0 M, u0 ) |= M θ. In addition, since θ 6∈ D, by the definition
of D, M |= L¬θ. Hence, M0 M <w,b M, implying, by Proposition 58,
M0 M <b M.
Finally, it follows from M0 |= A by Lemma 70 that M0 M |= A, which
contradicts the belief C-minimality of M (condition 3 of Definition 63).
This shows that condition (i) of Lemma 38 is satisfied in the case 1 of
Definition 65 and we turn to case 2 of that definition.
Case 2. Assume to the contrary that for some θ 6∈ D, A ∪ M D 6`S ¬M θ.
Then there exists a Kripke interpretation N ∈ C such that N |= A ∪ M D,
but N 6|= ¬M θ. We have N = M0 M00 , where M0 = hU 0 , R0 , I 0 i and M00 =
hU 00 , Uc00 , I 00 i are a Kripke interpretation and a KD45-model, respectively.
Thus, for some u0 ∈ U 0 ∪ U 00 , (M0 M00 , u0 ) |= M θ. Since M0 M00 ∈ C
and C satisfies the condition of case 2 of Definition 65, M0 M ∈ C as well.
In addition, by the definition of D, {M ϕ : M |= M ϕ} ⊆ {M ϕ : M00 |=
M ϕ}. Therefore, by Lemma 11, T h(Mc ) = T h(M00c ) which, together with
Lemma 71, implies that for each modal formula ϕ and each world u ∈ U 0 ,
(M0 M00 , u) |= ϕ if and only if (M0 M, u) |= ϕ. Consequently, M0 M |= A
and (M0 M, u0 ) |= M θ.
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Finally, since θ 6∈ D, by the definition of D, M |= L¬θ. Thus, u0 ∈ U 0 ,
implying M0 M <w,b M. Therefore, by Proposition 58, M0 M <b M,
which contradicts the belief C-minimality of M (condition 3 of Definition 63).
That is, condition (i) of Lemma 38 is satisfied in case 2 of Definition 65
as well and it remains to verify condition (ii) of the lemma.
Since E = T h(M), this condition “consists” of two inclusions:
T hS (A ∪ {M ϕ : M |= M ϕ}) ⊆ T h(M)
and
T h(M) ⊆ T hS (A ∪ {M ϕ : M |= M ϕ}).
(17)
The former inclusion is immediate and for the proof of the latter we proceed
as follows. E 0 = T hS (A ∪ {M ϕ : M |= M ϕ}) is consistent because it is valid
in M, and A, D satisfy condition (i) of Lemma 38. Thus, by Lemma 38, E 0
is a belief S-extension for A. Let M0 = hU 0 , Uc0 , I 0 i be a KD45-model of E 0 .
Since E 0 ⊆ E, we have {M ϕ : M0 |= M ϕ} ⊆ {M ϕ : M |= M ϕ}. Therefore,
{M ϕ : M0 c |= M ϕ} ⊆ {M ϕ : Mc |= M ϕ}, and, by Lemma 11,
T h(hUc , I|Uc i) = T h(hUc0 , I 0 |Uc0 i).
(18)
Consequently, by Lemma 12, for the proof of (17) it suffices to show that for
each ϕ ∈ GF m, M |= ϕ implies M0 |= ϕ.
So, let ϕ ∈ GF m be such that M |= ϕ and assume to the contrary that
0
M 6|= ϕ. Then it follows from (18) that for some u0 ∈ U 0 \ Uc0 , I 0 (u0 ) 6|=
ϕ. Renaming the elements of U if necessary, we may assume that u0 6∈ U .
Consider a KD45-model N = hU ∪ {u0 }, Uc , Ji, where J is an extension of I
onto U ∪ {u0 } defined by J(u0 ) = I 0 (u0 ). Then N ≺ M and, by Lemma 50,
N |= A, which contradicts the belief C-minimality of M (condition 2 of
Definition 63). This proves (17) and completes the proof of the “if” part of
the theorem.
4
Concluding remarks
Non-monotonic modal logic of belief proposed in this paper naturally extends
the classical non-monotonic modal logic and, in addition to a desirable behavior given by Corollary 68, possesses many nice formal properties. All this
provides a strong intuitive support to our approach. However, at this stage it
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is still not clear whether non-monotonic modal logic of belief is better suited
for study of knowledge presentation. Even though the new logic seems to
be a quite reasonable non-monotonic formalism, there are no (and cannot
be any) formal criteria for accepting our approach (or any other) as the ultimate right one. Therefore, the question “which logic is the right one?” can
be answered only by a field test.
We conclude the paper with some problems which, on the one hand, are
of interest in their own right, and, on the other hand, might give a better
insight into non-monotonic modal logic of belief.
• How is Moore’s autoepistemic logic related to non-monotonic modal
logic of belief?
• What is the power of introspection in non-monotonic modal logic of
belief, cf. [14]?
• Can non-monotonic modal logic and non-monotonic modal logic of belief be interpreted in each other, cf. Propositions 52, 53 and 44?
• How can propositional non-monotonic modal logic of belief be extended
to the first-order case, cf. [6]?
• What is the ground non-monotonic modal logic of belief, cf. [2]?
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