Bulletin of the Section of Logic
Volume 41:3/4 (2012), pp. 155–172
Michal Zawidzki
ADEQUACY OF THE LOGIC K(En )
Abstract
In this paper we characterize syntax and semantics of the logic K(En ) - modal
logic with global counting operators E>n , E=n and E<n . We fill the gap announced
in [1] by providing the axiomatization of this logic. Taking advantage of the
combinatorial parts of completeness proofs for graded modal logics to be found
in [4] and [6], we show that K(En ) is strongly adequate with respect to the class
of all frames F = ⟨W, R◇ ⟩.
1.
Introduction
Modal logics with counting operators were first introduced by Fine ([7])
under the name of graded modal logics. GMLs involved new type of modalities, namely modalities with cardinality constraints. A formula of the
form ◇>n ϕ expressed the fact that in at least n successors of the current
world formula ϕ holds. The expressibility of GMLs extended, therefore,
the expressibility of ordinary modal logics. Regarding the intrinsically local
character of graded modalities, it seemed natural to expand considerations
onto logics with global counting operators, which would allow us to state in
how many worlds in a whole model a certain formula holds. Surprisingly,
whereas GMLs were widely investigated, modal logics with counting operators remained an area of research rather neglected by modal logicians.
In most cases it aroused interest of description logicians and most of the
results were carried in the field of description logics.
In [1] Areces, Hoffmann and Denis present the topic of modal logics
with global counting operators from modal perspective. Not only do they
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describe mutual connections between graded logics and logics with global
counting operators or between the latter and description logics, but also
contribute with some technical results. Firstly, they state the conditions
of bisimulation between models for logics with global counting operators.
Secondly, they identify the translation function between modal logic with
global counting operators and the hybrid logic with global modality E H(E),
stating that the former can, in fact, be looked upon as hybrid logic.
The problem that remains open in [1] is axiomatization of modal logic
with global counting operators. In this article we will provide the axiomatization for the logic K(En ) and prove its strong adequacy with respect to
the class of all frames. The proof is of rather technical character. It utilises
the completeness results established for graded modal logics in [4] and [6].
2.
Syntax and semantics of K(En )
Syntax
Definition 1 (Syntax). Let prop = {p1 , p2 , . . .} be a denumerable set
of propositional letters. We define a set form of formulas of K(En ) as
follows:
form ∶∶= ⊺ ∣ p ∣ ¬ϕ ∣ ϕ ∧ ψ ∣ ◇ ϕ ∣ E>n ϕ,
(K(En ))
where p ∈ prop, ϕ ∈ form, n ∈ N.
We define other Boolean operators in a standard manner. ◻ is defined as
¬ ◇ ¬.
Notation 1. We read E>n ϕ as ”there are more than n worlds in a model,
in which formula ϕ is satisfied”. We define the operator E<n (that should
be read as ”there are less than n worlds in a model”) in the following way:
E<n ϕ = ¬E>n−1 ϕ.
Additionaly, we introduce the operator E=n (”there are exactly n worlds in
which formula ϕ is satisfied”) defined as follows:
⎧
⎪
⎪E>n−1 ϕ ∧ ¬E>n ϕ for n > 0
E=n ϕ = ⎨
⎪
¬E ϕ
for n = 0.
⎪
⎩ >0
Adequacy of the Logic K(En )
157
The reader might be unfamiliar with the above notational convention. Nevertheless, it has been proposed for two reasons. Firstly, it is more contemporary than the notation of graded modalities introduced by Fine in [7]
and Fattorosi-Barnaba, Caro in [6], even though the latter allows to define dual operator for Mn being an analog for E>n unambiguously, whereas
the former does not. Secondly, the notation used in this article is closer
to the counting properties of modalities than Fine’s notation. Due to the
global character of counting modalities we decided to express it with global
modality E supplied with cardinality constraints.
Remark 1. Note that E>0 is the ordinary global modality E.
Semantics
Definition 2 (Semantics). A model for K(En ) is a triple ⟨W, R◇ , V ⟩
where W is a non-empty set, R◇ is a binary relation on W , V ∶ prop →
P(W ) is a valuation function assigning to each p ∈ prop a set of worlds
w ∈ W in which p holds.
Given the model ⟨W, R◇ , V ⟩ and w ∈ W , the semantics for K(En ) is defined
as follows:
3.
M, w ⊧ p
iff
w ∈ V (p), p ∈ prop
M, w ⊧ ¬ϕ
iff
M, w ⊭ ϕ
M, w ⊧ ϕ ∧ ψ
iff
M, w ⊧ ϕ and M, w ⊧ ψ
M, w ⊧ ◇ϕ
iff
there is v such that wR◇ v and M, v ⊧ ϕ
M, w ⊧ E>n ϕ
iff
∥{w ∶ M, w ⊧ ϕ}∥ > n.
Axiomatization of K(En )
Now we give axiomatization for the logic K(En ).
Definition 3 (K(En ) axiomatization). The axioms of K(En ) are all
instances of the following schemata:
CT
K
Ax.1
Ax.2
All classical tautologies;
◻(ϕ → ψ) → (◻ϕ → ◻ψ);
E=0 ¬(ϕ → ψ) → (E>n ϕ → E>n ψ), for n ∈ N;
E=0 (ϕ ∧ ψ) → ((E=n ϕ ∧ E=m ψ) → E=n+m (ϕ ∨ ψ)), for n, m ∈ N;
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Ax.3
Ax.4
Ax.5
Ax.6
Michal Zawidzki
ϕ → E>0 ϕ;
E>n ϕ → E=0 (¬E>n ϕ), for n ∈ N;
E>n ϕ → E>m ϕ, for n, m ∈ N, n > m;
◇ϕ → E>0 ϕ.
The rules of inference of K(En ) are:
MP
NE
From ϕ → ψ and ϕ derive ψ;
From ϕ derive E=0 ¬ϕ.
Note that the classical inference rule, necessitation rule, is derivable
from NE and contrapositive of axiom 6.
Having formulated the axiom schemata and inference rules of K(En ),
we now formalise the notion of the consequence relation for K(En ), which
is utilised in following parts of the paper. Let L be a set of K(En )-formulas
and ϕ - a K(En )-formula.
Definition 4 (Syntactic and semantic consequence for K(En )).
The logic K(En ) is the least set of formulas containing the axiom schemata
from Def. 3 and closed under the rules MP and NE .
We say that ϕ is a (local) syntactical K(En )-consequence of L, formally
L ⊢K(En ) ϕ, iff (ψ1 ∧ ⋅ ⋅ ⋅ ∧ ψn ) → ϕ ∈ K(En ) for some {ψi , . . . , ψn } ⊆ L.
We say that ϕ is a (local) semantic K(En )-consequence of L, formally
L ⊧K(En ) ϕ, iff for all models M and all points w ∈ WM if M, w ⊧ L then
M, w ⊧ ϕ.
4.
Adequacy of K(En )
Soundness
We will now prove that K(En ) is complete with respect to the class of all
frames. We will proceed in two steps. First comes the proof of the easy
part.
Theorem 1. The axiomatization from definition 3 is sound with respect
to the class of all frames, i.e. if L ⊢K(En ) ϕ then L ⊧K(En ) ϕ.
Proof:
By easy verification of validity of the axioms and validitypreservation of the inference rules.
Adequacy of the Logic K(En )
159
The following two claims state certain syntactical properties of the logic
K(En ), which will be exploited in the latter part of the paper.
Claim 1. The logic K(En ) is closed under the replacement rule, i.e. if
⊢ ϕ ↔ ψ then ⊢ χ ↔ χ[ϕ/ψ].
Proof:
The case where χ contains no occurrence of E>n is trivial and
follows directly from CT and MP. Without loss of generality, assume
that χ = E>n ϕ. We have that ⊢ ϕ ↔ ψ, whence by CT and MP we
obtain ⊢ ϕ → ψ and ⊢ ψ → ϕ. Hence, by NE we have ⊢ E=0 ¬(ϕ → ψ) and
⊢ E=0 ¬(ψ → ϕ). By axiom 1 we obtain ⊢ E>n ϕ → ψ and ⊢ E>n ψ → ϕ. By
CT we get the conclusion.
Claim 2. Let ϕ and ψ be K(En ) formulas. Then:
1.
2.
3.
if ⊢ ϕ → ψ then ⊢ E>n ϕ → E>n ψ;
⊢ ϕ ↔ ψ then ⊢ E=n ϕ ↔ E=n ψ;
⊢ E>0 (ϕ ∨ ψ) then ⊢ E>0 ϕ ∨ E>0 ψ.
Proof: 1. Suppose that ⊢ ϕ → ψ. By NE we obtain ⊢ E=0 ¬(ϕ → ψ). By
applying MP to it and axiom 1 we obtain the conclusion.
2. Suppose that ⊢ ϕ ↔ ψ. By Claim 1 we obtain ⊢ (E>n−1 ϕ∧¬E>n ϕ) ↔
(E>n−1 ψ ∧ ¬E>n ψ). Hence conclusion.
3. Suppose that ⊢ E>0 (ϕ ∨ ψ). For the sake of contradiction suppose
also that ⊬ E>0 ϕ ∨ E>0 ψ. By CT we derive ⊢ ¬E>0 ϕ and ⊢ ¬E>0 ψ. Taking
contrapositive of axiom 3 we obtain ⊢ ¬ϕ and ⊢ ¬ψ. By CT we get
⊢ ¬(ϕ ∨ ψ) which leads to contradiction.
Completeness
Preliminaries
Proof of the second part of the adequacy theorem, namely the completeness
theorem, is, as usually, much more complicated. To establish it, we will
first need the following fact:
Fact 1. A logic L is strongly complete with respect to a class of frames F
iff every L-consistent set of formulas is satisfiable on some F ∈ F.
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Michal Zawidzki
The above fact allows us to conduct a proof by constructing model.
Namely, we will show that for every K(En )-consistent set of formulas there
is a model (since we deal with the logic K no particular properties of a
model are required) which satisfies this set. Major part of the proof and
preliminary lemmas comes from [6] and, in a simplified and more general
form, from [4]. Fattorosi-Barnaba and Caro proved completeness for several
normal modal logics with graded modalities. We can use their results for
the ”counting” part of our work and recall them in this paper for the
purpose of clarity of the argument.
We will start with several introductory definition and lemmas concerning properties of K(En )-maximal consistent sets [abbrev. MCS].
Definition 5. A finite set Γ of K(En ) formulas is called K(En )-consistent
iff Γ ⊢
/ K(En ) –. If Γ is an infinite set of K(En ) formulas, it is K(En )consistent iff all its finite subsets are K(En )-consistent. Γ is a K(En )maximal consistent set iff it is consistent and for all sets Σ, if Γ ⊊ Σ, then
Σ is inconsistent.
Fact 2. Let Γ be a K(En )-maximal consistent set. Then:
1.
2.
for all K(En ) formulas ϕ, ϕ ∈ Γ iff ¬ϕ ∉ Γ;
for all K(En ) formulas ϕ, ψ, ϕ ∨ ψ ∈ Γ iff ϕ ∈ Γ or ψ ∈ Γ.
Now, it appears to be useful to generalize axiom 2.
Lemma 1. Let ϕ1 , ⋯, ϕk+1 ∈ form. If ⊢ ⋀i≠j
i,j≤k+1 ¬(ϕi ∧ ϕj ) then
⊢ ⋀i≤k+1 E=ni ϕi → E=∑k+1 ni ⋁i≤k+1 ϕi , ni ∈ N.
i=1
Proof:
By induction on k.
Base: k = 1.
We have ⊢ ¬(ϕ0 ∧ ϕ1 ). Hence, by NE , we obtain ⊢ E=0 (ϕ0 ∧ ϕ1 ) and
by axiom 5 we get the conclusion: E=n0 ϕ0 ∧ E=n1 ϕ1 → E=n0 +n1 (ϕ0 ∨ ϕ1 ).
Inductive step: suppose that the conclusion holds for k.
i≠j
⊢ ⋀i≠j
i,j≤k ¬(ϕi ∧ϕj ) iff ⊢ ⋀i,j≤k ¬(ϕi ∧ϕj )∧⋀i≤k ¬(ϕi ∧ϕk+1 ). By inductive
hypothesis and basic case we obtain: ⊢ ⋀i≤k+1 E=ni ϕi → E=∑ki=1 ni ⋁i≤k ϕi ∧
E=nk+1 ϕk+1 . By CT, ⊢ ⋀i≤k ¬(ϕi ∧ ϕk+1 ) iff ⊢ ¬ ⋁i≤k (ϕi ∧ ϕk+1 ) iff
Adequacy of the Logic K(En )
161
⊢ ¬(⋁i≤k (ϕi ) ∧ ϕk+1 ), so we obtain ⊢ E=∑ki=1 ni ⋁i≤k (ϕi ) ∧ E=nk+1 ϕk+1 →
→ E∑k+1 ni ⋁i≤k+1 ϕi . Hence we have ⊢ ⋀i≤k+1 E=ni ϕi → E=∑k+1 ni ⋁i≤k+1 ϕi .
i=1
i=1
Therefore, by the proof of base and inductive step, the conclusion follows for every finite k.
Lemma 2. Let Γ be a MCS of formulas of K(En ). We have:
1.
2.
3.
If E=n ϕ ∈ Γ then E=m ϕ ∉ Γ for all m ∈ N such that n ≠ m.
For every ϕ ∈ form exactly one of the following conditions holds:
a. for every n ∈ N: E>n ϕ ∈ Γ;
b. there is n ∈ N such that E=n ϕ ∈ Γ.
If ⊢ ϕ → ψ and E=n ψ then there is m ∈ N such that E=m ϕ ∈ Γ and
necessarily m ≤ n.
Proof:
1. Suppose conversely that there is such formula ϕ that E=n ϕ ∈
Γ, E=m ϕ ∈ Γ and n ≠ m. Suppose without loss of generality that n > m. We
have then that n − 1 ≥ m and, by definition of E=n and maximality of Γ,
that E>n−1 ϕ ∈ Γ and thus that E>m ϕ ∈ Γ. But we also have that ¬E>m ϕ ∈ Γ.
Contradiction.
2. Both conditions are complementary. Suppose that a. holds. Then,
if there exists n such that E=n ϕ ∈ Γ, we have ¬E>n ϕ ∈ Γ, and by Fact 2.1
it follows that E>n ϕ ∉ Γ which contradicts the assumption. Thus, b. does
not hold. Suppose that a. does not hold. Then it is the case that there
is such n ∈ N that E>n ϕ ∉ Γ. Let n0 be the least such n. We have that
E>n0 ϕ ∉ Γ, so ¬E>n0 ϕ ∈ Γ. By assumption of minimality of n0 we also have
that E>n0 −1 ϕ ∈ Γ. Thus we obtain E=n0 ϕ ∈ Γ. By 1. we get uniqueness of
n0 .
3. Suppose that ⊢ ϕ → ψ and E=n ψ ∈ Γ. By Claim 2.1 we obtain
⊢ E>n ϕ → E>n ψ and by definition of E=n we have that ¬E>n ψ ∈ Γ. Therefore,
¬E>n ϕ ∈ Γ and by axiom 5 ¬E>m ϕ ∈ Γ for all m > n (¬E>n ϕ = E<n+1 ϕ). By
2. it follows that there is m ∈ N such that E=m ϕ ∈ Γ. By definition of E=n
m is not greater than n, so we have E=m ϕ ∈ Γ, m ≤ n.
Lemma 3. Let Γ be a MCS and ϕ0 , ⋯, ϕk - formulas of K(En ). Let ψi be
a formula of the form ϕ0 ∧ ϕ¯1 ∧ ⋯ ∧ ϕ¯k where ϕ¯j = ϕj or ¬ϕj . Then we
have a set of distinct formulas {ψi ∶ 0 ≤ i < 2k }.
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Michal Zawidzki
1.
If there is such n ∈ N that E=n ϕ0 ∈ Γ then there are also n0 , ⋯, n2k −1
k
such that ∑2i=0−1 ni = n and E=ni ψi ∈ Γ, (0 ≤ i < 2k ).
If E>n ϕ0 ∈ Γ for all n ∈ N then there is such i, 0 ≤ i < 2k that E>n ψi ∈ Γ
for all n ∈ N.
2.
Proof:
1. Assume that E=n ϕ0 ∈ Γ. By CT we have ⊢ ¬(ψi ∧ ψj ) for i ≠ j,
0 ≤ i, j < 2k . We have ⊢ ϕ0 ↔ ⋁i<2k ψi . Since ⊢ ψi → ϕ0 for all 0 ≤ i < 2k ,
by Lemma 2 we have E=mi ψi ∈ Γ such that mi ≤ n. Then by Lemma 1,
Lemma 2.1 Claim 2.2 we have that there is a set of indexes {ni ∶ 0 ≤ i < 2k }
k
such that E=ni ψi and ∑2i=0−1 ni = n.
2. Suppose conversely that for each i ∈ {0, . . . , 2k − 1} there exists ni
such that E=ni ψi ∈ Γ. Obviously, ⊢ ¬(ψi ∧ψj ) for i ≠ j, 0 ≤ i, j < 2k so we can
apply Lemma 1 and obtain ⊢ E=Σ2k −1 n ⋁i<2k ψi ∈ Γ. Since ⊢ ϕ0 ↔ ⋁i<2k ψi ,
i
i=0
by Claim 2.2 we obtain E=Σ2k −1 n ϕ0 ∈ Γ which contradicts the assumption
i
i=0
that for all n ∈ N E>n ϕ0 ∈ Γ.
We can now turn to the proper part of our work, namely the construction of a model that will satisfy arbitrarily chosen K(En )-consistent set of
formulas.
Let Γ− be our K(En )-consistent set of formulas. Let us introduce a
following notation:
Notation 2. We will denote a family of all K(En )-MCSs by G.
By Lindenbaum’s lemma there exists a K(En )-MCS, name it Γ, that
includes Γ− (in fact, if Γ− is finite then there are infinitely many such MCSs.
We can arbitrarily choose one). Obviously Γ ∈ G.
We will define a function that will help us set the universe of our intended model.
Definition 6. Let Γ1 , Γ2 ∈ G. The function λ ∶ G×G Ð→ N∪{ω} is defined
as follows:
⎧
⎪
⎪ω, if for any ϕ ∈ Γ2 , E>n ϕ ∈ Γ1 for any n ∈ N;
λ(Γ1 , Γ2 ) = ⎨
⎪
h = min{n ∈ N ∶ E=n ϕ ∈ Γ1 and ϕ ∈ Γ2 }, otherwise.
⎪
⎩
Adequacy of the Logic K(En )
163
By Lemma 2.2 the above function is well defined. It informs us how
many copies of each MCS we should paste in the universe of created model.
Here we state and prove a useful lemma.
Lemma 4. Let Γ1 , Γ2 ∈ G. The following conditions are equivalent:
1.
2.
3.
λ(Γ1 , Γ2 ) ≠ 0;
for any ϕ ∈ form: if ϕ ∈ Γ2 then E>0 ϕ ∈ Γ1 ;
for any ϕ ∈ form: if E=0 ¬ϕ ∈ Γ1 then ϕ ∈ Γ2 .
Proof:
(1 ⇒ 2) Suppose there exists such ϕ0 ∈ form that ϕ0 ∈ Γ2
but E>0 ϕ0 ∉ Γ1 . By maximality of Γ1 we have that ¬E>0 ϕ0 ∈ Γ1 , so by
definition of E=0 E=0 ϕ ∈ Γ1 . Thus, by definition of the function λ we obtain
λ(Γ1 , Γ2 ) = 0.
(2 ⇒ 3) Suppose there exists such ϕ0 ∈ form that E=0 ¬ϕ0 ∈ Γ1 and
ϕ0 ∉ Γ2 . By maximality of Γ2 ¬ϕ0 ∈ Γ2 and by consistency of Γ1 it is not
the case that E>0 ϕ0 ∈ Γ1 .
(3 ⇒ 1) Suppose λ(Γ1 , Γ2 ) = 0. Then, by definition of the function λ
there exists such ϕ0 ∈ form that E=0 ϕ0 ∈ Γ1 and ϕ0 ∈ Γ2 whence follows
¬ϕ0 ∉ Γ2 .
It is easily noticeable that the function λ will support our effort to
establish a universe in constructed model. Namely, a set Γ2 ∈ G will be a
member of the universe if and only if λ(Γ1 , Γ2 ) ≠ 0.
The following fact is an analog of Lemma 4 but it refers to ordinary
modalities.
Fact 3. Let Γ1 , Γ2 ∈ G. The following two conditions are equivalent:
1.
2.
for any ϕ ∈ form if ϕ ∈ Γ2 then ◇ϕ ∈ Γ1 ;
for any ϕ ∈ form if ◻ϕ ∈ Γ1 then ϕ ∈ Γ2 .
The next fact states an important property of the consistency of a set of
formulas provided that these formulas preceded by appropriate modalities
build a consistent set. It will help in finding MCSs related to our set Γ
with respect to R̂◇ .
Fact 4. Let {◻ϕ1 , ⋯, ◻ϕn , ◇ψ} be the consistent set of formulas. Then
the set {ϕ1 , ⋯, ϕn , ψ} is also consistent.
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Michal Zawidzki
We now give a lemma which will be helpful in setting the universe of
our model. Namely it assures that given a MCS Γ ∈ G we will find MCSs
Γ′ ∈ G satisfying counting modal formulas of Γ.
Lemma 5. Let Γ ∈ G and ϕ ∈ form.
1.
2.
If E>0 ϕ ∈ Γ then there exists Γ′ ∈ G such that ϕ ∈ Γ′ and λ(Γ, Γ′ ) > 0;
If E>n ϕ for all n ∈ N then there exists Γ′ ∈ G such that ϕ ∈ Γ′ and
λ(Γ, Γ′ ) = ω.
Proof:
We will show that such MCSs exist by constructing them stepby-step. Let ϕ = ϕ1 , ⋯, ϕn , ⋯ be an ordered list of formulas of form. We
will define a chain of sets of formulas ∆0 ⊂ ⋯ ⊂ ∆n ⊂ ⋯. Let ψi = ⋀{ϕ̄ ∶ ϕ̄ ∈
∆i } where ϕ̄i = ϕi or ¬ϕi . Each ∆i is such that:
i. E>0 ψi ∈ Γ;
ii. E>n ψi ∈ Γ for every n ∈ N.
Gradual saturating of the set ∆ resembles the one from a proof of Lindenbaum’s lemma. We will proceed by induction on the degree of saturation.
Base: n = 1
1. Let ∆1 = {ϕ}. Then i. is the same as 1. from the lemma.
2. Let ∆1 = {ϕ}. Then ii. is the same as 2. from the lemma.
Inductive step: from n to n + 1
1. Suppose that we have ∆n such that E>0 ψn ∈ Γ. By maximality and
consistency of Γ we have that E>0 (ψn ∧ ϕn+1 ) ∈ Γ or E>0 (ψn ∧ ¬ϕn+1 ) ∈ Γ
- for we have E>0 ψn ∈ Γ iff E>0 (ψn ∧ ⊺) ∈ Γ iff E>0 (ψn ∧ (ϕn+1 ∨ ¬ϕn+1 )) ∈
Γ iff E>0 ((ψn ∧ ϕn+1 ) ∨ (ψn ∧ ¬ϕn+1 )) ∈ Γ. By Claim 2.3 we obtain E>0 (ψn ∧
ϕn+1 ) ∨ E>0 (ψn ∧ ¬ϕn+1 ) ∈ Γ. So we define ∆n+1 as ∆n ∪ {ϕn+1 } if E>0 (ψn ∧
ϕn+1 ) ∈ Γ or as ∆n ∪ {¬ϕn+1 } if E>0 (ψn ∧ ¬ϕn+1 ) ∈ Γ.
2. Suppose that we have ∆n such that E>m ψn ∈ Γ for each m ∈ N. By
CT we know that ψn ↔ (ψn ∧ ϕn+1 ) ∨ (ψn ∧ ¬ϕn+1 ) and by Lemma 3.2 we
have E>n (ψn ∧ ϕn+1 ) ∈ Γ for each n ∈ N or E>m (ψn ∧ ¬ϕn+1 ) ∈ Γ for each
m ∈ N. We therefore define ∆n+1 as ∆n ∪ ϕn+1 if the former is the case and
as ∆n ∪ ¬ϕn+1 otherwise.
Now we must show that ∆i is consistent for every i ∈ N. If it were not
the case, then ⊢ ¬ψi and by NE ⊢ E=0 ψi , so E=0 ψi ∈ Γ which contradicts
the consistency of Γ. Let Γ′ = ⋃n∈N ∆n . Since all ∆i are consistent, Γ′ is
Adequacy of the Logic K(En )
165
a MCS. We will show that λ(Γ, Γ′ ) > 0 in the first case and λ(Γ, Γ′ ) = ω
in the second case. Take arbitrary χ ∈ Γ′ . It must have been included in
a certain link of our chain of sets - ∆m . So if E>0 ψm it follows that E>0 χ.
Since χ was arbitrary, by Lemma 4 we obtain λ(Γ, Γ′ ) > 0. Proof for the
second case is analogous.
The following lemma states an interesting property of each MCS, namely
the existence of a labelling formula for such a set - a formula that is included uniquely in a set labelled by it. Properties of so-defined labelling
formulas strongly resemble properties of nominals from hybrid logic (for
more information on translation of hybrid logic with global modality to
modal logic with counting operators see [1]).
Lemma 6. Let Γ1 , ⋯, Γn be distinct MCSs. Then there exist labelling formulas ϕ1 , ⋯, ϕn ∈ form such that ϕi ∈ Γj iff i = j and ⊢ ¬(ϕi ∧ ϕj ), i, j ≤ n
iff i ≠ j.
Proof:
We proceed by induction on n.
Base: n = 2
Let Γ1 , Γ2 be 2 distinct MCSs. Since they are distinct, there exists a
formula ϕ such that ϕ ∈ Γ1 but ϕ ∉ Γ2 , so by maximality of Γ2 ¬ϕ ∈ Γ2 . By
CT we also obtain ⊢ ¬(ϕ ∧ ¬ϕ).
Inductive step: from n to n + 1
Let Γ1 , ⋯, Γn be distinct MCSs and ϕ1 , ⋯, ϕn be formulas satisfying
the conditions of the lemma. We show that such a formula exists also for
Γn+1 being an MCS distinct from the former ones. From the basic case we
know that there exist such formulas ψ1 , ⋯, ψn that ψi ∈ Γi and ¬ψi ∈ Γn+1
for i ≤ n. Let χi be a formula of the form ϕi ∧ ψi for i ≤ n and ⋀i≤n ¬ψi for
i = n + 1. By CT we have that χi ∈ Γj , i, j ≤ n + 1 iff i = j. Since ⊢ χi → ϕi
for i ≤ n, by CT we obtain ⊢ (χi ∧ χj ) → (ϕi ∧ ϕj ), i, j ≤ n and therefore,
by taking contrapositive ⊢ ¬(χi ∧ χj ), i, j ≤ n. Since ⊢ χn+1 → ¬ψi , i ≤ n
and ⊢ χi → ψi , i ≤ n, by CT we have ⊢ (χn+1 ∧ χi ) → (¬ψi ∧ ψi ), i ≤ n, so
by taking contrapositive we obtain ⊢ ¬(χn+1 ∧ χi ), i ≤ n.
At the end of preliminary part of completeness proof of K(En ) we
introduce a notion that will be crucial in building a canonical model for
K(En ).
166
Michal Zawidzki
Definition 7. We will call the satisfying family of Γ, denoted SF (Γ), the
family of MCSs defined as follows:
SF (Γ) = {Γ′ × {1, ⋯, λ(Γ, Γ′ )} ∶ Γ′ ∈ G}.
Roughly speaking, the satisfying family of a set Γ is the family of MCSs,
each of which was copied exactly λ(Γ, Γ′ ) times. Formally speaking each
member of SF (Γ) is an ordered pair with an MCS in the first place and
an integer in the second. We identify these pairs with their first element,
namely a MCS, taking integers as indexes of the copies of particular MCSs.
For each Γ ∈ G we will call Γ1 a primary copy of Γ. It is easy to note that no
MCS Γ′ ∈ G such that λ(Γ, Γ′ ) = 0 is a member of SF (Γ). It is congruent
with an intuition that a set Γ′ including a certain formula ϕ such that
E=0 ϕ ∈ Γ cannot be in accessibility relation (R̂E ) with Γ because it does not
satisfy all its counting formulas.
Finally, all tools necessary for building a canonical model for K(En )
are at our disposal.
Constructing model
Now we construct a model for the arbitrarily chosen K(En )-MCS.
Definition 8. We have the logic K(En ) and the K(En )-MCS Γ. We
construct a model M̂ = ⟨Ŵ , R̂◇ , V̂ ⟩ for it in the following way:
1.
2.
3.
Ŵ = SF (Γ), where Γ is arbitrarily chosen K(En )-maximal consistent
extension for Γ− ;
R̂◇ ={⟨Γ′ , Γ′′ ⟩ ∶ Γ′ , Γ′′ ∈ Ŵ and if ϕ ∈ Γ′′ then ◇ϕ ∈ Γ′ for all ϕ ∈ Γ′′ };
V̂ ∶ prop Ð→ P(Ŵ ): V̂ (p) = {Γ′ ∈ Ŵ ∶ p ∈ Γ′ }.
Now, before we state and prove existence and truth lemma, we need
to show that the universe of the model, namely SF (Γ) is not only the
satisfaction family for Γ itself, but also for each member of the universe.
In order to prove this, it suffices to show that the satisfaction relation for
Γ defined as {Γ × SF (Γ) ∶ Γ ∈ Ŵ } is the equivalence relation and, therefore,
that SF (Γ) is a cluster with respect to this relation.
Lemma 7. {{Γ′ } × SF (Γ′ ) ∶ Γ′ ∈ G} is an equivalence relation.
Adequacy of the Logic K(En )
167
Proof:
We prove that {{Γ′ } × SF (Γ′ ) ∶ Γ′ ∈ G} is reflexive, transitive
and symmetric.
Reflexivity. We show that for each Γ′ ∈ G Γ′ ∈ SF (Γ′ ). It suffices to
prove that λ(Γ′ , Γ′ ) > 0, Γ′ ∈ G. By axiom 3 we have E>0 ϕ ∈ Γ′ for each
ϕ ∈ Γ′ . By Lemma 4 we obtain λ(Γ′ , Γ′ ) > 0.
Transitivity. Let Γ′ , Γ′′ , Γ′′′ ∈ G. We show that if Γ′′ ∈ SF (Γ′ ) and Γ′′′ ∈
SF (Γ′′ ) then Γ′′′ ∈ SF (Γ′ ). Suppose that λ(Γ′ , Γ′′ ) > 0 and λ(Γ′′ , Γ′′′ ) > 0.
By Lemma 4 we have E>0 ϕ ∈ Γ′ for each ϕ ∈ Γ′′ and E>0 ψ ∈ Γ′′ for each
ψ ∈ Γ′′′ . Assume that λ(Γ′ , Γ′′′ ) = 0. Then there is such ψ0 ∈ Γ′′′ that
E=0 ψ0 ∈ Γ′ . By axiom 3 we obtain E>0 (E=0 ψ0 ) ∈ Γ′ . By axiom 4 we have
E=0 ¬(E>0 (E=0 ψ0 )) ∈ Γ′ . Since λ(Γ′ , Γ′′ ) > 0, we have E>0 (E=0 ψ0 ) ∈ Γ′′ .
Since λ(Γ′′ , Γ′′′ ) > 0, E>0 ψ0 ∈ Γ′′ . By axiom 4 we obtain E=0 (E=0 ψ0 ) ∈ Γ′′ .
Contradiction.
Symmetry. Let Γ′ , Γ′′ ∈ G. We show that if Γ′′ ∈ SF (Γ′ ) then Γ′ ∈
SF (Γ′′ ), so λ(Γ′′ , Γ′ ) > 0 whenever λ(Γ′ , Γ′′ ) > 0. Assume that λ(Γ′ , Γ′′ ) >
0 and λ(Γ′′ , Γ′ ) = 0. Hence we have such ϕ0 ∈ Γ′ that E=0 ϕ0 ∈ Γ′′ . Since
λ(Γ′ , Γ′′ ) > 0 we have E>0 (E=0 ϕ0 ) ∈ Γ′ . By axiom 3 we obtain E>0 ϕ0 ∈ Γ′
and by axiom 4 we have E=0 (E=0 ϕ0 ) ∈ Γ′ . Contradiction.
Since we proved that {{Γ′ }×SF (Γ′ ) ∶ Γ′ ∈ G} is the equivalence relation,
it becomes obvious that Ŵ is the equivalence class of {{Γ′ } × SF (Γ′ ) ∶ Γ′ ∈
G} and hence that Ŵ = SF (Γ) = SF (Γ′ ) for all Γ′ ∈ Ŵ .
Now we have all that we need to state and prove two lemmas of crucial
importance for the whole argument.
Lemma 8 (Existence Lemma). Let M̂ = ⟨Ŵ , R̂◇ , V̂ ⟩ be a model for
K(En ) and Γ0 ∈ Ŵ . Then the following conditions hold:
1.
2.
E>n ϕ ∈ Γ0 iff ∥{Γ′ ∈ Ŵ ∶ ϕ ∈ Γ′ }∥ > n;
If ◇ ϕ ∈ Γ0 then there exists such Γ′ ∈ Ŵ that Γ0 R◇ Γ′ and ϕ ∈ Γ′ .
Proof:
Let Γ0 ∈ Ŵ .
1. (⇐) Assume that ∥{Γ′ ∈ Ŵ ∶ ϕ ∈ Γ′ }∥ > n. We show that E>n ϕ ∈ Γ0 .
If there exist such Γ′ ∈ Ŵ that ϕ ∈ Γ′ and λ(Γ0 , Γ′ ) > n then the case is
trivial. Indeed, since λ(Γ0 , Γ′ ) = m iff E>m−1 ϕ ∈ Γ0 for all ϕ ∈ Γ′ and since
m − 1 ≥ n, by axiom 5 we obtain E>n ϕ ∈ Γ0 . Suppose then that for all such
Γ′ ∈ Ŵ that ϕ ∈ Γ′ λ(Γ0 , Γ′ ) ≤ n. We name the family Γ̄ϕ = {Γ′ ∈ Ŵ ∶ ϕ ∈
Γ′ }. Without loss of generality we assume that Γ̄ϕ is finite and therefore,
includes finite subfamily of distinct primary copies, name them Γ1 , ⋯, Γm .
168
Michal Zawidzki
The rest of members of Γ̄ϕ are copies of one of Γ1 , ⋯, Γm (see Fig. 1). Let
∥Γ̄ϕ ∥ = k. We have that λ(Γ0 , Γi ) = ni for i ≤ m. By Definition 2 it follows
that we have ni copies of each Γi , i ≤ m. Hence, we obtain ∑m
i=1 ni = k
and k > n. Now, by Definition 6 each ni is determined by such formula
ψi ∈ Γi that E=ni ψi ∈ Γ0 and ni = min{n ∶ E=n ψ ∈ Γ0 and ψ ∈ Γi }. Let
ψ1 ∈ Γ1 , ⋯, ψm ∈ Γm be such formulas that E=ni ψi ∈ Γ0 , i ≤ m. Lemma 6
guarantees the existence of the so-called labelling formula for each Γi , i ≤
m. Let χ1 ∈ Γ1 , ⋯, χm ∈ Γm be such labelling formulas. Note that they
distinguish primary copies from each other but do not distinguish copies of
certain primary copy. Let ϑi = ϕ ∧ ψi ∧ χi , i ≤ m. It is easy to notice that
by maximality of Γi , i ≤ m, ϑi ∈ Γi , i ≤ m. ϑi , i ≤ m also work as labelling
formulas, so we have ⊢ ¬(ϑi ∧ ϑj ) for i, j ≤ m, i ≠ j. Indeed, since ⊢ ϑi → ψi ,
and therefore ⊢ (ϑi ∧ϑj ) → (ψi ∧ψj ), we have ⊢ ¬(ϑi ∧ϑj ) for i ≠ j. The last
property of ϑi , i ≤ m that we are going to use is that E=ni ϑi ∈ Γ0 . In fact,
since ⊢ ϑi → ψi and E=ni ψi ∈ Γ0 by Lemma 2 we obtain E=l ϑi ∈ Γ0 , l ≤ ni .
By definition 6 we have that for all formulas ϕ ∈ Γi , i ≤ m, E>ni −1 ϕ ∈ Γ0 .
By definition of E=n we get E=ni ϑi ∈ Γ0 , i ≤ m. Let µ = ⋁i≤m ϑi . Obviously,
µ ∈ Γ0 . Since ϑ1 , ⋯, ϑm ∈ Γ0 and µ ∈ Γ0 fulfil the conditions of Lemma 1,
it is the case that E=∑m
µ = E=k µ ∈ Γ0 . Since ⊢ µ ↔ ϕ ∧ ⋁i≤m (ψi ∧ χi ),
i=1 ni
we obtain ⊢ µ → ϕ and hence by NE ⊢ E=0 ¬(µ → ϕ). Thus, by axiom
1 ⊢ E>k−1 µ → E>k−1 ϕ. Again, by definition of E=n and by the fact that
E=k ∈ Γ0 we have E>k−1 µ ∈ Γ0 , so by MP E=k−1 ϕ ∈ Γ0 . Since k − 1 ≥ n, by
axiom 5 we obtain E>n ϕ ∈ Γ0 .
(⇒) Assume that E>n ϕ ∈ Γ0 . What we have to show is that ∥{Γ′ ∈
Ŵ ∶ ϕ ∈ Γ′ }∥ > n. Since E>n ϕ ∈ Γ0 , by Lemma 5.1 we obtain that there
exists such Γ′ ∈ Ŵ that ϕ ∈ Γ′ and λ(Γ0 , Γ′ ) > 0. Without loss of generality
suppose that the family of such Γ′ , name it Γ̄ϕ is finite. We distinguish
subfamily Γ1 , ⋯, Γm of Γ̄ϕ consisting of primary copies only, the rest of sets
being copies of primary copies. Finiteness of Γ̄ϕ entails λ(Γ0 , Γi ) < ω, i ≤ m.
We prove that k > n, where k = ∑m
i=1 ni . Analogously to the first part of
the proof, we define formulas ϑi = ϕ ∧ ψi ∧ χi for i ≤ m. Let µ = ⋁i≤m ϑi .
We have ⊢ µ ↔ ϕ ∧ ⋁i≤m (ψi ∧ ϑi ). Let ν = ⋁i≤m (ψi ∧ ϑi ). We have then
that µ ↔ ϕ ∧ ν. Therefore, by CT we have that ⊢ (µ ∨ (ϕ ∧ ¬ν)) ↔ ϕ.
Additionally, by CT, for all i ≤ m we have that ⊢ ¬(ϑi ∧ (ϕ ∧ ¬ν)). Finally,
we state that E=0 (ϕ ∧ ¬ν) ∈ Γ0 . Indeed, if it were not the case we would
have E>0 (ϕ ∧ ¬ν) ∈ Γ0 . Hence, by Lemma 4 there would exist such Γ′ ∈ Ŵ
that ϕ∧¬ν ∈ Γ′ and λ(Γ0 , Γ′ ) > 0. It would be the case that Γ′ ∈ Γ̄ϕ , name it
Γh , h ≤ m (without loss of generality, we can take it as one of the primary
Adequacy of the Logic K(En )
169
copies). So we would have ϑh ∈ Γh . Together with ⊢ ¬(ϑh ∧ (ϕ ∧ ¬ν))
it would contradict the consistency of Γh . Let ξ = µ ∨ (ϕ ∧ ¬ν). Since
⊢ ¬(ϑi ∧ ϑj for i, j ≤ m, i ≠ j, ⊢ ¬(ϑi ∧ (ϕ ∧ ¬ν)) for i ≤ m, E=ni ϑ ∈ Γ0 for
i ≤ m and E=0 (ϕ ∧ ¬ν) ∈ Γ0 , this set of formulas satisfies the conditions of
Lemma 1, so we have E=k ξ ∈ Γ0 . Since ⊢ ξ ↔ ϕ and the fact that E=n is
monotone over equivalence, we have E=k ϕ ∈ Γ0 . Hence k > n for it if were
not the case, by definition of E=n we would have ¬E>l ϕ ∈ Γ0 for some l ≤ n
and at the same time, by axiom 5 E>l ϕ ∈ Γ0 which would contradict the
consistency of Γ0 .
2. Assume that ◇ϕ ∈ Γ0 . We will show that there exists such Γ′ ∈ Ŵ
that Γ0 R̂◇ Γ′ and ϕ ∈ Γ′ . Let Σ = {ψ ∶ ◻ψ ∈ Γ0 }. By Fact 4 the set Σ ∪ {ϕ}
is consistent and by Lindenbaum’s lemma it can be extended to a MCS.
We name this extension Γ′ . Now we must show that Γ0 R̂◇ Γ′ . Since Σ ⊆ Γ′ ,
Fact 3 and definition of R◇ assure that Γ0 R̂◇ Γ′ . The only thing that left
is to show that Γ′ ∈ Ŵ . It suffices to prove that Γ′ ∈ SF (Γ0 ), so that
λ(Γ0 , Γ′ ) > 0. Since Γ0 R̂◇ Γ′ it is the case that ◇ψ ∈ Γ0 for all ψ ∈ Γ′ . By
axiom 6 we have that E>0 ψ ∈ Γ0 for all ◇ψ ∈ Γ0 . Therefore for all ψ ∈ Γ′
we have E>0 ψ ∈ Γ0 , which drives to the conclusion.
λ(Γ0 , Γji ) = ni , 1 < i < m, 1 ≤ j ≤ ni
Γ1i
⋰
⋯
∑i=1 ni = k > n
m
Γni i
Primary copy
λ(Γ0 , Γj1 ) = n1 , 1 ≤ j ≤ n1
⋱
λ(Γ0 , Γjm ) = nm , 1 ≤ j ≤ nm
Γn1 1
Γ1m
Primary copy
⋮
⋮
Γnmm
Γ11
Primary copy
Γ0
Fig. 1. The family Γ̄ϕ .
170
Michal Zawidzki
Since the first part of the Existence Lemma was of the form of equivalence, we therefore assured satisfiability of all formulas of one of the forms:
E<n ϕ, E=n ϕ, E>n ϕ for ϕ ∈ form. Proof of the first part of the above lemma
is much more complicated then the second and it is of great importance to
grasp the idea standing behind it. The sole definition of satisfaction family
of a set Γ (Def. 2) does not assure that the condition 1 of the Existence
Lemma holds. Thus, a reader might state such a problem: Let Γ1 , Γ2 ∈ G
and λ(Γ1 , Γ2 ) = n. It means that we copy Γ2 n times. But, the reader
might ask, what guarantee do we have that there is no such Γ3 ∈ G that
λ(Γ1 , Γ3 ) = m > 0 and ϕ ∈ Γ3 . It would mean that there are at least n + m
such sets Γ′ in SF (Γ1 ) that ϕ ∈ Γ′ and that would affect satisfaction of
the formula E=n ϕ ∈ Γ′ . Now, the proof of the latter part of the Existence
Lemma states that it cannot be the case. For if it were, it would also be
the case that E>n+m−1 ϕ ∈ Γ1 (we showed this using labelling formulas) and
that would contradict the consistency of Γ1 .
Now we can prove a lemma that is an easy consequence of the former
definitions and results.
Lemma 9 (Truth Lemma). Let M̂ = ⟨Ŵ , R̂◇ , V̂ ⟩ be a model constructed
for K(En )-MCS Γ. We have M̂, Γ′ ⊧ ϕ iff ϕ ∈ Γ′ for all Γ′ ∈ Ŵ .
Proof:
We proceed by induction on the degree of ϕ. Basic case holds
due to the definition of V̂ , Boolean cases follow from maximality of Γ.
We now consider ϕ = ◇ψ.
(⇒) Assume that M̂, Γ′ ⊧ ◇ψ. By Definition 2 there is such Γ′′ ∈ Ŵ
that Γ′ R̂◇ Γ′′ and M̂, Γ′′ ⊧ ψ. By induction hypothesis ψ ∈ Γ′′ . Hence, by
definition of R̂◇ , ◇ψ ∈ Γ′ .
(⇐) Assume that ◇ψ ∈ Γ′ . Then, by the Existence Lemma, there is
such Γ′′ ∈ Ŵ that Γ′ R̂◇ Γ′′ and ψ ∈ Γ′′ . By induction hypothesis M̂, Γ′′ ⊧ ψ
and since Γ′ R̂◇ Γ′′ , M̂, Γ′ ⊧ ◇ψ.
Now consider ϕ = E>n ψ.
(⇒) Assume that M̂, Γ′ ⊧ E>n ψ. By Definition 2 ∥{Γ ∈ Ŵ ∶ M, Γ ⊧
ψ}∥ > n. By induction hypothesis ∥{Γ ∈ Ŵ ∶ ψ ∈ Γ}∥ > n. By the Existence
Lemma we obtain E>n ψ ∈ Γ.
(⇐) Assume that E>n ψ ∈ Γ′ . By the Existence Lemma we have ∥{Γ ∈
Ŵ ∶ ψ ∈ Γ}∥ > n and by induction hypothesis ∥{Γ ∈ Ŵ ∶ M, Γ ⊧ ψ}∥ > n.
Hence, by Definition 2, M̂, Γ′ ⊧ E>n ψ.
Adequacy of the Logic K(En )
171
We can now conclude our considerations with two theorems.
Theorem 2 (Strong Completeness of K(En )). Logic K(En ) is strongly
complete with respect to the class of all frames F = ⟨W, R◇ ⟩.
Proof:
By Lemma 1 it suffices to show that every K(En )-consistent set
has a model based on a frame of the class of all frames. Given the K(En )consistent set Γ− , we can extend it to a MCS Γ and build a canonical
model for it in a way showed above. As we proved, a K(En )-model M̂ =
⟨Ŵ , R̂◇ , V̂ ⟩ is based on an arbitrary frame. Since Γ− ⊆ Γ, by the Truth
Lemma we obtain M̂, Γ ⊧ Γ− .
Theorem 3 (Adequacy of K(En )). Logic K(En ) is strongly adequate
with respect to the class of all frames F = ⟨W, R◇ ⟩.
Proof:
Soundness follows from Theorem 1. Strong completeness follows from Theorem 2.
The reader has probably already noticed that even though the proof
of adequacy was conducted for the logic K with global counting operators,
it can be automatically adapted for all axiomatizable normal modal logics
supplied with these operators. It is so due to the axiom 6 which makes the
second part of the Existence Lemma always hold regardless of the character
of R◇ relation.
Acknowledgements
The research reported in this paper is a part of the project financed from
the funds supplied by the National Science Centre, Poland (decision
no. DEC-2011/01/N/HS1/01979).
The author thanks to the anonymous reviewers for their valuable comments which helped significantly improve this paper.
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Department of Logic
University of Lodz
Kopcińskiego 16/18
90-232 Lódź
Poland
e-mail: [email protected]