(5) quantifier: ∀(all).
The definition of formulas (expressed by α, β, γ) and the substitution of
formulas α(y/x) (y is substitution free for x in α) are as usual, so are the
definitions of ∧, ∨, ↔ and ◇.
On the basis of a modal propositional calculus S, we can attain the modal
predicate calculus LS with axioms scheme:
(1) ∀x(α→β)→(∀xα→∀xβ);
Neighborhood Semantics of
Modal Predicate Logic*
Liu Zhuang-hu
(2) ∀xα→α(y/x), y is substitution free for x in α;
Department of Philosophy of Peking University, 100871
On the basis of a modal propositional logic S, we can naturally establish the
predicate logic LS by supplementing S with axioms and deduction rules about
predicates. However, there has been no satisfactory for LS yet. The difficulty lies
in the Barcan Formula, ∀x□α→□∀xα, which will be abbreviated as BF in the
following context. One simple semantics is to attach an independent domain D of
individuals, and to get a predicate relation frame <D, W, R>. But all these frames
satisfy BF while in general BF is not a theorem of LS. So, it is LS +BF that this
semantics corresponds. In widely used Kripke Semantics, though BF is not valid,
the axioms and deduction rules about predicate also are no longer valid. Therefore,
this semantics corresponds to a more complicated modal predicate logic whose
axioms and deduction rules about predicate are changed.
In this paper a neighborhood semantics for modal predicate logic is
constructed on the basis of modal propositional logic. The completeness with
respect to both model and frame under the neighbourhood semantics are given.
Also some discussion is given to the meaning of BF.
1. Neighborhood Semantics for Modal predicate Logic
The formal language of modal predicate logic is as follows :
(1) Regular System
LC = L+K( |− □(α→β)→(□α→□β))+RM(if |− α→β, then |− □α→□β)
(2) Normal System
LK = L+K+RN(if |− α, then |− □α)
(3) LT = LK+T( |− □α→α)
(4) LD = LK+D( |− □α→◇α)
We can get RM from K and RN, D from T. So, LC ⊆ LK ⊆ LD ⊆ LT.
Now we construct the neighborhood semantics for modal predicate logic.
Definition 2 (Frame)
K = <D, W, N> is called a frame, where D is a
non-empty set (of individuals), W a non-empty set (of possible worlds), and N:
W→ P(P(W)).
Definition 3 (Value-Assignment and Model)
Let K = <D, W, N> be a
frame, σ, τ functions on X and P×W respectively where X is the set of vasiables
and P the set of predicates (σ is called individual function and τ called relation
function) and V = <σ, τ>. V is called a K-assignment if
(1) individual variables: x, y, z,…;
(2) predicates: F, G, …, with every predicate has n arguments and n≥1;
(3) connectives: ¬(negative), →(implication);
(4) propositional operator: □(necessity);
*
(3) α→∀xα, x is not free in α
and deduction rule (generalization):
if |− α, then |− ∀xα
Proof and theoremhood in LS are defined as usual. |− α means that α is a
theorem. The set of all the theorems is denoted by Th(LS). The deduction of LS,
however is different for that of classical logic.
Definition 1 (Deduction)
Let A be a set of formulas, and α a formula.
α is deducted from A, written as A |− α, if there are formulas α1,…, αn∈A, such
that |− α1→…→αn→α.
The following is some systems which will be discussed in this paper. They
are obtained by adding some axiom scheme and deduction rules about the
necessity operator to the classical predicate calculus L.
本文原载于《南京大学学报》1998年数学半年刊，略有修改。
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(1) for all x∈X, σ(x)∈D;
(2) for any F∈P and u∈W, τ(F, u) is a n-ary relation on D, i.e., τ(F, u) ⊆ Dn.
and M = <K, V> is called a model
Definition 4
Let K = <D, W, N> be a frame, and V a K-Assignment.
For any variable x, for any a∈D, we may define a new assignment V(a/x) =
<σ(a/x), τ>, where
⎧σ ( y )
y≠x
⎩ a
y=x
σ(a/x)(y) = ⎨
2. Canonical Model and Model-Completeness
Definition 5 (Value of Formulas on Assignment)
Let K = <D, W, N>
be a frame, and V = <σ, τ> a K-assignment. The value V(α) ⊆ W of formula α on
the assignment V defined by induction:
(1) V(Fx1…xn) = {u | <σ(x1),…, σ(xn)>∈τ(F,u)}(i.e. u∈V(Fx1…xn) iff
<σ(x1),…, σ(xn)>∈τ(F,u));
(2) V(¬α) = W \ V(α)(i.e., u∈V(¬α) iff u∉V(α));
(3) V(α→β) = (W \ V(α))∪V(β)(i.e., u∈V(α→β) iff u∉V(α) or u∈V(β));
(4) V(□α) = {u | V(α)∈N(u)}(i.e., u∈V(□α) iff V(α)∈N(u));
(5) V(∀xα) = ∩{V(a/x)(α) | a∈D}(i.e., u∈V(∀xα) iff u∈V(a/x)(α) for
any a∈D).
Intuitively, V(α) is the set of all the possible worlds in which α is true. So,
α is true on u iff u∈V(α).
Definition 6 (Satisfication)
Let K = <D, W, N> be a frame, and M =
<K, V> a model.
(1) (M satisfying α) M |= α =df u∈V(α) for any u∈W (i.e.V(α) = W).
(2) (K satisfying α) K |= α =df <K,V> |=α for any K-assignment V.
The definitions of consistent, maximal consistent and Henkin Set are as
usual. They have some property similar to the classical predicate calculus.
Lemma 1
If A is a maximal consistent set of formulas, then
(1) (deduction closed property) if A |− α, then α∈A.
(2) ¬α∈A iff α∉A.
(3) α→β∈A iff α∉A or β∈A.
(4) Th(LS) ⊆ A.
Lemma 2
If B is a consistent finite set of formulas, then there exists a
maximal consistent Henkin Set (abbreviated as H-set below) A such that B ⊆ A.
Lemma 3
If |−/ α, then there exists a H-set A such that α∉A.
Lemma 4
If |−/ α→β, then there exists a H-set A such that α∈A and
β∉B.
Definition 9 (Canonical Model)
Suppose that LS is a modal predicate
calculus, K = <D, W, N> is a frame, where D is the set of all variables, W is the
set of all H-sets, and V is a K-assignation. For any formula α, let
|α| = {A | A∈W and α∈A}.
K is called a canonical frame if
(1) |α|∈N(A) iff □α∈A for any formula α;
(3) (M satisfying A) M |= A =df M |= α for any α∈A.
(4) (K satisfying A) K |= A =df K |= α for any α∈A.
All the frames satisfy the theorems of classical predicate calculus.
Theorem 1 (The Consistency of Classical Predicate Calculus)
any frame K,
(1) If α is an axiom of classical predicate calculus, then K |= α.
Definition 7 (Model-Completeness)
Let LS be a modal predicate
calculus. LS is called complete with respect to model if there exists a model M
such that |− α iff M |= α.
Definition 8 (Frame-Completeness) Let LS be a modal predicate calculus.
LS is called complete with respect to frame if there exists a frame K such that
|− α iff K |= α.
(2) V(Fx1…xn) = |Fx1…xn| for any F∈P and x1,…, xn∈X.
For
and M = <K, V> is called a canonical model of LS.
Lemma 5
(1) |¬α| = W \ |α|.
(2) |α→β| = (W \ |α|)∪|β|.
(3) |∀xα| = ∩{|α(y/x)| | y is a variable}.
(2) If K |= α and K |= α→β, then K |= β
(3) If K |= α, then K |= ∀xα.
There are two kinds of completeness in to the neighborhood semantics.
(4) |α| ⊆ |β| iff |− α→β.
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(5) |α| = W iff |− α.
Theorem 2
If M = <K, V> is a canonical model, then for any formula
α, V(α) = |α| (i.e., A∈V(α) iff α∈A).
Proof
Inductive proof
(1) α = Fx1…xn, by definition;
(2) α = ¬β, α = β→γ and α = ∀xβ, by (1),(2),(3) of lemma 5;
(3) α = □β, then V(β) = |β| by inductive hypothesis, so
A∈V(α) iff V(β)∈N(A) iff |β|∈N(A) iff □β∈A iff α∈A.
Theorem 3
Let LS be a modal predicate calculus. If LS has canonical
model then LS is complete with respect to model.
Proof
For any formula α, |− α iff |α| = W iff V(α) = W iff M |= α.
Theorem 4 Let LS be a modal predicate calculus. If RM holds in LS,
then LS has canonical model, so LS is complete with respect to model.
Proof Let N(A) = {S | □β∈A and |β| ⊆ S for some β}. We justify that
N(A) satisfies the condition of (1) for canonical model.
If □α∈A, then |α|∈N(A) follows from □α∈A and |α| ⊆ |α|.
If |α|∈N(A), then
□β∈A
and |β| ⊆ |α| for some formula β,
so
|− β→α,
by Lemma 5(4)
|− □β→□α,
by RM
(2) if S∈Σ and Q∈Σ, then S∩Q∈Σ;
(3) W∈Σ.
Lemma 6
K = <D, W, N> is a frame, F is a unary predicate, x and y is a
variable. Then for any S,Q ⊆ W, there exists a K-assignment V such that V(Fx) =
S and V(Fy) = Q.
Proof
Given an individual function σ such that σ(x)≠σ(y), we define
the relation function τ as following:
⎧{σ ( x), σ ( y )}
⎪
τ(F,u) = ⎨ {σ ( x)}
⎪
⎩
{σ ( y )}
∅
u ∈ S and u ∈ Q
u ∈ S and u ∉ Q
u ∉ S and u ∈ Q
u ∉ S and u ∉ Q
then
σ(x)∈τ(F,u) iff u∈S and σ(y)∈τ(F,u) iff u∈Q.
Let V = <σ, τ> be a K-assignment, then
V(Fx) = {u | σ(x)∈τ(F,u)}={u | u∈S} = S
and
V(Fy) = {u | σ(y)∈τ(F,u)} = {u | u∈Q} = Q.
Lemma 7
If K is a frame, then
(1) K |= α iff V(α) = W for any K-assignment V.
(2) K |= α1→…→αn→β iff V(α1)∩…∩V(αn) ⊆ V(β) for any
K-assignment V.
(3) V(α)∩V(α→β) = V(α∧β) for any K-assignment V.
Lemma 8
If K is a LK-frame, then
(1) K |= Fx∨¬Fx,
□α∈A.
by □β∈A and the deduction closed property of A
By Theorem 4 we can easily obtain:
Theorem 5 LC, LK, LD, LT are complete with respect to model.
(2) K |= □(Fx∨¬Fx),
3. Characterizing Frame and Frame-Completeness
(3) K |= □(α∧β)→□β,
Now we use canonical frame and characterizing frame to discuss the framecompleteness of LS.
Definition 10 (Characterizing Frame)
Let LS be a modal predicate
calculus, and K a frame. K is called a Characterizing Frame of LS, abbreviated as
LS-frame, if K |= Th(LS).
LK-frame can be simply described.
Definition 11 (Filter)
Σ is a subset of P(W). Σ is called a filter on W if
(1) if S∈Σ and S ⊆ Q, then Q∈Σ;
(4) K |= □α→(□β→□(α∧β)).
Theorem 6
Let K = <D, W, N> be a frame. If K is a LK-frame, then
N(u) is a filter on W for any u∈W.
Proof We justify that N(u) satisfies the there requirements of filter for any
u∈W.
(1) If S∈N(u) and S ⊆ Q, we take a K-assignment V such that
V(Fx) = S and V(Fy) = Q,
then
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by lemma 6
V(Fx∧Fy) = S∩Q = S∈N(u) and u∈V(□(Fx∧Fy)),
for any K-assignment V, V(α)=W
by K |= □(Fx∧Fy)→□Fy, we get
if u∈W, then
W∈N(u), V(α)∈N(u) and u∈V(□α),
V(□(Fx∧Fy)) ⊆ V(□Fy)
so
filter
u∈V(□Fy), V(Fy)∈N(u) and Q∈N(u).
(2) If S, Q∈N(u), we take a K-assignment V such that
So
V(Fx) = S and V(Fy) = Q,
V(□α) = W,
K |= □α.
by lemma 7(1)
For LC, LD and LT, the correspondent characterizing filters satisfy similar
descriptive condition.
Theorem 8
If K = <D, W, N> is a frame, then
by lemma 6
then
V(Fx),V(Fy)∈N(u) and V(Fx∧Fy) = S∩Q
so
(1) K is LC-frame iff N(u) satisfies (1),(2) of definition 11 for any u∈W.
u∈V(□Fx), u∈V(□Fy) and u∈V(□Fx)∩V(□Fy),
(2) K is LD-frame iff N(u) is a proper filter (which satisfies ∅∉N(u)) on W
for any u∈W.
by K|= □Fx→(□Fy→□(Fx∧Fy)), we get
V(□Fx)∩V(□Fy) ⊆ V(□(Fx∧Fy)),
so
u∈V(□(Fx∧Fy)), V(Fx∧Fy)∈N(u) and S∩Q∈N(u).
(3) Given a assignment V, then
V(Fx∨¬Fx) = W,
by K |= Fx∨¬Fx
V(□(Fx∨¬Fx)) = W,
by K |= □(Fx∨¬Fx)
so
u∈V(□(Fx∨¬Fx)), V(Fx∨¬Fx)∈N(u) and W∈N(u).
Theorem 7
Let K = <D, W, N> be a frame. If N(u) is a filter on W for
any u∈W, then K is a LK-frame.
Proof
It is only necessary to show (1) K|= □(α→β)→(□α→□β)) and (2)
if K |= α, then K |= □α.
(1) Given any K-assignment V. If u∈V(□(α→β))∩V(□α), then
V(α→β),V(α)∈N(u)
V(α→β)∩V(α)∈N(u),
V(α∧β)∈N(u),
V(β)∈N(u) and u∈V(□β),
by the condition (3) of
by the condition (2) of filter
(3) K is LT-frame iff for any u∈W, N(u) is a proper filter on W and u∈S for
any S∈N(u).
By means of the definitions of characterizing filter and canonical filter, we
can prove the completeness with respect to frame of various modal predicate
calculus LS by showing that the canonical frame of LS is a LS -frame.
Theorem 9
LK is complete with respect to frame.
Proof
It is sufficient to show that in K = <D, W, N>, the canonical
frame of LK, N(A) is a filter on W for any A∈W.
(1) If S∈N(A) and S ⊆ Q
then
□α∈A
and |α| ⊆ S for some formula α
so
|α| ⊆ Q and Q∈N(u).
(2) If S, Q∈N(A)
then
□α, □β∈A,
by lemma 7(3)
by the condition (3) of filter
|α| ⊆ S and |β| ⊆ Q for some formulas α, β
so
|α∧β| = |α|∩|β| ⊆ S∩Q
So
V(□(α→β))∩V(□α)⊆V(β)
K |= □(α→β)→(□α→□β).
(2) If K |= α, then
□(α∧β)∈A,
S∩Q∈N(u),
(3) Given α, we get
by lemma 7(2)
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by |−(□α∧□β)→□(α∧β) and □α, □β∈A
by |α∧β| ⊆ S∩Q.
□(α∨¬α)∈A,
by |− □(α∨¬α)
Theorem 12
Let K = <D, W, N> be a frame, and λ = card(D). If N(u) is
λ+-complete for any u∈W then K |= BF.
Proof
The elements of D can be well ordered into {ai | i<λ} since
card(D) = λ.
For any K-assignment V, for any u∈V(∀x□α), we get
so
W∈N(A).
by |α∨¬α| ⊆ W.
It is similar to prove:
Theorem 10
LC, LD and LT are complete with respect to frame.
u∈V(ai/x)(α) for any ai∈D,
4. Barcan Formula
so
We will discuss the meaning of Barcan Formula only in the characterizing
frame of LK. Here in this section all the frames mean LK-frame.
The cardinal number of a set X is written as card(X). The successor of
cardinal number λ is denoted as λ+.
Definition 12 (λ-complete)
Let Σ be a filter on W, and λ a cardinal
number. Σ is called λ-complete if ∩Γ∈Σ for any Γ ⊆ Σ such that card(Γ)<λ.
Theorem 11
K = <D, W, N> is a frame, λ = card(D). If K |= BF then
N(u) is λ+-complete for any u∈W.
Proof
We will prove for any Γ ⊆ N(u), if card(Γ)<λ+ then ∩Γ∈N(u).
The elements of D can be well ordered into {ai | i<λ} since card(D) = λ, Γ can
also be well ordered into {Si | i<λ} since card(Γ)<λ+.
Given an individual function σ and a unary predicate F, we may define a
relation function τ as τ(F, u) = {ai | u∈Si} for any u∈W, also a K-assignment V =
<σ, τ>. Then
by card({V(ai/x)(α) | ai∈D})≤λ and λ+-complete of N(u), we get
V(ai/x)(Fx) = {u | σ(ai/x)(x)∈τ(F,u)} = {u | ai∈τ(F,u)}
= {u | u∈Si} = Si∈N(u)
∩{V(ai/x)(α) | ai∈D}∈N(u), V(∀xα)∈N(u) and u∈V(□∀xα),
so
V(∀x□α) ⊆ V(□∀xα),
K |= ∀x□α→□∀xα.
by lemma 7(2)
We can prove that every relation frame satisfies Barcan Formula with the
help of this result. We call frames in the following context neighborhood frames
in order not to be confused with relational frame.
Definition 11 (Principal Filter)
Σ is a filter on W. Σ is called principal
filter if there exists S ⊆ W such that Σ = {Q | S ⊆ Q}.
Lemma 9
If Σ is principal filter, then Σ is λ+-complete for any cardinal
number λ.
Theorem 13
If K0 = <D, W, R> is a relation frame, then there exists a
neighborhood frame K = <D, W, N> such that
(1) K0 |= α iff K |= α for any formula α.
(2) N(u) is principal filter for any u∈W.
Proof
Let
N: W→ P(P(W))
N(u) = {S | R(u) ⊆ S}, where R(u) = {v | uRv},
and
V(∀xFx) = ∩{V(ai/x)(Fx) | ai∈D} = ∩{u | u∈Si} = ∩Γ.
For any ai∈D, we get
u∈V(ai/x)(□Fx),
V(ai/x)(α)∈N(u),
then K= <D, W, N> is a neighborhood frame, which satisfy (1) and (2).
Theorem 14
K0 |= BF for any relation frame K0.
Theorem 15
If modal predicate calculus LS is complete with respect to
relation frame, then BF∈Th(LS).
by V(ai/x)(Fx)∈N(u)
so
u∈∩{V(ai/x)(□Fx) | ai∈D} and u∈V(∀x□Fx),
by K |= ∀x□Fx→□∀xFx, we get
References
V(∀x□Fx) ⊆ V(□∀xFx)
[1] Hughes,G and Cresswell,M.J. An Introduction to Modal Logic.
so
[2] Segerberg,K. An Essay in Classical Modal Logic.
u∈V(□∀xFx), V(□∀xFx)∈N(u) and∩Γ∈N(u).
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[3] Scott,D. Advice on Modal Logic, Philosophy Problems in Logic.
[4] Liu Zhuanghu Neighborhood semantics and completeness with respect to model,
Journal of Peking University (Philosophy and Social Sciences), No.3, 1995, pp52-56.
[5] Mao yi
Neighborhood semantics, Information of Philosophy (Supplement), 1994, 10,
pp152-155.
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