Kripke submodel

Kripke submodel

Let

ℱ

=

(

W

,

R

)

be a

Kripke frame

. A

subframe

of

ℱ

is a pair

ℱ

′

=

(

W

′

,

R

′

)

such that

W

′

is a subset of

W

and

R

′

=

R

∩

(

W

′

×

W

′

)

. A

submodel

of a Kripke model

M

=

(

W

,

R

,

V

)

of a modal

propositional logic

PL

M

is a triple

(

W

′

,

R

′

,

V

′

)

where

(

W

′

,

R

′

)

is a subframe of

(

W

,

R

)

, and

V

′

⁢

(

p

)

=

V

⁢

(

p

)

∩

W

′

for each

propositional variable

p

in PL

M

.

The most common submodel of a Kripke model

M

=

(

W

,

R

,

V

)

is constructed by taking

W

w

:=

{

u

∣

w

⁢

R

*

⁢

u

}

, where

R

*

is the

reflexive transitive closure

of

R

, for any

w

∈

W

. The submodel

M

w

:=

(

W

w

,

R

w

,

V

w

)

is called the submodel of

M

generated

by the world

w

, and

ℱ

w

the subframe of

ℱ

generated by

w

. A submodel of

M

is called a

generated submodel

if it is generated by some world

w

in

M

.

Proposition 1

.

For any wff

A

and any

u

∈

W

w

,

M

⊧

u

A

iff

M

w

⊧

u

A

.

Proof.

We do

induction

on the number

n

of

connectives

in

A

.

If

n

=

0

, then

A

is either

⟂

or a propositional variable. Clearly,

M

⊧̸

u

⟂

iff

M

w

⊧̸

u

⟂

, or

M

⊧

u

⟂

iff

M

w

⊧

u

⟂

. If

A

is some propositional variable

p

, then

M

⊧

u

p

iff

u

∈

V

⁢

(

p

)

iff

u

∈

V

⁢

(

p

)

iff

u

∈

V

⁢

(

p

)

and

u

∈

W

w

(by

assumption

) iff

u

∈

V

⁢

(

p

)

∩

M

w

iff

u

∈

V

w

⁢

(

p

)

iff

M

w

⊧

u

p

.

Next, suppose

A

is

B

→

C

. Then

M

⊧

u

A

iff

u

∈

V

⁢

(

B

)

c

∪

V

⁢

(

C

)

iff

u

∈

(

W

-

V

⁢

(

B

)

)

∪

V

⁢

(

C

)

and

u

∈

W

w

(by assumption) iff

u

∈

W

w

-

V

⁢

(

B

)

or

u

∈

V

w

⁢

(

C

)

iff

u

∈

W

w

-

V

w

⁢

(

B

)

or

u

∈

V

w

⁢

(

C

)

iff

M

w

⊧

u

A

.

Finally, suppose

A

is

□

⁢

B

. First let

M

⊧

u

A

. To show

M

w

⊧

u

A

, pick any

v

∈

W

w

such that

u

⁢

R

w

⁢

v

. Then

u

⁢

R

⁢

v

, so that

M

⊧

v

B

, or

v

∈

V

⁢

(

B

)

. Since

v

∈

W

w

,

v

∈

V

w

⁢

(

B

)

, or

M

w

⊧

v

B

, and thus

M

w

⊧

u

A

.

Conversely

, let

M

w

⊧

u

A

. To show

M

⊧

u

A

, pick any

v

∈

W

such that

u

⁢

R

⁢

v

. Since

w

⁢

R

*

⁢

u

,

w

⁢

R

*

⁢

v

so that

v

∈

W

w

. Furthermore

(

u

,

v

)

∈

R

∩

(

W

w

×

W

w

)

=

R

w

. So

M

w

⊧

u

B

, or

v

∈

V

w

⁢

(

B

)

⊆

V

⁢

(

B

)

, which means

M

⊧

u

A

.
∎

Corollary 1

.

M

⊧

A

iff

M

w

⊧

A

for all

w

∈

W

.

Proof.

If

M

⊧

A

, then

M

⊧

u

A

for all

u

∈

W

, so that

M

w

⊧

u

A

for all

u

∈

W

w

and all

w

∈

W

, or

M

w

⊧

A

for all

w

∈

W

. Conversely, if

M

w

⊧

A

for all

w

∈

W

, then in particular

M

w

⊧

w

A

(since

R

*

is

reflexive

) for all

w

∈

W

, or

M

⊧

w

A

for all

w

∈

W

, or

M

⊧

A

.
∎

Corollary 2

.

ℱ

⊧

A

iff

F

w

⊧

A

for all

w

∈

W

.

Proof.

If

ℱ

⊧

A

, then

M

⊧

A

for all

M

based on

ℱ

, or

M

w

⊧

A

, where

M

w

is based on

ℱ

w

, for all

w

∈

W

by the last

corollary

. Since any model based on

ℱ

w

is of the form

M

w

,

ℱ

w

⊧

A

. Conversely, suppose

ℱ

w

⊧

A

for all

w

∈

W

. Let

M

be any model based on

ℱ

. Then

M

w

is based on

ℱ

w

, and therefore

M

w

⊧

A

. Since

w

is arbitrary,

M

⊧

A

by the last corollary, so

ℱ

⊧

A

.
∎

Title

Kripke submodel

Canonical

name

KripkeSubmodel

Date of creation

2013-03-22 19:34:50

Last modified on

2013-03-22 19:34:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Defines

generated submodel

\@unrecurse

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