Kripke semantics for modal propositional logic

Kripke semantics for modal propositional logic

A

Kripe model

for a modal

propositional logic

PL

M

is a triple

M

:=

(

W

,

R

,

V

)

, where

1.

W

is a set, whose

elements

are called

possible worlds

,

2.

R

is a

binary relation

on

W

,

3.

V

is a

function

that takes each wff (

well-formed formula

)

A

in PL

M

to a subset

V

⁢

(

A

)

of

W

, such that

–

V

⁢

(

⟂

)

=

∅

,

–

V

(

A

→

B

)

=

V

(

A

)

c

∪

V

(

B

)

,

–

V

⁢

(

□

⁢

A

)

=

V

⁢

(

A

)

□

, where

S

□

:=

{

s

∣

↑

s

⊆

S

}

, and

↑

s

:=

{

t

∣

s

⁢

R

⁢

t

}

.

For derived

connectives

, we also have

V

⁢

(

A

∧

B

)

=

V

⁢

(

A

)

∩

V

⁢

(

B

)

,

V

⁢

(

A

∨

B

)

=

V

⁢

(

A

)

∪

V

⁢

(

B

)

,

V

⁢

(

¬

⁢

A

)

=

V

⁢

(

A

)

c

, the

complement

of

V

⁢

(

A

)

, and

V

⁢

(

⋄

A

)

=

V

⁢

(

A

)

⋄

:=

V

⁢

(

A

)

c

⁢

□

⁢

c

.

One can also define a

satisfaction relation

⊧

between

W

and the set

L

of wff’s so that

⊧

w

A

iff

w

∈

V

⁢

(

A

)

for any

w

∈

W

and

A

∈

L

. It’s easy to see that

•

⊧̸

w

⟂

for any

w

∈

W

,

•

⊧

w

A

→

B

iff

⊧

w

A

implies

⊧

w

B

,

•

⊧

w

A

∧

B

iff

⊧

w

A

and

⊧

w

B

,

•

⊧

w

A

∨

B

iff

⊧

w

A

or

⊧

w

B

,

•

⊧

w

¬

⁢

A

iff

⊧̸

w

A

,

•

⊧

w

□

⁢

A

iff for all

u

such that

w

⁢

R

⁢

u

, we have

⊧

u

A

,

•

⊧

w

⋄

A

iff there is a

u

such that

w

⁢

R

⁢

u

and

⊧

u

A

.

When

⊧

w

A

, we say that

A

is true at world

w

.

The pair

ℱ

:=

(

W

,

R

)

in a Kripke model

M

:=

(

W

,

R

,

V

)

is also called a (Kripke) frame, and

M

is said to be a model based on the frame

ℱ

. The

validity

of a wff

A

at different levels (in a model, a frame, a

collection

of frames) is defined in the parent entry (

http://planetmath.org/KripkeSemantics

).

For example, any

tautology

is valid in any model.

Now, to prove that any tautology is valid, by the completeness of PL

c

, every tautology is a

theorem

, which is in turn the result of a

deduction

from axioms using

modus ponens

.

First, modus ponens preserves validity: for suppose

⊧

w

A

and

⊧

w

A

→

B

. Since

⊧

w

A

implies

⊧

w

B

, and

⊧

w

A

by assumption, we have

⊧

w

B

. Now,

w

is arbitrary, the result follows.

Next, we show that each axiom of PL

c

is valid:

•

A

→

(

B

→

A

)

: If

⊧

w

A

and

⊧

w

B

, then

⊧

w

A

, so

⊧

w

B

→

A

.

•

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

: Suppose

⊧

w

A

→

(

B

→

C

)

,

⊧

w

A

→

B

, and

⊧

w

A

. Then

⊧

w

B

→

C

and

⊧

w

B

, and therefore

⊧

w

C

.

•

(

¬

A

→

¬

B

)

→

(

B

→

A

)

: we use a different approach to show this:

V

(

(

¬

A

→

¬

B

)

→

(

B

→

A

)

)

=

V

(

¬

A

→

¬

B

)

c

∪

V

(

B

→

A

)

=

(

V

⁢

(

¬

⁢

A

)

∩

V

⁢

(

¬

⁢

B

)

c

)

∪

V

⁢

(

B

)

c

∪

V

⁢

(

A

)

=

(

V

⁢

(

A

)

c

∩

V

⁢

(

B

)

)

∪

V

⁢

(

B

)

c

∪

V

⁢

(

A

)

=

(

V

⁢

(

A

)

c

∪

V

⁢

(

B

)

c

)

∪

V

⁢

(

A

)

=

W

.

In

addition

, the rule of

necessitation

preserves validity as well: suppose

⊧

w

A

for all

w

, then certainly

⊧

u

A

for all

u

such that

w

⁢

R

⁢

u

, and therefore

⊧

w

□

⁢

A

.

There are also valid

formulas

that are not tautologies. Here’s one:

□

(

A

→

B

)

→

(

□

A

→

□

B

)

Proof.

Let

w

be any world in

M

. Suppose

⊧

w

□

(

A

→

B

)

. Then for all

u

such that

w

⁢

R

⁢

u

,

⊧

u

A

→

B

, or

⊧

u

A

implies

⊧

u

B

, or for all

u

such that

w

⁢

R

⁢

u

,

⊧

u

A

, implies that for all

u

such that

w

⁢

R

⁢

u

,

⊧

u

B

, or

⊧

w

□

⁢

A

implies

⊧

w

□

⁢

B

, or

⊧

w

(

□

A

→

□

B

)

. Therefore,

⊧

w

□

(

A

→

B

)

→

(

□

A

→

□

B

)

.
∎

From this, we see that

Kripke semantics

is appropriate only for normal modal logics.

Below are some examples of Kripke frames and their corresponding validating logics:

1.

A

→

□

⁢

A

is valid in a frame

(

W

,

R

)

iff

R

weak

identity

:

w

⁢

R

⁢

u

implies

w

=

u

.

Proof.

Let

(

W

,

R

)

be a frame validating

A

→

□

⁢

A

, and

M

a model based on

(

W

,

R

)

, with

V

⁢

(

p

)

=

{

w

}

. Then

⊧

w

p

. So

⊧

w

□

⁢

p

, or

⊧

u

p

for all

u

such that

w

⁢

R

⁢

u

. But then

u

∈

V

⁢

(

p

)

, or

u

=

w

. Hence

R

is the

relation

: if

w

⁢

R

⁢

u

, then

w

=

u

.

Conversely

, suppose

(

W

,

R

)

is weak identity,

M

based on

(

W

,

R

)

, and

w

a world in

M

with

⊧

w

A

. If

w

⁢

R

⁢

u

, then

u

=

w

, which means

⊧

u

A

for all

u

such that

w

⁢

R

⁢

u

. In other words,

⊧

w

□

⁢

A

, and therefore,

⊧

w

A

→

□

⁢

A

.
∎

2.

□

⁢

A

is valid in a frame

(

W

,

R

)

iff

R

=

∅

.

Proof.

First, suppose

□

⁢

A

is valid in

(

W

,

R

)

, and

M

a model based on

(

W

,

R

)

, with

V

⁢

(

p

)

=

∅

. Since

⊧

w

□

⁢

p

,

⊧

u

p

for any

u

such that

w

⁢

R

⁢

u

. Since no such

u

exists, and

w

is arbitrary,

R

=

∅

.

Conversely, given a model

M

based on

(

W

,

∅

)

, and a world

w

in

M

, it is

vacuously true

that

⊧

u

A

for any

u

such that

w

⁢

R

⁢

u

, since no such

u

exists. Therefore

⊧

w

□

⁢

A

.
∎

A logic is said to be sound if every theorem is valid, and

complete

if every valid wff is a theorem. Furthermore, a logic is said to have the finite model property if any wff in the class of finite frames is a theorem.

Title

Kripke semantics for modal propositional logic

Canonical name

KripkeSemanticsForModalPropositionalLogic

Date of creation

2013-03-22 19:33:22

Last modified on

2013-03-22 19:33:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

30

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Related topic

ModalLogic

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