Kripke semantics

Kripke semantics

A

Kripke frame

(or simply a frame)

ℱ

is a pair

(

W

,

R

)

where

W

is a non-empty set whose

elements

are called

worlds

or

possible worlds

,

R

is a

binary relation

on

W

called the

accessibility relation

. When

v

⁢

R

⁢

w

, we say that

w

is

accessible

from

v

. A Kripke frame is said to have

property

P

if

R

has the property

P

. For example, a

symmetric

frame is a frame whose accessibility relation is symmetric.

A

Kripke model

(or simply a model)

M

for a propositional

logical system

(classical, intuitionistic, or modal)

Λ

is a pair

(

ℱ

,

V

)

, where

ℱ

:=

(

W

,

R

)

is a Kripke frame, and

V

is a

function

that takes each

atomic formula

of

Λ

to a subset of

W

. If

w

∈

V

⁢

(

p

)

, we say that

p

is

true

at world

w

. We say that

M

is a

Λ

-model

based on

the frame

ℱ

if

M

=

(

ℱ

,

V

)

is a model for the logic

Λ

.

Remark

. Associated with each world

w

, we may also define a Boolean-valued

valuation

V

w

on the set of all wff’s of

Λ

, so that

V

w

⁢

(

p

)

=

1

iff

w

∈

V

⁢

(

p

)

. In this sense, the Kripke semantics can be thought of as a

generalization

of the truth-value semantics for classical propositional logic. The truth-value semantics is just a Kripke model based on a frame with one world.

Conversely

, given a

collection

of valuations

{

V

w

∣

w

∈

W

}

, we have model

(

ℱ

,

V

)

where

w

∈

V

⁢

(

p

)

iff

V

w

⁢

(

p

)

=

1

.

Since the

well-formed formulas

(wff’s) of

Λ

are uniquely readable,

V

may be inductively extended so it is defined on all wff’s. The following are some examples:

•

in classical

propositional logic

PL

c

,

V

(

A

→

B

)

:=

V

(

A

)

c

∪

V

(

B

)

, where

S

c

:=

W

-

S

,

•

in the modal propositional logic

K

,

V

⁢

(

□

⁢

A

)

:=

V

⁢

(

A

)

□

, where

S

□

:=

{

u

∣

↑

u

⊆

S

}

, and

↑

u

:=

{

w

∣

u

⁢

R

⁢

w

}

, and

•

in intuitionistic propositional logic PL

i

,

V

(

A

→

B

)

:=

(

V

(

A

)

-

V

(

B

)

)

#

, where

S

#

:=

(

↓

S

)

c

, and

↓

S

:=

{

u

∣

u

⁢

R

⁢

w

,

w

∈

S

}

.

Truth at a world can now be defined for wff’s: a wff

A

is

true

at world

w

if

w

∈

V

⁢

(

A

)

, and we write

M

⊧

w

A

or

⊧

w

A

if no confusion arises. If

w

∉

V

⁢

(

A

)

, we write

M

⊧̸

w

A

. The three examples above can be now interpreted as:

•

⊧

w

A

→

B

means

⊧

w

A

implies

⊧

w

B

in PL

c

,

•

⊧

w

□

⁢

A

means for all worlds

v

with

w

⁢

R

⁢

v

, we have

⊧

v

A

in

K

, and

•

⊧

w

A

→

B

means for all worlds

v

with

w

⁢

R

⁢

v

,

⊧

v

A

implies

⊧

v

B

in PL

i

.

A wff

A

is said to be

valid

•

in a model

M

if

A

in true at all possible worlds

w

in

M

,

•

in a frame if

A

is valid in all models

M

based on

ℱ

,

•

in a collection

𝒞

of frames if

A

is valid in all frames in

𝒞

.

We denote

M

⊧

A

,

ℱ

⊧

A

,

or

𝒞

⊧

A

if

A

is valid in

M

,

ℱ

, or

𝒞

respectively.

A logic

Λ

, equipped with a

deductive system

, is

sound

in

𝒞

if

⊢

A

implies

𝒞

⊧

A

.

Here,

⊢

A

means that wff

A

is a

theorem

deducible

from the deductive system of

Λ

. Conversely, if

𝒞

⊧

A

implies

⊢

A

,

we say that

Λ

is

complete

in

𝒞

.

Title

Kripke semantics

Canonical

name

KripkeSemantics

Date of creation

2013-03-22 19:31:15

Last modified on

2013-03-22 19:31:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B48

Classification

msc 03B20

Classification

msc 03B45

Defines

Kripke frame

Defines

possible world

Defines

accessibility relation

Defines

accessible

Defines

valid

Defines

sound

Defines

complete

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