Kripke semantics for intuitionistic propositional logic

Kripke semantics for intuitionistic propositional logic

A

Kripe model

for intuitionistic

propositional logic

PL

i

is a triple

M

:=

(

W

,

≤

,

V

)

, where

1.

W

is a set, whose

elements

are called

possible worlds

,

2.

≤

is a

preorder

on

W

,

3.

V

is a

function

that takes each wff (

well-formed formula

)

A

in PL

i

to a subset

V

⁢

(

A

)

of

W

, such that

–

if

p

is a

propositional variable

,

V

⁢

(

p

)

is

upper closed

,

–

V

⁢

(

A

∧

B

)

=

V

⁢

(

A

)

∩

V

⁢

(

B

)

,

–

V

⁢

(

A

∨

B

)

=

V

⁢

(

A

)

∪

V

⁢

(

B

)

,

–

V

⁢

(

¬

⁢

A

)

=

V

⁢

(

A

)

#

,

–

V

(

A

→

B

)

=

(

V

(

A

)

-

V

(

B

)

)

#

,

where

S

#

:=

(

↓

S

)

c

, the

complement

of the lower closure of any

S

⊆

W

.

Remarks

.

•

If

⟂

were used as a

primitive

symbol instead of

¬

, then we require that

V

⁢

(

⟂

)

=

∅

. Then introducing

¬

by

¬

⁢

A

:=

A

→

⟂

, we get

V

⁢

(

¬

⁢

A

)

=

V

⁢

(

A

)

#

.

•

Some simple

properties

of

#

: for any subset

S

of

W

,

S

#

is upper closed. This means that for any wff

A

,

V

⁢

(

A

)

is upper closed. Also,

S

and

S

#

are

disjoint

, which means that

V

⁢

(

A

∧

¬

⁢

A

)

=

∅

for any

A

.

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

•

for any propositional variable

p

, if

⊧

w

p

and

w

≤

u

, then

⊧

u

p

,

•

⊧

w

A

∧

B

iff

⊧

w

A

and

⊧

w

B

,

•

⊧

w

A

∨

B

iff

⊧

w

A

or

⊧

w

B

,

•

⊧

w

¬

⁢

A

iff for all

u

such that

w

≤

u

, we have

⊧̸

u

A

•

⊧

w

A

→

B

iff for all

u

such that

w

≤

u

, we have

⊧

u

A

implies

⊧

u

B

.

When

⊧

w

A

, we say that

A

is true at world

w

.

Remark

. Since

V

⁢

(

A

)

is upper closed,

⊧

w

A

implies

⊧

u

A

for any

u

such that

w

≤

u

. Now suppose

w

≤

u

and

u

≤

w

, then

⊧

w

A

iff

⊧

u

A

. This shows that, as far as

validity

of

formulas

is concerned, we can take

≤

to be a

partial order

in the

definition

above.

Some examples of Kripke models:

1.

Let

M

1

be the model consisting of

W

=

{

w

,

u

}

,

≤

=

{

(

w

,

w

)

,

(

u

,

u

)

,

(

w

,

u

)

}

, with

V

⁢

(

p

)

=

{

u

}

and

V

⁢

(

q

)

=

W

. Then

V

⁢

(

p

)

#

=

V

⁢

(

q

)

#

=

∅

, and we have the following:

–

V

⁢

(

p

∨

¬

⁢

p

)

=

{

u

}

.

–

V

(

q

→

p

)

=

V

(

p

)

, and

V

(

¬

p

→

¬

q

)

=

W

, so

V

(

(

¬

p

→

¬

q

)

→

(

q

→

p

)

)

=

{

w

}

#

=

{

u

}

.

–

V

(

p

→

q

)

=

V

(

¬

q

→

¬

p

)

=

W

, so

V

(

(

¬

q

→

¬

p

)

→

(

p

→

q

)

)

=

∅

#

=

W

.

–

V

(

(

p

→

q

)

∨

(

q

→

p

)

)

=

W

.

–

In fact, for any wff’s

A

,

B

, either

V

⁢

(

A

)

⊆

V

⁢

(

B

)

or

V

⁢

(

B

)

⊆

V

⁢

(

A

)

, since

≤

is

linearly ordered

, so that

V

(

(

A

→

B

)

∨

(

B

→

A

)

)

=

V

(

A

→

B

)

∪

V

(

B

→

A

)

=

W

,

assuming

V

⁢

(

A

)

⊆

V

⁢

(

B

)

.

2.

Let

M

2

be the model consisting of

W

=

{

w

,

u

,

v

}

,

≤

=

{

(

w

,

w

)

,

(

u

,

u

)

,

(

v

,

v

)

,

(

w

,

u

)

,

(

w

,

v

)

}

, with

V

⁢

(

p

)

=

{

u

}

and

V

⁢

(

q

)

=

{

v

}

. Then

–

V

⁢

(

¬

⁢

p

)

=

V

⁢

(

p

)

#

=

{

v

}

,

–

V

⁢

(

¬

⁢

¬

⁢

p

)

=

V

⁢

(

¬

⁢

p

)

#

=

{

u

}

,

–

so

V

⁢

(

¬

⁢

p

∨

¬

⁢

¬

⁢

p

)

=

{

u

,

v

}

.

–

V

(

p

→

q

)

=

V

(

p

)

#

=

{

v

}

,

–

V

(

q

→

p

)

=

V

(

q

)

#

=

{

u

}

,

–

so

V

(

(

p

→

q

)

∨

(

q

→

p

)

)

=

{

u

,

v

}

.

3.

Let

M

be an arbitrary model. Then

–

V

(

A

∧

B

→

A

)

=

(

V

(

A

∧

B

)

-

V

(

A

)

)

#

=

W

,

–

V

(

A

→

A

∨

B

)

=

(

V

(

A

)

-

V

(

A

∨

B

)

)

#

=

W

,

–

V

(

A

→

(

B

→

A

)

)

=

(

V

(

A

)

-

V

(

B

→

A

)

)

#

=

(

V

(

A

)

-

(

V

(

B

)

-

V

(

A

)

)

#

)

#

=

W

. The last

equation

comes from the fact that for any upper set

S

,

S

⊆

S

c

⁢

#

.

–

Suppose

V

(

A

)

=

V

(

A

→

B

)

=

W

. Then

∅

=

↓

(

V

(

A

)

-

V

(

B

)

)

=

↓

(

V

(

B

)

c

)

. Since

V

⁢

(

B

)

is upper,

V

⁢

(

B

)

c

is lower, so

∅

=

↓

(

V

(

B

)

c

)

=

V

(

B

)

c

, or

W

=

V

⁢

(

B

)

. This shows that

modus ponens

preserves

validity.

4.

Let

W

be any set and

≤

=

W

2

. Then for any wff

A

, either

V

⁢

(

A

)

=

W

or

V

⁢

(

A

)

=

∅

. Therefore,

V

⁢

(

¬

⁢

¬

⁢

A

)

=

V

⁢

(

A

)

, and

V

(

¬

¬

A

→

A

)

=

W

.

The pair

ℱ

:=

(

W

,

≤

)

in a Kripke model

M

:=

(

W

,

≤

,

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 various levels can be found in the parent entry. Furthermore,

A

is valid (with respect to Krikpe

semantics

) for PL

i

if it is valid in the class of all frames.

Based on the examples above, we see that

1.

(

¬

q

→

¬

p

)

→

(

p

→

q

)

is valid in

M

1

, while

(

¬

p

→

¬

q

)

→

(

q

→

p

)

is not.

2.

(

p

→

q

)

∨

(

q

→

p

)

is valid in the class of linearly ordered frames, while it is not valid in

M

2

, and neither is

¬

⁢

p

∨

¬

⁢

¬

⁢

p

.

3.

It is not hard to see that

¬

⁢

A

∨

¬

⁢

¬

⁢

A

is valid in any weakly

connected

frame, that is, for any

w

∈

W

, the set

{

u

∣

w

≤

u

}

is linear.

4.

Any wff in any of the schemas

A

∧

B

→

A

,

A

→

A

∨

B

, or

A

→

(

B

→

A

)

is valid in PL

i

. See remark below for more detail.

5.

Any

theorem

in the classical propositional logic is valid in any

universal

frame, that is, a frame with a universal relation.

Remark

. It can be shown that every theorem of PL

i

is valid. This is the soundness theorem of PL

i

.

Conversely

, every valid wff is a theorem. This is known as the

completeness theorem

of PL

i

. Furthermore, a wff valid in the class of finite frames is a theorem. This is the finite model property of PL

i

.

Title

Kripke semantics for intuitionistic propositional logic

Canonical

name

KripkeSemanticsForIntuitionisticPropositionalLogic

Date of creation

2013-03-22 19:33:19

Last modified on

2013-03-22 19:33:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

35

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B55

Classification

msc 03B20

Related topic

AxiomSystemForIntuitionisticLogic

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