modal logic S4

modal logic S4

The

modal logic

S4

is the smallest

normal modal logic

containing the following schemas:

•

(T)

□

⁢

A

→

A

, and

•

(4)

□

⁢

A

→

□

⁢

□

⁢

A

.

In this entry (

http://planetmath.org/ModalLogicT

), we show that T is valid in a frame iff the frame is

reflexive

.

Proposition 1

.

4 is valid in a frame

F

iff

F

is

transitive

.

Proof.

First, suppose

ℱ

is a frame validating 4, with

w

⁢

R

⁢

u

and

u

⁢

R

⁢

t

. Let

M

be a model with

V

⁢

(

p

)

=

{

v

∣

w

⁢

R

⁢

v

}

, where

p

a

propositional variable

. So

⊧

w

□

⁢

p

. By

assumption

, we have

⊧

w

□

⁢

p

→

□

⁢

□

⁢

p

. Then

⊧

w

□

⁢

□

⁢

p

. This means

⊧

v

□

⁢

p

for all

v

such that

w

⁢

R

⁢

v

. Since

w

⁢

R

⁢

u

,

⊧

u

□

⁢

p

, which means

⊧

s

p

for all

s

such that

u

⁢

R

⁢

s

. Since

u

⁢

R

⁢

t

, we have

⊧

t

p

, or

t

∈

V

⁢

(

p

)

, or

w

⁢

R

⁢

t

. Hence

R

is transitive.

Conversely

, let

ℱ

be a transitive frame,

M

a model based on

ℱ

, and

w

any world in

M

. Suppose

⊧

w

□

⁢

A

. We want to show

⊧

w

□

⁢

□

⁢

A

, or for all

u

with

w

⁢

R

⁢

u

, we have

⊧

u

□

⁢

A

, or for all

u

with

w

⁢

R

⁢

u

and all

t

with

u

⁢

R

⁢

t

, we have

⊧

t

A

. If

w

⁢

R

⁢

u

and

u

⁢

R

⁢

t

,

w

⁢

R

⁢

t

since

R

is transitive. Then

⊧

t

A

by assumption. Therefore,

⊧

w

□

⁢

A

→

□

⁢

□

⁢

A

.
∎

As a result,

Proposition 2

.

S4

is sound in the class of

preordered

frames.

Proof.

Since any

theorem

in

S4

is

deducible

from a

finite sequence

consisting of

tautologies

, which are valid in any frame,

instances

of T, which are valid in reflexive frames, instances of 4, which are valid in transitive frames by the

proposition

above, and applications of

modus ponens

and

necessitation

, both of which preserve

validity

in any frame, whence the result.
∎

In

addition

, using the

canonical model

of

S4

, which is preordered, we have

Proposition 3

.

S4

is

complete

in the class of serial frames.

Proof.

Since

S4

contains T, its canonical frame

ℱ

𝐒𝟒

is reflexive. We next show that the canonical frame

ℱ

Λ

of any

consistent

normal logic

Λ

containing the schema 4 must be transitive. So suppose

w

⁢

R

Λ

⁢

u

and

u

⁢

R

Λ

⁢

v

. If

A

∈

Δ

w

:=

{

B

∣

□

⁢

B

∈

w

}

, then

□

⁢

A

∈

w

, or

□

⁢

□

⁢

A

∈

w

by modus ponens on 4 and the fact that

w

is

closed under

modus ponens. Hence

□

⁢

A

∈

Δ

w

, or

□

⁢

A

∈

u

since

w

⁢

R

Λ

⁢

u

, or

A

∈

Δ

u

, or

A

∈

v

since

u

⁢

R

Λ

⁢

v

. As a result,

w

⁢

R

Λ

⁢

v

, and therefore

ℱ

𝐒𝟒

is a preordered frame.
∎

By a proper

translation

, one can map intuitionistic

propositional logic

PL

i

into

S4

, so that a wff of PL

i

is a theorem iff its

translate

is a theorem of

S4

.

Title

modal logic S4

Canonical name

ModalLogicS4

Date of creation

2013-03-22 19:34:01

Last modified on

2013-03-22 19:34:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B42

Classification

msc 03B45

Defines

S4

Defines

4

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