modal logic S5

modal logic S5

The

modal logic

S5

is the smallest

normal modal logic

containing the following schemas:

•

(T)

□

⁢

A

→

A

, and

•

(5)

⋄

A

→

□

⋄

A

.

S5

is also denoted by

KT5

, where

T

and

5

correspond to the schemas T and 5 respectively.

In this entry (

http://planetmath.org/ModalLogicT

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

reflexive

.

A

binary relation

R

on a set

W

is said to be

Euclidean

iff for any

u

,

v

,

w

,

u

⁢

R

⁢

v

and

u

⁢

R

⁢

w

imply

v

⁢

R

⁢

w

.

R

being Euclidean is first-order

definable

:

∀

u

∀

v

∀

w

(

(

u

R

v

∧

u

R

w

)

→

v

R

w

)

.

Proposition 1

.

5 is valid in a frame

F

iff

F

is Euclidean.

Proof.

First, let

ℱ

be a frame validating 5. Suppose

w

⁢

R

⁢

x

and

w

⁢

R

⁢

y

. Let

M

be a model based on

ℱ

, with

V

⁢

(

p

)

=

{

x

}

. Since

⊧

x

p

, we have

⊧

w

⋄

p

, and so

⊧

w

□

⋄

p

, or

⊧

u

⋄

p

for all

u

such that

w

⁢

R

⁢

u

. In particular,

⊧

y

⋄

p

. So there is a

z

such that

y

⁢

R

⁢

z

and

⊧

z

p

. But this means

z

=

x

, whence

y

⁢

R

⁢

x

, meaning

R

is Euclidean.

Conversely

, suppose

ℱ

is a Euclidean frame, and

M

a model based on

ℱ

. Suppose

⊧

w

⋄

A

. Then there is a

v

such that

w

⁢

R

⁢

v

and

⊧

v

A

. Now, for any

u

with

w

⁢

R

⁢

u

, we have

u

⁢

R

⁢

v

since

R

is Euclidean. So

⊧

u

⋄

A

. Since

u

is arbitrary,

⊧

w

□

⋄

A

, and therefore

⊧

w

⋄

A

→

□

⋄

A

.
∎

Now, a

relation

is both reflexive and Euclidean iff it is an

equivalence relation

:

Proof.

Suppose

R

is both reflexive and Euclidean. If

a

⁢

R

⁢

b

, since

a

⁢

R

⁢

a

,

b

⁢

R

⁢

a

so

R

is

symmetric

. If

a

⁢

R

⁢

b

and

b

⁢

R

⁢

c

, then

b

⁢

R

⁢

a

since

R

has just been proven symmetric, and therefore

a

⁢

R

⁢

c

, or

R

is

transitive

. Conversely, suppose

R

is an equivalence relation. If

a

⁢

R

⁢

b

and

a

⁢

R

⁢

c

, then

b

⁢

R

⁢

a

since

R

is symmetric, so that

b

⁢

R

⁢

c

since

R

is transitive. Hence

R

is Euclidean.
∎

This also shows that

S5

=

KTB4

,

where B is the schema

A

→

□

⋄

A

, valid in any symmetric frame (see here (

http://planetmath.org/ModalLogicB

)), and 4 is the schema

□

⁢

A

→

□

⁢

□

⁢

A

, valid in any transitive frame (see here (

http://planetmath.org/ModalLogicS4

)). It is also not hard to show that

S5

=

KDB4

=

KDB5

,

where

D

is the schema

□

⁢

A

→

⋄

A

, valid in any serial frame (see here (

http://planetmath.org/ModalLogicD

)).

As a result,

Proposition 2

.

S5

is sound in the class of equivalence frames.

Proof.

Since any

theorem

A

in

S5

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 5, which are valid in Euclidean frames by the

proposition

above, and applications of

modus ponens

and

necessitation

, both of which

preserve

validity

in any frame,

A

is valid in any frame which is both reflexive and Euclidean, and hence an equivalence frame.
∎

In

addition

, using the

canonical model

of

S5

, which is based on an equivalence frame, we have

Proposition 3

.

S5

is

complete

in the class of equivalence frames.

Proof.

By the discussion above, it is enough to show that the canonical frame of

S5

is reflexive, symmetric, and transitive. Since

S5

contains T, B, and 4,

ℱ

𝐒𝟓

is reflexive, symmetric, and transitive respectively, the proofs of which can be found in the corresponding entries on

T

,

B

, and

S4

.
∎

Remark

. Alternatively, one can also show that the canonical frame of the

consistent

normal logic containing 5 must be Euclidean.

Proof.

Let

Λ

be such a logic. Suppose

u

⁢

R

Λ

⁢

v

and

u

⁢

R

Λ

⁢

w

. We want to show that

v

⁢

R

Λ

⁢

w

, or

Δ

v

:=

{

B

∣

□

⁢

B

∈

v

}

⊆

w

. Let

A

be any wff. If

A

∉

w

,

A

∉

Δ

u

since

u

⁢

R

Λ

⁢

v

, so

□

⁢

A

∉

u

by the

definition

of

Δ

u

, or

¬

⁢

□

⁢

A

∈

u

since

u

is maximal, or

⋄

¬

⁢

A

∈

u

by

substitution theorem

on

¬

⁢

□

⁢

A

↔

⋄

¬

⁢

A

, or

□

⋄

¬

⁢

A

∈

u

by modus ponens on 5 and the fact that

u

is

closed under

modus ponens. This means that

⋄

¬

⁢

A

∈

Δ

u

by the definition of

Δ

u

, or

⋄

¬

⁢

A

∈

v

since

u

⁢

R

Λ

⁢

v

, so that

¬

⁢

□

⁢

A

∈

v

by the substitution theorem on

⋄

¬

⁢

A

↔

¬

⁢

□

⁢

A

, which means

□

⁢

A

∉

v

since

v

is maximal, or

A

∉

Δ

v

by the definition of

Δ

v

.
∎

Title

modal logic S5

Canonical

name

ModalLogicS5

Date of creation

2013-03-22 19:34:04

Last modified on

2013-03-22 19:34:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Classification

msc 03B42

Defines

S5

Defines

5

Defines

Euclidean

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