modal logic GL

modal logic GL

The

modal logic

GL

(after Gödel and Löb) is the smallest

normal modal logic

containing the following schema:

•

W:

□

(

□

A

→

A

)

→

□

A

.

GL

is also known as

provability logic

, because it is used to study the provability and

consistency

of

first order Peano arithmetic

.

Recall that 4 is the schema

□

⁢

A

→

□

⁢

□

⁢

A

.

Proposition 1

.

In any normal modal logic,

⊢

W

implies

⊢

4

.

The proof of this requires some

theorems

(

http://planetmath.org/SomeTheoremSchemasOfNormalModalLogic

) and

meta-theorems

(

http://planetmath.org/SyntacticPropertiesOfANormalModalLogic

) of a normal modal logic.

Proof.

We start with the

tautology

A

→

(

(

□

□

A

∧

□

A

)

→

(

□

A

∧

A

)

)

, which an

instance

of the schema

X

→

(

(

Y

∧

Z

)

→

(

Z

∧

X

)

)

. Since

□

⁢

(

□

⁢

A

∧

A

)

↔

□

⁢

□

⁢

A

∧

□

⁢

A

is a theorem in any normal modal logic,

A

→

(

□

(

□

A

∧

A

)

→

(

□

A

∧

A

)

)

is a theorem by the

substitution theorem

. By the syntactic property RM,

□

A

→

□

(

□

(

□

A

∧

A

)

→

(

□

A

∧

A

)

)

is a theorem. Since

□

(

□

(

□

A

∧

A

)

→

(

□

A

∧

A

)

)

→

□

(

□

A

∧

A

)

is an instance of W, by

law of syllogism

,

□

⁢

A

→

□

⁢

(

□

⁢

A

∧

A

)

is a theorem.

Next, from the tautology

□

⁢

A

∧

A

→

□

⁢

A

, we have the theorem

□

⁢

(

□

⁢

A

∧

A

)

→

□

⁢

□

⁢

A

by RM. Combining this with the last theorem in the previous paragraph, we see that, by law of syllogism,

□

⁢

A

→

□

⁢

□

⁢

A

, or 4, is a theorem.
∎

Corollary 1

.

4 is a theorem of

GL

.

A

binary relation

is said to be

converse

well-founded

iff its

inverse

is well-founded.

Proposition 2

.

W is valid in a frame

F

iff

F

is

transitive

and converse well-founded.

Proof.

Suppose first that the schema W is valid in

ℱ

=

(

U

,

R

)

, then any theorem of

GL

is valid in

ℱ

, so in particular 4 is valid in

ℱ

, and hence

ℱ

is transitive (see here (

http://planetmath.org/ModalLogicS4

)). We next show that

R

is converse well-founded. Suppose not. Then there is a non-empty subset

S

⊆

U

such that

S

has no

R

-

maximal element

. We want to find a model

(

U

,

R

,

V

)

such that, for some

propositional variable

p

and some world

u

in

U

,

⊧̸

u

□

(

□

p

→

p

)

→

□

p

, or equivalently,

⊧

u

□

(

□

p

→

p

)

and

⊧̸

u

□

⁢

p

. Let

V

be the

valuation

such that

V

⁢

(

p

)

:=

{

w

∈

U

∣

w

∉

S

}

. Pick any

u

∈

S

. Suppose

u

⁢

R

⁢

v

. To show that

⊧

u

□

(

□

p

→

p

)

, we want to show that

⊧

v

□

⁢

p

→

p

. There are two cases:

•

If

v

∈

S

, then

⊧̸

v

p

. Furthermore, since

S

does not contain an

R

-maximal element, there is a

w

∈

S

such that

v

⁢

R

⁢

w

. Since

w

∈

S

,

⊧̸

w

p

. Since

v

⁢

R

⁢

w

,

⊧̸

v

□

⁢

p

. As a result,

⊧

v

□

⁢

p

→

p

.

•

If

v

∉

S

, then

⊧

v

p

, so that

⊧

v

□

⁢

p

→

p

.

Next, we want to show that

⊧̸

u

□

⁢

p

. Since

u

∈

S

, and

S

does not have an

R

-maximal element, there is a

w

∈

S

such that

u

⁢

R

⁢

w

. Since

w

∈

S

,

⊧̸

w

p

. But since

u

⁢

R

⁢

w

,

⊧̸

u

□

⁢

p

.

Conversely, let

ℱ

be a transitive and converse well-founded frame,

M

a model based on

ℱ

, and

u

a world in

M

. We want to show that

⊧

u

□

(

□

p

→

p

)

→

□

p

. So suppose

⊧̸

u

□

⁢

p

. Then the set

S

:=

{

v

∣

u

⁢

R

⁢

v

⁢

and

⊧̸

v

p

}

is not empty. Since

R

is converse well-founded,

S

has a

R

-maximal element, say

w

. So

u

⁢

R

⁢

w

and

⊧̸

w

p

. Now, if

⊧

w

□

⁢

p

→

p

, then

⊧̸

w

□

⁢

p

, which means there is a

v

such that

w

⁢

R

⁢

v

and

⊧̸

v

p

. But since

R

is transitive and

u

⁢

R

⁢

w

, we get

u

⁢

R

⁢

v

, implying

v

∈

S

, contradicting the

R

-maximality of

w

. Therefore,

⊧̸

w

□

⁢

p

→

p

, or

⊧̸

u

□

(

□

p

→

p

)

. As a result,

⊧

u

□

(

□

p

→

p

)

→

□

p

.
∎

Proposition

2 immediately implies

Corollary 2

.

GL

is sound in the class of transitive and converse well-founded frames.

Remark

. However, unlike many other modal logics,

GL

is not

complete

in the class of transitive and converse well-founded frames. While its

canonical model

(hence the corresponding canonical frame) is transitive (because 4 is valid in it), it is not converse well-founded.

Instead, it can be shown that

GL

is

complete in the restricted class of finite transitive and converse well-founded frames, or equivalently, finite transitive and

irreflexive

frames.

Title

modal logic GL

Canonical name

ModalLogicGL

Date of creation

2013-03-22 19:35:33

Last modified on

2013-03-22 19:35:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Classification

msc 03F45

\@unrecurse

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