canonical model

canonical model

Recall that a logic is a set of wff’s containing all

tautologies

and

closed under

modus ponens

. Given a logic

Λ

, a set

Δ

of wff’s is

Λ

-

consistent

if

⟂

can not be deduced from

Δ

given

Λ

.

Λ

itself is said to be

Λ

-consistent if

⟂

can not be deduced from the

empty set

. Let

Λ

be a consistent

normal modal logic

. The

canonical frame

for

Λ

is the

Kripke frame

ℱ

Λ

:=

(

W

Λ

,

R

Λ

)

, where

1.

W

Λ

is the set of all

maximally consistent

sets, and

2.

w

⁢

R

Λ

⁢

u

iff

□

⁢

A

∈

w

implies

A

∈

u

for any wff

A

.

If we define

Δ

w

:=

{

B

∣

□

⁢

B

∈

w

}

, then the second condition above reads

w

⁢

R

Λ

⁢

u

iff

Δ

w

⊆

u

.

The

canonical model

of

Λ

based on

ℱ

Λ

is the pair

M

Λ

:=

(

ℱ

Λ

,

V

Λ

)

, where

•

V

⁢

(

p

)

:=

{

w

∈

W

Λ

∣

p

∈

w

}

.

The main result regarding the canonical model of

Λ

is:

Theorem 1

.

M

Λ

⊧

A

iff

Λ

⊢

A

, where

A

is any wff.

Since the logic

Λ

is the

intersection

of all maximally consistent sets (see here (

http://planetmath.org/LindenbaumsLemma

)), the

theorem

is the result of the following:

Proposition 1

.

M

Λ

⊧

w

A

iff

A

∈

w

.

which is the result of the following:

Lemma 1

.

For any world

w

in

M

Λ

,

□

⁢

A

∈

w

iff

A

∈

u

for all worlds

u

such that

w

⁢

R

Λ

⁢

u

.

Proof.

Suppose

□

⁢

A

∈

w

and

w

⁢

R

Λ

⁢

u

. Then

A

∈

u

by the definition of

R

Λ

.

Conversely

, suppose

A

∈

u

for all

u

such that

w

⁢

R

Λ

⁢

u

. In other words,

A

∈

u

for all

u

such that

Δ

w

⊆

u

, or

A

∈

⋂

{

u

∣

Δ

w

⊆

u

}

. But

⋂

{

u

∣

Δ

w

⊆

u

}

=

Ded

⁢

(

Δ

w

)

, the

deductive closure

of

Δ

w

, so

Δ

w

⊢

A

, and therefore

□

⁢

Δ

w

⊢

□

⁢

A

(see here (

http://planetmath.org/SyntacticPropertiesOfANormalModalLogic

)), or

w

⊢

□

⁢

A

(since

Δ

w

⊆

w

), or

□

⁢

A

∈

w

(since

w

is maximally consistent).
∎

Proof of

Proposition

1

. We do

induction

on the number

n

of

logical connectives

in

A

. If

n

=

0

, then

A

is either a

propositional variable

or

⟂

. The former is just the definition of

V

Λ

. The later case is just the definition of

Λ

-

consistency

. Next, if

A

is

B

→

C

, then

M

Λ

⊧

w

A

iff either

M

Λ

⊧̸

w

B

or

M

Λ

⊧

w

C

iff

B

∉

w

or

C

∈

w

iff

¬

⁢

B

∈

w

or

C

∈

w

iff

¬

⁢

B

∨

C

∈

w

iff

A

∈

w

. Finally, if

A

is

□

⁢

B

, then

M

Λ

⊧

w

□

⁢

B

iff

□

⁢

B

∈

w

iff

B

∈

u

for all

u

such that

w

⁢

R

Λ

⁢

u

iff

M

Λ

⊧

u

B

for all

u

such that

w

⁢

R

Λ

⁢

u

.

Recall that a logic is

complete

in a frame if it is complete in every model based on the frame. As a corollary to Theorem 1, we have

Corollary 1

.

Λ

is complete in its canonical frame

F

Λ

.

Proof.

Any wff valid in every model based on

ℱ

Λ

is valid in

M

Λ

in particular, and therefore a theorem of

Λ

by Theorem 1. ∎

The

converse

is not true. There are in fact normal modal logics that are sound in no frames at all.

Canonical models are useful in proving the

completeness theorems

for many common normal modal logics. To prove that a logic is complete in a class of frames, by the corollary above, it is enough to show that the canonical frame is in the class. Here are two examples:

1.

Let

Λ

be the smallest normal logic containing the schema

□

⁢

A

. Then

Λ

is complete in the class of null frames.

Proof.

By the discussion above, it is enough to show that

ℱ

Λ

is a null frame: the

assumption

∃

w

⁢

∃

u

⁢

(

w

⁢

R

Λ

⁢

u

)

leads to a

contradiction

. Suppose

w

⁢

R

Λ

⁢

u

. Then

Δ

w

⊆

u

. This means that if

□

⁢

A

∈

w

, then

A

∈

u

. But

□

⁢

A

is a theorem (in

Λ

),

□

⁢

A

∈

w

for any wff

A

. This means that

A

∈

u

for any wff

A

, or that

u

is inconsistent, a contradiction.
∎

2.

Let

Λ

be the smallest normal logic containing the schema

A

→

□

⁢

A

. Then

Λ

is complete in the class of weak

identity

frames (a

binary relation

R

is weak identity it is

satisfies

the condition

∀

x

∀

y

(

x

R

y

→

x

=

y

)

).

Proof.

Again, we show that

R

Λ

is weak identity. Suppose

w

⁢

R

Λ

⁢

u

. Then for any

A

,

□

⁢

A

∈

w

implies that

A

∈

u

. Now, if

A

∈

w

, then applying modus ponens to

A

→

□

⁢

A

, we get that

□

⁢

A

∈

w

since

w

is closed under modus ponens. But this means that

A

∈

u

. So

w

⊆

u

. But since both

w

and

u

are maximal, they are the same.
∎

Title

canonical model

Canonical name

CanonicalModel

Date of creation

2013-03-22 19:35:01

Last modified on

2013-03-22 19:35:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B45

Defines

canonical frame

\@unrecurse

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