compactness theorem for classical propositional logic

compactness theorem for classical propositional logic

Call a set

Δ

of wff’s of (classical)

propositional logic

sound

if there is an

interpretation

(or valuation)

v

such that

v

⁢

(

A

)

=

1

for every

A

∈

Δ

.

Theorem 1

.

(

Compactness Theorem

)

Δ

is sound iff every finite subset of it is sound.

One direction is

trivial

. The

converse

is trivial too if

Δ

is finite. We now prove that if every finite subset of an

infinite set

Δ

is sound, so is

Δ

. There are two well-known proofs of this, one assumes that the set

V

of

propositional variables

is

countable

, and the other does not. The one that does uses Konig’s lemma, and the other one uses

Tychonoff

’s

theorem

(hence the name compactness theorem).

Proof using Konig’s lemma

.
Let

A

1

,

…

be an enumeration of all the wff’s in

Δ

(this assumes that the set

V

of all propositional variables is countable). Let

i

1

≤

…

be positive integers such that

p

1

,

…

,

p

i

n

are the propositional variables that

occur in

wff’s

A

1

,

…

,

A

n

. Since

{

A

1

,

…

,

A

n

}

is sound, the set

S

n

:=

{

v

∈

2

V

∣

v

⁢

(

A

j

)

=

1

,

j

=

1

,

…

,

n

}

≠

∅

for each

n

. Furthermore,

S

n

+

1

⊆

S

n

. Let

U

n

:=

{

v

⁢

(

p

1

)

⁢

⋯

⁢

v

⁢

(

p

n

)

∣

v

∈

S

n

}

. In other words, each

U

n

is a set of words over

{

0

,

1

}

of length

n

. Then

U

n

is finite and non-empty (because

S

n

≠

∅

) for each

n

. Since

∅

≠

S

n

+

1

⊆

S

n

, there is a word in

U

n

that is a prefix of a word in

U

n

+

1

for each

n

.

Let

T

be the tree with root

r

(some arbitrary symbol) such that

1.

its vertices other than

r

are elements of

U

1

,

…

,

U

n

,

…

.

2.

the

children

of

r

are elements of

U

1

,

3.

if

ℓ

∈

U

n

is a vertex, then its children are elements of

U

n

+

1

with prefix

ℓ

It’s easy to see that

T

is finitely

branching

, since each

U

n

is finite. Furthermore, by

induction

, if

ℓ

∈

U

n

+

1

, then its prefix of length

n

, an element of

U

n

, is a vertex

ℓ

′

of

T

. So

ℓ

, as a child of

ℓ

′

, is a vertex

T

. Since each

U

n

≠

∅

for each

n

, the

T

has infinitely many vertices. As a result, we apply Konig’s lemma, so that

T

has an

infinite

branch

T

′

. It is easy to see that

T

′

corresponds to an infinite word

α

over

{

0

,

1

}

, which in turn corresponds to an interpretation

v

such that

v

⁢

(

p

i

)

is the

i

’th letter in

α

. It is now clear that

v

⁢

(

A

i

)

=

1

for each

A

i

∈

Δ

, hence

Δ

is sound.

□

Proof using Tychonoff’s theorem

.
For any set

Δ

of wff’s, define

S

⁢

(

Δ

)

:=

{

v

∈

2

V

∣

v

⁢

(

A

)

=

1

,

for all

⁢

A

∈

Δ

}

. It is easy to see that

S

⁢

(

⋃

{

Δ

i

∣

i

∈

I

}

)

=

⋂

{

S

⁢

(

Δ

i

)

∣

i

∈

I

}

(1)

for

v

is in the former set iff

v

⁢

(

A

)

=

1

for all

A

∈

⋃

{

Δ

i

∣

i

∈

I

}

iff

v

⁢

(

A

i

)

=

1

for all

A

i

∈

Δ

i

,

i

∈

I

, iff

v

in the later set.

Next, equip

2

:=

{

0

,

1

}

with the

discrete topology

, and

2

V

the

product topology

. Since

2

is

compact

, so is

2

V

by the Tychonoff’s theorem, and therefore the

finite intersection property

applies. In

addition

, for any wff

A

,

S

⁢

(

{

A

}

)

is closed in

2

V

, since there are only a finite number of propositional variables occurring in

A

. As a result, by equation

(

1

)

,

S

⁢

(

Δ

)

is closed for any set

Δ

of wff’s.

Now, suppose every finite subset of

Δ

is sound. In particular, every singleton subset of

Δ

is sound. This means that for each

A

∈

Δ

,

S

⁢

(

{

A

}

)

≠

∅

. Let

𝔇

:=

{

S

(

{

A

}

)

∣

A

∈

Δ

)

. Then every member of

𝔇

is closed, and, by

assumption

and equation

(

1

)

, the

intersection

of any finite family of members of

𝔇

is non-empty, which means that

⋂

𝔇

≠

∅

. But

⋂

𝔇

=

S

⁢

(

Δ

)

by equation

(

1

)

, the result follows.

□

Remark

. It can be shown that the general version of the compactness theorem is

equivalent

to the prime ideal theorem in

Boolean algebra

.

By the compactness theorem, one can show that soundness and

consistency

are the semantical and syntactical sides of the same idea.

Lemma 1

.

Soundness and consistency are the same on

finite sets

.

Proof.

If

A

1

,

…

,

A

n

⊢

⟂

, then

⊢

B

, where

B

is

A

1

→

(

A

2

→

(

⋯

→

(

A

n

→

⟂

)

⋯

)

, or

v

⁢

(

B

)

=

1

for all

v

, or

v

⁢

(

A

1

)

=

0

or

v

(

A

2

→

(

⋯

→

(

A

n

→

⟂

)

⋯

)

)

=

1

, or

v

⁢

(

A

1

)

=

0

or

v

⁢

(

A

2

)

=

0

or

⋯

or

v

⁢

(

A

n

)

=

0

for all

v

, by induction, or

v

⁢

(

A

i

)

=

0

for some

i

, where

i

=

1

,

…

,

n

, which means

{

A

1

,

…

,

A

n

}

is not sound.

Conversely, if

{

A

1

,

…

,

A

n

}

is not sound, then

v

⁢

(

A

i

)

=

0

for some

i

, where

i

=

1

,

…

,

n

for all

v

. Renumber if necessary, so that the indices

1

and

i

are switched, and

v

⁢

(

A

1

)

=

0

for all

v

. Let

B

be the

formula

A

2

→

(

⋯

→

(

A

n

→

⟂

)

⋯

)

Then

v

(

A

1

→

B

)

=

1

for all

v

. By the

completeness theorem

,

⊢

A

1

→

B

, and therefore

A

1

,

…

,

A

n

⊢

⟂

by the

deduction theorem

n

times.
∎

Proposition 1

.

Soundness and consistency are the same on all sets.

Proof.

For one direction, see here (

http://planetmath.org/PropertiesOfConsistency

). Now, suppose

Δ

is

consistent

. Then every finite subset of it is consistent, and therefore sound by the last

proposition

, hence

Δ

itself is sound by the compactness theorem. ∎

A set

Δ

of wff’s is

uniquely sound

if there is a unique interpretation

v

such that

v

⁢

(

A

)

=

1

for every

A

∈

Δ

.

Corollary 1

.

A set is

maximally consistent

iff it is a uniquely sound theory.

Proof.

If

Δ

is maximally consistent, then it is a theory (see here (

http://planetmath.org/MaximallyConsistent

)), and sound by the last corollary. Suppose

v

1

⁢

(

A

)

=

v

2

⁢

(

A

)

=

1

for all

A

∈

Δ

. If

v

1

≠

v

2

, then there is some wff

B

∉

Δ

such that

v

1

⁢

(

B

)

≠

v

2

⁢

(

B

)

. So one of them is

1

, say

v

1

⁢

(

B

)

=

1

. Then

Δ

∪

{

B

}

is sound, and therefore consistent by the last corollary, contradicting the maximality of

Δ

.

We actually proved more: any maximally consistent

Δ

is

{

A

∣

v

⁢

(

A

)

=

1

}

, where

v

is the unique interpretation such that

v

⁢

(

A

)

=

1

for all

A

∈

Δ

. One direction is obvious. The other direction comes from the fact that

{

A

∣

v

⁢

(

A

)

=

1

}

is consistent and

Δ

is maximal.

Conversely, suppose a theory

Δ

is uniquely sound. Then

Δ

=

Ded

⁢

(

Δ

)

=

⋂

{

Γ

∈

W

L

∣

Δ

⊆

Γ

}

, where

W

L

is the set of all maximally consistent sets of the logic

L

. Suppose

{

Γ

∈

W

L

∣

Δ

⊆

Γ

}

has at least two

elements

, say,

Γ

1

,

Γ

2

. Then by the last paragraph, they are both uniquely sound. Let

v

1

,

v

2

be the two respective unique interpretations such that

v

i

⁢

(

A

)

=

1

for every

A

∈

Γ

i

,

i

=

1

,

2

. Since

Γ

1

≠

Γ

2

, there is a wff

B

∈

Γ

1

-

Γ

2

such that

v

1

⁢

(

B

)

=

1

and

v

2

⁢

(

B

)

=

0

. In addition,

v

1

⁢

(

A

)

=

v

2

⁢

(

A

)

=

1

for all

A

∈

Δ

, which forces

v

1

=

v

2

because

Δ

is uniquely sound. But this contradicts the

inequality

v

2

⁢

(

B

)

≠

v

2

⁢

(

B

)

. Therefore,

{

Γ

∈

W

L

∣

Δ

⊆

Γ

}

has exactly one element, which is

Δ

.
∎

Remark

. It is easy to see that

Δ

v

:=

{

A

∣

v

⁢

(

A

)

=

1

}

is maximally consistent for any interpretation

v

: it is sound, and therefore consistent; it is maximal, for given any wff

B

, either

v

⁢

(

B

)

=

1

or

v

⁢

(

B

)

=

0

, so that either

B

∈

Δ

v

or

¬

⁢

B

∈

Δ

v

. Coupled with the corollary above, there is a

one-to-one correspondence

v

↦

Δ

v

between interpretations and maximally consistent sets of a logic.

Title

compactness theorem for classical propositional logic

Canonical name

CompactnessTheoremForClassicalPropositionalLogic

Date of creation

2013-03-22 19:35:21

Last modified on

2013-03-22 19:35:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03B05

Defines

sound

Defines

uniquely sound

\@unrecurse

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