maximally consistent

maximally consistent

A set

Δ

of

well-formed formulas

(wff) is

maximally consistent

if

Δ

is

consistent

and any consistent

superset

of it is itself:

Δ

⊆

Γ

with

Γ

consistent implies

Γ

=

Δ

.

Below are some basic

properties

of a maximally consistent set

Δ

:

1.

Δ

is

deductively closed

(

Δ

is a theory):

Δ

⊢

A

iff

A

∈

Δ

.

2.

Δ

is

complete

:

Δ

⊢

A

or

Δ

⊢

¬

⁢

A

for any wff

A

.

3.

for any wff

A

, either

A

∈

Δ

or

¬

⁢

A

∈

Δ

.

4.

If

A

∉

Δ

, then

Δ

∪

{

A

}

is not consistent.

5.

Δ

is a logic:

Δ

contains all

theorems

and is

closed under

modus ponens

.

6.

⟂

∉

Δ

.

7.

A

→

B

∈

Δ

iff

A

∈

Δ

implies

B

∈

Δ

.

8.

A

∧

B

∈

Δ

iff

A

∈

Δ

and

B

∈

Δ

.

9.

A

∨

B

∈

Δ

iff

A

∈

Δ

or

B

∈

Δ

.

Proof.

1.

If

A

∈

Δ

, then clearly

Δ

⊢

A

.

Conversely

, suppose

Δ

⊢

A

. Let

ℰ

be a

deduction

of

A

from

Δ

, and

Γ

:=

Δ

∪

{

A

}

. Suppose

Γ

⊢

B

. Let

ℰ

1

be a deduction of

B

from

Γ

, then

ℰ

,

ℰ

1

is a deduction of

B

from

Δ

, so

Δ

⊢

B

. Since

Δ

⊬

⟂

,

Γ

⊬

⟂

, so

Γ

is consistent. Since

Δ

is maximal,

Γ

=

Δ

, or

A

∈

Δ

.

2.

Suppose

Δ

⊬

A

, then

A

∉

Δ

by 1. Then

Δ

∪

{

A

}

is not consistent (since

Δ

is maximal), which means

Δ

,

A

⊢

⟂

, or

Δ

⊢

A

→

⟂

, or

Δ

⊢

¬

⁢

A

.

3.

If

A

∉

Δ

, then

Δ

⊬

A

by 1, so

Δ

⊢

¬

⁢

A

by 2, and therefore

¬

⁢

A

∈

Δ

by 1 again.

4.

If

A

∉

Δ

, then

¬

⁢

A

∈

Δ

by 3., so that

¬

⁢

A

,

A

,

⟂

is a deduction of

⟂

from

Δ

∪

{

A

}

, showing that

Δ

∪

{

A

}

is not consistent.

5.

If

A

is a theorem, then

Δ

⊢

A

, so that

A

∈

Δ

by 1. If

A

∈

Δ

and

A

→

B

∈

Δ

, then

A

,

A

→

B

,

B

is a deduction of

B

from

Δ

, so

B

∈

Δ

by 1.

6.

This is true for any consistent set.

7.

Suppose

A

→

B

∈

Δ

. If

A

∈

Δ

, then

B

∈

Δ

since

Δ

is closed under modus ponens. Conversely, suppose

A

∈

Δ

implies

B

∈

Δ

. This means that

Δ

,

A

⊢

B

. Then

Δ

⊢

A

→

B

by the

deduction theorem

, and therefore

A

→

B

∈

Δ

by 1.

8.

Suppose

A

∧

B

∈

Δ

, then by modus ponens on theorems

A

∧

B

→

A

and

A

∧

B

→

B

, we get

A

,

B

∈

Δ

, since

Δ

is a logic by 5. Conversely, suppose

A

,

B

∈

Δ

, then by modus ponens twice on theorem

A

→

(

B

→

A

∧

B

)

, we get

A

∧

B

∈

Δ

by 5.

9.

Suppose

A

∨

B

∈

Δ

. Then

¬

⁡

(

¬

⁢

A

∧

¬

⁢

B

)

∈

Δ

by the

definition

of

∨

, so

¬

⁢

A

∧

¬

⁢

B

∉

Δ

by 3., which means

¬

⁢

A

∉

Δ

or

¬

⁢

B

∉

Δ

by the

contrapositive

of 8, or

A

∈

Δ

or

B

∈

Δ

by 3. Conversely, suppose

A

∈

Δ

or

B

∈

Δ

. Then by modus ponens on theorems

A

→

A

∨

B

or

B

→

A

∨

B

respectively, we get

A

∨

B

∈

Δ

by 5.

∎

The

converses

of 2 and 3 above are true too, and they provide alternative definitions of maximal

consistency

.

1.

any complete consistent theory is maximally consistent.

2.

any consistent set satisfying the condition in 3 above is maximally consistent.

Proof.

Suppose

Δ

is complete consistent. Let

Γ

be a consistent superset of

Δ

.

Γ

is also complete. If

A

∈

Γ

-

Δ

, then

Γ

⊢

A

, so

Γ

⊬

¬

⁢

A

since

Γ

is consistent. But then

Δ

⊬

¬

⁢

A

since

Γ

is a superset of

Δ

, which means

Δ

⊢

A

since

Δ

is complete. But then

A

∈

Δ

since

Δ

is deductively closed, which is a

contradiction

. Hence

Δ

is maximal.

Next, suppose

Δ

is consistent satisfying the condition: either

A

∈

Δ

or

¬

⁢

A

∈

Δ

for any wff

A

. Suppose

Γ

is a consistent superset of

Δ

. If

A

∈

Γ

-

Δ

, then

¬

⁢

A

∈

Δ

by assumption, which means

¬

⁢

A

∈

Γ

since

Γ

is a superset of

Δ

. But then both

A

and

¬

⁢

A

are

deducible

from

Γ

, contradicting the assumption that

Γ

is consistent. Therefore,

Γ

is not a

proper superset

of

Δ

, or

Γ

=

Δ

.
∎

Remarks

.

•

In the converse of 2, we require that

Δ

be a theory, for there are complete consistent sets that are not deductively closed. One such an example is the set

V

of all

propositional variables

: it can be shown that for every wff

A

, exactly one of

V

⊢

A

or

V

⊢

¬

⁢

A

holds.

•

So far, none of the above actually tell us that a maximally consistent set exists. However, by Zorn’s lemma, it is not hard to see that such a set does exist. For more detail, see here (

http://planetmath.org/LindenbaumsLemma

).

•

There is also a

semantic

characterization

of a maximally consistent set: a set is maximally consistent iff there is a unique

valuation

v

such that

v

⁢

(

A

)

=

1

for every wff

A

in the set (see here (

http://planetmath.org/CompactnessTheoremForClassicalPropositionalLogic

)).

Title

maximally consistent

Canonical

name

MaximallyConsistent

Date of creation

2013-03-22 19:35:13

Last modified on

2013-03-22 19:35:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

Classification

msc 03B10

Classification

msc 03B99

Classification

msc 03B45

Related topic

FirstOrderTheories

Defines

complete

\@unrecurse

Generated on Thu Feb 8 19:23:43 2018 by

LaTeXML