consequence operator

consequence operator

1

Definition

Let

L

be a set. A

consequence operator

on

L

is a

mapping

1

1

Here, “

𝒫

” denotes the

power set

and

ℱ

denotes the finite power set.

C

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

which

satisfies

the following three properties:

1.

For all

X

⊆

L

, it happens that

X

⊆

C

⁢

(

X

)

.

2.

C

∘

C

=

C

3.

For all

X

,

Y

⊆

L

, if

X

⊆

Y

, then

C

⁢

(

X

)

⊆

C

⁢

(

Y

)

If, in

addition

, the following condition is satisfied, then a consequence operator

C

is known as

finitary

. (Synonyms are “finite consequence operator” and “algebraic consequence operator”.)

•

For all

X

∈

L

, it happens that

C

⁢

(

X

)

=

⋃

Y

∈

ℱ

⁢

(

X

)

C

⁢

(

Y

)

.

It is worth noting that, if the above condition is satisfied, then the third
condition of last paragraph becomes superfluous — as shown in

http://planetmath.org/node/ 8678

an attachment, it automatically follows
from conitions 1 and 2 of last paragraph and the condition stated above.

A consequence operator

C

such that

C

⁢

(

∅

)

=

∅

is called

axiomless

. A consequence operator

C

such that

C

⁢

(

∅

)

≠

∅

is called

axiomatic

.

2

Motivation

Alfred Tarksi introduced consequence operators as a way of discussing the
notion of

conclusions

following from

premises

in a general fashion. Suppose
that the set

L

consists of statements in some

language

. Then, given a
set of statements

X

, let

C

⁢

(

X

)

be the set of all statements which can be
inferred form statements of

X

.

The defining properties of “consequence operator” given above then express
some fundamental facts about the process of inferring conclusions from
premises: Any statement can be concluded from itself. If a statement

s

follows from a set of premises

X

and

Y

is a

superset

of

X

, then

s

also follows from

Y

. If one augments a set of premises by conclusions
derived from those premises, then one can only draw conclusions from the
larger set which could have been drawn from the original set of premises.
Note that these conditions hold for a large class of logics, not just

classical logic

of Aristotle, Boole, and Frege. However, they do not hold
for all logics — in particular, there are the so-called nonmonotonic
logics in which it is not always the case that, if

X

⊂

Y

, then

C

⁢

(

X

)

⊆

C

⁢

(

Y

)

.

In terms of this usage in logic, it is easy to understand the origin of the
terms ”axiomatic” and ”axiomless”. An axiom in a logical theory is a
statement which is assumed true without having to prove it from any other
statement. Hence, an axiom is a

consequence

of the

empty set

, so we call
consequence operators which allow one to deduce conclusions from an empty
set of premises axiomatic.

The distinction of finitary consequence operators has to do with whether one
is permitted to draw a conclusion from an

infinite set

of premises which
could not be drawn from any finite subset thereof. As for why one might
want to do this, consider the following example. Suppose

X

consists of
the following statements:

•

1

=

1

2

•

2

=

1

2

+

1

2

•

3

=

1

2

+

1

2

+

1

2

•

4

=

2

2

•

5

=

2

2

+

1

2

•

6

=

2

2

+

1

2

+

1

2

•

7

=

2

2

+

1

2

+

1

2

+

1

2

•

8

=

2

2

+

2

2

•

9

=

3

2

•

10

=

3

2

+

1

2

•

11

=

3

2

+

1

2

+

1

2

•

12

=

2

2

+

2

2

+

2

2

•

13

=

3

2

+

2

2

•

14

=

3

2

+

2

2

+

1

2

•

⋯

From

X

one would like to be able to draw the conclusion “Any

positive
integer

can be expressed as the sum of at most four squares.”. This
conclusion, however, cannot be inferred from any

proper subset

of

X

, in
particular, from any finite subset of

X

. To make this conclusion would
require a consequence operator which is not finitary.

3

Examples

1.

To begin, there are two

trivial

consequence operators defined on any set.
One is the

identity operator

I

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

defined as

I

⁢

(

X

)

=

X

. The other is the

constant operator

U

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

defined as

U

⁢

(

X

)

=

L

. It is perfectly
straightforward to check that these two

operators

satisfy the defining
properties of consequence operator and, furthermore, that they are both
finitary consequence operators and that

I

is axiomless whilst

U

is
axiomatic. Trivial though they may be, these operators play an important
role as exrtremal

elements

in the

lattice

of all consequence operators
over a given set.

2.

Next, we consider some less trivial consequence operators which can be
defined over an arbitrary set. Let

X

and

Y

be any two subsets of

L

.
Then we may define operators

C

∩

⁢

(

X

,

Y

)

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

and

C

∪

⁢

(

X

,

Y

)

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

as follows:

C

∩

⁢

(

X

,

Y

)

⁢

(

Z

)

=

{

X

∪

Z

Y

∩

Z

≠

∅

Z

Y

∩

Z

=

∅

C

∪

⁢

(

X

,

Y

)

⁢

(

Z

)

=

{

X

∪

Z

Y

∪

Z

=

Z

Z

Y

∪

Z

≠

Z

It is shown that these are indeed consequence operators in an attachment to
this entry.

3.

A much larger class of consequence operators may be defined as follows.
Let

K

be a subset of

𝒫

⁢

(

L

)

which includes

L

. Then, as
shown in an

http://planetmath.org/node/8671

attachment, the
map

C

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

, defined as

C

⁢

(

X

)

=

∩

{

Y

∈

K

∣

X

⊆

Y

}

,

is a consequence operator. As we shall see, all consequence operators
can be obtained by this construction. In particular, the examples discussed
above can be obtained as follows: To obtain

I

, set

K

=

𝐏

⁢

(

L

)

; to
obtain

U

, set

K

=

{

L

}

; to obtain

C

∩

⁢

(

X

,

Y

)

, set

K

=

{

Z

⊆

L

∣

Y

∩

Z

=

∅

}

∪

{

X

∪

Z

∣

Z

⊆

L

∧

Y

∩

Z

≠

∅

}

;

to obtain

C

∪

⁢

(

X

,

Y

)

, set

K

=

{

Z

⊆

L

∣

Y

∪

Z

≠

∅

}

∪

{

X

∪

Z

∣

Z

⊆

L

∧

Y

∪

Z

=

∅

}

.

4.

Turning to more specific examples, we have the example which inspired the
definintion in the first place. Let

L

be a set of logical expressions
constructed from some set of

sentence

letters and

predicate

letters and
the usual

connectives

and

quantifiers

. Given a subset

X

⊆

L

,
let

C

⁢

(

X

)

be the set of all expressions

ψ

for which there exists
a

finite set

of expressions

ϕ

1

,

…

,

ϕ

n

such that

⌜

⁢

ϕ

1

∧

⋯

∧

ϕ

n

⇒

ψ

⁢

⌝

is a

tautology

. Note that this is a finitary consequence operator — it
does not enable one to make the sort of

deductions

from infinite sets
of premises described above.

5.

This notion of consequence operator also applies to areas of mathematics
other than logic. For

instance

, suppose that

L

is a

vector space

. Then
the operator which assigns to a subset of

L

the

vector subspace

which it
spans is a consequence operator. This particular consequence operator is
finitary because if a vector

v

belongs to the span of a set

X

, then

v

can be expressed as a

linear combination

of a finite number of elements
of

X

.

6.

The

closure operator

in

topology

is a
consequence operator. It is worth pointing out that not every consequence
operator can be expressed as the closure operator for some topology because
the closure operator satisfies some extra conditions beyond those which
define consequence operators. Typically, the closure operator is not
finitary because infinite subsets of

topological spaces

may have

limit
points

.

4

Alternative Definition and Generalization

A consequence operator can be characterized by its

fixed points

. Given
a consequence operator

C

:

𝒫

⁢

(

L

)

→

𝒫

⁢

(

L

)

,
set

K

=

{

X

⊆

L

∣

C

⁢

(

X

)

=

X

}

. By the second defining
property of consequence operator, we have

K

=

{

C

⁢

(

X

)

∣

X

∈

L

}

.
One can show that

C

⁢

(

X

)

=

∩

{

Y

∈

K

∣

X

⊆

Y

}

.

Conversely

, suppose that

K

is a subset of

L

with the following minimum property:

•

For every

X

∈

L

, there exists a

Y

∈

K

such
that

X

⊆

Y

and if, for any

Z

∈

K

, if

X

⊆

Z

, then

Y

⊆

Z

.

Then the operator

C

defined as

C

⁢

(

X

)

=

∩

{

Y

∈

K

∣

X

⊆

Y

}

is a consequence operator with

K

as its set of fixed points.

One may also define consequence operators in the more general

context

of
a

partially ordered set

which may not be the power set of any set. Suppose
that

⟨

S

,

≤

⟩

is a partially ordered set. Then we may define
a consequence operator on this

ordered set

to be a map

C

:

S

→

S

which satisfies the following three properties:

1.

For all

X

∈

S

, it happens that

X

≤

C

⁢

(

X

)

.

2.

C

∘

C

=

C

3.

For all

X

,

Y

∈

S

, if

X

≤

Y

, then

C

⁢

(

X

)

≤

C

⁢

(

Y

)

Such more general consequence operators arise frequently when we restrict
attention to distinguished subsets of a set. As an example, we may consider
the following situation. Let

S

be the set of linear subspaces of a

Banach
space

, ordered by

inclusion

. Then the operator

C

:

S

→

S

which
assigns to each

subspace

its Cauchy completion is a consequence operator.

As an example which does not arise this way, let

S

=

ℝ

with the
usual order. Then the

ceiling function

⌈

⋅

⌉

:

ℝ

→

ℝ

is a consequence operator.

For another example, let

S

be the set of all fields
with a

countable

number of elements. This set may be ordered as follows:

E

≤

F

if and only if there exists a non-trivial

morphism

of

E

into

F

. Then the operator which sends each field to its algebraic closure is a
consequence operator.

Title

consequence operator

Canonical name

ConsequenceOperator

Date of creation

2013-03-22 16:28:48

Last modified on

2013-03-22 16:28:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

40

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03G25

Classification

msc 03G10

Classification

msc 03B22

Synonym

closure operator

Defines

finitary consequence operator

Defines

finite consequence operator

Defines

algebraic consequence operator

Defines

axiomatic consequence operator

Defines

axiomless consequence operator

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