quantifier algebra

quantifier algebra

A

quantifier algebra

is a triple

(

B

,

V

,

∃

)

, where

B

is a

Boolean algebra

,

V

is a set, and

∃

is a

function

∃

:

P

⁢

(

V

)

→

B

B

from the

power set

of

V

to the set of functions on

B

, such that

1.

the pair

(

B

,

∃

(

I

)

)

is a

monadic algebra

for each subset

I

⊆

V

,

2.

∃

(

∅

)

=

I

B

, the

identity function

on

B

, and

3.

∃

(

I

∪

J

)

=

∃

(

I

)

∘

∃

(

J

)

, for any

I

,

J

∈

P

⁢

(

V

)

.

The

cardinality

of

V

is called the

degree

of the quantifier algebra

(

B

,

V

,

∃

)

.

Think of

V

as a set of

variables

and

B

a set of

propositional functions

closed under

the usual

logical connectives

. From this,

∃

(

I

)

in the first condition can be viewed as the

existential quantifier

∃

bounding a set

I

of variables. The second condition stipulates that, when no variables are bound by

∃

, then

∃

has no effect on the propositional functions. The last condition states that the order and frequency of the variables bound by

∃

does not affect the outcome (

∃

x

2

,

x

1

,

x

2

is the same as

∃

x

1

⁢

∃

x

2

).

Remarks

•

A monadic algebra is a quantifier algebra where

V

=

{

x

}

, a

singleton

, and a Boolean algebra is just a quantifier algebra with

V

=

∅

.

•

In

classical first order logic

, the set of variables bound by a

quantifier

appearing in a

formula

is finite. Any variable not bound by the quantifier is considered free, as far as the scope of the quantifier is concerned. This basically says that every propositional function in the classical first order logic has a finite number variables. Translated into the

language

of quantifier algebras, this means that

for each

p

∈

B

, there is a finite

I

⊆

V

, such that

∃

(

V

-

I

)

⁢

(

p

)

=

p

.

Any quantifier algebra satisfying the above condition is said to be

locally finite

.

Alternatively, a set

I

⊆

V

is called a

support

of

p

∈

B

if

∃

(

V

-

I

)

⁢

(

p

)

=

p

. The

intersection

of all supports of

p

is called

the

support of

p

, denoted by

Supp

⁡

(

p

)

.

B

is locally finite iff every element of

B

has a

finite support

, or that

Supp

⁡

(

p

)

is finite.

•

Quantifier algebras are a step closer in fully characterizing the “

algebra

” of predicate logic than monadic algebras. However, it is not powerful enough to address situations where a “change of variable”

occurs in

a propositional function, such as

∃

x

(

x

2

+

z

2

=

1

)

versus

∃

y

(

y

2

+

z

2

=

1

)

. In a quatifier algebra, these two formulas are distinct, even though they are the same semantically in logic. In order take into account these additional considerations,

polyadic algebras

are needed.

References

1

P. Halmos,

Algebraic Logic

, Chelsea Publishing Co. New York (1962).

2

B. Plotkin,

Universal Algebra

, Algebraic Logic, and Databases

, Kluwer Academic Publishers (1994).

Title

quantifier algebra

Canonical

name

QuantifierAlgebra

Date of creation

2013-03-22 17:49:04

Last modified on

2013-03-22 17:49:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Related topic

MonadicAlgebra

Related topic

PolyadicAlgebra

Defines

locally finite

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