Lindenbaum-Tarski algebra

Lindenbaum-Tarski algebra

Lindenbaum-Tarski algebra of a propositional langauge

Let

L

be a classical propositional

language

. We define the

equivalence relation

∼

over

formulas

of

L

by

φ

∼

ψ

if and only if

⊢

φ

⇔

ψ

. Let

B

=

L

/

∼

be the set of

equivalence classes

. We define the

operations

join

∨

, meet

∧

, and

complementation

, denoted

[

φ

]

′

on

B

by :

[

φ

]

∨

[

ψ

]

:=

[

φ

∨

ψ

]

∧

[

ψ

]

:=

[

φ

∧

ψ

]

′

:=

[

¬

⁢

φ

]

We let

0

=

[

φ

∧

¬

⁢

φ

]

and

1

=

[

φ

∨

¬

⁢

φ

]

. Then the

structure

(

B

,

∨

,

∧

,

′

,

0

,

1

)

is a

Boolean algebra

, called the

Lindenbaum-Tarski algebra

of the propositional language

L

.

Intuitively, this

algebra

is an algebra of logical statements in which

logically equivalent

formulations of the same statement are not distinguished. One can develop intuition for this algrebra by considering a simple case. Suppose our language consists of a number of statement symbols

P

i

and the

connectives

∨

,

∧

,

¬

and that

⊢

denotes

tautologies

. Then our algebra consists of statements formed from these connectives with tautologously

equivalent

satements reckoned as the same element of the algebra. For

instance

, “

¬

⁡

(

P

1

∧

P

2

)

” is
considered the same as “

¬

⁢

P

1

∨

¬

⁢

P

2

”. Furthermore, since any statement of

propositional calculus

may be recast in

disjunctive normal form

, we may view this particular Lindenbaum-Tarski algebra as a

Boolean

analogue of

polynomials

in the

P

i

’s and their

negations

.

It can be shown that the Lindenbaum-Tarski algebra of the propositional language

L

is a

free Boolean algebra

freely generated by the set of all

elements

[

p

]

, where each

p

is a

propositional variable

of

L

Lindenbaum-Tarski algebra of a first order langauge

Now, let

L

be a first order language. As before, we define the equivalence relation

∼

over formulas of

L

by

φ

∼

ψ

if and only if

⊢

φ

⇔

ψ

. Let

B

=

L

/

∼

be the set of equivalence classes. The operations

∨

and

∧

and complementation on

B

are defined exactly the same way as previously. The resulting algebra is the Lindenbaum-Tarski algebra of the first order language

L

. It may be shown that

⋁

t

∈

T

[

φ

⁢

(

t

)

]

:=

[

∃

x

⁢

φ

⁢

(

x

)

]

⋀

t

∈

T

[

φ

⁢

(

t

)

]

:=

[

∀

x

⁢

φ

⁢

(

x

)

]

where

T

is the set of all terms in the language

L

. Basically, these results say that the statement

∃

x

⁢

φ

⁢

(

x

)

is equivalent to taking the

supremum

of all statements

φ

⁢

(

x

)

where

x

ranges over the entire set

V

of

variables

. In other words, if one of these statements is true (with truth value

1

, as opposed to

0

), then

∃

x

⁢

φ

⁢

(

x

)

is true. The statement

∀

x

⁢

φ

⁢

(

x

)

can be similarly analyzed.

Remark

. It may possible to define the Lindenbaum-Tarski algebra on logical languages other than the classical ones mentioned above, as long as there is a notion of formal proof that can allow the definition of the equivalence relation. For example, one may form the Lindenbaum-Tarski algebra of an intuitionistic propositional language (or

predicate

language) or a normal modal propositional language. The resulting algebra is a

Heyting algebra

(or a

complete Heyting algebra

) for intuitionistic propositional language (or predicate language), or a Boolean algebras with an

operator

for normal modal propositional languages.

Title

Lindenbaum-Tarski algebra

Canonical name

LindenbaumTarskiAlgebra

Date of creation

2013-03-22 12:42:40

Last modified on

2013-03-22 12:42:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

31

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G05

Classification

msc 03B05

Classification

msc 03B10

Synonym

Lindenbaum algebra

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