realization of a formula by a truth function

realization of a formula by a truth function

Fix a

countable set

V

=

{

v

1

,

v

2

,

…

}

of

propositional variables

. Let

p

be a

well-formed formula

over

V

constructed by a set

F

of

logical connectives

. Let

S

:=

{

v

k

1

,

…

,

v

k

n

}

be the set of variables occurring in

p

(

S

is finite as

p

is a string of finite length). Fix the

n

-tuple

𝒗

:=

(

v

k

1

,

…

,

v

k

n

)

. Every

valuation

ν

on

V

, when restricted to

S

, determines an

n

-tupe of zeros and ones:

ν

⁢

(

𝒗

)

:=

(

ν

⁢

(

v

k

1

)

,

…

,

ν

⁢

(

v

k

n

)

)

∈

{

0

,

1

}

n

. For this

ν

⁢

(

𝒗

)

, we

associate

the

interpretation

ν

¯

⁢

(

p

)

∈

{

0

,

1

}

.

Two valuations on

V

determine the same

a

∈

{

0

,

1

}

n

iff they agree on every

v

k

i

. If we set

ν

1

∼

ν

2

iff they determine the same

a

∈

{

0

,

1

}

n

, then

∼

is an

equivalence relation

on the set of all valuations on

V

. As there are

2

n

elements

in

{

0

,

1

}

n

, there are

2

n

equivalence classes

.

From the two paragraphs above, we see that there is a

truth function

ϕ

:

{

0

,

1

}

n

→

{

0

,

1

}

such that

ϕ

⁢

(

ν

⁢

(

𝒗

)

)

=

ν

¯

⁢

(

p

)

for any valuation

ν

on

V

. This

function

is called a

realization

of the wff

p

. Since

p

is arbitrary, it is easy to see that every wff admits a realization. It is also not hard to see that a realization of

p

is unique up to the order of the variables in the

n

-tuple

𝒗

. From now only, we make the

assumption

that every

n

-tuple

(

v

k

1

,

…

,

v

k

n

)

has the property that

k

1

<

⋯

<

k

n

. Let us write

ϕ

p

the

realization of

p

.

Realizations of wffs are closely related to semantical

implications

and equivalences:

1.

p

⊧

q

(

p

semantically implies

q

, or

p

entails

q

) iff

ϕ

p

≤

ϕ

q

;

2.

p

≡

q

iff

ϕ

p

=

ϕ

q

, where

≡

denotes

semantical equivalence

;

3.

p

is a

tautology

iff

ϕ

p

=

1

, the

constant function

whose value is

1

∈

{

0

,

1

}

.

If

F

=

{

¬

,

∨

,

∧

}

, then every wff

p

over

V

corresponds to a realization

[

p

]

that “looks” exactly likes

p

. We do this by

induction

:

•

if

p

is a propositional variable

v

i

, let

[

v

i

]

be the

identity function

on

{

0

,

1

}

;

•

if

p

has the form

¬

⁢

q

, define

[

p

]

:=

¬

⁡

[

q

]

;

•

if

p

has the form

q

∨

r

, define

[

p

]

:=

[

q

]

∨

[

r

]

;

•

if

p

has the form

q

∧

r

, define

[

p

]

:=

[

q

]

∧

[

r

]

;

where the

¬

,

∨

,

and

∧

on the

right hand side

of the definitions are the

Boolean

complementation

, join and meet

operations

on the

Boolean algebra

{

0

,

1

}

. Again by an easy induction, for each wff

p

, the function

[

p

]

is the realization of

p

(a function written in terms of symbols in

F

is called a

polynomial

).

Conversely

, every

n

-ary truth function

ϕ

:

{

0

,

1

}

n

→

{

0

,

1

}

is the realization of some wff

p

. This is true because every

n

-ary operation on

{

0

,

1

}

has a

conjunctive normal form

. Suppose

ϕ

is a function in variables

x

1

,

…

,

x

n

, with the form

α

1

∧

⋯

∧

α

m

, where each

α

i

is the join of the variables in

x

i

. If

α

i

is a function in

x

k

1

,

…

,

x

k

m

(each

k

j

∈

{

1

,

…

,

n

}

), then let

p

i

be the

disjunction

of propositional variables

v

k

1

,

…

,

v

k

m

. Then

ϕ

is the realization of wff

p

:=

p

1

∧

⋯

∧

p

n

. Notice that we have omitted parenthesis, and

p

1

∧

⋯

∧

p

n

is an abbreviation of

(

⋯

(

p

1

∧

p

2

)

∧

⋯

)

∧

p

n

)

.

Since every wff, regardless of logical connectives, has a realization, what we have just proved in fact is the following:

Theorem 1

.

{

¬

,

∨

,

∧

}

is

functionally complete

.

References

1

H. Enderton:

A Mathematical

Introduction

to Logic

, Academic Press, San Diego (1972).

Title

realization of a

formula

by a truth function

Canonical name

RealizationOfAFormulaByATruthFunction

Date of creation

2013-03-22 18:52:53

Last modified on

2013-03-22 18:52:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

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