first order language

first order language

Let

Σ

be a

signature

. The

first order language

FO

⁡

(

Σ

)

on

Σ

contains the following:

1.

the set

S

⁢

(

Σ

)

of

symbols

of

FO

⁡

(

Σ

)

, which is the

disjoint union

of the following sets:

(a)

Σ

(the

non-logical symbols

),

(b)

a

countably infinite

set

V

of variables,

(c)

the set of

logical symbols

{

∧

,

∨

,

¬

,

⇒

,

⇔

,

∀

,

∃

}

,

(d)

the singleton consisting of the

equality

symbol

{

=

}

, and

(e)

the set of parentheses (left and right)

{

(

,

)

}

;

2.

the set

T

⁢

(

Σ

)

of

terms

of

FO

⁡

(

Σ

)

, which is built inductively from

S

⁢

(

Σ

)

, as follows:

(a)

Any variable

v

∈

V

is a term;

(b)

Any constant symbol in

Σ

is a term;

(c)

If

f

is an

n

-ary function symbol in

Σ

, and

t

1

,

…

,

t

n

are
terms, then

f

⁢

(

t

1

,

…

,

t

n

)

is a term.

3.

the set

F

⁢

(

Σ

)

of

formulas

of

FO

⁡

(

Σ

)

, which is built inductively from

T

⁢

(

Σ

)

, as follows:

(a)

If

t

1

and

t

2

are terms, then

(

t

1

=

t

2

)

is a formula;

(b)

If

R

is an

n

-ary relation symbol and

t

1

,

…

,

t

n

are
terms, then

(

R

⁢

(

t

1

,

…

,

t

n

)

)

is a formula;

(c)

If

φ

is a formula, then so is

(

¬

⁢

φ

)

;

(d)

If

φ

and

ψ

are formulas, then so is

(

φ

∨

ψ

)

;

(e)

If

φ

is a formula, and

x

is a variable, then

(

∃

x

⁢

(

φ

)

)

is a formula.

In other words,

T

⁢

(

Σ

)

and

F

⁢

(

Σ

)

are the smallest sets, among all sets satisfying the conditions given for terms and formulas, respectively.

Formulas in 3(a) and 3(b), which do not contain any logical connectives, are called the

atomic formulas

.

For example, in the first order language of

partially ordered rings

, expressions such as

0

,

x

2

,

and

y

+

z

⁢

x

are terms, while

(

x

=

x

y

)

,

(

x

+

y

≤

y

z

)

,

and

(

∃

x

(

(

x

≤

0

)

∨

(

0

≤

x

)

)

)

are formulas, and the first two of which are atomic.

Remarks

.

1.

Generally, one omits parentheses in formulas, when there is no ambiguity. For example, a formula

(

φ

)

can be simply written

φ

. As such, the parentheses are also called the

auxiliary symbols

.

2.

The other logical symbols are obtained in the following way :

φ

∧

ψ

:=

def

⁢

¬

⁡

(

¬

⁢

φ

∨

¬

⁢

ψ

)

⁢

φ

⇒

ψ

:=

def

⁢

¬

⁢

φ

∨

ψ

φ

⇔

ψ

:=

def

(

φ

⇒

ψ

)

∧

(

ψ

⇒

φ

)

⁢

∀

x

⁢

(

φ

)

:=

def

⁢

¬

⁡

(

∃

x

⁢

(

¬

⁢

φ

)

)

where

φ

and

ψ

are formulas. All logical symbols are used when building formulas.

3.

In the literature, it is a common practice to write

Σ

ω

⁢

ω

for

FO

⁡

(

Σ

)

. The first

subscript

means that every formula in

FO

⁡

(

Σ

)

contains a finite number of

∨

’s (less than

ω

), while the second subscript signifies that every formula has a finite number of

∃

’s. In general,

Σ

α

⁢

β

denotes a

language

built from

Σ

such that, in any given formula, the number of

occurrences

of

∨

is less than

α

and the number of occurrences of

∃

is less than

β

. When the number of occurrences of

∨

(or

∃

) in a formula is not limited, we use the symbol

∞

in place of

α

(or

β

). Clearly, if

α

and

β

are not

ω

, we get a language that is not first-order.

First Order Languages as Formal Languages

If the signature

Σ

and the set

V

of variables are

countable

, then

S

⁢

(

Σ

)

,

T

⁢

(

Σ

)

, and

F

⁢

(

Σ

)

can be viewed as

formal languages

over a certain (finite)

alphabet

Γ

. The set

Γ

should include all of the logical connectives, the equality symbol, and the parentheses, as well as the following symbols

R

,

F

,

V

,

I

,

#

,

where they are used to form words for

relation

, formula, and variable symbols. More precisely,

•

V

⁢

I

n

⁢

#

stands for the variable

v

n

,

•

R

⁢

I

n

⁢

#

⁢

I

m

⁢

#

stands for the

m

-th relation symbol of arity

n

, and

•

F

⁢

I

n

⁢

#

⁢

I

m

⁢

#

stands for the

m

-th function symbol of arity

n

,

where

m

,

n

≥

0

are integers. The symbol

#

is used as a delimiter or separator. Note that the constant symbols are then words of the form

F

⁢

#

⁢

I

m

⁢

#

. It can shown that

S

⁢

(

Σ

)

,

T

⁢

(

Σ

)

and

F

⁢

(

Σ

)

are

context-free

over

Γ

, and in fact

unambiguous

.

References

1

W. Hodges,

A Shorter

Model Theory

, Cambridge University Press, (1997).

2

D. Marker,

Model Theory, An

Introduction

, Springer, (2002).

Title

first order language

Canonical name

FirstOrderLanguage

Date of creation

2013-03-22 12:42:46

Last modified on

2013-03-22 12:42:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

28

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03B10

Synonym

auxiliary symbol

Synonym

first-order language

Related topic

Type2

Related topic

Language

Related topic

AtomicFormula

Defines

first order language

Defines

term

Defines

formula

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