axiom system for first order logic

axiom system for first order logic

Let FO

(

Σ

)

be a

first order language

over

signature

Σ

. Furthermore, let

V

be a

countably infinite

set of

variables

and

V

⁢

(

Σ

)

the set extending

V

to include the

constants

in

Σ

. Before describing the

axiom system

for FO

(

Σ

)

, we need the following

definition

:

Definition

. Let

A

be a wff. A

generalization

of

A

is a wff having the form

∀

x

1

⁢

∀

x

2

⁢

⋯

⁢

∀

x

n

⁢

A

, for any

n

≥

0

. Thus,

A

is a generalization of itself.

The axiom system of FO

(

Σ

)

consists of any wff that is a generalization of a wff belonging to any one of the following six schemas:

1.

A

→

(

B

→

A

)

2.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

3.

¬

⁢

¬

⁢

A

→

A

4.

∀

x

(

A

→

B

)

→

(

∀

x

A

→

∀

x

B

)

, where

x

∈

V

5.

A

→

∀

x

⁢

A

, where

x

∈

V

is not free in

A

6.

∀

x

⁢

A

→

A

⁢

[

a

/

x

]

, where

x

∈

V

,

a

∈

V

⁢

(

Σ

)

, and

a

is

free for

x

in

A

and

modus ponens

(from

A

and

A

→

B

we may infer

B

) as its only

rule of inference

.

In schema 6, the wff

A

⁢

[

a

/

x

]

is obtained by replacing all

free occurrences

of

x

in

A

by

a

.

Logical symbols

∧

,

∨

, and

∃

are derived:

A

∨

B

:=

¬

⁢

A

→

B

;

A

∧

B

:=

¬

⁡

(

¬

⁢

A

∨

¬

⁢

B

)

; and

∃

x

⁢

A

:=

¬

⁢

∀

x

⁢

¬

⁢

A

.

Remark

. Again, in schema 6, the reason why we want

a

free for

x

in

A

is to keep the “intended meaning” of

A

intact. For example, suppose

A

is

∀

y

(

x

<

y

)

. Then

A

⁢

[

y

/

x

]

is

∀

y

(

y

<

y

)

. Obviously the two do not have the same “intended meaning”. In fact,

A

⁢

[

y

/

x

]

is not valid in any model. Similarly, in the schema 5, we want

x

not occur free in

A

to maintain its “intended meaning”. For example, if

A

is the

formula

x

=

0

, then

∀

x

⁢

A

is

∀

x

(

x

=

0

)

, which is not true in any model with at least two

elements

.

The

concepts

of

deductions

and

theorems

are exactly the same as those found in

propositional logic

. Here is an example: if

⊢

A

, then

⊢

∀

x

⁢

A

. To see this, we induct on the length

n

of the deduction of

A

. If

n

=

1

, then

A

is an axiom, and therefore so is its generalization

∀

x

⁢

A

, so that

⊢

∀

x

⁢

A

. Suppose now that the length is

n

+

1

. If

A

is an axiom, we are back to the previous

argument

. Otherwise,

A

is obtained from earlier formulas

B

and

B

→

A

via modus ponens. Since the deduction of

B

and

B

→

A

are at most

n

, by

induction

, we have

⊢

∀

x

⁢

B

and

⊢

∀

x

(

B

→

A

)

. Since

∀

x

(

B

→

A

)

→

(

∀

x

B

→

∀

x

A

)

is an axiom, we have

⊢

∀

x

⁢

B

→

∀

x

⁢

A

by modus ponens, and then

⊢

∀

x

⁢

A

by modus ponens again.

Remark

. Another popular axiom system for first order logic has 1, 2, 3, 6 above as its

axiom schemas

, plus the following schema

•

∀

x

(

A

→

B

)

→

(

A

→

∀

x

B

)

, where

x

∈

V

is not free in

A

and two rules of inferences: modus ponens, and

universal generalization

:

•

from

A

we may infer

∀

x

⁢

A

, where

x

∈

V

Note the similarity between the rule of generalization and axiom schema 5 above. However, the important

difference

is that

x

is not permitted to be free in

A

in the axiom schema.

With this change, the concept of deductions needs to be modified: we say that a wff

A

is

deducible

from a set

Γ

of wff’s, if there is a

finite sequence

A

1

,

…

,

A

n

where

A

=

A

n

, such that for each

i

,

i

=

1

,

…

,

n

1.

A

i

is either an axiom or in

Γ

2.

A

i

is obtained by an application of modus ponens

3.

A

i

is obtained by an application of generalization:

A

i

is

∀

x

⁢

A

j

for some

j

<

i

.

Title

axiom system for first order logic

Canonical

name

AxiomSystemForFirstOrderLogic

Date of creation

2013-03-22 19:32:22

Last modified on

2013-03-22 19:32:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Synonym

universal generalization

Defines

generalization

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