deduction theorem holds for first order logic

deduction theorem holds for first order logic

In this entry, we show that the

deduction theorem holds for first order logic

. Actually, depending on the

axiom systems

, some modifications to the

deduction theorem

may be necessary. If we use the system given in the beginning of this entry (

http://planetmath.org/AxiomSystemForFirstOrderLogic

), no modifications are necessary, and the proof is mutatis mutandis the same as the one for

propositional logic

. However, if we instead use the system given in the remark of that entry, the deduction theorem needs to be revised. For convenience, we briefly state the axiom system here:

1.

tautologies

2.

∀

x

⁢

A

→

A

⁢

[

a

/

x

]

, where

x

∈

V

,

a

∈

V

⁢

(

Σ

)

, and

a

is

free for

x

in

A

3.

∀

x

(

A

→

B

)

→

(

A

→

∀

x

B

)

, where

x

∈

V

is not free in

A

with

modus ponens

and

generalization

as

inference rules

. A

deduction

of a wff

A

from a set

Δ

is a

finite sequence

of wff’s, where each wff is either an axiom, an element of

Δ

, or the result of an application of an inference rule.

Theorem 1

.

(Modified Deduction Theorem). If

Δ

,

A

⊢

B

, then

Δ

⊢

A

→

B

, provided that there is a deduction

E

of

B

from

Δ

∪

{

A

}

that contains no applications of universal generalization

•

either to a

variable

occurring free in

A

, or

•

to a wff whose deduction in

ℰ

contains

A

as a

premise

A big portion of the proof is the same as the one given for the deduction theorem for propositional logic, namely, the cases that

B

is an axiom, an element of

Δ

, or as the result of an application of modus ponens. We will not repeat the proof here, but refer the reader to this entry (

http://planetmath.org/DeductionTheoremHoldsForClassicalPropositionalLogic

) for that. In this entry, we will concentrate on the proof of the last case: when

B

is the result of an application of universal generalization.

Proof.

Let us first rephrase the conditions in the

theorem

. We are given a deduction

ℰ

of

B

from

Δ

∪

{

A

}

, that is, sequence of wff’s

A

1

,

…

,

A

n

such that if

A

i

is

∀

x

⁢

A

j

(where

j

<

i

), then either

•

the variable

x

is not free in

A

, or

•

the deduction of

A

j

in the sequence

A

1

,

…

,

A

j

does not contain

A

As mentioned, we assume that

B

is

∀

x

⁢

A

k

, where

A

k

is in

ℰ

.

We induct on the length

n

of

ℰ

. By the assumption on

B

,

n

is at least

2

. If

n

=

2

, then

ℰ

is

A

1

,

∀

x

⁢

A

1

.

Then

A

1

is either an axiom, or an element of

Δ

∪

{

A

}

. Furthermore,

x

does not

occur free

in

A

1

. If

A

1

is either an axiom or in

Δ

, then

A

1

,

∀

x

A

1

,

∀

x

A

1

→

(

A

→

∀

x

A

1

)

,

A

→

∀

x

A

1

(1)

is a deduction of

A

→

B

from

Δ

. If

A

1

is

A

, then

⋯

,

A

→

A

,

∀

x

(

A

→

A

)

,

∀

x

(

A

→

A

)

→

(

A

→

∀

x

A

)

,

A

→

∀

x

A

(2)

is a deduction of

A

→

B

from

Δ

, where

⋯

,

A

→

A

is a deduction of the tautology

A

→

A

(see here (

http://planetmath.org/AxiomSystemForPropositionalLogic

)).

Now assume the length of

ℰ

is

n

>

2

. Furthermore, we may assume that it has the form

A

1

,

…

,

A

n

-

1

,

∀

x

⁢

A

n

-

1

from

Δ

∪

{

A

}

. Then none of

A

1

,

…

,

A

n

-

1

contain

x

free. Now, let us look at the conditions above:

•

x

is not free in

A

. As

A

1

,

…

,

A

n

-

1

is a deduction of

A

n

-

1

of length

n

-

1

, by

induction

, there is a deduction of

B

1

,

…

,

B

m

of

A

→

A

n

-

1

from

Δ

. Since

x

does not occur free in

A

,

B

1

,

…

,

B

m

,

∀

x

(

A

→

A

n

-

1

)

,

∀

x

(

A

→

A

n

-

1

)

→

(

A

→

∀

x

A

n

-

1

)

,

A

→

∀

x

A

n

-

1

(3)

is a deduction of

A

→

∀

x

⁢

A

n

-

1

.

•

the deduction of

A

j

in the sequence

A

1

,

…

,

A

j

does not contain

A

. Let

B

1

,

…

,

B

m

be this deduction of

A

j

in

ℰ

. Then

B

1

,

…

,

B

m

,

∀

x

A

n

-

1

,

∀

x

A

n

-

1

→

(

A

→

∀

x

A

n

-

1

)

,

A

→

∀

x

A

n

-

1

(4)

is a deduction of

A

→

∀

x

⁢

A

n

-

1

.

In both cases, since

A

does not

occur in

the deduction,

Δ

⊢

A

→

B

.
∎

Note that the last three steps of deductions

(

1

)

and

(

4

)

use generalization, the

axiom schema

A

→

(

B

→

A

)

, and modus ponens, and the last steps of deductions

(

2

)

and

(

3

)

use generalization, the axiom schema

∀

x

(

A

→

B

)

→

(

A

→

∀

x

B

)

, and modus ponens.

As a corollary, we have

Corollary 1

.

If

Δ

,

A

⊢

B

, and

A

a

sentence

, then

Δ

⊢

A

→

B

.

Remark

. If we want to restore the modified deduction theorem to its original version (Theorem 2 below), and we want keep the axiom system stated above, the way deductions are constructed needs to be modified. In this case, the modification is: if

A

1

,

…

,

A

n

is a deduction of

A

, and

A

is

∀

x

⁢

A

i

, the result of an application of generalization to an earlier wff

A

i

, then

x

does not occur free in any of the wff’s in the deduction. With this change, we have the original version:

Theorem 2

.

(Deduction Theorem). If

Δ

,

A

⊢

B

, then

Δ

⊢

A

→

B

.

This is an immediate

consequence

of the following fact: if

B

can be deduced with

A

as a

hypothesis

, then

A

→

B

can be deduced without

A

as a hypothesis, which can be easily proved using induction.

Title

deduction theorem holds for first order logic

Canonical

name

DeductionTheoremHoldsForFirstOrderLogic

Date of creation

2013-03-22 19:32:31

Last modified on

2013-03-22 19:32:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

26

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

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