some theorem schemas of intuitionistic propositional logic

some theorem schemas of intuitionistic propositional logic

We present some

theorem

schemas of intuitionistic

propositional logic

and their

deductions

, based on the

axiom system

given in this entry (

http://planetmath.org/AxiomSystemForIntuitionisticLogic

).

1.

A

∨

B

→

B

∨

A

Proof.

From the deduction

1.

A

→

B

∨

A

,

2.

B

→

B

∨

A

,

3.

(

A

→

B

∨

A

)

→

(

(

B

→

B

∨

A

)

→

(

A

∨

B

→

B

∨

A

)

)

,

4.

(

B

→

B

∨

A

)

→

(

A

∨

B

→

B

∨

A

)

,

5.

A

∨

B

→

B

∨

A

,

so we get

A

∨

B

⊢

i

B

∨

A

, and therefore

⊢

i

A

∨

B

→

B

∨

A

by the

deduction theorem

.
∎

2.

(

A

∧

B

)

∧

C

→

A

∧

(

B

∧

C

)

Proof.

From the deduction

1.

(

A

∧

B

)

∧

C

→

A

→

B

,

2.

(

A

∧

B

)

∧

C

→

C

,

3.

(

A

∧

B

)

∧

C

,

4.

A

∧

B

,

5.

C

,

6.

A

∧

B

→

A

,

7.

A

∧

B

→

B

,

8.

A

,

9.

B

,

10.

B

→

(

C

→

B

∧

C

)

,

11.

C

→

B

∧

C

,

12.

B

∧

C

,

13.

A

→

(

B

∧

C

→

A

∧

(

B

∧

C

)

)

,

14.

B

∧

C

→

A

∧

(

B

∧

C

)

,

15.

A

∧

(

B

∧

C

)

,

so

(

A

∧

B

)

∧

C

⊢

i

A

∧

(

B

∧

C

)

, and therefore

⊢

i

(

A

∧

B

)

∧

C

→

A

∧

(

B

∧

C

)

by the deduction theorem.
∎

3.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

Proof.

From the deduction

1.

(

A

→

B

)

→

(

(

A

→

(

B

→

C

)

)

→

(

A

→

C

)

)

,

2.

A

→

B

,

3.

(

A

→

(

B

→

C

)

)

→

(

A

→

C

)

,

4.

A

→

(

B

→

C

)

,

5.

A

→

C

,

so

A

→

(

B

→

C

)

,

A

→

B

⊢

i

A

→

C

. By two applications of the deduction theorem,

⊢

i

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

.
∎

4.

A

∧

¬

⁢

A

→

B

Proof.

From the deduction

1.

A

∧

¬

⁢

A

→

A

,

2.

A

∧

¬

⁢

A

→

¬

⁢

A

,

3.

A

∧

¬

⁢

A

,

4.

¬

⁢

A

,

5.

A

,

6.

¬

A

→

(

A

→

B

)

,

7.

A

→

B

,

8.

B

so

A

∧

¬

A

⊢

i

B

. By the deduction theorem,

⊢

i

A

∧

¬

A

→

B

.
∎

5.

A

→

¬

⁢

¬

⁢

A

Proof.

From the deduction

1.

A

→

(

¬

A

→

A

)

,

2.

A

,

3.

¬

⁢

A

→

A

,

4.

(

¬

A

→

A

)

→

(

(

¬

A

→

¬

A

)

→

¬

¬

A

)

,

5.

(

¬

A

→

¬

A

)

→

¬

¬

A

,

6.

¬

⁢

A

→

¬

⁢

A

,

7.

¬

⁢

¬

⁢

A

,

so

A

⊢

i

¬

¬

A

. By the deduction theorem,

⊢

i

A

→

¬

A

¬

A

.
∎

In the proof above, we use the schema

B

→

B

in step 6 of the deduction, because

⊢

i

B

→

B

, as a result of applying the deduction theorem to

B

⊢

i

B

.

6.

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

Proof.

From the deduction

1.

(

A

→

¬

¬

A

)

→

(

(

A

→

¬

¬

¬

A

)

→

¬

A

)

,

2.

A

→

¬

⁢

¬

⁢

A

,

3.

(

A

→

¬

¬

¬

A

)

→

¬

A

,

4.

¬

¬

¬

A

→

(

A

→

¬

¬

¬

A

)

,

5.

¬

⁢

¬

⁢

¬

⁢

A

,

6.

A

→

¬

⁢

¬

⁢

¬

⁢

A

,

7.

¬

⁢

A

,

so

A

→

¬

¬

A

,

¬

¬

¬

A

⊢

i

¬

A

. By the deduction theorem,

A

→

¬

¬

A

,

⊢

i

¬

¬

¬

A

→

¬

A

. Since

⊢

i

A

→

¬

A

¬

A

,

⊢

i

¬

¬

¬

A

→

¬

A

as a result.
∎

In the above proof, we use the fact that if

⊢

i

C

and

C

⊢

i

D

, then

⊢

i

D

. This is the result of the following fact: if

⊢

i

C

and

⊢

i

C

→

D

, then

⊢

i

D

.

7.

(

A

→

B

)

→

(

¬

B

→

¬

A

)

Proof.

From the deduction

1.

(

A

→

B

)

→

(

(

A

→

¬

B

)

→

¬

A

)

,

2.

A

→

B

,

3.

(

A

→

¬

B

)

→

¬

A

,

4.

¬

B

→

(

A

→

¬

B

)

,

5.

¬

⁢

B

,

6.

A

→

¬

⁢

B

,

7.

¬

⁢

A

,

so

A

→

B

,

¬

B

⊢

i

¬

A

. Applying the deduction theorem twice gives us

⊢

i

(

A

→

B

)

→

(

¬

B

→

¬

A

)

.
∎

8.

¬

⁡

(

A

∧

¬

⁢

A

)

Proof.

From the deduction

1.

(

A

∧

¬

A

→

B

)

→

(

(

A

∧

¬

A

→

¬

B

)

→

¬

(

A

∧

¬

A

)

)

,

2.

A

∧

¬

⁢

A

→

B

,

3.

(

A

∧

¬

A

→

¬

B

)

→

¬

(

A

∧

¬

A

)

,

4.

A

∧

¬

⁢

A

→

¬

⁢

B

,

5.

¬

⁡

(

A

∧

¬

⁢

A

)

.

Since

⊢

i

A

∧

¬

A

→

B

and

⊢

i

A

∧

¬

A

→

¬

B

are

instances

of theorem schema 4 above,

⊢

i

¬

(

A

∧

¬

A

)

as a result.
∎

This also shows that

⊢

i

B

→

¬

(

A

∧

¬

A

)

, which is the result of applying

modus ponens

to

¬

⁡

(

A

∧

¬

⁢

A

)

to

¬

(

A

∧

¬

A

)

→

(

B

→

¬

(

A

∧

¬

A

)

)

.

9.

(

B

→

A

∧

¬

A

)

→

¬

B

Proof.

From the deduction

1.

B

→

A

∧

¬

⁢

A

,

2.

(

B

→

A

∧

¬

A

)

→

(

(

B

→

¬

(

A

∧

¬

A

)

)

→

¬

B

)

,

3.

(

B

→

¬

(

A

∧

¬

A

)

)

→

¬

B

,

4.

B

→

¬

⁡

(

A

∧

¬

⁢

A

)

,

5.

¬

⁢

B

,

so

B

→

A

∧

¬

A

⊢

i

¬

B

. Applying the deduction theorem gives us

⊢

i

(

B

→

A

∧

¬

A

)

→

¬

B

.
∎

10.

¬

⁢

¬

⁡

(

A

∨

¬

⁢

A

)

Proof.

From the deduction

1.

A

→

(

A

∨

¬

⁢

A

)

,

2.

(

A

→

(

A

∨

¬

A

)

)

→

(

¬

(

A

∨

¬

A

)

→

¬

A

)

,

3.

¬

⁡

(

A

∨

¬

⁢

A

)

→

¬

⁢

A

,

4.

¬

⁢

A

→

(

A

∨

¬

⁢

A

)

,

5.

(

¬

A

→

(

A

∨

¬

A

)

)

→

(

¬

(

A

∨

¬

A

)

→

¬

¬

A

)

,

6.

¬

⁡

(

A

∨

¬

⁢

A

)

→

¬

⁢

¬

⁢

A

,

7.

(

¬

(

A

∨

¬

A

)

→

¬

A

)

→

(

(

¬

(

A

∨

¬

A

)

→

¬

¬

A

)

→

¬

¬

(

A

∨

¬

A

)

)

,

8.

(

¬

(

A

∨

¬

A

)

→

¬

¬

A

)

→

¬

¬

(

A

∨

¬

A

)

,

9.

¬

⁢

¬

⁡

(

A

∨

¬

⁢

A

)

,

∎

Remark

. Again from this entry (

http://planetmath.org/AxiomSystemForIntuitionisticLogic

), if we use the second axiom system instead, keeping in mind that

¬

⁢

A

means

A

→

⟂

, the following are theorem schemas:

1.

(

A

→

B

)

→

(

(

A

→

(

B

→

C

)

)

→

(

A

→

C

)

)

. The proof of this is essentially the same as the proof of the third theorem schema above.

2.

(

A

→

B

)

→

(

(

A

→

¬

B

)

→

¬

A

)

. This is just

(

A

→

B

)

→

(

(

A

→

(

B

→

⟂

)

)

→

(

A

→

⟂

)

)

, an instance of the theorem schema above.

3.

¬

A

→

(

A

→

B

)

.

Proof.

From the deduction

1.

¬

⁢

A

(which is

A

→

⟂

),

2.

A

,

3.

⟂

,

4.

⟂

→

B

,

5.

B

so

¬

A

,

A

⊢

i

B

. By two applications of the deduction theorem, we get

⊢

i

¬

A

→

(

A

→

B

)

.
∎

Title

some theorem schemas of intuitionistic propositional logic

Canonical name

SomeTheoremSchemasOfIntuitionisticPropositionalLogic

Date of creation

2013-03-22 19:31:21

Last modified on

2013-03-22 19:31:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B20

Classification

msc 03F55

Related topic

AxiomSystemForIntuitionisticLogic

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