some theorem schemas of propositional logic

some theorem schemas of propositional logic

Based on the

axiom system

in this entry (

http://

planetmath

.org/AxiomSystemForPropositionalLogic

), we will exhibit some

theorem

schemas, as well as prove some

meta-theorems

of

propositional logic

. All of these are based on the important

deduction theorem

, which is proved here (

http://planetmath.org/DeductionTheoremHoldsForClassicalPropositionalLogic

).

First, some theorem schemas:

1.

A

→

¬

⁢

¬

⁢

A

2.

¬

⁢

¬

⁢

A

→

A

3.

(law of the excluded middle)

A

∨

¬

⁢

A

4.

(ex falso quodlibet)

⟂

→

A

5.

A

↔

A

6.

(

law of double negation

)

A

↔

¬

⁢

¬

⁢

A

7.

A

∧

B

→

A

and

A

∧

B

→

B

8.

(

absorption law

for

∧

)

A

↔

A

∧

A

9.

(

commutative law

for

∧

)

A

∧

B

↔

B

∧

A

10.

(

associative

law for

∧

)

(

A

∧

B

)

∧

C

↔

A

∧

(

B

∧

C

)

11.

(

law of syllogism

)

(

A

→

B

)

→

(

(

B

→

C

)

→

(

A

→

C

)

)

12.

(law of importation)

(

A

→

(

B

→

C

)

)

→

(

A

∧

B

→

C

)

13.

(

A

→

B

)

→

(

(

B

→

(

C

→

D

)

)

→

(

A

∧

C

→

D

)

)

14.

(

A

→

B

)

↔

(

A

→

(

A

→

B

)

)

Proof.

Many of these can be easily proved using the deduction theorem:

1.

we need to show

A

⊢

¬

⁢

¬

⁢

A

, which means we need to show

A

,

¬

⁢

A

⊢

⟂

. Since

¬

⁢

A

is

A

→

⟂

, by

modus ponens

,

A

,

¬

⁢

A

⊢

⟂

.

2.

we observe first that

¬

⁢

A

→

¬

⁢

¬

⁢

¬

⁢

A

is an

instance

of the above theorem schema, since

(

¬

A

→

¬

¬

¬

A

)

→

(

¬

¬

A

→

A

)

is an instance of one of the

axiom schemas

, we have

⊢

¬

⁢

¬

⁢

A

→

A

as a result.

3.

since

A

∨

¬

⁢

A

is

¬

⁢

A

→

¬

⁢

A

, to show

⊢

A

∨

¬

⁢

A

, we need to show

¬

⁢

A

⊢

¬

⁢

A

, but this is obvious.

4.

we need to show

⟂

⊢

A

. Since

⟂

,

⟂

→

(

A

→

⟂

)

,

A

→

⟂

is a

deduction

of

A

→

⟂

from

⟂

, and the result follows.

5.

this is because

⊢

A

→

A

, so

⊢

(

A

→

A

)

∧

(

A

→

A

)

.

6.

this is the result of the first two theorem schemas above.

For the next four schemas, we need the the following meta-theorems (see here (

http://planetmath.org/SomeMetatheoremsOfPropositionalLogic

) for proofs):

M1.

Δ

⊢

A

and

Δ

⊢

B

iff

Δ

⊢

A

∧

B

M2.

Δ

⊢

A

implies

Δ

⊢

B

iff

Δ

⊢

A

→

B

7.

If

⊢

A

∧

B

, then

⊢

A

by M1, so

⊢

A

∧

B

→

A

by M2. Similarly,

⊢

A

∧

B

→

B

.

8.

⊢

A

∧

A

→

A

comes from 7, and since

⊢

A

implies

⊢

A

∧

A

by M1,

⊢

A

→

A

∧

A

by M2. Therefore,

⊢

A

↔

A

∧

A

by M1.

9.

If

⊢

A

∧

B

, then

⊢

A

and

⊢

B

by M1, so

⊢

B

∧

A

by M1 again, and therefore

⊢

A

∧

B

→

B

∧

A

by M2. Similarly,

⊢

B

∧

A

→

A

∧

B

. Combining the two and apply M1, we have the result.

10.

If

⊢

(

A

∧

B

)

∧

C

, then

⊢

A

∧

B

and

⊢

C

, so

⊢

A

,

⊢

B

, and

⊢

C

by M1. By M1 again, we have

⊢

A

and

⊢

B

∧

C

, and another application of M1,

⊢

A

∧

(

B

∧

C

)

. Therefore, by M2,

⊢

(

A

∧

B

)

∧

C

→

A

∧

(

B

∧

C

)

, Similarly,

⊢

A

∧

(

B

∧

C

)

→

(

A

∧

B

)

∧

C

. Combining the two and applying M1, we have the result.

11.

A

→

B

,

B

→

C

,

A

⊢

C

by modus ponens 3 times.

12.

A

→

(

B

→

C

)

,

A

∧

B

,

A

∧

B

→

A

,

A

,

B

→

C

,

A

∧

B

→

B

,

B

,

C

is a deduction of

C

from

A

→

(

B

→

C

)

and

A

∧

B

.

13.

A

→

B

,

B

→

(

C

→

D

)

,

A

∧

C

,

A

∧

C

→

A

,

A

,

B

,

C

→

D

,

A

∧

C

→

C

,

C

,

D

is a deduction of

D

from

A

→

B

,

B

→

(

C

→

D

)

, and

A

∧

C

.

14.

(

A

→

B

)

→

(

A

→

(

A

→

B

)

)

is just an axiom, while

⊢

(

A

→

(

A

→

B

)

)

→

(

A

→

B

)

comes from two applications of the deduction theorem to

A

→

(

A

→

B

)

,

A

⊢

B

, which is the result of the deduction

A

→

(

A

→

B

)

,

A

,

A

→

B

,

B

of

B

from

A

→

(

A

→

B

)

and

A

.

∎

Title

some theorem schemas of propositional logic

Canonical name

SomeTheoremSchemasOfPropositionalLogic

Date of creation

2013-03-22 19:33:04

Last modified on

2013-03-22 19:33:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

40

Author

CWoo (3771)

Entry type

Result

Classification

msc 03B05

Defines

law of double negation

Defines

law of the excluded middle

Defines

ex falso quodlibet

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