some meta-theorems of propositional logic

some meta-theorems of propositional logic

Based on the

axiom system

in this entry (

http://

planetmath

.org/AxiomSystemForPropositionalLogic

), we will prove some

meta-theorems

of

propositional logic

. In the discussion below,

Δ

and

Γ

are sets of

well-formed formulas

(wff’s), and

A

,

B

,

C

,

…

are wff’s.

1.

(

Deduction Theorem

)

Δ

,

A

⊢

B

iff

Δ

⊢

A

→

B

.

2.

(

Proof by Contradiction

)

Δ

,

A

⊢

⟂

iff

Δ

⊢

¬

⁢

A

.

3.

(Proof by Contrapositive)

Δ

,

A

⊢

¬

⁢

B

iff

Δ

,

B

⊢

¬

⁢

A

.

4.

(

Law of Syllogism

) If

Δ

⊢

A

→

B

and

Γ

⊢

B

→

C

, then

Δ

,

Γ

⊢

A

→

C

.

5.

Δ

⊢

A

and

Δ

⊢

B

iff

Δ

⊢

A

∧

B

.

6.

Δ

⊢

A

↔

B

iff

Δ

,

A

⊢

B

and

Δ

,

B

⊢

A

.

7.

If

Δ

⊢

A

↔

B

, then

Δ

⊢

B

↔

A

.

8.

If

Δ

⊢

A

↔

B

and

Δ

⊢

B

↔

C

, then

Δ

⊢

A

↔

C

.

9.

Δ

⊢

A

∧

B

→

C

iff

Δ

⊢

A

→

(

B

→

C

)

.

10.

Δ

⊢

A

implies

Δ

⊢

B

iff

Δ

⊢

A

→

B

. This is a useful restatement of the deduction theorem.

11.

(

Substitution Theorem

) If

⊢

B

i

↔

C

i

, then

⊢

A

[

B

¯

/

p

¯

]

↔

A

[

C

¯

/

p

¯

]

.

12.

Δ

⊢

⟂

iff there is a wff

A

such that

Δ

⊢

A

and

Δ

⊢

¬

⁢

A

.

13.

If

Δ

,

A

⊢

B

and

Δ

,

¬

⁢

A

⊢

B

, then

Δ

⊢

B

.

Remark

. The

theorem

schema

A

→

¬

⁢

¬

⁢

A

is used in the proofs below.

Proof.

The first three are proved here (

http://planetmath.org/DeductionTheoremHoldsForClassicalPropositionalLogic

), and the last three are proved here (

http://planetmath.org/SubstitutionTheoremForPropositionalLogic

). We will prove the rest here, some of which relies on the deduction theorem.

4.

From

Δ

⊢

A

→

B

, by the deduction theorem, we have

Δ

,

A

⊢

B

. Let

ℰ

1

be a

deduction

of

B

from

Δ

∪

{

A

}

, and

ℰ

2

be a deduction of

B

→

C

from

Γ

, then

ℰ

1

,

ℰ

2

,

C

is a deduction of

C

from

Δ

∪

{

A

}

∪

Γ

, so

Δ

,

A

,

Γ

⊢

C

, and by the deduction theorem again, we get

Δ

,

Γ

⊢

A

→

C

.

5.

(

⇒

)

. Since

A

∧

B

is

¬

⁡

(

A

→

¬

⁢

B

)

, by the deduction theorem, it is enough to show

Δ

,

A

→

¬

⁢

B

⊢

⟂

. Suppose

ℰ

1

is a deduction of

A

from

Δ

and

ℰ

2

is a deduction of

B

from

Δ

, then

ℰ

1

,

ℰ

2

,

A

→

¬

⁢

B

,

¬

⁢

B

,

⟂

is a deduction of

⟂

from

Δ

∪

{

A

→

¬

B

}

.

(

⇐

)

. We first show that

Δ

⊢

B

. Now,

¬

B

→

(

A

→

¬

B

)

is an axiom and

⊢

(

A

→

¬

B

)

→

¬

¬

(

A

→

¬

B

)

is a theorem,

⊢

¬

⁢

B

→

¬

⁢

¬

⁡

(

A

→

¬

⁢

B

)

, so that by

modus ponens

,

⊢

¬

⁡

(

A

→

¬

⁢

B

)

→

B

, using

axiom schema

(

¬

C

→

¬

D

)

→

(

D

→

C

)

. Since by assumption

Δ

⊢

¬

⁡

(

A

→

¬

⁢

B

)

, by modus ponens again, we get

Δ

⊢

B

.

We next show that

Δ

⊢

A

. From the deduction

A

,

A

→

⟂

,

⟂

, we have

A

,

¬

⁢

A

⊢

⟂

, so certainly

Δ

,

¬

⁢

A

,

A

,

B

⊢

⟂

. By three applications of the deduction theorem, we get

Δ

⊢

¬

A

→

(

A

→

¬

B

)

. By theorem

(

A

→

¬

B

)

→

¬

¬

(

A

→

¬

B

)

,

Δ

⊢

¬

⁢

A

→

¬

⁢

¬

⁡

(

A

→

¬

⁢

B

)

. By axiom schema

(

¬

C

→

¬

D

)

→

(

D

→

C

)

and modus ponens, we get

Δ

⊢

¬

⁡

(

A

→

¬

⁢

B

)

→

A

. Since

Δ

⊢

¬

⁢

A

→

¬

⁢

B

by assumption,

Δ

→

A

as a result.

6.

Δ

⊢

A

↔

B

iff

Δ

⊢

A

→

B

and

Δ

⊢

B

→

A

iff

Δ

,

A

⊢

B

and

Δ

,

B

⊢

A

.

7.

Apply 6 to

Δ

⊢

A

→

B

and

Δ

⊢

B

→

A

.

8.

Apply 5 and 6.

9.

Since

Δ

,

A

⊢

B

→

A

∧

B

by the theorem schema

⊢

A

→

(

B

→

A

∧

B

)

, together with

Δ

⊢

A

∧

B

→

C

, we have

Δ

,

A

⊢

B

→

C

by law of syllogism, or equivalently

Δ

⊢

A

→

(

B

→

C

)

, by the deduction theorem.

Conversely

,

Δ

,

A

⊢

B

→

C

and theorem schema

A

∧

B

→

B

result in

Δ

,

A

⊢

A

∧

B

→

C

by law of syllogism. So

Δ

⊢

A

→

(

A

∧

B

→

C

)

by the deduction theorem. But

A

∧

B

→

A

is a theorem schema,

Δ

⊢

A

∧

B

→

(

A

∧

B

→

C

)

, and therefore

Δ

⊢

A

∧

B

→

C

by the theorem schema

(

X

→

(

X

→

Y

)

)

↔

(

X

→

Y

)

.

10.

Assume the former. Then a deduction of

B

from

Δ

may or may not contain

A

. In either case,

Δ

,

A

⊢

B

, so

Δ

⊢

A

→

B

by the deduction theorem. Next, assume the later. Let

ℰ

1

be a deduction of

A

→

B

from

Δ

. Then if

ℰ

2

is a deduction of

A

from

Δ

, then

ℰ

1

,

ℰ

2

,

B

is a deduction of

B

from

Δ

, and therefore

Δ

⊢

B

.

To see the last meta-theorem implies the deduction theorem, assume

Δ

,

A

⊢

B

. Suppose

Δ

⊢

A

. Let

ℰ

1

be a deduction of

A

from

Δ

, and

ℰ

2

a deduction of

B

from

Δ

∪

{

A

}

. Then

ℰ

1

,

ℰ

2

is a deduction of

B

from

Δ

. So

Δ

⊢

B

. As a result

Δ

⁢

A

→

B

by the statement of the meta-theorem.
∎

Remark

. Meta-theorems 7 and 8, together with the theorem schema

A

↔

A

, show that

↔

defines an

equivalence relation

on the set of all wff’s of propositional logic. Formally, for any two wff’s

A

,

B

, if we define

A

∼

B

iff

⊢

A

↔

B

, then

∼

is an equivalence relation.

Title

some meta-theorems of propositional logic

Canonical name

SomeMetatheoremsOfPropositionalLogic

Date of creation

2013-03-22 19:34:29

Last modified on

2013-03-22 19:34:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Result

Classification

msc 03B05

Defines

law of syllogism

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