axiom system for intuitionistic propositional logic

axiom system for intuitionistic propositional logic

There are several Hilbert-style

axiom systems for intuitionistic propositional logic

, or PL

i

for short. One such a system is by Heyting, and is presented in this entry (

http://planetmath.org/IntuitionisticLogic

). Here, we describe another, based on the one by Kleene. The

language

of the logic consists of a

countably infinite

set of propositional letters

p

,

q

,

r

,

…

, and symbols for

logical connectives

→

,

∧

,

∨

.

Well-formed formulas

(wff) are defined recursively as follows:

•

propositional letters are wff

•

if

A

,

B

are wff, so are

A

→

B

,

A

∧

B

,

A

∨

B

, and

¬

⁢

A

.

In

addition

,

A

↔

B

(

biconditional

) is the abbreviation for

(

A

→

B

)

∧

(

B

→

A

)

, like PL

c

(classical

propositional logic

),

We also use parentheses to avoid ambiguity. The

axiom schemas

for PL

i

are

1.

A

→

(

B

→

A

)

.

2.

A

→

(

B

→

A

∧

B

)

.

3.

A

∧

B

→

A

.

4.

A

∧

B

→

B

.

5.

A

→

A

∨

B

.

6.

B

→

A

∨

B

.

7.

(

A

→

C

)

→

(

(

B

→

C

)

→

(

A

∨

B

→

C

)

)

.

8.

(

A

→

B

)

→

(

(

A

→

(

B

→

C

)

)

→

(

A

→

C

)

)

.

9.

(

A

→

B

)

→

(

(

A

→

¬

B

)

→

¬

A

)

.

10.

¬

A

→

(

A

→

B

)

.

where

A

,

B

, and

C

are wff’s. In addition,

modus ponens

is the only

rule of inference

for PL

i

.

As usual, given a set

Σ

of wff’s, a

deduction

of a wff

A

from

Σ

is a

finite sequence

of wff’s

A

1

,

…

,

A

n

such that

A

n

is

A

, and

A

i

is either an axiom, a wff in

Σ

, or is obtained by an application of modus ponens on

A

j

to

A

k

where

j

,

k

<

i

. In other words,

A

k

is the wff

A

j

→

A

i

. We write

Σ

⊢

i

A

to mean that

A

is a deduction from

Σ

. When

Σ

is the

empty set

, we say that

A

is a

theorem

(of PL

i

), and simply write

⊢

i

A

to mean that

A

is a theorem.

As with PL

c

, the

deduction theorem

holds for PL

i

. Using the deduction theorem, one can derive the well-known theorem schemas listed below:

1.

A

∧

B

→

B

∧

A

.

2.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

3.

A

∧

¬

⁢

A

→

B

4.

A

→

¬

⁢

¬

⁢

A

5.

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

6.

(

A

→

B

)

→

(

¬

B

→

¬

A

)

7.

¬

⁢

A

∧

¬

⁢

A

8.

¬

⁢

¬

⁡

(

A

∨

¬

⁢

A

)

For example, the first schema can be proved as follows:

Proof.

From the deduction,

1.

A

∧

B

→

A

,

2.

A

∧

B

→

B

,

3.

A

∧

B

,

4.

A

,

5.

B

,

6.

B

→

(

A

→

B

∧

A

)

,

7.

A

→

B

∧

A

,

8.

B

∧

A

,

we have

A

∧

B

⊢

i

B

∧

A

, and therefore

⊢

i

(

A

∧

B

)

→

(

B

∧

A

)

by the deduction theorem.
∎

Deductions of the other theorem schemas can be found here (

http://planetmath.org/SomeTheoremSchemasOfIntuitionisticPropositionalLogic

). In fact, it is not hard to see that

⊢

i

X

implies

⊢

c

X

(that

X

is a theorem of PL

c

). The

converse

is false. The following are theorems of PL

c

, not PL

i

:

1.

A

∨

¬

⁢

A

.

2.

¬

⁢

¬

⁢

A

→

A

3.

(

¬

A

→

¬

B

)

→

(

B

→

A

)

4.

(

(

A

→

B

)

→

A

)

→

A

5.

(

¬

A

→

B

)

→

(

(

¬

A

→

¬

B

)

→

A

)

Remark

. It is interesting to note, however, if any one of the above schemas were added to the list of axioms for PL

i

, then the resulting system is an

axiom system

for PL

c

. In notation,

PL

+

i

X

=

PL

c

,

where

X

is one of the schemas above. When this

equation

holds for some

X

, it is necessary that

⊢

c

X

and

⊬

i

X

. However, this condition is not

sufficient

to imply the equation, even if PL

+

i

X

is

consistent

(that is, the schema

C

∧

¬

⁢

C

of wff’s are not theorems). One such schema is

¬

⁢

¬

⁢

A

∨

¬

⁢

A

. A

logical system

PL such that PL

≤

i

PL

≤

PL

c

is called an

intermediate logic

. It can be shown that there are infinitely many such intermediate logics.

Remark

. Yet another popular axiom system for PL

i

uses the symbol

⟂

(for falsity) instead of

¬

. The wff’s in this language consists of all propositional letters, the symbol

⟂

, and

A

∧

B

,

A

∨

B

, and

A

→

B

, whenever

A

and

B

are wff’s. The axiom schemas consist of the first seven axiom schemas in the first system above, as well as

1.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

(the second theorem schema above)

2.

⟂

→

A

.

¬

⁢

A

is the abbreviation for

A

→

⟂

. The only rule of inference is modus ponens. Deductions and theorems are defined in the exact same way as above. Let us write

⊢

i

⁢

1

A

to mean wff

A

is a theorem in this axiom system. As mentioned, both axiom systems are

equivalent

, in that

⊢

i

⁢

1

A

implies

⊢

i

A

, and

⊢

i

A

implies

⊢

i

⁢

1

A

*

, where

A

*

is the wff obtained from

A

by replacing every occurrence of

⟂

by the wff

(

p

∧

¬

⁢

p

)

, where

p

is a propositional letter not occurring in

A

.

Title

axiom system for intuitionistic propositional logic

Canonical name

AxiomSystemForIntuitionisticPropositionalLogic

Date of creation

2013-03-22 19:30:58

Last modified on

2013-03-22 19:30:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

38

Author

CWoo (3771)

Entry type

Axiom

Classification

msc 03F55

Classification

msc 03B20

Classification

msc 03B55

Related topic

TruthValueSemanticsForIntuitionisticPropositionalLogic

Related topic

SomeTheoremSchemasOfIntuitionisticPropositionalLogic

Related topic

KripkeSemanticsForIntuitionisticPropositionalLogic

Related topic

DeductionTheoremForIntuitionisticPropositionalLogic

Related topic

DisjunctionProperty

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