intuitionistic propositional logic is a subsystem of classical propositional logic

intuitionistic propositional logic is a subsystem of classical propositional logic

Suppose

logical systems

ℒ

1

and

ℒ

2

share the same set

L

of

well-formed formulas

(wff’s). Let

F

1

be the set of all

theorems

of

ℒ

1

, and

F

2

the set of all theorems of

ℒ

2

. We say that

ℒ

1

is a subsystem of

ℒ

2

if

F

1

⊆

F

2

, and a proper subsystem if

F

1

⊂

F

2

.

In the subsequent discussion, PL

c

stands for classical

propositional logic

, and PL

i

for intuitionistic propositional logic. We write PL

≤

i

PL

c

to mean that PL

i

is a subsystem of PL

c

, or

<

if it is proper.

Proposition 1

.

PL

<

i

PL

c

.

Unless otherwise specified,

⊢

A

means the wff

A

is a theorem of PL

c

.

Before proving this, we need the following facts about PL

c

:

1.

⊢

A

and

⊢

B

iff

⊢

A

∧

B

.

2.

⊢

A

implies

⊢

B

iff

⊢

A

→

B

.

3.

A

→

B

,

B

→

C

⊢

A

→

C

4.

A

→

B

,

¬

⁢

A

→

B

⊢

B

5.

if

Δ

⊢

A

and

Γ

,

A

⊢

B

, then

Δ

,

Γ

⊢

B

.

Proof.

The first two facts are proved

http://planetmath.org/node/12533

here, and the third is

http://planetmath.org/node/12561

here.

Fact 4 is

equivalent

to

⊢

(

A

→

B

)

→

(

(

¬

A

→

B

)

→

B

)

by the

deduction theorem

twice. Use the theorem schemas

⊢

(

A

→

B

)

↔

(

¬

B

→

¬

A

)

and

⊢

A

↔

¬

¬

A

(

law of double negation

), and the

substitution theorem

twice, it is enough to show

⊢

(

¬

B

→

¬

A

)

→

(

(

¬

B

→

A

)

→

¬

¬

B

)

.

By the deduction theorem three times, it is enough to show

¬

⁢

B

→

¬

⁢

A

,

¬

⁢

B

→

A

,

¬

⁢

B

⊢

⟂

, which is provided by the

deduction

¬

⁢

B

→

¬

⁢

A

,

¬

⁢

B

→

A

,

¬

⁢

B

,

¬

⁢

A

,

A

,

⟂

.

For the last fact, given deduction

ℰ

1

of

A

from

Δ

, and deduction

ℰ

2

of

A

→

B

from

Γ

,

ℰ

1

,

ℰ

2

,

B

is a deduction of

B

from

Δ

∪

Γ

.
∎

Proof.

Since in both systems, the only

inference rule

is

modus ponens

, and theoremhood of wff’s is preserved by the inference rule (that is, if

⊢

A

and

⊢

A

→

B

, then

⊢

B

), all we need to show is that every axiom of PL

i

is a theorem of PL

c

.

1.

A

→

(

B

→

A

)

. This is just an

axiom schema

for PL

c

.

2.

A

→

(

B

→

A

∧

B

)

.

A

,

B

,

A

→

¬

⁢

B

,

¬

⁢

B

,

⟂

leads to

A

,

B

,

A

→

¬

⁢

B

⊢

⟂

. Applying the deduction theorem three times, we get

⊢

A

→

(

B

→

(

(

A

→

¬

B

)

→

⟂

)

)

, or

⊢

A

→

(

B

→

A

∧

B

)

.

3.

A

∧

B

→

A

and

A

∧

B

→

B

. See

http://planetmath.org/node/12533

here.

4.

A

→

A

∨

B

.

A

,

¬

A

,

⟂

,

⟂

→

B

,

B

results in

A

,

¬

⁢

A

⊢

B

, since

⟂

→

B

is a theorem. By applying the deduction theorem twice, we get

⊢

A

→

(

¬

A

→

B

)

, or

⊢

A

→

A

∨

B

.

5.

B

→

A

∨

B

.

Clearly,

B

,

¬

⁢

A

⊢

B

. By the deduction theorem,

B

⊢

¬

⁢

A

→

B

, or

B

⊢

A

∨

B

. By the deduction theorem again,

⊢

B

→

A

∨

B

.

6.

(

A

→

C

)

→

(

(

B

→

C

)

→

(

A

∨

B

→

C

)

)

.

By fact 3,

A

→

C

,

B

→

C

,

¬

⁢

A

→

B

⊢

¬

⁢

A

→

C

. With fact 4:

A

→

C

,

¬

⁢

A

→

C

⊢

C

, so

A

→

C

,

B

→

C

,

¬

⁢

A

→

B

⊢

C

by fact 5. Now apply the deduction theorem 3 times.

7.

(

A

→

B

)

→

(

(

A

→

(

B

→

C

)

)

→

(

A

→

C

)

)

.

From

A

→

B

,

A

→

(

B

→

C

)

,

A

,

B

→

C

,

B

,

C

, we get

A

→

B

,

A

→

(

B

→

C

)

,

A

⊢

C

. Applying the deduction theorem three times, we have the result.

8.

(

A

→

B

)

→

(

(

A

→

¬

B

)

→

¬

A

)

. This is just 7, where

C

is

⟂

.

9.

¬

A

→

(

A

→

B

)

.

From

¬

⁢

A

,

A

⊢

⟂

and

⟂

⊢

B

, we get

¬

⁢

A

,

A

⊢

B

by fact 5, and the result follows with two applications of the deduction theorem.

Since

A

∨

¬

⁢

A

is a theorem of PL

c

and not of PL

i

(see

http://planetmath.org/node/12493

here), we conclude that PL

i

is a proper subsystem of PL

c

.
∎

Remarks

.

1.

Strictly

speaking, the

language

L

i

of PL

i

under

http://planetmath.org/node/12491

this system is different from the language

L

c

of PL

c

under

http://planetmath.org/node/12507

this system. In

L

i

, the

logical connectives

consist of

→

,

¬

,

∧

,

∨

, whereas in

L

c

, only

→

is used. The other

connectives

are introduced as abbreviational devices:

¬

⁢

A

is

A

→

⟂

,

A

∨

B

is

¬

⁢

A

→

B

, and

A

∧

B

is

¬

⁡

(

A

→

¬

⁢

B

)

. So it doesn’t make much sense to say that PL

<

i

PL

c

.

2.

One way to get around this issue is to re-define what it means for one logical system to be a subsystem of another. For example, one can define a

mapping

τ

:

L

i

→

L

c

such that

–

takes

propositional variables

to propositional variables

–

τ

preserves

the formation rules of wff’s

A mapping satisfying these conditions is called a

translation

. Then, a system PL

i

is a subsystem of PL

c

if there is a translation

τ

:

L

i

→

L

c

such that

–

for any wff

A

in

L

i

, we have

⊢

i

A

implies

⊢

c

τ

(

A

)

.

To further require that PL

i

be a proper subsystem of PL

c

, we also need

–

there is an wff

B

in

L

i

such that

⊬

i

B

and

⊢

c

τ

(

B

)

.

3.

Using this

definition

, define

τ

:

L

i

→

L

c

by

τ

⁢

(

A

)

:=

{

A

if

A

is a propositional variable

τ

⁢

(

B

)

→

τ

⁢

(

C

)

if

A

is

B

→

C

¬

⁢

τ

⁢

(

B

)

if

A

is

¬

⁢

B

τ

⁢

(

B

)

∧

τ

⁢

(

C

)

if

A

is

B

∧

C

τ

⁢

(

B

)

∨

τ

⁢

(

C

)

if

A

is

B

∨

C

.

where the symbols

¬

,

∧

, and

∨

in

L

c

are used as abbreviational tools (see the first remark). Then we see that the

proposition

makes sense.

4.

Another way of getting around this issue is to come up with another

axiom system

for PL

c

that uses

→

,

¬

,

∧

, and

∨

as

primitive

logical connectives. Such an axiom system exists. In fact, add

¬

⁢

¬

⁢

A

→

A

to the list of axiom schemas for PL

i

, and we get an axiom system for PL

c

.

5.

Using the idea in 2 above further, it can be shown that PL

c

is a subsystem of PL

i

(under the right translation)! The

existence

of such a translation

ν

is known as the Glivenko’s theorem. This translation

ν

has the further

property

that for any wff

A

in

L

c

,

⊢

c

A

iff

⊢

i

ν

(

A

)

.

Title

intuitionistic propositional logic is a subsystem of classical propositional logic

Canonical

name

IntuitionisticPropositionalLogicIsASubsystemOfClassicalPropositionalLogic

Date of creation

2013-03-22 19:33:12

Last modified on

2013-03-22 19:33:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

31

Author

CWoo (3771)

Entry type

Result

Classification

msc 03B20

Classification

msc 03F55

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