natural deduction for intuitionistic propositional logic

natural deduction for intuitionistic propositional logic

The

natural deduction

system for intuitionistic

propositional logic

described here is based on the

language

that consists of a set of

propositional variables

, a symbol

⟂

for falsity, and

logical connectives

∧

,

∨

,

→

. In this system, there are no axioms, only

rules of inference

.

1.

for

∧

, there are three rules, one

introduction

rule

∧

I

, and two elimination rules

∧

E

L

and

∧

E

R

:

A

⁢

B

∧

A

B

(

∧

I

)

∧

A

B

A

(

∧

E

L

)

∧

A

B

B

(

∧

E

R

)

2.

for

→

, there are two rules, one introduction

→

I

and one elimination

→

E

(

modus ponens

):

[

A

]

.

.

.

.

B

→

A

B

(

→

I

)

A

→

⁢

A

B

B

(

→

E

)

3.

for

∨

, there are three rules, two introduction rules

∨

I

E

,

∨

I

L

, and one elimination rule

∨

E

:

A

∨

A

B

(

∨

I

L

)

B

∨

A

B

(

∨

I

R

)

∨

A

B

⁢

[

A

]

.

.

.

.

C

[

B

]

.

.

.

.

C

C

(

∨

E

)

4.

for

⟂

, there is one rule, an introduction rule

⟂

I

(ex falso quodlibet):

⟂

A

(

⟂

I

)

In short, intuitionistic propositional logic is classical propositional logic without the rule RAA (

reductio ad absurdum

). Furthermore,

¬

is a defined one-place logical symbol:

¬

⁢

A

:=

A

→

⟂

.

Remark

. In the rules

→

I

and

∨

E

, the square brackets around the top

formula

denote that the formula is removed, or

discharged

. In other words,

“once the

conclusion

is reached, the

assumption

can be dropped.”

In

→

I

, when the formula

B

is deduced from the assumption or

hypothesis

A

, we conclude with the formula

A

→

B

. Once this conclusion is reached,

A

is superfluous and therefore removed, as it is embodied in the formula

A

→

B

. This is often encountered in mathematical proofs: if we want to prove

A

→

B

, we first assume

A

, then we proceed with the proof and reach

B

, and therefore

A

→

B

. Simiarly, in

∨

E

, if

C

can be concluded from

A

and from

B

individually, then

C

can be concluded from anyone of them, or

A

∨

B

, without the assumptions

A

and

B

individually.

Intuitionistic propositional logic as defined by the natural deduction system above is termed NJ. Derivations and

theorems

for NJ are defined in the usual manner like all natural deduction systems, which can be found here (

http://planetmath.org/DeductionsFromNaturalDeduction

). Some of the theorems of NJ are listed below:

•

A

→

(

B

→

A

)

•

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

•

(

A

→

B

)

→

(

¬

B

→

¬

A

)

•

A

∧

B

→

B

∧

A

•

¬

⁡

(

A

∧

¬

⁢

A

)

•

A

→

¬

⁢

¬

⁢

A

•

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

For example,

⊢

A

→

¬

⁢

¬

⁢

A

as

[

A

]

2

[

¬

⁢

A

]

1

(

→

E

)

⟂

(

→

I

)

1

¬

⁢

A

→

⟂

(

→

I

)

2

A

→

(

¬

A

→

⟂

)

and

A

→

(

¬

A

→

⟂

)

is just

A

→

¬

⁢

¬

⁢

A

. Also,

⊢

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

, as

[

A

]

1

A→¬¬A

(

→

E

)

¬

⁢

¬

⁢

A

[

¬

⁢

¬

⁢

¬

⁢

A

]

2

(

→

E

)

⟂

(

→

I

)

1

¬

⁢

A

(

→

I

)

2

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

The

subscripts

indicate that the discharging of the assumptions at the top correspond to the applications of the inference rules

→

I

at the bottom. The box around

A

→

¬

⁢

¬

⁢

A

indicates that the derivation of

A

→

¬

⁢

¬

⁢

A

has been embedded (as a

subtree

) into the derivation of

¬

⁢

¬

⁢

¬

⁢

A

→

¬

⁢

A

.

Remark

. If

¬

were introduced as a primitive logical symbol instead of it being as “defined”, then we need to have inference rules for

¬

as well, one of which is introduction

¬

⁢

I

, and the other elimination

¬

⁢

E

:

[

A

]

.

.

.

.

⟂

⁢

¬

A

(

¬

⁢

I

)

A

⁢

¬

A

⟂

(

¬

⁢

E

)

Note that in the original NJ system,

¬

⁢

I

is just an

instance

of

→

I

, and

¬

⁢

E

is just an instance of

→

E

.

Title

natural deduction for intuitionistic propositional logic

Canonical

name

NaturalDeductionForIntuitionisticPropositionalLogic

Date of creation

2013-03-22 19:32:11

Last modified on

2013-03-22 19:32:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B20

Classification

msc 03F03

Classification

msc 03F55

Related topic

DisjunctionProperty

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