natural deduction for propositional logic

natural deduction for propositional logic

The

natural deduction

system for classical

propositional logic

described here is based on the

language

that consists of a set of

propositional variables

, a symbol

⟂

for falsity, and two

logical connectives

∧

and

→

. The other

connectives

are defined as follows:

•

¬

⁢

A

:=

A

→

⟂

,

•

A

∨

B

:=

¬

⁡

(

¬

⁢

A

∧

¬

⁢

B

)

,

•

A

↔

B

:=

(

A

→

B

)

∧

(

B

→

A

)

.

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 two rules, one introduction

⟂

I

(ex falso quodlibet), and one RAA (

reductio ad absurdum

):

⟂

A

(

⟂

I

)

[

⁢

¬

A

]

.

.

.

.

⟂

A

(

R

⁢

A

⁢

A

)

Remark

. In the rules

→

I

and RAA, 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 RAA, to prove

A

, we start by assuming

¬

⁢

A

, and then reach a

contradiction

(

⟂

), and hence we conclude with

A

. Of course,

¬

⁢

A

is no longer needed at this point.

Classical propositional logic as defined by the natural deduction system above is termed NK. Derivations and

theorems

for NK 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 NK 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

For example,

⊢

¬

⁡

(

A

∧

¬

⁢

A

)

as

[

A

∧

¬

⁢

A

]

1

(

∧

E

L

)

A

[

A

∧

¬

⁢

A

]

1

(

∧

E

R

)

¬

⁢

A

(

→

E

)

⟂

(

→

I

)

1

(

A

∧

¬

⁢

A

)

→

⟂

is a derivation of

(

A

∧

¬

⁢

A

)

→

⟂

, which is

¬

⁡

(

A

∧

¬

⁢

A

)

, where all of the assumptions have been discharged. The

subscript

indicates that the discharging of the assumptions at the top correspond to the application of the inference rule

→

I

at the bottom.

Note also that the first three theorems are the

axiom schemas

for the Łukasiewicz’s

axiom system for propositional logic

, and it is easy to see that NK and the

axiom system

are in fact

deductively equivalent

: the theorems in both systems are precisely the same. In

addition

to the theorems above, we also have the following

meta-theorems

:

1.

A

,

¬

⁢

A

⊢

⟂

,

2.

if

Σ

∪

{

A

}

⊢

⟂

, then

Σ

⊢

¬

⁢

A

,

3.

A

⊢

A

∨

B

and

B

⊢

A

∨

B

,

4.

if

Σ

∪

{

A

}

⊢

C

and

Σ

∪

{

B

}

⊢

C

, then

Σ

∪

{

A

∨

B

}

⊢

C

.

Remark

. If

¬

and

∨

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

¬

and

∨

as well:

1.

for

¬

, there are two rules, one introduction

¬

⁢

I

and one elimination

¬

⁢

E

:

[

A

]

.

.

.

.

⟂

⁢

¬

A

(

¬

⁢

I

)

A

⁢

¬

A

⟂

(

¬

⁢

E

)

2.

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

)

Note that the inference rules correspond to the meta-theorems above: rule

¬

⁢

E

corresponds to meta-theorem 1,

¬

⁢

I

corresponds to 2,

∨

I

L

and

∨

I

R

correspond to 3, and

∨

E

to 4. The resulting system is deductively equivalent to original NK system.

Title

natural deduction for propositional logic

Canonical

name

NaturalDeductionForPropositionalLogic

Date of creation

2013-03-22 19:31:59

Last modified on

2013-03-22 19:31:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03F03

Classification

msc 03B05

Synonym

natural deduction for classical propositional logic

Defines

NK

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