axiom system for propositional logic

axiom system for propositional logic

The

language

of (classical)

propositional logic

PL

c

consists of a set of propositional letters or

variables

, the symbol

⟂

(for falsity), together with two symbols for

logical connectives

¬

and

→

. The

well-formed formulas

(wff’s) of PL

c

are inductively defined as follows:

•

each propositional letter is a wff

•

⟂

is a wff

•

if

A

and

B

are wff’s, then

A

→

B

is a wff

We also use parentheses

(

and

)

to remove ambiguities. The other familiar logical connectives may be defined in terms of

→

:

¬

⁢

A

is

A

→

⟂

,

A

∨

B

is the abbreviation for

¬

⁢

A

→

B

,

A

∧

B

is the abbreviation for

¬

⁡

(

A

→

¬

⁢

B

)

, and

A

↔

B

is the abbreviation for

(

A

→

B

)

∧

(

B

→

A

)

.

The

axiom system

for PL

c

consists of sets of wffs called

axiom schemas

together with a

rule of inference

. The axiom schemas are:

1.

A

→

(

B

→

A

)

,

2.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

,

3.

(

¬

A

→

¬

B

)

→

(

B

→

A

)

,

and the rule of inference is

modus ponens

(MP): from

A

→

B

and

A

, we may infer

B

.

A

deduction

is a

finite sequence

of wff’s

A

1

,

…

,

A

n

such that each

A

i

is either an

instance

of one of the axiom schemas above, or as a result of applying rule MP to earlier wff’s in the sequence. In other words, there are

j

,

k

<

i

such that

A

k

is the wff

A

j

→

A

i

. The last wff

A

n

in the deduction is called a

theorem

of PL

c

. When

A

is a theorem of PL

c

, we write

⊢

c

A

or simply

⊢

A

.

For example,

⊢

A

→

A

, whose deduction is

1.

(

A

→

(

(

B

→

A

)

→

A

)

)

→

(

(

A

→

(

B

→

A

)

)

→

(

A

→

A

)

)

by Axiom II,

2.

A

→

(

(

B

→

A

)

→

A

)

by Axiom I,

3.

(

A

→

(

B

→

A

)

)

→

(

A

→

A

)

by modus ponens on

2

to

1

,

4.

A

→

(

B

→

A

)

by Axiom I,

5.

A

→

A

by modus ponens on

4

to

3

.

More generally, given a set

Σ

of wff’s, we write

Σ

⊢

A

if there is a finite sequence of wff’s such that each wff is either an axiom, a member of

Σ

, or as a result of applying MP to earlier wff’s in the sequence. An important (meta-)theorem called the

deduction theorem

, states: if

Σ

,

A

⊢

B

, then

Σ

⊢

A

→

B

. The deduction theorem holds for PL

c

(proof here (

http://planetmath.org/deductiontheoremholdsforclassicalpropositionallogic

))

Remark

. The axiom system above was first introduced by Polish logician Jan Łukasiewicz. Two axiom systems are said to be

deductively equivalent

if every theorem in one system is also a theorem in the other system. There are many axiom systems for PL

c

that are deductively equivalent to Łukasiewicz’s system. One such system consists of the first two axiom schemas above, but the third axiom schema is

¬

⁢

¬

⁢

A

→

A

, with MP its sole inference rule.

Title

axiom system for propositional logic

Canonical

name

AxiomSystemForPropositionalLogic

Date of creation

2013-03-22 19:31:50

Last modified on

2013-03-22 19:31:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

Synonym

axiom system for classical propositional logic

Related topic

DeductionTheoremHoldsForClassicalPropositionalLogic

Related topic

SubstitutionTheoremForPropositionalLogic

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