logical axiom

logical axiom

In a

logical system

, a

logical axiom

(sometimes called an axiom for short) is a logically valid (well-formed)

formula

used in a

deductive system

(particularly an

axiom system

) to deduce other logically valid formulas. By a logically valid formula, we mean the formula is true in every

interpretation

of the logical system. For example, in the classical

first-order logic

with equality, the formula

x

=

x

may be considered a logical axiom, since it is true in every model. Similarly, the formula

∀

x

⁢

A

→

A

⁢

[

x

/

y

]

may also be considered a logical axiom, since its interpretation “if for all

x

the formula

A

is true, then the formula

A

is true for an arbitrarily picked

x

” is also logically valid.

A logical axiom is to be contrasted from a

non-logical axiom

, which is a formula that is valid depending on its interpretation. For example, in the

first-order theory

of

abelian groups

(with one binary function symbol

⋅

), the formula

x

⋅

y

=

y

⋅

x

is a non-logical axiom. But it fails to be an axiom in the first-order theory of groups.

As discussed above, logical axioms are logically valid formulas set up to deduce other logically valid formulas (called

theorems

of the system) via certain rules, called

rules of inference

, for the logical system. These rules have the

property

that they

preserve

logical validity. A logical system in which all theorems are logically valid is said to be sound.

Conversely

, if a logical system in which any logically valid formula is a theorem is said to be

complete

.

Example

. (Classical

propositional logic

) Formulas of the following forms can be easily verified to be logicaly valid using the truth-valuation

semantics

of the logical system:

1.

a

→

(

b

→

a

)

2.

(

a

→

(

b

→

c

)

)

→

(

(

a

→

b

)

→

(

a

→

c

)

)

3.

(

¬

a

→

¬

b

)

→

(

b

→

a

)

where

→

is a binary

logical connective

and

¬

is a unary logical connective, and

a

,

b

,

c

are any (well-formed) formulas. Let us take these formulas as axioms.

Next, we pick a rule of inference. The popular choice is the rule “

modus ponens

”, which states that from formulas

A

and

A

→

B

, one my deduce

B

. It is easy to see that this rule preserves logical validity.

The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In

addition

, it is also complete.

Note that in the above set, we are actually looking at three smaller sets, each set containing formulas of a specific form. For example, the first sets of axioms above is:

{

a

→

(

b

→

a

)

∣

a

,

b

are well-formed formulas of classical propositional language.

}

.

Any such a set of axioms is called an

axiom scheme

, or

axiom schema

, and a

collection

of axiom schemas is called the

axiom schemata

.

Remark

. Note that the deductive system in the example above is not unique for classical propositional logic. The idea is that one may take an axiom schema (or schemata), and swap it with a

logically equivalent

set of schema (or schemata). The resulting schemata, together with the rules, gives a deductive system

equivalent

to the original one. The following schemata, together with modus ponens, also produces a sound and complete deductive system:

1.

a

→

(

b

→

a

)

2.

(

a

→

(

b

→

c

)

)

→

(

(

a

→

b

)

→

(

a

→

c

)

)

3.

a

→

(

a

∨

b

)

4.

b

→

(

a

∨

b

)

5.

(

a

→

c

)

→

(

(

b

→

c

)

→

(

(

a

∨

b

)

→

c

)

)

6.

(

a

∧

b

)

→

a

7.

(

a

∧

b

)

→

b

8.

(

c

→

a

)

→

(

(

c

→

b

)

→

(

c

→

(

a

∧

b

)

)

)

9.

(

a

→

¬

b

)

→

(

b

→

¬

a

)

10.

¬

⁡

(

a

→

a

)

→

b

11.

a

∨

¬

⁢

a

(law of the excluded middle)

Note that the

difference

between these axioms and the ones in the previous example is the appearance of the logical connectives

∨

and

∧

. Here,

∨

and

∧

are treated as

primitive

logical symbols, where as in the previous example, formulas of the forms

a

∨

b

and

a

∧

b

may be “defined” in terms of

a

,

b

,

→

and

¬

.

It is interesting to note that it is even possible to form a deductive system using just one axiom schema and modus ponens. The following example was discovered by C. Meredith:

[

(

(

(

a

→

b

)

→

(

¬

c

→

¬

d

)

)

→

c

)

→

e

]

→

(

(

e

→

a

)

→

(

d

→

a

)

)

.

References

1

D. Monk:

Mathematical Logic

, Springer-Verlag, New York (1976).

2

H. Enderton:

A Mathematical

Introduction

to Logic

, Academic Press, San Diego (1972).

3

J. R. Shoenfield,

Mathematical Logic

, AK Peters (2001).

4

H. Rasiowa:

Post Algebras as a Semantic

Foundation

of

m

-Valued Logics, Studies in

Algebraic Logic

, The

Mathematical Association of America

, (1974).

Title

logical axiom

Canonical

name

LogicalAxiom

Date of creation

2013-03-22 16:57:58

Last modified on

2013-03-22 16:57:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03B22

Classification

msc 03B05

Synonym

axiom scheme

Related topic

InferenceRule

Related topic

LogicalConnective

Related topic

DeductiveSystem

Defines

axiom schema

Defines

axiom schemata

Defines

law of the excluded middle

Defines

non-logical axiom

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