Hilbert system

Hilbert system

A

Hilbert system

is a style (formulation) of

deductive system

that emphasizes the role played by the axioms in the system. Typically, a Hilbert system has many

axiom schemes

, but only a few, sometimes one,

rules of inference

. As such, a Hilbert system is also called an

axiom system

. Below we list three examples of axiom systems in mathematical logic:

•

(intuitionistic

propositional logic

)

–

axiom schemes:

i.

A

→

(

B

→

A

)

ii.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

iii.

A

→

A

∨

B

iv.

B

→

A

∨

B

v.

(

A

→

C

)

→

(

(

B

→

C

)

→

(

A

∨

B

→

C

)

)

vi.

A

∧

B

→

A

vii.

A

∧

B

→

B

viii.

A

→

(

B

→

(

A

∧

B

)

)

ix.

⟂

→

A

–

rule of inference: (

modus ponens

): from

A

→

B

and

A

, we may infer

B

•

(classical predicate logic without equality)

–

axiom schemes:

i.

all of the axiom schemes above, and

ii.

law of double negation

:

¬

⁡

(

¬

⁢

A

)

→

A

iii.

∀

x

⁢

A

→

A

⁢

[

x

/

y

]

iv.

∀

x

(

A

→

B

)

→

(

A

→

∀

y

B

[

x

/

y

]

)

In the last two axiom schemes, we require that

y

is

free for

x

in

A

, and in the last axiom scheme, we also require that

x

does not

occur free

in

A

.

–

rules of inference:

i.

modus ponens, and

ii.

generalization

: from

A

, we may infer

∀

y

⁢

A

⁢

[

x

/

y

]

, where

y

is free for

x

in

A

•

(S4 modal propositional logic)

–

axiom schemes:

i.

all of the axiom schemes in intuitionistic propositional logic, as well as the law of double negation, and

ii.

Axiom K, or the

normality

axiom:

□

(

A

→

B

)

→

(

□

A

→

□

B

)

iii.

Axiom T:

□

⁢

A

→

A

iv.

Axiom 4:

□

⁢

A

→

□

⁢

(

□

⁢

A

)

–

rules of inference:

i.

modus ponens, and

ii.

necessitation

: from

A

, we may infer

□

⁢

A

where

A

,

B

,

C

above are

well-formed formulas

,

x

,

y

are individual variables, and

→

,

∨

,

∧

are binary,

□

unary, and

⟂

nullary

logical connectives

in the respective

logical systems

. The

connective

¬

may be defined as

¬

⁢

A

:=

A

→

⟂

for any

formula

A

.

Remarks

•

Hilbert systems need not be unique for a given logical system. For example, see this link (

http://planetmath.org/LogicalAxiom

).

•

For a given logical system, every Hilbert system is deductively equivalent to a

Gentzen system

: for any axiom

A

in a Hilbert system

H

, convert it to the

sequent

⇒

A

, and for any rule: from

A

1

,

…

,

A

n

we may deduce

B

, convert it to the rule: from

Δ

⇒

A

1

,

…

,

A

n

, we may infer

Δ

⇒

B

.

•

Since axioms are semantically valid statements, the use of Hilbert systems is more about deriving other semantically valid statements, or

theorems

, and less about the syntactical

analysis

of

deductions

themselves. Outside of structural proof theory, deductive systems a la Hilbert style are used almost exclusively everywhere in mathematics.

References

1

H. Enderton:

A Mathematical

Introduction

to Logic

, Academic Press, San Diego (1972).

2

A. S. Troelstra, H. Schwichtenberg,

Basic

Proof Theory

, 2nd Edition, Cambridge University Press (2000)

3

B. F. Chellas,

Modal Logic

, An Introduction

, Cambridge University Press (1980)

Title

Hilbert system

Canonical

name

HilbertSystem

Date of creation

2013-03-22 19:13:14

Last modified on

2013-03-22 19:13:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03F03

Classification

msc 03B99

Classification

msc 03B22

Synonym

axiom system

Related topic

GentzenSystem

Defines

generalization

Defines

necessitation

Defines

double negation

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