Gentzen system

Gentzen system

Introduction

A

Gentzen system

, attributed to the German logician Gerhard Gentzen, is a variant form of a

deductive system

. Like a deductive system, a Gentzen system has axioms and

inference rules

. But, unlike a deductive system, the basic building blocks in a Gentzen system are expressions called

sequents

, not

formulas

.

More precisely, given a

language

L

of

well-formed formulas

, a deductive system consists of a set of formulas called axioms, and a set of inference rules, which are pairs of sets of formulas. In a Gentzen system, the formulas are replaced by sequents, which are defined as expressions of the form

Δ

⇒

Γ

,

where

Δ

and

Γ

are

finite sequences

of formulas in

L

. The empty

sequence

is allowed, and is usually denoted by

∅

,

λ

, or blank space. In the sequent above,

Δ

is called the

antecedent

, and

Γ

the

succedent

. A

formula

in a sequent is a formula that occurs either in the antecedent or the succedent of the sequent, and a

subformula

in a sequent is a subformula of some formula in the sequent.

Notation: for any sequence

Δ

of formulas, we write

Δ

:=

Δ

1

,

A

,

Δ

2

to mean that

A

is a formula in

Δ

.

Δ

1

and

Δ

2

are

subsequences

of

Δ

, one before

A

, and the other after

A

, both of which may be empty.

Axioms

As discussed above, axioms of a Gentzen system are sequents. Typically, they are of the following form:

Δ

1

,

A

,

Δ

2

⇒

A

In the case of classical propositional or predicate logic where

⟂

is the nullary

logical connective

denoting

falsity

,

⟂

⇒

is also an axiom. In

addition

, when converting a

Hilbert system

into a Gentzen system, axioms take the form

⇒

B

,

where

B

is an axiom in the Hilbert system.

Rules of Inference

Rules of inferences

have the form

X

1

X

2

⋯

X

n

Y

where

X

1

,

…

,

X

n

and

Y

are sequents of the rule. The

X

’s are called the

premises

, and

Y

the

conclusion

. The inference rules of a Gentzen system can be grouped into two main kinds:

•

structural rule

: a rule is structural if either,

(a)

given any premise, every formula in it is also a formula in the conclusion, or

(b)

every formula in the conclusion is also a formula in some premise.

In the former case, if there is a formula

B

in the conclusion not in any of the premises, then

B

is said to be

introduced

by the rule. In the later case, if there is a formula

A

in one of the premises not in the conclusion, then

A

is said to be

eliminated

. Some examples of this kind of rules are:

–

weakening rules

Δ

1

,

Δ

2

⇒

Γ

Δ

1

,

A

,

Δ

2

⇒

Γ

or

Δ

⇒

Γ

1

,

Γ

2

Δ

⇒

Γ

1

,

B

,

Γ

2

–

contraction rules

Δ

1

,

A

,

A

,

Δ

2

⇒

Γ

Δ

1

,

A

,

Δ

2

⇒

Γ

or

Δ

⇒

Γ

1

,

B

,

B

,

Γ

2

Δ

⇒

Γ

1

,

B

,

Γ

2

–

exchange rules

Δ

1

,

A

,

B

,

Δ

2

⇒

Γ

Δ

1

,

B

,

A

,

Δ

2

⇒

Γ

or

Δ

⇒

Γ

1

,

A

,

B

,

Γ

2

Δ

⇒

Γ

1

,

B

,

A

,

Γ

2

–

cut rule

Δ

1

⇒

Γ

1

,

A

,

Γ

2

Δ

2

,

A

,

Δ

3

⇒

Γ

3

Δ

1

,

Δ

2

,

Δ

3

⇒

Γ

1

,

Γ

2

,

Γ

3

where

A

is called a

cut formula

.

•

logical rule

: if it is not a structural rule. In other words, for every premise

X

i

, there is at least one formula, say

A

i

, in it not in the conclusion

Y

, and a formula

B

in

Y

not in any of

X

i

’s. Typically,

Y

is obtained from the

X

i

’s via a logical connective. An example of a logical rule is the following:

A

,

Δ

⇒

Γ

,

B

Δ

⇒

Γ

,

A

→

B

Deductions

Deductions

in a Gentzen system

G

are finite trees, whose nodes are labeled by sequents. In any deduction, the label of any of its leaves is an axiom. In addition, given any node with label

Y

, its immediate

predecessors

are nodes with labels

X

1

,

…

,

X

n

, such that

X

1

X

2

⋯

X

n

Y

is a rule of inference in

G

. A sequent

X

is said to be

deducible

if there is a deduction whose root has label

X

. A formula

A

is called a

theorem

if the sequent

⇒

A

is deducible.

Remarks

.

•

Initially, Gentzen invented the sequent system to analyze the other deductive system he introduced:

natural deduction

. In fact, sequents can be thought of as abbreviated forms of natural deductions, where the

left hand side

stands for assumptions (leaves), and right hand side the conclusion (root), and the body of the deduction tree is ignored. Furthermore, if we interpret sequents as formulas themselves, a Gentzen system is really just a deductive system for natural deductions.

•

In some

logical systems

, where the exchange rules are automatically assumed, the antecedent and succedent that make up s sequent can be thought of as

multisets

instead of finite sequences, of formulas, since multisets are just finite sequences

modulo order

. Furthermore, if the weakening and contraction rules are automatically assumed, then the multisets can be

reduced

to sets (where

multiplicities

of

elements

are forgotten). In classical

propositional logic

, for example, the sequent

Δ

⇒

Γ

can be thought of as the formula

⋀

Δ

→

⋁

Γ

, where

⋀

Δ

is the

conjunction

of formulas in

Δ

, and

⋁

Γ

is the

disjunction

of formulas in

Γ

.

•

A given logical system may have several distinct but deductively equivalent Gentzen systems. For example, any Gentzen system for classical propositional logic with structural rules can be converted into one without any structural rules.

References

1

A. S. Troelstra, H. Schwichtenberg,

Basic

Proof Theory

, 2nd Edition, Cambridge University Press (2000)

Title

Gentzen system

Canonical name

GentzenSystem

Date of creation

2013-03-22 19:13:20

Last modified on

2013-03-22 19:13:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Topic

Classification

msc 03F07

Classification

msc 03F03

Classification

msc 03B99

Classification

msc 03F05

Synonym

sequent system

Related topic

Sequent

Related topic

HilbertSystem

Related topic

NaturalDeduction

Defines

antecedent

Defines

succedent

Defines

structural rule

Defines

logical rule

Defines

weakening rule

Defines

contraction rule

Defines

cut rule

Defines

exchange rule

Defines

cut formula

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