deduction

deduction

Deductions

are formalizations of proofs. Informally, they can be viewed as arriving at a

conclusion

from a set of assumptions, via applications of some prescribed rules in a finite number of steps.

There are several ways to formally define what a deduction is, given a

deductive system

. We will explore some of them here.

Deduction as a Sequence

Given a deductive system, a deduction from a set

Δ

of (well-formed)

formulas

can be thought of as a

finite sequence

of wff’s

A

1

,

A

2

,

⋯

,

A

n

such that

A

i

is either an axiom, an element of

Δ

, or as the result of an application of an

inference rule

R

to earlier wff’s

A

i

1

,

…

,

A

i

k

, where

i

1

,

…

,

i

k

<

i

. If

A

i

is an element of

Δ

, we call

A

i

an

assumption

or a

hypothesis

. This way of presenting a deduction is said to be

linear

.

Deductions presented this way are also called

deduction sequences

or

proof sequences

. Deductions in Hilbert-style

axiom systems

are presented this way.

Deduction as a Tree

Alternatively, a deduction can be thought of as a labeled

rooted tree

. The nodes are labeled by formulas, and for each node with label, say

Y

, its immediate

predecessors

(

children

) are nodes whose labels are, say

X

1

,

…

,

X

n

which are

premises

of some inference rule

R

. Deductions presented this way are also called

deduction trees

or

proof trees

. In a deduction tree, the labels of the leaves are either the axioms, or assumptions (from some set

Δ

), and the label of the root of the tree is the conclusion of the deduction. Below is an example of a deduction tree:

.25.25

⁢

Z

.75.75

⁢

Y

X

1

⁢

.75.75.85826772

⁢

X

2

Here,

X

1

and

X

2

are assumptions and

Y

the corresponding conclusion of some (binary) rule

R

, and

Y

is the assumption and

Z

is the conclusion of some (unary) rule

S

. Inference rules may be specified as labels of the edges.

Deductions in deduction systems such as

natural deduction

or

sequent calculus

are presented this way. In those systems, the edges of the trees are replaced by horizontal lines, dividing premises and conclusions, and rules are sometimes specified next to the lines, in the following fashion:

X

1

X

2

R

Y

S

Z

Deducibility Relation

Given a deductive system

D

, if

B

is the conclusion of a deduction from a set

Δ

, we write

Δ

⊢

D

B

.

If

Γ

is another set of wff’s, we also write

Δ

,

Γ

⊢

D

B

to mean

Δ

∪

Γ

⊢

D

B

. In

addition

, if

Γ

=

{

A

1

,

…

,

A

n

}

, we write

A

1

,

…

,

A

n

⊢

D

B

and

Δ

,

A

1

,

…

,

A

n

⊢

D

B

to mean

Γ

⊢

D

B

and

Δ

∪

Γ

⊢

D

B

respectively. Furthermore, we may also remove the

subscript

D

and write

Δ

⊢

B

if no confusion arises. When

Δ

⊢

B

, we also say that

B

is

deducible

from

Δ

. The turnstile symbol

⊢

is also called the

deducibility relation

.

If a formula

A

is deducible from

∅

, we say that

A

is a

theorem

(of the corresponding deductive system). In symbol, we write

⊢

D

A

or

⊢

A

.

A

property

of

⊢

is the following:

if

⊢

A

and

A

⊢

B

, then

⊢

B

.

In other words, if

A

is a theorem, and

B

can be deduced from

A

, then

B

is also a theorem, which conforms with our intuition. More generally, we have

if

Δ

⊢

A

and

Γ

,

A

⊢

B

, then

Δ

,

Γ

⊢

B

.

Remark

. We refrain from calling a deduction a proof. As we have seen earlier, deductions are mathematical abstractions of proofs. So they are mathematical constructed, viewed either as sequences or trees (of formulas). On the other hand, in the literature, “proof” is usually reserved to be used in the meta-language, as in the “proof” of Fermat’s Last Theorem, etc…

References

1

A. S. Troelstra, H. Schwichtenberg,

Basic

Proof Theory

, 2nd Edition, Cambridge University Press, 2000

2

D. Dalen,

Logic and

Structure

, 4th Edition, Springer, 2008

Title

deduction

Canonical

name

Deduction

Date of creation

2013-03-22 19:12:53

Last modified on

2013-03-22 19:12:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03F03

Classification

msc 03B99

Classification

msc 03B22

Synonym

proof tree

Synonym

proof sequence

Synonym

hypothesis

Defines

deduction tree

Defines

deduction sequence

Defines

deducible

Defines

deducibility relation

Defines

assumption

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