deduction theorem

deduction theorem

In mathematical logic, the

deduction theorem

is the following (meta-)statement:

Δ

,

A

⊢

B

iff

Δ

⊢

A

→

B

,

where

Δ

is a set of

formulas

, and

A

,

B

are formulas in a

logical system

where

→

is a (binary)

logical connective

denoting

implication

or entailment. In words, the statement says that if the formula

B

is

deducible

from a set

Δ

of

assumptions

, together with the assumption

A

, then the formula

A

→

B

is deducible from

Δ

alone.

Conversely

, if we can deduce

A

→

B

from

Δ

, and if in

addition

we assume

A

, then

B

can be deduced. The deduction theorem conforms with our intuitive understanding of how mathematical proofs work: if we want to prove the statement “

A

implies

B

”, then by assuming

A

, if we can prove

B

, we have established “

A

implies

B

”.

The

converse

statement of the deduction theorem turns out to be a

trivial

consequence

of

modus ponens

: if

Δ

⊢

A

→

B

, then certainly

Δ

,

A

⊢

A

→

B

. Since

Δ

,

A

⊢

A

, we get, via modus ponens,

Δ

,

A

⊢

B

as a result.

An apparently weaker version of the deduction theorem is to restrict

Δ

to a

finite set

. However, this version is actually

equivalent

the original version:

Proof.

Clearly, the original version implies the version where

Δ

is finite. Now assume the “weaker” version and that

Δ

,

A

⊢

B

, where

Δ

is an arbitrary set of formulas. Then there is a

deduction

(

finite sequence

of formulas)

A

1

,

…

,

A

n

+

1

=

B

such that each

A

i

(where

i

≤

n

) is either an axiom,

A

itself, a formula in

Δ

, or a formula obtained from an application of an

inference rule

to earlier formula(s). Let

Δ

′

be the set of

A

i

’s which are in

Δ

. Then

Δ

′

is finite, and clearly the sequence above is a deduction with assumptions in

Δ

′

∪

{

A

}

and

conclusion

B

. Thus

Δ

′

,

A

⊢

B

. By our (meta)-assumption,

Δ

′

⊢

A

→

B

, which means that there is a deduction

B

1

,

…

,

B

m

+

1

=

A

→

B

such that each

B

j

(where

j

≤

m

) is either an axiom, or a formula in

Δ

′

. Then certainly this is also a deduction with assumptions in

Δ

and conclusion

A

→

B

. Therefore,

Δ

⊢

A

→

B

.
∎

The deduction theorem holds in most of the widely studied logical systems, such as classical

propositional logic

and predicate logic,

intuitionistic logic

,

normal modal logics

, to name a few. On the other hand, the deduction theorem fails for other systems such as

fuzzy logic

. A modified version of the deduction theorem is usually available, however.

Remark

. In the statement of the deduction theorem,

Δ

is taken to be a (finite) set, meaning that some of the proof-structural rules, such as the

exchange rules

, weakening rules, etc…, are assumed automatically. In substructural logics where some of these structural rules are absent, Gentzen’s sequent systems are employed. The form of the deduction theorem then becomes

Δ

,

A

⇒

B

,

Γ

Δ

⇒

A

→

B

,

Γ

This is often taken as an inference rule, because the “deduction theorem” usually fails in these logics. Here,

Δ

is a finite sequence of formulas (as is

Γ

). The “deduction theorem” in this

context

is generally called the

right

→

-

introduction

rule

(since the

connective

→

is introduced on the

right hand side

of the succedent).

References

1

J. W. Robbin,

Mathematical Logic, A First Course

, Dover Publication (2006)

2

A. S. Troelstra, H. Schwichtenberg,

Basic

Proof Theory

, 2nd Edition, Cambridge University Press (2000)

3

G. Restall,

An Introduction to Substructural Logics

, Routledge, (2000)

4

M. Bergmann,

An Introduction to Many-Valued and Fuzzy Logic

, Cambridge University Press (2008)

Title

deduction theorem

Canonical name

DeductionTheorem

Date of creation

2013-03-22 19:13:30

Last modified on

2013-03-22 19:13:30

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Feature

Classification

msc 03F03

Classification

msc 03B99

Classification

msc 03B22

Classification

msc 03B47

Generated on Thu Feb 8 20:22:52 2018 by

LaTeXML