reductio ad absurdum

reductio ad absurdum

Reductio ad Absurdum

is a process of inference by means of which one can derive a

proposition

¬

⁢

X

from the fact the the

assumption

of

X

leads to a

contradiction

.

The underlying ideia is the following: if a contradiction can be deduced from a proposition

X

then

X

can not be true and one can therefore assert

¬

⁢

X

. It is a useful method to derive

negative

statements. The

hypothesis

from which the contradiction is derived is known as ”the reductio’s hypothesis”.

In Gentzen’s system of

natural deduction

the reductio’s hypotheses differs from the other hypotheses by the fact that is not included in the set of premisses on which the

conclusion

depends and thus behaves like the assumption of a conditional proof.

Assume that one has as hypothesis

x

1

→

¬

⁢

x

1

and one aims at deriving

¬

⁢

x

1

.

Using

Reductio ad Absurdum

one can assume

x

1

as the reductio’s hypothesis and therefore by

Modus Ponens

one gets

¬

⁢

x

1

. Thus one has

x

1

∧

¬

⁢

x

1

which is the contradiction that one is led to. Therefore one can assert

¬

⁢

x

1

.

The derivation looks liked this:

1

(1)

x

1

→

¬

⁢

x

1

Hyp.

2

(2)

x

1

Red. Hyp.

1, 2

(3)

¬

⁢

x

1

1, 2, MP

1, 2

(4)

x

1

∧

¬

⁢

x

1

1, 2, 3,

∧

-Int.

1

(5)

¬

⁢

x

1

Red.

ad

Abs.

In the

Prior Analytics

, I, 23 (41-26) Aristotle compares the method of the

Reductio ad Absurdum

, used by Euclid in his proof that the Square root of 2 is irrational with his own method of the

Reductio ad Impossibile

, used in the

reduction

of the

syllogisms

𝐁𝐚𝐫𝐨𝐜𝐨

and

𝐁𝐨𝐜𝐚𝐫𝐝𝐨

to the First Figure.

For these two syllogisms Aristotle’s problem consists in the fact that both have a type 0 premiss, which converts neither by simple conversion nor by conversion per accidens. Both

modi

must be

reduced

by the method of indirect reduction or reduction ad impossibile.

The

𝐁𝐚𝐫𝐨𝐜𝐨

syllogism has the form:

(S1)

All

X

is

M

Some

Y

is

¬

⁢

M

Ergo Some

Y

is

¬

⁢

X

.

To carry out the reduction one takes as premiss the

negation

of the conclusion of S1:

All

Y

is

X

together with the Major Premiss of S1:

All

X

is

M

.

One gets then a First Figure (

𝐁𝐚𝐫𝐛𝐚𝐫𝐚

) syllogism:

(S2)

All

X

is

M

All

Y

is

X

Ergo All

Y

is

M

.

But the conclusion of S2 is the negation of the Minor Premiss of S1. Therefore the hypothesis that the conclusion of S1 is false leads to a contradiction and is considered to be established indirectly via the Barbara syllogism.

This is a new meaning of the

concept

of reduction and it is in this new meaning that one says that Baroco syllogism can be reduced to Barbara. The same

argument

applies to the Bocardo type of syllogism.

A very common form of the use of the

Reductio ad Absurdum

consists in a proof of

X

assuming as Reductio Hypothesis

¬

⁢

X

. When the contradiction is reached one is then entitled to reject the Reductio Hypothesis thus obtaining

¬

⁢

¬

⁢

X

.

In

classical logic

this

Double Negation

implies

X

and we have reached our goal. In Heyting’s system this form of Double Negation is not accepted and therefore the

Reductio ad Absurdum

or ”Indirect Method of Proof” is not allowed in intuitionistic mathematics.

It is instructive to consider a misformed proof by

Reductio ad Absurdum

, such as John Kelley’ proof that in 2-space two lines that do not

intersect

are parallel, on pg. 295 of his

Algebra

a Modern

Introduction

. Due to a recurring misprint the negation signs have not been inserted and the proof seems to use as hypothesis the very assertion that he wants to prove.

References

1

Aristotle,

Prior and Posterior Analytics

, ed. by W.D. Ross, Oxford, 1949.

2

Heyting, A.,

Intuitionism: An Introduction

, North Holland, Amsterdam, 1956.

3

Kneale, W., M.,

The Development of Logic

, Oxford, 1962.

Title

reductio ad absurdum

Canonical

name

ReductioAdAbsurdum

Date of creation

2013-03-22 18:20:18

Last modified on

2013-03-22 18:20:18

Owner

gribskoff (21395)

Last modified by

gribskoff (21395)

Numerical id

10

Author

gribskoff (21395)

Entry type

Definition

Classification

msc 03-01

Synonym

Indirect Method of Proof

Related topic

NotUniformlyContinuousFunction

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