Information Technology Reference
In-Depth Information
1. There are numbers in [0,10] that are small. For instance, at least 0 is small.
2. Not all the numbers in [0,10] are small. For instance, at least 10 is not small.
3. If
x
is small, and
y
<
x
,also
y
is small
4. There exists some number
e
>
|
−
|≤
0suchthatif
x
y
e
, and one of the elements
x
or
y
is small, the other is also small.
Notice that from (4) follows immediatelly that if
x
is small, then it is also
x
+
e
small,
since
e
.
Can it exist a subset
S
of [0,10] containing all the small numbers and only them?
In other words: Is the classical axiom of specification applicable to the predicate
'small'?
Obviously, and provided it exists the set
S
|
x
+
e
−
x
|
=
=
{
x
∈
[
0
,
10
]
;
x
is small
}
, it should be
S
=[
, the complement of
S
in [0,10]. From
(1) and (2), 0 is in
S
, and 10 is in
S
, that is, both sets are not-empty.
Since [0,10] is a compact set in the usual topology of the real line, and
S
is
bounded by 10, it should exist
s
0
,
10
]
−
S
=
{
x
∈
[
0
,
10
]
;
x
is not small
}
, with elements of
S
arbitrarily
close to s by the left. Thus, by (4) it should be
s
in
S
, and by (3) all numbers in
S
are small. But
s
=
SupS
∈
[
0
,
10
]
e
in
S
is small, that by the set theoretic
+
e
>
s
implies that also
s
+
property
S
S
=
0, is a contradiction! Hence, there cannot exist any subset like
S
.
The predicate 'small', as used by a layperson, is not representable by any subset of
[0,10] since with it the axiom of specificity fails, and consequently Boolean algebra
cannot be used to represent commonsense reasonings involving the linguistic term
'small'.
>
(
−
)
/
=
Remarks 1.
a) Notice that, once e
0
is fixed, with the number
10
s
e
t, it is
=
+
·
obtained
10
e. It should be pointed out that if both s and e are with a finite
number of decimal digits, then t is a positive integer. For instance, if e
s
t
=
0
.
01
and
s
10
. Keeping e as
a negative power of 10, for instance if it were s = the number pi, it will suffice
to take s
=
3
.
4
, it results
(
10
−
3
.
4
)
/
0
.
01
=
660
, thus
3
.
4
+
660
.
0
.
01
=
∗
=
3
.
14
<
stoalsohaves
∗
in S, and since
(
10
−
3
.
14
)
/
0
.
01
=
686
it
follows
3
S. Consequently, provided
e is a negative power of 10, it always follows that 10 is small, and S
.
14
+
686
e
=
10
, that implies the absurd
10
∈
=[
0
,
10
]
:all
numbers in [0,10] are actually small, that is the 'paradox'.
This is exactly what is called the Sorites Paradox, that was classically posed
with the number of hairs in a human head (In Greek, Sorites means 'bald man').
Nevertheless, and from a layperson point of view, what Sorites just shows is that
there are some linguistic terms, like 'small', for which the axiom of specification
fails. Thus, it is not possible to represent them by means of classical sets or
elements in a Boolean algebra, in the corresponding universe of discourse. This
conclusion is at the same origin of fuzzy sets.
b) To clarify the inference done to state that s
te is in S, let's notice that it is a
deductive process reached by a t times reiteration of 'Modus Ponens':
+
