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1.2 Opposite, Negate, and Middle
Very often the meaning of a predicate
P
is not captured without simultaneously
capturing one of its opposites
aP
(
a
for
antonym
, a synonym of
opposite
). How
can I recognize that John is
young
without the possibility of recognizing that Peter
is
old
? Can someone recognize that a person is
tall
but not that other person is
short
?
The mastering of perception-based predicates shows this kind of polarity: we
jointly learn the meaning of P and some of its opposites
aP
. Even more, without
knowing how to use
young
and
old
it is not possible to know how to use
middle-
aged
, that is equivalent to '
not young and not old
'. The same could be said with
respect to
warm
that is equivalent with '
not cold and not hot
', in relation with
water's temperature. Composite predicates of this type are very frequent, for example
medium
, actually equivalent to '
not big and not small
'.
It should be noticed that a 'middle' predicate only exists with imprecise predicates,
but not with precise ones. For example, in the set of natural numbers, if
P
=
even
,
it is
aP
=
odd
, and
a
(
odd
)
=
even
, thus (
not even
)
and
(
not odd
)
=
odd and even
,
but for no
n
it can be stated '
n
is
odd and even
'.
Let us remark that, although
P
and
aP
are linguistic terms,
not P
is not a linguistic
term. For example, in all dictionary we will find
poor
and
rich
, but neither
not poor
nor
not rich
. The negate of
P
, not
P
, is more a logical concept than a linguistic one. Our
current problem is how to find the uses of
aP
(
aP
, μ
aP
)
, and
not P
(
not P
, μ
not P
)
,
(
P
, μ
P
)
given a use
of
P
.
1.2.1 Antonyms
Concerning every opposite
aP
of
P
, this opposition is translated by
aP
=
−
1
P
since '
x
is
less a P than y
' should be equivalent to '
y
is
less P than x
'.
Hence,
a
(
aP
)
=
−
1
(
−
1
P
)
−
1
aP
=
=
P
, that reflects
a
(
aP
)
=
P
.For
example, with
P
=
tall
,itis
aP
=
short
and
a
(
aP
)
=
tall
.
This property of
aP
shows a way for obtaining
μ
aP
once
μ
P
is known. Let it
A
:
X
ₒ
X
be a symmetry on
X
, that is a function such that
•
If
x
P
y
, then
A
(
y
)
P
A
(
x
)
•
A
ⓦ
A
=
id
X
,
and, once
μ
P
:
X
ₒ
L
is known, take
μ
aP
(
x
)
=
μ
P
(
A
(
x
))
, for all
x
in
X
, that is,
μ
aP
=
μ
P
ⓦ
A
.
Function
μ
aP
=
μ
P
ⓦ
A
is a degree for
aP
, since:
•
x
aP
y
⃔
y
P
x
⃒
A
(
x
)
P
A
(
y
)
⃒
μ
P
(
A
(
x
))
μ
P
(
A
(
y
)),
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