Information Technology Reference
In-Depth Information
and NINCDS-ARDRA. Early detection of dementia is important e.g. to achieve
favorable effects of pharmacologic treatment by cholinesterase inhibitors (for
Alzheimer's disease and Lewy body dementia).
26.5
Generalized General Logic
A signature
Σ
=(
S
,
Ω
)
consists of sorts (types) in
S
and operators in
Ω
.Here
S
is a
set
in the sense of ZFC. On the other hand,
is not just a set, but in fact an object
in an underlying category. If this category is
Set
, the ordinary category of sets and
functions, in the sense of ZFC, then
Ω
Ω
is a set just like
S
also is just a set.
→
answer
be two constant (0-ary
operators) of sort
answer
. In GDS, the first question is "
Are you basically satisfied
with your life?
". The observation about the target older person may be
Ye s
.This
gives no room for representing uncertainty about the observation.
For each sort
s
∈
Now let
answer
∈
S
be a sort, and let
no
,
yes
:
S
, the algebra
A
provides the sort with a domain
A(
s
)
,which
typically is seen as a set, i.e. an object in
Set
. Operators
ω
:
s
1
×···×
s
n
→
s
are
then provided with a meaning
, i.e. a morphism
in
Set
. Again, there is no a priori reason why
Set
must be fixed as the underlying
category for algebras of signatures.
In order to see the difference in using other underlying categories than
Set
,letus
first look at the term functor
T
Σ
:
Set
A(
ω
)
:
A(
s
1
)
×···×
A(
s
n
)
→
A(
s
)
Set
. The term functor can be constructed
in a strict categorical fashion [9], so that
T
Σ
X
becomes the set of all terms over the
set X of variables, i.e.
X
being an object of
Set
.
To continue the example above, the term yes is recorded as the observation. Let
now the underlying category be changed to
Set
→
(
)
L
, the Goguen category, where
(
)
(
,
α
)
→
L
is a suitable lattice.
Objects in
Set
L
are pairs
A
where
α
:
A
L
is a
mapping. Morphisms
f
:
(
A
1
,
α
1
)
→
(
A
2
,
α
2
)
are mappings
f
:
A
1
→
A
2
such that
α
2
(
f
(
a
))
≥
α
1
(
a
)
for all
a
∈
A
1
. Note that
Set
is not isomorphic to
Set
(
{
0
,
1
}
)
.
Assume for instance that
L
, with the
names for the elements in
L
really being just names or symbols for points in
L
.The
set
S
of sorts remains a set, but the 'set' of operators becomes an object of
Set
(
=
{
absent
,
possible
,
probable
,
present
}
L
)
,
so we now have
(
Ω
,
ϑ
)
,forsome
ϑ
:
Ω
→
L
, as the operator object in
Set
(
L
)
.The
constant
no
:
should now
be seen as
specific for an observer
. There may indeed be (at least) two observers,
Flo
and
Rence
,sothat
→
answer
is now recorded as
ϑ
(
no
)
∈
L
. Even more so,
ϑ
ϑ
Flo
and
ϑ
Rence
bind uncertainty values of
yes
to the specific
observer.
ϑ
Rence
=
absent
.
Flo
,an
experienced home carer may have recorded the observation after having cared for
the patient over the past five years, whereas
Rence
a primary care physician may
have seen the patient for the first time in relation to updating a prescription for
hypertension.
The term functor is now
T
Σ
:
Set
(
Thus, we may have
ϑ
Flo
=
present
and
, and in fact, these functors can
be extended to monads, and we speak of the term monad over
Set
(
L
)
→
Set
(
L
)
L
)
.
