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have something here that falls under the umbrella of depression, and we know that
depression accelerates memory loss.
We will not formally define sentences, but here only draw the attention to sen-
tences being defined by a sentence functor Sen with domain being the category
of monads over a fixed underlying category.
The typical example is
Sen(T
Σ
)=
id
2
◦
T
Σ
in the case of producing sentence for equational logic. First-order logic
and various extensions can be defined in the context of generalized general logic,
and this first-order logic must not be confused with the first-order logic appearing
together with ZFC. Note also that natural numbers can be made to “reappear” e.g. in
equational logic as defined above, and this Peano arithmetics is then not to be con-
fused with Peano's arithmetic as appearing in the realm of ZFC, for which Gödel's
self-referential numbering approaches are accepted.
At this point we can proceed to be extremely formal, but this requires writing
space far beyond what is available for this paper. We may for the purpose of this pa-
per, a bit informally, picture sentences like GDS-4(number_of_'positive'_answers)
≥
2, and then we intuitively see how the 'truth' of this sentence is related to our
observations.
Suppose we are faced with a dementia case, and we want to differentiate between
Alzheimer's Disease (AD) and a Vascular Dementia (VaD). Inhibiting drugs may
be used for AD, but they are not suitable for VaD. Also the progrediation of AD
differs from that of VaD, so a long-term treatment plan for AD may differ from
a corresponding plan for VaD, since e.g. behavioral syndromes related to a VaD
patient may be more clear than for a AD patient. We may then have sentences
formulated based on observations of depression (e.g. by a GDS scale), hypertension,
information about a previous stroke, and memory loss (e.g. by the MMSE scale).
Depending on how many of these sentences tend to show 'truth', we will make a
basic judgment about VaD and AD, which may be very useful e.g. for home care
decision-making before possible neurological statements are at hand.
Concerning entailment, and even more informally speaking, we then have con-
clusions like
{
depression
,
stroke
,
hypertension
}
VaD
Now the premises, called
theoremata
, fit into a logic shared by a particular profes-
sional. Back to
Flo
and
Rence
, we can then imagine the respective conclusions
{
depression
Flo
,
stroke
Flo
,
hypertension
Flo
}
Flo
VaD
Flo
{
depression
Rence
,
stroke
Rence
,
hypertension
Rence
}
Rence
VaD
Rence
where
depression
Flo
and
depression
Rence
embrace observations
ς
Flo
(
no
)=
present
and
ς
Rence
(
no
)=
absent
for the first question in GDS-4. Important here
is that the entailment
Rence
resides in
the logic adopted by
Rence
. The question of which conclusion is 'correct' is rather
irrelevant. The interesting aspect is whether or not there is a mapping
Flo
resides in the logic adopted by
Flo
,and
Flo
→
Rence
,
or in general a morphism between the logics adopted, respectively, by
Flo
and
Rence
.
