Information Technology Reference
In-Depth Information
To summarize our main claim in this section, we see logic for mathematics being
the first-order approach developing hand in hand with axiomatic set theory, pro-
viding ZFC, Zermelo-Fraenkel's set theory including the Axiom of Choice, as the
metalanguage (including appropriate intuitions for
conglomerates
and
universe
)for
the object language
category theory
. In turn, when we move over to defining
formal
logic
, category theory becomes the metalanguage for the object language
general-
ized general logic
.
In this strictly hierarchical approach we forbid moving back and forth, as Gödel
frequently did, and indeed remain strict when representation terms, sentences and
proofs in logic. Gödel's numbering indeed comes to a proof calculus, using proof
trees and provides numberings for sentences appearing in proof trees, then produc-
ing predicates involving these numberings, and goes back to the set of sentences
and throws this new sentences into the bag of old sentences. This was allowed one
hundred years ago, and some still allow it. We don't, and indeed for our families of
logic enabled by the framework of generalized general logic. Whatever happened
before ZFC became ZFC, is here not of our concern. We trust ZFC and we trust
ZFC as the metalanguage, not
a
metalanguage, for category theory. And general-
ized general logic must use categorical notions only, and as such based on ZFC. No
by-passing of this principle is allowed.
26.3
Ontology in the Medical Domain
Obviously, ontology as traditionally known e.g. in the medical domain as built upon
standards like HL7, SNOMED CT and openEHR, or OWL and RDF for web ontol-
ogy, are not fully logical. They are only partially logical in that even the underlying
signature for encoding their vocabularies is treated informally with concepts being
more like atoms, and sentence constructions as typically represented by relations,
like
IS_A
in SNOMED CT, e.g. in statements like
open fracture of foot
IS_A fracture of foot IS_A injury of foot IS_A disorder
of foot
.Theterm
open fracture of foot
is then more typically used in
first response for decision-making related to pre-hospital interventions for open frac-
tures,
whereas
disorder of foot
more
levels
involving
expertise
for
orthopedics.
OWL and SNOMED CT have adopted variants of description logics for their
partial ontologies. The variant EL++ is favored in OWL, and has recently (because
of that use within OWL) also been adopted for SNOMED CT. However, OWL is
more tightly bound to EL++, whereas SNOMED CT is still only intentionally bound
to EL++.
It should also be noted that description logic is not a formal logic as described
above. Description logic doesn't even have a formal involvement of signature as
they use
concept
as a primitive notion. Concepts are used as terms and sentences
are relational only, which means description logics appear (intentionally) as kind
of an informal subset of first-order logic. Description logic further does not re-
ally recognize the distinction between logic as the basis for mathematical reasoning
