Information Technology Reference
In-Depth Information
•
all sentences having terms as building blocks, and again equipped with the cor-
responding meaning (algebras) of these sentences,
•
all theoremata being conglomerates of sentences as used in reasoning,
•
entailment as the relation between theoremata representing what we already
know, and sentences representing knowledge we are trying to arrive at,
•
satisfaction as the semantic counterpart to entailment providing the notion of
valid conclusions,
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axioms saying what we take for granted at start,
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inference rules saying how we can jump to conclusions in a chain of entail-
ments, and these rules being selected so as to ensure equality (i.e. the so called
soundness and completeness of the logic) between the entailment and satisfac-
tion relations (equality cannot always be achieved as the completeness part of
logic is sometimes difficult to reach).
In a subsequent section we will make all these notions precise using category theory
as metalanguage.
At this point, note how the signature and terms are ingredients for information (as
“data”) in databases and database theory, where further inclusions of sentence and
theoremata are ingredients for knowledge (“representation”). Entailment and infer-
ence rules are then finally need in order to “compute” of “infer” with knowledge.
This means that we must be careful when we speak about “guidelines” since we
must reveal whether we speak only about the knowledge representations involved
or also about how to deduce new knowledge using these known representations.
Unfortunately, most guidelines are only knowledge representative.
For this notions to enable unambiguous formalism, category theory plays a fun-
damental role as the metalanguage (in turn with the Zermelo-Fraenkel set theory as
the metalanguage for category theory, and so on, hierarchically) for logic formalism,
as the formal (and computable) notions of
term
,
sentence
and
theoremata
are given
by functors extendable to
monads
, and in the case of theoremata even to
partially
ordered monads
[9].
An important part of this approach to ontology is also its capacity and capability
to embrace modelling of uncertainty and non-determinism.
At this point we should again underline that we do not have a single logic, as
dedicated to theory development, covering reasoning within all applications. The
situation is very much the opposite, namely, in that site response and site manage-
ment must be allowed to use different logics, and crisis response and management
logic again differs rather significantly from site response and management logics.
The important property in these respects is that there are mappings between these
logics so that knowledge, represented in a particular logic, can be carried over to
be represented in another logic, understood by other users and stakeholders. The
categorical and monadic approach to logic is critical in particular for these homo-
morphic transformations as represented enabled by functors and represented mostly
by natural transformations between them.
We thus define ontology as information and knowledge encoded using a
particular logic.
