So this gives me this idea of exploring the space between propositional logic and first

order logic with translations into sets is actually something we want to explore.

Remember A boxes and T boxes and all of those kind of things we want to build.

And that's something we want to come to next.

And all of this is being called under the name of ontologies.

Now ontologies is a big word and if you use big words you better look them up and so I've

done that.

So the classical definition of this is an ontology is a representation of the types,

the properties and interrelationships of the entities that really or fundamentally exist

for a particular domain of discourse.

Wow that's exactly what the kind of language philosophers or Wikipedia give you.

A very general account that really depends on what we mean by representation, entities

and types and interrelationships.

So since none of that is actually really being given there we can kind of make it so.

So this allows us to give the following technical definition namely an ontology and note that

this is kind of the definition the old Greeks came up with 2000 years ago.

And this is what the modern computer scientists came up a couple of decades ago that it basically

says an ontology is a formal system where we have a language, a class of models that

gives us a semantics and a satisfaction relation and a set of concept axioms that we can use

to express knowledge about a domain and we want to be able to talk about individuals

which are basically objects in the domain, concepts, classes of individuals that share

properties and aspects and relations which tell us in which way classes and individuals

can be related to one another.

And we've seen already those things.

Semantic networks are ontologies.

We have a language, we have a notion of meaning possibly just by translating into a formal

system that we've already seen and yeah a satisfiability relation.

Our set description language PL0DL is an ontology.

This is a formal one as opposed to the semantic networks which are inherently informal but

is relatively weak.

We cannot say many of the things we would like to say.

And of course personal logic is also an ontology format.

It's also formal.

It's very expressive.

It doesn't have decision procedures.

We have already three ontology formats that definitely adhere to our technical definition

and possibly also is what we mean by the classical definition of ontologies.

And this idea that we have formal ontologies that are formal systems and we have these

three examples where we basically have first-order logic and propositional logic which are kind

of the delineating examples basically gives rise to what we call now the description logic

paradigm.

So the idea is that a description logic is a formal system for talking about sets, their

relationships which is at least as expressive as PL0 with set theoretic semantics and also

offers individuals and relations.

So we have the following four components.

We have a formal language.

We have a set theoretical semantics.

We have a translation into first-order logic typically and we have a calculus.

So this is kind of the idea here.

We have first-order logic which is kind of the upper bound of description logics which