A formal background to unify triples and triple terms

[franconi]

We can make use of the following definitions to introduce the unified view between triple terms and triples, corresponding to their semantic counterparts propositions and facts respectively.

For each interpretation I we define the set IPR of propositions (i.e., denotation of triple terms) of the interpretation as
IPR={ <s, p, o> | s ∈ IR, p ∈ IP, o ∈ IR },
and the set F of all facts of the interpretation as
F = { <s, p, o> | <s, o> ∈ IEXT(p) },
so that the following hold:
I(<<(s p o)>>) ∈ IPR.
F ⊆ IPR,

Given a ground graph G and an interpretation I, the set of facts asserted by G is
GEXT(G) = { (I(s),I(p),I(o)) | (s p o.) ∈ G and <I(s), I(o)> ∈ IEXT(I(p)) }.

An interpretation I is a model of a graph G if and only if the following holds:
I ⊨ G iff GEXT(G) ⊆ F ⊆ IPR

We can discuss here whether this is correct, meaningful, and/or useful.

[niklasl]

I definitely find this correct, meaningful, and useful!

A detail: is it clearer to keep <<(s p o)>> above, where s, p, o are terms, but replace those in the comprehensions with x, y, z? (Given that in the sets they are resources/properties; such that x = [I+A](s). y = I(p). z = [I+A](o)..)

[franconi]

OK. Moreover, I changed GEXT to FEXT -- the set of facts asserted by G. I also dealt with non-ground graphs.

For each interpretation I we define the set IPR of propositions (i.e., denotation of triple terms) of the interpretation as
IPR={ <x, y, z> | x ∈ IR, y ∈ IP, z ∈ IR },
and the set F of all facts of the interpretation as
F = { <x, y, z> | <x, z> ∈ IEXT(y) };
the following hold for each interpretation I and each triple term <<(s p o)>>:
I(<<(s p o)>>) ∈ IPR,
F ⊆ IPR.

Given a ground graph G and an interpretation I, the set of facts asserted by G is
FEXT(G) = { (I(s),I(p),I(o)) | (s p o.) ∈ G and <I(s), I(o)> ∈ IEXT(I(p)) }.

An interpretation I is a model of a ground graph G if and only if the following holds:
I ⊨ G iff FEXT(G) ⊆ F ⊆ IPR.

We can easily extend the above to non-ground graphs:
FEXT(G,A) = { ([I+A](s),I(p),[I+A](o)) | (s p o.) ∈ G and <[I+A](s), [I+A](o)> ∈ IEXT(I(p)) },
I ⊨ G iff ∃ A.FEXT(G,A) ⊆ F ⊆ IPR.

[franconi]

We could add these definitions in RDF-semantics, and then use them in RDF-concepts.

[TallTed]

I mostly like the way this is going. I am OK with the term "propositions" replacing what had been referred to as "quoted triples", but I would like to avoid "facts", "truths", and similar absolutisms. "Assertions", "claims", "statements", "postulates" — something like these would be better.

[gkellogg]

We could add these definitions in RDF-semantics, and then use them in RDF-concepts.

Generally, layering of specs would favor definitions that are used in concepts to be defined in concepts, and referenced elsewhere. Semantics layers on top of Concepts.

[franconi]

We could add these definitions in RDF-semantics, and then use them in RDF-concepts.

Generally, layering of specs would favor definitions that are used in concepts to be defined in concepts, and referenced elsewhere. Semantics layers on top of Concepts.

Still, these are formal definitions which belong to semantics. As usual, these notion will be introduced in Concepts, but their formal definition should be in Semantics, I believe.

[franconi]

I would like to avoid "facts", "truths", and similar absolutisms. "Assertions", "claims", "statements", "postulates" — something like these would be better.

You suggest not to use the word facts.
We cannot use statements, since it clashes with the legacy rdf:Statement.
We cannot use assertions since we cannot say asserted assertions.
Similarly, asserted claims or asserted postulates do not work.

Remember what are we talking about here:

• F is the set of everything which is true in an interpretation, namely in a world, and I call it the set of facts true in that interpretation -- clearly F ⊆ IPR, since IPR is the set of all possible truths;
• FEXT(G) is the set of whatever is true in an interpretation because it has been stated in the graph, and I call it the set of facts asserted to be true in that interpretation -- clearly FEXT(G) ⊆ F if the interpretation is a model of G.

I am calling facts the set F, and asserted facts by G the set FEXT(G).

Open to any reasonable alternative name.

[afs]

We could add these definitions in RDF-semantics, and then use them in RDF-concepts.

Generally, layering of specs would favor definitions that are used in concepts to be defined in concepts, and referenced elsewhere. Semantics layers on top of Concepts.

Still, these are formal definitions which belong to semantics. As usual, these notion will be introduced in Concepts, but their formal definition should be in Semantics, I believe.

Concretely, which items of terminology are being proposed for RDF Concepts?

To semantics, the text in RDF Concept would be description. To someone reading RDF Concepts, they should be able to get a working understanding of the concept (:smile:) without needing to go elsewhere.

A <dfn> gives the wording of terminology. e.g. <dfn>triple term</dfn> and a natural language definition. RDF Concepts could explain "facts asserted by G"; but writing GEXT(G) = is something for semantics.

See section https://www.w3.org/TR/rdf12-semantics/#notation where there are cross-links.

See also w3c/rdf-concepts#118