# Under which theories can a subclass be a member of its superclass, and a member of itself?

I am relearning semantic web technologies and foundational mathematics. Currently, I am trying to understand the nature of whatever theory underpins RDFS: is it a set theory (and if so, what kind?), a type theory (likewise?), a description logic (ditto?), or something else?

The relations that hold in RDFS include the following:

• (1) rdfs:Class is a subclass of rdfs:Resource;
• (2) rdfs:Resource is a member of rdfs:Class; and
• (3) rdfs:Class is a member of rdfs:Class.

• (4) the rdf:type relation "indicates that a resource is a member of a class"; and
• (5) the rdfs:subClassOf relation "specifies a subset/superset relation between classes".

See the upper-left nodes in this diagram from the RDFS specification:

If I understand correctly, then whatever theory underpins RDFS, it has the following properties:

• if it is a set theory, then it is non-well-founded, due to (3).
• if it is a set theory, then, due to (5) it could be said to follow Set Theory and its Logic (Quine, 1969) in taking it to be the case that (pp.1-4, emphasis mine):

Sets are classes. ... Basically 'set' is a synonym of 'class' [because] the distinction [between sets (classes capable of being members) and 'proper classes' (classes not capable of being members)] emerges only in systems that admit [proper] classes, and even in such systems the classes we have to do with tend to be sets rather than [proper] classes until we get pretty far out.

My main question is: am I correct? If not, please explain the way in which I am mistaken.

My secondary question follows. RDFS has one standout elegance: its "property-centric approach ... allows anyone to extend the description of existing resources, one of the architectural principles of the [Semantic] Web." Beyond that, it seems to be a mongrel application of various theories. Perhaps there is an underlying consistency and elegance I am failing to notice, that falls under some cohesive heading. Is there a name for the branch of mathematical theory underpinning RDFS?

• Admittedly, my respect to Quine as a logician has just diminished after reading his quote. – Asaf Karagila Jul 26 '17 at 9:45
• Also, you seem to be actually interested in type theory and not at all in set theory. – Asaf Karagila Jul 26 '17 at 9:46
• Is there a reason that you are using a rather out-dated version of the RDF Schema specification? Also, note that in, at least, the current specification, it is meant as a less formal presentation of the formal semantics given in the RDF Semantics specification. The RDF Semantics specification takes priority over the the RDF Schema specification. – Derek Elkins Jul 28 '17 at 1:15
• @DerekElkins, thanks for your points :) I used w3.org/TR/PR-rdf-schema as the basis for my question primarily because it includes the diagram that I used above, a visualisation that I hoped would improve the clarity of my question. I used it secondarily because, unlike later RDFS specs, it is presented as a single cohesive document. This helped me to keep my question concise. I understand and agree that the RDF Semantics specification takes priority over the the RDF Schema specification, but AFAICT, nothing in RDF Semantics contradicts (1)...(5) above. – sampablokuper Jul 30 '17 at 12:03