My summary of discussion with Pat was: modal logic, rdf and category theory from Henry Story on 2018-09-19 (semantic-web@w3.org from September 2018)

From: Henry Story <henry.story@bblfish.net>
Date: Wed, 19 Sep 2018 12:50:53 +0200
To: SW-forum Web <semantic-web@w3.org>
Cc: "public-philoweb@w3.o" <public-philoweb@w3.org>
Message-Id: <1FE216CD-4F35-46AC-B2FC-0C9DFC2A0E9E@bblfish.net>

Here is my (subjective) summary of the long thread on modal logic, rdf and CT.
I have very likely missed a lot of details, but I hope it can help see the wood
for the trees.

(A) CT Structure of RDF
-------------------

For a while I thought Pat Hayes did not want one to talk about sets of possible models,
as I think one can use that to then build a definition of the meaning of a graph as the
set of models in which it is true, and that then seems to me very close to modal logic.
Since the RDF1.1 spec mixes model and interpretation I wanted to get a clear view on this.
So I asked on MathExchange a simple question on RDF

"What kind of Categorical object is an RDF Model?"
It got what looks to me like an excellent answer
https://math.stackexchange.com/questions/2905226/what-kind-of-categorical-object-is-an-rdf-model/2905287#2905287
It would be interesting to see how to extend this into the notion of RDF Datasets
found in the RDF1.1 spec, and that comes from the SPARQL query language

This is useful as it allows me then to speak of a category of RDF models.

(B) The Misunderstanding on "Context"
--------------------------------

There was a bit of a misunderstanding on the word "context". I had a minimalist
view of contexts as a quotation mechanism, whereas Pat Hayes was referring to
a well developed notion of context that has a long history in AI. It turns out that
Pat actually has written a lot on this topic.

1) I was thinking in terms of simple quotation where one can have an object of
a relation be a graph. In N3 one writes it with {... } so one can write
S believes { a r b }
The structure that allows one to write that is known as RDF Datasets, it has
a 4 th element for the graph name, so that one can specify for each triple the graph
in which it can be found. (There is a default graph )

2) I have been basing myself (and in my paper on foaf+ssl) on Tim Berners Lee's work on N3
which he wrote up
Berners-Lee, T., Connolly, D., Kagal, L., Scharf, Y., & Hendler, J. (2008).
N3logic: A logical framework for the world wide web.
Theory and Practice of Logic Programming, 8(3), 249-269.
http://dig.csail.mit.edu/2006/Papers/TPLP/n3logic-tplp.pdf

2) Pat Hayes has developed this notion of context in a number of articles of which the following:

⚬ This give an interesting history of the concepts from problems in both external language
and internal language views of language analysis.
Hayes, P. (1997, November).
Contexts in context.
In Context in knowledge representation and natural language, AAAI Fall Symposium.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.98.4812&rep=rep1&type=pdf

⚬ The following article was very influential. It also is very practical and gives a lot
of good examples including cryptography for the use of named graphs.

Bizer, C., Carroll, J. J., Hayes, P., & Stickler, P. (2005).
Named Graphs, Provenance and Trust.
In Proceedings of the 14th international conference on World Wide Web.
http://wwwconference.org/2005a/cdrom/docs/p613.pdf

It is not clear why these are better than DataSets though. I think in a
discussion Tim Berners Lee pointed out that if something has a name, then
it can also fail to have a name. So there is something that needs explaining
as to why the naming was important. I remember a long time ago at TPAC
Jeremy Carroll said he thought there was a theoretic problem with non named
graphs.

⚬ Hayes, P. (2007). Context Mereology. In AAAI Spring Symposium: Logical Formalizations of Commonsense Reasoning (pp. 59-64).
https://pdfs.semanticscholar.org/4899/493c11d2e803bb86ef6b849fb7b3185be1e3.pdf
This is I thought a really neat paper.

3) One should also note that Guha who was somehow related from the beginning to RDF
wrote his thesis on Contexts
Guha, R. V. (1991).
Contexts: a formalization and some applications (Vol. 101).
Stanford, CA: Stanford University.
http://www-formal.stanford.edu/guha/
I read it quickly a long time ago, and interpreted it as a way to formalize the quotation notion that appears in Tim Berners Lee's N3, with quotes within quotes within quotes. That is why I put the RDF/XML example in the mail, since it shows how this has been there since the beginning of rdf - just very unclearly. (But I would need to re-read this)

4) Amazingly Paul Smart of our Social Machines group wrote a paper where
they link both the views of concepts and tie in the Category Theory notion of Institutions
Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010).
Context representation for the semantic web.
https://eprints.soton.ac.uk/270829/

5) Searching further I found this thesis from 2013 relating context to DL & RDF

Klarman, S. (2013). Reasoning with contexts in description logics.
Now what is really interesting here is that it is a 2 dimensional Description Logics.
I remember you saying that CT could help merge such logics. This seems to be exactly
what he is doing.

6) Pat Hayes worked on a language IKL that has a well developed notion
of contexts
https://www.slideshare.net/PatHayes/ikl-survey
https://www.ihmc.us/users/phayes/IKL/GUIDE/GUIDE.html

Because I was worried about Pat Hayes' position I read up on Institutions to see
if that would give me tools in advance to see what would be needed to extend the
semantics to modal logics and show it could then be compatible. As it happens
there was a very nice paper on Institutions and the various RDF languages,
which for me was great to learn about this construct (which I don't find that
difficult now)
Lucanu, D., Li, Y. F., & Dong, J. S. (2006).
Semantic web languages–towards an institutional perspective.
In Algebra, Meaning, and Computation (pp. 99-123). Springer, Berlin, Heidelberg.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf