Re: modal logic, rdf and category theory

jus snipping this subdiscussion related to category theory and rdf, as I need to
 a bit of  time to read the articles Pat sent there.

> On 4 Sep 2018, at 09:46, Pat Hayes <phayes@ihmc.us> wrote:
> 
>> David Lewis showed that one can map counterfactual statements to first order logic as
>> long as one can quantify over possible worlds. Translated to this context this would mean
>> that we can quantify over interpretations.
> 
> Hmm. I don't think this actually makes sense. Try to sketch what such a logic would look like. To quantify over interpretations, you need a way to /refer/ to interpretations. I don't think it is internally coherent to have a logic which has names which refer to the interpretations of that very logic, so that the universe of an interpretation includes ... interpretations? Maybe Aczel's set theory can handle this, but its going to get very strange.

Ok, yes, I meant we quantify over models.  It's just that they are merged in the RDF1.1 spec with
Interpretations so I was not sure how to speak about them anymore.

[snip]

>>> 
>>> I don't think the semantic interpretation mapping is a functor, because I don't believe that the real world is a category :-)
>> Does the semantics not require sets? Is the world composed of sets?
> 
> No, it is composed of things with relations holding between them. Calling this a 'set' is the minimal amount of mathematics necessary to describe it at all; seeing it as having any further structure is a form of mathematical hallucination, IMO. But I know I am out on a lonely limb here.

So that is what category theorist call a Category. Objects and arrows between them. (with
an extra notion of a path which you can get for free if you start from objects and arrows).
It's easier if you start with a Quiver (the notion of a graph that corresponds to ours in RDF),
which is just arrows with start and end points. These automatically give rise to a Free
Category of the Quiver. 

So here is how Category Theory works: since I can always map your talk of objects and
relations to a Quiver, and back without loss, we are speaking of isomorphic things, and
so we can always translate one talk to the other. There is then no need to have disputes
about who has the best ontology. If you can talk of things with relations between them then I
should be able to talk of the free category generated by that quiver, and if I want
to translate back to your talk I just drop the path requirements of those categories.

Thinking abaut it I wondered if Quivers were the right abstraction. So I asked 
what Category Theorists thought on Stack Exchange.

https://math.stackexchange.com/questions/2905226/what-kind-of-categorical-object-is-an-rdf-model/2905287

The serious answer I got was one can create a category of models from subsets 
of IR×IP×IR and morphisms between them, and that these have some very nice properties
in Set.

The advantage of the CT abstraction is that it can help one locate RDF in the space
of mathematical structures, and so find ways of elegantly shifting to another model
domain if needed, but also bring proofs across from another part of mathematics to
bear on this.

So we can create the category of RDF models it seems. But when we add RDFS our models
are then reduced to those that have some extra properties. What happens to the old
models then? Are they impossible models? 

The paper on institutions I quite earlier shows how this works.
Lucanu, Dorel, Yuan Fang Li, and Jin Song Dong. "Semantic web languages–towards an institutional perspective." Algebra, Meaning, and Computation. Springer, Berlin, Heidelberg, 2006. 99-123.

Btw, I want to thank Google for creating Google Scholar, which makes researching 
advanced topics like this possible from outside the university
https://scholar.google.co.uk

> 
>> But seriously I am only putting that forward as a thought experiment to see where
>> it fails, in order to understand where people coming from category theory may be mislead
>> by trying to apply categories in an obvious way, but also to see why one may need more
>> complex structures like Institutions.
> 
> Fair enough :-)

And now I have a precise answer on StackExchange that I will need to spend
some time researching to fully understand...

> 
>>>> 
>>>> I give a simplistic but at least intuitive view of how such a functorial notion of semantics can
>>>> be understood to work in the math exchange question
>>>> https://math.stackexchange.com/questions/2896172/how-should-one-model-rdf-semantics-in-terms-of-category-theory
>>>> I need to develop that a lot more of course...
>>>>> 
>>>>>> 
>>>>>> So if this still needs to be proven
>>>>> 
>>>>> What exactly "needs to be proven" ?
>>>> I suppose that RDF1.1 with datasets is compatible with modal logic. Though I
>>>> have a feeling that Kripke modal logic is too simple and even David Lewisian
>>>> modal logic which is a neighborhood semantics based one is not quite right.
>>>> In the newly published book "Category Theory for the Working Philosopher"
>>>> https://books.google.de/books?id=RIM8DwAAQBAJ
>>>> there are many very intersting articles. One by Abramski on Contextuality and
>>>> Paradox. But also the one by Kohei Kishida on "Categories and Modalities"
>>>> which looks a neighborhood semantics with impossible worlds and shows
>>>> how that can be understood in terms of category theory.
>>>> I have not yet fully digested all these different pieces. But I hope this
>>>> gives some idea as to the work one could draw on to further the semantic web
>>>> and the web in general by placing it on even firmer formal foundations.
>>> 
>>> Well, good luck. I confess to not, myself, finding Category Theory much use in providing any useful insights; it seems to be a whole lot of jargon describing very little, compared to the simplicity and elegance of the usual set-theoretic picture. The Wikipedia article on Coalgebras (which I looked at to help me understand what you were talking about earlier) is a good example. What in this
>>> https://en.wikipedia.org/wiki/Coalgebra
>>> provides ANY useful insight AT ALL into what we are discussing? It defines a coalgebra as a vector space, for a start. What do vector spaces have to do with RDF, modal logic or the Web?
>> yes, that is not a very good introduction.
>> Corina Cirstea's article is much better and so is
>> "Universal Coalgebras: A theory of Systems"
>> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.2020&rep=rep1&type=pdf
>> as well as Bart Jacobs, Jan Rutten "A tutorial on (co) algebras and (co) induction"
>> https://pdfs.semanticscholar.org/40bb/e9978e2c4080740f55634ac58033bfb37d36.pdf
>> He has a lot of excellent articles from the 1990ies showing how OO programming
>> is coalgebraic. But he also has an article showing how there is a duality between
>> OO programming and modal logics with operators
>> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.7008
>> ( much more difficult but it shows how this can help bridge branches
>> that would seem incompatible)
>> Benjamin Braatz' thesis is an algebraic approach to RDF, and the first half
>> would be close to your heart, as he has blank nodes tied to graphs, which is
>> a way to make your metaphorical idea of surfaces real.
>> One of the key things of Category Theory is that it emphasizes structure above
>> elements. And most amazingly it is based on the same notion of a graph that
>> RDF uses. That is what is so weird about it. Category theory is less interested
>> in identity as it is in translation or isomorphism. So that is why it is very good
>> at finding deep symmetries between very different parts of mathematics, as well
>> as showing how the same structure is found across mathematical and logical domains.
> 
> OK, I know it is foundational in mathematics, but Web logic isn't primarily a mathematical topic. The actual metamathematics of logic (certainly of RDF) is very simple, almost trivial.
> It doesnt need anything high-powered to grasp it.

Absolutely. RDF is brilliantly simple.

What is mind boggling is that it is using a structure very close to the
foundational structure of category theory.  What we call Graphs they 
call Quivers (because graph theorists can't agree on a 
clear meaning of graph). As a result you have Graph Theorists like Benjamin Braatz 
who can write a thesis that explains RDF in category theoretical terms in terms of Graph transformations, or morphisms.

So RDF is brilliantly simple, and Category Theory is just that plus paths in graphs. 
How come such a simple notions has helped reveal so much in mathematics? 
 
>  And the subject-matter of Web logic isn't mathematical at all. The worlds that linked data describes have essentially no generalizable mathematical structure.

Well that is what I think I am putting into question when I point out that coalgebras
give you the mathematics of streams, processes etc... These are the dual of algebras.
In a way it looks like category theory has proven in the past 20 years that Heraclitus and Plato's
philosophy were duals, mirror images of one another. The unchanging space of what we know may 
call algebraic ideas is the dual to the space of processes and streams, where it may be
felt that one can never enter the same river twice.

> 
> But whatever, I don't mean to have an argument about this. If you can find insight in category theory, good luck with it :-) Thanks for the pointers, in any case.

Thanks for the very helpful discussion. I will still answer the part about modal logic, 
but then I need to give in my second year report. So this discussion has been very helpful
for that.

> 
> Pat

Received on Wednesday, 5 September 2018 08:05:04 UTC