Re: modal logic, rdf and category theory

Hi Simon,

In ISO 15926 we don't need any reification because we use N-ary 
relations as described by Natasha Noy and Alan Rector, with 
contributions of Pat Hayes and Chris Welty, in their 2006 Working Group 
Note "Defining N-ary Relations on the Semantic Web 
<https://www.w3.org/TR/swbp-n-aryRelations/>".

We make a distinction between Relation and Relationship. Our upper 
ontology knows many Relationships and ClassOfRelationships that are 
treated as full-fledged classes.

Relationships have two relations, MultidimensionalObjects have N relations.

For instance:  MyCar <<< AssemblyOfIndividual >>> MyEngine  where 
AssemblyOfIndividual is such a Relationship, that can be typed with an 
instance of ClassOfAssemblyOfIndividual (e.g. "Any Car has one Engine") 
if necessary, so including cardinalities for validation purposes.

or, a bit more complicated, to model the calculation of the head of a 
centrifugal pump <http://www.pumpfundamentals.com/what%20is%20head.htm>:



For the full story and code see the topic "Equations and Variables 
<http://15926.org/topics/equations-and-variables/index.htm>" and, if you 
have the stamina, "Process Design 
<http://15926.org/topics/process-design/index.htm>".

I'll get off my soap box :-)

Regards, Hans

------------------------------------------------------------------------

On 5-9-2018 02:35, Krzysztof Janowicz wrote:
> On 09/04/2018 05:27 PM, Simon.Cox@csiro.au wrote:
>>
>> Øthe awkward reification structure survived (sadly, in retrospect).
>>
>> Never replaced by anything better though, so any application that 
>> needs to make statements about statements is driven back to it.
>>
>> In the last couple of weeks in my case …
>>
>
> You can also hash a triple and use the resulting URI as the subject of 
> the triple about this hashed triple. Of course, this comes with some 
> limitations wrt the SPARQL queries you can run.
>
>> *From:*Dan Brickley [mailto:danbri@danbri.org]
>> *Sent:* Tuesday, 4 September, 2018 18:32
>> *To:* Pat Hayes <phayes@ihmc.us>
>> *Cc:* Story Henry <henry.story@bblfish.net>; Semantic Web 
>> <semantic-web@w3.org>; Benjamin Braatz <bb@bbraatz.eu>; 
>> public-philoweb@w3.org
>> *Subject:* Re: modal logic, rdf and category theory
>>
>> On Tue, 4 Sep 2018, 08:51 Pat Hayes, <phayes@ihmc.us 
>> <mailto:phayes@ihmc.us>> wrote:
>>
>>     On 9/3/18 3:27 PM, Henry Story wrote:
>>
>>
>>
>>     The paper you cite below,
>>
>>     2010 Context Representation on the Semantic Web
>>     Bao, Jie, Tao, Jiao, McGuinness, Deborah L. and Smart, Paul (2010)
>>     https://eprints.soton.ac.uk/270829/
>>
>>     was the earlier work I was thinking of here.
>>     >
>>     > I started out very early being made aware of Guha's Phd thesis
>>     > "Contexts: A Formalization and Some Applications"
>>     > http://www-formal.stanford.edu/guha/
>>     > But I read it quickly 15 years ago, and should perhaps study it
>>     more carefully now. As I
>>     > understand he was one of the people at the origin of RDF.
>>
>>     True, but...
>>
>>       So it looks to me like contexts
>>     > are there from the beginning.
>>
>>     ...not so. That is, nothing in the genesis of RDF or the first
>>     Working Group efforts (resulting in the 2004 standard) considered
>>     contexts or tried to get any context mechanism into RDF. RDF
>>     might have been more useful if we had, in retrospect.
>>
>> The first first RDF group, fwiw, ran 1997-99 and gave us a Model and 
>> Syntax specification with a notion of reification supported both 
>> within the abstract graph but also syntactically. It had a ton of 
>> problems, hence the rdfcore WG charter which followed. We constrained 
>> it to be a cleanup rather than total reinvention, so the awkward 
>> reification structure survived (sadly, in retrospect).
>>
>>     What pressure there was to 'expand' RDF was in the direction of
>>     making it as expressive as conventional FOL rather than a context
>>     logic. Guha and I wrote the L-base proposal with this in mind,
>>     for example. TimBL's N3 is in the same spirit, with explicit
>>     quantifiers and scope markers.
>>
>>     >
>>     > In fact I always supposed that the semantic web was going in
>>     that direction, and
>>     > this intuition was confirmed when I discovered Tim
>>     Berners-Lee's and Dan Connolly's
>>     > N3 language very early one, which already at the time allowed
>>     one to be more elegant
>>     > about context.
>>
>>     ? N3 has no context machinery in it at all. It is basically
>>     a(nother!) syntax for FOL.
>>
>>     >
>>     > Indeed in April 2006, I wrote a blog post showing how one could
>>     deal with temporally
>>     > constrained graphs by using an N3 rule to rewrite them.
>>     > "Keeping track of Context in Life and on the Web"
>>     >
>>     https://web.archive.org/web/20060626031531/http://blogs.sun.com:80/roller/page/bblfish?entry=it_s_all_about_context
>>     <https://web.archive.org/web/20060626031531/http:/blogs.sun.com:80/roller/page/bblfish?entry=it_s_all_about_context>
>>     >
>>     > I actually show some N3 rules being applied by CWM in that blog
>>     post to a context
>>     > in order to transform a graph with a temporal relation that
>>     depends on the context
>>     > into one that does not depend on that temporal context.
>>
>>     You keep talking about 'context' here, but that does not make any
>>     of this into anything like a context Logic. N3 has no LOGICAL
>>     MACHINERY for talking about contexts (contrast McCarthy's context
>>     logic, Guha's thesis and its realization in CycL, or the ICL
>>     logic developed for use in the IKRIS project.)  Just using the
>>     C-word when talking about collections of ordinary logical
>>     sentences muddles the issue. To reason with and about contexts
>>     requires /some/ kind of actual context logic, where contexts are
>>     real entities which are described, or at least referred to, in
>>     the logic itself. Without that, the word 'context' really has no
>>     clear meaning at all. For more on this general topic, see
>>
>>     http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.98.4812&rep=rep1&type=pdf
>>
>>     and
>>
>>     https://pdfs.semanticscholar.org/4899/493c11d2e803bb86ef6b849fb7b3185be1e3.pdf
>>
>>     >
>>     > Looking for documentation on N3 I just discovered
>>     > that Tim Berners, Lee Dan Connolly, Lalana Kagal, Yosi Scharf
>>     and Jim Hendler
>>     > wrote a paper that  same year
>>     > "N3Logic: A Logical Framework For the World Wide Web"
>>     > http://dig.csail.mit.edu/2006/Papers/TPLP/n3logic-tplp.pdf
>>     >
>>     > SPARQL seems to have formalized (in what to me seems often a
>>     much less elegant way)
>>     > the pattern matching mechanism of N3. It is also known that
>>     SPARQL can be used
>>     > as a rule language in a way very similar to the log:implies of
>>     N3. The nice thing
>>     > about N3 is that one always sticks to the triple structure of
>>     rdf, which makes it
>>     > very elegant.
>>     >
>>     > There is an interesting question if one can ever escape
>>     contexts at all. It certainly
>>     > looks like it will be very unlikely that people will work out
>>     the right abstractions
>>     > that take all contexts into account.
>>     >
>>     > Thanks to Google Scholar I followed up easily on the article by
>>     Guha you mentioned
>>     > and found some interesting papers.
>>     >
>>     > 2010 Context Representation on the Semantic Web
>>     > Bao, Jie, Tao, Jiao, McGuinness, Deborah L. and Smart, Paul (2010)
>>     > https://eprints.soton.ac.uk/270829/
>>     >
>>     > Is very interesting not least because it shows how one could
>>     build a more
>>     > sophisticated notion of contexts on top of N3 or Quad Stores.
>>     Most interesting
>>     > of all to this thread is that it argues that one could help
>>     specify logical
>>     > levels using Institution Theory that I mentioned earlier. (And
>>     it does so quickly
>>     > while explaining in plain english what some of the formulas mean)
>>     >
>>     >
>>     >>> If the semantics of current RDF has not got this part quite
>>     right it seems to be there in the syntax from the beginning,
>>     since a RDF/XML document can contain another RDF/XML document
>>     >> ?In what sense of "contain"?
>>     >
>>     > Well a predicate can be related to an RDF Literal which of
>>     course needs to be interpreted.
>>
>>     But RDF literals are typed, and the type - in all cases but one,
>>     a datatype - specifies the interpretation.
>>
>>     <Later> OK, I see where you might be going with this. If we say
>>     that an RDF/XML literal denotes an RDF graph, then a triple with
>>     such a literal as object could encode an assertion about that
>>     graph. Yes, that could be done. But it hasn't been done, I should
>>     perhaps emphasize.</Later>
>>
>>     >
>>     > <?xml version="1.0" encoding="utf-8" ?>
>>     > <rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#
>>     <http://www.w3.org/1999/02/22-rdf-syntax-ns>"
>>     >           xmlns:ns0="http://example.com/">
>>     > <rdf:Description rdf:about="http://example.com/LauraLane#i">
>>     >     <ns0:says rdf:parseType="Literal">
>>     >        <rdf:RDF
>>     xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#
>>     <http://www.w3.org/1999/02/22-rdf-syntax-ns>"
>>     >                xmlns:ns0="http://example.com/">
>>     >            <rdf:Description
>>     rdf:about="http://example.com/LauraLane#i">
>>     >                <ns0:disklikes
>>     rdf:resource="http://example.com/ClarkKent#geek"/>
>>     >            </rdf:Description>
>>     >       </rdf:RDF>
>>     >     </ns0:says>
>>     > </rdf:Description>
>>     > </rdf:RDF>
>>     >
>>     >
>>     >
>>     >>
>>     >>> , and presumably the point is not to automatically merge the
>>     embedded document with the one it is contained in. So what is the
>>     meaning of the embedded graph? Well it has to be some
>>     interpretation that satisfies the graph. But there could be many
>>     full interpretations in part because
>>     >>> one can only ever work out Finite interpretations (because
>>     the semantic web is incomplete) and these are compatible with
>>     very many full interpretations.
>>     >>
>>     >> I confess to not following your thinking here. AFAIK, the RDF
>>     WG discussions never considered the idea of one RDF document
>>     "containing" another in any semantically meaningful way.
>>     >
>>     > It allows a document to contain an XML Literal. If that XML
>>     Literal is an RDF/XML literal
>>     > it follows that it can be interpreted just the same way as any
>>     other document. But if one is
>>     > to map that intelligently into a quad store one would map that
>>     as a graph linked to from the first one,
>>     > such as
>>     >
>>     > :LauraLane :says { :LauraLane :dislikes :ClarkKent } .
>>
>>     But this is nowhere mandated by /any/ RDF standard, and I suspect
>>     it never will be.
>>
>>     >
>>     > At the time there were no quad stores to do this type of
>>     transformation, but it would not
>>     > be wrong.
>>
>>     Well, 'wrong' is a strong word. But it would be unjustified, and
>>     not used in any extant RDF tool, and would not transfer to any
>>     other RDF surface syntax.
>>
>>     And well it would not take a lot to create a literal that had
>>     that interpretation
>>     > for RDF/XML.
>>
>>     There is an issue, though. The same content - same RDF graph -
>>     could be represented in, say, Turtle or N-triples or even
>>     JSON-LD. But the corresponding literals would look entirely
>>     different. They would all have to have different datatypes. Which
>>     is possible, but kind of clunky.
>>
>>     I suggest that this line of thinking is going to get lost in
>>     syntactic weeds. You would do better to just start with quads and
>>     try to make them into what you are looking for without going via
>>     embedded literals. Or even adopting some version of the named
>>     graph convention. Just my 2c.
>>
>>
>>     >
>>     >>> But also because documents can be published to mislead ("fake
>>     news") software into believing the universe is other than it is.
>>     >>
>>     >> Well, of course. Publication on the Web has never been a
>>     guarantee of truth or accuracy, and RDF doesn't change that fact
>>     of social life.
>>     >>
>>     >>> I have a short blog post "Phishing in Context - Epistemology
>>     of the Screen" that goes into this
>>     >>> where I make clear the importance of context, and how some
>>     good salesman arbitrarily named Donald
>>     >>> can try to use confusions of context as bait
>>     >>> https://medium.com/cybersoton/phishing-in-context-9c84ca451314
>>     >>> Another simple logic of Contexts is the well known work by
>>     Mike Burrows (who wrote the AltaVista search engine), Martin
>>     Abadi, Butler Lampson, Gordon Plotkin from 1993 "A calculus for
>>     access control in distributed systems"
>>     https://dl.acm.org/citation.cfm?id=155225
>>     >>> Quad stores (RDF Datasets) can clearly be used as a
>>     foundation to build such things on.
>>     >>
>>     >> True. However, quad stores have no universally accepted
>>     semantics, so it will not be easy to standardize any of this.
>>     >
>>     > It's actually quite easy.
>>
>>     You misunderstand me. It may be fairly easy to invent a set of
>>     conventions for quads which do what you want them to do. But quad
>>     stores are an established, even by now an old, technology, and
>>     they are out there being used in many different ways. Large
>>     amounts of developer time and money have been committed to these
>>     uses. To get a STANDARD which assigns a SINGLE semantics to quad
>>     stores is now effectively impossible. (It was already impossible
>>     in 2012, when the RDF 1.1 WG was chartered.) You will not get the
>>     W3C to charter it; and even if someone did that, and it became a
>>     W3C Recommendation, nobody would pay that any real attention. It
>>     is a social problem, not a technical one; and far from being
>>     easy, it is effectively impossible.
>>
>>     The semantics of a graph is the set of interpretations for which
>>     it is
>>     > true. There are a few variables there:
>>     >
>>     > 1) the set IR of resources can be different. For example in our
>>     world the set may contain
>>     > books about Superman, and Laura Lane but no agents with those
>>     properties.
>>     > Whereas in the fictional spaces of those books those Characters
>>     do exist as resources
>>     > with special properties.  But it may also contain resources
>>     about Hillary Clinton and
>>     > weird pizzarias with basements that in our world don't exist -
>>     but that do in the imagination
>>     > of quite a few americans, that have lead some to act on those
>>     beliefs, and many to vote.
>>
>>     Indeed. How, or indeed whether, to deal with imaginary entities
>>     in formal ontologies is a much debated topic. My own view - in
>>     sharp contrast to the more established view often identified as
>>     Quinean - is that the logic must treat all entities similarly, so
>>     that to be in the universe of discourse means only that something
>>     has been /referred to by someone/, and that this is recognized as
>>     not /necessarily/ implying real existence. So we can all talk
>>     about what other people are talking about, and even perhaps
>>     debate with them, without thereby committing ourselves to agree
>>     that what they are talking about is in fact real. (The Horatio
>>     Principle: there are more things in heaven and earth than are
>>     dreamt of in /your/ ontology.) But I admit this is a
>>     controversial position. And I expect that I agree with you that
>>     having an explicit treatment of contexts would be a good first
>>     step towards a more sophisticated approach to this whole issue.
>>
>>     >
>>     > 2) the conventional interpretations accepted for the URIs used
>>     as subject and object.
>>     >    A URI could mean anything before it is coined. For example
>>     the URL for owl:sameAs could have
>>     >    been any number of other URIs.
>>     >
>>     > 3) the various interpretations due to the possibility of blank
>>     nodes being assigned to different
>>     >     resources in each of those universes.
>>     >
>>     > The set of those interpretations is the meaning of the graph.
>>     If the Interpretation considered to
>>     > be the actual one is in there then the graph is true.
>>     >
>>     > So we can understand what
>>     >
>>     > :HillaryClinton :in :Paris;
>>     >      :hears { :HillaryClinton :in :Boston }
>>     >
>>     > And even though the graph quoted contradicts the external one,
>>     we know what it would
>>     > mean for it to be true, but we (may) also know that it is not.
>>     >
>>     >>
>>     >>> If you think as agents that write RDF graphs as processes,
>>     then you can see how this is coalgebraic.
>>     >>> Different such processes are writing from partial information
>>     situations in different contexts, and
>>     >>> with potentially antagonistic aims. Context matters.
>>     >>
>>     >> I think that you are here using "context" informally, whereas
>>     I was using it rather more formally, referring to context logics
>>     such as CycL.
>>     >
>>     > yes, I may have been thinking all one needs is graphs for
>>     contexts. It certainly
>>     > seems to be an essential ingredient, as the article "Context
>>     Representation on the Semantic Web"
>>     > argues. And I now see there is a whole literature on more
>>     sophisticated notions of context.
>>
>>     I think the notion is similar, but the issue is that if you want
>>     to have engines drawing conclusions about contexts, you need some
>>     actual context /machinery/, and probably some logical context
>>     machinery.
>>
>>     >
>>     >>
>>     >>>> Right at the end of the RDF 1.1 semantics document there is
>>     indeed a tiny mention of the possibility of some future extension
>>     of RDF using a modal interpretation of RDF datasets (basically
>>     quad stores). But any such interpretation would require a major
>>     change to the semantics (analogous to the extension of Tarskian
>>     model theory to the Kripke semantics for modal logics) and some
>>     kind of enrichment of the RDF syntax to provide some way to
>>     indicate the syntactic scope of any modal operator. RDF graph
>>     syntax has no scope marking, a fact that gave the RDF and OWL WGs
>>     many technical challenges.
>>     >>>> (For further discussion of this point and what could be done
>>     about it, see
>>     >>>>
>>     https://www.slideshare.net/PatHayes/blogic-iswc-2009-invited-talk
>>     >>>> starting from slide 15.)
>>     >>> Very interesting set of slides. Here are some thoughts from
>>     my research in CT.
>>     >>> • Slide 10 on names
>>     >>>    The difference here is between names as syntactic,
>>     denotational semantic and operations semantic
>>     >>> terms.
>>     >>
>>     >> If I understand you here (I might well not...) then I
>>     disagree. The issue I was trying to describe in that slide is the
>>     fact that referring names on the Web are what one might call
>>     socially non-arbitrary. They are more like names in natural
>>     language than logical identifiers, in this regard. But this has
>>     nothing to do with the nature of the entities they refer to.
>>     >
>>     > yes, David Lewis in "Convention" explains how languages gain
>>     their meanings using co-operative Game Theory. It is a very nice
>>     read. In the case of the web the game is partly determined by the
>>     fact that
>>     > one can dereference the URI. That is the fastest, easiest
>>     method to find out something about it.
>>
>>     Careful. About what, exactly? What an IRI /denotes/ might be
>>     completely different from what you get when you dereference it.
>>     Google "HttpRange-14" for an amazing amount of debate and
>>     discussion about this. Many IRIs - I think most IRIs used in RDF
>>     - do not denote time-dependent things.
>>
>>       And
>>     > practical considerations are not unimportant in helping people
>>     choose between equivalent conventions.
>>     >
>>     >> The syntactic view of names that algebraic views of logic have
>>     is the one espoused by the RDF specs. These are indeed
>>     interchangeable.
>>     >
>>     > Well that is also true. At the beginning of the Web any URI can
>>     mean anything. Then the process of
>>     > convention starts and determines the actual language of the web.
>>     >
>>     >>
>>     >> But they aren't, because the same IRIs get used in non-logical
>>     contexts as well, and while a renaming might preserve the purely
>>     logical meaning it will not preserve the meaningful relationships
>>     to these other uses. But in any case, logical renaming requires
>>     ALL uses of the name to be replaced in one step, and the Web -
>>     even the purely formal part of it encoded in RDF - is too large
>>     and scattered for this to ever be possible.
>>     >
>>     > exactly. That is the process that stabilizes the language -
>>     David Lewis uses the word Metastable 3 times in his book. But
>>     that is true of all languages, not just the web.
>>
>>     Indeed. My point in that slide is that the IRIs used in RDF are
>>     now words in a human language in Lewis' sense, but the formal
>>     semantics does not face up to the reality of this.
>>     >
>>     >>
>>     >>>    But the Semantic web is an evolutionary *process* that
>>     starts at a certain stage and develops,
>>     >>> where different players have only a partial view on its
>>     evolution. Coalgebras represent the mathematics of states (that
>>     evolve) and observation as the subtitle of Bart Jacobs' recent
>>     book on Coalgebras points out clearly
>>     >>>
>>     https://www.cambridge.org/core/books/introduction-to-coalgebra/0D508876D20D95E17871320EADC185C6
>>     >>>    The Web itself is coalgebraic, since resources that URIs
>>     refer to are things that change *State*.
>>     >>
>>     >> Not all of them. And indeed, using a URI to /refer to/ (as
>>     opposed to /identify/, using RESTful HTTP) something with a
>>     changing state is in many ways non-"cool", in TimBL's phrase.
>>     >
>>     > URLs (without the hash and ignoring redirects) refer to
>>     information resources that have
>>     > state and that can change over time. Their referent are states
>>     of objects that can be observed
>>     > through representations that can change over time.
>>
>>     Wrong. Or at any rate, wrong in many cases. This is not true of
>>     pretty much any IRI of the form dbpedia.org/resource/.
>>     <http://dbpedia.org/resource/>..  for
>>     example. It is also not true for any of the XML Schema datatype
>>     IRIs. Nor for https://www.ihmc.us/users/phayes/PatHayesAbout.html
>>     <https://www.ihmc.us/users/phayes/PatHayesAbout.html>
>>
>>     To repeat: what an IRI denotes, and what you get when you
>>     dereference it, are two very different things. They might be the
>>     same in some cases, but in most cases they are not.
>>
>>       That is what I mean by these being
>>     > coalgebraic. As such the referent of those names don't change:
>>     they  refer to the stream
>>     > - the changing stream.
>>     >
>>     > What Tim means by cool URIs don't change is not that the
>>     representation does not change - but
>>     > that they don't change in ways that change the topic so that
>>     people referring to them could
>>     > argue that you have changed the meaning of the documents
>>     linking to your content. Ie: the identity
>>     > criterion for what constitutes representations that are the
>>     same as the previous ones is a socially
>>     > constructed notion of identity. Seriously changing the meaning
>>     is to let down those linking to you.
>>     > And it could have legal consequences.
>>
>>     Or, in a nutshell, the 'S' in 'REST" means 'state'. Right?
>>
>>     >
>>     > I have an illustration of how this works here
>>     >
>>     https://math.stackexchange.com/questions/2896172/how-should-one-model-rdf-semantics-in-terms-of-category-theory
>>     >
>>     >>
>>     >>> These are things that return representations (algebraically
>>     interpretable objects).
>>     >>>    Hence the fight between many logicians and Linked Data
>>     folks may just be a case of a categorical
>>     >>> misapprehension between people working in a dual category.
>>     >>
>>     >> Perhaps, but I think it goes deeper than that.
>>     >>>       Anyway we agree here about names and the importance of
>>     reference. And I think the
>>     >>> categorical duality here can help give us a mathematical
>>     representation of the web that shows how
>>     >>> these two sides can work together.
>>     >>> • Slide 18 on blank nodes
>>     >>>       Interestingly here you note that the problem is with a
>>     set theoretic definition of blank nodes
>>     >>> that are global. Benjamin Braatz in his 2009 thesis
>>     >>> "Formal Modelling and Application of Graph Transformations in
>>     the Resource Description Framework"
>>     >>>
>>     https://www.depositonce.tu-berlin.de/bitstream/11303/2617/2/Dokument_29.pdf
>>     >>> of which the first half is very readable for someone with
>>     knowledge of RDF.
>>     >>> gave a Category Theoretic model of RDF where each graph comes
>>     with its own blank node set, and
>>     >>> the only way to identify such nodes is by morphisms. This is
>>     actually an advantage of a category
>>     >>> theoretic way of looking things that tends to put less
>>     emphasis on identity and a lot more on morphisms.
>>     >>>     Still it looks like RDF1.1 allows blank nodes across contexts
>>     >>
>>     >> It doesn't mention contexts but it does allow for graphs to
>>     share bnodes, in particular, graphs in a single dataset. So yes,
>>     in effect.
>>     >>
>>     >>
>>     >>> (as I gather from 5.1.1),
>>     >>> which would be a way to make statements de Re about someone's
>>     beliefs, eg:
>>     >>>    Laura Lane believes of Superman that he cannot fly.
>>     >>> _:superman = :SuperMan .
>>     >>> :LL believes { _:superman a :NonFlyingPerson;
>>     >>>                       foaf:name "Clark Kent" } .
>>     >>> I wonder how much the blank node sharing would require
>>     changes to Benjamin Braatz'
>>     >>> thesis.
>>     >>>>
>>     >>>>>
>>     >>>>> It is to me very clear that RDF has a modal aspect to it,
>>     which comes out very clearly
>>     >>>>> with Quad stores.
>>     >>>>
>>     >>>> That is totally unclear to me. Quad stores can be, and have
>>     been, used to represent all kinds of 'extra' content, including
>>     graphs with time-stamping or location-stamping or representing
>>     states of something or linking information about a person or
>>     topic to that person or topic. None of this is modal.
>>     >>> Well all of these are thought of as modal logics.
>>     >>
>>     >> No no no! Please don't get this muddled. A logic with times in
>>     it is NOT a modal tense logic, and a logic which mentions belief
>>     contexts explicitly is NOT a modal belief logic. The modalities
>>     occur, and are required, when the logic does NOT mention the
>>     'parameters' of truth explicitly. So, for example, modal TENSE
>>     logic does not have expressions denoting times, but instead has
>>     modal operators for future and past tenses. There is an unspoken
>>     convention that any plain assertion made without the modalites is
>>     supposed to be true 'now'. As soon as you put times into the mix,
>>     the modalities become redundant and can be expalined away as
>>     simple quantified assertions, so that
>>     >>
>>     >> PAST(Full-Professor(PatHayes))
>>     >>
>>     >> would turn into something like
>>     >>
>>     >> (exists (T)(Earlier(T, NOW) & Full-Professor(PatHayes, T) )
>>     >>
>>     >> where we mention the 'now' explicitly. And this is no longer a
>>     modal logic: it's just conventional logic with an ontology of
>>     ties embedded into it. The same kind of thing happens with all
>>     the other modalities (though de dicto stuff in quantified belief
>>     logics does get a bit hairy.) In fact, context logic can be seen
>>     as a general-purpose device for /eliminating/ modalities and
>>     reducing all modal constructions to a non-modal framework. For
>>     lots more on this, see
>>     >> https://www.ihmc.us/users/phayes/IKL/GUIDE/GUIDE.html
>>     >> especially ...#ContextsModalities.
>>     >
>>     > I will need to look at this more closely.
>>     >
>>     > But I think there is an indexical element with any
>>     statement/graph in RDF, namely it's truth depends
>>     > on what the actual world is. This is of course impossible to
>>     completely describe, and
>>     > furthermore there are important disagreements as to its
>>     description. So though I think we
>>     > should agree that the actual world is objectively decidable, it
>>     won't help to ignore the indexical
>>     > role it plays. And this does not alter the meaning of a graph:
>>     its meaning is objective
>>     > (if  we idealise here as before, and ignore potential
>>     disagreements as to the meaning of terms),
>>     > and is the set of interpretations as argued above.
>>     >
>>     > The advantage of this is that I can argue and make sense of
>>     Sherlock Holmes by taking the
>>     > set IR of resources that best describe his partial world as the
>>     actual one when he speaks.
>>     > That will help me make sense of the story.
>>     > See "Truth in Fiction" http://andrewmbailey.com/dkl/
>>     <http://andrewmbailey.com/dkl/>
>>     >
>>     > But it also helps make sense of contemporary politics, as well
>>     as of thieves, liars and professional
>>     > con men/women, Phishers and others whose aim is to change their
>>     victim's perception
>>     > of the actual world enough to entrap them.
>>     >
>>     >
>>     >>
>>     >>> My guess is that the concept that
>>     >>> ties all modal logics together is the concept of context.
>>     >>
>>     >> That's about as wrong as it can get, in fact. See above.
>>     >>
>>     >>
>>     >>> It would be interesting
>>     >>> to see if there is a proof of that...
>>     >>>>
>>     >>>>> But it looks like this may need proving - or perhaps
>>     someone has already
>>     >>>>> done so? Modal logic need not I suppose involve possible
>>     worlds, and the interesting thing
>>     >>>>> is that Category Theories believe to have proven that modal
>>     logic is to coalgebras what
>>     >>>>> equational reasoning is to algebras.  See "Modal Logics are
>>     Coalgebraic" for a summary
>>     >>>>> https://academic.oup.com/comjnl/article-abstract/54/1/31/336864
>>     >>
>>     >> That is, unfortunately, behind a rather high paywall. If you
>>     have a link to an open published version, please give it.
>>     >
>>     > Sorry. Here it is available for all
>>     > https://eprints.soton.ac.uk/267144/1/ModalCoalgRev.pdf
>>
>>     Thanks.
>>
>>     >
>>     >>
>>     >>>>> Coalgebras give us the mathematics of infinite streams,
>>     processes, a notion of co-induction,
>>     >>>>> and are to semantics what algebra is to syntax.
>>     >>>>>
>>     >>>>> All RDF semantics tells us is how to merge two graphs when
>>     one believes them both
>>     >>>>> to be true.
>>     >>>>
>>     >>>> Not quite "all", but your introduction of "believes" is
>>     gratuitous. The RDF (and OWL) semantics saying nothing about
>>     believing or beliefs.
>>     >>> Yes, that's because I am thinking in terms of using these
>>     specs by writing User Agents that need to
>>     >>> help the User work with the published graphs encountered on
>>     the web in order to arrange meetings,
>>     >>> buy products, etc... So I take a pragmatic action oriented
>>     view of these specs.
>>     >>> Here R. Brandom, a student of David Lewis, whose thesis was
>>     on Impossible Worlds, and who went on
>>     >>> from there to develop a philosophy of Analytic Pragmatism, is
>>     well worth looking at. He uses pragmatism
>>     >>> I think to sidestep the idealisation of possible worlds, by
>>     instead speaking of the logical consequences an actor is bound to
>>     by making a statement. Such statements can of course be
>>     contradictory, which is why this is a process oriented view of
>>     possibilities, as I understand.
>>     >>> But nevertheless, the spec would say that the two graphs have
>>     compatible interpretations. Since any graph can have any number
>>     of interpretations, this is saying that there some number of
>>     models that makes them true. But then if there is set of model
>>     that makes them true, that may not be a model the actor dealing
>>     with that graph believes to be true
>>     >>
>>     >> Um... that does not make sense. Models(in this sense) aren't
>>     true or not true, they are representations of the way that the
>>     world could be arranged so as to make some /sentences/ (RDF
>>     graphs) true.
>>     >
>>     > yes :-)
>>     >
>>     >>
>>     >>> - ie one he would act on. Though he may be keen to use the
>>     misapprehension of the actor with that belief to take advantage
>>     of that situation.
>>     >>> Which is pretty much how Phishing works.
>>     >>>>
>>     >>>>> But what if one believes that someone else believes them to
>>     be true?
>>     >>>>
>>     >>>> And how is that nested modality to be represented in a form
>>     that can be transmitted across the Web? You need to explain how
>>     RDF syntax can be extended to cover this kind of assertion.
>>     >>> One does not need to transmit it over the web for it to be
>>     useful.
>>     >>
>>     >> But we are here talking about extending RDF (aren't we?) in
>>     some way, and that extension /does/ need to be transmittable over
>>     the Web. That is the whole point of defining these languages.
>>     >
>>     > yes. Just pointing out that I can gather a number of simple RDF
>>     graphs from the web
>>     > and already start using them using evidence logic, which are
>>     described in a chapter of
>>     > Eric Pacuit's recent book "Neighborhood Semantics for Modal
>>     Logics" in 1.4.4 a Logic of Evidence and
>>     > Belief. The idea is that every graph should count just as
>>     evidence for various propositions and
>>     > actions.
>>
>>     Well, you CAN do that, but that isn't how the vast bulk of
>>     RDF-coded linked data is in fact treated, as far as I know. Maybe
>>     this will some to be needed when, if ever, RDF is used to encode
>>     something more than simple data.
>>
>>     >
>>     >>
>>     >>> If I have an
>>     >>> application that merges different graphs and presents this to
>>     the user it should be
>>     >>> possible for the user to be surprised about a conclusion
>>     reached, ask then where
>>     >>> the information came from, and potentially remove some graphs
>>     that he finds dubious.
>>     >>
>>     >> All true, but AFAI can see, has nothing to do with extending
>>     RDF to be modal.
>>     >
>>     > Does the argument that the actual world being indexical makes
>>     it interpretable as modal
>>     > help convince you?
>>
>>     Not really, but then I don't really understand that argument, or
>>     in what sense you say that the actual world is indexical.
>>
>>     >
>>     > 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.
>>
>>       Perhaps there is an isomorphism, in which case
>>     > it already is modal?
>>     >
>>     >>
>>     >>> But one can even with RDF/XML pass graphs inside graphs since
>>     one can pass an rdf literal
>>     >>> in an rdf graph, and that can contain an rdf literal too...
>>     >>> But there is also some basic ways this has been done since
>>     the blogosphere where they
>>     >>> invented the nofollow attribute when linking to something,
>>     the user wanted to speak about
>>     >>> whilst telling Google that he did not want his link to count
>>     as a +1 for that web page.
>>     >>> Similarly one could have a relation relating an Agent to a
>>     content such as :disagrees
>>     >>> to keep a distance between that content and ones affirmed by
>>     the agent.
>>     >>
>>     >> Oh sure, one can imagine all kinds of ontologies of
>>     propositional attitudes towards content. As well as degrees of
>>     belief, numerical confidence scores and so on. But none of this
>>     requires any changes to the /logic/. OUr old 'named graph' paper
>>     had some ideas in it along these lines, also with detailed
>>     semantics worked out: we had to introduce a notion of rigid
>>     identifier (for the graph names) into the model theory to do it
>>     properly.
>>     >
>>     > Do you have a link to it?
>>
>>     http://wwwconference.org/2005a/cdrom/docs/p613.pdf
>>
>>     see especially sections 8 and 9.
>>
>>     >
>>     >>
>>     >>>>
>>     >>>>> Then by merging them one can find out what they think is
>>     true, and one can model that
>>     >>>>> in terms of possible worlds, or for those more
>>     syntactically oriented sets of all the
>>     >>>>> ways of completing those graphs in ways that are consistent
>>     (or sets of maximally complete
>>     >>>>> such graphs). There is a clear modal element to that, in so
>>     far as one cannot
>>     >>>>> merge graphs of what one believes to be true into someone
>>     else's belief store without getting
>>     >>>>> a wrong idea of what they believe.
>>     >>>>
>>     >>>> But one can say all of this without mentioning the modal
>>     notion of belief. You are here simply talking about truth,
>>     consistency and validity (or otherwise) of inference on RDF
>>     graphs, but adding 'believes' instead of 'true' throughout.
>>     >>> yes, I am speaking of actors that use these graphs in order
>>     to act in the world. I don't believe
>>     >>> and I don't think anyone here believes that software has to
>>     be written that maps an rdf graph
>>     >>> to the interpretation in the world. What happens is that
>>     software developers map graphs to
>>     >>> User Interfaces in a functorial way, and these user
>>     interfaces are then mapped by humans in the
>>     >>> end to things in the world. The humans complete the
>>     interpretation functor by composing with the
>>     >>> initial one designed by the software developer.
>>     >>
>>     >> 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.
>>
>>     > 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 :-)
>>
>>     >
>>     >>
>>     >>> 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. And the
>>     subject-matter of Web logic isn't mathematical at all. The worlds
>>     that linked data describes have essentially no generalizable
>>     mathematical structure.
>>
>>     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.
>>
>>     Pat
>>
>>     > For example it turns out that one can think of programming with
>>     types in ways
>>     > that are very similar to basic algebras one learns in high
>>     school. It used to be
>>     > abstract nonsense. Now category theoreticians are doing
>>     keynotes at programming
>>     > language conferences:
>>     https://skillsmatter.com/skillscasts/10179-the-maths-behind-types
>>     >
>>     >>
>>     >> But YMMV, as I am sure it does.
>>     >
>>     > :-)
>>     >
>>     >>
>>     >>>>
>>     >>>>> it seems like Institution theory may help to do
>>     >>>>> so. In a very interesting paper from 2006 by Dorel Lucanu,
>>     Yuan Fang Li, and Jin Song Dong
>>     >>>>> entitled "Semantic Web Languages – Towards an Institutional
>>     Perspective" show how one can
>>     >>>>> use the theory of institutions to show how RDF, RDFS, OWL
>>     (light, DL,...,Full), ... that
>>     >>>>> seem to have very different semantics can in fact be seen
>>     to be consistent.
>>     >>>>
>>     >>>> The OWL specification documents show this already, in almost
>>     painful detail. (Well, insofar as it is correct. Some RDFS
>>     tautologies are not valid in any OWL dialect, for example.)
>>     >>> Thanks for pointing that out.
>>     >>> In fairness, the article "Semantic Web Languages - Towards an
>>     Institutional Perspective" was
>>     >>> published in 2006 while the document "OWL 2 Web Ontology
>>     Language Mapping to RDF Graphs"
>>     >>> https://www.w3.org/TR/2012/REC-owl2-mapping-to-rdf-20121211/
>>     >>> came out in Dec 2012 so over 6 years later.
>>     >>
>>     >> Well, sure, but the same job was done in 2004 for the earlier
>>     versions of OWL and RDF.
>>     >>
>>     >>> I remember in the early days people doubting that these
>>     languages could have the same
>>     >>> semantics, and using that as an argument for the
>>     infeasibility of the semantic web.
>>     >>
>>     >> Yes. It was a very contentious matter for quite a while. The
>>     split between OWL-DL and OWL-Full was the product of those
>>     energetic debates.
>>     >>
>>     >> Pat
>>     >>
>>     >>>>>
>>     http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf
>>     >>>>> So if someone tells you that these are incompatible
>>     semantics point them to that paper.
>>     >>>>
>>     >>>> Or read the specifications themselves :-)
>>     >>> Yes, now I can point people to two such documents, and most
>>     interestingly for me
>>     >>> is I can see how the two methodologies overlap or diverge.
>>     >>>> Pat
>>     >>>>
>>     >>>>>
>>     >>>>> It looks like work needs to be done to show that these are
>>     also compatible with
>>     >>>>> modal logics (with neighborhood semantics is my guess: ie
>>     coalgebras of the form
>>     >>>>>      S -> S^2^2
>>     >>>>> a.k.a
>>     >>>>>      S -> 𝒫𝒫(S)
>>     >>>>> where 𝒫(S) is a predicate and 𝒫𝒫(S) is a set of
>>     predicates. Now if one thinks
>>     >>>>> of a graph as a predicate on possible worlds, one sees why
>>     this is similar to quad
>>     >>>>> stores. Those are known as a hyper-system as explained in
>>     "Universal Coalgebra: A Theory
>>     >>>>> of Systems"
>>     http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.2020&rep=rep1&type=pdf
>>     >>>>>
>>     >>>>> As for good introductions to CT, since that was part of the
>>     topic 4 years ago,
>>     >>>>> I think the best online intro (and more) for programmers
>>     are Bart Milewski's
>>     >>>>> ( https://bartoszmilewski.com/ ) videos on youtube
>>     >>>>> https://www.youtube.com/user/DrBartosz/playlists
>>     <https://www.youtube.com/user/DrBartosz/playlists>
>>     >>>>> I really recommend it. He is extremely clear without being
>>     boring.
>>     >>>>>
>>     >>>>> I also liked a lot "Category Theory for Computing Science"
>>     by Michael Barr and
>>     >>>>> Charles Wells  (online
>>     http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf )
>>     >>>>> because they make the relation of categories to Graphs so
>>     clear.
>>     >>>>>
>>     >>>>> Indeed just because the relation is so striking I asked a
>>     question on Math
>>     >>>>> Stackexchange to illustrate how one could be (mis?)lead
>>     into a simple pattern
>>     >>>>> of thinking of the relationship
>>     >>>>>
>>     >>>>>
>>     https://math.stackexchange.com/questions/2896172/how-should-one-model-rdf-semantics-in-terms-of-category-theory
>>     >>>>>
>>     >>>>> Has anyone come across further developments in this space
>>     since then?
>>     >>>>>
>>     >>>>> Henry Story
>>     >>>>>
>>     >>>>>
>>     >>>>>
>>     >>>>>> On 17 Apr 2014, at 20:02, Obrst, Leo J. <lobrst@mitre.org
>>     <mailto:lobrst@mitre.org>> wrote:
>>     >>>>>>
>>     >>>>>> Back a few years, emerging from the old IEEE Standard
>>     Upper Ontology group’s work was Bob Kent’s Information Flow
>>     Framework, an ontology framework (a meta-level framework) based
>>     on Barwise & Seligman’s Information Flow Theory, itself an
>>     application of Category Theory. See, for example:
>>     http://arxiv.org/pdf/1109.0983v1 <http://arxiv.org/pdf/1109.0983v1>.
>>     >>>>>>
>>     >>>>>> Mainly folks have used Information Flow Theory or Goguen’s
>>     notion of institutions as springboards from category theory to
>>     ontologies, especially for so-called “lattice of theories”,
>>     ontology mapping, and semantic interoperability applications.
>>     Work includes Mossakowski’s various papers:
>>     http://iws.cs.uni-magdeburg.de/~mossakow/
>>     <http://iws.cs.uni-magdeburg.de/%7Emossakow/>.
>>     >>>>>>
>>     >>>>>> For a short “position” paper, see:
>>     >>>>>> Markus Kr¨otzsch, Pascal Hitzler, Marc Ehrig, York Sure.
>>     2005. Category Theory in Ontology Research: Concrete Gain from an
>>     Abstract Approach. http://www.aifb.kit.edu/web/Techreport893.
>>     >>>>>>
>>     >>>>>> For RDF and category theory, the only paper I know of
>>     addresses graph transformations for RDF:
>>     >>>>>> Benjamin Braatz; Christoph Brandt. 2008. Graph
>>     Transformations for the Resource Description Framework.
>>     Proceedings of the Seventh International Workshop on Graph
>>     Transformation and Visual Modeling Techniques (GT-VMT 2008).
>>     http://journal.ub.tu-berlin.de/eceasst/article/view/158/142.
>>     >>>>>>
>>     >>>>>> Admittedly most of the above are applications beyond logic
>>     itself and RDF, but might shed some light on how category theory
>>     is being used for ontologies.
>>     >>>>>>
>>     >>>>>> Thanks,
>>     >>>>>> Leo
>>     >>>>>>
>>     >>>>>> From: henry.story@bblfish.net
>>     <mailto:henry.story@bblfish.net> [mailto:henry.story@bblfish.net
>>     <mailto:henry.story@bblfish.net>]
>>     >>>>>> Sent: Wednesday, April 16, 2014 6:09 PM
>>     >>>>>> To: Gregg Reynolds
>>     >>>>>> Cc: Antoine Zimmermann; SW-forum Web;
>>     public-philoweb@w3.org <mailto:public-philoweb@w3.org>
>>     >>>>>> Subject: Re: rdf and category theory
>>     >>>>>>
>>     >>>>>>
>>     >>>>>> On 11 Apr 2014, at 16:32, Gregg Reynolds
>>     <dev@mobileink.com <mailto:dev@mobileink.com>> wrote:
>>     >>>>>>
>>     >>>>>>
>>     >>>>>> On Fri, Apr 11, 2014 at 8:30 AM, Antoine Zimmermann
>>     <antoine.zimmermann@emse.fr <mailto:antoine.zimmermann@emse.fr>>
>>     wrote:
>>     >>>>>> There're a lot of resources available online and for free
>>     about category theory.
>>     >>>>>>
>>     >>>>>> Some examples:
>>     >>>>>>    - Jirí Adámek, Horst Herrlich, George E. Strecker.
>>     Abstract and Concrete Categories: The Joy of Cats (524 pages).
>>     http://katmat.math.uni-bremen.de/acc/acc.pdf
>>     >>>>>>    - Maarten M. Fokkinga. A Gentle Introduction to
>>     Category Theory: the calculational
>>     approach.http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf
>>     <http://wwwhome.ewi.utwente.nl/%7Efokkinga/mmf92b.pdf> (80 pages).
>>     >>>>>>    - Jaap van Oosten. Basic Category Theory (88 pages).
>>     http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf
>>     <http://www.staff.science.uu.nl/%7Eooste110/syllabi/catsmoeder.pdf>
>>     >>>>>>
>>     >>>>>>
>>     >>>>>> One of the best is Robert Goldblatt's Topoi : The
>>     Categorial Analysis of Logic .  He pays special attention to
>>     linking CT concepts to both classic math and ordinary intuition.
>>     >>>>>>
>>     >>>>>> I looked through Robert Goldblatt's Topoi quickly [1] and
>>     indeed it is the book that covers the subject probably most
>>     relevant to the semantic web community, since it aims to show how
>>     logic can be derived from Category Theory. In this area I found
>>     reading through the first part of Ralf Krömer's "Tool and Object:
>>     A History and Philosophy of Category Theory" to also be very
>>     interesting, as it gives an overview of the foundational debate
>>     in Mathematics  started by CT.
>>     >>>>>>
>>     >>>>>> It's so odd that RDF is entirely about relations just as
>>     CT is ( except that RDF is one to many whereas CT arrows are
>>     functions). So I really look forward to understanding how these
>>     two domains fit together, and perhaps how they complement each other.
>>     >>>>>>
>>     >>>>>>
>>     >>>>>> Henry
>>     >>>>>>
>>     >>>>>> [1] Having read through half of  "Conceptual Mathematics"
>>     by Willima Lawvere and done most of the exercises there, I am
>>     starting to be able to read a lot of these books much more easily.
>>     >>>>>>
>>     >>>>>>
>>     >>>>>>
>>     >>>>>>
>>     >>>>>> -Gregg
>>     >>>>>>
>>     >>>>>> Social Web Architect
>>     >>>>>> http://bblfish.net/
>>     >>>>>
>>     >>>>>
>>     >>>>>
>>     >>>>>
>>     >>>>>
>>     >>>> --
>>     >>>> -----------------------------------
>>     >>>> call or text to 850 291 0667
>>     >>>> www.ihmc.us/groups/phayes/ <http://www.ihmc.us/groups/phayes/>
>>     >>>> www.facebook.com/the.pat.hayes
>>     <http://www.facebook.com/the.pat.hayes>
>>     >>
>>     >> --
>>     >> -----------------------------------
>>     >> call or text to 850 291 0667
>>     >> www.ihmc.us/groups/phayes/ <http://www.ihmc.us/groups/phayes/>
>>     >> www.facebook.com/the.pat.hayes
>>     <http://www.facebook.com/the.pat.hayes>
>>     >
>>     >
>>     >
>>     >
>>     >
>>
>>     -- 
>>     -----------------------------------
>>     call or text to 850 291 0667
>>     www.ihmc.us/groups/phayes/ <http://www.ihmc.us/groups/phayes/>
>>     www.facebook.com/the.pat.hayes
>>     <http://www.facebook.com/the.pat.hayes>
>>
>>
>
> -- 
> Krzysztof Janowicz
>
> Geography Department, University of California, Santa Barbara
> 4830 Ellison Hall, Santa Barbara, CA 93106-4060
>
> Email:jano@geog.ucsb.edu
> Webpage:http://geog.ucsb.edu/~jano/
> Semantic Web Journal:http://www.semantic-web-journal.net



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Received on Wednesday, 5 September 2018 08:54:37 UTC