Re: modal logic, rdf and category theory

On Tue, 4 Sep 2018, 08:51 Pat Hayes, <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
> >
> > 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#"
> >           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#"
> >                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/...  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
>
> 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/
> >
> > 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
> >>>>> 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> 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.
> >>>>>>
> >>>>>> 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/.
> >>>>>>
> >>>>>> 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]
> >>>>>> Sent: Wednesday, April 16, 2014 6:09 PM
> >>>>>> To: Gregg Reynolds
> >>>>>> Cc: Antoine Zimmermann; SW-forum Web; public-philoweb@w3.org
> >>>>>> Subject: Re: rdf and category theory
> >>>>>>
> >>>>>>
> >>>>>> On 11 Apr 2014, at 16:32, Gregg Reynolds <dev@mobileink.com> wrote:
> >>>>>>
> >>>>>>
> >>>>>> On Fri, Apr 11, 2014 at 8:30 AM, Antoine Zimmermann <
> 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 (80 pages).
> >>>>>>    - Jaap van Oosten. Basic Category Theory (88 pages).
> http://www.staff.science.uu.nl/~ooste110/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/
> >>>> www.facebook.com/the.pat.hayes
> >>
> >> --
> >> -----------------------------------
> >> call or text to 850 291 0667
> >> www.ihmc.us/groups/phayes/
> >> www.facebook.com/the.pat.hayes
> >
> >
> >
> >
> >
>
> --
> -----------------------------------
> call or text to 850 291 0667
> www.ihmc.us/groups/phayes/
> www.facebook.com/the.pat.hayes
>
>
>
>