Re: modal logic, rdf and category theory

Just to add a little to the key passage of the previous e-mail (that was getting a bit long)
I have to write my second year report now so I need to be a little less involved. But I will
try to cite as many of the papers mentioned in the thread :-)

> On 6 Sep 2018, at 17:43, Henry Story <> wrote:
> I don't think I need to change anything to quad stores. All one needs is to define a vocabulary.
> For example
> :believes a rdf:Property
>  rdfs:domain foaf:Agent;
>  rdf:range rdf11:Graph;
>  rdfs:comment "the relation between an agent and the statements it believes that will guide
>         its action. All beliefs by the same agent should be mergeable into a larger graph
>         of beliefs by that agent" .
> So this far all I need is RDFS with Quads.
> So then I can write
> :LauraLane believes { :Superman a FlyingBeing } .
> :LauraLane believes { :KlarkKent a NonFlyingBeing } .
> one can add a few rules which in N3 would be 
>   see docs:
> <>
> <>
> { ?s :believes ?p, ?q } => { 
>  (?p, ?q) log:conjunction ?union.
>  ?s :believes ?union .
> }
> (I guess there is a way of stating that in SPARQL, but it is nice to see that
> N3 is quite elegant for writing things out like this)
> The rule says that one believes the conjunction of ones beliefs. This is a bit of 
> an idealization of course, but this type of doxastic logic tends to be. 
> We also explain her astonishment as she discovers that :Superman = :KlarkKent ( 
> Given that FlyingBeing and NonFlyingBeing are declared to be non overlapping 
> This proves that Laura Lane is subdoxastically an owl reasoner ;-) 
> And she is a fictional being, so we have fictional subdoxastic owl reasoners!
> The mind boggles.
> The rule allows the store to evolve from one state to the next. (So it is coalgebraic).
> There is an (Structural?) Operational Semantics describing how this evolution is done.
> <> (I have not read in detail)
> (If it is a SOS then it is bialgebraic.)
> Because it is possible to end up in a dead end - a reasoning process that leads to
> a contradiction - I suppose it would be useful for the process to keep track of the
> evolution of the state transformation in order to backtrack. I was told that
> CWM uses quints for something like such a reason.
> The above rule thus points out that one criterion of belief of p is that the
> subject try his best to take the union of all his existing beliefs and check new ones 
> for inconsistency.  The reason being that belief is tied to action via desire. The aim of belief is
> to make it possible to act in the world, and so to have the clearest picture when doing 
> so. The belief may be false as for the man who took an assault rifle to save
> children (who did not exist) in the Pizzagate affair, but he did the right thing given his beliefs.
> <>
> So I suppose the question in denotational semantics is what makes a graph
> true in a model. M |= S believes G
> The tricky thing here is that if one gives a purely extensional version of propositions
> where only models come in, then one has problems with co-referring terms like above,
> where :Superman = :KlarkKent since those will co-refer whatever the model, if they co-refer
> at all.  (This problem is explained in many places. I found this article here, that because
> it deals with natural language is more complex that what we need. <>)
> Let us take S to be a software agent instead of a human, then we can 
> work M |= S believes G iff
>  1) S's data store contains a graph of which G is a subgraph
>  2) that graph is in the belief relation to S, that is S will act on G by using the 
>     interpretation I on G to guide his actions in the world.
> So as a result people who use their Quad store to only place beliefs in them are right to
> have the default graph be the union of all the others.
> People who are more skeptical of the data they have (perhaps because they get it by
> following links randomly on the web) may only place in the default graph the information about
> the metadata of where they got graphs from.
> I am not sure if the above definition of belief is quite right, but the point is that it shows
> one I can use a quad store for this type of modal logic without changing anything to the RDF
> semantics, but just by defining a new relation between an agent and a believer.
> { S knows G } requires in addition that the actual world of S be a model for G.

The difference between belief and knowledge on this last point is a little subtle. A believer believes
his beliefs to be true. [1] So that a believer would not state of his beliefs that they are beliefs, he would just 
state them. At least in everyday life that is so.

On the other hand for hyper apps  as developed by the SoLiD project, and which I gave a modal
justification that is behind this thread [2] things are different. A Hyper-App cannot be written
for one web server in mind where all the data is known to be data if it is in folder x, but knowledge
if in folder y. Each person can organize his data as he wishes with names from his natural language, 
on his own server with his own domain.

The app needs, starting from the personal profile of the user, find out what data is believed by the agent, 
what the agent believes  others say ( a proxy ), or said (a version controlled proxy/cache).
(:said is much weaker than belief as there is no logical requirement to merge graphs from a user who
said various things) The hyper app would then present that information to the user 
(in potentially a graphical user interface) that is  the first part of an Interpretation functor to be 
completed by the human (since functors compose) to reach the full interpretation defined by the 
RDF spec. This first interpretation will be presented to the user as the truth (revisable by the user).
On the other hand if the app comes across a link to a belief store by another agent, the app will not 
be required to merge the information from the other in his own belief store as a belief, but rather 
can store it as a belief of the other agent, and merge all the beliefs of the other agent in a separate graph
in order  to get a complete view of that agents view on what actuality is, if needed.

If an epistemic agent such as the merereological office, and given it's apparatus of satellites, engineers,
procedures and so on..., published data it may state what information resources are knowledge (to what percentage), 
etc... Knowing this institution to be a knowledge institution would allow citizens and companies of that 
country to legally defend themselves from having merged that information in their belief store when trying
to make plans about when to take actions (eg. when to harvest, when to run an advertising campaign,...)

All of this seems compatible with web semantics. I have not looked at the state of inference rules standards
developed by the W3C, so I am not sure if any allows one to have rules as the N3 rules I wrote up above
to quantify into contexts. But if there were one may then try to use those to define such vocabularies.
Otherwise these relations can also be defined in english with logical rules as the other specs are.

Henry Story

[1] A point developed in
  Stalnaker, R. (2006). On Logics of Knowledge and Belief. Philosophical Studies, 1(128), 169-199.
  goes into the details of various Kripke modal logics on this 
  sadly I can't find a non paywalled version right now.,5 <,5>

Received on Friday, 7 September 2018 07:16:15 UTC