Possible Worlds and the Web

Tim Berner's Lee pressed to define his creation said recently (from memory): "...my short definition is that the web is a mapping from URI's onto meaning".

Meaning is defined in terms of possible interpretations of sentences, also known as possible worlds. Possible Worlds under the guise of the 5th and higher dimensions are fundamental components of contemporary physics. When logic and physics meet we are in the realm of metaphysics. To find these two meet the basic architecture of the web should give anyone pause for thought.

The following extract from RDF Semantics spec is a good starting point:

The basic intuition of model-theoretic semantics is that asserting a sentence makes a claim about the world: it is another way of saying that the world is, in fact, so arranged as to be an interpretation which makes the sentence true. In other words, an assertion amounts to stating a constraint on the possible ways the world might be. Notice that there is no presumption here that any assertion contains enough information to specify a single unique interpretation. It is usually impossible to assert enough in any language to completely constrain the interpretations to a single possible world, so there is no such thing as 'the' unique interpretation of an RDF graph. In general, the larger an RDF graph is - the more it says about the world - then the smaller the set of interpretations that an assertion of the graph allows to be true - the fewer the ways the world could be, while making the asserted graph true of it.

A few examples may help here. Take the sentence "Barack Obama is the 44th president of the U.S.A". There are many many ways the world/universe/complete 4 dimensional space time continuum from the beginning of the universe to the end if there is one, yes, there are many ways the world could be and that sentence be true. For example I could not have bothered to write this article now, I could have written it just a little later, or perhaps even not at all. There is a world in which you did not read it. There is a world in which I went out this morning to get a baguette from one of the many delicious local french bakeries. The world could be all these ways and yet still Barack Obama be the 44th president of the United States.

In N3 we speak about the meaning of a sentence by quoting it with '{' '}'. So for our example we can write:

@prefix dbpedia: <http://dbpedia.org/resource/> .
{ dbpedia:Barack_Obama a dbpedia:President_of_the_United_States . } = :g1 .

:g1 is the set of all possible worlds in which Obama is president of the USA. The only worlds that are not part of that set, are the worlds where Obama is not President, but say McCain or Sarah Palin is. That McCain might have become president of the United States is quite conceivable. Both those meanings are understandable, and we can speak about both of them

@prefix dbpedia: <http://dbpedia.org/resource/> .
{ dbpedia:Barack_Obama a dbpedia:President_of_the_United_States . } = :g1 .
{ dbpedia:John_McCain a dbpedia:President_of_the_United_States . } = :g2 .
:g1 hopedBy :george .
:g2 feardedBy :george .
:g1 fearedBy :jane .

Ie. we can say that George hopes Barack Obama to be the 44th president of the United States, but that Jane fears it.

Assume wikipedia had a resource for each member of the list of presidents of the USA, and that we were pointing to the 44th element above. Then even though we can speak about :g1 and :g2, there is no world that fits them both: The intersection of both :g1 and :g2 is { } , the empty set, whose extension according to David Lewis' book on Mereology is the fusion of absolutely all possibilities. The thing that is everything and everywhere and around at all times. Ie. you don't make any distinction when you say that: you don't say anything.

The definition of meaning in terms of possible worlds, make a few things very simple to explain. Implication being one of them. If every president has to be human, then


@prefix log: <http://www.w3.org/2000/10/swap/log#> .
{ dbpedia:Barack_Obama a dbpedia:President_of_the_United_States . } log:implies { dbpedia:Barack_Obama a dbpedia:Human . }

Ie the set of possible worlds in which Obama is a president of the United States is a subset of the set of worlds in which he is Human. There are worlds after all where Barack is just living a normal Lawyer's life.

So what is this mapping from URIs to meaning that Tim Berners Lee is talking about? I interpret him as speaking of the log:semantics relation.


@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
log:semantics a rdf:Property;
         :label "semantics";
         :comment """The log:semantics of a document is the formula.
         achieved by parsing representation of the document.
          For a document in Notation3, log:semantics is the
          log:parsedAsN3 of the log:contents of the document.
          For a document in RDF/XML, it is parsed according to the
          RDF/XML specification to yield an RDF formula [snip]""";
         :domain foaf:Document;
         :range log:Formula .

Of course it is easier to automate the mapping from resources that return RDF based representations, but log:semantics can be applied to any document. Any web page, even those written in natural languages, have some semantics. It is just that they currently require very advanced wetware processors to interpret them. These can indeed be very specialised wetware processors, as for example those that one meets at air ports.

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