# Antoine's semantics as a model theory. (Was: Re: Islands (ACTION-148))

```On Feb 29, 2012, at 5:28 AM, Antoine Zimmermann wrote:

>
>
> Le 29/02/2012 04:51, Pat Hayes a écrit :
>>
>> On Feb 28, 2012, at 11:39 AM, Antoine Zimmermann wrote:
>>
>>> ....
>>>> If I put two copies of a graph into a single trig document with
>>>> two different lables, one of the copies does not entail the
>>>> other, even though they are the same graph.
>>>
>>> Again, no, the RDF graphs entail each others. It's the pairs
>>> (n1,g), (n2,g) that do not entail each others. Which is exactly
>>> what I would like to have. Otherwise, I would not have made two
>>> copies in the first place.
>>
>> So the added label *changes the meaning* of the graph it is attached
>> to. (It must do so, since otherwise the graph must entail itself,
>> trivially.)
>
> No and no. g entails g. (n1,g) does not entail (n2,g)

Entailment is defined by the dictionary. A entails B when every interpretation which makes A true, makes B true. So, you are telling me that there is an interpretation (in your sense) which makes (n1, g) true but makes (n2, g) false. Please expand on how exactly that happens. (Actually, dont bother, as I have already worked it out, see below.)

> It does not change at all the meaning of g. The meaning of g is not the meaning of (n1,g), just like the meaning of a quad is not the meaning of a triple.

Well, not *just* like that, but OK.
>
> So, the natural question to ask is, how does the meaning
>> change? That is, where in the semantics of these graph/name pairs, is
>> there something that makes it mean something different from what the
>> bare graph would mean? One wants to see a specification of an
>> interpretation structure (like the INT/EXT mappings in the RDF
>> semantics, but probably involving the labels in some way) which
>> assigns meanings to the basic symbols, and then precisely given truth
>> conditions which specify, given such a structure, what the larger
>> syntactic objects - triples, graphs, named graphs and datasets - mean
>> in that structure. So far I dont see that in your proposal.
>
> It's a complete model theory for datasets.

Actually its not a model theory at all, but that can be fixed. A model theory is a formal specification of truth in an interpretation. It defines interpretation structures and truth conditions in enough detail to specify the truthvalue (usually eiher true or false) of sentences in a given interpretation. (Tarski's original paper defining the subject had the title "A theory of truth for formalized languages") All the other notions - entailment, consistency - then *follow* from the basic definition of truth. One does not get to stipulate what "entailment" means by adding little extra conditions on the side.

> Given any dataset, consistency is fully specified, entailment is fully specified, and interpretations are fully specified. What is missing?

See above.

> If you don't like the fact that I do not define what I(D) means for a dataset D, this is another matter. I(D) can be trivially defined, but does not need to.

But it does need to, in order to be an honest model theory and to be properly compared to alternatives. So, in the spirit of cooperation, let me re-write your wiki proposal as a Tarskian model-theoretic semantics. The key point is that the assignment of RDFinterpretations to graph labels is fixed, and entailment relies upon this mapping. So it should be defined as part of the truth conditions rather than hidden in a subscript convention. Here goes.

An RDFinterpretation is an interpretation of RDF graphs as defined in the 2004 semantics.

Given a vocabulary V, let K be the set of all RDFinterpretations on V. An <it>AZinterpretation</it> (on V) is a pair <I, CON> where I is an RDFinterpretation (of V) and CON is a mapping (which we can call the context mapping) from V to K, ie an assignment of an RDFinterpretation to each element of the vocabulary. (Question: should these RDFinterpretations all have the same universe??). Then a dataset (D, (n1, D1)...(nm, Dm)) is true in the AZinterpretation <I, CON> just when I(D)=true and for each j in 1...m, CON(nj)(Dj)=true; otherwise it is false.

This makes the standard definition of entailment match up with your definition: A entails B just when every AZinterpretation which makes A true, also makes B true. (To get stronger entailment regimes, just say that K is the set of RDFS/D-/OWL/OWL2 -interpretations of V rather than RDFinterpretations.  BTW, when I was trashing your semantics, I was using the standard notion of entailment with your truthconditions as stated, which in effect allows the CON mapping to change between the antecedent and the conclusion, which makes entailment almost always fail.)

This is not exactly equivalent to the quads proposal. Intuitively, the quads allows the interpretations of properties to change, while this allows *anything* to change, depending on the context. I dont think this construction can be expressed using quads, in fact.

FWIW, this is actually rather similar to a simplified modal logic semantics in which the contexts are 'possible worlds'. So one could view this as treating a graph with a label as saying "possibly G" where G is what the graph would say if you were to assert it directly, which is kind of cute.

The CON mapping is distinguished from the I mapping of an RDF interpretation, so the graph labels do not denote or refer to a graph in any RDF triples, according to these semantics. So metadata using the graph labels cannot be expressed in RDF in a datastore, unless we extend the semantics in some way.

OK, given this, I will try to summarize the various pros and cons of the three semantic ideas we now seem to have (the third being Sandro's reification idea). This will take a while.

Pat

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Received on Thursday, 1 March 2012 16:48:44 UTC