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No connection between propositions and facts in model-theoretic semantics #144

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no connection between propositions and facts in model-theoretic seman…
rat10 Jul 17, 2025
48d6170
Merge pull request #1 from rat10/rat10-patch-1
rat10 Jul 17, 2025
1624502
fixed wording
rat10 Jul 17, 2025
8505cea
add definition of reification, move comment into appendix
rat10 Jul 24, 2025
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60 changes: 57 additions & 3 deletions spec/index.html
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Expand Up @@ -710,7 +710,7 @@ <h3>Properties of simple entailment and satisfiability</h3>
<p class="fact"> If E contains an IRI which does not occur anywhere in S,
then S does not simply entail E.</p>

<p>The following semantic properties relate triple terms and triples asserted in a graph, and they introduce a general definition of satisfiability.</p>
<p>The following semantic properties relate triple terms, triples asserted in a graph and reified triples, and they introduce a general definition of satisfiability.</p>
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<p>The following semantic properties relate triple terms, triples asserted in a graph and reified triples, and they introduce a general definition of satisfiability.</p>
<p>The following semantic properties relate triple terms, triples asserted in a graph, and reified triples, and they introduce a general definition of satisfiability.</p>


<p>We define the <dfn>set of propositions</dfn> in an interpretation as follows:</p>

Expand All @@ -720,10 +720,20 @@ <h3>Properties of simple entailment and satisfiability</h3>

<p class="fact"> The set F of facts in an interpretation I is F(I) = {&nbsp;RE(x, y, z)&#65372;&lt;x, z&gt; is in IEXT(y)&nbsp;}. The set of facts is the set of propositions which are true in the interpretation. </p>

<p>We define the <dfn>set of reifications</dfn> in an interpretation as follows:</p>

<p class="fact"> The set R of reifications in an interpretation I is R(I) = {&nbsp;RE(x, y, z)&#65372;
x is in IR,
y is rdf:reifies,
z is a triple term and
&lt;x, z&gt; is in IEXT(y)&nbsp;}.
The set of reifications is the multi-set of propositions which are reified in an interpretation. </p>


<p>Given a blank node mapping, we define the <dfn>set of facts asserted by a graph</dfn> in an interpretation as follows:</p>

<p class="fact">Given a blank node mapping A, the set of all facts asserted by a graph G in an interpretation I is FEXT(G, I, A) = {&nbsp;RE(&nbsp;[I+A](s), I(p), [I+A](o)&nbsp;)&#65372;`s p o.` is in G&nbsp;}. We then observe that given a blank node mapping, the asserted facts of a graph with respect to an interpretation may not necessarily be among the facts of the interpretation.</p>
<p>We introduce a <dfn>general definition of satisfiability</dfn> of a graph in an interpretation as follows:</p>

<p class="fact">An interpretation (simply) satisfies a graph if and only if there exists a blank node mapping such that the facts asserted by the graph in the interpretation are among the facts of the interpretation.</p>
Expand Down Expand Up @@ -1959,8 +1969,52 @@ <h2 id="non_semantics">RDF reification, containers and collections</h2>
processes to check formal RDF entailment. For example, implementations may decide
to use special procedural techniques to implement the RDF collection vocabulary.</p>


<section id="TTerms">
<h3>RDF 1.2 reification - triple terms and reifiers</h3>
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<h3>RDF 1.2 reification - triple terms and reifiers</h3>
<h3>RDF 1.2 reification &mdash; triple terms and reifiers</h3>

<p>
To repeat nomenclatura:
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To repeat nomenclatura:
To repeat our nomenclature:

<ul>
<li>an `rdfs:Proposition` subsumes multiple kinds of triples:
abstract triples, asserted triples and reified triples
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abstract triples, asserted triples and reified triples
abstract triples, asserted triples, and reified triples

</li>
<li>an abstract triple is encoded as a triple term</li>
<li>an asserted triple is also called a fact,
and also known as an RDF statement
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and also known as an RDF statement
and is also known as an RDF statement,

or simply as a triple in the graph</li>
<li>a reified triple is represented by a reifier and defined by
a fact with `rdf:reifies` as the predicate,
the reified triple term as the object
Comment on lines +1985 to +1987
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<li>a reified triple is represented by a reifier and defined by
a fact with `rdf:reifies` as the predicate,
the reified triple term as the object
<li>a reified triple is represented by a reifier and is defined by
a fact with `rdf:reifies` as the predicate,
the reified triple term as the object,

and a reifier, denoting the reification, as the subject.</li>
</ul>.

Reifying an abstract proposition, encoded as a triple term,
never entails that triple as a fact.
Neither does a fact entail a reification of that triple.
Comment on lines +1992 to +1993
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never entails that triple as a fact.
Neither does a fact entail a reification of that triple.
never entails that triple as a fact,
nor does a fact entail a reification of that triple.


From that follows that in a strict interpretation of the model-theoretic semantics of RDF 1.2
an assertion on a reified triple (denoted by a reifier)
can never be an assertion on a fact asserting that same triple.
Comment on lines +1995 to +1997
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From that follows that in a strict interpretation of the model-theoretic semantics of RDF 1.2
an assertion on a reified triple (denoted by a reifier)
can never be an assertion on a fact asserting that same triple.
It follows that, in a strict interpretation of the model-theoretic semantics of RDF 1.2,
an assertion on a reified triple (denoted by a reifier)
can never be an assertion on a fact asserting that same triple.

The connection between a reification and an assertion of the same triple,
even if they occur in the same graph,
can only be understood as being merely coincidental.

A looser interpretation of that connection
as one of <a href="https://w3c.github.io/rdf-semantics/spec/#dfn-identify">identification</a>,
not <a href="https://www.w3.org/TR/rdf12-concepts/#dfn-denote">denotation</a>,
as applied in RDF 1.2 Concepts, RDF 1.2 Primer and the RDF 1.2 note on triple terms (tbd),
establishes an operational semantics of such a connection between reification and fact
as convention and best practice.
Comment on lines +2002 to +2007
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A looser interpretation of that connection
as one of <a href="https://w3c.github.io/rdf-semantics/spec/#dfn-identify">identification</a>,
not <a href="https://www.w3.org/TR/rdf12-concepts/#dfn-denote">denotation</a>,
as applied in RDF 1.2 Concepts, RDF 1.2 Primer and the RDF 1.2 note on triple terms (tbd),
establishes an operational semantics of such a connection between reification and fact
as convention and best practice.
A looser interpretation of that connection
as one of <a href="https://w3c.github.io/rdf-semantics/spec/#dfn-identify">identification</a>,
not of <a href="https://www.w3.org/TR/rdf12-concepts/#dfn-denote">denotation</a>,
as applied in RDF 1.2 Concepts, RDF 1.2 Primer, and the RDF 1.2 note on triple terms (tbd),
establishes an operational semantics of such a connection between reification and fact
as convention and best practice.

The semantics of RDF 1.2's triple term-based reification mechanism thus diverges
from RDF 1.0/1.1 reification which strictly upholds the model-theoretic interpretation
that reified and asserted triple have no connection beyond mere coincidence.
This design was chosen to facilitate assertions on asserted triples, a.k.a. "statements about statements",
while keeping the model-theoretic semantics of RDF 1.2 simple
and upholding a safe distance from modal logic complications.
Comment on lines +2011 to +2013
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This design was chosen to facilitate assertions on asserted triples, a.k.a. "statements about statements",
while keeping the model-theoretic semantics of RDF 1.2 simple
and upholding a safe distance from modal logic complications.
This design was chosen to facilitate assertions on asserted triples,
also known as "statements about statements",
while keeping the model-theoretic semantics of RDF 1.2 simple
and upholding a safe distance from modal-logic complications.

</p>

<section id="Reif">
<h3>Reification</h3>
<h3>RDF 1.0/1.1 reification - statement quad reification </h3>
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<h3>RDF 1.0/1.1 reification - statement quad reification </h3>
<h3>RDF 1.0/1.1 reification &mdash; statement quad reification </h3>


<div class="c1">
<table>
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