@@ -107,15 +107,15 @@ This means, "There is something that is a city AND There is something that is a
**Positive surface**
-There are special sheets of paper that can be interpreted as a logical statement.
+The default surface on which RDF graphs are written is called the **positive surface**.
For instance, the sheet of paper in Example 1 is an example of a **positive surface**
-(we use in this document a paper with a black border as a positive surface). Any
-RDF graph written on this surface is interpreted as logical assertion (true). An empty
+(in this document, we use a paper with a black border as a positive surface). Any
+RDF triple written on this surface is interpreted as logical assertion (true). An empty
sheet of positive paper is an empty claim and is treated as a logical tautology (true).
-When there is there is more than one RDF graph on the surface, then it is a logical conjunction (AND).
+When there is more than one RDF triple on the surface, it is a logical conjunction (AND).
If we interpret the sheets of paper with the black border in the examples above as
-positive surfaces, then they express:
+positive surfaces, then they express:
- Example 1: true
- Example 2: It is true that: Ghent a City.
@@ -125,16 +125,14 @@ positive surfaces, then they express:
The [blank nodes](https://www.w3.org/TR/rdf11-mt/#blank-nodes) need to be interpreted
as existential quantified variables on logical paper.
-A positive surface is the default surface, when no other surface information is provided.
-
**Negative surface**
-In the same way as a *positive surface* asserts a logical truth, a **negative surface**
+In the same way that a *positive surface* asserts a logical truth, a **negative surface**
asserts a logical negation. The examples below will use a sheet of paper with a red border
as a negative surface.
-An empty *negative surface* on the default positive surface expresses a logical contradicton. When there are one
-or more RDF graphs written on an *negative* surface they mean the negation of those RDF graphs.
+An empty *negative surface* on the default positive surface expresses a logical contradiction. When one
+or more RDF graphs are written on a *negative* surface, they mean the negation of those RDF graphs.
A blank node on a negative surface is interpreted as a universal quantified variable. The reason is
that:
@@ -164,7 +162,7 @@ A *negative surface* with one triple `[] a :City` means
**Propositional logic**
-With combination of conjunction, by putting triples on a *positive surface*,
+With combinations of conjunction, by putting triples on a *positive surface*,
and negation, by putting triples on a *negative surface*, any **compound
truth-functional statement** can be symbolized with positive and negative sheets
of paper:
@@ -185,14 +183,10 @@ Propositional logic using *positive* and *negative* surfaces.
**First-order logic**
-First-order logic can be added to the RDF surfaces by interpreting the blank node as
-an existential quantified variable and using the rule that a universal quantified variable can made from an
+First-order logic can be added to the RDF surfaces by interpreting a blank node as
+an existential quantified variable and using the rule that a universal quantified variable can be made from an
existential quantified variable by placing it in an enclosing negative surface:
-```
-∀ x : P(x) ⇔ NOT(∃ x : NOT P(x)):
-```
-
- A blank node on a *positive surface* references an existential quantified variable.
- A blank node on a *negative surface* references a universal quantified variable.
@@ -204,22 +198,27 @@ First-order logic using *positive* and *negative* surfaces.
**First-order logic in Notation3**
-RDF Surfaces provides a way to express the notion of *positive* and *negative surfaces*
-in Notation3 with help of the built-ins `log:onPositiveSurface` and `log:onNegativeSurface`.
+RDF Surfaces provide a way to express the notion of *positive* and *negative surfaces*
+in Notation3 with help of the built-in `log:onNegativeSurface`. A `log:onNegativeSurface`
+is true when at least one of the nested triples in the object is false. A *positive* surface
+can be constructed from double nested `log:onNegativeSurface` built-ins. A *negative* surface
+can be constructed from a single `log:onNegativeSurface` built-in.
-The subject of both built-ins are the blank nodes that need to be put on the
-surface (so that they become existential or universal quantified variables depending on the type
-of surface).
+The subjects of both built-ins are the blank nodes that need to be put on the
+surfaces (so that they become existential or universal quantified variables,
+depending on the nesting level of the surface).
-*"Something is a city"* in Notation3 .
+*"Something is a city"* in Notation3.
+
@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-(_:X) log:onPositiveSurface {
+() log:onNegativeSurface {
+ (_:X) log:onNegativeSurface {
_:X a :City .
+ }
} .
@@ -238,6 +237,23 @@ _:X a :City .
+*"Everything is a city"* in Notation3. Notice how we can make universal or existential
+quantified variables by placing blank nodes on an odd or even nested surface.
+
+
+
+@prefix : <http://example.org/ns#> .
+@prefix log: <http://www.w3.org/2000/10/swap/log#> .
+
+(_:X) log:onNegativeSurface {
+ () log:onNegativeSurface {
+ _:X a :City .
+ }
+} .
+
+
+
*"Nothing is a problem"* in Notation3 .
@@ -293,14 +309,15 @@ mean alive, or dead, or both alive and dead.
-A surface can be queried by providing a query surface. The results of this
-query will be the final result of the reasoning engine interpreting the surfaces.
+A surface can be queried by providing a query surface with the `log:onNegativeAnswerSurface`
+built-in. The results of this query will be the final result of the reasoning engine
+interpreting the surfaces.
`:Ghent a :City` is a triple on the implicit positive surface. The two nested
negative surfaces express that for any subject `_:S` on the positive service
-that is a `:City`, it implies that `_:S` needs be be also a `:HumanCommunity`.
+that is a `:City`, it is implied that `_:S` if also a `:HumanCommunity`.
-More test cases can be found at [https://github.com/w3c-cg/rdfsurfaces/blob/master/etc.md](https://github.com/w3c-cg/rdfsurfaces/blob/master/etc.md).
+More test cases can be found at
+[https://github.com/eyereasoner/rdfsurfaces-tests](https://github.com/eyereasoner/rdfsurfaces-tests).
Document Conventions
=====================
-Within this document, the following namespace prefix bindings to [[URI]]-s are used:
+Within this document, the following namespace prefix bindings to [[URI]]s are used:
@prefix ex: <http://example.org/ns#>.
@@ -317,12 +334,15 @@ ex:Ghent a ex:City.
}.
# Query
-(_:S _:C) log:onQuerySurface {
+(_:S _:C) log:onNegativeSurface {
_:S a _:C.
+ () log:onNegativeAnswerSurface {
+ _:S a _:C.
+ } .
}.
-When this surface is executed by a reasoner the result would be:
+When this surface is executed by a reasoner, the result would be:
@prefix ex: <http://example.org/ns#>. @@ -333,12 +353,13 @@ ex:Ghent a ex:HumanCommunity.
| `log:onPositiveSurface` - | Positive surfaces claim that an RDF graph on them is true | |
| `log:onNegativeSurface` - | Negative surfaces claim that an RDF graph on them is false - | |
| `log:onNeutralSurface` - | Neutral surfaces don't claim that the RDF graph on them is true or false - | |
| `log:onQuerySurface` - | Query surfaces use the RDF graph on them as a query + | Negative surface claim that any RDF graph in the object position is false. |
An `ex:myFirstSurface` is created which contains two triples and a nested surface. The
-blank node in `_:X a ex:OtherStatement` is a coreference to the `_:X` graffiti on
+blank node in `_:X a ex:OtherStatement` is a co-reference to the `_:X` graffiti on
`ex:myFirstSurface`.
In the nested surface `ex:mySecondSurface`, the blank node in `_:X ex:is true` is also a
reference to the `_:X` blank node graffiti on `ex:myFirstSurface`.
-The blank node `_:Y` coreference in `ex:mySecondSurface` doesn't refer to a parent blank
+The blank node `_:Y` co-reference in `ex:mySecondSurface` doesn't refer to a parent blank
node graffiti. It is assumed that the default positive surface contains that graffiti.
In the nested surface `ex:myThirdSurface`, the blank node in `_:X ex:is false` is a
-coreference to the `_:X` blank node graffiti on `ex:myThirdSurface`. The new graffiti
-shields the `_:X` that was defined on `ex:myFirstSurface`: the scope of `_:X` coreferences is now
-limited to `ex:myThirdSurface`.
+co-reference to the `_:X` blank node graffiti on `ex:myThirdSurface`. The new graffiti
+shields the `_:X` that was defined on `ex:myFirstSurface`: the scope of `_:X`
+co-references is now limited to `ex:myThirdSurface`.
@prefix ex: <http://example.org/ns#> .
@@ -444,42 +456,45 @@ limited to `ex:myThirdSurface`.
## Positive Surface ## {#PositiveSurface}
-A positive surface is an RDF graph which claims that an RDF Graph on it is true.
-This is the current default interperation of [RDF Semantics](https://www.w3.org/TR/rdf11-mt/).
+A positive surface is an RDF Graph which claims that an RDF Graph on it is true.
+This is the current default interpretation of [RDF Semantics](https://www.w3.org/TR/rdf11-mt/).
When no surfaces are provided, an implicit positive surface is assumed in the RDF document.
-The two surfaces below are equal.
+The two surfaces below are equal (because a double negation of a statement is the same as
+claiming the statement is true: positive):
@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-() log:onPositiveSurface {
- :Alice a :Person .
-}
+:Alice a :Person .
-and
+and
@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-:Alice a :Person .
+() log:onNegativeSurface {
+ () log:onNegativeSurface {
+ :Alice a :Person .
+ } .
+}
-The semantics of a positive surface is interpreted as a logical truth:
+The semantics of a positive surface are interpreted as a logical truth:
-- An empty positive surface is a tautology
-- When a triple is placed on a positive surface, then the surface is valid when the triple is true.
-- When two or more triples are placed on a positive surface, then the surface is valid when the logical conjunction of all triples is true.
+- An empty positive surface is a tautology.
+- When a triple is placed on a positive surface, the surface is valid when the triple is true.
+- When two or more triples are placed on a positive surface, the surface is valid when the logical conjunction of all triples is true.
The surface below should be interpreted as: "Alice is a person and knows Bob, and Bob is a
-person that knows Alice".
+person and knows Alice".
As a logical statement:
@@ -500,43 +515,55 @@ As RDF Surface:
:Alice a :Person .
:Alice :knows :Bob .
-() log:onPositiveSurface {
+() log:onNegativeSurface {
+ () log:onNegativeSurface {
:Bob a :Person .
:Bob :knows :Alice .
+ } .
} .
-When a blank node is marked on a positive surface, then it is interpreted as an existential
-variable in the scope of the positive surface.
+When a blank node is marked on an even-nested negative surface, it is interpreted
+as an existential quantified variable in the scope of the nested surface.
-The surface below should be interpreted as : "There is a person that
-knows Alice".
+The surface below should be interpreted as: "There is a person that knows Alice".
As a logical statement:
```
- ∃ _:X : _:X a :Person AND _:X :knows :Alice
+∃ _:X : _:X a :Person AND _:X :knows :Alice
+```
+
+Or, rewritten using NOT:
+
+```
+NOT( NOT(∃ _:X : _:X a :Person AND _:X :knows :Alice) )
```
+We make use of the logical rule that a negation of a negation cancels, and creates a positive statement.
+
As RDF Surface:
@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-(_:X) log:onPositiveSurface {
+() log:onNegativeSurface {
+ (_:X) log:onNegativeSurface {
_:X a :Person .
_:X :knows :Alice .
+ } .
} .
-In RDF Surfaces an default positive surface is implicitly assumed for each RDF document. On this default positive surface all
-existential variables are implicitly quantified. Example 19 can be rewritten as:
+In RDF Surfaces, a default positive surface is implicitly assumed for each
+RDF document. On this default positive surface, all existential variables
+are implicitly quantified. Example 20 can be rewritten as:
A default positive surface with an implicit `_:X` existential variable.
@@ -551,23 +578,56 @@ _:X :knows :Alice .
-Note: The use of a positive surface is deprecated and will be removed from future versions of the RDF Surface Primer.
-A deeply nested positive surface is syntactic sugar for a double nested negative surface.
+When a blank node is marked on an odd-nested negative surface, it is interpreted
+as a universal quantified variable in the scope of the nested surface.
+
+
+The surface below should be interpreted as: "Everything is a person and knows Alice".
+
+As a logical statement:
+
+```
+∀ _:X : _:X a :Person AND _:X :knows :Alice
+```
+
+Or, rewritten using NOT:
+
+```
+NOT( ∃ _:X : NOT(_:X a :Person AND _:X :knows :Alice) )
+```
+
+We make here use of the local rule that `NOT( ∃ _:X ... )` is equal to `∀ _:X NOT( ...)`.
+
+As RDF Surface:
+
+
+@prefix : <http://example.org/ns#> .
+@prefix log: <http://www.w3.org/2000/10/swap/log#> .
+
+(_:X) log:onNegativeSurface {
+ () log:onNegativeSurface {
+ _:X a :Person .
+ _:X :knows :Alice .
+ } .
+} .
+
+
+
## Negative Surface ## {#NegativeSurface}
A negative surface is an RDF graph which claims that an RDF Graph on it is false. The
-intepretation of the negative surface is the negation of RDF Graph on it.
+interpretation of the negative surface is the negation of the RDF Graph on it.
-The semantics of a negative surface is interpreted as a logical falsehood:
+The semantics of a negative surface are interpreted as a logical falsehood:
- An empty negative surface is a contradiction.
-- When a triple is placed on a negative surface, then the surface is valid when the triple is false.
-- When two or more triples are placed on a positive surface, then the surface is valid when the logical conjunction of all triples is false.
+- When a triple is placed on a negative surface, the surface is valid when the triple is false.
+- When two or more triples are placed on a positive surface, the surface is valid when the logical conjunction of all triples is false.
The surface below should be interpreted as: "Alice is a person and knows Bob and Bob is
-not a person that knows Alice".
+not a person and does not know Alice".
As a logical statement:
@@ -596,7 +656,7 @@ As RDF Surface:
-When a blank node is marked on a negative surface, then it should be interpreted as an existential
+When a blank node is marked on a negative surface, it should be interpreted as an existential
quantified variable in the scope of the negative surface.
@@ -645,7 +705,7 @@ As RDF Surface:
-When two or more negative surfaces are nested in a parent negative surface, then it should
+When two or more negative surfaces are nested in a parent negative surface, it should
be interpreted as a logical disjunction.
@@ -696,7 +756,7 @@ As RDF Surface:
-With combinations of AND and NOT other logical truth function can be build. E.g.
+Other logical truth functions can be built with combinations of AND and NOT, such as the following:
- Disjunction: `P ∨ Q` : `NOT( NOT(P) AND NOT(Q))`
- Material implication `P → Q` : `NOT( P AND NOT Q )` .
@@ -705,160 +765,10 @@ With combinations of AND and NOT other logical truth function can be build. E.g.
Negative surfaces should also follow the law of double negation elimination: `P = NOT(NOT(P))`.
-## Neutral Surface ## {#NeutralSurface}
-
-A neutral surface makes no claims at all about the graphs written on it. Neutral surfaces can be used, for instance,
-to store information about RDF Collections. Triples on a neutral surface will not be "executed"
-in any way by a reasoner. Unless, the triples are copied to a positive or negative surface.
-
-Nesting a neutral surface in another neutral surface will create a nesting of two surfaces.
-
-Writing blank node graffiti on a neutral surface doesn't have specific semantic meaning.
-
-
-The surface below contains on the (default) positive surface factual information about
-`http://example.org/ns#Alice` and `http://example.org/ns#Bob` and two books:
-`http://example.org/ns#Book1` and `http://example.org/ns#Book2`. The positive surface
-also contains structural information to group Books and Authors in an arbitrary author and
-book list.
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-@prefix ldp: <http://www.w3.org/ns/ldp#> .
-
-:Alice a :Person , :BookerPrizeWinner ;
- :name "Alice" .
-
-:Bob a :Person ;
- :name "Bob" .
-
-:Book1 a :Book ;
- :title "Learn RDF in 21 days" ;
- :author :Alice .
-
-:Book2 a :Book ;
- :title "The RDF Surface programming language" ;
- :author :Bob .
-
-() log:onNeutralSurface {
- :BookList1 a ldp:Container ;
- ldp:contains :Book1 , :Book2 .
-
- :AuthorList1 a ldp:Container ;
- ldp:contains :Alice , :Bob .
-} .
-
-
-
-## Query Surface ## {#QuerySurface}
-
-A query surface is used to return to external applications a single (positive) RDF Surface
-specified by a graph template. The result is an RDF Surface formed by taking each query
-solution in the solution sequence, substituting for the variables in the graph template,
-and combining the triples into a single RDF Surface by set union.
-
-The semantics of a query surface is as follows:
-
-- An empty query surface is an empty template which doesn't match any triple on the default surface.
-- A query surface which contains one or more triples, the triples will create a basic graph pattern that will be matched against all entailments that can be
-concluded from the default positive surface.
-- Blank nodes on a query surface are interpreted as universal quantified variables.
-
-
-A query surface that exposes all the triples on the default surface, plus all its entailments.
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-
-(_:S _:P _:O) log:onQuerySurface {
- _:S _:P _:O .
-} .
-
-
-The query above mimics the result of a SPARQL query (except for the added entailments):
-
-
-CONSTRUCT {
- ?S ?P ?O .
-}
-WHERE {
- ?S ?P ?O .
-}
-
-
-
-
-
-Return the MyExample surface which only contains one triple:
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-
-(_:S _:P _:O) log:onQuerySurface {
- () log:onMyExampleSurface {
- _:S _:P _:O .
- } .
-} .
-
-
-The query above mimics the result of a SPARQL query (except for the added entailments):
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-
-CONSTRUCT {
- () log:onMyExampleSurface {
- ?S ?P ?O .
- } .
-}
-WHERE {
- () log:onMyExampleSurface {
- ?S ?P ?O .
- } .
-}
-
-
-Return all triples on MyExample surfaces:
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-
-(_:G) log:onQuerySurface {
- () log:onMyExampleSurface _:G .
-} .
-
-
-
-
-
-Return all negative surfaces (without blank node graffiti) which contain one single triple about `http://example.org/ns#Alice`.
-
-
-@prefix : <http://example.org/ns#> .
-@prefix log: <http://www.w3.org/2000/10/swap/log#> .
-
-(_:P _:O) log:onQuerySurface {
- () log:onNegativeSurface {
- :Alice _:P _:O .
- } .
-} .
-
-
-
-
-## Question and Answer Surface
+## Answer Surface
Issue: TODO
-## Other Surfaces ## {#OtherSurface}
-
-Issue: TODO
-
Examples {#Examples}
================
@@ -866,10 +776,8 @@ Issue: TODO
See:
-- [https://github.com/eyereasoner/eye/tree/master/reasoning/blogic](https://github.com/eyereasoner/eye/tree/master/reasoning/blogic)
-- [https://github.com/eyereasoner/Notation3-By-Example/tree/main/blogic](https://github.com/eyereasoner/Notation3-By-Example/tree/main/blogic)
-- [https://github.com/eyereasoner/Notation3-By-Example/tree/main/examples](https://github.com/eyereasoner/Notation3-By-Example/tree/main/examples)
-
+- [https://github.com/eyereasoner/eye/tree/master/reasoning/rdfsurfaces](https://github.com/eyereasoner/eye/tree/master/reasoning/rdfsurfaces)
+- [https://github.com/eyereasoner/rdfsurfaces-tests](https://github.com/eyereasoner/rdfsurfaces-tests)
{
diff --git a/index.html b/index.html
index 6535cf7..ccbd42e 100644
--- a/index.html
+++ b/index.html
@@ -1488,7 +1488,8 @@
}
}
-
+
+