Quantifying cwm Variables…

Pipian — Thu, 18 Dec 2008 18:04:38 +0000

Mostly for my benefit, but here are a few examples of how cwm’s N3Rules translate into formal logic:

Global universal quantification:
```
@prefix : <#> .
@forAll   .

{   :a :b . } => {   :c :d . } .

:someValue :a :b .
```
âˆ€x (a(x, b) â†’ c(x, d))

Therefore the above entails the additional statement :someValue :c :d . as is bound to :someValue on the RHS.
Global existential quantification:
```
@prefix : <#> .
@forSome   .

{   :a :b . } => {   :c :d . } .

:someValue :a :b .
```
âˆƒx (a(x, b) â†’ c(x, d))

Therefore the above entails no additional statements.
LHS universal quantification:
```
@prefix : <#> .

{ @forAll   .   :a :b . } => {   :c :d . } .

:someValue :a :b .
```
(âˆ€x a(x, b)) â†’ c(x, d)

Therefore the above entails no additional statements.
LHS existential quantification:
```
@prefix : <#> .

{ @forSome   .   :a :b . } => {   :c :d . } .

:someValue :a :b .
```
(âˆƒx a(x, b)) â†’ c(x, d)

Therefore the above entails the additional statement :c :d . as is unbound on the RHS.
RHS universal quantification:
```
@prefix : <#> .

{ :someValue :a :b . } => { @forAll   .   :c :d . } .

:someValue :a :b .
```
a(someValue, b) â†’ (âˆ€x c(x, d))

Therefore the above entails (generally) @forAll :z . :z :c :d .
RHS existential quantification:
```
@prefix : <#> .

{ :someValue :a :b . } => { @forSome   .   :c :d . } .

:someValue :a :b .
```
a(someValue, b) â†’ (âˆƒx c(x, d))

Therefore the above entails the additional statement [ :c :d ] .

Finally, two trickier specific examples: “If there exists a foaf:Person that all (known) foaf:Persons foaf:know, then there exists a opularPerson” and, “any foaf:Person that is foaf:knows of all (known) foaf:Persons in a opularPerson” can’t be done properly without completely closing the world. cwm cannot do this without artificially closing the world through built-ins.

It Is A Mystery... » logic

Quantifying cwm Variables…