Ed and Mark,
EJB
> The ternary predicate approach that John describes is viable,
> but clumsy. The ternary predicate approach requires a different
> ternary predicate if you want to specify location instead of time,
> or a quaternary predicate to specify both, etc.
MLH
> This is why I claimed that IKL “that” (or Sowa “describes”) is
> necessary. It is just plain unrealistic to expect every predicate
> to have variants for time and location – not to mention other
> modifiers that English would express using adverbs.
I agree that adding an extra argument to every relation is clumsy.
And I was *not* recommending it. The points I wanted to emphasize
are theoretical, but with practical implications:
1. Quantifying over time can be done in a purely FOL semantics.
There is no need to introduce the 'that' operator and all
the semantic issues it entails. (I cited a book that uses
a sorted FOL with time as one of the sorts.)
2. If you want to factor out those references (to time, location,
or whatever), you can introduce a purely-syntactic notion of
context.
3. That syntactic notion does not affect the semantics in any way,
since every use of a context box (or other grouping markers)
can be eliminated by a syntactic translation that adds an
extra argument to every relation.
4. After the translation in #3, you can use an ordinary Tarski-style
model theory for the semantics. There is no need to go to the
more complex semantics of IKL in the model theory.
5. But as a practical KR notation, you don't have to translate
the context boxes (or other grouping markers) to the form
with extra relations or arguments.
For the details of how to represent a kind of context, attach time
and place references to that context, and then eliminate the contexts
by a translation as above, see the following article:
http://www.jfsowa.com/pubs/laws.htm
Laws, facts, and contexts
To represent a Tarski-style model, I use *nested graph models* (NGMs),
which have "context boxes" that allow graphs nested within graphs.
For the logic, I use conceptual graph notation, but you could use
just as well use CLIF notation. (The article was published in 2003
while the CL standard project was in its early days.)
To see the differences between the three kinds of notations, Figure 9
shows a conceptual graph in its nested form with type labels. Fig. 10
shows a nested graph model for which the denotation of Fig. 9 is true.
Then Fig. 11 shows the translation of the NGM in Fig. 10 to a "flat"
form with no nesting. An 'isin' relation links a node for each relation
to a node that represents the context in which the relation occurs.
This flattened graph could be represented as an ordinary Tarski model.
Figures 9 to 11 have proposition nodes. They raise issues that IKL
dealt with in more detail about 3 years later. If you're not using
verbs like 'believe', you don't need proposition nodes. Those boxes
could represent situations at particular times and places.
John
