Re: Issues about contexts and quantifiers
Date: Thu, 8 Apr 1993 11:32:00 -0700
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Subject: Re: Issues about contexts and quantifiers
>However, Mike said that a simple comma is not sufficient and
>that the last line of this statement should be
>          (believes Tom `(smaller_than ,(name ?a) ,(name ?b))) ))
>The reason for the extra "name" functions is that ,?a would refer
>to an actual physical block, and a physical thing cannot be present
>inside a belief.  But ,(name ?a) would produce a lexical identifier
>that could occur inside a belief.
>However, I was not happy with that analysis.  In my ontology for
>representing statements about the world, I make a sharp distinction
>between Tarski-style models, which are abstract data structures,
>and the real world, which is definitely NOT a data structure.
>I would say that the denotation of a term is always a data structure
>in some model; it is never an actual physical (i.e. nonlexical)

No, this is simply a mistake (one related, indeed, to the trickiness I
mentioned in my last message). A Tarskian model has a mathematical
description - it is a set, the domain, and associated sets of functions and
relations over that set, and so forth - but not (usually) a data structure.
Data structures are stored inside computers, and so must be finite; but a
Tarskian domain may be infinite, even uncountably infinite. For example, it
may consist of the real numbers.

>  In the Conceptual Structures book, I defined models to be
>constructed from sets of "individual markers" such as #1, #2, #3...,
>which are effectively GENSYM constants that serve as surrogates
>for the physical objects.  With this construction, ,?a would produce
>some individual marker like #805514, which would be a surrogate for
>the block, not the block itself.

Thats an interesting idea, but not Tarskian model theory. Treating model
domains as consisting of lexical objects amounts to treating the
quantifiers as lexical, so that  forall FOO is shorthand for a (possibly
countably infinite) conjunction of ground instances of FOO. This idea has
been thoroughly critiqued by Kripke.

>That was a disagreement about ontologies rather than a disagreement
>about the way the formalism would be used for any particular ontology.
>I would agree with Mike that ,(name ?a) would be necessary if one
>were to consider denotations to be physical things.  But I believe
>that a clear separation between models and reality is necessary for
>many reasons:  chief among them is the need to talk about multiple
>models of the same physical situation.  In one model, for example,
>Tom might be considered a single individual represented by one
>surrogate #459433.  But in another model, Tom might be represented
>by distinct surrogates at different stages in his lifetime.

This is not a problem. Consider three blocks on a table. How many sets of
blocks are there? Well, one can categorise these blocks into sets in many
ways, or perhaps consider moments of the histories of the blocks as
individuals and consider sets of these, or... Each way of individuating the
world might give a different Tarskian model; but that there are these
various styles of individuation does not mean that any of them are less
real, or that any of the sets thus considered somehow can't be allowed to
consist of real physical things. 

Different ways of 'carving up ' the world give different conceptual
frameworks, but they can all be perfectly real: there is no need to retreat
into the use of lexical substitutes for things. Apart from its inelegance,
the chief problem with that strategy is that it leaves no way to refer to
the actual world, so that the ways in which different styles of
invididuation might interact become unsayable. And it increases the
ever-present risk of confusion between assertions (about something) and
programs (which affect the state of something).

>That issue about surrogates can lead to a lot of discussion, but
>I also wanted to mention another topic we discussed, which is the
>treatment of quantifiers and sets.

I don't speak for the KIF group here, but I think this discussion misses an
important point about KIF. It wasnt intended to be all that readable: its
the machine's interchange language. KIF's role is to provide the expressive
power needed to encode anything anyone wants to express. One would expect
other notations, or perhaps extended notations, to provide ease of
readability. Thus, your suggestions for extended quantifiers are not really
relevant to KIF, but to a hypothetical human-interface to KIF.

Having said that, however, I can't help noting that what seems intuitive is
very much a function of what one is used to. Personally, I find such things

>       (exists (?b block)
>               (and (on ?p ?b)
>                    (forall (?x pyramid)
>                            (=> (on ?x ?b) (= ?x ?p)) ))))))))

not all that unpleasant to write and a real doddle to read: they tend to
fall into one of a small number of simple patterns which can be taken in
with a glance. But everyone to his own syntactic sugar.

More seriously, I would tend to avoid the use of small sets like these, and
rather write three pyramid quantifiers:

        (exists ((?b block)(?x ?y ?z pyramid))
                (and (on ?x ?b) (on ?y ?b) (on ?z ?b))

with suitable =/= additions if necessary. (Here's a question: how can one
automatically translate between this way of expressing the situation and
any of your exists-set-count-three methods?)

This suggests another useful quantifier, with the syntax

        (quant ((dist ?x1...?xn foo))---

and meaning

        (quant ?x1 foo)...(quant ?xn foo)
                (and (=/= ?x1 ?x2) ...(=/= ?xn-1 ?xn) --- )

Where quant is any other quantifier. We can go on finding and defining
'useful' quantifiers for ever, of course, to fit our various styles of
formalisation. Perhaps the most useful facility would be a
quantifier-defining ability, which would presumably be a sophisticated
macro-expander outputting (maybe moderately unreadable) KIF.

Pat Hayes

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