Pat, (01)
Innumerable proper tests and analyses have been performed over
the past two millennia. (02)
PC> It seems to me to be counterproductive to assume an answer
> before a proper test is performed. (03)
Before getting into any further discussion, it is essential to
distinguish two kinds of definitions in mathematics: (04)
1. Closed-form definitions, in which one concept or predicate P
is defined as a synonym for a particular expression X (possibly
with free variables in X that can be replaced by arguments of P).
Such a definition would allow any occurrence of P to be replaced
by X with arguments substituted for the free variables. (05)
2. Implicit definition, in which some concept or predicate P is
used in various axioms, but never defined by a closed-form
definition. (06)
Whenever possible, mathematicians like to find closed-form definitions
because they simplify the computation. To evaluate some predicate P
for some list of arguments, they can simply compute the expression.
Unfortunately, only a tiny fraction of interesting functions and
predicates have closed-form definitions. There are uncountable
infinities of functions that cannot be defined in closed form. (07)
PC> The relevant precedent of work with the Longman defining vocabulary
> is, I believe, highly suggestive of a reasonably small (<10,000)
> concept set that can specify the meaning of anything of interest,
> by combinations of the primitives. (08)
First of all, you are assuming that something that is impossible for
the much narrower language of arithmetic could be accomplished for
the vastly larger and more complex natural languages. (09)
Second, the Longman editors never accomplished that task. They
never even attempted to give complete, closed-form definitions
for all the words in their dictionaries. For example, look at
their definition of 'whist': (010)
a type of card game for 2 pairs of players (011)
Then look at one of the definitions for 'bridge': (012)
a type of card game for 4 players developed from the game
of WHIST and usually played as CONTRACT BRIDGE or sometimes
as AUCTION BRIDGE. (013)
Then look for the definitions of 'auction bridge' and 'contract
bridge', and you will find the note "see BRIDGE". You can also
look at the definition of 'poker': (014)
a type of card game usually played for money. (015)
Since bridge and whist are sometimes played for money, this
doesn't say much to distinguish them. I'd love to play for
money against somebody whose only knowledge of poker or bridge
is what they read in that dictionary. (016)
If you browse through the Longman's dictionary, you'll find an
enormous number of such examples. (017)
In short, they are just specifying a type hierarchy with very
few properties that are woefully insufficient to distinguish
many terms from one another. For physical things, they
supplement the written definitions with pictures, but those
won't provide much help for most of our purposes. (018)
If you want to see the complete axioms (rules) for contract bridge,
you can download a 76-page PDF from www.acbl.org -- but it's written
in English, not a formal logic. That is the kind of information
that is needed to supplement a very large number, probably most
of the so-called definitions given in the Longman's dictionary. (019)
I'll admit that the kind of type hierarchy specified in Longman's
can be useful for many of the same purposes as WordNet. But that
is far less than what is necessary for reasoning. (020)
As I've said many times, study Cyc. That is the best example of
what you're asking for. They have over 600,000 concepts defined
by over 2 million axioms, yet they're very far from providing
the kinds of definitions you're hoping to find in Longman's. (021)
John (022)
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