On Oct 10, 2008, at 8:01 PM, John F. Sowa wrote:
> Rob and Chris,
>
> RF>> How many [relations] are needed for English?
>
> CM> Good heavens, the question doesn't even make sense.
>
> The main sense I would make of it is "How many different kinds of
> relations might be expressed in English?"
>
> My interpretation of that question would be: What is the set of all
> possible relations that might be mentioned or defined in any version
> of any formal logic or any programming language or any similar
> formalism. (01)
Well, *sure*, but I don't see this as so much an interpretation of the
original question but as the rather different the question: How many
relations are needed for a formal representation of a fragment of
English? That question is of course welldefined for a given language
+semantics. My whole point was that English itself is not
sufficiently welldefined for the original question to have any
purchase short of such an "interpretation". (02)
> Since any of those relations could also be mentioned or defined in
> English sentences, (03)
Only by introducing enough technical vocabulary to make the notion of
relation precise. (04)
> the number for English would be the cardinal number for the totality
> of all those for any formal language. (05)
Right. (06)
> The number of relations that are possible would be uncountably
> infinite, (07)
Only if we assume (1) that there are infinitely many individual
objects and (2) that, for every set of objects there is a relation
whose extension is that set. (1) is of course quite reasonable if we
include abstract objects like numbers in our domain. (2) is *highly*
controversial. It means, for example, that there are properties and
relations that cannot be expressed by any recursively defined
language, even if we start with a countably infinite number of
primitive predicates. For example, on this view, there are properties
whose extensions are utterly random sets of natural numbers, ones
whose members cannot be picked out by any finite expression or any
computable process. If ever there were a definition of a
"meaningless" property or relation in the sense that Rob seems to be
groping for, that would be it. This is one very good reason why
semantic theories that include only expressible properties and
relations make for very good models of natural language.
Inexpressible properties and relations are simply beyond our
linguistic ken and hence play no role in both ordinary discourse and
the rather more refined discourse of ontological engineering. (08)
> but the number that could actually be defined or specified in a
> finite statement (in any natural or artificial language) would be
> merely countably infinite. (09)
Right. (010)
> That is still very large. (011)
Indeed. (012)
chris (013)
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