Barry, (01)
Substituting the metalanguage used to define mathematical
structures for the English terms that refer to them is
not likely to generate normal English sentences. (02)
For example, it would be ludicrous to take Frege's
definition of the number five as "the set of all sets
of cardinality 5" and insert that phrase into any
normal English sentence in place of the word "five". (03)
BS> A dog is an instance of function from entities to truth values
>
> Still does not sound right to me. (04)
That was my earlier definition, which I replaced with a
later definition, which added the term "abstract entity"
after "is". If you insist on doing the substitution
you would get (05)
A dog is an instance of an abstract entity, which may be
defined by a rule that is used for classifying entities. (06)
If you don't like my definition of _type_, please state your
preferred version. (07)
BS> Good to follow common sense, too, I think. (08)
I agree. But I don't think it's good common sense to insert
the metalanguage of mathematics into normal English sentences. (09)
> Then this would mean that we would have to accept not merely
> the statement
>
> dog is_a animal
>
> but also the instantiation statement
>
> dog <dog, animal> animal. (010)
This substitution is not equivalent to replacing a term
with its definition: the extension of "is_a" is not the
definition of "is_a", and the pair "<dog, animal>" is only
one element of the extension. (011)
If you insist on making a meaningful substitution, you
should replace the entire sentence "dog is_a animal"
with the pair "<dog, animal>". (012)
John (013)
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