On Fri, 2 May 2008, John F. Sowa wrote:
> I sympathize with your point of view, but...
>
> CM> I like to follow fairly standard practice and reserve
> > "theory" for deductively closed sets of sentences.
>
> Your use of the term 'fairly standard practice' usually means
> fairly standard among that subset of humanity that reads and
> writes papers on formal logic and related topics. (01)
No, within that subset it is a nearly universal standard. Outside that
subset one commonly finds it among those with some background in logic. (02)
> Since that subset is tiny, even within the IT community, that usage,
> without explanation, can cause misunderstanding. (03)
I'm puzzled -- I provided a precise explanation when I said I reserved
"theory" to mean "deductively closed set of sentences" and noted that
this definition is in fact a common convention, didn't I? The subset in
question may be small, but it is the subset from which the term comes. (04)
> When the distinction is important, it's helpful to expand the word
> 'theory' to either
>
> (1) the set of axioms of a theory, or
>
> (2) the set of implications of some axioms. (05)
But that's out of step with practice in mathematical logic. (I assume
that by the axioms of a theory you mean (again, following standard
conventions) a *decidable* set, i.e., that there is a way of
distinguishing axioms from non-axioms.) For it is not uncommon in logic
to define a theory first and only then ask whether it is axiomatizable.
For example, "the theory of arithmetic" is usually defined to be the set
of all true sentences in the language of arithmetic. (This set is
obviously deductively closed.) G?el's first incompleteness theorem can
then be stated very succinctly as the proposition that the theory of
arithmetic is not axiomatizable. (06)
> A set of statements that people can read, write, or store in a
> computer corresponds to version #1. That is more likely to be the
> object of attention in most discussions on this list. (07)
Agreed. As with "ontology", there is no overriding reason why "theory"
couldn't be appopriated from its discipline of origin and defined to
mean "set of axioms", so long as that definition is clear and
well-advertized. (08)
-chris (09)
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