SUO: *Date 15 Apr 2002 -- Theory Query
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SUO WG Members,
With special reference to those speculating
about sundry "lattices of theories" (LOT's):
What is your definition of a "theory"?
More acutely to the purpose:
What is a theory that it should latticed be?
I have asked this question many times before,
and all I've gotten is lots of gesticulation.
In particular, I am skeptical that you can even find,
without begging the question of intercommunicability,
any definition of a lattice of theories that remains
invariant over the choice of a language in which the
theories are expressed, indeed, where the finding of
the envisioned common language of comparison is just
another way of stating the initial problem to be met.
Consider this standard definition of a theory, the only one I know,
such as makes sense within the favored frame of first-order logics:
| A '(first-order) theory' T of $L$ is a collection of sentences of $L$.
|
| T is said to be 'closed' iff it is closed under the |= relation. Etc.
|
| Chang & Keisler, 'Model Theory', page 36.
Notice that the very definition of a theory is stated
relative to a given first-order predicate language $L$.
Until such a common language has been established, all
talk of lattices or other orders is just so much Babel.
Jon Awbrey
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