Re: SUO: Question
At 22:32 2002-04-08, Robert Grayson Spillers wrote:
>John,
>
>In your proposal of an (infinite) lattice of all possible theories, do
>modules that are ill formed also fit into this lattice (e.g. do not have
>any stated or consistent methodology, are defective or inappropriate in
>some other way)? I assume so since it is "all possible theories" even
>poorly formed or simply false theories ( I am using what I believe is a
>widely accepted definition of a theory - something that can be assigned a
>truth value). If one can have both p and ~p in the lattice how will one
>decide which to use? How will one know that both p and ~p are present (or
>not present) in the module of theories one selects to use? It seems to me
>that one must have some test of compatibility if one is asked to select
>some theories from a much larger (infinite?) number to use.
Robert,
It sounds to me like you're equating a formula with a theory.
I understand "theory" to be what Enderton defines in "A Mathematical
Introduction to Logic:"
Section 2.6, page 144
"Theories
"We define a theory to be a set of sentences closed under logical
implication. That is, T is a theory iff T is a set of sentences such that
for any sentence sigma of the language,
T |= sigma ==> sigma elementOf T."
In general a theory is not a finite set, so computer representations must
remain somehow incomplete for theories whose T has a transfinite cardinality.
I don't see the point in even contemplating admitting non-well-formed
formulas into any logical system. What interpretation can they be given?
None--they're not well-formed.
You may see it as quibbling, but a theory cannot be "false," but it can be
unsatisfiable. (A sentence can have a truth value, however.)
That said, you're correct, conjoining theories requires some sort of "truth
maintenance" to ensure that consistency remains after merging them. It's
also necessary that the theories' models are identical or are somehow
consistently mergeable as well.
Randall Schulz
Mountain View, CA USA
>...
>
>A tutorial would be very welcome.
>
>Bob