John wrote: (01)
> The point I have been emphasizing even more than the
> lattice of all possible theories is the need for a
> *registry* of actually specified and tested theories.
>
> Among the relationships to be recorded in the registry
> are implication/entailment, analogy (or whatever you
> want to call it), and many, many others, including
> who defined them, used them, tested them, etc. (02)
An excellent idea. (03)
> We should also agree on some standard terminology
> for the relations of theories:
>
> 1. I have never liked the word "subsumption" -- partly
> for the reason that it can be interpreted in many
> ways (as you did in your note). I'd rather just
> call it implication or entailment. (04)
But that's too narrow, as it only works for theories that share a
common language (or where the language of one is properly included in
that of another). (05)
> 2. That's an excellent reason for getting rid of the
> word "subsumption": "... ZF subsumes Peano Arithmetic".
> By using Goedel numbering, you could also say that
> arithmetic subsumes all sorts of theories. (06)
?? In the sense I intended, ZF subsumes PA in that, under an
appropriate mapping of the language of PA into that of ZF, every
axiom of PA is a theorem of ZF. Sure enough you can represent the
axioms of any theory T you please in PA as Gödel numbers, and
moreover you can even represent *that* a given sentence A is a
theorem of T (by representing proofs in T as Gödel numbers), but you
aren't thereby guaranteed (indeed it will almost never be the case)
that there is a reasonable translation of A into the language of PA
that is *itself* a theorem of PA. That's what's special about
relative interpretability. (07)
> That may
> be theoretically interesting, but hopelessly confusing
> for any serious discussion of practical problems. (08)
Well, I certainly agree with that. Of course, it has nothing to do
with relative interpretability, either. (09)
> 3. Relative interpretability is a big mouthful to say or type, (010)
Well, I have no particular affection for the *term*, it's the *idea*
I'm concerned about. (011)
> and it's hard to explain without getting
> into lots of technical issues. (012)
I guess you could say the same thing about the calculus vis-a-vis
engineering! (013)
> I use analogy because
> that's what Bohr did when he renamed the sun and earth
> the proton and electron for his model of the H atom. (014)
Sure, but now we're back to quibbling. :-) (015)
> Just one other point:
>
> CM> Is not subset the arc relation in the lattice under that
> > understanding?
>
> It is true that if theory T1 implies T2, then T2 is a subset
> of T1 and vice versa. (016)
Vice versa?? (017)
> However, that can be confusing, since
> (a) the directions of the two relations are opposite, and
> (b) implication is a logical relation (which is the main point)
> and subset is a secondary issue that distracts attention. (018)
Well, I was only talking extensionally among friends. :-) (019)
-chris (020)
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