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Peter_Smith wrote:
> Of course PRA is to be presented either explicitly with quantifiers or
> at least free variables (" [the equivalent of] initial universal
> quantifers") in the object-language.
If you (entirely belatedly) agree that this is reasonable and
that "of course" this [free variables in the object language]
is one way it gets done, then I suddenly
see why it was so easy for you to think that I had mistakenly
thought you WERE doing it that way (when you weren't).
But by the SAME token, IF doing that is so normal, reasonable,
and usual, then I certainly did NOT deserve to be so savagely
ATTACKED for thinking you were doing it that way, EVEN if I
had wrongly thought that (which I have never conceded I did).
In any case, intellectually, the burden remains on you to explain
why the difference(which is, orthographically, almost unrecognizable --
how is anyone supposed to be able to TELL that a variable in a sentence
is being-implicitly-quantified-over at the meta-level as opposed to the
object-
level, IF THE DOMAINS OF QUANTIFICATION ARE THE *SAME*? -- the
whole POINT of axiom-schemata, USUALLY, is that the meta-language
domain-of-quantification and the object-language
domain-of-quantification
ARE DIFFERENT; that's why one NEEDS to use an axiom SCHEMA as
OPPOSED to another AXIOM) has significant content, as opposed to being
a merel textual variation in presentation of the SAME content.
> The other theory I described wasn't PRA
Oh.
Well, I'll cop to THAT mistake.
> and I never said it was.
But you don't get off THAT easy.
We were DISCUSSING PRA.
You DID say that you were talking about a theory that
could "define" and primitively recursively enumerate ALL p.r.
functions. I'm not going to apologize for having thought (EVEN if
I was wrong) that any theory that could do ALL THAT *had* to be
PRA (or better).
> Rather
> it was a *different*, weaker, theory which I mentioned in response to a
> remark of yours,
That's the whole point. It was in response to a remark OF MINE.
I THEREFORE thought it was going to be about the topic *I* was asking
about, which was PRA.
> in order to show that there could be a theory which is
> quantifier free in the object language, but which defines the p.r.
> functions,
This is a misuse of "theory". That was a propositional theory.
PRA is just inherently NOT THAT weak. Almost EVERY first-
order theory (if the signature includes both a constant and a function)
has a propositional subtheory that you can get (after skolemizing
away the existentially-quantified axioms) just by conjoining all
the instances of its non-propositional (i.e. universally quantified)
sentences, where by "all" I mean NOT the usual first-order-
semantic instantiation to objects of the interpreted model/domain,
but rather instantiations to all the terms of the language.
What NEEDS to be discussed is the relationship between THAT
propositional theory and the parent 1st-order theory, and it would
be better if that were NOT limited to the specific case of PRA (PA
would be better) because PRA is just unusual anyhow.
> at least in the sense of allowing us to calculate the values
> of any p.r. function for any argument(s).
But my whole point is that EVERY 1st-order theory (again,
subject to the richness requirements above) has a propositional
subtheory that will get THAT much (or its analogue, in those
other theories) done. In a very real sense, the universally quantified
AXIOMS of those theories are going to be similarly isomorphic to the
trick you just pulled by making your axiom-schemata for your "not-PRA"
theory. People DON'T GENERALLY do that.
> So as others have already plainly said,
That's just a lie. THE ONLY other person who has
said this is Aatu Koskensilta.
> there are two different theories in play in this thread
More than two. There's the "usual" PRA that you allege is usually
at least implicitly, initially, universally quantified, and the weaker
propositional
one that you axiomatized with schemata, AND the one AK presented with
inference rules.
> (and I'm not the one who is confusing them).
I'm not either. I wasn't confusing two theories.
I was simply not being aware that you knew that one of the
theories about which you were talking was not PRA.
I don't feel bad about that because you introduced it by way
of talking about its ADEQUACY to do what PRA does.
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