Re: computation and cardinality

From: Chris Menzel (cmenzel_at_remove-this.tamu.edu)
Date: 01/29/05 

Date: 29 Jan 2005 01:05:51 GMT

On 21 Jan 2005 15:08:18 -0800, george <greeneg@cs.unc.edu> said:
>> > Inconsistent theories PROVE everything (and so make everything look
>> > dependent, as opposed to independent), but they DECIDE *nothing*.
>
> Chris Menzel replied:
>> You're flat wrong. They DECIDE *everything*.
>
> Under normal circumstances, at this point, the prosecution would
> simply rest. It is obvious to the world at large that Chris is wrong
> here. The real question is NOT why *I* am so self-destructive as to
> trash my own reputation by cursing so much in public. It is why an
> intellect as august and distinguished as that of Dr. Menzel would
> stoop to trying to publicly get away with so blatant a perversion of
> the meaning of "decide" as to try to pull THIS.

Well, I'm neither intentionally stooping nor pulling (and am certainly
neither august nor distinguished).

>> YOU don't get to decide what "decide" means around here.
>
> I'm NOT TRYING to do that. I am USING "decide" EXACTLY THE WAY THE
> DICTIONARY uses it.

What on earth has that got to do with anything, George? This is
sci.logic, not alt.websters. "decides" in such contexts as "theory T
decides sentence A" has a fixed meaning around here.

> For consistent theories, the technical term and the natural-language
> term ARE LOGICALLY EQUIVALENT in any case, so why am I getting accused
> of attempting re-definition??

Because you are claiming that inconsistent theories don't decide
anything, which, in the only sense that matter here is simply false. So
the only way your claim can be rendered true is to redefine "decides" --
which you don't get to do.

> For inconsistent theories, IT DOESN'T MATTER BECAUSE NOBODY GIVES A
> *** ABOUT INCONSISTENT THEORIES ANYWAY -- they are just worthless.

Their value is hardly the issue -- though what you say is quite false.
Naive set theory, for example, is valuable both pedagogically -- it's
still a good way to learn basic set theory (your former teacher uses it
in his intro text) -- and theoretically: it is an extremely valuable
exercise to diagnose the source of its inconsistency.

Be that as it may, the point you are curiously missing here is a simple
mathematical one. Here is the definition in question: A theory T
*decides* a sentence F (or, F is *decidable in* T) iff F is provable or
refutable in T (i.e., iff either F or its negation ~F is a theorem of
T). It follows immediately that an inconsistent theory decides every
sentence in its language. You may not like that, but as you yourself
well know, logic isn't about what you like. (You are so obviously wrong
here that I'm wondering if perhaps you are not aware of this
definition.)

> Speaking of blundering on the basics, even that wasn't strictly true;
> if Th is consistent and Th |- P, then it both decides and proves P,
> but it decides ~P as well, even though it doesn't prove it.

Just so. Do you think I said something to the contrary?

Chris Menzel

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