[hist-analytic] Elizabeth Anscombe's Intention (New "Look Inside" feature)
Baynesr at comcast.net
Baynesr at comcast.net
Wed Jun 16 21:41:22 EDT 2010
Well, I don't think we disagree on a lot. Where you seem to
disagree is, I think, a misunderstanding of what I'm saying here.
Yes, I understand (I think) the logic. Nothing you say surprises
me, that is. But my concern with existential quantification is not
so much confusion on my part, as skepticism as to the role of
quantification is a logical description of the concept of a limit
in calculus. I
You say that
Lim f(x) = fa
x -> a
Is just a definition of continuity. Here we disagree. I don't see this
as a definition, at all. Do you mean 'continuity' of a function, here?
If so, I don't think so. This is a statement about some function 'f', not
just any function that is continuous. It says that the limit of this function
as x goes to a is f(a), that's all. Not all continuous functions have limits
which are such that you can substitute 'a' for 'x' in deriving the limit. So,
for example, you may have a function such as f(x)=1/x, and this function
will not be a function; where the limit of the function does not exist you
do NOT have a limit that conforms to the "definition" above and yet,
contrary to what you appear to suggest, f(x)1/x is a continuous function.
As long as 'a' falls within the value of the function then the quoted expression
obtains, but in cases where it holds for all numbers BUT 'a' then it does
not hold and you can't simply substitute 'a' for 'x' in deriving the value of
the limit from the function.
Nice to hear from you. I mention you in the acknowledgments by the way.
Amazon has expeditiously execute every contractual obligation in a
timely fashion. I've benefited by your advice here as well as your thought
provoking comments on my last post.
Regards
Steve
----- Original Message -----
From: "Roger Bishop Jones" <rbj at rbjones.com>
To: Baynesr at comcast.net, hist-analytic at simplelists.com
Sent: Wednesday, June 16, 2010 2:51:09 PM GMT -06:00 US/Canada Central
Subject: Re: Elizabeth Anscombe's Intention (New "Look Inside" feature)
On Wednesday 16 Jun 2010 17:27, you wrote:
> Amazon has introduced the "Look Inside" feature of my
> book _Elizabeth Anscombe's Intention_.
Looks good!
> One thing on my mind as I do some
> math needed for economics is the idea of a Limit in
> calculus. You can simply substitute 'a' for 'x' in Lim
> f(x) when f(a) is defined;
> x->a
>
> That is,
>
> Lim f(x) = fa
> x -> a
This is a definition of continuity, it won't hold, even if
f(a) is defined, if there is a discontinuty at a.
Also, there may not be a limit either,
Lim f(x)
x -> a
does not always exist.
> But it is defined when x goes to a, where *a* is never
> reached.
The limit may then be defined, but won't necessarily = f(a)
and definitely won't, of course, if f is not defined at a.
> So the limit may be defined even when 'a'
> doesn't exist (as what x goes to), or so it seems.
I presume you mean here f is not defined at a, rather than
that a does not exist.
> My concern here is that quantification, e.g. UG, may be
> possible where a does not exist.
This doesn't happen in any logic I know of, i.e. you only
quantify over things which do exist.
(I dare say if you were keen you could formalise a
Meinongian logic in which unusual things happen.)
However, you can write something like:
for all x such that P x then Q x.
which is a way of quantifying over P's whether or not there
are any.
In that case you might say, if P is always false, that you
have quantified over something which does not exist, though I
think that's a confusing way of putting it. Really you did
a restricted quantification the effect of which was to
quantify over nothing.
Roger Jones
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