[hist-analytic] Clarification: Re: Elizabeth Anscombe's Intention (New "Look Inside" feature)

Baynesr at comcast.net Baynesr at comcast.net
Wed Jun 16 22:21:42 EDT 2010


There is a simpler way I should have replied to your statement 

"Lim f(x) = fa 
x -> a 

This is a definition of continuity" 

What you say looks right, but here is where, I think, it's wrong. It is 
true that in the case of 'Lim f(x) = f(a) ' 
x->a 

the function IS continuous, BUT only at that point, a. But for a function 
to be continuous it must be continuous at all points. So the formula 
entails continuity but only at a point; it does not therefore define 
'continuity'. Continuity of the function requires of the function 
continuity at all points.The formula is sufficient but not necessary 
for continuity, so it's not a definition of continuity. I think that is the 
best way to state where we might disagree. On quantification that 
is another story. Getting the facts straight on that without getting 
the calculus straight might not be possible; a lot depends on what 
work one thinks can really be achieved with 'Ex'. I'm not so sure 
it is the alpha and omega of 'existence', another topic. 

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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