Working with the set of real numbers
Ian Kelly
ian.g.kelly at gmail.com
Wed Feb 12 21:45:28 CET 2014
On Wed, Feb 12, 2014 at 7:11 AM, Rustom Mody <rustompmody at gmail.com> wrote:
> On Wednesday, February 12, 2014 3:37:04 PM UTC+5:30, Ben Finney wrote:
>> Chris Angelico writes:
>
>> > On Wed, Feb 12, 2014 at 7:56 PM, Ben Finney wrote:
>> > > So, if I understand you right, you want to say that you've not found
>> > > a computer that works with the *complete* set of real numbers. Yes?
>> > Correct. [...] My point is that computers *do not* work with real
>> > numbers, but only ever with some subset thereof [...]
>
>> You've done it again: by saying that "computers *do not* work with real
>> numbers", that if I find a real number - e.g. the number 4 - your
>> position is that, since it's a real number, computers don't work with
>> that number.
>
> There is a convention in logic called the implicit universal quantifier
> convention: when a bald unqualified reference is in a statement it means
> it is universally quantified. eg
> "A triangle is a polygon with 3 sides"
> really means
> "ALL polygons with 3 sides are triangles" ie the ALL is implied
>
> Now when for-all is inverted by de Morgan it becomes "for-some not..."
>
> So "computers work with real numbers" really means "computers work with
> all real numbers" and that is not true
I take exception whenever I see somebody trying to use predicate logic
to determine the meaning of an English sentence. English does not
follow the rules of predicate logic, and English sentences do not map
consistently to logical sentences.
To me, the meaning of "computers do not work with X" depends upon the
domain of X. "Computers do not work with real numbers" implies that
computers do not work with the set of real numbers (but implies
nothing about subsets). "Computers do not work with keyboards" on the
other hand would imply that no computer works with any keyboard (which
of course is demonstrably false).
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