Working with the set of real numbers

Rustom Mody rustompmody at gmail.com
Thu Feb 13 02:47:19 CET 2014


On Thursday, February 13, 2014 2:15:28 AM UTC+5:30, Ian wrote:
> On Wed, Feb 12, 2014 at 7:11 AM, Rustom Mody  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.

Ok See below.

> English does not follow the rules of predicate logic,

Agreed

> and English sentences do not map consistently to logical sentences.

Agreed

> To me, the meaning of "computers do not work with X" depends upon the
> domain of X.

Agreed

> "Computers do not work with real numbers" implies that
> computers do not work with the set of real numbers (but implies
> nothing about subsets).

How come?

> "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).

The example is the other way. If one says:
"Computers have keyboards"
and then we have the demonstratation of say
- a cloud server
- a android phone

which are computers that have no keyboards, then that demonstrates that
"(ALL) computers have keyboards" is false"

Two things therefore come into play here:
1. "All computers have keyboards" is falsified by predicate logic
2. Modelling the English "Computers have keyboards" to the above sentence
needs: grammar, context, good-sense, good-will and a lot of other
good (and soft) stuff.

tl;dr Predicate logic can help to gain some clarity about where
the implied but unstated quantifiers lie.



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