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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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