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Re: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Posted:
Jan 27, 2012 1:46 PM
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Tony Orlow wrote:
> On Jan 27, 10:31 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> > I'm going by Mendelson's characterization of first-order theory. He
> > said that all variables in any "interpretation" of a first-order
> > language range over some set. I took that to mean ONLY ONE set. If you
> > want to expand that set, supposedly you are talking about a different
> > interpretation.
>
> Hmmm... "All" is sometimes a vague word. Are you sure he didn't mean
> that each variable ranges over some set, and that the union of sets so
> employed constitutes the domain of discourse? He probably meant it as
> you read it, but it seems to me that one first order theory may speak
> both of, say, sets and quantities, the combination of which make up
> the entire domain addressed.
A logic in which some first order quantifiers range over one set and
others range over another set, is known as many-sorted. If Mendelson
meant to write of many-sorted logic, he would surely have said so.
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
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