tech:logic matters

[ucleaar <ucleaar@UCL.AC.UK>]

pc:
> Cowan:
> The fact that "all men are mortal" is equivalent to "for all X, if
> X is a man then X is mortal" is a theorem, not a mere convention of
> rewriting. My recent proposal that "ro prenu" means "ro da poi prenu"
> (and not "ro lo prenu") restores the original pre-Lojban situation.
> pc:
> Far from being a theorem, it is not even (uniformly) true.  It is, in
> fact, a convention introduced during the 19th century (with some
> precursors, but nailed down by Frege in the late 1880's).

If "all men are mortal" and "for all X, if X is a man then X is mortal"
are meant to be logical formulae (with the former representing restricted
quantification), then the question comes down to whether the truth "Ax Fx"
entails that the universe (of discourse) is nonempty. Do we really care
about that? I don't.
If "all men are mortal" and "for all X, if X is a man then X is mortal"
are meant to be english sentences, then I think we're wasting our
time discussing them on lojban list.

> > &: But why should {suo no lo ro broda} mean that there are brodas?
> > pc: Because the internal _ro_, properly understood, says that there
> > are some brodas, even if none of them do whatever the predication goes
> > on to claim.
> That's what I don't understand.
> pc:
> Well, the internal quantifier says how big the referred-to set is.  In
> this case, it has all members.  That implies, as "all" does, that it has
> some members, i.e., at least one.  What is the problem?  (This is not
> even about restricted quantifiers this time.)

So how do you make sense of {no lo ro broda}? And must {lo no broda
e lo ro broda cu brode} be false?

> pc: Logic is only relatively well understood and is often interestingly
> understood in relation to (not too surprisingly) natural languages.
> One of the discoveries that has emerged  in this relationship is that
> all the items in natural languages that function as quantifiers in a
> relatively specific sense have a clearly defined logical form in
> second order logic.

That rather depends what we mean by "function as quantifiers"; it's
a circular definition (not that I object to that).

> & That's a virtue only if you belive all those other so-called quantifiers
> are a good thing. I don't. (I mean I think it's fine for all these
> words to be in PA, but the formal metalanguage should contain only
> existential and universal quantifiers, in order to minimize the number
> of primitives.)
> pc: Unfortunately, with only those two quantifiers there are things that
> cannot be said ("many", "most," "non-denumerably many," to name
> the most commonly discussed cases).  Or, rather, they can be said
> only at the cost of considerbly complicating other areas.

"Many" can be treated as a number. "Most" can be done as a predicate
taking sets as arguments. (I recognize that you're being more orthodox
than me here, but I'm too much of an outsider to understand why the
orthodoxy is the orthodoxy.)

> Notice that, in fact, the uniform treatment of quantifiers does not
> require any quantifiers in the metalanguage other than the universal and
> particular (and those only of a much higher order) since the usual
> quantifiers are all second order relations in the metalanguage
> (roughly what Aristotle said of them in -340 or so).

Weird. As far as I'm concerned, all we need is (i) lexicosyntax,
and (ii) this metalanguage (which shd be a simple as possible). If
in (ii) we can get away with just two quantifiers then we should.
When I talk about "semantics", I mean (ii).

> whatever is the denial of _ro_ (?_ronai_? _nairo_? something else
> altogether?).

{na (ku) ... ro}

coo, mie And