tech:logic matters

["John E. Clifford" <pcliffje@CRL.COM>]

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).  It is
unAristotelian (the square of opposition collapses to its diagonals)
and violates one demarcated subset of English usage.  Its chief
virtues are that it makes Frege's task of reducing mathematics to
logic (which is going to fail regardless -- but he did not know that) a
lot easier, and that it unites the Aristotelian and Stoic strains in the
history of Logic (but there may have been a good reason why they
stayed apart for two millennia).  As noted later on this thread, this
had nothing to do with Cowan's reducing _ro broda_ to _ro da poi
broda_ since he wants there only to restrict the scope of the
quantifier, not give it existential import, whereas the original _ro da
poi broda_ had existential import and no particular scope
restrictions.

&
> &: 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.)

&:
> They are the quantifiers of natural language, the ones grammars are
> designed to deal with (arguable for the universal, the particular,
> finite numerals and the plurative -- though less in the last case; not
> for the majoritive or any of the rest)
That's open to debate, and decisions for lojban shouldn't depend on
resolution of that debate. I (naively?) thought that logic is relatively
well understood, while natural language is relatively ill understood,
and hence some of the appeal of lojban is that it is based principally
on logic.
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.  This form fits naturally with restricted
quantifiers and assigns to the unrestricted quantifiers of basic logic
courses a degenerate status (corresponding to their history, in fact) as
restricted to an unspcified (usually "universe of dioscourse") class.
This kind of interface theorem ought to be the most interesting for
Lojban, since it is where the idea of a language based on logic makes
the most sense.

&
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.  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).

&:
> They can fit most neatly into (Lojban's -- but most languages')
> syntax, forming a unit that occupies the place of  an argument, rather
> than a functionally fractionated and incomplete creature like the
> standard logical correlate (a quantifier, a predicate, and half a
> conditional)
That's not a pure semantics argument. It's an argument about correspondence
between syntax and semantics; but if the correspondence is bad, it's the
syntax we should blame. Meaning comes first.
pc:
Right!  So, the standard for Lojban syntax for quantifiers ought to be
_Q broda cu brode_  matching the semantics as closely as feasible.
this form claims there are brodas except for the cases of _no_ and
whatever is the denial of _ro_ (?_ronai_? _nairo_? something else
altogether?).  The unrestricted quantifiers are then explicitly set out
with a variable and all the propositional apparatus.

&:
Its logicality is the existence of rules for going from syntax to a
relatively well-understood and well-specified logic. Any logicality
arising from the regularity of those rules is an added bonus.
pc:
Well, we disagree on that, since I think we can always get SOME rules for
going from syntax to logic (look how well we do with English after all),
but that a logical language should have systematic rules, treating overtly
like constructions alike and representing similarly in the langauge what
are similar in the logic.