[SCL] Re: slightly free logic

[Chris Menzel]

On Wed, Dec 10, 2003 at 12:35:11AM -0600, Pat Hayes wrote:
> Chris, I'd be grateful if you could take a look at the current document Im
> trying to write  and particularly at
> http://www.ihmc.us/users/phayes/SCL-december.html#entailment

I'm back!  Am now reviewing the document.  Sorry for the dropping so
badly out of the picture; I really got swamped with teaching/admin/other
duties.  I still have to grade the final exam for my computability class
(only 7 students, though) and a few admin things (5 letters of
recommendations plus about 40 recommendation forms to fill out), but I
can spread those duties out over the coming week; I'm back to the point
where I can put in some reasonable SCL time.

> The issue is this. This now covers the full CL wild west syntax, and the
> model theory is unconventional but I think OK, except that quantification
> over functions could be empty. I am willing to live with this, myself, but
> it does have a kind of a smell of free logics about it that I know you
> have strong feelings about.

I do, mostly because they are a cheat.  The usual semantics for free
logic, by assigning denotations (in some ghostly "outer domain") to
"nondenoting" terms, runs roughshod over free logic's central motivating
intuition, that the terms in some valid arguments really don't denote,
period.  But if the semantics here only involves functions, that strikes
me as a rather different kettle o' fish.

> We could always retreat to the no-function-variables position, or the
> other alternative would be to insist that every interpretation has at
> least one function of every arity,  but that seems totally ad-hoc. 

Agreed, then again...

> I would be willing to put up with this slight oddity if we can retain
> the generality of the syntax.

Right.  From one perspective this is not really so nasty -- after all,
if there is one thing (as there must be in classical logic), basic set
theory immediately yields the existence of (extensional) functions of
every arity.

> THis isnt written up yet, but theres a general way to translate any
> ontology into one without a header, by restricting the quantifiers
> suitably. THis adds a lot of extra stuff to the universe, of course, but
> it also makes all the quantifiers 'normal' since they range over named
> sets, and so if there are no 'functions' (they are individuals in the
> translation) then the empty quantifiers just have the form (forall
> (?x)(false ?x) implies ...) which of course makes perfect sense.
> Anyway, gotta go to bed. Hopefully will get some more done tomorrow.

Will respond to existing document soon.

-c