Re: Primitives, definitions, and metalanguage

schubert@cs.rochester.edu
Date: Fri, 10 Jan 92 17:12:59 EST
From: schubert@cs.rochester.edu
Message-id: <9201102212.AA07438@ash.cs.rochester.edu>
To: interlingua@ISI.EDU, krss@ISI.EDU, sowa@watson.ibm.com
Subject: Re:  Primitives, definitions, and metalanguage

Some comments on John Sowa's comments:

>The question of familiarity is irrelevant for the choice of primitives.

If for all practical purposes two ways of doing things are equivalent,
then the more familiar way is to be preferred. What I was pointing out
was that for the particular use of Hilbert's epsilon suggested by John,
the same result could be achieved by conventional means. That's not to
say that epsilon won't buy us something important, so I, for one, am
open to the idea.


>But when [logicians or lexicographers] state their definitions, both 
>of them use exactly the same form:  Aristotle's style of definition by 
>genus and differentiae.

I don't see that logical definitions like

  S(x) <=>def member(x,x)
or
  P<=>Q <=>def P=>Q & Q=>P

are definitions by genus and differentiae. (Though I grant that
logicians and mathematicians often do say, "Let <symbol> be a <type>
such that <distinctive-property>", in which case they need to prove
there is such a thing, as JMc pointed out.) In any case, the point I
was getting at was that I think the definitional syntax should clearly
distinguish between complete and (potentially) partial definitions.

>I would say that any sentence that uses quoted expressions is a
>sentence in the metalanguage.  You can only consider modal logic
>as a kind of first-order logic if you treat the modal operators as
>special kinds of quantifiers over possible worlds.  But as soon as
>you treat them as operators on quoted expressions, you are treating
>them as metalanguage operators.  (And by the way, Quine considered
>metalanguage to be the most coherent way of thinking about modality.
>I tend to agree with him.)

As just one example of a logic with quotation (at the object level),
let me point to De Rivieres & Levesque, "The consistency of syntactical
treatments of knowledge", Comp. Int. 4, 1988. It is true that quotation
has sometimes been taken to automatically boost a sentence to the
metalevel, e.g., in the Genesereth and Nilsson book. But that's only
reasonable as long as we can avoid mixing sentences containing quotes
with others. Let me quote Stuart Russell on this: "[Genesereth &
Nilsson's] analysis of metalevel systems does not, however, allow for
mixed-level sentences, which refer to objects from more than one level.
Such sentences often arise in descriptions of sensing: 'If I open the
window I will know if the birds are singing" refers to an external
action (opening the window) with an internal effect (knowledge of a
proposition." (Do the Right Thing, p32)

As for Quine's view on modality, profound and provocative though
it may be, its current status seems to be something like Einstein's 
view on quantum uncertainty.

>   A = (lambda x)B(x).
>
> ...  But just because it uses a lambda expression,
>that does not make it a definition.   It only becomes a definition
>when you add that magic metalinguistic word "definition":
>
>   Definition:  A = (lambda x)B(x).

Agreed. That a definition is more than its defining axiom is a point
I have been pushing hard. John's remaining remarks about object-level
and meta-level truth also made perfect sense to me. Where things get
tricky is when one "pulls down" the metalanguage into the object
language using quotation ...

Len Schubert