logical symbols

RE01 Rice Brian T. EM2 BRice@vinson.navy.mil
Thu, 29 Oct 1998 12:34:44 -0800

>> This also explains my distinction between "extension" and "intension",
>> in that the extension of a symbol which is used any number of times
>> within an expression is the entity which remains "the same" for each
>> occurrence.
>You explained 'extension' but not 'intension'.

Well, I think that intensionality in terms of a first-order language
would be expressible as an arrow graph with objects being nodes or a
system where each object is a node with various spaces defined for the
system in which relationships are encoded or properties are "defined".
However, for the "full intensionality" that I have in mind, the system
would have to totally consist of "shapes" which describe the interaction
of other "shapes" in order to create what we would call definitions,
partial evaluations, contexts, closures, (and on and on).  Oh dear!  The
previous sentence will be terrible for you guys to sort out in your
minds, as it was for me when I first conceived of the idea.  I will
elaborate in this thread, pending your feedback.  It's basically a
universal homo-iconic system in which to implement logics.

>> [uncountable stuff]
>Ok, you're talking about infinite loops, infinite sets, etc.  I agree
>these have to be recognized as infinite and not fully evaluated.  It
>should be easy to prove loops/sets are infinite by detecting whether there
>is recursion in the definition.

I'm not so sure, since a formal definition may be infinite itself,
requiring a meta- or meta-meta- definition, or may be completely
undefineable or something else quite weird.  I really don't want it just
to _detect_ infinite extensions of definitions, but to avoid having to
deal with the issue at all, using some logical form which doesn't deal
with recursion or something like that which speaks of these infinities
with as much ease as it speaks of the natural numbers, etc.  I'm
thinking in terms of traditional extensional systems right now, so a
precise idea is not yet forthcoming.  Any suggestions?