Re: blow firstname.lastname@example.org
Date: Tue, 14 Sep 1993 15:43:29 +0000
To: email@example.com (Chris Menzel)
Subject: Re: blow away?
Chris Menzel writes
>Pat Hayes writes:
>:...... I bet we need some higher-order
>Certainly true in the sense that we need to be able to talk about
>"higher-order" objects like properties- and relations-in-intension in
>addition to more mundane entities. What is less clear is whether it
>is necessary to go all the way to a full-blown higher-order logic,
>i.e., with quantification over (some intensional counterpart of) the
>full power set of (in general, the nth cartesian product of) the
>domain of individuals, where we lose completeness and most of those
>other nice, friendly properties of first-order logic.
I agree, its not clear. And I don't think we need 'full-blown'
higher-order logic. But consider the possibility of a second-order
logic which is full-blown in this sense but restricted in other ways;
a linear second-order logic, say; or a second-order logic whose
quantification over predicates is restricted by some kind of context
mechanism but otherwise full-blown (full-blown in a box, it might
be called airbag logic). I'm making these up as I go along, but my
point was that we need to be experimenting, not standardising.
Incidentally, what do you think of Henkin semantics for higher-order
logic? Theres a sense in which higher-order logic with Henkins semantics
is first-order, but it sure doesn't FEEL like FOL when you are using
it. And we get completeness back, so its fairly friendly.
> At any rate,
>because of this it would seem to be sound methodologically as far as
>possible to try to satisfy the needs of KR with first-order extensions
>to classical FOL.
Well I agree, in fact. FOL seems the best foundation we have at the
present, and it keeps getting re-invented. But look at some of the
problems we have had working within it.
I guess my most recurrent worry is this. One of the reasons we feel
'safe' with FOL is that while acknowledging that we will need extensions
to it, we all feel that they will indeed be extensions, ie they can
hardly be weaker than FOL. (After all, what would you take out? 'NOT',
say? Nah...) But maybe this intuition is mistaken. Maybe in order to
get the comfort back we will need to sacrifice some expressive power
of FOL for something else, and do so in a systematic way which forces
us to alter the semantic base significantly.
This is related to a worry I expressed in earlier correspondence. We
seem to need ways of describing sets of sentences which are different
>From any of the usual semantic frameworks, because these always give
us (sub)languages, ie are always closed under some kind of inference.
What could these be? If we find them, perhaps they will be more like
what we need for the semantics of a representational language. Ie,
perhaps thinking of assertional logics (of any kind) as the appropriate
vehicles for representing thought may be misleading us. This has often
been expressed before, but the replacements offered have usually turned
out to be just FOL in disguise. I don't have any replacements,
just a feeling that we may need a major conceptual leap, and a concern
that talk of standardisation in Krep is more likely to smother than
encourage the strength of imagination required. If we feel too
comfortable we might doze off :-)
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