Chris, (01)
That may be true from a purely logical sense, but only for that sense. HOLs
have a long and valued history, from Fortran I through SQL and further, and
they do much more than mere logic. They represent aspects of reality that
model what we believe to be valid, real world mechanisms. (02)
For example, global warmers are using simulation of weather processes that
would be completely meaningless to observers in a pure FOL expression. They
compare the simulated results to slices of real data to see how far off they
are, and they now seem to have developed an understanding of about 1 percent
of the weather at the equator. (03)
IBM taught people to program in RPG for their GSD machines because, when RPG
was developed, it related well to the accounting books, inventory files and
other business practices the same people had been using previously. (04)
Prolog, and equivalent representations of interpreted logic, add sequence
and structure to the messy FOL content-only representation. (05)
A well developed HOL is far more expressive than logic, more readable,
communicable, and therefore more able to be tested for validity in reality
as compared to simple expression of theories that may or may not be
validated by experience. (06)
Pure FOL, in its original, very limited form, is fit for mathematical
expositions of very small issues. And yes, theoretically it can do anything
that the other languages above can do. But that is not the reason to stick
to FOL when more expressive and communicable languages abound. (07)
So, strictly speaking, the "subset" isn't well chosen as a description of
the difference, as you mentioned and explored in your email. But very, very
little work is done in pure FOL anymore. (08)
A better description would be "FOL is less expressive than many HOLs". (09)
HTH,
-Rich (010)
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher
Menzel
Sent: Monday, October 11, 2010 9:43 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] using SKOS for controlled values for
controlledvocabulary (011)
On Oct 8, 2010, at 6:48 PM, John Bottoms wrote:
> Can we say:
> FOL is a subset of HOL. (012)
Not really, at least under most definitions of the notion of a logic. This
is a bit better: Let L be a first-order language and let L' be the result of
adding second-order (and even higher-order, if you please) quantifiers to L.
Then every logical truth of L, under a standard first-order semantics, will
be a logical truth of L', under a standard second-order semantics. More
generally, for any set S of sentences of L and any particular sentence A of
L, if A is a logical consequence of S under a standard first-order semantics
for L, then A is a logical consequence of S under a standard second-order
semantics for L'. (013)
Chris Menzel (014)
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