Re: second-order aint reflection

From: Dieter Fensel (
Date: 10/14/00

Hi Pat,

sorry for the delay in my answer (your email reached me
late and I was quit busy). Actually we seem to agree in
the message of the paragraph. Many frameworks and applications
in knowledge engineering ask for a second-order "flavour" in
a sense that one has predicates over predicates (i.e., meta
classes). In its full and naive expressiveness this would
amount to second-order logic. However, there are other techniques
(you also mention them, another example is Frame logic of Kiefer et al.
that syntactically enrich the first-order semantics of the
language) that provide such a syntactical construct
without leaving the framework of first-order semantics.

Using such mechanisms in a future version of OIL is
one of our goals. I agree with you that the paragraph you refer
is rather short and un-technical in the way it discuss this issue.

Thanks for very helpful input,


At 05:08 PM 10/10/00 -0700, pat hayes wrote:
><This message has been sent to all authors of the paper, but I
>misspelt your email address so am resending it to you. -PH>
>Greetings, authors of "An informal description of OIL...". Thanks for
>writing this very readable paper, but you make one serious error in
>the discussion towards the end, under the heading "heavy oil", when
>you identify second-order expressivity with reification. These are
>not the same thing, and it would be a pity to have this widespread
>confusion made even worse by its being given the considerable
>authority of your collective imprimateur.
>Second-order refers to a language which can quantify over properties
>and relations as well as the individuals to which those properties
>and relations apply. Reification refers to the ability of a langauge
>to describe its own syntax, ie as you correctly observe, with
>"statements of the language as objects". But statements are not the
>same kind of thing as properties and relations, and it is important
>to keep these distinctions clear. For example, you say that
>unification is undecideable in second-order logic. This is true in a
>second-order logic where the quantification over predicates and
>relations is understood to range over all mathematically expressible
>relations; the reason is that such a unification must allow for
>arbitrary amounts of lambda-conversion.  But the second-order
>quantifiers can be understood in other ways. If the quantifiers are
>understood to range over all mathematically possible relations, then
>theorem-hood is not even recursively enumerable, so no complete logic
>is possible; on the other hand, if it can be understood to range over
>only relations which are named in the language, then first-order
>unification is complete. But, to emphasise my main point, none of
>this interpretive variation is possible for reification, since the
>meaning of a quantifier ranging over all *expressions* is fixed by
>the syntax of the langauge itself.  Reification in itself does not
>require any kind of higher-order construction; after all, syntax
>consists of tree structures (there are some technical issues here, I
>know, if one wants the trees to be finite.) . But reification brings
>its own problems which are quite different; notably, being too
>generous with reification allows the language to become paradoxical,
>which is widely thought of as an undesirable trait in a logic. There
>is nothing paradoxical in second-order logic.
>This confusion may have arisen from the observation that what makes
>second-order unification difficult is lambda-conversion, and one can
>think of lambda-conversion as an operation on expressions. But it is
>important to keep clear the difference between an expression, on the
>one hand, and the relation or function denoted by that expression, on
>the other. Putting the latter into the semantics might force the
>former into the syntax, but putting the former into the semantics
>gets one into a completely different kettle of fish.
>Since there is such a wide interest in 'web logic', and since so much
>of this interest is from people without a deep technical background
>in logic, I wonder if I could urge you to revise your paper to
>correct this misleading and potentially confusing remark?
>Pat Hayes
>IHMC                                    (850)434 8903   home
>40 South Alcaniz St.                    (850)202 4416   office
>Pensacola,  FL 32501                    (850)202 4440   fax

Dieter Fensel
Division of Mathematics & Computer Science,
Vrije Universiteit Amsterdam,
De Boelelaan 1081a, 1081 HV Amsterdam, NL
The Netherlands
Room number U3.25.
Tel. (mobil): +31-(0)6-51850619,
Fax and Answering machine: +31-(0)20-872 27 22
Privat: Liendenhof 64, NL-1108 HB Amsterdam, The Netherlands.
Tel.: +31-(0)20-365 52 60.
The information transmitted is intended only for the person or entity to which
it is addressed and may contain confidential and/or privileged material. Any
review, retransmission, dissemination or other use of, or taking of any action
in reliance upon, this information by persons or entities other than the
intended recipient is prohibited. If you received this in error, please
contact the sender and delete the material from any computer.

This archive was generated by hypermail 2.1.4 : 03/26/03 EST