**From:** pat hayes (*phayes@ai.uwf.edu*)

**Date:** 10/10/00

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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? Thanks. Pat Hayes --------------------------------------------------------------------- IHMC (850)434 8903 home 40 South Alcaniz St. (850)202 4416 office Pensacola, FL 32501 (850)202 4440 fax phayes@ai.uwf.edu http://www.coginst.uwf.edu/~phayes

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