On Mar 29, 2010, at 10:46 PM, John F. Sowa wrote:
> ...
> KDK> Combinatory logic, by the way, is even "simpler" than FOL:
>> the whole thing can be defined with only two operators, named
>> "S" and "K". And you can define FOL in combinatory logic.
>> So why don't we adopt that as the one, true uber-standard?
>> What could be simpler?
>
> Combinatory logic is just a different syntax for FOL. It could be
> defined as a dialect of Common Logic, if anyone wanted to do so. (01)
Whoa, combinatory logic is way stronger than ordinary FOL. (I suspect you have
in mind syntactically (and historically) related systems like the predicate
functor calculi of Quine and Schönfinkel that do away with the apparatus of
quantifiers and variables.) Combinatory logic is expressive enough to
formalize recursive function theory and, hence, exhibits Gödel-style
incompleteness. Its power aside, it would be a terrible choice as an
"über-standard" for knowledge engineering. Number of primitives is no measure
of simplicity. Combinatory logic is intuitively very difficult to grasp and
use -- its basic ontology consists of functions rather than objects, properties
and relations and, concomitantly, its syntax is based on function application
as opposed to FOL's predication, which arises naturally out of the Noun Phrase
/ Verb Phrase structure of ordinary language -- perhaps the central reason why
it is so easy to learn. A more meaningful measure of the intuitive difficulty
of combinatory logic is seen in the fact that, although it traces its origins
to at least the 1920s (with even deeper roots in the 19th century), it didn't
have a model theory until Scott's (very beautiful, but very complex) fixed
point construction in (IIRC) the early 1960s. (02)
Chris Menzel (03)
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