On Mar 30, 2010, at 12:49 PM, John F. Sowa wrote:
> CM> 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.) ...
> Yes. I agree that combinatory logic in its full glory goes
> far beyond the bounds of FOL. I was indeed thinking of the
> use of combinators for eliminating quantifiers and variables.
> In any case, simplicity is not a simple notion that can be
> characterized in any simple way. Reducing the number of axioms
> and primitives can sometimes make a system simpler. But if you
> go too far, the overall structure can become more complex.
> For example, all the operators of Boolean algebra can be
> defined in terms of just one: either Nand or Nor. Those
> definitions are actually useful for reducing the number of
> transistors in logic circuits.
> But the definitions of the most common operators, such as
> And, Or, and Not become more complex. When you combine a
> Boolean algebra based on Nand or Nor with the quantifiers,
> it can obscure rather than clarify many relationships. (01)
Just so. Those are good simple examples to illustrate the point. (02)
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