Tuesday, August 5, 2014

Leibniz and the Gaifman-Hales Theorem

The basic idea behind Leibniz's characteristique is that all concepts are generated out of simple concepts, and there are no non-trivial logical relations between the simple concepts.

Today I was thinking about how to model this mathematically. Concepts presumably form a Boolean algebra. But infinities are very important to Leibniz. For instance, to each individual there corresponds a complete individual concept, which is an infinite concept specifying everything the individual does. So an ordinary Boolean algebra with binary conjunction and disjunction won't be good enough for Leibniz. We need concepts to form a complete Boolean algebra, one where an arbitrary set of elements has a conjunction and a disjunction.

So we want the space of concepts to be a complete Boolean algebra. We also want it to be generated by—built up out of—the set of simple concepts. Finally, we don't want there to be any nontrivial logical relations between the simple concepts. We want the theory to be entirely formal. This is one of Leibniz's basic intuitions. It seems to me that the way to formalize this condition is to say that the complete Boolean algebra is freely generated by the simple concepts.

Pity, though. The Gaifman-Hales Theorem implies that if there are infinitely many simple concepts, there is no complete Boolean algebra generated by them (this assumes a quite weak version of the Axiom of Choice, namely that every infinite set contains a countably infinite subset).

It looks, thus, like the Leibniz project is provably a failure.

Perhaps not, though. Apparently if one relaxes the requirement that the complete Boolean algebra be a set and allows it to be a proper class, but keeps the idea that the simple concepts form a set, one can get a complete Boolean algebra freely generated by the simple concepts.

Still, it's interesting that from an infinite set of simple concepts, one generates a proper class of concepts.


MiloŇ° said...

Does his observation can be extended to similar projects run by Carnap and most recently David Chalmers?

Alexander R Pruss said...

Not necessarily. The details will be important.

Alexander R Pruss said...

On reflection, I think "freely generated" in the category theoretic sense may not capture the notion of independence that Leibniz is after. I need to think more.