Re: Did Natural Selection extend to the laws of logic?

From: John Bailey <jmb184.domain.name.hidden>
Date: Tue, 26 Sep 2000 10:41:44 -0400

Brent Meeker wrote:

> I guess I don't understand your question. First order predicate logic applies to propositions (or declarative sentences) which have values of 'true' and 'false'. The rules of logical inference propagate 'true'.

In a classical physical theory the subsets of phase space are correlated
with propositions about the ranges of values so that there is an obvious
correspondence between the set theoretic operators of union and intersec-
tion and the logical connectives `and' and `or' - with set theoretic inclusion
corresponding to logical implication.
In a quantum mechanical system the propositions concerning ranges of
the observables are correlated with the subspaces of a Hilbert space and the
logical connectives between propositions correspond to set products, sums
and complements with, once again, inclusion corresponding to implication.
Obviously therefore the quantum mechanical and classical propositional
calculi differ from an algebraic standpoint. This difference is precisely (that in)
the classical propositional calculus the logical connectives between
propositions obey the distributive law and in the modular lattice of
quantum mechancs, they do not (Above quoted from Foy, quant-ph/0001074)

> Now I imagine a different universe...what elements of this differen universe does logic apply to? It applies to propositions. Am I to imagine that 'propositions' are something different in this different universe. If so, then I'm at a loss...I don't know what to imagine!?

You actually restate my question quite well. Starting with the abstract statement: logic is empirical, especially the distributive law, I too cannot conceive of how the logic of a different universe could be different. You and I are apparently on the same channel.

> Of course there are different logics that have been invented to apply to
> sentences that are not simple declarative. But I don't see what they have to
> do with other universes.

We are only discussing the logical structures of the theories of classical physics and
quantum mechanics: boolean and modular lattices respectively.

> Please explain.

My question asks if there is a bridge between abstract claims and concrete conceptualization of a possible result.
Taken in smaller steps: logic of the macroworld, generally described by classical physics breaks down when applied to quantum phenomenon. For example, applying Bayes theorem to electron paths will get you hopelessly lost. Foy and others (starting with Birkhoff and von Neumann) have addressed the logic of the two domains and concluded, the logics are different. Evans(?) even concludes (or speculates) they are not only different, they are empirical. To me, that means they must be
experimentally determined. If experimentally determined, then there must be more than one outcome of which the experimenter can conceive. I have enough trouble with only the two logics that seem to be there. I was asking whether anyone could extend my imagination to a broader reach.
It may be that even if empirical, logic is the ultimate application of the anthropic principle--only universes with classical or quantum logic need apply.

John Bailey
Received on Tue Sep 26 2000 - 07:54:38 PDT