Re: When is a logic a quantum logic?

From: Mike Carroll (mcarroll_at_pobox.com)
Date: 06/25/04 

Date: Fri, 25 Jun 2004 20:52:54 +0000 (UTC)

Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message news:<cavj25$i5o$1@lfa222122.richmond.edu>...
> ... In
> mathematics we seek the simplest consistent axiom set, and reduce
> everything to that. In mathematics we want to be able to show that a
> many valued logic is a consistent mathematical structure by reducing it
> to Boolean logic. In physics we only want to use it when appropriate,
> having been assured that it is consistent to do so.

This gives Boolean logic an authority, independent of physics, which
it does not merit. We invented Boolean logic, and we may decide that
it is too simple. One of the best kinds of evidence for its being too
simple is that we have trouble mapping quantum logic to Boolean logic.

> I see quantum logic as giving truth values for these hypothetical, or
> counterfactual statements (in Boolean logic you always have p->q is true
> when p is false, which does not fit an intuitive notion there is some
> truth in describing what would happen in other circumstances.

This is presented as a difference between Boolean and quantum logic. I
would take it instead as a criticism of Boolean logic, that it fails
to provide a logic for counterfactual statements. Many logicians would
agree that this is a problem with Boolean logic, apart from any
specific concerns about quantum mechanics.

How could Boolean logic be "too simple"? It is possible to add a
closure operator to the language of Boolean algebra. We can then add
postulates for the closure operator. Linguistically, this adds to
Boolean logic some of the expressive power of topological spaces. This
has already been investigated to some extent: Sikorski, "Boolean
Algegras", section 41, "Topology in Boolean algebras. Applications to
non-classical logic."

The "non-classical" logic that Sikorski refers to are modal logics,
which are also the logics of hypotheticals and counterfactuals. This
may be a coincidence but I doubt it.

To summarize, it seems to me that the current state of quantum logic
indicates that Boolean logic is in need of revision. I will however
have to learn considerably more than what I now know about Hilbert
spaces, before I can see whether there is any truth to my conjecture.

(A longer version of these comments was posted earier, but seems to
have disappeared.)

Mike Carroll
Oro Valley, AZ, USA

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