Re: When is a logic a quantum logic?
From: Charles Francis (charles_at_lluestfarmpoultry.co.uk)
Date: 07/02/04
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Date: 2 Jul 2004 05:31:51 -0400
In message <6df6a403.0406182209.2d17faed@posting.google.com>, Mike
Carroll <mcarroll@pobox.com> writes
>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message
>news:<cavj25$i5o$1@lfa222122.richmond.edu>...
>>
>> Another might be that it does not meet the needs of mathematics. 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.
>
>Yes, this takes us closer to the nub.
>
>What exactly should we think about boolean logic? "...we want to be
>able to show that a many valued logic is a consistent mathematical
>structure by reducing it to Boolean logic." This suggests that boolean
>logic provides us with something like a touchstone or litmus test, by
>which to judge other systems.
Yes.
>This point of view seems implicit also
>in Svozil's "Quantum logic. A brief outline". Section 7 of that paper,
>which discusses "embeddings of quantum logics into classical logics",
>seems to take classical logic as given, and as providing some sort of
>standard.
>
>George Boole's "An Investigation of the Laws of Thought" was published
>150 years ago, in 1854. Boolean logic as we have it today is
>essentially the same as what he presented then. To me it seems
>unlikely that he got everything right the first time, down to the last
>detail.
If he hadn't got everything right he wouldn't be considered the genius
he is.
>
>More specifically, though, it may be that Boole omitted some operators
>from what we now call boolean algebra, and axioms for those operators.
What he didn't do was provide a mathematical structure for all of
thought, or all of language.
>The operators I have in mind are operators for closure and interior,
>and correspond to the similarly-named topological operations. An
>overview of "closure algebras" is given in Sikorski's "Boolean
>Algebras", section 41. These algebraic operators are related to the
>logical operators for possibility and necessity. Not by coincidence,
>in my opinion, these operators are involved in describing "what would
>happen in other circumstances", which you seem to consider an
>inadequacy [not your term!] of boolean logic. What is necessary is
>what would happen in all circumstances, and what is possible, in some
>but not others.
Simply because we need a more elaborate structure to model this. But it
is wrong to call this an inadequacy of Boolean logic, as though there
was something wrong with Boolean logic. Boolean logic does a specific
job, for which it is perfectly adequate.
>"In mathematics we seek the seek the simplest consistent axiom set..."
>This is true so far as it goes, but seems to be missing something. The
>calculus of 1-place predicates, for example, being decidable, is
>arguably simpler than full-fledged first order logic with relations.
>The monadic predicate calculus is simpler but also inadequate.
>
>Boolean logic is simple, but is it adequate? If inadequate, why map
>quantum logic it to it?
The point is that it is adequate. Adequate to show the consistency of
mathematical structures, whereas these other calculi are not.
>This again is too general to prove anything, and I would not mention
>it had I not something more specific in mind.
>
>Returning to Svozil's paper, on pages 2 and 3 we define elementary
>propositions and logical operations on elementary propositions by
>reference to linear subspaces of Hilbert spaces, and operations on
>those subspaces. We select "closed" linear subspaces to represent
>elementary propositions, and "the closure of" the linear span to
>represent the logical 'or' operation.
>
>It appears that the closure operator that Boole omitted from boolean
>algebra is highly relevant to our attempt to define a quantum logic.
>If our "classical logic" included closure and interior operators, the
>problem of mapping quantum logic to it might be radically transformed.
>
>Perhaps you noticed that I have not spelled out what exactly my
>proposed boolean logic with a closure operator looks like, that will
>provide this supposed panacea. Too true. I have a pretty good idea,
>but am to some extent trying to determine what the benefits might be,
>prior to incurring the full cost. Thanks for your assistance.
I can't see any benefits, or that there could be. Quantum logic seems to
me to model the hypothetical perfectly, both in theory and practice,
just as Bayesian logic models the future and Boolean the actual. What
other type of modelling do you think we need for physics?
-- Charles Francis
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