View Full Version : When is a logic a quantum logic?
Mike Carroll
Jun16-04, 05:27 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I\'ve been led by a long and circuitous route through logic, topology,\nand set theory, to a nonstandard logic. For now, I\'m calling this\nYANL, for "yet another nonstandard logic", as there are too many\nnonstandard logics already.\n\nYANL shares at least one feature with systems that have been developed\nas "quantum logics": YANL algebras are not boolean algebras. YANL was\nnot intended as a quantum logic. But that doesn\'t mean it isn\'t one.\n\nDoes anyone know if anyone has attempted to state criteria of adequacy\nfor a quantum logic? I suspect not. But I would appreciate any\nrecommendations anyone can make along these lines, to save me from\nsifting through the quantum logic literature from von Neumann to the\npresent.\n\nStatements of the form "If L is a quantum logic then ...", which allow\none to show that L is *not* a quantum logic, would be especially\nwelcome.\n\nThanks for any help.\n\nMike Carroll\nOro Valley, AZ\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I've been led by a long and circuitous route through logic, topology,
and set theory, to a nonstandard logic. For now, I'm calling this
YANL, for "yet another nonstandard logic", as there are too many
nonstandard logics already.
YANL shares at least one feature with systems that have been developed
as "quantum logics": YANL algebras are not boolean algebras. YANL was
not intended as a quantum logic. But that doesn't mean it isn't one.
Does anyone know if anyone has attempted to state criteria of adequacy
for a quantum logic? I suspect not. But I would appreciate any
recommendations anyone can make along these lines, to save me from
sifting through the quantum logic literature from von Neumann to the
present.
Statements of the form "If L is a quantum logic then ...", which allow
one to show that L is *not* a quantum logic, would be especially
welcome.
Thanks for any help.
Mike Carroll
Oro Valley, AZ
Charles Francis
Jun17-04, 02:40 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn message <6df6a403.0406151343.357ca7f@posting.google.com> , Mike\nCarroll <mcarroll@pobox.com> writes\n>I\'ve been led by a long and circuitous route through logic, topology,\n>and set theory, to a nonstandard logic. For now, I\'m calling this\n>YANL, for "yet another nonstandard logic", as there are too many\n>nonstandard logics already.\n>\n>YANL shares at least one feature with systems that have been developed\n>as "quantum logics": YANL algebras are not boolean algebras. YANL was\n>not intended as a quantum logic. But that doesn\'t mean it isn\'t one.\n>\n>Does anyone know if anyone has attempted to state criteria of adequacy\n>for a quantum logic? I suspect not. But I would appreciate any\n>recommendations anyone can make along these lines, to save me from\n>sifting through the quantum logic literature from von Neumann to the\n>present.\n>\n>Statements of the form "If L is a quantum logic then ...", which allow\n>one to show that L is *not* a quantum logic, would be especially\n>welcome.\n\n\nA quantum logic has the precisely defined mathematical structure of a\nHilbert space, and it should be very straightforward to check if YANL\nhas such a structure. See Karl Svozil Quantum logic. A brief outline, in\nwhich there are plenty of other refs.\n\nhttp://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf\n--\nCharles Francis\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In message <6df6a403.0406151343.357ca7f@posting.google.com>, Mike
Carroll <mcarroll@pobox.com> writes
>I've been led by a long and circuitous route through logic, topology,
>and set theory, to a nonstandard logic. For now, I'm calling this
>YANL, for "yet another nonstandard logic", as there are too many
>nonstandard logics already.
>
>YANL shares at least one feature with systems that have been developed
>as "quantum logics": YANL algebras are not boolean algebras. YANL was
>not intended as a quantum logic. But that doesn't mean it isn't one.
>
>Does anyone know if anyone has attempted to state criteria of adequacy
>for a quantum logic? I suspect not. But I would appreciate any
>recommendations anyone can make along these lines, to save me from
>sifting through the quantum logic literature from von Neumann to the
>present.
>
>Statements of the form "If L is a quantum logic then ...", which allow
>one to show that L is *not* a quantum logic, would be especially
>welcome.
A quantum logic has the precisely defined mathematical structure of a
Hilbert space, and it should be very straightforward to check if YANL
has such a structure. See Karl Svozil Quantum logic. A brief outline, in
which there are plenty of other refs.
http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf
--
Charles Francis
Arkadiusz Jadczyk
Jun17-04, 06:14 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 16 Jun 2004 22:27:40 +0000 (UTC), mcarroll@pobox.com (Mike\nCarroll) wrote:\n\n>Does anyone know if anyone has attempted to state criteria of adequacy\n>for a quantum logic?\n\n1) The orthomodular lattice of projection in a complex Hilbert space\nshould\nsatisfy the axioms.\n\n2) The "logic" should be "enough" states (i.e. measures)\n\n3) It should be "useful", that is it should allow us to "compute"\nor to "understand" something new.\n\n4) It is useful to have some kind of "Noether\'s theorem" holding, that\nwould guarantee that there is a relation between one parameter groups of\nauthomorphisms and "conserved quantities", at least for some\n"representations" (or "presentations" or "realizations") of you logic\n\nThat is about all what can be said in general. The rest of the devil is\nin the details.\n\nark\n--\n\nArkadiusz Jadczyk\nhttp://www.cassiopaea.org/quantum_future/homepage.htm\n\n--\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 16 Jun 2004 22:27:40 +0000 (UTC), mcarroll@pobox.com (Mike
Carroll) wrote:
>Does anyone know if anyone has attempted to state criteria of adequacy
>for a quantum logic?
1) The orthomodular lattice of projection in a complex Hilbert space
should
satisfy the axioms.
2) The "logic" should be "enough" states (i.e. measures)
3) It should be "useful", that is it should allow us to "compute"
or to "understand" something new.
4) It is useful to have some kind of "Noether's theorem" holding, that
would guarantee that there is a relation between one parameter groups of
authomorphisms and "conserved quantities", at least for some
"representations" (or "presentations" or "realizations") of you logic
That is about all what can be said in general. The rest of the devil is
in the details.
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Mike Carroll
Jun18-04, 04:40 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nCharles Francis <charles@lluestfarmpoultry.co.uk> wrote in message news:<z\\$E\\$k1r58e0AFw\\$0@clef.demon.co.uk>...\ n>\n> A quantum logic has the precisely defined mathematical structure of a\n> Hilbert space, and it should be very straightforward to check if YANL\n> has such a structure. See Karl Svozil Quantum logic. A brief outline, in\n> which there are plenty of other refs.\n>\n> http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf\n\nThank you very much for providing a reference which is both recent and\navailable online.\n\nThe paper, together with some further reflection, has helped me\nunderstand what bothers me about quantum logic as currently\nunderstood.\n\n"The starting point [of quantum logic] is von Neumann\'s Hilbert space\nformalism of quantum mechanics", the paper says on page 1. We then use\nthis formalism to provide an interpretation of logical operations (pp.\n2-3). After examining the resulting logic, we find that "The\npropositional system obtained is _not_ a classical Boolean\nalgebra...." (p.5).\n\nWhat bothers me is that a Hilbert space is a topology, constructed\nusing set theory, which in turn relies upon a logic whose\npropositional system is in fact a classical boolean algebra. Why not\nreplace classical boolean algebra to begin with, rather than accept it\nfirst and then replace or reject it later upon discovering that it\ndoes not meet our needs for physics?\n\n("Easier said than done," might be one reply.)\n\nThe moderator, however, may feel I am straying too far afield. Thanks\nin any case for the reference and its excellent bibliography.\n\nMike Carroll\nOro Valley, AZ\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message news:<z$E$k1r58e0AFw$0@clef.demon.co.uk>...
>
> A quantum logic has the precisely defined mathematical structure of a
> Hilbert space, and it should be very straightforward to check if YANL
> has such a structure. See Karl Svozil Quantum logic. A brief outline, in
> which there are plenty of other refs.
>
> http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf
Thank you very much for providing a reference which is both recent and
available online.
The paper, together with some further reflection, has helped me
understand what bothers me about quantum logic as currently
understood.
"The starting point [of quantum logic] is von Neumann's Hilbert space
formalism of quantum mechanics", the paper says on page 1. We then use
this formalism to provide an interpretation of logical operations (pp.2-3). After examining the resulting logic, we find that "The
propositional system obtained is _not_ a classical Boolean
algebra...." (p.5).
What bothers me is that a Hilbert space is a topology, constructed
using set theory, which in turn relies upon a logic whose
propositional system is in fact a classical boolean algebra. Why not
replace classical boolean algebra to begin with, rather than accept it
first and then replace or reject it later upon discovering that it
does not meet our needs for physics?
("Easier said than done," might be one reply.)
The moderator, however, may feel I am straying too far afield. Thanks
in any case for the reference and its excellent bibliography.
Mike Carroll
Oro Valley, AZ
Arnold Neumaier
Jun18-04, 01:26 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Mike Carroll wrote:\n>\n> "The starting point [of quantum logic] is von Neumann\'s Hilbert space\n> formalism of quantum mechanics", the paper says on page 1. We then use\n> this formalism to provide an interpretation of logical operations (pp.\n> 2-3). After examining the resulting logic, we find that "The\n> propositional system obtained is _not_ a classical Boolean\n> algebra...." (p.5).\n>\n> What bothers me is that a Hilbert space is a topology, constructed\n> using set theory, which in turn relies upon a logic whose\n> propositional system is in fact a classical boolean algebra. Why not\n> replace classical boolean algebra to begin with, rather than accept it\n> first and then replace or reject it later upon discovering that it\n> does not meet our needs for physics?\n\nThis is because quantum logic is clumsy to use and does not help to solve\nany significant quantum application. Almost everything done to apply\nquantum mechanics is based on classical logic for the Schroedinger equation.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mike Carroll wrote:
>
> "The starting point [of quantum logic] is von Neumann's Hilbert space
> formalism of quantum mechanics", the paper says on page 1. We then use
> this formalism to provide an interpretation of logical operations (pp.
> 2-3). After examining the resulting logic, we find that "The
> propositional system obtained is _not_ a classical Boolean
> algebra...." (p.5).
>
> What bothers me is that a Hilbert space is a topology, constructed
> using set theory, which in turn relies upon a logic whose
> propositional system is in fact a classical boolean algebra. Why not
> replace classical boolean algebra to begin with, rather than accept it
> first and then replace or reject it later upon discovering that it
> does not meet our needs for physics?
This is because quantum logic is clumsy to use and does not help to solve
any significant quantum application. Almost everything done to apply
quantum mechanics is based on classical logic for the Schroedinger equation.
Arnold Neumaier
Charles Francis
Jun18-04, 03:24 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In message <6df6a403.0406172152.528b390@posting.google.com> , Mike\nCarroll <mcarroll@pobox.com> writes\n>\n>\n>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message\n>news:<z\\$E\\$k1r58e0AFw\\$0@clef.demon. co.uk>...\n>>\n>> A quantum logic has the precisely defined mathematical structure of a\n>> Hilbert space, and it should be very straightforward to check if YANL\n>> has such a structure. See Karl Svozil Quantum logic. A brief outline, in\n>> which there are plenty of other refs.\n>>\n>> http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf\n>\n>Thank you very much for providing a reference which is both recent and\n>available online.\n>\n>The paper, together with some further reflection, has helped me\n>understand what bothers me about quantum logic as currently\n>understood.\n>\n>"The starting point [of quantum logic] is von Neumann\'s Hilbert space\n>formalism of quantum mechanics", the paper says on page 1. We then use\n>this formalism to provide an interpretation of logical operations (pp.\n>2-3). After examining the resulting logic, we find that "The\n>propositional system obtained is _not_ a classical Boolean\n>algebra...." (p.5).\n>\n>What bothers me is that a Hilbert space is a topology, constructed\n>using set theory, which in turn relies upon a logic whose\n>propositional system is in fact a classical boolean algebra. Why not\n>replace classical boolean algebra to begin with, rather than accept it\n>first and then replace or reject it later upon discovering that it\n>does not meet our needs for physics?\n\n\n>\n>("Easier said than done," might be one reply.)\n\nAnother might be that it does not meet the needs of mathematics. In\nmathematics we seek the simplest consistent axiom set, and reduce\neverything to that. In mathematics we want to be able to show that a\nmany valued logic is a consistent mathematical structure by reducing it\nto Boolean logic. In physics we only want to use it when appropriate,\nhaving been assured that it is consistent to do so.\n\nAnother is that it does a different job from Boolean algebra. In physics\nwe can understand definite statements\n\n"the particle is measured at position x", which has a Boolean truth\nvalue provided we have actually done a measurement\n\nBayesian statements, with probabilistic truth values\n\n"If I do a measurement I will measure the particle at x"\n\nMuch less understood are hypothetical statements\n\n"If I were to do a measurement I would measure the particle at x"\n\nwhich we apply to the case where we know perfectly well that we are not\ngoing to do a measurement (e.g. asking which slit did the particle come\nthrough in a Young\'s slit experiment)\n\nI see quantum logic as giving truth values for these hypothetical, or\ncounterfactual statements (in Boolean logic you always have p->q is true\nwhen p is false, which does not fit an intuitive notion there is some\ntruth in describing what would happen in other circumstances.\n\nI don\'t know that physics requires any other form of statement, or any\nother logic other than Boolean, Probability theory and quantum logic.\nAnd although I do see some value in Fuzzy Logic, I think what is being\ndescribed there is psychology rather than physics.\n\n>The moderator, however, may feel I am straying too far afield.\n\nI would think not. Foundations of quantum theory and the application of\nmathematics to physics are both very much on topic.\n\n--\nCharles Francis\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In message <6df6a403.0406172152.528b390@posting.google.com>, Mike
Carroll <mcarroll@pobox.com> writes
>
>
>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message
>news:<z$E$k1r58e0AFw$0@clef.demon.co.uk>...
>>
>> A quantum logic has the precisely defined mathematical structure of a
>> Hilbert space, and it should be very straightforward to check if YANL
>> has such a structure. See Karl Svozil Quantum logic. A brief outline, in
>> which there are plenty of other refs.
>>
>> http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf
>
>Thank you very much for providing a reference which is both recent and
>available online.
>
>The paper, together with some further reflection, has helped me
>understand what bothers me about quantum logic as currently
>understood.
>
>"The starting point [of quantum logic] is von Neumann's Hilbert space
>formalism of quantum mechanics", the paper says on page 1. We then use
>this formalism to provide an interpretation of logical operations (pp.>2-3). After examining the resulting logic, we find that "The
>propositional system obtained is _not_ a classical Boolean
>algebra...." (p.5).
>
>What bothers me is that a Hilbert space is a topology, constructed
>using set theory, which in turn relies upon a logic whose
>propositional system is in fact a classical boolean algebra. Why not
>replace classical boolean algebra to begin with, rather than accept it
>first and then replace or reject it later upon discovering that it
>does not meet our needs for physics?
>
>("Easier said than done," might be one reply.)
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.
Another is that it does a different job from Boolean algebra. In physics
we can understand definite statements
"the particle is measured at position x", which has a Boolean truth
value provided we have actually done a measurement
Bayesian statements, with probabilistic truth values
"If I do a measurement I will measure the particle at x"
Much less understood are hypothetical statements
"If I were to do a measurement I would measure the particle at x"
which we apply to the case where we know perfectly well that we are not
going to do a measurement (e.g. asking which slit did the particle come
through in a Young's slit experiment)
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.
I don't know that physics requires any other form of statement, or any
other logic other than Boolean, Probability theory and quantum logic.
And although I do see some value in Fuzzy Logic, I think what is being
described there is psychology rather than physics.
>The moderator, however, may feel I am straying too far afield.
I would think not. Foundations of quantum theory and the application of
mathematics to physics are both very much on topic.
--
Charles Francis
Charles Francis
Jun22-04, 01:05 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In message <cavc55\\$i3e\\$1@lfa222122.richmond.edu>, Arnold Neumaier\n<Arnold.Neumaier@univie.ac.at> writes\n>Mike Carroll wrote:\n>> "The starting point [of quantum logic] is von Neumann\'s Hilbert\n>>space\n>> formalism of quantum mechanics", the paper says on page 1. We then use\n>> this formalism to provide an interpretation of logical operations (pp.\n>> 2-3). After examining the resulting logic, we find that "The\n>> propositional system obtained is _not_ a classical Boolean\n>> algebra...." (p.5).\n>> What bothers me is that a Hilbert space is a topology, constructed\n>> using set theory, which in turn relies upon a logic whose\n>> propositional system is in fact a classical boolean algebra. Why not\n>> replace classical boolean algebra to begin with, rather than accept it\n>> first and then replace or reject it later upon discovering that it\n>> does not meet our needs for physics?\n>\n>This is because quantum logic is clumsy to use and does not help to solve\n>any significant quantum application. Almost everything done to apply\n>quantum mechanics is based on classical logic for the Schroedinger equation.\n>\nWith the notable exception of understanding\n--\nCharles Francis\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In message <cavc55$i3e$1@lfa222122.richmond.edu>, Arnold Neumaier
<Arnold.Neumaier@univie.ac.at> writes
>Mike Carroll wrote:
>> "The starting point [of quantum logic] is von Neumann's Hilbert
>>space
>> formalism of quantum mechanics", the paper says on page 1. We then use
>> this formalism to provide an interpretation of logical operations (pp.>> 2-3). After examining the resulting logic, we find that "The
>> propositional system obtained is _not_ a classical Boolean
>> algebra...." (p.5).
>> What bothers me is that a Hilbert space is a topology, constructed
>> using set theory, which in turn relies upon a logic whose
>> propositional system is in fact a classical boolean algebra. Why not
>> replace classical boolean algebra to begin with, rather than accept it
>> first and then replace or reject it later upon discovering that it
>> does not meet our needs for physics?
>
>This is because quantum logic is clumsy to use and does not help to solve
>any significant quantum application. Almost everything done to apply
>quantum mechanics is based on classical logic for the Schroedinger equation.
>
With the notable exception of understanding
--
Charles Francis
Mike Carroll
Jun25-04, 03:52 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message news:<cavj25\\$i5o\\$1@lfa222122.richmond.edu>...\ n> ... In\n> mathematics we seek the simplest consistent axiom set, and reduce\n> everything to that. In mathematics we want to be able to show that a\n> many valued logic is a consistent mathematical structure by reducing it\n> to Boolean logic. In physics we only want to use it when appropriate,\n> having been assured that it is consistent to do so.\n\nThis gives Boolean logic an authority, independent of physics, which\nit does not merit. We invented Boolean logic, and we may decide that\nit is too simple. One of the best kinds of evidence for its being too\nsimple is that we have trouble mapping quantum logic to Boolean logic.\n\n> I see quantum logic as giving truth values for these hypothetical, or\n> counterfactual statements (in Boolean logic you always have p->q is true\n> when p is false, which does not fit an intuitive notion there is some\n> truth in describing what would happen in other circumstances.\n\nThis is presented as a difference between Boolean and quantum logic. I\nwould take it instead as a criticism of Boolean logic, that it fails\nto provide a logic for counterfactual statements. Many logicians would\nagree that this is a problem with Boolean logic, apart from any\nspecific concerns about quantum mechanics.\n\nHow could Boolean logic be "too simple"? It is possible to add a\nclosure operator to the language of Boolean algebra. We can then add\npostulates for the closure operator. Linguistically, this adds to\nBoolean logic some of the expressive power of topological spaces. This\nhas already been investigated to some extent: Sikorski, "Boolean\nAlgegras", section 41, "Topology in Boolean algebras. Applications to\nnon-classical logic."\n\nThe "non-classical" logic that Sikorski refers to are modal logics,\nwhich are also the logics of hypotheticals and counterfactuals. This\nmay be a coincidence but I doubt it.\n\nTo summarize, it seems to me that the current state of quantum logic\nindicates that Boolean logic is in need of revision. I will however\nhave to learn considerably more than what I now know about Hilbert\nspaces, before I can see whether there is any truth to my conjecture.\n\n(A longer version of these comments was posted earier, but seems to\nhave disappeared.)\n\nMike Carroll\nOro Valley, AZ, USA\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>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
Mike Carroll
Jun27-04, 05:56 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message news:<cavj25\\$i5o\\$1@lfa222122.richmond.edu>...\ n>\n> Another might be that it does not meet the needs of mathematics. In\n> mathematics we seek the simplest consistent axiom set, and reduce\n> everything to that. In mathematics we want to be able to show that a\n> many valued logic is a consistent mathematical structure by reducing it\n> to Boolean logic. In physics we only want to use it when appropriate,\n> having been assured that it is consistent to do so.\n\nYes, this takes us closer to the nub.\n\nWhat exactly should we think about boolean logic? "...we want to be\nable to show that a many valued logic is a consistent mathematical\nstructure by reducing it to Boolean logic." This suggests that boolean\nlogic provides us with something like a touchstone or litmus test, by\nwhich to judge other systems. This point of view seems implicit also\nin Svozil\'s "Quantum logic. A brief outline". Section 7 of that paper,\nwhich discusses "embeddings of quantum logics into classical logics",\nseems to take classical logic as given, and as providing some sort of\nstandard.\n\nGeorge Boole\'s "An Investigation of the Laws of Thought" was published\n150 years ago, in 1854. Boolean logic as we have it today is\nessentially the same as what he presented then. To me it seems\nunlikely that he got everything right the first time, down to the last\ndetail.\n\nMore specifically, though, it may be that Boole omitted some operators\nfrom what we now call boolean algebra, and axioms for those operators.\nThe operators I have in mind are operators for closure and interior,\nand correspond to the similarly-named topological operations. An\noverview of "closure algebras" is given in Sikorski\'s "Boolean\nAlgebras", section 41. These algebraic operators are related to the\nlogical operators for possibility and necessity. Not by coincidence,\nin my opinion, these operators are involved in describing "what would\nhappen in other circumstances", which you seem to consider an\ninadequacy [not your term!] of boolean logic. What is necessary is\nwhat would happen in all circumstances, and what is possible, in some\nbut not others.\n\n"In mathematics we seek the seek the simplest consistent axiom set..."\nThis is true so far as it goes, but seems to be missing something. The\ncalculus of 1-place predicates, for example, being decidable, is\narguably simpler than full-fledged first order logic with relations.\nThe monadic predicate calculus is simpler but also inadequate.\n\nBoolean logic is simple, but is it adequate? If inadequate, why map\nquantum logic it to it?\n\nThis again is too general to prove anything, and I would not mention\nit had I not something more specific in mind.\n\nReturning to Svozil\'s paper, on pages 2 and 3 we define elementary\npropositions and logical operations on elementary propositions by\nreference to linear subspaces of Hilbert spaces, and operations on\nthose subspaces. We select "closed" linear subspaces to represent\nelementary propositions, and "the closure of" the linear span to\nrepresent the logical \'or\' operation.\n\nIt appears that the closure operator that Boole omitted from boolean\nalgebra is highly relevant to our attempt to define a quantum logic.\nIf our "classical logic" included closure and interior operators, the\nproblem of mapping quantum logic to it might be radically transformed.\n\nPerhaps you noticed that I have not spelled out what exactly my\nproposed boolean logic with a closure operator looks like, that will\nprovide this supposed panacea. Too true. I have a pretty good idea,\nbut am to some extent trying to determine what the benefits might be,\nprior to incurring the full cost. Thanks for your assistance.\n\nMike Carroll\nOro Valley, AZ\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>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. 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.
More specifically, though, it may be that Boole omitted some operators
from what we now call boolean algebra, and axioms for those operators.
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.
"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?
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.
Mike Carroll
Oro Valley, AZ
Charles Francis
Jun29-04, 04:42 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>[Moderator\'s note: Unless the connection to physics reasserts itself,\nfurther discussion of the nature of logic in general should probably\nbe taken to e-mail. -TB]\n\nIn message <6df6a403.0406232053.499c2e52@posting.google.com >, Mike\nCarroll <mcarroll@pobox.com> writes\n>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message\n>news:<cavj25\\$i5o\\$1@lfa222122.richmon d.edu>...\n>> ... In\n>> mathematics we seek the simplest consistent axiom set, and reduce\n>> everything to that. In mathematics we want to be able to show that a\n>> many valued logic is a consistent mathematical structure by reducing it\n>> to Boolean logic. In physics we only want to use it when appropriate,\n>> having been assured that it is consistent to do so.\n>\n>This gives Boolean logic an authority, independent of physics, which\n>it does not merit.\n\nI am sorry, but the notion that mathematics or logic in some way depend\non physics always strikes me as an unholy conceit. Logic and mathematics\nstudy thought, and their authority is that of consistent thought. This\nis an authority well above anything in physics, and it is utterly\nmerited, since without consistent thought it is not even meaningful to\nanalyse experimental results. To say otherwise is simply a license for\ninconsistent thinking.\n\n> We invented Boolean logic,\n\nOr at least Aristotle and Boole did, the work of genius should always be\ncorrectly attributed.\n\n> and we may decide that\n>it is too simple.\n\nToo simple for what? Certainly it is too simple to apply universally to\nlanguage structures, as Aristotle pointed out with his discussion of the\nsentence "there will be a sea battle tomorrow", hence the further\ninvention of many valued logic.\n\n> One of the best kinds of evidence for its being too\n>simple is that we have trouble mapping quantum logic to Boolean logic.\n\nYou have already commented that we can reduce the mathematical structure\nof Hilbert to that of Boolean logic. Boolean logic is what it is and\ndoes what it should do to perfection. The problem, if there is one,\ncomes in the misapplication of Boolean logic to language structures\nwhich it does not model.\n\n\n>> I see quantum logic as giving truth values for these hypothetical, or\n>> counterfactual statements (in Boolean logic you always have p->q is true\n>> when p is false, which does not fit an intuitive notion there is some\n>> truth in describing what would happen in other circumstances.\n>\n>This is presented as a difference between Boolean and quantum logic. I\n>would take it instead as a criticism of Boolean logic, that it fails\n>to provide a logic for counterfactual statements.\n\nIt was never supposed to. As discussed by Aristotle. Criticising it for\nthat is like criticising a car because it is not a bus.\n\n>Many logicians would\n>agree that this is a problem with Boolean logic, apart from any\n>specific concerns about quantum mechanics.\n\nIt is not a problem with the logic. It may be a problem with the\nmisapplication of the logic by particular individuals.\n\n>How could Boolean logic be "too simple"? It is possible to add a\n>closure operator to the language of Boolean algebra. We can then add\n>postulates for the closure operator.\n\n\nOne does not achieve anything by arbitrary complexity. You have to\nunderstand why something is as it is..\n\n>Linguistically, this adds to\n>Boolean logic some of the expressive power of topological spaces. This\n>has already been investigated to some extent: Sikorski, "Boolean\n>Algegras", section 41, "Topology in Boolean algebras. Applications to\n>non-classical logic."\n\n\n>The "non-classical" logics that Sikorski refers to are modal logics, which\n>are also the logics of hypotheticals and counterfactuals.\n\nDo we know this? Can you describe what you mean by a modal logic, and\njustify the claim that they are appropriate to hypotheticals and\ncounterfactuals?\n\n>This may be a\n>coincidence but I doubt it.\n\nWe have a logic for hypotheticals and counterfactuals, one which is\nquite justifiable and which works perfectly in experiment, namely\nquantum logic. If Sikorski\'s logics have the same mathematical\nstructure, then this is not a coincidence but nor is it interesting or\nnew. If they have a different structure and lead to different\npredictions of probabilities, then it is again not a coincidence, but it\na wild speculation which should be ignored.\n\n\n>I will however have to\n>learn considerably more than what I now know about Hilbert spaces, before I\n>can see whether there is any truth to my conjecture.\n\nYou should always assume that there is no truth in a conjecture which\nsuggests that one of the great mathematical genii of history was on the\nwrong track or understood what he was doing rather less well than many\nwho come after.\n\n\nRegards\n\n--\nCharles Francis\n\n\n\nRegards\n\n--\nCharles Francis\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>[Moderator's note: Unless the connection to physics reasserts itself,
further discussion of the nature of logic in general should probably
be taken to e-mail. -TB]
In message <6df6a403.0406232053.499c2e52@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>...
>> ... 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.
I am sorry, but the notion that mathematics or logic in some way depend
on physics always strikes me as an unholy conceit. Logic and mathematics
study thought, and their authority is that of consistent thought. This
is an authority well above anything in physics, and it is utterly
merited, since without consistent thought it is not even meaningful to
analyse experimental results. To say otherwise is simply a license for
inconsistent thinking.
> We invented Boolean logic,
Or at least Aristotle and Boole did, the work of genius should always be
correctly attributed.
> and we may decide that
>it is too simple.
Too simple for what? Certainly it is too simple to apply universally to
language structures, as Aristotle pointed out with his discussion of the
sentence "there will be a sea battle tomorrow", hence the further
invention of many valued logic.
> One of the best kinds of evidence for its being too
>simple is that we have trouble mapping quantum logic to Boolean logic.
You have already commented that we can reduce the mathematical structure
of Hilbert to that of Boolean logic. Boolean logic is what it is and
does what it should do to perfection. The problem, if there is one,
comes in the misapplication of Boolean logic to language structures
which it does not model.
>> 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.
It was never supposed to. As discussed by Aristotle. Criticising it for
that is like criticising a car because it is not a bus.
>Many logicians would
>agree that this is a problem with Boolean logic, apart from any
>specific concerns about quantum mechanics.
It is not a problem with the logic. It may be a problem with the
misapplication of the logic by particular individuals.
>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.
One does not achieve anything by arbitrary complexity. You have to
understand why something is as it is..
>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" logics that Sikorski refers to are modal logics, which
>are also the logics of hypotheticals and counterfactuals.
Do we know this? Can you describe what you mean by a modal logic, and
justify the claim that they are appropriate to hypotheticals and
counterfactuals?
>This may be a
>coincidence but I doubt it.
We have a logic for hypotheticals and counterfactuals, one which is
quite justifiable and which works perfectly in experiment, namely
quantum logic. If Sikorski's logics have the same mathematical
structure, then this is not a coincidence but nor is it interesting or
new. If they have a different structure and lead to different
predictions of probabilities, then it is again not a coincidence, but it
a wild speculation which should be ignored.
>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.
You should always assume that there is no truth in a conjecture which
suggests that one of the great mathematical genii of history was on the
wrong track or understood what he was doing rather less well than many
who come after.
Regards
--
Charles Francis
Regards
--
Charles Francis
Charles Francis
Jul2-04, 04:31 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn message <6df6a403.0406182209.2d17faed@posting.google.com >, Mike\nCarroll <mcarroll@pobox.com> writes\n>Charles Francis <charles@lluestfarmpoultry.co.uk> wrote in message\n>news:<cavj25\\$i5o\\$1@lfa222122.richmon d.edu>...\n>>\n>> Another might be that it does not meet the needs of mathematics. In\n>> mathematics we seek the simplest consistent axiom set, and reduce\n>> everything to that. In mathematics we want to be able to show that a\n>> many valued logic is a consistent mathematical structure by reducing it\n>> to Boolean logic. In physics we only want to use it when appropriate,\n>> having been assured that it is consistent to do so.\n>\n>Yes, this takes us closer to the nub.\n>\n>What exactly should we think about boolean logic? "...we want to be\n>able to show that a many valued logic is a consistent mathematical\n>structure by reducing it to Boolean logic." This suggests that boolean\n>logic provides us with something like a touchstone or litmus test, by\n>which to judge other systems.\n\nYes.\n\n>This point of view seems implicit also\n>in Svozil\'s "Quantum logic. A brief outline". Section 7 of that paper,\n>which discusses "embeddings of quantum logics into classical logics",\n>seems to take classical logic as given, and as providing some sort of\n>standard.\n>\n>George Boole\'s "An Investigation of the Laws of Thought" was published\n>150 years ago, in 1854. Boolean logic as we have it today is\n>essentially the same as what he presented then. To me it seems\n>unlikely that he got everything right the first time, down to the last\n>detail.\n\nIf he hadn\'t got everything right he wouldn\'t be considered the genius\nhe is.\n\n>\n>More specifically, though, it may be that Boole omitted some operators\n>from what we now call boolean algebra, and axioms for those operators.\n\nWhat he didn\'t do was provide a mathematical structure for all of\nthought, or all of language.\n\n>The operators I have in mind are operators for closure and interior,\n>and correspond to the similarly-named topological operations. An\n>overview of "closure algebras" is given in Sikorski\'s "Boolean\n>Algebras", section 41. These algebraic operators are related to the\n>logical operators for possibility and necessity. Not by coincidence,\n>in my opinion, these operators are involved in describing "what would\n>happen in other circumstances", which you seem to consider an\n>inadequacy [not your term!] of boolean logic. What is necessary is\n>what would happen in all circumstances, and what is possible, in some\n>but not others.\n\nSimply because we need a more elaborate structure to model this. But it\nis wrong to call this an inadequacy of Boolean logic, as though there\nwas something wrong with Boolean logic. Boolean logic does a specific\njob, for which it is perfectly adequate.\n\n>"In mathematics we seek the seek the simplest consistent axiom set..."\n>This is true so far as it goes, but seems to be missing something. The\n>calculus of 1-place predicates, for example, being decidable, is\n>arguably simpler than full-fledged first order logic with relations.\n>The monadic predicate calculus is simpler but also inadequate.\n>\n>Boolean logic is simple, but is it adequate? If inadequate, why map\n>quantum logic it to it?\n\nThe point is that it is adequate. Adequate to show the consistency of\nmathematical structures, whereas these other calculi are not.\n\n>This again is too general to prove anything, and I would not mention\n>it had I not something more specific in mind.\n>\n>Returning to Svozil\'s paper, on pages 2 and 3 we define elementary\n>propositions and logical operations on elementary propositions by\n>reference to linear subspaces of Hilbert spaces, and operations on\n>those subspaces. We select "closed" linear subspaces to represent\n>elementary propositions, and "the closure of" the linear span to\n>represent the logical \'or\' operation.\n>\n>It appears that the closure operator that Boole omitted from boolean\n>algebra is highly relevant to our attempt to define a quantum logic.\n>If our "classical logic" included closure and interior operators, the\n>problem of mapping quantum logic to it might be radically transformed.\n>\n>Perhaps you noticed that I have not spelled out what exactly my\n>proposed boolean logic with a closure operator looks like, that will\n>provide this supposed panacea. Too true. I have a pretty good idea,\n>but am to some extent trying to determine what the benefits might be,\n>prior to incurring the full cost. Thanks for your assistance.\n\nI can\'t see any benefits, or that there could be. Quantum logic seems to\nme to model the hypothetical perfectly, both in theory and practice,\njust as Bayesian logic models the future and Boolean the actual. What\nother type of modelling do you think we need for physics?\n\n--\nCharles Francis\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>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
Mike Carroll
Jul6-04, 01:47 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nCharles Francis <charles@clef.demon.co.uk> wrote in message news:<cbsnnd\\$dh9\\$1@lfa222122.richmond.edu>...\ n>\n> ... Can you describe what you mean by a modal logic, and\n> justify the claim that they are appropriate to hypotheticals and\n> counterfactuals?\n\nAs it turns out, quite a bit of work has already been done on modal\nlogic and quantum theory. The article "The Modal Interpretations of\nQuantum Theory" at:\nhttp://plato.stanford.edu/entries/qm-modal/\ngives an overview and a bibliography.\n\nI was not aware of this when I posted previously.\n\nMike Carroll\nOro Valley, AZ\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Charles Francis <charles@clef.demon.co.uk> wrote in message news:<cbsnnd$dh9$1@lfa222122.richmond.edu>...
>
> ... Can you describe what you mean by a modal logic, and
> justify the claim that they are appropriate to hypotheticals and
> counterfactuals?
As it turns out, quite a bit of work has already been done on modal
logic and quantum theory. The article "The Modal Interpretations of
Quantum Theory" at:
http://plato.stanford.edu/entries/qm-modal/
gives an overview and a bibliography.
I was not aware of this when I posted previously.
Mike Carroll
Oro Valley, AZ
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