Thinking Logically About Quantum Probabilities

7/29/2019 Thinking Logically About Quantum Probabilities

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Thinking!logically!about!quantum!probabilities!!

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!Dr!Rachel!Garden!

!March!2013!

[email protected]!

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!

!

!In!honour!of!!

Evigshedfiord,!Greenland,!

a!beautiful!place!to!think!!

!!

!Abstract!

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Probabilities!are!derived!from!logical!first!principles.!Propositional!logic!expressing!Booles!Laws!of!Thought!is!a!slightly!generalised!Boolean!algebra.!

Probabilities!are!measures!over!Boolean!fields!of!filters!of!this!algebra,!representing!measures!of!maximal!valuations!that!find!a!proposition!true.!Classical!

and!quantum!theories!use!the!same!propositional!logic!and!same!definition!of!a!

probability.!Differences!arise!because!their!maximal!valuations!are!different,!their!most!comprehensive!assignments!of!truthvalues!to!propositions.!Where!

propositions!are!all!compatible!truthvalues!can!be!assigned!to!them!all,!and!a!

single!probability!space!expresses!all!the!probabilities,!which!are!entirely!classical.!!Generally!however!since!propositions!may!be!incompatible,!maximal!valuations!do!

not!assign!truthvalues!to!all!the!propositions.!Now!different!probability!spaces!generate!probabilities!with!different!initial!conditions!and!so!in!general!

probabilities!are!strongly!conditional.!These!general!probabilities!have!quantum!

peculiarities:!nonzero!assignments!for!incompatible!descriptions,!drastic!changes!after!physical!measurement!and!unfamiliar!property!entanglements.!Since!these!

are!logical!features!of!probabilities!however!they!do!not!describe!weird!or!non

local!reality.!Instead!they!express!reconditionalising!required!when!new!information!is!added!to!a!theory!lacking!comprehensive!descriptions.!The!use!of!

Hilbert!Space!to!represent!quantum!mechanics!is!briefly!discussed,!and!a!logical!reason!for!preferring!classical!theories.! !


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1.#Introduction!!

The!difference!between!classical!and!quantum!probabilities!lies!at!the!heart!of!interpreting!quantum!theory.!!!

!

Classical!probabilities!can!be!expressed!as!measures!over!a!Boolean!field!of!sets,!the!logical!event!space!over!which!probabilities!measure!events!in!which!a!

proposition!is!true.!Quantum!probabilities!on!the!other!hand!are!expressed!by!

projections!in!complexvalued!innerproduct!vector!space!we!call!Hilbert!Space,!an!entirely!different!mathematical!representation.!Yet!quantum!probabilities,!like!

those!of!classical!theories,!make!predictions!about!the!outcomes!of!measurement!that!are!well!supported!by!experiment.!

!

This!difference!has!particular!significance!to!logic!because!the!lack!of!a!probability!space!in!quantum!theories!seems!to!show!the!lack!of!traditional!logic!too,!since!

this!is!assumed!represented!by!a!Boolean!algebra.!Since!any!Boolean!algebra!is!isomorphic!to!its!field!of!ultrafilters,!this!logic!also!provides!the!space!over!which!

its!probabilities!are!defined.!Yet!Hilbert!Space!is!nothing!like!a!Boolean!algebra!or!

its!field,!nor!can!it!apparently!be!augmented!by!hidden!variables!nor!transformed!in!any!other!plausible!way!into!one.!It!seems!therefore!that!quantum!theories!

being!represented!in!Hilbert!Space,!lack!both!a!traditional!logic!and!a!probability!

space,!leaving!the!foundations!of!their!logic!and!their!probabilities!obscure.!!!

In!this!paper!traditional!methods!of!formal!logic!are!applied!to!this!problem.!Here!logic!is!used!the!oldfashioned!way!to!mean!not!simply!formal!methods,!but!

specifically!a!formal!system!that!can!express!Booles!!Laws!of!Thought.!!

!Traditional!first!principles!are!presented!at!the!start!of!section!2!and!used!to!

determine!the!truthfunctional!logic!Lthat!expresses!Booles!Laws!of!Thought.!

This!logic!is!not!assumed!bivalent!so!is!represented!by!a!slight!generalisation!of!

Boolean!algebra,!a!distributive!lattice!with!orthogonal!and!complement.!In!section!3!a!modal!concept!of!uncertainty!is!introduced!into!this!logic,!then!elaborated!to!

numerical!degrees!of!uncertainty!which!are!mathematical!measures!of!sets!in!

which!a!proposition!is!true.!Probabilities!are!introduced!in!section!4!as!a!special!case!of!these!degrees!which!makes!their!logical!foundation!clear:!a!probability!is!a!

welldefined!mathematical!measure!over!a!Boolean!field!of!maximal!filters!of!its!

algebra,!representing!measures!of!maximal!valuations!in!which!a!proposition!is!true.!It!is!shown!that!classical!probabilities!arise!in!one!special!case!where!a!single!

probability!space!expresses!all!the!probabilities.!In!general!many!different!probability!spaces!are!required.!!!

!

Section!5!considers!classical!and!quantum!theories!of!mechanics.!Classical!mechanics!uses!a!logic!that!has!classical!probabilities,!while!quantum!theories!do!

not.!In!fact!any!nonclassical!logic!of!mechanics!has!the!peculiarities!of!quantum!

probabilities,!including!nonzero!assignments!to!incompatible!propositions,!drastic!changes!after!measurement!and!Belltype!entanglements.!!The!resulting!

logical!interpretation!of!quantum!theories!is!considered!in!conclusion,!section!6,!

including!a!brief!consideration!of!Hilbert!Space,!as!well!as!a!logical!reason!to!prefer!classical!theories.!


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2.#Logical#foundations#

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Formal!logic!is!derived!from!a!few!primitive!concepts.!These!include!a!formal!language!L!of!simple!propositions!p,!q,!!and!two!primitive!truthvalues!true!and!

false,!represented!by!t!and!f!respectively,!that!are!assigned!to!these!propositions!to!

express!their!meaning.!!The!notion!of!a!simple!valuation!is!derived!from!these!primitives.!

!

Defn:!(valuation!h,!set!H!of!logic!L)!!

i)!A!(simple)!valuation!h!is!a!structurepreserving!map!from!propositions!to!

truthvalues,!h:!L!=!{p,!q,!..}!>!{t,!f}.!!ii)!The!set!of!all!valuations!is!H!=!{h:!h!is!a!valuation!of!L}!!

!A!valuation!assigns!truthvalues!to!simple!propositions!in!a!structure!preserving!

way,!meaning!that!primitive!relations!over!L!are!respected!by!these!assignments.!

Valuations!of!simple!mechanical!propositions!are!discussed!in!section!5!below,!but!in!general!no!assumptions!are!made!about!relations!over!L.!!!

!Valuation!rules!defining!truthfunctional!connectives!extend!any!simple!valuation!h!

to!an!assignment!of!truthvalues!to!all!complex!propositions,!represented!by!well

formed!formulae!(wffs)!,!,!!!that!are!derived!from!simple!propositions!by!logical!connection.!For!simplicity!the!same!symbols!h!and!L!are!used!for!the!

extended!language!and!its!valuations,!so!h:!L!=!{,!,!!...}!>!{t,!f}.!!Logical!connectives!in!the!logic!L!derived!from!L!are!determined!by!Booles!Laws!of!

Thought!that!are!assumed!to!express!accepted!rules!of!reasoning.!Binary!

implication!therefore!has!the!properties!of!setinclusion,!which!means!that!

connective!!of!logic!L!generates!a!partial!ordering!on!the!Lindenbaum!algebra!A!

of!equivalence!classes!that!represent!it,!generated!by!the!biconditional.!Binary!

connectives!conjunction!and!disjunction,!,!!are!represented!by!lattice!meet!and!

join!respectively!of!this!algebra.!It!follows!from!Booles!laws!that!the!truthfunctional!logic!L!has!a!Lindenbaum!algebra!A!that!is!a!distributive!lattice.i!So!far!

this!account!of!formal!logic!is!entirely!traditional.!!

Unary!connectives!are!not!traditionally!treated!here,!because!it!is!not!assumed!that!

every!valuation!assigns!a!truthvalue!to!every!proposition.!This!assumption!is!not!justified!by!the!first!principles!of!logic!or!by!ordinary!reasoning.!A!language!L!of!

propositions!expresses!meaning!when!only!some!truthvalues!are!assigned,!just!as!in!ordinary!language.!For!example!I!can!describe!my!dog!by!claiming!Jack!is!fat!is!

true,!without!making!any!other!claims!about!his!colour!or!his!breed.!Quantum!

theories!too!use!descriptions!in!which!only!some!truthvalues!are!assigned.!For!these!theories!use!incompatible!magnitudes!with!corresponding!incompatible!

descriptions!of!reality!which!means!that!truthvalues!cannot!be!assigned!to!all!

their!propositions:!a!precise!position!and!a!precise!momentum!proposition!for!example!are!never!both!true!in!the!same!description.!Lastly!this!assumption!is!far!

too!strong!for!investigations!into!uncertainty!since!it!is!the!propositions!without!truthvalues!that!are!of!particular!interest,!for!example!because!these!are!assigned!

probabilities.!!


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For!all!these!reasons!it!is!not!assumed!here!that!every!valuation!of!logic!L!assigns!a!

truthvalue!to!every!proposition!in!the!language!L.!It!follows!that!logic!L,!like!

ordinary!reasoning,!can!distinguish!two!senses!of!not,!called!here!negation!!and!denial!~,!each!of!which!satisfies!a!distinct!Boolean!Law.!Negation!expresses!not!

in!the!sense!of!an!opposite!,!so!this!connective!is!defined!by!the!rule!that!a!

proposition!and!its!negation!always!take!opposite!truthvalues:!h(p)!=!t!iff!!

h(p)!=!f!for!any!h!in!H!and!p!in!L.!This!connective!satisfies!the!Law!of!Double!Negation!because!the!opposite!of!an!opposite!truthvalue!comes!back!to!the!

original:!p!!p!is!logically!true!in!L.!!Denial!~!on!the!other!hand!expresses!not!

in!the!sense!of!failure!to!be!true,!and!is!defined!by!the!rule!that!a!denial!is!true!

whenever!a!proposition!fails!to!be!true:!h(~p)!=!t!!iff!!h(p)!!t.!This!connective!satisfies!the!Law!of!Excluded!Middle!since!a!proposition!or!its!denial!must!always!

hold,!p!!~p!is!logically!true!in!logic!L!ii.!It!follows!that!!generates!an!involution!!

(an!orthogonal!operation)!on!the!Lindenbaum!algebra!A!of!logic!L,!while!~!is!

generates!a!lattice!complement!.iii!!!

!The!Lindenbaum!algebra!A!of!logic!Lis!a!therefore!a!distributive!lattice!with!

orthogonal!and!complement,!here!called!a!global!algebra.!A!Boolean!algebra!of!

course!is!a!distributive!lattice!with!single!orthogonalcomplement,!and!so!is!the!special!case!of!a!global!algebra!where!the!unary!operations!coincide.!A!global!

algebra!is!therefore!more!general!that!a!Boolean!algebra!but!as!discussed!above!is!also!Boolean!in!the!sense!that!it!expresses!Booles!Laws!of!Thought.!!!Elsewhere!

it!has!been!shown!that!a!global!algebra!is!always!relatively!Boolean!in!the!sense!

that!any!element!a!of!global!algebra!A!has!a!relative!orthogonalcomplement,!for!

the!orthogonal!of!a!is!also!a!lattice!complement!with!respect!to!a!subalgebra!of!A.iv!!

In!a!sense!the!connectives!of!L!are!traditional.!!!!

3.#Uncertainty#

!

Probabilities!are!introduced!in!the!next!section!as!special!degrees!of!uncertainty;!a!

logical!concept!expressing!how!close!a!proposition!is!to!being!certain.!In!this!section!the!modal!connectives!possible!and!certain!are!introduced!into!

propositional!logic!in!a!new!and!simple!way.!TruthLsystems!are!then!defined!and!shown!to!be!Boolean!fields!of!sets!so!that!finally!degrees!of!uncertainty!can!be!

introduced!as!welldefined!mathematical!measures!over!these!fields!of!sets.!!

!Modalities!are!logical!connectives!whose!valuation!rules!are!not!truthfunctional,!

which!means!their!value!in!any!valuation!h!depends!not!on!other!assignments!made!by!h!but!on!assignments!made!by!other!valuations!as!well.v!!The!simplest!

case!is!where!proposition!p!is!(simply)!possible!in!h,!defined!by!the!condition!that!

some!valuation!finds!p!true.!Proposition!p!is!(simply)!certain!in!h!when!all!valuations!in!H!find!p!true.!More!interesting!than!these!however!and!more!general,!

are!modalities!derived!from!some!binary!relation!R!on!H.!A!proposition!is!now!RhL

possible!when!some!valuation!related!by!R!to!h!finds!it!true,!and!RhLcertain!when!all!such!valuations!do.!

!


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Defn:!(Rpossible!MR,!Rcertain!LR)!

i)!Proposition!p!of!L!is!Rpossible!in!h!(or!Rhpossible)!in!the!extension!ML!

of!logic!L, h(MRp)!=!t!!iff!!h(p)!=!t!!for!some!h!in!H!such!that!h!R!h;!!

ii)!p!is!Rcertain!in!h!ML, h(LRp)!=!t!!iff!!h(p)!=!t!for!all!h!where!h!R!h.!!

!

The!simple!modalities!discussed!above!are!M = MRand!L = LRfor!trivial!relation!R!

over!H!that!relates!all!valuations.vi !In!general!either!one!of!i)!or!ii)!is!regarded!as!fundamental!with!the!other!modality!derived.!For!example!certainty!is!derived!from!possibility!by!setting!LR =df ~MR~, and so!only!one!connective!generates!a!modal!extension!of!logic!L. vi iThe!language!of!this!extension!includes!all!propositions!in!L!as!well!as!modal!wffs!of!form!MRfor!!in!L,!and!the!valuation!rule!i)!above!extends!any!h!in!H!of!logic!L!to!a!valuation!of!this!modal!extension!

MLas!well.!For!simplicity!however!symbols!h,!H!and!L!may!be!used!for!the!modal!

versions!as!well.!!

!

Degrees!of!uncertainty!will!be!numbers!expressing!how!close!a!proposition!p!is!to!being!RLcertain!in!h!and!will!be!defined!using!the!mathematical!theory!of!

measures,!to!measure!how!many!valuations!related!by!R!to!h!find!p!true.!For!example!the!RLuncertainty!of!p!in!h!is!1!when!all!the!measured!valuations!find!p!

true,!so!in!this!case!p!is!RLcertain!in!h;!if!this!uncertainty!is!0.5!then!half!the!related!

valuations!find!p!true,!and!if!it!is!0.37!then!37%!of!valuations!do!so;!and!where!p!is!RhLimpossible!with!a!Rhdegree!of!0!then!no!related!valuations!find!p!true.!!These!

measures!are!intuitive!but!care!is!needed!to!show!they!are!welldefined!measures!over!a!Boolean!field!of!sets.!!

!

An!appropriate!field!is!now!identified.!First!filters!of!the!Lindenbaum!algebra!A!of!L !are!shown!to!represent!valuations!of!L ,!and!second!truthLsystems!of!the!logic!L !are!constructed!to!represent!systems!of!maximal!valuations!that!find!each!

proposition!true.!Lastly!these!truthsystems!are!shown!to!be!a!Boolean!field!of!

sets.!The!following!definitions!are!standard,!apart!from!definition!iii):!!!

Defn:!!(filter,!maximal!filter,!exhaustive!filter,!ultrafilter)!

i).!A!filter!of!lattice!A!is!a!nonempty!subset!F,!F!!A,!where!i)!if!a!!b!!F!

then!a!!F!and!b!!F,!and!ii)!if!a!!F!and!a!!c!in!Athen!c!!F.!!

ii).!Filter!S!of!A!is!maximal!if!there!is!no!other!filter!S!of!A!such!that!S!!S!

iii)!Filter!S!of!global!algebra!A!=!!is!exhaustive!if!for!every!a!in!A,!either!a!!F!or!a!!!F.!

iv).!S!is!an!ultrafilter!of!Boolean!algebra!A!=!!if!for!every!a!in!A,!

either!a!!F!or!a!!F.!

!The!truthLset!of!any!valuation!contains!the!propositions!it!finds!true.!!

!Defn:!!(truthset!T!,!maximal,!exhaustive)!

i).!The!truthLset!of!h!T!!=!{!!L:!h()!=!t}.!!ii).!Truthset!Th!is!maximal!if!there!is!no!h!in!H!such!that!T! !T!".!In!this!case!we!can!also!say!valuation!h!is!maximal.!


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iii).!Truthset!Th!or!its!valuation!is!exhaustive!if!for!every!proposition!!

!in!L,!either!!or!!is!in!Th!i.e.!either!h()!=!t!or!h()!=!f!for!all!!in!L.!!

Maximal!valuations!are!those!with!maximal!truthsets:!no!other!valuation!makes!the!same!truthassignments!and!yet!makes!more.!A!valuation!and!its!truthset!are!

exhaustive!when!true!or!false!are!assigned!to!every!proposition.!!!

Truthsets,!valuations!and!filters!of!the!Lindenbaum!algebra!A!exactly!correspond.!!

It!is!easily!seen!that!any!valuation!determines!its!truthset.!Any!truthset!

determines!the!valuation!because!the!set!of!propositions!true!in!h!also!determines!

those!false!in!h!(the!negations!of!the!true!propositions),!and!these!sets!determine!the!propositions!left!without!a!truthvalue,!and!so!we!can!conclude!that!Th!=!Th!iff!!

h!=!h.!It!is!also!easy!to!appreciate!that!truthsets!and!filters!correspond!because!of!

their!properties.!For!example!for!any!h!in!H!the!set![Th]!=!{[]:!!!Th}!is!a!proper!subset!of!A!because!Th!is!a!proper!subset!of!L,!containing!the!logical!truths!of!L!but!

excluding!their!negations.![Th]!has!the!other!properties!of!a!filter!because!of!corresponding!logical!properties!and!so![Th]!=!F,!a!filter!of!the!algebra!Afor!any!

valuation!h.viii!To!show!each!filter!F!of!the!Lindenbaum!algebraArepresents!the!

truthset!of!some!valuation!h!in!H,!suppose!on!the!contrary!it!does!not,!so!F!![Th]!for!any!h!in!H.!Then!either!F!contains!equivalence!classes!of!contradictory!

propositions!or!else!it!fails!to!contain!classes!of!logical!consequences!of!

propositions!represented!in!F,!or!classes!of!their!conjuncts.!For!example!the!first!

case!would!mean!for!some!!in!L,![]!and!either![]!or![~]!is!in!F.!But!this!would!

mean!0!=![!!]!=![!!~]!is!also!in!F!by!the!property!of!filters,!and!so!any![]!is!

in!F!because!0!!!is!logically!true!in!L!for!any!!in!L!so!0!![]!and!any![]!in!A!is!

in!F,!which!contradicts!the!assumption!that!F!is!filter!and!thus!a!proper!subset!of!this!algebra.!So!no!contradiction!is!contained!in!F.!The!other!cases!are!easily!established!since!they!similarly!conflict!with!the!other!properties!of!filters!and!so!F!

=![Th]!after!all,!every!filter!of!A!represents!a!truthset!Th!of!some!h!in!H.!

!

So!truthsets,!valuations!and!filters!of!Aall!exactly!correspond!and!so!do!maximal!

filters,!maximal!valuations!and!maximal!truthsets.!It!follows!that!any!relation!R!

over!H!has!a!corresponding!relation!R!over!filters!of!A.!Note!that!exhaustive!filters,!

truthsets!and!valuations!are!always!maximal,!because!no!valuation!can!make!

more!truthassignments!than!one!that!assigns!truthvalues!to!every!proposition.!

However!maximal!filters,!truthsets!and!valuations!may!not!be!exhaustive!for!there!may!be!no!consistent!assignment!of!truthvalues!to!every!proposition!in!L.!Being!

maximal!is!therefore!more!general!than!being!exhaustive:!every!logic!has!maximal!valuations!but!these!are!only!sometimes!exhaustive.!

!

A!degree!of!uncertainty!is!to!measure!the!set!of!valuations!in!which!a!proposition!is!true,!and!by!discussion!above!this!will!be!represented!by!a!measure!of!the!set!of!

filters!on!algebra!Acontaining!this!equivalence!class.!Since!every!truthset!is!

contained!in!a!maximal!one,!just!as!every!filter!is!contained!in!a!maximal!one,!this!

measure!can!be!restricted!to!the!maximal!valuations!of!Land!hence!to!the!maximal!

filters!of!A.!TruthLsystems!of!logic!L!can!now!be!defined!as!systems!of!related!sets!

of!maximal!filters!containing!each!equivalence!class.!!!


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Defn:!(Rhtruthsystem!SRh!of!L, initial!condition!h)!

i)!For!h!!H,!R!a!binary!relation!over!H,!SRh!is!the!RhtruthLsystem!of!L!!

SRh!=!< S!!!",!,, >!for!S!!!"!=!{[Th]:!h!is!maximal,!h!R!h!&!h()!=!t},!!!

S!!!"!!S!!!"!=!S!!!!!" ,!and!S!!!"!!S!!!"!=!S!!!!!" ,!!and!!!S!!!"!=!S~!!!" .!!ii)!h!is!called!the!initial!valuation!(or!initial!condition)!of!truthsystem!S

Rh

.!!

Truthsystem!SRh!of!logic!L!is!the!system!of!sets!of!maximal!filters!related!by!R!to!h!

containing!each!equivalence!class,!representing!sets!of!maximal!valuations!related!

to!the!initial!condition!in!which!each!proposition!is!true.!!Operations!on!truth

systems!are!generated!by!operations!of!the!algebra!A!and!hence!by!the!logical!

connectives!and!these!are!now!shown!to!be!Boolean!set!operations.!!

!

Result!1:!!Each!truthsystem!SRh!of!logic!L!is!a!Boolean!field!of!sets.!

Proof!follows!from!properties!of!logical!connectives!generating!the!

operations!on!SRh

.!It!is!easy!to!see!that!the!binary!connectives!generate!set!operations!over!SRh!so!the!structure!is!a!distributive!lattice.!To!see!it!is!a!

Boolean!field!note!that!denial!generates!the!set!complement!operation!on!

these!systems.!This!connective!obviously!generates!lattice!complement!(because!it!satisfies!Excluded!Middle)!but!in!addition!generates!an!

involution!(orthogonal)!by!this!argument:!!

S~!!!" !=!!{[Th]:!h!R!h!&!h(~)!=!t}!(by!the!definition!of!S!!!")!!=!{[Th]:!h!R!h!&!h()!!t}!(by!the!definition!of!~)!!!

=!{[Th]:!h!R!h!&!h()!=!t!}!(by!the!definition!of!)!!!

=!!S!!!"!(by!the!definition!of!set!complement!and!S!!!").!

!

So!the!structure!of!the!algebra!Aof!logic!L!differs!from!that!of!its!truthsystems:!

the!logic!is!a!global!while!the!truthsystems!are!Boolean!algebras.!The!difference!

between!these!arises!because!denial!does!not!generate!an!involution!on!the!Lindenbaum!algebra!Aof!logic!Lbut!it!does!generate!an!involution!on!the!truth

systems.Denial!fails!to!do!so!on!the!logic!becausewhen!!has!no!truthvalue!~!is!

true,!so!~~!is!false!which!means!these!two!propositions!have!different!values!in!

some!valuations!and!so!are!not!logically!equivalent:!h((~~!!))!!t!for!all!h!in!H.!

It!follows!that![~~]!![]!on!Aand!so!the!operation!!representing!~!on!A!is!not!

an!involution,!a!!a.!On!the!truthsystems!however!things!are!different.!Wff!!is!

true!in!h!if!and!only!~!fails!to!be!true!in!h,!and!so!the!denial!of!this!denial,!~~,!

must!be!true.!That!is,!h()!=!t!iff!!h(~)!!t!!iff!!h(~~)!=!t!and!so!!!Th!!iff!!

~~!!Th!which!means!that!!S~~!!!" = !S!!!"!and!denial!generates!a!full!set!complement!on!any!truthsystem,!!S

!!!"!=!S~!!!" !.!Each!truthsystem!is!a!Boolean!algebra!even!though!the!algebra!Aof!logic!L!is!not.!!

!At!last!numerical!uncertainties!can!be!introduced!as!welldefined!measures!over!

these!truthsystems.!!

!

Defn:!(degree!of!uncertainty!degRh(),!Rh)!

For!any!!of!logic!L!and!truthsystem!SRh!the!(degree!of)!uncertainty!of!!

given!R!and!initial!h,!degRh()!!=!!!!!"(S!!!")!where!!!!"!is!a!measure!over!SRh.!


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!

Degrees!of!uncertainty,!or!uncertainties,!are!measures!over!the!truthsystems!of!the!set!of!maximal!filters!containing!an!equivalence!class,!which!represents!a!measure!

of!the!maximal!valuations!of!logic!L!in!which!these!propositions!are!true.!By!Result!

1!above!these!are!welldefined!mathematical!measures,!because!the!truthsystems!

over!which!they!are!defined!are!Boolean!fields!of!setsix.!!!

Uncertainties!can!be!expressed!in!terms!of!propositions!or!of!valuations!because!of!

a!natural!correspondence!between!these.!The!characteristic!proposition!of!valuation!h!is!logically!equivalent!to!the!conjunction!of!all!propositions!true!in!h.!

The!characteristic!valuation!of!a!wff!finds!this!proposition!true!and!every!logical!

consequence!of!it!true!but!makes!no!other!truthassignments.!!!

Defn:!(characteristic!wff!h,!characteristic!valuation!h)!

i).!For!any!valuation!h!in!H,!wff!h!in!L!is!the!characteristic!proposition!for!h!

if!!h!=!!where!!!(!i!)!is!logically!true,!for!any!i!!Th!ii).!For!any!wff!!of!logic!L,!h!in!H!is!the!characteristic!valuation!for!!if!!

!!!=!{:!!!!is!logically!true!in!L!}.!

!

This!correspondence!means!that!any!relation!R!over!H,!which!is!also!relation!R!

over!the!filters!of!A!,!generates!a!corresponding!relation!R!over!L:!!!R!!iff!!!h!R!h.!

Uncertainties!can!now!be!expressed!in!terms!of!either.!

!

Defn:!!(degR(),!degRh(h))!

i)!Uncertainty!in!terms!of!wffs:!!degR()!=df!!!!!!(S!!!!)!=!!!!"!(S!!!"!).!!ii)!Uncertainty!in!terms!of!valuations:!degRh(h)!=df!!degRh(!").!

!

The!uncertainty!with!initial!proposition!has!its!characteristic!valuation!as!initial!condition,!while!the!uncertainty!of!a!valuation!is!the!uncertainty!of!its!

characteristic!proposition.!!!

4.#Probabilities#

!

Probabilities!are!special!uncertainties!in!a!logic!whose!propositions!describe!reality.!No!metaphysical!assumptions!are!introduced!about!the!nature!of!reality!or!

of!the!descriptive!relation!between!language!and!reality,!or!about!the!nature!of!probabilities!beyond!their!logical!definition.!The!probability!of!a!proposition,!given!

some!initial!condition,!expresses!how!close!this!description!is!to!being!certain!in!

any!subsequent!description!of!the!same!reality.!This!is!an!uncertainty!measuring!how!many!maximal!valuations!that!might!next!describe!the!same!reality!as!initial!h,!

find!this!proposition!true.!So!probabilities!are!uncertainties!generated!by!a!successor!relation!over!H!that!relates!each!valuation!h!to!other!maximal!

valuations!that!might!next!describe!the!same!reality.!!Logical!properties!of!the!

successor!relation!are!suggested!that!generates!classical!probabilities!in!the!special!case!of!a!classical!logic,!as!well!as!general!probabilities!with!the!

characteristic!quantum!peculiarities.!Some!new!terminology!is!useful!to!this!discussion.!


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!

Defn:!(weak,!strong!consistency,!classical,!bivalent,!context,!(in)compatible)!!

For!h,!h!in!H,!proposition!p!and!wffs!,!!in!L!

i).!h!is!weakly!consistent!with!h,!h!W!h,!if!there!is!no!!in!L!such!that!!

h()!=!t!and!h()!=!f.!h!is!strongly!consistent!with!h,!h!!h!if!Th!!Th.!!ii).!The!context!of!h!Ch!=!{:!h()!=!t!or!h()!=!f}.!iii)!Logic!Lis!classical!if!all!its!maximal!valuations!are!exhaustive,!and!is!

bivalent!if!all!valuations!are!exhaustive.!

iv)!Propositions!,!!are!compatible!if!there!is!an!h!in!H!such!that!,!!!Ch,!and!they!are!incompatible!if!there!is!no!such!valuation!h!in!H.!!!

!One!valuation!is!weakly!consistent!with!another!if!it!does!not!contradict!it,!i.e.!does!

not!assign!a!conflicting!truthvalue.!It!is!strongly!consistent!when!it!makes!at!least!

the!same!assignments!of!truthvalues,!in!which!case!we!may!also!say!it!agrees!with!or!includes!the!other.!The!context!of!any!valuation!contains!the!propositions!it!

assigns!a!truthvalue.!In!a!classical!logic!all!the!maximal!valuations!assign!truthvalues!to!all!the!propositions,!while!in!a!bivalent!logic!all!the!valuations!do!so.!This!

distinction!is!not!traditional!because!classical!logic!is!traditionally!assumed!to!be!

bivalent.!However!this!distinction!and!terminology!are!justified!later!when!classical!logic!is!shown!to!have!classical!probabilities.!Two!propositions!are!

compatible!if!the!same!valuation!assigns!them!each!a!truthvalue!and!incompatible!

if!no!valuation!does!so.!!!!

The!following!Principle!generates!the!fundamental!definition!of!a!probability.!!

Principle!of!Non!Contradiction:!Probabilities!are!uncertainties!derived!from!

weak!consistency.!!!

Defn:!(probability!probh(),!probability!space!Ph)!

i)!For!any!h,!in!H!!in!L!of!logic!L,!the!(general)!probability!of!!given!initial!

h,!probh()!=!!degWh()!where!W!is!weak!consistency.!!

ii)!The!hLprobabilityLspace!Ph!of!logic!L!is!the!successor!truthsystem!with!

initial!condition!h,!Ph!=!SWh!

!

A!probability!measures!the!maximal!valuations!that!do!not!contradict!initial!truthassignments!and!in!which!a!proposition!is!true.!Since!the!same!relation!W!is!used!

for!all!probabilities!this!may!be!omitted!in!the!formalism.!!

Traditionally!we!assume!that!probabilities!are!based!on!strong!consistency!for!it!is!

assumed!that!all!the!valuations!measured!in!a!probability!agree!with!initial!truthvalues.!This!turns!out!to!be!a!suitable!assumption!in!classical!logic!but!in!general!it!

is!not.!For!where!propositions!may!be!incompatible!and!so!maximal!valuations!are!not!exhaustive,!then!if!probabilities!were!to!measure!only!valuations!that!agree!

with!an!initial!valuation,!other!maximal!valuations!assigning!truthvalues!to!

descriptions!incompatible!with!the!initial!condition!would!be!ignored,!which!is!clearly!inappropriate.!The!weaker!relation!is!used!in!this!general!case!and!this!

turns!out!to!coincide!with!strong!consistency!in!any!logic!that!is!classical.!For!where!all!maximal!valuations!are!exhaustive!they!must!always!agree!with!any!


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initial!truthassignment!they!do!not!contradict,!and!so!it!turns!out!that!the!

successor!in!any!classical!logic!is!strong!consistency.!Our!traditional!assumption!that!probabilities!measure!valuations!that!agree!with!the!initial!condition!is!

therefore!preserved!in!a!logic!that!is!classical.x!!In!fact!classical!logic!generates!probabilities!that!are!entirely!classical.!To!show!this!we!first!define!for!any!logic!L!

its!basic!probability!space,!which!is!generated!from!all!maximal!valuations.!!!

Defn:!(trivial!valuation,!basic!probability!space)!

i).!The!trivial!valuation!h0!of!any!logic!L!is!the!valuation!included!in!every!

other,!i.e.!h0!!h!for!all!h!in!H!(and!so!T!! !=!{!in!L:!!is!logically!true}).!

ii)!The!basic!probabilityLspace!P!of!logic!L!is!that!which!has!h0!as!initial!

condition,!i.e.!P!=!Ph!for!h!=!h0.!

!

This!basic!probability!space!is!generated!from!a!trivial!initial!condition.!This!has!a!special!role!in!classical!logic.!

!Result!2:!A!classical!logic!L!has!classical!probabilities!in!the!sense!that!every!

probability!can!be!expressed!in!terms!of!measures!over!the!basic!probability!space.!

Proof:!In!a!classical!logic!L!the!successor!relation!between!any!initial!

valuation!and!a!maximal!one!coincides!with!agreement,!as!discussed!above.!

To!show!the!resulting!probabilities!are!classical!requires!showing!they!are!

all!measures!over!the!basic!probability!space!Pof!classical!logic!L.!This!

follows!because!for!any!wffs!,!,!in!classical!L,!!

prob()!=!deg!()!(because!in!this!special!case!W!=!!)!!=!!

!

(!!!

)!(by!the!definition!of!deg)!!=!!!(!

!!)!(by!properties!of!)!!=!(S

!!)!/!(S!)!(by!properties!of!!!!where!!is!a!measure!over!the!basic!truthsystem!S = P).!!

!The!traditional!account!of!classical!probabilities!is!almost!entirely!retained!in!this!

special!case:!probabilities!are!measures!of!sets!of!exhaustive!filters!in!the!algebra!A

of!L!that!contain!an!equivalence!class,!representing!measures!over!the!maximal!

valuations!in!L!which!find!these!and!all!initial!propositions!true.!The!only!

departure!from!tradition!is!that!the!basic!probability!space!P !of!classical!logic!L!does!not!represent!the!Lindenbaum!algebra!A!of!L. The!exhaustive!filters!

correspond!to!ultrafilters!of!an!associated!Boolean!algebra!A * derived!from!logic!

L, but!this!algebra!represents!a!bivalent!version!of!Lnot!this!logic!itself.xi

It!was!earlier!argued!that!any!logic!L!based!on!Boolean!laws!has!a!distributive!

lattice!of!equivalence!classes!with!an!involution!and!complement,!a!slight!

generalisation!of!a!Boolean!algebra.!!Both!classical!and!nonclassical!logic!also!have!

truthsystems!that!are!Boolean!fields!of!sets,!including!the!basic!probability!space!system!P !generated!from!all!its!maximal!valuations.!Any!Boolean!field!is!isomorphic!to!a!Boolean!algebra,!a!distributive!lattice!with!orthogonal

complement.xii!It!follows!that!in!both!classical!and!nonclassical!logic!the!algebra!A!

of!L!is!distinct!from!its!basic!probability!space!P .!!!


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!

The!key!difference!between!classical!and!nonclassical!logic!is!that!only!the!former!has!all!propositions!compatible,!all!maximal!valuations!exhaustive,!and!so!only!this!

logic!can!use!its!basic!probability!space!P !to!generate!all!its!probabilities.!In!general!Result!2!does!not!hold!and!so!in!general!the!basic!probability!space!P !has!no!special!significance.!In!general!logic!L!does!not!have!maximal!exhaustive!

valuations,!succession!is!weak!consistency!and!many!different!truthsystems!

generate!its!probabilities,!each!associated!with!a!different!initial!condition.!!!

Any!probability!in!any!logic!L!is!a!welldefined!mathematical!measure!over!a!

Boolean!field!of!sets.!However!these!form!different!systems!in!classical!or!general!

logic:!classical!logic!can!express!all!it!probabilities!over!a!single!probability!space!

while!in!general!a!logic!cannot.!In!general!a!logic!requires!a!family!of!different!spaces,!since!the!probability!space!used!to!generate!a!probability!depends!on!its!

initial!condition.!In!the!next!section!it!is!shown!that!this!conditional!nature!of!

general!probabilities!is!what!gives!rise!to!the!peculiarities!characteristic!of!quantum!probabilities.xiii!!

!

5.#Probabilities#in#mechanics#

!

The!simple!propositions!of!any!mechanical!theory!share!a!common!form.!Each!simple!mechanical!proposition!associates!a!Borel!subset!of!real!numbers!with!a!

magnitude!of!the!theory:!p!=!(m,!)!for!m!a!magnitude!of!the!mechanical!theory!T!

and!!a!Borel!subset!of!values!of!this!magnitude,!m!!MT!and!!!Vm.!!Informally!p!asserts!The!value!of!m!lies!in!.!Each!magnitude!m!thus!generates!a!family!of!mL

propositions!in!the!language!LT!of!theory!T.!!Simple!propositions!of!form!(m,!r)!for!r!a!real!number!in!Vm!are!called!atomic!propositions.!!

!Valuations!of!LT!preserve!the!primitive!relations!generated!by!set!operations!on!

each!valueset!Vm!and!these!are!expressed!by!the!logical!connectives.!Implication!

expresses!set!inclusion!on!Vm!so!if!p!=!(m,!),!p!=!(m,!)!and!!!,!then!(p!!p)!is!logically!true!in!any!logic!LT!of!mechanical!theory!T.!Informally!this!expresses!our!assumption!that!When!the!value!of!m!is!in!!and!!!!then!the!value!of!m!is!in!

.!!Disjunction!and!conjunction!similarly!express!set!union!and!set!intersection!on!Vm.!Negation!represents!the!set!complement,!for!when!p!=!(m,!)!is!true,!then!

p!=!(m,!Vm!!)!is!false,!expressing!our!assumption!When!the!value!of!m!is!in!!this!value!does!not!lie!outside!this!set!and!so!is!not!in!Vm!!.!It!follows!that!when!

any!atomic!mproposition!is!found!true!in!any!h,!all!other!atomic!mpropositions!

will!be!false!in!h!and!each!mproposition!has!a!truthvalue.!!!!

Probabilities!in!mechanical!theories!are!always!expressed!as!conditional!on!the!physical!measurement!of!a!magnitude,!assumed!to!be!some!procedure!that!is!in!

principle!describable,!assigning!a!proper!subset!of!values!to!the!measured!

magnitude.!A!physical!measurement!of!magnitude!m!thus!finds!some!proper!mproposition!true,!p!=!(m,!)!in!LT!for!!!Vm.!The!least!such!mproposition!with!

respect!to!set!inclusion!among!the!valuesets!is!called!the!measurement!outcome.!

An!ideally!accurate!physical!measurement!is!one!that!has!an!atomic!proposition!as!outcome,!so!after!this!truthvalues!are!assigned!to!all!the!mpropositions.!


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!

aws!of!a!mechanical!theory!establish!relations!among!its!magnitudes!and!these!generate!relations!among!the!simple!propositions!that!are!preserved!by!

valuations.!Since!the!laws!of!classical!and!quantum!theories!differ,!so!too!do!primitive!relations!among!their!simple!propositions!and!so!too!does!the!logic!of!

these!two!different!kinds!of!theory.!Classical!mechanics!can!describe!a!real!system!by!assigning!precise!values!to!every!magnitude,!which!means!its!maximal!valuations!assign!truthvalues!to!all!the!propositions,!and!so!the!maximal!

valuations!of!the!logic!used!by!these!theories!are!always!exhaustive.!The!logic!of!

classical!mechanics!is!therefore!classical.!Quantum!mechanics!on!the!other!hand!has!incompatible!magnitudes!that!generate!incompatible!propositions!and!these!

cannot!all!be!assigned!truthvalues!in!a!single!valuation.!This!means!that!maximal!valuations!of!quantum!theories!are!not!exhaustive!and!so!their!logic!is!not!

classical.!We!say!in!this!case!that!the!theory!too!is!nonclassical.!!

!It!follows!from!earlier!discussion!that!the!propositional!logic!of!both!classical!and!

quantum!theories!is!the!same!logic!L!whose!Lindenbaum!algebra!A!of!equivalence!

classes!is!a!distributive!lattice!with!involution!and!complement,!a!global!algebra.!

However!the!maximal!valuations!of!these!logics!differ,!since!in!the!logic!of!classical!mechanics!these!are!always!exhaustive!while!in!the!logic!of!quantum!mechanics!

they!are!not.!As!discussed!in!the!last!section!this!means!that!their!probabilities!

differ,!because!by!Result!2!classical!theories!use!probabilities!generated!from!a!single!probability!space!while!quantum!theories!do!not.!!In!fact!any!nonclassical!

theory!of!mechanics!has!probabilities!with!the!peculiarities!of!quantum!theory:!

these!probabilities!are!irreducibly!statistical;!may!be!nonzero!even!when!corresponding!joint!probabilities!are!zero;!alter!drastically!after!physical!

measurement;!and!exhibit!nonclassical!entanglements!in!the!sense!that!Belllike!classical!inequalities!fail!in!a!nonclassical!sequence!of!probabilities.!Contradictory!

descriptions!may!even!be!used!in!describing!a!sequence!of!measurements.!Each!

peculiarity!is!now!briefly!discussed.!!!

Irreducibly!statistical!probabilities!arise!in!any!nonclassical!theory.!For!since!the!

maximal!valuations!are!not!exhaustive!it!follows!that!there!must!be!propositions!p,!q!in!the!logic!where!p!is!true!in!maximal!valuation!h!but!q!has!no!truthvalue!in!h:!

for!if!there!were!no!such!pair!of!propositions!the!logic!would!be!classical.!Suppose!then!that!h!assigns!certainty!to!p,!probh(p)!=!1.!This!means!all!maximal!successor!

valuations!of!h!find!p!true.!Since!q!has!no!truthvalue!in!h,!and!h!is!weakly!

consistent!with!itself,!it!follows!that!q!is!not!true!in!every!successor!of!h!and!therefore!is!not!certain!in!h,!i.e.!probh(q)!!1.!But!neither!is!q!impossible!according!

to!h,!for!although!q!is!not!true!in!h!neither!is!it!false!in!h!and!so!q!may!be!true!in!a!weakly!consistent!maximal!successor!h!of!h!that!makes!no!truthvalue!assignment!

to!p.!It!follows!that!the!probability!of!q!according!to!h!is!not!zero,!probh(q)!!0.!So!

nonclassical!conditional!probabilities!do!not!assign!probabilities!of!1!or!0!to!every!proposition!but!are!instead!irreducibly!statistical.!!

!

When!a!logic!is!not!classical,!nonzero!conditional!probabilities!arise!where!corresponding!joint!probabilities!are!always!zero.!For!by!definition!in!nonclassical!

logic!there!are!incompatible!propositions!p,!q!that!share!no!common!context,!and!so!p,!q,!cannot!both!be!true!in!some!maximal!valuation!h,!which!means!their!joint!


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probability!is!always!zero,!probh(p!!q)!=!0!for!all!h!in!H.!However!the!

corresponding!conditional!probability!probp(q)!measures!the!maximal!successors!of!a!valuation!characteristic!for!p!that!find!q!true!and!this!is!generally!a!nonempty!

set!by!the!reasoning!above:!a!valuation!h!may!be!weakly!consistent!with!hp!yet!find!q!true,!and!so!the!conditional!probability!probp(q)!!0.!This!key!difference!

between!classical!and!nonclassical!probabilities!shows!that!that!nonclassical!conditional!probabilities!are!not!expressed!in!terms!of!corresponding!joint!probabilities!as!classical!conditionals!are.!Instead!these!are!measures!over!a!

probability!space!that!depends!on!the!initial!condition.!!

!The!probabilities!of!nonclassical!theories!of!mechanics!also!depend!on!the!

magnitude!of!the!initial!physical!measurement.!Magnitudes!m!and!n!of!theory!T!are!incompatible!when!their!corresponding!simple!propositions!are,!and!these!

generate!probabilities!that!depend!on!an!initial!physical!measurement.!Let!

probm(p)!be!the!probability!of!p!given!an!initial!physical!measurement!of!magnitude!m,!so!!probm(p)!=!probq(p)!!where!q!is!some!outcome!of!measuring!m.!It!

is!easy!to!see!that!in!general!probm(p)!!!probn(p)!in!any!nonclassical!logic!of!

mechanical!propositions.!For!on!the!left!is!the!probability!of!proposition!p!given!a!physical!measurement!of!magnitude!m,!which!is!a!mathematical!measure!of!the!set!

of!maximal!valuations!weakly!consistent!with!an!mLoutcome!that!find!p!true.!On!the!right!however!is!the!probability!of!p!given!a!physical!measurement!of!n,!a!

mathematical!measure!of!the!set!of!maximal!valuations!weakly!consistent!with!an!

nLoutcome!that!find!p!true.!These!are!mathematical!measures!of!distinct!sets!in!distinct!truthsystems!and!so!these!probabilities!are!distinct.!Only!in!the!special!

case!of!a!classical!logic,!where!all!maximal!valuations!assign!truthvalues!to!all!mechanical!propositions,!will!these!two!probabilities!be!mathematical!measures!of!

the!same!set!and!hence!will!always!coincide.!In!general!the!probabilities!are!measurementdependent.!!!

By!similar!reasoning!drastic!changes!in!valuation!generally!occur!after!physical!

measurement!in!a!nonclassical!theory!of!mechanics.!For!a!physical!measurement!of!magnitude!m!must!result!in!some!moutcome!p!=!(m,!)!being!true,!for!!!Vm!

and!therefore!after!such!a!measurement!the!system!is!described!by!valuation!h!where!h(p)!=!t!for!some!such!p.!!Since!proposition!p!may!not!have!been!true!in!the!

initial!valuation!h,!it!follows!that!h!is!distinct!from!h!which!is!to!say!that!the!

description!of!the!real!system!has!changed.!Indeed!if!a!physical!measurement!of!magnitude!n!incompatible!with!m!next!takes!place,!then!there!must!be!drastic!

change!of!description.!For!now!an!noutcome!is!true!of!the!system!that!could!not!have!been!true!in!h,!because!m!and!n!are!incompatible!and!mproposition!p!was!

true!in!h.!So!the!new!valuation!h!finds!this!noutcome!q!true!while!the!previous!

valuation!h!does!not,!and!so!these!two!valuations!are!radically!distinct.!This!is!a!drastic!change!imposed!solely!by!logical!features!of!the!theory!that!is!irrespective!

of!any!physical!considerations!about!the!system!or!its!measurement.!

!A!kind!of!entanglement!of!probabilities!evident!in!quantum!theories!similarly!

occurs!in!any!nonclassical!logic!of!mechanics.!!Let!+!and!!be!values!up!and!down!of!spin!in!each!of!the!directions!a,!b!and!c,!and!consider!a!sequence!of!

measurements!of!these!three!magnitudes.!Simple!propositions!describing!this!system!are!pairs!such!as!(a,!+),!(b,!),!(c,!+),!informally!understood!as!spin!in!


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direction!a!is!up,!spin!in!direction!b!is!down!and!!spin!in!direction!c!is!up!

respectively.!If!these!are!propositions!of!a!classical!logic,!then!the!probability!of!any!sequence!of!measurement!outcomes!is!expressed!in!terms!of!a!joint!

probability!on!the!basic!probability!space!P:!for!example!the!probability!that!

direction!c!is!up!on!a!system!where!b!was!down!after!a!was!initially!up!is!expressed!

in!terms!of!!prob((a,!+)!!(b,!)!!(c,!+))!expressed!by!a!corresponding!set!intersection!on!the!basic!probability!space.!Other!expressions!also!hold!aong!the!

classical!probabilities!because!they!also!express!set!operations!on!P,!for!example!

prob(a,!+)!=!prob((a,!+)!!(b,!+))!+!prob((a,!+)!!(b,!)),!informally!!Any!a+!event!is!

also!either!b+!or!b.!From!similar!expressions!for!other!magnitudes,!Belltype!

inequalities!are!derived,!for!example!prob((a,!+)!!(b,!)!!!prob(a,!+).!These!hold!

in!any!classical!logic!along!with!other!similar!inequalities!involving!all!three!pairs!of!values.!These!expressions!all!seem!almost!selfevident!because!they!express!set!

relations!on!P!that!informally!express!relations!among!properties!on!classical!

event!space.!!

!However!such!relations!and!the!inequalities!derived!from!them!fail!in!quantum!

mechanics,!a!failure!often!called!quantum!entanglement.!!It!is!easy!to!see!these!do!

not!hold!in!a!nonclassical!logic!of!mechanics!because!here!set!relations!on!the!basic!probability!space!P!do!not!express!conditional!probabilities.!!In!general!the!

magnitudes!like!spin!in!direction!a,!b!and!c!are!incompatible,!as!they!are!in!quantum!theory,!and!so!their!simple!propositions!are!also!incompatible,!with!zero!

joint!probabilities:!prob((a,+)!!(b,!))!=!0!!and!this!is!not!prob(a,!+)(b,!).!!In!a!nonclassical!logic!the!equality!that!was!derived!from!set!operations!on!classical!P!fails:!

prob(a,!+)!!prob((a,!+)!!(b,!+))!+!prob((a,!+)!!(b,!))!because!proposition!(a,!+)!

may!be!true!while!both!propositions!(b,!+)!and!(b,!)!lack!a!truthvalue.!It!follows!that!inequalities!derived!from!these!equations!also!fail,!so!for!example!prob((a,!+)!

!(b,!))!!!prob(a,!+)!in!a!logic!that!is!not!classical.!!However!this!failure!does!not!

express!entanglement!of!properties,!for!in!nonclassical!logic!true!propositions!no!longer!correspond!to!properties!nor!do!maximal!valuations!express!events.!

Proposition!(a,!+)!can!be!true!but!this!is!not!a!property!in!the!classical!sense!of!partially!expressing!an!event!because!as!we!have!seen,!!maximal!valuations!are!

not!exhaustive!so!neither!(b,!+)!nor!(b,!)!may!have!truthvalues,!and!the!informal!

assumption!that!Each!a+!event!is!either!b+!or!b!does!not!hold.!!!!

In!nonclassical!theories!a!sequence!of!outcomes!such!as!(a,!+),!(b,!),!(c,!+)!is!

expressed!not!by!set!relations!on!one!basic!probability!space,!but!instead!by!combining!two!distinct!probabilities!defined!over!two!different!probability!spaces:!

first!prob(a,+)(b,)!that!b!is!down!after!a!was!initially!up,!and!second!!prob(b,)(c,+)!!that!c!is!found!up!on!a!system!where!b!was!initially!down.!!These!are!

two!different!measures!over!two!different!truthsystems!spaces,!and!their!

combination!therefore!does!not!correspond!to!set!combination!on!any!probability!space.!In!this!sense!there!is!no!entanglement!of!properties!in!a!mechanical!

theory!that!is!not!classical,!just!a!combination!of!distinct!probability!measures!that!cannot!be!expressed!on!a!single!probability!space.!

!

Contradictory!probability!assignments!can!arise!in!such!a!sequence!of!!nonclassical!probabilities!even!though!the!Principle!of!Noncontradiction!is!used!at!

each!stage!of!the!calculation.!Because!weak!consistency!is!not!a!transitive!relation!


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a!probability!early!in!a!sequence!may!be!contradicted!by!one!that!is!later,!just!as!

we!find!in!quantum!theories.!!Suppose!again!that!proposition!(a,!+)!is!found!true!after!a!measurement!of!a.!This!outcome!is!certain!if!the!measurement!of!magnitude!

a!is!immediately!repeated!in!which!case!it!is!also!impossible!that!spin!in!direction!

a!has!value!down:!for!prob(a,+)(a,!+)!=!1!and!prob(a,+)(a,!)!=!prob(a,+)(a,!+)!=!0!

follow!from!the!definitions.!However!if!instead!of!repeating!this!measurement!an!intervening!measurement!of!an!incompatible!magnitude!takes!place,!then!value!

down!may!now!be!found.!For!probabilities!are!now!conditional!on!the!outcome!of!

this!measurement,!so!if!magnitude!b!is!measured!after!the!initial!measurement!of!a,!with!outcome!up!so!(b,!+)!is!now!true!then!the!probability!of!another!

measurement!of!a!finding!down!in!this!direction!is!no!longer!0,!for!prob(b,+)(a,!)!!

0,which!means!that!even!though!(a,!)!was!impossible!after!the!initial!measurement!of!a,!prob(a,+)(a,!)!=!0,!and!even!though!an!immediate!remeasuring!must!find!

value!up!for!this!magnitude,!an!intervening!measurement!of!b!changes!the!situation!and!now!value!up!for!a!is!not!certain,!and!value!down!is!not!impossible.!

Quantum!systems!with!correlated!spin!values!are!known!to!exhibit!such!a!loss!of!correlation!after!intervening!measurements!and!fluctuations!of!quantum!values!are!also!well!known.!These!are!features!of!any!nonclassical!mechanical!theory.!!

!

6.#Conclusion#

!

A!general!definition!of!probability!has!been!derived!from!logical!first!principles,!

that!has!quantum!peculiarities!in!general!and!classical!probabilities!as!a!special!case.!This!definition!preserves!traditional!deduction!and!also!the!traditional!

understanding!of!probabilities!as!mathematical!measures!over!a!Boolean!field!of!

sets,!measures!of!sets!of!maximal!valuations!that!find!a!proposition!true.!!!

It!is!not!assumed!here!that!logic!is!bivalent!which!means!the!role!of!Boolean!algebras!in!logic!was!carefully!reassessed.!Bivalence!is!not!supported!by!logical!

first!principles,!by!our!use!of!ordinary!language,!or!by!quantum!theories,!since!

these!manifestly!use!incompatible!descriptions!that!cannot!be!assigned!truthvalues!together.!Assuming!bivalence!is!also!inappropriate!for!the!logical!

foundations!of!uncertainty!since!probabilities!are!assigned!to!propositions!when!truthvalues!are!lacking.!Both!classical!and!quantum!theories!of!mechanics!

therefore!use!the!same!nonbivalent!logic,!where!negation!is!distinguished!from!

denial.!However!bivalent!logic!that!satisfies!Booles!Laws!of!thought!is!a!Boolean!algebra,!and!this!is!isomorphic!to!its!Boolean!field!of!ultrafilters,!which!forms!a!

natural!event!space.!Without!bivalence!this!correspondence!between!the!

structure!of!a!logic!and!of!its!probability!space!fails.!This!breakdown!is!perhaps!the!reason!bivalence!is!usually!assumed!to!hold!in!logic!even!when!this!assumption!is!

inappropriate.!!

Boolean!nonbivalent!logic!was!distinguished!here!from!its!probability!spaces.!The!

Lindenbaum!algebra!of!nonbivalent!logic!L!is!a!slight!generalisation!of!a!Boolean!

algebra,!a!distributive!lattice!with!orthogonal!and!complement.!Probabilities!of!L!

however!are!derived!from!truthLsystems!of!this!logic!that!are!Boolean!fields!of!

maximal!filters!of!this!algebra!containing!each!equivalence!class.!This!preserves!


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our!traditional!understanding!of!probabilities!as!measures!over!a!Boolean!field!of!

maximal!valuations!of!the!logic,!in!which!a!proposition!is!true.!!!

Though!classical!and!quantum!theories!of!mechanics!share!propositional!logic!L!

and!use!the!same!definition!of!a!probability,!differences!arise!because!of!

differences!in!their!maximal!valuations.!The!logic!of!classical!mechanics!has!all!magnitudes!and!propositions!compatible!while!quantum!theories!do!not.!!Maximal!

valuations!of!classical!logic!thus!assign!truthvalues!to!all!the!propositions,!their!

successor!relation!is!therefore!strong!consistency!and!a!single!probability!space!can!express!all!the!probabilities.!However!the!maximal!valuations!of!the!logic!of!

quantum!theories!do!not!assign!truthvalues!to!all!the!propositions,!their!successor!

relation!is!weak!consistency!and!different!probability!spaces!generally!express!probabilities!with!different!initial!conditions.!This!strongly!conditional!nature!of!

general!probabilities!gives!rise!to!their!characteristic!peculiarities.!!

This!difference!also!means!classical!and!nonclassical!theories!of!mechanics!require!different!mathematical!expression.!!Classical!probabilities!can!all!be!expressed!on!the!basic!probability!space!P!which!therefore!provides!a!natural!

representation!of!the!theories:!maximal!valuations!correspond!to!precise!values!for!every!magnitude!that!can!be!represented!as!events!in!an!event!space!or!

points!in!phase!space.!Probabilities!measure!this!space!and!so!operations!among!these!probabilities!are!represented!by!set!operations!on!P!or!correspondingly!by!

logical!connection.!For!example!the!logical!combination!of!probabilities!expressed!by!classical!valuations!hp,!hq!can!be!defined!in!terms!of!a!corresponding!logical!

combination!of!valuations,!where!for!example!hp!!hq!=df!!hpq!and!!hp!!hq!=df!!hpq!!

where!T!!!!!!=!T!!! !T!!! !and!T!!!!!!=!T!!! !T!!! ,!or!by!corresponding!set!relations!among!the!maximal!filters,!Sp!!Sq,!Sp!!Sq!!on!P.!!Quantum!theory!too!can!combine!

its!valuations!and!its!probability!measures!in!this!way!when!they!are!expressed!on!the!same!probability!space.!However!in!addition!any!general!theory!of!mechanics!

has!another!operation!of!probability!generation!derived!from!the!conditional!

probabilities!that!cannot!be!so!expressed.!For!example!operation!!might!be!defined!by!setting!!hp!!hq!=df!!!probp(q),!and!although!in!any!classical!logic!this!can!

be!expressed!in!terms!of!logical!combination!(by!Result!2!of!section!4),!in!quantum!

theory!it!cannot.!!Classical!representations!in!terms!of!event!space!or!phase!space!are!inadequate!for!any!mechanical!theory!without!exhaustive!maximal!

valuations,!and!a!new!representation!is!required.!!

!Detailed!discussion!of!Hilbert!Space!lies!beyond!the!scope!of!this!paper,!but!in!

principle!an!inner!product!vector!space!seems!appropriate!for!this!representation.!!!Such!a!space!has!two!fundamentally!different!operations!among!vectors!

representing!its!probability!descriptions.!The!linear!combination!of!vectors,!generated!by!vector!addition!and!scalar!multiplication!express!a!kind!of!logical!

combination.!But!in!addition!the!operation!of!inner!product,!by!Borns!Projection!

Postulate!can!express!probability!generation.!!!

The!onedimensional!vectors!or!rays!of!a!Hilbert!Space!representing!a!quantum!

theory,!are!eigenvectors!in!the!basis!of!an!observable!operator!that!represents!a!magnitude!of!this!theory.!Eigenvalues!corresponding!to!the!eigenvectors!express!

the!values!of!this!magnitude!that!may!be!discovered!by!physical!measurement,!


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according!to!the!theory.!Rays!i!therefore!correspond!to!atomic!propositions!p!=!

(m,!ri)!in!the!logic!of!this!quantum!theory,!or!to!valuations!hp!characteristic!of!these!propositions.!The!linear!combination!of!vectors!and!the!vector!subspace!

operations!derived!from!this!combination!represent!the!logical!combination!of!propositions!or!valuations.!For!example!if!h1!is!characteristic!for!p1!=!(m,!r1),!and!

h2!for!p2!=!(m,!r2)!then!another!valuation!h3!can!be!derived!characteristic!for!p3!=!(m,!r1!!r2),!which!is!welldefined!because!the!subsets!are!assumed!Borel.!!But!in!

addition!to!this!kind!of!combination!is!the!inner!product!p1!!p2!that!provides!a!scalar,!the!projection!of!one!vector!in!the!direction!of!the!other,!its!component!in!

this!direction.!Where!the!pi!are!atomic!mpropositions!this!inner!product!is!always!

0!as!we!would!expect!since!prob!!(p!)!=!0!for!j!!k.!!In!general!vector!!of!the!

Hilbert!Space!generates!probability!assignments!with!these!rays!in!accordance!

with!Borns!Projection!Postulate:!the!component!of!vector!!in!the!direction!of!

the!eigenvector!i!of!operator!M!gives!the!probability!that!the!corresponding!eigenvalue!ai!=!ri!of!the!magnitude!m!represented!by!M!is!the!value!discovered!by!

physical!measurement!of!this!magnitude!on!a!system!initially!described!by!.!!!

All!vectors!of!the!Hilbert!Space!assign!probabilities!to!the!atomic!propositions!in!

this!way,!generating!the!theorys!probability!assignments.!Where!vector!!coincides!with!a!ray,!!!=!k!say,!then!eigenvalue!rk!is!predicted!with!certainty!and!

p!=!(m,!rk)!is!true.!A!vector!may!represent!a!probability!assignment!in!which!no!simple!proposition!is!predicted!with!certainty!and!so!vectors!are!more!general!

than!valuations!of!the!theory:!they!do!not!assign!true!to!any!proposition.!These!

general!probability!assignments!are!however!logical!descriptions!provided!by!the!theory,!measures!of!the!maximal!valuations!of!the!logic!of!this!theory!in!which!

propositions!are!true.!Hilbert!Space!on!this!interpretation!is!a!generalisation!of!

classical!phase!space!required!to!represent!not!only!the!logical!combinations!expressed!by!classical!and!quantum!theories,!but!also!the!strongly!conditional!

probabilities!used!in!any!mechanical!theory!that!is!not!classical.!!!

This!logical!interpretation!of!quantum!mechanics!is!not!quantum!logic,!the!claim!

that!deduction!in!quantum!mechanics!is!nondistributive!and!therefore!radically!nonBoolean.!xiv!Birkhoff!and!von!Neumann!proposed!this!view!in!1936!and!it!has!

proved!surprisingly!popular,!given!that!ordinary!thought!and!reasoning!about!

quantum!theory!would!be!impossible!if!they!were!correct.!In!fact!their!argument!rests!on!an!example!of!quantum!descriptions!that!involves!negation!that!is!also!

resolved!by!distinguishing!negation!from!denial,!which!of!course!can!be!achieved!by!this!analysis!without!rejecting!laws!of!traditional!reasoning.!xv!!!

!

This!interpretation!also!differs!from!current!views!that!quantum!realities!are!weird!because!they!depend!on!observation,!are!strangely!entangled,!or!are!non

local!and!therefore!conflict!with!Relativity.!The!changes!in!quantum!states!after!physical!measurement!have!been!explained!as!logical!changes!required!by!the!use!

of!incompatible!descriptions!and!a!weak!successor!relation.!These!changes!do!not!

describe!physically!weird!disturbances!nor!indicate!that!properties!combine!in!new!nonsettheoretic!ways,!nor!do!they!describe!messages!traveling!faster!than!

light!in!conflict!with!Relativity!theory.!Instead!these!changes!express!a!logical!re

conditionalising!that!is!required!when!new!information!is!accepted!in!a!logic!that!lacks!compatible!comprehensive!descriptions.!Without!maximal!exhaustive!


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valuations!the!descriptions!of!a!theory!must!be!drastically!changed!when!new!

truthvalues!are!determined.!While!it!may!be!the!case!that!physical!measurements!of!very!small!systems!often!disturb!the!measured!system,!these!disturbances!will!

be!described!by!additional!description!as!sometimes!occurs!in!classical!mechanics.!!!!

This!interpretation!shows!that!probabilities!with!quantum!peculiarities!are!understandable,!useful!descriptions!of!reality!and!are!based!on!traditional!deduction!and!measure!theory.!The!merits!of!classical!and!quantum!theories!are!

now!briefly!considered.!If!observation!supports!both!a!classical!and!a!nonclassical!

description!of!reality,!which!theory!should!be!preferred?!!!

The!Principle!of!Noncontradiction!suggests!one!reason!to!favour!classical!over!nonclassical!descriptions.!This!Principle,!apparently!fundamental!to!both!theories!

since!it!generates!both!classical!and!nonclassical!probabilities!with!quantum!

properties,!excludes!contradictory!valuations!from!their!calculation:!only!maximal!valuations!that!do!not!contradict!initial!conditions!are!measured!in!any!

probability.!This!suggests!a!stronger!underlying!Metaphysical!Assumption!that!

Contradictory!valuations!describe!different!realities.!However!only!classical!logic!can!admit!this!stronger!rule!because!only!here!is!the!successor!relation!strong!

consistency.!Because!this!relation!is!transitive,!contradictory!valuations!are!never!used!to!describe!the!same!reality!in!a!classical!theory,!even!in!a!sequence!of!

descriptions.!In!classical!theories!distinct!maximal!valuations,!which!are!always!

contradictory,!can!be!assumed!unique!characterisations!of!the!reality!they!describe,!so!any!change!in!description!is!assumed!to!be!a!change!in!reality.!In!

general!however!where!maximal!valuations!are!not!exhaustive,!the!weaker!successor!relation!must!be!used!and!this!is!not!transitive.!There!is!in!these!theories!

no!assurance!that!contradictory!valuations!are!not!used!to!describe!the!same!reality!in!a!sequence!of!descriptions,!and!indeed!there!will!be!since!the!unique!relation!between!maximal!descriptions!and!realities!is!lost.!Although!nonclassical!

logic!excludes!contradictory!valuations!from!being!measured!in!each!calculation!of!

a!probability!they!cannot!be!excluded!from!describing!the!same!reality.!Nonclassical!theories!cannot!accept!the!Metaphysical!assumption!and!instead!exhibit!a!

kind!of!descriptive!slippage,!a!loss!of!information!in!a!sequence!of!different!measurements!that!never!classically!occurs.!!

!

The!nature!of!language,!description!and!reality!lie!outside!of!logic!in!metaphysics.!However!an!ancient!story!from!India!helps!us!appreciate!nonclassical!

descriptions.!In!this!story!blind!sages!describe!an!elephant,!something!they!have!not!encountered!before.!Each!feels!a!different!part!of!the!animal!and!each!

describes!it!differently.!One!feels!a!leg!and!declares!Its!a!tree,!another!an!ear!

insisting,!Its!a!fan!while!the!third!feels!the!trunk!and!disagrees:!This!is!an!eel.!This!story!reminds!us!that!descriptions!are!not!realities!and!propositions!even!

when!true!are!not!properties.!Each!sage!assigns!true!to!a!different!proposition,!

expressing!a!meaningful!and!useful!description!of!this!reality,!which!are!however!incompatible:!these!descriptions!cannot!be!combined!into!one!comprehensive!

description.!We,!who!have!encountered!elephants!before!and!describe!them!differently,!might!judge!the!sages!descriptions!inadequate,!an!imperfect!fit!with!

the!reality!described.!!And!just!as!alternative!compatible!descriptions!deliver!a!


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more!elegant!and!useful!theory!of!elephants,!so!too!new!compatible!descriptions!

might!generate!a!better,!classical!theory!of!the!very!small.!!

According!to!this!version!of!this!logical!interpretation!of!quantum!theories,!incompatible!propositions!are!imperfections!that!might!one!day!be!replaced.!!Since!

the!form!of!simple!quantum!descriptions,!and!to!some!extent!their!magnitudes,!have!been!derived!from!classical!mechanics!it!is!not!surprising!to!suppose!that!these!might!one!day!be!improved.!Infinitely!divisible!real!numbers!for!example!

may!be!inappropriate!as!values!for!magnitudes!for!the!extremely!small.!Changing!

this,!or!changing!the!magnitudes!used!by!quantum!theory!would!change!the!form!of!simple!propositions!and!hence!their!primitive!relations,!fundamentally!altering!

the!logic!of!the!theory.!Eventually!such!changes!might!allow!compatible!descriptions!that!can!be!combined!in!comprehensive!assignments!of!truthvalues!

to!all!of!the!propositions.!In!this!case!microsystems!would!be!described!using!

classical!probabilities,!where!true!propositions!are!assumed!to!correspond!to!properties!and!probabilities!range!over!events.!

!

Footnotes##

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!i!The!Lindenbaum!algebra!(also!called!LindenbuamTarski!as!it!was!devised!by!Tarski)!represents!

the!logic!L!by!regarding!all!logically!equivalent!propositions!as!the!same!element,!so!each!a!in!A!is!

an!equivalence!class![]!=!{!!L:!!!!and!!!!are!both!logically!true!in!L}.!To!say!implication!is!

represented!by!a!partial!ordering!!on!A!is!to!say!the!operation!!generated!by!logical!implication,!

i.e.![]!![]!=df!(!!)!is!logically!true,!is!reflexive,!antisymmetric!and!transitive:!!is!reflexive!

when!a!!a!for!all!elements!a!in!A;!antiLsymmetric!when!if!a!!b!and!b!!a!then!a!=!b;!and!transitive!

when!if!a!!b!and!b!!c!then!a!!c.!A!lattice!has!meet!and!join!(least!upper!and!greatest!lower!bound!

with!respect!to!this!partial!ordering)!defined!every!where,!and!a!distributive!lattice!in!addition!satisfies!the!two!Laws!of!Distribution.!!

ii!This!result!is!shown!in!the!truthtable!below,!where!value!u!indicates!a!proposition!is!unassigned,!

without!a!truthvalue!in!this!valuation.!!

!

i) ii) ~

t f t t t t t f t

f t f t f f t t f

u u u t u u t t u

!iii

!An!orthogonal!or!involution!!is!has!the!property!that!for!any!element!a!of!an!algebra!A

,!"

a!!!!=!a;!lattice!complement!!has!the!property!that!a!!a!=!1!where!1!is!the!universal!bound!of!

algebra!A.!iv!The!orthogonal!of!a!is!a!lattice!complement!with!respect!to!the!logical!context!of!a,!the!set!of!

equivalence!classes!representing!propositions!logically!implying!or!logically!implied!by! a,!see!Garden![1984!]!or![1992]!for!more.!v!Traditionally!these!are!defined!in!terms!of!Kripkes!possible!worlds!(see!e.g.!Hughes!and!

Cresswell![1972])!but!here!a!simpler!method!is!used,!shown!in!Garden![1984]!to!be!a!more!general!

definition!that!has!Kripke!modal!logics!as!a!special!case.!vi!For!h(MR)!=!t!iff!!h()!for!some!h!such!that!h!R!h,!but!where!R!is!trivial,!so!h!R!h!for!all!h!in!H,!

h(M)!=!t!iff!!h()!for!some!h!in!H.!vii!For!h(~MR ~)!=!t!iff!!h(MR~)!!t!(by!definition!~)!iff!there!is!no!h!in!H!such!that!!

h!R!h!and!h(~)!=!t!(by!definition!MR

)!iff!for!all!h!in!H,!h()!=!t,!(by!definition!~)!


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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!viii!For!example!if!!!!is!logically!true!and!h()!=!t!then![]!![]!(by!property!of!)!and!![]!!![Th],!(by!defn!Th)!but!in!this!case!also!by!properties!of!!h()!=!t!and!so![]!![Th]!and!this!property!of!a!filter!holds.!The!other!property!follows!from!properties!of!. !ix!I!have!elsewhere!called!these!Kolmogorov!measures!but!here!simply!mathematical!measures!to!

avoid!unnecessary!distractions!see!footnote!xi)!below.!x!Suppose!h!W!h!for!h!a!maximal!valuation!in!classical!logic!L!and!h!any!valuation!in!H!of!L.!Then!

for!all!p,!h(p)!=!t!or!h(p)!=!f!(because!h!is!exhaustive).!Suppose!h(p)!=!t,!then!h(p)!!f!!(because!h!

W!h)!and!so!h(p)!=!t!and!h!includes!h.!xi!Algebra!A* is!the!algebra!representing!the!bivalent!logic!L*!derived!from!L!which!has!each!h!in!H!exhaustive.!Maximal!filters!of!A!are!represented!by!ultrafilters!of!A*.!xii!Algebra!A!of!L!and!probability!space!P!of!L!coincide!only!when!logic!L!is!bivalent,!where!negation!and!denial!cannot!be!distinguished!and!are!represented!by!a!single!operation!on!A.!!xiii!It!is!this!conditional!nature!of!general!probabilities,!the!fact!that!many!probability!spaces!are!

required!by!a!logic,!that!obscures!the!traditional!nature!of!these!nonclassical!probabilities.!Streater!

for!example!writes!of!my!work:!Her!conclusion,!that![quantum]!theory!leads!to!a!Kolmogorovian!

probability!theory,!is!bizarre;!This!may!accord!with!his!own!definition!of!Kolmogorovian!but!

misses!the!point!that!each!probability!is!a!welldefined!and!traditional!mathematical!measure!over!

a!traditionally!Boolean!field!of!sets.!Hopefully!this!paper!makes!these!issues!clearer!than!did!my!

work!of!30!years!ago.!!xiv!Birkhoff!and!Von!Neumann![!1936]!xv!Garden![1984!]!and![1992!].!!!

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Boole,!George!An!Investigation!of!The!Laws!Of!Thought,!1853!available!online!from!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Rasiowa,!Helena,!An!Algebraic!Approach!to!NonClassical!Logics,!vol!78!of!Studies!in!Logic!and!the!

Foundation!of!Mathematics,!NorthHolland!Publishing!Company!1974!

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Sikorski,!Roman,!Boolean!Algebras,!Springer!Verlag,!1969!

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Streater,!R.!F.!Lost!Causes!in!Theoretical!Physics,!available!in!2013!from!http://www.mth.kcl.ac.uk/~streater/lostcauses.html#III !

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Thinking Logically About Quantum Probabilities