Operational Meanings of Orders of Observables Defined through Quantum Set Theories with Different Conditionals

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by  Masanao Ozawa

Operational Meanings of Orders of Observables Defined
through Quantum Set Theories with Different Conditionals
MasanaoOzawa∗
GraduateSchoolofInformationScience,NagoyaUniversity
Nagoya,Japan
[email protected]
In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional.
Currently,wehaveatleastthreecandidates,calledtheSasakiconditional,thecontrapositiveSasaki
conditional,andtherelevanceconditional. Afundamentalproblemistoshowhowtheformofthe
conditionalfollowsfroman analysis of operationalconceptsin quantumtheory. Here, we attempt
such an analysis throughquantumset theory(QST). In this paper, we developquantumset theory
based on quantum logics with those three conditionals, each of which defines different quantum
logical truth value assignment. We show that those three models satisfy the transfer principle of
the same form to determine the quantum logical truth values of theorems of the ZFC set theory.
Wealsoshowthattherealsinthemodelandthetruthvaluesoftheirequalityarethesameforthose
models.Interestingly,however,theorderrelationbetweenquantumrealssignificantlydependsonthe
underlyingconditionals. Wecharacterizetheoperationalmeaningsofthoseorderrelationsinterms
ofjointprobabilityobtainedbythesuccessiveprojectivemeasurementsofarbitrarytwoobservables.
Those characterizations clearly show their individual features and will play a fundamental role in
futureapplicationstoquantumphysics.
1 Introduction
Quantumsettheorycrossesovertwodifferentfieldsofmathematics,namely,foundationsofmathematics
andfoundationsofquantummechanics,andoriginatedfromthemethodsofforcingintroducedbyCohen
[5,6]fortheindependenceproofofthecontinuumhypothesisandquantumlogicintroducedbyBirkhoff
and von Neumann [2]. After Cohen’s work, the forcing subsequently became a central method in set
theory and also incorporated with various notions in mathematics, in particular, the notion of sheaves
[8] and notions of sets in nonstandard logics such as Boolean-valued set theory [1], by which Scott
and Solovay [22] reformulated the method of forcing, topos [12], and intuitionistic set theory [9]. As
a successor of those attempts, quantum set theory, a set theory based on the Birkhoff-von Neumann
quantumlogic,wasintroducedbyTakeuti[23],whoestablishedtheone-to-onecorrespondence between
realsinthemodel(quantumreals)andquantumobservables. Quantumsettheorywasrecentlydeveloped
by the present author [18, 19] to obtain the transfer principle to determine quantum truth values of
theorems of the ZFC set theory, and clarify the operational meaning of the equality between quantum
reals,whichextendstheprobabilistic interpretation ofquantum theory,
In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional.
Hardegree[11]definedamaterialconditional onanorthomodular latticeasapolynomially definablebi-
naryoperationsatisfyingthreefundamentalrequirements, andshowedthatthereareexactlythreebinary
operations satisfyingthoseconditions: theSasakiconditional, thecontrapositive Sasakiconditional, and
the relevance conditional. Naturally, afundamental problem istoshow howthe form ofthe conditional
∗SupportedbyJSPSKAKENHI,No.26247016,No.15K13456.
(cid:13)c M.Ozawa
R.DuncanandC.Heunen(Eds.):QuantumPhysicsandLogic(QPL)2016
Thisworkislicensedunderthe
EPTCS236,2017,pp.127–144,doi:10.4204/EPTCS.236.9
CreativeCommonsAttributionLicense.
128 QuantumSetTheory
followsfromananalysisoftheoperationalconceptinquantumtheorytestablebyexperiments. Here,we
attemptsuchananalysis through quantum settheory. Inquantum settheory(QST),thequantum logical
truth values oftwoatomic formulas, equality and membership relations, depend crucially on thechoice
of conditional. In the previous investigations, we have adopted only the Sasaki conditional, proved the
transfer principle to determine quantum truth values of theorems of the ZFC set theory, established the
one-to-one correspondence between reals in the model, or “quantum reals”, and quantum observables,
andclarifiedtheoperationalmeaningoftheequalitybetweenquantumreals. Inthispaper,westudyQST
based ontheabovethree materialconditional together. Weconstruct theuniversal QSTmodelbased on
the logic of the projection lattice of avon Neumann algebra witheach conditional. Then, weshow that
this new model satisfies the transfer principle of the same form as the old model based on the Sasaki
conditional. Wealso show that thereals in themodel and the truth values oftheir equality arethe same
forthosethreemodels. Uptothispoint,thosemodelsbehaveindistinguishably. However,werevealthat
theorderrelationbetweenquantumrealsdependcruciallyontheunderlyingconditionals. Wecharacter-
izetheoperational meaningsofthoseorderrelations, whichturnoutcloselyrelatedtothespectral order
introducedbyOlson[16]playingasignificantroleinthetoposapproachtoquantumtheory[7],interms
ofjointprobabilityoftheoutcomesofthesuccessiveprojectivemeasurementsoftwoobservables. Those
characterizations clarify theirindividual features andwillplay afundamental roleinfuture applications
toquantum physics.
2 Preliminaries
2.1 Complete orthomodularlattices
Acomplete orthomodular lattice isacomplete lattice Q withanorthocomplementation, aunary opera-
tion ⊥ on Q satisfying (C1) if P≤Q then Q⊥ ≤P⊥, (C2) P⊥⊥ =P, (C3) P∨P⊥=1 and P∧P⊥=0,
where0= Qand1= Q,thatsatisfiestheorthomodularlaw(OM)ifP≤QthenP∨(P⊥∧Q)=Q.
In this paper, any complete orthomodular lattice is called a logic. A non-empty subset of a logic Q is
V W
called a subalgebra iff it is closed under ∧, ∨, and ⊥. A subalgebra A of Q is said to be complete iff
it has the supremum and the infimum in Q of an arbitrary subset of A. For any subset A of Q, the
subalgebrageneratedbyA isdenotedbyG A. WereferthereadertoKalmbach[13]forastandardtext
0
onorthomodular lattices.
We say that P and Q in a logic Q commute, in symbols P | Q, iff P=(P∧Q)∨(P∧Q⊥). A logic
◦
Q isaBooleanalgebraifandonlyifP| QforallP,Q∈Q [13,pp.24–25]. ForanysubsetA ⊆Q,we
◦
denotebyA! thecommutantofA inQ[13,p.23],i.e.,
A!={P∈Q|P| QforallQ∈A}.
◦
Then,A! isacomplete subalgebra ofQ. Asublogic ofQ isasubsetA ofQ satisfying A =A!!. For
anysubset A ⊆Q, thesmallest logic including A isA!! called thesublogic generated byA. Then, it
iseasytoseethatasubsetA isaBooleansublogic, orequivalently adistributivesublogic, ifandonlyif
A =A!!⊆A!.
Thefollowingproposition isusefulinlaterdiscussions.
Proposition2.1. LetQ bealogiconH . IfPa ∈Q andPa |◦ Qforalla ,then( a Pa )|◦ Q, a Pa |◦ Q,
Q∧( a Pa )= a (Q∧Pa )CQ∧( a Pa )= a (Q∧Pa ).
W V
PWroof. SuppWosethatPa ∈Q anVdPa |◦ QhVoldforeverya . From
Pa ∧Q≤Q, Pa ∧Q⊥≤Q⊥,
a a
_ _
M.Ozawa 129
wehave
Pa ∧Q|◦ Q, Pa ∧Q⊥ |◦ Q. (1)
a a
_ _
Bytheassumption, wehavePa =(Pa ∧Q)∨(Pa ∧Q⊥)foreverya . Since
Pa = (Pa ∧Q)∨(Pa ∧Q⊥)
a a
_ _
= ( Pa ∧Q)∨( Pa ∧Q⊥),
a a
_ _
byEq.(1)wehave a Pa |◦ Q. ByEq.(1)thedistributive lawholdsandwehave
W Q∧ Pa = Q∧[( Pa ∧Q)∨( Pa ∧Q⊥)]
a a a
_ _ _
= (Pa ∧Q).
a
_
Thus,wehaveQ∧ a Pa = a (Q∧Pa ). TherestoftheassertionsfollowsfromtheDeMorganlaw.
W W
2.2 Logics onHilbertspaces
Let H be a Hilbert space. For any subset S ⊆ H , we denote by S⊥ the orthogonal complement of
S. Then, S⊥⊥ is the closed linear span of S. Let C(H ) be the set of all closed linear subspaces in
H . With the set inclusion ordering, the set C(H )is a complete lattice. The operation M 7→M⊥ is an
orthocomplementation onthelatticeC(H ),withwhichC(H )isalogic.
Denote by B(H )the algebra of bounded linear operators on H and Q(H )the set of projections
onH . WedefinetheoperatororderingonB(H )byA≤Biff(y ,Ay )≤(y ,By )forally ∈H . For
anyA∈B(H ),denotebyR(A)∈C(H )theclosureoftherangeofA,i.e.,R(A)=(AH )⊥⊥. Forany
M∈C(H ),denotebyP(M)∈Q(H )theprojection operatorofH ontoM. Then,RP(M)=M for
all M ∈C(H ) and PR(P)=P for all P∈Q(H ), and we have P≤Q if and only if R(P)⊆R(Q)
for all P,Q∈Q(H ), so that Q(H ) with the operator ordering is also a logic isomorphic to C(H ).
AnysublogicofQ(H )willbecalledalogiconH . ThelatticeoperationsarecharacterizedbyP∧Q=
weak-limn→¥ (PQ)n,P⊥=1−PforallP,Q∈Q(H ).
LetA ⊆B(H ). WedenotebyA′thecommutantofA inB(H ). Aself-adjointsubalgebraM of
B(H )iscalledavonNeumannalgebraonH iffM′′ =M. Foranyself-adjoint subsetA ⊆B(H ),
A′′ isthe von Neumann algebra generated by A. Wedenote by P(M) the set of projections in avon
Neumann algebra M. For any P,Q∈Q(H ), we have P | Q iff [P,Q]=0, where [P,Q]=PQ−QP.
◦
For any subset A ⊆Q(H ), we denote by A! the commutant of A in Q(H ). For any subset A ⊆
Q(H ),thesmallestlogicincluding A isthelogicA!! calledthelogicgenerated byA. Then,asubset
Q⊆Q(H )isalogiconH ifandonlyifQ=P(M)forsomevonNeumannalgebraM onH [18,
Proposition 2.1].
2.3 Commutators
Marsden[14]hasintroduced thecommutator ⊥⊥(P,Q)oftwoelementsPandQofalogicQ by
⊥⊥(P,Q)=(P∧Q)∨(P∧Q⊥)∨(P⊥∧Q)∨(P⊥∧Q⊥).
130 QuantumSetTheory
BrunsandKalmbach[3]havegeneralized thisnotiontofinitesubsets ofQ by
⊥⊥(F)= Pa (P)
a :F→_{id,⊥}P^∈F
for allF ∈Pw (Q), where Pw (Q)stands for theset offinite subsets ofQ, and {id,⊥}stands for the
set consisting of the identity operation id and the orthocomplementation ⊥. Generalizing this notion to
arbitrary subsetsA ofQ,Takeuti[23]defined⊥⊥(A)by
⊥⊥(A)= {E ∈A!|P ∧E | P ∧E forallP ,P ∈A},
1 ◦ 2 1 2
_
ofanyA ∈P(Q),whereP(Q)stands forthepowersetofQ. Takeuti’sdefinition hasbeenreformu-
latedinseveralmoreconvenient forms[21,4,19].
We have the following characterizations of commutators in logics on Hilbert spaces [18, Theorems
2.5,2.6,Proposition 2.2].
Theorem2.2. LetQbealogiconH andletA ⊆Q. Then,wehavethefollowingrelations.
(i) ⊥⊥(A)=P{y ∈H |[P ,P ]P y =0forallP ,P ,P ∈A}.
1 2 3 1 2 3
(ii) ⊥⊥(A)=P{y ∈H |[A,B]y =0forallA,B∈A′′}.
3 Conditionals
In classical logic, the conditional operation → is defined by negation ⊥ and disjunction ∨ as P→Q=
P⊥∨Q. In quantum logic there is a well-known arbitrariness in choosing a binary operation for condi-
tional. Hardegree [11] defined a material conditional on an orthomodular lattice Q as a polynomially
definablebinaryoperation →onQsatisfying thefollowing“minimumimplicativeconditions”:
(LB) IfP| Q,thenP→Q=P⊥∨QforallP,Q∈Q.
◦
(E) P→Q=1ifandonlyifP≤Q.
(MP) (modusponens)P∧(P→Q)≤Q.
(MT) (modustollens)Q⊥∧(P→Q)≤P⊥.
Then,heprovedthatthereareexactlythreematerialconditionals:
(S) (Sasakiconditional) P→ Q:=P⊥∨(P∧Q),
S
(C) (Contrapositive Sasakiconditional) P→ Q:=(P∨Q)⊥∨Q,
C
(R) (Relevanceconditional) P→ Q:=(P∧Q)∨(P⊥∧Q)∨(P⊥∧Q⊥).
R
Weshalldenote by→ with j=S,C,Ranyoneoftheabovematerialconditionals. Oncethecondi-
j
tional→ isspecified, thelogicalequivalence ↔ isdefinedby
j j
P↔ Q:=(P→ Q)∧(Q→ P).
j j j
Then,itiseasytoseethatwehave
P↔ Q=P↔ Q=P↔ Q=(P∧Q)∨(P⊥∧Q⊥).
S C R
Thus,wewrite↔for↔ forall j=S,C,R.
j
Intheprevious investigations [23,18,19]onquantum settheoryonlytheSasakiarrowwasadopted
astheconditional. Inthispaper, wedevelopaquantum settheorybasedontheabovethreeconditionals
M.Ozawa 131
together and show that they equally ensure the transfer principle for quantum set theory. We shall also
show that the notions of equality defined through those three are the same, but that the order relations
definedthroughthemaredifferent.
Wehavethefollowingcharacterizations ofconditionals inlogicsonHilbertspaces.
Theorem3.1. LetQbealogiconH andletP,Q∈Q. Then,wehavethefollowing relations.
(i) P→ Q=P{y ∈H |Q⊥Py =0}.
S
(ii) P→ Q=P{y ∈H |PQ⊥y =0}.
C
(iii) P→ Q=P{y ∈H |Q⊥Py =PQ⊥y =0}.
R
(iv) P↔Q=P{y ∈H |Py =Qy }.
Proof. Toshow(i)supposey ∈R(P→ Q). Then,wehavey =P⊥y +(P∧Q)y ,sothatwehave
S
Q⊥Py =0. Conversely,supposeQ⊥Py =0. Then,wehavePy =QPy +Q⊥Py =QPy ∈R(Q).Since
Py ∈R(P), wehave Py ∈R(P)∩R(Q)=R(P∧Q). It follows that y =P⊥y +Py =P⊥y +(P∧
Q)y ∈R(P→ Q). Thus,relation(i)holds. Relation(ii)followsfromtherelationP→ Q=Q⊥→ P⊥.
S C S
Relation (iii) follows from the relation P→ Q=(P→ Q)∧(P→ Q). To show relation (iv), suppose
R S C
y ∈ R(P↔Q). Then, y ∈ R(P→ Q)∩R(Q→ P), and hence PQ⊥y = 0 and P⊥Qy = 0, so that
R R
Py =PQy =Qy . Conversely,ifPy =Qy ,wehaveQ⊥Py =0andP⊥Qy =0,sothaty ∈R(P↔Q).
Thus,relation(iv)follows.
Thefollowingtheoremshowsimportantpropertiesofmaterialconditionals inestablishing thetrans-
ferprinciple forquantumsettheory.
Proposition 3.2. Thematerialconditionals → with j=S,C,Rsatisfy thefollowing properties.
j
(i) P→ Q∈{P,Q}!! forallP,Q∈Q.
j
(ii) (P→ Q)∧E =[(P∧E)→(Q∧E)]∧E ifP,Q| E forallP,Q,E ∈Q.
j ◦
Proof. Assertions followfromtheLemmabelow.
Lemma 3.3. Let f be a two-variable ortholattice polynomial on a logic Q on H . Then, we have the
following statements.
(i) f(P,Q)∈{P,Q}!! forallP,Q∈Q.
(ii) f(P,Q)∧E = f(P∧E,Q∧E)∧EifP,Q| E forallP,Q,E ∈Q.
◦
Proof. Since f(P,Q) is in the ortholattice G {P,G} generated by P and Q and we have G {P,G}⊆
0 0
{P,Q}!!,sothatstatement(i)follows. Theproofof(ii)iscarriedoutbyinductiononthecomplexityofthe
polynomial f(P,Q). First, note that from P,Q| E wehave g(P,Q)| E forany two-variable polynomial
◦ ◦
g. If f(P,Q)=P or f(P,Q)=Q, assertion (ii) holds obviously. If f(P,Q)=g (P,Q)∧g (P,Q) with
1 2
two-variablepolynomialsg ,g ,theassertionholdsfromassociativity. Supposethat f(P,Q)=g (P,Q)∨
1 2 1
g (P,Q) with two-variable polynomials g ,g . Since g (P,Q),g (P,Q) | E, the assertion follows from
2 1 2 1 2 ◦
thedistributivelawfocusingonE. Suppose f(P,Q)=g(P,Q)⊥withatwo-variablepolynomialg. Forthe
casewheregisatomic,theassertionfollows;forinstance,ifg(P,Q)=P,wehave f(P∧E,Q∧E)∧E=
(P∧E)⊥∧E =(P⊥∨E⊥)∧E =P⊥∧E = f(P,Q)∧E. Then,weassumeg(P,Q)=g (P,Q)∧g (P,Q)
1 2
or g(P,Q)=g (P,Q)∨g (P,Q) with two-variable polynomials g ,g . If g(P,Q)=g (P,Q)∧g (P,Q),
1 2 1 2 1 2
132 QuantumSetTheory
bytheinduction hypothesis andthedistributivity wehave
f(P,Q)∧E = g(P,Q)⊥∧E
= (g (P,Q)⊥∨g (P,Q)⊥)∧E
1 2
= (g (P,Q)⊥∧E)∨(g (P,Q)⊥∧E)
1 2
= (g (P∧E,Q∧E)⊥∧E)∨(g (P∧E,Q∧E)⊥∧E)
1 2
= (g (P∧E,Q∧E)⊥∨g (P∧E,Q∧E)⊥)∧E)
1 2
= (g (P∧E,Q∧E)∧g (P∧E,Q∧E))⊥∧E
1 2
= g(P∧E,Q∧E)⊥∧E
= f(P∧E,Q∧E)∧E.
Thus,theassertionfollowsifg(P,Q)=g (P,Q)∧g (P,Q),andsimilarlytheassertionfollowsifg(P,Q)=
1 2
g (P,Q)∨g (P,Q). Thus, the assertion generally follows from the induction on the complexity of the
1 2
polynomial f.
4 Quantum set theory
We denote byV the universe of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Let
L(∈) be the language for first-order theory with equality augmented by a binary relation symbol ∈,
bounded quantifier symbols ∀x ∈y, ∃x∈ y, and no constant symbols. For any class U, the language
L(∈,U)istheoneobtained byaddinganameforeachelementofU.
LetQ bealogiconH . Foreachordinala ,let
Va(Q)={u|u:dom(u)→Q and(∃b <a )dom(u)⊆Vb(Q)}.
TheQ-valued universeV(Q) isdefinedby
V(Q)= Va(Q),
a ∈[On
whereOnistheclassofallordinals. Foreveryu∈V(Q),therankofu,denotedbyrank(u),isdefinedas
theleasta suchthatu∈V(Q). Itiseasytoseethatifu∈dom(v)thenrank(u)<rank(v).
a +1
In what follows → generally denotes one of the Sasaki conditional → , the contrapositive Sasaki
j S
conditional → , and the relevance conditional → . For any u,v∈V(Q), the Q-valued truth values of
C R
atomicformulasu=vandu∈vareassigned bythefollowingrulesrecursiveinrank.
(i) [[u=v]]j,Q = u′∈dom(u)(u(u′)→j [[u′∈v]]j,Q)∧ v′∈dom(v)(v(v′)→j [[v′∈u]]j,Q).
(ii) [[u∈v]]j,Q = Vv′∈dom(v)(v(v′)∧[[u=v′]]j,Q). V
To each statemWent f of L(∈,V(Q)) we assign the Q-valued truth value [[f ]]j,Q by the following
rules.
(iii) [[¬f ]]j,Q =[[f ]]⊥j,Q.
(iv) [[f 1∧f 2]]j,Q =[[f 1]]j,Q∧[[f 2]]j,Q.
(v) [[f 1∨f 2]]j,Q =[[f 1]]j,Q∨[[f 2]]j,Q.
(vi) [[f 1→f 2]]j,Q =[[f 1]]j,Q →j [[f 2]]j,Q.
M.Ozawa 133
(vii) [[f 1↔f 2]]j,Q =[[f 1]]j,Q ↔[[f 2]]j,Q.
(viii) [[(∀x∈u)f (x)]]j,Q = u′∈dom(u)(u(u′)→j [[f (u′)]]j,Q).
(ix) [[(∃x∈u)f (x)]]j,Q =Vu′∈dom(u)(u(u′)∧[[f (u′)]]j,Q).
(x) [[(∀x)f (x)]]j,Q = u∈VW(Q)[[f (u)]]j,Q.
(xi) [[(∃x)f (x)]]j,Q =Vu∈V(Q)[[f (u)]]j,Q.
AformulainL(∈)WiscalledaD -formulaifithasnounboundedquantifiers∀xor∃x. Thefollowing
0
theorem holds.
Theorem 4.1 (D -Absoluteness Principle). For any D -formula f (x ,...,x ) of L(∈) and u ,...,u ∈
0 0 1 n 1 n
V(Q),wehave
[[f (u1,...,un)]]j,Q =[[f (u1,...,un)]]j,Q(H).
Proof. The assertion is proved by the induction on the complexity of formulas and the rank of ele-
mentsofV(Q). Letu,v∈V(Q). Weassumethattheassertionholdsforallu′∈dom(u)andv′∈dom(v).
Then, we have [[u′ ∈ v]]j,Q = [[u′ ∈ v]]j,Q(H), [[v′ ∈ u]]j,Q = [[v′ ∈ u]]j,Q(H), and [[u = v′]]j,Q = [[u =
v′]]j,Q(H). Thus,
[[u=v]]j,Q = (u(u′)→j [[u′∈v]]j,Q)∧ (v(v′)→j [[v′∈u]]j,Q)
u′∈d^om(u) v′∈d^om(v)
= (u(u′)→j [[u′∈v]]j,Q(H))∧ (v(v′)→j [[v′∈u]]j,Q(H))
u′∈d^om(u) v′∈d^om(v)
= [[u=v]]j,Q(H),
andwealsohave
[[u∈v]]j,Q = (v(v′)∧[[u=v′]]j,Q)
v′∈d_om(v)
= (v(v′)∧[[u=v′]]j,Q(H))
v′∈d_om(v)
= [[u∈v]]j,Q(H).
Thus,theassertion holdsforatomicformulas. Anyinduction stepaddingalogicalsymbolworkseasily,
even when bounded quantifiers are concerned, since the ranges of the supremum and the infimum are
commonforevaluating [[···]]j,Q and[[···]]j,Q(H).
Henceforth,foranyD -formulaf (x ,...,x )andu ,...,u ∈V(Q),weabbreviate[[f (u ,...,u )]] =
0 1 n 1 n 1 n j
[[f (u1,...,un)]]j,Q,whichisthecommonQ(H )-valued truthvalueforu1,...,un∈V(Q).
The universe V can be embedded in V(Q) by the following operation ∨:v7→vˇ defined by the ∈-
recursion: foreachv∈V,vˇ={uˇ|u∈v}×{1}. Thenwehavethefollowing.
Theorem 4.2 (D -Elementary Equivalence Principle). For any D -formula f (x ,...,x ) of L(∈) and
0 0 1 n
u ,...,u ∈V,wehavehV,∈i|=f (u ,...,u )ifandonlyif[[f (uˇ ,...,uˇ )]] =1.
1 n 1 n 1 n j
Proof. Let2 be the sublogic such that 2={0,1}. Then, by induction it is easy to see that hV,∈i|=
f (u ,...,u )ifandonlyif[[f (uˇ ,...,uˇ )]] =1foranyf (x ,...,x )inL(∈),andthisisequivalent to
1 n 1 n j,2 1 n
[[f (uˇ ,...,uˇ )]] =1foranyD -formulaf (x ,...,x )bytheD -absoluteness principle.
1 n j 0 1 n 0
134 QuantumSetTheory
5 Transfer principle
In this section, we investigate the transfer principle that transfers any D -formula provable in ZFC to a
0
truestatement aboutelementsofV(Q).
Theresultsinthissectionhavebeenobtainedfor j=SinRef.[18]. Here,wegeneralizethemtothe
case j=C,R. Foru∈V(Q),wedefinethesupport ofu,denoted byL(u),bytransfinite recursion onthe
rankofubytherelation
L(u)= L(x)∪{u(x)|x∈dom(u)}.
x∈d[om(u)
For A ⊆ V(Q) we write L(A) = L(u) and for u ,...,u ∈ V(Q) we write L(u ,...,u ) =
u∈A 1 n 1 n
L({u ,...,u }).
1 n S
For u∈V(Q), wedefine the support of u, denoted by L(u), by transfinite recursion on the rank of u
bytherelation
L(u)= L(x)∪{u(x)|x∈dom(u)}.
x∈d[om(u)
For A ⊆ V(Q) we write L(A) = L(u) and for u ,...,u ∈ V(Q) we write L(u ,...,u ) =
u∈A 1 n 1 n
L({u ,...,u }). Then,weobtainthefollowingcharacterization ofsubuniverses ofV(Q(H)).
1 n S
Proposition 5.1. Let Q be alogic on H and a an ordinal. For any u∈V(Q(H)), wehave u∈Va(Q) if
andonlyifu∈Va(Q(H))andL(u)⊆Q. Inparticular,u∈V(Q)ifandonlyifu∈V(Q(H))andL(u)⊆Q.
Moreover, rank(u)istheleasta suchthatu∈Va(Q(H)) foranyu∈V(Q).
Proof. Immediatefromtransfinite induction ona .
LetA ⊆V(Q). ThecommutatorofA,denotedby∨(A),isdefinedby
∨(A)=⊥⊥(L(A)).
Foranyu ,...,u ∈V(Q),wewrite∨(u ,...,u )=∨({u ,...,u }).
1 n 1 n 1 n
Letu∈V(Q)and p∈Q. Therestrictionu| ofuto pisdefinedbythefollowingtransfiniterecursion:
p
dom(u| ) = {x| |x∈dom(u)},
p p
u| (x| ) = u(x)∧p
p p
foranyx∈dom(u). Byinduction, itiseasytoseethatifq,p∈Q,then(u| )| =u| forallu∈V(Q).
p q p∧q
Proposition 5.2. ForanyA ⊆V(Q) and p∈Q,wehave
L({u| |u∈A})=L(A)∧p.
p
Proof. By induction, it is easy to see the relation L(u| )=L(u)∧p, so that the assertion follows
p
easily.
LetA ⊆V(Q). Thelogicgenerated byA,denoted byQ(A),isdefineby
Q(A)=L(A)!!.
Foru ,...,u ∈V(Q),wewriteQ(u ,...,u )=Q({u ,...,u }).
1 n 1 n 1 n
M.Ozawa 135
Proposition 5.3. For any D -formula f (x ,...,x ) in L(∈) and u ,···,u ∈ V(Q), we have
0 1 n 1 n
[[f (u ,...,u )]] ∈Q(u ,...,u ).
1 n j 1 n
Proof. LetA ={u ,...,u }. SinceL(A)⊆Q(A),itfollowsfromProposition5.1thatu ,...,u ∈
1 n 1 n
V(Q(A)). By the D 0-absoluteness principle, we have [[f (u1,...,un)]]j =[[f (u1,...,un)]]jQ(A) ∈Q(A).
Proposition 5.4. For anyD -formula f (x ,...,x )inL(∈)and u ,...,u ∈V(Q),if p∈L(u ,...,u )!,
0 1 n 1 n 1 n
then p| [[f (u ,...,u )]] and p| [[f (u | ,...,u | )]] .
◦ 1 n j ◦ 1 p n p j
Proof. Let u ,...,u ∈V(Q). If p∈L(u ,...,u )!, then p∈Q(u ,...,u )!. From Proposition 5.3,
1 n 1 n 1 n
[[f (u ,...,u )]] ∈Q(u ,...,u ),sothat p| [[f (u ,...,u )]] . FromProposition 5.2,L(u | ,...,u | )=
1 n j 1 n ◦ 1 n j 1 p n p
L(u ,...,u )∧p,andhence p∈L(u | ,...,u | )!,sothat p| [[f (u | ,...,u | )]] .
1 n 1 p n p ◦ 1 p n p j
Wedefinethebinary relation x ⊆x by∀x∈x (x∈x ). Then, bydefinition foranyu,v∈V(Q) we
1 2 1 2
have
[[u⊆v]] = u(u′)→ [[u′∈v]] ,
j j j
u′∈d^om(u)
andwehave[[u=v]] =[[u⊆v]] ∧[[v⊆u]] .
j j j
Proposition 5.5. Foranyu,v∈V(Q) and p∈L(u,v)!,wehavethefollowing relations.
(i)[[u| ∈v| ]] =[[u∈v]] ∧p.
p p j j
(ii)[[u| ⊆v| ]] ∧p=[[u⊆ p]] ∧p.
p p j j
(iii)[[u| =v| ]] ∧p=[[u= p]] ∧p
p p j j
Proof. We prove the relations by induction on the ranks of u,v. If rank(u) = rank(v) = 0, then
dom(u)=dom(v)=0/,sothattherelations trivially hold. Letu,v∈V(Q) and p∈L(u,v)!. Toprove(i),
letv′∈dom(v). Then,wehave p| v(v′)bytheassumption on p. Byinduction hypothesis, wehavealso
◦
[[u| =v′| ]] ∧p=[[u=v′]] ∧p. ByProposition5.4,wehave p| [[u=v′]] ,sothatv(v′),[[u=v′]] ∈{p}!,
p p j j ◦ j j
andhencev(v′)∧[[u=v′]] ∈{p}!. Thus,wehave
j
[[u| ∈v| ]] = v| (v′)∧[[u| =v′]]
p p j p p j
v′∈do_m(v|p)
= v| (v′| )∧[[u| =v′| ]]
p p p p j
v′∈d_om(v)
= (v(v′)∧p)∧([[u=v′]] ∧p)
j
v′∈d_om(v)
= (v(v′)∧[[u=v′]] )∧p
j
 
v′∈d_om(v)
 
= v(v′)∧[[u=v′]] ∧p,
j
 
v′∈d_om(v)
 
where the last equality follows from Proposition 2.1. Thus, by definition of [[u = v]] we obtain the
j
relation [[u| ∈v| ]] =[[u=v]] ∧p, and relation (i) has been proved. To prove (ii), let u′ ∈dom(u).
p p j j
136 QuantumSetTheory
Then,wehave[[u′| ∈v| ]] =[[u′ ∈v]] ∧pbyinduction hypothesis. Thus,wehave
p p j j
[[u| ⊆v| ]] = (u| (u′)→ [[u′ ∈v| ]] )
p p j p j p j
u′∈do^m(u|p)
= (u| (u′| )→ [[u′| ∈v| ]] )
p p j p p j
u′∈d^om(u)
= (u(u′)∧p)→ ([[u′∈v]] ∧p).
j j
u′∈d^om(u)
Wehave p| u(u′)byassumptionon p,and p| [[u′∈v]] byProposition5.4,sothat p| u(u′)→ [[u′∈v]]
◦ ◦ j ◦ j j
and p| (u(u′)∧p)→ ([[u′∈v]] ∧p). Thus,byProposition 2.1andProposition 3.2(ii)wehave
◦ j j
p∧[[u| ⊆v| ]] = p∧ (u(u′)∧p)→ ([[u′∈v]] ∧p)
p p j j j
u′∈d^om(u)
= p∧ (u(u′)→ [[u′ ∈v]] )
j j
u′∈d^om(u)
= p∧[[u⊆v]] .
j
Thus,wehaveprovedrelation(ii). Relation(iii)followseasilyfromrelation(ii).
Wehavethefollowingtheorem.
Theorem 5.6 (D -Restriction Principle). For any D -formula f (x ,...,x ) in L(∈) and u ,...,u ∈
0 0 1 n 1 n
V(Q),if p∈L(u ,...,u )!,then[[f (u ,...,u )]] ∧p=[[f (u | ,...,u | )]] ∧p.
1 n 1 n j 1 p n p j
Proof. Weprovetheassertionbyinductiononthecomplexityoff (x ,...,x ). FromProposition5.5,
1 n
the assertion holds for atomic formulas. Then, the verification of every induction step follows from the
fact that (i) the function a7→a∧p of all a∈{p}! preserves the supremum and the infimum as shown
in Proposition 2.1, (ii) it satisfies (a→ b)∧p=[(a∧p)→ (b∧p)]∧p for all a,b∈{p}! from the
j j
defining property ofgeneralized implications, (iii)itsatisfiesrelation (ii)ofTheorem??,andthat(iv)it
satisfiestherelationa⊥∧p=(a∧p)⊥∧pforalla,b∈{p}!.
Now, we obtain the following transfer principle for bounded theorems of ZFC with respect to any
materialconditionals → ,whichisofthesameformasTheorem4.6ofRef.[18]obtainedfortheSasaki
j
conditional → .
S
Theorem 5.7 (D -ZFC Transfer Principle). For any D -formula f (x ,...,x ) of L(∈) and u ,...,u ∈
0 0 1 n 1 n
V(Q),iff (x ,...,x )isprovable inZFC,thenwehave
1 n
∨(u ,...,u )≤[[f (u ,...,u )]] .
1 n 1 n j
Proof. Let p = ∨(u ,...,u ). Then, we have a∧ p | b∧ p for any a,b ∈ L(u ,...,u ), and
1 n ◦ 1 n
hence there is a Boolean sublogic B such that L(u ,...,u )∧ p ⊆ B. From Proposition 5.2, we
1 n
have L(u | ,...,u | )⊆B. From Proposition 5.1, we have u | ,...,u | ∈V(B). By the ZFC trans-
1 p n p 1 p n p
fer principle of the Boolean-valued universe [1, Theorem 1.33], we have [[f (u1|p,...,un|p)]]jB = 1.
By the D -absoluteness principle, we have [[f (u | ,...,u | )]] = 1. From Proposition 5.6, we have
0 1 p n p j
[[f (u ,...,u )]] ∧p=[[f (u | ,...,u | )]] ∧p= p,andtheassertion follows.
1 n j 1 p n p j