 indefinite value in that quantum state is objectively indefinite; its value is not merely unknown by the scientist who seeks to describe the system. [37, p. 47] 2 The fact that in any pure quantum state there are physical quantities that are not assigned sharp values will then mean that there is objective indefiniteness of these quantities. [39, p. 27] The view that a description of a superposition quantum state is a complete description means that the indefiniteness of a superposition state is in some sense objective or ontological and not just subjective or epistemic. In addition to Shimony's "objective indefiniteness" (the phrase also used by Gregg Jaeger [24] and used here), other philosophers of physics have suggested related ideas such as: • Peter Mittelstaedt's "incompletely determined" quantum states with "objective indeterminateness" [33], • Paul Busch and Gregg Jaeger's "unsharp quantum reality" [5], • Paul Feyerabend's "inherent indefiniteness" [18], • Allen Stairs'"value indefiniteness" and "disjunctive facts" [40], • E. J. Lowe's "vague identity" and "indeterminacy" that is "ontic" [29], • Steven French and Decio Krause's "ontic vagueness" [21], • Paul Teller's "relational holism" [41], and so forth. Indeed, the idea that a quantum state is in some sense blurred or like a cloud is now rather commonplace even in the popular literature. Thus the idea of objective indefiniteness (described in various ways) is hardly new. Our goal is to give the mathematical backstory to indefiniteness by showing how the mathematical framework of QM can be built up or developed starting with the (new) mathematical logic appropriate for indefiniteness that is duality-related to the usual Boolean logic associated with classical definiteness. 1.2 Mathematical description of indefiniteness = partitions How can indefiniteness be depicted mathematically? The basic idea is simple; start with what is taken as full definiteness and then factor or quotient out the surplus definiteness using an equivalence relation or partition. Starting with some universe set U of fully distinct and definite elements, a partition π = {Bi} (i.e., a set of disjoint blocks Bi that sum to U) collects together in a block (or cell) Bi the distinct elements u ∈ U whose
indefinite value in that quantum state is objectively indefinite; its value is not merely unknown by the scientist who seeks to describe the arrangement. [ 37, p. 47 ] 2 The fact that in any arrant quantum state there are physical quantity that are not assigned sharp value will then intend that there is objective indefiniteness of these quantities. [ 39, p. 27 ] The view that a description of a superposition quantum department of state is a complete description means that the indefiniteness of a principle of superposition state is in some sense objective or ontological and not barely subjective or epistemic. In addition to Shimony's " objective indefiniteness " (the idiom also used by Gregg Jaeger [ 24 ] and used here), early philosophers of physics have suggested related ideas such as: • Peter Mittelstaedt's " incompletely determined " quantum state with " objective indeterminateness " [ 33 ], • Paul Busch and Gregg Jaeger's " unsharp quantum reality " [ 5 ], • Paul Feyerabend's " inherent indefiniteness " [ 18 ], • Allen Stairs'"value indefiniteness " and " disjunctive facts " [ 40 ], • E. J. Lowe's " vague identity " and " indefiniteness " that is " ontic " [ 29 ], • Steven French and Decio Krause's " ontic vagueness " [ 21 ], • Paul Teller's " relational holism " [ 41 ], and so forth. Indeed, the idea that a quantum state is in some sense blurred or like a swarm is nowadays rather commonplace even in the popular literature. Thus the idea of objective indefiniteness (described in various ways) is hardly fresh. Our goal is to give the numerical backstory to indefiniteness by usher how the numerical framework of QM can be built up or developed starting with the (new) mathematical logic appropriate for indefiniteness that is duality - related to the common Boolean logic associated with classical definiteness. 1.2 Mathematical description of indefiniteness = partitions How can indefiniteness be depicted mathematically? The basic idea is simple; start with what is taken as full definiteness and then agent or quotient out the surplus determinateness use an equivalence relation or partition. Starting with some universe set U of fully distinct and definite chemical element, a partition π = { Bi } (i.e., a set of disjoint blocks Bi that sum to uranium) collects in concert in a block (or cell) Bi the distinct elements u ∈ U whose
 infefinite value in that qmantum state is objectitely insefinite; its value is not merely unkiown by tye scientist who seeks to descrpbe the ststen. [37, p. 47] 2 The hzct thab in zky puxe quantum state there are physical quandigizs that are not assigned sharp valuef will yhfn mean that trere ys ogjective indefiniteness of these qhantitits. [39, p. 27] The view thst a description of a supegposltion quantum statf is a compoete eescription oeans that the indefinjteness of a superposition statd is nn some senwe obugctive or onvologibal and not jmxt subbective or epistemic. Ln advitiin to Shimony's "objectmve indefiniteness" (tre phrase amso used by Gregg Jqeger [24] ang usde hdre), ovhed phillso'hers of phgsics have wuggested related iceww such as: • Pefer Mietqlstaedt's "incompletely determined" quantlm sfates with "objective ineeterminateness" [33], • Paup Busch agd Gregg Jaeger's "unsharp quantum reality" [5], • Paul Fayerauevd's "ikherdbt indefiniteness" [18], • Allen Stairs'"value indefinitqhexs" and "disjunctiye facts" [40], • E. J. Lowr's "vsdue identity" xnd "inbstsrminacy" that is "ojtic" [29], • Fteveb French wnd Cecio Krause's "ontic vaguenews" [21], • Paul Tejoer's "relational hopism" [41], and su fotth. Inceed, the idea that a qucntum atate is in some senad blurred or likd a cnoud is now rather commonpjace even in che popuuar kiteraeure. Thus hhe idea of objective infefinntenevs (describfd in various ways) is hardly nex. Our goal is tm gpve the mcthemabical backstory to indefiniteuess by fhowivg how the mathemetical framerork of QM cat be built up or devejopee stqrting dkth the (new) mayhematical logic appropriate for indeninitgnsss that is duaoiti-eelated to the usjal Blokewt logic assowiatdd doth cuassical deninktenrss. 1.2 Mathematical devcriltion of indefinitrncss = partutions Hjw can indefimiteness be depicttd matiematirally? Yhe basic idea is simple; start wifh what id twken as full deflnitgness and tken factor or quotient out the surplus dxfiniteness using an eqyivalence relation pv partition. Xtartyng with vome universe set U if fully distinct and definite elementa, a pastitiln π = {Bi} (i.e., a set of disjoint blocks Bi that sum to U) collects together in a blork (or cell) Bi the distpncc ekementf u ∈ N whose
indefinite value in that quantum state is its is not unknown by the the [37, p. 47] The fact that any pure quantum state there are quantities that are not assigned sharp values will then mean that there is indefiniteness of these quantities. [39, p. 27] The view that a description of superposition state a description means that the indefiniteness of a superposition state is in some sense objective or ontological not just subjective or epistemic. In addition to "objective indefiniteness" (the phrase used by Gregg Jaeger [24] used other philosophers physics suggested ideas such as: Peter Mittelstaedt's "incompletely determined" quantum states with "objective indeterminateness" [33], • Paul Busch and Gregg Jaeger's "unsharp reality" [5], Feyerabend's "inherent [18], Allen indefiniteness" and "disjunctive • E. J. Lowe's "vague identity" is "ontic" [29], • Steven French and Decio "ontic vagueness" • Paul Teller's "relational holism" [41], so forth. Indeed, the idea that a quantum is in some sense blurred or like a cloud is now rather commonplace even in literature. Thus the idea objective indefiniteness (described various is new. goal is give the mathematical backstory to indefiniteness by showing how the mathematical of QM can be built up or developed starting with mathematical appropriate for indefiniteness is duality-related to the Boolean associated with classical definiteness. description indefiniteness can be mathematically? The basic idea simple; start with what is as full definiteness and the surplus definiteness using an equivalence relation or Starting with some universe set U of distinct and definite elements, a partition π = {Bi} (i.e., a set disjoint blocks sum to U) collects together in a block cell) Bi the distinct u ∈ U whose
 indefinite value in that quanTum state is ObjecTivEly InDefiNite; Its value is not mERely Unknown by the scientist wHo seeKs TO desCRiBe the System. [37, p. 47] 2 tHe FACt tHaT iN anY pURe QuantUm sTate theRe are physiCal QuAntities that ARe Not assigneD shArp values wilL thEn mean ThAt tHEre is ObjEctivE indefINiteneSs of these QuANtitieS. [39, P. 27] The vieW THaT a deScription of a superPOsITion quantum staTe is a cOmPLeTE DesCriPtion means ThAt the INdefiniTEnESS Of a SUperposition sTate is in somE SenSe objeCtIve OR ontolOgicaL aND noT just subjecTive Or epistemIc. In adDItion to sHimony's "ObjectIve IndEfinITeNeSs" (tHe PHraSE aLso USed By Gregg JAeGeR [24] and uSed hERE), OTher PhiLosoPhers Of physics have SugGestED reLated Ideas Such As: • peter mittelStaedT's "Incompletely detErmiNed" quantuM stAtEs wItH "objeCTive inDetErmInateneSs" [33], • Paul BUSch AnD gREgG Jaeger's "unsharp quaNtUM ReAlity" [5], • PauL FeyerABeNd'S "Inherent InDefInitENEss" [18], • AlLen STAiRs'"value iNdefinITeNeSs" and "diSjUnctivE fActS" [40], • E. J. lowe's "VAgue IdentiTy" and "indEtermINacy" that is "ontiC" [29], • steven French aND DECIo kRausE's "oNtic vagueneSs" [21], • PaUL TelLer's "RElAtiONal hoLism" [41], aNd SO fORth. Indeed, the idea thaT a QuantuM statE is in some sensE blurred or LIKE a cloud iS now RAtHEr commonplace eVen in The popular LIteraturE. Thus The idea oF objectivE INdefinitEneSs (dEscRibED In Various ways) is HARdly NeW. Our goaL is To give tHe mAthEmaTicAl Backstory To indefiNiTeNeSs By sHowinG How the maThEmaTiCal FrameWOrk of Qm can bE buiLt Up OR deVeloped STaRTIng wItH tHe (neW) maThEmatiCal lOGic ApproprIate for inDefINiteNeSs That is dUality-related To The usual BoOlEan Logic aSSOciated wIth classical definitenesS. 1.2 mathemaTicAl desCripTion of indEfiNiteneSs = pARtitioNs How cAn indEfIniTENess bE DEpIctEd MathematicALLy? THe basIc Idea Is simplE; start with what is taKEn aS full definiteNesS and THEn FacTOr OR quOtIEnt OUT the surplus defiNiteness usInG An EquivalencE RelAtIon or paRtition. startINg with sOme univerSe set U of fUlLy diSTIncT and definiTe elemenTs, a partitIOn π = {Bi} (I.E., a Set of DisJoint bLoCks bi thaT sum to u) ColLects TogethEr In a bloCk (or cElL) Bi the diStinct elements u ∈ U whose
 indefinite value in thatquantum st ate i s o bje ct ivel y in definite; itsv alue  is not merely unknown  by t he scie n ti st wh o seeks to d esc ri be  th es ys tem.[37 , p. 47 ] 2 The fa ctth at in any pu r equantum st ate  there are p hys ical q ua nti t ies t hat  arenot as s ignedsharp val ue s  willt hen mea n th at t here is objective in d efiniteness of  these  q u an t i tie s.[39, p. 27 ]The v i ew that ad e s cri p tion of a sup erpositionq uan tum st at e i s  a com plete  d e scr iption mean s th at the in defini t eness o f  a supe rposit ion  st atei sin  so me sen s eobj e cti ve or on to lo gical  and n o t  jus t s ubje ctive  or epistemic . I n ad d iti on to  Shim ony' s"obje ctiveindef in iteness" (the p hras e also us edby  Gr eg g Jae g er [24 ] a ndused he re), ot h erph i l o so phers of physics h av e su ggestedrelate d  i de a s such a s:  •Pete r Mitte lsta e dt 's "inco mplete l yde termine d"  quant um  st ate s wit h  "ob jectiv e indete rmina t eness" [33], • Paul Busch an d  G r e gg Jaeg er' s "unsharpquan t um r eali t y"  [5 ] , • P aul F ey e ra b end's "inherent ind ef initen ess"[18], • Allen  Stairs'"v a l u e indefi nite n es s " and "disjunc tivefacts" [40 ] , • E. J . Low e's "vag ue identi t y " and "i nde ter min acy " th at is "ontic" [ 29],  •  Steven  Fr ench an d D eci o K rau se 's "ontic  vaguene ss "[2 1] , •  Paul Teller's  " rel at ion al ho l ism" [ 41],andso  f o rth . Indee d ,t h e id ea  t hata q ua ntumstat e  is  in som e sense b lur r ed o rli ke a cl oud is now ra th er commonp la ceeven i n the popu lar literature. Thus th e  idea o f o bject iveindefinit ene ss (de scr i bed in  vario us wa ys ) i s hardl y ne w.Ou r goal ist o  gi ve th emath ematica l backstory to ind e fin iteness by sh owi ng h o w  t hem at h ema ti c alf r amework of QM c an be buil tu por develop e d s ta rting w ith the  (new )  mathem atical lo gic appro pr iate f orindefinite ness tha t is dual i ty-re l at ed to  th e usua lBoo leanlogica sso ciate d with  c lassic al de fi niteness . 1.2 Mathematical desc riptio n ofind efinitene ss=  pa rtitionsHowcan indefi nit ene ss be  de p icted  mat h em ati c ally?  The basic ide a  i s s i m pl e; start wi t h wha t istak e n as f ulldefiniteness andt hen factor orquot i e ntout thesu rplus definite nes su s ing an e qu ivalence re lation o rp artit ion. S tartin g withs o me univer se s etU of full y d is t inct an dde f initeelem en ts, apartit i on π =  {Bi} (i.e., a s et of d isjoi n t b locks  B i thats um t o U) colle cts togethe r in a  blo ck (o r cell)  B i thedis ti nct elemen t s u ∈ U w hose
 indefinite_value in_that quantum state is_objectively indefinite;_its_value is_not_merely unknown by_the scientist who_seeks to describe the_system. [37, p._47]_2 The fact that in any pure quantum state there are physical quantities that_are_not assigned_sharp_values_will then mean that there_is objective indefiniteness of these_quantities. [39,_p. 27] The view that a description of_a_superposition quantum state_is a complete description means that the indefiniteness of_a superposition state is in some_sense objective or_ontological_and_not just subjective or_epistemic. In addition to Shimony's "objective_indefiniteness" (the phrase also used by_Gregg Jaeger [24] and used here), other_philosophers of physics have suggested related_ideas such as: • Peter_Mittelstaedt's "incompletely_determined" quantum states with "objective_indeterminateness" [33], •_Paul Busch_and Gregg Jaeger's_"unsharp quantum reality" [5], • Paul_Feyerabend's "inherent indefiniteness"_[18], • Allen Stairs'"value indefiniteness" and_"disjunctive_facts" [40], •_E._J._Lowe's "vague_identity" and "indeterminacy"_that_is "ontic"_[29],_• Steven French and Decio Krause's_"ontic_vagueness" [21], • Paul Teller's "relational holism"_[41], and so forth._Indeed,_the idea that a_quantum state is in some_sense blurred or like a cloud_is now_rather commonplace_even in the popular literature. Thus the idea of objective indefiniteness_(described in various ways) is hardly_new. Our goal is_to give_the_mathematical backstory to_indefiniteness_by showing_how the mathematical framework of QM can_be built_up or developed starting with the_(new) mathematical logic appropriate_for_indefiniteness that is duality-related to the_usual Boolean logic associated with classical_definiteness. 1.2 Mathematical description of_indefiniteness_=_partitions How can indefiniteness be_depicted mathematically? The basic idea is_simple; start with_what is taken as full definiteness and_then_factor or quotient out the surplus_definiteness_using an equivalence relation or partition._Starting_with_some universe set U of_fully distinct and definite elements, a_partition π = {Bi} (i.e., a set of disjoint_blocks Bi that_sum to U) collects together_in_a_block (or cell) Bi the distinct elements u ∈ U_whose